LMFDB - Embedded newform 6664.2.a.x.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - 3x^{4} + 27x^{3} + 6x^{2} - 12x - 4 $ x^7 - 11x^5 - 3x^4 + 27x^3 + 6x^2 - 12*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.51862$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.51862 q^{3} +0.798372 q^{5} -0.693805 q^{9} -5.64429 q^{11} -4.80031 q^{13} -1.21242 q^{15} +1.00000 q^{17} +5.07740 q^{19} +4.19007 q^{23} -4.36260 q^{25} +5.60947 q^{27} +1.85926 q^{29} -10.0901 q^{31} +8.57151 q^{33} -6.11378 q^{37} +7.28982 q^{39} -6.24965 q^{41} -0.575750 q^{43} -0.553915 q^{45} -5.74135 q^{47} -1.51862 q^{51} +4.50334 q^{53} -4.50625 q^{55} -7.71062 q^{57} -7.97849 q^{59} -8.93411 q^{61} -3.83243 q^{65} +10.6787 q^{67} -6.36311 q^{69} +3.73942 q^{71} +13.1786 q^{73} +6.62512 q^{75} +5.15610 q^{79} -6.43722 q^{81} -5.24724 q^{83} +0.798372 q^{85} -2.82350 q^{87} +0.528350 q^{89} +15.3230 q^{93} +4.05366 q^{95} -5.53836 q^{97} +3.91604 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{5} + q^{9} + 5 q^{11} - 11 q^{13} + 8 q^{15} + 7 q^{17} + 2 q^{19} + 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 2 q^{31} - 2 q^{33} + 8 q^{37} + q^{39} - 6 q^{41} + 10 q^{43} + 10 q^{45} + q^{47}+ \cdots - 12 q^{99}+O(q^{100}) $ 7 * q + q^5 + q^9 + 5 * q^11 - 11 * q^13 + 8 * q^15 + 7 * q^17 + 2 * q^19 + 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 2 * q^31 - 2 * q^33 + 8 * q^37 + q^39 - 6 * q^41 + 10 * q^43 + 10 * q^45 + q^47 + 3 * q^53 - 11 * q^55 - 5 * q^57 + 4 * q^59 + 32 * q^61 + 20 * q^65 + 31 * q^67 - 16 * q^69 + 3 * q^71 + 7 * q^73 + 19 * q^75 + 8 * q^79 - q^81 - 24 * q^83 + q^85 + 3 * q^87 - 4 * q^89 + 19 * q^93 + 23 * q^95 - 15 * q^97 - 12 * q^99

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
$3$	−1.51862	−0.876773	−0.438387	−	0.898786i	$-0.644450\pi$
$3$	−1.51862	−0.876773	−0.438387	+	0.898786i	$0.644450\pi$
$4$	0	0
$5$	0.798372	0.357043	0.178521	−	0.983936i	$-0.442869\pi$
$5$	0.798372	0.357043	0.178521	+	0.983936i	$0.442869\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.693805	−0.231268
$10$	0	0
$11$	−5.64429	−1.70182	−0.850909	−	0.525313i	$-0.823948\pi$
$11$	−5.64429	−1.70182	−0.850909	+	0.525313i	$0.823948\pi$
$12$	0	0
$13$	−4.80031	−1.33137	−0.665683	−	0.746235i	$-0.731860\pi$
$13$	−4.80031	−1.33137	−0.665683	+	0.746235i	$0.731860\pi$
$14$	0	0
$15$	−1.21242	−0.313046
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.07740	1.16484	0.582418	−	0.812890i	$-0.302107\pi$
$19$	5.07740	1.16484	0.582418	+	0.812890i	$0.302107\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.19007	0.873691	0.436845	−	0.899537i	$-0.356096\pi$
$23$	4.19007	0.873691	0.436845	+	0.899537i	$0.356096\pi$
$24$	0	0
$25$	−4.36260	−0.872520
$26$	0	0
$27$	5.60947	1.07954
$28$	0	0
$29$	1.85926	0.345256	0.172628	−	0.984987i	$-0.444774\pi$
$29$	1.85926	0.345256	0.172628	+	0.984987i	$0.444774\pi$
$30$	0	0
$31$	−10.0901	−1.81224	−0.906120	−	0.423020i	$-0.860970\pi$
$31$	−10.0901	−1.81224	−0.906120	+	0.423020i	$0.860970\pi$
$32$	0	0
$33$	8.57151	1.49211
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.11378	−1.00510	−0.502550	−	0.864548i	$-0.667605\pi$
$37$	−6.11378	−1.00510	−0.502550	+	0.864548i	$0.667605\pi$
$38$	0	0
$39$	7.28982	1.16731
$40$	0	0
$41$	−6.24965	−0.976032	−0.488016	−	0.872835i	$-0.662279\pi$
$41$	−6.24965	−0.976032	−0.488016	+	0.872835i	$0.662279\pi$
$42$	0	0
$43$	−0.575750	−0.0878010	−0.0439005	−	0.999036i	$-0.513978\pi$
$43$	−0.575750	−0.0878010	−0.0439005	+	0.999036i	$0.513978\pi$
$44$	0	0
$45$	−0.553915	−0.0825727
$46$	0	0
$47$	−5.74135	−0.837462	−0.418731	−	0.908110i	$-0.637525\pi$
$47$	−5.74135	−0.837462	−0.418731	+	0.908110i	$0.637525\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.51862	−0.212649
$52$	0	0
$53$	4.50334	0.618581	0.309291	−	0.950968i	$-0.399908\pi$
$53$	4.50334	0.618581	0.309291	+	0.950968i	$0.399908\pi$
$54$	0	0
$55$	−4.50625	−0.607622
$56$	0	0
$57$	−7.71062	−1.02130
$58$	0	0
$59$	−7.97849	−1.03871	−0.519355	−	0.854558i	$-0.673828\pi$
$59$	−7.97849	−1.03871	−0.519355	+	0.854558i	$0.673828\pi$
$60$	0	0
$61$	−8.93411	−1.14390	−0.571948	−	0.820290i	$-0.693812\pi$
$61$	−8.93411	−1.14390	−0.571948	+	0.820290i	$0.693812\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.83243	−0.475355
$66$	0	0
$67$	10.6787	1.30462	0.652308	−	0.757954i	$-0.273801\pi$
$67$	10.6787	1.30462	0.652308	+	0.757954i	$0.273801\pi$
$68$	0	0
$69$	−6.36311	−0.766029
$70$	0	0
$71$	3.73942	0.443787	0.221894	−	0.975071i	$-0.428776\pi$
$71$	3.73942	0.443787	0.221894	+	0.975071i	$0.428776\pi$
$72$	0	0
