LMFDB - Embedded newform 8092.2.a.z.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8092,2,Mod(1,8092)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8092, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8092.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8092 = 2^{2} \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8092.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20,0,8,0,8,0,20,0,28,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.6149453156$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 12 x^{18} + 240 x^{17} - 224 x^{16} - 2776 x^{15} + 5324 x^{14} + 15280 x^{13} + \cdots + 544 $ x^20 - 8x^19 - 12x^18 + 240x^17 - 224x^16 - 2776x^15 + 5324x^14 + 15280x^13 - 42463x^12 - 36576x^11 + 166532x^10 - 1048x^9 - 328476x^8 + 166744x^7 + 277764x^6 - 258464x^5 - 36367x^4 + 111904x^3 - 40080x^2 + 2688*x + 544
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 476)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.948608$ of defining polynomial
Character	$\chi$	$=$	8092.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.948608 q^{3} -3.07520 q^{5} +1.00000 q^{7} -2.10014 q^{9} +1.96405 q^{11} +2.55706 q^{13} +2.91716 q^{15} -5.17418 q^{19} -0.948608 q^{21} +4.86371 q^{23} +4.45685 q^{25} +4.83804 q^{27} -0.866107 q^{29} -1.50523 q^{31} -1.86311 q^{33} -3.07520 q^{35} -1.50842 q^{37} -2.42565 q^{39} -0.716268 q^{41} +1.87713 q^{43} +6.45836 q^{45} -7.11332 q^{47} +1.00000 q^{49} -8.36741 q^{53} -6.03984 q^{55} +4.90827 q^{57} -0.611228 q^{59} +5.37219 q^{61} -2.10014 q^{63} -7.86346 q^{65} +1.25625 q^{67} -4.61375 q^{69} +4.48800 q^{71} +5.55879 q^{73} -4.22781 q^{75} +1.96405 q^{77} -8.28300 q^{79} +1.71102 q^{81} -13.1152 q^{83} +0.821597 q^{87} -8.10832 q^{89} +2.55706 q^{91} +1.42787 q^{93} +15.9116 q^{95} -19.1555 q^{97} -4.12478 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 8 q^{3} + 8 q^{5} + 20 q^{7} + 28 q^{9} + 16 q^{11} + 8 q^{13} + 8 q^{15} + 8 q^{19} + 8 q^{21} + 40 q^{23} + 36 q^{25} + 32 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{33} + 8 q^{35} + 24 q^{37} + 16 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) $ 20 * q + 8 * q^3 + 8 * q^5 + 20 * q^7 + 28 * q^9 + 16 * q^11 + 8 * q^13 + 8 * q^15 + 8 * q^19 + 8 * q^21 + 40 * q^23 + 36 * q^25 + 32 * q^27 + 16 * q^29 + 16 * q^31 + 8 * q^33 + 8 * q^35 + 24 * q^37 + 16 * q^39 - 16 * q^41 + 16 * q^43 + 24 * q^45 + 20 * q^49 + 24 * q^53 - 8 * q^55 + 24 * q^57 + 8 * q^61 + 28 * q^63 + 8 * q^65 - 8 * q^67 + 8 * q^69 + 72 * q^71 + 40 * q^75 + 16 * q^77 + 60 * q^81 + 8 * q^83 - 24 * q^89 + 8 * q^91 + 32 * q^93 + 48 * q^95 - 16 * q^97 + 72 * 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
$3$	−0.948608	−0.547679	−0.273840	−	0.961775i	$-0.588294\pi$
$3$	−0.948608	−0.547679	−0.273840	+	0.961775i	$0.588294\pi$
$4$	0	0
$5$	−3.07520	−1.37527	−0.687636	−	0.726056i	$-0.741351\pi$
$5$	−3.07520	−1.37527	−0.687636	+	0.726056i	$0.741351\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.10014	−0.700047
$10$	0	0
$11$	1.96405	0.592183	0.296091	−	0.955160i	$-0.404317\pi$
$11$	1.96405	0.592183	0.296091	+	0.955160i	$0.404317\pi$
$12$	0	0
$13$	2.55706	0.709200	0.354600	−	0.935018i	$-0.384617\pi$
$13$	2.55706	0.709200	0.354600	+	0.935018i	$0.384617\pi$
$14$	0	0
$15$	2.91716	0.753208
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−5.17418	−1.18704	−0.593520	−	0.804820i	$-0.702262\pi$
$19$	−5.17418	−1.18704	−0.593520	+	0.804820i	$0.702262\pi$
$20$	0	0
$21$	−0.948608	−0.207003
$22$	0	0
$23$	4.86371	1.01415	0.507077	−	0.861901i	$-0.330726\pi$
$23$	4.86371	1.01415	0.507077	+	0.861901i	$0.330726\pi$
$24$	0	0
$25$	4.45685	0.891371
$26$	0	0
$27$	4.83804	0.931081
$28$	0	0
$29$	−0.866107	−0.160832	−0.0804161	−	0.996761i	$-0.525625\pi$
$29$	−0.866107	−0.160832	−0.0804161	+	0.996761i	$0.525625\pi$
$30$	0	0
$31$	−1.50523	−0.270347	−0.135173	−	0.990822i	$-0.543159\pi$
$31$	−1.50523	−0.270347	−0.135173	+	0.990822i	$0.543159\pi$
$32$	0	0
$33$	−1.86311	−0.324326
$34$	0	0
$35$	−3.07520	−0.519804
$36$	0	0
$37$	−1.50842	−0.247983	−0.123992	−	0.992283i	$-0.539570\pi$
$37$	−1.50842	−0.247983	−0.123992	+	0.992283i	$0.539570\pi$
$38$	0	0
$39$	−2.42565	−0.388414
$40$	0	0
$41$	−0.716268	−0.111862	−0.0559311	−	0.998435i	$-0.517813\pi$
$41$	−0.716268	−0.111862	−0.0559311	+	0.998435i	$0.517813\pi$
$42$	0	0
$43$	1.87713	0.286260	0.143130	−	0.989704i	$-0.454283\pi$
$43$	1.87713	0.286260	0.143130	+	0.989704i	$0.454283\pi$
$44$	0	0
$45$	6.45836	0.962755
$46$	0	0
$47$	−7.11332	−1.03758	−0.518792	−	0.854901i	$-0.673618\pi$
$47$	−7.11332	−1.03758	−0.518792	+	0.854901i	$0.673618\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.36741	−1.14935	−0.574676	−	0.818381i	$-0.694872\pi$
