LMFDB - Embedded newform 448.2.q.a.159.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(31,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("448.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.q (of order $6$, 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:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			159.3
Root			$-1.40126 - 0.809017i$ of defining polynomial
Character	$\chi$	$=$	448.159
Dual form			448.2.q.a.31.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.93649 - 1.11803i) q^{3} +(-1.11803 + 1.93649i) q^{5} +(1.73205 + 2.00000i) q^{7} +(1.00000 - 1.73205i) q^{9} +O(q^{10})$	\(q+(1.93649 - 1.11803i) q^{3} +(-1.11803 + 1.93649i) q^{5} +(1.73205 + 2.00000i) q^{7} +(1.00000 - 1.73205i) q^{9} +(1.93649 + 3.35410i) q^{11} +5.00000i q^{15} +(4.50000 - 2.59808i) q^{17} +(-5.80948 - 3.35410i) q^{19} +(5.59017 + 1.93649i) q^{21} +(2.59808 + 1.50000i) q^{23} +2.23607i q^{27} -7.74597i q^{29} +(0.866025 + 1.50000i) q^{31} +(7.50000 + 4.33013i) q^{33} +(-5.80948 + 1.11803i) q^{35} +(-10.0623 - 5.80948i) q^{37} -10.3923i q^{41} +(2.23607 + 3.87298i) q^{45} +(2.59808 - 4.50000i) q^{47} +(-1.00000 + 6.92820i) q^{49} +(5.80948 - 10.0623i) q^{51} +(3.35410 - 1.93649i) q^{53} -8.66025 q^{55} -15.0000 q^{57} +(1.93649 - 1.11803i) q^{59} +(-3.35410 + 5.80948i) q^{61} +(5.19615 - 1.00000i) q^{63} +(5.80948 + 10.0623i) q^{67} +6.70820 q^{69} +6.00000i q^{71} +(-4.50000 + 2.59808i) q^{73} +(-3.35410 + 9.68246i) q^{77} +(-11.2583 - 6.50000i) q^{79} +(5.50000 + 9.52628i) q^{81} +4.47214i q^{83} +11.6190i q^{85} +(-8.66025 - 15.0000i) q^{87} +(-4.50000 - 2.59808i) q^{89} +(3.35410 + 1.93649i) q^{93} +(12.9904 - 7.50000i) q^{95} -3.46410i q^{97} +7.74597 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 36 q^{17} + 60 q^{33} - 8 q^{49} - 120 q^{57} - 36 q^{73} + 44 q^{81} - 36 q^{89}+O(q^{100}) $ 8 * q + 8 * q^9 + 36 * q^17 + 60 * q^33 - 8 * q^49 - 120 * q^57 - 36 * q^73 + 44 * q^81 - 36 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$

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			0
$2$		0			0
$3$	1.93649	−	1.11803i	1.11803	−	0.645497i	0.177136	−	0.984186i	$-0.443317\pi$
$3$	1.93649	−	1.11803i	1.11803	−	0.645497i	0.940898	+	0.338689i	$0.109984\pi$
$4$		0			0
$5$	−1.11803	+	1.93649i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$5$	−1.11803	+	1.93649i	−0.500000	+	0.866025i	−1.00000			$\pi$
$6$		0			0
$7$	1.73205	+	2.00000i	0.654654	+	0.755929i
$7$	1.73205	+	2.00000i	0.654654	+	0.755929i
$8$		0			0
$9$	1.00000	−	1.73205i	0.333333	−	0.577350i
$10$		0			0
$11$	1.93649	+	3.35410i	0.583874	+	1.01130i	0.995015	+	0.0997278i	$0.0317972\pi$
$11$	1.93649	+	3.35410i	0.583874	+	1.01130i	−0.411141	+	0.911572i	$0.634869\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$			5.00000i			1.29099i
$16$		0			0
$17$	4.50000	−	2.59808i	1.09141	−	0.630126i	0.157459	−	0.987526i	$-0.449670\pi$
$17$	4.50000	−	2.59808i	1.09141	−	0.630126i	0.933952	+	0.357400i	$0.116337\pi$
$18$		0			0
$19$	−5.80948	−	3.35410i	−1.33278	−	0.769484i	−0.347059	−	0.937843i	$-0.612820\pi$
$19$	−5.80948	−	3.35410i	−1.33278	−	0.769484i	−0.985726	+	0.168359i	$0.946153\pi$
$20$		0			0
$21$	5.59017	+	1.93649i	1.21988	+	0.422577i
$22$		0			0
$23$	2.59808	+	1.50000i	0.541736	+	0.312772i	0.745782	−	0.666190i	$-0.232076\pi$
$23$	2.59808	+	1.50000i	0.541736	+	0.312772i	−0.204046	+	0.978961i	$0.565409\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			2.23607i			0.430331i
$28$		0			0
$29$		−	7.74597i		−	1.43839i	−0.694808	−	0.719195i	$-0.744511\pi$
$29$		−	7.74597i		−	1.43839i	0.694808	−	0.719195i	$-0.255489\pi$
$30$		0			0
$31$	0.866025	+	1.50000i	0.155543	+	0.269408i	0.933257	−	0.359211i	$-0.116954\pi$
$31$	0.866025	+	1.50000i	0.155543	+	0.269408i	−0.777714	+	0.628619i	$0.783621\pi$
$32$		0			0
$33$	7.50000	+	4.33013i	1.30558	+	0.753778i
$34$		0			0
$35$	−5.80948	+	1.11803i	−0.981981	+	0.188982i
$36$		0			0
$37$	−10.0623	−	5.80948i	−1.65423	−	0.955072i	−0.975304	−	0.220868i	$-0.929111\pi$
$37$	−10.0623	−	5.80948i	−1.65423	−	0.955072i	−0.678929	−	0.734204i	$-0.737556\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	10.3923i		−	1.62301i	−0.584349	−	0.811503i	$-0.698650\pi$
$41$		−	10.3923i		−	1.62301i	0.584349	−	0.811503i	$-0.301350\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	2.23607	+	3.87298i	0.333333	+	0.577350i
$46$		0			0
$47$	2.59808	−	4.50000i	0.378968	−	0.656392i	−0.611944	−	0.790901i	$-0.709612\pi$
$47$	2.59808	−	4.50000i	0.378968	−	0.656392i	0.990912	+	0.134509i	$0.0429456\pi$
$48$		0			0
$49$	−1.00000	+	6.92820i	−0.142857	+	0.989743i
$50$		0			0
$51$	5.80948	−	10.0623i	0.813489	−	1.40900i
$52$		0			0
$53$	3.35410	−	1.93649i	0.460721	−	0.265998i	−0.251626	−	0.967825i	$-0.580965\pi$
$53$	3.35410	−	1.93649i	0.460721	−	0.265998i	0.712348	+	0.701827i	$0.247632\pi$
$54$		0			0
$55$	−8.66025			−1.16775
$56$		0			0
$57$	−15.0000			−1.98680
$58$		0			0
$59$	1.93649	−	1.11803i	0.252110	−	0.145556i	−0.368620	−	0.929580i	$-0.620170\pi$
$59$	1.93649	−	1.11803i	0.252110	−	0.145556i	0.620730	+	0.784024i	$0.286836\pi$
$60$		0			0
$61$	−3.35410	+	5.80948i	−0.429449	+	0.743827i	−0.996824	−	0.0796321i	$-0.974625\pi$
$61$	−3.35410	+	5.80948i	−0.429449	+	0.743827i	0.567376	+	0.823459i	$0.307959\pi$
$62$		0			0
$63$	5.19615	−	1.00000i	0.654654	−	0.125988i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.80948	+	10.0623i	0.709740	+	1.22931i	0.964953	+	0.262421i	$0.0845211\pi$
$67$	5.80948	+	10.0623i	0.709740	+	1.22931i	−0.255213	+	0.966885i	$0.582146\pi$
$68$		0			0
$69$	6.70820			0.807573
$70$		0			0
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		0			0
$73$	−4.50000	+	2.59808i	−0.526685	+	0.304082i	−0.739666	−	0.672975i	$-0.765016\pi$
$73$	−4.50000	+	2.59808i	−0.526685	+	0.304082i	0.212980	+	0.977056i	$0.431683\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.35410	+	9.68246i	−0.382235	+	1.10342i
$78$		0			0
$79$	−11.2583	−	6.50000i	−1.26666	−	0.731307i	−0.292306	−	0.956325i	$-0.594423\pi$
$79$	−11.2583	−	6.50000i	−1.26666	−	0.731307i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$	5.50000	+	9.52628i	0.611111	+	1.05848i