$73$	13.1786	1.54244	0.771218	−	0.636572i	$-0.219648\pi$
$73$	13.1786	1.54244	0.771218	+	0.636572i	$0.219648\pi$
$74$	0	0
$75$	6.62512	0.765003
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.15610	0.580107	0.290053	−	0.957010i	$-0.406327\pi$
$79$	5.15610	0.580107	0.290053	+	0.957010i	$0.406327\pi$
$80$	0	0
$81$	−6.43722	−0.715247
$82$	0	0
$83$	−5.24724	−0.575960	−0.287980	−	0.957636i	$-0.592984\pi$
$83$	−5.24724	−0.575960	−0.287980	+	0.957636i	$0.592984\pi$
$84$	0	0
$85$	0.798372	0.0865956
$86$	0	0
$87$	−2.82350	−0.302711
$88$	0	0
$89$	0.528350	0.0560049	0.0280025	−	0.999608i	$-0.491085\pi$
$89$	0.528350	0.0560049	0.0280025	+	0.999608i	$0.491085\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	15.3230	1.58892
$94$	0	0
$95$	4.05366	0.415896
$96$	0	0
$97$	−5.53836	−0.562336	−0.281168	−	0.959659i	$-0.590722\pi$
$97$	−5.53836	−0.562336	−0.281168	+	0.959659i	$0.590722\pi$
$98$	0	0
$99$	3.91604	0.393577
$100$	0	0
$101$	−11.7142	−1.16560	−0.582801	−	0.812615i	$-0.698043\pi$
$101$	−11.7142	−1.16560	−0.582801	+	0.812615i	$0.698043\pi$
$102$	0	0
$103$	4.55841	0.449154	0.224577	−	0.974456i	$-0.427900\pi$
$103$	4.55841	0.449154	0.224577	+	0.974456i	$0.427900\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.6436	1.70567	0.852834	−	0.522182i	$-0.174882\pi$
$107$	17.6436	1.70567	0.852834	+	0.522182i	$0.174882\pi$
$108$	0	0
$109$	13.4212	1.28551	0.642757	−	0.766070i	$-0.277791\pi$
$109$	13.4212	1.28551	0.642757	+	0.766070i	$0.277791\pi$
$110$	0	0
$111$	9.28449	0.881245
$112$	0	0
$113$	−15.4940	−1.45755	−0.728775	−	0.684753i	$-0.759910\pi$
$113$	−15.4940	−1.45755	−0.728775	+	0.684753i	$0.759910\pi$
$114$	0	0
$115$	3.34524	0.311945
$116$	0	0
$117$	3.33048	0.307903
$118$	0	0
$119$	0	0
$120$	0	0
$121$	20.8580	1.89618
$122$	0	0
$123$	9.49082	0.855759
$124$	0	0
$125$	−7.47484	−0.668570
$126$	0	0
$127$	1.46849	0.130308	0.0651539	−	0.997875i	$-0.479246\pi$
$127$	1.46849	0.130308	0.0651539	+	0.997875i	$0.479246\pi$
$128$	0	0
$129$	0.874343	0.0769816
$130$	0	0
$131$	12.6991	1.10952	0.554761	−	0.832009i	$-0.312810\pi$
$131$	12.6991	1.10952	0.554761	+	0.832009i	$0.312810\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.47845	0.385443
$136$	0	0
$137$	−12.2219	−1.04419	−0.522093	−	0.852888i	$-0.674849\pi$
$137$	−12.2219	−1.04419	−0.522093	+	0.852888i	$0.674849\pi$
$138$	0	0
$139$	6.86885	0.582608	0.291304	−	0.956631i	$-0.405911\pi$
$139$	6.86885	0.582608	0.291304	+	0.956631i	$0.405911\pi$
$140$	0	0
$141$	8.71891	0.734265
$142$	0	0
$143$	27.0943	2.26574
$144$	0	0
$145$	1.48438	0.123271
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.94297	−0.568791	−0.284395	−	0.958707i	$-0.591793\pi$
$149$	−6.94297	−0.568791	−0.284395	+	0.958707i	$0.591793\pi$
$150$	0	0
$151$	−13.7436	−1.11844	−0.559220	−	0.829019i	$-0.688899\pi$
$151$	−13.7436	−1.11844	−0.559220	+	0.829019i	$0.688899\pi$
$152$	0	0
$153$	−0.693805	−0.0560908
$154$	0	0
$155$	−8.05568	−0.647048
$156$	0	0
$157$	3.89577	0.310916	0.155458	−	0.987842i	$-0.450315\pi$
$157$	3.89577	0.310916	0.155458	+	0.987842i	$0.450315\pi$
$158$	0	0
$159$	−6.83885	−0.542356
$160$	0	0
$161$	0	0
$162$	0	0
$163$	24.3662	1.90851	0.954254	−	0.298999i	$-0.0966526\pi$
$163$	24.3662	1.90851	0.954254	+	0.298999i	$0.0966526\pi$
$164$	0	0
$165$	6.84326	0.532747
$166$	0	0
$167$	−14.5608	−1.12675	−0.563374	−	0.826202i	$-0.690497\pi$
$167$	−14.5608	−1.12675	−0.563374	+	0.826202i	$0.690497\pi$
$168$	0	0
$169$	10.0429	0.772534
$170$	0	0
$171$	−3.52273	−0.269390
$172$	0	0
$173$	−13.9870	−1.06341	−0.531704	−	0.846930i	$-0.678448\pi$
$173$	−13.9870	−1.06341	−0.531704	+	0.846930i	$0.678448\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.1163	0.910714
$178$	0	0
$179$	22.8083	1.70477	0.852387	−	0.522912i	$-0.175154\pi$
$179$	22.8083	1.70477	0.852387	+	0.522912i	$0.175154\pi$
$180$	0	0
$181$	−0.886797	−0.0659151	−0.0329575	−	0.999457i	$-0.510493\pi$
$181$	−0.886797	−0.0659151	−0.0329575	+	0.999457i	$0.510493\pi$
$182$	0	0
$183$	13.5675	1.00294
$184$	0	0
$185$	−4.88108	−0.358864
$186$	0	0
$187$	−5.64429	−0.412751
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.77707	0.635086	0.317543	−	0.948244i	$-0.397142\pi$
$191$	8.77707	0.635086	0.317543	+	0.948244i	$0.397142\pi$
$192$	0	0
$193$	−17.2235	−1.23978	−0.619889	−	0.784690i	$-0.712822\pi$
$193$	−17.2235	−1.23978	−0.619889	+	0.784690i	$0.712822\pi$
$194$	0	0
$195$	5.81999	0.416778
$196$	0	0
$197$	20.3939	1.45300	0.726502	−	0.687164i	$-0.241145\pi$
$197$	20.3939	1.45300	0.726502	+	0.687164i	$0.241145\pi$
$198$	0	0
$199$	8.41576	0.596577	0.298289	−	0.954476i	$-0.403584\pi$
