LMFDB - Embedded newform 2475.4.a.s.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2475,4,Mod(1,2475)] 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("2475.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-4,0,22,0,0,4,-48,0,0,-33] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$146.029727264$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.23612.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 26 $ x^3 - x^2 - 20*x + 26
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.26150$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.26150 q^{2} +19.6833 q^{4} +10.3207 q^{7} -61.4719 q^{8} -11.0000 q^{11} -63.9817 q^{13} -54.3024 q^{14} +165.967 q^{16} +17.1461 q^{17} +90.2104 q^{19} +57.8765 q^{22} -212.605 q^{23} +336.639 q^{26} +203.146 q^{28} -57.5461 q^{29} -141.704 q^{31} -381.462 q^{32} -90.2140 q^{34} +257.963 q^{37} -474.642 q^{38} +225.914 q^{41} +347.445 q^{43} -216.517 q^{44} +1118.62 q^{46} +404.364 q^{47} -236.483 q^{49} -1259.37 q^{52} +259.568 q^{53} -634.433 q^{56} +302.779 q^{58} +853.067 q^{59} -203.699 q^{61} +745.573 q^{62} +679.320 q^{64} -266.890 q^{67} +337.492 q^{68} -92.4460 q^{71} +242.026 q^{73} -1357.27 q^{74} +1775.64 q^{76} -113.528 q^{77} -1021.60 q^{79} -1188.65 q^{82} +706.415 q^{83} -1828.08 q^{86} +676.191 q^{88} +440.218 q^{89} -660.336 q^{91} -4184.77 q^{92} -2127.56 q^{94} +197.761 q^{97} +1244.25 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{2} + 22 q^{4} + 4 q^{7} - 48 q^{8} - 33 q^{11} + 56 q^{14} + 50 q^{16} - 218 q^{17} + 146 q^{19} + 44 q^{22} - 200 q^{23} + 508 q^{26} + 340 q^{28} - 68 q^{29} - 68 q^{31} - 688 q^{32} - 176 q^{34}+ \cdots + 572 q^{98}+O(q^{100}) $ 3 * q - 4 * q^2 + 22 * q^4 + 4 * q^7 - 48 * q^8 - 33 * q^11 + 56 * q^14 + 50 * q^16 - 218 * q^17 + 146 * q^19 + 44 * q^22 - 200 * q^23 + 508 * q^26 + 340 * q^28 - 68 * q^29 - 68 * q^31 - 688 * q^32 - 176 * q^34 + 390 * q^37 - 316 * q^38 + 196 * q^41 + 524 * q^43 - 242 * q^44 + 1160 * q^46 - 60 * q^47 - 157 * q^49 - 1020 * q^52 - 158 * q^53 - 1368 * q^56 - 1092 * q^58 + 1044 * q^59 + 642 * q^61 + 88 * q^62 + 1166 * q^64 + 236 * q^67 + 144 * q^68 + 544 * q^71 - 900 * q^73 - 1536 * q^74 + 1996 * q^76 - 44 * q^77 - 1586 * q^79 - 380 * q^82 - 1582 * q^83 - 3568 * q^86 + 528 * q^88 + 2122 * q^89 - 8 * q^91 - 4128 * q^92 - 2152 * q^94 - 618 * q^97 + 572 * q^98

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 $

$2$	−5.26150	−1.86022	−0.930110	−	0.367281i	$-0.880289\pi$
$2$	−5.26150	−1.86022	−0.930110	+	0.367281i	$0.880289\pi$
$3$	0	0
$3$	0	0
$4$	19.6833	2.46042
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.3207	0.557266	0.278633	−	0.960398i	$-0.410119\pi$
$7$	10.3207	0.557266	0.278633	+	0.960398i	$0.410119\pi$
$8$	−61.4719	−2.71670
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−63.9817	−1.36502	−0.682512	−	0.730874i	$-0.739113\pi$
$13$	−63.9817	−1.36502	−0.682512	+	0.730874i	$0.739113\pi$
$14$	−54.3024	−1.03664
$15$	0	0
$16$	165.967	2.59324
$17$	17.1461	0.244620	0.122310	−	0.992492i	$-0.460970\pi$
$17$	17.1461	0.244620	0.122310	+	0.992492i	$0.460970\pi$
$18$	0	0
$19$	90.2104	1.08925	0.544623	−	0.838681i	$-0.316673\pi$
$19$	90.2104	1.08925	0.544623	+	0.838681i	$0.316673\pi$
$20$	0	0
$21$	0	0
$22$	57.8765	0.560877
$23$	−212.605	−1.92744	−0.963721	−	0.266913i	$-0.913996\pi$
$23$	−212.605	−1.92744	−0.963721	+	0.266913i	$0.913996\pi$
$24$	0	0
$25$	0	0
$26$	336.639	2.53925
$27$	0	0
$28$	203.146	1.37111
$29$	−57.5461	−0.368484	−0.184242	−	0.982881i	$-0.558983\pi$
$29$	−57.5461	−0.368484	−0.184242	+	0.982881i	$0.558983\pi$
$30$	0	0
$31$	−141.704	−0.820991	−0.410496	−	0.911863i	$-0.634644\pi$
$31$	−141.704	−0.820991	−0.410496	+	0.911863i	$0.634644\pi$
$32$	−381.462	−2.10730
$33$	0	0
$34$	−90.2140	−0.455046
$35$	0	0
$36$	0	0
$37$	257.963	1.14619	0.573093	−	0.819490i	$-0.305743\pi$
$37$	257.963	1.14619	0.573093	+	0.819490i	$0.305743\pi$
$38$	−474.642	−2.02624
$39$	0	0
$40$	0	0
$41$	225.914	0.860533	0.430266	−	0.902702i	$-0.358420\pi$
$41$	225.914	0.860533	0.430266	+	0.902702i	$0.358420\pi$
$42$	0	0
$43$	347.445	1.23221	0.616103	−	0.787666i	$-0.288711\pi$
$43$	347.445	1.23221	0.616103	+	0.787666i	$0.288711\pi$
$44$	−216.517	−0.741844
$45$	0	0
$46$	1118.62	3.58547
$47$	404.364	1.25495	0.627473	−	0.778638i	$-0.284089\pi$
$47$	404.364	1.25495	0.627473	+	0.778638i	$0.284089\pi$
$48$	0	0
$49$	−236.483	−0.689455
$50$	0	0
$51$	0	0
$52$	−1259.37	−3.35853
$53$	259.568	0.672726	0.336363	−	0.941732i	$-0.390803\pi$
$53$	259.568	0.672726	0.336363	+	0.941732i	$0.390803\pi$
$54$	0	0
$55$	0	0
$56$	−634.433	−1.51392
$57$	0	0
$58$	302.779	0.685462
$59$	853.067	1.88237	0.941185	−	0.337891i	$-0.109713\pi$
$59$	853.067	1.88237	0.941185	+	0.337891i	$0.109713\pi$
$60$	0	0
$61$	−203.699	−0.427558	−0.213779	−	0.976882i	$-0.568577\pi$
