LMFDB - Embedded newform 312.6.q.b.289.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [312,6,Mod(217,312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("312.217"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	312.q (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [18,0,-81] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$50.0397517816$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 9 x^{17} - 31561 x^{16} + 110968 x^{15} + 409124362 x^{14} + 1115716662 x^{13} + \cdots + 20\!\cdots\!67 $ x^18 - 9x^17 - 31561x^16 + 110968x^15 + 409124362x^14 + 1115716662x^13 - 2808048629982x^12 - 26499127774948x^11 + 10907873301973059x^10 + 176462680377440065x^9 - 23454619484036846171x^8 - 530331899073585858580x^7 + 24627493257146053334842x^6 + 709380046153262710357026x^5 - 8593193773667045133757410x^4 - 319119577053841589654097540x^3 + 438400952547001279651462761x^2 + 42503548768832626083162040851*x + 202050265707790330225564628667
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{40}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.5
Root			$-8.14732 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	312.289
Dual form			312.6.q.b.217.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.50000 - 7.79423i) q^{3} +13.6473 q^{5} +(22.5118 - 38.9916i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(300.003 + 519.621i) q^{11} +(-82.3237 - 603.751i) q^{13} +(-61.4129 - 106.370i) q^{15} +(-162.967 + 282.268i) q^{17} +(-466.609 + 808.191i) q^{19} -405.212 q^{21} +(-314.146 - 544.118i) q^{23} -2938.75 q^{25} +729.000 q^{27} +(3301.05 + 5717.59i) q^{29} -6012.14 q^{31} +(2700.03 - 4676.59i) q^{33} +(307.226 - 532.130i) q^{35} +(2146.74 + 3718.27i) q^{37} +(-4335.32 + 3358.53i) q^{39} +(3061.27 + 5302.28i) q^{41} +(9608.35 - 16642.1i) q^{43} +(-552.716 + 957.333i) q^{45} -3711.32 q^{47} +(7389.94 + 12799.7i) q^{49} +2933.41 q^{51} -9403.31 q^{53} +(4094.24 + 7091.43i) q^{55} +8398.96 q^{57} +(4159.52 - 7204.51i) q^{59} +(-27787.1 + 48128.6i) q^{61} +(1823.46 + 3158.32i) q^{63} +(-1123.50 - 8239.59i) q^{65} +(12435.0 + 21538.1i) q^{67} +(-2827.32 + 4897.06i) q^{69} +(-20857.0 + 36125.4i) q^{71} +80038.2 q^{73} +(13224.4 + 22905.3i) q^{75} +27014.4 q^{77} +60393.3 q^{79} +(-3280.50 - 5681.99i) q^{81} +118846. q^{83} +(-2224.07 + 3852.20i) q^{85} +(29709.5 - 51458.3i) q^{87} +(31102.3 + 53870.7i) q^{89} +(-25394.5 - 10381.6i) q^{91} +(27054.6 + 46860.0i) q^{93} +(-6367.96 + 11029.6i) q^{95} +(-37116.1 + 64287.0i) q^{97} -48600.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 81 q^{3} + 90 q^{5} - 48 q^{7} - 729 q^{9} - 602 q^{11} + 429 q^{13} - 405 q^{15} + 877 q^{17} - 594 q^{19} + 864 q^{21} - 3346 q^{23} + 7412 q^{25} + 13122 q^{27} + 6955 q^{29} - 2804 q^{31} - 5418 q^{33}+ \cdots + 97524 q^{99}+O(q^{100}) $ 18 * q - 81 * q^3 + 90 * q^5 - 48 * q^7 - 729 * q^9 - 602 * q^11 + 429 * q^13 - 405 * q^15 + 877 * q^17 - 594 * q^19 + 864 * q^21 - 3346 * q^23 + 7412 * q^25 + 13122 * q^27 + 6955 * q^29 - 2804 * q^31 - 5418 * q^33 - 4790 * q^35 - 8431 * q^37 + 1755 * q^39 - 7183 * q^41 + 1228 * q^43 - 3645 * q^45 + 8276 * q^47 - 28271 * q^49 - 15786 * q^51 + 33402 * q^53 + 19638 * q^55 + 10692 * q^57 - 16984 * q^59 + 20631 * q^61 - 3888 * q^63 + 15697 * q^65 - 77828 * q^67 - 30114 * q^69 - 34854 * q^71 - 18162 * q^73 - 33354 * q^75 + 28576 * q^77 + 112116 * q^79 - 59049 * q^81 + 72492 * q^83 + 27659 * q^85 + 62595 * q^87 - 4994 * q^89 - 114838 * q^91 + 12618 * q^93 + 14002 * q^95 - 31836 * q^97 + 97524 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/312\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$157$	$209$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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			0
$2$		0			0
$3$	−4.50000	−	7.79423i	−0.288675	−	0.500000i
$3$	−4.50000	−	7.79423i	−0.288675	−	0.500000i
$4$		0			0
$5$	13.6473			0.244131			0.122065	−	0.992522i	$-0.461048\pi$
$5$	13.6473			0.244131			0.122065	+	0.992522i	$0.461048\pi$
$6$		0			0
$7$	22.5118	−	38.9916i	0.173646	−	0.300764i	−0.766046	−	0.642786i	$-0.777778\pi$
$7$	22.5118	−	38.9916i	0.173646	−	0.300764i	0.939692	+	0.342022i	$0.111112\pi$
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	300.003	+	519.621i	0.747557	+	1.29481i	0.948991	+	0.315304i	$0.102106\pi$
$11$	300.003	+	519.621i	0.747557	+	1.29481i	−0.201434	+	0.979502i	$0.564560\pi$
$12$		0			0
$13$	−82.3237	−	603.751i	−0.135103	−	0.990831i
$13$	−82.3237	−	603.751i	−0.135103	−	0.990831i
$14$		0			0
$15$	−61.4129	−	106.370i	−0.0704745	−	0.122065i
$16$		0			0
$17$	−162.967	+	282.268i	−0.136766	+	0.236886i	−0.926271	−	0.376859i	$-0.877004\pi$
$17$	−162.967	+	282.268i	−0.136766	+	0.236886i	0.789505	+	0.613745i	$0.210338\pi$
$18$		0			0
$19$	−466.609	+	808.191i	−0.296530	+	0.513606i	−0.975340	−	0.220709i	$-0.929163\pi$
$19$	−466.609	+	808.191i	−0.296530	+	0.513606i	0.678809	+	0.734315i	$0.262496\pi$
$20$		0			0
$21$	−405.212			−0.200509
$22$		0			0
$23$	−314.146	−	544.118i	−0.123826	−	0.214473i	0.797447	−	0.603389i	$-0.206183\pi$
$23$	−314.146	−	544.118i	−0.123826	−	0.214473i	−0.921274	+	0.388915i	$0.872850\pi$
$24$		0			0
$25$	−2938.75			−0.940400
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	3301.05	+	5717.59i	0.728882	+	1.26246i	0.957356	+	0.288911i	$0.0932931\pi$
$29$	3301.05	+	5717.59i	0.728882	+	1.26246i	−0.228474	+	0.973550i	$0.573374\pi$
$30$		0			0
$31$	−6012.14			−1.12363			−0.561817	−	0.827262i	$-0.689897\pi$
$31$	−6012.14			−1.12363			−0.561817	+	0.827262i	$0.689897\pi$
$32$		0			0
$33$	2700.03	−	4676.59i	0.431602	−	0.747557i
$34$		0			0
$35$	307.226	−	532.130i	0.0423923	−	0.0734257i
$36$		0			0
$37$	2146.74	+	3718.27i	0.257796	+	0.446515i	0.965651	−	0.259842i	$-0.0836705\pi$
$37$	2146.74	+	3718.27i	0.257796	+	0.446515i	−0.707855	+	0.706357i	$0.750337\pi$
$38$		0			0
$39$	−4335.32	+	3358.53i	−0.456415	+	0.353580i
$40$		0			0
$41$	3061.27	+	5302.28i	0.284408	+	0.492610i	0.972466	−	0.233047i	$-0.0748695\pi$
$41$	3061.27	+	5302.28i	0.284408	+	0.492610i	−0.688057	+	0.725656i	$0.741536\pi$
$42$		0			0
$43$	9608.35	−	16642.1i	0.792460	−	1.37258i	−0.131979	−	0.991253i	$-0.542133\pi$
$43$	9608.35	−	16642.1i	0.792460	−	1.37258i	0.924439	−	0.381329i	$-0.124534\pi$
$44$		0			0
$45$	−552.716	+	957.333i	−0.0406884	+	0.0704745i
$46$		0			0
$47$	−3711.32			−0.245067			−0.122533	−	0.992464i	$-0.539102\pi$
$47$	−3711.32			−0.245067			−0.122533	+	0.992464i	$0.539102\pi$
$48$		0			0
$49$	7389.94	+	12799.7i	0.439694	+	0.761572i
$50$		0			0
$51$	2933.41			0.157924
$52$		0			0
$53$	−9403.31			−0.459823			−0.229912	−	0.973211i	$-0.573844\pi$
$53$	−9403.31			−0.459823			−0.229912	+	0.973211i	$0.573844\pi$
$54$		0			0
$55$	4094.24	+	7091.43i	0.182502	+	0.316102i