$199$	8.41576	0.596577	0.298289	+	0.954476i	$0.403584\pi$
$200$	0	0
$201$	−16.2169	−1.14385
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.98955	−0.348485
$206$	0	0
$207$	−2.90709	−0.202057
$208$	0	0
$209$	−28.6583	−1.98234
$210$	0	0
$211$	−22.8992	−1.57644	−0.788222	−	0.615391i	$-0.788998\pi$
$211$	−22.8992	−1.57644	−0.788222	+	0.615391i	$0.788998\pi$
$212$	0	0
$213$	−5.67874	−0.389101
$214$	0	0
$215$	−0.459663	−0.0313487
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−20.0132	−1.35237
$220$	0	0
$221$	−4.80031	−0.322904
$222$	0	0
$223$	9.97337	0.667866	0.333933	−	0.942597i	$-0.391624\pi$
$223$	9.97337	0.667866	0.333933	+	0.942597i	$0.391624\pi$
$224$	0	0
$225$	3.02679	0.201786
$226$	0	0
$227$	23.5144	1.56071	0.780353	−	0.625340i	$-0.215040\pi$
$227$	23.5144	1.56071	0.780353	+	0.625340i	$0.215040\pi$
$228$	0	0
$229$	10.4418	0.690013	0.345007	−	0.938600i	$-0.387877\pi$
$229$	10.4418	0.690013	0.345007	+	0.938600i	$0.387877\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.0907	1.57824	0.789118	−	0.614242i	$-0.210538\pi$
$233$	24.0907	1.57824	0.789118	+	0.614242i	$0.210538\pi$
$234$	0	0
$235$	−4.58374	−0.299010
$236$	0	0
$237$	−7.83014	−0.508622
$238$	0	0
$239$	29.7263	1.92283	0.961417	−	0.275095i	$-0.0887093\pi$
$239$	29.7263	1.92283	0.961417	+	0.275095i	$0.0887093\pi$
$240$	0	0
$241$	−4.06445	−0.261814	−0.130907	−	0.991395i	$-0.541789\pi$
$241$	−4.06445	−0.261814	−0.130907	+	0.991395i	$0.541789\pi$
$242$	0	0
$243$	−7.05275	−0.452434
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−24.3731	−1.55082
$248$	0	0
$249$	7.96854	0.504986
$250$	0	0
$251$	−2.80048	−0.176765	−0.0883823	−	0.996087i	$-0.528170\pi$
$251$	−2.80048	−0.176765	−0.0883823	+	0.996087i	$0.528170\pi$
$252$	0	0
$253$	−23.6500	−1.48686
$254$	0	0
$255$	−1.21242	−0.0759248
$256$	0	0
$257$	−3.79658	−0.236824	−0.118412	−	0.992965i	$-0.537780\pi$
$257$	−3.79658	−0.236824	−0.118412	+	0.992965i	$0.537780\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.28996	−0.0798468
$262$	0	0
$263$	−7.23328	−0.446023	−0.223012	−	0.974816i	$-0.571589\pi$
$263$	−7.23328	−0.446023	−0.223012	+	0.974816i	$0.571589\pi$
$264$	0	0
$265$	3.59534	0.220860
$266$	0	0
$267$	−0.802360	−0.0491036
$268$	0	0
$269$	−0.843371	−0.0514212	−0.0257106	−	0.999669i	$-0.508185\pi$
$269$	−0.843371	−0.0514212	−0.0257106	+	0.999669i	$0.508185\pi$
$270$	0	0
$271$	−2.41212	−0.146526	−0.0732630	−	0.997313i	$-0.523341\pi$
$271$	−2.41212	−0.146526	−0.0732630	+	0.997313i	$0.523341\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.6238	1.48487
$276$	0	0
$277$	23.7300	1.42580	0.712900	−	0.701266i	$-0.247381\pi$
$277$	23.7300	1.42580	0.712900	+	0.701266i	$0.247381\pi$
$278$	0	0
$279$	7.00058	0.419114
$280$	0	0
$281$	−2.87529	−0.171526	−0.0857629	−	0.996316i	$-0.527333\pi$
$281$	−2.87529	−0.171526	−0.0857629	+	0.996316i	$0.527333\pi$
$282$	0	0
$283$	26.2474	1.56025	0.780123	−	0.625626i	$-0.215157\pi$
$283$	26.2474	1.56025	0.780123	+	0.625626i	$0.215157\pi$
$284$	0	0
$285$	−6.15595	−0.364647
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	8.41065	0.493041
$292$	0	0
$293$	14.2126	0.830311	0.415156	−	0.909750i	$-0.363727\pi$
$293$	14.2126	0.830311	0.415156	+	0.909750i	$0.363727\pi$
$294$	0	0
$295$	−6.36980	−0.370864
$296$	0	0
$297$	−31.6615	−1.83719
$298$	0	0
$299$	−20.1136	−1.16320
$300$	0	0
$301$	0	0
$302$	0	0
$303$	17.7893	1.02197
$304$	0	0
$305$	−7.13275	−0.408420
$306$	0	0
$307$	−26.5976	−1.51800	−0.759002	−	0.651088i	$-0.774313\pi$
$307$	−26.5976	−1.51800	−0.759002	+	0.651088i	$0.774313\pi$
$308$	0	0
$309$	−6.92248	−0.393806
$310$	0	0
$311$	−21.2543	−1.20522	−0.602611	−	0.798035i	$-0.705873\pi$
$311$	−21.2543	−1.20522	−0.602611	+	0.798035i	$0.705873\pi$
$312$	0	0
$313$	17.7569	1.00368	0.501839	−	0.864961i	$-0.332657\pi$
$313$	17.7569	1.00368	0.501839	+	0.864961i	$0.332657\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.99053	0.111799	0.0558997	−	0.998436i	$-0.482197\pi$
$317$	1.99053	0.111799	0.0558997	+	0.998436i	$0.482197\pi$
$318$	0	0
$319$	−10.4942	−0.587563
$320$	0	0
$321$	−26.7938	−1.49548
$322$	0	0
$323$	5.07740	0.282514
$324$	0	0
$325$	20.9418	1.16164
$326$	0	0
$327$	−20.3816	−1.12710
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.3990	−0.846408	−0.423204	−	0.906034i	$-0.639095\pi$
$331$	−15.3990	−0.846408	−0.423204	+	0.906034i	$0.639095\pi$
$332$	0	0
$333$	4.24177	0.232448
$334$	0	0
$335$	8.52561	0.465804
$336$	0	0
$337$	−24.0830	−1.31188	−0.655942	−	0.754811i	$-0.727729\pi$
$337$	−24.0830	−1.31188	−0.655942	+	0.754811i	$0.727729\pi$
$338$	0	0
$339$	23.5294	1.27794
$340$	0	0
$341$	56.9516	3.08410