$53$	−8.36741	−1.14935	−0.574676	+	0.818381i	$0.694872\pi$
$54$	0	0
$55$	−6.03984	−0.814412
$56$	0	0
$57$	4.90827	0.650117
$58$	0	0
$59$	−0.611228	−0.0795751	−0.0397876	−	0.999208i	$-0.512668\pi$
$59$	−0.611228	−0.0795751	−0.0397876	+	0.999208i	$0.512668\pi$
$60$	0	0
$61$	5.37219	0.687838	0.343919	−	0.938999i	$-0.388245\pi$
$61$	5.37219	0.687838	0.343919	+	0.938999i	$0.388245\pi$
$62$	0	0
$63$	−2.10014	−0.264593
$64$	0	0
$65$	−7.86346	−0.975342
$66$	0	0
$67$	1.25625	0.153476	0.0767379	−	0.997051i	$-0.475550\pi$
$67$	1.25625	0.153476	0.0767379	+	0.997051i	$0.475550\pi$
$68$	0	0
$69$	−4.61375	−0.555431
$70$	0	0
$71$	4.48800	0.532628	0.266314	−	0.963886i	$-0.414194\pi$
$71$	4.48800	0.532628	0.266314	+	0.963886i	$0.414194\pi$
$72$	0	0
$73$	5.55879	0.650608	0.325304	−	0.945610i	$-0.394533\pi$
$73$	5.55879	0.650608	0.325304	+	0.945610i	$0.394533\pi$
$74$	0	0
$75$	−4.22781	−0.488185
$76$	0	0
$77$	1.96405	0.223824
$78$	0	0
$79$	−8.28300	−0.931910	−0.465955	−	0.884808i	$-0.654289\pi$
$79$	−8.28300	−0.931910	−0.465955	+	0.884808i	$0.654289\pi$
$80$	0	0
$81$	1.71102	0.190114
$82$	0	0
$83$	−13.1152	−1.43958	−0.719788	−	0.694194i	$-0.755761\pi$
$83$	−13.1152	−1.43958	−0.719788	+	0.694194i	$0.755761\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.821597	0.0880844
$88$	0	0
$89$	−8.10832	−0.859480	−0.429740	−	0.902953i	$-0.641395\pi$
$89$	−8.10832	−0.859480	−0.429740	+	0.902953i	$0.641395\pi$
$90$	0	0
$91$	2.55706	0.268052
$92$	0	0
$93$	1.42787	0.148063
$94$	0	0
$95$	15.9116	1.63250
$96$	0	0
$97$	−19.1555	−1.94495	−0.972474	−	0.233011i	$-0.925142\pi$
$97$	−19.1555	−1.94495	−0.972474	+	0.233011i	$0.925142\pi$
$98$	0	0
$99$	−4.12478	−0.414556
$100$	0	0
$101$	18.3302	1.82392	0.911961	−	0.410278i	$-0.134568\pi$
$101$	18.3302	1.82392	0.911961	+	0.410278i	$0.134568\pi$
$102$	0	0
$103$	−9.14293	−0.900879	−0.450440	−	0.892807i	$-0.648733\pi$
$103$	−9.14293	−0.900879	−0.450440	+	0.892807i	$0.648733\pi$
$104$	0	0
$105$	2.91716	0.284686
$106$	0	0
$107$	3.48588	0.336993	0.168497	−	0.985702i	$-0.446109\pi$
$107$	3.48588	0.336993	0.168497	+	0.985702i	$0.446109\pi$
$108$	0	0
$109$	−3.05289	−0.292414	−0.146207	−	0.989254i	$-0.546707\pi$
$109$	−3.05289	−0.292414	−0.146207	+	0.989254i	$0.546707\pi$
$110$	0	0
$111$	1.43090	0.135815
$112$	0	0
$113$	15.8578	1.49178	0.745890	−	0.666069i	$-0.232024\pi$
$113$	15.8578	1.49178	0.745890	+	0.666069i	$0.232024\pi$
$114$	0	0
$115$	−14.9569	−1.39474
$116$	0	0
$117$	−5.37018	−0.496474
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.14252	−0.649320
$122$	0	0
$123$	0.679458	0.0612646
$124$	0	0
$125$	1.67028	0.149394
$126$	0	0
$127$	−21.0365	−1.86668	−0.933342	−	0.358988i	$-0.883122\pi$
$127$	−21.0365	−1.86668	−0.933342	+	0.358988i	$0.883122\pi$
$128$	0	0
$129$	−1.78067	−0.156779
$130$	0	0
$131$	7.22592	0.631332	0.315666	−	0.948870i	$-0.397772\pi$
$131$	7.22592	0.631332	0.315666	+	0.948870i	$0.397772\pi$
$132$	0	0
$133$	−5.17418	−0.448659
$134$	0	0
$135$	−14.8779	−1.28049
$136$	0	0
$137$	21.8370	1.86566	0.932831	−	0.360314i	$-0.117330\pi$
$137$	21.8370	1.86566	0.932831	+	0.360314i	$0.117330\pi$
$138$	0	0
$139$	4.52664	0.383945	0.191972	−	0.981400i	$-0.438512\pi$
$139$	4.52664	0.383945	0.191972	+	0.981400i	$0.438512\pi$
$140$	0	0
$141$	6.74775	0.568263
$142$	0	0
$143$	5.02218	0.419976
$144$	0	0
$145$	2.66345	0.221188
$146$	0	0
$147$	−0.948608	−0.0782399
$148$	0	0
$149$	−8.42472	−0.690180	−0.345090	−	0.938570i	$-0.612152\pi$
$149$	−8.42472	−0.690180	−0.345090	+	0.938570i	$0.612152\pi$
$150$	0	0
$151$	12.3210	1.00267	0.501333	−	0.865254i	$-0.332843\pi$
$151$	12.3210	1.00267	0.501333	+	0.865254i	$0.332843\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.62887	0.371800
$156$	0	0
$157$	−6.13536	−0.489655	−0.244827	−	0.969567i	$-0.578731\pi$
$157$	−6.13536	−0.489655	−0.244827	+	0.969567i	$0.578731\pi$
$158$	0	0
$159$	7.93739	0.629476
$160$	0	0
$161$	4.86371	0.383314
$162$	0	0
$163$	13.6914	1.07239	0.536197	−	0.844093i	$-0.319860\pi$
$163$	13.6914	1.07239	0.536197	+	0.844093i	$0.319860\pi$
$164$	0	0
$165$	5.72944	0.446037
$166$	0	0
$167$	−6.50679	−0.503511	−0.251755	−	0.967791i	$-0.581008\pi$
$167$	−6.50679	−0.503511	−0.251755	+	0.967791i	$0.581008\pi$
$168$	0	0
$169$	−6.46146	−0.497035
$170$	0	0
$171$	10.8665	0.830984
$172$	0	0
$173$	21.1169	1.60549	0.802744	−	0.596323i	$-0.203372\pi$
$173$	21.1169	1.60549	0.802744	+	0.596323i	$0.203372\pi$
$174$	0	0
$175$	4.45685	0.336907
$176$	0	0
$177$	0.579816	0.0435816
$178$	0	0
$179$	−9.46818	−0.707685	−0.353843	−	0.935305i	$-0.615125\pi$
$179$	−9.46818	−0.707685	−0.353843	+	0.935305i	$0.615125\pi$
$180$	0	0
$181$	−16.8825	−1.25486	−0.627432	−	0.778671i	$-0.715894\pi$
$181$	−16.8825	−1.25486	−0.627432	+	0.778671i	$0.715894\pi$
$182$	0	0
