LMFDB - Embedded newform 3332.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.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:	$26.6061539535$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 8x^{6} + 36x^{5} + 17x^{4} - 76x^{3} - 20x^{2} + 44x + 17 $ x^8 - 4x^7 - 8x^6 + 36x^5 + 17x^4 - 76x^3 - 20x^2 + 44*x + 17
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.453360$ of defining polynomial
Character	$\chi$	$=$	3332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.453360 q^{3} -3.09710 q^{5} -2.79446 q^{9} -3.54035 q^{11} +6.65819 q^{13} +1.40410 q^{15} -1.00000 q^{17} -3.06981 q^{19} -2.80826 q^{23} +4.59204 q^{25} +2.62698 q^{27} +1.12602 q^{29} -8.64360 q^{31} +1.60505 q^{33} -3.44940 q^{37} -3.01856 q^{39} -4.53841 q^{41} -10.2580 q^{43} +8.65474 q^{45} +6.83631 q^{47} +0.453360 q^{51} -6.24289 q^{53} +10.9648 q^{55} +1.39173 q^{57} +4.06149 q^{59} +12.4207 q^{61} -20.6211 q^{65} -6.60650 q^{67} +1.27315 q^{69} +6.60825 q^{71} +5.57268 q^{73} -2.08185 q^{75} +4.57456 q^{79} +7.19243 q^{81} +8.30865 q^{83} +3.09710 q^{85} -0.510494 q^{87} +2.87836 q^{89} +3.91866 q^{93} +9.50750 q^{95} +4.43114 q^{97} +9.89338 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 + 4 * q^5 + 8 * q^9 - 4 * q^11 + 20 * q^13 + 12 * q^15 - 8 * q^17 + 8 * q^19 + 4 * q^23 + 8 * q^25 + 28 * q^27 - 16 * q^29 - 8 * q^31 + 16 * q^33 + 8 * q^37 + 20 * q^39 + 12 * q^41 - 4 * q^43 + 20 * q^45 + 4 * q^47 - 4 * q^51 + 12 * q^55 - 16 * q^59 + 32 * q^61 + 44 * q^69 + 24 * q^73 + 24 * q^75 + 4 * q^79 + 36 * q^81 + 28 * q^83 - 4 * q^85 - 40 * q^87 + 20 * q^89 - 16 * q^93 - 20 * q^95 + 56 * q^97 - 8 * 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
$3$	−0.453360	−0.261748	−0.130874	−	0.991399i	$-0.541778\pi$
$3$	−0.453360	−0.261748	−0.130874	+	0.991399i	$0.541778\pi$
$4$	0	0
$5$	−3.09710	−1.38507	−0.692533	−	0.721386i	$-0.743505\pi$
$5$	−3.09710	−1.38507	−0.692533	+	0.721386i	$0.743505\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.79446	−0.931488
$10$	0	0
$11$	−3.54035	−1.06746	−0.533728	−	0.845656i	$-0.679209\pi$
$11$	−3.54035	−1.06746	−0.533728	+	0.845656i	$0.679209\pi$
$12$	0	0
$13$	6.65819	1.84665	0.923324	−	0.384021i	$-0.125461\pi$
$13$	6.65819	1.84665	0.923324	+	0.384021i	$0.125461\pi$
$14$	0	0
$15$	1.40410	0.362538
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−3.06981	−0.704262	−0.352131	−	0.935951i	$-0.614543\pi$
$19$	−3.06981	−0.704262	−0.352131	+	0.935951i	$0.614543\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.80826	−0.585562	−0.292781	−	0.956180i	$-0.594581\pi$
$23$	−2.80826	−0.585562	−0.292781	+	0.956180i	$0.594581\pi$
$24$	0	0
$25$	4.59204	0.918407
$26$	0	0
$27$	2.62698	0.505562
$28$	0	0
$29$	1.12602	0.209097	0.104549	−	0.994520i	$-0.466660\pi$
$29$	1.12602	0.209097	0.104549	+	0.994520i	$0.466660\pi$
$30$	0	0
$31$	−8.64360	−1.55244	−0.776218	−	0.630465i	$-0.782864\pi$
$31$	−8.64360	−1.55244	−0.776218	+	0.630465i	$0.782864\pi$
$32$	0	0
$33$	1.60505	0.279404
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.44940	−0.567079	−0.283539	−	0.958961i	$-0.591509\pi$
$37$	−3.44940	−0.567079	−0.283539	+	0.958961i	$0.591509\pi$
$38$	0	0
$39$	−3.01856	−0.483356
$40$	0	0
$41$	−4.53841	−0.708781	−0.354391	−	0.935097i	$-0.615312\pi$
$41$	−4.53841	−0.708781	−0.354391	+	0.935097i	$0.615312\pi$
$42$	0	0
$43$	−10.2580	−1.56433	−0.782165	−	0.623071i	$-0.785885\pi$
$43$	−10.2580	−1.56433	−0.782165	+	0.623071i	$0.785885\pi$
$44$	0	0
$45$	8.65474	1.29017
$46$	0	0
$47$	6.83631	0.997179	0.498589	−	0.866838i	$-0.333852\pi$
$47$	6.83631	0.997179	0.498589	+	0.866838i	$0.333852\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.453360	0.0634831
$52$	0	0
$53$	−6.24289	−0.857527	−0.428763	−	0.903417i	$-0.641051\pi$
$53$	−6.24289	−0.857527	−0.428763	+	0.903417i	$0.641051\pi$
$54$	0	0
$55$	10.9648	1.47850
$56$	0	0
$57$	1.39173	0.184339
$58$	0	0
$59$	4.06149	0.528761	0.264381	−	0.964418i	$-0.414833\pi$
$59$	4.06149	0.528761	0.264381	+	0.964418i	$0.414833\pi$
$60$	0	0
$61$	12.4207	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$61$	12.4207	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−20.6211	−2.55773
$66$	0	0
$67$	−6.60650	−0.807112	−0.403556	−	0.914955i	$-0.632226\pi$
$67$	−6.60650	−0.807112	−0.403556	+	0.914955i	$0.632226\pi$
$68$	0	0
$69$	1.27315	0.153269
$70$	0	0
$71$	6.60825	0.784255	0.392128	−	0.919911i	$-0.371739\pi$
$71$	6.60825	0.784255	0.392128	+	0.919911i	$0.371739\pi$
$72$	0	0
$73$	5.57268	0.652232	0.326116	−	0.945330i	$-0.394260\pi$
