LMFDB - Embedded newform 976.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [976,2,Mod(1,976)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(976, 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("976.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 976 = 2^{4} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	976.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:	$7.79339923728$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.643168996.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 9x^{4} + 20x^{3} + 25x^{2} - 13x - 5 $ x^6 - 3x^5 - 9x^4 + 20x^3 + 25x^2 - 13*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 488)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.28207$ of defining polynomial
Character	$\chi$	$=$	976.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.28207 q^{3} -1.70864 q^{5} -2.72208 q^{7} +7.77196 q^{9} +O(q^{10})$	$q-3.28207 q^{3} -1.70864 q^{5} -2.72208 q^{7} +7.77196 q^{9} -2.67795 q^{11} -6.46335 q^{13} +5.60786 q^{15} -4.81609 q^{17} -2.01724 q^{19} +8.93406 q^{21} -0.767808 q^{23} -2.08056 q^{25} -15.6619 q^{27} +0.835899 q^{29} +3.28207 q^{31} +8.78920 q^{33} +4.65105 q^{35} +11.1419 q^{37} +21.2132 q^{39} -1.80936 q^{41} -5.60786 q^{43} -13.2794 q^{45} +8.53724 q^{47} +0.409742 q^{49} +15.8067 q^{51} -6.88992 q^{53} +4.57564 q^{55} +6.62072 q^{57} +10.6140 q^{59} +1.00000 q^{61} -21.1559 q^{63} +11.0435 q^{65} +2.07677 q^{67} +2.52000 q^{69} +13.4000 q^{71} +11.6275 q^{73} +6.82854 q^{75} +7.28960 q^{77} -11.5394 q^{79} +28.0875 q^{81} -11.2747 q^{83} +8.22895 q^{85} -2.74348 q^{87} -17.4037 q^{89} +17.5938 q^{91} -10.7720 q^{93} +3.44673 q^{95} -15.8966 q^{97} -20.8129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} + 5 q^{5} - 6 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 + 5 * q^5 - 6 * q^7 + 9 * q^9	$ 6 q - 3 q^{3} + 5 q^{5} - 6 q^{7} + 9 q^{9} - 3 q^{11} + 7 q^{13} + 6 q^{15} + 6 q^{17} - 5 q^{19} + 12 q^{21} + 6 q^{23} + 17 q^{25} - 12 q^{27} + 15 q^{29} + 3 q^{31} + 8 q^{33} + 5 q^{35} + 21 q^{37} + 20 q^{39} + 4 q^{41} - 6 q^{43} + q^{45} + 8 q^{47} + 8 q^{49} + 16 q^{51} + 3 q^{53} + 17 q^{55} + 4 q^{57} + 3 q^{59} + 6 q^{61} + 3 q^{63} - q^{65} - 5 q^{67} - 21 q^{69} + 57 q^{71} + 14 q^{73} + q^{75} - 9 q^{77} - 5 q^{79} + 2 q^{81} - q^{83} + 2 q^{85} + 28 q^{87} + 3 q^{91} - 27 q^{93} + 52 q^{95} - 19 q^{97} - 13 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 + 5 * q^5 - 6 * q^7 + 9 * q^9 - 3 * q^11 + 7 * q^13 + 6 * q^15 + 6 * q^17 - 5 * q^19 + 12 * q^21 + 6 * q^23 + 17 * q^25 - 12 * q^27 + 15 * q^29 + 3 * q^31 + 8 * q^33 + 5 * q^35 + 21 * q^37 + 20 * q^39 + 4 * q^41 - 6 * q^43 + q^45 + 8 * q^47 + 8 * q^49 + 16 * q^51 + 3 * q^53 + 17 * q^55 + 4 * q^57 + 3 * q^59 + 6 * q^61 + 3 * q^63 - q^65 - 5 * q^67 - 21 * q^69 + 57 * q^71 + 14 * q^73 + q^75 - 9 * q^77 - 5 * q^79 + 2 * q^81 - q^83 + 2 * q^85 + 28 * q^87 + 3 * q^91 - 27 * q^93 + 52 * q^95 - 19 * q^97 - 13 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.28207	−1.89490	−0.947451	−	0.319901i	$-0.896350\pi$
$3$	−3.28207	−1.89490	−0.947451	+	0.319901i	$0.896350\pi$
$4$	0	0
$5$	−1.70864	−0.764125	−0.382063	−	0.924136i	$-0.624786\pi$
$5$	−1.70864	−0.764125	−0.382063	+	0.924136i	$0.624786\pi$
$6$	0	0
$7$	−2.72208	−1.02885	−0.514426	−	0.857535i	$-0.671995\pi$
$7$	−2.72208	−1.02885	−0.514426	+	0.857535i	$0.671995\pi$
$8$	0	0
$9$	7.77196	2.59065
$10$	0	0
$11$	−2.67795	−0.807431	−0.403716	−	0.914884i	$-0.632282\pi$
$11$	−2.67795	−0.807431	−0.403716	+	0.914884i	$0.632282\pi$
$12$	0	0
$13$	−6.46335	−1.79261	−0.896306	−	0.443437i	$-0.853759\pi$
$13$	−6.46335	−1.79261	−0.896306	+	0.443437i	$0.853759\pi$
$14$	0	0
$15$	5.60786	1.44794
$16$	0	0
$17$	−4.81609	−1.16807	−0.584037	−	0.811727i	$-0.698528\pi$
$17$	−4.81609	−1.16807	−0.584037	+	0.811727i	$0.698528\pi$
$18$	0	0
$19$	−2.01724	−0.462787	−0.231393	−	0.972860i	$-0.574328\pi$
$19$	−2.01724	−0.462787	−0.231393	+	0.972860i	$0.574328\pi$
$20$	0	0
$21$	8.93406	1.94957
$22$	0	0
$23$	−0.767808	−0.160099	−0.0800495	−	0.996791i	$-0.525508\pi$
$23$	−0.767808	−0.160099	−0.0800495	+	0.996791i	$0.525508\pi$
$24$	0	0
$25$	−2.08056	−0.416113
$26$	0	0
$27$	−15.6619	−3.01413
$28$	0	0
$29$	0.835899	0.155223	0.0776113	−	0.996984i	$-0.475271\pi$
$29$	0.835899	0.155223	0.0776113	+	0.996984i	$0.475271\pi$
$30$	0	0
$31$	3.28207	0.589476	0.294738	−	0.955578i	$-0.404768\pi$
$31$	3.28207	0.589476	0.294738	+	0.955578i	$0.404768\pi$
$32$	0	0
$33$	8.78920	1.53000
$34$	0	0
$35$	4.65105	0.786171
$36$	0	0
$37$	11.1419	1.83171	0.915857	−	0.401504i	$-0.131512\pi$
$37$	11.1419	1.83171	0.915857	+	0.401504i	$0.131512\pi$
$38$	0	0
$39$	21.2132	3.39682
$40$	0	0
$41$	−1.80936	−0.282575	−0.141287	−	0.989969i	$-0.545124\pi$
$41$	−1.80936	−0.282575	−0.141287	+	0.989969i	$0.545124\pi$
$42$	0	0
$43$	−5.60786	−0.855190	−0.427595	−	0.903970i	$-0.640639\pi$