$82$		0			0
$83$			4.47214i			0.490881i	0.969412	+	0.245440i	$0.0789325\pi$
$83$			4.47214i			0.490881i	−0.969412	+	0.245440i	$0.921067\pi$
$84$		0			0
$85$			11.6190i			1.26025i
$86$		0			0
$87$	−8.66025	−	15.0000i	−0.928477	−	1.60817i
$88$		0			0
$89$	−4.50000	−	2.59808i	−0.476999	−	0.275396i	0.242166	−	0.970235i	$-0.422142\pi$
$89$	−4.50000	−	2.59808i	−0.476999	−	0.275396i	−0.719165	+	0.694839i	$0.755475\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	3.35410	+	1.93649i	0.347804	+	0.200805i
$94$		0			0
$95$	12.9904	−	7.50000i	1.33278	−	0.769484i
$96$		0			0
$97$		−	3.46410i		−	0.351726i	−0.984415	−	0.175863i	$-0.943728\pi$
$97$		−	3.46410i		−	0.351726i	0.984415	−	0.175863i	$-0.0562716\pi$
$98$		0			0
$99$	7.74597			0.778499
$100$		0			0
$101$	−5.59017	−	9.68246i	−0.556243	−	0.963441i	−0.997806	−	0.0662104i	$-0.978909\pi$
$101$	−5.59017	−	9.68246i	−0.556243	−	0.963441i	0.441563	−	0.897230i	$-0.354424\pi$
$102$		0			0
$103$	7.79423	−	13.5000i	0.767988	−	1.33019i	−0.170664	−	0.985329i	$-0.554591\pi$
$103$	7.79423	−	13.5000i	0.767988	−	1.33019i	0.938652	−	0.344865i	$-0.112075\pi$
$104$		0			0
$105$	−10.0000	+	8.66025i	−0.975900	+	0.845154i
$106$		0			0
$107$	−1.93649	+	3.35410i	−0.187208	+	0.324253i	−0.944318	−	0.329034i	$-0.893277\pi$
$107$	−1.93649	+	3.35410i	−0.187208	+	0.324253i	0.757111	+	0.653287i	$0.226610\pi$
$108$		0			0
$109$	10.0623	−	5.80948i	0.963794	−	0.556447i	0.0664554	−	0.997789i	$-0.478831\pi$
$109$	10.0623	−	5.80948i	0.963794	−	0.556447i	0.897339	+	0.441343i	$0.145498\pi$
$110$		0			0
$111$	−25.9808			−2.46598
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$	−5.80948	+	3.35410i	−0.541736	+	0.312772i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.9904	+	4.50000i	1.19083	+	0.412514i
$120$		0			0
$121$	−2.00000	+	3.46410i	−0.181818	+	0.314918i
$122$		0			0
$123$	−11.6190	−	20.1246i	−1.04765	−	1.81458i
$124$		0			0
$125$	−11.1803			−1.00000
$126$		0			0
$127$			10.0000i			0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$			10.0000i			0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.93649	−	1.11803i	−0.169192	−	0.0976831i	0.413013	−	0.910725i	$-0.364476\pi$
$131$	−1.93649	−	1.11803i	−0.169192	−	0.0976831i	−0.582205	+	0.813042i	$0.697810\pi$
$132$		0			0
$133$	−3.35410	−	17.4284i	−0.290838	−	1.51124i
$134$		0			0
$135$	−4.33013	−	2.50000i	−0.372678	−	0.215166i
$136$		0			0
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	0.607233	−	0.794524i	$-0.292279\pi$
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	−0.991694	+	0.128618i	$0.958946\pi$
$138$		0			0
$139$		−	13.4164i		−	1.13796i	−0.822350	−	0.568982i	$-0.807337\pi$
$139$		−	13.4164i		−	1.13796i	0.822350	−	0.568982i	$-0.192663\pi$
$140$		0			0
$141$		−	11.6190i		−	0.978492i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	15.0000	+	8.66025i	1.24568	+	0.719195i
$146$		0			0
$147$	5.80948	+	14.5344i	0.479157	+	1.19878i
$148$		0			0
$149$	10.0623	+	5.80948i	0.824336	+	0.475931i	0.851909	−	0.523689i	$-0.175445\pi$
$149$	10.0623	+	5.80948i	0.824336	+	0.475931i	−0.0275733	+	0.999620i	$0.508778\pi$
$150$		0			0
$151$	−11.2583	+	6.50000i	−0.916190	+	0.528962i	−0.882418	−	0.470467i	$-0.844085\pi$
$151$	−11.2583	+	6.50000i	−0.916190	+	0.528962i	−0.0337724	+	0.999430i	$0.510752\pi$
$152$		0			0
$153$		−	10.3923i		−	0.840168i
$154$		0			0
$155$	−3.87298			−0.311086
$156$		0			0
$157$	−3.35410	−	5.80948i	−0.267686	−	0.463647i	0.700577	−	0.713576i	$-0.252926\pi$
$157$	−3.35410	−	5.80948i	−0.267686	−	0.463647i	−0.968264	+	0.249930i	$0.919592\pi$
$158$		0			0
$159$	4.33013	−	7.50000i	0.343401	−	0.594789i
$160$		0			0
$161$	1.50000	+	7.79423i	0.118217	+	0.614271i
$162$		0			0
$163$	−5.80948	+	10.0623i	−0.455033	+	0.788141i	−0.998690	−	0.0511669i	$-0.983706\pi$
$163$	−5.80948	+	10.0623i	−0.455033	+	0.788141i	0.543657	+	0.839308i	$0.317039\pi$
$164$		0			0
$165$	−16.7705	+	9.68246i	−1.30558	+	0.753778i
$166$		0			0
$167$	10.3923			0.804181			0.402090	−	0.915600i	$-0.368284\pi$
$167$	10.3923			0.804181			0.402090	+	0.915600i	$0.368284\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$	−11.6190	+	6.70820i	−0.888523	+	0.512989i
$172$		0			0
$173$	5.59017	−	9.68246i	0.425013	−	0.736144i	−0.571409	−	0.820666i	$-0.693603\pi$
$173$	5.59017	−	9.68246i	0.425013	−	0.736144i	0.996422	+	0.0845218i	$0.0269363\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2.50000	−	4.33013i	0.187912	−	0.325472i
$178$		0			0
$179$	−5.80948	−	10.0623i	−0.434221	−	0.752092i	0.563011	−	0.826449i	$-0.309643\pi$
$179$	−5.80948	−	10.0623i	−0.434221	−	0.752092i	−0.997232	+	0.0743573i	$0.976309\pi$
$180$		0			0
$181$	13.4164			0.997234			0.498617	−	0.866822i	$-0.333841\pi$
$181$	13.4164			0.997234			0.498617	+	0.866822i	$0.333841\pi$
$182$		0			0
$183$			15.0000i			1.10883i
$184$		0			0
$185$	22.5000	−	12.9904i	1.65423	−	0.955072i
$186$		0			0
$187$	17.4284	+	10.0623i	1.27449	+	0.735829i
$188$		0			0
$189$	−4.47214	+	3.87298i	−0.325300	+	0.281718i
$190$		0			0
$191$	18.1865	+	10.5000i	1.31593	+	0.759753i	0.983071	−	0.183223i	$-0.0586530\pi$
$191$	18.1865	+	10.5000i	1.31593	+	0.759753i	0.332860	+	0.942976i	$0.391986\pi$
$192$		0			0
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	0.964059	−	0.265689i	$-0.0855996\pi$
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	−0.712123	+	0.702055i	$0.752266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			23.2379i			1.65563i	0.561000	+	0.827816i	$0.310417\pi$
$197$			23.2379i			1.65563i	−0.561000	+	0.827816i	$0.689583\pi$
$198$		0			0
$199$	−2.59808	−	4.50000i	−0.184173	−	0.318997i	0.759125	−	0.650945i	$-0.225627\pi$
$199$	−2.59808	−	4.50000i	−0.184173	−	0.318997i	−0.943297	+	0.331949i	$0.892294\pi$
$200$		0			0
$201$	22.5000	+	12.9904i	1.58703	+	0.916271i
$202$		0			0
$203$	15.4919	−	13.4164i	1.08732	−	0.941647i
$204$		0			0
$205$	20.1246	+	11.6190i	1.40556	+	0.811503i
$206$		0			0
$207$	5.19615	−	3.00000i	0.361158	−	0.208514i
$208$		0			0
$209$		−	25.9808i		−	1.79713i
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$	6.70820	+	11.6190i	0.459639	+	0.796117i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.50000	+	4.33013i	−0.101827	+	0.293948i
