LMFDB - Embedded newform 3312.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3312,2,Mod(1,3312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3312 = 2^{4} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3312.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:	$26.4464531494$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.44688.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 6x + 2 $ x^4 - 2x^3 - 5x^2 + 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1656)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.277334$ of defining polynomial
Character	$\chi$	$=$	3312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64575 q^{5} -4.65684 q^{7} -3.56576 q^{11} -0.554669 q^{13} -5.21151 q^{17} -6.20042 q^{19} +1.00000 q^{23} -2.29150 q^{25} +4.55467 q^{29} -4.55467 q^{31} +7.66401 q^{35} +3.54358 q^{37} +7.84617 q^{41} +9.33194 q^{43} +11.3137 q^{47} +14.6862 q^{49} -3.66794 q^{53} +5.86836 q^{55} -11.1599 q^{59} -1.56576 q^{61} +0.912847 q^{65} -8.38259 q^{67} -3.84617 q^{71} -8.97769 q^{73} +16.6052 q^{77} -9.76618 q^{79} +0.434239 q^{83} +8.57685 q^{85} +7.02935 q^{89} +2.58301 q^{91} +10.2043 q^{95} +4.02218 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 2 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 12 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} + 14 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 28 q^{49} + 8 q^{53} - 16 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 2 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 8 * q^19 + 4 * q^23 + 12 * q^25 + 12 * q^29 - 12 * q^31 + 12 * q^35 + 14 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 28 * q^49 + 8 * q^53 - 16 * q^55 + 16 * q^59 + 6 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^71 + 16 * q^73 + 12 * q^77 - 10 * q^79 + 14 * q^83 + 16 * q^85 + 14 * q^89 - 32 * q^91 + 20 * q^95 + 4 * 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	0
$3$	0	0
$4$	0	0
$5$	−1.64575	−0.736002	−0.368001	−	0.929825i	$-0.619958\pi$
$5$	−1.64575	−0.736002	−0.368001	+	0.929825i	$0.619958\pi$
$6$	0	0
$7$	−4.65684	−1.76012	−0.880061	−	0.474861i	$-0.842498\pi$
$7$	−4.65684	−1.76012	−0.880061	+	0.474861i	$0.842498\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.56576	−1.07512	−0.537559	−	0.843226i	$-0.680653\pi$
$11$	−3.56576	−1.07512	−0.537559	+	0.843226i	$0.680653\pi$
$12$	0	0
$13$	−0.554669	−0.153837	−0.0769187	−	0.997037i	$-0.524508\pi$
$13$	−0.554669	−0.153837	−0.0769187	+	0.997037i	$0.524508\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.21151	−1.26398	−0.631989	−	0.774978i	$-0.717761\pi$
$17$	−5.21151	−1.26398	−0.631989	+	0.774978i	$0.717761\pi$
$18$	0	0
$19$	−6.20042	−1.42247	−0.711237	−	0.702952i	$-0.751865\pi$
$19$	−6.20042	−1.42247	−0.711237	+	0.702952i	$0.751865\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.29150	−0.458301
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.55467	0.845781	0.422890	−	0.906181i	$-0.361016\pi$
$29$	4.55467	0.845781	0.422890	+	0.906181i	$0.361016\pi$
$30$	0	0
$31$	−4.55467	−0.818043	−0.409021	−	0.912525i	$-0.634130\pi$
$31$	−4.55467	−0.818043	−0.409021	+	0.912525i	$0.634130\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7.66401	1.29545
$36$	0	0
$37$	3.54358	0.582560	0.291280	−	0.956638i	$-0.405919\pi$
$37$	3.54358	0.582560	0.291280	+	0.956638i	$0.405919\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.84617	1.22537	0.612683	−	0.790329i	$-0.290090\pi$
$41$	7.84617	1.22537	0.612683	+	0.790329i	$0.290090\pi$
$42$	0	0
$43$	9.33194	1.42311	0.711554	−	0.702632i	$-0.247992\pi$
$43$	9.33194	1.42311	0.711554	+	0.702632i	$0.247992\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.3137	1.65027	0.825135	−	0.564935i	$-0.191099\pi$
$47$	11.3137	1.65027	0.825135	+	0.564935i	$0.191099\pi$
$48$	0	0
$49$	14.6862	2.09803
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.66794	−0.503830	−0.251915	−	0.967749i	$-0.581060\pi$
$53$	−3.66794	−0.503830	−0.251915	+	0.967749i	$0.581060\pi$
$54$	0	0
$55$	5.86836	0.791289
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.1599	−1.45289	−0.726445	−	0.687225i	$-0.758829\pi$
$59$	−11.1599	−1.45289	−0.726445	+	0.687225i	$0.758829\pi$
$60$	0	0
$61$	−1.56576	−0.200475	−0.100238	−	0.994964i	$-0.531960\pi$
$61$	−1.56576	−0.200475	−0.100238	+	0.994964i	$0.531960\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.912847	0.113225
$66$	0	0
$67$	−8.38259	−1.02410	−0.512048	−	0.858957i	$-0.671113\pi$