$43$	−5.60786	−0.855190	−0.427595	+	0.903970i	$0.640639\pi$
$44$	0	0
$45$	−13.2794	−1.97958
$46$	0	0
$47$	8.53724	1.24528	0.622642	−	0.782507i	$-0.286059\pi$
$47$	8.53724	1.24528	0.622642	+	0.782507i	$0.286059\pi$
$48$	0	0
$49$	0.409742	0.0585345
$50$	0	0
$51$	15.8067	2.21339
$52$	0	0
$53$	−6.88992	−0.946404	−0.473202	−	0.880954i	$-0.656902\pi$
$53$	−6.88992	−0.946404	−0.473202	+	0.880954i	$0.656902\pi$
$54$	0	0
$55$	4.57564	0.616979
$56$	0	0
$57$	6.62072	0.876936
$58$	0	0
$59$	10.6140	1.38183	0.690913	−	0.722938i	$-0.257209\pi$
$59$	10.6140	1.38183	0.690913	+	0.722938i	$0.257209\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	0	0
$63$	−21.1559	−2.66540
$64$	0	0
$65$	11.0435	1.36978
$66$	0	0
$67$	2.07677	0.253718	0.126859	−	0.991921i	$-0.459510\pi$
$67$	2.07677	0.253718	0.126859	+	0.991921i	$0.459510\pi$
$68$	0	0
$69$	2.52000	0.303372
$70$	0	0
$71$	13.4000	1.59029	0.795146	−	0.606419i	$-0.207394\pi$
$71$	13.4000	1.59029	0.795146	+	0.606419i	$0.207394\pi$
$72$	0	0
$73$	11.6275	1.36089	0.680445	−	0.732799i	$-0.261786\pi$
$73$	11.6275	1.36089	0.680445	+	0.732799i	$0.261786\pi$
$74$	0	0
$75$	6.82854	0.788492
$76$	0	0
$77$	7.28960	0.830727
$78$	0	0
$79$	−11.5394	−1.29828	−0.649141	−	0.760668i	$-0.724872\pi$
$79$	−11.5394	−1.29828	−0.649141	+	0.760668i	$0.724872\pi$
$80$	0	0
$81$	28.0875	3.12083
$82$	0	0
$83$	−11.2747	−1.23756	−0.618780	−	0.785564i	$-0.712373\pi$
$83$	−11.2747	−1.23756	−0.618780	+	0.785564i	$0.712373\pi$
$84$	0	0
$85$	8.22895	0.892555
$86$	0	0
$87$	−2.74348	−0.294131
$88$	0	0
$89$	−17.4037	−1.84479	−0.922396	−	0.386245i	$-0.873772\pi$
$89$	−17.4037	−1.84479	−0.922396	+	0.386245i	$0.873772\pi$
$90$	0	0
$91$	17.5938	1.84433
$92$	0	0
$93$	−10.7720	−1.11700
$94$	0	0
$95$	3.44673	0.353627
$96$	0	0
$97$	−15.8966	−1.61406	−0.807028	−	0.590514i	$-0.798925\pi$
$97$	−15.8966	−1.61406	−0.807028	+	0.590514i	$0.798925\pi$
$98$	0	0
$99$	−20.8129	−2.09177
$100$	0	0
$101$	0.753130	0.0749392	0.0374696	−	0.999298i	$-0.488070\pi$
$101$	0.753130	0.0749392	0.0374696	+	0.999298i	$0.488070\pi$
$102$	0	0
$103$	−2.58016	−0.254231	−0.127116	−	0.991888i	$-0.540572\pi$
$103$	−2.58016	−0.254231	−0.127116	+	0.991888i	$0.540572\pi$
$104$	0	0
$105$	−15.2651	−1.48972
$106$	0	0
$107$	7.47627	0.722758	0.361379	−	0.932419i	$-0.382306\pi$
$107$	7.47627	0.722758	0.361379	+	0.932419i	$0.382306\pi$
$108$	0	0
$109$	1.30902	0.125381	0.0626906	−	0.998033i	$-0.480032\pi$
$109$	1.30902	0.125381	0.0626906	+	0.998033i	$0.480032\pi$
$110$	0	0
$111$	−36.5684	−3.47092
$112$	0	0
$113$	−5.98511	−0.563032	−0.281516	−	0.959556i	$-0.590837\pi$
$113$	−5.98511	−0.563032	−0.281516	+	0.959556i	$0.590837\pi$
$114$	0	0
$115$	1.31190	0.122336
$116$	0	0
$117$	−50.2329	−4.64403
$118$	0	0
$119$	13.1098	1.20177
$120$	0	0
$121$	−3.82860	−0.348055
$122$	0	0
$123$	5.93844	0.535451
$124$	0	0
$125$	12.0981	1.08209
$126$	0	0
$127$	3.48951	0.309644	0.154822	−	0.987942i	$-0.450520\pi$
$127$	3.48951	0.309644	0.154822	+	0.987942i	$0.450520\pi$
$128$	0	0
$129$	18.4054	1.62050
$130$	0	0
$131$	11.0865	0.968633	0.484316	−	0.874893i	$-0.339068\pi$
$131$	11.0865	0.968633	0.484316	+	0.874893i	$0.339068\pi$
$132$	0	0
$133$	5.49110	0.476139
$134$	0	0
$135$	26.7605	2.30317
$136$	0	0
$137$	20.7252	1.77067	0.885337	−	0.464950i	$-0.153928\pi$
$137$	20.7252	1.77067	0.885337	+	0.464950i	$0.153928\pi$
$138$	0	0
$139$	−6.69398	−0.567776	−0.283888	−	0.958857i	$-0.591624\pi$
$139$	−6.69398	−0.567776	−0.283888	+	0.958857i	$0.591624\pi$
$140$	0	0
$141$	−28.0198	−2.35969
$142$	0	0
$143$	17.3085	1.44741
$144$	0	0
$145$	−1.42825	−0.118609
$146$	0	0
$147$	−1.34480	−0.110917
$148$	0	0
$149$	12.7033	1.04069	0.520347	−	0.853955i	$-0.325803\pi$
$149$	12.7033	1.04069	0.520347	+	0.853955i	$0.325803\pi$
$150$	0	0
$151$	−11.3823	−0.926282	−0.463141	−	0.886285i	$-0.653278\pi$
$151$	−11.3823	−0.926282	−0.463141	+	0.886285i	$0.653278\pi$
$152$	0	0
$153$	−37.4305	−3.02608
$154$	0	0
$155$	−5.60786	−0.450434
$156$	0	0
$157$	−15.4083	−1.22972	−0.614859	−	0.788637i	$-0.710787\pi$
$157$	−15.4083	−1.22972	−0.614859	+	0.788637i	$0.710787\pi$
$158$	0	0
$159$	22.6132	1.79334
$160$	0	0
$161$	2.09004	0.164718
$162$	0	0
$163$	−20.5901	−1.61274	−0.806369	−	0.591413i	$-0.798570\pi$
$163$	−20.5901	−1.61274	−0.806369	+	0.591413i	$0.798570\pi$
$164$	0	0
$165$	−15.0175	−1.16911
$166$	0	0
$167$	−1.89769	−0.146848	−0.0734238	−	0.997301i	$-0.523393\pi$
$167$	−1.89769	−0.146848	−0.0734238	+	0.997301i	$0.523393\pi$
$168$	0	0
$169$	28.7749	2.21346
$170$	0	0
$171$	−15.6779	−1.19892
$172$	0	0
$173$	4.05500	0.308296	0.154148	−	0.988048i	$-0.450737\pi$
$173$	4.05500	0.308296	0.154148	+	0.988048i	$0.450737\pi$
$174$	0	0
$175$	5.66347	0.428118
$176$	0	0
$177$	−34.8359	−2.61842
$178$	0	0
$179$	−17.1662	−1.28306	−0.641532	−	0.767097i	$-0.721701\pi$
$179$	−17.1662	−1.28306	−0.641532	+	0.767097i	$0.721701\pi$
$180$	0	0