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5.08013	−0.273505
$346$	0	0
$347$	−18.3933	−0.987402	−0.493701	−	0.869632i	$-0.664356\pi$
$347$	−18.3933	−0.987402	−0.493701	+	0.869632i	$0.664356\pi$
$348$	0	0
$349$	−22.6309	−1.21140	−0.605701	−	0.795692i	$-0.707107\pi$
$349$	−22.6309	−1.21140	−0.605701	+	0.795692i	$0.707107\pi$
$350$	0	0
$351$	−26.9272	−1.43727
$352$	0	0
$353$	−5.51846	−0.293718	−0.146859	−	0.989157i	$-0.546916\pi$
$353$	−5.51846	−0.293718	−0.146859	+	0.989157i	$0.546916\pi$
$354$	0	0
$355$	2.98545	0.158451
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.6668	0.985197	0.492599	−	0.870257i	$-0.336047\pi$
$359$	18.6668	0.985197	0.492599	+	0.870257i	$0.336047\pi$
$360$	0	0
$361$	6.78000	0.356842
$362$	0	0
$363$	−31.6753	−1.66252
$364$	0	0
$365$	10.5214	0.550716
$366$	0	0
$367$	−25.6477	−1.33880	−0.669398	−	0.742904i	$-0.733448\pi$
$367$	−25.6477	−1.33880	−0.669398	+	0.742904i	$0.733448\pi$
$368$	0	0
$369$	4.33604	0.225725
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5.04036	−0.260980	−0.130490	−	0.991450i	$-0.541655\pi$
$373$	−5.04036	−0.260980	−0.130490	+	0.991450i	$0.541655\pi$
$374$	0	0
$375$	11.3514	0.586185
$376$	0	0
$377$	−8.92502	−0.459662
$378$	0	0
$379$	22.9762	1.18021	0.590105	−	0.807326i	$-0.299086\pi$
$379$	22.9762	1.18021	0.590105	+	0.807326i	$0.299086\pi$
$380$	0	0
$381$	−2.23008	−0.114250
$382$	0	0
$383$	28.9346	1.47849	0.739245	−	0.673436i	$-0.235182\pi$
$383$	28.9346	1.47849	0.739245	+	0.673436i	$0.235182\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.399458	0.0203056
$388$	0	0
$389$	7.51392	0.380971	0.190486	−	0.981690i	$-0.438994\pi$
$389$	7.51392	0.380971	0.190486	+	0.981690i	$0.438994\pi$
$390$	0	0
$391$	4.19007	0.211901
$392$	0	0
$393$	−19.2850	−0.972800
$394$	0	0
$395$	4.11649	0.207123
$396$	0	0
$397$	26.1059	1.31022	0.655109	−	0.755534i	$-0.272622\pi$
$397$	26.1059	1.31022	0.655109	+	0.755534i	$0.272622\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.72306	0.335733	0.167867	−	0.985810i	$-0.446312\pi$
$401$	6.72306	0.335733	0.167867	+	0.985810i	$0.446312\pi$
$402$	0	0
$403$	48.4357	2.41275
$404$	0	0
$405$	−5.13930	−0.255374
$406$	0	0
$407$	34.5080	1.71050
$408$	0	0
$409$	−4.89731	−0.242156	−0.121078	−	0.992643i	$-0.538635\pi$
$409$	−4.89731	−0.242156	−0.121078	+	0.992643i	$0.538635\pi$
$410$	0	0
$411$	18.5604	0.915515
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.18925	−0.205642
$416$	0	0
$417$	−10.4311	−0.510815
$418$	0	0
$419$	4.81341	0.235151	0.117575	−	0.993064i	$-0.462488\pi$
$419$	4.81341	0.235151	0.117575	+	0.993064i	$0.462488\pi$
$420$	0	0
$421$	−0.402770	−0.0196298	−0.00981490	−	0.999952i	$-0.503124\pi$
$421$	−0.402770	−0.0196298	−0.00981490	+	0.999952i	$0.503124\pi$
$422$	0	0
$423$	3.98338	0.193678
$424$	0	0
$425$	−4.36260	−0.211617
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−41.1459	−1.98654
$430$	0	0
$431$	8.06403	0.388431	0.194215	−	0.980959i	$-0.437784\pi$
$431$	8.06403	0.388431	0.194215	+	0.980959i	$0.437784\pi$
$432$	0	0
$433$	31.8869	1.53238	0.766192	−	0.642611i	$-0.222149\pi$
$433$	31.8869	1.53238	0.766192	+	0.642611i	$0.222149\pi$
$434$	0	0
$435$	−2.25421	−0.108081
$436$	0	0
$437$	21.2747	1.01771
$438$	0	0
$439$	−18.9637	−0.905086	−0.452543	−	0.891742i	$-0.649483\pi$
$439$	−18.9637	−0.905086	−0.452543	+	0.891742i	$0.649483\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10.9216	0.518899	0.259449	−	0.965757i	$-0.416459\pi$
$443$	10.9216	0.518899	0.259449	+	0.965757i	$0.416459\pi$
$444$	0	0
$445$	0.421820	0.0199962
$446$	0	0
$447$	10.5437	0.498700
$448$	0	0
$449$	3.09341	0.145987	0.0729936	−	0.997332i	$-0.476745\pi$
$449$	3.09341	0.145987	0.0729936	+	0.997332i	$0.476745\pi$
$450$	0	0
$451$	35.2749	1.66103
$452$	0	0
$453$	20.8713	0.980618
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.04923	0.0490810	0.0245405	−	0.999699i	$-0.492188\pi$
$457$	1.04923	0.0490810	0.0245405	+	0.999699i	$0.492188\pi$
$458$	0	0
$459$	5.60947	0.261828
$460$	0	0
$461$	29.5804	1.37770	0.688848	−	0.724906i	$-0.258117\pi$
$461$	29.5804	1.37770	0.688848	+	0.724906i	$0.258117\pi$
$462$	0	0
$463$	1.85379	0.0861529	0.0430764	−	0.999072i	$-0.486284\pi$
$463$	1.85379	0.0861529	0.0430764	+	0.999072i	$0.486284\pi$
$464$	0	0
$465$	12.2335	0.567314
$466$	0	0
$467$	−31.8357	−1.47318	−0.736591	−	0.676338i	$-0.763566\pi$
$467$	−31.8357	−1.47318	−0.736591	+	0.676338i	$0.763566\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.91618	−0.272603
$472$	0	0
$473$	3.24970	0.149421
$474$	0	0
$475$	−22.1507	−1.01634
$476$	0	0
$477$	−3.12444	−0.143058
$478$	0	0
