LMFDB - Embedded newform 3332.2.a.t.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:	$1$
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			$1.27952$ 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)

$f(q)$	$=$	$q-1.27952 q^{3} -4.13895 q^{5} -1.36282 q^{9} +O(q^{10})$	$q-1.27952 q^{3} -4.13895 q^{5} -1.36282 q^{9} +1.53680 q^{11} -0.0160623 q^{13} +5.29588 q^{15} +1.00000 q^{17} -0.519822 q^{19} +0.958750 q^{23} +12.1309 q^{25} +5.58233 q^{27} +5.79146 q^{29} -1.29639 q^{31} -1.96637 q^{33} +2.76827 q^{37} +0.0205521 q^{39} +0.769697 q^{41} -4.93770 q^{43} +5.64063 q^{45} +1.60825 q^{47} -1.27952 q^{51} +5.00816 q^{53} -6.36074 q^{55} +0.665125 q^{57} -5.50390 q^{59} -14.5818 q^{61} +0.0664811 q^{65} +5.13657 q^{67} -1.22674 q^{69} +13.4293 q^{71} +3.16522 q^{73} -15.5218 q^{75} -10.3220 q^{79} -3.05428 q^{81} -16.1399 q^{83} -4.13895 q^{85} -7.41032 q^{87} +10.5848 q^{89} +1.65876 q^{93} +2.15152 q^{95} -10.0542 q^{97} -2.09438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 4 q^{5} + 8 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 - 4 * q^5 + 8 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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
$3$	−1.27952	−0.738734	−0.369367	−	0.929284i	$-0.620425\pi$
$3$	−1.27952	−0.738734	−0.369367	+	0.929284i	$0.620425\pi$
$4$	0	0
$5$	−4.13895	−1.85099	−0.925497	−	0.378756i	$-0.876352\pi$
$5$	−4.13895	−1.85099	−0.925497	+	0.378756i	$0.876352\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.36282	−0.454272
$10$	0	0
$11$	1.53680	0.463363	0.231681	−	0.972792i	$-0.425577\pi$
$11$	1.53680	0.463363	0.231681	+	0.972792i	$0.425577\pi$
$12$	0	0
$13$	−0.0160623	−0.00445489	−0.00222744	−	0.999998i	$-0.500709\pi$
$13$	−0.0160623	−0.00445489	−0.00222744	+	0.999998i	$0.500709\pi$
$14$	0	0
$15$	5.29588	1.36739
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−0.519822	−0.119255	−0.0596277	−	0.998221i	$-0.518991\pi$
$19$	−0.519822	−0.119255	−0.0596277	+	0.998221i	$0.518991\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.958750	0.199913	0.0999566	−	0.994992i	$-0.468130\pi$
$23$	0.958750	0.199913	0.0999566	+	0.994992i	$0.468130\pi$
$24$	0	0
$25$	12.1309	2.42618
$26$	0	0
$27$	5.58233	1.07432
$28$	0	0
$29$	5.79146	1.07545	0.537724	−	0.843121i	$-0.319284\pi$
$29$	5.79146	1.07545	0.537724	+	0.843121i	$0.319284\pi$
$30$	0	0
$31$	−1.29639	−0.232838	−0.116419	−	0.993200i	$-0.537141\pi$
$31$	−1.29639	−0.232838	−0.116419	+	0.993200i	$0.537141\pi$
$32$	0	0
$33$	−1.96637	−0.342302
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.76827	0.455101	0.227550	−	0.973766i	$-0.426928\pi$
$37$	2.76827	0.455101	0.227550	+	0.973766i	$0.426928\pi$
$38$	0	0
$39$	0.0205521	0.00329098
$40$	0	0
$41$	0.769697	0.120207	0.0601033	−	0.998192i	$-0.480857\pi$
$41$	0.769697	0.120207	0.0601033	+	0.998192i	$0.480857\pi$
$42$	0	0
$43$	−4.93770	−0.752992	−0.376496	−	0.926418i	$-0.622871\pi$
$43$	−4.93770	−0.752992	−0.376496	+	0.926418i	$0.622871\pi$
$44$	0	0
$45$	5.64063	0.840855
$46$	0	0
$47$	1.60825	0.234587	0.117294	−	0.993097i	$-0.462578\pi$
$47$	1.60825	0.234587	0.117294	+	0.993097i	$0.462578\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1.27952	−0.179169
$52$	0	0
$53$	5.00816	0.687923	0.343962	−	0.938984i	$-0.388231\pi$
$53$	5.00816	0.687923	0.343962	+	0.938984i	$0.388231\pi$
$54$	0	0
$55$	−6.36074	−0.857682
$56$	0	0
$57$	0.665125	0.0880980
$58$	0	0
$59$	−5.50390	−0.716547	−0.358273	−	0.933617i	$-0.616634\pi$
$59$	−5.50390	−0.716547	−0.358273	+	0.933617i	$0.616634\pi$
$60$	0	0
$61$	−14.5818	−1.86701	−0.933506	−	0.358562i	$-0.883267\pi$
$61$	−14.5818	−1.86701	−0.933506	+	0.358562i	$0.883267\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.0664811	0.00824597
$66$	0	0
$67$	5.13657	0.627532	0.313766	−	0.949500i	$-0.398409\pi$
$67$	5.13657	0.627532	0.313766	+	0.949500i	$0.398409\pi$
$68$	0	0
$69$	−1.22674	−0.147683
$70$	0	0
$71$	13.4293	1.59377	0.796883	−	0.604133i	$-0.206481\pi$
$71$	13.4293	1.59377	0.796883	+	0.604133i	$0.206481\pi$
$72$	0	0
$73$	3.16522	0.370461	0.185230	−	0.982695i	$-0.440697\pi$
$73$	3.16522	0.370461	0.185230	+	0.982695i	$0.440697\pi$
$74$	0	0
$75$	−15.5218	−1.79230
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.3220	−1.16131	−0.580657	−	0.814148i	$-0.697204\pi$
$79$	−10.3220	−1.16131	−0.580657	+	0.814148i	$0.697204\pi$
$80$	0	0
$81$	−3.05428	−0.339364
$82$	0	0
$83$	−16.1399	−1.77159	−0.885793	−	0.464080i	$-0.846385\pi$
$83$	−16.1399	−1.77159	−0.885793	+	0.464080i	$0.846385\pi$
