LMFDB - Embedded newform 3332.2.a.u.1.3

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			1.3
Root			$-0.703456$ 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)

$f(q)$	$=$	$q-0.703456 q^{3} -0.200537 q^{5} -2.50515 q^{9} +0.369152 q^{11} -3.09466 q^{13} +0.141069 q^{15} -1.00000 q^{17} +3.18269 q^{19} -3.84786 q^{23} -4.95979 q^{25} +3.87263 q^{27} -2.87291 q^{29} +6.70249 q^{31} -0.259682 q^{33} -4.84269 q^{37} +2.17695 q^{39} +10.7673 q^{41} +12.5862 q^{43} +0.502375 q^{45} -0.142451 q^{47} +0.703456 q^{51} +7.07619 q^{53} -0.0740286 q^{55} -2.23888 q^{57} -6.71357 q^{59} -5.43405 q^{61} +0.620592 q^{65} +1.03265 q^{67} +2.70680 q^{69} -2.83998 q^{71} +9.84364 q^{73} +3.48899 q^{75} -12.3647 q^{79} +4.79123 q^{81} -5.51678 q^{83} +0.200537 q^{85} +2.02096 q^{87} +2.57375 q^{89} -4.71490 q^{93} -0.638245 q^{95} +14.8270 q^{97} -0.924782 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.703456	−0.406140	−0.203070	−	0.979164i	$-0.565092\pi$
$3$	−0.703456	−0.406140	−0.203070	+	0.979164i	$0.565092\pi$
$4$	0	0
$5$	−0.200537	−0.0896827	−0.0448414	−	0.998994i	$-0.514278\pi$
$5$	−0.200537	−0.0896827	−0.0448414	+	0.998994i	$0.514278\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.50515	−0.835050
$10$	0	0
$11$	0.369152	0.111304	0.0556518	−	0.998450i	$-0.482276\pi$
$11$	0.369152	0.111304	0.0556518	+	0.998450i	$0.482276\pi$
$12$	0	0
$13$	−3.09466	−0.858303	−0.429151	−	0.903233i	$-0.641187\pi$
$13$	−3.09466	−0.858303	−0.429151	+	0.903233i	$0.641187\pi$
$14$	0	0
$15$	0.141069	0.0364238
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	3.18269	0.730158	0.365079	−	0.930977i	$-0.381042\pi$
$19$	3.18269	0.730158	0.365079	+	0.930977i	$0.381042\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.84786	−0.802334	−0.401167	−	0.916005i	$-0.631395\pi$
$23$	−3.84786	−0.802334	−0.401167	+	0.916005i	$0.631395\pi$
$24$	0	0
$25$	−4.95979	−0.991957
$26$	0	0
$27$	3.87263	0.745288
$28$	0	0
$29$	−2.87291	−0.533486	−0.266743	−	0.963768i	$-0.585947\pi$
$29$	−2.87291	−0.533486	−0.266743	+	0.963768i	$0.585947\pi$
$30$	0	0
$31$	6.70249	1.20380	0.601901	−	0.798571i	$-0.294410\pi$
$31$	6.70249	1.20380	0.601901	+	0.798571i	$0.294410\pi$
$32$	0	0
$33$	−0.259682	−0.0452049
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.84269	−0.796134	−0.398067	−	0.917356i	$-0.630319\pi$
$37$	−4.84269	−0.796134	−0.398067	+	0.917356i	$0.630319\pi$
$38$	0	0
$39$	2.17695	0.348591
$40$	0	0
$41$	10.7673	1.68157	0.840786	−	0.541367i	$-0.182093\pi$
$41$	10.7673	1.68157	0.840786	+	0.541367i	$0.182093\pi$
$42$	0	0
$43$	12.5862	1.91938	0.959691	−	0.281057i	$-0.0906850\pi$
$43$	12.5862	1.91938	0.959691	+	0.281057i	$0.0906850\pi$
$44$	0	0
$45$	0.502375	0.0748896
$46$	0	0
$47$	−0.142451	−0.0207786	−0.0103893	−	0.999946i	$-0.503307\pi$
$47$	−0.142451	−0.0207786	−0.0103893	+	0.999946i	$0.503307\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.703456	0.0985035
$52$	0	0
$53$	7.07619	0.971989	0.485995	−	0.873962i	$-0.338457\pi$
$53$	7.07619	0.971989	0.485995	+	0.873962i	$0.338457\pi$
$54$	0	0
$55$	−0.0740286	−0.00998201
$56$	0	0
$57$	−2.23888	−0.296547
$58$	0	0
$59$	−6.71357	−0.874032	−0.437016	−	0.899454i	$-0.643965\pi$
$59$	−6.71357	−0.874032	−0.437016	+	0.899454i	$0.643965\pi$
$60$	0	0
$61$	−5.43405	−0.695759	−0.347879	−	0.937539i	$-0.613098\pi$
$61$	−5.43405	−0.695759	−0.347879	+	0.937539i	$0.613098\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.620592	0.0769750
$66$	0	0
$67$	1.03265	0.126159	0.0630793	−	0.998009i	$-0.479908\pi$
$67$	1.03265	0.126159	0.0630793	+	0.998009i	$0.479908\pi$
$68$	0	0
$69$	2.70680	0.325860
$70$	0	0
$71$	−2.83998	−0.337043	−0.168522	−	0.985698i	$-0.553899\pi$
$71$	−2.83998	−0.337043	−0.168522	+	0.985698i	$0.553899\pi$
$72$	0	0
$73$	9.84364	1.15211	0.576055	−	0.817411i	$-0.304591\pi$
$73$	9.84364	1.15211	0.576055	+	0.817411i	$0.304591\pi$
$74$	0	0
$75$	3.48899	0.402874
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.3647	−1.39114	−0.695569	−	0.718459i	$-0.744848\pi$
$79$	−12.3647	−1.39114	−0.695569	+	0.718459i	$0.744848\pi$
$80$	0	0
$81$	4.79123	0.532358
$82$	0	0
$83$	−5.51678	−0.605545	−0.302773	−	0.953063i	$-0.597912\pi$
$83$	−5.51678	−0.605545	−0.302773	+	0.953063i	$0.597912\pi$
$84$	0	0
$85$	0.200537	0.0217513
$86$	0	0
$87$	2.02096	0.216670
$88$	0	0
$89$	2.57375	0.272817	0.136408	−	0.990653i	$-0.456444\pi$
$89$	2.57375	0.272817	0.136408	+	0.990653i	$0.456444\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.71490	−0.488913
