LMFDB - Embedded newform 3150.3.c.f.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(449,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("3150.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.c (of order $2$, degree $1$, not 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:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.9671731157401600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 $ x^16 + 7x^12 + 753x^8 + 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$-0.941471 + 2.08559i$ of defining polynomial
Character	$\chi$	$=$	3150.449
Dual form			3150.3.c.f.449.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} -2.82843 q^{8} +6.61683i q^{11} -6.21640i q^{13} +3.74166i q^{14} +4.00000 q^{16} -6.35806 q^{17} +23.5709 q^{19} -9.35761i q^{22} -12.8687 q^{23} +8.79132i q^{26} -5.29150i q^{28} +21.7071i q^{29} -39.5362 q^{31} -5.65685 q^{32} +8.99165 q^{34} -56.9502i q^{37} -33.3343 q^{38} -44.4993i q^{41} +60.1402i q^{43} +13.2337i q^{44} +18.1991 q^{46} +20.6191 q^{47} -7.00000 q^{49} -12.4328i q^{52} +40.0788 q^{53} +7.48331i q^{56} -30.6985i q^{58} +25.4866i q^{59} -0.743719 q^{61} +55.9127 q^{62} +8.00000 q^{64} -68.8315i q^{67} -12.7161 q^{68} +56.6306i q^{71} -13.1000i q^{73} +80.5398i q^{74} +47.1418 q^{76} +17.5065 q^{77} +22.2403 q^{79} +62.9315i q^{82} -151.071 q^{83} -85.0511i q^{86} -18.7152i q^{88} -6.08695i q^{89} -16.4471 q^{91} -25.7374 q^{92} -29.1598 q^{94} -135.146i q^{97} +9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) $ 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	−2.82843	−0.353553
$9$	0	0
$10$	0	0
$11$	6.61683i	0.601530i	0.953698	+	0.300765i	$0.0972420\pi$
$11$	6.61683i	0.601530i	−0.953698	+	0.300765i	$0.902758\pi$
$12$	0	0
$13$	− 6.21640i	− 0.478185i	−0.970997	−	0.239092i	$-0.923150\pi$
$13$	− 6.21640i	− 0.478185i	0.970997	−	0.239092i	$-0.0768499\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	−6.35806	−0.374003	−0.187002	−	0.982360i	$-0.559877\pi$
$17$	−6.35806	−0.374003	−0.187002	+	0.982360i	$0.559877\pi$
$18$	0	0
$19$	23.5709	1.24057	0.620287	−	0.784375i	$-0.287016\pi$
$19$	23.5709	1.24057	0.620287	+	0.784375i	$0.287016\pi$
$20$	0	0
$21$	0	0
$22$	− 9.35761i	− 0.425346i
$23$	−12.8687	−0.559508	−0.279754	−	0.960072i	$-0.590253\pi$
$23$	−12.8687	−0.559508	−0.279754	+	0.960072i	$0.590253\pi$
$24$	0	0
$25$	0	0
$26$	8.79132i	0.338128i
$27$	0	0
$28$	− 5.29150i	− 0.188982i
$29$	21.7071i	0.748522i	0.927323	+	0.374261i	$0.122104\pi$
$29$	21.7071i	0.748522i	−0.927323	+	0.374261i	$0.877896\pi$
$30$	0	0
$31$	−39.5362	−1.27536	−0.637681	−	0.770300i	$-0.720106\pi$
$31$	−39.5362	−1.27536	−0.637681	+	0.770300i	$0.720106\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	8.99165	0.264460
$35$	0	0
$36$	0	0
$37$	− 56.9502i	− 1.53920i	−0.638529	−	0.769598i	$-0.720457\pi$
$37$	− 56.9502i	− 1.53920i	0.638529	−	0.769598i	$-0.279543\pi$
$38$	−33.3343	−0.877219
$39$	0	0
$40$	0	0
$41$	− 44.4993i	− 1.08535i	−0.839943	−	0.542674i	$-0.817412\pi$
$41$	− 44.4993i	− 1.08535i	0.839943	−	0.542674i	$-0.182588\pi$
$42$	0	0
$43$	60.1402i	1.39861i	0.714824	+	0.699305i	$0.246507\pi$
$43$	60.1402i	1.39861i	−0.714824	+	0.699305i	$0.753493\pi$
$44$	13.2337i	0.300765i
$45$	0	0
$46$	18.1991	0.395632
$47$	20.6191	0.438705	0.219352	−	0.975646i	$-0.429606\pi$
$47$	20.6191	0.438705	0.219352	+	0.975646i	$0.429606\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 12.4328i	− 0.239092i
$53$	40.0788	0.756204	0.378102	−	0.925764i	$-0.376577\pi$
$53$	40.0788	0.756204	0.378102	+	0.925764i	$0.376577\pi$
$54$	0	0
$55$	0	0
$56$	7.48331i	0.133631i
$57$	0	0
$58$	− 30.6985i	− 0.529285i
$59$	25.4866i	0.431976i	0.976396	+	0.215988i	$0.0692971\pi$
$59$	25.4866i	0.431976i	−0.976396	+	0.215988i	$0.930703\pi$
$60$	0	0
$61$	−0.743719	−0.0121921	−0.00609606	−	0.999981i	$-0.501940\pi$
$61$	−0.743719	−0.0121921	−0.00609606	+	0.999981i	$0.501940\pi$
$62$	55.9127	0.901818
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	− 68.8315i	− 1.02734i	−0.857989	−	0.513668i	$-0.828286\pi$
$67$	− 68.8315i	− 1.02734i	0.857989	−	0.513668i	$-0.171714\pi$
$68$	−12.7161	−0.187002
$69$	0	0
$70$	0	0
$71$	56.6306i	0.797614i	0.917035	+	0.398807i	$0.130576\pi$
$71$	56.6306i	0.797614i	−0.917035	+	0.398807i	$0.869424\pi$
$72$	0	0
$73$	− 13.1000i	− 0.179452i	−0.995966	−	0.0897261i	$-0.971401\pi$
$73$	− 13.1000i	− 0.179452i	0.995966	−	0.0897261i	$-0.0285992\pi$
$74$	80.5398i	1.08838i
$75$	0	0
$76$	47.1418	0.620287
$77$	17.5065	0.227357
$78$	0	0
$79$	22.2403	0.281522	0.140761	−	0.990044i	$-0.455045\pi$
$79$	22.2403	0.281522	0.140761	+	0.990044i	$0.455045\pi$
$80$	0	0
$81$	0	0
$82$	62.9315i	0.767458i
$83$	−151.071	−1.82013	−0.910066	−	0.414463i	$-0.863969\pi$
$83$	−151.071	−1.82013	−0.910066	+	0.414463i	$0.863969\pi$