$218$		0			0
$219$	−5.80948	+	10.0623i	−0.392568	+	0.679948i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−6.92820			−0.463947			−0.231973	−	0.972722i	$-0.574518\pi$
$223$	−6.92820			−0.463947			−0.231973	+	0.972722i	$0.574518\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.5554	−	7.82624i	0.899706	−	0.519446i	0.0226014	−	0.999745i	$-0.492805\pi$
$227$	13.5554	−	7.82624i	0.899706	−	0.519446i	0.877105	+	0.480299i	$0.159472\pi$
$228$		0			0
$229$	3.35410	−	5.80948i	0.221645	−	0.383901i	−0.733662	−	0.679514i	$-0.762191\pi$
$229$	3.35410	−	5.80948i	0.221645	−	0.383901i	0.955308	+	0.295613i	$0.0955240\pi$
$230$		0			0
$231$	4.33013	+	22.5000i	0.284901	+	1.48039i
$232$		0			0
$233$	7.50000	−	12.9904i	0.491341	−	0.851028i	−0.508609	−	0.860998i	$-0.669840\pi$
$233$	7.50000	−	12.9904i	0.491341	−	0.851028i	0.999950	+	0.00996947i	$0.00317343\pi$
$234$		0			0
$235$	5.80948	+	10.0623i	0.378968	+	0.656392i
$236$		0			0
$237$	−29.0689			−1.88823
$238$		0			0
$239$			12.0000i			0.776215i	0.921614	+	0.388108i	$0.126871\pi$
$239$			12.0000i			0.776215i	−0.921614	+	0.388108i	$0.873129\pi$
$240$		0			0
$241$	7.50000	−	4.33013i	0.483117	−	0.278928i	−0.238597	−	0.971119i	$-0.576688\pi$
$241$	7.50000	−	4.33013i	0.483117	−	0.278928i	0.721715	+	0.692191i	$0.243354\pi$
$242$		0			0
$243$	15.4919	+	8.94427i	0.993808	+	0.573775i
$244$		0			0
$245$	−12.2984	−	9.68246i	−0.785714	−	0.618590i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	5.00000	+	8.66025i	0.316862	+	0.548821i
$250$		0			0
$251$			8.94427i			0.564557i	0.959332	+	0.282279i	$0.0910903\pi$
$251$			8.94427i			0.564557i	−0.959332	+	0.282279i	$0.908910\pi$
$252$		0			0
$253$			11.6190i			0.730477i
$254$		0			0
$255$	12.9904	+	22.5000i	0.813489	+	1.40900i
$256$		0			0
$257$	−13.5000	−	7.79423i	−0.842107	−	0.486191i	0.0158730	−	0.999874i	$-0.494947\pi$
$257$	−13.5000	−	7.79423i	−0.842107	−	0.486191i	−0.857980	+	0.513683i	$0.828281\pi$
$258$		0			0
$259$	−5.80948	−	30.1869i	−0.360983	−	1.87572i
$260$		0			0
$261$	−13.4164	−	7.74597i	−0.830455	−	0.479463i
$262$		0			0
$263$	−2.59808	+	1.50000i	−0.160204	+	0.0924940i	−0.577959	−	0.816066i	$-0.696151\pi$
$263$	−2.59808	+	1.50000i	−0.160204	+	0.0924940i	0.417755	+	0.908560i	$0.362817\pi$
$264$		0			0
$265$			8.66025i			0.531995i
$266$		0			0
$267$	−11.6190			−0.711068
$268$		0			0
$269$	5.59017	+	9.68246i	0.340839	+	0.590350i	0.984589	−	0.174886i	$-0.0559557\pi$
$269$	5.59017	+	9.68246i	0.340839	+	0.590350i	−0.643750	+	0.765236i	$0.722622\pi$
$270$		0			0
$271$	−9.52628	+	16.5000i	−0.578680	+	1.00230i	0.416951	+	0.908929i	$0.363099\pi$
$271$	−9.52628	+	16.5000i	−0.578680	+	1.00230i	−0.995631	+	0.0933746i	$0.970235\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−10.0623	+	5.80948i	−0.604585	+	0.349058i	−0.770843	−	0.637025i	$-0.780165\pi$
$277$	−10.0623	+	5.80948i	−0.604585	+	0.349058i	0.166258	+	0.986082i	$0.446832\pi$
$278$		0			0
$279$	3.46410			0.207390
$280$		0			0
$281$	24.0000			1.43172			0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000			1.43172			0.715860	+	0.698244i	$0.246035\pi$
$282$		0			0
$283$	5.80948	−	3.35410i	0.345337	−	0.199381i	−0.317292	−	0.948328i	$-0.602774\pi$
$283$	5.80948	−	3.35410i	0.345337	−	0.199381i	0.662630	+	0.748947i	$0.269440\pi$
$284$		0			0
$285$	16.7705	−	29.0474i	0.993399	−	1.72062i
$286$		0			0
$287$	20.7846	−	18.0000i	1.22688	−	1.06251i
$288$		0			0
$289$	5.00000	−	8.66025i	0.294118	−	0.509427i
$290$		0			0
$291$	−3.87298	−	6.70820i	−0.227038	−	0.393242i
$292$		0			0
$293$	22.3607			1.30632			0.653162	−	0.757218i	$-0.273442\pi$
$293$	22.3607			1.30632			0.653162	+	0.757218i	$0.273442\pi$
$294$		0			0
$295$			5.00000i			0.291111i
$296$		0			0
$297$	−7.50000	+	4.33013i	−0.435194	+	0.251259i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−21.6506	−	12.5000i	−1.24380	−	0.718106i
$304$		0			0
$305$	−7.50000	−	12.9904i	−0.429449	−	0.743827i
$306$		0			0
$307$		−	13.4164i		−	0.765715i	−0.923807	−	0.382857i	$-0.874940\pi$
$307$		−	13.4164i		−	0.765715i	0.923807	−	0.382857i	$-0.125060\pi$
$308$		0			0
$309$		−	34.8569i		−	1.98294i
$310$		0			0
$311$	−7.79423	−	13.5000i	−0.441970	−	0.765515i	0.555865	−	0.831272i	$-0.312387\pi$
$311$	−7.79423	−	13.5000i	−0.441970	−	0.765515i	−0.997836	+	0.0657575i	$0.979054\pi$
$312$		0			0
$313$	1.50000	+	0.866025i	0.0847850	+	0.0489506i	0.541793	−	0.840512i	$-0.317746\pi$
$313$	1.50000	+	0.866025i	0.0847850	+	0.0489506i	−0.457008	+	0.889463i	$0.651079\pi$
$314$		0			0
$315$	−3.87298	+	11.1803i	−0.218218	+	0.629941i
$316$		0			0
$317$	3.35410	+	1.93649i	0.188385	+	0.108764i	0.591226	−	0.806506i	$-0.298644\pi$
$317$	3.35410	+	1.93649i	0.188385	+	0.108764i	−0.402841	+	0.915270i	$0.631977\pi$
$318$		0			0
$319$	25.9808	−	15.0000i	1.45464	−	0.839839i
$320$		0			0
$321$			8.66025i			0.483368i
$322$		0			0
$323$	−34.8569			−1.93949
$324$		0			0
$325$		0			0
$326$		0			0
$327$	12.9904	−	22.5000i	0.718370	−	1.24425i
$328$		0			0
$329$	13.5000	−	2.59808i	0.744279	−	0.143237i
$330$		0			0
$331$	−5.80948	+	10.0623i	−0.319318	+	0.553074i	−0.980346	−	0.197287i	$-0.936787\pi$
$331$	−5.80948	+	10.0623i	−0.319318	+	0.553074i	0.661028	+	0.750361i	$0.270120\pi$
$332$		0			0
$333$	−20.1246	+	11.6190i	−1.10282	+	0.636715i
$334$		0			0
$335$	−25.9808			−1.41948
$336$		0			0
$337$	−16.0000			−0.871576			−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000			−0.871576			−0.435788	+	0.900049i	$0.643530\pi$
$338$		0			0
$339$	−23.2379	+	13.4164i	−1.26211	+	0.728679i
$340$		0			0
$341$	−3.35410	+	5.80948i	−0.181635	+	0.314601i
$342$		0			0
$343$	−15.5885	+	10.0000i	−0.841698	+	0.539949i
$344$		0			0
$345$	−7.50000	+	12.9904i	−0.403786	+	0.699379i
$346$		0			0
$347$	5.80948	+	10.0623i	0.311869	+	0.540173i	0.978767	−	0.204976i	$-0.0657117\pi$
$347$	5.80948	+	10.0623i	0.311869	+	0.540173i	−0.666898	+	0.745149i	$0.732378\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.5000	+	7.79423i	−0.718532	+	0.414845i	−0.814212	−	0.580567i	$-0.802831\pi$
$353$	−13.5000	+	7.79423i	−0.718532	+	0.414845i	0.0956798	+	0.995412i	$0.469498\pi$
$354$		0			0