$94$	0	0
$95$	−0.638245	−0.0654826
$96$	0	0
$97$	14.8270	1.50545	0.752726	−	0.658334i	$-0.228739\pi$
$97$	14.8270	1.50545	0.752726	+	0.658334i	$0.228739\pi$
$98$	0	0
$99$	−0.924782	−0.0929441
$100$	0	0
$101$	14.3476	1.42764	0.713819	−	0.700331i	$-0.246964\pi$
$101$	14.3476	1.42764	0.713819	+	0.700331i	$0.246964\pi$
$102$	0	0
$103$	9.67665	0.953468	0.476734	−	0.879048i	$-0.341820\pi$
$103$	9.67665	0.953468	0.476734	+	0.879048i	$0.341820\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.53652	−0.921930	−0.460965	−	0.887418i	$-0.652497\pi$
$107$	−9.53652	−0.921930	−0.460965	+	0.887418i	$0.652497\pi$
$108$	0	0
$109$	8.57234	0.821081	0.410541	−	0.911842i	$-0.365340\pi$
$109$	8.57234	0.821081	0.410541	+	0.911842i	$0.365340\pi$
$110$	0	0
$111$	3.40662	0.323342
$112$	0	0
$113$	10.8275	1.01856	0.509281	−	0.860600i	$-0.329911\pi$
$113$	10.8275	1.01856	0.509281	+	0.860600i	$0.329911\pi$
$114$	0	0
$115$	0.771637	0.0719555
$116$	0	0
$117$	7.75257	0.716726
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.8637	−0.987612
$122$	0	0
$123$	−7.57433	−0.682954
$124$	0	0
$125$	1.99730	0.178644
$126$	0	0
$127$	5.02711	0.446084	0.223042	−	0.974809i	$-0.428401\pi$
$127$	5.02711	0.446084	0.223042	+	0.974809i	$0.428401\pi$
$128$	0	0
$129$	−8.85386	−0.779539
$130$	0	0
$131$	13.7709	1.20317	0.601583	−	0.798811i	$-0.294537\pi$
$131$	13.7709	1.20317	0.601583	+	0.798811i	$0.294537\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.776604	−0.0668395
$136$	0	0
$137$	10.3085	0.880716	0.440358	−	0.897822i	$-0.354852\pi$
$137$	10.3085	0.880716	0.440358	+	0.897822i	$0.354852\pi$
$138$	0	0
$139$	9.17849	0.778509	0.389255	−	0.921130i	$-0.372733\pi$
$139$	9.17849	0.778509	0.389255	+	0.921130i	$0.372733\pi$
$140$	0	0
$141$	0.100208	0.00843903
$142$	0	0
$143$	−1.14240	−0.0955322
$144$	0	0
$145$	0.576124	0.0478445
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.95204	−0.323764	−0.161882	−	0.986810i	$-0.551756\pi$
$149$	−3.95204	−0.323764	−0.161882	+	0.986810i	$0.551756\pi$
$150$	0	0
$151$	−2.34019	−0.190442	−0.0952208	−	0.995456i	$-0.530356\pi$
$151$	−2.34019	−0.190442	−0.0952208	+	0.995456i	$0.530356\pi$
$152$	0	0
$153$	2.50515	0.202529
$154$	0	0
$155$	−1.34410	−0.107960
$156$	0	0
$157$	−7.47884	−0.596876	−0.298438	−	0.954429i	$-0.596466\pi$
$157$	−7.47884	−0.596876	−0.298438	+	0.954429i	$0.596466\pi$
$158$	0	0
$159$	−4.97779	−0.394764
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.20394	0.720908	0.360454	−	0.932777i	$-0.382622\pi$
$163$	9.20394	0.720908	0.360454	+	0.932777i	$0.382622\pi$
$164$	0	0
$165$	0.0520758	0.00405410
$166$	0	0
$167$	4.25555	0.329304	0.164652	−	0.986352i	$-0.447350\pi$
$167$	4.25555	0.329304	0.164652	+	0.986352i	$0.447350\pi$
$168$	0	0
$169$	−3.42311	−0.263316
$170$	0	0
$171$	−7.97310	−0.609719
$172$	0	0
$173$	5.40778	0.411146	0.205573	−	0.978642i	$-0.434094\pi$
$173$	5.40778	0.411146	0.205573	+	0.978642i	$0.434094\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.72270	0.354980
$178$	0	0
$179$	−10.1938	−0.761922	−0.380961	−	0.924591i	$-0.624407\pi$
$179$	−10.1938	−0.761922	−0.380961	+	0.924591i	$0.624407\pi$
$180$	0	0
$181$	0.0382153	0.00284052	0.00142026	−	0.999999i	$-0.499548\pi$
$181$	0.0382153	0.00284052	0.00142026	+	0.999999i	$0.499548\pi$
$182$	0	0
$183$	3.82261	0.282576
$184$	0	0
$185$	0.971138	0.0713995
$186$	0	0
$187$	−0.369152	−0.0269951
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.2538	−1.75494	−0.877472	−	0.479629i	$-0.840771\pi$
$191$	−24.2538	−1.75494	−0.877472	+	0.479629i	$0.840771\pi$
$192$	0	0
$193$	12.7938	0.920917	0.460459	−	0.887681i	$-0.347685\pi$
$193$	12.7938	0.920917	0.460459	+	0.887681i	$0.347685\pi$
$194$	0	0
$195$	−0.436559	−0.0312626
$196$	0	0
$197$	9.97518	0.710702	0.355351	−	0.934733i	$-0.384361\pi$
$197$	9.97518	0.710702	0.355351	+	0.934733i	$0.384361\pi$
$198$	0	0
$199$	−6.68593	−0.473953	−0.236976	−	0.971515i	$-0.576156\pi$
$199$	−6.68593	−0.473953	−0.236976	+	0.971515i	$0.576156\pi$
$200$	0	0
$201$	−0.726426	−0.0512381
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.15924	−0.150808
$206$	0	0
$207$	9.63946	0.669989
$208$	0	0
$209$	1.17490	0.0812692
$210$	0	0
$211$	9.63425	0.663249	0.331624	−	0.943411i	$-0.392403\pi$
$211$	9.63425	0.663249	0.331624	+	0.943411i	$0.392403\pi$
$212$	0	0
$213$	1.99780	0.136887
$214$	0	0
$215$	−2.52400	−0.172135
