LMFDB - Embedded newform 975.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(1,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.a (trivial)

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:	yes
Analytic conductor:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -3.00000 q^{14} -1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} -3.00000 q^{21} +1.00000 q^{22} -3.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} +1.00000 q^{29} +3.00000 q^{31} -5.00000 q^{32} +1.00000 q^{33} -5.00000 q^{34} -1.00000 q^{36} +8.00000 q^{37} +8.00000 q^{38} +1.00000 q^{39} -2.00000 q^{41} +3.00000 q^{42} -8.00000 q^{43} +1.00000 q^{44} +11.0000 q^{47} +1.00000 q^{48} +2.00000 q^{49} -5.00000 q^{51} +1.00000 q^{52} +11.0000 q^{53} +1.00000 q^{54} +9.00000 q^{56} +8.00000 q^{57} -1.00000 q^{58} +5.00000 q^{59} +1.00000 q^{61} -3.00000 q^{62} +3.00000 q^{63} +7.00000 q^{64} -1.00000 q^{66} -3.00000 q^{67} -5.00000 q^{68} +16.0000 q^{71} +3.00000 q^{72} +4.00000 q^{73} -8.00000 q^{74} +8.00000 q^{76} -3.00000 q^{77} -1.00000 q^{78} +12.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +3.00000 q^{83} +3.00000 q^{84} +8.00000 q^{86} -1.00000 q^{87} -3.00000 q^{88} -3.00000 q^{91} -3.00000 q^{93} -11.0000 q^{94} +5.00000 q^{96} +2.00000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.00000	−0.801784
$15$	0	0
$16$	−1.00000	−0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	−1.00000	−0.235702
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	−3.00000	−0.566947
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−5.00000	−0.883883
$33$	1.00000	0.174078
$34$	−5.00000	−0.857493
$35$	0	0
$36$	−1.00000	−0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	8.00000	1.29777
$39$	1.00000	0.160128
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	3.00000	0.462910
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	0	0
$47$	11.0000	1.60451	0.802257	−	0.596978i	$-0.203632\pi$
$47$	11.0000	1.60451	0.802257	+	0.596978i	$0.203632\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	−5.00000	−0.700140
$52$	1.00000	0.138675
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	9.00000	1.20268
$57$	8.00000	1.05963
$58$	−1.00000	−0.131306
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−3.00000	−0.381000
$63$	3.00000	0.377964
$64$	7.00000	0.875000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	3.00000	0.353553
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	8.00000	0.917663
$77$	−3.00000	−0.341882
$78$	−1.00000	−0.113228
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	8.00000	0.862662
$87$	−1.00000	−0.107211
$88$	−3.00000	−0.319801
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	−3.00000	−0.311086
$94$	−11.0000	−1.13456
$95$	0	0
$96$	5.00000	0.510310
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−2.00000	−0.202031
$99$	−1.00000	−0.100504
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	5.00000	0.495074
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−11.0000	−1.06841
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	1.00000	0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	−3.00000	−0.283473
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	−1.00000	−0.0924500
$118$	−5.00000	−0.460287
$119$	15.0000	1.37505
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−1.00000	−0.0905357
$123$	2.00000	0.180334
$124$	−3.00000	−0.269408
$125$	0	0
$126$	−3.00000	−0.267261
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	3.00000	0.265165
$129$	8.00000	0.704361
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	−1.00000	−0.0870388
$133$	−24.0000	−2.08106
$134$	3.00000	0.259161
$135$	0	0
$136$	15.0000	1.28624
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	−11.0000	−0.926367
$142$	−16.0000	−1.34269
$143$	1.00000	0.0836242