$84$	0	0
$85$	0	0
$86$	− 85.0511i	− 0.988966i
$87$	0	0
$88$	− 18.7152i	− 0.212673i
$89$	− 6.08695i	− 0.0683927i	−0.999415	−	0.0341963i	$-0.989113\pi$
$89$	− 6.08695i	− 0.0683927i	0.999415	−	0.0341963i	$-0.0108872\pi$
$90$	0	0
$91$	−16.4471	−0.180737
$92$	−25.7374	−0.279754
$93$	0	0
$94$	−29.1598	−0.310211
$95$	0	0
$96$	0	0
$97$	− 135.146i	− 1.39326i	−0.717431	−	0.696630i	$-0.754682\pi$
$97$	− 135.146i	− 1.39326i	0.717431	−	0.696630i	$-0.245318\pi$
$98$	9.89949	0.101015
$99$	0	0
$100$	0	0
$101$	99.7356i	0.987481i	0.869609	+	0.493741i	$0.164371\pi$
$101$	99.7356i	0.987481i	−0.869609	+	0.493741i	$0.835629\pi$
$102$	0	0
$103$	− 92.9283i	− 0.902216i	−0.892469	−	0.451108i	$-0.851029\pi$
$103$	− 92.9283i	− 0.902216i	0.892469	−	0.451108i	$-0.148971\pi$
$104$	17.5826i	0.169064i
$105$	0	0
$106$	−56.6800	−0.534717
$107$	41.9615	0.392164	0.196082	−	0.980588i	$-0.437178\pi$
$107$	41.9615	0.392164	0.196082	+	0.980588i	$0.437178\pi$
$108$	0	0
$109$	−145.762	−1.33727	−0.668635	−	0.743591i	$-0.733121\pi$
$109$	−145.762	−1.33727	−0.668635	+	0.743591i	$0.733121\pi$
$110$	0	0
$111$	0	0
$112$	− 10.5830i	− 0.0944911i
$113$	−29.6161	−0.262089	−0.131045	−	0.991376i	$-0.541833\pi$
$113$	−29.6161	−0.262089	−0.131045	+	0.991376i	$0.541833\pi$
$114$	0	0
$115$	0	0
$116$	43.4143i	0.374261i
$117$	0	0
$118$	− 36.0434i	− 0.305453i
$119$	16.8218i	0.141360i
$120$	0	0
$121$	77.2176	0.638162
$122$	1.05178	0.00862113
$123$	0	0
$124$	−79.0725	−0.637681
$125$	0	0
$126$	0	0
$127$	− 213.731i	− 1.68292i	−0.540319	−	0.841460i	$-0.681696\pi$
$127$	− 213.731i	− 1.68292i	0.540319	−	0.841460i	$-0.318304\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	0	0
$131$	136.249i	1.04007i	0.854146	+	0.520033i	$0.174080\pi$
$131$	136.249i	1.04007i	−0.854146	+	0.520033i	$0.825920\pi$
$132$	0	0
$133$	− 62.3628i	− 0.468893i
$134$	97.3424i	0.726436i
$135$	0	0
$136$	17.9833	0.132230
$137$	−5.36713	−0.0391761	−0.0195881	−	0.999808i	$-0.506235\pi$
$137$	−5.36713	−0.0391761	−0.0195881	+	0.999808i	$0.506235\pi$
$138$	0	0
$139$	−135.187	−0.972571	−0.486286	−	0.873800i	$-0.661649\pi$
$139$	−135.187	−0.972571	−0.486286	+	0.873800i	$0.661649\pi$
$140$	0	0
$141$	0	0
$142$	− 80.0878i	− 0.563998i
$143$	41.1329	0.287643
$144$	0	0
$145$	0	0
$146$	18.5262i	0.126892i
$147$	0	0
$148$	− 113.900i	− 0.769598i
$149$	102.343i	0.686864i	0.939178	+	0.343432i	$0.111589\pi$
$149$	102.343i	0.686864i	−0.939178	+	0.343432i	$0.888411\pi$
$150$	0	0
$151$	38.7465	0.256599	0.128300	−	0.991735i	$-0.459048\pi$
$151$	38.7465	0.256599	0.128300	+	0.991735i	$0.459048\pi$
$152$	−66.6686	−0.438609
$153$	0	0
$154$	−24.7579	−0.160766
$155$	0	0
$156$	0	0
$157$	− 2.44059i	− 0.0155451i	−0.999970	−	0.00777257i	$-0.997526\pi$
$157$	− 2.44059i	− 0.0155451i	0.999970	−	0.00777257i	$-0.00247411\pi$
$158$	−31.4525	−0.199066
$159$	0	0
$160$	0	0
$161$	34.0473i	0.211474i
$162$	0	0
$163$	134.270i	0.823742i	0.911242	+	0.411871i	$0.135125\pi$
$163$	134.270i	0.823742i	−0.911242	+	0.411871i	$0.864875\pi$
$164$	− 88.9986i	− 0.542674i
$165$	0	0
$166$	213.647	1.28703
$167$	−130.041	−0.778691	−0.389346	−	0.921092i	$-0.627299\pi$
$167$	−130.041	−0.778691	−0.389346	+	0.921092i	$0.627299\pi$
$168$	0	0
$169$	130.356	0.771339
$170$	0	0
$171$	0	0
$172$	120.280i	0.699305i
$173$	−63.1882	−0.365250	−0.182625	−	0.983183i	$-0.558459\pi$
$173$	−63.1882	−0.365250	−0.182625	+	0.983183i	$0.558459\pi$
$174$	0	0
$175$	0	0
$176$	26.4673i	0.150382i
$177$	0	0
$178$	8.60825i	0.0483609i
$179$	− 302.826i	− 1.69177i	−0.533368	−	0.845883i	$-0.679074\pi$
$179$	− 302.826i	− 1.69177i	0.533368	−	0.845883i	$-0.320926\pi$
$180$	0	0
$181$	49.3100	0.272431	0.136216	−	0.990679i	$-0.456506\pi$
$181$	49.3100	0.272431	0.136216	+	0.990679i	$0.456506\pi$
$182$	23.2597	0.127800
$183$	0	0
$184$	36.3981	0.197816
$185$	0	0
$186$	0	0
$187$	− 42.0702i	− 0.224974i
$188$	41.2382	0.219352
$189$	0	0
$190$	0	0
$191$	42.4520i	0.222262i	0.993806	+	0.111131i	$0.0354473\pi$
$191$	42.4520i	0.222262i	−0.993806	+	0.111131i	$0.964553\pi$
$192$	0	0
$193$	84.2418i	0.436486i	0.975894	+	0.218243i	$0.0700325\pi$
$193$	84.2418i	0.436486i	−0.975894	+	0.218243i	$0.929967\pi$
$194$	191.126i	0.985183i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	−188.098	−0.954814	−0.477407	−	0.878682i	$-0.658423\pi$
$197$	−188.098	−0.954814	−0.477407	+	0.878682i	$0.658423\pi$
$198$	0	0
$199$	−36.9792	−0.185825	−0.0929127	−	0.995674i	$-0.529618\pi$
$199$	−36.9792	−0.185825	−0.0929127	+	0.995674i	$0.529618\pi$
$200$	0	0
$201$	0	0
$202$	− 141.047i	− 0.698255i
$203$	57.4317	0.282915
$204$	0	0
$205$	0	0
$206$	131.420i	0.637963i
$207$	0	0
$208$	− 24.8656i	− 0.119546i
$209$	155.965i	0.746243i
$210$	0	0
$211$	−142.836	−0.676948	−0.338474	−	0.940976i	$-0.609911\pi$