$56$		0			0
$57$	8398.96			0.342404
$58$		0			0
$59$	4159.52	−	7204.51i	0.155566	−	0.269448i	−0.777699	−	0.628637i	$-0.783613\pi$
$59$	4159.52	−	7204.51i	0.155566	−	0.269448i	0.933265	+	0.359189i	$0.116947\pi$
$60$		0			0
$61$	−27787.1	+	48128.6i	−0.956132	+	1.65607i	−0.224377	+	0.974502i	$0.572035\pi$
$61$	−27787.1	+	48128.6i	−0.956132	+	1.65607i	−0.731755	+	0.681568i	$0.761298\pi$
$62$		0			0
$63$	1823.46	+	3158.32i	0.0578820	+	0.100255i
$64$		0			0
$65$	−1123.50	−	8239.59i	−0.0329829	−	0.241892i
$66$		0			0
$67$	12435.0	+	21538.1i	0.338423	+	0.586166i	0.984136	−	0.177414i	$-0.0567731\pi$
$67$	12435.0	+	21538.1i	0.338423	+	0.586166i	−0.645713	+	0.763580i	$0.723440\pi$
$68$		0			0
$69$	−2827.32	+	4897.06i	−0.0714911	+	0.123826i
$70$		0			0
$71$	−20857.0	+	36125.4i	−0.491028	+	0.850486i	−0.999947	−	0.0103287i	$-0.996712\pi$
$71$	−20857.0	+	36125.4i	−0.491028	+	0.850486i	0.508918	+	0.860815i	$0.330046\pi$
$72$		0			0
$73$	80038.2			1.75788			0.878942	−	0.476929i	$-0.158250\pi$
$73$	80038.2			1.75788			0.878942	+	0.476929i	$0.158250\pi$
$74$		0			0
$75$	13224.4	+	22905.3i	0.271470	+	0.470200i
$76$		0			0
$77$	27014.4			0.519241
$78$		0			0
$79$	60393.3			1.08873			0.544366	−	0.838848i	$-0.316770\pi$
$79$	60393.3			1.08873			0.544366	+	0.838848i	$0.316770\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	118846.			1.89360			0.946801	−	0.321820i	$-0.104295\pi$
$83$	118846.			1.89360			0.946801	+	0.321820i	$0.104295\pi$
$84$		0			0
$85$	−2224.07	+	3852.20i	−0.0333888	+	0.0578310i
$86$		0			0
$87$	29709.5	−	51458.3i	0.420820	−	0.728882i
$88$		0			0
$89$	31102.3	+	53870.7i	0.416214	+	0.720904i	0.995555	−	0.0941812i	$-0.0300233\pi$
$89$	31102.3	+	53870.7i	0.416214	+	0.720904i	−0.579341	+	0.815085i	$0.696690\pi$
$90$		0			0
$91$	−25394.5	−	10381.6i	−0.321467	−	0.131420i
$92$		0			0
$93$	27054.6	+	46860.0i	0.324365	+	0.561817i
$94$		0			0
$95$	−6367.96	+	11029.6i	−0.0723922	+	0.125387i
$96$		0			0
$97$	−37116.1	+	64287.0i	−0.400528	+	0.693735i	−0.993790	−	0.111275i	$-0.964507\pi$
$97$	−37116.1	+	64287.0i	−0.400528	+	0.693735i	0.593261	+	0.805010i	$0.297840\pi$
$98$		0			0
$99$	−48600.5			−0.498371
$100$		0			0
$101$	48374.5	+	83787.1i	0.471860	+	0.817286i	0.999482	−	0.0321939i	$-0.0102494\pi$
$101$	48374.5	+	83787.1i	0.471860	+	0.817286i	−0.527622	+	0.849480i	$0.676916\pi$
$102$		0			0
$103$	−121468.			−1.12815			−0.564076	−	0.825723i	$-0.690768\pi$
$103$	−121468.			−1.12815			−0.564076	+	0.825723i	$0.690768\pi$
$104$		0			0
$105$	−5530.06			−0.0489505
$106$		0			0
$107$	16644.9	+	28829.9i	0.140547	+	0.243435i	0.927703	−	0.373319i	$-0.121780\pi$
$107$	16644.9	+	28829.9i	0.140547	+	0.243435i	−0.787155	+	0.616755i	$0.788447\pi$
$108$		0			0
$109$	145820.			1.17557			0.587787	−	0.809016i	$-0.299999\pi$
$109$	145820.			1.17557			0.587787	+	0.809016i	$0.299999\pi$
$110$		0			0
$111$	19320.7	−	33464.4i	0.148838	−	0.257796i
$112$		0			0
$113$	−36099.6	+	62526.4i	−0.265954	+	0.460646i	−0.967813	−	0.251670i	$-0.919020\pi$
$113$	−36099.6	+	62526.4i	−0.265954	+	0.460646i	0.701859	+	0.712316i	$0.252354\pi$
$114$		0			0
$115$	−4287.26	−	7425.75i	−0.0302298	−	0.0523595i
$116$		0			0
$117$	45686.1	+	18677.1i	0.308546	+	0.126138i
$118$		0			0
$119$	7337.37	+	12708.7i	0.0474978	+	0.0822685i
$120$		0			0
$121$	−99478.3	+	172301.i	−0.617682	+	1.06986i
$122$		0			0
$123$	27551.4	−	47720.5i	0.164203	−	0.284408i
$124$		0			0
$125$	−82753.9			−0.473711
$126$		0			0
$127$	128247.	+	222130.i	0.705566	+	1.22208i	0.966487	+	0.256716i	$0.0826404\pi$
$127$	128247.	+	222130.i	0.705566	+	1.22208i	−0.260921	+	0.965360i	$0.584026\pi$
$128$		0			0
$129$	−172950.			−0.915054
$130$		0			0
$131$	−4949.20			−0.0251975			−0.0125987	−	0.999921i	$-0.504010\pi$
$131$	−4949.20			−0.0251975			−0.0125987	+	0.999921i	$0.504010\pi$
$132$		0			0
$133$	21008.4	+	36387.6i	0.102983	+	0.178371i
$134$		0			0
$135$	9948.90			0.0469830
$136$		0			0
$137$	110143.	−	190773.i	0.501365	−	0.868389i	−0.498634	−	0.866813i	$-0.666165\pi$
$137$	110143.	−	190773.i	0.501365	−	0.868389i	0.999999	−	0.00157651i	$-0.000501820\pi$
$138$		0			0
$139$	6838.36	−	11844.4i	0.0300203	−	0.0519967i	−0.850625	−	0.525773i	$-0.823776\pi$
$139$	6838.36	−	11844.4i	0.0300203	−	0.0519967i	0.880645	+	0.473776i	$0.157109\pi$
$140$		0			0
$141$	16700.9	+	28926.9i	0.0707446	+	0.122533i
$142$		0			0
$143$	289024.	−	223904.i	1.18194	−	0.915635i
$144$		0			0
$145$	45050.5	+	78029.8i	0.177942	+	0.308205i
$146$		0			0
$147$	66509.4	−	115198.i	0.253857	−	0.439694i
$148$		0			0
$149$	−203698.	+	352815.i	−0.751659	+	1.30191i	0.195359	+	0.980732i	$0.437413\pi$
$149$	−203698.	+	352815.i	−0.751659	+	1.30191i	−0.947018	+	0.321180i	$0.895921\pi$
$150$		0			0
$151$	253796.			0.905822			0.452911	−	0.891556i	$-0.350385\pi$
$151$	253796.			0.905822			0.452911	+	0.891556i	$0.350385\pi$
$152$		0			0
$153$	−13200.3	−	22863.7i	−0.0455887	−	0.0789619i
$154$		0			0
$155$	−82049.6			−0.274313
$156$		0			0
$157$	−378401.			−1.22519			−0.612595	−	0.790397i	$-0.709874\pi$
$157$	−378401.			−1.22519			−0.612595	+	0.790397i	$0.709874\pi$
$158$		0			0
$159$	42314.9	+	73291.5i	0.132740	+	0.229912i
$160$		0			0
$161$	−28288.0			−0.0860078
$162$		0			0
$163$	−102604.	+	177715.i	−0.302479	+	0.523909i	−0.976697	−	0.214623i	$-0.931148\pi$
$163$	−102604.	+	177715.i	−0.302479	+	0.523909i	0.674218	+	0.738533i	$0.264481\pi$
$164$		0			0
$165$	36848.1	−	63822.9i	0.105367	−	0.182502i
$166$		0			0
$167$	−139499.	−	241620.i	−0.387062	−	0.670411i	0.604991	−	0.796232i	$-0.293177\pi$
$167$	−139499.	−	241620.i	−0.387062	−	0.670411i	−0.992053	+	0.125821i	$0.959843\pi$
$168$		0			0
$169$	−357739.	+	99406.1i	−0.963494	+	0.267729i
$170$		0			0
$171$	−37795.3	−	65463.4i	−0.0988435	−	0.171202i
$172$		0			0
$173$	184310.	−	319235.i	0.468203	−	0.810952i	−0.531136	−	0.847286i	$-0.678235\pi$
$173$	184310.	−	319235.i	0.468203	−	0.810952i	0.999340	+	0.0363345i	$0.0115682\pi$
$174$		0			0
$175$	−66156.6	+	114587.i	−0.163297	+	0.282838i
$176$		0			0
$177$	−74871.4			−0.179632
$178$		0			0
$179$	−298985.	−	517857.i	−0.697456	−	1.20803i	−0.969346	−	0.245700i	$-0.920982\pi$
$179$	−298985.	−	517857.i	−0.697456	−	1.20803i	0.271890	−	0.962328i	$-0.412351\pi$
$180$		0			0
$181$	−154591.			−0.350742			−0.175371	−	0.984502i	$-0.556112\pi$
$181$	−154591.			−0.350742			−0.175371	+	0.984502i	$0.556112\pi$
$182$		0			0
$183$	500167.			1.10405
$184$		0			0
$185$	29297.3	+	50744.4i	0.0629358	+	0.109008i
$186$		0			0
$187$	−195563.			−0.408961
$188$		0			0
$189$	16411.1	−	28424.9i	0.0334182	−	0.0578820i
$190$		0			0
$191$	209118.	−	362203.i	0.414771	−	0.718404i	−0.580634	−	0.814165i	$-0.697195\pi$
$191$	209118.	−	362203.i	0.414771	−	0.718404i	0.995404	+	0.0957611i	$0.0305285\pi$