$181$	−10.1279	−0.752797	−0.376399	−	0.926458i	$-0.622838\pi$
$181$	−10.1279	−0.752797	−0.376399	+	0.926458i	$0.622838\pi$
$182$	0	0
$183$	−3.28207	−0.242617
$184$	0	0
$185$	−19.0374	−1.39966
$186$	0	0
$187$	12.8972	0.943140
$188$	0	0
$189$	42.6330	3.10109
$190$	0	0
$191$	2.18523	0.158117	0.0790587	−	0.996870i	$-0.474809\pi$
$191$	2.18523	0.158117	0.0790587	+	0.996870i	$0.474809\pi$
$192$	0	0
$193$	13.5649	0.976425	0.488212	−	0.872725i	$-0.337649\pi$
$193$	13.5649	0.976425	0.488212	+	0.872725i	$0.337649\pi$
$194$	0	0
$195$	−36.2456	−2.59560
$196$	0	0
$197$	18.4030	1.31116	0.655580	−	0.755126i	$-0.272424\pi$
$197$	18.4030	1.31116	0.655580	+	0.755126i	$0.272424\pi$
$198$	0	0
$199$	3.38279	0.239800	0.119900	−	0.992786i	$-0.461743\pi$
$199$	3.38279	0.239800	0.119900	+	0.992786i	$0.461743\pi$
$200$	0	0
$201$	−6.81609	−0.480770
$202$	0	0
$203$	−2.27539	−0.159701
$204$	0	0
$205$	3.09154	0.215922
$206$	0	0
$207$	−5.96737	−0.414761
$208$	0	0
$209$	5.40206	0.373669
$210$	0	0
$211$	13.2132	0.909631	0.454815	−	0.890586i	$-0.349705\pi$
$211$	13.2132	0.909631	0.454815	+	0.890586i	$0.349705\pi$
$212$	0	0
$213$	−43.9798	−3.01345
$214$	0	0
$215$	9.58179	0.653472
$216$	0	0
$217$	−8.93406	−0.606483
$218$	0	0
$219$	−38.1621	−2.57875
$220$	0	0
$221$	31.1281	2.09390
$222$	0	0
$223$	−4.55602	−0.305093	−0.152547	−	0.988296i	$-0.548747\pi$
$223$	−4.55602	−0.305093	−0.152547	+	0.988296i	$0.548747\pi$
$224$	0	0
$225$	−16.1700	−1.07800
$226$	0	0
$227$	−14.6862	−0.974760	−0.487380	−	0.873190i	$-0.662047\pi$
$227$	−14.6862	−0.974760	−0.487380	+	0.873190i	$0.662047\pi$
$228$	0	0
$229$	−1.15209	−0.0761323	−0.0380661	−	0.999275i	$-0.512120\pi$
$229$	−1.15209	−0.0761323	−0.0380661	+	0.999275i	$0.512120\pi$
$230$	0	0
$231$	−23.9249	−1.57415
$232$	0	0
$233$	−3.60786	−0.236359	−0.118179	−	0.992992i	$-0.537706\pi$
$233$	−3.60786	−0.236359	−0.118179	+	0.992992i	$0.537706\pi$
$234$	0	0
$235$	−14.5870	−0.951553
$236$	0	0
$237$	37.8730	2.46012
$238$	0	0
$239$	4.72393	0.305566	0.152783	−	0.988260i	$-0.451176\pi$
$239$	4.72393	0.305566	0.152783	+	0.988260i	$0.451176\pi$
$240$	0	0
$241$	−19.2473	−1.23983	−0.619915	−	0.784669i	$-0.712833\pi$
$241$	−19.2473	−1.23983	−0.619915	+	0.784669i	$0.712833\pi$
$242$	0	0
$243$	−45.1992	−2.89953
$244$	0	0
$245$	−0.700099	−0.0447277
$246$	0	0
$247$	13.0381	0.829597
$248$	0	0
$249$	37.0043	2.34506
$250$	0	0
$251$	−7.64634	−0.482633	−0.241316	−	0.970447i	$-0.577579\pi$
$251$	−7.64634	−0.482633	−0.241316	+	0.970447i	$0.577579\pi$
$252$	0	0
$253$	2.05615	0.129269
$254$	0	0
$255$	−27.0080	−1.69130
$256$	0	0
$257$	0.414443	0.0258522	0.0129261	−	0.999916i	$-0.495885\pi$
$257$	0.414443	0.0258522	0.0129261	+	0.999916i	$0.495885\pi$
$258$	0	0
$259$	−30.3291	−1.88456
$260$	0	0
$261$	6.49657	0.402128
$262$	0	0
$263$	−8.64396	−0.533009	−0.266505	−	0.963834i	$-0.585869\pi$
$263$	−8.64396	−0.533009	−0.266505	+	0.963834i	$0.585869\pi$
$264$	0	0
$265$	11.7724	0.723171
$266$	0	0
$267$	57.1202	3.49570
$268$	0	0
$269$	12.8310	0.782317	0.391159	−	0.920323i	$-0.372074\pi$
$269$	12.8310	0.782317	0.391159	+	0.920323i	$0.372074\pi$
$270$	0	0
$271$	−29.2658	−1.77777	−0.888884	−	0.458131i	$-0.848519\pi$
$271$	−29.2658	−1.77777	−0.888884	+	0.458131i	$0.848519\pi$
$272$	0	0
$273$	−57.7440	−3.49482
$274$	0	0
$275$	5.57164	0.335982
$276$	0	0
$277$	4.30590	0.258717	0.129358	−	0.991598i	$-0.458708\pi$
$277$	4.30590	0.258717	0.129358	+	0.991598i	$0.458708\pi$
$278$	0	0
$279$	25.5081	1.52713
$280$	0	0
$281$	−26.0249	−1.55252	−0.776258	−	0.630416i	$-0.782884\pi$
$281$	−26.0249	−1.55252	−0.776258	+	0.630416i	$0.782884\pi$
$282$	0	0
$283$	−5.84737	−0.347590	−0.173795	−	0.984782i	$-0.555603\pi$
$283$	−5.84737	−0.347590	−0.173795	+	0.984782i	$0.555603\pi$
$284$	0	0
$285$	−11.3124	−0.670089
$286$	0	0
$287$	4.92523	0.290727
$288$	0	0
$289$	6.19477	0.364398
$290$	0	0
$291$	52.1737	3.05848
$292$	0	0
$293$	13.0131	0.760233	0.380117	−	0.924939i	$-0.375884\pi$
$293$	13.0131	0.760233	0.380117	+	0.924939i	$0.375884\pi$
$294$	0	0
$295$	−18.1355	−1.05589
$296$	0	0
$297$	41.9417	2.43370
$298$	0	0
$299$	4.96261	0.286995
$300$	0	0
$301$	15.2651	0.879863
$302$	0	0
$303$	−2.47182	−0.142003
$304$	0	0
$305$	−1.70864	−0.0978362
$306$	0	0
$307$	1.74600	0.0996497	0.0498249	−	0.998758i	$-0.484134\pi$
$307$	1.74600	0.0996497	0.0498249	+	0.998758i	$0.484134\pi$
$308$	0	0
$309$	8.46827	0.481743
$310$	0	0
$311$	31.7472	1.80022	0.900110	−	0.435663i	$-0.143486\pi$
$311$	31.7472	1.80022	0.900110	+	0.435663i	$0.143486\pi$
$312$	0	0
$313$	23.2980	1.31688	0.658441	−	0.752632i	$-0.271216\pi$
$313$	23.2980	1.31688	0.658441	+	0.752632i	$0.271216\pi$
$314$	0	0
$315$	36.1478	2.03670
$316$	0	0
$317$	11.0950	0.623155	0.311577	−	0.950221i	$-0.399143\pi$
$317$	11.0950	0.623155	0.311577	+	0.950221i	$0.399143\pi$
$318$	0	0
$319$	−2.23849	−0.125332
$320$	0	0
$321$	−24.5376	−1.36956
$322$	0	0
$323$	9.71522	0.540570