$67$	−8.38259	−1.02410	−0.512048	+	0.858957i	$0.671113\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.84617	−0.456457	−0.228228	−	0.973608i	$-0.573293\pi$
$71$	−3.84617	−0.456457	−0.228228	+	0.973608i	$0.573293\pi$
$72$	0	0
$73$	−8.97769	−1.05076	−0.525380	−	0.850868i	$-0.676077\pi$
$73$	−8.97769	−1.05076	−0.525380	+	0.850868i	$0.676077\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.6052	1.89234
$78$	0	0
$79$	−9.76618	−1.09878	−0.549391	−	0.835566i	$-0.685140\pi$
$79$	−9.76618	−1.09878	−0.549391	+	0.835566i	$0.685140\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.434239	0.0476639	0.0238320	−	0.999716i	$-0.492413\pi$
$83$	0.434239	0.0476639	0.0238320	+	0.999716i	$0.492413\pi$
$84$	0	0
$85$	8.57685	0.930290
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.02935	0.745109	0.372555	−	0.928010i	$-0.378482\pi$
$89$	7.02935	0.745109	0.372555	+	0.928010i	$0.378482\pi$
$90$	0	0
$91$	2.58301	0.270773
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.2043	1.04694
$96$	0	0
$97$	4.02218	0.408391	0.204195	−	0.978930i	$-0.434542\pi$
$97$	4.02218	0.408391	0.204195	+	0.978930i	$0.434542\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.8684	1.77797	0.888984	−	0.457938i	$-0.151412\pi$
$101$	17.8684	1.77797	0.888984	+	0.457938i	$0.151412\pi$
$102$	0	0
$103$	−15.7884	−1.55567	−0.777837	−	0.628466i	$-0.783683\pi$
$103$	−15.7884	−1.55567	−0.777837	+	0.628466i	$0.783683\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.58794	0.926902	0.463451	−	0.886123i	$-0.346611\pi$
$107$	9.58794	0.926902	0.463451	+	0.886123i	$0.346611\pi$
$108$	0	0
$109$	9.01724	0.863695	0.431848	−	0.901947i	$-0.357862\pi$
$109$	9.01724	0.863695	0.431848	+	0.901947i	$0.357862\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.39368	0.319250	0.159625	−	0.987178i	$-0.448971\pi$
$113$	3.39368	0.319250	0.159625	+	0.987178i	$0.448971\pi$
$114$	0	0
$115$	−1.64575	−0.153467
$116$	0	0
$117$	0	0
$118$	0	0
$119$	24.2692	2.22475
$120$	0	0
$121$	1.71465	0.155877
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−2.73068	−0.242309	−0.121154	−	0.992634i	$-0.538660\pi$
$127$	−2.73068	−0.242309	−0.121154	+	0.992634i	$0.538660\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.97782	0.172803	0.0864013	−	0.996260i	$-0.472463\pi$
$131$	1.97782	0.172803	0.0864013	+	0.996260i	$0.472463\pi$
$132$	0	0
$133$	28.8744	2.50373
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.7439	−1.43053	−0.715263	−	0.698856i	$-0.753693\pi$
$137$	−16.7439	−1.43053	−0.715263	+	0.698856i	$0.753693\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.97782	0.165393
$144$	0	0
$145$	−7.49585	−0.622497
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.2003	−1.24526	−0.622628	−	0.782518i	$-0.713935\pi$
$149$	−15.2003	−1.24526	−0.622628	+	0.782518i	$0.713935\pi$
$150$	0	0
$151$	−1.82399	−0.148434	−0.0742170	−	0.997242i	$-0.523646\pi$
$151$	−1.82399	−0.148434	−0.0742170	+	0.997242i	$0.523646\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.49585	0.602081
$156$	0	0
$157$	7.03943	0.561808	0.280904	−	0.959736i	$-0.409366\pi$
$157$	7.03943	0.561808	0.280904	+	0.959736i	$0.409366\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.65684	−0.367011
$162$	0	0
$163$	−14.9715	−1.17266	−0.586331	−	0.810072i	$-0.699428\pi$
$163$	−14.9715	−1.17266	−0.586331	+	0.810072i	$0.699428\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.9999	1.31549	0.657745	−	0.753241i	$-0.271510\pi$
$167$	16.9999	1.31549	0.657745	+	0.753241i	$0.271510\pi$
$168$	0	0
$169$	−12.6923	−0.976334
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.7084	1.95457	0.977286	−	0.211926i	$-0.0679735\pi$
$173$	25.7084	1.95457	0.977286	+	0.211926i	$0.0679735\pi$
$174$	0	0
$175$	10.6712	0.806665
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.7084	−1.17410	−0.587050	−	0.809551i	$-0.699711\pi$
$179$	−15.7084	−1.17410	−0.587050	+	0.809551i	$0.699711\pi$
$180$	0	0
$181$	−7.58794	−0.564008	−0.282004	−	0.959413i	$-0.590999\pi$
$181$	−7.58794	−0.564008	−0.282004	+	0.959413i	$0.590999\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.83185	−0.428766
$186$	0	0
$187$	18.5830	1.35892
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.6052	1.78037	0.890185	−	0.455600i	$-0.150575\pi$
$191$	24.6052	1.78037	0.890185	+	0.455600i	$0.150575\pi$
$192$	0	0
$193$	−19.3785	−1.39490	−0.697449	−	0.716635i	$-0.745681\pi$
$193$	−19.3785	−1.39490	−0.697449	+	0.716635i	$0.745681\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.46752	0.389544	0.194772	−	0.980849i	$-0.437603\pi$
$197$	5.46752	0.389544	0.194772	+	0.980849i	$0.437603\pi$
$198$	0	0
