LMFDB - Embedded newform 832.2.b.c.417.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [832,2,Mod(417,832)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("832.417"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-12,0,0,0,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$6.64355344817$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.195105024.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 $ x^8 - 4x^7 + 5x^6 + 4x^5 - 20x^4 + 12x^3 + 45x^2 - 108*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.1
Root			$0.560908 + 1.63871i$ of defining polynomial
Character	$\chi$	$=$	832.417
Dual form			832.2.b.c.417.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.27743i q^{3} -3.27743i q^{5} +1.61023 q^{7} -7.74153 q^{9} -1.66719i q^{11} -1.00000i q^{13} -10.7415 q^{15} -3.52106 q^{17} +3.79691i q^{19} -5.27743i q^{21} +8.79849 q^{23} -5.74153 q^{25} +15.5400i q^{27} -4.24363i q^{29} +7.68615 q^{31} -5.46410 q^{33} -5.27743i q^{35} -3.94304i q^{37} -3.27743 q^{39} +10.7985 q^{41} +2.49790i q^{43} +25.3723i q^{45} -8.94462 q^{47} -4.40714 q^{49} +11.5400i q^{51} +2.12972i q^{53} -5.46410 q^{55} +12.4441 q^{57} -4.88766i q^{59} +11.3533i q^{61} -12.4657 q^{63} -3.27743 q^{65} -0.0891755i q^{67} -28.8364i q^{69} +3.05538 q^{71} -10.1329 q^{73} +18.8174i q^{75} -2.68457i q^{77} -0.292266 q^{79} +27.7067 q^{81} +4.08918i q^{83} +11.5400i q^{85} -13.9082 q^{87} +1.22047 q^{89} -1.61023i q^{91} -25.1908i q^{93} +12.4441 q^{95} -14.0190 q^{97} +12.9066i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{9} - 36 q^{15} - 4 q^{17} + 24 q^{23} + 4 q^{25} - 20 q^{31} - 16 q^{33} - 4 q^{39} + 40 q^{41} - 40 q^{47} - 4 q^{49} - 16 q^{55} - 8 q^{57} - 44 q^{63} - 4 q^{65} + 56 q^{71} - 16 q^{73} - 32 q^{79}+ \cdots - 40 q^{97}+O(q^{100}) $ 8 * q - 12 * q^9 - 36 * q^15 - 4 * q^17 + 24 * q^23 + 4 * q^25 - 20 * q^31 - 16 * q^33 - 4 * q^39 + 40 * q^41 - 40 * q^47 - 4 * q^49 - 16 * q^55 - 8 * q^57 - 44 * q^63 - 4 * q^65 + 56 * q^71 - 16 * q^73 - 32 * q^79 + 40 * q^81 + 24 * q^87 - 16 * q^89 - 8 * q^95 - 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 3.27743i	− 1.89222i	−0.323840	−	0.946112i	$-0.604974\pi$
$3$	− 3.27743i	− 1.89222i	0.323840	−	0.946112i	$-0.395026\pi$
$4$	0	0
$5$	− 3.27743i	− 1.46571i	−0.680385	−	0.732855i	$-0.738187\pi$
$5$	− 3.27743i	− 1.46571i	0.680385	−	0.732855i	$-0.261813\pi$
$6$	0	0
$7$	1.61023	0.608612	0.304306	−	0.952574i	$-0.401576\pi$
$7$	1.61023	0.608612	0.304306	+	0.952574i	$0.401576\pi$
$8$	0	0
$9$	−7.74153	−2.58051
$10$	0	0
$11$	− 1.66719i	− 0.502677i	−0.967899	−	0.251339i	$-0.919129\pi$
$11$	− 1.66719i	− 0.502677i	0.967899	−	0.251339i	$-0.0808708\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−10.7415	−2.77345