$61$	−203.699	−0.427558	−0.213779	+	0.976882i	$0.568577\pi$
$62$	745.573	1.52722
$63$	0	0
$64$	679.320	1.32680
$65$	0	0
$66$	0	0
$67$	−266.890	−0.486653	−0.243327	−	0.969944i	$-0.578239\pi$
$67$	−266.890	−0.486653	−0.243327	+	0.969944i	$0.578239\pi$
$68$	337.492	0.601866
$69$	0	0
$70$	0	0
$71$	−92.4460	−0.154526	−0.0772629	−	0.997011i	$-0.524618\pi$
$71$	−92.4460	−0.154526	−0.0772629	+	0.997011i	$0.524618\pi$
$72$	0	0
$73$	242.026	0.388040	0.194020	−	0.980998i	$-0.437847\pi$
$73$	242.026	0.388040	0.194020	+	0.980998i	$0.437847\pi$
$74$	−1357.27	−2.13216
$75$	0	0
$76$	1775.64	2.68000
$77$	−113.528	−0.168022
$78$	0	0
$79$	−1021.60	−1.45492	−0.727460	−	0.686150i	$-0.759299\pi$
$79$	−1021.60	−1.45492	−0.727460	+	0.686150i	$0.759299\pi$
$80$	0	0
$81$	0	0
$82$	−1188.65	−1.60078
$83$	706.415	0.934206	0.467103	−	0.884203i	$-0.345298\pi$
$83$	706.415	0.934206	0.467103	+	0.884203i	$0.345298\pi$
$84$	0	0
$85$	0	0
$86$	−1828.08	−2.29217
$87$	0	0
$88$	676.191	0.819116
$89$	440.218	0.524304	0.262152	−	0.965027i	$-0.415568\pi$
$89$	440.218	0.524304	0.262152	+	0.965027i	$0.415568\pi$
$90$	0	0
$91$	−660.336	−0.760682
$92$	−4184.77	−4.74231
$93$	0	0
$94$	−2127.56	−2.33448
$95$	0	0
$96$	0	0
$97$	197.761	0.207006	0.103503	−	0.994629i	$-0.466995\pi$
$97$	197.761	0.207006	0.103503	+	0.994629i	$0.466995\pi$
$98$	1244.25	1.28254
$99$	0	0
$100$	0	0
$101$	−1400.62	−1.37987	−0.689937	−	0.723870i	$-0.742362\pi$
$101$	−1400.62	−1.37987	−0.689937	+	0.723870i	$0.742362\pi$
$102$	0	0
$103$	1345.70	1.28734	0.643669	−	0.765304i	$-0.277411\pi$
$103$	1345.70	1.28734	0.643669	+	0.765304i	$0.277411\pi$
$104$	3933.07	3.70836
$105$	0	0
$106$	−1365.72	−1.25142
$107$	−889.178	−0.803366	−0.401683	−	0.915779i	$-0.631575\pi$
$107$	−889.178	−0.803366	−0.401683	+	0.915779i	$0.631575\pi$
$108$	0	0
$109$	1256.29	1.10395	0.551974	−	0.833861i	$-0.313875\pi$
$109$	1256.29	1.10395	0.551974	+	0.833861i	$0.313875\pi$
$110$	0	0
$111$	0	0
$112$	1712.90	1.44512
$113$	−2394.01	−1.99301	−0.996504	−	0.0835448i	$-0.973376\pi$
$113$	−2394.01	−1.99301	−0.996504	+	0.0835448i	$0.973376\pi$
$114$	0	0
$115$	0	0
$116$	−1132.70	−0.906626
$117$	0	0
$118$	−4488.41	−3.50162
$119$	176.960	0.136318
$120$	0	0
$121$	121.000	0.0909091
$122$	1071.76	0.795352
$123$	0	0
$124$	−2789.20	−2.01998
$125$	0	0
$126$	0	0
$127$	−2065.57	−1.44322	−0.721612	−	0.692298i	$-0.756598\pi$
$127$	−2065.57	−1.44322	−0.721612	+	0.692298i	$0.756598\pi$
$128$	−522.548	−0.360837
$129$	0	0
$130$	0	0
$131$	−785.526	−0.523907	−0.261953	−	0.965081i	$-0.584367\pi$
$131$	−785.526	−0.523907	−0.261953	+	0.965081i	$0.584367\pi$
$132$	0	0
$133$	931.035	0.607000
$134$	1404.24	0.905282
$135$	0	0
$136$	−1054.00	−0.664558
$137$	1276.24	0.795885	0.397942	−	0.917410i	$-0.369724\pi$
$137$	1276.24	0.795885	0.397942	+	0.917410i	$0.369724\pi$
$138$	0	0
$139$	−2703.21	−1.64952	−0.824760	−	0.565482i	$-0.808690\pi$
$139$	−2703.21	−1.64952	−0.824760	+	0.565482i	$0.808690\pi$
$140$	0	0
$141$	0	0
$142$	486.405	0.287452
$143$	703.798	0.411570
$144$	0	0
$145$	0	0
$146$	−1273.42	−0.721840
$147$	0	0
$148$	5077.58	2.82010
$149$	2400.99	1.32011	0.660056	−	0.751217i	$-0.270533\pi$
$149$	2400.99	1.32011	0.660056	+	0.751217i	$0.270533\pi$
$150$	0	0
$151$	−2517.30	−1.35665	−0.678326	−	0.734761i	$-0.737294\pi$
$151$	−2517.30	−1.35665	−0.678326	+	0.734761i	$0.737294\pi$
$152$	−5545.40	−2.95916
$153$	0	0
$154$	597.326	0.312558
$155$	0	0
$156$	0	0
$157$	−1391.42	−0.707310	−0.353655	−	0.935376i	$-0.615061\pi$
$157$	−1391.42	−0.707310	−0.353655	+	0.935376i	$0.615061\pi$
$158$	5375.13	2.70647
$159$	0	0
$160$	0	0
$161$	−2194.23	−1.07410
$162$	0	0
$163$	2720.53	1.30729	0.653644	−	0.756802i	$-0.273239\pi$
$163$	2720.53	1.30729	0.653644	+	0.756802i	$0.273239\pi$
$164$	4446.74	2.11727
$165$	0	0
$166$	−3716.80	−1.73783
$167$	2950.25	1.36705	0.683525	−	0.729927i	$-0.260446\pi$
$167$	2950.25	1.36705	0.683525	+	0.729927i	$0.260446\pi$
$168$	0	0
$169$	1896.65	0.863292
$170$	0	0
$171$	0	0
$172$	6838.88	3.03174
$173$	537.049	0.236018	0.118009	−	0.993013i	$-0.462349\pi$
$173$	537.049	0.236018	0.118009	+	0.993013i	$0.462349\pi$
$174$	0	0
$175$	0	0
$176$	−1825.64	−0.781891
$177$	0	0
$178$	−2316.21	−0.975320
$179$	−2891.25	−1.20728	−0.603638	−	0.797259i	$-0.706283\pi$
$179$	−2891.25	−1.20728	−0.603638	+	0.797259i	$0.706283\pi$
$180$	0	0
$181$	435.209	0.178723	0.0893615	−	0.995999i	$-0.471517\pi$
$181$	435.209	0.178723	0.0893615	+	0.995999i	$0.471517\pi$
$182$	3474.36	1.41504
$183$	0	0
$184$	13069.2	5.23628
$185$	0	0
$186$	0	0
$187$	−188.607	−0.0737556
$188$	7959.23	3.08769
$189$	0	0
$190$	0	0
$191$	3779.49	1.43180	0.715901	−	0.698202i	$-0.246016\pi$
$191$	3779.49	1.43180	0.715901	+	0.698202i	$0.246016\pi$
$192$	0	0
$193$	−3751.91	−1.39932	−0.699660	−	0.714476i	$-0.746665\pi$