$199$	25.5029	1.80785	0.903926	−	0.427689i	$-0.140672\pi$
$199$	25.5029	1.80785	0.903926	+	0.427689i	$0.140672\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−21.2104	−1.48868
$204$	0	0
$205$	−12.9128	−0.901872
$206$	0	0
$207$	0	0
$208$	0	0
$209$	22.1092	1.52933
$210$	0	0
$211$	15.3359	1.05576	0.527882	−	0.849317i	$-0.322986\pi$
$211$	15.3359	1.05576	0.527882	+	0.849317i	$0.322986\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−15.3581	−1.04741
$216$	0	0
$217$	21.2104	1.43985
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.89066	0.194447
$222$	0	0
$223$	19.3420	1.29524	0.647619	−	0.761964i	$-0.275765\pi$
$223$	19.3420	1.29524	0.647619	+	0.761964i	$0.275765\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.00494	0.199445	0.0997224	−	0.995015i	$-0.468205\pi$
$227$	3.00494	0.199445	0.0997224	+	0.995015i	$0.468205\pi$
$228$	0	0
$229$	−5.01724	−0.331549	−0.165774	−	0.986164i	$-0.553012\pi$
$229$	−5.01724	−0.331549	−0.165774	+	0.986164i	$0.553012\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.6274	−1.74442	−0.872209	−	0.489134i	$-0.837313\pi$
$233$	−26.6274	−1.74442	−0.872209	+	0.489134i	$0.837313\pi$
$234$	0	0
$235$	−18.6195	−1.21460
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−18.6274	−1.20490	−0.602452	−	0.798155i	$-0.705810\pi$
$239$	−18.6274	−1.20490	−0.602452	+	0.798155i	$0.705810\pi$
$240$	0	0
$241$	−17.4959	−1.12701	−0.563503	−	0.826114i	$-0.690547\pi$
$241$	−17.4959	−1.12701	−0.563503	+	0.826114i	$0.690547\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−24.1698	−1.54415
$246$	0	0
$247$	3.43918	0.218830
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.1266	1.01790	0.508950	−	0.860796i	$-0.330034\pi$
$251$	16.1266	1.01790	0.508950	+	0.860796i	$0.330034\pi$
$252$	0	0
$253$	−3.56576	−0.224177
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.5829	0.722520	0.361260	−	0.932465i	$-0.382347\pi$
$257$	11.5829	0.722520	0.361260	+	0.932465i	$0.382347\pi$
$258$	0	0
$259$	−16.5019	−1.02538
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.89066	−0.178246	−0.0891229	−	0.996021i	$-0.528406\pi$
$263$	−2.89066	−0.178246	−0.0891229	+	0.996021i	$0.528406\pi$
$264$	0	0
$265$	6.03651	0.370820
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.10318	−0.0672624	−0.0336312	−	0.999434i	$-0.510707\pi$
$269$	−1.10318	−0.0672624	−0.0336312	+	0.999434i	$0.510707\pi$
$270$	0	0
$271$	9.31369	0.565766	0.282883	−	0.959154i	$-0.408709\pi$
$271$	9.31369	0.565766	0.282883	+	0.959154i	$0.408709\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.17095	0.492727
$276$	0	0
$277$	−6.86848	−0.412687	−0.206343	−	0.978480i	$-0.566156\pi$
$277$	−6.86848	−0.412687	−0.206343	+	0.978480i	$0.566156\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.66300	0.278171	0.139086	−	0.990280i	$-0.455584\pi$
$281$	4.66300	0.278171	0.139086	+	0.990280i	$0.455584\pi$
$282$	0	0
$283$	−12.4270	−0.738706	−0.369353	−	0.929289i	$-0.620421\pi$
$283$	−12.4270	−0.738706	−0.369353	+	0.929289i	$0.620421\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−36.5384	−2.15679
$288$	0	0
$289$	10.1599	0.597639
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.60138	0.327236	0.163618	−	0.986524i	$-0.447684\pi$
$293$	5.60138	0.327236	0.163618	+	0.986524i	$0.447684\pi$
$294$	0	0
$295$	18.3664	1.06933
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.554669	−0.0320773
$300$	0	0
$301$	−43.4574	−2.50484
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.57685	0.147550
$306$	0	0
$307$	17.5102	0.999359	0.499679	−	0.866210i	$-0.333451\pi$
$307$	17.5102	0.999359	0.499679	+	0.866210i	$0.333451\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.6985	−1.23041	−0.615204	−	0.788368i	$-0.710927\pi$
$311$	−21.6985	−1.23041	−0.615204	+	0.788368i	$0.710927\pi$
$312$	0	0
$313$	−26.6496	−1.50632	−0.753161	−	0.657836i	$-0.771472\pi$
$313$	−26.6496	−1.50632	−0.753161	+	0.657836i	$0.771472\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.8097	0.887959	0.443980	−	0.896037i	$-0.353566\pi$
$317$	15.8097	0.887959	0.443980	+	0.896037i	$0.353566\pi$
$318$	0	0
$319$	−16.2409	−0.909314
$320$	0	0
$321$	0	0
$322$	0	0
$323$	32.3136	1.79797
$324$	0	0
$325$	1.27102	0.0705038
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−52.6861	−2.90468
$330$	0	0
$331$	25.8745	1.42219	0.711096	−	0.703095i	$-0.248199\pi$
$331$	25.8745	1.42219	0.711096	+	0.703095i	$0.248199\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.7957	0.753737
$336$	0	0
$337$	21.7811	1.18649	0.593245	−	0.805022i	$-0.297846\pi$
