LMFDB - Embedded newform 976.2.a.g.1.2

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:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 122)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ 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)

$f(q)$	$=$	$q-0.462598 q^{3} +1.39821 q^{5} +3.18421 q^{7} -2.78600 q^{9} +O(q^{10})$	$q-0.462598 q^{3} +1.39821 q^{5} +3.18421 q^{7} -2.78600 q^{9} -0.323404 q^{11} +2.32340 q^{13} -0.646809 q^{15} +1.72161 q^{17} +2.86081 q^{19} -1.47301 q^{21} +6.50761 q^{23} -3.04502 q^{25} +2.67660 q^{27} +0.0643910 q^{29} -5.10941 q^{31} +0.149606 q^{33} +4.45219 q^{35} +8.98062 q^{37} -1.07480 q^{39} -2.65722 q^{41} -11.4432 q^{43} -3.89541 q^{45} +4.79641 q^{47} +3.13919 q^{49} -0.796415 q^{51} +12.9598 q^{53} -0.452186 q^{55} -1.32340 q^{57} +6.60179 q^{59} -1.00000 q^{61} -8.87122 q^{63} +3.24860 q^{65} +4.69182 q^{67} -3.01041 q^{69} -3.41758 q^{71} +13.9806 q^{73} +1.40862 q^{75} -1.02979 q^{77} -4.19462 q^{79} +7.11982 q^{81} +16.1544 q^{83} +2.40717 q^{85} -0.0297872 q^{87} +2.51803 q^{89} +7.39821 q^{91} +2.36360 q^{93} +4.00000 q^{95} -6.18421 q^{97} +0.901005 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 3 * q + q^3 + q^5 - 4 * q^7 + 2 * q^9	$ 3 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9} + 7 q^{11} - q^{13} + 14 q^{15} - 6 q^{17} + 3 q^{19} - 6 q^{21} - 2 q^{23} + 10 q^{25} + 16 q^{27} + q^{29} + 3 q^{31} + 10 q^{33} + 7 q^{35} + 7 q^{37} - 8 q^{39} + 4 q^{41} - 12 q^{43} + 17 q^{45} + 8 q^{47} + 15 q^{49} + 4 q^{51} + 11 q^{53} + 5 q^{55} + 4 q^{57} + 23 q^{59} - 3 q^{61} - 25 q^{63} - 3 q^{65} - 21 q^{67} - 13 q^{69} - 27 q^{71} + 22 q^{73} + 5 q^{75} - 27 q^{77} - 3 q^{79} + 7 q^{81} + 11 q^{83} + 20 q^{85} - 24 q^{87} - 10 q^{89} + 19 q^{91} + 27 q^{93} + 12 q^{95} - 5 q^{97} + 25 q^{99}+O(q^{100}) $ 3 * q + q^3 + q^5 - 4 * q^7 + 2 * q^9 + 7 * q^11 - q^13 + 14 * q^15 - 6 * q^17 + 3 * q^19 - 6 * q^21 - 2 * q^23 + 10 * q^25 + 16 * q^27 + q^29 + 3 * q^31 + 10 * q^33 + 7 * q^35 + 7 * q^37 - 8 * q^39 + 4 * q^41 - 12 * q^43 + 17 * q^45 + 8 * q^47 + 15 * q^49 + 4 * q^51 + 11 * q^53 + 5 * q^55 + 4 * q^57 + 23 * q^59 - 3 * q^61 - 25 * q^63 - 3 * q^65 - 21 * q^67 - 13 * q^69 - 27 * q^71 + 22 * q^73 + 5 * q^75 - 27 * q^77 - 3 * q^79 + 7 * q^81 + 11 * q^83 + 20 * q^85 - 24 * q^87 - 10 * q^89 + 19 * q^91 + 27 * q^93 + 12 * q^95 - 5 * q^97 + 25 * 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
$3$	−0.462598	−0.267081	−0.133541	−	0.991043i	$-0.542635\pi$
$3$	−0.462598	−0.267081	−0.133541	+	0.991043i	$0.542635\pi$
$4$	0	0
$5$	1.39821	0.625297	0.312649	−	0.949869i	$-0.398784\pi$
$5$	1.39821	0.625297	0.312649	+	0.949869i	$0.398784\pi$
$6$	0	0
$7$	3.18421	1.20352	0.601759	−	0.798678i	$-0.294467\pi$
$7$	3.18421	1.20352	0.601759	+	0.798678i	$0.294467\pi$
$8$	0	0
$9$	−2.78600	−0.928668
$10$	0	0
$11$	−0.323404	−0.0975101	−0.0487550	−	0.998811i	$-0.515525\pi$
$11$	−0.323404	−0.0975101	−0.0487550	+	0.998811i	$0.515525\pi$
$12$	0	0
$13$	2.32340	0.644396	0.322198	−	0.946672i	$-0.395578\pi$
$13$	2.32340	0.644396	0.322198	+	0.946672i	$0.395578\pi$
$14$	0	0
$15$	−0.646809	−0.167005
$16$	0	0
$17$	1.72161	0.417552	0.208776	−	0.977963i	$-0.433052\pi$
$17$	1.72161	0.417552	0.208776	+	0.977963i	$0.433052\pi$
$18$	0	0
$19$	2.86081	0.656314	0.328157	−	0.944623i	$-0.393573\pi$
$19$	2.86081	0.656314	0.328157	+	0.944623i	$0.393573\pi$
$20$	0	0
$21$	−1.47301	−0.321437
$22$	0	0
$23$	6.50761	1.35693	0.678466	−	0.734632i	$-0.262645\pi$
$23$	6.50761	1.35693	0.678466	+	0.734632i	$0.262645\pi$
$24$	0	0
$25$	−3.04502	−0.609003
$26$	0	0
$27$	2.67660	0.515111
$28$	0	0
$29$	0.0643910	0.0119571	0.00597855	−	0.999982i	$-0.498097\pi$
$29$	0.0643910	0.0119571	0.00597855	+	0.999982i	$0.498097\pi$
$30$	0	0
$31$	−5.10941	−0.917677	−0.458838	−	0.888520i	$-0.651734\pi$
$31$	−5.10941	−0.917677	−0.458838	+	0.888520i	$0.651734\pi$
$32$	0	0
$33$	0.149606	0.0260431
$34$	0	0
$35$	4.45219	0.752557
$36$	0	0
$37$	8.98062	1.47641	0.738203	−	0.674579i	$-0.235675\pi$
$37$	8.98062	1.47641	0.738203	+	0.674579i	$0.235675\pi$
$38$	0	0
$39$	−1.07480	−0.172106
$40$	0	0
$41$	−2.65722	−0.414988	−0.207494	−	0.978236i	$-0.566531\pi$
$41$	−2.65722	−0.414988	−0.207494	+	0.978236i	$0.566531\pi$
$42$	0	0
$43$	−11.4432	−1.74508	−0.872538	−	0.488547i	$-0.837527\pi$
$43$	−11.4432	−1.74508	−0.872538	+	0.488547i	$0.837527\pi$
$44$	0	0
$45$	−3.89541	−0.580693
$46$	0	0
$47$	4.79641	0.699629	0.349815	−	0.936819i	$-0.386245\pi$
$47$	4.79641	0.699629	0.349815	+	0.936819i	$0.386245\pi$
$48$	0	0
$49$	3.13919	0.448456
$50$	0	0
$51$	−0.796415	−0.111520
$52$	0	0
$53$	12.9598	1.78017	0.890083	−	0.455799i	$-0.150646\pi$
$53$	12.9598	1.78017	0.890083	+	0.455799i	$0.150646\pi$
$54$	0	0
$55$	−0.452186	−0.0609728
$56$	0	0
$57$	−1.32340	−0.175289
$58$	0	0
$59$	6.60179	0.859480	0.429740	−	0.902953i	$-0.358605\pi$