$73$	5.57268	0.652232	0.326116	+	0.945330i	$0.394260\pi$
$74$	0	0
$75$	−2.08185	−0.240391
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.57456	0.514678	0.257339	−	0.966321i	$-0.417154\pi$
$79$	4.57456	0.514678	0.257339	+	0.966321i	$0.417154\pi$
$80$	0	0
$81$	7.19243	0.799159
$82$	0	0
$83$	8.30865	0.911993	0.455997	−	0.889982i	$-0.349283\pi$
$83$	8.30865	0.911993	0.455997	+	0.889982i	$0.349283\pi$
$84$	0	0
$85$	3.09710	0.335928
$86$	0	0
$87$	−0.510494	−0.0547307
$88$	0	0
$89$	2.87836	0.305106	0.152553	−	0.988295i	$-0.451251\pi$
$89$	2.87836	0.305106	0.152553	+	0.988295i	$0.451251\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.91866	0.406346
$94$	0	0
$95$	9.50750	0.975449
$96$	0	0
$97$	4.43114	0.449914	0.224957	−	0.974369i	$-0.427776\pi$
$97$	4.43114	0.449914	0.224957	+	0.974369i	$0.427776\pi$
$98$	0	0
$99$	9.89338	0.994322
$100$	0	0
$101$	3.07544	0.306018	0.153009	−	0.988225i	$-0.451104\pi$
$101$	3.07544	0.306018	0.153009	+	0.988225i	$0.451104\pi$
$102$	0	0
$103$	15.6959	1.54656	0.773279	−	0.634066i	$-0.218615\pi$
$103$	15.6959	1.54656	0.773279	+	0.634066i	$0.218615\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.32316	−0.417935	−0.208968	−	0.977923i	$-0.567010\pi$
$107$	−4.32316	−0.417935	−0.208968	+	0.977923i	$0.567010\pi$
$108$	0	0
$109$	16.9900	1.62735	0.813674	−	0.581322i	$-0.197464\pi$
$109$	16.9900	1.62735	0.813674	+	0.581322i	$0.197464\pi$
$110$	0	0
$111$	1.56382	0.148431
$112$	0	0
$113$	−9.49175	−0.892909	−0.446454	−	0.894806i	$-0.647313\pi$
$113$	−9.49175	−0.892909	−0.446454	+	0.894806i	$0.647313\pi$
$114$	0	0
$115$	8.69745	0.811042
$116$	0	0
$117$	−18.6061	−1.72013
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.53408	0.139461
$122$	0	0
$123$	2.05754	0.185522
$124$	0	0
$125$	1.26351	0.113011
$126$	0	0
$127$	14.3430	1.27274	0.636368	−	0.771386i	$-0.280436\pi$
$127$	14.3430	1.27274	0.636368	+	0.771386i	$0.280436\pi$
$128$	0	0
$129$	4.65057	0.409460
$130$	0	0
$131$	6.78080	0.592441	0.296221	−	0.955120i	$-0.404274\pi$
$131$	6.78080	0.592441	0.296221	+	0.955120i	$0.404274\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−8.13602	−0.700237
$136$	0	0
$137$	9.46127	0.808331	0.404165	−	0.914686i	$-0.367562\pi$
$137$	9.46127	0.808331	0.404165	+	0.914686i	$0.367562\pi$
$138$	0	0
$139$	−2.03353	−0.172482	−0.0862409	−	0.996274i	$-0.527485\pi$
$139$	−2.03353	−0.172482	−0.0862409	+	0.996274i	$0.527485\pi$
$140$	0	0
$141$	−3.09931	−0.261009
$142$	0	0
$143$	−23.5723	−1.97122
$144$	0	0
$145$	−3.48741	−0.289613
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−22.3323	−1.82953	−0.914766	−	0.403984i	$-0.867625\pi$
$149$	−22.3323	−1.82953	−0.914766	+	0.403984i	$0.867625\pi$
$150$	0	0
$151$	19.6431	1.59853	0.799265	−	0.600979i	$-0.205223\pi$
$151$	19.6431	1.59853	0.799265	+	0.600979i	$0.205223\pi$
$152$	0	0
$153$	2.79446	0.225919
$154$	0	0
$155$	26.7701	2.15023
$156$	0	0
$157$	24.4250	1.94933	0.974664	−	0.223672i	$-0.0718044\pi$
$157$	24.4250	1.94933	0.974664	+	0.223672i	$0.0718044\pi$
$158$	0	0
$159$	2.83028	0.224455
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−4.63623	−0.363137	−0.181569	−	0.983378i	$-0.558117\pi$
$163$	−4.63623	−0.363137	−0.181569	+	0.983378i	$0.558117\pi$
$164$	0	0
$165$	−4.97101	−0.386993
$166$	0	0
$167$	−18.7982	−1.45465	−0.727323	−	0.686295i	$-0.759236\pi$
$167$	−18.7982	−1.45465	−0.727323	+	0.686295i	$0.759236\pi$
$168$	0	0
$169$	31.3315	2.41011
$170$	0	0
$171$	8.57847	0.656012
$172$	0	0
$173$	−3.68628	−0.280263	−0.140131	−	0.990133i	$-0.544752\pi$
$173$	−3.68628	−0.280263	−0.140131	+	0.990133i	$0.544752\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.84132	−0.138402
$178$	0	0
$179$	18.0582	1.34973	0.674865	−	0.737941i	$-0.264202\pi$
$179$	18.0582	1.34973	0.674865	+	0.737941i	$0.264202\pi$
$180$	0	0
$181$	8.56085	0.636323	0.318162	−	0.948036i	$-0.396935\pi$
$181$	8.56085	0.636323	0.318162	+	0.948036i	$0.396935\pi$
$182$	0	0
$183$	−5.63106	−0.416260
$184$	0	0
$185$	10.6832	0.785441
$186$	0	0
$187$	3.54035	0.258896
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.88693	−0.281248	−0.140624	−	0.990063i	$-0.544911\pi$
$191$	−3.88693	−0.281248	−0.140624	+	0.990063i	$0.544911\pi$
$192$	0	0
$193$	−9.12362	−0.656733	−0.328366	−	0.944550i	$-0.606498\pi$
$193$	−9.12362	−0.656733	−0.328366	+	0.944550i	$0.606498\pi$
$194$	0	0
$195$	9.34877	0.669480
$196$	0	0
$197$	−22.2054	−1.58207	−0.791036	−	0.611770i	$-0.790458\pi$
$197$	−22.2054	−1.58207	−0.791036	+	0.611770i	$0.790458\pi$
$198$	0	0