$84$	0	0
$85$	−4.13895	−0.448932
$86$	0	0
$87$	−7.41032	−0.794470
$88$	0	0
$89$	10.5848	1.12199	0.560995	−	0.827819i	$-0.310419\pi$
$89$	10.5848	1.12199	0.560995	+	0.827819i	$0.310419\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.65876	0.172005
$94$	0	0
$95$	2.15152	0.220741
$96$	0	0
$97$	−10.0542	−1.02085	−0.510427	−	0.859921i	$-0.670513\pi$
$97$	−10.0542	−1.02085	−0.510427	+	0.859921i	$0.670513\pi$
$98$	0	0
$99$	−2.09438	−0.210493
$100$	0	0
$101$	11.5236	1.14665	0.573323	−	0.819330i	$-0.305654\pi$
$101$	11.5236	1.14665	0.573323	+	0.819330i	$0.305654\pi$
$102$	0	0
$103$	7.09694	0.699283	0.349641	−	0.936884i	$-0.386303\pi$
$103$	7.09694	0.699283	0.349641	+	0.936884i	$0.386303\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.9252	−1.24953	−0.624764	−	0.780814i	$-0.714805\pi$
$107$	−12.9252	−1.24953	−0.624764	+	0.780814i	$0.714805\pi$
$108$	0	0
$109$	−8.79060	−0.841987	−0.420993	−	0.907064i	$-0.638318\pi$
$109$	−8.79060	−0.841987	−0.420993	+	0.907064i	$0.638318\pi$
$110$	0	0
$111$	−3.54207	−0.336198
$112$	0	0
$113$	−8.29520	−0.780347	−0.390174	−	0.920741i	$-0.627585\pi$
$113$	−8.29520	−0.780347	−0.390174	+	0.920741i	$0.627585\pi$
$114$	0	0
$115$	−3.96822	−0.370038
$116$	0	0
$117$	0.0218900	0.00202373
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.63824	−0.785295
$122$	0	0
$123$	−0.984846	−0.0888006
$124$	0	0
$125$	−29.5144	−2.63984
$126$	0	0
$127$	11.1242	0.987110	0.493555	−	0.869715i	$-0.335697\pi$
$127$	11.1242	0.987110	0.493555	+	0.869715i	$0.335697\pi$
$128$	0	0
$129$	6.31790	0.556260
$130$	0	0
$131$	20.2010	1.76497	0.882486	−	0.470338i	$-0.155868\pi$
$131$	20.2010	1.76497	0.882486	+	0.470338i	$0.155868\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−23.1050	−1.98856
$136$	0	0
$137$	17.9159	1.53066	0.765331	−	0.643637i	$-0.222575\pi$
$137$	17.9159	1.53066	0.765331	+	0.643637i	$0.222575\pi$
$138$	0	0
$139$	0.0526564	0.00446626	0.00223313	−	0.999998i	$-0.499289\pi$
$139$	0.0526564	0.00446626	0.00223313	+	0.999998i	$0.499289\pi$
$140$	0	0
$141$	−2.05779	−0.173297
$142$	0	0
$143$	−0.0246846	−0.00206423
$144$	0	0
$145$	−23.9706	−1.99065
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.18629	0.179108	0.0895538	−	0.995982i	$-0.471456\pi$
$149$	2.18629	0.179108	0.0895538	+	0.995982i	$0.471456\pi$
$150$	0	0
$151$	−17.9310	−1.45920	−0.729601	−	0.683873i	$-0.760294\pi$
$151$	−17.9310	−1.45920	−0.729601	+	0.683873i	$0.760294\pi$
$152$	0	0
$153$	−1.36282	−0.110177
$154$	0	0
$155$	5.36567	0.430981
$156$	0	0
$157$	1.46355	0.116804	0.0584020	−	0.998293i	$-0.481399\pi$
$157$	1.46355	0.116804	0.0584020	+	0.998293i	$0.481399\pi$
$158$	0	0
$159$	−6.40806	−0.508192
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.5954	−1.29986	−0.649928	−	0.759996i	$-0.725201\pi$
$163$	−16.5954	−1.29986	−0.649928	+	0.759996i	$0.725201\pi$
$164$	0	0
$165$	8.13872	0.633599
$166$	0	0
$167$	−13.3783	−1.03524	−0.517621	−	0.855610i	$-0.673182\pi$
$167$	−13.3783	−1.03524	−0.517621	+	0.855610i	$0.673182\pi$
$168$	0	0
$169$	−12.9997	−0.999980
$170$	0	0
$171$	0.708423	0.0541744
$172$	0	0
$173$	−3.90950	−0.297234	−0.148617	−	0.988895i	$-0.547482\pi$
$173$	−3.90950	−0.297234	−0.148617	+	0.988895i	$0.547482\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.04237	0.529337
$178$	0	0
$179$	−24.4163	−1.82496	−0.912480	−	0.409121i	$-0.865835\pi$
$179$	−24.4163	−1.82496	−0.912480	+	0.409121i	$0.865835\pi$
$180$	0	0
$181$	−15.2516	−1.13364	−0.566821	−	0.823841i	$-0.691827\pi$
$181$	−15.2516	−1.13364	−0.566821	+	0.823841i	$0.691827\pi$
$182$	0	0
$183$	18.6578	1.37922
$184$	0	0
$185$	−11.4577	−0.842389
$186$	0	0
$187$	1.53680	0.112382
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.35507	0.0980492	0.0490246	−	0.998798i	$-0.484389\pi$
$191$	1.35507	0.0980492	0.0490246	+	0.998798i	$0.484389\pi$
$192$	0	0
$193$	−21.5938	−1.55436	−0.777178	−	0.629281i	$-0.783349\pi$
$193$	−21.5938	−1.55436	−0.777178	+	0.629281i	$0.783349\pi$
$194$	0	0
$195$	−0.0850642	−0.00609158
$196$	0	0
$197$	0.342141	0.0243765	0.0121883	−	0.999926i	$-0.496120\pi$
$197$	0.342141	0.0243765	0.0121883	+	0.999926i	$0.496120\pi$
$198$	0	0
$199$	18.5461	1.31470	0.657348	−	0.753587i	$-0.271678\pi$
$199$	18.5461	1.31470	0.657348	+	0.753587i	$0.271678\pi$
$200$	0	0
$201$	−6.57237	−0.463579
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.18574	−0.222501
$206$	0	0
$207$	−1.30660	−0.0908150
$208$	0	0
$209$	−0.798863	−0.0552585
$210$	0	0
$211$	3.63388	0.250166	0.125083	−	0.992146i	$-0.460080\pi$