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	−2.00000	−0.164957
$148$	−8.00000	−0.657596
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	−24.0000	−1.94666
$153$	5.00000	0.404226
$154$	3.00000	0.241747
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	−12.0000	−0.954669
$159$	−11.0000	−0.872357
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−3.00000	−0.232845
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−9.00000	−0.694365
$169$	1.00000	0.0769231
$170$	0	0
$171$	−8.00000	−0.611775
$172$	8.00000	0.609994
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	1.00000	0.0753778
$177$	−5.00000	−0.375823
$178$	0	0
$179$	26.0000	1.94333	0.971666	−	0.236360i	$-0.0759544\pi$
$179$	26.0000	1.94333	0.971666	+	0.236360i	$0.0759544\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	3.00000	0.222375
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	0	0
$186$	3.00000	0.219971
$187$	−5.00000	−0.365636
$188$	−11.0000	−0.802257
$189$	−3.00000	−0.218218
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	−7.00000	−0.505181
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−2.00000	−0.142857
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	1.00000	0.0710669
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	−9.00000	−0.633238
$203$	3.00000	0.210559
$204$	5.00000	0.350070
$205$	0	0
$206$	−14.0000	−0.975426
$207$	0	0
$208$	1.00000	0.0693375
$209$	8.00000	0.553372
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−11.0000	−0.755483
$213$	−16.0000	−1.09630
$214$	18.0000	1.23045
$215$	0	0
$216$	−3.00000	−0.204124
$217$	9.00000	0.610960
$218$	10.0000	0.677285
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−5.00000	−0.336336
$222$	8.00000	0.536925
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−15.0000	−1.00223
$225$	0	0
$226$	−2.00000	−0.133038
$227$	−23.0000	−1.52656	−0.763282	−	0.646066i	$-0.776413\pi$
$227$	−23.0000	−1.52656	−0.763282	+	0.646066i	$0.776413\pi$
$228$	−8.00000	−0.529813
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	3.00000	0.196960
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−5.00000	−0.325472
$237$	−12.0000	−0.779484
$238$	−15.0000	−0.972306
$239$	−19.0000	−1.22901	−0.614504	−	0.788914i	$-0.710644\pi$
$239$	−19.0000	−1.22901	−0.614504	+	0.788914i	$0.710644\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	10.0000	0.642824
$243$	−1.00000	−0.0641500
$244$	−1.00000	−0.0640184
$245$	0	0
$246$	−2.00000	−0.127515
$247$	8.00000	0.509028
$248$	9.00000	0.571501
$249$	−3.00000	−0.190117
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	−3.00000	−0.188982
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	−8.00000	−0.498058
$259$	24.0000	1.49129
$260$	0	0
$261$	1.00000	0.0618984
$262$	−6.00000	−0.370681
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	24.0000	1.47153
$267$	0	0
$268$	3.00000	0.183254
$269$	17.0000	1.03651	0.518254	−	0.855227i	$-0.326582\pi$
$269$	17.0000	1.03651	0.518254	+	0.855227i	$0.326582\pi$
$270$	0	0
$271$	−5.00000	−0.303728	−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000	−0.303728	−0.151864	+	0.988401i	$0.548528\pi$
$272$	−5.00000	−0.303170
$273$	3.00000	0.181568
$274$	−12.0000	−0.724947
$275$	0	0
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	10.0000	0.599760
$279$	3.00000	0.179605
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	11.0000	0.655040
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	−6.00000	−0.354169
$288$	−5.00000	−0.294628
$289$	8.00000	0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	−4.00000	−0.234082
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	24.0000	1.39497
$297$	1.00000	0.0580259
$298$	−10.0000	−0.579284
$299$	0	0
$300$	0	0
$301$	−24.0000	−1.38334
$302$	17.0000	0.978240
$303$	−9.00000	−0.517036
$304$	8.00000	0.458831
$305$	0	0
$306$	−5.00000	−0.285831