$211$	−142.836	−0.676948	−0.338474	+	0.940976i	$0.609911\pi$
$212$	80.1576	0.378102
$213$	0	0
$214$	−59.3425	−0.277302
$215$	0	0
$216$	0	0
$217$	104.603i	0.482042i
$218$	206.139	0.945593
$219$	0	0
$220$	0	0
$221$	39.5243i	0.178843i
$222$	0	0
$223$	69.1669i	0.310165i	0.987901	+	0.155083i	$0.0495644\pi$
$223$	69.1669i	0.310165i	−0.987901	+	0.155083i	$0.950436\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	41.8835	0.185325
$227$	−212.932	−0.938025	−0.469012	−	0.883192i	$-0.655390\pi$
$227$	−212.932	−0.938025	−0.469012	+	0.883192i	$0.655390\pi$
$228$	0	0
$229$	−386.569	−1.68807	−0.844036	−	0.536286i	$-0.819827\pi$
$229$	−386.569	−1.68807	−0.844036	+	0.536286i	$0.819827\pi$
$230$	0	0
$231$	0	0
$232$	− 61.3970i	− 0.264642i
$233$	−220.391	−0.945886	−0.472943	−	0.881093i	$-0.656808\pi$
$233$	−220.391	−0.945886	−0.472943	+	0.881093i	$0.656808\pi$
$234$	0	0
$235$	0	0
$236$	50.9731i	0.215988i
$237$	0	0
$238$	− 23.7897i	− 0.0999566i
$239$	174.401i	0.729711i	0.931064	+	0.364856i	$0.118882\pi$
$239$	174.401i	0.729711i	−0.931064	+	0.364856i	$0.881118\pi$
$240$	0	0
$241$	383.572	1.59159	0.795793	−	0.605569i	$-0.207054\pi$
$241$	383.572	1.59159	0.795793	+	0.605569i	$0.207054\pi$
$242$	−109.202	−0.451249
$243$	0	0
$244$	−1.48744	−0.00609606
$245$	0	0
$246$	0	0
$247$	− 146.526i	− 0.593224i
$248$	111.825	0.450909
$249$	0	0
$250$	0	0
$251$	− 289.914i	− 1.15503i	−0.816378	−	0.577517i	$-0.804022\pi$
$251$	− 289.914i	− 1.15503i	0.816378	−	0.577517i	$-0.195978\pi$
$252$	0	0
$253$	− 85.1498i	− 0.336561i
$254$	302.261i	1.19000i
$255$	0	0
$256$	16.0000	0.0625000
$257$	−335.260	−1.30451	−0.652257	−	0.757998i	$-0.726178\pi$
$257$	−335.260	−1.30451	−0.652257	+	0.757998i	$0.726178\pi$
$258$	0	0
$259$	−150.676	−0.581761
$260$	0	0
$261$	0	0
$262$	− 192.685i	− 0.735437i
$263$	365.898	1.39125	0.695623	−	0.718407i	$-0.255128\pi$
$263$	365.898	1.39125	0.695623	+	0.718407i	$0.255128\pi$
$264$	0	0
$265$	0	0
$266$	88.1943i	0.331558i
$267$	0	0
$268$	− 137.663i	− 0.513668i
$269$	− 319.390i	− 1.18732i	−0.804715	−	0.593661i	$-0.797682\pi$
$269$	− 319.390i	− 1.18732i	0.804715	−	0.593661i	$-0.202318\pi$
$270$	0	0
$271$	69.4293	0.256197	0.128098	−	0.991761i	$-0.459113\pi$
$271$	69.4293	0.256197	0.128098	+	0.991761i	$0.459113\pi$
$272$	−25.4322	−0.0935008
$273$	0	0
$274$	7.59026	0.0277017
$275$	0	0
$276$	0	0
$277$	− 544.473i	− 1.96561i	−0.184657	−	0.982803i	$-0.559118\pi$
$277$	− 544.473i	− 1.96561i	0.184657	−	0.982803i	$-0.440882\pi$
$278$	191.184	0.687712
$279$	0	0
$280$	0	0
$281$	− 54.2597i	− 0.193095i	−0.995328	−	0.0965476i	$-0.969220\pi$
$281$	− 54.2597i	− 0.193095i	0.995328	−	0.0965476i	$-0.0307800\pi$
$282$	0	0
$283$	− 256.916i	− 0.907829i	−0.891045	−	0.453914i	$-0.850027\pi$
$283$	− 256.916i	− 0.907829i	0.891045	−	0.453914i	$-0.149973\pi$
$284$	113.261i	0.398807i
$285$	0	0
$286$	−58.1707	−0.203394
$287$	−117.734	−0.410223
$288$	0	0
$289$	−248.575	−0.860121
$290$	0	0
$291$	0	0
$292$	− 26.2000i	− 0.0897261i
$293$	−118.382	−0.404035	−0.202017	−	0.979382i	$-0.564750\pi$
$293$	−118.382	−0.404035	−0.202017	+	0.979382i	$0.564750\pi$
$294$	0	0
$295$	0	0
$296$	161.080i	0.544188i
$297$	0	0
$298$	− 144.734i	− 0.485686i
$299$	79.9969i	0.267548i
$300$	0	0
$301$	159.116	0.528625
$302$	−54.7958	−0.181443
$303$	0	0
$304$	94.2837	0.310144
$305$	0	0
$306$	0	0
$307$	− 144.092i	− 0.469355i	−0.972073	−	0.234677i	$-0.924597\pi$
$307$	− 144.092i	− 0.469355i	0.972073	−	0.234677i	$-0.0754034\pi$
$308$	35.0130	0.113678
$309$	0	0
$310$	0	0
$311$	126.216i	0.405839i	0.979195	+	0.202920i	$0.0650430\pi$
$311$	126.216i	0.405839i	−0.979195	+	0.202920i	$0.934957\pi$
$312$	0	0
$313$	− 112.210i	− 0.358498i	−0.983804	−	0.179249i	$-0.942633\pi$
$313$	− 112.210i	− 0.358498i	0.983804	−	0.179249i	$-0.0573668\pi$
$314$	3.45151i	0.0109921i
$315$	0	0
$316$	44.4805	0.140761
$317$	454.598	1.43406	0.717032	−	0.697040i	$-0.245500\pi$
$317$	454.598	1.43406	0.717032	+	0.697040i	$0.245500\pi$
$318$	0	0
$319$	−143.632	−0.450258
$320$	0	0
$321$	0	0
$322$	− 48.1502i	− 0.149535i
$323$	−149.865	−0.463979
$324$	0	0
$325$	0	0
$326$	− 189.886i	− 0.582473i
$327$	0	0
$328$	125.863i	0.383729i
$329$	− 54.5530i	− 0.165815i
$330$	0	0
$331$	−453.034	−1.36868	−0.684342	−	0.729161i	$-0.739910\pi$
$331$	−453.034	−1.36868	−0.684342	+	0.729161i	$0.739910\pi$
$332$	−302.142	−0.910066
$333$	0	0
$334$	183.906	0.550618
$335$	0	0
$336$	0	0
$337$	201.485i	0.597878i	0.954272	+	0.298939i	$0.0966328\pi$
$337$	201.485i	0.597878i	−0.954272	+	0.298939i	$0.903367\pi$
$338$	−184.352	−0.545419
$339$	0	0
$340$	0	0
$341$	− 261.604i	− 0.767169i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	− 170.102i	− 0.494483i
$345$	0	0
$346$	89.3616	0.258271
$347$	−138.431	−0.398936	−0.199468	−	0.979904i	$-0.563921\pi$