$211$	3.63388	0.250166	0.125083	+	0.992146i	$0.460080\pi$
$212$	0	0
$213$	−17.1831	−1.17737
$214$	0	0
$215$	20.4369	1.39378
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.04997	−0.273672
$220$	0	0
$221$	−0.0160623	−0.00108047
$222$	0	0
$223$	22.8440	1.52975	0.764875	−	0.644179i	$-0.222801\pi$
$223$	22.8440	1.52975	0.764875	+	0.644179i	$0.222801\pi$
$224$	0	0
$225$	−16.5322	−1.10215
$226$	0	0
$227$	−6.58399	−0.436995	−0.218497	−	0.975838i	$-0.570116\pi$
$227$	−6.58399	−0.436995	−0.218497	+	0.975838i	$0.570116\pi$
$228$	0	0
$229$	−5.61189	−0.370844	−0.185422	−	0.982659i	$-0.559365\pi$
$229$	−5.61189	−0.370844	−0.185422	+	0.982659i	$0.559365\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.8916	−1.23763	−0.618817	−	0.785536i	$-0.712388\pi$
$233$	−18.8916	−1.23763	−0.618817	+	0.785536i	$0.712388\pi$
$234$	0	0
$235$	−6.65646	−0.434219
$236$	0	0
$237$	13.2072	0.857902
$238$	0	0
$239$	6.79748	0.439693	0.219846	−	0.975535i	$-0.429444\pi$
$239$	6.79748	0.439693	0.219846	+	0.975535i	$0.429444\pi$
$240$	0	0
$241$	−15.8682	−1.02216	−0.511082	−	0.859532i	$-0.670755\pi$
$241$	−15.8682	−1.02216	−0.511082	+	0.859532i	$0.670755\pi$
$242$	0	0
$243$	−12.8390	−0.823620
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.00834956	0.000531270 0
$248$	0	0
$249$	20.6514	1.30873
$250$	0	0
$251$	8.71087	0.549825	0.274913	−	0.961469i	$-0.411351\pi$
$251$	8.71087	0.549825	0.274913	+	0.961469i	$0.411351\pi$
$252$	0	0
$253$	1.47341	0.0926324
$254$	0	0
$255$	5.29588	0.331641
$256$	0	0
$257$	7.07610	0.441395	0.220697	−	0.975342i	$-0.429167\pi$
$257$	7.07610	0.441395	0.220697	+	0.975342i	$0.429167\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−7.89271	−0.488546
$262$	0	0
$263$	18.4753	1.13923	0.569617	−	0.821910i	$-0.307092\pi$
$263$	18.4753	1.13923	0.569617	+	0.821910i	$0.307092\pi$
$264$	0	0
$265$	−20.7285	−1.27334
$266$	0	0
$267$	−13.5435	−0.828851
$268$	0	0
$269$	−18.8617	−1.15002	−0.575010	−	0.818146i	$-0.695002\pi$
$269$	−18.8617	−1.15002	−0.575010	+	0.818146i	$0.695002\pi$
$270$	0	0
$271$	29.0252	1.76316	0.881579	−	0.472037i	$-0.156481\pi$
$271$	29.0252	1.76316	0.881579	+	0.472037i	$0.156481\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.6428	1.12420
$276$	0	0
$277$	−0.756258	−0.0454391	−0.0227196	−	0.999742i	$-0.507232\pi$
$277$	−0.756258	−0.0454391	−0.0227196	+	0.999742i	$0.507232\pi$
$278$	0	0
$279$	1.76674	0.105772
$280$	0	0
$281$	20.2702	1.20922	0.604609	−	0.796523i	$-0.293330\pi$
$281$	20.2702	1.20922	0.604609	+	0.796523i	$0.293330\pi$
$282$	0	0
$283$	10.9667	0.651902	0.325951	−	0.945387i	$-0.394316\pi$
$283$	10.9667	0.651902	0.325951	+	0.945387i	$0.394316\pi$
$284$	0	0
$285$	−2.75292	−0.163069
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.8646	0.754139
$292$	0	0
$293$	5.00489	0.292389	0.146194	−	0.989256i	$-0.453298\pi$
$293$	5.00489	0.292389	0.146194	+	0.989256i	$0.453298\pi$
$294$	0	0
$295$	22.7804	1.32632
$296$	0	0
$297$	8.57893	0.497800
$298$	0	0
$299$	−0.0153998	−0.000890591 0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−14.7448	−0.847066
$304$	0	0
$305$	60.3534	3.45583
$306$	0	0
$307$	23.1563	1.32160	0.660799	−	0.750563i	$-0.270217\pi$
$307$	23.1563	1.32160	0.660799	+	0.750563i	$0.270217\pi$
$308$	0	0
$309$	−9.08071	−0.516584
$310$	0	0
$311$	−27.4088	−1.55421	−0.777105	−	0.629371i	$-0.783313\pi$
$311$	−27.4088	−1.55421	−0.777105	+	0.629371i	$0.783313\pi$
$312$	0	0
$313$	−22.8081	−1.28919	−0.644595	−	0.764524i	$-0.722974\pi$
$313$	−22.8081	−1.28919	−0.644595	+	0.764524i	$0.722974\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.6005	−1.26937	−0.634685	−	0.772771i	$-0.718870\pi$
$317$	−22.6005	−1.26937	−0.634685	+	0.772771i	$0.718870\pi$
$318$	0	0
$319$	8.90033	0.498323
$320$	0	0
$321$	16.5381	0.923069
$322$	0	0
$323$	−0.519822	−0.0289237
$324$	0	0
$325$	−0.194850	−0.0108083
$326$	0	0
$327$	11.2478	0.622004
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−25.5591	−1.40486	−0.702428	−	0.711755i	$-0.747901\pi$
$331$	−25.5591	−1.40486	−0.702428	+	0.711755i	$0.747901\pi$
$332$	0	0
$333$	−3.77265	−0.206740
$334$	0	0
$335$	−21.2600	−1.16156
$336$	0	0
$337$	23.1633	1.26179	0.630893	−	0.775870i	$-0.282689\pi$
$337$	23.1633	1.26179	0.630893	+	0.775870i	$0.282689\pi$
$338$	0	0
$339$	10.6139	0.576469
$340$	0	0
$341$	−1.99229	−0.107888
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5.07743	0.273360
$346$	0	0
$347$	−36.4373	−1.95606	−0.978029	−	0.208468i	$-0.933152\pi$