$355$	−11.6190	−	6.70820i	−0.616670	−	0.356034i
$356$		0			0
$357$	30.1869	−	5.80948i	1.59766	−	0.307470i
$358$		0			0
$359$	18.1865	+	10.5000i	0.959849	+	0.554169i	0.896126	−	0.443799i	$-0.146370\pi$
$359$	18.1865	+	10.5000i	0.959849	+	0.554169i	0.0637221	+	0.997968i	$0.479703\pi$
$360$		0			0
$361$	13.0000	+	22.5167i	0.684211	+	1.18509i
$362$		0			0
$363$			8.94427i			0.469453i
$364$		0			0
$365$		−	11.6190i		−	0.608164i
$366$		0			0
$367$	−0.866025	−	1.50000i	−0.0452062	−	0.0782994i	0.842537	−	0.538639i	$-0.181061\pi$
$367$	−0.866025	−	1.50000i	−0.0452062	−	0.0782994i	−0.887743	+	0.460339i	$0.847728\pi$
$368$		0			0
$369$	−18.0000	−	10.3923i	−0.937043	−	0.541002i
$370$		0			0
$371$	9.68246	+	3.35410i	0.502688	+	0.174136i
$372$		0			0
$373$	−10.0623	−	5.80948i	−0.521006	−	0.300803i	0.216340	−	0.976318i	$-0.430588\pi$
$373$	−10.0623	−	5.80948i	−0.521006	−	0.300803i	−0.737346	+	0.675515i	$0.763921\pi$
$374$		0			0
$375$	−21.6506	+	12.5000i	−1.11803	+	0.645497i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	23.2379			1.19365			0.596825	−	0.802371i	$-0.296429\pi$
$379$	23.2379			1.19365			0.596825	+	0.802371i	$0.296429\pi$
$380$		0			0
$381$	11.1803	+	19.3649i	0.572786	+	0.992095i
$382$		0			0
$383$	−7.79423	+	13.5000i	−0.398266	+	0.689818i	−0.993512	−	0.113726i	$-0.963721\pi$
$383$	−7.79423	+	13.5000i	−0.398266	+	0.689818i	0.595246	+	0.803544i	$0.297055\pi$
$384$		0			0
$385$	−15.0000	−	17.3205i	−0.764471	−	0.882735i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.0623	−	5.80948i	0.510179	−	0.294552i	−0.222728	−	0.974881i	$-0.571496\pi$
$389$	10.0623	−	5.80948i	0.510179	−	0.294552i	0.732907	+	0.680329i	$0.238163\pi$
$390$		0			0
$391$	15.5885			0.788342
$392$		0			0
$393$	−5.00000			−0.252217
$394$		0			0
$395$	25.1744	−	14.5344i	1.26666	−	0.731307i
$396$		0			0
$397$	−10.0623	+	17.4284i	−0.505013	+	0.874708i	0.494971	+	0.868910i	$0.335179\pi$
$397$	−10.0623	+	17.4284i	−0.505013	+	0.874708i	−0.999983	+	0.00579782i	$0.998154\pi$
$398$		0			0
$399$	−25.9808	−	30.0000i	−1.30066	−	1.50188i
$400$		0			0
$401$	−16.5000	+	28.5788i	−0.823971	+	1.42716i	0.0787327	+	0.996896i	$0.474913\pi$
$401$	−16.5000	+	28.5788i	−0.823971	+	1.42716i	−0.902703	+	0.430263i	$0.858421\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−24.5967			−1.22222
$406$		0			0
$407$		−	45.0000i		−	2.23057i
$408$		0			0
$409$	−19.5000	+	11.2583i	−0.964213	+	0.556689i	−0.897467	−	0.441081i	$-0.854595\pi$
$409$	−19.5000	+	11.2583i	−0.964213	+	0.556689i	−0.0667458	+	0.997770i	$0.521262\pi$
$410$		0			0
$411$	−17.4284	−	10.0623i	−0.859681	−	0.496337i
$412$		0			0
$413$	5.59017	+	1.93649i	0.275074	+	0.0952885i
$414$		0			0
$415$	−8.66025	−	5.00000i	−0.425115	−	0.245440i
$416$		0			0
$417$	−15.0000	−	25.9808i	−0.734553	−	1.27228i
$418$		0			0
$419$			4.47214i			0.218478i	0.994016	+	0.109239i	$0.0348414\pi$
$419$			4.47214i			0.218478i	−0.994016	+	0.109239i	$0.965159\pi$
$420$		0			0
$421$			23.2379i			1.13255i	0.824218	+	0.566273i	$0.191615\pi$
$421$			23.2379i			1.13255i	−0.824218	+	0.566273i	$0.808385\pi$
$422$		0			0
$423$	−5.19615	−	9.00000i	−0.252646	−	0.437595i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−17.4284	+	3.35410i	−0.843421	+	0.162316i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.79423	+	4.50000i	−0.375435	+	0.216757i	−0.675830	−	0.737057i	$-0.736215\pi$
$431$	−7.79423	+	4.50000i	−0.375435	+	0.216757i	0.300395	+	0.953815i	$0.402881\pi$
$432$		0			0
$433$			10.3923i			0.499422i	0.968320	+	0.249711i	$0.0803357\pi$
$433$			10.3923i			0.499422i	−0.968320	+	0.249711i	$0.919664\pi$
$434$		0			0
$435$	38.7298			1.85695
$436$		0			0
$437$	−10.0623	−	17.4284i	−0.481345	−	0.833715i
$438$		0			0
$439$	6.06218	−	10.5000i	0.289332	−	0.501138i	−0.684318	−	0.729183i	$-0.739900\pi$
$439$	6.06218	−	10.5000i	0.289332	−	0.501138i	0.973650	+	0.228046i	$0.0732335\pi$
$440$		0			0
$441$	11.0000	+	8.66025i	0.523810	+	0.412393i
$442$		0			0
$443$	13.5554	−	23.4787i	0.644038	−	1.11551i	−0.340484	−	0.940250i	$-0.610591\pi$
$443$	13.5554	−	23.4787i	0.644038	−	1.11551i	0.984523	−	0.175257i	$-0.0560757\pi$
$444$		0			0
$445$	10.0623	−	5.80948i	0.476999	−	0.275396i
$446$		0			0
$447$	25.9808			1.22885
$448$		0			0
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		0			0
$451$	34.8569	−	20.1246i	1.64134	−	0.947631i
$452$		0			0
$453$	−14.5344	+	25.1744i	−0.682888	+	1.18280i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.50000	−	4.33013i	0.116945	−	0.202555i	−0.801611	−	0.597847i	$-0.796023\pi$
$457$	2.50000	−	4.33013i	0.116945	−	0.202555i	0.918556	+	0.395292i	$0.129357\pi$
$458$		0			0
$459$	5.80948	+	10.0623i	0.271163	+	0.469668i
$460$		0			0
$461$	8.94427			0.416576			0.208288	−	0.978068i	$-0.433211\pi$
$461$	8.94427			0.416576			0.208288	+	0.978068i	$0.433211\pi$
$462$		0			0
$463$		−	22.0000i		−	1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$		−	22.0000i		−	1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$		0			0
$465$	−7.50000	+	4.33013i	−0.347804	+	0.200805i
$466$		0			0
$467$	−1.93649	−	1.11803i	−0.0896101	−	0.0517364i	0.454525	−	0.890734i	$-0.349809\pi$
$467$	−1.93649	−	1.11803i	−0.0896101	−	0.0517364i	−0.544135	+	0.838997i	$0.683142\pi$
$468$		0			0
$469$	−10.0623	+	29.0474i	−0.464634	+	1.34128i
$470$		0			0
$471$	−12.9904	−	7.50000i	−0.598565	−	0.345582i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	7.74597i		−	0.354663i
$478$		0			0
$479$	2.59808	+	4.50000i	0.118709	+	0.205610i	0.919256	−	0.393659i	$-0.128791\pi$
$479$	2.59808	+	4.50000i	0.118709	+	0.205610i	−0.800547	+	0.599270i	$0.795458\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	11.6190	+	13.4164i	0.528681	+	0.610468i
$484$		0			0
$485$	6.70820	+	3.87298i	0.304604	+	0.175863i
$486$		0			0
$487$	−16.4545	+	9.50000i	−0.745624	+	0.430486i	−0.824110	−	0.566429i	$-0.808325\pi$
$487$	−16.4545	+	9.50000i	−0.745624	+	0.430486i	0.0784867	+	0.996915i	$0.474991\pi$
$488$		0			0
$489$			25.9808i			1.17489i
$490$		0			0
$491$	−7.74597			−0.349571			−0.174785	−	0.984607i	$-0.555923\pi$
$491$	−7.74597			−0.349571			−0.174785	+	0.984607i	$0.555923\pi$
$492$		0			0