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	3.00000	0.170941
$309$	−14.0000	−0.796432
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	3.00000	0.169842
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	−11.0000	−0.620766
$315$	0	0
$316$	−12.0000	−0.675053
$317$	16.0000	0.898650	0.449325	−	0.893368i	$-0.351665\pi$
$317$	16.0000	0.898650	0.449325	+	0.893368i	$0.351665\pi$
$318$	11.0000	0.616849
$319$	−1.00000	−0.0559893
$320$	0	0
$321$	18.0000	1.00466
$322$	0	0
$323$	−40.0000	−2.22566
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	10.0000	0.553001
$328$	−6.00000	−0.331295
$329$	33.0000	1.81935
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−3.00000	−0.164646
$333$	8.00000	0.438397
$334$	12.0000	0.656611
$335$	0	0
$336$	3.00000	0.163663
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	−1.00000	−0.0543928
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−3.00000	−0.162459
$342$	8.00000	0.432590
$343$	−15.0000	−0.809924
$344$	−24.0000	−1.29399
$345$	0	0
$346$	−21.0000	−1.12897
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	1.00000	0.0536056
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	5.00000	0.266501
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	5.00000	0.265747
$355$	0	0
$356$	0	0
$357$	−15.0000	−0.793884
$358$	−26.0000	−1.37414
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	−5.00000	−0.262794
$363$	10.0000	0.524864
$364$	3.00000	0.157243
$365$	0	0
$366$	1.00000	0.0522708
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	33.0000	1.71327
$372$	3.00000	0.155543
$373$	31.0000	1.60512	0.802560	−	0.596572i	$-0.203471\pi$
$373$	31.0000	1.60512	0.802560	+	0.596572i	$0.203471\pi$
$374$	5.00000	0.258544
$375$	0	0
$376$	33.0000	1.70185
$377$	−1.00000	−0.0515026
$378$	3.00000	0.154303
$379$	−35.0000	−1.79783	−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000	−1.79783	−0.898915	+	0.438124i	$0.855643\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	−10.0000	−0.511645
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	10.0000	0.508987
$387$	−8.00000	−0.406663
$388$	−2.00000	−0.101535
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	6.00000	0.303046
$393$	−6.00000	−0.302660
$394$	−6.00000	−0.302276
$395$	0	0
$396$	1.00000	0.0502519
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−10.0000	−0.501255
$399$	24.0000	1.20150
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	−3.00000	−0.149626
$403$	−3.00000	−0.149441
$404$	−9.00000	−0.447767
$405$	0	0
$406$	−3.00000	−0.148888
$407$	−8.00000	−0.396545
$408$	−15.0000	−0.742611
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	−14.0000	−0.689730
$413$	15.0000	0.738102
$414$	0	0
$415$	0	0
$416$	5.00000	0.245145
$417$	10.0000	0.489702
$418$	−8.00000	−0.391293
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	16.0000	0.778868
$423$	11.0000	0.534838
$424$	33.0000	1.60262
$425$	0	0
$426$	16.0000	0.775203
$427$	3.00000	0.145180
$428$	18.0000	0.870063
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	1.00000	0.0481125
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	−9.00000	−0.432014
$435$	0	0
$436$	10.0000	0.478913
$437$	0	0
$438$	4.00000	0.191127
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	5.00000	0.237826
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	−8.00000	−0.378811
$447$	−10.0000	−0.472984
$448$	21.0000	0.992157
$449$	−28.0000	−1.32140	−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000	−1.32140	−0.660701	+	0.750649i	$0.729741\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	−2.00000	−0.0940721
$453$	17.0000	0.798730
$454$	23.0000	1.07944
$455$	0	0
$456$	24.0000	1.12390
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	18.0000	0.841085
$459$	−5.00000	−0.233380
$460$	0	0
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	−3.00000	−0.139573