$347$	−36.4373	−1.95606	−0.978029	+	0.208468i	$0.933152\pi$
$348$	0	0
$349$	0.0585067	0.00313179	0.00156590	−	0.999999i	$-0.499502\pi$
$349$	0.0585067	0.00313179	0.00156590	+	0.999999i	$0.499502\pi$
$350$	0	0
$351$	−0.0896653	−0.00478598
$352$	0	0
$353$	−23.5207	−1.25188	−0.625939	−	0.779872i	$-0.715284\pi$
$353$	−23.5207	−1.25188	−0.625939	+	0.779872i	$0.715284\pi$
$354$	0	0
$355$	−55.5832	−2.95005
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.6575	0.562483	0.281242	−	0.959637i	$-0.409254\pi$
$359$	10.6575	0.562483	0.281242	+	0.959637i	$0.409254\pi$
$360$	0	0
$361$	−18.7298	−0.985778
$362$	0	0
$363$	11.0528	0.580124
$364$	0	0
$365$	−13.1007	−0.685721
$366$	0	0
$367$	−29.9738	−1.56462	−0.782309	−	0.622890i	$-0.785958\pi$
$367$	−29.9738	−1.56462	−0.782309	+	0.622890i	$0.785958\pi$
$368$	0	0
$369$	−1.04896	−0.0546065
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−12.7252	−0.658888	−0.329444	−	0.944175i	$-0.606861\pi$
$373$	−12.7252	−0.658888	−0.329444	+	0.944175i	$0.606861\pi$
$374$	0	0
$375$	37.7643	1.95014
$376$	0	0
$377$	−0.0930244	−0.00479100
$378$	0	0
$379$	16.3107	0.837824	0.418912	−	0.908027i	$-0.362411\pi$
$379$	16.3107	0.837824	0.418912	+	0.908027i	$0.362411\pi$
$380$	0	0
$381$	−14.2336	−0.729211
$382$	0	0
$383$	23.8797	1.22020	0.610099	−	0.792325i	$-0.291130\pi$
$383$	23.8797	1.22020	0.610099	+	0.792325i	$0.291130\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.72918	0.342063
$388$	0	0
$389$	−2.75525	−0.139697	−0.0698485	−	0.997558i	$-0.522252\pi$
$389$	−2.75525	−0.139697	−0.0698485	+	0.997558i	$0.522252\pi$
$390$	0	0
$391$	0.958750	0.0484861
$392$	0	0
$393$	−25.8477	−1.30384
$394$	0	0
$395$	42.7222	2.14959
$396$	0	0
$397$	−21.4903	−1.07857	−0.539283	−	0.842124i	$-0.681305\pi$
$397$	−21.4903	−1.07857	−0.539283	+	0.842124i	$0.681305\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.3195	1.76377	0.881886	−	0.471462i	$-0.156274\pi$
$401$	35.3195	1.76377	0.881886	+	0.471462i	$0.156274\pi$
$402$	0	0
$403$	0.0208230	0.00103727
$404$	0	0
$405$	12.6415	0.628161
$406$	0	0
$407$	4.25428	0.210877
$408$	0	0
$409$	−3.10362	−0.153464	−0.0767321	−	0.997052i	$-0.524449\pi$
$409$	−3.10362	−0.153464	−0.0767321	+	0.997052i	$0.524449\pi$
$410$	0	0
$411$	−22.9239	−1.13075
$412$	0	0
$413$	0	0
$414$	0	0
$415$	66.8023	3.27919
$416$	0	0
$417$	−0.0673752	−0.00329938
$418$	0	0
$419$	34.8546	1.70276	0.851379	−	0.524551i	$-0.175767\pi$
$419$	34.8546	1.70276	0.851379	+	0.524551i	$0.175767\pi$
$420$	0	0
$421$	−20.4225	−0.995334	−0.497667	−	0.867368i	$-0.665810\pi$
$421$	−20.4225	−0.995334	−0.497667	+	0.867368i	$0.665810\pi$
$422$	0	0
$423$	−2.19175	−0.106566
$424$	0	0
$425$	12.1309	0.588434
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.0315846	0.00152492
$430$	0	0
$431$	3.36333	0.162006	0.0810030	−	0.996714i	$-0.474188\pi$
$431$	3.36333	0.162006	0.0810030	+	0.996714i	$0.474188\pi$
$432$	0	0
$433$	11.1308	0.534914	0.267457	−	0.963570i	$-0.413817\pi$
$433$	11.1308	0.534914	0.267457	+	0.963570i	$0.413817\pi$
$434$	0	0
$435$	30.6709	1.47056
$436$	0	0
$437$	−0.498379	−0.0238407
$438$	0	0
$439$	8.96569	0.427909	0.213955	−	0.976844i	$-0.431366\pi$
$439$	8.96569	0.427909	0.213955	+	0.976844i	$0.431366\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.94610	0.187485	0.0937423	−	0.995596i	$-0.470117\pi$
$443$	3.94610	0.187485	0.0937423	+	0.995596i	$0.470117\pi$
$444$	0	0
$445$	−43.8100	−2.07679
$446$	0	0
$447$	−2.79741	−0.132313
$448$	0	0
$449$	23.8289	1.12455	0.562277	−	0.826949i	$-0.309926\pi$
$449$	23.8289	1.12455	0.562277	+	0.826949i	$0.309926\pi$
$450$	0	0
$451$	1.18287	0.0556992
$452$	0	0
$453$	22.9431	1.07796
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.01424	−0.141000	−0.0705000	−	0.997512i	$-0.522459\pi$
$457$	−3.01424	−0.141000	−0.0705000	+	0.997512i	$0.522459\pi$
$458$	0	0
$459$	5.58233	0.260561
$460$	0	0
$461$	−8.45569	−0.393821	−0.196910	−	0.980421i	$-0.563091\pi$
$461$	−8.45569	−0.393821	−0.196910	+	0.980421i	$0.563091\pi$
$462$	0	0
$463$	−24.2074	−1.12501	−0.562506	−	0.826793i	$-0.690163\pi$
$463$	−24.2074	−1.12501	−0.562506	+	0.826793i	$0.690163\pi$
$464$	0	0
$465$	−6.86551	−0.318380
$466$	0	0
$467$	8.28824	0.383534	0.191767	−	0.981440i	$-0.438578\pi$
$467$	8.28824	0.383534	0.191767	+	0.981440i	$0.438578\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.87265	−0.0862871
$472$	0	0
$473$	−7.58826	−0.348908
$474$	0	0
$475$	−6.30590	−0.289335
$476$	0	0
$477$	−6.82520	−0.312505
$478$	0	0
$479$	22.1591	1.01248	0.506238	−	0.862394i	$-0.331036\pi$
