LMFDB - Embedded newform 3328.2.a.bp.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,8,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.10323968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 25x^{2} - 8 $ x^6 - 10x^4 + 25x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.611393$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.611393 q^{3} -1.62620 q^{5} +3.10261 q^{7} -2.62620 q^{9} +5.31965 q^{11} +1.00000 q^{13} +0.994247 q^{15} -1.89692 q^{17} +0.885578 q^{19} -1.89692 q^{21} -1.22279 q^{23} -2.35548 q^{25} +3.43982 q^{27} +2.00000 q^{29} -3.71400 q^{31} -3.25240 q^{33} -5.04546 q^{35} +11.1493 q^{37} -0.611393 q^{39} +5.79383 q^{41} +3.43982 q^{43} +4.27072 q^{45} +0.274184 q^{47} +2.62620 q^{49} +1.15976 q^{51} +7.79383 q^{53} -8.65080 q^{55} -0.541436 q^{57} +1.56000 q^{59} -7.04623 q^{61} -8.14807 q^{63} -1.62620 q^{65} -12.7477 q^{67} +0.747604 q^{69} -8.08505 q^{71} -16.2986 q^{73} +1.44012 q^{75} +16.5048 q^{77} +9.41650 q^{79} +5.77551 q^{81} -10.9765 q^{83} +3.08476 q^{85} -1.22279 q^{87} +5.25240 q^{89} +3.10261 q^{91} +2.27072 q^{93} -1.44012 q^{95} -8.50479 q^{97} -13.9704 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{5} + 2 q^{9} + 6 q^{13} - 4 q^{17} - 4 q^{21} + 14 q^{25} + 12 q^{29} + 16 q^{33} + 24 q^{37} + 20 q^{41} + 36 q^{45} - 2 q^{49} + 32 q^{53} - 24 q^{57} + 8 q^{61} + 8 q^{65} + 40 q^{69} - 12 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) $ 6 * q + 8 * q^5 + 2 * q^9 + 6 * q^13 - 4 * q^17 - 4 * q^21 + 14 * q^25 + 12 * q^29 + 16 * q^33 + 24 * q^37 + 20 * q^41 + 36 * q^45 - 2 * q^49 + 32 * q^53 - 24 * q^57 + 8 * q^61 + 8 * q^65 + 40 * q^69 - 12 * q^73 + 28 * q^77 - 26 * q^81 - 20 * q^85 - 4 * q^89 + 24 * q^93 + 20 * q^97

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
$3$	−0.611393	−0.352988	−0.176494	−	0.984302i	$-0.556476\pi$
$3$	−0.611393	−0.352988	−0.176494	+	0.984302i	$0.556476\pi$
$4$	0	0
$5$	−1.62620	−0.727258	−0.363629	−	0.931544i	$-0.618462\pi$
$5$	−1.62620	−0.727258	−0.363629	+	0.931544i	$0.618462\pi$
$6$	0	0
$7$	3.10261	1.17268	0.586338	−	0.810066i	$-0.300569\pi$
$7$	3.10261	1.17268	0.586338	+	0.810066i	$0.300569\pi$
$8$	0	0
$9$	−2.62620	−0.875399
$10$	0	0
$11$	5.31965	1.60393	0.801967	−	0.597369i	$-0.203787\pi$
$11$	5.31965	1.60393	0.801967	+	0.597369i	$0.203787\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.994247	0.256713
$16$	0	0
$17$	−1.89692	−0.460070	−0.230035	−	0.973182i	$-0.573884\pi$
$17$	−1.89692	−0.460070	−0.230035	+	0.973182i	$0.573884\pi$
$18$	0	0
$19$	0.885578	0.203165	0.101583	−	0.994827i	$-0.467609\pi$
$19$	0.885578	0.203165	0.101583	+	0.994827i	$0.467609\pi$
$20$	0	0
$21$	−1.89692	−0.413941
$22$	0	0
$23$	−1.22279	−0.254969	−0.127484	−	0.991841i	$-0.540690\pi$
$23$	−1.22279	−0.254969	−0.127484	+	0.991841i	$0.540690\pi$
$24$	0	0
$25$	−2.35548	−0.471096
$26$	0	0
$27$	3.43982	0.661994
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−3.71400	−0.667055	−0.333527	−	0.942740i	$-0.608239\pi$
$31$	−3.71400	−0.667055	−0.333527	+	0.942740i	$0.608239\pi$
$32$	0	0
$33$	−3.25240	−0.566169
$34$	0	0
$35$	−5.04546	−0.852839
$36$	0	0
$37$	11.1493	1.83294	0.916468	−	0.400108i	$-0.131028\pi$
$37$	11.1493	1.83294	0.916468	+	0.400108i	$0.131028\pi$
$38$	0	0
$39$	−0.611393	−0.0979013
$40$	0	0
$41$	5.79383	0.904845	0.452422	−	0.891804i	$-0.350560\pi$
$41$	5.79383	0.904845	0.452422	+	0.891804i	$0.350560\pi$
$42$	0	0
$43$	3.43982	0.524568	0.262284	−	0.964991i	$-0.415524\pi$
$43$	3.43982	0.524568	0.262284	+	0.964991i	$0.415524\pi$
$44$	0	0
$45$	4.27072	0.636641
$46$	0	0
$47$	0.274184	0.0399939	0.0199970	−	0.999800i	$-0.493634\pi$
$47$	0.274184	0.0399939	0.0199970	+	0.999800i	$0.493634\pi$
$48$	0	0
$49$	2.62620	0.375171
$50$	0	0
$51$	1.15976	0.162399
$52$	0	0
$53$	7.79383	1.07057	0.535283	−	0.844673i	$-0.320205\pi$
$53$	7.79383	1.07057	0.535283	+	0.844673i	$0.320205\pi$
$54$	0	0
$55$	−8.65080	−1.16647
$56$	0	0
$57$	−0.541436	−0.0717150
$58$	0	0
$59$	1.56000	0.203094	0.101547	−	0.994831i	$-0.467621\pi$
$59$	1.56000	0.203094	0.101547	+	0.994831i	$0.467621\pi$
$60$	0	0
$61$	−7.04623	−0.902177	−0.451089	−	0.892479i	$-0.648964\pi$
$61$	−7.04623	−0.902177	−0.451089	+	0.892479i	$0.648964\pi$
$62$	0	0
$63$	−8.14807	−1.02656
$64$	0	0
$65$	−1.62620	−0.201705
$66$	0	0
$67$	−12.7477	−1.55737	−0.778687	−	0.627413i	$-0.784114\pi$
$67$	−12.7477	−1.55737	−0.778687	+	0.627413i	$0.784114\pi$
$68$	0	0
$69$	0.747604	0.0900009
$70$	0	0
$71$	−8.08505	−0.959519	−0.479759	−	0.877400i	$-0.659276\pi$
$71$	−8.08505	−0.959519	−0.479759	+	0.877400i	$0.659276\pi$
$72$	0	0