$16$	0	0
$17$	−3.52106	−0.853982	−0.426991	−	0.904256i	$-0.640426\pi$
$17$	−3.52106	−0.853982	−0.426991	+	0.904256i	$0.640426\pi$
$18$	0	0
$19$	3.79691i	0.871071i	0.900172	+	0.435535i	$0.143441\pi$
$19$	3.79691i	0.871071i	−0.900172	+	0.435535i	$0.856559\pi$
$20$	0	0
$21$	− 5.27743i	− 1.15163i
$22$	0	0
$23$	8.79849	1.83461	0.917306	−	0.398184i	$-0.130359\pi$
$23$	8.79849	1.83461	0.917306	+	0.398184i	$0.130359\pi$
$24$	0	0
$25$	−5.74153	−1.14831
$26$	0	0
$27$	15.5400i	2.99068i
$28$	0	0
$29$	− 4.24363i	− 0.788023i	−0.919106	−	0.394011i	$-0.871087\pi$
$29$	− 4.24363i	− 0.788023i	0.919106	−	0.394011i	$-0.128913\pi$
$30$	0	0
$31$	7.68615	1.38047	0.690236	−	0.723584i	$-0.257506\pi$
$31$	7.68615	1.38047	0.690236	+	0.723584i	$0.257506\pi$
$32$	0	0
$33$	−5.46410	−0.951178
$34$	0	0
$35$	− 5.27743i	− 0.892048i
$36$	0	0
$37$	− 3.94304i	− 0.648232i	−0.946017	−	0.324116i	$-0.894933\pi$
$37$	− 3.94304i	− 0.648232i	0.946017	−	0.324116i	$-0.105067\pi$
$38$	0	0
$39$	−3.27743	−0.524808
$40$	0	0
$41$	10.7985	1.68644	0.843220	−	0.537568i	$-0.180657\pi$
$41$	10.7985	1.68644	0.843220	+	0.537568i	$0.180657\pi$
$42$	0	0
$43$	2.49790i	0.380926i	0.981694	+	0.190463i	$0.0609989\pi$
$43$	2.49790i	0.380926i	−0.981694	+	0.190463i	$0.939001\pi$
$44$	0	0
$45$	25.3723i	3.78228i
$46$	0	0
$47$	−8.94462	−1.30471	−0.652353	−	0.757915i	$-0.726218\pi$
$47$	−8.94462	−1.30471	−0.652353	+	0.757915i	$0.726218\pi$
$48$	0	0
$49$	−4.40714	−0.629592
$50$	0	0
$51$	11.5400i	1.61593i
$52$	0	0
$53$	2.12972i	0.292539i	0.989245	+	0.146270i	$0.0467267\pi$
$53$	2.12972i	0.292539i	−0.989245	+	0.146270i	$0.953273\pi$
$54$	0	0
$55$	−5.46410	−0.736779
$56$	0	0
$57$	12.4441	1.64826
$58$	0	0
$59$	− 4.88766i	− 0.636319i	−0.948037	−	0.318160i	$-0.896935\pi$
$59$	− 4.88766i	− 0.636319i	0.948037	−	0.318160i	$-0.103065\pi$
$60$	0	0
$61$	11.3533i	1.45365i	0.686825	+	0.726823i	$0.259004\pi$
$61$	11.3533i	1.45365i	−0.686825	+	0.726823i	$0.740996\pi$
$62$	0	0
$63$	−12.4657	−1.57053
$64$	0	0
$65$	−3.27743	−0.406515
$66$	0	0
$67$	− 0.0891755i	− 0.0108945i	−0.999985	−	0.00544726i	$-0.998266\pi$
$67$	− 0.0891755i	− 0.0108945i	0.999985	−	0.00544726i	$-0.00173393\pi$
$68$	0	0
$69$	− 28.8364i	− 3.47149i
$70$	0	0
$71$	3.05538	0.362607	0.181303	−	0.983427i	$-0.441968\pi$
$71$	3.05538	0.362607	0.181303	+	0.983427i	$0.441968\pi$
$72$	0	0
$73$	−10.1329	−1.18596	−0.592981	−	0.805216i	$-0.702049\pi$
$73$	−10.1329	−1.18596	−0.592981	+	0.805216i	$0.702049\pi$
$74$	0	0
$75$	18.8174i	2.17285i
$76$	0	0
$77$	− 2.68457i	− 0.305935i
$78$	0	0
$79$	−0.292266	−0.0328825	−0.0164413	−	0.999865i	$-0.505234\pi$
$79$	−0.292266	−0.0328825	−0.0164413	+	0.999865i	$0.505234\pi$