$199$	−22.3615	−1.58516	−0.792582	−	0.609765i	$-0.791264\pi$
$199$	−22.3615	−1.58516	−0.792582	+	0.609765i	$0.791264\pi$
$200$	0	0
$201$	2.99512	0.211260
$202$	0	0
$203$	0	0
$204$	0	0
$205$	14.0559	0.981708
$206$	0	0
$207$	7.84757	0.545444
$208$	0	0
$209$	10.8682	0.751769
$210$	0	0
$211$	19.8752	1.36826	0.684131	−	0.729359i	$-0.260182\pi$
$211$	19.8752	1.36826	0.684131	+	0.729359i	$0.260182\pi$
$212$	0	0
$213$	−2.99592	−0.205277
$214$	0	0
$215$	31.7701	2.16670
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2.52643	−0.170720
$220$	0	0
$221$	−6.65819	−0.447878
$222$	0	0
$223$	−6.56638	−0.439717	−0.219859	−	0.975532i	$-0.570560\pi$
$223$	−6.56638	−0.439717	−0.219859	+	0.975532i	$0.570560\pi$
$224$	0	0
$225$	−12.8323	−0.855485
$226$	0	0
$227$	−25.3441	−1.68215	−0.841073	−	0.540922i	$-0.818076\pi$
$227$	−25.3441	−1.68215	−0.841073	+	0.540922i	$0.818076\pi$
$228$	0	0
$229$	15.0996	0.997813	0.498906	−	0.866656i	$-0.333735\pi$
$229$	15.0996	0.997813	0.498906	+	0.866656i	$0.333735\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.96743	0.259915	0.129957	−	0.991520i	$-0.458516\pi$
$233$	3.96743	0.259915	0.129957	+	0.991520i	$0.458516\pi$
$234$	0	0
$235$	−21.1727	−1.38116
$236$	0	0
$237$	−2.07392	−0.134716
$238$	0	0
$239$	−12.6784	−0.820100	−0.410050	−	0.912063i	$-0.634489\pi$
$239$	−12.6784	−0.820100	−0.410050	+	0.912063i	$0.634489\pi$
$240$	0	0
$241$	−16.8444	−1.08504	−0.542521	−	0.840042i	$-0.682530\pi$
$241$	−16.8444	−1.08504	−0.542521	+	0.840042i	$0.682530\pi$
$242$	0	0
$243$	−11.1417	−0.714740
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−20.4394	−1.30052
$248$	0	0
$249$	−3.76681	−0.238712
$250$	0	0
$251$	−29.0437	−1.83322	−0.916610	−	0.399782i	$-0.869086\pi$
$251$	−29.0437	−1.83322	−0.916610	+	0.399782i	$0.869086\pi$
$252$	0	0
$253$	9.94221	0.625061
$254$	0	0
$255$	−1.40410	−0.0879283
$256$	0	0
$257$	13.2769	0.828193	0.414096	−	0.910233i	$-0.364098\pi$
$257$	13.2769	0.828193	0.414096	+	0.910233i	$0.364098\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.14663	−0.194772
$262$	0	0
$263$	−9.26082	−0.571047	−0.285523	−	0.958372i	$-0.592167\pi$
$263$	−9.26082	−0.571047	−0.285523	+	0.958372i	$0.592167\pi$
$264$	0	0
$265$	19.3349	1.18773
$266$	0	0
$267$	−1.30494	−0.0798607
$268$	0	0
$269$	−3.05810	−0.186456	−0.0932278	−	0.995645i	$-0.529718\pi$
$269$	−3.05810	−0.186456	−0.0932278	+	0.995645i	$0.529718\pi$
$270$	0	0
$271$	13.9100	0.844974	0.422487	−	0.906369i	$-0.361157\pi$
$271$	13.9100	0.844974	0.422487	+	0.906369i	$0.361157\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16.2574	−0.980359
$276$	0	0
$277$	20.6937	1.24337	0.621683	−	0.783269i	$-0.286449\pi$
$277$	20.6937	1.24337	0.621683	+	0.783269i	$0.286449\pi$
$278$	0	0
$279$	24.1542	1.44608
$280$	0	0
$281$	−10.5352	−0.628477	−0.314238	−	0.949344i	$-0.601749\pi$
$281$	−10.5352	−0.628477	−0.314238	+	0.949344i	$0.601749\pi$
$282$	0	0
$283$	20.4887	1.21793	0.608963	−	0.793199i	$-0.291586\pi$
$283$	20.4887	1.21793	0.608963	+	0.793199i	$0.291586\pi$
$284$	0	0
$285$	−4.31032	−0.255321
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.00890	−0.117764
$292$	0	0
$293$	27.5224	1.60788	0.803938	−	0.594714i	$-0.202735\pi$
$293$	27.5224	1.60788	0.803938	+	0.594714i	$0.202735\pi$
$294$	0	0
$295$	−12.5788	−0.732369
$296$	0	0
$297$	−9.30042	−0.539665
$298$	0	0
$299$	−18.6979	−1.08133
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.39428	−0.0800994
$304$	0	0
$305$	−38.4682	−2.20268
$306$	0	0
$307$	7.76987	0.443450	0.221725	−	0.975109i	$-0.428831\pi$
$307$	7.76987	0.443450	0.221725	+	0.975109i	$0.428831\pi$
$308$	0	0
$309$	−7.11587	−0.404808
$310$	0	0
$311$	22.2882	1.26385	0.631924	−	0.775030i	$-0.282265\pi$
$311$	22.2882	1.26385	0.631924	+	0.775030i	$0.282265\pi$
$312$	0	0
$313$	−20.1306	−1.13785	−0.568926	−	0.822389i	$-0.692641\pi$
$313$	−20.1306	−1.13785	−0.568926	+	0.822389i	$0.692641\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.82760	0.271145	0.135573	−	0.990767i	$-0.456713\pi$
$317$	4.82760	0.271145	0.135573	+	0.990767i	$0.456713\pi$
$318$	0	0
$319$	−3.98652	−0.223202
$320$	0	0
$321$	1.95995	0.109394
$322$	0	0
$323$	3.06981	0.170809
$324$	0	0
$325$	30.5746	1.69598
$326$	0	0
$327$	−7.70259	−0.425954
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.5175	1.40257	0.701284	−	0.712882i	$-0.252610\pi$
$331$	25.5175	1.40257	0.701284	+	0.712882i	$0.252610\pi$
$332$	0	0
$333$	9.63924	0.528227
$334$	0	0
$335$	20.4610	1.11790
$336$	0	0
$337$	−6.38558	−0.347845	−0.173922	−	0.984759i	$-0.555644\pi$