$337$	21.7811	1.18649	0.593245	+	0.805022i	$0.297846\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.2409	0.879492
$342$	0	0
$343$	−35.7934	−1.93266
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.9837	1.39488	0.697440	−	0.716644i	$-0.254323\pi$
$347$	25.9837	1.39488	0.697440	+	0.716644i	$0.254323\pi$
$348$	0	0
$349$	22.3136	1.19442	0.597209	−	0.802086i	$-0.296276\pi$
$349$	22.3136	1.19442	0.597209	+	0.802086i	$0.296276\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.204349	0.0108764	0.00543821	−	0.999985i	$-0.498269\pi$
$353$	0.204349	0.0108764	0.00543821	+	0.999985i	$0.498269\pi$
$354$	0	0
$355$	6.32984	0.335953
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.7145	−1.35716	−0.678580	−	0.734526i	$-0.737405\pi$
$359$	−25.7145	−1.35716	−0.678580	+	0.734526i	$0.737405\pi$
$360$	0	0
$361$	19.4452	1.02343
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.7750	0.773361
$366$	0	0
$367$	12.8168	0.669033	0.334516	−	0.942390i	$-0.391427\pi$
$367$	12.8168	0.669033	0.334516	+	0.942390i	$0.391427\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	17.0810	0.886801
$372$	0	0
$373$	−28.3896	−1.46996	−0.734980	−	0.678089i	$-0.762808\pi$
$373$	−28.3896	−1.46996	−0.734980	+	0.678089i	$0.762808\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.52633	−0.130113
$378$	0	0
$379$	4.41909	0.226994	0.113497	−	0.993538i	$-0.463795\pi$
$379$	4.41909	0.226994	0.113497	+	0.993538i	$0.463795\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.10934	−0.465465	−0.232733	−	0.972541i	$-0.574767\pi$
$383$	−9.10934	−0.465465	−0.232733	+	0.972541i	$0.574767\pi$
$384$	0	0
$385$	−27.3280	−1.39276
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.6011	1.19663	0.598313	−	0.801263i	$-0.295838\pi$
$389$	23.6011	1.19663	0.598313	+	0.801263i	$0.295838\pi$
$390$	0	0
$391$	−5.21151	−0.263557
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.0727	0.808706
$396$	0	0
$397$	13.6923	0.687199	0.343599	−	0.939116i	$-0.388354\pi$
$397$	13.6923	0.687199	0.343599	+	0.939116i	$0.388354\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.21151	−0.260250	−0.130125	−	0.991498i	$-0.541538\pi$
$401$	−5.21151	−0.260250	−0.130125	+	0.991498i	$0.541538\pi$
$402$	0	0
$403$	2.52633	0.125846
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.6355	−0.626321
$408$	0	0
$409$	−24.8399	−1.22825	−0.614127	−	0.789207i	$-0.710492\pi$
$409$	−24.8399	−1.22825	−0.614127	+	0.789207i	$0.710492\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	51.9697	2.55726
$414$	0	0
$415$	−0.714650	−0.0350808
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.3674	−1.19043	−0.595214	−	0.803567i	$-0.702933\pi$
$419$	−24.3674	−1.19043	−0.595214	+	0.803567i	$0.702933\pi$
$420$	0	0
$421$	−30.1710	−1.47044	−0.735221	−	0.677827i	$-0.762922\pi$
$421$	−30.1710	−1.47044	−0.735221	+	0.677827i	$0.762922\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	11.9422	0.579281
$426$	0	0
$427$	7.29150	0.352861
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.8239	0.521367	0.260684	−	0.965424i	$-0.416052\pi$
$431$	10.8239	0.521367	0.260684	+	0.965424i	$0.416052\pi$
$432$	0	0
$433$	40.5162	1.94708	0.973542	−	0.228507i	$-0.0733843\pi$
$433$	40.5162	1.94708	0.973542	+	0.228507i	$0.0733843\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.20042	−0.296606
$438$	0	0
$439$	−4.04437	−0.193027	−0.0965136	−	0.995332i	$-0.530769\pi$
$439$	−4.04437	−0.193027	−0.0965136	+	0.995332i	$0.530769\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.52633	−0.120030	−0.0600148	−	0.998197i	$-0.519115\pi$
$443$	−2.52633	−0.120030	−0.0600148	+	0.998197i	$0.519115\pi$
$444$	0	0
$445$	−11.5686	−0.548402
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−32.6479	−1.54075	−0.770374	−	0.637593i	$-0.779930\pi$
$449$	−32.6479	−1.54075	−0.770374	+	0.637593i	$0.779930\pi$
$450$	0	0
$451$	−27.9776	−1.31741
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.25098	−0.199289
$456$	0	0
$457$	−23.9778	−1.12163	−0.560817	−	0.827939i	$-0.689513\pi$
$457$	−23.9778	−1.12163	−0.560817	+	0.827939i	$0.689513\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.32186	−0.154714	−0.0773572	−	0.997003i	$-0.524648\pi$
$461$	−3.32186	−0.154714	−0.0773572	+	0.997003i	$0.524648\pi$
$462$	0	0
$463$	16.4087	0.762577	0.381288	−	0.924456i	$-0.375480\pi$
$463$	16.4087	0.762577	0.381288	+	0.924456i	$0.375480\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.5131	1.59708	0.798538	−	0.601945i	$-0.205607\pi$
$467$	34.5131	1.59708	0.798538	+	0.601945i	$0.205607\pi$
$468$	0	0
$469$	39.0364	1.80253