$192$		0			0
$193$	405258.	+	701928.i	0.783138	+	1.35644i	0.930105	+	0.367294i	$0.119716\pi$
$193$	405258.	+	701928.i	0.783138	+	1.35644i	−0.146967	+	0.989141i	$0.546951\pi$
$194$		0			0
$195$	−59165.5	+	45834.9i	−0.111425	+	0.0863198i
$196$		0			0
$197$	59938.4	+	103816.i	0.110037	+	0.190590i	0.915785	−	0.401669i	$-0.131570\pi$
$197$	59938.4	+	103816.i	0.110037	+	0.190590i	−0.805748	+	0.592259i	$0.798236\pi$
$198$		0			0
$199$	194477.	−	336844.i	0.348125	−	0.602970i	−0.637791	−	0.770209i	$-0.720152\pi$
$199$	194477.	−	336844.i	0.348125	−	0.602970i	0.985916	+	0.167239i	$0.0534851\pi$
$200$		0			0
$201$	111915.	−	193843.i	0.195389	−	0.338423i
$202$		0			0
$203$	297250.			0.506270
$204$		0			0
$205$	41778.2	+	72361.9i	0.0694328	+	0.120261i
$206$		0			0
$207$	50891.7			0.0825508
$208$		0			0
$209$	−559937.			−0.886693
$210$		0			0
$211$	−242831.	−	420595.i	−0.375489	−	0.650366i	0.614911	−	0.788597i	$-0.289192\pi$
$211$	−242831.	−	420595.i	−0.375489	−	0.650366i	−0.990400	+	0.138230i	$0.955859\pi$
$212$		0			0
$213$	375427.			0.566991
$214$		0			0
$215$	131128.	−	227121.i	0.193464	−	0.335089i
$216$		0			0
$217$	−135344.	+	234423.i	−0.195115	+	0.337948i
$218$		0			0
$219$	−360172.	−	623836.i	−0.507457	−	0.878942i
$220$		0			0
$221$	183836.	+	75154.4i	0.253191	+	0.103508i
$222$		0			0
$223$	−442680.	−	766745.i	−0.596112	−	1.03250i	−0.993389	−	0.114798i	$-0.963378\pi$
$223$	−442680.	−	766745.i	−0.596112	−	1.03250i	0.397277	−	0.917699i	$-0.369955\pi$
$224$		0			0
$225$	119019.	−	206148.i	0.156733	−	0.271470i
$226$		0			0
$227$	346046.	−	599369.i	0.445727	−	0.772022i	−0.552376	−	0.833595i	$-0.686279\pi$
$227$	346046.	−	599369.i	0.445727	−	0.772022i	0.998103	+	0.0615735i	$0.0196119\pi$
$228$		0			0
$229$	−1.50110e6			−1.89156			−0.945779	−	0.324809i	$-0.894700\pi$
$229$	−1.50110e6			−1.89156			−0.945779	+	0.324809i	$0.894700\pi$
$230$		0			0
$231$	−121565.	−	210557.i	−0.149892	−	0.259621i
$232$		0			0
$233$	622467.			0.751149			0.375575	−	0.926792i	$-0.377445\pi$
$233$	622467.			0.751149			0.375575	+	0.926792i	$0.377445\pi$
$234$		0			0
$235$	−50649.6			−0.0598283
$236$		0			0
$237$	−271770.	−	470719.i	−0.314290	−	0.544366i
$238$		0			0
$239$	−249504.			−0.282542			−0.141271	−	0.989971i	$-0.545119\pi$
$239$	−249504.			−0.282542			−0.141271	+	0.989971i	$0.545119\pi$
$240$		0			0
$241$	−516792.	+	895110.i	−0.573156	+	0.992736i	0.423083	+	0.906091i	$0.360948\pi$
$241$	−516792.	+	895110.i	−0.573156	+	0.992736i	−0.996239	+	0.0866448i	$0.972385\pi$
$242$		0			0
$243$	−29524.5	+	51137.9i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	100853.	+	174682.i	0.107343	+	0.185923i
$246$		0			0
$247$	526359.	+	215183.i	0.548959	+	0.224422i
$248$		0			0
$249$	−534806.	−	926311.i	−0.546636	−	0.946801i
$250$		0			0
$251$	−888477.	+	1.53889e6i	−0.890147	+	1.54178i	−0.0504493	+	0.998727i	$0.516065\pi$
$251$	−888477.	+	1.53889e6i	−0.890147	+	1.54178i	−0.839698	+	0.543054i	$0.817268\pi$
$252$		0			0
$253$	188490.	−	326474.i	0.185134	−	0.320662i
$254$		0			0
$255$	40033.2			0.0385540
$256$		0			0
$257$	−747404.	−	1.29454e6i	−0.705867	−	1.22260i	−0.966378	−	0.257126i	$-0.917224\pi$
$257$	−747404.	−	1.29454e6i	−0.705867	−	1.22260i	0.260511	−	0.965471i	$-0.416109\pi$
$258$		0			0
$259$	193308.			0.179061
$260$		0			0
$261$	−534770.			−0.485921
$262$		0			0
$263$	−1.02688e6	−	1.77860e6i	−0.915438	−	1.58558i	−0.806259	−	0.591562i	$-0.798511\pi$
$263$	−1.02688e6	−	1.77860e6i	−0.915438	−	1.58558i	−0.109178	−	0.994022i	$-0.534822\pi$
$264$		0			0
$265$	−128330.			−0.112257
$266$		0			0
$267$	279920.	−	484836.i	0.240301	−	0.416214i
$268$		0			0
$269$	6435.48	−	11146.6i	0.00542251	−	0.00939207i	−0.863301	−	0.504689i	$-0.831607\pi$
$269$	6435.48	−	11146.6i	0.00542251	−	0.00939207i	0.868724	+	0.495297i	$0.164941\pi$
$270$		0			0
$271$	25304.7	+	43829.0i	0.0209304	+	0.0362525i	0.876301	−	0.481764i	$-0.160004\pi$
$271$	25304.7	+	43829.0i	0.0209304	+	0.0362525i	−0.855370	+	0.518017i	$0.826670\pi$
$272$		0			0
$273$	33358.6	+	244648.i	0.0270895	+	0.198671i
$274$		0			0
$275$	−881634.	−	1.52704e6i	−0.703002	−	1.21764i
$276$		0			0
$277$	−787453.	+	1.36391e6i	−0.616631	+	1.06804i	0.373465	+	0.927644i	$0.378170\pi$
$277$	−787453.	+	1.36391e6i	−0.616631	+	1.06804i	−0.990096	+	0.140392i	$0.955164\pi$
$278$		0			0
$279$	243492.	−	421740.i	0.187272	−	0.324365i
$280$		0			0
$281$	1.78516e6			1.34869			0.674344	−	0.738417i	$-0.264426\pi$
$281$	1.78516e6			1.34869			0.674344	+	0.738417i	$0.264426\pi$
$282$		0			0
$283$	141327.	+	244785.i	0.104896	+	0.181685i	0.913696	−	0.406399i	$-0.133216\pi$
$283$	141327.	+	244785.i	0.104896	+	0.181685i	−0.808800	+	0.588084i	$0.799882\pi$
$284$		0			0
$285$	114623.			0.0835913
$286$		0			0
$287$	275659.			0.197546
$288$		0			0
$289$	656812.	+	1.13763e6i	0.462590	+	0.801230i
$290$		0			0
$291$	668090.			0.462490
$292$		0			0
$293$	609936.	−	1.05644e6i	0.415064	−	0.718912i	−0.580371	−	0.814352i	$-0.697092\pi$
$293$	609936.	−	1.05644e6i	0.415064	−	0.718912i	0.995435	+	0.0954401i	$0.0304258\pi$
$294$		0			0
$295$	56766.4	−	98322.2i	0.0379783	−	0.0657804i
$296$		0			0
$297$	218702.	+	378803.i	0.143867	+	0.249186i
$298$		0			0
$299$	−302650.	+	234460.i	−0.195778	+	0.151667i
$300$		0			0
$301$	−432602.	−	749289.i	−0.275215	−	0.476687i
$302$		0			0
$303$	435371.	−	754084.i	0.272429	−	0.471860i
$304$		0			0
$305$	−379219.	+	656826.i	−0.233421	+	0.404298i
$306$		0			0
$307$	−501934.			−0.303949			−0.151975	−	0.988384i	$-0.548563\pi$
$307$	−501934.			−0.303949			−0.151975	+	0.988384i	$0.548563\pi$
$308$		0			0
$309$	546605.	+	946747.i	0.325670	+	0.564076i
$310$		0			0
$311$	−227712.			−0.133501			−0.0667505	−	0.997770i	$-0.521263\pi$
$311$	−227712.			−0.133501			−0.0667505	+	0.997770i	$0.521263\pi$
$312$		0			0
$313$	−1.16967e6			−0.674840			−0.337420	−	0.941354i	$-0.609554\pi$
$313$	−1.16967e6			−0.674840			−0.337420	+	0.941354i	$0.609554\pi$
$314$		0			0
$315$	24885.3	+	43102.6i	0.0141308	+	0.0244752i
$316$		0			0
$317$	−621813.			−0.347546			−0.173773	−	0.984786i	$-0.555596\pi$
$317$	−621813.			−0.347546			−0.173773	+	0.984786i	$0.555596\pi$
$318$		0			0
$319$	−1.98065e6	+	3.43059e6i	−1.08976	+	1.88752i
$320$		0			0
$321$	149805.	−	259469.i	0.0811451	−	0.140547i
$322$		0			0
$323$	−152084.	−	263417.i	−0.0811105	−	0.140488i
$324$		0			0
$325$	241929.	+	1.77427e6i	0.127051	+	0.931778i
$326$		0			0
$327$	−656189.	−	1.13655e6i	−0.339359	−	0.587787i
$328$		0			0
$329$	−83548.5	+	144710.i	−0.0425548	+	0.0737072i
$330$		0			0
$331$	−495102.	+	857541.i	−0.248384	+	0.430215i	−0.963078	−	0.269223i	$-0.913233\pi$
$331$	−495102.	+	857541.i	−0.248384	+	0.430215i	0.714693	+	0.699438i	$0.246566\pi$
$332$		0			0
$333$	−347772.			−0.171864
$334$		0			0