$73$	−16.2986	−1.90761	−0.953805	−	0.300427i	$-0.902871\pi$
$73$	−16.2986	−1.90761	−0.953805	+	0.300427i	$0.902871\pi$
$74$	0	0
$75$	1.44012	0.166291
$76$	0	0
$77$	16.5048	1.88090
$78$	0	0
$79$	9.41650	1.05944	0.529720	−	0.848173i	$-0.322297\pi$
$79$	9.41650	1.05944	0.529720	+	0.848173i	$0.322297\pi$
$80$	0	0
$81$	5.77551	0.641723
$82$	0	0
$83$	−10.9765	−1.20483	−0.602414	−	0.798184i	$-0.705794\pi$
$83$	−10.9765	−1.20483	−0.602414	+	0.798184i	$0.705794\pi$
$84$	0	0
$85$	3.08476	0.334589
$86$	0	0
$87$	−1.22279	−0.131097
$88$	0	0
$89$	5.25240	0.556753	0.278376	−	0.960472i	$-0.410204\pi$
$89$	5.25240	0.556753	0.278376	+	0.960472i	$0.410204\pi$
$90$	0	0
$91$	3.10261	0.325242
$92$	0	0
$93$	2.27072	0.235463
$94$	0	0
$95$	−1.44012	−0.147754
$96$	0	0
$97$	−8.50479	−0.863531	−0.431765	−	0.901986i	$-0.642109\pi$
$97$	−8.50479	−0.863531	−0.431765	+	0.901986i	$0.642109\pi$
$98$	0	0
$99$	−13.9704	−1.40408
$100$	0	0
$101$	11.4586	1.14017	0.570085	−	0.821586i	$-0.306910\pi$
$101$	11.4586	1.14017	0.570085	+	0.821586i	$0.306910\pi$
$102$	0	0
$103$	13.6332	1.34332	0.671661	−	0.740858i	$-0.265581\pi$
$103$	13.6332	1.34332	0.671661	+	0.740858i	$0.265581\pi$
$104$	0	0
$105$	3.08476	0.301042
$106$	0	0
$107$	17.6844	1.70962	0.854810	−	0.518941i	$-0.173674\pi$
$107$	17.6844	1.70962	0.854810	+	0.518941i	$0.173674\pi$
$108$	0	0
$109$	14.6079	1.39918	0.699590	−	0.714544i	$-0.253366\pi$
$109$	14.6079	1.39918	0.699590	+	0.714544i	$0.253366\pi$
$110$	0	0
$111$	−6.81662	−0.647005
$112$	0	0
$113$	10.2986	0.968813	0.484407	−	0.874843i	$-0.339036\pi$
$113$	10.2986	0.968813	0.484407	+	0.874843i	$0.339036\pi$
$114$	0	0
$115$	1.98849	0.185428
$116$	0	0
$117$	−2.62620	−0.242792
$118$	0	0
$119$	−5.88539	−0.539513
$120$	0	0
$121$	17.2986	1.57260
$122$	0	0
$123$	−3.54231	−0.319399
$124$	0	0
$125$	11.9615	1.06987
$126$	0	0
$127$	10.0909	0.895424	0.447712	−	0.894178i	$-0.352239\pi$
$127$	10.0909	0.895424	0.447712	+	0.894178i	$0.352239\pi$
$128$	0	0
$129$	−2.10308	−0.185166
$130$	0	0
$131$	5.21098	0.455285	0.227643	−	0.973745i	$-0.426898\pi$
$131$	5.21098	0.455285	0.227643	+	0.973745i	$0.426898\pi$
$132$	0	0
$133$	2.74760	0.238247
$134$	0	0
$135$	−5.59383	−0.481440
$136$	0	0
$137$	12.5048	1.06836	0.534178	−	0.845372i	$-0.320621\pi$
$137$	12.5048	1.06836	0.534178	+	0.845372i	$0.320621\pi$
$138$	0	0
$139$	−17.4559	−1.48059	−0.740295	−	0.672282i	$-0.765314\pi$
$139$	−17.4559	−1.48059	−0.740295	+	0.672282i	$0.765314\pi$
$140$	0	0
$141$	−0.167635	−0.0141174
$142$	0	0
$143$	5.31965	0.444851
$144$	0	0
$145$	−3.25240	−0.270097
$146$	0	0
$147$	−1.60564	−0.132431
$148$	0	0
$149$	−4.50479	−0.369047	−0.184523	−	0.982828i	$-0.559074\pi$
$149$	−4.50479	−0.369047	−0.184523	+	0.982828i	$0.559074\pi$
$150$	0	0
$151$	14.5818	1.18665	0.593326	−	0.804962i	$-0.297814\pi$
$151$	14.5818	1.18665	0.593326	+	0.804962i	$0.297814\pi$
$152$	0	0
$153$	4.98168	0.402745
$154$	0	0
$155$	6.03971	0.485121
$156$	0	0
$157$	−1.79383	−0.143163	−0.0715817	−	0.997435i	$-0.522805\pi$
$157$	−1.79383	−0.143163	−0.0715817	+	0.997435i	$0.522805\pi$
$158$	0	0
$159$	−4.76510	−0.377897
$160$	0	0
$161$	−3.79383	−0.298996
$162$	0	0
$163$	−7.85651	−0.615369	−0.307685	−	0.951488i	$-0.599554\pi$
$163$	−7.85651	−0.615369	−0.307685	+	0.951488i	$0.599554\pi$
$164$	0	0
$165$	5.28904	0.411751
$166$	0	0
$167$	14.2620	1.10363	0.551814	−	0.833967i	$-0.313936\pi$
$167$	14.2620	1.10363	0.551814	+	0.833967i	$0.313936\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.32570	−0.177851
$172$	0	0
$173$	−2.29862	−0.174761	−0.0873806	−	0.996175i	$-0.527850\pi$
$173$	−2.29862	−0.174761	−0.0873806	+	0.996175i	$0.527850\pi$
$174$	0	0
$175$	−7.30814	−0.552443
$176$	0	0
$177$	−0.953771	−0.0716898
$178$	0	0
$179$	15.5587	1.16291	0.581456	−	0.813578i	$-0.302483\pi$
$179$	15.5587	1.16291	0.581456	+	0.813578i	$0.302483\pi$
$180$	0	0
$181$	1.04623	0.0777656	0.0388828	−	0.999244i	$-0.487620\pi$
$181$	1.04623	0.0777656	0.0388828	+	0.999244i	$0.487620\pi$
$182$	0	0
$183$	4.30802	0.318458
$184$	0	0
$185$	−18.1310	−1.33302
$186$	0	0
$187$	−10.0909	−0.737921
$188$	0	0
$189$	10.6724	0.776305
$190$	0	0
$191$	8.65080	0.625950	0.312975	−	0.949761i	$-0.398674\pi$
$191$	8.65080	0.625950	0.312975	+	0.949761i	$0.398674\pi$
$192$	0	0
$193$	12.5048	0.900115	0.450057	−	0.893000i	$-0.351404\pi$
$193$	12.5048	0.900115	0.450057	+	0.893000i	$0.351404\pi$
$194$	0	0
$195$	0.994247	0.0711995
$196$	0	0
$197$	15.3555	1.09403	0.547016	−	0.837122i	$-0.315764\pi$
$197$	15.3555	1.09403	0.547016	+	0.837122i	$0.315764\pi$
$198$	0	0
$199$	14.9821	1.06205	0.531025	−	0.847356i	$-0.321807\pi$
$199$	14.9821	1.06205	0.531025	+	0.847356i	$0.321807\pi$