$59$	6.60179	0.859480	0.429740	+	0.902953i	$0.358605\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0	0
$63$	−8.87122	−1.11767
$64$	0	0
$65$	3.24860	0.402939
$66$	0	0
$67$	4.69182	0.573198	0.286599	−	0.958051i	$-0.407475\pi$
$67$	4.69182	0.573198	0.286599	+	0.958051i	$0.407475\pi$
$68$	0	0
$69$	−3.01041	−0.362411
$70$	0	0
$71$	−3.41758	−0.405592	−0.202796	−	0.979221i	$-0.565003\pi$
$71$	−3.41758	−0.405592	−0.202796	+	0.979221i	$0.565003\pi$
$72$	0	0
$73$	13.9806	1.63631	0.818154	−	0.574999i	$-0.194998\pi$
$73$	13.9806	1.63631	0.818154	+	0.574999i	$0.194998\pi$
$74$	0	0
$75$	1.40862	0.162653
$76$	0	0
$77$	−1.02979	−0.117355
$78$	0	0
$79$	−4.19462	−0.471932	−0.235966	−	0.971761i	$-0.575825\pi$
$79$	−4.19462	−0.471932	−0.235966	+	0.971761i	$0.575825\pi$
$80$	0	0
$81$	7.11982	0.791091
$82$	0	0
$83$	16.1544	1.77318	0.886589	−	0.462558i	$-0.153068\pi$
$83$	16.1544	1.77318	0.886589	+	0.462558i	$0.153068\pi$
$84$	0	0
$85$	2.40717	0.261094
$86$	0	0
$87$	−0.0297872	−0.00319352
$88$	0	0
$89$	2.51803	0.266910	0.133455	−	0.991055i	$-0.457393\pi$
$89$	2.51803	0.266910	0.133455	+	0.991055i	$0.457393\pi$
$90$	0	0
$91$	7.39821	0.775543
$92$	0	0
$93$	2.36360	0.245094
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−6.18421	−0.627911	−0.313956	−	0.949438i	$-0.601654\pi$
$97$	−6.18421	−0.627911	−0.313956	+	0.949438i	$0.601654\pi$
$98$	0	0
$99$	0.901005	0.0905544
$100$	0	0
$101$	−4.31299	−0.429159	−0.214579	−	0.976707i	$-0.568838\pi$
$101$	−4.31299	−0.429159	−0.214579	+	0.976707i	$0.568838\pi$
$102$	0	0
$103$	−11.5720	−1.14022	−0.570112	−	0.821567i	$-0.693100\pi$
$103$	−11.5720	−1.14022	−0.570112	+	0.821567i	$0.693100\pi$
$104$	0	0
$105$	−2.05957	−0.200994
$106$	0	0
$107$	2.34278	0.226485	0.113243	−	0.993567i	$-0.463876\pi$
$107$	2.34278	0.226485	0.113243	+	0.993567i	$0.463876\pi$
$108$	0	0
$109$	−2.32340	−0.222542	−0.111271	−	0.993790i	$-0.535492\pi$
$109$	−2.32340	−0.222542	−0.111271	+	0.993790i	$0.535492\pi$
$110$	0	0
$111$	−4.15442	−0.394320
$112$	0	0
$113$	−4.90582	−0.461501	−0.230750	−	0.973013i	$-0.574118\pi$
$113$	−4.90582	−0.461501	−0.230750	+	0.973013i	$0.574118\pi$
$114$	0	0
$115$	9.09899	0.848486
$116$	0	0
$117$	−6.47301	−0.598430
$118$	0	0
$119$	5.48197	0.502532
$120$	0	0
$121$	−10.8954	−0.990492
$122$	0	0
$123$	1.22923	0.110836
$124$	0	0
$125$	−11.2486	−1.00611
$126$	0	0
$127$	−1.87122	−0.166044	−0.0830219	−	0.996548i	$-0.526457\pi$
$127$	−1.87122	−0.166044	−0.0830219	+	0.996548i	$0.526457\pi$
$128$	0	0
$129$	5.29362	0.466077
$130$	0	0
$131$	6.14961	0.537294	0.268647	−	0.963239i	$-0.413424\pi$
$131$	6.14961	0.537294	0.268647	+	0.963239i	$0.413424\pi$
$132$	0	0
$133$	9.10941	0.789886
$134$	0	0
$135$	3.74244	0.322098
$136$	0	0
$137$	−16.5076	−1.41034	−0.705170	−	0.709038i	$-0.749129\pi$
$137$	−16.5076	−1.41034	−0.705170	+	0.709038i	$0.749129\pi$
$138$	0	0
$139$	10.9702	0.930481	0.465241	−	0.885184i	$-0.345968\pi$
$139$	10.9702	0.930481	0.465241	+	0.885184i	$0.345968\pi$
$140$	0	0
$141$	−2.21881	−0.186858
$142$	0	0
$143$	−0.751399	−0.0628351
$144$	0	0
$145$	0.0900320	0.00747675
$146$	0	0
$147$	−1.45219	−0.119774
$148$	0	0
$149$	−21.2099	−1.73758	−0.868789	−	0.495182i	$-0.835101\pi$
$149$	−21.2099	−1.73758	−0.868789	+	0.495182i	$0.835101\pi$
$150$	0	0
$151$	−21.0048	−1.70935	−0.854674	−	0.519165i	$-0.826243\pi$
$151$	−21.0048	−1.70935	−0.854674	+	0.519165i	$0.826243\pi$
$152$	0	0
$153$	−4.79641	−0.387767
$154$	0	0
$155$	−7.14401	−0.573821
$156$	0	0
$157$	−5.13919	−0.410152	−0.205076	−	0.978746i	$-0.565744\pi$
$157$	−5.13919	−0.410152	−0.205076	+	0.978746i	$0.565744\pi$
$158$	0	0
$159$	−5.99518	−0.475449
$160$	0	0
$161$	20.7216	1.63309
$162$	0	0
$163$	−10.9806	−0.860069	−0.430034	−	0.902812i	$-0.641499\pi$
$163$	−10.9806	−0.860069	−0.430034	+	0.902812i	$0.641499\pi$
$164$	0	0
$165$	0.209181	0.0162847
$166$	0	0
$167$	−24.6081	−1.90423	−0.952114	−	0.305742i	$-0.901095\pi$
$167$	−24.6081	−1.90423	−0.952114	+	0.305742i	$0.901095\pi$
$168$	0	0
$169$	−7.60179	−0.584753
$170$	0	0
$171$	−7.97021	−0.609497
$172$	0	0
$173$	1.65722	0.125996	0.0629981	−	0.998014i	$-0.479934\pi$
$173$	1.65722	0.125996	0.0629981	+	0.998014i	$0.479934\pi$
$174$	0	0
$175$	−9.69597	−0.732946
$176$	0	0
$177$	−3.05398	−0.229551
$178$	0	0
$179$	−5.75622	−0.430240	−0.215120	−	0.976588i	$-0.569014\pi$
$179$	−5.75622	−0.430240	−0.215120	+	0.976588i	$0.569014\pi$
$180$	0	0
$181$	−25.8760	−1.92335	−0.961675	−	0.274191i	$-0.911590\pi$
$181$	−25.8760	−1.92335	−0.961675	+	0.274191i	$0.911590\pi$
$182$	0	0
$183$	0.462598	0.0341963
$184$	0	0
$185$	12.5568	0.923193
$186$	0	0
$187$	−0.556777	−0.0407155
$188$	0	0
$189$	8.52284	0.619946
$190$	0	0
$191$	2.13919	0.154787	0.0773933	−	0.997001i	$-0.475340\pi$
$191$	2.13919	0.154787	0.0773933	+	0.997001i	$0.475340\pi$
$192$	0	0
$193$	25.4224	1.82994	0.914972	−	0.403517i	$-0.132212\pi$