$337$	−6.38558	−0.347845	−0.173922	+	0.984759i	$0.555644\pi$
$338$	0	0
$339$	4.30318	0.233717
$340$	0	0
$341$	30.6014	1.65716
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.94308	−0.212288
$346$	0	0
$347$	23.5121	1.26220	0.631098	−	0.775703i	$-0.282604\pi$
$347$	23.5121	1.26220	0.631098	+	0.775703i	$0.282604\pi$
$348$	0	0
$349$	6.73510	0.360522	0.180261	−	0.983619i	$-0.442306\pi$
$349$	6.73510	0.360522	0.180261	+	0.983619i	$0.442306\pi$
$350$	0	0
$351$	17.4909	0.933596
$352$	0	0
$353$	−5.65720	−0.301102	−0.150551	−	0.988602i	$-0.548105\pi$
$353$	−5.65720	−0.301102	−0.150551	+	0.988602i	$0.548105\pi$
$354$	0	0
$355$	−20.4664	−1.08624
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−34.5038	−1.82104	−0.910519	−	0.413467i	$-0.864318\pi$
$359$	−34.5038	−1.82104	−0.910519	+	0.413467i	$0.864318\pi$
$360$	0	0
$361$	−9.57628	−0.504015
$362$	0	0
$363$	−0.695488	−0.0365037
$364$	0	0
$365$	−17.2591	−0.903385
$366$	0	0
$367$	−21.1712	−1.10513	−0.552565	−	0.833470i	$-0.686351\pi$
$367$	−21.1712	−1.10513	−0.552565	+	0.833470i	$0.686351\pi$
$368$	0	0
$369$	12.6824	0.660221
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.6581	1.53564	0.767819	−	0.640666i	$-0.221342\pi$
$373$	29.6581	1.53564	0.767819	+	0.640666i	$0.221342\pi$
$374$	0	0
$375$	−0.572823	−0.0295805
$376$	0	0
$377$	7.49727	0.386129
$378$	0	0
$379$	8.67022	0.445359	0.222680	−	0.974892i	$-0.428520\pi$
$379$	8.67022	0.445359	0.222680	+	0.974892i	$0.428520\pi$
$380$	0	0
$381$	−6.50254	−0.333135
$382$	0	0
$383$	−30.2331	−1.54484	−0.772421	−	0.635111i	$-0.780954\pi$
$383$	−30.2331	−1.54484	−0.772421	+	0.635111i	$0.780954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	28.6656	1.45716
$388$	0	0
$389$	15.8089	0.801543	0.400771	−	0.916178i	$-0.368742\pi$
$389$	15.8089	0.801543	0.400771	+	0.916178i	$0.368742\pi$
$390$	0	0
$391$	2.80826	0.142020
$392$	0	0
$393$	−3.07414	−0.155070
$394$	0	0
$395$	−14.1679	−0.712863
$396$	0	0
$397$	−14.6825	−0.736893	−0.368446	−	0.929649i	$-0.620110\pi$
$397$	−14.6825	−0.736893	−0.368446	+	0.929649i	$0.620110\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	39.4024	1.96766	0.983831	−	0.179097i	$-0.0573176\pi$
$401$	39.4024	1.96766	0.983831	+	0.179097i	$0.0573176\pi$
$402$	0	0
$403$	−57.5507	−2.86680
$404$	0	0
$405$	−22.2757	−1.10689
$406$	0	0
$407$	12.2121	0.605331
$408$	0	0
$409$	12.4277	0.614508	0.307254	−	0.951627i	$-0.400590\pi$
$409$	12.4277	0.614508	0.307254	+	0.951627i	$0.400590\pi$
$410$	0	0
$411$	−4.28936	−0.211579
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−25.7327	−1.26317
$416$	0	0
$417$	0.921922	0.0451467
$418$	0	0
$419$	−11.5824	−0.565838	−0.282919	−	0.959144i	$-0.591303\pi$
$419$	−11.5824	−0.565838	−0.282919	+	0.959144i	$0.591303\pi$
$420$	0	0
$421$	−7.22185	−0.351971	−0.175986	−	0.984393i	$-0.556311\pi$
$421$	−7.22185	−0.351971	−0.175986	+	0.984393i	$0.556311\pi$
$422$	0	0
$423$	−19.1038	−0.928860
$424$	0	0
$425$	−4.59204	−0.222746
$426$	0	0
$427$	0	0
$428$	0	0
$429$	10.6867	0.515961
$430$	0	0
$431$	37.0625	1.78524	0.892618	−	0.450813i	$-0.148866\pi$
$431$	37.0625	1.78524	0.892618	+	0.450813i	$0.148866\pi$
$432$	0	0
$433$	5.21558	0.250645	0.125322	−	0.992116i	$-0.460003\pi$
$433$	5.21558	0.250645	0.125322	+	0.992116i	$0.460003\pi$
$434$	0	0
$435$	1.58105	0.0758056
$436$	0	0
$437$	8.62081	0.412389
$438$	0	0
$439$	−4.68381	−0.223546	−0.111773	−	0.993734i	$-0.535653\pi$
$439$	−4.68381	−0.223546	−0.111773	+	0.993734i	$0.535653\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.16753	−0.150494	−0.0752471	−	0.997165i	$-0.523975\pi$
$443$	−3.16753	−0.150494	−0.0752471	+	0.997165i	$0.523975\pi$
$444$	0	0
$445$	−8.91458	−0.422592
$446$	0	0
$447$	10.1246	0.478876
$448$	0	0
$449$	−7.42717	−0.350510	−0.175255	−	0.984523i	$-0.556075\pi$
$449$	−7.42717	−0.350510	−0.175255	+	0.984523i	$0.556075\pi$
$450$	0	0
$451$	16.0676	0.756592
$452$	0	0
$453$	−8.90538	−0.418411
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.799652	0.0374061	0.0187031	−	0.999825i	$-0.494046\pi$
$457$	0.799652	0.0374061	0.0187031	+	0.999825i	$0.494046\pi$
$458$	0	0
$459$	−2.62698	−0.122617
$460$	0	0
$461$	2.54474	0.118520	0.0592602	−	0.998243i	$-0.481126\pi$
$461$	2.54474	0.118520	0.0592602	+	0.998243i	$0.481126\pi$
$462$	0	0
$463$	−21.5866	−1.00321	−0.501607	−	0.865096i	$-0.667258\pi$
$463$	−21.5866	−1.00321	−0.501607	+	0.865096i	$0.667258\pi$
$464$	0	0
$465$	−12.1365	−0.562816
$466$	0	0
$467$	6.02385	0.278751	0.139375	−	0.990240i	$-0.455491\pi$