$200$	0	0
$201$	7.79383	0.549735
$202$	0	0
$203$	6.20522	0.435521
$204$	0	0
$205$	−9.42192	−0.658055
$206$	0	0
$207$	3.21128	0.223199
$208$	0	0
$209$	4.71096	0.325864
$210$	0	0
$211$	9.77109	0.672670	0.336335	−	0.941742i	$-0.390813\pi$
$211$	9.77109	0.672670	0.336335	+	0.941742i	$0.390813\pi$
$212$	0	0
$213$	4.94315	0.338699
$214$	0	0
$215$	−5.59383	−0.381496
$216$	0	0
$217$	−11.5231	−0.782240
$218$	0	0
$219$	9.96487	0.673364
$220$	0	0
$221$	−1.89692	−0.127600
$222$	0	0
$223$	18.4675	1.23668	0.618339	−	0.785912i	$-0.287806\pi$
$223$	18.4675	1.23668	0.618339	+	0.785912i	$0.287806\pi$
$224$	0	0
$225$	6.18596	0.412397
$226$	0	0
$227$	15.7416	1.04481	0.522403	−	0.852699i	$-0.325036\pi$
$227$	15.7416	1.04481	0.522403	+	0.852699i	$0.325036\pi$
$228$	0	0
$229$	17.0848	1.12899	0.564496	−	0.825436i	$-0.309070\pi$
$229$	17.0848	1.12899	0.564496	+	0.825436i	$0.309070\pi$
$230$	0	0
$231$	−10.0909	−0.663934
$232$	0	0
$233$	−13.4200	−0.879175	−0.439588	−	0.898200i	$-0.644875\pi$
$233$	−13.4200	−0.879175	−0.439588	+	0.898200i	$0.644875\pi$
$234$	0	0
$235$	−0.445878	−0.0290859
$236$	0	0
$237$	−5.75719	−0.373970
$238$	0	0
$239$	−17.5016	−1.13208	−0.566041	−	0.824377i	$-0.691525\pi$
$239$	−17.5016	−1.13208	−0.566041	+	0.824377i	$0.691525\pi$
$240$	0	0
$241$	−4.84006	−0.311776	−0.155888	−	0.987775i	$-0.549824\pi$
$241$	−4.84006	−0.311776	−0.155888	+	0.987775i	$0.549824\pi$
$242$	0	0
$243$	−13.8506	−0.888515
$244$	0	0
$245$	−4.27072	−0.272846
$246$	0	0
$247$	0.885578	0.0563480
$248$	0	0
$249$	6.71096	0.425290
$250$	0	0
$251$	−1.05727	−0.0667344	−0.0333672	−	0.999443i	$-0.510623\pi$
$251$	−1.05727	−0.0667344	−0.0333672	+	0.999443i	$0.510623\pi$
$252$	0	0
$253$	−6.50479	−0.408953
$254$	0	0
$255$	−1.88600	−0.118106
$256$	0	0
$257$	−0.644520	−0.0402041	−0.0201020	−	0.999798i	$-0.506399\pi$
$257$	−0.644520	−0.0402041	−0.0201020	+	0.999798i	$0.506399\pi$
$258$	0	0
$259$	34.5920	2.14944
$260$	0	0
$261$	−5.25240	−0.325115
$262$	0	0
$263$	−26.2610	−1.61932	−0.809662	−	0.586897i	$-0.800349\pi$
$263$	−26.2610	−1.61932	−0.809662	+	0.586897i	$0.800349\pi$
$264$	0	0
$265$	−12.6743	−0.778577
$266$	0	0
$267$	−3.21128	−0.196527
$268$	0	0
$269$	27.7572	1.69239	0.846193	−	0.532877i	$-0.178889\pi$
$269$	27.7572	1.69239	0.846193	+	0.532877i	$0.178889\pi$
$270$	0	0
$271$	−26.6613	−1.61956	−0.809778	−	0.586737i	$-0.800412\pi$
$271$	−26.6613	−1.61956	−0.809778	+	0.586737i	$0.800412\pi$
$272$	0	0
$273$	−1.89692	−0.114807
$274$	0	0
$275$	−12.5303	−0.755607
$276$	0	0
$277$	−14.5048	−0.871509	−0.435754	−	0.900066i	$-0.643518\pi$
$277$	−14.5048	−0.871509	−0.435754	+	0.900066i	$0.643518\pi$
$278$	0	0
$279$	9.75371	0.583939
$280$	0	0
$281$	−16.7110	−0.996892	−0.498446	−	0.866921i	$-0.666096\pi$
$281$	−16.7110	−0.996892	−0.498446	+	0.866921i	$0.666096\pi$
$282$	0	0
$283$	−18.3589	−1.09132	−0.545661	−	0.838006i	$-0.683721\pi$
$283$	−18.3589	−1.09132	−0.545661	+	0.838006i	$0.683721\pi$
$284$	0	0
$285$	0.880483	0.0521553
$286$	0	0
$287$	17.9760	1.06109
$288$	0	0
$289$	−13.4017	−0.788336
$290$	0	0
$291$	5.19977	0.304816
$292$	0	0
$293$	33.1127	1.93446	0.967231	−	0.253897i	$-0.0817123\pi$
$293$	33.1127	1.93446	0.967231	+	0.253897i	$0.0817123\pi$
$294$	0	0
$295$	−2.53686	−0.147702
$296$	0	0
$297$	18.2986	1.06179
$298$	0	0
$299$	−1.22279	−0.0707156
$300$	0	0
$301$	10.6724	0.615148
$302$	0	0
$303$	−7.00569	−0.402466
$304$	0	0
$305$	11.4586	0.656115
$306$	0	0
$307$	26.7243	1.52523	0.762617	−	0.646850i	$-0.223914\pi$
$307$	26.7243	1.52523	0.762617	+	0.646850i	$0.223914\pi$
$308$	0	0
$309$	−8.33527	−0.474177
$310$	0	0
$311$	−18.0673	−1.02450	−0.512251	−	0.858836i	$-0.671188\pi$
$311$	−18.0673	−1.02450	−0.512251	+	0.858836i	$0.671188\pi$
$312$	0	0
$313$	15.2139	0.859938	0.429969	−	0.902844i	$-0.358524\pi$
$313$	15.2139	0.859938	0.429969	+	0.902844i	$0.358524\pi$
$314$	0	0
$315$	13.2504	0.746574
$316$	0	0
$317$	−5.58767	−0.313835	−0.156917	−	0.987612i	$-0.550156\pi$
$317$	−5.58767	−0.313835	−0.156917	+	0.987612i	$0.550156\pi$
$318$	0	0
$319$	10.6393	0.595686
$320$	0	0
$321$	−10.8122	−0.603476
$322$	0	0
$323$	−1.67987	−0.0934703
$324$	0	0
$325$	−2.35548	−0.130659
$326$	0	0
$327$	−8.93116	−0.493894
$328$	0	0
$329$	0.850688	0.0468999
$330$	0	0
$331$	−19.0789	−1.04867	−0.524336	−	0.851511i	$-0.675687\pi$
$331$	−19.0789	−1.04867	−0.524336	+	0.851511i	$0.675687\pi$
$332$	0	0
$333$	−29.2803	−1.60455
$334$	0	0
$335$	20.7302	1.13261
$336$	0	0
$337$	−6.33716	−0.345207	−0.172603	−	0.984991i	$-0.555218\pi$
$337$	−6.33716	−0.345207	−0.172603	+	0.984991i	$0.555218\pi$
$338$	0	0
$339$	−6.29651	−0.341980
$340$	0	0