$193$	25.4224	1.82994	0.914972	+	0.403517i	$0.132212\pi$
$194$	0	0
$195$	−1.50280	−0.107618
$196$	0	0
$197$	22.6918	1.61673	0.808363	−	0.588685i	$-0.200354\pi$
$197$	22.6918	1.61673	0.808363	+	0.588685i	$0.200354\pi$
$198$	0	0
$199$	23.6620	1.67736	0.838679	−	0.544627i	$-0.183329\pi$
$199$	23.6620	1.67736	0.838679	+	0.544627i	$0.183329\pi$
$200$	0	0
$201$	−2.17043	−0.153090
$202$	0	0
$203$	0.205034	0.0143906
$204$	0	0
$205$	−3.71535	−0.259491
$206$	0	0
$207$	−18.1302	−1.26014
$208$	0	0
$209$	−0.925197	−0.0639972
$210$	0	0
$211$	−9.72161	−0.669263	−0.334632	−	0.942349i	$-0.608612\pi$
$211$	−9.72161	−0.669263	−0.334632	+	0.942349i	$0.608612\pi$
$212$	0	0
$213$	1.58097	0.108326
$214$	0	0
$215$	−16.0000	−1.09119
$216$	0	0
$217$	−16.2694	−1.10444
$218$	0	0
$219$	−6.46742	−0.437027
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	16.2847	1.09050	0.545251	−	0.838273i	$-0.316435\pi$
$223$	16.2847	1.09050	0.545251	+	0.838273i	$0.316435\pi$
$224$	0	0
$225$	8.48342	0.565562
$226$	0	0
$227$	14.4134	0.956653	0.478327	−	0.878182i	$-0.341243\pi$
$227$	14.4134	0.956653	0.478327	+	0.878182i	$0.341243\pi$
$228$	0	0
$229$	−1.37738	−0.0910200	−0.0455100	−	0.998964i	$-0.514491\pi$
$229$	−1.37738	−0.0910200	−0.0455100	+	0.998964i	$0.514491\pi$
$230$	0	0
$231$	0.476378	0.0313434
$232$	0	0
$233$	−26.1801	−1.71511	−0.857557	−	0.514390i	$-0.828018\pi$
$233$	−26.1801	−1.71511	−0.857557	+	0.514390i	$0.828018\pi$
$234$	0	0
$235$	6.70638	0.437476
$236$	0	0
$237$	1.94043	0.126044
$238$	0	0
$239$	−4.02082	−0.260086	−0.130043	−	0.991508i	$-0.541511\pi$
$239$	−4.02082	−0.260086	−0.130043	+	0.991508i	$0.541511\pi$
$240$	0	0
$241$	−18.3282	−1.18062	−0.590312	−	0.807175i	$-0.700995\pi$
$241$	−18.3282	−1.18062	−0.590312	+	0.807175i	$0.700995\pi$
$242$	0	0
$243$	−11.3234	−0.726397
$244$	0	0
$245$	4.38924	0.280419
$246$	0	0
$247$	6.64681	0.422926
$248$	0	0
$249$	−7.47301	−0.473583
$250$	0	0
$251$	19.5124	1.23161	0.615807	−	0.787897i	$-0.288830\pi$
$251$	19.5124	1.23161	0.615807	+	0.787897i	$0.288830\pi$
$252$	0	0
$253$	−2.10459	−0.132314
$254$	0	0
$255$	−1.11355	−0.0697334
$256$	0	0
$257$	−18.9508	−1.18212	−0.591060	−	0.806627i	$-0.701291\pi$
$257$	−18.9508	−1.18212	−0.591060	+	0.806627i	$0.701291\pi$
$258$	0	0
$259$	28.5962	1.77688
$260$	0	0
$261$	−0.179393	−0.0111042
$262$	0	0
$263$	−28.1801	−1.73766	−0.868829	−	0.495113i	$-0.835127\pi$
$263$	−28.1801	−1.73766	−0.868829	+	0.495113i	$0.835127\pi$
$264$	0	0
$265$	18.1205	1.11313
$266$	0	0
$267$	−1.16484	−0.0712868
$268$	0	0
$269$	28.0900	1.71268	0.856340	−	0.516413i	$-0.172733\pi$
$269$	28.0900	1.71268	0.856340	+	0.516413i	$0.172733\pi$
$270$	0	0
$271$	−6.73684	−0.409234	−0.204617	−	0.978842i	$-0.565595\pi$
$271$	−6.73684	−0.409234	−0.204617	+	0.978842i	$0.565595\pi$
$272$	0	0
$273$	−3.42240	−0.207133
$274$	0	0
$275$	0.984771	0.0593839
$276$	0	0
$277$	0.0643910	0.00386888	0.00193444	−	0.999998i	$-0.499384\pi$
$277$	0.0643910	0.00386888	0.00193444	+	0.999998i	$0.499384\pi$
$278$	0	0
$279$	14.2348	0.852216
$280$	0	0
$281$	4.55678	0.271835	0.135917	−	0.990720i	$-0.456602\pi$
$281$	4.55678	0.271835	0.135917	+	0.990720i	$0.456602\pi$
$282$	0	0
$283$	−18.3130	−1.08859	−0.544297	−	0.838892i	$-0.683204\pi$
$283$	−18.3130	−1.08859	−0.544297	+	0.838892i	$0.683204\pi$
$284$	0	0
$285$	−1.85039	−0.109608
$286$	0	0
$287$	−8.46115	−0.499446
$288$	0	0
$289$	−14.0361	−0.825650
$290$	0	0
$291$	2.86081	0.167703
$292$	0	0
$293$	13.4432	0.785361	0.392681	−	0.919675i	$-0.371548\pi$
$293$	13.4432	0.785361	0.392681	+	0.919675i	$0.371548\pi$
$294$	0	0
$295$	9.23068	0.537431
$296$	0	0
$297$	−0.865623	−0.0502285
$298$	0	0
$299$	15.1198	0.874402
$300$	0	0
$301$	−36.4376	−2.10023
$302$	0	0
$303$	1.99518	0.114620
$304$	0	0
$305$	−1.39821	−0.0800611
$306$	0	0
$307$	−17.8954	−1.02134	−0.510672	−	0.859775i	$-0.670604\pi$
$307$	−17.8954	−1.02134	−0.510672	+	0.859775i	$0.670604\pi$
$308$	0	0
$309$	5.35319	0.304532
$310$	0	0
$311$	−19.2742	−1.09294	−0.546471	−	0.837478i	$-0.684029\pi$
$311$	−19.2742	−1.09294	−0.546471	+	0.837478i	$0.684029\pi$
$312$	0	0
$313$	−11.2936	−0.638353	−0.319176	−	0.947695i	$-0.603406\pi$
$313$	−11.2936	−0.638353	−0.319176	+	0.947695i	$0.603406\pi$
$314$	0	0
$315$	−12.4038	−0.698875
$316$	0	0
$317$	3.05398	0.171529	0.0857643	−	0.996315i	$-0.472667\pi$
$317$	3.05398	0.171529	0.0857643	+	0.996315i	$0.472667\pi$
$318$	0	0
$319$	−0.0208243	−0.00116594
$320$	0	0
$321$	−1.08377	−0.0604899
$322$	0	0
$323$	4.92520	0.274045
$324$	0	0
$325$	−7.07480	−0.392439
$326$	0	0
$327$	1.07480	0.0594368
$328$	0	0
$329$	15.2728	0.842016
$330$	0	0
$331$	−11.4674	−0.630306	−0.315153	−	0.949041i	$-0.602056\pi$
$331$	−11.4674	−0.630306	−0.315153	+	0.949041i	$0.602056\pi$
$332$	0	0
$333$	−25.0200	−1.37109
$334$	0	0
$335$	6.56014	0.358419
$336$	0	0
$337$	29.1440	1.58758	0.793788	−	0.608195i	$-0.208106\pi$