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−33.2755	−1.53001
$474$	0	0
$475$	14.2083	0.651921
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.63567	0.348883	0.174441	−	0.984668i	$-0.444188\pi$
$479$	7.63567	0.348883	0.174441	+	0.984668i	$0.444188\pi$
$480$	0	0
$481$	−1.96551	−0.0896196
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.61951	−0.300577
$486$	0	0
$487$	2.53864	0.115037	0.0575183	−	0.998344i	$-0.481681\pi$
$487$	2.53864	0.115037	0.0575183	+	0.998344i	$0.481681\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14.5547	−0.656843	−0.328422	−	0.944531i	$-0.606517\pi$
$491$	−14.5547	−0.656843	−0.328422	+	0.944531i	$0.606517\pi$
$492$	0	0
$493$	−23.7367	−1.06905
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.9110	0.803419
$498$	0	0
$499$	33.2913	1.49032	0.745161	−	0.666885i	$-0.232373\pi$
$499$	33.2913	1.49032	0.745161	+	0.666885i	$0.232373\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.23281	−0.0549682	−0.0274841	−	0.999622i	$-0.508750\pi$
$503$	−1.23281	−0.0549682	−0.0274841	+	0.999622i	$0.508750\pi$
$504$	0	0
$505$	−29.4069	−1.30859
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.6113	−1.44547	−0.722736	−	0.691124i	$-0.757116\pi$
$509$	−32.6113	−1.44547	−0.722736	+	0.691124i	$0.757116\pi$
$510$	0	0
$511$	41.8077	1.84946
$512$	0	0
$513$	0	0
$514$	0	0
$515$	25.9837	1.14498
$516$	0	0
$517$	−40.3419	−1.77423
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11.7501	−0.514783	−0.257392	−	0.966307i	$-0.582863\pi$
$521$	−11.7501	−0.514783	−0.257392	+	0.966307i	$0.582863\pi$
$522$	0	0
$523$	1.69627	0.0741728	0.0370864	−	0.999312i	$-0.488192\pi$
$523$	1.69627	0.0741728	0.0370864	+	0.999312i	$0.488192\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23.7367	1.03399
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.35203	−0.188507
$534$	0	0
$535$	−15.7794	−0.682202
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−52.3674	−2.25563
$540$	0	0
$541$	−28.4879	−1.22479	−0.612395	−	0.790552i	$-0.709794\pi$
$541$	−28.4879	−1.22479	−0.612395	+	0.790552i	$0.709794\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.8401	−0.635682
$546$	0	0
$547$	11.2792	0.482264	0.241132	−	0.970492i	$-0.422481\pi$
$547$	11.2792	0.482264	0.241132	+	0.970492i	$0.422481\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−28.2409	−1.20310
$552$	0	0
$553$	45.4796	1.93399
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.55860	0.277897	0.138948	−	0.990300i	$-0.455628\pi$
$557$	6.55860	0.277897	0.138948	+	0.990300i	$0.455628\pi$
$558$	0	0
$559$	−5.17614	−0.218927
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.3674	0.521225	0.260613	−	0.965443i	$-0.416075\pi$
$563$	12.3674	0.521225	0.260613	+	0.965443i	$0.416075\pi$
$564$	0	0
$565$	−5.58515	−0.234969
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.41586	−0.394733	−0.197367	−	0.980330i	$-0.563239\pi$
$569$	−9.41586	−0.394733	−0.197367	+	0.980330i	$0.563239\pi$
$570$	0	0
$571$	4.38259	0.183405	0.0917027	−	0.995786i	$-0.470769\pi$
$571$	4.38259	0.183405	0.0917027	+	0.995786i	$0.470769\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.29150	−0.0955623
$576$	0	0
$577$	−2.22653	−0.0926918	−0.0463459	−	0.998925i	$-0.514758\pi$
$577$	−2.22653	−0.0926918	−0.0463459	+	0.998925i	$0.514758\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.02218	−0.0838943
$582$	0	0
$583$	13.0790	0.541676
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.9332	−1.31802	−0.659012	−	0.752132i	$-0.729025\pi$
$587$	−31.9332	−1.31802	−0.659012	+	0.752132i	$0.729025\pi$
$588$	0	0
$589$	28.2409	1.16364
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.16613	0.335343	0.167671	−	0.985843i	$-0.446375\pi$
$593$	8.16613	0.335343	0.167671	+	0.985843i	$0.446375\pi$
$594$	0	0
$595$	−39.9411	−1.63742
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.4170	−0.711639	−0.355820	−	0.934555i	$-0.615798\pi$
$599$	−17.4170	−0.711639	−0.355820	+	0.934555i	$0.615798\pi$
$600$	0	0
$601$	26.9395	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$601$	26.9395	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.82189	−0.114726
$606$	0	0
$607$	−0.394688	−0.0160199	−0.00800994	−	0.999968i	$-0.502550\pi$
$607$	−0.394688	−0.0160199	−0.00800994	+	0.999968i	$0.502550\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.27535	−0.253873
$612$	0	0
$613$	26.6751	1.07740	0.538699	−	0.842499i	$-0.318916\pi$
$613$	26.6751	1.07740	0.538699	+	0.842499i	$0.318916\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.4443	−0.903574	−0.451787	−	0.892126i	$-0.649213\pi$