$493$	−20.1246	−	34.8569i	−0.906367	−	1.56987i
$494$		0			0
$495$	−8.66025	+	15.0000i	−0.389249	+	0.674200i
$496$		0			0
$497$	−12.0000	+	10.3923i	−0.538274	+	0.466159i
$498$		0			0
$499$	−5.80948	+	10.0623i	−0.260068	+	0.450451i	−0.966260	−	0.257570i	$-0.917078\pi$
$499$	−5.80948	+	10.0623i	−0.260068	+	0.450451i	0.706192	+	0.708021i	$0.250412\pi$
$500$		0			0
$501$	20.1246	−	11.6190i	0.899101	−	0.519096i
$502$		0			0
$503$	20.7846			0.926740			0.463370	−	0.886165i	$-0.346640\pi$
$503$	20.7846			0.926740			0.463370	+	0.886165i	$0.346640\pi$
$504$		0			0
$505$	25.0000			1.11249
$506$		0			0
$507$	−25.1744	+	14.5344i	−1.11803	+	0.645497i
$508$		0			0
$509$	−19.0066	+	32.9204i	−0.842452	+	1.45917i	0.0453642	+	0.998971i	$0.485555\pi$
$509$	−19.0066	+	32.9204i	−0.842452	+	1.45917i	−0.887816	+	0.460199i	$0.847778\pi$
$510$		0			0
$511$	−12.9904	−	4.50000i	−0.574661	−	0.199068i
$512$		0			0
$513$	7.50000	−	12.9904i	0.331133	−	0.573539i
$514$		0			0
$515$	17.4284	+	30.1869i	0.767988	+	1.33019i
$516$		0			0
$517$	20.1246			0.885079
$518$		0			0
$519$		−	25.0000i		−	1.09738i
$520$		0			0
$521$	13.5000	−	7.79423i	0.591446	−	0.341471i	−0.174223	−	0.984706i	$-0.555741\pi$
$521$	13.5000	−	7.79423i	0.591446	−	0.341471i	0.765669	+	0.643235i	$0.222408\pi$
$522$		0			0
$523$	−29.0474	−	16.7705i	−1.27015	−	0.733323i	−0.295136	−	0.955455i	$-0.595365\pi$
$523$	−29.0474	−	16.7705i	−1.27015	−	0.733323i	−0.975017	+	0.222132i	$0.928698\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.79423	+	4.50000i	0.339522	+	0.196023i
$528$		0			0
$529$	−7.00000	−	12.1244i	−0.304348	−	0.527146i
$530$		0			0
$531$		−	4.47214i		−	0.194074i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−4.33013	−	7.50000i	−0.187208	−	0.324253i
$536$		0			0
$537$	−22.5000	−	12.9904i	−0.970947	−	0.560576i
$538$		0			0
$539$	−25.1744	+	10.0623i	−1.08434	+	0.433414i
$540$		0			0
$541$	−30.1869	−	17.4284i	−1.29784	−	0.749307i	−0.317807	−	0.948155i	$-0.602946\pi$
$541$	−30.1869	−	17.4284i	−1.29784	−	0.749307i	−0.980030	+	0.198849i	$0.936280\pi$
$542$		0			0
$543$	25.9808	−	15.0000i	1.11494	−	0.643712i
$544$		0			0
$545$			25.9808i			1.11289i
$546$		0			0
$547$	−23.2379			−0.993581			−0.496790	−	0.867871i	$-0.665488\pi$
$547$	−23.2379			−0.993581			−0.496790	+	0.867871i	$0.665488\pi$
$548$		0			0
$549$	6.70820	+	11.6190i	0.286299	+	0.495885i
$550$		0			0
$551$	−25.9808	+	45.0000i	−1.10682	+	1.91706i
$552$		0			0
$553$	−6.50000	−	33.7750i	−0.276408	−	1.43626i
$554$		0			0
$555$	29.0474	−	50.3115i	1.23299	−	2.13561i
$556$		0			0
$557$	36.8951	−	21.3014i	1.56330	−	0.902570i	0.566376	−	0.824147i	$-0.308345\pi$
$557$	36.8951	−	21.3014i	1.56330	−	0.902570i	0.996920	−	0.0784229i	$-0.0249884\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	45.0000			1.89990
$562$		0			0
$563$	−36.7933	+	21.2426i	−1.55065	+	0.895271i	−0.552566	+	0.833469i	$0.686351\pi$
$563$	−36.7933	+	21.2426i	−1.55065	+	0.895271i	−0.998088	+	0.0618017i	$0.980315\pi$
$564$		0			0
$565$	13.4164	−	23.2379i	0.564433	−	0.977626i
$566$		0			0
$567$	−9.52628	+	27.5000i	−0.400066	+	1.15489i
$568$		0			0
$569$	−13.5000	+	23.3827i	−0.565949	+	0.980253i	0.431011	+	0.902347i	$0.358157\pi$
$569$	−13.5000	+	23.3827i	−0.565949	+	0.980253i	−0.996961	+	0.0779066i	$0.975176\pi$
$570$		0			0
$571$	−5.80948	−	10.0623i	−0.243119	−	0.421094i	0.718482	−	0.695545i	$-0.244837\pi$
$571$	−5.80948	−	10.0623i	−0.243119	−	0.421094i	−0.961601	+	0.274451i	$0.911504\pi$
$572$		0			0
$573$	46.9574			1.96167
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.50000	−	0.866025i	0.0624458	−	0.0360531i	−0.468452	−	0.883489i	$-0.655188\pi$
$577$	1.50000	−	0.866025i	0.0624458	−	0.0360531i	0.530898	+	0.847436i	$0.321855\pi$
$578$		0			0
$579$	13.5554	+	7.82624i	0.563345	+	0.325247i
$580$		0			0
$581$	−8.94427	+	7.74597i	−0.371071	+	0.321357i
$582$		0			0
$583$	12.9904	+	7.50000i	0.538007	+	0.310618i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			4.47214i			0.184585i	0.995732	+	0.0922924i	$0.0294195\pi$
$587$			4.47214i			0.184585i	−0.995732	+	0.0922924i	$0.970581\pi$
$588$		0			0
$589$		−	11.6190i		−	0.478750i
$590$		0			0
$591$	25.9808	+	45.0000i	1.06871	+	1.85105i
$592$		0			0
$593$	−31.5000	−	18.1865i	−1.29355	−	0.746831i	−0.314268	−	0.949334i	$-0.601759\pi$
$593$	−31.5000	−	18.1865i	−1.29355	−	0.746831i	−0.979282	+	0.202503i	$0.935092\pi$
$594$		0			0
$595$	−23.2379	+	20.1246i	−0.952661	+	0.825029i
$596$		0			0
$597$	−10.0623	−	5.80948i	−0.411823	−	0.237766i
$598$		0			0
$599$	−23.3827	+	13.5000i	−0.955391	+	0.551595i	−0.894751	−	0.446565i	$-0.852647\pi$
$599$	−23.3827	+	13.5000i	−0.955391	+	0.551595i	−0.0606393	+	0.998160i	$0.519314\pi$
$600$		0			0
$601$			17.3205i			0.706518i	0.935526	+	0.353259i	$0.114927\pi$
$601$			17.3205i			0.706518i	−0.935526	+	0.353259i	$0.885073\pi$
$602$		0			0
$603$	23.2379			0.946320
$604$		0			0
$605$	−4.47214	−	7.74597i	−0.181818	−	0.314918i
$606$		0			0
$607$	6.06218	−	10.5000i	0.246056	−	0.426182i	−0.716372	−	0.697719i	$-0.754199\pi$
$607$	6.06218	−	10.5000i	0.246056	−	0.426182i	0.962428	+	0.271537i	$0.0875319\pi$
$608$		0			0
$609$	15.0000	−	43.3013i	0.607831	−	1.75466i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	10.0623	−	5.80948i	0.406413	−	0.234642i	−0.282835	−	0.959169i	$-0.591275\pi$
$613$	10.0623	−	5.80948i	0.406413	−	0.234642i	0.689247	+	0.724526i	$0.257941\pi$
$614$		0			0
$615$	51.9615			2.09529
$616$		0			0
$617$	36.0000			1.44931			0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000			1.44931			0.724653	+	0.689114i	$0.242000\pi$
$618$		0			0
$619$	−5.80948	+	3.35410i	−0.233503	+	0.134813i	−0.612187	−	0.790713i	$-0.709710\pi$
$619$	−5.80948	+	3.35410i	−0.233503	+	0.134813i	0.378684	+	0.925526i	$0.376377\pi$
$620$		0			0
$621$	−3.35410	+	5.80948i	−0.134595	+	0.233126i
$622$		0			0
$623$	−2.59808	−	13.5000i	−0.104090	−	0.540866i
$624$		0			0
$625$	12.5000	−	21.6506i	0.500000	−	0.866025i
$626$		0			0
$627$	−29.0474	−	50.3115i	−1.16004	−	2.00925i
$628$		0			0
$629$	−60.3738			−2.40726
$630$		0			0
$631$			26.0000i			1.03504i	0.855670	+	0.517522i	$0.173145\pi$