$183$	−5.09610	−0.376715
$184$	0	0
$185$	4.63870	0.341044
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4.83804	0.351915
$190$	0	0
$191$	−9.52990	−0.689559	−0.344780	−	0.938684i	$-0.612046\pi$
$191$	−9.52990	−0.689559	−0.344780	+	0.938684i	$0.612046\pi$
$192$	0	0
$193$	2.12347	0.152851	0.0764253	−	0.997075i	$-0.475649\pi$
$193$	2.12347	0.152851	0.0764253	+	0.997075i	$0.475649\pi$
$194$	0	0
$195$	7.45934	0.534175
$196$	0	0
$197$	3.67515	0.261844	0.130922	−	0.991393i	$-0.458206\pi$
$197$	3.67515	0.261844	0.130922	+	0.991393i	$0.458206\pi$
$198$	0	0
$199$	−8.93341	−0.633272	−0.316636	−	0.948547i	$-0.602553\pi$
$199$	−8.93341	−0.633272	−0.316636	+	0.948547i	$0.602553\pi$
$200$	0	0
$201$	−1.19169	−0.0840555
$202$	0	0
$203$	−0.866107	−0.0607888
$204$	0	0
$205$	2.20267	0.153841
$206$	0	0
$207$	−10.2145	−0.709955
$208$	0	0
$209$	−10.1623	−0.702944
$210$	0	0
$211$	27.3255	1.88116	0.940581	−	0.339570i	$-0.110282\pi$
$211$	27.3255	1.88116	0.940581	+	0.339570i	$0.110282\pi$
$212$	0	0
$213$	−4.25736	−0.291709
$214$	0	0
$215$	−5.77256	−0.393686
$216$	0	0
$217$	−1.50523	−0.102181
$218$	0	0
$219$	−5.27312	−0.356324
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.1460	1.14818	0.574092	−	0.818791i	$-0.305355\pi$
$223$	17.1460	1.14818	0.574092	+	0.818791i	$0.305355\pi$
$224$	0	0
$225$	−9.36003	−0.624002
$226$	0	0
$227$	−4.45185	−0.295479	−0.147740	−	0.989026i	$-0.547200\pi$
$227$	−4.45185	−0.295479	−0.147740	+	0.989026i	$0.547200\pi$
$228$	0	0
$229$	26.7608	1.76841	0.884203	−	0.467103i	$-0.154702\pi$
$229$	26.7608	1.76841	0.884203	+	0.467103i	$0.154702\pi$
$230$	0	0
$231$	−1.86311	−0.122584
$232$	0	0
$233$	9.71746	0.636612	0.318306	−	0.947988i	$-0.396886\pi$
$233$	9.71746	0.636612	0.318306	+	0.947988i	$0.396886\pi$
$234$	0	0
$235$	21.8749	1.42696
$236$	0	0
$237$	7.85732	0.510388
$238$	0	0
$239$	30.5463	1.97588	0.987940	−	0.154840i	$-0.0494862\pi$
$239$	30.5463	1.97588	0.987940	+	0.154840i	$0.0494862\pi$
$240$	0	0
$241$	14.9545	0.963306	0.481653	−	0.876362i	$-0.340037\pi$
$241$	14.9545	0.963306	0.481653	+	0.876362i	$0.340037\pi$
$242$	0	0
$243$	−16.1372	−1.03520
$244$	0	0
$245$	−3.07520	−0.196467
$246$	0	0
$247$	−13.2307	−0.841848
$248$	0	0
$249$	12.4412	0.788426
$250$	0	0
$251$	−17.3960	−1.09802	−0.549011	−	0.835815i	$-0.684996\pi$
$251$	−17.3960	−1.09802	−0.549011	+	0.835815i	$0.684996\pi$
$252$	0	0
$253$	9.55256	0.600564
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.8982	−0.991701	−0.495850	−	0.868408i	$-0.665144\pi$
$257$	−15.8982	−0.991701	−0.495850	+	0.868408i	$0.665144\pi$
$258$	0	0
$259$	−1.50842	−0.0937288
$260$	0	0
$261$	1.81895	0.112590
$262$	0	0
$263$	27.7848	1.71329	0.856643	−	0.515910i	$-0.172546\pi$
$263$	27.7848	1.71329	0.856643	+	0.515910i	$0.172546\pi$
$264$	0	0
$265$	25.7314	1.58067
$266$	0	0
$267$	7.69162	0.470719
$268$	0	0
$269$	0.386116	0.0235419	0.0117709	−	0.999931i	$-0.496253\pi$
$269$	0.386116	0.0235419	0.0117709	+	0.999931i	$0.496253\pi$
$270$	0	0
$271$	−19.9108	−1.20949	−0.604746	−	0.796418i	$-0.706726\pi$
$271$	−19.9108	−1.20949	−0.604746	+	0.796418i	$0.706726\pi$
$272$	0	0
$273$	−2.42565	−0.146807
$274$	0	0
$275$	8.75348	0.527855
$276$	0	0
$277$	24.3362	1.46222	0.731110	−	0.682260i	$-0.239003\pi$
$277$	24.3362	1.46222	0.731110	+	0.682260i	$0.239003\pi$
$278$	0	0
$279$	3.16119	0.189255
$280$	0	0
$281$	−11.7032	−0.698157	−0.349078	−	0.937094i	$-0.613505\pi$
$281$	−11.7032	−0.698157	−0.349078	+	0.937094i	$0.613505\pi$
$282$	0	0
$283$	−20.9171	−1.24339	−0.621697	−	0.783257i	$-0.713557\pi$
$283$	−20.9171	−1.24339	−0.621697	+	0.783257i	$0.713557\pi$
$284$	0	0
$285$	−15.0939	−0.894087
$286$	0	0
$287$	−0.716268	−0.0422800
$288$	0	0
$289$	0	0
$290$	0	0
$291$	18.1711	1.06521
$292$	0	0
$293$	9.01821	0.526849	0.263425	−	0.964680i	$-0.415148\pi$
$293$	9.01821	0.526849	0.263425	+	0.964680i	$0.415148\pi$
$294$	0	0
$295$	1.87965	0.109437
$296$	0	0
$297$	9.50214	0.551370
$298$	0	0
$299$	12.4368	0.719238
$300$	0	0
$301$	1.87713	0.108196
$302$	0	0
$303$	−17.3882	−0.998924
$304$	0	0
$305$	−16.5205	−0.945964
$306$	0	0
$307$	−12.6478	−0.721847	−0.360924	−	0.932595i	$-0.617538\pi$
$307$	−12.6478	−0.721847	−0.360924	+	0.932595i	$0.617538\pi$
$308$	0	0
$309$	8.67306	0.493393
$310$	0	0
$311$	17.0239	0.965338	0.482669	−	0.875803i	$-0.339667\pi$
$311$	17.0239	0.965338	0.482669	+	0.875803i	$0.339667\pi$
$312$	0	0
$313$	−22.1457	−1.25175	−0.625873	−	0.779925i	$-0.715257\pi$
$313$	−22.1457	−1.25175	−0.625873	+	0.779925i	$0.715257\pi$
$314$	0	0
$315$	6.45836	0.363887
$316$	0	0
$317$	−25.8353	−1.45105	−0.725526	−	0.688194i	$-0.758404\pi$
$317$	−25.8353	−1.45105	−0.725526	+	0.688194i	$0.758404\pi$
$318$	0	0
$319$	−1.70108	−0.0952420
$320$	0	0
$321$	−3.30674	−0.184564