$324$	0	0
$325$	13.4474	0.745928
$326$	0	0
$327$	−4.29628	−0.237585
$328$	0	0
$329$	−23.2391	−1.28121
$330$	0	0
$331$	−7.12035	−0.391370	−0.195685	−	0.980667i	$-0.562693\pi$
$331$	−7.12035	−0.391370	−0.195685	+	0.980667i	$0.562693\pi$
$332$	0	0
$333$	86.5943	4.74534
$334$	0	0
$335$	−3.54844	−0.193872
$336$	0	0
$337$	−17.0669	−0.929693	−0.464846	−	0.885391i	$-0.653890\pi$
$337$	−17.0669	−0.929693	−0.464846	+	0.885391i	$0.653890\pi$
$338$	0	0
$339$	19.6435	1.06689
$340$	0	0
$341$	−8.78920	−0.475962
$342$	0	0
$343$	17.9392	0.968628
$344$	0	0
$345$	−4.30575	−0.231814
$346$	0	0
$347$	16.7187	0.897507	0.448753	−	0.893656i	$-0.351868\pi$
$347$	16.7187	0.897507	0.448753	+	0.893656i	$0.351868\pi$
$348$	0	0
$349$	24.6573	1.31988	0.659938	−	0.751320i	$-0.270583\pi$
$349$	24.6573	1.31988	0.659938	+	0.751320i	$0.270583\pi$
$350$	0	0
$351$	101.228	5.40316
$352$	0	0
$353$	27.0850	1.44159	0.720794	−	0.693149i	$-0.243777\pi$
$353$	27.0850	1.44159	0.720794	+	0.693149i	$0.243777\pi$
$354$	0	0
$355$	−22.8958	−1.21518
$356$	0	0
$357$	−43.0273	−2.27725
$358$	0	0
$359$	0.805415	0.0425082	0.0212541	−	0.999774i	$-0.493234\pi$
$359$	0.805415	0.0425082	0.0212541	+	0.999774i	$0.493234\pi$
$360$	0	0
$361$	−14.9307	−0.785828
$362$	0	0
$363$	12.5657	0.659529
$364$	0	0
$365$	−19.8671	−1.03989
$366$	0	0
$367$	−34.2154	−1.78603	−0.893014	−	0.450029i	$-0.851414\pi$
$367$	−34.2154	−1.78603	−0.893014	+	0.450029i	$0.851414\pi$
$368$	0	0
$369$	−14.0623	−0.732052
$370$	0	0
$371$	18.7549	0.973708
$372$	0	0
$373$	10.8822	0.563458	0.281729	−	0.959494i	$-0.409092\pi$
$373$	10.8822	0.563458	0.281729	+	0.959494i	$0.409092\pi$
$374$	0	0
$375$	−39.7068	−2.05045
$376$	0	0
$377$	−5.40271	−0.278254
$378$	0	0
$379$	3.32612	0.170851	0.0854257	−	0.996345i	$-0.472775\pi$
$379$	3.32612	0.170851	0.0854257	+	0.996345i	$0.472775\pi$
$380$	0	0
$381$	−11.4528	−0.586745
$382$	0	0
$383$	−4.93768	−0.252304	−0.126152	−	0.992011i	$-0.540263\pi$
$383$	−4.93768	−0.252304	−0.126152	+	0.992011i	$0.540263\pi$
$384$	0	0
$385$	−12.4553	−0.634779
$386$	0	0
$387$	−43.5840	−2.21550
$388$	0	0
$389$	−5.86574	−0.297405	−0.148702	−	0.988882i	$-0.547510\pi$
$389$	−5.86574	−0.297405	−0.148702	+	0.988882i	$0.547510\pi$
$390$	0	0
$391$	3.69783	0.187007
$392$	0	0
$393$	−36.3866	−1.83546
$394$	0	0
$395$	19.7166	0.992050
$396$	0	0
$397$	6.13558	0.307936	0.153968	−	0.988076i	$-0.450795\pi$
$397$	6.13558	0.307936	0.153968	+	0.988076i	$0.450795\pi$
$398$	0	0
$399$	−18.0222	−0.902236
$400$	0	0
$401$	−6.59114	−0.329146	−0.164573	−	0.986365i	$-0.552625\pi$
$401$	−6.59114	−0.329146	−0.164573	+	0.986365i	$0.552625\pi$
$402$	0	0
$403$	−21.2132	−1.05670
$404$	0	0
$405$	−47.9912	−2.38470
$406$	0	0
$407$	−29.8374	−1.47898
$408$	0	0
$409$	38.6619	1.91171	0.955854	−	0.293843i	$-0.0949344\pi$
$409$	38.6619	1.91171	0.955854	+	0.293843i	$0.0949344\pi$
$410$	0	0
$411$	−68.0215	−3.35525
$412$	0	0
$413$	−28.8922	−1.42169
$414$	0	0
$415$	19.2644	0.945651
$416$	0	0
$417$	21.9701	1.07588
$418$	0	0
$419$	−16.2591	−0.794309	−0.397154	−	0.917752i	$-0.630002\pi$
$419$	−16.2591	−0.794309	−0.397154	+	0.917752i	$0.630002\pi$
$420$	0	0
$421$	−25.1680	−1.22661	−0.613306	−	0.789845i	$-0.710161\pi$
$421$	−25.1680	−1.22661	−0.613306	+	0.789845i	$0.710161\pi$
$422$	0	0
$423$	66.3510	3.22610
$424$	0	0
$425$	10.0202	0.486050
$426$	0	0
$427$	−2.72208	−0.131731
$428$	0	0
$429$	−56.8077	−2.74270
$430$	0	0
$431$	−1.22918	−0.0592077	−0.0296038	−	0.999562i	$-0.509425\pi$
$431$	−1.22918	−0.0592077	−0.0296038	+	0.999562i	$0.509425\pi$
$432$	0	0
$433$	10.8662	0.522194	0.261097	−	0.965313i	$-0.415916\pi$
$433$	10.8662	0.522194	0.261097	+	0.965313i	$0.415916\pi$
$434$	0	0
$435$	4.68760	0.224753
$436$	0	0
$437$	1.54885	0.0740917
$438$	0	0
$439$	25.2820	1.20664	0.603322	−	0.797498i	$-0.293843\pi$
$439$	25.2820	1.20664	0.603322	+	0.797498i	$0.293843\pi$
$440$	0	0
$441$	3.18449	0.151643
$442$	0	0
$443$	30.5911	1.45343	0.726713	−	0.686941i	$-0.241047\pi$
$443$	30.5911	1.45343	0.726713	+	0.686941i	$0.241047\pi$
$444$	0	0
$445$	29.7366	1.40965
$446$	0	0
$447$	−41.6930	−1.97201
$448$	0	0
$449$	28.5146	1.34569	0.672843	−	0.739785i	$-0.265073\pi$
$449$	28.5146	1.34569	0.672843	+	0.739785i	$0.265073\pi$
$450$	0	0
$451$	4.84537	0.228160
$452$	0	0
$453$	37.3576	1.75521
$454$	0	0
$455$	−30.0614	−1.40930
$456$	0	0
$457$	−10.5866	−0.495219	−0.247610	−	0.968860i	$-0.579645\pi$
$457$	−10.5866	−0.495219	−0.247610	+	0.968860i	$0.579645\pi$
$458$	0	0
$459$	75.4291	3.52073
$460$	0	0
$461$	10.0198	0.466667	0.233334	−	0.972397i	$-0.425037\pi$
$461$	10.0198	0.466667	0.233334	+	0.972397i	$0.425037\pi$
$462$	0	0
$463$	−32.7324	−1.52120	−0.760602	−	0.649219i	$-0.775096\pi$
$463$	−32.7324	−1.52120	−0.760602	+	0.649219i	$0.775096\pi$
$464$	0	0
$465$	18.4054	0.853528
$466$	0	0
$467$	25.8250	1.19504	0.597519	−	0.801854i	$-0.296153\pi$
$467$	25.8250	1.19504	0.597519	+	0.801854i	$0.296153\pi$