$463$	9.00000	0.418265	0.209133	−	0.977887i	$-0.432936\pi$
$463$	9.00000	0.418265	0.209133	+	0.977887i	$0.432936\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	1.00000	0.0462250
$469$	−9.00000	−0.415581
$470$	0	0
$471$	−11.0000	−0.506853
$472$	15.0000	0.690431
$473$	8.00000	0.367840
$474$	12.0000	0.551178
$475$	0	0
$476$	−15.0000	−0.687524
$477$	11.0000	0.503655
$478$	19.0000	0.869040
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	−10.0000	−0.455488
$483$	0	0
$484$	10.0000	0.454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	−37.0000	−1.67663	−0.838315	−	0.545186i	$-0.816459\pi$
$487$	−37.0000	−1.67663	−0.838315	+	0.545186i	$0.816459\pi$
$488$	3.00000	0.135804
$489$	−20.0000	−0.904431
$490$	0	0
$491$	−26.0000	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$492$	−2.00000	−0.0901670
$493$	5.00000	0.225189
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−3.00000	−0.134704
$497$	48.0000	2.15309
$498$	3.00000	0.134433
$499$	−41.0000	−1.83541	−0.917706	−	0.397260i	$-0.869961\pi$
$499$	−41.0000	−1.83541	−0.917706	+	0.397260i	$0.869961\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	30.0000	1.33897
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	9.00000	0.400892
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	8.00000	0.354943
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	11.0000	0.486136
$513$	8.00000	0.353209
$514$	−3.00000	−0.132324
$515$	0	0
$516$	−8.00000	−0.352180
$517$	−11.0000	−0.483779
$518$	−24.0000	−1.05450
$519$	−21.0000	−0.921798
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	−1.00000	−0.0437688
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	14.0000	0.610429
$527$	15.0000	0.653410
$528$	−1.00000	−0.0435194
$529$	−23.0000	−1.00000
$530$	0	0
$531$	5.00000	0.216982
$532$	24.0000	1.04053
$533$	2.00000	0.0866296
$534$	0	0
$535$	0	0
$536$	−9.00000	−0.388741
$537$	−26.0000	−1.12198
$538$	−17.0000	−0.732922
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	5.00000	0.214768
$543$	−5.00000	−0.214571
$544$	−25.0000	−1.07187
$545$	0	0
$546$	−3.00000	−0.128388
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−12.0000	−0.512615
$549$	1.00000	0.0426790
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	36.0000	1.53088
$554$	6.00000	0.254916
$555$	0	0
$556$	10.0000	0.424094
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−3.00000	−0.127000
$559$	8.00000	0.338364
$560$	0	0
$561$	5.00000	0.211100
$562$	18.0000	0.759284
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	11.0000	0.463184
$565$	0	0
$566$	−18.0000	−0.756596
$567$	3.00000	0.125988
$568$	48.0000	2.01404
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−1.00000	−0.0418121
$573$	−10.0000	−0.417756
$574$	6.00000	0.250435
$575$	0	0
$576$	7.00000	0.291667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	−8.00000	−0.332756
$579$	10.0000	0.415586
$580$	0	0
$581$	9.00000	0.373383
$582$	2.00000	0.0829027
$583$	−11.0000	−0.455573
$584$	12.0000	0.496564
$585$	0	0
$586$	30.0000	1.23929
$587$	45.0000	1.85735	0.928674	−	0.370896i	$-0.120949\pi$
$587$	45.0000	1.85735	0.928674	+	0.370896i	$0.120949\pi$
$588$	2.00000	0.0824786
$589$	−24.0000	−0.988903
$590$	0	0
$591$	−6.00000	−0.246807
$592$	−8.00000	−0.328798
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−10.0000	−0.409273
$598$	0	0
$599$	46.0000	1.87951	0.939755	−	0.341850i	$-0.111053\pi$
$599$	46.0000	1.87951	0.939755	+	0.341850i	$0.111053\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	24.0000	0.978167
$603$	−3.00000	−0.122169
$604$	17.0000	0.691720
$605$	0	0
$606$	9.00000	0.365600
$607$	−38.0000	−1.54237	−0.771186	−	0.636610i	$-0.780336\pi$
$607$	−38.0000	−1.54237	−0.771186	+	0.636610i	$0.780336\pi$
$608$	40.0000	1.62221
$609$	−3.00000	−0.121566
$610$	0	0
$611$	−11.0000	−0.445012
$612$	−5.00000	−0.202113