$631$			26.0000i			1.03504i	−0.855670	+	0.517522i	$0.826855\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−19.3649	−	11.1803i	−0.768473	−	0.443678i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	10.3923	+	6.00000i	0.411113	+	0.237356i
$640$		0			0
$641$	10.5000	+	18.1865i	0.414725	+	0.718325i	0.995400	−	0.0958109i	$-0.0305444\pi$
$641$	10.5000	+	18.1865i	0.414725	+	0.718325i	−0.580674	+	0.814136i	$0.697211\pi$
$642$		0			0
$643$			40.2492i			1.58727i	0.608391	+	0.793637i	$0.291815\pi$
$643$			40.2492i			1.58727i	−0.608391	+	0.793637i	$0.708185\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.1865	−	31.5000i	−0.714986	−	1.23839i	−0.962965	−	0.269627i	$-0.913100\pi$
$647$	−18.1865	−	31.5000i	−0.714986	−	1.23839i	0.247978	−	0.968766i	$-0.420234\pi$
$648$		0			0
$649$	7.50000	+	4.33013i	0.294401	+	0.169972i
$650$		0			0
$651$	1.93649	+	10.0623i	0.0758971	+	0.394373i
$652$		0			0
$653$	−36.8951	−	21.3014i	−1.44382	−	0.833589i	−0.445716	−	0.895174i	$-0.647051\pi$
$653$	−36.8951	−	21.3014i	−1.44382	−	0.833589i	−0.998102	+	0.0615859i	$0.980384\pi$
$654$		0			0
$655$	4.33013	−	2.50000i	0.169192	−	0.0976831i
$656$		0			0
$657$			10.3923i			0.405442i
$658$		0			0
$659$	46.4758			1.81044			0.905220	−	0.424943i	$-0.139706\pi$
$659$	46.4758			1.81044			0.905220	+	0.424943i	$0.139706\pi$
$660$		0			0
$661$	−3.35410	−	5.80948i	−0.130459	−	0.225962i	0.793394	−	0.608708i	$-0.208312\pi$
$661$	−3.35410	−	5.80948i	−0.130459	−	0.225962i	−0.923854	+	0.382746i	$0.874979\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	37.5000	+	12.9904i	1.45419	+	0.503745i
$666$		0			0
$667$	11.6190	−	20.1246i	0.449888	−	0.779228i
$668$		0			0
$669$	−13.4164	+	7.74597i	−0.518708	+	0.299476i
$670$		0			0
$671$	−25.9808			−1.00298
$672$		0			0
$673$	32.0000			1.23351			0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000			1.23351			0.616755	+	0.787155i	$0.288447\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.82624	−	13.5554i	0.300787	−	0.520978i	−0.675528	−	0.737335i	$-0.736084\pi$
$677$	7.82624	−	13.5554i	0.300787	−	0.520978i	0.976314	+	0.216357i	$0.0694174\pi$
$678$		0			0
$679$	6.92820	−	6.00000i	0.265880	−	0.230259i
$680$		0			0
$681$	17.5000	−	30.3109i	0.670601	−	1.16152i
$682$		0			0
$683$	9.68246	+	16.7705i	0.370489	+	0.641706i	0.989641	−	0.143566i	$-0.0458568\pi$
$683$	9.68246	+	16.7705i	0.370489	+	0.641706i	−0.619152	+	0.785271i	$0.712523\pi$
$684$		0			0
$685$	20.1246			0.768922
$686$		0			0
$687$		−	15.0000i		−	0.572286i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	29.0474	+	16.7705i	1.10501	+	0.637980i	0.937533	−	0.347895i	$-0.113103\pi$
$691$	29.0474	+	16.7705i	1.10501	+	0.637980i	0.167481	+	0.985875i	$0.446437\pi$
$692$		0			0
$693$	13.4164	+	15.4919i	0.509647	+	0.588490i
$694$		0			0
$695$	25.9808	+	15.0000i	0.985506	+	0.568982i
$696$		0			0
$697$	−27.0000	−	46.7654i	−1.02270	−	1.77136i
$698$		0			0
$699$		−	33.5410i		−	1.26864i
$700$		0			0
$701$			30.9839i			1.17024i	0.810945	+	0.585122i	$0.198953\pi$
$701$			30.9839i			1.17024i	−0.810945	+	0.585122i	$0.801047\pi$
$702$		0			0
$703$	38.9711	+	67.5000i	1.46982	+	2.54581i
$704$		0			0
$705$	22.5000	+	12.9904i	0.847399	+	0.489246i
$706$		0			0
$707$	9.68246	−	27.9508i	0.364146	−	1.05120i
$708$		0			0
$709$	30.1869	+	17.4284i	1.13369	+	0.654538i	0.944861	−	0.327471i	$-0.106196\pi$
$709$	30.1869	+	17.4284i	1.13369	+	0.654538i	0.188832	+	0.982009i	$0.439530\pi$
$710$		0			0
$711$	−22.5167	+	13.0000i	−0.844441	+	0.487538i
$712$		0			0
$713$			5.19615i			0.194597i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	13.4164	+	23.2379i	0.501045	+	0.867835i
$718$		0			0
$719$	−23.3827	+	40.5000i	−0.872027	+	1.51040i	−0.0121307	+	0.999926i	$0.503861\pi$
$719$	−23.3827	+	40.5000i	−0.872027	+	1.51040i	−0.859896	+	0.510469i	$0.829472\pi$
$720$		0			0
$721$	40.5000	−	7.79423i	1.50830	−	0.290272i
$722$		0			0
$723$	9.68246	−	16.7705i	0.360095	−	0.623702i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	48.4974			1.79867			0.899335	−	0.437260i	$-0.144051\pi$
$727$	48.4974			1.79867			0.899335	+	0.437260i	$0.144051\pi$
$728$		0			0
$729$	7.00000			0.259259
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−16.7705	+	29.0474i	−0.619433	+	1.07289i	0.370156	+	0.928969i	$0.379304\pi$
$733$	−16.7705	+	29.0474i	−0.619433	+	1.07289i	−0.989589	+	0.143920i	$0.954029\pi$
$734$		0			0
$735$	−34.6410	−	5.00000i	−1.27775	−	0.184428i
$736$		0			0
$737$	−22.5000	+	38.9711i	−0.828798	+	1.43552i
$738$		0			0
$739$	17.4284	+	30.1869i	0.641115	+	1.11044i	0.985184	+	0.171499i	$0.0548612\pi$
$739$	17.4284	+	30.1869i	0.641115	+	1.11044i	−0.344069	+	0.938944i	$0.611805\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	18.0000i		−	0.660356i	−0.943919	−	0.330178i	$-0.892891\pi$
$743$		−	18.0000i		−	0.660356i	0.943919	−	0.330178i	$-0.107109\pi$
$744$		0			0
$745$	−22.5000	+	12.9904i	−0.824336	+	0.475931i
$746$		0			0
$747$	7.74597	+	4.47214i	0.283410	+	0.163627i
$748$		0			0
$749$	−10.0623	+	1.93649i	−0.367669	+	0.0707579i
$750$		0			0
$751$	−6.06218	−	3.50000i	−0.221212	−	0.127717i	0.385299	−	0.922792i	$-0.374098\pi$
$751$	−6.06218	−	3.50000i	−0.221212	−	0.127717i	−0.606511	+	0.795075i	$0.707432\pi$
$752$		0			0
$753$	10.0000	+	17.3205i	0.364420	+	0.631194i
$754$		0			0
$755$		−	29.0689i		−	1.05792i
$756$		0			0
$757$		−	23.2379i		−	0.844596i	−0.906457	−	0.422298i	$-0.861224\pi$
$757$		−	23.2379i		−	0.844596i	0.906457	−	0.422298i	$-0.138776\pi$
$758$		0			0
$759$	12.9904	+	22.5000i	0.471521	+	0.816698i
$760$		0			0
$761$	−31.5000	−	18.1865i	−1.14187	−	0.659261i	−0.194980	−	0.980807i	$-0.562464\pi$
$761$	−31.5000	−	18.1865i	−1.14187	−	0.659261i	−0.946894	+	0.321546i	$0.895798\pi$
$762$		0			0
$763$	29.0474	+	10.0623i	1.05159	+	0.364280i
$764$		0			0
$765$	20.1246	+	11.6190i	0.727607	+	0.420084i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	17.3205i		−	0.624593i	−0.949985	−	0.312297i	$-0.898902\pi$
$769$		−	17.3205i		−	0.624593i	0.949985	−	0.312297i	$-0.101098\pi$
$770$		0			0
$771$	−34.8569			−1.25534
$772$		0			0
$773$	−1.11803	−	1.93649i	−0.0402129	−	0.0696508i	0.845218	−	0.534421i	$-0.179470\pi$