$80$	0	0
$81$	27.7067	3.07852
$82$	0	0
$83$	4.08918i	0.448845i	0.974492	+	0.224423i	$0.0720496\pi$
$83$	4.08918i	0.448845i	−0.974492	+	0.224423i	$0.927950\pi$
$84$	0	0
$85$	11.5400i	1.25169i
$86$	0	0
$87$	−13.9082	−1.49112
$88$	0	0
$89$	1.22047	0.129370	0.0646848	−	0.997906i	$-0.479396\pi$
$89$	1.22047	0.129370	0.0646848	+	0.997906i	$0.479396\pi$
$90$	0	0
$91$	− 1.61023i	− 0.168798i
$92$	0	0
$93$	− 25.1908i	− 2.61216i
$94$	0	0
$95$	12.4441	1.27674
$96$	0	0
$97$	−14.0190	−1.42341	−0.711705	−	0.702479i	$-0.752077\pi$
$97$	−14.0190	−1.42341	−0.711705	+	0.702479i	$0.752077\pi$
$98$	0	0
$99$	12.9066i	1.29716i
$100$	0	0
$101$	1.33123i	0.132462i	0.997804	+	0.0662312i	$0.0210975\pi$
$101$	1.33123i	0.132462i	−0.997804	+	0.0662312i	$0.978903\pi$
$102$	0	0
$103$	3.62665	0.357345	0.178672	−	0.983909i	$-0.442820\pi$
$103$	3.62665	0.357345	0.178672	+	0.983909i	$0.442820\pi$
$104$	0	0
$105$	−17.2964	−1.68795
$106$	0	0
$107$	− 14.8174i	− 1.43246i	−0.697866	−	0.716228i	$-0.745867\pi$
$107$	− 14.8174i	− 1.43246i	0.697866	−	0.716228i	$-0.254133\pi$
$108$	0	0
$109$	− 2.72573i	− 0.261077i	−0.991443	−	0.130539i	$-0.958329\pi$
$109$	− 2.72573i	− 0.261077i	0.991443	−	0.130539i	$-0.0416707\pi$
$110$	0	0
$111$	−12.9230	−1.22660
$112$	0	0
$113$	1.03896	0.0977375	0.0488688	−	0.998805i	$-0.484438\pi$
$113$	1.03896	0.0977375	0.0488688	+	0.998805i	$0.484438\pi$
$114$	0	0
$115$	− 28.8364i	− 2.68901i
$116$	0	0
$117$	7.74153i	0.715705i
$118$	0	0
$119$	−5.66973	−0.519743
$120$	0	0
$121$	8.22047	0.747315
$122$	0	0
$123$	− 35.3913i	− 3.19112i
$124$	0	0
$125$	2.43031i	0.217373i
$126$	0	0
$127$	13.9082	1.23415	0.617076	−	0.786903i	$-0.288317\pi$
$127$	13.9082	1.23415	0.617076	+	0.786903i	$0.288317\pi$
$128$	0	0
$129$	8.18667	0.720796
$130$	0	0
$131$	− 10.5790i	− 0.924290i	−0.886804	−	0.462145i	$-0.847080\pi$
$131$	− 10.5790i	− 0.924290i	0.886804	−	0.462145i	$-0.152920\pi$
$132$	0	0
$133$	6.11392i	0.530144i
$134$	0	0
$135$	50.9313	4.38347
$136$	0	0
$137$	−5.93787	−0.507307	−0.253653	−	0.967295i	$-0.581632\pi$
$137$	−5.93787	−0.507307	−0.253653	+	0.967295i	$0.581632\pi$
$138$	0	0
$139$	11.6508i	0.988206i	0.869403	+	0.494103i	$0.164503\pi$
$139$	11.6508i	0.988206i	−0.869403	+	0.494103i	$0.835497\pi$
$140$	0	0
$141$	29.3153i	2.46880i
$142$	0	0
$143$	−1.66719	−0.139418
$144$	0	0
$145$	−13.9082	−1.15501
$146$	0	0
$147$	14.4441i	1.19133i
$148$	0	0
$149$	0.668770i	0.0547877i	0.999625	+	0.0273939i	$0.00872083\pi$
$149$	0.668770i	0.0547877i	−0.999625	+	0.0273939i	$0.991279\pi$
$150$	0	0
$151$	9.98358	0.812453	0.406226	−	0.913773i	$-0.366844\pi$
$151$	9.98358	0.812453	0.406226	+	0.913773i	$0.366844\pi$
$152$	0	0
$153$	27.2584	2.20371