$341$	−19.7572	−1.06991
$342$	0	0
$343$	−13.5702	−0.732722
$344$	0	0
$345$	−1.21575	−0.0654539
$346$	0	0
$347$	−5.04546	−0.270855	−0.135427	−	0.990787i	$-0.543241\pi$
$347$	−5.04546	−0.270855	−0.135427	+	0.990787i	$0.543241\pi$
$348$	0	0
$349$	25.4758	1.36369	0.681845	−	0.731496i	$-0.261178\pi$
$349$	25.4758	1.36369	0.681845	+	0.731496i	$0.261178\pi$
$350$	0	0
$351$	3.43982	0.183604
$352$	0	0
$353$	14.5414	0.773963	0.386981	−	0.922088i	$-0.373518\pi$
$353$	14.5414	0.773963	0.386981	+	0.922088i	$0.373518\pi$
$354$	0	0
$355$	13.1479	0.697817
$356$	0	0
$357$	3.59829	0.190442
$358$	0	0
$359$	28.8783	1.52414	0.762069	−	0.647496i	$-0.224184\pi$
$359$	28.8783	1.52414	0.762069	+	0.647496i	$0.224184\pi$
$360$	0	0
$361$	−18.2158	−0.958724
$362$	0	0
$363$	−10.5763	−0.555110
$364$	0	0
$365$	26.5048	1.38732
$366$	0	0
$367$	3.57707	0.186722	0.0933608	−	0.995632i	$-0.470239\pi$
$367$	3.57707	0.186722	0.0933608	+	0.995632i	$0.470239\pi$
$368$	0	0
$369$	−15.2158	−0.792100
$370$	0	0
$371$	24.1812	1.25543
$372$	0	0
$373$	−20.7110	−1.07237	−0.536186	−	0.844100i	$-0.680136\pi$
$373$	−20.7110	−1.07237	−0.536186	+	0.844100i	$0.680136\pi$
$374$	0	0
$375$	−7.31316	−0.377650
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−23.9701	−1.23126	−0.615630	−	0.788035i	$-0.711098\pi$
$379$	−23.9701	−1.23126	−0.615630	+	0.788035i	$0.711098\pi$
$380$	0	0
$381$	−6.16952	−0.316074
$382$	0	0
$383$	−30.6606	−1.56669	−0.783343	−	0.621590i	$-0.786487\pi$
$383$	−30.6606	−1.56669	−0.783343	+	0.621590i	$0.786487\pi$
$384$	0	0
$385$	−26.8401	−1.36790
$386$	0	0
$387$	−9.03365	−0.459206
$388$	0	0
$389$	27.3449	1.38644	0.693220	−	0.720726i	$-0.256192\pi$
$389$	27.3449	1.38644	0.693220	+	0.720726i	$0.256192\pi$
$390$	0	0
$391$	2.31952	0.117303
$392$	0	0
$393$	−3.18596	−0.160710
$394$	0	0
$395$	−15.3131	−0.770486
$396$	0	0
$397$	8.09246	0.406149	0.203074	−	0.979163i	$-0.434907\pi$
$397$	8.09246	0.406149	0.203074	+	0.979163i	$0.434907\pi$
$398$	0	0
$399$	−1.67987	−0.0840985
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	−3.71400	−0.185008
$404$	0	0
$405$	−9.39212	−0.466698
$406$	0	0
$407$	59.3104	2.93991
$408$	0	0
$409$	−8.09246	−0.400146	−0.200073	−	0.979781i	$-0.564118\pi$
$409$	−8.09246	−0.400146	−0.200073	+	0.979781i	$0.564118\pi$
$410$	0	0
$411$	−7.64535	−0.377117
$412$	0	0
$413$	4.84006	0.238164
$414$	0	0
$415$	17.8500	0.876220
$416$	0	0
$417$	10.6724	0.522631
$418$	0	0
$419$	9.93661	0.485435	0.242718	−	0.970097i	$-0.421961\pi$
$419$	9.93661	0.485435	0.242718	+	0.970097i	$0.421961\pi$
$420$	0	0
$421$	9.96147	0.485492	0.242746	−	0.970090i	$-0.421952\pi$
$421$	9.96147	0.485492	0.242746	+	0.970090i	$0.421952\pi$
$422$	0	0
$423$	−0.720062	−0.0350106
$424$	0	0
$425$	4.46815	0.216737
$426$	0	0
$427$	−21.8617	−1.05796
$428$	0	0
$429$	−3.25240	−0.157027
$430$	0	0
$431$	10.1083	0.486900	0.243450	−	0.969913i	$-0.421721\pi$
$431$	10.1083	0.486900	0.243450	+	0.969913i	$0.421721\pi$
$432$	0	0
$433$	9.83237	0.472513	0.236257	−	0.971691i	$-0.424079\pi$
$433$	9.83237	0.472513	0.236257	+	0.971691i	$0.424079\pi$
$434$	0	0
$435$	1.98849	0.0953410
$436$	0	0
$437$	−1.08287	−0.0518008
$438$	0	0
$439$	−30.7298	−1.46666	−0.733328	−	0.679875i	$-0.762034\pi$
$439$	−30.7298	−1.46666	−0.733328	+	0.679875i	$0.762034\pi$
$440$	0	0
$441$	−6.89692	−0.328425
$442$	0	0
$443$	−4.62785	−0.219876	−0.109938	−	0.993938i	$-0.535065\pi$
$443$	−4.62785	−0.219876	−0.109938	+	0.993938i	$0.535065\pi$
$444$	0	0
$445$	−8.54144	−0.404903
$446$	0	0
$447$	2.75420	0.130269
$448$	0	0
$449$	35.1387	1.65830	0.829149	−	0.559028i	$-0.188826\pi$
$449$	35.1387	1.65830	0.829149	+	0.559028i	$0.188826\pi$
$450$	0	0
$451$	30.8211	1.45131
$452$	0	0
$453$	−8.91524	−0.418874
$454$	0	0
$455$	−5.04546	−0.236535
$456$	0	0
$457$	−0.840061	−0.0392964	−0.0196482	−	0.999807i	$-0.506255\pi$
$457$	−0.840061	−0.0392964	−0.0196482	+	0.999807i	$0.506255\pi$
$458$	0	0
$459$	−6.52505	−0.304563
$460$	0	0
$461$	−32.4942	−1.51340	−0.756702	−	0.653760i	$-0.773191\pi$
$461$	−32.4942	−1.51340	−0.756702	+	0.653760i	$0.773191\pi$
$462$	0	0
$463$	−6.06829	−0.282017	−0.141009	−	0.990008i	$-0.545035\pi$
$463$	−6.06829	−0.282017	−0.141009	+	0.990008i	$0.545035\pi$
$464$	0	0
$465$	−3.69264	−0.171242
$466$	0	0
$467$	−35.0773	−1.62319	−0.811593	−	0.584224i	$-0.801399\pi$
$467$	−35.0773	−1.62319	−0.811593	+	0.584224i	$0.801399\pi$
$468$	0	0
$469$	−39.5510	−1.82630
$470$	0	0
$471$	1.09674	0.0505350
$472$	0	0
$473$	18.2986	0.841372
$474$	0	0
$475$	−2.08596	−0.0957104
$476$	0	0
$477$	−20.4681	−0.937172
$478$	0	0
$479$	−8.50266	−0.388497	−0.194248	−	0.980952i	$-0.562227\pi$