$337$	29.1440	1.58758	0.793788	+	0.608195i	$0.208106\pi$
$338$	0	0
$339$	2.26943	0.123258
$340$	0	0
$341$	1.65240	0.0894827
$342$	0	0
$343$	−12.2936	−0.663793
$344$	0	0
$345$	−4.20918	−0.226615
$346$	0	0
$347$	−4.29921	−0.230794	−0.115397	−	0.993319i	$-0.536814\pi$
$347$	−4.29921	−0.230794	−0.115397	+	0.993319i	$0.536814\pi$
$348$	0	0
$349$	−8.53885	−0.457074	−0.228537	−	0.973535i	$-0.573394\pi$
$349$	−8.53885	−0.457074	−0.228537	+	0.973535i	$0.573394\pi$
$350$	0	0
$351$	6.21881	0.331936
$352$	0	0
$353$	25.9002	1.37853	0.689265	−	0.724509i	$-0.257934\pi$
$353$	25.9002	1.37853	0.689265	+	0.724509i	$0.257934\pi$
$354$	0	0
$355$	−4.77849	−0.253616
$356$	0	0
$357$	−2.53595	−0.134217
$358$	0	0
$359$	15.4689	0.816415	0.408208	−	0.912889i	$-0.366154\pi$
$359$	15.4689	0.816415	0.408208	+	0.912889i	$0.366154\pi$
$360$	0	0
$361$	−10.8158	−0.569252
$362$	0	0
$363$	5.04020	0.264542
$364$	0	0
$365$	19.5478	1.02318
$366$	0	0
$367$	−11.2936	−0.589522	−0.294761	−	0.955571i	$-0.595240\pi$
$367$	−11.2936	−0.589522	−0.294761	+	0.955571i	$0.595240\pi$
$368$	0	0
$369$	7.40302	0.385386
$370$	0	0
$371$	41.2667	2.14246
$372$	0	0
$373$	10.4834	0.542811	0.271406	−	0.962465i	$-0.412512\pi$
$373$	10.4834	0.542811	0.271406	+	0.962465i	$0.412512\pi$
$374$	0	0
$375$	5.20359	0.268712
$376$	0	0
$377$	0.149606	0.00770512
$378$	0	0
$379$	−3.25005	−0.166944	−0.0834719	−	0.996510i	$-0.526601\pi$
$379$	−3.25005	−0.166944	−0.0834719	+	0.996510i	$0.526601\pi$
$380$	0	0
$381$	0.865623	0.0443472
$382$	0	0
$383$	13.3338	0.681326	0.340663	−	0.940185i	$-0.389348\pi$
$383$	13.3338	0.681326	0.340663	+	0.940185i	$0.389348\pi$
$384$	0	0
$385$	−1.43986	−0.0733819
$386$	0	0
$387$	31.8809	1.62059
$388$	0	0
$389$	−10.9508	−0.555230	−0.277615	−	0.960692i	$-0.589544\pi$
$389$	−10.9508	−0.555230	−0.277615	+	0.960692i	$0.589544\pi$
$390$	0	0
$391$	11.2036	0.566590
$392$	0	0
$393$	−2.84480	−0.143501
$394$	0	0
$395$	−5.86495	−0.295098
$396$	0	0
$397$	−24.3345	−1.22131	−0.610656	−	0.791896i	$-0.709094\pi$
$397$	−24.3345	−1.22131	−0.610656	+	0.791896i	$0.709094\pi$
$398$	0	0
$399$	−4.21400	−0.210964
$400$	0	0
$401$	−6.51803	−0.325495	−0.162747	−	0.986668i	$-0.552036\pi$
$401$	−6.51803	−0.325495	−0.162747	+	0.986668i	$0.552036\pi$
$402$	0	0
$403$	−11.8712	−0.591347
$404$	0	0
$405$	9.95498	0.494667
$406$	0	0
$407$	−2.90437	−0.143964
$408$	0	0
$409$	36.0305	1.78159	0.890796	−	0.454404i	$-0.150148\pi$
$409$	36.0305	1.78159	0.890796	+	0.454404i	$0.150148\pi$
$410$	0	0
$411$	7.63640	0.376676
$412$	0	0
$413$	21.0215	1.03440
$414$	0	0
$415$	22.5872	1.10876
$416$	0	0
$417$	−5.07480	−0.248514
$418$	0	0
$419$	−24.3505	−1.18960	−0.594800	−	0.803874i	$-0.702769\pi$
$419$	−24.3505	−1.18960	−0.594800	+	0.803874i	$0.702769\pi$
$420$	0	0
$421$	9.87603	0.481328	0.240664	−	0.970608i	$-0.422635\pi$
$421$	9.87603	0.481328	0.240664	+	0.970608i	$0.422635\pi$
$422$	0	0
$423$	−13.3628	−0.649723
$424$	0	0
$425$	−5.24234	−0.254291
$426$	0	0
$427$	−3.18421	−0.154095
$428$	0	0
$429$	0.347596	0.0167821
$430$	0	0
$431$	−15.4224	−0.742871	−0.371435	−	0.928459i	$-0.621134\pi$
$431$	−15.4224	−0.742871	−0.371435	+	0.928459i	$0.621134\pi$
$432$	0	0
$433$	−23.5124	−1.12994	−0.564968	−	0.825113i	$-0.691111\pi$
$433$	−23.5124	−1.12994	−0.564968	+	0.825113i	$0.691111\pi$
$434$	0	0
$435$	−0.0416486	−0.00199690
$436$	0	0
$437$	18.6170	0.890573
$438$	0	0
$439$	−1.05398	−0.0503037	−0.0251518	−	0.999684i	$-0.508007\pi$
$439$	−1.05398	−0.0503037	−0.0251518	+	0.999684i	$0.508007\pi$
$440$	0	0
$441$	−8.74580	−0.416467
$442$	0	0
$443$	39.2590	1.86525	0.932626	−	0.360844i	$-0.117511\pi$
$443$	39.2590	1.86525	0.932626	+	0.360844i	$0.117511\pi$
$444$	0	0
$445$	3.52072	0.166898
$446$	0	0
$447$	9.81164	0.464075
$448$	0	0
$449$	14.9356	0.704855	0.352427	−	0.935839i	$-0.385356\pi$
$449$	14.9356	0.704855	0.352427	+	0.935839i	$0.385356\pi$
$450$	0	0
$451$	0.859357	0.0404655
$452$	0	0
$453$	9.71680	0.456535
$454$	0	0
$455$	10.3442	0.484945
$456$	0	0
$457$	−8.55678	−0.400269	−0.200135	−	0.979768i	$-0.564138\pi$
$457$	−8.55678	−0.400269	−0.200135	+	0.979768i	$0.564138\pi$
$458$	0	0
$459$	4.60806	0.215086
$460$	0	0
$461$	3.66204	0.170558	0.0852790	−	0.996357i	$-0.472822\pi$
$461$	3.66204	0.170558	0.0852790	+	0.996357i	$0.472822\pi$
$462$	0	0
$463$	19.6829	0.914740	0.457370	−	0.889276i	$-0.348791\pi$
$463$	19.6829	0.914740	0.457370	+	0.889276i	$0.348791\pi$
$464$	0	0
$465$	3.30481	0.153257
$466$	0	0
$467$	−18.2251	−0.843356	−0.421678	−	0.906746i	$-0.638559\pi$
$467$	−18.2251	−0.843356	−0.421678	+	0.906746i	$0.638559\pi$
$468$	0	0
$469$	14.9398	0.689854
$470$	0	0
$471$	2.37738	0.109544
$472$	0	0
$473$	3.70079	0.170162
$474$	0	0
$475$	−8.71120	−0.399697
$476$	0	0
$477$	−36.1060	−1.65318
$478$	0	0
$479$	31.2936	1.42984	0.714921	−	0.699205i	$-0.246463\pi$
$479$	31.2936	1.42984	0.714921	+	0.699205i	$0.246463\pi$