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−9.00000	−0.362620
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	14.0000	0.563163
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	−6.00000	−0.240578
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	0	0
$626$	−29.0000	−1.15907
$627$	−8.00000	−0.319489
$628$	−11.0000	−0.438948
$629$	40.0000	1.59490
$630$	0	0
$631$	−44.0000	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$632$	36.0000	1.43200
$633$	16.0000	0.635943
$634$	−16.0000	−0.635441
$635$	0	0
$636$	11.0000	0.436178
$637$	−2.00000	−0.0792429
$638$	1.00000	0.0395904
$639$	16.0000	0.632950
$640$	0	0
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	−18.0000	−0.710403
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	0	0
$646$	40.0000	1.57378
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	3.00000	0.117851
$649$	−5.00000	−0.196267
$650$	0	0
$651$	−9.00000	−0.352738
$652$	−20.0000	−0.783260
$653$	−35.0000	−1.36966	−0.684828	−	0.728705i	$-0.740123\pi$
$653$	−35.0000	−1.36966	−0.684828	+	0.728705i	$0.740123\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	2.00000	0.0780869
$657$	4.00000	0.156055
$658$	−33.0000	−1.28647
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	12.0000	0.466393
$663$	5.00000	0.194184
$664$	9.00000	0.349268
$665$	0	0
$666$	−8.00000	−0.309994
$667$	0	0
$668$	12.0000	0.464294
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−1.00000	−0.0386046
$672$	15.0000	0.578638
$673$	−45.0000	−1.73462	−0.867311	−	0.497766i	$-0.834154\pi$
$673$	−45.0000	−1.73462	−0.867311	+	0.497766i	$0.834154\pi$
$674$	5.00000	0.192593
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	2.00000	0.0768095
$679$	6.00000	0.230259
$680$	0	0
$681$	23.0000	0.881362
$682$	3.00000	0.114876
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	8.00000	0.305888
$685$	0	0
$686$	15.0000	0.572703
$687$	18.0000	0.686743
$688$	8.00000	0.304997
$689$	−11.0000	−0.419067
$690$	0	0
$691$	−29.0000	−1.10321	−0.551606	−	0.834105i	$-0.685985\pi$
$691$	−29.0000	−1.10321	−0.551606	+	0.834105i	$0.685985\pi$
$692$	−21.0000	−0.798300
$693$	−3.00000	−0.113961
$694$	−2.00000	−0.0759190
$695$	0	0
$696$	−3.00000	−0.113715
$697$	−10.0000	−0.378777
$698$	−16.0000	−0.605609
$699$	−6.00000	−0.226941
$700$	0	0
$701$	31.0000	1.17085	0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000	1.17085	0.585427	+	0.810725i	$0.300927\pi$
$702$	−1.00000	−0.0377426
$703$	−64.0000	−2.41381
$704$	−7.00000	−0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	27.0000	1.01544
$708$	5.00000	0.187912
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	0	0
$714$	15.0000	0.561361
$715$	0	0
$716$	−26.0000	−0.971666
$717$	19.0000	0.709568
$718$	−15.0000	−0.559795
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	−45.0000	−1.67473
$723$	−10.0000	−0.371904
$724$	−5.00000	−0.185824
$725$	0	0
$726$	−10.0000	−0.371135
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	−9.00000	−0.333562
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.0000	−1.47945
$732$	1.00000	0.0369611
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	0	0
$737$	3.00000	0.110506
$738$	2.00000	0.0736210
$739$	7.00000	0.257499	0.128750	−	0.991677i	$-0.458904\pi$
$739$	7.00000	0.257499	0.128750	+	0.991677i	$0.458904\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	−33.0000	−1.21147
$743$	−13.0000	−0.476924	−0.238462	−	0.971152i	$-0.576643\pi$
$743$	−13.0000	−0.476924	−0.238462	+	0.971152i	$0.576643\pi$
$744$	−9.00000	−0.329956
$745$	0	0
$746$	−31.0000	−1.13499
$747$	3.00000	0.109764
$748$	5.00000	0.182818
$749$	−54.0000	−1.97312
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	−11.0000	−0.401129
$753$	30.0000	1.09326
$754$	1.00000	0.0364179
$755$	0	0
$756$	3.00000	0.109109
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	35.0000	1.27126