$773$	−1.11803	−	1.93649i	−0.0402129	−	0.0696508i	−0.885431	+	0.464770i	$0.846137\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−45.0000	−	51.9615i	−1.61437	−	1.86411i
$778$		0			0
$779$	−34.8569	+	60.3738i	−1.24888	+	2.16312i
$780$		0			0
$781$	−20.1246	+	11.6190i	−0.720115	+	0.415759i
$782$		0			0
$783$	17.3205			0.618984
$784$		0			0
$785$	15.0000			0.535373
$786$		0			0
$787$	17.4284	−	10.0623i	0.621256	−	0.358682i	−0.156102	−	0.987741i	$-0.549893\pi$
$787$	17.4284	−	10.0623i	0.621256	−	0.358682i	0.777358	+	0.629059i	$0.216559\pi$
$788$		0			0
$789$	−3.35410	+	5.80948i	−0.119409	+	0.206823i
$790$		0			0
$791$	−20.7846	−	24.0000i	−0.739016	−	0.853342i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	9.68246	+	16.7705i	0.343401	+	0.594789i
$796$		0			0
$797$	17.8885			0.633645			0.316822	−	0.948485i	$-0.397384\pi$
$797$	17.8885			0.633645			0.316822	+	0.948485i	$0.397384\pi$
$798$		0			0
$799$		−	27.0000i		−	0.955191i
$800$		0			0
$801$	−9.00000	+	5.19615i	−0.317999	+	0.183597i
$802$		0			0
$803$	−17.4284	−	10.0623i	−0.615036	−	0.355091i
$804$		0			0
$805$	−16.7705	−	5.80948i	−0.591083	−	0.204757i
$806$		0			0
$807$	21.6506	+	12.5000i	0.762138	+	0.440021i
$808$		0			0
$809$	4.50000	+	7.79423i	0.158212	+	0.274030i	0.934224	−	0.356687i	$-0.116094\pi$
$809$	4.50000	+	7.79423i	0.158212	+	0.274030i	−0.776012	+	0.630718i	$0.782761\pi$
$810$		0			0
$811$			40.2492i			1.41334i	0.707543	+	0.706671i	$0.249804\pi$
$811$			40.2492i			1.41334i	−0.707543	+	0.706671i	$0.750196\pi$
$812$		0			0
$813$			42.6028i			1.49415i
$814$		0			0
$815$	−12.9904	−	22.5000i	−0.455033	−	0.788141i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	36.8951	+	21.3014i	1.28765	+	0.743424i	0.978234	−	0.207503i	$-0.0665337\pi$
$821$	36.8951	+	21.3014i	1.28765	+	0.743424i	0.309414	+	0.950927i	$0.399867\pi$
$822$		0			0
$823$	4.33013	−	2.50000i	0.150939	−	0.0871445i	−0.422628	−	0.906303i	$-0.638892\pi$
$823$	4.33013	−	2.50000i	0.150939	−	0.0871445i	0.573567	+	0.819159i	$0.305559\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.4919			−0.538707			−0.269354	−	0.963041i	$-0.586810\pi$
$827$	−15.4919			−0.538707			−0.269354	+	0.963041i	$0.586810\pi$
$828$		0			0
$829$	−10.0623	−	17.4284i	−0.349478	−	0.605314i	0.636679	−	0.771129i	$-0.280308\pi$
$829$	−10.0623	−	17.4284i	−0.349478	−	0.605314i	−0.986157	+	0.165815i	$0.946975\pi$
$830$		0			0
$831$	−12.9904	+	22.5000i	−0.450631	+	0.780516i
$832$		0			0
$833$	13.5000	+	33.7750i	0.467747	+	1.17023i
$834$		0			0
$835$	−11.6190	+	20.1246i	−0.402090	+	0.696441i
$836$		0			0
$837$	−3.35410	+	1.93649i	−0.115935	+	0.0669349i
$838$		0			0
$839$	−20.7846			−0.717564			−0.358782	−	0.933421i	$-0.616808\pi$
$839$	−20.7846			−0.717564			−0.358782	+	0.933421i	$0.616808\pi$
$840$		0			0
$841$	−31.0000			−1.06897
$842$		0			0
$843$	46.4758	−	26.8328i	1.60071	−	0.924171i
$844$		0			0
$845$	14.5344	−	25.1744i	0.500000	−	0.866025i
$846$		0			0
$847$	−10.3923	+	2.00000i	−0.357084	+	0.0687208i
$848$		0			0
$849$	7.50000	−	12.9904i	0.257399	−	0.445829i
$850$		0			0
$851$	−17.4284	−	30.1869i	−0.597439	−	1.03479i
$852$		0			0
$853$	−26.8328			−0.918738			−0.459369	−	0.888246i	$-0.651924\pi$
$853$	−26.8328			−0.918738			−0.459369	+	0.888246i	$0.651924\pi$
$854$		0			0
$855$		−	30.0000i		−	1.02598i
$856$		0			0
$857$	31.5000	−	18.1865i	1.07602	−	0.621240i	0.146200	−	0.989255i	$-0.453296\pi$
$857$	31.5000	−	18.1865i	1.07602	−	0.621240i	0.929820	+	0.368015i	$0.119962\pi$
$858$		0			0
$859$	17.4284	+	10.0623i	0.594650	+	0.343321i	0.766934	−	0.641726i	$-0.221781\pi$
$859$	17.4284	+	10.0623i	0.594650	+	0.343321i	−0.172284	+	0.985047i	$0.555115\pi$
$860$		0			0
$861$	20.1246	−	58.0948i	0.685845	−	1.97986i
$862$		0			0
$863$	−28.5788	−	16.5000i	−0.972835	−	0.561667i	−0.0727356	−	0.997351i	$-0.523173\pi$
$863$	−28.5788	−	16.5000i	−0.972835	−	0.561667i	−0.900099	+	0.435685i	$0.856506\pi$
$864$		0			0
$865$	12.5000	+	21.6506i	0.425013	+	0.736144i
$866$		0			0
$867$		−	22.3607i		−	0.759408i
$868$		0			0
$869$		−	50.3488i		−	1.70797i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−6.00000	−	3.46410i	−0.203069	−	0.117242i
$874$		0			0
$875$	−19.3649	−	22.3607i	−0.654654	−	0.755929i
$876$		0			0
$877$	10.0623	+	5.80948i	0.339780	+	0.196172i	0.660175	−	0.751112i	$-0.270482\pi$
$877$	10.0623	+	5.80948i	0.339780	+	0.196172i	−0.320395	+	0.947284i	$0.603816\pi$
$878$		0			0
$879$	43.3013	−	25.0000i	1.46052	−	0.843229i
$880$		0			0
$881$			41.5692i			1.40050i	0.713896	+	0.700251i	$0.246929\pi$
$881$			41.5692i			1.40050i	−0.713896	+	0.700251i	$0.753071\pi$
$882$		0			0
$883$	−46.4758			−1.56404			−0.782018	−	0.623256i	$-0.785809\pi$
$883$	−46.4758			−1.56404			−0.782018	+	0.623256i	$0.785809\pi$
$884$		0			0
$885$	5.59017	+	9.68246i	0.187912	+	0.325472i
$886$		0			0
$887$	−7.79423	+	13.5000i	−0.261705	+	0.453286i	−0.966695	−	0.255931i	$-0.917618\pi$
$887$	−7.79423	+	13.5000i	−0.261705	+	0.453286i	0.704990	+	0.709217i	$0.250951\pi$
$888$		0			0
$889$	−20.0000	+	17.3205i	−0.670778	+	0.580911i
$890$		0			0
$891$	−21.3014	+	36.8951i	−0.713624	+	1.23603i
$892$		0			0
$893$	−30.1869	+	17.4284i	−1.01017	+	0.583220i
$894$		0			0
$895$	25.9808			0.868441
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11.6190	−	6.70820i	0.387514	−	0.223731i
$900$		0			0
$901$	10.0623	−	17.4284i	0.335224	−	0.580625i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.0000	+	25.9808i	−0.498617	+	0.863630i
$906$		0			0
$907$	−17.4284	−	30.1869i	−0.578701	−	1.00234i	−0.995629	−	0.0934003i	$-0.970226\pi$
$907$	−17.4284	−	30.1869i	−0.578701	−	1.00234i	0.416927	−	0.908940i	$-0.363107\pi$
$908$		0			0
$909$	−22.3607			−0.741657
$910$		0			0
$911$		−	6.00000i		−	0.198789i	−0.995048	−	0.0993944i	$-0.968309\pi$
$911$		−	6.00000i		−	0.198789i	0.995048	−	0.0993944i	$-0.0316906\pi$
$912$		0			0
$913$	−15.0000	+	8.66025i	−0.496428	+	0.286613i
$914$		0			0
$915$	−29.0474	−	16.7705i	−0.960277	−	0.554416i
$916$		0			0
$917$	−1.11803	−	5.80948i	−0.0369207	−	0.191846i
$918$		0			0
$919$	−37.2391	−	21.5000i	−1.22840	−	0.709220i	−0.261708	−	0.965147i	$-0.584286\pi$
$919$	−37.2391	−	21.5000i	−1.22840	−	0.709220i	−0.966696	+	0.255927i	$0.917619\pi$