$154$	0	0
$155$	− 25.1908i	− 2.02337i
$156$	0	0
$157$	− 23.4209i	− 1.86919i	−0.355709	−	0.934597i	$-0.615760\pi$
$157$	− 23.4209i	− 1.86919i	0.355709	−	0.934597i	$-0.384240\pi$
$158$	0	0
$159$	6.97999	0.553549
$160$	0	0
$161$	14.1676	1.11657
$162$	0	0
$163$	6.99842i	0.548159i	0.961707	+	0.274079i	$0.0883731\pi$
$163$	6.99842i	0.548159i	−0.961707	+	0.274079i	$0.911627\pi$
$164$	0	0
$165$	17.9082i	1.39415i
$166$	0	0
$167$	10.0892	0.780724	0.390362	−	0.920661i	$-0.372350\pi$
$167$	10.0892	0.780724	0.390362	+	0.920661i	$0.372350\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 29.3939i	− 2.24781i
$172$	0	0
$173$	− 10.5928i	− 0.805353i	−0.915342	−	0.402677i	$-0.868080\pi$
$173$	− 10.5928i	− 0.805353i	0.915342	−	0.402677i	$-0.131920\pi$
$174$	0	0
$175$	−9.24521	−0.698872
$176$	0	0
$177$	−16.0190	−1.20406
$178$	0	0
$179$	− 0.316391i	− 0.0236482i	−0.999930	−	0.0118241i	$-0.996236\pi$
$179$	− 0.316391i	− 0.0236482i	0.999930	−	0.0118241i	$-0.00376381\pi$
$180$	0	0
$181$	− 9.41547i	− 0.699846i	−0.936778	−	0.349923i	$-0.886208\pi$
$181$	− 9.41547i	− 0.699846i	0.936778	−	0.349923i	$-0.113792\pi$
$182$	0	0
$183$	37.2097	2.75062
$184$	0	0
$185$	−12.9230	−0.950120
$186$	0	0
$187$	5.87028i	0.429278i
$188$	0	0
$189$	25.0231i	1.82016i
$190$	0	0
$191$	−19.0611	−1.37921	−0.689605	−	0.724185i	$-0.742216\pi$
$191$	−19.0611	−1.37921	−0.689605	+	0.724185i	$0.742216\pi$
$192$	0	0
$193$	−24.8850	−1.79126	−0.895632	−	0.444796i	$-0.853276\pi$
$193$	−24.8850	−1.79126	−0.895632	+	0.444796i	$0.853276\pi$
$194$	0	0
$195$	10.7415i	0.769217i
$196$	0	0
$197$	24.7933i	1.76645i	0.468949	+	0.883225i	$0.344633\pi$
$197$	24.7933i	1.76645i	−0.468949	+	0.883225i	$0.655367\pi$
$198$	0	0
$199$	−6.81429	−0.483052	−0.241526	−	0.970394i	$-0.577648\pi$
$199$	−6.81429	−0.483052	−0.241526	+	0.970394i	$0.577648\pi$
$200$	0	0
$201$	−0.292266	−0.0206149
$202$	0	0
$203$	− 6.83324i	− 0.479600i
$204$	0	0
$205$	− 35.3913i	− 2.47183i
$206$	0	0
$207$	−68.1137	−4.73423
$208$	0	0
$209$	6.33018	0.437868
$210$	0	0
$211$	− 4.72257i	− 0.325115i	−0.986699	−	0.162558i	$-0.948026\pi$
$211$	− 4.72257i	− 0.325115i	0.986699	−	0.162558i	$-0.0519744\pi$
$212$	0	0
$213$	− 10.0138i	− 0.686133i
$214$	0	0
$215$	8.18667	0.558327
$216$	0	0
$217$	12.3765	0.840172
$218$	0	0
$219$	33.2097i	2.24411i
$220$	0	0
$221$	3.52106i	0.236852i
$222$	0	0
$223$	−8.24617	−0.552204	−0.276102	−	0.961128i	$-0.589043\pi$
$223$	−8.24617	−0.552204	−0.276102	+	0.961128i	$0.589043\pi$
$224$	0	0
$225$	44.4482	2.96321
$226$	0	0
$227$	− 7.81587i	− 0.518757i	−0.965776	−	0.259379i	$-0.916482\pi$
$227$	− 7.81587i	− 0.518757i	0.965776	−	0.259379i	$-0.0835177\pi$
$228$	0	0