$193$	−3751.91	−1.39932	−0.699660	+	0.714476i	$0.746665\pi$
$194$	−1040.52	−0.385076
$195$	0	0
$196$	−4654.78	−1.69635
$197$	−3920.73	−1.41797	−0.708986	−	0.705223i	$-0.750847\pi$
$197$	−3920.73	−1.41797	−0.708986	+	0.705223i	$0.750847\pi$
$198$	0	0
$199$	−597.084	−0.212694	−0.106347	−	0.994329i	$-0.533915\pi$
$199$	−597.084	−0.212694	−0.106347	+	0.994329i	$0.533915\pi$
$200$	0	0
$201$	0	0
$202$	7369.37	2.56687
$203$	−593.917	−0.205344
$204$	0	0
$205$	0	0
$206$	−7080.40	−2.39473
$207$	0	0
$208$	−10618.9	−3.53984
$209$	−992.314	−0.328420
$210$	0	0
$211$	−4384.55	−1.43054	−0.715272	−	0.698846i	$-0.753697\pi$
$211$	−4384.55	−1.43054	−0.715272	+	0.698846i	$0.753697\pi$
$212$	5109.17	1.65519
$213$	0	0
$214$	4678.41	1.49444
$215$	0	0
$216$	0	0
$217$	−1462.48	−0.457510
$218$	−6609.95	−2.05359
$219$	0	0
$220$	0	0
$221$	−1097.03	−0.333912
$222$	0	0
$223$	2333.03	0.700587	0.350294	−	0.936640i	$-0.386082\pi$
$223$	2333.03	0.700587	0.350294	+	0.936640i	$0.386082\pi$
$224$	−3936.95	−1.17433
$225$	0	0
$226$	12596.1	3.70743
$227$	−2120.00	−0.619864	−0.309932	−	0.950759i	$-0.600306\pi$
$227$	−2120.00	−0.619864	−0.309932	+	0.950759i	$0.600306\pi$
$228$	0	0
$229$	2347.12	0.677301	0.338651	−	0.940912i	$-0.390030\pi$
$229$	2347.12	0.677301	0.338651	+	0.940912i	$0.390030\pi$
$230$	0	0
$231$	0	0
$232$	3537.47	1.00106
$233$	−375.499	−0.105578	−0.0527891	−	0.998606i	$-0.516811\pi$
$233$	−375.499	−0.105578	−0.0527891	+	0.998606i	$0.516811\pi$
$234$	0	0
$235$	0	0
$236$	16791.2	4.63142
$237$	0	0
$238$	−931.072	−0.253582
$239$	1428.15	0.386524	0.193262	−	0.981147i	$-0.438093\pi$
$239$	1428.15	0.386524	0.193262	+	0.981147i	$0.438093\pi$
$240$	0	0
$241$	190.819	0.0510032	0.0255016	−	0.999675i	$-0.491882\pi$
$241$	190.819	0.0510032	0.0255016	+	0.999675i	$0.491882\pi$
$242$	−636.641	−0.169111
$243$	0	0
$244$	−4009.49	−1.05197
$245$	0	0
$246$	0	0
$247$	−5771.81	−1.48685
$248$	8710.79	2.23039
$249$	0	0
$250$	0	0
$251$	6294.80	1.58297	0.791483	−	0.611191i	$-0.209309\pi$
$251$	6294.80	1.58297	0.791483	+	0.611191i	$0.209309\pi$
$252$	0	0
$253$	2338.65	0.581145
$254$	10868.0	2.68471
$255$	0	0
$256$	−2685.18	−0.655561
$257$	4459.44	1.08238	0.541191	−	0.840900i	$-0.317974\pi$
$257$	4459.44	1.08238	0.541191	+	0.840900i	$0.317974\pi$
$258$	0	0
$259$	2662.36	0.638731
$260$	0	0
$261$	0	0
$262$	4133.04	0.974581
$263$	−4416.65	−1.03552	−0.517761	−	0.855525i	$-0.673234\pi$
$263$	−4416.65	−1.03552	−0.517761	+	0.855525i	$0.673234\pi$
$264$	0	0
$265$	0	0
$266$	−4898.64	−1.12915
$267$	0	0
$268$	−5253.28	−1.19737
$269$	−1914.86	−0.434020	−0.217010	−	0.976169i	$-0.569630\pi$
$269$	−1914.86	−0.434020	−0.217010	+	0.976169i	$0.569630\pi$
$270$	0	0
$271$	6088.34	1.36472	0.682362	−	0.731014i	$-0.260953\pi$
$271$	6088.34	1.36472	0.682362	+	0.731014i	$0.260953\pi$
$272$	2845.69	0.634357
$273$	0	0
$274$	−6714.91	−1.48052
$275$	0	0
$276$	0	0
$277$	832.321	0.180539	0.0902696	−	0.995917i	$-0.471227\pi$
$277$	832.321	0.180539	0.0902696	+	0.995917i	$0.471227\pi$
$278$	14222.9	3.06847
$279$	0	0
$280$	0	0
$281$	2545.32	0.540360	0.270180	−	0.962810i	$-0.412917\pi$
$281$	2545.32	0.540360	0.270180	+	0.962810i	$0.412917\pi$
$282$	0	0
$283$	5911.71	1.24175	0.620874	−	0.783911i	$-0.286778\pi$
$283$	5911.71	1.24175	0.620874	+	0.783911i	$0.286778\pi$
$284$	−1819.65	−0.380198
$285$	0	0
$286$	−3703.03	−0.765611
$287$	2331.59	0.479545
$288$	0	0
$289$	−4619.01	−0.940161
$290$	0	0
$291$	0	0
$292$	4763.87	0.954742
$293$	−6871.03	−1.37000	−0.685000	−	0.728543i	$-0.740198\pi$
$293$	−6871.03	−1.37000	−0.685000	+	0.728543i	$0.740198\pi$
$294$	0	0
$295$	0	0
$296$	−15857.5	−3.11384
$297$	0	0
$298$	−12632.8	−2.45570
$299$	13602.8	2.63100
$300$	0	0
$301$	3585.88	0.686666
$302$	13244.7	2.52367
$303$	0	0
$304$	14972.0	2.82468
$305$	0	0
$306$	0	0
$307$	−200.179	−0.0372144	−0.0186072	−	0.999827i	$-0.505923\pi$
$307$	−200.179	−0.0372144	−0.0186072	+	0.999827i	$0.505923\pi$
$308$	−2234.61	−0.413404
$309$	0	0
$310$	0	0
$311$	−5734.93	−1.04565	−0.522827	−	0.852439i	$-0.675122\pi$
$311$	−5734.93	−1.04565	−0.522827	+	0.852439i	$0.675122\pi$
$312$	0	0
$313$	3077.36	0.555727	0.277864	−	0.960621i	$-0.410374\pi$
$313$	3077.36	0.555727	0.277864	+	0.960621i	$0.410374\pi$
$314$	7320.97	1.31575
$315$	0	0
$316$	−20108.5	−3.57971
$317$	2142.38	0.379584	0.189792	−	0.981824i	$-0.439219\pi$
$317$	2142.38	0.379584	0.189792	+	0.981824i	$0.439219\pi$
$318$	0	0
$319$	633.007	0.111102
$320$	0	0
$321$	0	0
$322$	11544.9	1.99806
$323$	1546.75	0.266451
$324$	0	0
$325$	0	0
$326$	−14314.0	−2.43184
$327$	0	0
$328$	−13887.4	−2.33781
$329$	4173.32	0.699339
$330$	0	0
$331$	−1618.23	−0.268719	−0.134359	−	0.990933i	$-0.542898\pi$
$331$	−1618.23	−0.268719	−0.134359	+	0.990933i	$0.542898\pi$
$332$	13904.6	2.29854
$333$	0	0
$334$	−15522.7	−2.54301