$468$	0	0
$469$	−5.65314	−0.261038
$470$	0	0
$471$	50.5712	2.33020
$472$	0	0
$473$	15.0175	0.690507
$474$	0	0
$475$	4.19700	0.192571
$476$	0	0
$477$	−53.5482	−2.45180
$478$	0	0
$479$	−17.4960	−0.799413	−0.399706	−	0.916643i	$-0.630888\pi$
$479$	−17.4960	−0.799413	−0.399706	+	0.916643i	$0.630888\pi$
$480$	0	0
$481$	−72.0139	−3.28355
$482$	0	0
$483$	−6.85964	−0.312124
$484$	0	0
$485$	27.1615	1.23334
$486$	0	0
$487$	35.6461	1.61528	0.807640	−	0.589676i	$-0.200745\pi$
$487$	35.6461	1.61528	0.807640	+	0.589676i	$0.200745\pi$
$488$	0	0
$489$	67.5779	3.05598
$490$	0	0
$491$	−17.3473	−0.782873	−0.391436	−	0.920205i	$-0.628022\pi$
$491$	−17.3473	−0.782873	−0.391436	+	0.920205i	$0.628022\pi$
$492$	0	0
$493$	−4.02577	−0.181311
$494$	0	0
$495$	35.5617	1.59838
$496$	0	0
$497$	−36.4760	−1.63617
$498$	0	0
$499$	−3.75328	−0.168020	−0.0840099	−	0.996465i	$-0.526773\pi$
$499$	−3.75328	−0.168020	−0.0840099	+	0.996465i	$0.526773\pi$
$500$	0	0
$501$	6.22834	0.278262
$502$	0	0
$503$	−1.99309	−0.0888676	−0.0444338	−	0.999012i	$-0.514148\pi$
$503$	−1.99309	−0.0888676	−0.0444338	+	0.999012i	$0.514148\pi$
$504$	0	0
$505$	−1.28683	−0.0572630
$506$	0	0
$507$	−94.4412	−4.19428
$508$	0	0
$509$	11.0683	0.490595	0.245297	−	0.969448i	$-0.421114\pi$
$509$	11.0683	0.490595	0.245297	+	0.969448i	$0.421114\pi$
$510$	0	0
$511$	−31.6509	−1.40015
$512$	0	0
$513$	31.5938	1.39490
$514$	0	0
$515$	4.40856	0.194264
$516$	0	0
$517$	−22.8623	−1.00548
$518$	0	0
$519$	−13.3088	−0.584191
$520$	0	0
$521$	21.9359	0.961031	0.480516	−	0.876986i	$-0.340450\pi$
$521$	21.9359	0.961031	0.480516	+	0.876986i	$0.340450\pi$
$522$	0	0
$523$	19.9780	0.873579	0.436790	−	0.899564i	$-0.356115\pi$
$523$	19.9780	0.873579	0.436790	+	0.899564i	$0.356115\pi$
$524$	0	0
$525$	−18.5879	−0.811241
$526$	0	0
$527$	−15.8067	−0.688552
$528$	0	0
$529$	−22.4105	−0.974368
$530$	0	0
$531$	82.4916	3.57983
$532$	0	0
$533$	11.6945	0.506546
$534$	0	0
$535$	−12.7742	−0.552278
$536$	0	0
$537$	56.3407	2.43128
$538$	0	0
$539$	−1.09727	−0.0472626
$540$	0	0
$541$	−1.67851	−0.0721648	−0.0360824	−	0.999349i	$-0.511488\pi$
$541$	−1.67851	−0.0721648	−0.0360824	+	0.999349i	$0.511488\pi$
$542$	0	0
$543$	33.2403	1.42648
$544$	0	0
$545$	−2.23664	−0.0958069
$546$	0	0
$547$	20.8557	0.891724	0.445862	−	0.895102i	$-0.352897\pi$
$547$	20.8557	0.891724	0.445862	+	0.895102i	$0.352897\pi$
$548$	0	0
$549$	7.77196	0.331699
$550$	0	0
$551$	−1.68621	−0.0718349
$552$	0	0
$553$	31.4112	1.33574
$554$	0	0
$555$	62.4821	2.65222
$556$	0	0
$557$	4.84438	0.205263	0.102632	−	0.994719i	$-0.467274\pi$
$557$	4.84438	0.205263	0.102632	+	0.994719i	$0.467274\pi$
$558$	0	0
$559$	36.2456	1.53302
$560$	0	0
$561$	−42.3296	−1.78716
$562$	0	0
$563$	−6.34412	−0.267373	−0.133686	−	0.991024i	$-0.542682\pi$
$563$	−6.34412	−0.267373	−0.133686	+	0.991024i	$0.542682\pi$
$564$	0	0
$565$	10.2264	0.430227
$566$	0	0
$567$	−76.4564	−3.21087
$568$	0	0
$569$	−26.6646	−1.11784	−0.558919	−	0.829222i	$-0.688784\pi$
$569$	−26.6646	−1.11784	−0.558919	+	0.829222i	$0.688784\pi$
$570$	0	0
$571$	−13.9276	−0.582852	−0.291426	−	0.956593i	$-0.594130\pi$
$571$	−13.9276	−0.582852	−0.291426	+	0.956593i	$0.594130\pi$
$572$	0	0
$573$	−7.17206	−0.299617
$574$	0	0
$575$	1.59747	0.0666192
$576$	0	0
$577$	−12.6719	−0.527539	−0.263769	−	0.964586i	$-0.584966\pi$
$577$	−12.6719	−0.527539	−0.263769	+	0.964586i	$0.584966\pi$
$578$	0	0
$579$	−44.5210	−1.85023
$580$	0	0
$581$	30.6907	1.27327
$582$	0	0
$583$	18.4508	0.764156
$584$	0	0
$585$	85.8297	3.54862
$586$	0	0
$587$	−15.4357	−0.637100	−0.318550	−	0.947906i	$-0.603196\pi$
$587$	−15.4357	−0.637100	−0.318550	+	0.947906i	$0.603196\pi$
$588$	0	0
$589$	−6.62072	−0.272802
$590$	0	0
$591$	−60.3999	−2.48452
$592$	0	0
$593$	−31.3316	−1.28663	−0.643317	−	0.765600i	$-0.722442\pi$
$593$	−31.3316	−1.28663	−0.643317	+	0.765600i	$0.722442\pi$
$594$	0	0
$595$	−22.3999	−0.918306
$596$	0	0
$597$	−11.1025	−0.454397
$598$	0	0
$599$	19.8702	0.811872	0.405936	−	0.913901i	$-0.366946\pi$
$599$	19.8702	0.811872	0.405936	+	0.913901i	$0.366946\pi$
$600$	0	0
$601$	−15.2192	−0.620802	−0.310401	−	0.950606i	$-0.600463\pi$
$601$	−15.2192	−0.620802	−0.310401	+	0.950606i	$0.600463\pi$
$602$	0	0
$603$	16.1406	0.657295
$604$	0	0
$605$	6.54168	0.265957
$606$	0	0
$607$	−16.1226	−0.654398	−0.327199	−	0.944955i	$-0.606105\pi$
$607$	−16.1226	−0.654398	−0.327199	+	0.944955i	$0.606105\pi$
$608$	0	0
$609$	7.46797	0.302617
$610$	0	0
$611$	−55.1792	−2.23231
$612$	0	0
$613$	−7.61748	−0.307667	−0.153834	−	0.988097i	$-0.549162\pi$
$613$	−7.61748	−0.307667	−0.153834	+	0.988097i	$0.549162\pi$
$614$	0	0
$615$	−10.1466	−0.409152
$616$	0	0
$617$	−12.9318	−0.520613	−0.260307	−	0.965526i	$-0.583824\pi$
$617$	−12.9318	−0.520613	−0.260307	+	0.965526i	$0.583824\pi$
$618$	0	0
$619$	44.3329	1.78189	0.890944	−	0.454113i	$-0.150044\pi$
$619$	44.3329	1.78189	0.890944	+	0.454113i	$0.150044\pi$