$335$	169705.	+	293938.i	0.0826195	+	0.143101i
$336$		0			0
$337$	322683.			0.154775			0.0773875	−	0.997001i	$-0.475342\pi$
$337$	322683.			0.154775			0.0773875	+	0.997001i	$0.475342\pi$
$338$		0			0
$339$	649794.			0.307097
$340$		0			0
$341$	−1.80366e6	−	3.12403e6i	−0.839980	−	1.45489i
$342$		0			0
$343$	1.42215e6			0.652697
$344$		0			0
$345$	−38585.3	+	66831.7i	−0.0174532	+	0.0302298i
$346$		0			0
$347$	576417.	−	998384.i	0.256988	−	0.445117i	−0.708445	−	0.705766i	$-0.750603\pi$
$347$	576417.	−	998384.i	0.256988	−	0.445117i	0.965434	+	0.260649i	$0.0839365\pi$
$348$		0			0
$349$	−775854.	−	1.34382e6i	−0.340970	−	0.590578i	0.643643	−	0.765326i	$-0.277422\pi$
$349$	−775854.	−	1.34382e6i	−0.340970	−	0.590578i	−0.984613	+	0.174748i	$0.944089\pi$
$350$		0			0
$351$	−60014.0	−	440135.i	−0.0260007	−	0.190686i
$352$		0			0
$353$	2.04013e6	+	3.53362e6i	0.871409	+	1.50932i	0.860540	+	0.509384i	$0.170127\pi$
$353$	2.04013e6	+	3.53362e6i	0.871409	+	1.50932i	0.0108693	+	0.999941i	$0.496540\pi$
$354$		0			0
$355$	−284643.	+	493015.i	−0.119875	+	0.207630i
$356$		0			0
$357$	66036.3	−	114378.i	0.0274228	−	0.0474978i
$358$		0			0
$359$	1.80155e6			0.737751			0.368875	−	0.929479i	$-0.379743\pi$
$359$	1.80155e6			0.737751			0.368875	+	0.929479i	$0.379743\pi$
$360$		0			0
$361$	802601.	+	1.39015e6i	0.324140	+	0.561426i
$362$		0			0
$363$	1.79061e6			0.713238
$364$		0			0
$365$	1.09231e6			0.429153
$366$		0			0
$367$	1.29159e6	+	2.23710e6i	0.500564	+	0.867003i	1.00000		0.000651930i	$0.000207516\pi$
$367$	1.29159e6	+	2.23710e6i	0.500564	+	0.867003i	−0.499435	+	0.866351i	$0.666459\pi$
$368$		0			0
$369$	−495926.			−0.189606
$370$		0			0
$371$	−211685.	+	366650.i	−0.0798465	+	0.138298i
$372$		0			0
$373$	−1.13590e6	+	1.96744e6i	−0.422735	+	0.732198i	−0.996206	−	0.0870275i	$-0.972263\pi$
$373$	−1.13590e6	+	1.96744e6i	−0.422735	+	0.732198i	0.573471	+	0.819226i	$0.305597\pi$
$374$		0			0
$375$	372393.	+	645003.i	0.136749	+	0.236856i
$376$		0			0
$377$	3.18025e6	−	2.46371e6i	1.15241	−	0.892762i
$378$		0			0
$379$	−259442.	−	449366.i	−0.0927773	−	0.160695i	0.815901	−	0.578191i	$-0.196241\pi$
$379$	−259442.	−	449366.i	−0.0927773	−	0.160695i	−0.908679	+	0.417496i	$0.862908\pi$
$380$		0			0
$381$	1.15422e6	−	1.99917e6i	0.407359	−	0.705566i
$382$		0			0
$383$	−597621.	+	1.03511e6i	−0.208175	+	0.360570i	−0.951140	−	0.308761i	$-0.900086\pi$
$383$	−597621.	+	1.03511e6i	−0.208175	+	0.360570i	0.742964	+	0.669331i	$0.233419\pi$
$384$		0			0
$385$	368675.			0.126763
$386$		0			0
$387$	778276.	+	1.34801e6i	0.264153	+	0.457527i
$388$		0			0
$389$	4.47762e6			1.50028			0.750141	−	0.661277i	$-0.229985\pi$
$389$	4.47762e6			1.50028			0.750141	+	0.661277i	$0.229985\pi$
$390$		0			0
$391$	204782.			0.0677409
$392$		0			0
$393$	22271.4	+	38575.2i	0.00727388	+	0.0125987i
$394$		0			0
$395$	824206.			0.265793
$396$		0			0
$397$	2.06631e6	−	3.57895e6i	0.657989	−	1.13967i	−0.323146	−	0.946349i	$-0.604740\pi$
$397$	2.06631e6	−	3.57895e6i	0.657989	−	1.13967i	0.981135	−	0.193322i	$-0.0619262\pi$
$398$		0			0
$399$	189076.	−	327489.i	0.0594571	−	0.102983i
$400$		0			0
$401$	155894.	+	270016.i	0.0484137	+	0.0838549i	0.889217	−	0.457486i	$-0.151250\pi$
$401$	155894.	+	270016.i	0.0484137	+	0.0838549i	−0.840803	+	0.541341i	$0.817917\pi$
$402$		0			0
$403$	494941.	+	3.62984e6i	0.151807	+	1.11333i
$404$		0			0
$405$	−44770.0	−	77544.0i	−0.0135628	−	0.0234915i
$406$		0			0
$407$	−1.28806e6	+	2.23098e6i	−0.385434	+	0.667591i
$408$		0			0
$409$	−434561.	+	752681.i	−0.128452	+	0.222486i	−0.923077	−	0.384615i	$-0.874334\pi$
$409$	−434561.	+	752681.i	−0.128452	+	0.222486i	0.794625	+	0.607101i	$0.207668\pi$
$410$		0			0
$411$	−1.98257e6			−0.578926
$412$		0			0
$413$	−187277.	−	324373.i	−0.0540267	−	0.0935770i
$414$		0			0
$415$	1.62193e6			0.462286
$416$		0			0
$417$	−123090.			−0.0346644
$418$		0			0
$419$	2.90752e6	+	5.03598e6i	0.809074	+	1.40136i	0.913506	+	0.406825i	$0.133364\pi$
$419$	2.90752e6	+	5.03598e6i	0.809074	+	1.40136i	−0.104432	+	0.994532i	$0.533303\pi$
$420$		0			0
$421$	−187958.			−0.0516838			−0.0258419	−	0.999666i	$-0.508227\pi$
$421$	−187958.			−0.0516838			−0.0258419	+	0.999666i	$0.508227\pi$
$422$		0			0
$423$	150309.	−	260342.i	0.0408444	−	0.0707446i
$424$		0			0
$425$	478920.	−	829514.i	0.128615	−	0.222767i
$426$		0			0
$427$	1.25107e6	+	2.16692e6i	0.332057	+	0.575140i
$428$		0			0
$429$	−3.04577e6	−	1.24515e6i	−0.799014	−	0.326647i
$430$		0			0
$431$	2.09639e6	+	3.63106e6i	0.543600	+	0.941543i	0.998694	+	0.0510995i	$0.0162726\pi$
$431$	2.09639e6	+	3.63106e6i	0.543600	+	0.941543i	−0.455093	+	0.890444i	$0.650394\pi$
$432$		0			0
$433$	−2.10751e6	+	3.65031e6i	−0.540194	+	0.935644i	0.458698	+	0.888592i	$0.348316\pi$
$433$	−2.10751e6	+	3.65031e6i	−0.540194	+	0.935644i	−0.998892	+	0.0470517i	$0.985017\pi$
$434$		0			0
$435$	405454.	−	702268.i	0.102735	−	0.177942i
$436$		0			0
$437$	586334.			0.146873
$438$		0			0
$439$	1.56199e6	+	2.70544e6i	0.386827	+	0.670003i	0.992021	−	0.126075i	$-0.0402379\pi$
$439$	1.56199e6	+	2.70544e6i	0.386827	+	0.670003i	−0.605194	+	0.796078i	$0.706905\pi$
$440$		0			0
$441$	−1.19717e6			−0.293129
$442$		0			0
$443$	−7.92957e6			−1.91973			−0.959865	−	0.280463i	$-0.909512\pi$
$443$	−7.92957e6			−1.91973			−0.959865	+	0.280463i	$0.909512\pi$
$444$		0			0
$445$	424463.	+	735191.i	0.101611	+	0.175995i
$446$		0			0
$447$	3.66656e6			0.867941
$448$		0			0
$449$	881072.	−	1.52606e6i	0.206251	−	0.357237i	−0.744280	−	0.667868i	$-0.767207\pi$
$449$	881072.	−	1.52606e6i	0.206251	−	0.357237i	0.950531	+	0.310631i	$0.100540\pi$
$450$		0			0
$451$	−1.83678e6	+	3.18140e6i	−0.425223	+	0.736507i
$452$		0			0
$453$	−1.14208e6	−	1.97815e6i	−0.261488	−	0.452911i
$454$		0			0
$455$	−346566.	−	141681.i	−0.0784798	−	0.0320836i
$456$		0			0
$457$	−1.02583e6	−	1.77680e6i	−0.229766	−	0.397967i	0.727972	−	0.685606i	$-0.240463\pi$
$457$	−1.02583e6	−	1.77680e6i	−0.229766	−	0.397967i	−0.957739	+	0.287639i	$0.907130\pi$
$458$		0			0
$459$	−118803.	+	205773.i	−0.0263206	+	0.0455887i
$460$		0			0
$461$	−97067.1	+	168125.i	−0.0212726	+	0.0368452i	−0.876466	−	0.481464i	$-0.840105\pi$
$461$	−97067.1	+	168125.i	−0.0212726	+	0.0368452i	0.855193	+	0.518309i	$0.173438\pi$
$462$		0			0
$463$	−1.93631e6			−0.419781			−0.209891	−	0.977725i	$-0.567311\pi$
$463$	−1.93631e6			−0.419781			−0.209891	+	0.977725i	$0.567311\pi$
$464$		0			0
$465$	369223.	+	639513.i	0.0791874	+	0.137157i
$466$		0			0
$467$	−4.97796e6			−1.05623			−0.528116	−	0.849173i	$-0.677101\pi$
$467$	−4.97796e6			−1.05623			−0.528116	+	0.849173i	$0.677101\pi$
$468$		0			0
$469$	1.11974e6			0.235063
$470$		0			0
$471$	1.70280e6	+	2.94934e6i	0.353682	+	0.612595i
$472$		0			0
$473$	1.15301e7			2.36964
$474$		0			0