$479$	−8.50266	−0.388497	−0.194248	+	0.980952i	$0.562227\pi$
$480$	0	0
$481$	11.1493	0.508365
$482$	0	0
$483$	2.31952	0.105542
$484$	0	0
$485$	13.8305	0.628010
$486$	0	0
$487$	0.171694	0.00778019	0.00389009	−	0.999992i	$-0.498762\pi$
$487$	0.171694	0.00778019	0.00389009	+	0.999992i	$0.498762\pi$
$488$	0	0
$489$	4.80342	0.217218
$490$	0	0
$491$	2.85669	0.128921	0.0644603	−	0.997920i	$-0.479467\pi$
$491$	2.85669	0.128921	0.0644603	+	0.997920i	$0.479467\pi$
$492$	0	0
$493$	−3.79383	−0.170866
$494$	0	0
$495$	22.7187	1.02113
$496$	0	0
$497$	−25.0848	−1.12521
$498$	0	0
$499$	−18.2785	−0.818256	−0.409128	−	0.912477i	$-0.634167\pi$
$499$	−18.2785	−0.818256	−0.409128	+	0.912477i	$0.634167\pi$
$500$	0	0
$501$	−8.71970	−0.389567
$502$	0	0
$503$	24.1465	1.07664	0.538319	−	0.842741i	$-0.319060\pi$
$503$	24.1465	1.07664	0.538319	+	0.842741i	$0.319060\pi$
$504$	0	0
$505$	−18.6339	−0.829197
$506$	0	0
$507$	−0.611393	−0.0271529
$508$	0	0
$509$	−34.5972	−1.53350	−0.766748	−	0.641948i	$-0.778126\pi$
$509$	−34.5972	−1.53350	−0.766748	+	0.641948i	$0.778126\pi$
$510$	0	0
$511$	−50.5683	−2.23701
$512$	0	0
$513$	3.04623	0.134494
$514$	0	0
$515$	−22.1703	−0.976942
$516$	0	0
$517$	1.45856	0.0641476
$518$	0	0
$519$	1.40536	0.0616886
$520$	0	0
$521$	−10.3372	−0.452879	−0.226440	−	0.974025i	$-0.572709\pi$
$521$	−10.3372	−0.452879	−0.226440	+	0.974025i	$0.572709\pi$
$522$	0	0
$523$	−23.5239	−1.02863	−0.514314	−	0.857602i	$-0.671953\pi$
$523$	−23.5239	−1.02863	−0.514314	+	0.857602i	$0.671953\pi$
$524$	0	0
$525$	4.46815	0.195006
$526$	0	0
$527$	7.04516	0.306892
$528$	0	0
$529$	−21.5048	−0.934991
$530$	0	0
$531$	−4.09686	−0.177789
$532$	0	0
$533$	5.79383	0.250959
$534$	0	0
$535$	−28.7584	−1.24333
$536$	0	0
$537$	−9.51249	−0.410494
$538$	0	0
$539$	13.9704	0.601750
$540$	0	0
$541$	−35.1493	−1.51119	−0.755593	−	0.655041i	$-0.772651\pi$
$541$	−35.1493	−1.51119	−0.755593	+	0.655041i	$0.772651\pi$
$542$	0	0
$543$	−0.639657	−0.0274503
$544$	0	0
$545$	−23.7553	−1.01757
$546$	0	0
$547$	37.6378	1.60927	0.804637	−	0.593767i	$-0.202360\pi$
$547$	37.6378	1.60927	0.804637	+	0.593767i	$0.202360\pi$
$548$	0	0
$549$	18.5048	0.789765
$550$	0	0
$551$	1.77116	0.0754538
$552$	0	0
$553$	29.2158	1.24238
$554$	0	0
$555$	11.0852	0.470539
$556$	0	0
$557$	9.39212	0.397957	0.198979	−	0.980004i	$-0.436238\pi$
$557$	9.39212	0.397957	0.198979	+	0.980004i	$0.436238\pi$
$558$	0	0
$559$	3.43982	0.145489
$560$	0	0
$561$	6.16952	0.260477
$562$	0	0
$563$	−12.4735	−0.525694	−0.262847	−	0.964838i	$-0.584661\pi$
$563$	−12.4735	−0.525694	−0.262847	+	0.964838i	$0.584661\pi$
$564$	0	0
$565$	−16.7476	−0.704577
$566$	0	0
$567$	17.9192	0.752534
$568$	0	0
$569$	32.1589	1.34817	0.674086	−	0.738653i	$-0.264538\pi$
$569$	32.1589	1.34817	0.674086	+	0.738653i	$0.264538\pi$
$570$	0	0
$571$	−3.73139	−0.156154	−0.0780768	−	0.996947i	$-0.524878\pi$
$571$	−3.73139	−0.156154	−0.0780768	+	0.996947i	$0.524878\pi$
$572$	0	0
$573$	−5.28904	−0.220953
$574$	0	0
$575$	2.88025	0.120115
$576$	0	0
$577$	30.3911	1.26520	0.632599	−	0.774480i	$-0.281988\pi$
$577$	30.3911	1.26520	0.632599	+	0.774480i	$0.281988\pi$
$578$	0	0
$579$	−7.64535	−0.317730
$580$	0	0
$581$	−34.0558	−1.41287
$582$	0	0
$583$	41.4604	1.71712
$584$	0	0
$585$	4.27072	0.176572
$586$	0	0
$587$	30.8150	1.27187	0.635935	−	0.771743i	$-0.280615\pi$
$587$	30.8150	1.27187	0.635935	+	0.771743i	$0.280615\pi$
$588$	0	0
$589$	−3.28904	−0.135523
$590$	0	0
$591$	−9.38824	−0.386181
$592$	0	0
$593$	−42.4681	−1.74396	−0.871979	−	0.489543i	$-0.837163\pi$
$593$	−42.4681	−1.74396	−0.871979	+	0.489543i	$0.837163\pi$
$594$	0	0
$595$	9.57082	0.392365
$596$	0	0
$597$	−9.15994	−0.374891
$598$	0	0
$599$	−40.7860	−1.66647	−0.833236	−	0.552918i	$-0.813514\pi$
$599$	−40.7860	−1.66647	−0.833236	+	0.552918i	$0.813514\pi$
$600$	0	0
$601$	17.7466	0.723897	0.361949	−	0.932198i	$-0.382112\pi$
$601$	17.7466	0.723897	0.361949	+	0.932198i	$0.382112\pi$
$602$	0	0
$603$	33.4779	1.36332
$604$	0	0
$605$	−28.1310	−1.14369
$606$	0	0
$607$	−23.0497	−0.935560	−0.467780	−	0.883845i	$-0.654946\pi$
$607$	−23.0497	−0.935560	−0.467780	+	0.883845i	$0.654946\pi$
$608$	0	0
$609$	−3.79383	−0.153734
$610$	0	0
$611$	0.274184	0.0110923
$612$	0	0
$613$	−22.5972	−0.912694	−0.456347	−	0.889802i	$-0.650842\pi$
$613$	−22.5972	−0.912694	−0.456347	+	0.889802i	$0.650842\pi$
$614$	0	0
$615$	5.76050	0.232286
$616$	0	0
$617$	19.2158	0.773597	0.386799	−	0.922164i	$-0.373581\pi$
$617$	19.2158	0.773597	0.386799	+	0.922164i	$0.373581\pi$
$618$	0	0
$619$	11.6162	0.466893	0.233446	−	0.972370i	$-0.425000\pi$
$619$	11.6162	0.466893	0.233446	+	0.972370i	$0.425000\pi$
$620$	0	0