$480$	0	0
$481$	20.8656	0.951390
$482$	0	0
$483$	−9.58578	−0.436168
$484$	0	0
$485$	−8.64681	−0.392631
$486$	0	0
$487$	−28.6773	−1.29949	−0.649745	−	0.760152i	$-0.725125\pi$
$487$	−28.6773	−1.29949	−0.649745	+	0.760152i	$0.725125\pi$
$488$	0	0
$489$	5.07962	0.229708
$490$	0	0
$491$	9.43841	0.425949	0.212975	−	0.977058i	$-0.431685\pi$
$491$	9.43841	0.425949	0.212975	+	0.977058i	$0.431685\pi$
$492$	0	0
$493$	0.110856	0.00499272
$494$	0	0
$495$	1.25979	0.0566234
$496$	0	0
$497$	−10.8823	−0.488138
$498$	0	0
$499$	−35.9162	−1.60783	−0.803916	−	0.594743i	$-0.797254\pi$
$499$	−35.9162	−1.60783	−0.803916	+	0.594743i	$0.797254\pi$
$500$	0	0
$501$	11.3836	0.508584
$502$	0	0
$503$	14.3684	0.640656	0.320328	−	0.947307i	$-0.396207\pi$
$503$	14.3684	0.640656	0.320328	+	0.947307i	$0.396207\pi$
$504$	0	0
$505$	−6.03046	−0.268352
$506$	0	0
$507$	3.51658	0.156177
$508$	0	0
$509$	−32.3047	−1.43188	−0.715940	−	0.698161i	$-0.754002\pi$
$509$	−32.3047	−1.43188	−0.715940	+	0.698161i	$0.754002\pi$
$510$	0	0
$511$	44.5172	1.96933
$512$	0	0
$513$	7.65722	0.338075
$514$	0	0
$515$	−16.1801	−0.712979
$516$	0	0
$517$	−1.55118	−0.0682209
$518$	0	0
$519$	−0.766628	−0.0336512
$520$	0	0
$521$	−19.8809	−0.870996	−0.435498	−	0.900190i	$-0.643428\pi$
$521$	−19.8809	−0.870996	−0.435498	+	0.900190i	$0.643428\pi$
$522$	0	0
$523$	2.08377	0.0911167	0.0455584	−	0.998962i	$-0.485493\pi$
$523$	2.08377	0.0911167	0.0455584	+	0.998962i	$0.485493\pi$
$524$	0	0
$525$	4.48534	0.195756
$526$	0	0
$527$	−8.79641	−0.383178
$528$	0	0
$529$	19.3490	0.841263
$530$	0	0
$531$	−18.3926	−0.798171
$532$	0	0
$533$	−6.17380	−0.267417
$534$	0	0
$535$	3.27569	0.141620
$536$	0	0
$537$	2.66282	0.114909
$538$	0	0
$539$	−1.01523	−0.0437290
$540$	0	0
$541$	24.0554	1.03422	0.517112	−	0.855918i	$-0.327007\pi$
$541$	24.0554	1.03422	0.517112	+	0.855918i	$0.327007\pi$
$542$	0	0
$543$	11.9702	0.513691
$544$	0	0
$545$	−3.24860	−0.139155
$546$	0	0
$547$	−12.0242	−0.514117	−0.257059	−	0.966396i	$-0.582753\pi$
$547$	−12.0242	−0.514117	−0.257059	+	0.966396i	$0.582753\pi$
$548$	0	0
$549$	2.78600	0.118904
$550$	0	0
$551$	0.184210	0.00784762
$552$	0	0
$553$	−13.3566	−0.567979
$554$	0	0
$555$	−5.80875	−0.246567
$556$	0	0
$557$	10.3476	0.438442	0.219221	−	0.975675i	$-0.429648\pi$
$557$	10.3476	0.438442	0.219221	+	0.975675i	$0.429648\pi$
$558$	0	0
$559$	−26.5872	−1.12452
$560$	0	0
$561$	0.257564	0.0108744
$562$	0	0
$563$	0.0900320	0.00379439	0.00189720	−	0.999998i	$-0.499396\pi$
$563$	0.0900320	0.00379439	0.00189720	+	0.999998i	$0.499396\pi$
$564$	0	0
$565$	−6.85936	−0.288575
$566$	0	0
$567$	22.6710	0.952093
$568$	0	0
$569$	24.5243	1.02811	0.514056	−	0.857757i	$-0.328142\pi$
$569$	24.5243	1.02811	0.514056	+	0.857757i	$0.328142\pi$
$570$	0	0
$571$	21.7979	0.912212	0.456106	−	0.889925i	$-0.349244\pi$
$571$	21.7979	0.912212	0.456106	+	0.889925i	$0.349244\pi$
$572$	0	0
$573$	−0.989588	−0.0413406
$574$	0	0
$575$	−19.8158	−0.826376
$576$	0	0
$577$	34.3088	1.42830	0.714148	−	0.699995i	$-0.246814\pi$
$577$	34.3088	1.42830	0.714148	+	0.699995i	$0.246814\pi$
$578$	0	0
$579$	−11.7604	−0.488744
$580$	0	0
$581$	51.4391	2.13405
$582$	0	0
$583$	−4.19125	−0.173584
$584$	0	0
$585$	−9.05061	−0.374197
$586$	0	0
$587$	32.8477	1.35577	0.677885	−	0.735168i	$-0.262897\pi$
$587$	32.8477	1.35577	0.677885	+	0.735168i	$0.262897\pi$
$588$	0	0
$589$	−14.6170	−0.602284
$590$	0	0
$591$	−10.4972	−0.431797
$592$	0	0
$593$	15.4640	0.635032	0.317516	−	0.948253i	$-0.397151\pi$
$593$	15.4640	0.635032	0.317516	+	0.948253i	$0.397151\pi$
$594$	0	0
$595$	7.66494	0.314232
$596$	0	0
$597$	−10.9460	−0.447991
$598$	0	0
$599$	25.7458	1.05195	0.525973	−	0.850502i	$-0.323701\pi$
$599$	25.7458	1.05195	0.525973	+	0.850502i	$0.323701\pi$
$600$	0	0
$601$	7.56159	0.308444	0.154222	−	0.988036i	$-0.450713\pi$
$601$	7.56159	0.308444	0.154222	+	0.988036i	$0.450713\pi$
$602$	0	0
$603$	−13.0714	−0.532310
$604$	0	0
$605$	−15.2340	−0.619352
$606$	0	0
$607$	−20.7160	−0.840837	−0.420419	−	0.907330i	$-0.638117\pi$
$607$	−20.7160	−0.840837	−0.420419	+	0.907330i	$0.638117\pi$
$608$	0	0
$609$	−0.0948486	−0.00384346
$610$	0	0
$611$	11.1440	0.450838
$612$	0	0
$613$	2.05957	0.0831854	0.0415927	−	0.999135i	$-0.486757\pi$
$613$	2.05957	0.0831854	0.0415927	+	0.999135i	$0.486757\pi$
$614$	0	0
$615$	1.71871	0.0693052
$616$	0	0
$617$	34.6773	1.39605	0.698027	−	0.716071i	$-0.254061\pi$
$617$	34.6773	1.39605	0.698027	+	0.716071i	$0.254061\pi$
$618$	0	0
$619$	−20.6635	−0.830536	−0.415268	−	0.909699i	$-0.636312\pi$
$619$	−20.6635	−0.830536	−0.415268	+	0.909699i	$0.636312\pi$
$620$	0	0
$621$	17.4183	0.698970
$622$	0	0
$623$	8.01793	0.321231
$624$	0	0
$625$	−0.502798	−0.0201119
$626$	0	0
$627$	0.427995	0.0170925
$628$	0	0
$629$	15.4611	0.616476
$630$	0	0
$631$	17.5914	0.700302	0.350151	−	0.936693i	$-0.386130\pi$
$631$	17.5914	0.700302	0.350151	+	0.936693i	$0.386130\pi$