$335$	0	0
$336$	0	0
$337$	−2406.47	−0.388988	−0.194494	−	0.980904i	$-0.562306\pi$
$337$	−2406.47	−0.388988	−0.194494	+	0.980904i	$0.562306\pi$
$338$	−9979.23	−1.60591
$339$	0	0
$340$	0	0
$341$	1558.74	0.247538
$342$	0	0
$343$	−5980.67	−0.941475
$344$	−21358.1	−3.34753
$345$	0	0
$346$	−2825.68	−0.439045
$347$	−6612.89	−1.02305	−0.511525	−	0.859268i	$-0.670919\pi$
$347$	−6612.89	−1.02305	−0.511525	+	0.859268i	$0.670919\pi$
$348$	0	0
$349$	349.871	0.0536623	0.0268311	−	0.999640i	$-0.491458\pi$
$349$	349.871	0.0536623	0.0268311	+	0.999640i	$0.491458\pi$
$350$	0	0
$351$	0	0
$352$	4196.08	0.635374
$353$	1723.29	0.259835	0.129917	−	0.991525i	$-0.458529\pi$
$353$	1723.29	0.259835	0.129917	+	0.991525i	$0.458529\pi$
$354$	0	0
$355$	0	0
$356$	8664.97	1.29001
$357$	0	0
$358$	15212.3	2.24580
$359$	−5875.74	−0.863816	−0.431908	−	0.901918i	$-0.642159\pi$
$359$	−5875.74	−0.863816	−0.431908	+	0.901918i	$0.642159\pi$
$360$	0	0
$361$	1278.92	0.186458
$362$	−2289.85	−0.332464
$363$	0	0
$364$	−12997.6	−1.87159
$365$	0	0
$366$	0	0
$367$	5368.28	0.763548	0.381774	−	0.924256i	$-0.375313\pi$
$367$	5368.28	0.763548	0.381774	+	0.924256i	$0.375313\pi$
$368$	−35285.5	−4.99832
$369$	0	0
$370$	0	0
$371$	2678.93	0.374887
$372$	0	0
$373$	−10393.9	−1.44282	−0.721412	−	0.692506i	$-0.756507\pi$
$373$	−10393.9	−1.44282	−0.721412	+	0.692506i	$0.756507\pi$
$374$	992.354	0.137202
$375$	0	0
$376$	−24857.0	−3.40931
$377$	3681.90	0.502990
$378$	0	0
$379$	10918.9	1.47986	0.739928	−	0.672686i	$-0.234860\pi$
$379$	10918.9	1.47986	0.739928	+	0.672686i	$0.234860\pi$
$380$	0	0
$381$	0	0
$382$	−19885.8	−2.66347
$383$	−11663.6	−1.55609	−0.778044	−	0.628210i	$-0.783788\pi$
$383$	−11663.6	−1.55609	−0.778044	+	0.628210i	$0.783788\pi$
$384$	0	0
$385$	0	0
$386$	19740.7	2.60304
$387$	0	0
$388$	3892.59	0.509321
$389$	5827.00	0.759487	0.379744	−	0.925092i	$-0.376012\pi$
$389$	5827.00	0.759487	0.379744	+	0.925092i	$0.376012\pi$
$390$	0	0
$391$	−3645.34	−0.471490
$392$	14537.1	1.87304
$393$	0	0
$394$	20628.9	2.63774
$395$	0	0
$396$	0	0
$397$	−7366.99	−0.931332	−0.465666	−	0.884961i	$-0.654185\pi$
$397$	−7366.99	−0.931332	−0.465666	+	0.884961i	$0.654185\pi$
$398$	3141.55	0.395658
$399$	0	0
$400$	0	0
$401$	−14604.2	−1.81870	−0.909349	−	0.416035i	$-0.863419\pi$
$401$	−14604.2	−1.81870	−0.909349	+	0.416035i	$0.863419\pi$
$402$	0	0
$403$	9066.43	1.12067
$404$	−27569.0	−3.39507
$405$	0	0
$406$	3124.89	0.381985
$407$	−2837.60	−0.345588
$408$	0	0
$409$	12581.2	1.52103	0.760515	−	0.649320i	$-0.224946\pi$
$409$	12581.2	1.52103	0.760515	+	0.649320i	$0.224946\pi$
$410$	0	0
$411$	0	0
$412$	26487.9	3.16739
$413$	8804.26	1.04898
$414$	0	0
$415$	0	0
$416$	24406.6	2.87651
$417$	0	0
$418$	5221.06	0.610934
$419$	3776.01	0.440263	0.220131	−	0.975470i	$-0.429351\pi$
$419$	3776.01	0.440263	0.220131	+	0.975470i	$0.429351\pi$
$420$	0	0
$421$	12683.4	1.46830	0.734148	−	0.678989i	$-0.237582\pi$
$421$	12683.4	1.46830	0.734148	+	0.678989i	$0.237582\pi$
$422$	23069.3	2.66113
$423$	0	0
$424$	−15956.2	−1.82759
$425$	0	0
$426$	0	0
$427$	−2102.32	−0.238263
$428$	−17502.0	−1.97662
$429$	0	0
$430$	0	0
$431$	−14152.6	−1.58168	−0.790841	−	0.612022i	$-0.790356\pi$
$431$	−14152.6	−1.58168	−0.790841	+	0.612022i	$0.790356\pi$
$432$	0	0
$433$	10950.2	1.21532	0.607661	−	0.794197i	$-0.292108\pi$
$433$	10950.2	1.21532	0.607661	+	0.794197i	$0.292108\pi$
$434$	7694.84	0.851070
$435$	0	0
$436$	24727.9	2.71618
$437$	−19179.2	−2.09946
$438$	0	0
$439$	−11221.0	−1.21993	−0.609964	−	0.792429i	$-0.708816\pi$
$439$	−11221.0	−1.21993	−0.609964	+	0.792429i	$0.708816\pi$
$440$	0	0
$441$	0	0
$442$	5772.04	0.621149
$443$	9647.11	1.03465	0.517323	−	0.855790i	$-0.326929\pi$
$443$	9647.11	1.03465	0.517323	+	0.855790i	$0.326929\pi$
$444$	0	0
$445$	0	0
$446$	−12275.2	−1.30325
$447$	0	0
$448$	7011.07	0.739379
$449$	−6482.03	−0.681305	−0.340652	−	0.940189i	$-0.610648\pi$
$449$	−6482.03	−0.681305	−0.340652	+	0.940189i	$0.610648\pi$
$450$	0	0
$451$	−2485.05	−0.259460
$452$	−47122.2	−4.90363
$453$	0	0
$454$	11154.4	1.15308
$455$	0	0
$456$	0	0
$457$	−11319.8	−1.15868	−0.579342	−	0.815085i	$-0.696690\pi$
$457$	−11319.8	−1.15868	−0.579342	+	0.815085i	$0.696690\pi$
$458$	−12349.4	−1.25993
$459$	0	0
$460$	0	0
$461$	8406.73	0.849329	0.424664	−	0.905351i	$-0.360392\pi$
$461$	8406.73	0.849329	0.424664	+	0.905351i	$0.360392\pi$
$462$	0	0
$463$	−9758.56	−0.979523	−0.489761	−	0.871857i	$-0.662916\pi$
$463$	−9758.56	−0.979523	−0.489761	+	0.871857i	$0.662916\pi$
$464$	−9550.78	−0.955568
$465$	0	0
$466$	1975.68	0.196399
$467$	−16388.7	−1.62394	−0.811969	−	0.583701i	$-0.801604\pi$
$467$	−16388.7	−1.62394	−0.811969	+	0.583701i	$0.801604\pi$
$468$	0	0
$469$	−2754.49	−0.271195
$470$	0	0
$471$	0	0
$472$	−52439.6	−5.11384
$473$	−3821.89	−0.371524
$474$	0	0
$475$	0	0