$475$	1.37125e6	−	2.37507e6i	0.278857	−	0.482995i
$476$		0			0
$477$	380834.	−	659624.i	0.0766372	−	0.132740i
$478$		0			0
$479$	−2.15777e6	−	3.73737e6i	−0.429701	−	0.744264i	0.567146	−	0.823618i	$-0.308048\pi$
$479$	−2.15777e6	−	3.73737e6i	−0.429701	−	0.744264i	−0.996847	+	0.0793538i	$0.974714\pi$
$480$		0			0
$481$	2.06818e6	−	1.60220e6i	0.407592	−	0.315758i
$482$		0			0
$483$	127296.	+	220483.i	0.0248283	+	0.0430039i
$484$		0			0
$485$	−506536.	+	877345.i	−0.0977812	+	0.169362i
$486$		0			0
$487$	−1.48661e6	+	2.57489e6i	−0.284038	+	0.491967i	−0.972375	−	0.233423i	$-0.925007\pi$
$487$	−1.48661e6	+	2.57489e6i	−0.284038	+	0.491967i	0.688338	+	0.725390i	$0.258341\pi$
$488$		0			0
$489$	1.84687e6			0.349273
$490$		0			0
$491$	−985768.	−	1.70740e6i	−0.184532	−	0.319618i	0.758887	−	0.651222i	$-0.225743\pi$
$491$	−985768.	−	1.70740e6i	−0.184532	−	0.319618i	−0.943419	+	0.331604i	$0.892410\pi$
$492$		0			0
$493$	−2.15185e6			−0.398745
$494$		0			0
$495$	−663267.			−0.121668
$496$		0			0
$497$	939058.	+	1.62650e6i	0.170530	+	0.295367i
$498$		0			0
$499$	9.15096e6			1.64519			0.822594	−	0.568629i	$-0.192526\pi$
$499$	9.15096e6			1.64519			0.822594	+	0.568629i	$0.192526\pi$
$500$		0			0
$501$	−1.25549e6	+	2.17458e6i	−0.223470	+	0.387062i
$502$		0			0
$503$	−4.15716e6	+	7.20041e6i	−0.732617	+	1.26893i	0.223145	+	0.974785i	$0.428368\pi$
$503$	−4.15716e6	+	7.20041e6i	−0.732617	+	1.26893i	−0.955761	+	0.294144i	$0.904966\pi$
$504$		0			0
$505$	660183.	+	1.14347e6i	0.115196	+	0.199525i
$506$		0			0
$507$	2.38462e6	+	2.34097e6i	0.412002	+	0.404460i
$508$		0			0
$509$	−4.99683e6	−	8.65476e6i	−0.854869	−	1.48068i	−0.876766	−	0.480917i	$-0.840304\pi$
$509$	−4.99683e6	−	8.65476e6i	−0.854869	−	1.48068i	0.0218966	−	0.999760i	$-0.493030\pi$
$510$		0			0
$511$	1.80180e6	−	3.12081e6i	0.305250	−	0.528708i
$512$		0			0
$513$	−340158.	+	589171.i	−0.0570673	+	0.0988435i
$514$		0			0
$515$	−1.65771e6			−0.275417
$516$		0			0
$517$	−1.11341e6	−	1.92848e6i	−0.183201	−	0.317314i
$518$		0			0
$519$	−3.31759e6			−0.540635
$520$		0			0
$521$	−4.63095e6			−0.747439			−0.373719	−	0.927542i	$-0.621918\pi$
$521$	−4.63095e6			−0.747439			−0.373719	+	0.927542i	$0.621918\pi$
$522$		0			0
$523$	−4.05821e6	−	7.02902e6i	−0.648754	−	1.12368i	−0.983421	−	0.181339i	$-0.941957\pi$
$523$	−4.05821e6	−	7.02902e6i	−0.648754	−	1.12368i	0.334666	−	0.942337i	$-0.391376\pi$
$524$		0			0
$525$	1.19082e6			0.188559
$526$		0			0
$527$	979782.	−	1.69703e6i	0.153675	−	0.266173i
$528$		0			0
$529$	3.02080e6	−	5.23217e6i	0.469334	−	0.812911i
$530$		0			0
$531$	336921.	+	583565.i	0.0518552	+	0.0898158i
$532$		0			0
$533$	2.94924e6	−	2.28475e6i	0.449669	−	0.348354i
$534$		0			0
$535$	227159.	+	393451.i	0.0343119	+	0.0594300i
$536$		0			0
$537$	−2.69086e6	+	4.66071e6i	−0.402676	+	0.697456i
$538$		0			0
$539$	−4.43401e6	+	7.67993e6i	−0.657392	+	1.13864i
$540$		0			0
$541$	−2.72696e6			−0.400576			−0.200288	−	0.979737i	$-0.564188\pi$
$541$	−2.72696e6			−0.400576			−0.200288	+	0.979737i	$0.564188\pi$
$542$		0			0
$543$	695660.	+	1.20492e6i	0.101250	+	0.175371i
$544$		0			0
$545$	1.99005e6			0.286994
$546$		0			0
$547$	−3.24480e6			−0.463681			−0.231841	−	0.972754i	$-0.574475\pi$
$547$	−3.24480e6			−0.463681			−0.231841	+	0.972754i	$0.574475\pi$
$548$		0			0
$549$	−2.25075e6	−	3.89842e6i	−0.318711	−	0.552023i
$550$		0			0
$551$	−6.16120e6			−0.864543
$552$		0			0
$553$	1.35956e6	−	2.35483e6i	0.189054	−	0.327451i
$554$		0			0
$555$	263676.	−	456699.i	0.0363360	−	0.0629358i
$556$		0			0
$557$	−4.44217e6	−	7.69407e6i	−0.606677	−	1.05080i	−0.991784	−	0.127924i	$-0.959169\pi$
$557$	−4.44217e6	−	7.69407e6i	−0.606677	−	1.05080i	0.385107	−	0.922872i	$-0.374165\pi$
$558$		0			0
$559$	−1.08387e7	−	4.43101e6i	−1.46706	−	0.599754i
$560$		0			0
$561$	880032.	+	1.52426e6i	0.118057	+	0.204481i
$562$		0			0
$563$	37328.4	−	64654.8i	0.00496328	−	0.00859666i	−0.863533	−	0.504292i	$-0.831753\pi$
$563$	37328.4	−	64654.8i	0.00496328	−	0.00859666i	0.868496	+	0.495696i	$0.165087\pi$
$564$		0			0
$565$	−492663.	+	853318.i	−0.0649276	+	0.112458i
$566$		0			0
$567$	−295400.			−0.0385880
$568$		0			0
$569$	5.84632e6	+	1.01261e7i	0.757011	+	1.31118i	0.944368	+	0.328890i	$0.106674\pi$
$569$	5.84632e6	+	1.01261e7i	0.757011	+	1.31118i	−0.187357	+	0.982292i	$0.559992\pi$
$570$		0			0
$571$	1.09629e7			1.40713			0.703566	−	0.710630i	$-0.251590\pi$
$571$	1.09629e7			1.40713			0.703566	+	0.710630i	$0.251590\pi$
$572$		0			0
$573$	−3.76412e6			−0.478936
$574$		0			0
$575$	923198.	+	1.59903e6i	0.116446	+	0.201691i
$576$		0			0
$577$	−1.06395e7			−1.33040			−0.665202	−	0.746663i	$-0.731655\pi$
$577$	−1.06395e7			−1.33040			−0.665202	+	0.746663i	$0.731655\pi$
$578$		0			0
$579$	3.64732e6	−	6.31735e6i	0.452145	−	0.783138i
$580$		0			0
$581$	2.67543e6	−	4.63398e6i	0.328817	−	0.569527i
$582$		0			0
$583$	−2.82102e6	−	4.88615e6i	−0.343744	−	0.595382i
$584$		0			0
$585$	623493.	+	254892.i	0.0753255	+	0.0307941i
$586$		0			0
$587$	−4.58015e6	−	7.93305e6i	−0.548636	−	0.950266i	−0.998368	−	0.0571023i	$-0.981814\pi$
$587$	−4.58015e6	−	7.93305e6i	−0.548636	−	0.950266i	0.449732	−	0.893163i	$-0.351519\pi$
$588$		0			0
$589$	2.80532e6	−	4.85895e6i	0.333191	−	0.577104i
$590$		0			0
$591$	539446.	−	934348.i	0.0635300	−	0.110037i
$592$		0			0
$593$	1.13631e7			1.32697			0.663484	−	0.748191i	$-0.269077\pi$
$593$	1.13631e7			1.32697			0.663484	+	0.748191i	$0.269077\pi$
$594$		0			0
$595$	100135.	+	173440.i	0.0115957	+	0.0200843i
$596$		0			0
$597$	−3.50058e6			−0.401980
$598$		0			0
$599$	41835.6			0.00476408			0.00238204	−	0.999997i	$-0.499242\pi$
$599$	41835.6			0.00476408			0.00238204	+	0.999997i	$0.499242\pi$
$600$		0			0
$601$	−973576.	−	1.68628e6i	−0.109947	−	0.190434i	0.805802	−	0.592186i	$-0.201735\pi$
$601$	−973576.	−	1.68628e6i	−0.109947	−	0.190434i	−0.915749	+	0.401752i	$0.868401\pi$
$602$		0			0
$603$	−2.01448e6			−0.225615
$604$		0			0
$605$	−1.35761e6	+	2.35145e6i	−0.150795	+	0.261185i
$606$		0			0
$607$	1.52565e6	−	2.64250e6i	0.168067	−	0.291101i	−0.769673	−	0.638438i	$-0.779581\pi$
$607$	1.52565e6	−	2.64250e6i	0.168067	−	0.291101i	0.937740	+	0.347337i	$0.112914\pi$
$608$		0			0
$609$	−1.33763e6	−	2.31684e6i	−0.146148	−	0.253135i
$610$		0			0
$611$	305530.	+	2.24072e6i	0.0331093	+	0.242820i
$612$		0			0
$613$	−6.76690e6	−	1.17206e7i	−0.727342	−	1.25979i	−0.958003	−	0.286759i	$-0.907422\pi$
$613$	−6.76690e6	−	1.17206e7i	−0.727342	−	1.25979i	0.230661	−	0.973034i	$-0.425911\pi$
$614$		0			0
$615$	376003.	−	651257.i	0.0400870	−	0.0694328i
$616$		0			0
$617$	−3.87947e6	+	6.71944e6i	−0.410260	+	0.710592i	−0.994918	−	0.100688i	$-0.967895\pi$
$617$	−3.87947e6	+	6.71944e6i	−0.410260	+	0.710592i	0.584658	+	0.811280i	$0.301229\pi$
$618$		0			0