$632$	0	0
$633$	4.49720	0.178748
$634$	0	0
$635$	−2.61635	−0.103827
$636$	0	0
$637$	7.29362	0.288984
$638$	0	0
$639$	9.52139	0.376661
$640$	0	0
$641$	22.8477	0.902430	0.451215	−	0.892415i	$-0.350991\pi$
$641$	22.8477	0.902430	0.451215	+	0.892415i	$0.350991\pi$
$642$	0	0
$643$	−31.0748	−1.22547	−0.612735	−	0.790288i	$-0.709931\pi$
$643$	−31.0748	−1.22547	−0.612735	+	0.790288i	$0.709931\pi$
$644$	0	0
$645$	7.40157	0.291437
$646$	0	0
$647$	−27.5187	−1.08187	−0.540936	−	0.841064i	$-0.681930\pi$
$647$	−27.5187	−1.08187	−0.540936	+	0.841064i	$0.681930\pi$
$648$	0	0
$649$	−2.13505	−0.0838080
$650$	0	0
$651$	7.52621	0.294975
$652$	0	0
$653$	−30.7804	−1.20453	−0.602265	−	0.798296i	$-0.705735\pi$
$653$	−30.7804	−1.20453	−0.602265	+	0.798296i	$0.705735\pi$
$654$	0	0
$655$	8.59843	0.335968
$656$	0	0
$657$	−38.9501	−1.51959
$658$	0	0
$659$	22.4238	0.873509	0.436755	−	0.899581i	$-0.356128\pi$
$659$	22.4238	0.873509	0.436755	+	0.899581i	$0.356128\pi$
$660$	0	0
$661$	15.1094	0.587688	0.293844	−	0.955853i	$-0.405065\pi$
$661$	15.1094	0.587688	0.293844	+	0.955853i	$0.405065\pi$
$662$	0	0
$663$	−1.85039	−0.0718633
$664$	0	0
$665$	12.7368	0.493913
$666$	0	0
$667$	0.419032	0.0162250
$668$	0	0
$669$	−7.53326	−0.291252
$670$	0	0
$671$	0.323404	0.0124849
$672$	0	0
$673$	−16.9073	−0.651727	−0.325864	−	0.945417i	$-0.605655\pi$
$673$	−16.9073	−0.651727	−0.325864	+	0.945417i	$0.605655\pi$
$674$	0	0
$675$	−8.15028	−0.313704
$676$	0	0
$677$	−40.6773	−1.56335	−0.781677	−	0.623683i	$-0.785636\pi$
$677$	−40.6773	−1.56335	−0.781677	+	0.623683i	$0.785636\pi$
$678$	0	0
$679$	−19.6918	−0.755703
$680$	0	0
$681$	−6.66763	−0.255504
$682$	0	0
$683$	26.9508	1.03125	0.515623	−	0.856816i	$-0.327561\pi$
$683$	26.9508	1.03125	0.515623	+	0.856816i	$0.327561\pi$
$684$	0	0
$685$	−23.0811	−0.881882
$686$	0	0
$687$	0.637175	0.0243098
$688$	0	0
$689$	30.1109	1.14713
$690$	0	0
$691$	20.8518	0.793241	0.396621	−	0.917983i	$-0.370183\pi$
$691$	20.8518	0.793241	0.396621	+	0.917983i	$0.370183\pi$
$692$	0	0
$693$	2.86899	0.108984
$694$	0	0
$695$	15.3386	0.581828
$696$	0	0
$697$	−4.57470	−0.173279
$698$	0	0
$699$	12.1109	0.458075
$700$	0	0
$701$	18.2951	0.690995	0.345498	−	0.938420i	$-0.387710\pi$
$701$	18.2951	0.690995	0.345498	+	0.938420i	$0.387710\pi$
$702$	0	0
$703$	25.6918	0.968986
$704$	0	0
$705$	−3.10236	−0.116842
$706$	0	0
$707$	−13.7335	−0.516500
$708$	0	0
$709$	41.3241	1.55196	0.775979	−	0.630759i	$-0.217256\pi$
$709$	41.3241	1.55196	0.775979	+	0.630759i	$0.217256\pi$
$710$	0	0
$711$	11.6862	0.438268
$712$	0	0
$713$	−33.2501	−1.24522
$714$	0	0
$715$	−1.05061	−0.0392906
$716$	0	0
$717$	1.86003	0.0694640
$718$	0	0
$719$	26.9765	1.00605	0.503026	−	0.864271i	$-0.332220\pi$
$719$	26.9765	1.00605	0.503026	+	0.864271i	$0.332220\pi$
$720$	0	0
$721$	−36.8477	−1.37228
$722$	0	0
$723$	8.47861	0.315323
$724$	0	0
$725$	−0.196072	−0.00728192
$726$	0	0
$727$	1.85039	0.0686273	0.0343137	−	0.999411i	$-0.489075\pi$
$727$	1.85039	0.0686273	0.0343137	+	0.999411i	$0.489075\pi$
$728$	0	0
$729$	−16.1213	−0.597084
$730$	0	0
$731$	−19.7008	−0.728660
$732$	0	0
$733$	−42.7639	−1.57952	−0.789761	−	0.613415i	$-0.789795\pi$
$733$	−42.7639	−1.57952	−0.789761	+	0.613415i	$0.789795\pi$
$734$	0	0
$735$	−2.03046	−0.0748946
$736$	0	0
$737$	−1.51736	−0.0558925
$738$	0	0
$739$	9.74580	0.358505	0.179253	−	0.983803i	$-0.442632\pi$
$739$	9.74580	0.358505	0.179253	+	0.983803i	$0.442632\pi$
$740$	0	0
$741$	−3.07480	−0.112956
$742$	0	0
$743$	33.5526	1.23093	0.615463	−	0.788166i	$-0.288969\pi$
$743$	33.5526	1.23093	0.615463	+	0.788166i	$0.288969\pi$
$744$	0	0
$745$	−29.6558	−1.08650
$746$	0	0
$747$	−45.0063	−1.64669
$748$	0	0
$749$	7.45990	0.272579
$750$	0	0
$751$	32.7756	1.19600	0.597999	−	0.801497i	$-0.295963\pi$
$751$	32.7756	1.19600	0.597999	+	0.801497i	$0.295963\pi$
$752$	0	0
$753$	−9.02642	−0.328941
$754$	0	0
$755$	−29.3691	−1.06885
$756$	0	0
$757$	6.28465	0.228420	0.114210	−	0.993457i	$-0.463566\pi$
$757$	6.28465	0.228420	0.114210	+	0.993457i	$0.463566\pi$
$758$	0	0
$759$	0.973580	0.0353387
$760$	0	0
$761$	2.86562	0.103879	0.0519394	−	0.998650i	$-0.483460\pi$
$761$	2.86562	0.103879	0.0519394	+	0.998650i	$0.483460\pi$
$762$	0	0
$763$	−7.39821	−0.267833
$764$	0	0
$765$	−6.70638	−0.242470
$766$	0	0
$767$	15.3386	0.553846
$768$	0	0
$769$	17.7812	0.641206	0.320603	−	0.947214i	$-0.396114\pi$
$769$	17.7812	0.641206	0.320603	+	0.947214i	$0.396114\pi$
$770$	0	0
$771$	8.76663	0.315722
$772$	0	0
$773$	20.1080	0.723233	0.361616	−	0.932327i	$-0.382225\pi$
$773$	20.1080	0.723233	0.361616	+	0.932327i	$0.382225\pi$
$774$	0	0
$775$	15.5582	0.558868
$776$	0	0
$777$	−13.2286	−0.474572
$778$	0	0
$779$	−7.60179	−0.272362
$780$	0	0
$781$	1.10526	0.0395493
$782$	0	0
$783$	0.172349	0.00615924
$784$	0	0
$785$	−7.18566	−0.256467
$786$	0	0
$787$	27.0665	0.964817	0.482408	−	0.875946i	$-0.339762\pi$