$620$	0	0
$621$	12.0253	0.482559
$622$	0	0
$623$	47.3744	1.89802
$624$	0	0
$625$	−10.2684	−0.410738
$626$	0	0
$627$	−17.7299	−0.708065
$628$	0	0
$629$	−53.6604	−2.13958
$630$	0	0
$631$	23.7875	0.946964	0.473482	−	0.880803i	$-0.342997\pi$
$631$	23.7875	0.946964	0.473482	+	0.880803i	$0.342997\pi$
$632$	0	0
$633$	−43.3664	−1.72366
$634$	0	0
$635$	−5.96231	−0.236607
$636$	0	0
$637$	−2.64830	−0.104930
$638$	0	0
$639$	104.144	4.11989
$640$	0	0
$641$	16.0122	0.632443	0.316222	−	0.948685i	$-0.397586\pi$
$641$	16.0122	0.632443	0.316222	+	0.948685i	$0.397586\pi$
$642$	0	0
$643$	43.6890	1.72293	0.861463	−	0.507820i	$-0.169549\pi$
$643$	43.6890	1.72293	0.861463	+	0.507820i	$0.169549\pi$
$644$	0	0
$645$	−31.4481	−1.23827
$646$	0	0
$647$	5.55942	0.218563	0.109282	−	0.994011i	$-0.465145\pi$
$647$	5.55942	0.218563	0.109282	+	0.994011i	$0.465145\pi$
$648$	0	0
$649$	−28.4237	−1.11573
$650$	0	0
$651$	29.3222	1.14923
$652$	0	0
$653$	−0.353890	−0.0138488	−0.00692439	−	0.999976i	$-0.502204\pi$
$653$	−0.353890	−0.0138488	−0.00692439	+	0.999976i	$0.502204\pi$
$654$	0	0
$655$	−18.9428	−0.740157
$656$	0	0
$657$	90.3681	3.52559
$658$	0	0
$659$	15.9843	0.622660	0.311330	−	0.950302i	$-0.399225\pi$
$659$	15.9843	0.622660	0.311330	+	0.950302i	$0.399225\pi$
$660$	0	0
$661$	−4.12174	−0.160317	−0.0801585	−	0.996782i	$-0.525543\pi$
$661$	−4.12174	−0.160317	−0.0801585	+	0.996782i	$0.525543\pi$
$662$	0	0
$663$	−102.165	−3.96774
$664$	0	0
$665$	−9.38229	−0.363830
$666$	0	0
$667$	−0.641809	−0.0248510
$668$	0	0
$669$	14.9531	0.578122
$670$	0	0
$671$	−2.67795	−0.103381
$672$	0	0
$673$	−26.3364	−1.01519	−0.507596	−	0.861595i	$-0.669466\pi$
$673$	−26.3364	−1.01519	−0.507596	+	0.861595i	$0.669466\pi$
$674$	0	0
$675$	32.5855	1.25422
$676$	0	0
$677$	−36.7845	−1.41374	−0.706871	−	0.707343i	$-0.749894\pi$
$677$	−36.7845	−1.41374	−0.706871	+	0.707343i	$0.749894\pi$
$678$	0	0
$679$	43.2719	1.66062
$680$	0	0
$681$	48.2012	1.84708
$682$	0	0
$683$	12.1866	0.466306	0.233153	−	0.972440i	$-0.425096\pi$
$683$	12.1866	0.466306	0.233153	+	0.972440i	$0.425096\pi$
$684$	0	0
$685$	−35.4118	−1.35302
$686$	0	0
$687$	3.78124	0.144263
$688$	0	0
$689$	44.5320	1.69653
$690$	0	0
$691$	−15.7794	−0.600278	−0.300139	−	0.953896i	$-0.597033\pi$
$691$	−15.7794	−0.600278	−0.300139	+	0.953896i	$0.597033\pi$
$692$	0	0
$693$	56.6544	2.15212
$694$	0	0
$695$	11.4376	0.433852
$696$	0	0
$697$	8.71405	0.330068
$698$	0	0
$699$	11.8412	0.447876
$700$	0	0
$701$	6.01612	0.227226	0.113613	−	0.993525i	$-0.463758\pi$
$701$	6.01612	0.227226	0.113613	+	0.993525i	$0.463758\pi$
$702$	0	0
$703$	−22.4759	−0.847693
$704$	0	0
$705$	47.8756	1.80310
$706$	0	0
$707$	−2.05008	−0.0771013
$708$	0	0
$709$	−26.6450	−1.00067	−0.500337	−	0.865831i	$-0.666791\pi$
$709$	−26.6450	−1.00067	−0.500337	+	0.865831i	$0.666791\pi$
$710$	0	0
$711$	−89.6836	−3.36340
$712$	0	0
$713$	−2.52000	−0.0943746
$714$	0	0
$715$	−29.5740	−1.10600
$716$	0	0
$717$	−15.5043	−0.579017
$718$	0	0
$719$	34.2234	1.27632	0.638159	−	0.769905i	$-0.279696\pi$
$719$	34.2234	1.27632	0.638159	+	0.769905i	$0.279696\pi$
$720$	0	0
$721$	7.02343	0.261566
$722$	0	0
$723$	63.1711	2.34936
$724$	0	0
$725$	−1.73914	−0.0645900
$726$	0	0
$727$	−6.91463	−0.256449	−0.128225	−	0.991745i	$-0.540928\pi$
$727$	−6.91463	−0.256449	−0.128225	+	0.991745i	$0.540928\pi$
$728$	0	0
$729$	64.0845	2.37350
$730$	0	0
$731$	27.0080	0.998926
$732$	0	0
$733$	−20.5802	−0.760145	−0.380073	−	0.924957i	$-0.624101\pi$
$733$	−20.5802	−0.760145	−0.380073	+	0.924957i	$0.624101\pi$
$734$	0	0
$735$	2.29777	0.0847546
$736$	0	0
$737$	−5.56148	−0.204860
$738$	0	0
$739$	−33.4347	−1.22992	−0.614958	−	0.788560i	$-0.710827\pi$
$739$	−33.4347	−1.22992	−0.614958	+	0.788560i	$0.710827\pi$
$740$	0	0
$741$	−42.7920	−1.57200
$742$	0	0
$743$	−37.2834	−1.36780	−0.683898	−	0.729578i	$-0.739716\pi$
$743$	−37.2834	−1.36780	−0.683898	+	0.729578i	$0.739716\pi$
$744$	0	0
$745$	−21.7053	−0.795220
$746$	0	0
$747$	−87.6266	−3.20609
$748$	0	0
$749$	−20.3510	−0.743611
$750$	0	0
$751$	12.1022	0.441614	0.220807	−	0.975317i	$-0.429131\pi$
$751$	12.1022	0.441614	0.220807	+	0.975317i	$0.429131\pi$
$752$	0	0
$753$	25.0958	0.914541
$754$	0	0
$755$	19.4483	0.707795
$756$	0	0
$757$	41.0751	1.49290	0.746449	−	0.665442i	$-0.231757\pi$
$757$	41.0751	1.49290	0.746449	+	0.665442i	$0.231757\pi$
$758$	0	0
$759$	−6.74841	−0.244952
$760$	0	0
$761$	19.1574	0.694454	0.347227	−	0.937781i	$-0.387123\pi$
$761$	19.1574	0.694454	0.347227	+	0.937781i	$0.387123\pi$
$762$	0	0
$763$	−3.56326	−0.128999
$764$	0	0
$765$	63.9551	2.31230
$766$	0	0
$767$	−68.6021	−2.47708
$768$	0	0
$769$	−0.628185	−0.0226529	−0.0113265	−	0.999936i	$-0.503605\pi$
$769$	−0.628185	−0.0226529	−0.0113265	+	0.999936i	$0.503605\pi$
$770$	0	0
$771$	−1.36023	−0.0489874
$772$	0	0
$773$	39.1120	1.40676	0.703381	−	0.710813i	$-0.251673\pi$
$773$	39.1120	1.40676	0.703381	+	0.710813i	$0.251673\pi$