$619$	7.15135e6			0.750173			0.375087	−	0.926990i	$-0.377613\pi$
$619$	7.15135e6			0.750173			0.375087	+	0.926990i	$0.377613\pi$
$620$		0			0
$621$	−229013.	−	396662.i	−0.0238304	−	0.0412754i
$622$		0			0
$623$	2.80067e6			0.289096
$624$		0			0
$625$	8.05423e6			0.824753
$626$		0			0
$627$	2.51972e6	+	4.36427e6i	0.255966	+	0.443346i
$628$		0			0
$629$	−1.39940e6			−0.141031
$630$		0			0
$631$	−1.05433e6	+	1.82615e6i	−0.105415	+	0.182584i	−0.913908	−	0.405922i	$-0.866950\pi$
$631$	−1.05433e6	+	1.82615e6i	−0.105415	+	0.182584i	0.808493	+	0.588506i	$0.200284\pi$
$632$		0			0
$633$	−2.18548e6	+	3.78536e6i	−0.216789	+	0.375489i
$634$		0			0
$635$	1.75023e6	+	3.03148e6i	0.172250	+	0.298346i
$636$		0			0
$637$	7.11950e6	−	5.51541e6i	0.695186	−	0.538554i
$638$		0			0
$639$	−1.68942e6	−	2.92616e6i	−0.163676	−	0.283495i
$640$		0			0
$641$	3.14729e6	−	5.45127e6i	0.302546	−	0.524026i	−0.674166	−	0.738580i	$-0.735497\pi$
$641$	3.14729e6	−	5.45127e6i	0.302546	−	0.524026i	0.976712	+	0.214555i	$0.0688300\pi$
$642$		0			0
$643$	4.28140e6	−	7.41560e6i	0.408374	−	0.707325i	−0.586334	−	0.810070i	$-0.699429\pi$
$643$	4.28140e6	−	7.41560e6i	0.408374	−	0.707325i	0.994708	+	0.102745i	$0.0327626\pi$
$644$		0			0
$645$	−2.36031e6			−0.223393
$646$		0			0
$647$	−7.15260e6	−	1.23887e7i	−0.671743	−	1.16349i	−0.977409	−	0.211354i	$-0.932213\pi$
$647$	−7.15260e6	−	1.23887e7i	−0.671743	−	1.16349i	0.305666	−	0.952139i	$-0.401121\pi$
$648$		0			0
$649$	4.99148e6			0.465176
$650$		0			0
$651$	2.43619e6			0.225299
$652$		0			0
$653$	5.63627e6	+	9.76231e6i	0.517260	+	0.895921i	0.999799	+	0.0200466i	$0.00638145\pi$
$653$	5.63627e6	+	9.76231e6i	0.517260	+	0.895921i	−0.482539	+	0.875875i	$0.660285\pi$
$654$		0			0
$655$	−67543.3			−0.00615147
$656$		0			0
$657$	−3.24155e6	+	5.61452e6i	−0.292981	+	0.507457i
$658$		0			0
$659$	2.30484e6	−	3.99210e6i	0.206741	−	0.358086i	−0.743945	−	0.668241i	$-0.767048\pi$
$659$	2.30484e6	−	3.99210e6i	0.206741	−	0.358086i	0.950686	+	0.310155i	$0.100381\pi$
$660$		0			0
$661$	−4.82633e6	−	8.35945e6i	−0.429648	−	0.744173i	0.567193	−	0.823585i	$-0.308029\pi$
$661$	−4.82633e6	−	8.35945e6i	−0.429648	−	0.744173i	−0.996842	+	0.0794117i	$0.974696\pi$
$662$		0			0
$663$	−241489.	−	1.77105e6i	−0.0213360	−	0.156476i
$664$		0			0
$665$	286709.	+	496594.i	0.0251412	+	0.0435459i
$666$		0			0
$667$	2.07403e6	−	3.59232e6i	0.180509	−	0.312651i
$668$		0			0
$669$	−3.98412e6	+	6.90070e6i	−0.344165	+	0.596112i
$670$		0			0
$671$	−3.33448e7			−2.85905
$672$		0			0
$673$	−4.53822e6	−	7.86042e6i	−0.386231	−	0.668972i	0.605708	−	0.795687i	$-0.292890\pi$
$673$	−4.53822e6	−	7.86042e6i	−0.386231	−	0.668972i	−0.991939	+	0.126715i	$0.959557\pi$
$674$		0			0
$675$	−2.14235e6			−0.180980
$676$		0			0
$677$	−4.71013e6			−0.394967			−0.197484	−	0.980306i	$-0.563277\pi$
$677$	−4.71013e6			−0.394967			−0.197484	+	0.980306i	$0.563277\pi$
$678$		0			0
$679$	1.67110e6	+	2.89443e6i	0.139100	+	0.240929i
$680$		0			0
$681$	−6.22883e6			−0.514681
$682$		0			0
$683$	−3.65350e6	+	6.32805e6i	−0.299680	+	0.519060i	−0.976063	−	0.217490i	$-0.930213\pi$
$683$	−3.65350e6	+	6.32805e6i	−0.299680	+	0.519060i	0.676383	+	0.736550i	$0.263546\pi$
$684$		0			0
$685$	1.50315e6	−	2.60353e6i	0.122398	−	0.212000i
$686$		0			0
$687$	6.75493e6	+	1.16999e7i	0.546046	+	0.945779i
$688$		0			0
$689$	774115.	+	5.67726e6i	0.0621237	+	0.455607i
$690$		0			0
$691$	−1.05478e7	−	1.82693e7i	−0.840361	−	1.45555i	−0.889589	−	0.456761i	$-0.849009\pi$
$691$	−1.05478e7	−	1.82693e7i	−0.840361	−	1.45555i	0.0492280	−	0.998788i	$-0.484324\pi$
$692$		0			0
$693$	−1.09408e6	+	1.89501e6i	−0.0865402	+	0.149892i
$694$		0			0
$695$	93325.3	−	161644.i	0.00732887	−	0.0126940i
$696$		0			0
$697$	−1.99555e6			−0.155589
$698$		0			0
$699$	−2.80110e6	−	4.85165e6i	−0.216838	−	0.375575i
$700$		0			0
$701$	9.13818e6			0.702368			0.351184	−	0.936306i	$-0.385779\pi$
$701$	9.13818e6			0.702368			0.351184	+	0.936306i	$0.385779\pi$
$702$		0			0
$703$	−4.00676e6			−0.305777
$704$		0			0
$705$	227923.	+	394775.i	0.0172709	+	0.0299141i
$706$		0			0
$707$	4.35599e6			0.327747
$708$		0			0
$709$	−2.21617e6	+	3.83852e6i	−0.165572	+	0.286780i	−0.936858	−	0.349709i	$-0.886280\pi$
$709$	−2.21617e6	+	3.83852e6i	−0.165572	+	0.286780i	0.771286	+	0.636489i	$0.219614\pi$
$710$		0			0
$711$	−2.44593e6	+	4.23647e6i	−0.181455	+	0.314290i
$712$		0			0
$713$	1.88869e6	+	3.27131e6i	0.139135	+	0.240989i
$714$		0			0
$715$	3.94441e6	−	3.05570e6i	0.288547	−	0.223535i
$716$		0			0
$717$	1.12277e6	+	1.94469e6i	0.0815629	+	0.141271i
$718$		0			0
$719$	7.35001e6	−	1.27306e7i	0.530232	−	0.918388i	−0.469146	−	0.883121i	$-0.655438\pi$
$719$	7.35001e6	−	1.27306e7i	0.530232	−	0.918388i	0.999378	−	0.0352678i	$-0.0112284\pi$
$720$		0			0
$721$	−2.73446e6	+	4.73622e6i	−0.195899	+	0.339308i
$722$		0			0
$723$	9.30225e6			0.661824
$724$		0			0
$725$	−9.70097e6	−	1.68026e7i	−0.685441	−	1.18722i
$726$		0			0
$727$	2.31830e7			1.62680			0.813400	−	0.581705i	$-0.197614\pi$
$727$	2.31830e7			1.62680			0.813400	+	0.581705i	$0.197614\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	3.13169e6	+	5.42425e6i	0.216763	+	0.375445i
$732$		0			0
$733$	−1.19022e7			−0.818215			−0.409108	−	0.912486i	$-0.634160\pi$
$733$	−1.19022e7			−0.818215			−0.409108	+	0.912486i	$0.634160\pi$
$734$		0			0
$735$	907676.	−	1.57214e6i	0.0619744	−	0.107343i
$736$		0			0
$737$	−7.46110e6	+	1.29230e7i	−0.505981	+	0.876385i
$738$		0			0
$739$	−5.58988e6	−	9.68195e6i	−0.376523	−	0.652157i	0.614031	−	0.789282i	$-0.289547\pi$
$739$	−5.58988e6	−	9.68195e6i	−0.376523	−	0.652157i	−0.990554	+	0.137125i	$0.956214\pi$
$740$		0			0
$741$	−691434.	−	5.07089e6i	−0.0462599	−	0.339264i
$742$		0			0
$743$	−3.16516e6	−	5.48222e6i	−0.210341	−	0.364321i	0.741480	−	0.670975i	$-0.234124\pi$
$743$	−3.16516e6	−	5.48222e6i	−0.210341	−	0.364321i	−0.951821	+	0.306653i	$0.900791\pi$
$744$		0			0
$745$	−2.77993e6	+	4.81498e6i	−0.183503	+	0.317837i
$746$		0			0
$747$	−4.81325e6	+	8.33680e6i	−0.315600	+	0.546636i
$748$		0			0
$749$	1.49883e6			0.0976221
$750$		0			0
$751$	−1.27302e6	−	2.20493e6i	−0.0823635	−	0.142658i	0.821901	−	0.569630i	$-0.192914\pi$
$751$	−1.27302e6	−	2.20493e6i	−0.0823635	−	0.142658i	−0.904265	+	0.426972i	$0.859580\pi$
$752$		0			0
$753$	1.59926e7			1.02785
$754$		0			0
$755$	3.46364e6			0.221139
$756$		0			0
$757$	−1.31648e6	−	2.28021e6i	−0.0834977	−	0.144622i	0.821252	−	0.570565i	$-0.193276\pi$
$757$	−1.31648e6	−	2.28021e6i	−0.0834977	−	0.144622i	−0.904750	+	0.425943i	$0.859942\pi$
$758$		0			0
$759$	−3.39282e6			−0.213775
$760$		0			0
$761$	−4.70923e6	+	8.15663e6i	−0.294774	+	0.510563i	−0.974932	−	0.222502i	$-0.928578\pi$
$761$	−4.70923e6	+	8.15663e6i	−0.294774	+	0.510563i	0.680159	+	0.733065i	$0.261911\pi$