$479$	22.1591	1.01248	0.506238	+	0.862394i	$0.331036\pi$
$480$	0	0
$481$	−0.0444649	−0.00202742
$482$	0	0
$483$	0	0
$484$	0	0
$485$	41.6140	1.88959
$486$	0	0
$487$	31.3324	1.41981	0.709904	−	0.704298i	$-0.248738\pi$
$487$	31.3324	1.41981	0.709904	+	0.704298i	$0.248738\pi$
$488$	0	0
$489$	21.2343	0.960247
$490$	0	0
$491$	5.58409	0.252007	0.126003	−	0.992030i	$-0.459785\pi$
$491$	5.58409	0.252007	0.126003	+	0.992030i	$0.459785\pi$
$492$	0	0
$493$	5.79146	0.260834
$494$	0	0
$495$	8.66852	0.389621
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−2.27863	−0.102005	−0.0510027	−	0.998699i	$-0.516242\pi$
$499$	−2.27863	−0.102005	−0.0510027	+	0.998699i	$0.516242\pi$
$500$	0	0
$501$	17.1178	0.764768
$502$	0	0
$503$	−36.8357	−1.64242	−0.821212	−	0.570624i	$-0.806701\pi$
$503$	−36.8357	−1.64242	−0.821212	+	0.570624i	$0.806701\pi$
$504$	0	0
$505$	−47.6958	−2.12243
$506$	0	0
$507$	16.6335	0.738719
$508$	0	0
$509$	39.3753	1.74528	0.872641	−	0.488363i	$-0.162406\pi$
$509$	39.3753	1.74528	0.872641	+	0.488363i	$0.162406\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.90182	−0.128118
$514$	0	0
$515$	−29.3739	−1.29437
$516$	0	0
$517$	2.47156	0.108699
$518$	0	0
$519$	5.00231	0.219577
$520$	0	0
$521$	−6.78449	−0.297234	−0.148617	−	0.988895i	$-0.547482\pi$
$521$	−6.78449	−0.297234	−0.148617	+	0.988895i	$0.547482\pi$
$522$	0	0
$523$	−35.5401	−1.55406	−0.777030	−	0.629464i	$-0.783274\pi$
$523$	−35.5401	−1.55406	−0.777030	+	0.629464i	$0.783274\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.29639	−0.0564714
$528$	0	0
$529$	−22.0808	−0.960035
$530$	0	0
$531$	7.50081	0.325507
$532$	0	0
$533$	−0.0123631	−0.000535507 0
$534$	0	0
$535$	53.4968	2.31287
$536$	0	0
$537$	31.2413	1.34816
$538$	0	0
$539$	0	0
$540$	0	0
$541$	28.3887	1.22053	0.610263	−	0.792199i	$-0.291064\pi$
$541$	28.3887	1.22053	0.610263	+	0.792199i	$0.291064\pi$
$542$	0	0
$543$	19.5148	0.837460
$544$	0	0
$545$	36.3838	1.55851
$546$	0	0
$547$	−25.3338	−1.08320	−0.541598	−	0.840638i	$-0.682180\pi$
$547$	−25.3338	−1.08320	−0.541598	+	0.840638i	$0.682180\pi$
$548$	0	0
$549$	19.8724	0.848132
$550$	0	0
$551$	−3.01053	−0.128253
$552$	0	0
$553$	0	0
$554$	0	0
$555$	14.6604	0.622301
$556$	0	0
$557$	−35.0713	−1.48602	−0.743010	−	0.669280i	$-0.766603\pi$
$557$	−35.0713	−1.48602	−0.743010	+	0.669280i	$0.766603\pi$
$558$	0	0
$559$	0.0793109	0.00335449
$560$	0	0
$561$	−1.96637	−0.0830204
$562$	0	0
$563$	37.1770	1.56682	0.783412	−	0.621503i	$-0.213477\pi$
$563$	37.1770	1.56682	0.783412	+	0.621503i	$0.213477\pi$
$564$	0	0
$565$	34.3334	1.44442
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.7661	−1.28978	−0.644891	−	0.764275i	$-0.723097\pi$
$569$	−30.7661	−1.28978	−0.644891	+	0.764275i	$0.723097\pi$
$570$	0	0
$571$	−17.1129	−0.716154	−0.358077	−	0.933692i	$-0.616568\pi$
$571$	−17.1129	−0.716154	−0.358077	+	0.933692i	$0.616568\pi$
$572$	0	0
$573$	−1.73384	−0.0724322
$574$	0	0
$575$	11.6305	0.485025
$576$	0	0
$577$	−18.7692	−0.781372	−0.390686	−	0.920524i	$-0.627762\pi$
$577$	−18.7692	−0.781372	−0.390686	+	0.920524i	$0.627762\pi$
$578$	0	0
$579$	27.6298	1.14825
$580$	0	0
$581$	0	0
$582$	0	0
$583$	7.69654	0.318758
$584$	0	0
$585$	−0.0906016	−0.00374592
$586$	0	0
$587$	−22.6989	−0.936884	−0.468442	−	0.883494i	$-0.655184\pi$
$587$	−22.6989	−0.936884	−0.468442	+	0.883494i	$0.655184\pi$
$588$	0	0
$589$	0.673890	0.0277671
$590$	0	0
$591$	−0.437778	−0.0180078
$592$	0	0
$593$	−36.9707	−1.51821	−0.759103	−	0.650970i	$-0.774362\pi$
$593$	−36.9707	−1.51821	−0.759103	+	0.650970i	$0.774362\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−23.7302	−0.971211
$598$	0	0
$599$	1.40362	0.0573505	0.0286753	−	0.999589i	$-0.490871\pi$
$599$	1.40362	0.0573505	0.0286753	+	0.999589i	$0.490871\pi$
$600$	0	0
$601$	−27.7980	−1.13390	−0.566951	−	0.823751i	$-0.691877\pi$
$601$	−27.7980	−1.13390	−0.566951	+	0.823751i	$0.691877\pi$
$602$	0	0
$603$	−7.00021	−0.285070
$604$	0	0
$605$	35.7532	1.45358
$606$	0	0
$607$	10.4113	0.422580	0.211290	−	0.977423i	$-0.432234\pi$
$607$	10.4113	0.422580	0.211290	+	0.977423i	$0.432234\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.0258322	−0.00104506
$612$	0	0
$613$	18.6101	0.751655	0.375828	−	0.926690i	$-0.377358\pi$
$613$	18.6101	0.751655	0.375828	+	0.926690i	$0.377358\pi$
$614$	0	0
$615$	4.07623	0.164369
$616$	0	0
$617$	−16.8106	−0.676771	−0.338386	−	0.941008i	$-0.609881\pi$
$617$	−16.8106	−0.676771	−0.338386	+	0.941008i	$0.609881\pi$
$618$	0	0