$347$	−138.431	−0.398936	−0.199468	+	0.979904i	$0.563921\pi$
$348$	0	0
$349$	−221.471	−0.634586	−0.317293	−	0.948327i	$-0.602774\pi$
$349$	−221.471	−0.634586	−0.317293	+	0.948327i	$0.602774\pi$
$350$	0	0
$351$	0	0
$352$	− 37.4304i	− 0.106336i
$353$	−644.762	−1.82652	−0.913260	−	0.407377i	$-0.866444\pi$
$353$	−644.762	−1.82652	−0.913260	+	0.407377i	$0.866444\pi$
$354$	0	0
$355$	0	0
$356$	− 12.1739i	− 0.0341963i
$357$	0	0
$358$	428.261i	1.19626i
$359$	429.577i	1.19659i	0.801275	+	0.598296i	$0.204155\pi$
$359$	429.577i	1.19659i	−0.801275	+	0.598296i	$0.795845\pi$
$360$	0	0
$361$	194.588	0.539026
$362$	−69.7349	−0.192638
$363$	0	0
$364$	−32.8941	−0.0903685
$365$	0	0
$366$	0	0
$367$	− 394.169i	− 1.07403i	−0.843573	−	0.537015i	$-0.819552\pi$
$367$	− 394.169i	− 1.07403i	0.843573	−	0.537015i	$-0.180448\pi$
$368$	−51.4747	−0.139877
$369$	0	0
$370$	0	0
$371$	− 106.039i	− 0.285818i
$372$	0	0
$373$	23.1268i	0.0620021i	0.999519	+	0.0310011i	$0.00986952\pi$
$373$	23.1268i	0.0620021i	−0.999519	+	0.0310011i	$0.990130\pi$
$374$	59.4962i	0.159081i
$375$	0	0
$376$	−58.3197	−0.155105
$377$	134.940	0.357932
$378$	0	0
$379$	−593.049	−1.56477	−0.782387	−	0.622793i	$-0.785998\pi$
$379$	−593.049	−1.56477	−0.782387	+	0.622793i	$0.785998\pi$
$380$	0	0
$381$	0	0
$382$	− 60.0362i	− 0.157163i
$383$	−116.922	−0.305280	−0.152640	−	0.988282i	$-0.548778\pi$
$383$	−116.922	−0.305280	−0.152640	+	0.988282i	$0.548778\pi$
$384$	0	0
$385$	0	0
$386$	− 119.136i	− 0.308642i
$387$	0	0
$388$	− 270.292i	− 0.696630i
$389$	105.357i	0.270841i	0.990788	+	0.135421i	$0.0432386\pi$
$389$	105.357i	0.270841i	−0.990788	+	0.135421i	$0.956761\pi$
$390$	0	0
$391$	81.8198	0.209258
$392$	19.7990	0.0505076
$393$	0	0
$394$	266.011	0.675156
$395$	0	0
$396$	0	0
$397$	353.467i	0.890346i	0.895445	+	0.445173i	$0.146858\pi$
$397$	353.467i	0.890346i	−0.895445	+	0.445173i	$0.853142\pi$
$398$	52.2965	0.131398
$399$	0	0
$400$	0	0
$401$	− 768.878i	− 1.91740i	−0.284417	−	0.958701i	$-0.591800\pi$
$401$	− 768.878i	− 1.91740i	0.284417	−	0.958701i	$-0.408200\pi$
$402$	0	0
$403$	245.773i	0.609859i
$404$	199.471i	0.493741i
$405$	0	0
$406$	−81.2206	−0.200051
$407$	376.830	0.925872
$408$	0	0
$409$	−585.131	−1.43064	−0.715319	−	0.698798i	$-0.753718\pi$
$409$	−585.131	−1.43064	−0.715319	+	0.698798i	$0.753718\pi$
$410$	0	0
$411$	0	0
$412$	− 185.857i	− 0.451108i
$413$	67.4311	0.163271
$414$	0	0
$415$	0	0
$416$	35.1653i	0.0845320i
$417$	0	0
$418$	− 220.567i	− 0.527673i
$419$	287.325i	0.685740i	0.939383	+	0.342870i	$0.111399\pi$
$419$	287.325i	0.685740i	−0.939383	+	0.342870i	$0.888601\pi$
$420$	0	0
$421$	−437.539	−1.03929	−0.519643	−	0.854383i	$-0.673935\pi$
$421$	−437.539	−1.03929	−0.519643	+	0.854383i	$0.673935\pi$
$422$	202.001	0.478674
$423$	0	0
$424$	−113.360	−0.267358
$425$	0	0
$426$	0	0
$427$	1.96770i	0.00460819i
$428$	83.9230	0.196082
$429$	0	0
$430$	0	0
$431$	− 51.0877i	− 0.118533i	−0.998242	−	0.0592665i	$-0.981124\pi$
$431$	− 51.0877i	− 0.118533i	0.998242	−	0.0592665i	$-0.0188762\pi$
$432$	0	0
$433$	− 124.021i	− 0.286422i	−0.989692	−	0.143211i	$-0.954257\pi$
$433$	− 124.021i	− 0.286422i	0.989692	−	0.143211i	$-0.0457428\pi$
$434$	− 147.931i	− 0.340855i
$435$	0	0
$436$	−291.525	−0.668635
$437$	−303.327	−0.694111
$438$	0	0
$439$	497.601	1.13349	0.566743	−	0.823894i	$-0.308203\pi$
$439$	497.601	1.13349	0.566743	+	0.823894i	$0.308203\pi$
$440$	0	0
$441$	0	0
$442$	− 55.8957i	− 0.126461i
$443$	−823.485	−1.85888	−0.929441	−	0.368971i	$-0.879710\pi$
$443$	−823.485	−1.85888	−0.929441	+	0.368971i	$0.879710\pi$
$444$	0	0
$445$	0	0
$446$	− 97.8167i	− 0.219320i
$447$	0	0
$448$	− 21.1660i	− 0.0472456i
$449$	583.811i	1.30025i	0.759828	+	0.650124i	$0.225283\pi$
$449$	583.811i	1.30025i	−0.759828	+	0.650124i	$0.774717\pi$
$450$	0	0
$451$	294.444	0.652870
$452$	−59.2322	−0.131045
$453$	0	0
$454$	301.131	0.663284
$455$	0	0
$456$	0	0
$457$	− 317.492i	− 0.694731i	−0.937730	−	0.347366i	$-0.887076\pi$
$457$	− 317.492i	− 0.694731i	0.937730	−	0.347366i	$-0.112924\pi$
$458$	546.691	1.19365
$459$	0	0
$460$	0	0
$461$	− 703.769i	− 1.52661i	−0.646036	−	0.763307i	$-0.723575\pi$
$461$	− 703.769i	− 1.52661i	0.646036	−	0.763307i	$-0.276425\pi$
$462$	0	0
$463$	296.981i	0.641428i	0.947176	+	0.320714i	$0.103923\pi$
$463$	296.981i	0.641428i	−0.947176	+	0.320714i	$0.896077\pi$
$464$	86.8285i	0.187130i
$465$	0	0
$466$	311.680	0.668842
$467$	−839.787	−1.79826	−0.899130	−	0.437682i	$-0.855800\pi$
$467$	−839.787	−1.79826	−0.899130	+	0.437682i	$0.855800\pi$
$468$	0	0
$469$	−182.111	−0.388296
$470$	0	0
$471$	0	0
$472$	− 72.0869i	− 0.152726i
$473$	−397.937	−0.841305
$474$	0	0
$475$	0	0
$476$	33.6437i	0.0706800i
$477$	0	0
$478$	− 246.640i	− 0.515984i
$479$	− 840.794i	− 1.75531i	−0.479292	−	0.877655i	$-0.659107\pi$
$479$	− 840.794i	− 1.75531i	0.479292	−	0.877655i	$-0.340893\pi$