$762$		0			0
$763$	3.28266e6	−	5.68574e6i	0.204134	−	0.353570i
$764$		0			0
$765$	−180149.	−	312028.i	−0.0111296	−	0.0192770i
$766$		0			0
$767$	−4.69216e6	−	1.91822e6i	−0.287995	−	0.117736i
$768$		0			0
$769$	581498.	+	1.00718e6i	0.0354595	+	0.0614176i	0.883211	−	0.468977i	$-0.155377\pi$
$769$	581498.	+	1.00718e6i	0.0354595	+	0.0614176i	−0.847751	+	0.530394i	$0.822044\pi$
$770$		0			0
$771$	−6.72664e6	+	1.16509e7i	−0.407532	+	0.705867i
$772$		0			0
$773$	4.67559e6	−	8.09835e6i	0.281441	−	0.487470i	−0.690299	−	0.723524i	$-0.742521\pi$
$773$	4.67559e6	−	8.09835e6i	0.281441	−	0.487470i	0.971740	+	0.236054i	$0.0758543\pi$
$774$		0			0
$775$	1.76682e7			1.05667
$776$		0			0
$777$	−869887.	−	1.50669e6i	−0.0516904	−	0.0895304i
$778$		0			0
$779$	−5.71367e6			−0.337343
$780$		0			0
$781$	−2.50287e7			−1.46829
$782$		0			0
$783$	2.40647e6	+	4.16812e6i	0.140273	+	0.242961i
$784$		0			0
$785$	−5.16416e6			−0.299106
$786$		0			0
$787$	−3.38009e6	+	5.85449e6i	−0.194532	+	0.336940i	−0.946747	−	0.321978i	$-0.895652\pi$
$787$	−3.38009e6	+	5.85449e6i	−0.194532	+	0.336940i	0.752215	+	0.658918i	$0.228986\pi$
$788$		0			0
$789$	−9.24189e6	+	1.60074e7i	−0.528528	+	0.915438i
$790$		0			0
$791$	1.62534e6	+	2.81516e6i	0.0923638	+	0.159979i
$792$		0			0
$793$	3.13452e7	+	1.28144e7i	1.77006	+	0.723625i
$794$		0			0
$795$	577485.	+	1.00023e6i	0.0324058	+	0.0561285i
$796$		0			0
$797$	−1.67981e6	+	2.90951e6i	−0.0936728	+	0.162246i	−0.909054	−	0.416679i	$-0.863194\pi$
$797$	−1.67981e6	+	2.90951e6i	−0.0936728	+	0.162246i	0.815381	+	0.578925i	$0.196527\pi$
$798$		0			0
$799$	604824.	−	1.04759e6i	0.0335168	−	0.0580527i
$800$		0			0
$801$	−5.03857e6			−0.277476
$802$		0			0
$803$	2.40117e7	+	4.15895e7i	1.31412	+	2.27612i
$804$		0			0
$805$	−386055.			−0.0209971
$806$		0			0
$807$	−115839.			−0.00626138
$808$		0			0
$809$	492148.	+	852425.i	0.0264377	+	0.0457915i	0.878942	−	0.476929i	$-0.158250\pi$
$809$	492148.	+	852425.i	0.0264377	+	0.0457915i	−0.852504	+	0.522721i	$0.824917\pi$
$810$		0			0
$811$	2.22480e7			1.18779			0.593893	−	0.804544i	$-0.297590\pi$
$811$	2.22480e7			1.18779			0.593893	+	0.804544i	$0.297590\pi$
$812$		0			0
$813$	227742.	−	394461.i	0.0120842	−	0.0209304i
$814$		0			0
$815$	−1.40027e6	+	2.42534e6i	−0.0738444	+	0.127902i
$816$		0			0
$817$	8.96668e6	+	1.55308e7i	0.469977	+	0.814024i
$818$		0			0
$819$	1.75673e6	−	1.36092e6i	0.0915154	−	0.0708961i
$820$		0			0
$821$	2.48133e6	+	4.29780e6i	0.128478	+	0.222530i	0.923087	−	0.384591i	$-0.125658\pi$
$821$	2.48133e6	+	4.29780e6i	0.128478	+	0.222530i	−0.794609	+	0.607121i	$0.792324\pi$
$822$		0			0
$823$	1.90316e7	−	3.29637e7i	0.979435	−	1.69643i	0.314987	−	0.949096i	$-0.398000\pi$
$823$	1.90316e7	−	3.29637e7i	0.979435	−	1.69643i	0.664448	−	0.747335i	$-0.268667\pi$
$824$		0			0
$825$	−7.93471e6	+	1.37433e7i	−0.405879	+	0.703002i
$826$		0			0
$827$	−2.49081e7			−1.26642			−0.633209	−	0.773980i	$-0.718263\pi$
$827$	−2.49081e7			−1.26642			−0.633209	+	0.773980i	$0.718263\pi$
$828$		0			0
$829$	−1.26064e7	−	2.18349e7i	−0.637095	−	1.10348i	−0.986067	−	0.166348i	$-0.946803\pi$
$829$	−1.26064e7	−	2.18349e7i	−0.637095	−	1.10348i	0.348972	−	0.937133i	$-0.386531\pi$
$830$		0			0
$831$	1.41742e7			0.712024
$832$		0			0
$833$	−4.81727e6			−0.240541
$834$		0			0
$835$	−1.90379e6	−	3.29746e6i	−0.0944937	−	0.163668i
$836$		0			0
$837$	−4.38285e6			−0.216243
$838$		0			0
$839$	−9.20499e6	+	1.59435e7i	−0.451459	+	0.781950i	−0.998477	−	0.0551710i	$-0.982430\pi$
$839$	−9.20499e6	+	1.59435e7i	−0.451459	+	0.781950i	0.547018	+	0.837121i	$0.315763\pi$
$840$		0			0
$841$	−1.15383e7	+	1.99849e7i	−0.562538	+	0.974344i
$842$		0			0
$843$	−8.03323e6	−	1.39140e7i	−0.389333	−	0.674344i
$844$		0			0
$845$	−4.88217e6	+	1.35663e6i	−0.235218	+	0.0653610i
$846$		0			0
$847$	4.47887e6	+	7.75763e6i	0.214516	+	0.371553i
$848$		0			0
$849$	1.27194e6	−	2.20306e6i	0.0605615	−	0.104896i
$850$		0			0
$851$	1.34878e6	−	2.33616e6i	0.0638437	−	0.110581i
$852$		0			0
$853$	1.07679e7			0.506708			0.253354	−	0.967374i	$-0.418466\pi$
$853$	1.07679e7			0.506708			0.253354	+	0.967374i	$0.418466\pi$
$854$		0			0
$855$	−515805.	−	893400.i	−0.0241307	−	0.0417956i
$856$		0			0
$857$	2.40885e7			1.12036			0.560179	−	0.828371i	$-0.310732\pi$
$857$	2.40885e7			1.12036			0.560179	+	0.828371i	$0.310732\pi$
$858$		0			0
$859$	−3.88080e6			−0.179448			−0.0897239	−	0.995967i	$-0.528598\pi$
$859$	−3.88080e6			−0.179448			−0.0897239	+	0.995967i	$0.528598\pi$
$860$		0			0
$861$	−1.24046e6	−	2.14855e6i	−0.0570265	−	0.0987728i
$862$		0			0
$863$	2.17440e7			0.993831			0.496916	−	0.867799i	$-0.334466\pi$
$863$	2.17440e7			0.993831			0.496916	+	0.867799i	$0.334466\pi$
$864$		0			0
$865$	2.51534e6	−	4.35670e6i	0.114303	−	0.197978i
$866$		0			0
$867$	5.91131e6	−	1.02387e7i	0.267077	−	0.462590i
$868$		0			0
$869$	1.81182e7	+	3.13816e7i	0.813889	+	1.40970i
$870$		0			0
$871$	1.19800e7	−	9.28077e6i	0.535070	−	0.414513i
$872$		0			0
$873$	−3.00641e6	−	5.20725e6i	−0.133509	−	0.231245i
$874$		0			0
$875$	−1.86294e6	+	3.22671e6i	−0.0822581	+	0.142475i
$876$		0			0
$877$	−9.62832e6	+	1.66767e7i	−0.422719	+	0.732170i	−0.996204	−	0.0870454i	$-0.972257\pi$
$877$	−9.62832e6	+	1.66767e7i	−0.422719	+	0.732170i	0.573486	+	0.819216i	$0.305591\pi$
$878$		0			0
$879$	−1.09788e7			−0.479275
$880$		0			0
$881$	1.73478e7	+	3.00472e7i	0.753015	+	1.30426i	0.946355	+	0.323128i	$0.104734\pi$
$881$	1.73478e7	+	3.00472e7i	0.753015	+	1.30426i	−0.193340	+	0.981132i	$0.561932\pi$
$882$		0			0
$883$	2.20047e7			0.949758			0.474879	−	0.880051i	$-0.342492\pi$
$883$	2.20047e7			0.949758			0.474879	+	0.880051i	$0.342492\pi$
$884$		0			0
$885$	−1.02179e6			−0.0438536
$886$		0			0
$887$	1.04067e7	+	1.80250e7i	0.444125	+	0.769246i	0.997991	−	0.0633594i	$-0.0201814\pi$
$887$	1.04067e7	+	1.80250e7i	0.444125	+	0.769246i	−0.553866	+	0.832606i	$0.686848\pi$
$888$		0			0
$889$	1.15483e7			0.490075
$890$		0			0
$891$	1.96832e6	−	3.40923e6i	0.0830618	−	0.143867i
$892$		0			0
$893$	1.73174e6	−	2.99946e6i	0.0726697	−	0.125868i
$894$		0			0
$895$	−4.08034e6	−	7.06736e6i	−0.170270	−	0.294917i
$896$		0			0
$897$	3.18936e6	+	1.30385e6i	0.132350	+	0.0541063i
$898$		0			0
$899$	−1.98464e7	−	3.43749e7i	−0.818996	−	1.41854i
$900$		0			0
$901$	1.53243e6	−	2.65425e6i	0.0628882	−	0.108926i
$902$		0			0
$903$	−3.89342e6	+	6.74360e6i	−0.158896	+	0.275215i
$904$		0			0
$905$	−2.10975e6			−0.0856269
$906$		0			0
$907$	2.55114e6	+	4.41870e6i	0.102971	+	0.178351i	0.912908	−	0.408166i	$-0.133832\pi$
$907$	2.55114e6	+	4.41870e6i	0.102971	+	0.178351i	−0.809936	+	0.586518i	$0.800498\pi$
$908$		0			0
$909$	−7.83667e6			−0.314573
$910$		0			0
$911$	3.41074e7			1.36161			0.680805	−	0.732465i	$-0.261630\pi$