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−6.92456	−0.467919
$220$	0	0
$221$	3.09466	0.208169
$222$	0	0
$223$	14.2325	0.953080	0.476540	−	0.879153i	$-0.341891\pi$
$223$	14.2325	0.953080	0.476540	+	0.879153i	$0.341891\pi$
$224$	0	0
$225$	12.4250	0.828334
$226$	0	0
$227$	−10.8629	−0.720994	−0.360497	−	0.932760i	$-0.617393\pi$
$227$	−10.8629	−0.720994	−0.360497	+	0.932760i	$0.617393\pi$
$228$	0	0
$229$	16.0833	1.06282	0.531408	−	0.847116i	$-0.321663\pi$
$229$	16.0833	1.06282	0.531408	+	0.847116i	$0.321663\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.90856	0.321570	0.160785	−	0.986989i	$-0.448597\pi$
$233$	4.90856	0.321570	0.160785	+	0.986989i	$0.448597\pi$
$234$	0	0
$235$	0.0285667	0.00186348
$236$	0	0
$237$	8.69802	0.564997
$238$	0	0
$239$	28.2990	1.83051	0.915255	−	0.402876i	$-0.131989\pi$
$239$	28.2990	1.83051	0.915255	+	0.402876i	$0.131989\pi$
$240$	0	0
$241$	12.0909	0.778840	0.389420	−	0.921060i	$-0.372675\pi$
$241$	12.0909	0.778840	0.389420	+	0.921060i	$0.372675\pi$
$242$	0	0
$243$	−14.9883	−0.961500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.84931	−0.626697
$248$	0	0
$249$	3.88081	0.245936
$250$	0	0
$251$	6.78218	0.428088	0.214044	−	0.976824i	$-0.431336\pi$
$251$	6.78218	0.428088	0.214044	+	0.976824i	$0.431336\pi$
$252$	0	0
$253$	−1.42045	−0.0893027
$254$	0	0
$255$	−0.141069	−0.00883407
$256$	0	0
$257$	−10.2859	−0.641616	−0.320808	−	0.947144i	$-0.603954\pi$
$257$	−10.2859	−0.641616	−0.320808	+	0.947144i	$0.603954\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	7.19707	0.445487
$262$	0	0
$263$	22.8609	1.40966	0.704832	−	0.709374i	$-0.251022\pi$
$263$	22.8609	1.40966	0.704832	+	0.709374i	$0.251022\pi$
$264$	0	0
$265$	−1.41904	−0.0871707
$266$	0	0
$267$	−1.81052	−0.110802
$268$	0	0
$269$	12.3958	0.755787	0.377894	−	0.925849i	$-0.376648\pi$
$269$	12.3958	0.755787	0.377894	+	0.925849i	$0.376648\pi$
$270$	0	0
$271$	11.8149	0.717705	0.358853	−	0.933394i	$-0.383168\pi$
$271$	11.8149	0.717705	0.358853	+	0.933394i	$0.383168\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.83092	−0.110408
$276$	0	0
$277$	−24.6663	−1.48205	−0.741026	−	0.671476i	$-0.765660\pi$
$277$	−24.6663	−1.48205	−0.741026	+	0.671476i	$0.765660\pi$
$278$	0	0
$279$	−16.7907	−1.00524
$280$	0	0
$281$	−26.1300	−1.55879	−0.779393	−	0.626536i	$-0.784472\pi$
$281$	−26.1300	−1.55879	−0.779393	+	0.626536i	$0.784472\pi$
$282$	0	0
$283$	5.61446	0.333745	0.166872	−	0.985979i	$-0.446633\pi$
$283$	5.61446	0.333745	0.166872	+	0.985979i	$0.446633\pi$
$284$	0	0
$285$	0.448977	0.0265951
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−10.4301	−0.611425
$292$	0	0
$293$	18.5564	1.08407	0.542037	−	0.840355i	$-0.317653\pi$
$293$	18.5564	1.08407	0.542037	+	0.840355i	$0.317653\pi$
$294$	0	0
$295$	1.34632	0.0783856
$296$	0	0
$297$	1.42959	0.0829532
$298$	0	0
$299$	11.9078	0.688645
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−10.0929	−0.579821
$304$	0	0
$305$	1.08973	0.0623976
$306$	0	0
$307$	−18.9093	−1.07921	−0.539604	−	0.841919i	$-0.681426\pi$
$307$	−18.9093	−1.07921	−0.539604	+	0.841919i	$0.681426\pi$
$308$	0	0
$309$	−6.80709	−0.387242
$310$	0	0
$311$	−29.1565	−1.65332	−0.826658	−	0.562705i	$-0.809761\pi$
$311$	−29.1565	−1.65332	−0.826658	+	0.562705i	$0.809761\pi$
$312$	0	0
$313$	26.3584	1.48987	0.744933	−	0.667139i	$-0.232481\pi$
$313$	26.3584	1.48987	0.744933	+	0.667139i	$0.232481\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.9968	1.17930	0.589649	−	0.807660i	$-0.299266\pi$
$317$	20.9968	1.17930	0.589649	+	0.807660i	$0.299266\pi$
$318$	0	0
$319$	−1.06054	−0.0593789
$320$	0	0
$321$	6.70852	0.374433
$322$	0	0
$323$	−3.18269	−0.177089
$324$	0	0
$325$	15.3488	0.851400
$326$	0	0
$327$	−6.03026	−0.333474
$328$	0	0
$329$	0	0
$330$	0	0
$331$	30.2063	1.66029	0.830144	−	0.557549i	$-0.188258\pi$
$331$	30.2063	1.66029	0.830144	+	0.557549i	$0.188258\pi$
$332$	0	0
$333$	12.1317	0.664812
$334$	0	0
$335$	−0.207085	−0.0113143
$336$	0	0
$337$	−12.7200	−0.692901	−0.346451	−	0.938068i	$-0.612613\pi$
$337$	−12.7200	−0.692901	−0.346451	+	0.938068i	$0.612613\pi$
$338$	0	0
$339$	−7.61664	−0.413679
$340$	0	0
$341$	2.47424	0.133988
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.542812	−0.0292240
$346$	0	0
$347$	27.1235	1.45607	0.728033	−	0.685542i	$-0.240435\pi$
$347$	27.1235	1.45607	0.728033	+	0.685542i	$0.240435\pi$
$348$	0	0
$349$	13.3003	0.711947	0.355973	−	0.934496i	$-0.384149\pi$
$349$	13.3003	0.711947	0.355973	+	0.934496i	$0.384149\pi$