$480$	0	0
$481$	−354.026	−0.736020
$482$	−542.453	−1.12542
$483$	0	0
$484$	154.435	0.319081
$485$	0	0
$486$	0	0
$487$	− 274.175i	− 0.562987i	−0.959563	−	0.281494i	$-0.909170\pi$
$487$	− 274.175i	− 0.562987i	0.959563	−	0.281494i	$-0.0908298\pi$
$488$	2.10356	0.00431057
$489$	0	0
$490$	0	0
$491$	165.218i	0.336494i	0.985745	+	0.168247i	$0.0538106\pi$
$491$	165.218i	0.336494i	−0.985745	+	0.168247i	$0.946189\pi$
$492$	0	0
$493$	− 138.015i	− 0.279950i
$494$	207.220i	0.419473i
$495$	0	0
$496$	−158.145	−0.318841
$497$	149.830	0.301470
$498$	0	0
$499$	−61.0172	−0.122279	−0.0611395	−	0.998129i	$-0.519473\pi$
$499$	−61.0172	−0.122279	−0.0611395	+	0.998129i	$0.519473\pi$
$500$	0	0
$501$	0	0
$502$	410.000i	0.816733i
$503$	−129.075	−0.256611	−0.128305	−	0.991735i	$-0.540954\pi$
$503$	−129.075	−0.256611	−0.128305	+	0.991735i	$0.540954\pi$
$504$	0	0
$505$	0	0
$506$	120.420i	0.237984i
$507$	0	0
$508$	− 427.462i	− 0.841460i
$509$	− 490.684i	− 0.964016i	−0.876167	−	0.482008i	$-0.839908\pi$
$509$	− 490.684i	− 0.964016i	0.876167	−	0.482008i	$-0.160092\pi$
$510$	0	0
$511$	−34.6594	−0.0678265
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	474.129	0.922430
$515$	0	0
$516$	0	0
$517$	136.433i	0.263894i
$518$	213.088	0.411367
$519$	0	0
$520$	0	0
$521$	− 770.179i	− 1.47827i	−0.673557	−	0.739135i	$-0.735234\pi$
$521$	− 770.179i	− 1.47827i	0.673557	−	0.739135i	$-0.264766\pi$
$522$	0	0
$523$	74.8406i	0.143099i	0.997437	+	0.0715494i	$0.0227943\pi$
$523$	74.8406i	0.143099i	−0.997437	+	0.0715494i	$0.977206\pi$
$524$	272.497i	0.520033i
$525$	0	0
$526$	−517.458	−0.983760
$527$	251.374	0.476990
$528$	0	0
$529$	−363.397	−0.686951
$530$	0	0
$531$	0	0
$532$	− 124.726i	− 0.234447i
$533$	−276.626	−0.518998
$534$	0	0
$535$	0	0
$536$	194.685i	0.363218i
$537$	0	0
$538$	451.685i	0.839564i
$539$	− 46.3178i	− 0.0859328i
$540$	0	0
$541$	−899.685	−1.66300	−0.831502	−	0.555522i	$-0.812519\pi$
$541$	−899.685	−1.66300	−0.831502	+	0.555522i	$0.812519\pi$
$542$	−98.1878	−0.181158
$543$	0	0
$544$	35.9666	0.0661151
$545$	0	0
$546$	0	0
$547$	− 92.3806i	− 0.168886i	−0.996428	−	0.0844430i	$-0.973089\pi$
$547$	− 92.3806i	− 0.168886i	0.996428	−	0.0844430i	$-0.0269111\pi$
$548$	−10.7343	−0.0195881
$549$	0	0
$550$	0	0
$551$	511.657i	0.928597i
$552$	0	0
$553$	− 58.8422i	− 0.106405i
$554$	770.001i	1.38989i
$555$	0	0
$556$	−270.375	−0.486286
$557$	−202.763	−0.364026	−0.182013	−	0.983296i	$-0.558261\pi$
$557$	−202.763	−0.364026	−0.182013	+	0.983296i	$0.558261\pi$
$558$	0	0
$559$	373.856	0.668794
$560$	0	0
$561$	0	0
$562$	76.7349i	0.136539i
$563$	445.660	0.791581	0.395791	−	0.918341i	$-0.370471\pi$
$563$	445.660	0.791581	0.395791	+	0.918341i	$0.370471\pi$
$564$	0	0
$565$	0	0
$566$	363.333i	0.641932i
$567$	0	0
$568$	− 160.176i	− 0.281999i
$569$	− 1063.41i	− 1.86891i	−0.356080	−	0.934456i	$-0.615887\pi$
$569$	− 1063.41i	− 1.86891i	0.356080	−	0.934456i	$-0.384113\pi$
$570$	0	0
$571$	−825.254	−1.44528	−0.722639	−	0.691226i	$-0.757071\pi$
$571$	−825.254	−1.44528	−0.722639	+	0.691226i	$0.757071\pi$
$572$	82.2658	0.143821
$573$	0	0
$574$	166.501	0.290072
$575$	0	0
$576$	0	0
$577$	1013.50i	1.75651i	0.478196	+	0.878253i	$0.341291\pi$
$577$	1013.50i	1.75651i	−0.478196	+	0.878253i	$0.658709\pi$
$578$	351.538	0.608198
$579$	0	0
$580$	0	0
$581$	399.696i	0.687945i
$582$	0	0
$583$	265.195i	0.454879i
$584$	37.0524i	0.0634459i
$585$	0	0
$586$	167.418	0.285696
$587$	−57.3023	−0.0976189	−0.0488095	−	0.998808i	$-0.515543\pi$
$587$	−57.3023	−0.0976189	−0.0488095	+	0.998808i	$0.515543\pi$
$588$	0	0
$589$	−931.906	−1.58218
$590$	0	0
$591$	0	0
$592$	− 227.801i	− 0.384799i
$593$	−684.040	−1.15352	−0.576762	−	0.816912i	$-0.695684\pi$
$593$	−684.040	−1.15352	−0.576762	+	0.816912i	$0.695684\pi$
$594$	0	0
$595$	0	0
$596$	204.685i	0.343432i
$597$	0	0
$598$	− 113.133i	− 0.189185i
$599$	− 715.347i	− 1.19423i	−0.802154	−	0.597117i	$-0.796313\pi$
$599$	− 715.347i	− 1.19423i	0.802154	−	0.597117i	$-0.203687\pi$
$600$	0	0
$601$	−333.692	−0.555227	−0.277614	−	0.960693i	$-0.589543\pi$
$601$	−333.692	−0.555227	−0.277614	+	0.960693i	$0.589543\pi$
$602$	−225.024	−0.373794
$603$	0	0
$604$	77.4930	0.128300
$605$	0	0
$606$	0	0
$607$	− 109.174i	− 0.179858i	−0.995948	−	0.0899290i	$-0.971336\pi$
$607$	− 109.174i	− 0.179858i	0.995948	−	0.0899290i	$-0.0286640\pi$
$608$	−133.337	−0.219305
$609$	0	0
$610$	0	0
$611$	− 128.177i	− 0.209782i
$612$	0	0
$613$	− 326.936i	− 0.533338i	−0.963788	−	0.266669i	$-0.914077\pi$
$613$	− 326.936i	− 0.533338i	0.963788	−	0.266669i	$-0.0859231\pi$
$614$	203.777i	0.331884i
$615$	0	0
$616$	−49.5158	−0.0803828
$617$	−525.085	−0.851030	−0.425515	−	0.904951i	$-0.639907\pi$
$617$	−525.085	−0.851030	−0.425515	+	0.904951i	$0.639907\pi$
$618$	0	0
