LMFDB - Embedded newform 144.3.m.b.19.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,3,Mod(19,144)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("144.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	144.m (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.92371580679$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536 $ x^16 - 6x^14 + 10x^12 + 88x^10 - 752x^8 + 1408x^6 + 2560x^4 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			19.4
Root			$1.99750 - 0.0999235i$ of defining polynomial
Character	$\chi$	$=$	144.19
Dual form			144.3.m.b.91.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0999235 - 1.99750i) q^{2} +(-3.98003 + 0.399195i) q^{4} +(-6.01265 + 6.01265i) q^{5} +8.23187 q^{7} +(1.19509 + 7.91023i) q^{8} +O(q^{10})$	\(q+(-0.0999235 - 1.99750i) q^{2} +(-3.98003 + 0.399195i) q^{4} +(-6.01265 + 6.01265i) q^{5} +8.23187 q^{7} +(1.19509 + 7.91023i) q^{8} +(12.6111 + 11.4095i) q^{10} +(6.51529 + 6.51529i) q^{11} +(8.82865 + 8.82865i) q^{13} +(-0.822557 - 16.4432i) q^{14} +(15.6813 - 3.17761i) q^{16} -14.1753 q^{17} +(-23.7488 + 23.7488i) q^{19} +(21.5303 - 26.3308i) q^{20} +(12.3633 - 13.6653i) q^{22} -9.42125 q^{23} -47.3040i q^{25} +(16.7531 - 18.5174i) q^{26} +(-32.7631 + 3.28612i) q^{28} +(23.7973 + 23.7973i) q^{29} +24.4148i q^{31} +(-7.91422 - 31.0059i) q^{32} +(1.41645 + 28.3153i) q^{34} +(-49.4954 + 49.4954i) q^{35} +(24.2052 - 24.2052i) q^{37} +(49.8113 + 45.0652i) q^{38} +(-54.7471 - 40.3758i) q^{40} +6.67771i q^{41} +(0.897918 + 0.897918i) q^{43} +(-28.5319 - 23.3302i) q^{44} +(0.941404 + 18.8190i) q^{46} -25.2401i q^{47} +18.7636 q^{49} +(-94.4898 + 4.72678i) q^{50} +(-38.6626 - 31.6139i) q^{52} +(32.6251 - 32.6251i) q^{53} -78.3484 q^{55} +(9.83783 + 65.1160i) q^{56} +(45.1573 - 49.9132i) q^{58} +(-8.31871 - 8.31871i) q^{59} +(-68.4028 - 68.4028i) q^{61} +(48.7687 - 2.43961i) q^{62} +(-61.1435 + 18.9069i) q^{64} -106.167 q^{65} +(-7.71922 + 7.71922i) q^{67} +(56.4183 - 5.65873i) q^{68} +(103.813 + 93.9213i) q^{70} +137.259 q^{71} +52.8655i q^{73} +(-50.7685 - 45.9312i) q^{74} +(85.0404 - 104.001i) q^{76} +(53.6330 + 53.6330i) q^{77} +87.2269i q^{79} +(-75.1802 + 113.392i) q^{80} +(13.3387 - 0.667260i) q^{82} +(9.53893 - 9.53893i) q^{83} +(85.2314 - 85.2314i) q^{85} +(1.70387 - 1.88332i) q^{86} +(-43.7511 + 59.3238i) q^{88} -146.488i q^{89} +(72.6762 + 72.6762i) q^{91} +(37.4969 - 3.76091i) q^{92} +(-50.4172 + 2.52208i) q^{94} -285.586i q^{95} +101.170 q^{97} +(-1.87493 - 37.4804i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 12 q^{4}+O(q^{10}) $ 16 * q - 12 * q^4	$ 16 q - 12 q^{4} + 16 q^{10} + 32 q^{16} - 32 q^{19} + 104 q^{22} - 24 q^{34} + 96 q^{37} - 312 q^{40} - 32 q^{43} - 224 q^{46} + 112 q^{49} - 264 q^{52} - 256 q^{55} + 312 q^{58} - 32 q^{61} + 456 q^{64} - 256 q^{67} + 744 q^{70} + 264 q^{76} - 280 q^{82} + 160 q^{85} - 912 q^{88} + 288 q^{91} - 1104 q^{94}+O(q^{100}) $ 16 * q - 12 * q^4 + 16 * q^10 + 32 * q^16 - 32 * q^19 + 104 * q^22 - 24 * q^34 + 96 * q^37 - 312 * q^40 - 32 * q^43 - 224 * q^46 + 112 * q^49 - 264 * q^52 - 256 * q^55 + 312 * q^58 - 32 * q^61 + 456 * q^64 - 256 * q^67 + 744 * q^70 + 264 * q^76 - 280 * q^82 + 160 * q^85 - 912 * q^88 + 288 * q^91 - 1104 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.0999235	−	1.99750i	−0.0499617	−	0.998751i
$2$	−0.0999235	−	1.99750i	−0.0499617	−	0.998751i
$3$		0			0
$3$		0			0
$4$	−3.98003	+	0.399195i	−0.995008	+	0.0997987i
$5$	−6.01265	+	6.01265i	−1.20253	+	1.20253i	−0.229136	+	0.973394i	$0.573590\pi$
$5$	−6.01265	+	6.01265i	−1.20253	+	1.20253i	−0.973394	+	0.229136i	$0.926410\pi$
$6$		0			0
$7$	8.23187			1.17598			0.587990	−	0.808868i	$-0.299919\pi$
$7$	8.23187			1.17598			0.587990	+	0.808868i	$0.299919\pi$
$8$	1.19509	+	7.91023i	0.149386	+	0.988779i
$9$		0			0
$10$	12.6111	+	11.4095i	1.26111	+	1.14095i
$11$	6.51529	+	6.51529i	0.592299	+	0.592299i	0.938252	−	0.345953i	$-0.112444\pi$
$11$	6.51529	+	6.51529i	0.592299	+	0.592299i	−0.345953	+	0.938252i	$0.612444\pi$
$12$		0			0
$13$	8.82865	+	8.82865i	0.679127	+	0.679127i	0.959803	−	0.280676i	$-0.0905586\pi$
$13$	8.82865	+	8.82865i	0.679127	+	0.679127i	−0.280676	+	0.959803i	$0.590559\pi$
$14$	−0.822557	−	16.4432i	−0.0587541	−	1.17451i
$15$		0			0
$16$	15.6813	−	3.17761i	0.980080	−	0.198601i
$17$	−14.1753			−0.833844			−0.416922	−	0.908942i	$-0.636891\pi$
$17$	−14.1753			−0.833844			−0.416922	+	0.908942i	$0.636891\pi$
$18$		0			0
$19$	−23.7488	+	23.7488i	−1.24994	+	1.24994i	−0.294187	+	0.955748i	$0.595049\pi$
$19$	−23.7488	+	23.7488i	−1.24994	+	1.24994i	−0.955748	+	0.294187i	$0.904951\pi$
$20$	21.5303	−	26.3308i	1.07652	−	1.31654i
$21$		0			0
$22$	12.3633	−	13.6653i	0.561967	−	0.621152i
$23$	−9.42125			−0.409619			−0.204810	−	0.978802i	$-0.565658\pi$
$23$	−9.42125			−0.409619			−0.204810	+	0.978802i	$0.565658\pi$
$24$		0			0
$25$		−	47.3040i		−	1.89216i
$26$	16.7531	−	18.5174i	0.644348	−	0.712209i
$27$		0			0
$28$	−32.7631	+	3.28612i	−1.17011	+	0.117361i
$29$	23.7973	+	23.7973i	0.820598	+	0.820598i	0.986194	−	0.165596i	$-0.0529547\pi$
$29$	23.7973	+	23.7973i	0.820598	+	0.820598i	−0.165596	+	0.986194i	$0.552955\pi$
$30$		0			0
$31$			24.4148i			0.787575i	0.919202	+	0.393787i	$0.128835\pi$
$31$			24.4148i			0.787575i	−0.919202	+	0.393787i	$0.871165\pi$
$32$	−7.91422	−	31.0059i	−0.247319	−	0.968934i
$33$		0			0
$34$	1.41645	+	28.3153i	0.0416603	+	0.832803i
$35$	−49.4954	+	49.4954i	−1.41415	+	1.41415i
$36$		0			0
$37$	24.2052	−	24.2052i	0.654193	−	0.654193i	−0.299807	−	0.954000i	$-0.596922\pi$
$37$	24.2052	−	24.2052i	0.654193	−	0.654193i	0.954000	+	0.299807i	$0.0969221\pi$
$38$	49.8113	+	45.0652i	1.31082	+	1.18593i
$39$		0			0
$40$	−54.7471	−	40.3758i	−1.36868	−	1.00940i
$41$			6.67771i			0.162871i	0.996679	+	0.0814355i	$0.0259505\pi$
$41$			6.67771i			0.162871i	−0.996679	+	0.0814355i	$0.974050\pi$
$42$		0			0
$43$	0.897918	+	0.897918i	0.0208818	+	0.0208818i	0.717471	−	0.696589i	$-0.245300\pi$
$43$	0.897918	+	0.897918i	0.0208818	+	0.0208818i	−0.696589	+	0.717471i	$0.745300\pi$
$44$	−28.5319	−	23.3302i	−0.648453	−	0.530232i
$45$		0			0
$46$	0.941404	+	18.8190i	0.0204653	+	0.409108i
$47$		−	25.2401i		−	0.537023i	−0.963276	−	0.268512i	$-0.913468\pi$
$47$		−	25.2401i		−	0.537023i	0.963276	−	0.268512i	$-0.0865318\pi$
$48$		0			0
$49$	18.7636			0.382931
$50$	−94.4898	+	4.72678i	−1.88980	+	0.0945356i
$51$		0			0
$52$	−38.6626	−	31.6139i	−0.743512	−	0.607960i
$53$	32.6251	−	32.6251i	0.615568	−	0.615568i	−0.328824	−	0.944391i	$-0.606652\pi$
$53$	32.6251	−	32.6251i	0.615568	−	0.615568i	0.944391	+	0.328824i	$0.106652\pi$
$54$		0			0
$55$	−78.3484			−1.42452
$56$	9.83783	+	65.1160i	0.175676	+	1.16279i
$57$		0			0
$58$	45.1573	−	49.9132i	0.778575	−	0.860572i
$59$	−8.31871	−	8.31871i	−0.140995	−	0.140995i	0.633086	−	0.774081i	$-0.281788\pi$
$59$	−8.31871	−	8.31871i	−0.140995	−	0.140995i	−0.774081	+	0.633086i	$0.781788\pi$
$60$		0			0
$61$	−68.4028	−	68.4028i	−1.12136	−	1.12136i	−0.991538	−	0.129820i	$-0.958560\pi$
$61$	−68.4028	−	68.4028i	−1.12136	−	1.12136i	−0.129820	−	0.991538i	$-0.541440\pi$
$62$	48.7687	−	2.43961i	0.786591	−	0.0393486i
$63$		0			0
$64$	−61.1435	+	18.9069i	−0.955367	+	0.295420i
$65$	−106.167			−1.63334
$66$		0			0
$67$	−7.71922	+	7.71922i	−0.115212	+	0.115212i	−0.762362	−	0.647150i	$-0.775961\pi$