$467$	6.02385	0.278751	0.139375	+	0.990240i	$0.455491\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−11.0733	−0.510232
$472$	0	0
$473$	36.3169	1.66985
$474$	0	0
$475$	−14.0967	−0.646799
$476$	0	0
$477$	17.4455	0.798776
$478$	0	0
$479$	−25.9297	−1.18476	−0.592378	−	0.805660i	$-0.701811\pi$
$479$	−25.9297	−1.18476	−0.592378	+	0.805660i	$0.701811\pi$
$480$	0	0
$481$	−22.9668	−1.04719
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.7237	−0.623160
$486$	0	0
$487$	−17.3380	−0.785659	−0.392830	−	0.919611i	$-0.628504\pi$
$487$	−17.3380	−0.785659	−0.392830	+	0.919611i	$0.628504\pi$
$488$	0	0
$489$	2.10188	0.0950503
$490$	0	0
$491$	−16.6262	−0.750332	−0.375166	−	0.926958i	$-0.622414\pi$
$491$	−16.6262	−0.750332	−0.375166	+	0.926958i	$0.622414\pi$
$492$	0	0
$493$	−1.12602	−0.0507135
$494$	0	0
$495$	−30.6408	−1.37720
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−11.1886	−0.500872	−0.250436	−	0.968133i	$-0.580574\pi$
$499$	−11.1886	−0.500872	−0.250436	+	0.968133i	$0.580574\pi$
$500$	0	0
$501$	8.52234	0.380750
$502$	0	0
$503$	−33.0057	−1.47165	−0.735826	−	0.677170i	$-0.763206\pi$
$503$	−33.0057	−1.47165	−0.735826	+	0.677170i	$0.763206\pi$
$504$	0	0
$505$	−9.52494	−0.423854
$506$	0	0
$507$	−14.2044	−0.630841
$508$	0	0
$509$	36.1597	1.60275	0.801374	−	0.598163i	$-0.204102\pi$
$509$	36.1597	1.60275	0.801374	+	0.598163i	$0.204102\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8.06432	−0.356048
$514$	0	0
$515$	−48.6116	−2.14209
$516$	0	0
$517$	−24.2029	−1.06444
$518$	0	0
$519$	1.67121	0.0733581
$520$	0	0
$521$	20.2824	0.888589	0.444294	−	0.895881i	$-0.353454\pi$
$521$	20.2824	0.888589	0.444294	+	0.895881i	$0.353454\pi$
$522$	0	0
$523$	−1.31283	−0.0574062	−0.0287031	−	0.999588i	$-0.509138\pi$
$523$	−1.31283	−0.0574062	−0.0287031	+	0.999588i	$0.509138\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.64360	0.376521
$528$	0	0
$529$	−15.1137	−0.657117
$530$	0	0
$531$	−11.3497	−0.492535
$532$	0	0
$533$	−30.2176	−1.30887
$534$	0	0
$535$	13.3892	0.578868
$536$	0	0
$537$	−8.18685	−0.353288
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.89150	0.382275	0.191138	−	0.981563i	$-0.438782\pi$
$541$	8.89150	0.382275	0.191138	+	0.981563i	$0.438782\pi$
$542$	0	0
$543$	−3.88115	−0.166556
$544$	0	0
$545$	−52.6198	−2.25398
$546$	0	0
$547$	10.5467	0.450944	0.225472	−	0.974250i	$-0.427608\pi$
$547$	10.5467	0.450944	0.225472	+	0.974250i	$0.427608\pi$
$548$	0	0
$549$	−34.7093	−1.48136
$550$	0	0
$551$	−3.45667	−0.147259
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.84331	−0.205587
$556$	0	0
$557$	36.6686	1.55370	0.776850	−	0.629686i	$-0.216816\pi$
$557$	36.6686	1.55370	0.776850	+	0.629686i	$0.216816\pi$
$558$	0	0
$559$	−68.2997	−2.88877
$560$	0	0
$561$	−1.60505	−0.0677654
$562$	0	0
$563$	−0.626188	−0.0263907	−0.0131953	−	0.999913i	$-0.504200\pi$
$563$	−0.626188	−0.0263907	−0.0131953	+	0.999913i	$0.504200\pi$
$564$	0	0
$565$	29.3969	1.23674
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.9728	0.711539	0.355769	−	0.934574i	$-0.384219\pi$
$569$	16.9728	0.711539	0.355769	+	0.934574i	$0.384219\pi$
$570$	0	0
$571$	−26.3060	−1.10087	−0.550437	−	0.834877i	$-0.685539\pi$
$571$	−26.3060	−1.10087	−0.550437	+	0.834877i	$0.685539\pi$
$572$	0	0
$573$	1.76218	0.0736160
$574$	0	0
$575$	−12.8956	−0.537784
$576$	0	0
$577$	36.2699	1.50994	0.754968	−	0.655762i	$-0.227653\pi$
$577$	36.2699	1.50994	0.754968	+	0.655762i	$0.227653\pi$
$578$	0	0
$579$	4.13629	0.171898
$580$	0	0
$581$	0	0
$582$	0	0
$583$	22.1020	0.915371
$584$	0	0
$585$	57.6249	2.38250
$586$	0	0
$587$	12.5726	0.518928	0.259464	−	0.965753i	$-0.416454\pi$
$587$	12.5726	0.518928	0.259464	+	0.965753i	$0.416454\pi$
$588$	0	0
$589$	26.5342	1.09332
$590$	0	0
$591$	10.0671	0.414103
$592$	0	0
$593$	16.1797	0.664420	0.332210	−	0.943205i	$-0.392206\pi$
$593$	16.1797	0.664420	0.332210	+	0.943205i	$0.392206\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.1378	0.414913
$598$	0	0
$599$	−42.6737	−1.74360	−0.871801	−	0.489861i	$-0.837048\pi$
$599$	−42.6737	−1.74360	−0.871801	+	0.489861i	$0.837048\pi$
$600$	0	0
$601$	−0.499863	−0.0203898	−0.0101949	−	0.999948i	$-0.503245\pi$
$601$	−0.499863	−0.0203898	−0.0101949	+	0.999948i	$0.503245\pi$
$602$	0	0
$603$	18.4616	0.751815
$604$	0	0
$605$	−4.75119	−0.193163
$606$	0	0
$607$	33.5454	1.36157	0.680783	−	0.732486i	$-0.261640\pi$
$607$	33.5454	1.36157	0.680783	+	0.732486i	$0.261640\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	45.5174	1.84144
$612$	0	0
$613$	−8.38832	−0.338801	−0.169401	−	0.985547i	$-0.554183\pi$