$619$	314.513	0.508099	0.254050	−	0.967191i	$-0.418237\pi$
$619$	314.513	0.508099	0.254050	+	0.967191i	$0.418237\pi$
$620$	0	0
$621$	0	0
$622$	− 178.496i	− 0.286972i
$623$	−16.1046	−0.0258500
$624$	0	0
$625$	0	0
$626$	158.689i	0.253497i
$627$	0	0
$628$	− 4.88118i	− 0.00777257i
$629$	362.093i	0.575664i
$630$	0	0
$631$	361.534	0.572954	0.286477	−	0.958087i	$-0.407516\pi$
$631$	361.534	0.572954	0.286477	+	0.958087i	$0.407516\pi$
$632$	−62.9049	−0.0995331
$633$	0	0
$634$	−642.899	−1.01404
$635$	0	0
$636$	0	0
$637$	43.5148i	0.0683121i
$638$	203.127	0.318381
$639$	0	0
$640$	0	0
$641$	299.674i	0.467510i	0.972296	+	0.233755i	$0.0751013\pi$
$641$	299.674i	0.467510i	−0.972296	+	0.233755i	$0.924899\pi$
$642$	0	0
$643$	276.617i	0.430198i	0.976592	+	0.215099i	$0.0690074\pi$
$643$	276.617i	0.430198i	−0.976592	+	0.215099i	$0.930993\pi$
$644$	68.0947i	0.105737i
$645$	0	0
$646$	211.941	0.328083
$647$	900.003	1.39104	0.695520	−	0.718507i	$-0.255174\pi$
$647$	900.003	1.39104	0.695520	+	0.718507i	$0.255174\pi$
$648$	0	0
$649$	−168.640	−0.259846
$650$	0	0
$651$	0	0
$652$	268.540i	0.411871i
$653$	173.928	0.266352	0.133176	−	0.991092i	$-0.457482\pi$
$653$	173.928	0.266352	0.133176	+	0.991092i	$0.457482\pi$
$654$	0	0
$655$	0	0
$656$	− 177.997i	− 0.271337i
$657$	0	0
$658$	77.1497i	0.117249i
$659$	− 238.864i	− 0.362465i	−0.983440	−	0.181232i	$-0.941991\pi$
$659$	− 238.864i	− 0.362465i	0.983440	−	0.181232i	$-0.0580086\pi$
$660$	0	0
$661$	852.168	1.28921	0.644605	−	0.764516i	$-0.277022\pi$
$661$	852.168	1.28921	0.644605	+	0.764516i	$0.277022\pi$
$662$	640.687	0.967806
$663$	0	0
$664$	427.293	0.643514
$665$	0	0
$666$	0	0
$667$	− 279.342i	− 0.418804i
$668$	−260.083	−0.389346
$669$	0	0
$670$	0	0
$671$	− 4.92106i	− 0.00733392i
$672$	0	0
$673$	159.472i	0.236957i	0.992957	+	0.118479i	$0.0378017\pi$
$673$	159.472i	0.236957i	−0.992957	+	0.118479i	$0.962198\pi$
$674$	− 284.943i	− 0.422764i
$675$	0	0
$676$	260.713	0.385670
$677$	−1100.80	−1.62600	−0.812999	−	0.582265i	$-0.802167\pi$
$677$	−1100.80	−1.62600	−0.812999	+	0.582265i	$0.802167\pi$
$678$	0	0
$679$	−357.563	−0.526603
$680$	0	0
$681$	0	0
$682$	369.965i	0.542470i
$683$	1305.66	1.91166	0.955830	−	0.293919i	$-0.0949596\pi$
$683$	1305.66	1.91166	0.955830	+	0.293919i	$0.0949596\pi$
$684$	0	0
$685$	0	0
$686$	− 26.1916i	− 0.0381802i
$687$	0	0
$688$	240.561i	0.349652i
$689$	− 249.146i	− 0.361605i
$690$	0	0
$691$	−322.487	−0.466696	−0.233348	−	0.972393i	$-0.574968\pi$
$691$	−322.487	−0.466696	−0.233348	+	0.972393i	$0.574968\pi$
$692$	−126.376	−0.182625
$693$	0	0
$694$	195.771	0.282091
$695$	0	0
$696$	0	0
$697$	282.929i	0.405924i
$698$	313.207	0.448720
$699$	0	0
$700$	0	0
$701$	− 574.441i	− 0.819459i	−0.912207	−	0.409730i	$-0.865623\pi$
$701$	− 574.441i	− 0.819459i	0.912207	−	0.409730i	$-0.134377\pi$
$702$	0	0
$703$	− 1342.37i	− 1.90949i
$704$	52.9346i	0.0751912i
$705$	0	0
$706$	911.831	1.29154
$707$	263.876	0.373233
$708$	0	0
$709$	1194.48	1.68474	0.842372	−	0.538896i	$-0.181158\pi$
$709$	1194.48	1.68474	0.842372	+	0.538896i	$0.181158\pi$
$710$	0	0
$711$	0	0
$712$	17.2165i	0.0241805i
$713$	508.779	0.713575
$714$	0	0
$715$	0	0
$716$	− 605.652i	− 0.845883i
$717$	0	0
$718$	− 607.513i	− 0.846119i
$719$	− 58.4935i	− 0.0813539i	−0.999172	−	0.0406770i	$-0.987049\pi$
$719$	− 58.4935i	− 0.0813539i	0.999172	−	0.0406770i	$-0.0129515\pi$
$720$	0	0
$721$	−245.865	−0.341006
$722$	−275.190	−0.381149
$723$	0	0
$724$	98.6201	0.136216
$725$	0	0
$726$	0	0
$727$	− 194.084i	− 0.266965i	−0.991051	−	0.133483i	$-0.957384\pi$
$727$	− 194.084i	− 0.266965i	0.991051	−	0.133483i	$-0.0426161\pi$
$728$	46.5193	0.0639002
$729$	0	0
$730$	0	0
$731$	− 382.375i	− 0.523085i
$732$	0	0
$733$	690.669i	0.942250i	0.882066	+	0.471125i	$0.156152\pi$
$733$	690.669i	0.942250i	−0.882066	+	0.471125i	$0.843848\pi$
$734$	557.439i	0.759453i
$735$	0	0
$736$	72.7963	0.0989080
$737$	455.446	0.617973
$738$	0	0
$739$	−0.443737	−0.000600456 0	−0.000300228	−	1.00000i	$-0.500096\pi$
$739$	−0.443737	−0.000600456 0	−0.000300228		1.00000i	$0.500096\pi$
$740$	0	0
$741$	0	0
$742$	149.961i	0.202104i
$743$	−723.574	−0.973855	−0.486927	−	0.873442i	$-0.661882\pi$
$743$	−723.574	−0.973855	−0.486927	+	0.873442i	$0.661882\pi$
$744$	0	0
$745$	0	0
$746$	− 32.7062i	− 0.0438421i
$747$	0	0
$748$	− 84.1403i	− 0.112487i
$749$	− 111.020i	− 0.148224i
$750$	0	0
$751$	923.228	1.22933	0.614666	−	0.788788i	$-0.289291\pi$
$751$	923.228	1.22933	0.614666	+	0.788788i	$0.289291\pi$
$752$	82.4765	0.109676
$753$	0	0
$754$	−190.834	−0.253096
$755$	0	0
$756$	0	0
$757$	1096.39i	1.44833i	0.689627	+	0.724165i	$0.257775\pi$
$757$	1096.39i	1.44833i	−0.689627	+	0.724165i	$0.742225\pi$
$758$	838.698	1.10646
$759$	0	0
$760$	0	0
$761$	913.964i	1.20100i	0.799623	+	0.600502i	$0.205033\pi$