$350$	0	0
$351$	−11.9845	−0.639683
$352$	0	0
$353$	20.6223	1.09762	0.548808	−	0.835948i	$-0.315082\pi$
$353$	20.6223	1.09762	0.548808	+	0.835948i	$0.315082\pi$
$354$	0	0
$355$	0.569520	0.0302270
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.35717	−0.282740	−0.141370	−	0.989957i	$-0.545151\pi$
$359$	−5.35717	−0.282740	−0.141370	+	0.989957i	$0.545151\pi$
$360$	0	0
$361$	−8.87051	−0.466869
$362$	0	0
$363$	7.64215	0.401109
$364$	0	0
$365$	−1.97401	−0.103324
$366$	0	0
$367$	−11.5081	−0.600716	−0.300358	−	0.953826i	$-0.597106\pi$
$367$	−11.5081	−0.600716	−0.300358	+	0.953826i	$0.597106\pi$
$368$	0	0
$369$	−26.9737	−1.40420
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5.85404	0.303111	0.151555	−	0.988449i	$-0.451572\pi$
$373$	5.85404	0.303111	0.151555	+	0.988449i	$0.451572\pi$
$374$	0	0
$375$	−1.40501	−0.0725546
$376$	0	0
$377$	8.89066	0.457892
$378$	0	0
$379$	−0.0146945	−0.000754805 0	−0.000377402	−	1.00000i	$-0.500120\pi$
$379$	−0.0146945	−0.000754805 0	−0.000377402		1.00000i	$0.500120\pi$
$380$	0	0
$381$	−3.53635	−0.181173
$382$	0	0
$383$	−9.21982	−0.471111	−0.235555	−	0.971861i	$-0.575691\pi$
$383$	−9.21982	−0.471111	−0.235555	+	0.971861i	$0.575691\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−31.5304	−1.60278
$388$	0	0
$389$	13.7712	0.698230	0.349115	−	0.937080i	$-0.386482\pi$
$389$	13.7712	0.698230	0.349115	+	0.937080i	$0.386482\pi$
$390$	0	0
$391$	3.84786	0.194595
$392$	0	0
$393$	−9.68719	−0.488654
$394$	0	0
$395$	2.47958	0.124761
$396$	0	0
$397$	−2.09627	−0.105209	−0.0526043	−	0.998615i	$-0.516752\pi$
$397$	−2.09627	−0.105209	−0.0526043	+	0.998615i	$0.516752\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.1428	−0.806131	−0.403065	−	0.915171i	$-0.632055\pi$
$401$	−16.1428	−0.806131	−0.403065	+	0.915171i	$0.632055\pi$
$402$	0	0
$403$	−20.7419	−1.03323
$404$	0	0
$405$	−0.960817	−0.0477434
$406$	0	0
$407$	−1.78769	−0.0886126
$408$	0	0
$409$	9.85765	0.487429	0.243715	−	0.969847i	$-0.421634\pi$
$409$	9.85765	0.487429	0.243715	+	0.969847i	$0.421634\pi$
$410$	0	0
$411$	−7.25159	−0.357694
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.10632	0.0543069
$416$	0	0
$417$	−6.45666	−0.316184
$418$	0	0
$419$	−29.2448	−1.42870	−0.714352	−	0.699787i	$-0.753278\pi$
$419$	−29.2448	−1.42870	−0.714352	+	0.699787i	$0.753278\pi$
$420$	0	0
$421$	11.5615	0.563471	0.281735	−	0.959492i	$-0.409090\pi$
$421$	11.5615	0.563471	0.281735	+	0.959492i	$0.409090\pi$
$422$	0	0
$423$	0.356861	0.0173512
$424$	0	0
$425$	4.95979	0.240585
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.803627	0.0387995
$430$	0	0
$431$	−32.7294	−1.57652	−0.788259	−	0.615344i	$-0.789017\pi$
$431$	−32.7294	−1.57652	−0.788259	+	0.615344i	$0.789017\pi$
$432$	0	0
$433$	−2.84522	−0.136733	−0.0683663	−	0.997660i	$-0.521779\pi$
$433$	−2.84522	−0.136733	−0.0683663	+	0.997660i	$0.521779\pi$
$434$	0	0
$435$	−0.405278	−0.0194316
$436$	0	0
$437$	−12.2465	−0.585831
$438$	0	0
$439$	4.92953	0.235274	0.117637	−	0.993057i	$-0.462468\pi$
$439$	4.92953	0.235274	0.117637	+	0.993057i	$0.462468\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.3649	0.539964	0.269982	−	0.962865i	$-0.412982\pi$
$443$	11.3649	0.539964	0.269982	+	0.962865i	$0.412982\pi$
$444$	0	0
$445$	−0.516131	−0.0244670
$446$	0	0
$447$	2.78009	0.131494
$448$	0	0
$449$	5.21674	0.246193	0.123097	−	0.992395i	$-0.460717\pi$
$449$	5.21674	0.246193	0.123097	+	0.992395i	$0.460717\pi$
$450$	0	0
$451$	3.97478	0.187165
$452$	0	0
$453$	1.64622	0.0773460
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.0627	0.891717	0.445859	−	0.895103i	$-0.352898\pi$
$457$	19.0627	0.891717	0.445859	+	0.895103i	$0.352898\pi$
$458$	0	0
$459$	−3.87263	−0.180759
$460$	0	0
$461$	−39.6079	−1.84472	−0.922361	−	0.386329i	$-0.873743\pi$
$461$	−39.6079	−1.84472	−0.922361	+	0.386329i	$0.873743\pi$
$462$	0	0
$463$	1.70237	0.0791158	0.0395579	−	0.999217i	$-0.487405\pi$
$463$	1.70237	0.0791158	0.0395579	+	0.999217i	$0.487405\pi$
$464$	0	0
$465$	0.945511	0.0438470
$466$	0	0
$467$	−16.5730	−0.766908	−0.383454	−	0.923560i	$-0.625266\pi$
$467$	−16.5730	−0.766908	−0.383454	+	0.923560i	$0.625266\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.26103	0.242416
$472$	0	0
$473$	4.64624	0.213634
$474$	0	0
$475$	−15.7854	−0.724286
$476$	0	0
$477$	−17.7269	−0.811660
$478$	0	0
$479$	−24.5762	−1.12291	−0.561457	−	0.827506i	$-0.689759\pi$
$479$	−24.5762	−1.12291	−0.561457	+	0.827506i	$0.689759\pi$