$322$	0	0
$323$	0	0
$324$	0	0
$325$	11.3964	0.632160
$326$	0	0
$327$	2.89600	0.160149
$328$	0	0
$329$	−7.11332	−0.392170
$330$	0	0
$331$	28.9367	1.59050	0.795252	−	0.606279i	$-0.207339\pi$
$331$	28.9367	1.59050	0.795252	+	0.606279i	$0.207339\pi$
$332$	0	0
$333$	3.16790	0.173600
$334$	0	0
$335$	−3.86323	−0.211071
$336$	0	0
$337$	32.9305	1.79384	0.896919	−	0.442194i	$-0.145800\pi$
$337$	32.9305	1.79384	0.896919	+	0.442194i	$0.145800\pi$
$338$	0	0
$339$	−15.0429	−0.817017
$340$	0	0
$341$	−2.95634	−0.160095
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	14.1882	0.763868
$346$	0	0
$347$	29.3061	1.57323	0.786617	−	0.617441i	$-0.211831\pi$
$347$	29.3061	1.57323	0.786617	+	0.617441i	$0.211831\pi$
$348$	0	0
$349$	19.1762	1.02648	0.513240	−	0.858245i	$-0.328445\pi$
$349$	19.1762	1.02648	0.513240	+	0.858245i	$0.328445\pi$
$350$	0	0
$351$	12.3711	0.660322
$352$	0	0
$353$	−6.23962	−0.332101	−0.166051	−	0.986117i	$-0.553102\pi$
$353$	−6.23962	−0.332101	−0.166051	+	0.986117i	$0.553102\pi$
$354$	0	0
$355$	−13.8015	−0.732508
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.5638	1.13810	0.569048	−	0.822304i	$-0.307312\pi$
$359$	21.5638	1.13810	0.569048	+	0.822304i	$0.307312\pi$
$360$	0	0
$361$	7.77218	0.409062
$362$	0	0
$363$	6.77545	0.355619
$364$	0	0
$365$	−17.0944	−0.894762
$366$	0	0
$367$	−3.95597	−0.206500	−0.103250	−	0.994655i	$-0.532924\pi$
$367$	−3.95597	−0.206500	−0.103250	+	0.994655i	$0.532924\pi$
$368$	0	0
$369$	1.50426	0.0783089
$370$	0	0
$371$	−8.36741	−0.434414
$372$	0	0
$373$	12.2776	0.635708	0.317854	−	0.948140i	$-0.397038\pi$
$373$	12.2776	0.635708	0.317854	+	0.948140i	$0.397038\pi$
$374$	0	0
$375$	−1.58444	−0.0818202
$376$	0	0
$377$	−2.21469	−0.114062
$378$	0	0
$379$	33.7224	1.73221	0.866103	−	0.499866i	$-0.166618\pi$
$379$	33.7224	1.73221	0.866103	+	0.499866i	$0.166618\pi$
$380$	0	0
$381$	19.9554	1.02234
$382$	0	0
$383$	20.4969	1.04734	0.523670	−	0.851921i	$-0.324562\pi$
$383$	20.4969	1.04734	0.523670	+	0.851921i	$0.324562\pi$
$384$	0	0
$385$	−6.03984	−0.307819
$386$	0	0
$387$	−3.94225	−0.200396
$388$	0	0
$389$	−13.1854	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$389$	−13.1854	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−6.85457	−0.345767
$394$	0	0
$395$	25.4719	1.28163
$396$	0	0
$397$	10.8140	0.542740	0.271370	−	0.962475i	$-0.412523\pi$
$397$	10.8140	0.542740	0.271370	+	0.962475i	$0.412523\pi$
$398$	0	0
$399$	4.90827	0.245721
$400$	0	0
$401$	−4.27206	−0.213336	−0.106668	−	0.994295i	$-0.534018\pi$
$401$	−4.27206	−0.213336	−0.106668	+	0.994295i	$0.534018\pi$
$402$	0	0
$403$	−3.84895	−0.191730
$404$	0	0
$405$	−5.26174	−0.261458
$406$	0	0
$407$	−2.96261	−0.146851
$408$	0	0
$409$	38.2296	1.89033	0.945167	−	0.326588i	$-0.105899\pi$
$409$	38.2296	1.89033	0.945167	+	0.326588i	$0.105899\pi$
$410$	0	0
$411$	−20.7148	−1.02178
$412$	0	0
$413$	−0.611228	−0.0300766
$414$	0	0
$415$	40.3318	1.97981
$416$	0	0
$417$	−4.29401	−0.210279
$418$	0	0
$419$	21.4262	1.04674	0.523369	−	0.852106i	$-0.324675\pi$
$419$	21.4262	1.04674	0.523369	+	0.852106i	$0.324675\pi$
$420$	0	0
$421$	−15.6311	−0.761815	−0.380907	−	0.924613i	$-0.624388\pi$
$421$	−15.6311	−0.761815	−0.380907	+	0.924613i	$0.624388\pi$
$422$	0	0
$423$	14.9390	0.726358
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.37219	0.259978
$428$	0	0
$429$	−4.76408	−0.230012
$430$	0	0
$431$	9.79841	0.471973	0.235986	−	0.971756i	$-0.424168\pi$
$431$	9.79841	0.471973	0.235986	+	0.971756i	$0.424168\pi$
$432$	0	0
$433$	29.4813	1.41678	0.708390	−	0.705822i	$-0.249422\pi$
$433$	29.4813	1.41678	0.708390	+	0.705822i	$0.249422\pi$
$434$	0	0
$435$	−2.52657	−0.121140
$436$	0	0
$437$	−25.1657	−1.20384
$438$	0	0
$439$	−26.2230	−1.25156	−0.625779	−	0.780001i	$-0.715219\pi$
$439$	−26.2230	−1.25156	−0.625779	+	0.780001i	$0.715219\pi$
$440$	0	0
$441$	−2.10014	−0.100007
$442$	0	0
$443$	23.6159	1.12203	0.561013	−	0.827807i	$-0.310412\pi$
$443$	23.6159	1.12203	0.561013	+	0.827807i	$0.310412\pi$
$444$	0	0
$445$	24.9347	1.18202
$446$	0	0
$447$	7.99176	0.377997
$448$	0	0
$449$	−22.3051	−1.05264	−0.526321	−	0.850286i	$-0.676429\pi$
$449$	−22.3051	−1.05264	−0.526321	+	0.850286i	$0.676429\pi$
$450$	0	0
$451$	−1.40678	−0.0662429
$452$	0	0
$453$	−11.6878	−0.549140
$454$	0	0
$455$	−7.86346	−0.368645
$456$	0	0
$457$	−5.00361	−0.234059	−0.117029	−	0.993128i	$-0.537337\pi$
$457$	−5.00361	−0.234059	−0.117029	+	0.993128i	$0.537337\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.6446	0.868366	0.434183	−	0.900825i	$-0.357037\pi$
$461$	18.6446	0.868366	0.434183	+	0.900825i	$0.357037\pi$
$462$	0	0
$463$	−33.5355	−1.55853	−0.779264	−	0.626696i	$-0.784407\pi$
$463$	−33.5355	−1.55853	−0.779264	+	0.626696i	$0.784407\pi$
$464$	0	0
$465$	−4.39099	−0.203627