$617$	−22.4443	−0.903574	−0.451787	+	0.892126i	$0.649213\pi$
$618$	0	0
$619$	−15.6618	−0.629500	−0.314750	−	0.949175i	$-0.601921\pi$
$619$	−15.6618	−0.629500	−0.314750	+	0.949175i	$0.601921\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−32.7346	−1.31148
$624$	0	0
$625$	−8.29150	−0.331660
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.4674	−0.736343
$630$	0	0
$631$	−7.15897	−0.284994	−0.142497	−	0.989795i	$-0.545513\pi$
$631$	−7.15897	−0.284994	−0.142497	+	0.989795i	$0.545513\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.49402	0.178340
$636$	0	0
$637$	−8.14597	−0.322755
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−31.5312	−1.24541	−0.622704	−	0.782457i	$-0.713966\pi$
$641$	−31.5312	−1.24541	−0.622704	+	0.782457i	$0.713966\pi$
$642$	0	0
$643$	−14.6681	−0.578452	−0.289226	−	0.957261i	$-0.593398\pi$
$643$	−14.6681	−0.578452	−0.289226	+	0.957261i	$0.593398\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.7512	0.737184	0.368592	−	0.929591i	$-0.379840\pi$
$647$	18.7512	0.737184	0.368592	+	0.929591i	$0.379840\pi$
$648$	0	0
$649$	39.7934	1.56203
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23.0221	0.900923	0.450461	−	0.892796i	$-0.351259\pi$
$653$	23.0221	0.900923	0.450461	+	0.892796i	$0.351259\pi$
$654$	0	0
$655$	−3.25499	−0.127183
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.1833	−0.786228	−0.393114	−	0.919490i	$-0.628602\pi$
$659$	−20.1833	−0.786228	−0.393114	+	0.919490i	$0.628602\pi$
$660$	0	0
$661$	32.9381	1.28115	0.640573	−	0.767898i	$-0.278697\pi$
$661$	32.9381	1.28115	0.640573	+	0.767898i	$0.278697\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−47.5201	−1.84275
$666$	0	0
$667$	4.55467	0.176357
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.58313	0.215534
$672$	0	0
$673$	−7.33587	−0.282777	−0.141389	−	0.989954i	$-0.545157\pi$
$673$	−7.33587	−0.282777	−0.141389	+	0.989954i	$0.545157\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.9088	1.68755	0.843776	−	0.536695i	$-0.180328\pi$
$677$	43.9088	1.68755	0.843776	+	0.536695i	$0.180328\pi$
$678$	0	0
$679$	−18.7307	−0.718818
$680$	0	0
$681$	0	0
$682$	0	0
$683$	35.9776	1.37664	0.688322	−	0.725406i	$-0.258348\pi$
$683$	35.9776	1.37664	0.688322	+	0.725406i	$0.258348\pi$
$684$	0	0
$685$	27.5563	1.05287
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.03449	0.0775079
$690$	0	0
$691$	32.1376	1.22257	0.611285	−	0.791411i	$-0.290653\pi$
$691$	32.1376	1.22257	0.611285	+	0.791411i	$0.290653\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.58301	0.249708
$696$	0	0
$697$	−40.8904	−1.54883
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.4046	−0.430748	−0.215374	−	0.976532i	$-0.569097\pi$
$701$	−11.4046	−0.430748	−0.215374	+	0.976532i	$0.569097\pi$
$702$	0	0
$703$	−21.9717	−0.828677
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−83.2101	−3.12944
$708$	0	0
$709$	3.30272	0.124036	0.0620181	−	0.998075i	$-0.480246\pi$
$709$	3.30272	0.124036	0.0620181	+	0.998075i	$0.480246\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−4.55467	−0.170574
$714$	0	0
$715$	−3.25499	−0.121730
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.79553	0.253430	0.126715	−	0.991939i	$-0.459557\pi$
$719$	6.79553	0.253430	0.126715	+	0.991939i	$0.459557\pi$
$720$	0	0
$721$	73.5239	2.73817
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10.4370	−0.387622
$726$	0	0
$727$	−38.5089	−1.42822	−0.714108	−	0.700035i	$-0.753168\pi$
$727$	−38.5089	−1.42822	−0.714108	+	0.700035i	$0.753168\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−48.6335	−1.79878
$732$	0	0
$733$	46.6970	1.72479	0.862397	−	0.506232i	$-0.168962\pi$
$733$	46.6970	1.72479	0.862397	+	0.506232i	$0.168962\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	29.8903	1.10102
$738$	0	0
$739$	51.2101	1.88380	0.941898	−	0.335900i	$-0.109040\pi$
$739$	51.2101	1.88380	0.941898	+	0.335900i	$0.109040\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−48.2751	−1.77104	−0.885521	−	0.464600i	$-0.846198\pi$
$743$	−48.2751	−1.77104	−0.885521	+	0.464600i	$0.846198\pi$
$744$	0	0
$745$	25.0159	0.916512
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−44.6496	−1.63146
$750$	0	0
$751$	−1.24973	−0.0456032	−0.0228016	−	0.999740i	$-0.507259\pi$
$751$	−1.24973	−0.0456032	−0.0228016	+	0.999740i	$0.507259\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.00183	0.109248
$756$	0	0
$757$	−27.0838	−0.984377	−0.492189	−	0.870489i	$-0.663803\pi$