$619$	−38.9688	−1.56629	−0.783145	−	0.621839i	$-0.786386\pi$
$619$	−38.9688	−1.56629	−0.783145	+	0.621839i	$0.786386\pi$
$620$	0	0
$621$	5.35206	0.214771
$622$	0	0
$623$	0	0
$624$	0	0
$625$	61.5039	2.46016
$626$	0	0
$627$	1.02217	0.0408213
$628$	0	0
$629$	2.76827	0.110378
$630$	0	0
$631$	2.53283	0.100830	0.0504152	−	0.998728i	$-0.483946\pi$
$631$	2.53283	0.100830	0.0504152	+	0.998728i	$0.483946\pi$
$632$	0	0
$633$	−4.64963	−0.184806
$634$	0	0
$635$	−46.0423	−1.82713
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−18.3017	−0.724004
$640$	0	0
$641$	−3.61319	−0.142713	−0.0713563	−	0.997451i	$-0.522733\pi$
$641$	−3.61319	−0.142713	−0.0713563	+	0.997451i	$0.522733\pi$
$642$	0	0
$643$	3.55385	0.140150	0.0700750	−	0.997542i	$-0.477676\pi$
$643$	3.55385	0.140150	0.0700750	+	0.997542i	$0.477676\pi$
$644$	0	0
$645$	−26.1495	−1.02963
$646$	0	0
$647$	38.9319	1.53057	0.765285	−	0.643692i	$-0.222598\pi$
$647$	38.9319	1.53057	0.765285	+	0.643692i	$0.222598\pi$
$648$	0	0
$649$	−8.45840	−0.332021
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23.5910	0.923189	0.461594	−	0.887091i	$-0.347278\pi$
$653$	23.5910	0.923189	0.461594	+	0.887091i	$0.347278\pi$
$654$	0	0
$655$	−83.6110	−3.26695
$656$	0	0
$657$	−4.31361	−0.168290
$658$	0	0
$659$	3.43190	0.133688	0.0668439	−	0.997763i	$-0.478707\pi$
$659$	3.43190	0.133688	0.0668439	+	0.997763i	$0.478707\pi$
$660$	0	0
$661$	−38.1243	−1.48286	−0.741431	−	0.671029i	$-0.765853\pi$
$661$	−38.1243	−1.48286	−0.741431	+	0.671029i	$0.765853\pi$
$662$	0	0
$663$	0.0205521	0.000798179 0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.55256	0.214996
$668$	0	0
$669$	−29.2295	−1.13008
$670$	0	0
$671$	−22.4094	−0.865104
$672$	0	0
$673$	−9.27831	−0.357653	−0.178826	−	0.983881i	$-0.557230\pi$
$673$	−9.27831	−0.357653	−0.178826	+	0.983881i	$0.557230\pi$
$674$	0	0
$675$	67.7186	2.60649
$676$	0	0
$677$	10.4284	0.400797	0.200398	−	0.979715i	$-0.435776\pi$
$677$	10.4284	0.400797	0.200398	+	0.979715i	$0.435776\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8.42438	0.322823
$682$	0	0
$683$	−33.6376	−1.28711	−0.643554	−	0.765401i	$-0.722541\pi$
$683$	−33.6376	−1.28711	−0.643554	+	0.765401i	$0.722541\pi$
$684$	0	0
$685$	−74.1532	−2.83325
$686$	0	0
$687$	7.18055	0.273955
$688$	0	0
$689$	−0.0804427	−0.00306462
$690$	0	0
$691$	5.83824	0.222097	0.111049	−	0.993815i	$-0.464579\pi$
$691$	5.83824	0.222097	0.111049	+	0.993815i	$0.464579\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.217942	−0.00826702
$696$	0	0
$697$	0.769697	0.0291544
$698$	0	0
$699$	24.1723	0.914281
$700$	0	0
$701$	−46.7733	−1.76660	−0.883302	−	0.468805i	$-0.844685\pi$
$701$	−46.7733	−1.76660	−0.883302	+	0.468805i	$0.844685\pi$
$702$	0	0
$703$	−1.43901	−0.0542732
$704$	0	0
$705$	8.51710	0.320772
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5.15814	−0.193718	−0.0968590	−	0.995298i	$-0.530880\pi$
$709$	−5.15814	−0.193718	−0.0968590	+	0.995298i	$0.530880\pi$
$710$	0	0
$711$	14.0670	0.527553
$712$	0	0
$713$	−1.24291	−0.0465473
$714$	0	0
$715$	0.102168	0.00382088
$716$	0	0
$717$	−8.69755	−0.324816
$718$	0	0
$719$	41.4647	1.54637	0.773186	−	0.634179i	$-0.218662\pi$
$719$	41.4647	1.54637	0.773186	+	0.634179i	$0.218662\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	20.3038	0.755107
$724$	0	0
$725$	70.2556	2.60923
$726$	0	0
$727$	2.82479	0.104766	0.0523828	−	0.998627i	$-0.483318\pi$
$727$	2.82479	0.104766	0.0523828	+	0.998627i	$0.483318\pi$
$728$	0	0
$729$	25.5906	0.947800
$730$	0	0
$731$	−4.93770	−0.182627
$732$	0	0
$733$	15.3118	0.565556	0.282778	−	0.959185i	$-0.408744\pi$
$733$	15.3118	0.565556	0.282778	+	0.959185i	$0.408744\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.89389	0.290775
$738$	0	0
$739$	−25.0298	−0.920737	−0.460369	−	0.887728i	$-0.652283\pi$
$739$	−25.0298	−0.920737	−0.460369	+	0.887728i	$0.652283\pi$
$740$	0	0
$741$	−0.0106835	−0.000392467 0
$742$	0	0
$743$	−16.1647	−0.593027	−0.296513	−	0.955029i	$-0.595824\pi$
$743$	−16.1647	−0.593027	−0.296513	+	0.955029i	$0.595824\pi$
$744$	0	0
$745$	−9.04893	−0.331527
$746$	0	0
$747$	21.9958	0.804783
$748$	0	0
$749$	0	0
$750$	0	0
$751$	44.4820	1.62317	0.811586	−	0.584233i	$-0.198604\pi$
$751$	44.4820	1.62317	0.811586	+	0.584233i	$0.198604\pi$
$752$	0	0
$753$	−11.1458	−0.406174
$754$	0	0
$755$	74.2154	2.70097
$756$	0	0
$757$	4.00941	0.145724	0.0728622	−	0.997342i	$-0.476787\pi$
$757$	4.00941	0.145724	0.0728622	+	0.997342i	$0.476787\pi$
$758$	0	0
$759$	−1.88526	−0.0684306
$760$	0	0
$761$	30.0752	1.09023	0.545113	−	0.838363i	$-0.316487\pi$