$613$	−8.38832	−0.338801	−0.169401	+	0.985547i	$0.554183\pi$
$614$	0	0
$615$	−6.37240	−0.256960
$616$	0	0
$617$	−22.1742	−0.892698	−0.446349	−	0.894859i	$-0.647276\pi$
$617$	−22.1742	−0.892698	−0.446349	+	0.894859i	$0.647276\pi$
$618$	0	0
$619$	8.15864	0.327924	0.163962	−	0.986467i	$-0.447573\pi$
$619$	8.15864	0.327924	0.163962	+	0.986467i	$0.447573\pi$
$620$	0	0
$621$	−7.37723	−0.296038
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−26.8734	−1.07494
$626$	0	0
$627$	−4.92720	−0.196774
$628$	0	0
$629$	3.44940	0.137537
$630$	0	0
$631$	−24.4102	−0.971754	−0.485877	−	0.874027i	$-0.661500\pi$
$631$	−24.4102	−0.971754	−0.485877	+	0.874027i	$0.661500\pi$
$632$	0	0
$633$	−9.01060	−0.358139
$634$	0	0
$635$	−44.4217	−1.76282
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−18.4665	−0.730524
$640$	0	0
$641$	−14.7994	−0.584541	−0.292271	−	0.956336i	$-0.594411\pi$
$641$	−14.7994	−0.584541	−0.292271	+	0.956336i	$0.594411\pi$
$642$	0	0
$643$	19.2503	0.759157	0.379578	−	0.925160i	$-0.376069\pi$
$643$	19.2503	0.759157	0.379578	+	0.925160i	$0.376069\pi$
$644$	0	0
$645$	−14.4033	−0.567129
$646$	0	0
$647$	13.4065	0.527064	0.263532	−	0.964651i	$-0.415113\pi$
$647$	13.4065	0.527064	0.263532	+	0.964651i	$0.415113\pi$
$648$	0	0
$649$	−14.3791	−0.564429
$650$	0	0
$651$	0	0
$652$	0	0
$653$	45.2896	1.77232	0.886160	−	0.463379i	$-0.153363\pi$
$653$	45.2896	1.77232	0.886160	+	0.463379i	$0.153363\pi$
$654$	0	0
$655$	−21.0008	−0.820570
$656$	0	0
$657$	−15.5726	−0.607547
$658$	0	0
$659$	12.8324	0.499878	0.249939	−	0.968262i	$-0.419589\pi$
$659$	12.8324	0.499878	0.249939	+	0.968262i	$0.419589\pi$
$660$	0	0
$661$	12.1691	0.473322	0.236661	−	0.971592i	$-0.423947\pi$
$661$	12.1691	0.473322	0.236661	+	0.971592i	$0.423947\pi$
$662$	0	0
$663$	3.01856	0.117231
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.16216	−0.122439
$668$	0	0
$669$	2.97693	0.115095
$670$	0	0
$671$	−43.9737	−1.69759
$672$	0	0
$673$	30.8271	1.18830	0.594148	−	0.804356i	$-0.297489\pi$
$673$	30.8271	1.18830	0.594148	+	0.804356i	$0.297489\pi$
$674$	0	0
$675$	12.0632	0.464312
$676$	0	0
$677$	−17.2671	−0.663630	−0.331815	−	0.943344i	$-0.607661\pi$
$677$	−17.2671	−0.663630	−0.331815	+	0.943344i	$0.607661\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	11.4900	0.440298
$682$	0	0
$683$	46.0135	1.76066	0.880328	−	0.474366i	$-0.157323\pi$
$683$	46.0135	1.76066	0.880328	+	0.474366i	$0.157323\pi$
$684$	0	0
$685$	−29.3025	−1.11959
$686$	0	0
$687$	−6.84558	−0.261175
$688$	0	0
$689$	−41.5663	−1.58355
$690$	0	0
$691$	46.2245	1.75846	0.879231	−	0.476396i	$-0.158057\pi$
$691$	46.2245	1.75846	0.879231	+	0.476396i	$0.158057\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.29805	0.238899
$696$	0	0
$697$	4.53841	0.171905
$698$	0	0
$699$	−1.79867	−0.0680320
$700$	0	0
$701$	49.9926	1.88819	0.944097	−	0.329668i	$-0.106937\pi$
$701$	49.9926	1.88819	0.944097	+	0.329668i	$0.106937\pi$
$702$	0	0
$703$	10.5890	0.399372
$704$	0	0
$705$	9.59888	0.361515
$706$	0	0
$707$	0	0
$708$	0	0
$709$	23.4690	0.881395	0.440697	−	0.897656i	$-0.354731\pi$
$709$	23.4690	0.881395	0.440697	+	0.897656i	$0.354731\pi$
$710$	0	0
$711$	−12.7834	−0.479417
$712$	0	0
$713$	24.2734	0.909047
$714$	0	0
$715$	73.0058	2.73026
$716$	0	0
$717$	5.74790	0.214659
$718$	0	0
$719$	−13.4216	−0.500543	−0.250271	−	0.968176i	$-0.580520\pi$
$719$	−13.4216	−0.500543	−0.250271	+	0.968176i	$0.580520\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.63658	0.284007
$724$	0	0
$725$	5.17074	0.192036
$726$	0	0
$727$	−28.7007	−1.06445	−0.532224	−	0.846603i	$-0.678644\pi$
$727$	−28.7007	−1.06445	−0.532224	+	0.846603i	$0.678644\pi$
$728$	0	0
$729$	−16.5261	−0.612077
$730$	0	0
$731$	10.2580	0.379406
$732$	0	0
$733$	5.32164	0.196559	0.0982795	−	0.995159i	$-0.468666\pi$
$733$	5.32164	0.196559	0.0982795	+	0.995159i	$0.468666\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	23.3893	0.861556
$738$	0	0
$739$	−24.1826	−0.889573	−0.444787	−	0.895637i	$-0.646720\pi$
$739$	−24.1826	−0.889573	−0.444787	+	0.895637i	$0.646720\pi$
$740$	0	0
$741$	9.26639	0.340409
$742$	0	0
$743$	−8.96281	−0.328813	−0.164407	−	0.986393i	$-0.552571\pi$
$743$	−8.96281	−0.328813	−0.164407	+	0.986393i	$0.552571\pi$
$744$	0	0
$745$	69.1653	2.53402
$746$	0	0
$747$	−23.2182	−0.849511
$748$	0	0
$749$	0	0
$750$	0	0
$751$	9.00923	0.328752	0.164376	−	0.986398i	$-0.447439\pi$
$751$	9.00923	0.328752	0.164376	+	0.986398i	$0.447439\pi$
$752$	0	0
$753$	13.1672	0.479841
$754$	0	0
$755$	−60.8365	−2.21407
$756$	0	0