$67$	−7.71922	+	7.71922i	−0.115212	+	0.115212i	0.647150	+	0.762362i	$0.275961\pi$
$68$	56.4183	−	5.65873i	0.829681	−	0.0832165i
$69$		0			0
$70$	103.813	+	93.9213i	1.48304	+	1.34173i
$71$	137.259			1.93322			0.966612	−	0.256245i	$-0.0824853\pi$
$71$	137.259			1.93322			0.966612	+	0.256245i	$0.0824853\pi$
$72$		0			0
$73$			52.8655i			0.724185i	0.932142	+	0.362093i	$0.117938\pi$
$73$			52.8655i			0.724185i	−0.932142	+	0.362093i	$0.882062\pi$
$74$	−50.7685	−	45.9312i	−0.686061	−	0.620692i
$75$		0			0
$76$	85.0404	−	104.001i	1.11895	−	1.36844i
$77$	53.6330	+	53.6330i	0.696533	+	0.696533i
$78$		0			0
$79$			87.2269i			1.10414i	0.833798	+	0.552069i	$0.186162\pi$
$79$			87.2269i			1.10414i	−0.833798	+	0.552069i	$0.813838\pi$
$80$	−75.1802	+	113.392i	−0.939753	+	1.41740i
$81$		0			0
$82$	13.3387	−	0.667260i	0.162668	−	0.00813732i
$83$	9.53893	−	9.53893i	0.114927	−	0.114927i	−0.647305	−	0.762231i	$-0.724104\pi$
$83$	9.53893	−	9.53893i	0.114927	−	0.114927i	0.762231	+	0.647305i	$0.224104\pi$
$84$		0			0
$85$	85.2314	−	85.2314i	1.00272	−	1.00272i
$86$	1.70387	−	1.88332i	0.0198124	−	0.0218990i
$87$		0			0
$88$	−43.7511	+	59.3238i	−0.497172	+	0.674135i
$89$		−	146.488i		−	1.64593i	−0.568089	−	0.822967i	$-0.692317\pi$
$89$		−	146.488i		−	1.64593i	0.568089	−	0.822967i	$-0.307683\pi$
$90$		0			0
$91$	72.6762	+	72.6762i	0.798640	+	0.798640i
$92$	37.4969	−	3.76091i	0.407574	−	0.0408795i
$93$		0			0
$94$	−50.4172	+	2.52208i	−0.536353	+	0.0268306i
$95$		−	285.586i		−	3.00617i
$96$		0			0
$97$	101.170			1.04298			0.521492	−	0.853256i	$-0.325376\pi$
$97$	101.170			1.04298			0.521492	+	0.853256i	$0.325376\pi$
$98$	−1.87493	−	37.4804i	−0.0191319	−	0.382453i
$99$		0			0
$100$	18.8835	+	188.271i	0.188835	+	1.88271i
$101$	47.4912	−	47.4912i	0.470210	−	0.470210i	−0.431773	−	0.901982i	$-0.642112\pi$
$101$	47.4912	−	47.4912i	0.470210	−	0.470210i	0.901982	+	0.431773i	$0.142112\pi$
$102$		0			0
$103$	7.58518			0.0736425			0.0368213	−	0.999322i	$-0.488277\pi$
$103$	7.58518			0.0736425			0.0368213	+	0.999322i	$0.488277\pi$
$104$	−59.2856	+	80.3877i	−0.570054	+	0.772958i
$105$		0			0
$106$	−68.4287	−	61.9087i	−0.645554	−	0.584044i
$107$	119.757	+	119.757i	1.11922	+	1.11922i	0.991856	+	0.127367i	$0.0406525\pi$
$107$	119.757	+	119.757i	1.11922	+	1.11922i	0.127367	+	0.991856i	$0.459347\pi$
$108$		0			0
$109$	61.0263	+	61.0263i	0.559874	+	0.559874i	0.929272	−	0.369397i	$-0.120436\pi$
$109$	61.0263	+	61.0263i	0.559874	+	0.559874i	−0.369397	+	0.929272i	$0.620436\pi$
$110$	7.82884	+	156.501i	0.0711713	+	1.42274i
$111$		0			0
$112$	129.086	−	26.1577i	1.15256	−	0.233551i
$113$	−31.6377			−0.279980			−0.139990	−	0.990153i	$-0.544707\pi$
$113$	−31.6377			−0.279980			−0.139990	+	0.990153i	$0.544707\pi$
$114$		0			0
$115$	56.6467	−	56.6467i	0.492580	−	0.492580i
$116$	−104.214	−	85.2144i	−0.898396	−	0.734607i
$117$		0			0
$118$	−15.7854	+	17.4479i	−0.133775	+	0.147863i
$119$	−116.690			−0.980585
$120$		0			0
$121$		−	36.1019i		−	0.298363i
$122$	−129.800	+	143.470i	−1.06393	+	1.17598i
$123$		0			0
$124$	−9.74627	−	97.1717i	−0.0785989	−	0.783643i
$125$	134.106	+	134.106i	1.07285	+	1.07285i
$126$		0			0
$127$		−	87.8736i		−	0.691918i	−0.938250	−	0.345959i	$-0.887554\pi$
$127$		−	87.8736i		−	0.691918i	0.938250	−	0.345959i	$-0.112446\pi$
$128$	43.8762	+	120.245i	0.342783	+	0.939415i
$129$		0			0
$130$	10.6086	+	212.069i	0.0816046	+	1.63130i
$131$	−176.351	+	176.351i	−1.34619	+	1.34619i	−0.456428	+	0.889761i	$0.650871\pi$
$131$	−176.351	+	176.351i	−1.34619	+	1.34619i	−0.889761	+	0.456428i	$0.849129\pi$
$132$		0			0
$133$	−195.497	+	195.497i	−1.46990	+	1.46990i
$134$	16.1905	+	14.6478i	0.120825	+	0.109312i
$135$		0			0
$136$	−16.9408	−	112.130i	−0.124565	−	0.824487i
$137$		−	97.0224i		−	0.708192i	−0.935209	−	0.354096i	$-0.884789\pi$
$137$		−	97.0224i		−	0.708192i	0.935209	−	0.354096i	$-0.115211\pi$
$138$		0			0
$139$	−130.221	−	130.221i	−0.936841	−	0.936841i	0.0612800	−	0.998121i	$-0.480482\pi$
$139$	−130.221	−	130.221i	−0.936841	−	0.936841i	−0.998121	+	0.0612800i	$0.980482\pi$
$140$	177.235	−	216.751i	1.26596	−	1.54822i
$141$		0			0
$142$	−13.7154	−	274.175i	−0.0965872	−	1.93081i
$143$			115.042i			0.804493i
$144$		0			0
$145$	−286.170			−1.97359
$146$	105.599	−	5.28251i	0.723281	−	0.0361816i
$147$		0			0
$148$	−86.6747	+	106.000i	−0.585640	+	0.716215i
$149$	26.3592	−	26.3592i	0.176907	−	0.176907i	−0.613099	−	0.790006i	$-0.710077\pi$
$149$	26.3592	−	26.3592i	0.176907	−	0.176907i	0.790006	+	0.613099i	$0.210077\pi$
$150$		0			0
$151$	196.107			1.29872			0.649360	−	0.760481i	$-0.275037\pi$
$151$	196.107			1.29872			0.649360	+	0.760481i	$0.275037\pi$
$152$	−216.240	−	159.476i	−1.42263	−	1.04919i
$153$		0			0
$154$	101.773	−	112.491i	0.660863	−	0.730463i
$155$	−146.798	−	146.798i	−0.947083	−	0.947083i
$156$		0			0
$157$	98.5323	+	98.5323i	0.627594	+	0.627594i	0.947462	−	0.319868i	$-0.103638\pi$
$157$	98.5323	+	98.5323i	0.627594	+	0.627594i	−0.319868	+	0.947462i	$0.603638\pi$
$158$	174.236	−	8.71602i	1.10276	−	0.0551647i
$159$		0			0
$160$	234.013	+	138.842i	1.46258	+	0.867764i
$161$	−77.5545			−0.481705
$162$		0			0
$163$	−90.6677	+	90.6677i	−0.556244	+	0.556244i	−0.928236	−	0.371992i	$-0.878675\pi$
$163$	−90.6677	+	90.6677i	−0.556244	+	0.556244i	0.371992	+	0.928236i	$0.378675\pi$
$164$	−2.66571	−	26.5775i	−0.0162543	−	0.162058i
$165$		0			0
$166$	−20.0072	−	18.1009i	−0.120525	−	0.109041i
$167$	181.602			1.08744			0.543719	−	0.839267i	$-0.317016\pi$
$167$	181.602			1.08744			0.543719	+	0.839267i	$0.317016\pi$
$168$		0			0
$169$		−	13.1100i		−	0.0775741i
$170$	−178.767	−	161.733i	−1.05157	−	0.951373i
$171$		0			0
$172$	−3.93219	−	3.21530i	−0.0228615	−	0.0186936i
$173$	2.66532	+	2.66532i	0.0154065	+	0.0154065i	0.714768	−	0.699362i	$-0.246532\pi$
$173$	2.66532	+	2.66532i	0.0154065	+	0.0154065i	−0.699362	+	0.714768i	$0.746532\pi$
$174$		0			0
$175$		−	389.400i		−	2.22514i
$176$	122.871	+	81.4651i	0.698132	+	0.462870i
$177$		0			0
$178$	−292.610	+	14.6376i	−1.64388	+	0.0822337i
$179$	12.7491	−	12.7491i	0.0712241	−	0.0712241i	−0.670597	−	0.741822i	$-0.733962\pi$
$179$	12.7491	−	12.7491i	0.0712241	−	0.0712241i	0.741822	+	0.670597i	$0.233962\pi$
$180$		0			0
$181$	−61.3009	+	61.3009i	−0.338679	+	0.338679i	−0.855870	−	0.517191i	$-0.826978\pi$
$181$	−61.3009	+	61.3009i	−0.338679	+	0.338679i	0.517191	+	0.855870i	$0.326978\pi$
$182$	137.909	−	152.433i	0.757741	−	0.837544i
$183$		0			0
$184$	−11.2592	−	74.5242i	−0.0611916	−	0.405023i
$185$			291.074i			1.57338i
$186$		0			0
$187$	−92.3566	−	92.3566i	−0.493885	−	0.493885i
$188$	10.0757	+	100.456i	0.0535942	+	0.534342i
$189$		0			0
$190$	−570.459	+	28.5368i	−3.00242	+	0.150194i
$191$		−	80.1204i		−	0.419479i	−0.977757	−	0.209739i	$-0.932738\pi$
$191$		−	80.1204i		−	0.419479i	0.977757	−	0.209739i	$-0.0672615\pi$
$192$		0			0
$193$	28.7555			0.148992			0.0744961	−	0.997221i	$-0.476265\pi$
$193$	28.7555			0.148992			0.0744961	+	0.997221i	$0.476265\pi$
$194$	−10.1092	−	202.086i	−0.0521093	−	1.04168i
$195$		0			0
$196$	−74.6798	+	7.49034i	−0.381019	+	0.0382160i
$197$	−92.8030	+	92.8030i	−0.471081	+	0.471081i	−0.902264	−	0.431183i	$-0.858096\pi$
$197$	−92.8030	+	92.8030i	−0.471081	+	0.471081i	0.431183	+	0.902264i	$0.358096\pi$
$198$		0			0
$199$	165.555			0.831934			0.415967	−	0.909380i	$-0.363443\pi$
$199$	165.555			0.831934			0.415967	+	0.909380i	$0.363443\pi$