$479$	37.4836	1.71267	0.856334	−	0.516422i	$-0.172736\pi$
$479$	37.4836	1.71267	0.856334	+	0.516422i	$0.172736\pi$
$480$	0	0
$481$	29.3480	1.33816
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.42168	−0.200778
$486$	0	0
$487$	31.0843	1.40857	0.704283	−	0.709920i	$-0.251269\pi$
$487$	31.0843	1.40857	0.704283	+	0.709920i	$0.251269\pi$
$488$	0	0
$489$	−37.0029	−1.67333
$490$	0	0
$491$	37.0322	1.67124	0.835621	−	0.549306i	$-0.185108\pi$
$491$	37.0322	1.67124	0.835621	+	0.549306i	$0.185108\pi$
$492$	0	0
$493$	1.85926	0.0837369
$494$	0	0
$495$	3.12646	0.140524
$496$	0	0
$497$	0	0
$498$	0	0
$499$	7.63405	0.341747	0.170873	−	0.985293i	$-0.445341\pi$
$499$	7.63405	0.341747	0.170873	+	0.985293i	$0.445341\pi$
$500$	0	0
$501$	22.1123	0.987903
$502$	0	0
$503$	−0.908307	−0.0404994	−0.0202497	−	0.999795i	$-0.506446\pi$
$503$	−0.908307	−0.0404994	−0.0202497	+	0.999795i	$0.506446\pi$
$504$	0	0
$505$	−9.35226	−0.416170
$506$	0	0
$507$	−15.2514	−0.677337
$508$	0	0
$509$	7.08235	0.313920	0.156960	−	0.987605i	$-0.449831\pi$
$509$	7.08235	0.313920	0.156960	+	0.987605i	$0.449831\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	28.4815	1.25749
$514$	0	0
$515$	3.63931	0.160367
$516$	0	0
$517$	32.4059	1.42521
$518$	0	0
$519$	21.2408	0.932369
$520$	0	0
$521$	16.9271	0.741589	0.370795	−	0.928715i	$-0.379085\pi$
$521$	16.9271	0.741589	0.370795	+	0.928715i	$0.379085\pi$
$522$	0	0
$523$	8.62332	0.377071	0.188536	−	0.982066i	$-0.439626\pi$
$523$	8.62332	0.377071	0.188536	+	0.982066i	$0.439626\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.0901	−0.439533
$528$	0	0
$529$	−5.44328	−0.236664
$530$	0	0
$531$	5.53552	0.240221
$532$	0	0
$533$	30.0002	1.29946
$534$	0	0
$535$	14.0861	0.608997
$536$	0	0
$537$	−34.6371	−1.49470
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−44.6120	−1.91802	−0.959009	−	0.283374i	$-0.908546\pi$
$541$	−44.6120	−1.91802	−0.959009	+	0.283374i	$0.908546\pi$
$542$	0	0
$543$	1.34670	0.0577926
$544$	0	0
$545$	10.7151	0.458983
$546$	0	0
$547$	34.2732	1.46542	0.732709	−	0.680543i	$-0.238256\pi$
$547$	34.2732	1.46542	0.732709	+	0.680543i	$0.238256\pi$
$548$	0	0
$549$	6.19853	0.264547
$550$	0	0
$551$	9.44021	0.402167
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.41248	0.314642
$556$	0	0
$557$	22.9262	0.971413	0.485707	−	0.874122i	$-0.338562\pi$
$557$	22.9262	0.971413	0.485707	+	0.874122i	$0.338562\pi$
$558$	0	0
$559$	2.76378	0.116895
$560$	0	0
$561$	8.57151	0.361890
$562$	0	0
$563$	−13.2963	−0.560370	−0.280185	−	0.959946i	$-0.590396\pi$
$563$	−13.2963	−0.560370	−0.280185	+	0.959946i	$0.590396\pi$
$564$	0	0
$565$	−12.3700	−0.520408
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.9671	0.669376	0.334688	−	0.942329i	$-0.391369\pi$
$569$	15.9671	0.669376	0.334688	+	0.942329i	$0.391369\pi$
$570$	0	0
$571$	−22.5937	−0.945517	−0.472759	−	0.881192i	$-0.656742\pi$
$571$	−22.5937	−0.945517	−0.472759	+	0.881192i	$0.656742\pi$
$572$	0	0
$573$	−13.3290	−0.556827
$574$	0	0
$575$	−18.2796	−0.762313
$576$	0	0
$577$	−37.1048	−1.54469	−0.772346	−	0.635202i	$-0.780917\pi$
$577$	−37.1048	−1.54469	−0.772346	+	0.635202i	$0.780917\pi$
$578$	0	0
$579$	26.1559	1.08700
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−25.4182	−1.05271
$584$	0	0
$585$	2.65896	0.109934
$586$	0	0
$587$	14.9886	0.618646	0.309323	−	0.950957i	$-0.399898\pi$
$587$	14.9886	0.618646	0.309323	+	0.950957i	$0.399898\pi$
$588$	0	0
$589$	−51.2316	−2.11096
$590$	0	0
$591$	−30.9705	−1.27396
$592$	0	0
$593$	−42.5518	−1.74739	−0.873697	−	0.486471i	$-0.838284\pi$
$593$	−42.5518	−1.74739	−0.873697	+	0.486471i	$0.838284\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12.7803	−0.523063
$598$	0	0
$599$	−5.39333	−0.220365	−0.110183	−	0.993911i	$-0.535144\pi$
$599$	−5.39333	−0.220365	−0.110183	+	0.993911i	$0.535144\pi$
$600$	0	0
$601$	39.2266	1.60009	0.800044	−	0.599942i	$-0.204809\pi$
$601$	39.2266	1.60009	0.800044	+	0.599942i	$0.204809\pi$
$602$	0	0
$603$	−7.40896	−0.301716
$604$	0	0
$605$	16.6525	0.677019
$606$	0	0
$607$	3.71236	0.150680	0.0753401	−	0.997158i	$-0.475996\pi$
$607$	3.71236	0.150680	0.0753401	+	0.997158i	$0.475996\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.5602	1.11497
$612$	0	0
$613$	−18.6584	−0.753605	−0.376803	−	0.926294i	$-0.622976\pi$
$613$	−18.6584	−0.753605	−0.376803	+	0.926294i	$0.622976\pi$
$614$	0	0
$615$	7.57721	0.305543
$616$	0	0
$617$	0.327324	0.0131776	0.00658878	−	0.999978i	$-0.497903\pi$
$617$	0.327324	0.0131776	0.00658878	+	0.999978i	$0.497903\pi$
$618$	0	0
$619$	−11.6615	−0.468716	−0.234358	−	0.972150i	$-0.575299\pi$