$761$	30.0752	1.09023	0.545113	+	0.838363i	$0.316487\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5.64063	0.203937
$766$	0	0
$767$	0.0884055	0.00319214
$768$	0	0
$769$	38.3403	1.38259	0.691293	−	0.722575i	$-0.257041\pi$
$769$	38.3403	1.38259	0.691293	+	0.722575i	$0.257041\pi$
$770$	0	0
$771$	−9.05404	−0.326073
$772$	0	0
$773$	−32.6953	−1.17597	−0.587983	−	0.808873i	$-0.700078\pi$
$773$	−32.6953	−1.17597	−0.587983	+	0.808873i	$0.700078\pi$
$774$	0	0
$775$	−15.7263	−0.564905
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.400106	−0.0143353
$780$	0	0
$781$	20.6382	0.738492
$782$	0	0
$783$	32.3299	1.15538
$784$	0	0
$785$	−6.05756	−0.216204
$786$	0	0
$787$	19.8276	0.706778	0.353389	−	0.935476i	$-0.385029\pi$
$787$	19.8276	0.706778	0.353389	+	0.935476i	$0.385029\pi$
$788$	0	0
$789$	−23.6396	−0.841591
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.234218	0.00831733
$794$	0	0
$795$	26.5226	0.940660
$796$	0	0
$797$	21.6470	0.766775	0.383387	−	0.923588i	$-0.374757\pi$
$797$	21.6470	0.766775	0.383387	+	0.923588i	$0.374757\pi$
$798$	0	0
$799$	1.60825	0.0568957
$800$	0	0
$801$	−14.4252	−0.509689
$802$	0	0
$803$	4.86431	0.171658
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.1341	0.849559
$808$	0	0
$809$	−2.54896	−0.0896166	−0.0448083	−	0.998996i	$-0.514268\pi$
$809$	−2.54896	−0.0896166	−0.0448083	+	0.998996i	$0.514268\pi$
$810$	0	0
$811$	−15.4648	−0.543042	−0.271521	−	0.962432i	$-0.587527\pi$
$811$	−15.4648	−0.543042	−0.271521	+	0.962432i	$0.587527\pi$
$812$	0	0
$813$	−37.1385	−1.30250
$814$	0	0
$815$	68.6877	2.40602
$816$	0	0
$817$	2.56672	0.0897983
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.2553	−1.50962	−0.754811	−	0.655942i	$-0.772271\pi$
$821$	−43.2553	−1.50962	−0.754811	+	0.655942i	$0.772271\pi$
$822$	0	0
$823$	−2.18064	−0.0760124	−0.0380062	−	0.999278i	$-0.512101\pi$
$823$	−2.18064	−0.0760124	−0.0380062	+	0.999278i	$0.512101\pi$
$824$	0	0
$825$	−23.8539	−0.830485
$826$	0	0
$827$	2.33083	0.0810508	0.0405254	−	0.999179i	$-0.487097\pi$
$827$	2.33083	0.0810508	0.0405254	+	0.999179i	$0.487097\pi$
$828$	0	0
$829$	10.6090	0.368466	0.184233	−	0.982883i	$-0.441020\pi$
$829$	10.6090	0.368466	0.184233	+	0.982883i	$0.441020\pi$
$830$	0	0
$831$	0.967650	0.0335674
$832$	0	0
$833$	0	0
$834$	0	0
$835$	55.3719	1.91623
$836$	0	0
$837$	−7.23685	−0.250142
$838$	0	0
$839$	−11.5111	−0.397408	−0.198704	−	0.980060i	$-0.563673\pi$
$839$	−11.5111	−0.397408	−0.198704	+	0.980060i	$0.563673\pi$
$840$	0	0
$841$	4.54104	0.156588
$842$	0	0
$843$	−25.9362	−0.893290
$844$	0	0
$845$	53.8052	1.85096
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−14.0321	−0.481582
$850$	0	0
$851$	2.65408	0.0909807
$852$	0	0
$853$	−3.81565	−0.130646	−0.0653228	−	0.997864i	$-0.520808\pi$
$853$	−3.81565	−0.130646	−0.0653228	+	0.997864i	$0.520808\pi$
$854$	0	0
$855$	−2.93212	−0.100277
$856$	0	0
$857$	−28.4078	−0.970393	−0.485197	−	0.874405i	$-0.661252\pi$
$857$	−28.4078	−0.970393	−0.485197	+	0.874405i	$0.661252\pi$
$858$	0	0
$859$	−21.7166	−0.740961	−0.370481	−	0.928840i	$-0.620807\pi$
$859$	−21.7166	−0.740961	−0.370481	+	0.928840i	$0.620807\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41.4622	−1.41139	−0.705694	−	0.708516i	$-0.749365\pi$
$863$	−41.4622	−1.41139	−0.705694	+	0.708516i	$0.749365\pi$
$864$	0	0
$865$	16.1812	0.550178
$866$	0	0
$867$	−1.27952	−0.0434549
$868$	0	0
$869$	−15.8628	−0.538110
$870$	0	0
$871$	−0.0825053	−0.00279559
$872$	0	0
$873$	13.7021	0.463746
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.6058	0.695809	0.347905	−	0.937530i	$-0.386893\pi$
$877$	20.6058	0.695809	0.347905	+	0.937530i	$0.386893\pi$
$878$	0	0
$879$	−6.40388	−0.215997
$880$	0	0
$881$	35.1391	1.18387	0.591934	−	0.805987i	$-0.298365\pi$
$881$	35.1391	1.18387	0.591934	+	0.805987i	$0.298365\pi$
$882$	0	0
$883$	2.56154	0.0862027	0.0431014	−	0.999071i	$-0.486276\pi$
$883$	2.56154	0.0862027	0.0431014	+	0.999071i	$0.486276\pi$
$884$	0	0
$885$	−29.1480	−0.979800
$886$	0	0
$887$	37.6761	1.26504	0.632520	−	0.774544i	$-0.282021\pi$
$887$	37.6761	1.26504	0.632520	+	0.774544i	$0.282021\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.69382	−0.157249
$892$	0	0
$893$	−0.836003	−0.0279758
$894$	0	0
$895$	101.058	3.37799
$896$	0	0
$897$	0.0197044	0.000657910 0
$898$	0	0
$899$	−7.50797	−0.250405
$900$	0	0
$901$	5.00816	0.166846
$902$	0	0
$903$	0	0
$904$	0	0
$905$	63.1256	2.09836
$906$	0	0
$907$	−28.4367	−0.944224	−0.472112	−	0.881539i	$-0.656508\pi$