$621$	−4.20617	−0.168788
$622$	0	0
$623$	16.2961	0.652891
$624$	0	0
$625$	−7.67432	−0.306973
$626$	0	0
$627$	−2.88025	−0.115026
$628$	0	0
$629$	−21.1493	−0.843278
$630$	0	0
$631$	−38.0887	−1.51629	−0.758143	−	0.652089i	$-0.773893\pi$
$631$	−38.0887	−1.51629	−0.758143	+	0.652089i	$0.773893\pi$
$632$	0	0
$633$	−5.97398	−0.237444
$634$	0	0
$635$	−16.4098	−0.651205
$636$	0	0
$637$	2.62620	0.104054
$638$	0	0
$639$	21.2329	0.839962
$640$	0	0
$641$	35.7205	1.41088	0.705438	−	0.708771i	$-0.250750\pi$
$641$	35.7205	1.41088	0.705438	+	0.708771i	$0.250750\pi$
$642$	0	0
$643$	−30.1405	−1.18863	−0.594313	−	0.804234i	$-0.702576\pi$
$643$	−30.1405	−1.18863	−0.594313	+	0.804234i	$0.702576\pi$
$644$	0	0
$645$	3.42003	0.134664
$646$	0	0
$647$	−20.1818	−0.793430	−0.396715	−	0.917942i	$-0.629850\pi$
$647$	−20.1818	−0.793430	−0.396715	+	0.917942i	$0.629850\pi$
$648$	0	0
$649$	8.29862	0.325750
$650$	0	0
$651$	7.04516	0.276121
$652$	0	0
$653$	3.83048	0.149898	0.0749491	−	0.997187i	$-0.476121\pi$
$653$	3.83048	0.149898	0.0749491	+	0.997187i	$0.476121\pi$
$654$	0	0
$655$	−8.47408	−0.331110
$656$	0	0
$657$	42.8034	1.66992
$658$	0	0
$659$	−7.71957	−0.300712	−0.150356	−	0.988632i	$-0.548042\pi$
$659$	−7.71957	−0.300712	−0.150356	+	0.988632i	$0.548042\pi$
$660$	0	0
$661$	−11.4952	−0.447112	−0.223556	−	0.974691i	$-0.571767\pi$
$661$	−11.4952	−0.447112	−0.223556	+	0.974691i	$0.571767\pi$
$662$	0	0
$663$	1.15976	0.0450414
$664$	0	0
$665$	−4.46815	−0.173267
$666$	0	0
$667$	−2.44557	−0.0946930
$668$	0	0
$669$	−11.2909	−0.436533
$670$	0	0
$671$	−37.4834	−1.44703
$672$	0	0
$673$	−0.345895	−0.0133333	−0.00666664	−	0.999978i	$-0.502122\pi$
$673$	−0.345895	−0.0133333	−0.00666664	+	0.999978i	$0.502122\pi$
$674$	0	0
$675$	−8.10243	−0.311863
$676$	0	0
$677$	19.2524	0.739930	0.369965	−	0.929046i	$-0.379370\pi$
$677$	19.2524	0.739930	0.369965	+	0.929046i	$0.379370\pi$
$678$	0	0
$679$	−26.3871	−1.01264
$680$	0	0
$681$	−9.62431	−0.368804
$682$	0	0
$683$	32.5072	1.24385	0.621926	−	0.783076i	$-0.286350\pi$
$683$	32.5072	1.24385	0.621926	+	0.783076i	$0.286350\pi$
$684$	0	0
$685$	−20.3353	−0.776971
$686$	0	0
$687$	−10.4455	−0.398521
$688$	0	0
$689$	7.79383	0.296921
$690$	0	0
$691$	44.5395	1.69436	0.847181	−	0.531305i	$-0.178298\pi$
$691$	44.5395	1.69436	0.847181	+	0.531305i	$0.178298\pi$
$692$	0	0
$693$	−43.3449	−1.64653
$694$	0	0
$695$	28.3868	1.07677
$696$	0	0
$697$	−10.9904	−0.416292
$698$	0	0
$699$	8.20492	0.310339
$700$	0	0
$701$	29.6435	1.11962	0.559809	−	0.828621i	$-0.310874\pi$
$701$	29.6435	1.11962	0.559809	+	0.828621i	$0.310874\pi$
$702$	0	0
$703$	9.87358	0.372389
$704$	0	0
$705$	0.272607	0.0102670
$706$	0	0
$707$	35.5515	1.33705
$708$	0	0
$709$	−21.5877	−0.810742	−0.405371	−	0.914152i	$-0.632858\pi$
$709$	−21.5877	−0.810742	−0.405371	+	0.914152i	$0.632858\pi$
$710$	0	0
$711$	−24.7296	−0.927433
$712$	0	0
$713$	4.54144	0.170078
$714$	0	0
$715$	−8.65080	−0.323521
$716$	0	0
$717$	10.7003	0.399611
$718$	0	0
$719$	−27.9756	−1.04332	−0.521658	−	0.853155i	$-0.674686\pi$
$719$	−27.9756	−1.04332	−0.521658	+	0.853155i	$0.674686\pi$
$720$	0	0
$721$	42.2986	1.57528
$722$	0	0
$723$	2.95918	0.110053
$724$	0	0
$725$	−4.71096	−0.174961
$726$	0	0
$727$	−19.4727	−0.722201	−0.361101	−	0.932527i	$-0.617599\pi$
$727$	−19.4727	−0.722201	−0.361101	+	0.932527i	$0.617599\pi$
$728$	0	0
$729$	−8.85838	−0.328088
$730$	0	0
$731$	−6.52505	−0.241338
$732$	0	0
$733$	−20.3372	−0.751170	−0.375585	−	0.926788i	$-0.622558\pi$
$733$	−20.3372	−0.751170	−0.375585	+	0.926788i	$0.622558\pi$
$734$	0	0
$735$	2.61109	0.0963115
$736$	0	0
$737$	−67.8130	−2.49792
$738$	0	0
$739$	−20.3017	−0.746811	−0.373405	−	0.927668i	$-0.621810\pi$
$739$	−20.3017	−0.746811	−0.373405	+	0.927668i	$0.621810\pi$
$740$	0	0
$741$	−0.541436	−0.0198902
$742$	0	0
$743$	−44.6373	−1.63758	−0.818791	−	0.574091i	$-0.805355\pi$
$743$	−44.6373	−1.63758	−0.818791	+	0.574091i	$0.805355\pi$
$744$	0	0
$745$	7.32568	0.268392
$746$	0	0
$747$	28.8265	1.05471
$748$	0	0
$749$	54.8680	2.00483
$750$	0	0
$751$	38.8323	1.41701	0.708505	−	0.705706i	$-0.249370\pi$
$751$	38.8323	1.41701	0.708505	+	0.705706i	$0.249370\pi$
$752$	0	0
$753$	0.646409	0.0235564
$754$	0	0
$755$	−23.7130	−0.863003
$756$	0	0
$757$	−18.1695	−0.660383	−0.330191	−	0.943914i	$-0.607113\pi$
$757$	−18.1695	−0.660383	−0.330191	+	0.943914i	$0.607113\pi$
$758$	0	0
$759$	3.97699	0.144355
$760$	0	0
$761$	−21.8496	−0.792049	−0.396025	−	0.918240i	$-0.629610\pi$
$761$	−21.8496	−0.792049	−0.396025	+	0.918240i	$0.629610\pi$
$762$	0	0
$763$	45.3226	1.64079
$764$	0	0
$765$	−8.10119	−0.292899
$766$	0	0
$767$	1.56000	0.0563282
$768$	0	0