$759$	0	0
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	−8.00000	−0.289809
$763$	−30.0000	−1.08607
$764$	−10.0000	−0.361787
$765$	0	0
$766$	−24.0000	−0.867155
$767$	−5.00000	−0.180540
$768$	17.0000	0.613435
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	−3.00000	−0.108042
$772$	10.0000	0.359908
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	6.00000	0.215387
$777$	−24.0000	−0.860995
$778$	30.0000	1.07555
$779$	16.0000	0.573259
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	6.00000	0.214013
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	−6.00000	−0.213741
$789$	14.0000	0.498413
$790$	0	0
$791$	6.00000	0.213335
$792$	−3.00000	−0.106600
$793$	−1.00000	−0.0355110
$794$	2.00000	0.0709773
$795$	0	0
$796$	−10.0000	−0.354441
$797$	43.0000	1.52314	0.761569	−	0.648084i	$-0.224429\pi$
$797$	43.0000	1.52314	0.761569	+	0.648084i	$0.224429\pi$
$798$	−24.0000	−0.849591
$799$	55.0000	1.94576
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.00000	−0.141157
$804$	−3.00000	−0.105802
$805$	0	0
$806$	3.00000	0.105670
$807$	−17.0000	−0.598428
$808$	27.0000	0.949857
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−33.0000	−1.15879	−0.579393	−	0.815048i	$-0.696710\pi$
$811$	−33.0000	−1.15879	−0.579393	+	0.815048i	$0.696710\pi$
$812$	−3.00000	−0.105279
$813$	5.00000	0.175358
$814$	8.00000	0.280400
$815$	0	0
$816$	5.00000	0.175035
$817$	64.0000	2.23908
$818$	−2.00000	−0.0699284
$819$	−3.00000	−0.104828
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	12.0000	0.418548
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	42.0000	1.46314
$825$	0	0
$826$	−15.0000	−0.521917
$827$	−1.00000	−0.0347734	−0.0173867	−	0.999849i	$-0.505535\pi$
$827$	−1.00000	−0.0347734	−0.0173867	+	0.999849i	$0.505535\pi$
$828$	0	0
$829$	51.0000	1.77130	0.885652	−	0.464350i	$-0.153712\pi$
$829$	51.0000	1.77130	0.885652	+	0.464350i	$0.153712\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	−7.00000	−0.242681
$833$	10.0000	0.346479
$834$	−10.0000	−0.346272
$835$	0	0
$836$	−8.00000	−0.276686
$837$	−3.00000	−0.103695
$838$	26.0000	0.898155
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−28.0000	−0.964944
$843$	18.0000	0.619953
$844$	16.0000	0.550743
$845$	0	0
$846$	−11.0000	−0.378188
$847$	−30.0000	−1.03081
$848$	−11.0000	−0.377742
$849$	−18.0000	−0.617758
$850$	0	0
$851$	0	0
$852$	16.0000	0.548151
$853$	−24.0000	−0.821744	−0.410872	−	0.911693i	$-0.634776\pi$
$853$	−24.0000	−0.821744	−0.410872	+	0.911693i	$0.634776\pi$
$854$	−3.00000	−0.102658
$855$	0	0
$856$	−54.0000	−1.84568
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	1.00000	0.0341394
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	12.0000	0.408722
$863$	7.00000	0.238283	0.119141	−	0.992877i	$-0.461986\pi$
$863$	7.00000	0.238283	0.119141	+	0.992877i	$0.461986\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−34.0000	−1.15537
$867$	−8.00000	−0.271694
$868$	−9.00000	−0.305480
$869$	−12.0000	−0.407072
$870$	0	0
$871$	3.00000	0.101651
$872$	−30.0000	−1.01593
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	4.00000	0.135147
$877$	−24.0000	−0.810422	−0.405211	−	0.914223i	$-0.632802\pi$
$877$	−24.0000	−0.810422	−0.405211	+	0.914223i	$0.632802\pi$
$878$	−4.00000	−0.134993
$879$	30.0000	1.01187
$880$	0	0
$881$	−25.0000	−0.842271	−0.421136	−	0.906998i	$-0.638368\pi$
$881$	−25.0000	−0.842271	−0.421136	+	0.906998i	$0.638368\pi$
$882$	−2.00000	−0.0673435
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	5.00000	0.168168
$885$	0	0
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	−24.0000	−0.805387
$889$	−24.0000	−0.804934
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−8.00000	−0.267860
$893$	−88.0000	−2.94481
$894$	10.0000	0.334450
$895$	0	0
$896$	9.00000	0.300669
$897$	0	0
$898$	28.0000	0.934372