$200$	374.185	−	56.5326i	1.87093	−	0.282663i
$201$		0			0
$202$	−99.6092	−	90.1183i	−0.493115	−	0.446130i
$203$	195.897	+	195.897i	0.965008	+	0.965008i
$204$		0			0
$205$	−40.1507	−	40.1507i	−0.195857	−	0.195857i
$206$	−0.757938	−	15.1514i	−0.00367931	−	0.0735506i
$207$		0			0
$208$	166.499	+	110.390i	0.800474	+	0.530724i
$209$	−309.460			−1.48067
$210$		0			0
$211$	187.769	−	187.769i	0.889902	−	0.889902i	−0.104611	−	0.994513i	$-0.533360\pi$
$211$	187.769	−	187.769i	0.889902	−	0.889902i	0.994513	+	0.104611i	$0.0333598\pi$
$212$	−116.825	+	142.873i	−0.551062	+	0.673928i
$213$		0			0
$214$	227.248	−	251.181i	1.06191	−	1.17374i
$215$	−10.7977			−0.0502220
$216$		0			0
$217$			200.980i			0.926173i
$218$	115.802	−	127.998i	0.531203	−	0.587147i
$219$		0			0
$220$	311.829	−	31.2763i	1.41740	−	0.142165i
$221$	−125.149	−	125.149i	−0.566286	−	0.566286i
$222$		0			0
$223$			307.192i			1.37754i	0.724979	+	0.688771i	$0.241849\pi$
$223$			307.192i			1.37754i	−0.724979	+	0.688771i	$0.758151\pi$
$224$	−65.1488	−	255.236i	−0.290843	−	1.13945i
$225$		0			0
$226$	3.16135	+	63.1964i	0.0139883	+	0.279630i
$227$	−112.960	+	112.960i	−0.497621	+	0.497621i	−0.910697	−	0.413075i	$-0.864455\pi$
$227$	−112.960	+	112.960i	−0.497621	+	0.497621i	0.413075	+	0.910697i	$0.364455\pi$
$228$		0			0
$229$	−7.32210	+	7.32210i	−0.0319742	+	0.0319742i	−0.722913	−	0.690939i	$-0.757197\pi$
$229$	−7.32210	+	7.32210i	−0.0319742	+	0.0319742i	0.690939	+	0.722913i	$0.257197\pi$
$230$	−118.812	−	107.492i	−0.516575	−	0.467355i
$231$		0			0
$232$	−159.802	+	216.682i	−0.688804	+	0.933976i
$233$			115.408i			0.495315i	0.968848	+	0.247657i	$0.0796607\pi$
$233$			115.408i			0.495315i	−0.968848	+	0.247657i	$0.920339\pi$
$234$		0			0
$235$	151.760	+	151.760i	0.645787	+	0.645787i
$236$	36.4295	+	29.7879i	0.154362	+	0.126220i
$237$		0			0
$238$	11.6600	+	233.088i	0.0489917	+	0.979360i
$239$			323.271i			1.35260i	0.736627	+	0.676300i	$0.236417\pi$
$239$			323.271i			1.35260i	−0.736627	+	0.676300i	$0.763583\pi$
$240$		0			0
$241$	118.056			0.489860			0.244930	−	0.969541i	$-0.421235\pi$
$241$	118.056			0.489860			0.244930	+	0.969541i	$0.421235\pi$
$242$	−72.1136	+	3.60743i	−0.297990	+	0.0149067i
$243$		0			0
$244$	299.551	+	244.939i	1.22767	+	1.00385i
$245$	−112.819	+	112.819i	−0.460486	+	0.460486i
$246$		0			0
$247$	−419.339			−1.69773
$248$	−193.127	+	29.1779i	−0.778737	+	0.117653i
$249$		0			0
$250$	254.477	−	281.278i	1.01791	−	1.12511i
$251$	−166.229	−	166.229i	−0.662265	−	0.662265i	0.293648	−	0.955913i	$-0.405130\pi$
$251$	−166.229	−	166.229i	−0.662265	−	0.662265i	−0.955913	+	0.293648i	$0.905130\pi$
$252$		0			0
$253$	−61.3822	−	61.3822i	−0.242617	−	0.242617i
$254$	−175.528	+	8.78064i	−0.691054	+	0.0345694i
$255$		0			0
$256$	235.806	−	99.6582i	0.921115	−	0.389290i
$257$	342.745			1.33364			0.666820	−	0.745219i	$-0.267655\pi$
$257$	342.745			1.33364			0.666820	+	0.745219i	$0.267655\pi$
$258$		0			0
$259$	199.254	−	199.254i	0.769319	−	0.769319i
$260$	422.549	−	42.3814i	1.62519	−	0.163005i
$261$		0			0
$262$	369.882	+	334.639i	1.41176	+	1.27725i
$263$	−219.637			−0.835123			−0.417562	−	0.908649i	$-0.637115\pi$
$263$	−219.637			−0.835123			−0.417562	+	0.908649i	$0.637115\pi$
$264$		0			0
$265$			392.327i			1.48048i
$266$	410.040	+	370.970i	1.54150	+	1.39463i
$267$		0			0
$268$	27.6413	−	33.8042i	0.103139	−	0.126135i
$269$	77.9061	+	77.9061i	0.289614	+	0.289614i	0.836928	−	0.547314i	$-0.184350\pi$
$269$	77.9061	+	77.9061i	0.289614	+	0.289614i	−0.547314	+	0.836928i	$0.684350\pi$
$270$		0			0
$271$		−	155.092i		−	0.572294i	−0.958186	−	0.286147i	$-0.907625\pi$
$271$		−	155.092i		−	0.572294i	0.958186	−	0.286147i	$-0.0923746\pi$
$272$	−222.288	+	45.0438i	−0.817234	+	0.165602i
$273$		0			0
$274$	−193.802	+	9.69481i	−0.707308	+	0.0353825i
$275$	308.199	−	308.199i	1.12072	−	1.12072i
$276$		0			0
$277$	−30.0601	+	30.0601i	−0.108520	+	0.108520i	−0.759282	−	0.650762i	$-0.774450\pi$
$277$	−30.0601	+	30.0601i	−0.108520	+	0.108520i	0.650762	+	0.759282i	$0.274450\pi$
$278$	−247.104	+	273.129i	−0.888864	+	0.982477i
$279$		0			0
$280$	−450.671	−	332.368i	−1.60954	−	1.18703i
$281$		−	265.590i		−	0.945162i	−0.881287	−	0.472581i	$-0.843323\pi$
$281$		−	265.590i		−	0.945162i	0.881287	−	0.472581i	$-0.156677\pi$
$282$		0			0
$283$	160.221	+	160.221i	0.566153	+	0.566153i	0.931049	−	0.364895i	$-0.118895\pi$
$283$	160.221	+	160.221i	0.566153	+	0.566153i	−0.364895	+	0.931049i	$0.618895\pi$
$284$	−546.295	+	54.7930i	−1.92357	+	0.192933i
$285$		0			0
$286$	229.798	−	11.4954i	0.803488	−	0.0401939i
$287$			54.9700i			0.191533i
$288$		0			0
$289$	−88.0595			−0.304704
$290$	28.5951	+	571.626i	0.0986039	+	1.97112i
$291$		0			0
$292$	−21.1036	−	210.406i	−0.0722728	−	0.720570i
$293$	173.762	−	173.762i	0.593044	−	0.593044i	−0.345408	−	0.938453i	$-0.612260\pi$
$293$	173.762	−	173.762i	0.593044	−	0.593044i	0.938453	+	0.345408i	$0.112260\pi$
$294$		0			0
$295$	100.035			0.339102
$296$	220.396	+	162.541i	0.744580	+	0.549125i
$297$		0			0
$298$	−55.2864	−	50.0186i	−0.185525	−	0.167848i
$299$	−83.1769	−	83.1769i	−0.278183	−	0.278183i
$300$		0			0
$301$	7.39154	+	7.39154i	0.0245566	+	0.0245566i
$302$	−19.5957	−	391.724i	−0.0648863	−	1.29710i
$303$		0			0
$304$	−296.947	+	447.876i	−0.976799	+	1.47328i
$305$	822.565			2.69693
$306$		0			0
$307$	136.842	−	136.842i	0.445741	−	0.445741i	−0.448195	−	0.893936i	$-0.647933\pi$
$307$	136.842	−	136.842i	0.445741	−	0.445741i	0.893936	+	0.448195i	$0.147933\pi$
$308$	−234.871	−	192.051i	−0.762569	−	0.623542i
$309$		0			0
$310$	−278.560	+	307.898i	−0.898582	+	0.993218i
$311$	−458.117			−1.47305			−0.736523	−	0.676412i	$-0.763534\pi$
$311$	−458.117			−1.47305			−0.736523	+	0.676412i	$0.763534\pi$
$312$		0			0
$313$		−	170.865i		−	0.545893i	−0.962029	−	0.272947i	$-0.912002\pi$
$313$		−	170.865i		−	0.545893i	0.962029	−	0.272947i	$-0.0879982\pi$
$314$	186.973	−	206.664i	0.595455	−	0.658166i
$315$		0			0
$316$	−34.8205	−	347.166i	−0.110192	−	1.09863i
$317$	−372.560	−	372.560i	−1.17527	−	1.17527i	−0.980935	−	0.194334i	$-0.937746\pi$
$317$	−372.560	−	372.560i	−1.17527	−	1.17527i	−0.194334	−	0.980935i	$-0.562254\pi$
$318$		0			0
$319$			310.093i			0.972079i
$320$	253.954	−	481.315i	0.793607	−	1.50411i
$321$		0			0
$322$	7.74951	+	154.915i	0.0240668	+	0.481103i
$323$	336.647	−	336.647i	1.04225	−	1.04225i
$324$		0			0
$325$	417.630	−	417.630i	1.28502	−	1.28502i
$326$	190.169	+	172.049i	0.583340	+	0.527758i
$327$		0			0
$328$	−52.8222	+	7.98047i	−0.161043	+	0.0243307i
$329$		−	207.773i		−	0.631529i
$330$		0			0
$331$	−177.685	−	177.685i	−0.536812	−	0.536812i	0.385779	−	0.922591i	$-0.373933\pi$
$331$	−177.685	−	177.685i	−0.536812	−	0.536812i	−0.922591	+	0.385779i	$0.873933\pi$
$332$	−34.1574	+	41.7731i	−0.102884	+	0.125823i
$333$		0			0
$334$	−18.1463	−	362.751i	−0.0543303	−	1.08608i
$335$		−	92.8260i		−	0.277092i
$336$		0			0
$337$	491.680			1.45899			0.729496	−	0.683985i	$-0.239755\pi$
$337$	491.680			1.45899			0.729496	+	0.683985i	$0.239755\pi$
$338$	−26.1873	+	1.31000i	−0.0774772	+	0.00387574i
$339$		0			0
$340$	−305.200	+	373.248i	−0.897646	+	1.09779i
$341$	−159.070	+	159.070i	−0.466480	+	0.466480i
$342$		0			0
$343$	−248.902			−0.725661
$344$	−6.02965	+	8.17583i	−0.0175280	+	0.0237670i
$345$		0			0
$346$	5.05766	−	5.59031i	0.0146175	−	0.0161570i