$774$	0	0
$775$	−6.82854	−0.245289
$776$	0	0
$777$	99.5423	3.57106
$778$	0	0
$779$	3.64992	0.130772
$780$	0	0
$781$	−35.8846	−1.28405
$782$	0	0
$783$	−13.0917	−0.467861
$784$	0	0
$785$	26.3272	0.939659
$786$	0	0
$787$	27.9705	0.997042	0.498521	−	0.866878i	$-0.333877\pi$
$787$	27.9705	0.997042	0.498521	+	0.866878i	$0.333877\pi$
$788$	0	0
$789$	28.3700	1.01000
$790$	0	0
$791$	16.2920	0.579276
$792$	0	0
$793$	−6.46335	−0.229520
$794$	0	0
$795$	−38.6377	−1.37034
$796$	0	0
$797$	16.0142	0.567254	0.283627	−	0.958935i	$-0.408462\pi$
$797$	16.0142	0.567254	0.283627	+	0.958935i	$0.408462\pi$
$798$	0	0
$799$	−41.1161	−1.45458
$800$	0	0
$801$	−135.261	−4.77922
$802$	0	0
$803$	−31.1377	−1.09883
$804$	0	0
$805$	−3.57111	−0.125865
$806$	0	0
$807$	−42.1120	−1.48241
$808$	0	0
$809$	−1.67095	−0.0587474	−0.0293737	−	0.999568i	$-0.509351\pi$
$809$	−1.67095	−0.0587474	−0.0293737	+	0.999568i	$0.509351\pi$
$810$	0	0
$811$	−29.0590	−1.02040	−0.510199	−	0.860056i	$-0.670428\pi$
$811$	−29.0590	−1.02040	−0.510199	+	0.860056i	$0.670428\pi$
$812$	0	0
$813$	96.0522	3.36870
$814$	0	0
$815$	35.1809	1.23233
$816$	0	0
$817$	11.3124	0.395771
$818$	0	0
$819$	136.738	4.77802
$820$	0	0
$821$	−7.90046	−0.275728	−0.137864	−	0.990451i	$-0.544024\pi$
$821$	−7.90046	−0.275728	−0.137864	+	0.990451i	$0.544024\pi$
$822$	0	0
$823$	−25.4724	−0.887913	−0.443956	−	0.896048i	$-0.646426\pi$
$823$	−25.4724	−0.887913	−0.443956	+	0.896048i	$0.646426\pi$
$824$	0	0
$825$	−18.2865	−0.636653
$826$	0	0
$827$	−15.4891	−0.538609	−0.269304	−	0.963055i	$-0.586794\pi$
$827$	−15.4891	−0.538609	−0.269304	+	0.963055i	$0.586794\pi$
$828$	0	0
$829$	25.4134	0.882644	0.441322	−	0.897349i	$-0.354510\pi$
$829$	25.4134	0.882644	0.441322	+	0.897349i	$0.354510\pi$
$830$	0	0
$831$	−14.1323	−0.490243
$832$	0	0
$833$	−1.97335	−0.0683727
$834$	0	0
$835$	3.24246	0.112210
$836$	0	0
$837$	−51.4033	−1.77676
$838$	0	0
$839$	5.44058	0.187830	0.0939148	−	0.995580i	$-0.470062\pi$
$839$	5.44058	0.187830	0.0939148	+	0.995580i	$0.470062\pi$
$840$	0	0
$841$	−28.3013	−0.975906
$842$	0	0
$843$	85.4154	2.94186
$844$	0	0
$845$	−49.1659	−1.69136
$846$	0	0
$847$	10.4218	0.358096
$848$	0	0
$849$	19.1915	0.658649
$850$	0	0
$851$	−8.55482	−0.293256
$852$	0	0
$853$	46.0438	1.57651	0.788255	−	0.615349i	$-0.210985\pi$
$853$	46.0438	1.57651	0.788255	+	0.615349i	$0.210985\pi$
$854$	0	0
$855$	26.7878	0.916125
$856$	0	0
$857$	−4.52511	−0.154575	−0.0772875	−	0.997009i	$-0.524626\pi$
$857$	−4.52511	−0.154575	−0.0772875	+	0.997009i	$0.524626\pi$
$858$	0	0
$859$	17.0282	0.580994	0.290497	−	0.956876i	$-0.406179\pi$
$859$	17.0282	0.580994	0.290497	+	0.956876i	$0.406179\pi$
$860$	0	0
$861$	−16.1649	−0.550899
$862$	0	0
$863$	5.75648	0.195953	0.0979765	−	0.995189i	$-0.468763\pi$
$863$	5.75648	0.195953	0.0979765	+	0.995189i	$0.468763\pi$
$864$	0	0
$865$	−6.92852	−0.235577
$866$	0	0
$867$	−20.3316	−0.690499
$868$	0	0
$869$	30.9019	1.04827
$870$	0	0
$871$	−13.4229	−0.454817
$872$	0	0
$873$	−123.548	−4.18146
$874$	0	0
$875$	−32.9321	−1.11331
$876$	0	0
$877$	0.808282	0.0272937	0.0136469	−	0.999907i	$-0.495656\pi$
$877$	0.808282	0.0272937	0.0136469	+	0.999907i	$0.495656\pi$
$878$	0	0
$879$	−42.7098	−1.44057
$880$	0	0
$881$	7.63093	0.257093	0.128546	−	0.991704i	$-0.458969\pi$
$881$	7.63093	0.257093	0.128546	+	0.991704i	$0.458969\pi$
$882$	0	0
$883$	31.2197	1.05063	0.525314	−	0.850908i	$-0.323948\pi$
$883$	31.2197	1.05063	0.525314	+	0.850908i	$0.323948\pi$
$884$	0	0
$885$	59.5218	2.00080
$886$	0	0
$887$	41.3888	1.38970	0.694849	−	0.719155i	$-0.255471\pi$
$887$	41.3888	1.38970	0.694849	+	0.719155i	$0.255471\pi$
$888$	0	0
$889$	−9.49875	−0.318578
$890$	0	0
$891$	−75.2167	−2.51985
$892$	0	0
$893$	−17.2217	−0.576301
$894$	0	0
$895$	29.3308	0.980421
$896$	0	0
$897$	−16.2876	−0.543828
$898$	0	0
$899$	2.74348	0.0915000
$900$	0	0
$901$	33.1825	1.10547
$902$	0	0
$903$	−50.1009	−1.66725
$904$	0	0
$905$	17.3048	0.575231
$906$	0	0
$907$	25.4850	0.846215	0.423108	−	0.906079i	$-0.360939\pi$
$907$	25.4850	0.846215	0.423108	+	0.906079i	$0.360939\pi$
$908$	0	0
$909$	5.85330	0.194142
$910$	0	0
$911$	51.3937	1.70275	0.851375	−	0.524558i	$-0.175769\pi$
$911$	51.3937	1.70275	0.851375	+	0.524558i	$0.175769\pi$
$912$	0	0
$913$	30.1931	0.999245
$914$	0	0
$915$	5.60786	0.185390
$916$	0	0
$917$	−30.1784	−0.996579
$918$	0	0
$919$	33.1524	1.09360	0.546798	−	0.837265i	$-0.315847\pi$
$919$	33.1524	1.09360	0.546798	+	0.837265i	$0.315847\pi$
$920$	0	0
$921$	−5.73050	−0.188826
$922$	0	0
$923$	−86.6091	−2.85077
$924$	0	0
$925$	−23.1814	−0.762199
$926$	0	0
$927$	−20.0529	−0.658625
$928$	0	0
$929$	−12.6107	−0.413742	−0.206871	−	0.978368i	$-0.566328\pi$
$929$	−12.6107	−0.413742	−0.206871	+	0.978368i	$0.566328\pi$
$930$	0	0
$931$	−0.826548	−0.0270890
$932$	0	0
$933$	−104.196	−3.41124
$934$	0	0
$935$	−22.0367	−0.720677
$936$	0	0