$619$	−11.6615	−0.468716	−0.234358	+	0.972150i	$0.575299\pi$
$620$	0	0
$621$	23.5041	0.943187
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15.8453	0.633812
$626$	0	0
$627$	43.5210	1.73806
$628$	0	0
$629$	−6.11378	−0.243773
$630$	0	0
$631$	10.1776	0.405162	0.202581	−	0.979266i	$-0.435067\pi$
$631$	10.1776	0.405162	0.202581	+	0.979266i	$0.435067\pi$
$632$	0	0
$633$	34.7750	1.38218
$634$	0	0
$635$	1.17241	0.0465255
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.59443	−0.102634
$640$	0	0
$641$	−16.9177	−0.668211	−0.334105	−	0.942536i	$-0.608434\pi$
$641$	−16.9177	−0.668211	−0.334105	+	0.942536i	$0.608434\pi$
$642$	0	0
$643$	17.1979	0.678220	0.339110	−	0.940747i	$-0.389874\pi$
$643$	17.1979	0.678220	0.339110	+	0.940747i	$0.389874\pi$
$644$	0	0
$645$	0.698051	0.0274857
$646$	0	0
$647$	30.8934	1.21455	0.607273	−	0.794493i	$-0.292263\pi$
$647$	30.8934	1.21455	0.607273	+	0.794493i	$0.292263\pi$
$648$	0	0
$649$	45.0329	1.76770
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.13997	0.279408	0.139704	−	0.990193i	$-0.455385\pi$
$653$	7.13997	0.279408	0.139704	+	0.990193i	$0.455385\pi$
$654$	0	0
$655$	10.1386	0.396147
$656$	0	0
$657$	−9.14336	−0.356716
$658$	0	0
$659$	31.5442	1.22879	0.614394	−	0.789000i	$-0.289401\pi$
$659$	31.5442	1.22879	0.614394	+	0.789000i	$0.289401\pi$
$660$	0	0
$661$	41.4489	1.61218	0.806088	−	0.591796i	$-0.201581\pi$
$661$	41.4489	1.61218	0.806088	+	0.591796i	$0.201581\pi$
$662$	0	0
$663$	7.28982	0.283113
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.79044	0.301647
$668$	0	0
$669$	−15.1457	−0.585567
$670$	0	0
$671$	50.4267	1.94670
$672$	0	0
$673$	−34.7524	−1.33961	−0.669804	−	0.742538i	$-0.733622\pi$
$673$	−34.7524	−1.33961	−0.669804	+	0.742538i	$0.733622\pi$
$674$	0	0
$675$	−24.4719	−0.941924
$676$	0	0
$677$	−5.27903	−0.202890	−0.101445	−	0.994841i	$-0.532347\pi$
$677$	−5.27903	−0.202890	−0.101445	+	0.994841i	$0.532347\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−35.7093	−1.36839
$682$	0	0
$683$	28.2578	1.08125	0.540627	−	0.841262i	$-0.318187\pi$
$683$	28.2578	1.08125	0.540627	+	0.841262i	$0.318187\pi$
$684$	0	0
$685$	−9.75762	−0.372819
$686$	0	0
$687$	−15.8571	−0.604985
$688$	0	0
$689$	−21.6174	−0.823558
$690$	0	0
$691$	−34.2871	−1.30434	−0.652172	−	0.758071i	$-0.726142\pi$
$691$	−34.2871	−1.30434	−0.652172	+	0.758071i	$0.726142\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.48390	0.208016
$696$	0	0
$697$	−6.24965	−0.236723
$698$	0	0
$699$	−36.5846	−1.38376
$700$	0	0
$701$	30.1766	1.13975	0.569877	−	0.821730i	$-0.306991\pi$
$701$	30.1766	1.13975	0.569877	+	0.821730i	$0.306991\pi$
$702$	0	0
$703$	−31.0421	−1.17078
$704$	0	0
$705$	6.96094	0.262164
$706$	0	0
$707$	0	0
$708$	0	0
$709$	37.1916	1.39676	0.698380	−	0.715727i	$-0.253905\pi$
$709$	37.1916	1.39676	0.698380	+	0.715727i	$0.253905\pi$
$710$	0	0
$711$	−3.57733	−0.134160
$712$	0	0
$713$	−42.2784	−1.58334
$714$	0	0
$715$	21.6314	0.808967
$716$	0	0
$717$	−45.1428	−1.68589
$718$	0	0
$719$	−25.4763	−0.950106	−0.475053	−	0.879957i	$-0.657571\pi$
$719$	−25.4763	−0.950106	−0.475053	+	0.879957i	$0.657571\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	6.17234	0.229552
$724$	0	0
$725$	−8.11121	−0.301243
$726$	0	0
$727$	−29.7250	−1.10244	−0.551220	−	0.834360i	$-0.685838\pi$
$727$	−29.7250	−1.10244	−0.551220	+	0.834360i	$0.685838\pi$
$728$	0	0
$729$	30.0221	1.11193
$730$	0	0
$731$	−0.575750	−0.0212949
$732$	0	0
$733$	−32.2085	−1.18965	−0.594824	−	0.803856i	$-0.702778\pi$
$733$	−32.2085	−1.18965	−0.594824	+	0.803856i	$0.702778\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−60.2739	−2.22022
$738$	0	0
$739$	−12.7400	−0.468649	−0.234325	−	0.972158i	$-0.575288\pi$
$739$	−12.7400	−0.468649	−0.234325	+	0.972158i	$0.575288\pi$
$740$	0	0
$741$	37.0134	1.35972
$742$	0	0
$743$	−37.8141	−1.38727	−0.693633	−	0.720328i	$-0.743991\pi$
$743$	−37.8141	−1.38727	−0.693633	+	0.720328i	$0.743991\pi$
$744$	0	0
$745$	−5.54308	−0.203083
$746$	0	0
$747$	3.64056	0.133201
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.2976	1.03259	0.516296	−	0.856410i	$-0.327310\pi$
$751$	28.2976	1.03259	0.516296	+	0.856410i	$0.327310\pi$
$752$	0	0
$753$	4.25285	0.154983
$754$	0	0
$755$	−10.9725	−0.399331
$756$	0	0
$757$	−27.2325	−0.989782	−0.494891	−	0.868955i	$-0.664792\pi$
$757$	−27.2325	−0.989782	−0.494891	+	0.868955i	$0.664792\pi$
$758$	0	0
$759$	35.9153	1.30364
$760$	0	0
$761$	−12.0865	−0.438136	−0.219068	−	0.975710i	$-0.570302\pi$
$761$	−12.0865	−0.438136	−0.219068	+	0.975710i	$0.570302\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.553915	−0.0200268