$476$	3483.16	0.335400
$477$	0	0
$478$	−7514.20	−0.719020
$479$	−13829.0	−1.31913	−0.659567	−	0.751646i	$-0.729260\pi$
$479$	−13829.0	−1.31913	−0.659567	+	0.751646i	$0.729260\pi$
$480$	0	0
$481$	−16504.9	−1.56457
$482$	−1004.00	−0.0948771
$483$	0	0
$484$	2381.68	0.223674
$485$	0	0
$486$	0	0
$487$	−13264.4	−1.23423	−0.617113	−	0.786875i	$-0.711698\pi$
$487$	−13264.4	−1.23423	−0.617113	+	0.786875i	$0.711698\pi$
$488$	12521.8	1.16155
$489$	0	0
$490$	0	0
$491$	7468.22	0.686428	0.343214	−	0.939257i	$-0.388484\pi$
$491$	7468.22	0.686428	0.343214	+	0.939257i	$0.388484\pi$
$492$	0	0
$493$	−986.690	−0.0901385
$494$	30368.4	2.76586
$495$	0	0
$496$	−23518.2	−2.12903
$497$	−954.109	−0.0861119
$498$	0	0
$499$	−5276.64	−0.473377	−0.236688	−	0.971586i	$-0.576062\pi$
$499$	−5276.64	−0.473377	−0.236688	+	0.971586i	$0.576062\pi$
$500$	0	0
$501$	0	0
$502$	−33120.1	−2.94466
$503$	−10956.8	−0.971253	−0.485626	−	0.874166i	$-0.661408\pi$
$503$	−10956.8	−0.971253	−0.485626	+	0.874166i	$0.661408\pi$
$504$	0	0
$505$	0	0
$506$	−12304.8	−1.08106
$507$	0	0
$508$	−40657.3	−3.55093
$509$	−12734.4	−1.10892	−0.554462	−	0.832209i	$-0.687076\pi$
$509$	−12734.4	−1.10892	−0.554462	+	0.832209i	$0.687076\pi$
$510$	0	0
$511$	2497.87	0.216242
$512$	18308.4	1.58032
$513$	0	0
$514$	−23463.3	−2.01347
$515$	0	0
$516$	0	0
$517$	−4448.00	−0.378381
$518$	−14008.0	−1.18818
$519$	0	0
$520$	0	0
$521$	5650.70	0.475166	0.237583	−	0.971367i	$-0.423645\pi$
$521$	5650.70	0.475166	0.237583	+	0.971367i	$0.423645\pi$
$522$	0	0
$523$	−14103.2	−1.17914	−0.589572	−	0.807716i	$-0.700703\pi$
$523$	−14103.2	−1.17914	−0.589572	+	0.807716i	$0.700703\pi$
$524$	−15461.8	−1.28903
$525$	0	0
$526$	23238.2	1.92630
$527$	−2429.66	−0.200830
$528$	0	0
$529$	33033.8	2.71503
$530$	0	0
$531$	0	0
$532$	18325.9	1.49347
$533$	−14454.4	−1.17465
$534$	0	0
$535$	0	0
$536$	16406.2	1.32209
$537$	0	0
$538$	10075.0	0.807372
$539$	2601.31	0.207878
$540$	0	0
$541$	−6391.90	−0.507965	−0.253983	−	0.967209i	$-0.581741\pi$
$541$	−6391.90	−0.507965	−0.253983	+	0.967209i	$0.581741\pi$
$542$	−32033.8	−2.53869
$543$	0	0
$544$	−6540.57	−0.515486
$545$	0	0
$546$	0	0
$547$	−20786.7	−1.62482	−0.812409	−	0.583088i	$-0.801844\pi$
$547$	−20786.7	−1.62482	−0.812409	+	0.583088i	$0.801844\pi$
$548$	25120.6	1.95821
$549$	0	0
$550$	0	0
$551$	−5191.26	−0.401370
$552$	0	0
$553$	−10543.6	−0.810777
$554$	−4379.26	−0.335843
$555$	0	0
$556$	−53208.2	−4.05851
$557$	−15125.9	−1.15064	−0.575320	−	0.817928i	$-0.695123\pi$
$557$	−15125.9	−1.15064	−0.575320	+	0.817928i	$0.695123\pi$
$558$	0	0
$559$	−22230.1	−1.68199
$560$	0	0
$561$	0	0
$562$	−13392.2	−1.00519
$563$	8706.42	0.651744	0.325872	−	0.945414i	$-0.394342\pi$
$563$	8706.42	0.651744	0.325872	+	0.945414i	$0.394342\pi$
$564$	0	0
$565$	0	0
$566$	−31104.4	−2.30992
$567$	0	0
$568$	5682.83	0.419800
$569$	−7067.76	−0.520731	−0.260366	−	0.965510i	$-0.583843\pi$
$569$	−7067.76	−0.520731	−0.260366	+	0.965510i	$0.583843\pi$
$570$	0	0
$571$	−3326.42	−0.243794	−0.121897	−	0.992543i	$-0.538898\pi$
$571$	−3326.42	−0.243794	−0.121897	+	0.992543i	$0.538898\pi$
$572$	13853.1	1.01264
$573$	0	0
$574$	−12267.7	−0.892060
$575$	0	0
$576$	0	0
$577$	3308.06	0.238676	0.119338	−	0.992854i	$-0.461923\pi$
$577$	3308.06	0.238676	0.119338	+	0.992854i	$0.461923\pi$
$578$	24302.9	1.74891
$579$	0	0
$580$	0	0
$581$	7290.70	0.520601
$582$	0	0
$583$	−2855.25	−0.202834
$584$	−14877.8	−1.05419
$585$	0	0
$586$	36151.9	2.54850
$587$	−5694.88	−0.400431	−0.200215	−	0.979752i	$-0.564164\pi$
$587$	−5694.88	−0.400431	−0.200215	+	0.979752i	$0.564164\pi$
$588$	0	0
$589$	−12783.1	−0.894262
$590$	0	0
$591$	0	0
$592$	42813.5	2.97234
$593$	−3907.69	−0.270606	−0.135303	−	0.990804i	$-0.543201\pi$
$593$	−3907.69	−0.270606	−0.135303	+	0.990804i	$0.543201\pi$
$594$	0	0
$595$	0	0
$596$	47259.5	3.24803
$597$	0	0
$598$	−71571.1	−4.89425
$599$	−10727.2	−0.731720	−0.365860	−	0.930670i	$-0.619225\pi$
$599$	−10727.2	−0.731720	−0.365860	+	0.930670i	$0.619225\pi$
$600$	0	0
$601$	3348.98	0.227301	0.113650	−	0.993521i	$-0.463746\pi$
$601$	3348.98	0.227301	0.113650	+	0.993521i	$0.463746\pi$
$602$	−18867.1	−1.27735
$603$	0	0
$604$	−49548.8	−3.33793
$605$	0	0
$606$	0	0
$607$	21539.9	1.44032	0.720162	−	0.693806i	$-0.244068\pi$
$607$	21539.9	1.44032	0.720162	+	0.693806i	$0.244068\pi$
$608$	−34411.8	−2.29537
$609$	0	0
$610$	0	0
$611$	−25871.9	−1.71303
$612$	0	0
$613$	9284.33	0.611730	0.305865	−	0.952075i	$-0.401054\pi$
$613$	9284.33	0.611730	0.305865	+	0.952075i	$0.401054\pi$
$614$	1053.24	0.0692270
$615$	0	0
$616$	6978.77	0.456465
$617$	−20711.3	−1.35139	−0.675695	−	0.737181i	$-0.736156\pi$
$617$	−20711.3	−1.35139	−0.675695	+	0.737181i	$0.736156\pi$
$618$	0	0
$619$	−13282.9	−0.862496	−0.431248	−	0.902233i	$-0.641927\pi$
$619$	−13282.9	−0.862496	−0.431248	+	0.902233i	$0.641927\pi$