$787$	27.0665	0.964817	0.482408	+	0.875946i	$0.339762\pi$
$788$	0	0
$789$	13.0361	0.464096
$790$	0	0
$791$	−15.6212	−0.555425
$792$	0	0
$793$	−2.32340	−0.0825065
$794$	0	0
$795$	−8.38251	−0.297297
$796$	0	0
$797$	−44.3330	−1.57036	−0.785178	−	0.619270i	$-0.787429\pi$
$797$	−44.3330	−1.57036	−0.785178	+	0.619270i	$0.787429\pi$
$798$	0	0
$799$	8.25756	0.292132
$800$	0	0
$801$	−7.01523	−0.247871
$802$	0	0
$803$	−4.52139	−0.159557
$804$	0	0
$805$	28.9731	1.02117
$806$	0	0
$807$	−12.9944	−0.457425
$808$	0	0
$809$	−5.21467	−0.183338	−0.0916690	−	0.995790i	$-0.529220\pi$
$809$	−5.21467	−0.183338	−0.0916690	+	0.995790i	$0.529220\pi$
$810$	0	0
$811$	−50.0963	−1.75912	−0.879559	−	0.475789i	$-0.842163\pi$
$811$	−50.0963	−1.75912	−0.879559	+	0.475789i	$0.842163\pi$
$812$	0	0
$813$	3.11645	0.109299
$814$	0	0
$815$	−15.3532	−0.537799
$816$	0	0
$817$	−32.7368	−1.14532
$818$	0	0
$819$	−20.6114	−0.720222
$820$	0	0
$821$	−23.2984	−0.813121	−0.406560	−	0.913624i	$-0.633272\pi$
$821$	−23.2984	−0.813121	−0.406560	+	0.913624i	$0.633272\pi$
$822$	0	0
$823$	10.2624	0.357724	0.178862	−	0.983874i	$-0.442758\pi$
$823$	10.2624	0.357724	0.178862	+	0.983874i	$0.442758\pi$
$824$	0	0
$825$	−0.455554	−0.0158603
$826$	0	0
$827$	1.77704	0.0617937	0.0308969	−	0.999523i	$-0.490164\pi$
$827$	1.77704	0.0617937	0.0308969	+	0.999523i	$0.490164\pi$
$828$	0	0
$829$	10.3539	0.359604	0.179802	−	0.983703i	$-0.442454\pi$
$829$	10.3539	0.359604	0.179802	+	0.983703i	$0.442454\pi$
$830$	0	0
$831$	−0.0297872	−0.00103331
$832$	0	0
$833$	5.40447	0.187254
$834$	0	0
$835$	−34.4072	−1.19071
$836$	0	0
$837$	−13.6758	−0.472705
$838$	0	0
$839$	−49.4529	−1.70730	−0.853651	−	0.520845i	$-0.825617\pi$
$839$	−49.4529	−1.70730	−0.853651	+	0.520845i	$0.825617\pi$
$840$	0	0
$841$	−28.9959	−0.999857
$842$	0	0
$843$	−2.10796	−0.0726019
$844$	0	0
$845$	−10.6289	−0.365645
$846$	0	0
$847$	−34.6933	−1.19207
$848$	0	0
$849$	8.47156	0.290743
$850$	0	0
$851$	58.4424	2.00338
$852$	0	0
$853$	3.07817	0.105395	0.0526973	−	0.998611i	$-0.483218\pi$
$853$	3.07817	0.105395	0.0526973	+	0.998611i	$0.483218\pi$
$854$	0	0
$855$	−11.1440	−0.381117
$856$	0	0
$857$	−38.7658	−1.32422	−0.662108	−	0.749408i	$-0.730338\pi$
$857$	−38.7658	−1.32422	−0.662108	+	0.749408i	$0.730338\pi$
$858$	0	0
$859$	−44.5318	−1.51941	−0.759703	−	0.650270i	$-0.774656\pi$
$859$	−44.5318	−1.51941	−0.759703	+	0.650270i	$0.774656\pi$
$860$	0	0
$861$	3.91411	0.133393
$862$	0	0
$863$	−6.34760	−0.216075	−0.108037	−	0.994147i	$-0.534457\pi$
$863$	−6.34760	−0.216075	−0.108037	+	0.994147i	$0.534457\pi$
$864$	0	0
$865$	2.31714	0.0787851
$866$	0	0
$867$	6.49306	0.220516
$868$	0	0
$869$	1.35656	0.0460181
$870$	0	0
$871$	10.9010	0.369366
$872$	0	0
$873$	17.2292	0.583121
$874$	0	0
$875$	−35.8179	−1.21087
$876$	0	0
$877$	−6.47446	−0.218627	−0.109313	−	0.994007i	$-0.534865\pi$
$877$	−6.47446	−0.218627	−0.109313	+	0.994007i	$0.534865\pi$
$878$	0	0
$879$	−6.21881	−0.209755
$880$	0	0
$881$	17.4209	0.586927	0.293463	−	0.955970i	$-0.405192\pi$
$881$	17.4209	0.586927	0.293463	+	0.955970i	$0.405192\pi$
$882$	0	0
$883$	−28.9798	−0.975249	−0.487625	−	0.873053i	$-0.662137\pi$
$883$	−28.9798	−0.975249	−0.487625	+	0.873053i	$0.662137\pi$
$884$	0	0
$885$	−4.27010	−0.143538
$886$	0	0
$887$	−38.4072	−1.28959	−0.644793	−	0.764357i	$-0.723057\pi$
$887$	−38.4072	−1.28959	−0.644793	+	0.764357i	$0.723057\pi$
$888$	0	0
$889$	−5.95835	−0.199837
$890$	0	0
$891$	−2.30258	−0.0771393
$892$	0	0
$893$	13.7216	0.459176
$894$	0	0
$895$	−8.04838	−0.269028
$896$	0	0
$897$	−6.99440	−0.233536
$898$	0	0
$899$	−0.329000	−0.0109728
$900$	0	0
$901$	22.3117	0.743312
$902$	0	0
$903$	16.8560	0.560932
$904$	0	0
$905$	−36.1801	−1.20267
$906$	0	0
$907$	10.8269	0.359500	0.179750	−	0.983712i	$-0.442471\pi$
$907$	10.8269	0.359500	0.179750	+	0.983712i	$0.442471\pi$
$908$	0	0
$909$	12.0160	0.398546
$910$	0	0
$911$	−31.9792	−1.05952	−0.529759	−	0.848148i	$-0.677718\pi$
$911$	−31.9792	−1.05952	−0.529759	+	0.848148i	$0.677718\pi$
$912$	0	0
$913$	−5.22441	−0.172903
$914$	0	0
$915$	0.646809	0.0213828
$916$	0	0
$917$	19.5816	0.646643
$918$	0	0
$919$	18.2188	0.600983	0.300492	−	0.953784i	$-0.402849\pi$
$919$	18.2188	0.600983	0.300492	+	0.953784i	$0.402849\pi$
$920$	0	0
$921$	8.27839	0.272782
$922$	0	0
$923$	−7.94043	−0.261362
$924$	0	0
$925$	−27.3461	−0.899136
$926$	0	0
$927$	32.2396	1.05889
$928$	0	0
$929$	47.3691	1.55413	0.777065	−	0.629421i	$-0.216708\pi$
$929$	47.3691	1.55413	0.777065	+	0.629421i	$0.216708\pi$
$930$	0	0
$931$	8.98062	0.294328
$932$	0	0
$933$	8.91623	0.291904
$934$	0	0
$935$	−0.778489	−0.0254593
$936$	0	0
$937$	−33.0830	−1.08077	−0.540387	−	0.841417i	$-0.681722\pi$
$937$	−33.0830	−1.08077	−0.540387	+	0.841417i	$0.681722\pi$
$938$	0	0
$939$	5.22441	0.170492
$940$	0	0
$941$	10.8560	0.353895	0.176948	−	0.984220i	$-0.443378\pi$