$480$	0	0
$481$	14.9865	0.683324
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.97335	−0.135013
$486$	0	0
$487$	−2.04497	−0.0926663	−0.0463331	−	0.998926i	$-0.514754\pi$
$487$	−2.04497	−0.0926663	−0.0463331	+	0.998926i	$0.514754\pi$
$488$	0	0
$489$	−6.47456	−0.292790
$490$	0	0
$491$	13.1994	0.595680	0.297840	−	0.954616i	$-0.403734\pi$
$491$	13.1994	0.595680	0.297840	+	0.954616i	$0.403734\pi$
$492$	0	0
$493$	2.87291	0.129389
$494$	0	0
$495$	0.185453	0.00833548
$496$	0	0
$497$	0	0
$498$	0	0
$499$	29.6114	1.32559	0.662795	−	0.748801i	$-0.269370\pi$
$499$	29.6114	1.32559	0.662795	+	0.748801i	$0.269370\pi$
$500$	0	0
$501$	−2.99359	−0.133744
$502$	0	0
$503$	−11.7752	−0.525028	−0.262514	−	0.964928i	$-0.584552\pi$
$503$	−11.7752	−0.525028	−0.262514	+	0.964928i	$0.584552\pi$
$504$	0	0
$505$	−2.87722	−0.128034
$506$	0	0
$507$	2.40801	0.106943
$508$	0	0
$509$	−17.0112	−0.754010	−0.377005	−	0.926211i	$-0.623046\pi$
$509$	−17.0112	−0.754010	−0.377005	+	0.926211i	$0.623046\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	12.3254	0.544178
$514$	0	0
$515$	−1.94052	−0.0855097
$516$	0	0
$517$	−0.0525861	−0.00231273
$518$	0	0
$519$	−3.80414	−0.166983
$520$	0	0
$521$	3.37095	0.147684	0.0738420	−	0.997270i	$-0.476474\pi$
$521$	3.37095	0.147684	0.0738420	+	0.997270i	$0.476474\pi$
$522$	0	0
$523$	4.63433	0.202645	0.101322	−	0.994854i	$-0.467693\pi$
$523$	4.63433	0.202645	0.101322	+	0.994854i	$0.467693\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.70249	−0.291965
$528$	0	0
$529$	−8.19399	−0.356260
$530$	0	0
$531$	16.8185	0.729861
$532$	0	0
$533$	−33.3211	−1.44330
$534$	0	0
$535$	1.91242	0.0826812
$536$	0	0
$537$	7.17090	0.309447
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4.17894	0.179667	0.0898334	−	0.995957i	$-0.471367\pi$
$541$	4.17894	0.179667	0.0898334	+	0.995957i	$0.471367\pi$
$542$	0	0
$543$	−0.0268828	−0.00115365
$544$	0	0
$545$	−1.71907	−0.0736368
$546$	0	0
$547$	19.1368	0.818229	0.409114	−	0.912483i	$-0.365838\pi$
$547$	19.1368	0.818229	0.409114	+	0.912483i	$0.365838\pi$
$548$	0	0
$549$	13.6131	0.580993
$550$	0	0
$551$	−9.14357	−0.389529
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−0.683153	−0.0289982
$556$	0	0
$557$	16.6431	0.705190	0.352595	−	0.935776i	$-0.385299\pi$
$557$	16.6431	0.705190	0.352595	+	0.935776i	$0.385299\pi$
$558$	0	0
$559$	−38.9500	−1.64741
$560$	0	0
$561$	0.259682	0.0109638
$562$	0	0
$563$	27.9120	1.17635	0.588175	−	0.808733i	$-0.299846\pi$
$563$	27.9120	1.17635	0.588175	+	0.808733i	$0.299846\pi$
$564$	0	0
$565$	−2.17130	−0.0913474
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.09195	0.381154	0.190577	−	0.981672i	$-0.438964\pi$
$569$	9.09195	0.381154	0.190577	+	0.981672i	$0.438964\pi$
$570$	0	0
$571$	−35.2749	−1.47621	−0.738105	−	0.674686i	$-0.764279\pi$
$571$	−35.2749	−1.47621	−0.738105	+	0.674686i	$0.764279\pi$
$572$	0	0
$573$	17.0615	0.712753
$574$	0	0
$575$	19.0845	0.795881
$576$	0	0
$577$	−9.34435	−0.389011	−0.194505	−	0.980901i	$-0.562310\pi$
$577$	−9.34435	−0.389011	−0.194505	+	0.980901i	$0.562310\pi$
$578$	0	0
$579$	−8.99987	−0.374022
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.61219	0.108186
$584$	0	0
$585$	−1.55468	−0.0642779
$586$	0	0
$587$	−13.7933	−0.569311	−0.284655	−	0.958630i	$-0.591879\pi$
$587$	−13.7933	−0.569311	−0.284655	+	0.958630i	$0.591879\pi$
$588$	0	0
$589$	21.3319	0.878966
$590$	0	0
$591$	−7.01710	−0.288645
$592$	0	0
$593$	−46.7655	−1.92043	−0.960214	−	0.279264i	$-0.909910\pi$
$593$	−46.7655	−1.92043	−0.960214	+	0.279264i	$0.909910\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.70326	0.192491
$598$	0	0
$599$	−4.08619	−0.166957	−0.0834786	−	0.996510i	$-0.526603\pi$
$599$	−4.08619	−0.166957	−0.0834786	+	0.996510i	$0.526603\pi$
$600$	0	0
$601$	−1.43949	−0.0587180	−0.0293590	−	0.999569i	$-0.509347\pi$
$601$	−1.43949	−0.0587180	−0.0293590	+	0.999569i	$0.509347\pi$
$602$	0	0
$603$	−2.58695	−0.105349
$604$	0	0
$605$	2.17858	0.0885717
$606$	0	0
$607$	−33.0234	−1.34038	−0.670189	−	0.742190i	$-0.733787\pi$
$607$	−33.0234	−1.34038	−0.670189	+	0.742190i	$0.733787\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.440837	0.0178343
$612$	0	0
$613$	29.3637	1.18599	0.592994	−	0.805207i	$-0.297946\pi$
$613$	29.3637	1.18599	0.592994	+	0.805207i	$0.297946\pi$
$614$	0	0
$615$	1.51893	0.0612492
$616$	0	0
$617$	−43.3541	−1.74537	−0.872685	−	0.488284i	$-0.837623\pi$
$617$	−43.3541	−1.74537	−0.872685	+	0.488284i	$0.837623\pi$