$757$	−27.0838	−0.984377	−0.492189	+	0.870489i	$0.663803\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.9411	−1.30286	−0.651431	−	0.758708i	$-0.725831\pi$
$761$	−35.9411	−1.30286	−0.651431	+	0.758708i	$0.725831\pi$
$762$	0	0
$763$	−41.9919	−1.52021
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.19002	0.223509
$768$	0	0
$769$	49.1882	1.77377	0.886886	−	0.461989i	$-0.152864\pi$
$769$	49.1882	1.77377	0.886886	+	0.461989i	$0.152864\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.90074	0.320138	0.160069	−	0.987106i	$-0.448828\pi$
$773$	8.90074	0.320138	0.160069	+	0.987106i	$0.448828\pi$
$774$	0	0
$775$	10.4370	0.374909
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−48.6496	−1.74305
$780$	0	0
$781$	13.7145	0.490744
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−11.5851	−0.413492
$786$	0	0
$787$	9.88046	0.352200	0.176100	−	0.984372i	$-0.443652\pi$
$787$	9.88046	0.352200	0.176100	+	0.984372i	$0.443652\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.8038	−0.561919
$792$	0	0
$793$	0.868478	0.0308406
$794$	0	0
$795$	0	0
$796$	0	0
$797$	55.3013	1.95887	0.979437	−	0.201750i	$-0.0646630\pi$
$797$	55.3013	1.95887	0.979437	+	0.201750i	$0.0646630\pi$
$798$	0	0
$799$	−58.9614	−2.08590
$800$	0	0
$801$	0	0
$802$	0	0
$803$	32.0123	1.12969
$804$	0	0
$805$	7.66401	0.270121
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−30.6863	−1.07887	−0.539437	−	0.842026i	$-0.681363\pi$
$809$	−30.6863	−1.07887	−0.539437	+	0.842026i	$0.681363\pi$
$810$	0	0
$811$	2.25518	0.0791902	0.0395951	−	0.999216i	$-0.487393\pi$
$811$	2.25518	0.0791902	0.0395951	+	0.999216i	$0.487393\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	24.6394	0.863082
$816$	0	0
$817$	−57.8620	−2.02433
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−20.3947	−0.711780	−0.355890	−	0.934528i	$-0.615822\pi$
$821$	−20.3947	−0.711780	−0.355890	+	0.934528i	$0.615822\pi$
$822$	0	0
$823$	−5.18229	−0.180643	−0.0903216	−	0.995913i	$-0.528789\pi$
$823$	−5.18229	−0.180643	−0.0903216	+	0.995913i	$0.528789\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.1934	−0.563099	−0.281550	−	0.959547i	$-0.590848\pi$
$827$	−16.1934	−0.563099	−0.281550	+	0.959547i	$0.590848\pi$
$828$	0	0
$829$	−26.0503	−0.904763	−0.452382	−	0.891824i	$-0.649426\pi$
$829$	−26.0503	−0.904763	−0.452382	+	0.891824i	$0.649426\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−76.5373	−2.65186
$834$	0	0
$835$	−27.9776	−0.968204
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−0.991950	−0.0342459	−0.0171230	−	0.999853i	$-0.505451\pi$
$839$	−0.991950	−0.0342459	−0.0171230	+	0.999853i	$0.505451\pi$
$840$	0	0
$841$	−8.25499	−0.284655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	20.8884	0.718584
$846$	0	0
$847$	−7.98486	−0.274363
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.54358	0.121472
$852$	0	0
$853$	−21.7811	−0.745770	−0.372885	−	0.927878i	$-0.621631\pi$
$853$	−21.7811	−0.745770	−0.372885	+	0.927878i	$0.621631\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.3197	1.17234	0.586170	−	0.810188i	$-0.300635\pi$
$857$	34.3197	1.17234	0.586170	+	0.810188i	$0.300635\pi$
$858$	0	0
$859$	11.3280	0.386507	0.193253	−	0.981149i	$-0.438096\pi$
$859$	11.3280	0.386507	0.193253	+	0.981149i	$0.438096\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.0059	−1.08949	−0.544747	−	0.838601i	$-0.683374\pi$
$863$	−32.0059	−1.08949	−0.544747	+	0.838601i	$0.683374\pi$
$864$	0	0
$865$	−42.3096	−1.43857
$866$	0	0
$867$	0	0
$868$	0	0
$869$	34.8239	1.18132
$870$	0	0
$871$	4.64956	0.157544
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−55.8821	−1.88916
$876$	0	0
$877$	24.4309	0.824972	0.412486	−	0.910964i	$-0.364661\pi$
$877$	24.4309	0.824972	0.412486	+	0.910964i	$0.364661\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0.0797445	0.00268666	0.00134333	−	0.999999i	$-0.499572\pi$
$881$	0.0797445	0.00268666	0.00134333	+	0.999999i	$0.499572\pi$
$882$	0	0
$883$	−15.0282	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$883$	−15.0282	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	50.7304	1.70336	0.851681	−	0.524061i	$-0.175584\pi$
$887$	50.7304	1.70336	0.851681	+	0.524061i	$0.175584\pi$
$888$	0	0
$889$	12.7164	0.426493
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−70.1496	−2.34747
$894$	0	0
$895$	25.8521	0.864140
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−20.7450	−0.691885
$900$	0	0
$901$	19.1155	0.636829
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.4879	0.415111
$906$	0	0
$907$	35.3664	1.17432	0.587162	−	0.809470i	$-0.300245\pi$