$766$	0	0
$767$	38.2992	1.38290
$768$	0	0
$769$	1.76850	0.0637739	0.0318870	−	0.999491i	$-0.489848\pi$
$769$	1.76850	0.0637739	0.0318870	+	0.999491i	$0.489848\pi$
$770$	0	0
$771$	5.76555	0.207641
$772$	0	0
$773$	−7.91241	−0.284589	−0.142295	−	0.989824i	$-0.545448\pi$
$773$	−7.91241	−0.284589	−0.142295	+	0.989824i	$0.545448\pi$
$774$	0	0
$775$	44.0192	1.58122
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−31.7320	−1.13692
$780$	0	0
$781$	−21.1064	−0.755245
$782$	0	0
$783$	10.4295	0.372719
$784$	0	0
$785$	3.11027	0.111010
$786$	0	0
$787$	−15.8449	−0.564810	−0.282405	−	0.959295i	$-0.591132\pi$
$787$	−15.8449	−0.564810	−0.282405	+	0.959295i	$0.591132\pi$
$788$	0	0
$789$	10.9846	0.391061
$790$	0	0
$791$	0	0
$792$	0	0
$793$	42.8865	1.52294
$794$	0	0
$795$	−5.45995	−0.193644
$796$	0	0
$797$	−7.19960	−0.255023	−0.127511	−	0.991837i	$-0.540699\pi$
$797$	−7.19960	−0.255023	−0.127511	+	0.991837i	$0.540699\pi$
$798$	0	0
$799$	−5.74135	−0.203114
$800$	0	0
$801$	−0.366572	−0.0129522
$802$	0	0
$803$	−74.3837	−2.62494
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.28076	0.0450848
$808$	0	0
$809$	−32.2702	−1.13456	−0.567280	−	0.823525i	$-0.692004\pi$
$809$	−32.2702	−1.13456	−0.567280	+	0.823525i	$0.692004\pi$
$810$	0	0
$811$	9.81258	0.344566	0.172283	−	0.985047i	$-0.444886\pi$
$811$	9.81258	0.344566	0.172283	+	0.985047i	$0.444886\pi$
$812$	0	0
$813$	3.66309	0.128470
$814$	0	0
$815$	19.4533	0.681419
$816$	0	0
$817$	−2.92331	−0.102274
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.27394	−0.149162	−0.0745808	−	0.997215i	$-0.523762\pi$
$821$	−4.27394	−0.149162	−0.0745808	+	0.997215i	$0.523762\pi$
$822$	0	0
$823$	−0.896382	−0.0312459	−0.0156230	−	0.999878i	$-0.504973\pi$
$823$	−0.896382	−0.0312459	−0.0156230	+	0.999878i	$0.504973\pi$
$824$	0	0
$825$	−37.3941	−1.30190
$826$	0	0
$827$	−19.0211	−0.661428	−0.330714	−	0.943731i	$-0.607290\pi$
$827$	−19.0211	−0.661428	−0.330714	+	0.943731i	$0.607290\pi$
$828$	0	0
$829$	−8.52244	−0.295997	−0.147998	−	0.988988i	$-0.547283\pi$
$829$	−8.52244	−0.295997	−0.147998	+	0.988988i	$0.547283\pi$
$830$	0	0
$831$	−36.0368	−1.25010
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−11.6249	−0.402298
$836$	0	0
$837$	−56.6003	−1.95639
$838$	0	0
$839$	16.8692	0.582389	0.291195	−	0.956664i	$-0.405947\pi$
$839$	16.8692	0.582389	0.291195	+	0.956664i	$0.405947\pi$
$840$	0	0
$841$	−25.5432	−0.880798
$842$	0	0
$843$	4.36647	0.150389
$844$	0	0
$845$	8.01800	0.275828
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−39.8597	−1.36798
$850$	0	0
$851$	−25.6172	−0.878147
$852$	0	0
$853$	−14.8586	−0.508750	−0.254375	−	0.967106i	$-0.581870\pi$
$853$	−14.8586	−0.508750	−0.254375	+	0.967106i	$0.581870\pi$
$854$	0	0
$855$	−2.81245	−0.0961837
$856$	0	0
$857$	−1.70855	−0.0583629	−0.0291815	−	0.999574i	$-0.509290\pi$
$857$	−1.70855	−0.0583629	−0.0291815	+	0.999574i	$0.509290\pi$
$858$	0	0
$859$	−0.749299	−0.0255658	−0.0127829	−	0.999918i	$-0.504069\pi$
$859$	−0.749299	−0.0255658	−0.0127829	+	0.999918i	$0.504069\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−23.0076	−0.783186	−0.391593	−	0.920138i	$-0.628076\pi$
$863$	−23.0076	−0.783186	−0.391593	+	0.920138i	$0.628076\pi$
$864$	0	0
$865$	−11.1668	−0.379683
$866$	0	0
$867$	−1.51862	−0.0515749
$868$	0	0
$869$	−29.1026	−0.987236
$870$	0	0
$871$	−51.2612	−1.73692
$872$	0	0
$873$	3.84254	0.130050
$874$	0	0
$875$	0	0
$876$	0	0
$877$	48.3027	1.63107	0.815534	−	0.578709i	$-0.196443\pi$
$877$	48.3027	1.63107	0.815534	+	0.578709i	$0.196443\pi$
$878$	0	0
$879$	−21.5835	−0.727995
$880$	0	0
$881$	52.2783	1.76130	0.880650	−	0.473767i	$-0.157106\pi$
$881$	52.2783	1.76130	0.880650	+	0.473767i	$0.157106\pi$
$882$	0	0
$883$	11.5340	0.388149	0.194075	−	0.980987i	$-0.437830\pi$
$883$	11.5340	0.388149	0.194075	+	0.980987i	$0.437830\pi$
$884$	0	0
$885$	9.67329	0.325164
$886$	0	0
$887$	−24.7707	−0.831717	−0.415859	−	0.909429i	$-0.636519\pi$
$887$	−24.7707	−0.831717	−0.415859	+	0.909429i	$0.636519\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	36.3335	1.21722
$892$	0	0
$893$	−29.1511	−0.975506
$894$	0	0
$895$	18.2095	0.608677
$896$	0	0
$897$	30.5449	1.01986
$898$	0	0
$899$	−18.7602	−0.625687
$900$	0	0
$901$	4.50334	0.150028
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.707994	−0.0235345
$906$	0	0
$907$	−11.6604	−0.387177	−0.193588	−	0.981083i	$-0.562013\pi$
$907$	−11.6604	−0.387177	−0.193588	+	0.981083i	$0.562013\pi$
$908$	0	0
$909$	8.12734	0.269567
$910$	0	0
$911$	−15.4049	−0.510389	−0.255194	−	0.966890i	$-0.582139\pi$
$911$	−15.4049	−0.510389	−0.255194	+	0.966890i	$0.582139\pi$