$620$	0	0
$621$	0	0
$622$	30174.3	1.94515
$623$	4543.36	0.292177
$624$	0	0
$625$	0	0
$626$	−16191.5	−1.03378
$627$	0	0
$628$	−27387.9	−1.74028
$629$	4423.06	0.280380
$630$	0	0
$631$	−14789.9	−0.933086	−0.466543	−	0.884498i	$-0.654501\pi$
$631$	−14789.9	−0.933086	−0.466543	+	0.884498i	$0.654501\pi$
$632$	62799.5	3.95258
$633$	0	0
$634$	−11272.1	−0.706109
$635$	0	0
$636$	0	0
$637$	15130.6	0.941123
$638$	−3330.57	−0.206675
$639$	0	0
$640$	0	0
$641$	5808.52	0.357914	0.178957	−	0.983857i	$-0.442728\pi$
$641$	5808.52	0.357914	0.178957	+	0.983857i	$0.442728\pi$
$642$	0	0
$643$	18891.2	1.15862	0.579311	−	0.815106i	$-0.303322\pi$
$643$	18891.2	1.15862	0.579311	+	0.815106i	$0.303322\pi$
$644$	−43189.8	−2.64273
$645$	0	0
$646$	−8138.24	−0.495657
$647$	−243.046	−0.0147684	−0.00738418	−	0.999973i	$-0.502350\pi$
$647$	−243.046	−0.0147684	−0.00738418	+	0.999973i	$0.502350\pi$
$648$	0	0
$649$	−9383.74	−0.567556
$650$	0	0
$651$	0	0
$652$	53549.0	3.21648
$653$	−15920.4	−0.954081	−0.477041	−	0.878881i	$-0.658291\pi$
$653$	−15920.4	−0.954081	−0.477041	+	0.878881i	$0.658291\pi$
$654$	0	0
$655$	0	0
$656$	37494.4	2.23157
$657$	0	0
$658$	−21957.9	−1.30092
$659$	7476.38	0.441940	0.220970	−	0.975281i	$-0.429078\pi$
$659$	7476.38	0.441940	0.220970	+	0.975281i	$0.429078\pi$
$660$	0	0
$661$	−14920.1	−0.877948	−0.438974	−	0.898500i	$-0.644658\pi$
$661$	−14920.1	−0.877948	−0.438974	+	0.898500i	$0.644658\pi$
$662$	8514.30	0.499876
$663$	0	0
$664$	−43424.7	−2.53796
$665$	0	0
$666$	0	0
$667$	12234.6	0.710232
$668$	58070.8	3.36351
$669$	0	0
$670$	0	0
$671$	2240.69	0.128914
$672$	0	0
$673$	−563.692	−0.0322864	−0.0161432	−	0.999870i	$-0.505139\pi$
$673$	−563.692	−0.0322864	−0.0161432	+	0.999870i	$0.505139\pi$
$674$	12661.6	0.723602
$675$	0	0
$676$	37332.5	2.12406
$677$	13280.2	0.753914	0.376957	−	0.926231i	$-0.376970\pi$
$677$	13280.2	0.753914	0.376957	+	0.926231i	$0.376970\pi$
$678$	0	0
$679$	2041.03	0.115357
$680$	0	0
$681$	0	0
$682$	−8201.30	−0.460475
$683$	6856.80	0.384141	0.192070	−	0.981381i	$-0.438480\pi$
$683$	6856.80	0.384141	0.192070	+	0.981381i	$0.438480\pi$
$684$	0	0
$685$	0	0
$686$	31467.3	1.75135
$687$	0	0
$688$	57664.5	3.19541
$689$	−16607.6	−0.918287
$690$	0	0
$691$	−28374.6	−1.56211	−0.781057	−	0.624460i	$-0.785319\pi$
$691$	−28374.6	−1.56211	−0.781057	+	0.624460i	$0.785319\pi$
$692$	10570.9	0.580702
$693$	0	0
$694$	34793.7	1.90310
$695$	0	0
$696$	0	0
$697$	3873.54	0.210503
$698$	−1840.84	−0.0998237
$699$	0	0
$700$	0	0
$701$	−667.753	−0.0359781	−0.0179891	−	0.999838i	$-0.505726\pi$
$701$	−667.753	−0.0359781	−0.0179891	+	0.999838i	$0.505726\pi$
$702$	0	0
$703$	23271.0	1.24848
$704$	−7472.52	−0.400044
$705$	0	0
$706$	−9067.10	−0.483349
$707$	−14455.4	−0.768956
$708$	0	0
$709$	23667.8	1.25368	0.626842	−	0.779147i	$-0.284347\pi$
$709$	23667.8	1.25368	0.626842	+	0.779147i	$0.284347\pi$
$710$	0	0
$711$	0	0
$712$	−27061.0	−1.42438
$713$	30126.9	1.58241
$714$	0	0
$715$	0	0
$716$	−56909.5	−2.97040
$717$	0	0
$718$	30915.2	1.60689
$719$	−11835.5	−0.613896	−0.306948	−	0.951726i	$-0.599308\pi$
$719$	−11835.5	−0.613896	−0.306948	+	0.951726i	$0.599308\pi$
$720$	0	0
$721$	13888.6	0.717390
$722$	−6729.01	−0.346853
$723$	0	0
$724$	8566.38	0.439733
$725$	0	0
$726$	0	0
$727$	−15633.2	−0.797530	−0.398765	−	0.917053i	$-0.630561\pi$
$727$	−15633.2	−0.797530	−0.398765	+	0.917053i	$0.630561\pi$
$728$	40592.1	2.06654
$729$	0	0
$730$	0	0
$731$	5957.31	0.301422
$732$	0	0
$733$	14870.7	0.749335	0.374668	−	0.927159i	$-0.377757\pi$
$733$	14870.7	0.749335	0.374668	+	0.927159i	$0.377757\pi$
$734$	−28245.2	−1.42037
$735$	0	0
$736$	81100.6	4.06169
$737$	2935.79	0.146731
$738$	0	0
$739$	20850.3	1.03788	0.518939	−	0.854812i	$-0.326327\pi$
$739$	20850.3	1.03788	0.518939	+	0.854812i	$0.326327\pi$
$740$	0	0
$741$	0	0
$742$	−14095.2	−0.697372
$743$	−29254.7	−1.44448	−0.722242	−	0.691641i	$-0.756888\pi$
$743$	−29254.7	−1.44448	−0.722242	+	0.691641i	$0.756888\pi$
$744$	0	0
$745$	0	0
$746$	54687.2	2.68397
$747$	0	0
$748$	−3712.41	−0.181470
$749$	−9176.95	−0.447688
$750$	0	0
$751$	10936.8	0.531411	0.265705	−	0.964054i	$-0.414395\pi$
$751$	10936.8	0.531411	0.265705	+	0.964054i	$0.414395\pi$
$752$	67111.2	3.25438
$753$	0	0
$754$	−19372.3	−0.935672
$755$	0	0
$756$	0	0
$757$	8476.34	0.406972	0.203486	−	0.979078i	$-0.434773\pi$
$757$	8476.34	0.406972	0.203486	+	0.979078i	$0.434773\pi$
$758$	−57449.6	−2.75286
$759$	0	0
$760$	0	0
$761$	−913.964	−0.0435364	−0.0217682	−	0.999763i	$-0.506930\pi$
$761$	−913.964	−0.0435364	−0.0217682	+	0.999763i	$0.506930\pi$
$762$	0	0
$763$	12965.8	0.615193
$764$	74393.0	3.52283
$765$	0	0
$766$	61367.9	2.89466
$767$	−54580.6	−2.56948
$768$	0	0
$769$	32215.2	1.51067	0.755337	−	0.655337i	$-0.227473\pi$
$769$	32215.2	1.51067	0.755337	+	0.655337i	$0.227473\pi$