$466$	0	0
$467$	1.10557	0.0511598	0.0255799	−	0.999673i	$-0.491857\pi$
$467$	1.10557	0.0511598	0.0255799	+	0.999673i	$0.491857\pi$
$468$	0	0
$469$	1.25625	0.0580084
$470$	0	0
$471$	5.82005	0.268174
$472$	0	0
$473$	3.68678	0.169518
$474$	0	0
$475$	−23.0606	−1.05809
$476$	0	0
$477$	17.5727	0.804601
$478$	0	0
$479$	28.1986	1.28843	0.644213	−	0.764846i	$-0.277185\pi$
$479$	28.1986	1.28843	0.644213	+	0.764846i	$0.277185\pi$
$480$	0	0
$481$	−3.85712	−0.175870
$482$	0	0
$483$	−4.61375	−0.209933
$484$	0	0
$485$	58.9070	2.67483
$486$	0	0
$487$	−41.9575	−1.90128	−0.950638	−	0.310301i	$-0.899570\pi$
$487$	−41.9575	−1.90128	−0.950638	+	0.310301i	$0.899570\pi$
$488$	0	0
$489$	−12.9878	−0.587328
$490$	0	0
$491$	27.4261	1.23772	0.618860	−	0.785501i	$-0.287595\pi$
$491$	27.4261	1.23772	0.618860	+	0.785501i	$0.287595\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	12.6845	0.570127
$496$	0	0
$497$	4.48800	0.201314
$498$	0	0
$499$	−9.09373	−0.407091	−0.203546	−	0.979065i	$-0.565247\pi$
$499$	−9.09373	−0.407091	−0.203546	+	0.979065i	$0.565247\pi$
$500$	0	0
$501$	6.17240	0.275762
$502$	0	0
$503$	11.9890	0.534565	0.267283	−	0.963618i	$-0.413874\pi$
$503$	11.9890	0.534565	0.267283	+	0.963618i	$0.413874\pi$
$504$	0	0
$505$	−56.3690	−2.50839
$506$	0	0
$507$	6.12939	0.272216
$508$	0	0
$509$	31.6070	1.40096	0.700479	−	0.713673i	$-0.252970\pi$
$509$	31.6070	1.40096	0.700479	+	0.713673i	$0.252970\pi$
$510$	0	0
$511$	5.55879	0.245907
$512$	0	0
$513$	−25.0329	−1.10523
$514$	0	0
$515$	28.1163	1.23895
$516$	0	0
$517$	−13.9709	−0.614439
$518$	0	0
$519$	−20.0317	−0.879293
$520$	0	0
$521$	28.9582	1.26868	0.634340	−	0.773054i	$-0.281272\pi$
$521$	28.9582	1.26868	0.634340	+	0.773054i	$0.281272\pi$
$522$	0	0
$523$	19.6802	0.860555	0.430278	−	0.902697i	$-0.358416\pi$
$523$	19.6802	0.860555	0.430278	+	0.902697i	$0.358416\pi$
$524$	0	0
$525$	−4.22781	−0.184517
$526$	0	0
$527$	0	0
$528$	0	0
$529$	0.655661	0.0285070
$530$	0	0
$531$	1.28367	0.0557064
$532$	0	0
$533$	−1.83154	−0.0793327
$534$	0	0
$535$	−10.7198	−0.463457
$536$	0	0
$537$	8.98160	0.387585
$538$	0	0
$539$	1.96405	0.0845975
$540$	0	0
$541$	17.9593	0.772133	0.386066	−	0.922471i	$-0.373834\pi$
$541$	17.9593	0.772133	0.386066	+	0.922471i	$0.373834\pi$
$542$	0	0
$543$	16.0149	0.687263
$544$	0	0
$545$	9.38826	0.402149
$546$	0	0
$547$	21.8256	0.933195	0.466598	−	0.884470i	$-0.345480\pi$
$547$	21.8256	0.933195	0.466598	+	0.884470i	$0.345480\pi$
$548$	0	0
$549$	−11.2824	−0.481519
$550$	0	0
$551$	4.48140	0.190914
$552$	0	0
$553$	−8.28300	−0.352229
$554$	0	0
$555$	−4.40031	−0.186783
$556$	0	0
$557$	11.0590	0.468586	0.234293	−	0.972166i	$-0.424723\pi$
$557$	11.0590	0.468586	0.234293	+	0.972166i	$0.424723\pi$
$558$	0	0
$559$	4.79994	0.203016
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.41320	−0.185994	−0.0929970	−	0.995666i	$-0.529645\pi$
$563$	−4.41320	−0.185994	−0.0929970	+	0.995666i	$0.529645\pi$
$564$	0	0
$565$	−48.7660	−2.05160
$566$	0	0
$567$	1.71102	0.0718563
$568$	0	0
$569$	14.8723	0.623480	0.311740	−	0.950167i	$-0.399088\pi$
$569$	14.8723	0.623480	0.311740	+	0.950167i	$0.399088\pi$
$570$	0	0
$571$	−22.6262	−0.946877	−0.473438	−	0.880827i	$-0.656987\pi$
$571$	−22.6262	−0.946877	−0.473438	+	0.880827i	$0.656987\pi$
$572$	0	0
$573$	9.04014	0.377657
$574$	0	0
$575$	21.6768	0.903987
$576$	0	0
$577$	−8.74877	−0.364216	−0.182108	−	0.983279i	$-0.558292\pi$
$577$	−8.74877	−0.364216	−0.182108	+	0.983279i	$0.558292\pi$
$578$	0	0
$579$	−2.01434	−0.0837131
$580$	0	0
$581$	−13.1152	−0.544109
$582$	0	0
$583$	−16.4340	−0.680626
$584$	0	0
$585$	16.5144	0.682786
$586$	0	0
$587$	14.8510	0.612966	0.306483	−	0.951876i	$-0.400848\pi$
$587$	14.8510	0.612966	0.306483	+	0.951876i	$0.400848\pi$
$588$	0	0
$589$	7.78832	0.320912
$590$	0	0
$591$	−3.48628	−0.143406
$592$	0	0
$593$	−38.4055	−1.57713	−0.788563	−	0.614954i	$-0.789175\pi$
$593$	−38.4055	−1.57713	−0.788563	+	0.614954i	$0.789175\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.47430	0.346830
$598$	0	0
$599$	7.91405	0.323359	0.161680	−	0.986843i	$-0.448309\pi$
$599$	7.91405	0.323359	0.161680	+	0.986843i	$0.448309\pi$
$600$	0	0
$601$	17.2672	0.704345	0.352173	−	0.935935i	$-0.385443\pi$
$601$	17.2672	0.704345	0.352173	+	0.935935i	$0.385443\pi$
$602$	0	0
$603$	−2.63831	−0.107440
$604$	0	0
$605$	21.9647	0.892991
$606$	0	0
$607$	−11.4062	−0.462964	−0.231482	−	0.972839i	$-0.574357\pi$
$607$	−11.4062	−0.462964	−0.231482	+	0.972839i	$0.574357\pi$
$608$	0	0
$609$	0.821597	0.0332928
$610$	0	0
$611$	−18.1892	−0.735854
$612$	0	0
$613$	−0.0371290	−0.00149962	−0.000749812	−	1.00000i	$-0.500239\pi$
$613$	−0.0371290	−0.00149962	−0.000749812		1.00000i	$0.500239\pi$
$614$	0	0
$615$	−2.08947	−0.0842555
$616$	0	0