$912$	0	0
$913$	29.6170	0.980178
$914$	0	0
$915$	10.8319	0.358092
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−44.9428	−1.48253	−0.741264	−	0.671214i	$-0.765773\pi$
$919$	−44.9428	−1.48253	−0.741264	+	0.671214i	$0.765773\pi$
$920$	0	0
$921$	40.3915	1.33095
$922$	0	0
$923$	−17.9504	−0.590843
$924$	0	0
$925$	26.6720	0.876970
$926$	0	0
$927$	−3.16265	−0.103875
$928$	0	0
$929$	35.0540	1.15008	0.575042	−	0.818124i	$-0.304986\pi$
$929$	35.0540	1.15008	0.575042	+	0.818124i	$0.304986\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	32.2771	1.05671
$934$	0	0
$935$	−4.50625	−0.147370
$936$	0	0
$937$	−9.30499	−0.303981	−0.151990	−	0.988382i	$-0.548568\pi$
$937$	−9.30499	−0.303981	−0.151990	+	0.988382i	$0.548568\pi$
$938$	0	0
$939$	−26.9659	−0.879999
$940$	0	0
$941$	10.3680	0.337986	0.168993	−	0.985617i	$-0.445948\pi$
$941$	10.3680	0.337986	0.168993	+	0.985617i	$0.445948\pi$
$942$	0	0
$943$	−26.1865	−0.852750
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.4557	0.372260	0.186130	−	0.982525i	$-0.440405\pi$
$947$	11.4557	0.372260	0.186130	+	0.982525i	$0.440405\pi$
$948$	0	0
$949$	−63.2612	−2.05354
$950$	0	0
$951$	−3.02286	−0.0980228
$952$	0	0
$953$	25.4052	0.822956	0.411478	−	0.911420i	$-0.365013\pi$
$953$	25.4052	0.822956	0.411478	+	0.911420i	$0.365013\pi$
$954$	0	0
$955$	7.00737	0.226753
$956$	0	0
$957$	15.9367	0.515159
$958$	0	0
$959$	0	0
$960$	0	0
$961$	70.8107	2.28422
$962$	0	0
$963$	−12.2412	−0.394467
$964$	0	0
$965$	−13.7508	−0.442654
$966$	0	0
$967$	29.5141	0.949108	0.474554	−	0.880226i	$-0.342609\pi$
$967$	29.5141	0.949108	0.474554	+	0.880226i	$0.342609\pi$
$968$	0	0
$969$	−7.71062	−0.247701
$970$	0	0
$971$	10.6885	0.343012	0.171506	−	0.985183i	$-0.445137\pi$
$971$	10.6885	0.343012	0.171506	+	0.985183i	$0.445137\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−31.8026	−1.01850
$976$	0	0
$977$	−52.1761	−1.66926	−0.834631	−	0.550810i	$-0.814319\pi$
$977$	−52.1761	−1.66926	−0.834631	+	0.550810i	$0.814319\pi$
$978$	0	0
$979$	−2.98216	−0.0953102
$980$	0	0
$981$	−9.31166	−0.297298
$982$	0	0
$983$	20.0698	0.640126	0.320063	−	0.947396i	$-0.396296\pi$
$983$	20.0698	0.640126	0.320063	+	0.947396i	$0.396296\pi$
$984$	0	0
$985$	16.2819	0.518785
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.41243	−0.0767110
$990$	0	0
$991$	−45.2564	−1.43762	−0.718809	−	0.695208i	$-0.755312\pi$
$991$	−45.2564	−1.43762	−0.718809	+	0.695208i	$0.755312\pi$
$992$	0	0
$993$	23.3852	0.742108
$994$	0	0
$995$	6.71891	0.213004
$996$	0	0
$997$	−26.4524	−0.837756	−0.418878	−	0.908042i	$-0.637577\pi$
$997$	−26.4524	−0.837756	−0.418878	+	0.908042i	$0.637577\pi$
$998$	0	0
$999$	−34.2951	−1.08505

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.x.1.2		7
7.3	odd	6		952.2.q.d.681.2	yes	14
7.5	odd	6		952.2.q.d.137.2	✓	14
7.6	odd	2		6664.2.a.w.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.q.d.137.2	✓	14	7.5	odd	6
952.2.q.d.681.2	yes	14	7.3	odd	6
6664.2.a.w.1.6		7	7.6	odd	2
6664.2.a.x.1.2		7	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:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q-1.51862 q^{3} +0.798372 q^{5} -0.693805 q^{9} -5.64429 q^{11} -4.80031 q^{13} -1.21242 q^{15} +1.00000 q^{17} +5.07740 q^{19} +4.19007 q^{23} -4.36260 q^{25} +5.60947 q^{27} +1.85926 q^{29} -10.0901 q^{31} +8.57151 q^{33} -6.11378 q^{37} +7.28982 q^{39} -6.24965 q^{41} -0.575750 q^{43} -0.553915 q^{45} -5.74135 q^{47} -1.51862 q^{51} +4.50334 q^{53} -4.50625 q^{55} -7.71062 q^{57} -7.97849 q^{59} -8.93411 q^{61} -3.83243 q^{65} +10.6787 q^{67} -6.36311 q^{69} +3.73942 q^{71} +13.1786 q^{73} +6.62512 q^{75} +5.15610 q^{79} -6.43722 q^{81} -5.24724 q^{83} +0.798372 q^{85} -2.82350 q^{87} +0.528350 q^{89} +15.3230 q^{93} +4.05366 q^{95} -5.53836 q^{97} +3.91604 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{5} + q^{9} + 5 q^{11} - 11 q^{13} + 8 q^{15} + 7 q^{17} + 2 q^{19} + 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 2 q^{31} - 2 q^{33} + 8 q^{37} + q^{39} - 6 q^{41} + 10 q^{43} + 10 q^{45} + q^{47}+ \cdots - 12 q^{99}+O(q^{100}) \) 7 * q + q^5 + q^9 + 5 * q^11 - 11 * q^13 + 8 * q^15 + 7 * q^17 + 2 * q^19 + 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 2 * q^31 - 2 * q^33 + 8 * q^37 + q^39 - 6 * q^41 + 10 * q^43 + 10 * q^45 + q^47 + 3 * q^53 - 11 * q^55 - 5 * q^57 + 4 * q^59 + 32 * q^61 + 20 * q^65 + 31 * q^67 - 16 * q^69 + 3 * q^71 + 7 * q^73 + 19 * q^75 + 8 * q^79 - q^81 - 24 * q^83 + q^85 + 3 * q^87 - 4 * q^89 + 19 * q^93 + 23 * q^95 - 15 * q^97 - 12 * q^99

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