$757$	−44.0715	−1.60181	−0.800903	−	0.598794i	$-0.795647\pi$
$757$	−44.0715	−1.60181	−0.800903	+	0.598794i	$0.795647\pi$
$758$	0	0
$759$	−4.50740	−0.163608
$760$	0	0
$761$	−13.2382	−0.479886	−0.239943	−	0.970787i	$-0.577129\pi$
$761$	−13.2382	−0.479886	−0.239943	+	0.970787i	$0.577129\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−8.65474	−0.312913
$766$	0	0
$767$	27.0422	0.976436
$768$	0	0
$769$	23.1833	0.836012	0.418006	−	0.908444i	$-0.362729\pi$
$769$	23.1833	0.836012	0.418006	+	0.908444i	$0.362729\pi$
$770$	0	0
$771$	−6.01923	−0.216777
$772$	0	0
$773$	6.52140	0.234558	0.117279	−	0.993099i	$-0.462583\pi$
$773$	6.52140	0.234558	0.117279	+	0.993099i	$0.462583\pi$
$774$	0	0
$775$	−39.6917	−1.42577
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.9321	0.499168
$780$	0	0
$781$	−23.3955	−0.837157
$782$	0	0
$783$	2.95804	0.105712
$784$	0	0
$785$	−75.6468	−2.69995
$786$	0	0
$787$	−43.1883	−1.53950	−0.769748	−	0.638348i	$-0.779618\pi$
$787$	−43.1883	−1.53950	−0.769748	+	0.638348i	$0.779618\pi$
$788$	0	0
$789$	4.19849	0.149470
$790$	0	0
$791$	0	0
$792$	0	0
$793$	82.6995	2.93675
$794$	0	0
$795$	−8.76565	−0.310886
$796$	0	0
$797$	−8.78993	−0.311355	−0.155678	−	0.987808i	$-0.549756\pi$
$797$	−8.78993	−0.311355	−0.155678	+	0.987808i	$0.549756\pi$
$798$	0	0
$799$	−6.83631	−0.241851
$800$	0	0
$801$	−8.04349	−0.284203
$802$	0	0
$803$	−19.7292	−0.696229
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.38642	0.0488043
$808$	0	0
$809$	−1.08320	−0.0380834	−0.0190417	−	0.999819i	$-0.506062\pi$
$809$	−1.08320	−0.0380834	−0.0190417	+	0.999819i	$0.506062\pi$
$810$	0	0
$811$	46.5032	1.63295	0.816474	−	0.577382i	$-0.195926\pi$
$811$	46.5032	1.63295	0.816474	+	0.577382i	$0.195926\pi$
$812$	0	0
$813$	−6.30625	−0.221170
$814$	0	0
$815$	14.3589	0.502969
$816$	0	0
$817$	31.4901	1.10170
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.1929	−0.949038	−0.474519	−	0.880245i	$-0.657378\pi$
$821$	−27.1929	−0.949038	−0.474519	+	0.880245i	$0.657378\pi$
$822$	0	0
$823$	28.2603	0.985092	0.492546	−	0.870286i	$-0.336066\pi$
$823$	28.2603	0.985092	0.492546	+	0.870286i	$0.336066\pi$
$824$	0	0
$825$	7.37046	0.256607
$826$	0	0
$827$	21.3385	0.742014	0.371007	−	0.928630i	$-0.379013\pi$
$827$	21.3385	0.742014	0.371007	+	0.928630i	$0.379013\pi$
$828$	0	0
$829$	0.399793	0.0138854	0.00694270	−	0.999976i	$-0.497790\pi$
$829$	0.399793	0.0138854	0.00694270	+	0.999976i	$0.497790\pi$
$830$	0	0
$831$	−9.38171	−0.325448
$832$	0	0
$833$	0	0
$834$	0	0
$835$	58.2199	2.01478
$836$	0	0
$837$	−22.7065	−0.784853
$838$	0	0
$839$	−8.76505	−0.302603	−0.151302	−	0.988488i	$-0.548346\pi$
$839$	−8.76505	−0.302603	−0.151302	+	0.988488i	$0.548346\pi$
$840$	0	0
$841$	−27.7321	−0.956278
$842$	0	0
$843$	4.77623	0.164502
$844$	0	0
$845$	−97.0367	−3.33816
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9.28875	−0.318789
$850$	0	0
$851$	9.68681	0.332060
$852$	0	0
$853$	26.7104	0.914547	0.457273	−	0.889326i	$-0.348826\pi$
$853$	26.7104	0.914547	0.457273	+	0.889326i	$0.348826\pi$
$854$	0	0
$855$	−26.5684	−0.908620
$856$	0	0
$857$	10.0031	0.341699	0.170849	−	0.985297i	$-0.445349\pi$
$857$	10.0031	0.341699	0.170849	+	0.985297i	$0.445349\pi$
$858$	0	0
$859$	−6.63127	−0.226256	−0.113128	−	0.993580i	$-0.536087\pi$
$859$	−6.63127	−0.226256	−0.113128	+	0.993580i	$0.536087\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.7849	−0.741565	−0.370783	−	0.928720i	$-0.620911\pi$
$863$	−21.7849	−0.741565	−0.370783	+	0.928720i	$0.620911\pi$
$864$	0	0
$865$	11.4168	0.388182
$866$	0	0
$867$	−0.453360	−0.0153969
$868$	0	0
$869$	−16.1955	−0.549396
$870$	0	0
$871$	−43.9873	−1.49045
$872$	0	0
$873$	−12.3827	−0.419089
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−52.8443	−1.78443	−0.892213	−	0.451615i	$-0.850848\pi$
$877$	−52.8443	−1.78443	−0.892213	+	0.451615i	$0.850848\pi$
$878$	0	0
$879$	−12.4776	−0.420857
$880$	0	0
$881$	32.2324	1.08594	0.542968	−	0.839753i	$-0.317300\pi$
$881$	32.2324	1.08594	0.542968	+	0.839753i	$0.317300\pi$
$882$	0	0
$883$	−16.6798	−0.561318	−0.280659	−	0.959807i	$-0.590553\pi$
$883$	−16.6798	−0.561318	−0.280659	+	0.959807i	$0.590553\pi$
$884$	0	0
$885$	5.70275	0.191696
$886$	0	0
$887$	−26.6367	−0.894372	−0.447186	−	0.894441i	$-0.647574\pi$
$887$	−26.6367	−0.894372	−0.447186	+	0.894441i	$0.647574\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−25.4637	−0.853066
$892$	0	0
$893$	−20.9862	−0.702275
$894$	0	0
$895$	−55.9279	−1.86946
$896$	0	0
$897$	8.47688	0.283035
$898$	0	0
$899$	−9.73289	−0.324610
$900$	0	0
$901$	6.24289	0.207981