$347$	176.788	+	176.788i	0.509477	+	0.509477i	0.914366	−	0.404889i	$-0.132690\pi$
$347$	176.788	+	176.788i	0.509477	+	0.509477i	−0.404889	+	0.914366i	$0.632690\pi$
$348$		0			0
$349$	−272.298	−	272.298i	−0.780223	−	0.780223i	0.199645	−	0.979868i	$-0.436021\pi$
$349$	−272.298	−	272.298i	−0.780223	−	0.780223i	−0.979868	+	0.199645i	$0.936021\pi$
$350$	−777.827	+	38.9102i	−2.22236	+	0.111172i
$351$		0			0
$352$	150.449	−	253.576i	0.427412	−	0.720386i
$353$	168.099			0.476200			0.238100	−	0.971241i	$-0.423475\pi$
$353$	168.099			0.476200			0.238100	+	0.971241i	$0.423475\pi$
$354$		0			0
$355$	−825.290	+	825.290i	−2.32476	+	2.32476i
$356$	58.4773	+	583.027i	0.164262	+	1.63772i
$357$		0			0
$358$	−26.7403	−	24.1925i	−0.0746937	−	0.0675767i
$359$	476.009			1.32593			0.662965	−	0.748651i	$-0.269298\pi$
$359$	476.009			1.32593			0.662965	+	0.748651i	$0.269298\pi$
$360$		0			0
$361$		−	767.008i		−	2.12468i
$362$	128.574	+	116.323i	0.355177	+	0.321335i
$363$		0			0
$364$	−318.266	−	260.242i	−0.874356	−	0.714950i
$365$	−317.862	−	317.862i	−0.870855	−	0.870855i
$366$		0			0
$367$			104.894i			0.285814i	0.989736	+	0.142907i	$0.0456450\pi$
$367$			104.894i			0.285814i	−0.989736	+	0.142907i	$0.954355\pi$
$368$	−147.737	+	29.9371i	−0.401460	+	0.0813508i
$369$		0			0
$370$	581.422	−	29.0852i	1.57141	−	0.0786086i
$371$	268.565	−	268.565i	0.723896	−	0.723896i
$372$		0			0
$373$	−122.301	+	122.301i	−0.327884	+	0.327884i	−0.851781	−	0.523898i	$-0.824477\pi$
$373$	−122.301	+	122.301i	−0.327884	+	0.327884i	0.523898	+	0.851781i	$0.324477\pi$
$374$	−175.254	+	193.711i	−0.468593	+	0.517944i
$375$		0			0
$376$	199.655	−	30.1642i	0.530997	−	0.0802240i
$377$			420.197i			1.11458i
$378$		0			0
$379$	75.5625	+	75.5625i	0.199373	+	0.199373i	0.799731	−	0.600358i	$-0.204975\pi$
$379$	75.5625	+	75.5625i	0.199373	+	0.199373i	−0.600358	+	0.799731i	$0.704975\pi$
$380$	114.005	+	1136.64i	0.300012	+	2.99116i
$381$		0			0
$382$	−160.041	+	8.00591i	−0.418955	+	0.0209579i
$383$			501.509i			1.30942i	0.755878	+	0.654712i	$0.227210\pi$
$383$			501.509i			1.30942i	−0.755878	+	0.654712i	$0.772790\pi$
$384$		0			0
$385$	−644.953			−1.67520
$386$	−2.87335	−	57.4392i	−0.00744391	−	0.148806i
$387$		0			0
$388$	−402.658	+	40.3863i	−1.03778	+	0.104089i
$389$	22.6053	−	22.6053i	0.0581114	−	0.0581114i	−0.677454	−	0.735565i	$-0.736917\pi$
$389$	22.6053	−	22.6053i	0.0581114	−	0.0581114i	0.735565	+	0.677454i	$0.236917\pi$
$390$		0			0
$391$	133.549			0.341559
$392$	22.4242	+	148.425i	0.0572047	+	0.378634i
$393$		0			0
$394$	194.647	+	176.101i	0.494029	+	0.446957i
$395$	−524.465	−	524.465i	−1.32776	−	1.32776i
$396$		0			0
$397$	−6.09563	−	6.09563i	−0.0153542	−	0.0153542i	0.699388	−	0.714742i	$-0.253456\pi$
$397$	−6.09563	−	6.09563i	−0.0153542	−	0.0153542i	−0.714742	+	0.699388i	$0.753456\pi$
$398$	−16.5428	−	330.696i	−0.0415649	−	0.830895i
$399$		0			0
$400$	−150.314	−	741.787i	−0.375785	−	1.85447i
$401$	−598.130			−1.49160			−0.745798	−	0.666172i	$-0.767932\pi$
$401$	−598.130			−1.49160			−0.745798	+	0.666172i	$0.767932\pi$
$402$		0			0
$403$	−215.550	+	215.550i	−0.534863	+	0.534863i
$404$	−170.058	+	207.975i	−0.420936	+	0.514789i
$405$		0			0
$406$	371.729	−	410.878i	0.915589	−	1.01202i
$407$	315.407			0.774957
$408$		0			0
$409$		−	271.241i		−	0.663181i	−0.943423	−	0.331591i	$-0.892415\pi$
$409$		−	271.241i		−	0.663181i	0.943423	−	0.331591i	$-0.107585\pi$
$410$	−76.1892	+	84.2132i	−0.185827	+	0.205398i
$411$		0			0
$412$	−30.1892	+	3.02796i	−0.0732749	+	0.00734943i
$413$	−68.4785	−	68.4785i	−0.165808	−	0.165808i
$414$		0			0
$415$			114.709i			0.276406i
$416$	203.868	−	343.612i	0.490068	−	0.825990i
$417$		0			0
$418$	30.9224	+	618.148i	0.0739769	+	1.47882i
$419$	412.163	−	412.163i	0.983681	−	0.983681i	−0.0161876	−	0.999869i	$-0.505153\pi$
$419$	412.163	−	412.163i	0.983681	−	0.983681i	0.999869	+	0.0161876i	$0.00515289\pi$
$420$		0			0
$421$	135.609	−	135.609i	0.322111	−	0.322111i	−0.527466	−	0.849576i	$-0.676858\pi$
$421$	135.609	−	135.609i	0.322111	−	0.322111i	0.849576	+	0.527466i	$0.176858\pi$
$422$	−393.832	−	356.307i	−0.933252	−	0.844330i
$423$		0			0
$424$	297.062	+	219.082i	0.700618	+	0.516703i
$425$			670.550i			1.57777i
$426$		0			0
$427$	−563.083	−	563.083i	−1.31869	−	1.31869i
$428$	−524.442	−	428.829i	−1.22533	−	1.00194i
$429$		0			0
$430$	1.07895	+	21.5685i	0.00250918	+	0.0501593i
$431$			204.169i			0.473710i	0.971545	+	0.236855i	$0.0761166\pi$
$431$			204.169i			0.473710i	−0.971545	+	0.236855i	$0.923883\pi$
$432$		0			0
$433$	317.530			0.733325			0.366663	−	0.930354i	$-0.380500\pi$
$433$	317.530			0.733325			0.366663	+	0.930354i	$0.380500\pi$
$434$	401.457	−	20.0826i	0.925016	−	0.0462732i
$435$		0			0
$436$	−267.248	−	218.525i	−0.612954	−	0.501204i
$437$	223.743	−	223.743i	0.511998	−	0.511998i
$438$		0			0
$439$	226.050			0.514921			0.257461	−	0.966289i	$-0.417114\pi$
$439$	226.050			0.514921			0.257461	+	0.966289i	$0.417114\pi$
$440$	−93.6335	−	619.754i	−0.212803	−	1.40853i
$441$		0			0
$442$	−237.480	+	262.491i	−0.537286	+	0.593871i
$443$	157.388	+	157.388i	0.355278	+	0.355278i	0.862069	−	0.506791i	$-0.169169\pi$
$443$	157.388	+	157.388i	0.355278	+	0.355278i	−0.506791	+	0.862069i	$0.669169\pi$
$444$		0			0
$445$	880.782	+	880.782i	1.97929	+	1.97929i
$446$	613.617	−	30.6957i	1.37582	−	0.0688244i
$447$		0			0
$448$	−503.325	+	155.639i	−1.12349	+	0.347409i
$449$	458.567			1.02131			0.510653	−	0.859787i	$-0.329404\pi$
$449$	458.567			1.02131			0.510653	+	0.859787i	$0.329404\pi$
$450$		0			0
$451$	−43.5072	+	43.5072i	−0.0964684	+	0.0964684i
$452$	125.919	−	12.6296i	0.278582	−	0.0279416i
$453$		0			0
$454$	236.925	+	214.351i	0.521862	+	0.472138i
$455$	−873.954			−1.92078
$456$		0			0
$457$		−	548.682i		−	1.20062i	−0.799768	−	0.600309i	$-0.795044\pi$
$457$		−	548.682i		−	1.20062i	0.799768	−	0.600309i	$-0.204956\pi$
$458$	15.3576	+	13.8943i	0.0335318	+	0.0303368i
$459$		0			0
$460$	−202.842	+	248.069i	−0.440962	+	0.539280i
$461$	−359.058	−	359.058i	−0.778868	−	0.778868i	0.200770	−	0.979638i	$-0.435656\pi$
$461$	−359.058	−	359.058i	−0.778868	−	0.778868i	−0.979638	+	0.200770i	$0.935656\pi$
$462$		0			0
$463$		−	126.697i		−	0.273643i	−0.990596	−	0.136822i	$-0.956311\pi$
$463$		−	126.697i		−	0.273643i	0.990596	−	0.136822i	$-0.0436887\pi$
$464$	448.792	+	297.554i	0.967223	+	0.641280i
$465$		0			0
$466$	230.528	−	11.5320i	0.494696	−	0.0247468i
$467$	−526.834	+	526.834i	−1.12812	+	1.12812i	−0.137643	+	0.990482i	$0.543953\pi$
$467$	−526.834	+	526.834i	−1.12812	+	1.12812i	−0.990482	+	0.137643i	$0.956047\pi$
$468$		0			0
$469$	−63.5436	+	63.5436i	−0.135487	+	0.135487i
$470$	287.976	−	318.305i	0.612716	−	0.677245i
$471$		0			0
$472$	55.8613	−	75.7445i	0.118350	−	0.160476i
$473$			11.7004i			0.0247366i
$474$		0			0
$475$	1123.41	+	1123.41i	2.36508	+	2.36508i
$476$	464.428	−	46.5819i	0.975689	−	0.0978611i
$477$		0			0
$478$	645.735	−	32.3024i	1.35091	−	0.0675782i
$479$			712.831i			1.48817i	0.668087	+	0.744083i	$0.267113\pi$
$479$			712.831i			1.48817i	−0.668087	+	0.744083i	$0.732887\pi$
$480$		0			0
$481$	427.398			0.888560
$482$	−11.7966	−	235.818i	−0.0244743	−	0.489248i
$483$		0			0
$484$	14.4117	+	143.687i	0.0297762	+	0.296873i
$485$	−608.297	+	608.297i	−1.25422	+	1.25422i
$486$		0			0
$487$	579.851			1.19066			0.595330	−	0.803481i	$-0.297021\pi$
$487$	579.851			1.19066			0.595330	+	0.803481i	$0.297021\pi$