$941$	10.8560	0.353895	0.176948	+	0.984220i	$0.443378\pi$
$942$	0	0
$943$	−17.2922	−0.563110
$944$	0	0
$945$	11.9167	0.387650
$946$	0	0
$947$	−31.9646	−1.03871	−0.519355	−	0.854558i	$-0.673828\pi$
$947$	−31.9646	−1.03871	−0.519355	+	0.854558i	$0.673828\pi$
$948$	0	0
$949$	32.4826	1.05443
$950$	0	0
$951$	−1.41277	−0.0458121
$952$	0	0
$953$	0.646809	0.0209522	0.0104761	−	0.999945i	$-0.496665\pi$
$953$	0.646809	0.0209522	0.0104761	+	0.999945i	$0.496665\pi$
$954$	0	0
$955$	2.99104	0.0967877
$956$	0	0
$957$	0.00963330	0.000311400 0
$958$	0	0
$959$	−52.5637	−1.69737
$960$	0	0
$961$	−4.89396	−0.157870
$962$	0	0
$963$	−6.52699	−0.210329
$964$	0	0
$965$	35.5458	1.14426
$966$	0	0
$967$	34.5068	1.10967	0.554833	−	0.831962i	$-0.312782\pi$
$967$	34.5068	1.10967	0.554833	+	0.831962i	$0.312782\pi$
$968$	0	0
$969$	−2.27839	−0.0731924
$970$	0	0
$971$	−0.504247	−0.0161821	−0.00809103	−	0.999967i	$-0.502575\pi$
$971$	−0.504247	−0.0161821	−0.00809103	+	0.999967i	$0.502575\pi$
$972$	0	0
$973$	34.9315	1.11985
$974$	0	0
$975$	3.27279	0.104813
$976$	0	0
$977$	17.9959	0.575738	0.287869	−	0.957670i	$-0.407053\pi$
$977$	17.9959	0.575738	0.287869	+	0.957670i	$0.407053\pi$
$978$	0	0
$979$	−0.814341	−0.0260264
$980$	0	0
$981$	6.47301	0.206667
$982$	0	0
$983$	25.8462	0.824367	0.412184	−	0.911101i	$-0.364766\pi$
$983$	25.8462	0.824367	0.412184	+	0.911101i	$0.364766\pi$
$984$	0	0
$985$	31.7279	1.01093
$986$	0	0
$987$	−7.06517	−0.224887
$988$	0	0
$989$	−74.4681	−2.36795
$990$	0	0
$991$	−14.6052	−0.463948	−0.231974	−	0.972722i	$-0.574518\pi$
$991$	−14.6052	−0.463948	−0.231974	+	0.972722i	$0.574518\pi$
$992$	0	0
$993$	5.30481	0.168343
$994$	0	0
$995$	33.0844	1.04885
$996$	0	0
$997$	−6.90023	−0.218532	−0.109266	−	0.994013i	$-0.534850\pi$
$997$	−6.90023	−0.218532	−0.109266	+	0.994013i	$0.534850\pi$
$998$	0	0
$999$	24.0375	0.760513

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	976.2.a.g.1.2		3
3.2	odd	2		8784.2.a.bm.1.2		3
4.3	odd	2		122.2.a.c.1.2	✓	3
8.3	odd	2		3904.2.a.u.1.2		3
8.5	even	2		3904.2.a.t.1.2		3
12.11	even	2		1098.2.a.p.1.2		3
20.3	even	4		3050.2.b.k.1099.2		6
20.7	even	4		3050.2.b.k.1099.5		6
20.19	odd	2		3050.2.a.t.1.2		3
28.27	even	2		5978.2.a.q.1.2		3
244.243	odd	2		7442.2.a.j.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
122.2.a.c.1.2	✓	3	4.3	odd	2
976.2.a.g.1.2		3	1.1	even	1	trivial
1098.2.a.p.1.2		3	12.11	even	2
3050.2.a.t.1.2		3	20.19	odd	2
3050.2.b.k.1099.2		6	20.3	even	4
3050.2.b.k.1099.5		6	20.7	even	4
3904.2.a.t.1.2		3	8.5	even	2
3904.2.a.u.1.2		3	8.3	odd	2
5978.2.a.q.1.2		3	28.27	even	2
7442.2.a.j.1.2		3	244.243	odd	2
8784.2.a.bm.1.2		3	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)

\(f(q)\)	\(=\)	\(q-0.462598 q^{3} +1.39821 q^{5} +3.18421 q^{7} -2.78600 q^{9} +O(q^{10})\)	\(q-0.462598 q^{3} +1.39821 q^{5} +3.18421 q^{7} -2.78600 q^{9} -0.323404 q^{11} +2.32340 q^{13} -0.646809 q^{15} +1.72161 q^{17} +2.86081 q^{19} -1.47301 q^{21} +6.50761 q^{23} -3.04502 q^{25} +2.67660 q^{27} +0.0643910 q^{29} -5.10941 q^{31} +0.149606 q^{33} +4.45219 q^{35} +8.98062 q^{37} -1.07480 q^{39} -2.65722 q^{41} -11.4432 q^{43} -3.89541 q^{45} +4.79641 q^{47} +3.13919 q^{49} -0.796415 q^{51} +12.9598 q^{53} -0.452186 q^{55} -1.32340 q^{57} +6.60179 q^{59} -1.00000 q^{61} -8.87122 q^{63} +3.24860 q^{65} +4.69182 q^{67} -3.01041 q^{69} -3.41758 q^{71} +13.9806 q^{73} +1.40862 q^{75} -1.02979 q^{77} -4.19462 q^{79} +7.11982 q^{81} +16.1544 q^{83} +2.40717 q^{85} -0.0297872 q^{87} +2.51803 q^{89} +7.39821 q^{91} +2.36360 q^{93} +4.00000 q^{95} -6.18421 q^{97} +0.901005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 3 * q + q^3 + q^5 - 4 * q^7 + 2 * q^9	\( 3 q + q^{3} + q^{5} - 4 q^{7} + 2 q^{9} + 7 q^{11} - q^{13} + 14 q^{15} - 6 q^{17} + 3 q^{19} - 6 q^{21} - 2 q^{23} + 10 q^{25} + 16 q^{27} + q^{29} + 3 q^{31} + 10 q^{33} + 7 q^{35} + 7 q^{37} - 8 q^{39} + 4 q^{41} - 12 q^{43} + 17 q^{45} + 8 q^{47} + 15 q^{49} + 4 q^{51} + 11 q^{53} + 5 q^{55} + 4 q^{57} + 23 q^{59} - 3 q^{61} - 25 q^{63} - 3 q^{65} - 21 q^{67} - 13 q^{69} - 27 q^{71} + 22 q^{73} + 5 q^{75} - 27 q^{77} - 3 q^{79} + 7 q^{81} + 11 q^{83} + 20 q^{85} - 24 q^{87} - 10 q^{89} + 19 q^{91} + 27 q^{93} + 12 q^{95} - 5 q^{97} + 25 q^{99}+O(q^{100}) \) 3 * q + q^3 + q^5 - 4 * q^7 + 2 * q^9 + 7 * q^11 - q^13 + 14 * q^15 - 6 * q^17 + 3 * q^19 - 6 * q^21 - 2 * q^23 + 10 * q^25 + 16 * q^27 + q^29 + 3 * q^31 + 10 * q^33 + 7 * q^35 + 7 * q^37 - 8 * q^39 + 4 * q^41 - 12 * q^43 + 17 * q^45 + 8 * q^47 + 15 * q^49 + 4 * q^51 + 11 * q^53 + 5 * q^55 + 4 * q^57 + 23 * q^59 - 3 * q^61 - 25 * q^63 - 3 * q^65 - 21 * q^67 - 13 * q^69 - 27 * q^71 + 22 * q^73 + 5 * q^75 - 27 * q^77 - 3 * q^79 + 7 * q^81 + 11 * q^83 + 20 * q^85 - 24 * q^87 - 10 * q^89 + 19 * q^91 + 27 * q^93 + 12 * q^95 - 5 * q^97 + 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more