$761$	913.964i	1.20100i	−0.799623	+	0.600502i	$0.794967\pi$
$762$	0	0
$763$	385.651i	0.505441i
$764$	84.9040i	0.111131i
$765$	0	0
$766$	165.353	0.215866
$767$	158.435	0.206564
$768$	0	0
$769$	−770.840	−1.00239	−0.501197	−	0.865333i	$-0.667107\pi$
$769$	−770.840	−1.00239	−0.501197	+	0.865333i	$0.667107\pi$
$770$	0	0
$771$	0	0
$772$	168.484i	0.218243i
$773$	250.584	0.324171	0.162085	−	0.986777i	$-0.448178\pi$
$773$	250.584	0.324171	0.162085	+	0.986777i	$0.448178\pi$
$774$	0	0
$775$	0	0
$776$	382.251i	0.492592i
$777$	0	0
$778$	− 148.998i	− 0.191514i
$779$	− 1048.89i	− 1.34646i
$780$	0	0
$781$	−374.715	−0.479789
$782$	−115.711	−0.147968
$783$	0	0
$784$	−28.0000	−0.0357143
$785$	0	0
$786$	0	0
$787$	2.48249i	0.00315437i	0.999999	+	0.00157719i	$0.000502034\pi$
$787$	2.48249i	0.00315437i	−0.999999	+	0.00157719i	$0.999498\pi$
$788$	−376.197	−0.477407
$789$	0	0
$790$	0	0
$791$	78.3568i	0.0990604i
$792$	0	0
$793$	4.62326i	0.00583009i
$794$	− 499.878i	− 0.629570i
$795$	0	0
$796$	−73.9585	−0.0929127
$797$	−269.762	−0.338471	−0.169236	−	0.985576i	$-0.554130\pi$
$797$	−269.762	−0.338471	−0.169236	+	0.985576i	$0.554130\pi$
$798$	0	0
$799$	−131.098	−0.164077
$800$	0	0
$801$	0	0
$802$	1087.36i	1.35581i
$803$	86.6805	0.107946
$804$	0	0
$805$	0	0
$806$	− 347.576i	− 0.431236i
$807$	0	0
$808$	− 282.095i	− 0.349127i
$809$	− 811.512i	− 1.00311i	−0.865127	−	0.501553i	$-0.832762\pi$
$809$	− 811.512i	− 1.00311i	0.865127	−	0.501553i	$-0.167238\pi$
$810$	0	0
$811$	891.278	1.09899	0.549493	−	0.835498i	$-0.314821\pi$
$811$	891.278	1.09899	0.549493	+	0.835498i	$0.314821\pi$
$812$	114.863	0.141457
$813$	0	0
$814$	−532.918	−0.654691
$815$	0	0
$816$	0	0
$817$	1417.56i	1.73508i
$818$	827.500	1.01161
$819$	0	0
$820$	0	0
$821$	814.161i	0.991670i	0.868417	+	0.495835i	$0.165138\pi$
$821$	814.161i	0.991670i	−0.868417	+	0.495835i	$0.834862\pi$
$822$	0	0
$823$	1497.09i	1.81906i	0.415640	+	0.909529i	$0.363558\pi$
$823$	1497.09i	1.81906i	−0.415640	+	0.909529i	$0.636442\pi$
$824$	262.841i	0.318982i
$825$	0	0
$826$	−95.3620	−0.115450
$827$	538.871	0.651597	0.325798	−	0.945439i	$-0.394367\pi$
$827$	538.871	0.651597	0.325798	+	0.945439i	$0.394367\pi$
$828$	0	0
$829$	1512.58	1.82459	0.912293	−	0.409539i	$-0.134310\pi$
$829$	1512.58	1.82459	0.912293	+	0.409539i	$0.134310\pi$
$830$	0	0
$831$	0	0
$832$	− 49.7312i	− 0.0597731i
$833$	44.5064	0.0534291
$834$	0	0
$835$	0	0
$836$	311.929i	0.373121i
$837$	0	0
$838$	− 406.339i	− 0.484892i
$839$	− 999.083i	− 1.19080i	−0.803429	−	0.595401i	$-0.796993\pi$
$839$	− 999.083i	− 1.19080i	0.803429	−	0.595401i	$-0.203007\pi$
$840$	0	0
$841$	369.801	0.439715
$842$	618.774	0.734886
$843$	0	0
$844$	−285.672	−0.338474
$845$	0	0
$846$	0	0
$847$	− 204.299i	− 0.241203i
$848$	160.315	0.189051
$849$	0	0
$850$	0	0
$851$	732.875i	0.861192i
$852$	0	0
$853$	94.6292i	0.110937i	0.998460	+	0.0554685i	$0.0176652\pi$
$853$	94.6292i	0.110937i	−0.998460	+	0.0554685i	$0.982335\pi$
$854$	− 2.78274i	− 0.00325848i
$855$	0	0
$856$	−118.685	−0.138651
$857$	1326.35	1.54766	0.773831	−	0.633392i	$-0.218338\pi$
$857$	1326.35	1.54766	0.773831	+	0.633392i	$0.218338\pi$
$858$	0	0
$859$	1366.84	1.59119	0.795597	−	0.605826i	$-0.207157\pi$
$859$	1366.84	1.59119	0.795597	+	0.605826i	$0.207157\pi$
$860$	0	0
$861$	0	0
$862$	72.2490i	0.0838155i
$863$	−367.560	−0.425909	−0.212955	−	0.977062i	$-0.568309\pi$
$863$	−367.560	−0.425909	−0.212955	+	0.977062i	$0.568309\pi$
$864$	0	0
$865$	0	0
$866$	175.392i	0.202531i
$867$	0	0
$868$	209.206i	0.241021i
$869$	147.160i	0.169344i
$870$	0	0
$871$	−427.884	−0.491256
$872$	412.278	0.472796
$873$	0	0
$874$	428.969	0.490811
$875$	0	0
$876$	0	0
$877$	− 252.803i	− 0.288259i	−0.989559	−	0.144129i	$-0.953962\pi$
$877$	− 252.803i	− 0.288259i	0.989559	−	0.144129i	$-0.0460382\pi$
$878$	−703.714	−0.801496
$879$	0	0
$880$	0	0
$881$	− 927.007i	− 1.05222i	−0.850416	−	0.526111i	$-0.823650\pi$
$881$	− 927.007i	− 1.05222i	0.850416	−	0.526111i	$-0.176350\pi$
$882$	0	0
$883$	965.503i	1.09343i	0.837317	+	0.546717i	$0.184123\pi$
$883$	965.503i	1.09343i	−0.837317	+	0.546717i	$0.815877\pi$
$884$	79.0485i	0.0894214i
$885$	0	0
$886$	1164.58	1.31443
$887$	41.9639	0.0473099	0.0236549	−	0.999720i	$-0.492470\pi$
$887$	41.9639	0.0473099	0.0236549	+	0.999720i	$0.492470\pi$
$888$	0	0
$889$	−565.479	−0.636084
$890$	0	0
$891$	0	0
$892$	138.334i	0.155083i
$893$	486.012	0.544246
$894$	0	0
$895$	0	0
$896$	29.9333i	0.0334077i
$897$	0	0
$898$	− 825.634i	− 0.919414i
$899$	− 858.218i	− 0.954637i
$900$	0	0
$901$	−254.823	−0.282823
$902$	−416.407	−0.461649
$903$	0	0
$904$	83.7669	0.0926625
$905$	0	0
$906$	0	0
$907$	− 356.488i	− 0.393040i	−0.980500	−	0.196520i	$-0.937036\pi$
$907$	− 356.488i	− 0.393040i	0.980500	−	0.196520i	$-0.0629641\pi$
$908$	−425.863	−0.469012
$909$	0	0