$488$	459.334	−	622.830i	0.941259	−	1.27629i
$489$		0			0
$490$	236.630	+	214.083i	0.482918	+	0.436905i
$491$	−131.309	−	131.309i	−0.267432	−	0.267432i	0.560632	−	0.828065i	$-0.310558\pi$
$491$	−131.309	−	131.309i	−0.267432	−	0.267432i	−0.828065	+	0.560632i	$0.810558\pi$
$492$		0			0
$493$	−337.336	−	337.336i	−0.684251	−	0.684251i
$494$	41.9018	+	837.630i	0.0848215	+	1.69561i
$495$		0			0
$496$	77.5809	+	382.856i	0.156413	+	0.771887i
$497$	1129.90			2.27343
$498$		0			0
$499$	329.208	−	329.208i	0.659736	−	0.659736i	−0.295582	−	0.955318i	$-0.595513\pi$
$499$	329.208	−	329.208i	0.659736	−	0.659736i	0.955318	+	0.295582i	$0.0955134\pi$
$500$	−587.281	−	480.212i	−1.17456	−	0.960424i
$501$		0			0
$502$	−315.432	+	348.652i	−0.628350	+	0.694526i
$503$	254.299			0.505564			0.252782	−	0.967523i	$-0.418654\pi$
$503$	254.299			0.505564			0.252782	+	0.967523i	$0.418654\pi$
$504$		0			0
$505$			571.096i			1.13088i
$506$	−116.478	+	128.745i	−0.230193	+	0.254436i
$507$		0			0
$508$	35.0787	+	349.740i	0.0690525	+	0.688464i
$509$	326.512	+	326.512i	0.641477	+	0.641477i	0.950918	−	0.309442i	$-0.100142\pi$
$509$	326.512	+	326.512i	0.641477	+	0.641477i	−0.309442	+	0.950918i	$0.600142\pi$
$510$		0			0
$511$			435.182i			0.851628i
$512$	−222.630	−	461.064i	−0.434824	−	0.900515i
$513$		0			0
$514$	−34.2483	−	684.634i	−0.0666309	−	1.33197i
$515$	−45.6071	+	45.6071i	−0.0885574	+	0.0885574i
$516$		0			0
$517$	164.447	−	164.447i	0.318079	−	0.318079i
$518$	−417.920	−	378.099i	−0.806795	−	0.729922i
$519$		0			0
$520$	−126.879	−	839.807i	−0.243999	−	1.61501i
$521$		−	775.042i		−	1.48760i	−0.668400	−	0.743802i	$-0.733020\pi$
$521$		−	775.042i		−	1.48760i	0.668400	−	0.743802i	$-0.266980\pi$
$522$		0			0
$523$	−461.875	−	461.875i	−0.883126	−	0.883126i	0.110725	−	0.993851i	$-0.464683\pi$
$523$	−461.875	−	461.875i	−0.883126	−	0.883126i	−0.993851	+	0.110725i	$0.964683\pi$
$524$	631.483	−	772.279i	1.20512	−	1.47382i
$525$		0			0
$526$	21.9469	+	438.726i	0.0417242	+	0.834080i
$527$		−	346.089i		−	0.656715i
$528$		0			0
$529$	−440.240			−0.832212
$530$	783.673	−	39.2027i	1.47863	−	0.0739673i
$531$		0			0
$532$	700.042	−	856.124i	1.31587	−	1.60926i
$533$	−58.9551	+	58.9551i	−0.110610	+	0.110610i
$534$		0			0
$535$	−1440.11			−2.69180
$536$	−70.2860	−	51.8356i	−0.131131	−	0.0967083i
$537$		0			0
$538$	147.833	−	163.402i	0.274783	−	0.303722i
$539$	122.251	+	122.251i	0.226810	+	0.226810i
$540$		0			0
$541$	−426.071	−	426.071i	−0.787563	−	0.787563i	0.193531	−	0.981094i	$-0.438006\pi$
$541$	−426.071	−	426.071i	−0.787563	−	0.787563i	−0.981094	+	0.193531i	$0.938006\pi$
$542$	−309.796	+	15.4973i	−0.571579	+	0.0285928i
$543$		0			0
$544$	112.187	+	439.519i	0.206226	+	0.807940i
$545$	−733.860			−1.34653
$546$		0			0
$547$	−111.138	+	111.138i	−0.203178	+	0.203178i	−0.801360	−	0.598182i	$-0.795890\pi$
$547$	−111.138	+	111.138i	−0.203178	+	0.203178i	0.598182	+	0.801360i	$0.295890\pi$
$548$	38.7308	+	386.152i	0.0706767	+	0.704657i
$549$		0			0
$550$	−646.425	−	584.832i	−1.17532	−	1.06333i
$551$	−1130.32			−2.05139
$552$		0			0
$553$			718.041i			1.29845i
$554$	63.0488	+	57.0414i	0.113807	+	0.102963i
$555$		0			0
$556$	570.266	+	466.299i	1.02566	+	0.838668i
$557$	233.731	+	233.731i	0.419625	+	0.419625i	0.885075	−	0.465449i	$-0.154107\pi$
$557$	233.731	+	233.731i	0.419625	+	0.419625i	−0.465449	+	0.885075i	$0.654107\pi$
$558$		0			0
$559$			15.8548i			0.0283628i
$560$	−618.874	+	933.428i	−1.10513	+	1.66684i
$561$		0			0
$562$	−530.517	+	26.5387i	−0.943981	+	0.0472219i
$563$	−261.362	+	261.362i	−0.464231	+	0.464231i	−0.900039	−	0.435809i	$-0.856462\pi$
$563$	−261.362	+	261.362i	−0.464231	+	0.464231i	0.435809	+	0.900039i	$0.356462\pi$
$564$		0			0
$565$	190.227	−	190.227i	0.336684	−	0.336684i
$566$	304.033	−	336.052i	0.537160	−	0.593732i
$567$		0			0
$568$	164.037	+	1085.75i	0.288797	+	1.91153i
$569$			453.881i			0.797682i	0.917020	+	0.398841i	$0.130587\pi$
$569$			453.881i			0.797682i	−0.917020	+	0.398841i	$0.869413\pi$
$570$		0			0
$571$	−289.009	−	289.009i	−0.506146	−	0.506146i	0.407195	−	0.913341i	$-0.366507\pi$
$571$	−289.009	−	289.009i	−0.506146	−	0.506146i	−0.913341	+	0.407195i	$0.866507\pi$
$572$	−45.9243	−	457.872i	−0.0802873	−	0.800476i
$573$		0			0
$574$	109.803	−	5.49280i	0.191294	−	0.00956933i
$575$			445.662i			0.775065i
$576$		0			0
$577$	588.041			1.01913			0.509567	−	0.860431i	$-0.329806\pi$
$577$	588.041			1.01913			0.509567	+	0.860431i	$0.329806\pi$
$578$	8.79921	+	175.899i	0.0152235	+	0.304324i
$579$		0			0
$580$	1138.97	−	114.238i	1.96374	−	0.196962i
$581$	78.5232	−	78.5232i	0.135152	−	0.135152i
$582$		0			0
$583$	425.124			0.729201
$584$	−418.179	+	63.1791i	−0.716059	+	0.108183i
$585$		0			0
$586$	−364.453	−	329.727i	−0.621933	−	0.562674i
$587$	392.596	+	392.596i	0.668818	+	0.668818i	0.957442	−	0.288624i	$-0.0931978\pi$
$587$	392.596	+	392.596i	0.668818	+	0.668818i	−0.288624	+	0.957442i	$0.593198\pi$
$588$		0			0
$589$	−579.822	−	579.822i	−0.984417	−	0.984417i
$590$	−9.99585	−	199.820i	−0.0169421	−	0.338678i
$591$		0			0
$592$	302.653	−	456.483i	0.511239	−	0.771086i
$593$	−1008.70			−1.70101			−0.850505	−	0.525967i	$-0.823703\pi$
$593$	−1008.70			−1.70101			−0.850505	+	0.525967i	$0.823703\pi$
$594$		0			0
$595$	701.614	−	701.614i	1.17918	−	1.17918i
$596$	−94.3879	+	115.433i	−0.158369	+	0.193679i
$597$		0			0
$598$	−157.835	+	174.457i	−0.263938	+	0.291735i
$599$	526.819			0.879498			0.439749	−	0.898121i	$-0.355067\pi$
$599$	526.819			0.879498			0.439749	+	0.898121i	$0.355067\pi$
$600$		0			0
$601$			867.891i			1.44408i	0.691853	+	0.722039i	$0.256795\pi$
$601$			867.891i			1.44408i	−0.691853	+	0.722039i	$0.743205\pi$
$602$	14.0260	−	15.5032i	0.0232991	−	0.0257528i
$603$		0			0
$604$	−780.511	+	78.2848i	−1.29224	+	0.129611i
$605$	217.068	+	217.068i	0.358790	+	0.358790i
$606$		0			0
$607$		−	98.8497i		−	0.162850i	−0.996679	−	0.0814248i	$-0.974053\pi$
$607$		−	98.8497i		−	0.162850i	0.996679	−	0.0814248i	$-0.0259470\pi$
$608$	924.305	+	548.399i	1.52024	+	0.901971i
$609$		0			0
$610$	−82.1935	−	1643.07i	−0.134743	−	2.69356i
$611$	222.836	−	222.836i	0.364707	−	0.364707i
$612$		0			0
$613$	−616.976	+	616.976i	−1.00649	+	1.00649i	−0.00650651	+	0.999979i	$0.502071\pi$
$613$	−616.976	+	616.976i	−1.00649	+	1.00649i	−0.999979	+	0.00650651i	$0.997929\pi$
$614$	−287.017	−	259.669i	−0.467454	−	0.422914i
$615$		0			0
$616$	−360.153	+	488.346i	−0.584664	+	0.792769i
$617$			1073.62i			1.74007i	0.492993	+	0.870033i	$0.335903\pi$
$617$			1073.62i			1.74007i	−0.492993	+	0.870033i	$0.664097\pi$
$618$		0			0
$619$	397.119	+	397.119i	0.641549	+	0.641549i	0.950936	−	0.309387i	$-0.100124\pi$
$619$	397.119	+	397.119i	0.641549	+	0.641549i	−0.309387	+	0.950936i	$0.600124\pi$
$620$	642.861	+	525.659i	1.03687	+	0.847837i
$621$		0			0
$622$	45.7767	+	915.091i	0.0735960	+	1.47121i
$623$		−	1205.87i		−	1.93559i
$624$		0			0
$625$	−430.067			−0.688107
$626$	−341.302	+	17.0734i	−0.545211	+	0.0272738i
$627$		0			0
$628$	−431.495	−	352.828i	−0.687094	−	0.561828i
$629$	−343.117	+	343.117i	−0.545495	+	0.545495i
$630$		0			0
$631$	−381.081			−0.603931			−0.301966	−	0.953319i	$-0.597643\pi$
$631$	−381.081			−0.603931			−0.301966	+	0.953319i	$0.597643\pi$
$632$	−689.985	+	104.244i	−1.09175	+	0.164943i
$633$		0			0
$634$	−706.963	+	781.418i	−1.11508	+	1.23252i