$920$		0			0
$921$	−15.0000	−	25.9808i	−0.494267	−	0.856095i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−15.5885	−	27.0000i	−0.511992	−	0.886796i
$928$		0			0
$929$	−22.5000	−	12.9904i	−0.738201	−	0.426201i	0.0832138	−	0.996532i	$-0.473482\pi$
$929$	−22.5000	−	12.9904i	−0.738201	−	0.426201i	−0.821415	+	0.570331i	$0.806815\pi$
$930$		0			0
$931$	29.0474	−	36.8951i	0.951989	−	1.20919i
$932$		0			0
$933$	−30.1869	−	17.4284i	−0.988275	−	0.570581i
$934$		0			0
$935$	−38.9711	+	22.5000i	−1.27449	+	0.735829i
$936$		0			0
$937$			27.7128i			0.905338i	0.891679	+	0.452669i	$0.149528\pi$
$937$			27.7128i			0.905338i	−0.891679	+	0.452669i	$0.850472\pi$
$938$		0			0
$939$	3.87298			0.126390
$940$		0			0
$941$	1.11803	+	1.93649i	0.0364469	+	0.0631278i	0.883673	−	0.468104i	$-0.155063\pi$
$941$	1.11803	+	1.93649i	0.0364469	+	0.0631278i	−0.847227	+	0.531232i	$0.821729\pi$
$942$		0			0
$943$	15.5885	−	27.0000i	0.507630	−	0.879241i
$944$		0			0
$945$	−2.50000	−	12.9904i	−0.0813250	−	0.422577i
$946$		0			0
$947$	−5.80948	+	10.0623i	−0.188783	+	0.326981i	−0.944845	−	0.327519i	$-0.893787\pi$
$947$	−5.80948	+	10.0623i	−0.188783	+	0.326981i	0.756062	+	0.654500i	$0.227121\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	8.66025			0.280828
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−40.6663	+	23.4787i	−1.31593	+	0.759753i
$956$		0			0
$957$	33.5410	−	58.0948i	1.08423	−	1.87794i
$958$		0			0
$959$	7.79423	−	22.5000i	0.251689	−	0.726563i
$960$		0			0
$961$	14.0000	−	24.2487i	0.451613	−	0.782216i
$962$		0			0
$963$	3.87298	+	6.70820i	0.124805	+	0.216169i
$964$		0			0
$965$	−15.6525			−0.503871
$966$		0			0
$967$		−	34.0000i		−	1.09337i	−0.837340	−	0.546683i	$-0.815890\pi$
$967$		−	34.0000i		−	1.09337i	0.837340	−	0.546683i	$-0.184110\pi$
$968$		0			0
$969$	−67.5000	+	38.9711i	−2.16841	+	1.25193i
$970$		0			0
$971$	44.5393	+	25.7148i	1.42933	+	0.825227i	0.997068	−	0.0765194i	$-0.0243807\pi$
$971$	44.5393	+	25.7148i	1.42933	+	0.825227i	0.432266	+	0.901746i	$0.357714\pi$
$972$		0			0
$973$	26.8328	−	23.2379i	0.860221	−	0.744973i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.50000	−	7.79423i	−0.143968	−	0.249359i	0.785020	−	0.619471i	$-0.212653\pi$
$977$	−4.50000	−	7.79423i	−0.143968	−	0.249359i	−0.928987	+	0.370111i	$0.879319\pi$
$978$		0			0
$979$		−	20.1246i		−	0.643185i
$980$		0			0
$981$		−	23.2379i		−	0.741929i
$982$		0			0
$983$	−23.3827	−	40.5000i	−0.745792	−	1.29175i	−0.949824	−	0.312785i	$-0.898738\pi$
$983$	−23.3827	−	40.5000i	−0.745792	−	1.29175i	0.204032	−	0.978964i	$-0.434595\pi$
$984$		0			0
$985$	−45.0000	−	25.9808i	−1.43382	−	0.827816i
$986$		0			0
$987$	23.2379	−	20.1246i	0.739671	−	0.640573i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−16.4545	+	9.50000i	−0.522694	+	0.301777i	−0.738036	−	0.674761i	$-0.764247\pi$
$991$	−16.4545	+	9.50000i	−0.522694	+	0.301777i	0.215342	+	0.976539i	$0.430913\pi$
$992$		0			0
$993$			25.9808i			0.824475i
$994$		0			0
$995$	11.6190			0.368345
$996$		0			0
$997$	23.4787	+	40.6663i	0.743578	+	1.28792i	0.950856	+	0.309633i	$0.100206\pi$
$997$	23.4787	+	40.6663i	0.743578	+	1.28792i	−0.207278	+	0.978282i	$0.566460\pi$
$998$		0			0
$999$	12.9904	−	22.5000i	0.410997	−	0.711868i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.q.a.159.3	yes	8
4.3	odd	2	inner	448.2.q.a.159.1	yes	8
7.2	even	3		3136.2.e.a.1567.7		8
7.3	odd	6	inner	448.2.q.a.31.4	yes	8
7.5	odd	6		3136.2.e.a.1567.1		8
8.3	odd	2	inner	448.2.q.a.159.4	yes	8
8.5	even	2	inner	448.2.q.a.159.2	yes	8
28.3	even	6	inner	448.2.q.a.31.2	yes	8
28.19	even	6		3136.2.e.a.1567.6		8
28.23	odd	6		3136.2.e.a.1567.4		8
56.3	even	6	inner	448.2.q.a.31.3	yes	8
56.5	odd	6		3136.2.e.a.1567.8		8
56.19	even	6		3136.2.e.a.1567.3		8
56.37	even	6		3136.2.e.a.1567.2		8
56.45	odd	6	inner	448.2.q.a.31.1	✓	8
56.51	odd	6		3136.2.e.a.1567.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.q.a.31.1	✓	8	56.45	odd	6	inner
448.2.q.a.31.2	yes	8	28.3	even	6	inner
448.2.q.a.31.3	yes	8	56.3	even	6	inner
448.2.q.a.31.4	yes	8	7.3	odd	6	inner
448.2.q.a.159.1	yes	8	4.3	odd	2	inner
448.2.q.a.159.2	yes	8	8.5	even	2	inner
448.2.q.a.159.3	yes	8	1.1	even	1	trivial
448.2.q.a.159.4	yes	8	8.3	odd	2	inner
3136.2.e.a.1567.1		8	7.5	odd	6
3136.2.e.a.1567.2		8	56.37	even	6
3136.2.e.a.1567.3		8	56.19	even	6
3136.2.e.a.1567.4		8	28.23	odd	6
3136.2.e.a.1567.5		8	56.51	odd	6
3136.2.e.a.1567.6		8	28.19	even	6
3136.2.e.a.1567.7		8	7.2	even	3
3136.2.e.a.1567.8		8	56.5	odd	6

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.93649 - 1.11803i) q^{3} +(-1.11803 + 1.93649i) q^{5} +(1.73205 + 2.00000i) q^{7} +(1.00000 - 1.73205i) q^{9} +O(q^{10})\)	\(q+(1.93649 - 1.11803i) q^{3} +(-1.11803 + 1.93649i) q^{5} +(1.73205 + 2.00000i) q^{7} +(1.00000 - 1.73205i) q^{9} +(1.93649 + 3.35410i) q^{11} +5.00000i q^{15} +(4.50000 - 2.59808i) q^{17} +(-5.80948 - 3.35410i) q^{19} +(5.59017 + 1.93649i) q^{21} +(2.59808 + 1.50000i) q^{23} +2.23607i q^{27} -7.74597i q^{29} +(0.866025 + 1.50000i) q^{31} +(7.50000 + 4.33013i) q^{33} +(-5.80948 + 1.11803i) q^{35} +(-10.0623 - 5.80948i) q^{37} -10.3923i q^{41} +(2.23607 + 3.87298i) q^{45} +(2.59808 - 4.50000i) q^{47} +(-1.00000 + 6.92820i) q^{49} +(5.80948 - 10.0623i) q^{51} +(3.35410 - 1.93649i) q^{53} -8.66025 q^{55} -15.0000 q^{57} +(1.93649 - 1.11803i) q^{59} +(-3.35410 + 5.80948i) q^{61} +(5.19615 - 1.00000i) q^{63} +(5.80948 + 10.0623i) q^{67} +6.70820 q^{69} +6.00000i q^{71} +(-4.50000 + 2.59808i) q^{73} +(-3.35410 + 9.68246i) q^{77} +(-11.2583 - 6.50000i) q^{79} +(5.50000 + 9.52628i) q^{81} +4.47214i q^{83} +11.6190i q^{85} +(-8.66025 - 15.0000i) q^{87} +(-4.50000 - 2.59808i) q^{89} +(3.35410 + 1.93649i) q^{93} +(12.9904 - 7.50000i) q^{95} -3.46410i q^{97} +7.74597 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 36 q^{17} + 60 q^{33} - 8 q^{49} - 120 q^{57} - 36 q^{73} + 44 q^{81} - 36 q^{89}+O(q^{100}) \) 8 * q + 8 * q^9 + 36 * q^17 + 60 * q^33 - 8 * q^49 - 120 * q^57 - 36 * q^73 + 44 * q^81 - 36 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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