$617$	15.8563	0.638351	0.319176	−	0.947696i	$-0.396594\pi$
$617$	15.8563	0.638351	0.319176	+	0.947696i	$0.396594\pi$
$618$	0	0
$619$	−12.9294	−0.519677	−0.259839	−	0.965652i	$-0.583669\pi$
$619$	−12.9294	−0.519677	−0.259839	+	0.965652i	$0.583669\pi$
$620$	0	0
$621$	23.5308	0.944259
$622$	0	0
$623$	−8.10832	−0.324853
$624$	0	0
$625$	−27.4207	−1.09683
$626$	0	0
$627$	9.64009	0.384988
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−43.4836	−1.73105	−0.865527	−	0.500862i	$-0.833016\pi$
$631$	−43.4836	−1.73105	−0.865527	+	0.500862i	$0.833016\pi$
$632$	0	0
$633$	−25.9211	−1.03027
$634$	0	0
$635$	64.6913	2.56720
$636$	0	0
$637$	2.55706	0.101314
$638$	0	0
$639$	−9.42544	−0.372865
$640$	0	0
$641$	17.3258	0.684328	0.342164	−	0.939640i	$-0.388840\pi$
$641$	17.3258	0.684328	0.342164	+	0.939640i	$0.388840\pi$
$642$	0	0
$643$	−30.6710	−1.20954	−0.604772	−	0.796399i	$-0.706736\pi$
$643$	−30.6710	−1.20954	−0.604772	+	0.796399i	$0.706736\pi$
$644$	0	0
$645$	5.47590	0.215613
$646$	0	0
$647$	9.53013	0.374668	0.187334	−	0.982296i	$-0.440015\pi$
$647$	9.53013	0.374668	0.187334	+	0.982296i	$0.440015\pi$
$648$	0	0
$649$	−1.20048	−0.0471230
$650$	0	0
$651$	1.42787	0.0559626
$652$	0	0
$653$	−32.4206	−1.26872	−0.634358	−	0.773040i	$-0.718735\pi$
$653$	−32.4206	−1.26872	−0.634358	+	0.773040i	$0.718735\pi$
$654$	0	0
$655$	−22.2212	−0.868252
$656$	0	0
$657$	−11.6743	−0.455456
$658$	0	0
$659$	23.1542	0.901958	0.450979	−	0.892534i	$-0.351075\pi$
$659$	23.1542	0.901958	0.450979	+	0.892534i	$0.351075\pi$
$660$	0	0
$661$	36.7821	1.43066	0.715330	−	0.698787i	$-0.246277\pi$
$661$	36.7821	1.43066	0.715330	+	0.698787i	$0.246277\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	15.9116	0.617027
$666$	0	0
$667$	−4.21249	−0.163108
$668$	0	0
$669$	−16.2649	−0.628836
$670$	0	0
$671$	10.5512	0.407326
$672$	0	0
$673$	−6.03779	−0.232740	−0.116370	−	0.993206i	$-0.537126\pi$
$673$	−6.03779	−0.232740	−0.116370	+	0.993206i	$0.537126\pi$
$674$	0	0
$675$	21.5624	0.829938
$676$	0	0
$677$	−49.4508	−1.90055	−0.950274	−	0.311414i	$-0.899197\pi$
$677$	−49.4508	−1.90055	−0.950274	+	0.311414i	$0.899197\pi$
$678$	0	0
$679$	−19.1555	−0.735121
$680$	0	0
$681$	4.22306	0.161828
$682$	0	0
$683$	5.11632	0.195770	0.0978852	−	0.995198i	$-0.468792\pi$
$683$	5.11632	0.195770	0.0978852	+	0.995198i	$0.468792\pi$
$684$	0	0
$685$	−67.1532	−2.56579
$686$	0	0
$687$	−25.3856	−0.968519
$688$	0	0
$689$	−21.3959	−0.815120
$690$	0	0
$691$	10.5046	0.399613	0.199807	−	0.979835i	$-0.435969\pi$
$691$	10.5046	0.399613	0.199807	+	0.979835i	$0.435969\pi$
$692$	0	0
$693$	−4.12478	−0.156687
$694$	0	0
$695$	−13.9203	−0.528028
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−9.21806	−0.348659
$700$	0	0
$701$	−29.3140	−1.10717	−0.553587	−	0.832791i	$-0.686742\pi$
$701$	−29.3140	−1.10717	−0.553587	+	0.832791i	$0.686742\pi$
$702$	0	0
$703$	7.80485	0.294366
$704$	0	0
$705$	−20.7507	−0.781516
$706$	0	0
$707$	18.3302	0.689377
$708$	0	0
$709$	−0.624981	−0.0234716	−0.0117358	−	0.999931i	$-0.503736\pi$
$709$	−0.624981	−0.0234716	−0.0117358	+	0.999931i	$0.503736\pi$
$710$	0	0
$711$	17.3955	0.652381
$712$	0	0
$713$	−7.32098	−0.274173
$714$	0	0
$715$	−15.4442	−0.577581
$716$	0	0
$717$	−28.9765	−1.08215
$718$	0	0
$719$	−27.7342	−1.03431	−0.517156	−	0.855891i	$-0.673009\pi$
$719$	−27.7342	−1.03431	−0.517156	+	0.855891i	$0.673009\pi$
$720$	0	0
$721$	−9.14293	−0.340500
$722$	0	0
$723$	−14.1860	−0.527583
$724$	0	0
$725$	−3.86011	−0.143361
$726$	0	0
$727$	−8.61018	−0.319334	−0.159667	−	0.987171i	$-0.551042\pi$
$727$	−8.61018	−0.319334	−0.159667	+	0.987171i	$0.551042\pi$
$728$	0	0
$729$	10.1748	0.376845
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−45.6053	−1.68447	−0.842235	−	0.539111i	$-0.818760\pi$
$733$	−45.6053	−1.68447	−0.842235	+	0.539111i	$0.818760\pi$
$734$	0	0
$735$	2.91716	0.107601
$736$	0	0
$737$	2.46734	0.0908857
$738$	0	0
$739$	24.5363	0.902583	0.451292	−	0.892376i	$-0.350963\pi$
$739$	24.5363	0.902583	0.451292	+	0.892376i	$0.350963\pi$
$740$	0	0
$741$	12.5507	0.461063
$742$	0	0
$743$	17.4318	0.639512	0.319756	−	0.947500i	$-0.396399\pi$
$743$	17.4318	0.639512	0.319756	+	0.947500i	$0.396399\pi$
$744$	0	0
$745$	25.9077	0.949184
$746$	0	0
$747$	27.5437	1.00777
$748$	0	0
$749$	3.48588	0.127371
$750$	0	0
$751$	38.4257	1.40217	0.701086	−	0.713076i	$-0.252699\pi$
$751$	38.4257	1.40217	0.701086	+	0.713076i	$0.252699\pi$
$752$	0	0
$753$	16.5019	0.601364
$754$	0	0
$755$	−37.8895	−1.37894
$756$	0	0
$757$	−11.8991	−0.432480	−0.216240	−	0.976340i	$-0.569379\pi$
$757$	−11.8991	−0.432480	−0.216240	+	0.976340i	$0.569379\pi$
$758$	0	0
$759$	−9.06164	−0.328917
$760$	0	0
$761$	21.6526	0.784906	0.392453	−	0.919772i	$-0.371627\pi$
$761$	21.6526	0.784906	0.392453	+	0.919772i	$0.371627\pi$