$635$	528.354	+	528.354i	0.832053	+	0.832053i
$636$		0			0
$637$	165.657	+	165.657i	0.260059	+	0.260059i
$638$	619.412	−	30.9856i	0.970865	−	0.0485668i
$639$		0			0
$640$	−986.804	−	459.179i	−1.54188	−	0.717468i
$641$	−485.443			−0.757321			−0.378661	−	0.925536i	$-0.623615\pi$
$641$	−485.443			−0.757321			−0.378661	+	0.925536i	$0.623615\pi$
$642$		0			0
$643$	457.641	−	457.641i	0.711728	−	0.711728i	−0.255168	−	0.966897i	$-0.582131\pi$
$643$	457.641	−	457.641i	0.711728	−	0.711728i	0.966897	+	0.255168i	$0.0821308\pi$
$644$	308.669	−	30.9593i	0.479300	−	0.0480735i
$645$		0			0
$646$	−706.092	−	638.814i	−1.09302	−	0.988877i
$647$	845.780			1.30723			0.653617	−	0.756826i	$-0.273251\pi$
$647$	845.780			1.30723			0.653617	+	0.756826i	$0.273251\pi$
$648$		0			0
$649$		−	108.398i		−	0.167023i
$650$	−875.948	−	792.486i	−1.34761	−	1.21921i
$651$		0			0
$652$	324.666	−	397.054i	0.497954	−	0.608979i
$653$	−11.1039	−	11.1039i	−0.0170044	−	0.0170044i	0.698553	−	0.715558i	$-0.253827\pi$
$653$	−11.1039	−	11.1039i	−0.0170044	−	0.0170044i	−0.715558	+	0.698553i	$0.753827\pi$
$654$		0			0
$655$		−	2120.67i		−	3.23766i
$656$	21.2192	+	104.715i	0.0323463	+	0.159627i
$657$		0			0
$658$	−415.027	+	20.7614i	−0.630741	+	0.0315523i
$659$	−296.711	+	296.711i	−0.450245	+	0.450245i	−0.895436	−	0.445191i	$-0.853136\pi$
$659$	−296.711	+	296.711i	−0.450245	+	0.450245i	0.445191	+	0.895436i	$0.353136\pi$
$660$		0			0
$661$	−0.564899	+	0.564899i	−0.000854613	+	0.000854613i	−0.707534	−	0.706679i	$-0.750192\pi$
$661$	−0.564899	+	0.564899i	−0.000854613	+	0.000854613i	0.706679	+	0.707534i	$0.250192\pi$
$662$	−337.171	+	372.680i	−0.509321	+	0.562961i
$663$		0			0
$664$	86.8551	+	64.0553i	0.130806	+	0.0964688i
$665$		−	2350.91i		−	3.53520i
$666$		0			0
$667$	−224.201	−	224.201i	−0.336133	−	0.336133i
$668$	−722.782	+	72.4946i	−1.08201	+	0.108525i
$669$		0			0
$670$	−185.420	+	9.27550i	−0.276746	+	0.0138440i
$671$		−	891.329i		−	1.32836i
$672$		0			0
$673$	−1091.24			−1.62146			−0.810731	−	0.585419i	$-0.800930\pi$
$673$	−1091.24			−1.62146			−0.810731	+	0.585419i	$0.800930\pi$
$674$	−49.1304	−	982.132i	−0.0728938	−	1.45717i
$675$		0			0
$676$	5.23345	+	52.1783i	0.00774180	+	0.0771868i
$677$	494.409	−	494.409i	0.730294	−	0.730294i	−0.240384	−	0.970678i	$-0.577273\pi$
$677$	494.409	−	494.409i	0.730294	−	0.730294i	0.970678	+	0.240384i	$0.0772733\pi$
$678$		0			0
$679$	832.814			1.22653
$680$	776.060	+	572.341i	1.14126	+	0.841678i
$681$		0			0
$682$	333.637	+	301.847i	0.489204	+	0.442591i
$683$	−845.268	−	845.268i	−1.23758	−	1.23758i	−0.960987	−	0.276594i	$-0.910794\pi$
$683$	−845.268	−	845.268i	−1.23758	−	1.23758i	−0.276594	−	0.960987i	$-0.589206\pi$
$684$		0			0
$685$	583.362	+	583.362i	0.851623	+	0.851623i
$686$	24.8711	+	497.182i	0.0362553	+	0.724755i
$687$		0			0
$688$	16.9338	+	11.2273i	0.0246130	+	0.0163187i
$689$	576.071			0.836097
$690$		0			0
$691$	457.715	−	457.715i	0.662395	−	0.662395i	−0.293549	−	0.955944i	$-0.594836\pi$
$691$	457.715	−	457.715i	0.662395	−	0.662395i	0.955944	+	0.293549i	$0.0948364\pi$
$692$	−11.6720	−	9.54407i	−0.0168671	−	0.0137920i
$693$		0			0
$694$	335.470	−	370.801i	0.483386	−	0.534295i
$695$	1565.95			2.25316
$696$		0			0
$697$		−	94.6589i		−	0.135809i
$698$	−516.707	+	571.125i	−0.740268	+	0.818230i
$699$		0			0
$700$	155.446	+	1549.82i	0.222066	+	2.21403i
$701$	−199.252	−	199.252i	−0.284239	−	0.284239i	0.550558	−	0.834797i	$-0.314415\pi$
$701$	−199.252	−	199.252i	−0.284239	−	0.284239i	−0.834797	+	0.550558i	$0.814415\pi$
$702$		0			0
$703$			1149.69i			1.63540i
$704$	−521.552	−	275.184i	−0.740841	−	0.390886i
$705$		0			0
$706$	−16.7970	−	335.777i	−0.0237918	−	0.475605i
$707$	390.941	−	390.941i	0.552958	−	0.552958i
$708$		0			0
$709$	−547.374	+	547.374i	−0.772036	+	0.772036i	−0.978462	−	0.206426i	$-0.933817\pi$
$709$	−547.374	+	547.374i	−0.772036	+	0.772036i	0.206426	+	0.978462i	$0.433817\pi$
$710$	1730.98	+	1566.05i	2.43801	+	2.20571i
$711$		0			0
$712$	1158.75	−	175.067i	1.62746	−	0.245880i
$713$		−	230.018i		−	0.322606i
$714$		0			0
$715$	−691.710	−	691.710i	−0.967427	−	0.967427i
$716$	−45.6525	+	55.8313i	−0.0637605	+	0.0779766i
$717$		0			0
$718$	−47.5644	−	950.828i	−0.0662457	−	1.32427i
$719$		−	90.5082i		−	0.125881i	−0.998017	−	0.0629403i	$-0.979952\pi$
$719$		−	90.5082i		−	0.125881i	0.998017	−	0.0629403i	$-0.0200478\pi$
$720$		0			0
$721$	62.4402			0.0866022
$722$	−1532.10	+	76.6421i	−2.12202	+	0.106153i
$723$		0			0
$724$	219.508	−	268.450i	0.303188	−	0.370788i
$725$	1125.71	−	1125.71i	1.55270	−	1.55270i
$726$		0			0
$727$	−222.068			−0.305459			−0.152729	−	0.988268i	$-0.548806\pi$
$727$	−222.068			−0.305459			−0.152729	+	0.988268i	$0.548806\pi$
$728$	−488.031	+	661.741i	−0.670372	+	0.908984i
$729$		0			0
$730$	−603.168	+	666.692i	−0.826258	+	0.913277i
$731$	−12.7283	−	12.7283i	−0.0174122	−	0.0174122i
$732$		0			0
$733$	201.142	+	201.142i	0.274410	+	0.274410i	0.830873	−	0.556463i	$-0.187842\pi$
$733$	201.142	+	201.142i	0.274410	+	0.274410i	−0.556463	+	0.830873i	$0.687842\pi$
$734$	209.526	−	10.4814i	0.285457	−	0.0142798i
$735$		0			0
$736$	74.5618	+	292.114i	0.101307	+	0.396894i
$737$	−100.586			−0.136480
$738$		0			0
$739$	−771.595	+	771.595i	−1.04411	+	1.04411i	−0.0451261	+	0.998981i	$0.514369\pi$
$739$	−771.595	+	771.595i	−1.04411	+	1.04411i	−0.998981	+	0.0451261i	$0.985631\pi$
$740$	−116.195	−	1158.48i	−0.157021	−	1.56552i
$741$		0			0
$742$	−563.296	−	509.624i	−0.759159	−	0.686825i
$743$	−754.171			−1.01503			−0.507517	−	0.861641i	$-0.669437\pi$
$743$	−754.171			−1.01503			−0.507517	+	0.861641i	$0.669437\pi$
$744$		0			0
$745$			316.977i			0.425473i
$746$	256.517	+	232.075i	0.343856	+	0.311093i
$747$		0			0
$748$	404.450	+	330.714i	0.540709	+	0.442131i
$749$	985.822	+	985.822i	1.31618	+	1.31618i
$750$		0			0
$751$		−	606.477i		−	0.807559i	−0.914856	−	0.403780i	$-0.867696\pi$
$751$		−	606.477i		−	0.807559i	0.914856	−	0.403780i	$-0.132304\pi$
$752$	−80.2033	−	395.797i	−0.106653	−	0.526326i
$753$		0			0
$754$	839.344	−	41.9875i	1.11319	−	0.0556863i
$755$	−1179.12	+	1179.12i	−1.56175	+	1.56175i
$756$		0			0
$757$	−852.254	+	852.254i	−1.12583	+	1.12583i	−0.134983	+	0.990848i	$0.543098\pi$
$757$	−852.254	+	852.254i	−1.12583	+	1.12583i	−0.990848	+	0.134983i	$0.956902\pi$
$758$	143.386	−	158.487i	0.189163	−	0.209085i
$759$		0			0
$760$	2259.05	−	341.301i	2.97244	−	0.449081i
$761$			1373.25i			1.80453i	0.431180	+	0.902266i	$0.358097\pi$
$761$			1373.25i			1.80453i	−0.431180	+	0.902266i	$0.641903\pi$
$762$		0			0
$763$	502.360	+	502.360i	0.658401	+	0.658401i
$764$	31.9837	+	318.882i	0.0418634	+	0.417385i
$765$		0			0
$766$	1001.77	−	50.1126i	1.30779	−	0.0654211i
$767$		−	146.886i		−	0.191507i
$768$		0			0
$769$	515.727			0.670647			0.335323	−	0.942103i	$-0.391154\pi$
$769$	515.727			0.670647			0.335323	+	0.942103i	$0.391154\pi$
$770$	64.4460	+	1288.30i	0.0836961	+	1.67311i
$771$		0			0
$772$	−114.448	+	11.4791i	−0.148248	+	0.0148692i
$773$	448.985	−	448.985i	0.580834	−	0.580834i	−0.354299	−	0.935132i	$-0.615280\pi$
$773$	448.985	−	448.985i	0.580834	−	0.580834i	0.935132	+	0.354299i	$0.115280\pi$
$774$		0			0
$775$	1154.92			1.49022
$776$	120.907	+	800.274i	0.155808	+	1.03128i
$777$		0			0
$778$	−47.4130	−	42.8954i	−0.0609422	−	0.0551355i
$779$	−158.587	−	158.587i	−0.203578	−	0.203578i
$780$		0			0
$781$	894.282	+	894.282i	1.14505	+	1.14505i