$618$	0	0
$619$	−14.9088	−0.599235	−0.299618	−	0.954059i	$-0.596859\pi$
$619$	−14.9088	−0.599235	−0.299618	+	0.954059i	$0.596859\pi$
$620$	0	0
$621$	−14.9013	−0.597970
$622$	0	0
$623$	0	0
$624$	0	0
$625$	24.3984	0.975936
$626$	0	0
$627$	−0.826487	−0.0330067
$628$	0	0
$629$	4.84269	0.193091
$630$	0	0
$631$	26.3812	1.05022	0.525110	−	0.851034i	$-0.324024\pi$
$631$	26.3812	1.05022	0.525110	+	0.851034i	$0.324024\pi$
$632$	0	0
$633$	−6.77727	−0.269372
$634$	0	0
$635$	−1.00812	−0.0400061
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7.11457	0.281448
$640$	0	0
$641$	−30.8365	−1.21797	−0.608985	−	0.793182i	$-0.708423\pi$
$641$	−30.8365	−1.21797	−0.608985	+	0.793182i	$0.708423\pi$
$642$	0	0
$643$	−30.1699	−1.18979	−0.594893	−	0.803805i	$-0.702805\pi$
$643$	−30.1699	−1.18979	−0.594893	+	0.803805i	$0.702805\pi$
$644$	0	0
$645$	1.77552	0.0699112
$646$	0	0
$647$	26.0146	1.02274	0.511370	−	0.859360i	$-0.329138\pi$
$647$	26.0146	1.02274	0.511370	+	0.859360i	$0.329138\pi$
$648$	0	0
$649$	−2.47833	−0.0972830
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.75285	−0.107727	−0.0538637	−	0.998548i	$-0.517154\pi$
$653$	−2.75285	−0.107727	−0.0538637	+	0.998548i	$0.517154\pi$
$654$	0	0
$655$	−2.76156	−0.107903
$656$	0	0
$657$	−24.6598	−0.962070
$658$	0	0
$659$	17.9141	0.697834	0.348917	−	0.937154i	$-0.386549\pi$
$659$	17.9141	0.697834	0.348917	+	0.937154i	$0.386549\pi$
$660$	0	0
$661$	30.3762	1.18150	0.590750	−	0.806855i	$-0.298832\pi$
$661$	30.3762	1.18150	0.590750	+	0.806855i	$0.298832\pi$
$662$	0	0
$663$	−2.17695	−0.0845458
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.0545	0.428034
$668$	0	0
$669$	−10.0119	−0.387084
$670$	0	0
$671$	−2.00599	−0.0774405
$672$	0	0
$673$	−45.4261	−1.75105	−0.875523	−	0.483176i	$-0.839483\pi$
$673$	−45.4261	−1.75105	−0.875523	+	0.483176i	$0.839483\pi$
$674$	0	0
$675$	−19.2074	−0.739294
$676$	0	0
$677$	−15.4002	−0.591879	−0.295939	−	0.955207i	$-0.595633\pi$
$677$	−15.4002	−0.591879	−0.295939	+	0.955207i	$0.595633\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	7.64155	0.292825
$682$	0	0
$683$	−38.9298	−1.48961	−0.744803	−	0.667284i	$-0.767457\pi$
$683$	−38.9298	−1.48961	−0.744803	+	0.667284i	$0.767457\pi$
$684$	0	0
$685$	−2.06724	−0.0789850
$686$	0	0
$687$	−11.3139	−0.431652
$688$	0	0
$689$	−21.8984	−0.834261
$690$	0	0
$691$	34.6743	1.31907	0.659537	−	0.751672i	$-0.270752\pi$
$691$	34.6743	1.31907	0.659537	+	0.751672i	$0.270752\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.84062	−0.0698188
$696$	0	0
$697$	−10.7673	−0.407841
$698$	0	0
$699$	−3.45296	−0.130603
$700$	0	0
$701$	−42.7812	−1.61582	−0.807912	−	0.589304i	$-0.799402\pi$
$701$	−42.7812	−1.61582	−0.807912	+	0.589304i	$0.799402\pi$
$702$	0	0
$703$	−15.4128	−0.581304
$704$	0	0
$705$	−0.0200954	−0.000756836 0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−9.23697	−0.346902	−0.173451	−	0.984843i	$-0.555492\pi$
$709$	−9.23697	−0.346902	−0.173451	+	0.984843i	$0.555492\pi$
$710$	0	0
$711$	30.9754	1.16167
$712$	0	0
$713$	−25.7902	−0.965851
$714$	0	0
$715$	0.229093	0.00856759
$716$	0	0
$717$	−19.9071	−0.743444
$718$	0	0
$719$	−30.5552	−1.13952	−0.569759	−	0.821812i	$-0.692963\pi$
$719$	−30.5552	−1.13952	−0.569759	+	0.821812i	$0.692963\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−8.50538	−0.316318
$724$	0	0
$725$	14.2490	0.529195
$726$	0	0
$727$	−45.8821	−1.70167	−0.850837	−	0.525430i	$-0.823904\pi$
$727$	−45.8821	−1.70167	−0.850837	+	0.525430i	$0.823904\pi$
$728$	0	0
$729$	−3.83007	−0.141854
$730$	0	0
$731$	−12.5862	−0.465519
$732$	0	0
$733$	24.5005	0.904948	0.452474	−	0.891778i	$-0.350542\pi$
$733$	24.5005	0.904948	0.452474	+	0.891778i	$0.350542\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.381206	0.0140419
$738$	0	0
$739$	−30.1296	−1.10833	−0.554167	−	0.832405i	$-0.686963\pi$
$739$	−30.1296	−1.10833	−0.554167	+	0.832405i	$0.686963\pi$
$740$	0	0
$741$	6.92856	0.254527
$742$	0	0
$743$	−37.7607	−1.38530	−0.692652	−	0.721272i	$-0.743558\pi$
$743$	−37.7607	−1.38530	−0.692652	+	0.721272i	$0.743558\pi$
$744$	0	0
$745$	0.792529	0.0290360
$746$	0	0
$747$	13.8204	0.505660
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7.56429	0.276025	0.138013	−	0.990430i	$-0.455929\pi$
$751$	7.56429	0.276025	0.138013	+	0.990430i	$0.455929\pi$
$752$	0	0
$753$	−4.77097	−0.173864
$754$	0	0
$755$	0.469293	0.0170793
$756$	0	0
$757$	−4.31815	−0.156946	−0.0784728	−	0.996916i	$-0.525004\pi$