$762$	0	0
$763$	−3.05289	−0.110522
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.56294	−0.0564347
$768$	0	0
$769$	−36.4694	−1.31512	−0.657560	−	0.753402i	$-0.728411\pi$
$769$	−36.4694	−1.31512	−0.657560	+	0.753402i	$0.728411\pi$
$770$	0	0
$771$	15.0811	0.543134
$772$	0	0
$773$	−0.315668	−0.0113538	−0.00567689	−	0.999984i	$-0.501807\pi$
$773$	−0.315668	−0.0113538	−0.00567689	+	0.999984i	$0.501807\pi$
$774$	0	0
$775$	−6.70857	−0.240979
$776$	0	0
$777$	1.43090	0.0513333
$778$	0	0
$779$	3.70610	0.132785
$780$	0	0
$781$	8.81465	0.315413
$782$	0	0
$783$	−4.19026	−0.149748
$784$	0	0
$785$	18.8675	0.673408
$786$	0	0
$787$	23.5689	0.840141	0.420070	−	0.907492i	$-0.362005\pi$
$787$	23.5689	0.840141	0.420070	+	0.907492i	$0.362005\pi$
$788$	0	0
$789$	−26.3569	−0.938331
$790$	0	0
$791$	15.8578	0.563840
$792$	0	0
$793$	13.7370	0.487815
$794$	0	0
$795$	−24.4091	−0.865700
$796$	0	0
$797$	−5.74105	−0.203358	−0.101679	−	0.994817i	$-0.532422\pi$
$797$	−5.74105	−0.203358	−0.101679	+	0.994817i	$0.532422\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	17.0286	0.601677
$802$	0	0
$803$	10.9177	0.385279
$804$	0	0
$805$	−14.9569	−0.527161
$806$	0	0
$807$	−0.366273	−0.0128934
$808$	0	0
$809$	−33.0934	−1.16350	−0.581751	−	0.813367i	$-0.697632\pi$
$809$	−33.0934	−1.16350	−0.581751	+	0.813367i	$0.697632\pi$
$810$	0	0
$811$	38.6316	1.35654	0.678269	−	0.734813i	$-0.262730\pi$
$811$	38.6316	1.35654	0.678269	+	0.734813i	$0.262730\pi$
$812$	0	0
$813$	18.8875	0.662414
$814$	0	0
$815$	−42.1038	−1.47483
$816$	0	0
$817$	−9.71264	−0.339802
$818$	0	0
$819$	−5.37018	−0.187649
$820$	0	0
$821$	27.7502	0.968490	0.484245	−	0.874933i	$-0.339094\pi$
$821$	27.7502	0.968490	0.484245	+	0.874933i	$0.339094\pi$
$822$	0	0
$823$	46.2818	1.61328	0.806640	−	0.591042i	$-0.201283\pi$
$823$	46.2818	1.61328	0.806640	+	0.591042i	$0.201283\pi$
$824$	0	0
$825$	−8.30362	−0.289095
$826$	0	0
$827$	26.5345	0.922694	0.461347	−	0.887220i	$-0.347366\pi$
$827$	26.5345	0.922694	0.461347	+	0.887220i	$0.347366\pi$
$828$	0	0
$829$	22.7980	0.791806	0.395903	−	0.918292i	$-0.370432\pi$
$829$	22.7980	0.791806	0.395903	+	0.918292i	$0.370432\pi$
$830$	0	0
$831$	−23.0855	−0.800827
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.0097	0.692464
$836$	0	0
$837$	−7.28234	−0.251715
$838$	0	0
$839$	35.3037	1.21882	0.609409	−	0.792856i	$-0.291407\pi$
$839$	35.3037	1.21882	0.609409	+	0.792856i	$0.291407\pi$
$840$	0	0
$841$	−28.2499	−0.974133
$842$	0	0
$843$	11.1018	0.382366
$844$	0	0
$845$	19.8703	0.683558
$846$	0	0
$847$	−7.14252	−0.245420
$848$	0	0
$849$	19.8422	0.680982
$850$	0	0
$851$	−7.33653	−0.251493
$852$	0	0
$853$	−15.7748	−0.540118	−0.270059	−	0.962844i	$-0.587043\pi$
$853$	−15.7748	−0.540118	−0.270059	+	0.962844i	$0.587043\pi$
$854$	0	0
$855$	−33.4167	−1.14283
$856$	0	0
$857$	−1.25187	−0.0427632	−0.0213816	−	0.999771i	$-0.506806\pi$
$857$	−1.25187	−0.0427632	−0.0213816	+	0.999771i	$0.506806\pi$
$858$	0	0
$859$	−7.33442	−0.250247	−0.125124	−	0.992141i	$-0.539933\pi$
$859$	−7.33442	−0.250247	−0.125124	+	0.992141i	$0.539933\pi$
$860$	0	0
$861$	0.679458	0.0231559
$862$	0	0
$863$	30.7762	1.04763	0.523817	−	0.851831i	$-0.324507\pi$
$863$	30.7762	1.04763	0.523817	+	0.851831i	$0.324507\pi$
$864$	0	0
$865$	−64.9387	−2.20798
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−16.2682	−0.551861
$870$	0	0
$871$	3.21231	0.108845
$872$	0	0
$873$	40.2293	1.36156
$874$	0	0
$875$	1.67028	0.0564658
$876$	0	0
$877$	19.0041	0.641723	0.320861	−	0.947126i	$-0.396028\pi$
$877$	19.0041	0.641723	0.320861	+	0.947126i	$0.396028\pi$
$878$	0	0
$879$	−8.55474	−0.288544
$880$	0	0
$881$	−17.4376	−0.587488	−0.293744	−	0.955884i	$-0.594901\pi$
$881$	−17.4376	−0.587488	−0.293744	+	0.955884i	$0.594901\pi$
$882$	0	0
$883$	36.2538	1.22004	0.610020	−	0.792386i	$-0.291162\pi$
$883$	36.2538	1.22004	0.610020	+	0.792386i	$0.291162\pi$
$884$	0	0
$885$	−1.78305	−0.0599366
$886$	0	0
$887$	−16.3488	−0.548937	−0.274469	−	0.961596i	$-0.588502\pi$
$887$	−16.3488	−0.548937	−0.274469	+	0.961596i	$0.588502\pi$
$888$	0	0
$889$	−21.0365	−0.705540
$890$	0	0
$891$	3.36053	0.112582
$892$	0	0
$893$	36.8056	1.23165
$894$	0	0
$895$	29.1166	0.973259
$896$	0	0
$897$	−11.7976	−0.393911
$898$	0	0
$899$	1.30369	0.0434804
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−1.78067	−0.0592568
$904$	0	0
$905$	51.9170	1.72578
$906$	0	0
$907$	27.4870	0.912692	0.456346	−	0.889802i	$-0.349158\pi$
$907$	27.4870	0.912692	0.456346	+	0.889802i	$0.349158\pi$
$908$	0	0
$909$	−38.4960	−1.27683
$910$	0	0
$911$	37.9207	1.25637	0.628184	−	0.778065i	$-0.283798\pi$
$911$	37.9207	1.25637	0.628184	+	0.778065i	$0.283798\pi$
$912$	0	0
$913$	−25.7588	−0.852492
$914$	0	0
$915$	15.6715	0.518085