$907$	35.3664	1.17432	0.587162	+	0.809470i	$0.300245\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	42.9126	1.42176	0.710879	−	0.703314i	$-0.248297\pi$
$911$	42.9126	1.42176	0.710879	+	0.703314i	$0.248297\pi$
$912$	0	0
$913$	−1.54839	−0.0512443
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.21038	−0.304154
$918$	0	0
$919$	−34.7278	−1.14557	−0.572783	−	0.819707i	$-0.694136\pi$
$919$	−34.7278	−1.14557	−0.572783	+	0.819707i	$0.694136\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.13335	0.0702201
$924$	0	0
$925$	−8.12012	−0.266988
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−36.7810	−1.20674	−0.603372	−	0.797460i	$-0.706177\pi$
$929$	−36.7810	−1.20674	−0.603372	+	0.797460i	$0.706177\pi$
$930$	0	0
$931$	−91.0605	−2.98439
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−30.5830	−1.00017
$936$	0	0
$937$	−2.36433	−0.0772393	−0.0386197	−	0.999254i	$-0.512296\pi$
$937$	−2.36433	−0.0772393	−0.0386197	+	0.999254i	$0.512296\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.82189	−0.0919909	−0.0459954	−	0.998942i	$-0.514646\pi$
$941$	−2.82189	−0.0919909	−0.0459954	+	0.998942i	$0.514646\pi$
$942$	0	0
$943$	7.84617	0.255506
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49.8950	1.62137	0.810685	−	0.585483i	$-0.199095\pi$
$947$	49.8950	1.62137	0.810685	+	0.585483i	$0.199095\pi$
$948$	0	0
$949$	4.97965	0.161646
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.6852	1.25314	0.626568	−	0.779367i	$-0.284459\pi$
$953$	38.6852	1.25314	0.626568	+	0.779367i	$0.284459\pi$
$954$	0	0
$955$	−40.4940	−1.31036
$956$	0	0
$957$	0	0
$958$	0	0
$959$	77.9736	2.51790
$960$	0	0
$961$	−10.2550	−0.330806
$962$	0	0
$963$	0	0
$964$	0	0
$965$	31.8922	1.02665
$966$	0	0
$967$	0.918999	0.0295530	0.0147765	−	0.999891i	$-0.495296\pi$
$967$	0.918999	0.0295530	0.0147765	+	0.999891i	$0.495296\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.6083	1.17482	0.587408	−	0.809291i	$-0.300148\pi$
$971$	36.6083	1.17482	0.587408	+	0.809291i	$0.300148\pi$
$972$	0	0
$973$	18.6274	0.597166
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.9502	0.734243	0.367122	−	0.930173i	$-0.380343\pi$
$977$	22.9502	0.734243	0.367122	+	0.930173i	$0.380343\pi$
$978$	0	0
$979$	−25.0650	−0.801080
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.3058	−0.807131	−0.403565	−	0.914951i	$-0.632229\pi$
$983$	−25.3058	−0.807131	−0.403565	+	0.914951i	$0.632229\pi$
$984$	0	0
$985$	−8.99817	−0.286706
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.33194	0.296738
$990$	0	0
$991$	−2.65596	−0.0843692	−0.0421846	−	0.999110i	$-0.513432\pi$
$991$	−2.65596	−0.0843692	−0.0421846	+	0.999110i	$0.513432\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−41.9714	−1.33058
$996$	0	0
$997$	−28.8966	−0.915164	−0.457582	−	0.889168i	$-0.651284\pi$
$997$	−28.8966	−0.915164	−0.457582	+	0.889168i	$0.651284\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.a.bh.1.1		4
3.2	odd	2		3312.2.a.bg.1.3		4
4.3	odd	2		1656.2.a.p.1.2	yes	4
12.11	even	2		1656.2.a.o.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1656.2.a.o.1.4	✓	4	12.11	even	2
1656.2.a.p.1.2	yes	4	4.3	odd	2
3312.2.a.bg.1.3		4	3.2	odd	2
3312.2.a.bh.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3312.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64575 q^{5} -4.65684 q^{7} -3.56576 q^{11} -0.554669 q^{13} -5.21151 q^{17} -6.20042 q^{19} +1.00000 q^{23} -2.29150 q^{25} +4.55467 q^{29} -4.55467 q^{31} +7.66401 q^{35} +3.54358 q^{37} +7.84617 q^{41} +9.33194 q^{43} +11.3137 q^{47} +14.6862 q^{49} -3.66794 q^{53} +5.86836 q^{55} -11.1599 q^{59} -1.56576 q^{61} +0.912847 q^{65} -8.38259 q^{67} -3.84617 q^{71} -8.97769 q^{73} +16.6052 q^{77} -9.76618 q^{79} +0.434239 q^{83} +8.57685 q^{85} +7.02935 q^{89} +2.58301 q^{91} +10.2043 q^{95} +4.02218 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 2 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 8 q^{19} + 4 q^{23} + 12 q^{25} + 12 q^{29} - 12 q^{31} + 12 q^{35} + 14 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 28 q^{49} + 8 q^{53} - 16 q^{55}+ \cdots + 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 2 * q^7 - 2 * q^11 + 4 * q^13 + 2 * q^17 - 8 * q^19 + 4 * q^23 + 12 * q^25 + 12 * q^29 - 12 * q^31 + 12 * q^35 + 14 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 28 * q^49 + 8 * q^53 - 16 * q^55 + 16 * q^59 + 6 * q^61 + 4 * q^65 - 8 * q^67 + 12 * q^71 + 16 * q^73 + 12 * q^77 - 10 * q^79 + 14 * q^83 + 16 * q^85 + 14 * q^89 - 32 * q^91 + 20 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more