$911$	3.41074e7			1.36161			0.680805	+	0.732465i	$0.261630\pi$
$912$		0			0
$913$	3.56541e7	+	6.17547e7i	1.41557	+	2.45185i
$914$		0			0
$915$	6.82594e6			0.269532
$916$		0			0
$917$	−111415.	+	192977.i	−0.00437544	+	0.00757849i
$918$		0			0
$919$	−3.79211e6	+	6.56812e6i	−0.148113	+	0.256538i	−0.930530	−	0.366216i	$-0.880653\pi$
$919$	−3.79211e6	+	6.56812e6i	−0.148113	+	0.256538i	0.782417	+	0.622754i	$0.213986\pi$
$920$		0			0
$921$	2.25870e6	+	3.91219e6i	0.0877426	+	0.151975i
$922$		0			0
$923$	2.35278e7	+	9.61848e6i	0.909028	+	0.371623i
$924$		0			0
$925$	−6.30874e6	−	1.09271e7i	−0.242431	−	0.419903i
$926$		0			0
$927$	4.91944e6	−	8.52073e6i	0.188025	−	0.325670i
$928$		0			0
$929$	−1.97048e7	+	3.41297e7i	−0.749088	+	1.29746i	0.199172	+	0.979965i	$0.436175\pi$
$929$	−1.97048e7	+	3.41297e7i	−0.749088	+	1.29746i	−0.948260	+	0.317494i	$0.897159\pi$
$930$		0			0
$931$	−1.37928e7			−0.521531
$932$		0			0
$933$	1.02470e6	+	1.77484e6i	0.0385384	+	0.0667505i
$934$		0			0
$935$	−2.66891e6			−0.0998400
$936$		0			0
$937$	1.25596e7			0.467334			0.233667	−	0.972317i	$-0.424927\pi$
$937$	1.25596e7			0.467334			0.233667	+	0.972317i	$0.424927\pi$
$938$		0			0
$939$	5.26349e6	+	9.11664e6i	0.194809	+	0.337420i
$940$		0			0
$941$	−7.71119e6			−0.283888			−0.141944	−	0.989875i	$-0.545335\pi$
$941$	−7.71119e6			−0.283888			−0.141944	+	0.989875i	$0.545335\pi$
$942$		0			0
$943$	1.92338e6	−	3.33138e6i	0.0704344	−	0.121996i
$944$		0			0
$945$	223968.	−	387923.i	0.00815841	−	0.0141308i
$946$		0			0
$947$	−2.33024e7	−	4.03609e7i	−0.844355	−	1.46247i	−0.886180	−	0.463340i	$-0.846651\pi$
$947$	−2.33024e7	−	4.03609e7i	−0.844355	−	1.46247i	0.0418255	−	0.999125i	$-0.486683\pi$
$948$		0			0
$949$	−6.58904e6	−	4.83232e7i	−0.237496	−	1.74177i
$950$		0			0
$951$	2.79816e6	+	4.84655e6i	0.100328	+	0.173773i
$952$		0			0
$953$	2.59113e7	−	4.48797e7i	0.924182	−	1.60073i	0.131309	−	0.991341i	$-0.458082\pi$
$953$	2.59113e7	−	4.48797e7i	0.924182	−	1.60073i	0.792872	−	0.609388i	$-0.208585\pi$
$954$		0			0
$955$	2.85390e6	−	4.94310e6i	0.101258	−	0.175384i
$956$		0			0
$957$	3.56517e7			1.25835
$958$		0			0
$959$	−4.95901e6	−	8.58926e6i	−0.174120	−	0.301585i
$960$		0			0
$961$	7.51664e6			0.262552
$962$		0			0
$963$	−2.69648e6			−0.0936983
$964$		0			0
$965$	5.53069e6	+	9.57943e6i	0.191188	+	0.331147i
$966$		0			0
$967$	−5.61771e7			−1.93194			−0.965968	−	0.258660i	$-0.916719\pi$
$967$	−5.61771e7			−1.93194			−0.965968	+	0.258660i	$0.916719\pi$
$968$		0			0
$969$	−1.36876e6	+	2.37075e6i	−0.0468292	+	0.0811105i
$970$		0			0
$971$	−7.55563e6	+	1.30867e7i	−0.257172	+	0.445434i	−0.965483	−	0.260466i	$-0.916124\pi$
$971$	−7.55563e6	+	1.30867e7i	−0.257172	+	0.445434i	0.708312	+	0.705900i	$0.249457\pi$
$972$		0			0
$973$	−307887.	−	533277.i	−0.0104258	−	0.0180580i
$974$		0			0
$975$	1.27404e7	−	9.86989e6i	0.429213	−	0.332507i
$976$		0			0
$977$	1.11361e7	+	1.92884e7i	0.373249	+	0.646485i	0.990063	−	0.140623i	$-0.0449105\pi$
$977$	1.11361e7	+	1.92884e7i	0.373249	+	0.646485i	−0.616815	+	0.787108i	$0.711577\pi$
$978$		0			0
$979$	−1.86616e7	+	3.23228e7i	−0.622287	+	1.07783i
$980$		0			0
$981$	−5.90570e6	+	1.02290e7i	−0.195929	+	0.339359i
$982$		0			0
$983$	4.00313e6			0.132134			0.0660671	−	0.997815i	$-0.478955\pi$
$983$	4.00313e6			0.132134			0.0660671	+	0.997815i	$0.478955\pi$
$984$		0			0
$985$	817999.	+	1.41682e6i	0.0268635	+	0.0465289i
$986$		0			0
$987$	1.50387e6			0.0491381
$988$		0			0
$989$	−1.20737e7			−0.392509
$990$		0			0
$991$	1.52042e7	+	2.63344e7i	0.491789	+	0.851803i	0.999955	−	0.00945598i	$-0.00300998\pi$
$991$	1.52042e7	+	2.63344e7i	0.491789	+	0.851803i	−0.508167	+	0.861259i	$0.669677\pi$
$992$		0			0
$993$	8.91183e6			0.286810
$994$		0			0
$995$	2.65409e6	−	4.59701e6i	0.0849880	−	0.147204i
$996$		0			0
$997$	−2.87718e7	+	4.98343e7i	−0.916705	+	1.58778i	−0.112318	+	0.993672i	$0.535828\pi$
$997$	−2.87718e7	+	4.98343e7i	−0.916705	+	1.58778i	−0.804386	+	0.594107i	$0.797506\pi$
$998$		0			0
$999$	1.56498e6	+	2.71062e6i	0.0496128	+	0.0859319i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.6.q.b.289.5	yes	18
13.9	even	3	inner	312.6.q.b.217.5	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.6.q.b.217.5	✓	18	13.9	even	3	inner
312.6.q.b.289.5	yes	18	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	312.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 - 7.79423i) q^{3} +13.6473 q^{5} +(22.5118 - 38.9916i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(300.003 + 519.621i) q^{11} +(-82.3237 - 603.751i) q^{13} +(-61.4129 - 106.370i) q^{15} +(-162.967 + 282.268i) q^{17} +(-466.609 + 808.191i) q^{19} -405.212 q^{21} +(-314.146 - 544.118i) q^{23} -2938.75 q^{25} +729.000 q^{27} +(3301.05 + 5717.59i) q^{29} -6012.14 q^{31} +(2700.03 - 4676.59i) q^{33} +(307.226 - 532.130i) q^{35} +(2146.74 + 3718.27i) q^{37} +(-4335.32 + 3358.53i) q^{39} +(3061.27 + 5302.28i) q^{41} +(9608.35 - 16642.1i) q^{43} +(-552.716 + 957.333i) q^{45} -3711.32 q^{47} +(7389.94 + 12799.7i) q^{49} +2933.41 q^{51} -9403.31 q^{53} +(4094.24 + 7091.43i) q^{55} +8398.96 q^{57} +(4159.52 - 7204.51i) q^{59} +(-27787.1 + 48128.6i) q^{61} +(1823.46 + 3158.32i) q^{63} +(-1123.50 - 8239.59i) q^{65} +(12435.0 + 21538.1i) q^{67} +(-2827.32 + 4897.06i) q^{69} +(-20857.0 + 36125.4i) q^{71} +80038.2 q^{73} +(13224.4 + 22905.3i) q^{75} +27014.4 q^{77} +60393.3 q^{79} +(-3280.50 - 5681.99i) q^{81} +118846. q^{83} +(-2224.07 + 3852.20i) q^{85} +(29709.5 - 51458.3i) q^{87} +(31102.3 + 53870.7i) q^{89} +(-25394.5 - 10381.6i) q^{91} +(27054.6 + 46860.0i) q^{93} +(-6367.96 + 11029.6i) q^{95} +(-37116.1 + 64287.0i) q^{97} -48600.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 81 q^{3} + 90 q^{5} - 48 q^{7} - 729 q^{9} - 602 q^{11} + 429 q^{13} - 405 q^{15} + 877 q^{17} - 594 q^{19} + 864 q^{21} - 3346 q^{23} + 7412 q^{25} + 13122 q^{27} + 6955 q^{29} - 2804 q^{31} - 5418 q^{33}+ \cdots + 97524 q^{99}+O(q^{100}) \) 18 * q - 81 * q^3 + 90 * q^5 - 48 * q^7 - 729 * q^9 - 602 * q^11 + 429 * q^13 - 405 * q^15 + 877 * q^17 - 594 * q^19 + 864 * q^21 - 3346 * q^23 + 7412 * q^25 + 13122 * q^27 + 6955 * q^29 - 2804 * q^31 - 5418 * q^33 - 4790 * q^35 - 8431 * q^37 + 1755 * q^39 - 7183 * q^41 + 1228 * q^43 - 3645 * q^45 + 8276 * q^47 - 28271 * q^49 - 15786 * q^51 + 33402 * q^53 + 19638 * q^55 + 10692 * q^57 - 16984 * q^59 + 20631 * q^61 - 3888 * q^63 + 15697 * q^65 - 77828 * q^67 - 30114 * q^69 - 34854 * q^71 - 18162 * q^73 - 33354 * q^75 + 28576 * q^77 + 112116 * q^79 - 59049 * q^81 + 72492 * q^83 + 27659 * q^85 + 62595 * q^87 - 4994 * q^89 - 114838 * q^91 + 12618 * q^93 + 14002 * q^95 - 31836 * q^97 + 97524 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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