$769$	41.7205	1.50448	0.752241	−	0.658888i	$-0.228973\pi$
$769$	41.7205	1.50448	0.752241	+	0.658888i	$0.228973\pi$
$770$	0	0
$771$	0.394055	0.0141916
$772$	0	0
$773$	2.91524	0.104854	0.0524269	−	0.998625i	$-0.483304\pi$
$773$	2.91524	0.104854	0.0524269	+	0.998625i	$0.483304\pi$
$774$	0	0
$775$	8.74826	0.314247
$776$	0	0
$777$	−21.1493	−0.758727
$778$	0	0
$779$	5.13089	0.183833
$780$	0	0
$781$	−43.0096	−1.53900
$782$	0	0
$783$	6.87964	0.245858
$784$	0	0
$785$	2.91713	0.104117
$786$	0	0
$787$	37.6033	1.34041	0.670207	−	0.742175i	$-0.266205\pi$
$787$	37.6033	1.34041	0.670207	+	0.742175i	$0.266205\pi$
$788$	0	0
$789$	16.0558	0.571602
$790$	0	0
$791$	31.9526	1.13610
$792$	0	0
$793$	−7.04623	−0.250219
$794$	0	0
$795$	7.74899	0.274828
$796$	0	0
$797$	−20.8401	−0.738193	−0.369096	−	0.929391i	$-0.620333\pi$
$797$	−20.8401	−0.738193	−0.369096	+	0.929391i	$0.620333\pi$
$798$	0	0
$799$	−0.520105	−0.0184000
$800$	0	0
$801$	−13.7938	−0.487381
$802$	0	0
$803$	−86.7029	−3.05968
$804$	0	0
$805$	6.16952	0.217447
$806$	0	0
$807$	−16.9706	−0.597392
$808$	0	0
$809$	−26.3372	−0.925965	−0.462983	−	0.886367i	$-0.653221\pi$
$809$	−26.3372	−0.925965	−0.462983	+	0.886367i	$0.653221\pi$
$810$	0	0
$811$	−20.9761	−0.736572	−0.368286	−	0.929713i	$-0.620055\pi$
$811$	−20.9761	−0.736572	−0.368286	+	0.929713i	$0.620055\pi$
$812$	0	0
$813$	16.3005	0.571684
$814$	0	0
$815$	12.7762	0.447532
$816$	0	0
$817$	3.04623	0.106574
$818$	0	0
$819$	−8.14807	−0.284717
$820$	0	0
$821$	14.4296	0.503597	0.251799	−	0.967780i	$-0.418978\pi$
$821$	14.4296	0.503597	0.251799	+	0.967780i	$0.418978\pi$
$822$	0	0
$823$	48.7971	1.70096	0.850481	−	0.526006i	$-0.176311\pi$
$823$	48.7971	1.70096	0.850481	+	0.526006i	$0.176311\pi$
$824$	0	0
$825$	7.66095	0.266720
$826$	0	0
$827$	−32.4601	−1.12875	−0.564373	−	0.825520i	$-0.690882\pi$
$827$	−32.4601	−1.12875	−0.564373	+	0.825520i	$0.690882\pi$
$828$	0	0
$829$	−22.8401	−0.793268	−0.396634	−	0.917977i	$-0.629822\pi$
$829$	−22.8401	−0.793268	−0.396634	+	0.917977i	$0.629822\pi$
$830$	0	0
$831$	8.86813	0.307632
$832$	0	0
$833$	−4.98168	−0.172605
$834$	0	0
$835$	−23.1928	−0.802622
$836$	0	0
$837$	−12.7755	−0.441586
$838$	0	0
$839$	52.3503	1.80733	0.903667	−	0.428235i	$-0.140865\pi$
$839$	52.3503	1.80733	0.903667	+	0.428235i	$0.140865\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	10.2170	0.351891
$844$	0	0
$845$	−1.62620	−0.0559429
$846$	0	0
$847$	53.6709	1.84415
$848$	0	0
$849$	11.2245	0.385223
$850$	0	0
$851$	−13.6332	−0.467341
$852$	0	0
$853$	−21.2909	−0.728988	−0.364494	−	0.931206i	$-0.618758\pi$
$853$	−21.2909	−0.728988	−0.364494	+	0.931206i	$0.618758\pi$
$854$	0	0
$855$	3.78205	0.129343
$856$	0	0
$857$	8.50479	0.290518	0.145259	−	0.989394i	$-0.453598\pi$
$857$	8.50479	0.290518	0.145259	+	0.989394i	$0.453598\pi$
$858$	0	0
$859$	43.8542	1.49629	0.748143	−	0.663538i	$-0.230946\pi$
$859$	43.8542	1.49629	0.748143	+	0.663538i	$0.230946\pi$
$860$	0	0
$861$	−10.9904	−0.374552
$862$	0	0
$863$	23.6155	0.803880	0.401940	−	0.915666i	$-0.368336\pi$
$863$	23.6155	0.803880	0.401940	+	0.915666i	$0.368336\pi$
$864$	0	0
$865$	3.73802	0.127096
$866$	0	0
$867$	8.19372	0.278273
$868$	0	0
$869$	50.0925	1.69927
$870$	0	0
$871$	−12.7477	−0.431938
$872$	0	0
$873$	22.3353	0.755934
$874$	0	0
$875$	37.1118	1.25461
$876$	0	0
$877$	35.1772	1.18785	0.593925	−	0.804520i	$-0.297578\pi$
$877$	35.1772	1.18785	0.593925	+	0.804520i	$0.297578\pi$
$878$	0	0
$879$	−20.2449	−0.682842
$880$	0	0
$881$	0.721586	0.0243108	0.0121554	−	0.999926i	$-0.496131\pi$
$881$	0.721586	0.0243108	0.0121554	+	0.999926i	$0.496131\pi$
$882$	0	0
$883$	−27.9297	−0.939909	−0.469954	−	0.882691i	$-0.655730\pi$
$883$	−27.9297	−0.939909	−0.469954	+	0.882691i	$0.655730\pi$
$884$	0	0
$885$	1.55102	0.0521370
$886$	0	0
$887$	2.86789	0.0962944	0.0481472	−	0.998840i	$-0.484668\pi$
$887$	2.86789	0.0962944	0.0481472	+	0.998840i	$0.484668\pi$
$888$	0	0
$889$	31.3082	1.05004
$890$	0	0
$891$	30.7237	1.02928
$892$	0	0
$893$	0.242812	0.00812538
$894$	0	0
$895$	−25.3015	−0.845737
$896$	0	0
$897$	0.747604	0.0249618
$898$	0	0
$899$	−7.42801	−0.247738
$900$	0	0
$901$	−14.7842	−0.492535
$902$	0	0
$903$	−6.52505	−0.217140
$904$	0	0
$905$	−1.70138	−0.0565556
$906$	0	0
$907$	4.53656	0.150634	0.0753170	−	0.997160i	$-0.476003\pi$
$907$	4.53656	0.150634	0.0753170	+	0.997160i	$0.476003\pi$
$908$	0	0
$909$	−30.0925	−0.998104
$910$	0	0
$911$	−8.39870	−0.278261	−0.139131	−	0.990274i	$-0.544431\pi$
$911$	−8.39870	−0.278261	−0.139131	+	0.990274i	$0.544431\pi$
$912$	0	0
$913$	−58.3911	−1.93246
$914$	0	0
$915$	−7.00569	−0.231601
$916$	0	0
$917$	16.1676	0.533902
$918$	0	0