$757$	−4.31815	−0.156946	−0.0784728	+	0.996916i	$0.525004\pi$
$758$	0	0
$759$	0.999221	0.0362694
$760$	0	0
$761$	26.4300	0.958088	0.479044	−	0.877791i	$-0.340984\pi$
$761$	26.4300	0.958088	0.479044	+	0.877791i	$0.340984\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.502375	−0.0181634
$766$	0	0
$767$	20.7762	0.750185
$768$	0	0
$769$	47.6141	1.71701	0.858503	−	0.512808i	$-0.171395\pi$
$769$	47.6141	1.71701	0.858503	+	0.512808i	$0.171395\pi$
$770$	0	0
$771$	7.23567	0.260586
$772$	0	0
$773$	−9.37262	−0.337110	−0.168555	−	0.985692i	$-0.553910\pi$
$773$	−9.37262	−0.337110	−0.168555	+	0.985692i	$0.553910\pi$
$774$	0	0
$775$	−33.2429	−1.19412
$776$	0	0
$777$	0	0
$778$	0	0
$779$	34.2690	1.22781
$780$	0	0
$781$	−1.04838	−0.0375141
$782$	0	0
$783$	−11.1257	−0.397601
$784$	0	0
$785$	1.49978	0.0535295
$786$	0	0
$787$	20.8406	0.742886	0.371443	−	0.928456i	$-0.378863\pi$
$787$	20.8406	0.742886	0.371443	+	0.928456i	$0.378863\pi$
$788$	0	0
$789$	−16.0816	−0.572522
$790$	0	0
$791$	0	0
$792$	0	0
$793$	16.8165	0.597172
$794$	0	0
$795$	0.998229	0.0354035
$796$	0	0
$797$	−34.2311	−1.21253	−0.606264	−	0.795264i	$-0.707332\pi$
$797$	−34.2311	−1.21253	−0.606264	+	0.795264i	$0.707332\pi$
$798$	0	0
$799$	0.142451	0.00503955
$800$	0	0
$801$	−6.44763	−0.227816
$802$	0	0
$803$	3.63380	0.128234
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.71992	−0.306956
$808$	0	0
$809$	−5.51489	−0.193893	−0.0969466	−	0.995290i	$-0.530908\pi$
$809$	−5.51489	−0.193893	−0.0969466	+	0.995290i	$0.530908\pi$
$810$	0	0
$811$	33.1850	1.16528	0.582642	−	0.812729i	$-0.302019\pi$
$811$	33.1850	1.16528	0.582642	+	0.812729i	$0.302019\pi$
$812$	0	0
$813$	−8.31128	−0.291489
$814$	0	0
$815$	−1.84573	−0.0646530
$816$	0	0
$817$	40.0580	1.40145
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.6110	1.69654	0.848268	−	0.529568i	$-0.177646\pi$
$821$	48.6110	1.69654	0.848268	+	0.529568i	$0.177646\pi$
$822$	0	0
$823$	−9.33485	−0.325392	−0.162696	−	0.986676i	$-0.552019\pi$
$823$	−9.33485	−0.325392	−0.162696	+	0.986676i	$0.552019\pi$
$824$	0	0
$825$	1.28797	0.0448413
$826$	0	0
$827$	−1.89827	−0.0660095	−0.0330047	−	0.999455i	$-0.510508\pi$
$827$	−1.89827	−0.0660095	−0.0330047	+	0.999455i	$0.510508\pi$
$828$	0	0
$829$	14.5849	0.506556	0.253278	−	0.967394i	$-0.418491\pi$
$829$	14.5849	0.506556	0.253278	+	0.967394i	$0.418491\pi$
$830$	0	0
$831$	17.3516	0.601921
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−0.853394	−0.0295329
$836$	0	0
$837$	25.9563	0.897179
$838$	0	0
$839$	−11.0257	−0.380648	−0.190324	−	0.981721i	$-0.560954\pi$
$839$	−11.0257	−0.380648	−0.190324	+	0.981721i	$0.560954\pi$
$840$	0	0
$841$	−20.7464	−0.715393
$842$	0	0
$843$	18.3813	0.633086
$844$	0	0
$845$	0.686459	0.0236149
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.94952	−0.135547
$850$	0	0
$851$	18.6340	0.638765
$852$	0	0
$853$	11.7576	0.402573	0.201286	−	0.979532i	$-0.435488\pi$
$853$	11.7576	0.402573	0.201286	+	0.979532i	$0.435488\pi$
$854$	0	0
$855$	1.59890	0.0546812
$856$	0	0
$857$	37.4622	1.27969	0.639843	−	0.768506i	$-0.278999\pi$
$857$	37.4622	1.27969	0.639843	+	0.768506i	$0.278999\pi$
$858$	0	0
$859$	−5.04132	−0.172008	−0.0860038	−	0.996295i	$-0.527410\pi$
$859$	−5.04132	−0.172008	−0.0860038	+	0.996295i	$0.527410\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36.4021	1.23914	0.619570	−	0.784941i	$-0.287307\pi$
$863$	36.4021	1.23914	0.619570	+	0.784941i	$0.287307\pi$
$864$	0	0
$865$	−1.08446	−0.0368727
$866$	0	0
$867$	−0.703456	−0.0238906
$868$	0	0
$869$	−4.56446	−0.154839
$870$	0	0
$871$	−3.19570	−0.108282
$872$	0	0
$873$	−37.1438	−1.25713
$874$	0	0
$875$	0	0
$876$	0	0
$877$	51.7835	1.74860	0.874302	−	0.485383i	$-0.161320\pi$
$877$	51.7835	1.74860	0.874302	+	0.485383i	$0.161320\pi$
$878$	0	0
$879$	−13.0536	−0.440286
$880$	0	0
$881$	−35.7933	−1.20591	−0.602954	−	0.797776i	$-0.706010\pi$
$881$	−35.7933	−1.20591	−0.602954	+	0.797776i	$0.706010\pi$
$882$	0	0
$883$	28.5377	0.960371	0.480186	−	0.877167i	$-0.340569\pi$
$883$	28.5377	0.960371	0.480186	+	0.877167i	$0.340569\pi$
$884$	0	0
$885$	−0.947075	−0.0318356
$886$	0	0
$887$	2.54617	0.0854919	0.0427459	−	0.999086i	$-0.486389\pi$
$887$	2.54617	0.0854919	0.0427459	+	0.999086i	$0.486389\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.76869	0.0592534
$892$	0	0
$893$	−0.453377	−0.0151717
$894$	0	0
$895$	2.04424	0.0683313
$896$	0	0
$897$	−8.37661	−0.279687
$898$	0	0
$899$	−19.2556	−0.642212