$907$	−28.4367	−0.944224	−0.472112	+	0.881539i	$0.656508\pi$
$908$	0	0
$909$	−15.7046	−0.520889
$910$	0	0
$911$	−9.68890	−0.321008	−0.160504	−	0.987035i	$-0.551312\pi$
$911$	−9.68890	−0.321008	−0.160504	+	0.987035i	$0.551312\pi$
$912$	0	0
$913$	−24.8038	−0.820887
$914$	0	0
$915$	−77.2237	−2.55294
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.22717	0.0404807	0.0202404	−	0.999795i	$-0.493557\pi$
$919$	1.22717	0.0404807	0.0202404	+	0.999795i	$0.493557\pi$
$920$	0	0
$921$	−29.6290	−0.976310
$922$	0	0
$923$	−0.215706	−0.00710005
$924$	0	0
$925$	33.5816	1.10416
$926$	0	0
$927$	−9.67184	−0.317665
$928$	0	0
$929$	−37.3244	−1.22457	−0.612287	−	0.790635i	$-0.709750\pi$
$929$	−37.3244	−1.22457	−0.612287	+	0.790635i	$0.709750\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	35.0702	1.14815
$934$	0	0
$935$	−6.36074	−0.208018
$936$	0	0
$937$	−11.8297	−0.386459	−0.193229	−	0.981154i	$-0.561896\pi$
$937$	−11.8297	−0.386459	−0.193229	+	0.981154i	$0.561896\pi$
$938$	0	0
$939$	29.1835	0.952368
$940$	0	0
$941$	1.74652	0.0569348	0.0284674	−	0.999595i	$-0.490937\pi$
$941$	1.74652	0.0569348	0.0284674	+	0.999595i	$0.490937\pi$
$942$	0	0
$943$	0.737947	0.0240309
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.3949	−0.597754	−0.298877	−	0.954292i	$-0.596612\pi$
$947$	−18.3949	−0.597754	−0.298877	+	0.954292i	$0.596612\pi$
$948$	0	0
$949$	−0.0508408	−0.00165036
$950$	0	0
$951$	28.9179	0.937727
$952$	0	0
$953$	−34.8806	−1.12989	−0.564947	−	0.825127i	$-0.691103\pi$
$953$	−34.8806	−1.12989	−0.564947	+	0.825127i	$0.691103\pi$
$954$	0	0
$955$	−5.60855	−0.181488
$956$	0	0
$957$	−11.3882	−0.368128
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.3194	−0.945787
$962$	0	0
$963$	17.6147	0.567626
$964$	0	0
$965$	89.3756	2.87710
$966$	0	0
$967$	45.2995	1.45674	0.728368	−	0.685187i	$-0.240279\pi$
$967$	45.2995	1.45674	0.728368	+	0.685187i	$0.240279\pi$
$968$	0	0
$969$	0.665125	0.0213669
$970$	0	0
$971$	8.93181	0.286635	0.143318	−	0.989677i	$-0.454223\pi$
$971$	8.93181	0.286635	0.143318	+	0.989677i	$0.454223\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.249316	0.00798449
$976$	0	0
$977$	24.4223	0.781338	0.390669	−	0.920531i	$-0.372244\pi$
$977$	24.4223	0.781338	0.390669	+	0.920531i	$0.372244\pi$
$978$	0	0
$979$	16.2668	0.519888
$980$	0	0
$981$	11.9800	0.382491
$982$	0	0
$983$	−40.0513	−1.27744	−0.638719	−	0.769440i	$-0.720535\pi$
$983$	−40.0513	−1.27744	−0.638719	+	0.769440i	$0.720535\pi$
$984$	0	0
$985$	−1.41610	−0.0451208
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.73402	−0.150533
$990$	0	0
$991$	−23.4318	−0.744335	−0.372168	−	0.928166i	$-0.621385\pi$
$991$	−23.4318	−0.744335	−0.372168	+	0.928166i	$0.621385\pi$
$992$	0	0
$993$	32.7035	1.03781
$994$	0	0
$995$	−76.7613	−2.43350
$996$	0	0
$997$	−25.5055	−0.807768	−0.403884	−	0.914810i	$-0.632340\pi$
$997$	−25.5055	−0.807768	−0.403884	+	0.914810i	$0.632340\pi$
$998$	0	0
$999$	15.4534	0.488924

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-1.27952 q^{3} -4.13895 q^{5} -1.36282 q^{9} +O(q^{10})\)	\(q-1.27952 q^{3} -4.13895 q^{5} -1.36282 q^{9} +1.53680 q^{11} -0.0160623 q^{13} +5.29588 q^{15} +1.00000 q^{17} -0.519822 q^{19} +0.958750 q^{23} +12.1309 q^{25} +5.58233 q^{27} +5.79146 q^{29} -1.29639 q^{31} -1.96637 q^{33} +2.76827 q^{37} +0.0205521 q^{39} +0.769697 q^{41} -4.93770 q^{43} +5.64063 q^{45} +1.60825 q^{47} -1.27952 q^{51} +5.00816 q^{53} -6.36074 q^{55} +0.665125 q^{57} -5.50390 q^{59} -14.5818 q^{61} +0.0664811 q^{65} +5.13657 q^{67} -1.22674 q^{69} +13.4293 q^{71} +3.16522 q^{73} -15.5218 q^{75} -10.3220 q^{79} -3.05428 q^{81} -16.1399 q^{83} -4.13895 q^{85} -7.41032 q^{87} +10.5848 q^{89} +1.65876 q^{93} +2.15152 q^{95} -10.0542 q^{97} -2.09438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 4 q^{5} + 8 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 - 4 * q^5 + 8 * q^9	\( 8 q - 4 q^{3} - 4 q^{5} + 8 q^{9} - 4 q^{11} - 20 q^{13} + 12 q^{15} + 8 q^{17} - 8 q^{19} + 4 q^{23} + 8 q^{25} - 28 q^{27} - 16 q^{29} + 8 q^{31} - 16 q^{33} + 8 q^{37} + 20 q^{39} - 12 q^{41} - 4 q^{43} - 20 q^{45} - 4 q^{47} - 4 q^{51} - 12 q^{55} + 16 q^{59} - 32 q^{61} - 44 q^{69} - 24 q^{73} - 24 q^{75} + 4 q^{79} + 36 q^{81} - 28 q^{83} - 4 q^{85} + 40 q^{87} - 20 q^{89} - 16 q^{93} - 20 q^{95} - 56 q^{97} - 8 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

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Database

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