$229$	16.6466i	1.10004i	0.835153	+	0.550018i	$0.185379\pi$
$229$	16.6466i	1.10004i	−0.835153	+	0.550018i	$0.814621\pi$
$230$	0	0
$231$	−8.79849	−0.578898
$232$	0	0
$233$	6.07276	0.397840	0.198920	−	0.980016i	$-0.436257\pi$
$233$	6.07276	0.397840	0.198920	+	0.980016i	$0.436257\pi$
$234$	0	0
$235$	29.3153i	1.91232i
$236$	0	0
$237$	0.957882i	0.0622211i
$238$	0	0
$239$	18.5088	1.19723	0.598616	−	0.801036i	$-0.295718\pi$
$239$	18.5088	1.19723	0.598616	+	0.801036i	$0.295718\pi$
$240$	0	0
$241$	30.3681	1.95618	0.978090	−	0.208181i	$-0.0667543\pi$
$241$	30.3681	1.95618	0.978090	+	0.208181i	$0.0667543\pi$
$242$	0	0
$243$	− 44.1866i	− 2.83457i
$244$	0	0
$245$	14.4441i	0.922799i
$246$	0	0
$247$	3.79691	0.241592
$248$	0	0
$249$	13.4020	0.849316
$250$	0	0
$251$	3.22362i	0.203473i	0.994811	+	0.101737i	$0.0324399\pi$
$251$	3.22362i	0.203473i	−0.994811	+	0.101737i	$0.967560\pi$
$252$	0	0
$253$	− 14.6688i	− 0.922218i
$254$	0	0
$255$	37.8216	2.36848
$256$	0	0
$257$	5.11488	0.319057	0.159529	−	0.987193i	$-0.449003\pi$
$257$	5.11488	0.319057	0.159529	+	0.987193i	$0.449003\pi$
$258$	0	0
$259$	− 6.34922i	− 0.394522i
$260$	0	0
$261$	32.8522i	2.03350i
$262$	0	0
$263$	18.9092	1.16599	0.582997	−	0.812474i	$-0.301880\pi$
$263$	18.9092	1.16599	0.582997	+	0.812474i	$0.301880\pi$
$264$	0	0
$265$	6.97999	0.428777
$266$	0	0
$267$	− 4.00000i	− 0.244796i
$268$	0	0
$269$	− 9.23943i	− 0.563338i	−0.959512	−	0.281669i	$-0.909112\pi$
$269$	− 9.23943i	− 0.563338i	0.959512	−	0.281669i	$-0.0908880\pi$
$270$	0	0
$271$	−16.0080	−0.972417	−0.486208	−	0.873843i	$-0.661620\pi$
$271$	−16.0080	−0.972417	−0.486208	+	0.873843i	$0.661620\pi$
$272$	0	0
$273$	−5.27743	−0.319404
$274$	0	0
$275$	9.57223i	0.577227i
$276$	0	0
$277$	− 9.65910i	− 0.580359i	−0.956972	−	0.290180i	$-0.906285\pi$
$277$	− 9.65910i	− 0.580359i	0.956972	−	0.290180i	$-0.0937150\pi$
$278$	0	0
$279$	−59.5025	−3.56232
$280$	0	0
$281$	17.9735	1.07221	0.536104	−	0.844152i	$-0.319896\pi$
$281$	17.9735	1.07221	0.536104	+	0.844152i	$0.319896\pi$
$282$	0	0
$283$	− 22.6391i	− 1.34575i	−0.739754	−	0.672877i	$-0.765058\pi$
$283$	− 22.6391i	− 1.34575i	0.739754	−	0.672877i	$-0.234942\pi$
$284$	0	0
$285$	− 40.7846i	− 2.41587i
$286$	0	0
$287$	17.3881	1.02639
$288$	0	0
$289$	−4.60214	−0.270714
$290$	0	0
$291$	45.9461i	2.69341i
$292$	0	0
$293$	17.4261i	1.01804i	0.860754	+	0.509022i	$0.169993\pi$
$293$	17.4261i	1.01804i	−0.860754	+	0.509022i	$0.830007\pi$
$294$	0	0
$295$	−16.0190	−0.932660
$296$	0	0
$297$	25.9082	1.50335
$298$	0	0
$299$	− 8.79849i	− 0.508830i
$300$	0	0
$301$	4.02220i	0.231836i
$302$	0	0
$303$	4.36301	0.250648
$304$	0	0
$305$	37.2097	2.13062
$306$	0	0
$307$	8.96874i	0.511873i	0.966694	+	0.255937i	$0.0823838\pi$