$937$	−24.8849	−0.812955	−0.406478	−	0.913661i	$-0.633243\pi$
$937$	−24.8849	−0.812955	−0.406478	+	0.913661i	$0.633243\pi$
$938$	0	0
$939$	−76.4657	−2.49536
$940$	0	0
$941$	11.0856	0.361381	0.180691	−	0.983540i	$-0.442167\pi$
$941$	11.0856	0.361381	0.180691	+	0.983540i	$0.442167\pi$
$942$	0	0
$943$	1.38924	0.0452399
$944$	0	0
$945$	−72.8442	−2.36962
$946$	0	0
$947$	−3.15395	−0.102490	−0.0512448	−	0.998686i	$-0.516319\pi$
$947$	−3.15395	−0.102490	−0.0512448	+	0.998686i	$0.516319\pi$
$948$	0	0
$949$	−75.1523	−2.43955
$950$	0	0
$951$	−36.4144	−1.18082
$952$	0	0
$953$	−10.4222	−0.337609	−0.168804	−	0.985650i	$-0.553991\pi$
$953$	−10.4222	−0.337609	−0.168804	+	0.985650i	$0.553991\pi$
$954$	0	0
$955$	−3.73376	−0.120821
$956$	0	0
$957$	7.34688	0.237491
$958$	0	0
$959$	−56.4157	−1.82176
$960$	0	0
$961$	−20.2280	−0.652517
$962$	0	0
$963$	58.1053	1.87242
$964$	0	0
$965$	−23.1775	−0.746111
$966$	0	0
$967$	−49.0860	−1.57850	−0.789250	−	0.614071i	$-0.789531\pi$
$967$	−49.0860	−1.57850	−0.789250	+	0.614071i	$0.789531\pi$
$968$	0	0
$969$	−31.8860	−1.02433
$970$	0	0
$971$	−39.6579	−1.27268	−0.636342	−	0.771407i	$-0.719553\pi$
$971$	−39.6579	−1.27268	−0.636342	+	0.771407i	$0.719553\pi$
$972$	0	0
$973$	18.2216	0.584157
$974$	0	0
$975$	−44.1353	−1.41346
$976$	0	0
$977$	40.4363	1.29367	0.646835	−	0.762630i	$-0.276092\pi$
$977$	40.4363	1.29367	0.646835	+	0.762630i	$0.276092\pi$
$978$	0	0
$979$	46.6063	1.48954
$980$	0	0
$981$	10.1736	0.324819
$982$	0	0
$983$	−11.6094	−0.370284	−0.185142	−	0.982712i	$-0.559274\pi$
$983$	−11.6094	−0.370284	−0.185142	+	0.982712i	$0.559274\pi$
$984$	0	0
$985$	−31.4440	−1.00189
$986$	0	0
$987$	76.2722	2.42777
$988$	0	0
$989$	4.30575	0.136915
$990$	0	0
$991$	33.3482	1.05934	0.529671	−	0.848203i	$-0.322315\pi$
$991$	33.3482	1.05934	0.529671	+	0.848203i	$0.322315\pi$
$992$	0	0
$993$	23.3695	0.741607
$994$	0	0
$995$	−5.77996	−0.183237
$996$	0	0
$997$	16.0436	0.508105	0.254053	−	0.967190i	$-0.418236\pi$
$997$	16.0436	0.508105	0.254053	+	0.967190i	$0.418236\pi$
$998$	0	0
$999$	−174.503	−5.52103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	976.2.a.k.1.1		6
3.2	odd	2		8784.2.a.by.1.5		6
4.3	odd	2		488.2.a.d.1.6	✓	6
8.3	odd	2		3904.2.a.bi.1.1		6
8.5	even	2		3904.2.a.bk.1.6		6
12.11	even	2		4392.2.a.q.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
488.2.a.d.1.6	✓	6	4.3	odd	2
976.2.a.k.1.1		6	1.1	even	1	trivial
3904.2.a.bi.1.1		6	8.3	odd	2
3904.2.a.bk.1.6		6	8.5	even	2
4392.2.a.q.1.5		6	12.11	even	2
8784.2.a.by.1.5		6	3.2	odd	2

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:	\( N \)	\(=\)	\( 976 = 2^{4} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	976.a (trivial)

\(f(q)\)	\(=\)	\(q-3.28207 q^{3} -1.70864 q^{5} -2.72208 q^{7} +7.77196 q^{9} +O(q^{10})\)	\(q-3.28207 q^{3} -1.70864 q^{5} -2.72208 q^{7} +7.77196 q^{9} -2.67795 q^{11} -6.46335 q^{13} +5.60786 q^{15} -4.81609 q^{17} -2.01724 q^{19} +8.93406 q^{21} -0.767808 q^{23} -2.08056 q^{25} -15.6619 q^{27} +0.835899 q^{29} +3.28207 q^{31} +8.78920 q^{33} +4.65105 q^{35} +11.1419 q^{37} +21.2132 q^{39} -1.80936 q^{41} -5.60786 q^{43} -13.2794 q^{45} +8.53724 q^{47} +0.409742 q^{49} +15.8067 q^{51} -6.88992 q^{53} +4.57564 q^{55} +6.62072 q^{57} +10.6140 q^{59} +1.00000 q^{61} -21.1559 q^{63} +11.0435 q^{65} +2.07677 q^{67} +2.52000 q^{69} +13.4000 q^{71} +11.6275 q^{73} +6.82854 q^{75} +7.28960 q^{77} -11.5394 q^{79} +28.0875 q^{81} -11.2747 q^{83} +8.22895 q^{85} -2.74348 q^{87} -17.4037 q^{89} +17.5938 q^{91} -10.7720 q^{93} +3.44673 q^{95} -15.8966 q^{97} -20.8129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} + 5 q^{5} - 6 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 + 5 * q^5 - 6 * q^7 + 9 * q^9	\( 6 q - 3 q^{3} + 5 q^{5} - 6 q^{7} + 9 q^{9} - 3 q^{11} + 7 q^{13} + 6 q^{15} + 6 q^{17} - 5 q^{19} + 12 q^{21} + 6 q^{23} + 17 q^{25} - 12 q^{27} + 15 q^{29} + 3 q^{31} + 8 q^{33} + 5 q^{35} + 21 q^{37} + 20 q^{39} + 4 q^{41} - 6 q^{43} + q^{45} + 8 q^{47} + 8 q^{49} + 16 q^{51} + 3 q^{53} + 17 q^{55} + 4 q^{57} + 3 q^{59} + 6 q^{61} + 3 q^{63} - q^{65} - 5 q^{67} - 21 q^{69} + 57 q^{71} + 14 q^{73} + q^{75} - 9 q^{77} - 5 q^{79} + 2 q^{81} - q^{83} + 2 q^{85} + 28 q^{87} + 3 q^{91} - 27 q^{93} + 52 q^{95} - 19 q^{97} - 13 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 + 5 * q^5 - 6 * q^7 + 9 * q^9 - 3 * q^11 + 7 * q^13 + 6 * q^15 + 6 * q^17 - 5 * q^19 + 12 * q^21 + 6 * q^23 + 17 * q^25 - 12 * q^27 + 15 * q^29 + 3 * q^31 + 8 * q^33 + 5 * q^35 + 21 * q^37 + 20 * q^39 + 4 * q^41 - 6 * q^43 + q^45 + 8 * q^47 + 8 * q^49 + 16 * q^51 + 3 * q^53 + 17 * q^55 + 4 * q^57 + 3 * q^59 + 6 * q^61 + 3 * q^63 - q^65 - 5 * q^67 - 21 * q^69 + 57 * q^71 + 14 * q^73 + q^75 - 9 * q^77 - 5 * q^79 + 2 * q^81 - q^83 + 2 * q^85 + 28 * q^87 + 3 * q^91 - 27 * q^93 + 52 * q^95 - 19 * q^97 - 13 * q^99

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