$782$	−13.3447	−	266.765i	−0.0170649	−	0.341132i
$783$		0			0
$784$	294.238	−	59.6236i	0.375303	−	0.0760505i
$785$	−1184.88			−1.50940
$786$		0			0
$787$	70.0485	−	70.0485i	0.0890070	−	0.0890070i	−0.661201	−	0.750208i	$-0.729953\pi$
$787$	70.0485	−	70.0485i	0.0890070	−	0.0890070i	0.750208	+	0.661201i	$0.229953\pi$
$788$	332.312	−	406.405i	0.421716	−	0.515743i
$789$		0			0
$790$	−995.214	+	1100.03i	−1.25976	+	1.39244i
$791$	−260.437			−0.329251
$792$		0			0
$793$		−	1207.81i		−	1.52309i
$794$	−11.5669	+	12.7851i	−0.0145679	+	0.0161022i
$795$		0			0
$796$	−658.914	+	66.0887i	−0.827781	+	0.0830260i
$797$	584.026	+	584.026i	0.732780	+	0.732780i	0.971170	−	0.238390i	$-0.0766195\pi$
$797$	584.026	+	584.026i	0.732780	+	0.732780i	−0.238390	+	0.971170i	$0.576620\pi$
$798$		0			0
$799$			357.787i			0.447794i
$800$	−1466.70	+	374.374i	−1.83338	+	0.467968i
$801$		0			0
$802$	59.7673	+	1194.77i	0.0745228	+	1.48973i
$803$	−344.434	+	344.434i	−0.428935	+	0.428935i
$804$		0			0
$805$	466.308	−	466.308i	0.579265	−	0.579265i
$806$	452.100	+	409.023i	0.560918	+	0.507472i
$807$		0			0
$808$	432.423	+	318.910i	0.535176	+	0.394691i
$809$		−	1252.15i		−	1.54778i	−0.633321	−	0.773889i	$-0.718309\pi$
$809$		−	1252.15i		−	1.54778i	0.633321	−	0.773889i	$-0.281691\pi$
$810$		0			0
$811$	−255.775	−	255.775i	−0.315382	−	0.315382i	0.531608	−	0.846990i	$-0.321588\pi$
$811$	−255.775	−	255.775i	−0.315382	−	0.315382i	−0.846990	+	0.531608i	$0.821588\pi$
$812$	−857.875	−	701.473i	−1.05650	−	0.863883i
$813$		0			0
$814$	−31.5166	−	630.027i	−0.0387182	−	0.773989i
$815$		−	1090.31i		−	1.33780i
$816$		0			0
$817$	−42.6489			−0.0522018
$818$	−541.805	+	27.1034i	−0.662353	+	0.0331337i
$819$		0			0
$820$	175.829	+	143.773i	0.214426	+	0.175333i
$821$	228.380	−	228.380i	0.278173	−	0.278173i	−0.554206	−	0.832379i	$-0.686978\pi$
$821$	228.380	−	228.380i	0.278173	−	0.278173i	0.832379	+	0.554206i	$0.186978\pi$
$822$		0			0
$823$	1050.55			1.27649			0.638246	−	0.769832i	$-0.279660\pi$
$823$	1050.55			1.27649			0.638246	+	0.769832i	$0.279660\pi$
$824$	9.06498	+	60.0005i	0.0110012	+	0.0728162i
$825$		0			0
$826$	−129.943	+	143.629i	−0.157316	+	0.173885i
$827$	78.0300	+	78.0300i	0.0943531	+	0.0943531i	0.752708	−	0.658355i	$-0.228747\pi$
$827$	78.0300	+	78.0300i	0.0943531	+	0.0943531i	−0.658355	+	0.752708i	$0.728747\pi$
$828$		0			0
$829$	−517.848	−	517.848i	−0.624665	−	0.624665i	0.322056	−	0.946721i	$-0.395626\pi$
$829$	−517.848	−	517.848i	−0.624665	−	0.624665i	−0.946721	+	0.322056i	$0.895626\pi$
$830$	229.131	−	11.4621i	0.276061	−	0.0138097i
$831$		0			0
$832$	−706.737	−	372.892i	−0.849443	−	0.448188i
$833$	−265.981			−0.319305
$834$		0			0
$835$	−1091.91	+	1091.91i	−1.30768	+	1.30768i
$836$	1231.66	−	123.535i	1.47328	−	0.147769i
$837$		0			0
$838$	−864.480	−	782.111i	−1.03160	−	0.933306i
$839$	649.581			0.774232			0.387116	−	0.922031i	$-0.373471\pi$
$839$	649.581			0.774232			0.387116	+	0.922031i	$0.373471\pi$
$840$		0			0
$841$			291.627i			0.346762i
$842$	−284.429	−	257.328i	−0.337802	−	0.305615i
$843$		0			0
$844$	−672.371	+	822.284i	−0.796648	+	0.974270i
$845$	78.8260	+	78.8260i	0.0932852	+	0.0932852i
$846$		0			0
$847$		−	297.186i		−	0.350869i
$848$	407.933	−	615.273i	0.481054	−	0.725558i
$849$		0			0
$850$	1339.43	−	67.0037i	1.57580	−	0.0788279i
$851$	−228.043	+	228.043i	−0.267970	+	0.267970i
$852$		0			0
$853$	1124.04	−	1124.04i	1.31775	−	1.31775i	0.402202	−	0.915551i	$-0.368245\pi$
$853$	1124.04	−	1124.04i	1.31775	−	1.31775i	0.915551	−	0.402202i	$-0.131755\pi$
$854$	−1068.49	+	1181.02i	−1.25116	+	1.38293i
$855$		0			0
$856$	−804.184	+	1090.42i	−0.939467	+	1.27386i
$857$		−	596.973i		−	0.696585i	−0.937386	−	0.348292i	$-0.886762\pi$
$857$		−	596.973i		−	0.696585i	0.937386	−	0.348292i	$-0.113238\pi$
$858$		0			0
$859$	701.595	+	701.595i	0.816758	+	0.816758i	0.985637	−	0.168879i	$-0.0540147\pi$
$859$	701.595	+	701.595i	0.816758	+	0.816758i	−0.168879	+	0.985637i	$0.554015\pi$
$860$	42.9753	−	4.31040i	0.0499713	−	0.00501210i
$861$		0			0
$862$	407.828	−	20.4013i	0.473118	−	0.0236674i
$863$		−	437.703i		−	0.507187i	−0.967311	−	0.253594i	$-0.918387\pi$
$863$		−	437.703i		−	0.507187i	0.967311	−	0.253594i	$-0.0816126\pi$
$864$		0			0
$865$	−32.0513			−0.0370535
$866$	−31.7287	−	634.266i	−0.0366382	−	0.732409i
$867$		0			0
$868$	−80.2300	−	799.905i	−0.0924309	−	0.921549i
$869$	−568.309	+	568.309i	−0.653981	+	0.653981i
$870$		0			0
$871$	−136.301			−0.156487
$872$	−409.800	+	555.664i	−0.469954	+	0.637229i
$873$		0			0
$874$	−469.284	−	424.570i	−0.536939	−	0.485778i
$875$	1103.94	+	1103.94i	1.26165	+	1.26165i
$876$		0			0
$877$	769.110	+	769.110i	0.876978	+	0.876978i	0.993221	−	0.116243i	$-0.0370850\pi$
$877$	769.110	+	769.110i	0.876978	+	0.876978i	−0.116243	+	0.993221i	$0.537085\pi$
$878$	−22.5877	−	451.536i	−0.0257264	−	0.514278i
$879$		0			0
$880$	−1228.60	+	248.961i	−1.39614	+	0.282910i
$881$	−351.923			−0.399459			−0.199729	−	0.979851i	$-0.564006\pi$
$881$	−351.923			−0.399459			−0.199729	+	0.979851i	$0.564006\pi$
$882$		0			0
$883$	661.014	−	661.014i	0.748601	−	0.748601i	−0.225616	−	0.974216i	$-0.572439\pi$
$883$	661.014	−	661.014i	0.748601	−	0.748601i	0.974216	+	0.225616i	$0.0724394\pi$
$884$	548.056	+	448.138i	0.619973	+	0.506944i
$885$		0			0
$886$	298.656	−	330.110i	0.337084	−	0.372584i
$887$	−794.853			−0.896114			−0.448057	−	0.894005i	$-0.647884\pi$
$887$	−794.853			−0.896114			−0.448057	+	0.894005i	$0.647884\pi$
$888$		0			0
$889$		−	723.364i		−	0.813683i
$890$	1671.35	−	1847.38i	1.87793	−	2.07570i
$891$		0			0
$892$	−122.629	−	1222.63i	−0.137477	−	1.37066i
$893$	599.421	+	599.421i	0.671244	+	0.671244i
$894$		0			0
$895$			153.312i			0.171298i
$896$	361.183	+	989.841i	0.403106	+	1.10473i
$897$		0			0
$898$	−45.8216	−	915.988i	−0.0510263	−	1.02003i
$899$	−581.008	+	581.008i	−0.646282	+	0.646282i
$900$		0			0
$901$	−462.472	+	462.472i	−0.513288	+	0.513288i
$902$	91.2532	+	82.5584i	0.101168	+	0.0915282i
$903$		0			0
$904$	−37.8099	−	250.262i	−0.0418252	−	0.276838i
$905$		−	737.162i		−	0.814544i
$906$		0			0
$907$	−943.805	−	943.805i	−1.04058	−	1.04058i	−0.999141	−	0.0414379i	$-0.986806\pi$
$907$	−943.805	−	943.805i	−1.04058	−	1.04058i	−0.0414379	−	0.999141i	$-0.513194\pi$
$908$	404.491	−	494.678i	0.445475	−	0.544799i
$909$		0			0
$910$	87.3285	+	1745.72i	0.0959654	+	1.91838i
$911$		−	190.842i		−	0.209487i	−0.994499	−	0.104743i	$-0.966598\pi$
$911$		−	190.842i		−	0.209487i	0.994499	−	0.104743i	$-0.0334021\pi$
$912$		0			0
$913$	124.298			0.136142
$914$	−1095.99	+	54.8262i	−1.19912	+	0.0599849i
$915$		0			0
$916$	26.2192	−	32.0651i	0.0286236	−	0.0350056i
$917$	−1451.70	+	1451.70i	−1.58309	+	1.58309i
$918$		0			0
$919$	−925.230			−1.00678			−0.503389	−	0.864060i	$-0.667914\pi$
$919$	−925.230			−1.00678			−0.503389	+	0.864060i	$0.667914\pi$
$920$	515.786	+	380.390i	0.560637	+	0.413468i
$921$		0			0
$922$	−681.341	+	753.098i	−0.738982	+	0.816809i
$923$	1211.81	+	1211.81i	1.31290	+	1.31290i
$924$		0			0
$925$	−1145.00	−	1145.00i	−1.23784	−	1.23784i
$926$	−253.077	+	12.6600i	−0.273301	+	0.0136717i
$927$		0			0
$928$	549.520	−	926.195i	0.592155	−	0.998055i
$929$	−91.8388			−0.0988577			−0.0494289	−	0.998778i	$-0.515740\pi$
$929$	−91.8388			−0.0988577			−0.0494289	+	0.998778i	$0.515740\pi$
$930$		0			0
$931$	−445.613	+	445.613i	−0.478639	+	0.478639i