$910$	0	0
$911$	277.860i	0.305006i	0.988303	+	0.152503i	$0.0487334\pi$
$911$	277.860i	0.305006i	−0.988303	+	0.152503i	$0.951267\pi$
$912$	0	0
$913$	− 999.611i	− 1.09486i
$914$	449.002i	0.491249i
$915$	0	0
$916$	−773.137	−0.844036
$917$	360.480	0.393108
$918$	0	0
$919$	271.756	0.295708	0.147854	−	0.989009i	$-0.452763\pi$
$919$	271.756	0.295708	0.147854	+	0.989009i	$0.452763\pi$
$920$	0	0
$921$	0	0
$922$	995.279i	1.07948i
$923$	352.039	0.381407
$924$	0	0
$925$	0	0
$926$	− 419.995i	− 0.453558i
$927$	0	0
$928$	− 122.794i	− 0.132321i
$929$	368.676i	0.396853i	0.980116	+	0.198426i	$0.0635831\pi$
$929$	368.676i	0.396853i	−0.980116	+	0.198426i	$0.936417\pi$
$930$	0	0
$931$	−164.996	−0.177225
$932$	−440.783	−0.472943
$933$	0	0
$934$	1187.64	1.27156
$935$	0	0
$936$	0	0
$937$	− 601.796i	− 0.642258i	−0.947035	−	0.321129i	$-0.895938\pi$
$937$	− 601.796i	− 0.642258i	0.947035	−	0.321129i	$-0.104062\pi$
$938$	257.544	0.274567
$939$	0	0
$940$	0	0
$941$	− 63.5278i	− 0.0675109i	−0.999430	−	0.0337554i	$-0.989253\pi$
$941$	− 63.5278i	− 0.0675109i	0.999430	−	0.0337554i	$-0.0107467\pi$
$942$	0	0
$943$	572.647i	0.607261i
$944$	101.946i	0.107994i
$945$	0	0
$946$	562.769	0.594893
$947$	−47.9984	−0.0506847	−0.0253424	−	0.999679i	$-0.508068\pi$
$947$	−47.9984	−0.0506847	−0.0253424	+	0.999679i	$0.508068\pi$
$948$	0	0
$949$	−81.4350	−0.0858113
$950$	0	0
$951$	0	0
$952$	− 47.5793i	− 0.0499783i
$953$	223.027	0.234026	0.117013	−	0.993130i	$-0.462668\pi$
$953$	223.027	0.234026	0.117013	+	0.993130i	$0.462668\pi$
$954$	0	0
$955$	0	0
$956$	348.802i	0.364856i
$957$	0	0
$958$	1189.06i	1.24119i
$959$	14.2001i	0.0148072i
$960$	0	0
$961$	602.114	0.626550
$962$	500.668	0.520445
$963$	0	0
$964$	767.144	0.795793
$965$	0	0
$966$	0	0
$967$	1382.98i	1.43017i	0.699036	+	0.715087i	$0.253613\pi$
$967$	1382.98i	1.43017i	−0.699036	+	0.715087i	$0.746387\pi$
$968$	−218.404	−0.225624
$969$	0	0
$970$	0	0
$971$	− 646.400i	− 0.665706i	−0.942979	−	0.332853i	$-0.891989\pi$
$971$	− 646.400i	− 0.665706i	0.942979	−	0.332853i	$-0.108011\pi$
$972$	0	0
$973$	357.672i	0.367597i
$974$	387.742i	0.398092i
$975$	0	0
$976$	−2.97488	−0.00304803
$977$	−265.222	−0.271466	−0.135733	−	0.990745i	$-0.543339\pi$
$977$	−265.222	−0.271466	−0.135733	+	0.990745i	$0.543339\pi$
$978$	0	0
$979$	40.2763	0.0411402
$980$	0	0
$981$	0	0
$982$	− 233.654i	− 0.237937i
$983$	−1376.44	−1.40024	−0.700121	−	0.714024i	$-0.746871\pi$
$983$	−1376.44	−1.40024	−0.700121	+	0.714024i	$0.746871\pi$
$984$	0	0
$985$	0	0
$986$	195.183i	0.197954i
$987$	0	0
$988$	− 293.053i	− 0.296612i
$989$	− 773.925i	− 0.782533i
$990$	0	0
$991$	−80.2405	−0.0809693	−0.0404846	−	0.999180i	$-0.512890\pi$
$991$	−80.2405	−0.0809693	−0.0404846	+	0.999180i	$0.512890\pi$
$992$	223.651	0.225454
$993$	0	0
$994$	−211.892	−0.213171
$995$	0	0
$996$	0	0
$997$	− 409.667i	− 0.410899i	−0.978668	−	0.205450i	$-0.934134\pi$
$997$	− 409.667i	− 0.410899i	0.978668	−	0.205450i	$-0.0658657\pi$
$998$	86.2913	0.0864643
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.c.f.449.4		16
3.2	odd	2	inner	3150.3.c.f.449.9		16
5.2	odd	4		3150.3.e.f.701.4		8
5.3	odd	4		630.3.e.b.71.5	yes	8
5.4	even	2	inner	3150.3.c.f.449.16		16
15.2	even	4		3150.3.e.f.701.7		8
15.8	even	4		630.3.e.b.71.3	✓	8
15.14	odd	2	inner	3150.3.c.f.449.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.3.e.b.71.3	✓	8	15.8	even	4
630.3.e.b.71.5	yes	8	5.3	odd	4
3150.3.c.f.449.4		16	1.1	even	1	trivial
3150.3.c.f.449.5		16	15.14	odd	2	inner
3150.3.c.f.449.9		16	3.2	odd	2	inner
3150.3.c.f.449.16		16	5.4	even	2	inner
3150.3.e.f.701.4		8	5.2	odd	4
3150.3.e.f.701.7		8	15.2	even	4

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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} -2.82843 q^{8} +6.61683i q^{11} -6.21640i q^{13} +3.74166i q^{14} +4.00000 q^{16} -6.35806 q^{17} +23.5709 q^{19} -9.35761i q^{22} -12.8687 q^{23} +8.79132i q^{26} -5.29150i q^{28} +21.7071i q^{29} -39.5362 q^{31} -5.65685 q^{32} +8.99165 q^{34} -56.9502i q^{37} -33.3343 q^{38} -44.4993i q^{41} +60.1402i q^{43} +13.2337i q^{44} +18.1991 q^{46} +20.6191 q^{47} -7.00000 q^{49} -12.4328i q^{52} +40.0788 q^{53} +7.48331i q^{56} -30.6985i q^{58} +25.4866i q^{59} -0.743719 q^{61} +55.9127 q^{62} +8.00000 q^{64} -68.8315i q^{67} -12.7161 q^{68} +56.6306i q^{71} -13.1000i q^{73} +80.5398i q^{74} +47.1418 q^{76} +17.5065 q^{77} +22.2403 q^{79} +62.9315i q^{82} -151.071 q^{83} -85.0511i q^{86} -18.7152i q^{88} -6.08695i q^{89} -16.4471 q^{91} -25.7374 q^{92} -29.1598 q^{94} -135.146i q^{97} +9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) \) 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

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Database

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