$916$	0	0
$917$	7.22592	0.238621
$918$	0	0
$919$	−27.3818	−0.903242	−0.451621	−	0.892210i	$-0.649154\pi$
$919$	−27.3818	−0.903242	−0.451621	+	0.892210i	$0.649154\pi$
$920$	0	0
$921$	11.9978	0.395341
$922$	0	0
$923$	11.4761	0.377740
$924$	0	0
$925$	−6.72282	−0.221045
$926$	0	0
$927$	19.2015	0.630658
$928$	0	0
$929$	44.5007	1.46002	0.730010	−	0.683437i	$-0.239515\pi$
$929$	44.5007	1.46002	0.730010	+	0.683437i	$0.239515\pi$
$930$	0	0
$931$	−5.17418	−0.169577
$932$	0	0
$933$	−16.1490	−0.528696
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.8667	−0.747023	−0.373512	−	0.927626i	$-0.621846\pi$
$937$	−22.8667	−0.747023	−0.373512	+	0.927626i	$0.621846\pi$
$938$	0	0
$939$	21.0075	0.685555
$940$	0	0
$941$	−37.8614	−1.23425	−0.617124	−	0.786866i	$-0.711702\pi$
$941$	−37.8614	−1.23425	−0.617124	+	0.786866i	$0.711702\pi$
$942$	0	0
$943$	−3.48372	−0.113445
$944$	0	0
$945$	−14.8779	−0.483979
$946$	0	0
$947$	−0.854066	−0.0277534	−0.0138767	−	0.999904i	$-0.504417\pi$
$947$	−0.854066	−0.0277534	−0.0138767	+	0.999904i	$0.504417\pi$
$948$	0	0
$949$	14.2142	0.461411
$950$	0	0
$951$	24.5075	0.794711
$952$	0	0
$953$	21.8074	0.706410	0.353205	−	0.935546i	$-0.385092\pi$
$953$	21.8074	0.706410	0.353205	+	0.935546i	$0.385092\pi$
$954$	0	0
$955$	29.3063	0.948331
$956$	0	0
$957$	1.61366	0.0521621
$958$	0	0
$959$	21.8370	0.705154
$960$	0	0
$961$	−28.7343	−0.926913
$962$	0	0
$963$	−7.32085	−0.235911
$964$	0	0
$965$	−6.53009	−0.210211
$966$	0	0
$967$	36.5451	1.17521	0.587606	−	0.809147i	$-0.300071\pi$
$967$	36.5451	1.17521	0.587606	+	0.809147i	$0.300071\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.7727	−1.21218	−0.606092	−	0.795395i	$-0.707264\pi$
$971$	−37.7727	−1.21218	−0.606092	+	0.795395i	$0.707264\pi$
$972$	0	0
$973$	4.52664	0.145118
$974$	0	0
$975$	−10.8107	−0.346221
$976$	0	0
$977$	−55.8437	−1.78660	−0.893300	−	0.449461i	$-0.851616\pi$
$977$	−55.8437	−1.78660	−0.893300	+	0.449461i	$0.851616\pi$
$978$	0	0
$979$	−15.9251	−0.508969
$980$	0	0
$981$	6.41151	0.204704
$982$	0	0
$983$	26.3263	0.839678	0.419839	−	0.907599i	$-0.362087\pi$
$983$	26.3263	0.839678	0.419839	+	0.907599i	$0.362087\pi$
$984$	0	0
$985$	−11.3018	−0.360106
$986$	0	0
$987$	6.74775	0.214783
$988$	0	0
$989$	9.12983	0.290312
$990$	0	0
$991$	−12.6119	−0.400632	−0.200316	−	0.979731i	$-0.564197\pi$
$991$	−12.6119	−0.400632	−0.200316	+	0.979731i	$0.564197\pi$
$992$	0	0
$993$	−27.4496	−0.871086
$994$	0	0
$995$	27.4720	0.870921
$996$	0	0
$997$	−20.3000	−0.642906	−0.321453	−	0.946926i	$-0.604171\pi$
$997$	−20.3000	−0.642906	−0.321453	+	0.946926i	$0.604171\pi$
$998$	0	0
$999$	−7.29780	−0.230892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8092.2.a.z.1.7		20
17.5	odd	16		476.2.u.a.365.7	yes	40
17.7	odd	16		476.2.u.a.253.7	✓	40
17.16	even	2		8092.2.a.y.1.14		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.u.a.253.7	✓	40	17.7	odd	16
476.2.u.a.365.7	yes	40	17.5	odd	16
8092.2.a.y.1.14		20	17.16	even	2
8092.2.a.z.1.7		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8092 = 2^{2} \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8092.a (trivial)

\(f(q)\)	\(=\)	\(q-0.948608 q^{3} -3.07520 q^{5} +1.00000 q^{7} -2.10014 q^{9} +1.96405 q^{11} +2.55706 q^{13} +2.91716 q^{15} -5.17418 q^{19} -0.948608 q^{21} +4.86371 q^{23} +4.45685 q^{25} +4.83804 q^{27} -0.866107 q^{29} -1.50523 q^{31} -1.86311 q^{33} -3.07520 q^{35} -1.50842 q^{37} -2.42565 q^{39} -0.716268 q^{41} +1.87713 q^{43} +6.45836 q^{45} -7.11332 q^{47} +1.00000 q^{49} -8.36741 q^{53} -6.03984 q^{55} +4.90827 q^{57} -0.611228 q^{59} +5.37219 q^{61} -2.10014 q^{63} -7.86346 q^{65} +1.25625 q^{67} -4.61375 q^{69} +4.48800 q^{71} +5.55879 q^{73} -4.22781 q^{75} +1.96405 q^{77} -8.28300 q^{79} +1.71102 q^{81} -13.1152 q^{83} +0.821597 q^{87} -8.10832 q^{89} +2.55706 q^{91} +1.42787 q^{93} +15.9116 q^{95} -19.1555 q^{97} -4.12478 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 8 q^{3} + 8 q^{5} + 20 q^{7} + 28 q^{9} + 16 q^{11} + 8 q^{13} + 8 q^{15} + 8 q^{19} + 8 q^{21} + 40 q^{23} + 36 q^{25} + 32 q^{27} + 16 q^{29} + 16 q^{31} + 8 q^{33} + 8 q^{35} + 24 q^{37} + 16 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) \) 20 * q + 8 * q^3 + 8 * q^5 + 20 * q^7 + 28 * q^9 + 16 * q^11 + 8 * q^13 + 8 * q^15 + 8 * q^19 + 8 * q^21 + 40 * q^23 + 36 * q^25 + 32 * q^27 + 16 * q^29 + 16 * q^31 + 8 * q^33 + 8 * q^35 + 24 * q^37 + 16 * q^39 - 16 * q^41 + 16 * q^43 + 24 * q^45 + 20 * q^49 + 24 * q^53 - 8 * q^55 + 24 * q^57 + 8 * q^61 + 28 * q^63 + 8 * q^65 - 8 * q^67 + 8 * q^69 + 72 * q^71 + 40 * q^75 + 16 * q^77 + 60 * q^81 + 8 * q^83 - 24 * q^89 + 8 * q^91 + 32 * q^93 + 48 * q^95 - 16 * q^97 + 72 * q^99

Introduction

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