$919$	−3.63360	−0.119861	−0.0599307	−	0.998203i	$-0.519088\pi$
$919$	−3.63360	−0.119861	−0.0599307	+	0.998203i	$0.519088\pi$
$920$	0	0
$921$	−16.3390	−0.538390
$922$	0	0
$923$	−8.08505	−0.266123
$924$	0	0
$925$	−26.2620	−0.863489
$926$	0	0
$927$	−35.8036	−1.17594
$928$	0	0
$929$	23.3449	0.765920	0.382960	−	0.923765i	$-0.374905\pi$
$929$	23.3449	0.765920	0.382960	+	0.923765i	$0.374905\pi$
$930$	0	0
$931$	2.32570	0.0762218
$932$	0	0
$933$	11.0462	0.361637
$934$	0	0
$935$	16.4098	0.536659
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	0	0
$939$	−9.30166	−0.303548
$940$	0	0
$941$	10.1676	0.331455	0.165728	−	0.986172i	$-0.447003\pi$
$941$	10.1676	0.331455	0.165728	+	0.986172i	$0.447003\pi$
$942$	0	0
$943$	−7.08462	−0.230707
$944$	0	0
$945$	−17.3555	−0.564574
$946$	0	0
$947$	−29.6269	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$947$	−29.6269	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$948$	0	0
$949$	−16.2986	−0.529076
$950$	0	0
$951$	3.41626	0.110780
$952$	0	0
$953$	41.1127	1.33177	0.665885	−	0.746054i	$-0.268054\pi$
$953$	41.1127	1.33177	0.665885	+	0.746054i	$0.268054\pi$
$954$	0	0
$955$	−14.0679	−0.455227
$956$	0	0
$957$	−6.50479	−0.210270
$958$	0	0
$959$	38.7975	1.25284
$960$	0	0
$961$	−17.2062	−0.555038
$962$	0	0
$963$	−46.4429	−1.49660
$964$	0	0
$965$	−20.3353	−0.654615
$966$	0	0
$967$	42.9056	1.37975	0.689875	−	0.723928i	$-0.257665\pi$
$967$	42.9056	1.37975	0.689875	+	0.723928i	$0.257665\pi$
$968$	0	0
$969$	1.02706	0.0329939
$970$	0	0
$971$	24.5011	0.786277	0.393138	−	0.919479i	$-0.371389\pi$
$971$	24.5011	0.786277	0.393138	+	0.919479i	$0.371389\pi$
$972$	0	0
$973$	−54.1589	−1.73625
$974$	0	0
$975$	1.44012	0.0461209
$976$	0	0
$977$	−15.0096	−0.480199	−0.240100	−	0.970748i	$-0.577180\pi$
$977$	−15.0096	−0.480199	−0.240100	+	0.970748i	$0.577180\pi$
$978$	0	0
$979$	27.9409	0.892995
$980$	0	0
$981$	−38.3632	−1.22484
$982$	0	0
$983$	13.9940	0.446339	0.223170	−	0.974780i	$-0.428360\pi$
$983$	13.9940	0.446339	0.223170	+	0.974780i	$0.428360\pi$
$984$	0	0
$985$	−24.9711	−0.795644
$986$	0	0
$987$	−0.520105	−0.0165551
$988$	0	0
$989$	−4.20617	−0.133748
$990$	0	0
$991$	46.7739	1.48582	0.742911	−	0.669390i	$-0.233445\pi$
$991$	46.7739	1.48582	0.742911	+	0.669390i	$0.233445\pi$
$992$	0	0
$993$	11.6647	0.370169
$994$	0	0
$995$	−24.3638	−0.772385
$996$	0	0
$997$	41.7572	1.32246	0.661232	−	0.750182i	$-0.270034\pi$
$997$	41.7572	1.32246	0.661232	+	0.750182i	$0.270034\pi$
$998$	0	0
$999$	38.3516	1.21339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bp.1.3		6
4.3	odd	2	inner	3328.2.a.bp.1.4		6
8.3	odd	2		3328.2.a.bo.1.3		6
8.5	even	2		3328.2.a.bo.1.4		6
16.3	odd	4		1664.2.b.k.833.8	yes	12
16.5	even	4		1664.2.b.k.833.7	yes	12
16.11	odd	4		1664.2.b.k.833.5	✓	12
16.13	even	4		1664.2.b.k.833.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.k.833.5	✓	12	16.11	odd	4
1664.2.b.k.833.6	yes	12	16.13	even	4
1664.2.b.k.833.7	yes	12	16.5	even	4
1664.2.b.k.833.8	yes	12	16.3	odd	4
3328.2.a.bo.1.3		6	8.3	odd	2
3328.2.a.bo.1.4		6	8.5	even	2
3328.2.a.bp.1.3		6	1.1	even	1	trivial
3328.2.a.bp.1.4		6	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-0.611393 q^{3} -1.62620 q^{5} +3.10261 q^{7} -2.62620 q^{9} +5.31965 q^{11} +1.00000 q^{13} +0.994247 q^{15} -1.89692 q^{17} +0.885578 q^{19} -1.89692 q^{21} -1.22279 q^{23} -2.35548 q^{25} +3.43982 q^{27} +2.00000 q^{29} -3.71400 q^{31} -3.25240 q^{33} -5.04546 q^{35} +11.1493 q^{37} -0.611393 q^{39} +5.79383 q^{41} +3.43982 q^{43} +4.27072 q^{45} +0.274184 q^{47} +2.62620 q^{49} +1.15976 q^{51} +7.79383 q^{53} -8.65080 q^{55} -0.541436 q^{57} +1.56000 q^{59} -7.04623 q^{61} -8.14807 q^{63} -1.62620 q^{65} -12.7477 q^{67} +0.747604 q^{69} -8.08505 q^{71} -16.2986 q^{73} +1.44012 q^{75} +16.5048 q^{77} +9.41650 q^{79} +5.77551 q^{81} -10.9765 q^{83} +3.08476 q^{85} -1.22279 q^{87} +5.25240 q^{89} +3.10261 q^{91} +2.27072 q^{93} -1.44012 q^{95} -8.50479 q^{97} -13.9704 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{5} + 2 q^{9} + 6 q^{13} - 4 q^{17} - 4 q^{21} + 14 q^{25} + 12 q^{29} + 16 q^{33} + 24 q^{37} + 20 q^{41} + 36 q^{45} - 2 q^{49} + 32 q^{53} - 24 q^{57} + 8 q^{61} + 8 q^{65} + 40 q^{69} - 12 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) \) 6 * q + 8 * q^5 + 2 * q^9 + 6 * q^13 - 4 * q^17 - 4 * q^21 + 14 * q^25 + 12 * q^29 + 16 * q^33 + 24 * q^37 + 20 * q^41 + 36 * q^45 - 2 * q^49 + 32 * q^53 - 24 * q^57 + 8 * q^61 + 8 * q^65 + 40 * q^69 - 12 * q^73 + 28 * q^77 - 26 * q^81 - 20 * q^85 - 4 * q^89 + 24 * q^93 + 20 * q^97

Introduction

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