$932$	−46.0704	−	459.329i	−0.0494318	−	0.492842i
$933$		0			0
$934$	1105.00	+	999.710i	1.18308	+	1.07035i
$935$	1110.62			1.18782
$936$		0			0
$937$			1615.44i			1.72406i	0.506857	+	0.862030i	$0.330807\pi$
$937$			1615.44i			1.72406i	−0.506857	+	0.862030i	$0.669193\pi$
$938$	133.278	+	120.579i	0.142087	+	0.128549i
$939$		0			0
$940$	−664.591	−	543.427i	−0.707012	−	0.578114i
$941$	−484.931	−	484.931i	−0.515336	−	0.515336i	0.400820	−	0.916157i	$-0.368725\pi$
$941$	−484.931	−	484.931i	−0.515336	−	0.515336i	−0.916157	+	0.400820i	$0.868725\pi$
$942$		0			0
$943$		−	62.9124i		−	0.0667151i
$944$	−156.882	−	104.014i	−0.166188	−	0.110185i
$945$		0			0
$946$	23.3716	−	1.16914i	0.0247057	−	0.00123588i
$947$	1053.02	−	1053.02i	1.11195	−	1.11195i	0.119064	−	0.992887i	$-0.462010\pi$
$947$	1053.02	−	1053.02i	1.11195	−	1.11195i	0.992887	−	0.119064i	$-0.0379895\pi$
$948$		0			0
$949$	−466.731	+	466.731i	−0.491814	+	0.491814i
$950$	2131.76	−	2356.27i	2.24396	−	2.48029i
$951$		0			0
$952$	−139.455	−	923.042i	−0.146486	−	0.969581i
$953$			1256.72i			1.31870i	0.751838	+	0.659348i	$0.229168\pi$
$953$			1256.72i			1.31870i	−0.751838	+	0.659348i	$0.770832\pi$
$954$		0			0
$955$	481.736	+	481.736i	0.504436	+	0.504436i
$956$	−129.048	−	1286.63i	−0.134988	−	1.34585i
$957$		0			0
$958$	1423.88	−	71.2286i	1.48631	−	0.0743514i
$959$		−	798.675i		−	0.832821i
$960$		0			0
$961$	364.917			0.379726
$962$	−42.7071	−	853.728i	−0.0443940	−	0.887451i
$963$		0			0
$964$	−469.868	+	47.1274i	−0.487414	+	0.0488874i
$965$	−172.897	+	172.897i	−0.179168	+	0.179168i
$966$		0			0
$967$	−1842.43			−1.90530			−0.952651	−	0.304066i	$-0.901656\pi$
$967$	−1842.43			−1.90530			−0.952651	+	0.304066i	$0.901656\pi$
$968$	285.574	−	43.1451i	0.295015	−	0.0445714i
$969$		0			0
$970$	1275.86	+	1154.29i	1.31532	+	1.18999i
$971$	−802.530	−	802.530i	−0.826498	−	0.826498i	0.160532	−	0.987031i	$-0.448679\pi$
$971$	−802.530	−	802.530i	−0.826498	−	0.826498i	−0.987031	+	0.160532i	$0.948679\pi$
$972$		0			0
$973$	−1071.96	−	1071.96i	−1.10171	−	1.10171i
$974$	−57.9408	−	1158.25i	−0.0594874	−	1.18917i
$975$		0			0
$976$	−1290.00	−	855.286i	−1.32172	−	0.876318i
$977$	−1505.64			−1.54109			−0.770544	−	0.637387i	$-0.780015\pi$
$977$	−1505.64			−1.54109			−0.770544	+	0.637387i	$0.780015\pi$
$978$		0			0
$979$	954.413	−	954.413i	0.974886	−	0.974886i
$980$	403.987	−	494.061i	0.412232	−	0.504143i
$981$		0			0
$982$	−249.170	+	275.411i	−0.253737	+	0.280460i
$983$	−764.708			−0.777933			−0.388966	−	0.921252i	$-0.627168\pi$
$983$	−764.708			−0.777933			−0.388966	+	0.921252i	$0.627168\pi$
$984$		0			0
$985$		−	1115.98i		−	1.13298i
$986$	−640.121	+	707.536i	−0.649210	+	0.717583i
$987$		0			0
$988$	1668.98	−	167.398i	1.68925	−	0.169431i
$989$	−8.45951	−	8.45951i	−0.00855360	−	0.00855360i
$990$		0			0
$991$			1683.22i			1.69851i	0.527984	+	0.849254i	$0.322948\pi$
$991$			1683.22i			1.69851i	−0.527984	+	0.849254i	$0.677052\pi$
$992$	757.003	−	193.224i	0.763108	−	0.194783i
$993$		0			0
$994$	−112.903	−	2256.97i	−0.113585	−	2.27060i
$995$	−995.424	+	995.424i	−1.00043	+	1.00043i
$996$		0			0
$997$	942.082	−	942.082i	0.944917	−	0.944917i	−0.0536431	−	0.998560i	$-0.517083\pi$
$997$	942.082	−	942.082i	0.944917	−	0.944917i	0.998560	+	0.0536431i	$0.0170833\pi$
$998$	−690.490	−	624.699i	−0.691874	−	0.625950i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.3.m.b.19.4	✓	16
3.2	odd	2	inner	144.3.m.b.19.5	yes	16
4.3	odd	2		576.3.m.b.559.1		16
8.3	odd	2		1152.3.m.d.991.8		16
8.5	even	2		1152.3.m.e.991.8		16
12.11	even	2		576.3.m.b.559.8		16
16.3	odd	4		1152.3.m.e.415.8		16
16.5	even	4		576.3.m.b.271.1		16
16.11	odd	4	inner	144.3.m.b.91.4	yes	16
16.13	even	4		1152.3.m.d.415.8		16
24.5	odd	2		1152.3.m.e.991.1		16
24.11	even	2		1152.3.m.d.991.1		16
48.5	odd	4		576.3.m.b.271.8		16
48.11	even	4	inner	144.3.m.b.91.5	yes	16
48.29	odd	4		1152.3.m.d.415.1		16
48.35	even	4		1152.3.m.e.415.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.m.b.19.4	✓	16	1.1	even	1	trivial
144.3.m.b.19.5	yes	16	3.2	odd	2	inner
144.3.m.b.91.4	yes	16	16.11	odd	4	inner
144.3.m.b.91.5	yes	16	48.11	even	4	inner
576.3.m.b.271.1		16	16.5	even	4
576.3.m.b.271.8		16	48.5	odd	4
576.3.m.b.559.1		16	4.3	odd	2
576.3.m.b.559.8		16	12.11	even	2
1152.3.m.d.415.1		16	48.29	odd	4
1152.3.m.d.415.8		16	16.13	even	4
1152.3.m.d.991.1		16	24.11	even	2
1152.3.m.d.991.8		16	8.3	odd	2
1152.3.m.e.415.1		16	48.35	even	4
1152.3.m.e.415.8		16	16.3	odd	4
1152.3.m.e.991.1		16	24.5	odd	2
1152.3.m.e.991.8		16	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	144.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.0999235 - 1.99750i) q^{2} +(-3.98003 + 0.399195i) q^{4} +(-6.01265 + 6.01265i) q^{5} +8.23187 q^{7} +(1.19509 + 7.91023i) q^{8} +O(q^{10})\)	\(q+(-0.0999235 - 1.99750i) q^{2} +(-3.98003 + 0.399195i) q^{4} +(-6.01265 + 6.01265i) q^{5} +8.23187 q^{7} +(1.19509 + 7.91023i) q^{8} +(12.6111 + 11.4095i) q^{10} +(6.51529 + 6.51529i) q^{11} +(8.82865 + 8.82865i) q^{13} +(-0.822557 - 16.4432i) q^{14} +(15.6813 - 3.17761i) q^{16} -14.1753 q^{17} +(-23.7488 + 23.7488i) q^{19} +(21.5303 - 26.3308i) q^{20} +(12.3633 - 13.6653i) q^{22} -9.42125 q^{23} -47.3040i q^{25} +(16.7531 - 18.5174i) q^{26} +(-32.7631 + 3.28612i) q^{28} +(23.7973 + 23.7973i) q^{29} +24.4148i q^{31} +(-7.91422 - 31.0059i) q^{32} +(1.41645 + 28.3153i) q^{34} +(-49.4954 + 49.4954i) q^{35} +(24.2052 - 24.2052i) q^{37} +(49.8113 + 45.0652i) q^{38} +(-54.7471 - 40.3758i) q^{40} +6.67771i q^{41} +(0.897918 + 0.897918i) q^{43} +(-28.5319 - 23.3302i) q^{44} +(0.941404 + 18.8190i) q^{46} -25.2401i q^{47} +18.7636 q^{49} +(-94.4898 + 4.72678i) q^{50} +(-38.6626 - 31.6139i) q^{52} +(32.6251 - 32.6251i) q^{53} -78.3484 q^{55} +(9.83783 + 65.1160i) q^{56} +(45.1573 - 49.9132i) q^{58} +(-8.31871 - 8.31871i) q^{59} +(-68.4028 - 68.4028i) q^{61} +(48.7687 - 2.43961i) q^{62} +(-61.1435 + 18.9069i) q^{64} -106.167 q^{65} +(-7.71922 + 7.71922i) q^{67} +(56.4183 - 5.65873i) q^{68} +(103.813 + 93.9213i) q^{70} +137.259 q^{71} +52.8655i q^{73} +(-50.7685 - 45.9312i) q^{74} +(85.0404 - 104.001i) q^{76} +(53.6330 + 53.6330i) q^{77} +87.2269i q^{79} +(-75.1802 + 113.392i) q^{80} +(13.3387 - 0.667260i) q^{82} +(9.53893 - 9.53893i) q^{83} +(85.2314 - 85.2314i) q^{85} +(1.70387 - 1.88332i) q^{86} +(-43.7511 + 59.3238i) q^{88} -146.488i q^{89} +(72.6762 + 72.6762i) q^{91} +(37.4969 - 3.76091i) q^{92} +(-50.4172 + 2.52208i) q^{94} -285.586i q^{95} +101.170 q^{97} +(-1.87493 - 37.4804i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 12 q^{4}+O(q^{10}) \) 16 * q - 12 * q^4	\( 16 q - 12 q^{4} + 16 q^{10} + 32 q^{16} - 32 q^{19} + 104 q^{22} - 24 q^{34} + 96 q^{37} - 312 q^{40} - 32 q^{43} - 224 q^{46} + 112 q^{49} - 264 q^{52} - 256 q^{55} + 312 q^{58} - 32 q^{61} + 456 q^{64} - 256 q^{67} + 744 q^{70} + 264 q^{76} - 280 q^{82} + 160 q^{85} - 912 q^{88} + 288 q^{91} - 1104 q^{94}+O(q^{100}) \) 16 * q - 12 * q^4 + 16 * q^10 + 32 * q^16 - 32 * q^19 + 104 * q^22 - 24 * q^34 + 96 * q^37 - 312 * q^40 - 32 * q^43 - 224 * q^46 + 112 * q^49 - 264 * q^52 - 256 * q^55 + 312 * q^58 - 32 * q^61 + 456 * q^64 - 256 * q^67 + 744 * q^70 + 264 * q^76 - 280 * q^82 + 160 * q^85 - 912 * q^88 + 288 * q^91 - 1104 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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