$770$	0	0
$771$	0	0
$772$	−73850.2	−3.44291
$773$	72.6900	0.00338225	0.00169112	−	0.999999i	$-0.499462\pi$
$773$	72.6900	0.00338225	0.00169112	+	0.999999i	$0.499462\pi$
$774$	0	0
$775$	0	0
$776$	−12156.7	−0.562373
$777$	0	0
$778$	−30658.7	−1.41281
$779$	20379.8	0.937332
$780$	0	0
$781$	1016.91	0.0465913
$782$	19179.9	0.877075
$783$	0	0
$784$	−39248.5	−1.78792
$785$	0	0
$786$	0	0
$787$	−487.318	−0.0220724	−0.0110362	−	0.999939i	$-0.503513\pi$
$787$	−487.318	−0.0220724	−0.0110362	+	0.999939i	$0.503513\pi$
$788$	−77173.1	−3.48880
$789$	0	0
$790$	0	0
$791$	−24707.9	−1.11064
$792$	0	0
$793$	13033.0	0.583627
$794$	38761.4	1.73248
$795$	0	0
$796$	−11752.6	−0.523317
$797$	31379.9	1.39465	0.697324	−	0.716756i	$-0.254374\pi$
$797$	31379.9	1.39465	0.697324	+	0.716756i	$0.254374\pi$
$798$	0	0
$799$	6933.25	0.306985
$800$	0	0
$801$	0	0
$802$	76839.8	3.38318
$803$	−2662.28	−0.116999
$804$	0	0
$805$	0	0
$806$	−47703.0	−2.08470
$807$	0	0
$808$	86099.0	3.74870
$809$	−1824.26	−0.0792802	−0.0396401	−	0.999214i	$-0.512621\pi$
$809$	−1824.26	−0.0792802	−0.0396401	+	0.999214i	$0.512621\pi$
$810$	0	0
$811$	−4364.52	−0.188976	−0.0944878	−	0.995526i	$-0.530121\pi$
$811$	−4364.52	−0.188976	−0.0944878	+	0.995526i	$0.530121\pi$
$812$	−11690.3	−0.505232
$813$	0	0
$814$	14930.0	0.642870
$815$	0	0
$816$	0	0
$817$	31343.1	1.34218
$818$	−66196.1	−2.82945
$819$	0	0
$820$	0	0
$821$	−3306.16	−0.140543	−0.0702714	−	0.997528i	$-0.522387\pi$
$821$	−3306.16	−0.140543	−0.0702714	+	0.997528i	$0.522387\pi$
$822$	0	0
$823$	19183.7	0.812519	0.406259	−	0.913758i	$-0.366833\pi$
$823$	19183.7	0.812519	0.406259	+	0.913758i	$0.366833\pi$
$824$	−82722.8	−3.49731
$825$	0	0
$826$	−46323.6	−1.95134
$827$	20646.4	0.868131	0.434066	−	0.900881i	$-0.357079\pi$
$827$	20646.4	0.868131	0.434066	+	0.900881i	$0.357079\pi$
$828$	0	0
$829$	5345.44	0.223950	0.111975	−	0.993711i	$-0.464282\pi$
$829$	5345.44	0.223950	0.111975	+	0.993711i	$0.464282\pi$
$830$	0	0
$831$	0	0
$832$	−43464.0	−1.81111
$833$	−4054.75	−0.168654
$834$	0	0
$835$	0	0
$836$	−19532.1	−0.808051
$837$	0	0
$838$	−19867.5	−0.818985
$839$	−29284.9	−1.20504	−0.602519	−	0.798105i	$-0.705836\pi$
$839$	−29284.9	−1.20504	−0.602519	+	0.798105i	$0.705836\pi$
$840$	0	0
$841$	−21077.4	−0.864219
$842$	−66733.8	−2.73135
$843$	0	0
$844$	−86302.6	−3.51974
$845$	0	0
$846$	0	0
$847$	1248.81	0.0506605
$848$	43079.9	1.74454
$849$	0	0
$850$	0	0
$851$	−54844.2	−2.20921
$852$	0	0
$853$	8070.62	0.323954	0.161977	−	0.986795i	$-0.448213\pi$
$853$	8070.62	0.323954	0.161977	+	0.986795i	$0.448213\pi$
$854$	11061.4	0.443222
$855$	0	0
$856$	54659.5	2.18250
$857$	−11344.2	−0.452169	−0.226085	−	0.974108i	$-0.572593\pi$
$857$	−11344.2	−0.452169	−0.226085	+	0.974108i	$0.572593\pi$
$858$	0	0
$859$	25470.6	1.01170	0.505848	−	0.862623i	$-0.331180\pi$
$859$	25470.6	1.01170	0.505848	+	0.862623i	$0.331180\pi$
$860$	0	0
$861$	0	0
$862$	74463.6	2.94228
$863$	−14558.4	−0.574243	−0.287122	−	0.957894i	$-0.592698\pi$
$863$	−14558.4	−0.574243	−0.287122	+	0.957894i	$0.592698\pi$
$864$	0	0
$865$	0	0
$866$	−57614.6	−2.26077
$867$	0	0
$868$	−28786.5	−1.12567
$869$	11237.6	0.438675
$870$	0	0
$871$	17076.0	0.664294
$872$	−77226.3	−2.99910
$873$	0	0
$874$	100911.	3.90546
$875$	0	0
$876$	0	0
$877$	−185.528	−0.00714350	−0.00357175	−	0.999994i	$-0.501137\pi$
$877$	−185.528	−0.00714350	−0.00357175	+	0.999994i	$0.501137\pi$
$878$	59039.1	2.26933
$879$	0	0
$880$	0	0
$881$	−18950.1	−0.724681	−0.362340	−	0.932046i	$-0.618022\pi$
$881$	−18950.1	−0.724681	−0.362340	+	0.932046i	$0.618022\pi$
$882$	0	0
$883$	−21258.3	−0.810189	−0.405095	−	0.914275i	$-0.632761\pi$
$883$	−21258.3	−0.810189	−0.405095	+	0.914275i	$0.632761\pi$
$884$	−21593.3	−0.821563
$885$	0	0
$886$	−50758.2	−1.92467
$887$	35707.4	1.35168	0.675839	−	0.737049i	$-0.263781\pi$
$887$	35707.4	1.35168	0.675839	+	0.737049i	$0.263781\pi$
$888$	0	0
$889$	−21318.1	−0.804259
$890$	0	0
$891$	0	0
$892$	45921.8	1.72374
$893$	36477.8	1.36695
$894$	0	0
$895$	0	0
$896$	−5393.06	−0.201082
$897$	0	0
$898$	34105.2	1.26738
$899$	8154.49	0.302522
$900$	0	0
$901$	4450.58	0.164562
$902$	13075.1	0.482653
$903$	0	0
$904$	147165.	5.41440
$905$	0	0
$906$	0	0
$907$	6542.52	0.239516	0.119758	−	0.992803i	$-0.461788\pi$
$907$	6542.52	0.239516	0.119758	+	0.992803i	$0.461788\pi$
$908$	−41728.6	−1.52512
$909$	0	0
$910$	0	0
$911$	−31171.2	−1.13364	−0.566821	−	0.823841i	$-0.691827\pi$
$911$	−31171.2	−1.13364	−0.566821	+	0.823841i	$0.691827\pi$
$912$	0	0
$913$	−7770.56	−0.281674
$914$	59559.2	2.15541
$915$	0	0
$916$	46199.2	1.66644
$917$	−8107.19	−0.291955
$918$	0	0
$919$	12031.9	0.431877	0.215938	−	0.976407i	$-0.430719\pi$
$919$	12031.9	0.431877	0.215938	+	0.976407i	$0.430719\pi$
$920$	0	0
$921$	0	0
$922$	−44232.0	−1.57994
$923$	5914.85	0.210931
$924$	0	0