$900$	0	0
$901$	−7.07619	−0.235742
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.00766358	−0.000254746 0
$906$	0	0
$907$	−36.7520	−1.22033	−0.610165	−	0.792274i	$-0.708897\pi$
$907$	−36.7520	−1.22033	−0.610165	+	0.792274i	$0.708897\pi$
$908$	0	0
$909$	−35.9428	−1.19215
$910$	0	0
$911$	−18.1727	−0.602088	−0.301044	−	0.953610i	$-0.597335\pi$
$911$	−18.1727	−0.602088	−0.301044	+	0.953610i	$0.597335\pi$
$912$	0	0
$913$	−2.03653	−0.0673994
$914$	0	0
$915$	−0.766574	−0.0253422
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−3.85580	−0.127191	−0.0635956	−	0.997976i	$-0.520257\pi$
$919$	−3.85580	−0.127191	−0.0635956	+	0.997976i	$0.520257\pi$
$920$	0	0
$921$	13.3018	0.438310
$922$	0	0
$923$	8.78875	0.289285
$924$	0	0
$925$	24.0187	0.789731
$926$	0	0
$927$	−24.2415	−0.796194
$928$	0	0
$929$	−7.24726	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$929$	−7.24726	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	20.5103	0.671478
$934$	0	0
$935$	0.0740286	0.00242099
$936$	0	0
$937$	31.7443	1.03704	0.518520	−	0.855065i	$-0.326483\pi$
$937$	31.7443	1.03704	0.518520	+	0.855065i	$0.326483\pi$
$938$	0	0
$939$	−18.5420	−0.605095
$940$	0	0
$941$	23.6609	0.771322	0.385661	−	0.922641i	$-0.373973\pi$
$941$	23.6609	0.771322	0.385661	+	0.922641i	$0.373973\pi$
$942$	0	0
$943$	−41.4311	−1.34918
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.0969	−0.653062	−0.326531	−	0.945187i	$-0.605880\pi$
$947$	−20.0969	−0.653062	−0.326531	+	0.945187i	$0.605880\pi$
$948$	0	0
$949$	−30.4627	−0.988860
$950$	0	0
$951$	−14.7703	−0.478960
$952$	0	0
$953$	27.5224	0.891538	0.445769	−	0.895148i	$-0.352930\pi$
$953$	27.5224	0.891538	0.445769	+	0.895148i	$0.352930\pi$
$954$	0	0
$955$	4.86378	0.157388
$956$	0	0
$957$	0.746044	0.0241162
$958$	0	0
$959$	0	0
$960$	0	0
$961$	13.9234	0.449140
$962$	0	0
$963$	23.8904	0.769858
$964$	0	0
$965$	−2.56562	−0.0825904
$966$	0	0
$967$	−11.5594	−0.371726	−0.185863	−	0.982576i	$-0.559508\pi$
$967$	−11.5594	−0.371726	−0.185863	+	0.982576i	$0.559508\pi$
$968$	0	0
$969$	2.23888	0.0719231
$970$	0	0
$971$	−6.13380	−0.196843	−0.0984215	−	0.995145i	$-0.531379\pi$
$971$	−6.13380	−0.196843	−0.0984215	+	0.995145i	$0.531379\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−10.7972	−0.345788
$976$	0	0
$977$	11.5275	0.368797	0.184399	−	0.982852i	$-0.440966\pi$
$977$	11.5275	0.368797	0.184399	+	0.982852i	$0.440966\pi$
$978$	0	0
$979$	0.950105	0.0303655
$980$	0	0
$981$	−21.4750	−0.685644
$982$	0	0
$983$	−55.7362	−1.77771	−0.888854	−	0.458191i	$-0.848498\pi$
$983$	−55.7362	−1.77771	−0.888854	+	0.458191i	$0.848498\pi$
$984$	0	0
$985$	−2.00039	−0.0637377
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48.4300	−1.53999
$990$	0	0
$991$	−12.1311	−0.385359	−0.192679	−	0.981262i	$-0.561718\pi$
$991$	−12.1311	−0.385359	−0.192679	+	0.981262i	$0.561718\pi$
$992$	0	0
$993$	−21.2488	−0.674310
$994$	0	0
$995$	1.34077	0.0425054
$996$	0	0
$997$	51.3299	1.62563	0.812817	−	0.582520i	$-0.197933\pi$
$997$	51.3299	1.62563	0.812817	+	0.582520i	$0.197933\pi$
$998$	0	0
$999$	−18.7540	−0.593349

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.u.1.3	yes	8
7.6	odd	2		3332.2.a.t.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.t.1.6	✓	8	7.6	odd	2
3332.2.a.u.1.3	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)

\(f(q)\)	\(=\)	\(q-0.703456 q^{3} -0.200537 q^{5} -2.50515 q^{9} +0.369152 q^{11} -3.09466 q^{13} +0.141069 q^{15} -1.00000 q^{17} +3.18269 q^{19} -3.84786 q^{23} -4.95979 q^{25} +3.87263 q^{27} -2.87291 q^{29} +6.70249 q^{31} -0.259682 q^{33} -4.84269 q^{37} +2.17695 q^{39} +10.7673 q^{41} +12.5862 q^{43} +0.502375 q^{45} -0.142451 q^{47} +0.703456 q^{51} +7.07619 q^{53} -0.0740286 q^{55} -2.23888 q^{57} -6.71357 q^{59} -5.43405 q^{61} +0.620592 q^{65} +1.03265 q^{67} +2.70680 q^{69} -2.83998 q^{71} +9.84364 q^{73} +3.48899 q^{75} -12.3647 q^{79} +4.79123 q^{81} -5.51678 q^{83} +0.200537 q^{85} +2.02096 q^{87} +2.57375 q^{89} -4.71490 q^{93} -0.638245 q^{95} +14.8270 q^{97} -0.924782 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 4 q^{5} + 8 q^{9} - 4 q^{11} + 20 q^{13} + 12 q^{15} - 8 q^{17} + 8 q^{19} + 4 q^{23} + 8 q^{25} + 28 q^{27} - 16 q^{29} - 8 q^{31} + 16 q^{33} + 8 q^{37} + 20 q^{39} + 12 q^{41} - 4 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 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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