$899$	3.00000	0.100056
$900$	0	0
$901$	55.0000	1.83232
$902$	−2.00000	−0.0665927
$903$	24.0000	0.798670
$904$	6.00000	0.199557
$905$	0	0
$906$	−17.0000	−0.564787
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	23.0000	0.763282
$909$	9.00000	0.298511
$910$	0	0
$911$	56.0000	1.85536	0.927681	−	0.373373i	$-0.121799\pi$
$911$	56.0000	1.85536	0.927681	+	0.373373i	$0.121799\pi$
$912$	−8.00000	−0.264906
$913$	−3.00000	−0.0992855
$914$	−34.0000	−1.12462
$915$	0	0
$916$	18.0000	0.594737
$917$	18.0000	0.594412
$918$	5.00000	0.165025
$919$	22.0000	0.725713	0.362857	−	0.931845i	$-0.381802\pi$
$919$	22.0000	0.725713	0.362857	+	0.931845i	$0.381802\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	−42.0000	−1.38320
$923$	−16.0000	−0.526646
$924$	−3.00000	−0.0986928
$925$	0	0
$926$	−9.00000	−0.295758
$927$	14.0000	0.459820
$928$	−5.00000	−0.164133
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	−16.0000	−0.524379
$932$	−6.00000	−0.196537
$933$	−6.00000	−0.196431
$934$	32.0000	1.04707
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	−45.0000	−1.47009	−0.735043	−	0.678021i	$-0.762838\pi$
$937$	−45.0000	−1.47009	−0.735043	+	0.678021i	$0.762838\pi$
$938$	9.00000	0.293860
$939$	−29.0000	−0.946379
$940$	0	0
$941$	−56.0000	−1.82555	−0.912774	−	0.408465i	$-0.866064\pi$
$941$	−56.0000	−1.82555	−0.912774	+	0.408465i	$0.866064\pi$
$942$	11.0000	0.358399
$943$	0	0
$944$	−5.00000	−0.162736
$945$	0	0
$946$	−8.00000	−0.260102
$947$	−21.0000	−0.682408	−0.341204	−	0.939989i	$-0.610835\pi$
$947$	−21.0000	−0.682408	−0.341204	+	0.939989i	$0.610835\pi$
$948$	12.0000	0.389742
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−16.0000	−0.518836
$952$	45.0000	1.45846
$953$	−37.0000	−1.19855	−0.599274	−	0.800544i	$-0.704544\pi$
$953$	−37.0000	−1.19855	−0.599274	+	0.800544i	$0.704544\pi$
$954$	−11.0000	−0.356138
$955$	0	0
$956$	19.0000	0.614504
$957$	1.00000	0.0323254
$958$	15.0000	0.484628
$959$	36.0000	1.16250
$960$	0	0
$961$	−22.0000	−0.709677
$962$	8.00000	0.257930
$963$	−18.0000	−0.580042
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	−37.0000	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$967$	−37.0000	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$968$	−30.0000	−0.964237
$969$	40.0000	1.28499
$970$	0	0
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	1.00000	0.0320750
$973$	−30.0000	−0.961756
$974$	37.0000	1.18556
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	20.0000	0.639529
$979$	0	0
$980$	0	0
$981$	−10.0000	−0.319275
$982$	26.0000	0.829693
$983$	21.0000	0.669796	0.334898	−	0.942254i	$-0.391298\pi$
$983$	21.0000	0.669796	0.334898	+	0.942254i	$0.391298\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	−5.00000	−0.159232
$987$	−33.0000	−1.05040
$988$	−8.00000	−0.254514
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	−15.0000	−0.476250
$993$	12.0000	0.380808
$994$	−48.0000	−1.52247
$995$	0	0
$996$	3.00000	0.0950586
$997$	−3.00000	−0.0950110	−0.0475055	−	0.998871i	$-0.515127\pi$
$997$	−3.00000	−0.0950110	−0.0475055	+	0.998871i	$0.515127\pi$
$998$	41.0000	1.29783
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.a.d.1.1	✓	1
3.2	odd	2		2925.2.a.o.1.1		1
5.2	odd	4		975.2.c.g.274.1		2
5.3	odd	4		975.2.c.g.274.2		2
5.4	even	2		975.2.a.k.1.1	yes	1
15.2	even	4		2925.2.c.i.2224.2		2
15.8	even	4		2925.2.c.i.2224.1		2
15.14	odd	2		2925.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.a.d.1.1	✓	1	1.1	even	1	trivial
975.2.a.k.1.1	yes	1	5.4	even	2
975.2.c.g.274.1		2	5.2	odd	4
975.2.c.g.274.2		2	5.3	odd	4
2925.2.a.b.1.1		1	15.14	odd	2
2925.2.a.o.1.1		1	3.2	odd	2
2925.2.c.i.2224.1		2	15.8	even	4
2925.2.c.i.2224.2		2	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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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