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−26.5138	−0.881350
$906$	0	0
$907$	−23.5861	−0.783165	−0.391583	−	0.920143i	$-0.628072\pi$
$907$	−23.5861	−0.783165	−0.391583	+	0.920143i	$0.628072\pi$
$908$	0	0
$909$	−8.59420	−0.285052
$910$	0	0
$911$	−27.6670	−0.916648	−0.458324	−	0.888785i	$-0.651550\pi$
$911$	−27.6670	−0.916648	−0.458324	+	0.888785i	$0.651550\pi$
$912$	0	0
$913$	−29.4155	−0.973512
$914$	0	0
$915$	17.4400	0.576547
$916$	0	0
$917$	0	0
$918$	0	0
$919$	24.4123	0.805286	0.402643	−	0.915357i	$-0.368092\pi$
$919$	24.4123	0.805286	0.402643	+	0.915357i	$0.368092\pi$
$920$	0	0
$921$	−3.52255	−0.116072
$922$	0	0
$923$	43.9990	1.44824
$924$	0	0
$925$	−15.8398	−0.520809
$926$	0	0
$927$	−43.8615	−1.44060
$928$	0	0
$929$	−30.0344	−0.985397	−0.492698	−	0.870200i	$-0.663989\pi$
$929$	−30.0344	−0.985397	−0.492698	+	0.870200i	$0.663989\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−10.1046	−0.330809
$934$	0	0
$935$	−10.9648	−0.358588
$936$	0	0
$937$	−11.7364	−0.383412	−0.191706	−	0.981452i	$-0.561402\pi$
$937$	−11.7364	−0.383412	−0.191706	+	0.981452i	$0.561402\pi$
$938$	0	0
$939$	9.12643	0.297830
$940$	0	0
$941$	−23.9485	−0.780700	−0.390350	−	0.920667i	$-0.627646\pi$
$941$	−23.9485	−0.780700	−0.390350	+	0.920667i	$0.627646\pi$
$942$	0	0
$943$	12.7450	0.415035
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.6896	1.09477	0.547383	−	0.836882i	$-0.315624\pi$
$947$	33.6896	1.09477	0.547383	+	0.836882i	$0.315624\pi$
$948$	0	0
$949$	37.1039	1.20444
$950$	0	0
$951$	−2.18864	−0.0709716
$952$	0	0
$953$	0.853762	0.0276561	0.0138280	−	0.999904i	$-0.495598\pi$
$953$	0.853762	0.0276561	0.0138280	+	0.999904i	$0.495598\pi$
$954$	0	0
$955$	12.0382	0.389547
$956$	0	0
$957$	1.80733	0.0584226
$958$	0	0
$959$	0	0
$960$	0	0
$961$	43.7118	1.41006
$962$	0	0
$963$	12.0809	0.389302
$964$	0	0
$965$	28.2568	0.909618
$966$	0	0
$967$	−5.38932	−0.173309	−0.0866544	−	0.996238i	$-0.527618\pi$
$967$	−5.38932	−0.173309	−0.0866544	+	0.996238i	$0.527618\pi$
$968$	0	0
$969$	−1.39173	−0.0447087
$970$	0	0
$971$	57.1550	1.83419	0.917096	−	0.398667i	$-0.130527\pi$
$971$	57.1550	1.83419	0.917096	+	0.398667i	$0.130527\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−13.8613	−0.443917
$976$	0	0
$977$	−55.3682	−1.77139	−0.885693	−	0.464271i	$-0.846316\pi$
$977$	−55.3682	−1.77139	−0.885693	+	0.464271i	$0.846316\pi$
$978$	0	0
$979$	−10.1904	−0.325687
$980$	0	0
$981$	−47.4780	−1.51585
$982$	0	0
$983$	16.9700	0.541259	0.270630	−	0.962684i	$-0.412768\pi$
$983$	16.9700	0.541259	0.270630	+	0.962684i	$0.412768\pi$
$984$	0	0
$985$	68.7725	2.19127
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28.8071	0.916012
$990$	0	0
$991$	−55.4986	−1.76297	−0.881485	−	0.472211i	$-0.843456\pi$
$991$	−55.4986	−1.76297	−0.881485	+	0.472211i	$0.843456\pi$
$992$	0	0
$993$	−11.5686	−0.367119
$994$	0	0
$995$	69.2558	2.19556
$996$	0	0
$997$	−47.2359	−1.49598	−0.747988	−	0.663713i	$-0.768980\pi$
$997$	−47.2359	−1.49598	−0.747988	+	0.663713i	$0.768980\pi$
$998$	0	0
$999$	−9.06151	−0.286694

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.u.1.4	yes	8
7.6	odd	2		3332.2.a.t.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.t.1.5	✓	8	7.6	odd	2
3332.2.a.u.1.4	yes	8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

\(f(q)\)	\(=\)	\(q-0.453360 q^{3} -3.09710 q^{5} -2.79446 q^{9} -3.54035 q^{11} +6.65819 q^{13} +1.40410 q^{15} -1.00000 q^{17} -3.06981 q^{19} -2.80826 q^{23} +4.59204 q^{25} +2.62698 q^{27} +1.12602 q^{29} -8.64360 q^{31} +1.60505 q^{33} -3.44940 q^{37} -3.01856 q^{39} -4.53841 q^{41} -10.2580 q^{43} +8.65474 q^{45} +6.83631 q^{47} +0.453360 q^{51} -6.24289 q^{53} +10.9648 q^{55} +1.39173 q^{57} +4.06149 q^{59} +12.4207 q^{61} -20.6211 q^{65} -6.60650 q^{67} +1.27315 q^{69} +6.60825 q^{71} +5.57268 q^{73} -2.08185 q^{75} +4.57456 q^{79} +7.19243 q^{81} +8.30865 q^{83} +3.09710 q^{85} -0.510494 q^{87} +2.87836 q^{89} +3.91866 q^{93} +9.50750 q^{95} +4.43114 q^{97} +9.89338 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 + 4 * q^5 + 8 * q^9 - 4 * q^11 + 20 * q^13 + 12 * q^15 - 8 * q^17 + 8 * q^19 + 4 * q^23 + 8 * q^25 + 28 * q^27 - 16 * q^29 - 8 * q^31 + 16 * q^33 + 8 * q^37 + 20 * q^39 + 12 * q^41 - 4 * q^43 + 20 * q^45 + 4 * q^47 - 4 * q^51 + 12 * q^55 - 16 * q^59 + 32 * q^61 + 44 * q^69 + 24 * q^73 + 24 * q^75 + 4 * q^79 + 36 * q^81 + 28 * q^83 - 4 * q^85 - 40 * q^87 + 20 * q^89 - 16 * q^93 - 20 * q^95 + 56 * q^97 - 8 * q^99

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