$925$	0	0
$926$	51344.7	1.82213
$927$	0	0
$928$	21951.6	0.776506
$929$	12546.2	0.443085	0.221542	−	0.975151i	$-0.428891\pi$
$929$	12546.2	0.443085	0.221542	+	0.975151i	$0.428891\pi$
$930$	0	0
$931$	−21333.2	−0.750986
$932$	−7391.07	−0.259767
$933$	0	0
$934$	86229.1	3.02088
$935$	0	0
$936$	0	0
$937$	−17909.8	−0.624427	−0.312214	−	0.950012i	$-0.601070\pi$
$937$	−17909.8	−0.624427	−0.312214	+	0.950012i	$0.601070\pi$
$938$	14492.7	0.504483
$939$	0	0
$940$	0	0
$941$	−829.893	−0.0287500	−0.0143750	−	0.999897i	$-0.504576\pi$
$941$	−829.893	−0.0287500	−0.0143750	+	0.999897i	$0.504576\pi$
$942$	0	0
$943$	−48030.4	−1.65863
$944$	141581.	4.88144
$945$	0	0
$946$	20108.9	0.691116
$947$	−17654.9	−0.605814	−0.302907	−	0.953020i	$-0.597957\pi$
$947$	−17654.9	−0.605814	−0.302907	+	0.953020i	$0.597957\pi$
$948$	0	0
$949$	−15485.2	−0.529685
$950$	0	0
$951$	0	0
$952$	−10878.0	−0.370335
$953$	30736.6	1.04476	0.522380	−	0.852713i	$-0.325044\pi$
$953$	30736.6	1.04476	0.522380	+	0.852713i	$0.325044\pi$
$954$	0	0
$955$	0	0
$956$	28110.8	0.951012
$957$	0	0
$958$	72761.5	2.45388
$959$	13171.7	0.443519
$960$	0	0
$961$	−9711.08	−0.325974
$962$	86840.6	2.91045
$963$	0	0
$964$	3755.97	0.125489
$965$	0	0
$966$	0	0
$967$	−23645.2	−0.786327	−0.393163	−	0.919469i	$-0.628619\pi$
$967$	−23645.2	−0.786327	−0.393163	+	0.919469i	$0.628619\pi$
$968$	−7438.10	−0.246973
$969$	0	0
$970$	0	0
$971$	−27402.0	−0.905635	−0.452818	−	0.891603i	$-0.649581\pi$
$971$	−27402.0	−0.905635	−0.452818	+	0.891603i	$0.649581\pi$
$972$	0	0
$973$	−27899.1	−0.919222
$974$	69790.6	2.29593
$975$	0	0
$976$	−33807.5	−1.10876
$977$	−49118.7	−1.60844	−0.804220	−	0.594332i	$-0.797416\pi$
$977$	−49118.7	−1.60844	−0.804220	+	0.594332i	$0.797416\pi$
$978$	0	0
$979$	−4842.40	−0.158084
$980$	0	0
$981$	0	0
$982$	−39294.0	−1.27691
$983$	52630.5	1.70768	0.853842	−	0.520533i	$-0.174267\pi$
$983$	52630.5	1.70768	0.853842	+	0.520533i	$0.174267\pi$
$984$	0	0
$985$	0	0
$986$	5191.47	0.167677
$987$	0	0
$988$	−113609.	−3.65827
$989$	−73868.4	−2.37500
$990$	0	0
$991$	45472.7	1.45761	0.728803	−	0.684724i	$-0.240077\pi$
$991$	45472.7	1.45761	0.728803	+	0.684724i	$0.240077\pi$
$992$	54054.5	1.73007
$993$	0	0
$994$	5020.04	0.160187
$995$	0	0
$996$	0	0
$997$	−55068.1	−1.74927	−0.874637	−	0.484779i	$-0.838900\pi$
$997$	−55068.1	−1.74927	−0.874637	+	0.484779i	$0.838900\pi$
$998$	27763.0	0.880585
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.s.1.1		3
3.2	odd	2		825.4.a.s.1.3		3
5.4	even	2		495.4.a.l.1.3		3
15.2	even	4		825.4.c.l.199.6		6
15.8	even	4		825.4.c.l.199.1		6
15.14	odd	2		165.4.a.d.1.1	✓	3
165.164	even	2		1815.4.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.d.1.1	✓	3	15.14	odd	2
495.4.a.l.1.3		3	5.4	even	2
825.4.a.s.1.3		3	3.2	odd	2
825.4.c.l.199.1		6	15.8	even	4
825.4.c.l.199.6		6	15.2	even	4
1815.4.a.s.1.3		3	165.164	even	2
2475.4.a.s.1.1		3	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-5.26150 q^{2} +19.6833 q^{4} +10.3207 q^{7} -61.4719 q^{8} -11.0000 q^{11} -63.9817 q^{13} -54.3024 q^{14} +165.967 q^{16} +17.1461 q^{17} +90.2104 q^{19} +57.8765 q^{22} -212.605 q^{23} +336.639 q^{26} +203.146 q^{28} -57.5461 q^{29} -141.704 q^{31} -381.462 q^{32} -90.2140 q^{34} +257.963 q^{37} -474.642 q^{38} +225.914 q^{41} +347.445 q^{43} -216.517 q^{44} +1118.62 q^{46} +404.364 q^{47} -236.483 q^{49} -1259.37 q^{52} +259.568 q^{53} -634.433 q^{56} +302.779 q^{58} +853.067 q^{59} -203.699 q^{61} +745.573 q^{62} +679.320 q^{64} -266.890 q^{67} +337.492 q^{68} -92.4460 q^{71} +242.026 q^{73} -1357.27 q^{74} +1775.64 q^{76} -113.528 q^{77} -1021.60 q^{79} -1188.65 q^{82} +706.415 q^{83} -1828.08 q^{86} +676.191 q^{88} +440.218 q^{89} -660.336 q^{91} -4184.77 q^{92} -2127.56 q^{94} +197.761 q^{97} +1244.25 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{2} + 22 q^{4} + 4 q^{7} - 48 q^{8} - 33 q^{11} + 56 q^{14} + 50 q^{16} - 218 q^{17} + 146 q^{19} + 44 q^{22} - 200 q^{23} + 508 q^{26} + 340 q^{28} - 68 q^{29} - 68 q^{31} - 688 q^{32} - 176 q^{34}+ \cdots + 572 q^{98}+O(q^{100}) \) 3 * q - 4 * q^2 + 22 * q^4 + 4 * q^7 - 48 * q^8 - 33 * q^11 + 56 * q^14 + 50 * q^16 - 218 * q^17 + 146 * q^19 + 44 * q^22 - 200 * q^23 + 508 * q^26 + 340 * q^28 - 68 * q^29 - 68 * q^31 - 688 * q^32 - 176 * q^34 + 390 * q^37 - 316 * q^38 + 196 * q^41 + 524 * q^43 - 242 * q^44 + 1160 * q^46 - 60 * q^47 - 157 * q^49 - 1020 * q^52 - 158 * q^53 - 1368 * q^56 - 1092 * q^58 + 1044 * q^59 + 642 * q^61 + 88 * q^62 + 1166 * q^64 + 236 * q^67 + 144 * q^68 + 544 * q^71 - 900 * q^73 - 1536 * q^74 + 1996 * q^76 - 44 * q^77 - 1586 * q^79 - 380 * q^82 - 1582 * q^83 - 3568 * q^86 + 528 * q^88 + 2122 * q^89 - 8 * q^91 - 4128 * q^92 - 2152 * q^94 - 618 * q^97 + 572 * q^98

Introduction

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