$307$	8.96874i	0.511873i	−0.966694	+	0.255937i	$0.917616\pi$
$308$	0	0
$309$	− 11.8861i	− 0.676176i
$310$	0	0
$311$	16.3630	0.927861	0.463931	−	0.885872i	$-0.346439\pi$
$311$	16.3630	0.927861	0.463931	+	0.885872i	$0.346439\pi$
$312$	0	0
$313$	−13.7026	−0.774515	−0.387257	−	0.921972i	$-0.626578\pi$
$313$	−13.7026	−0.774515	−0.387257	+	0.921972i	$0.626578\pi$
$314$	0	0
$315$	40.8554i	2.30194i
$316$	0	0
$317$	15.4502i	0.867771i	0.900968	+	0.433886i	$0.142858\pi$
$317$	15.4502i	0.867771i	−0.900968	+	0.433886i	$0.857142\pi$
$318$	0	0
$319$	−7.07495	−0.396121
$320$	0	0
$321$	−48.5631	−2.71053
$322$	0	0
$323$	− 13.3691i	− 0.743879i
$324$	0	0
$325$	5.74153i	0.318483i
$326$	0	0
$327$	−8.93337	−0.494017
$328$	0	0
$329$	−14.4029	−0.794060
$330$	0	0
$331$	22.6175i	1.24317i	0.783347	+	0.621585i	$0.213511\pi$
$331$	22.6175i	1.24317i	−0.783347	+	0.621585i	$0.786489\pi$
$332$	0	0
$333$	30.5252i	1.67277i
$334$	0	0
$335$	−0.292266	−0.0159682
$336$	0	0
$337$	17.7026	0.964320	0.482160	−	0.876083i	$-0.339852\pi$
$337$	17.7026	0.964320	0.482160	+	0.876083i	$0.339852\pi$
$338$	0	0
$339$	− 3.40513i	− 0.184941i
$340$	0	0
$341$	− 12.8143i	− 0.693933i
$342$	0	0
$343$	−18.3682	−0.991788
$344$	0	0
$345$	−94.5092	−5.08820
$346$	0	0
$347$	13.8323i	0.742556i	0.928522	+	0.371278i	$0.121080\pi$
$347$	13.8323i	0.742556i	−0.928522	+	0.371278i	$0.878920\pi$
$348$	0	0
$349$	− 1.79437i	− 0.0960504i	−0.998846	−	0.0480252i	$-0.984707\pi$
$349$	− 1.79437i	− 0.0960504i	0.998846	−	0.0480252i	$-0.0152928\pi$
$350$	0	0
$351$	15.5400	0.829465
$352$	0	0
$353$	7.81849	0.416136	0.208068	−	0.978114i	$-0.433282\pi$
$353$	7.81849	0.416136	0.208068	+	0.978114i	$0.433282\pi$
$354$	0	0
$355$	− 10.0138i	− 0.531477i
$356$	0	0
$357$	18.5821i	0.983471i
$358$	0	0
$359$	−6.08918	−0.321374	−0.160687	−	0.987005i	$-0.551371\pi$
$359$	−6.08918	−0.321374	−0.160687	+	0.987005i	$0.551371\pi$
$360$	0	0
$361$	4.58348	0.241236
$362$	0	0
$363$	− 26.9420i	− 1.41409i
$364$	0	0
$365$	33.2097i	1.73828i
$366$	0	0
$367$	6.73321	0.351470	0.175735	−	0.984437i	$-0.443770\pi$
$367$	6.73321	0.351470	0.175735	+	0.984437i	$0.443770\pi$
$368$	0	0
$369$	−83.5968	−4.35188
$370$	0	0
$371$	3.42934i	0.178043i
$372$	0	0
$373$	− 12.5549i	− 0.650066i	−0.945703	−	0.325033i	$-0.894625\pi$
$373$	− 12.5549i	− 0.650066i	0.945703	−	0.325033i	$-0.105375\pi$
$374$	0	0
$375$	7.96515	0.411319
$376$	0	0
$377$	−4.24363	−0.218558
$378$	0	0
$379$	19.5615i	1.00481i	0.864633	+	0.502404i	$0.167551\pi$
$379$	19.5615i	1.00481i	−0.864633	+	0.502404i	$0.832449\pi$
$380$	0	0
$381$	− 45.5831i	− 2.33529i
$382$	0	0
$383$	10.9478	0.559405	0.279703	−	0.960087i	$-0.409764\pi$
$383$	10.9478	0.559405	0.279703	+	0.960087i	$0.409764\pi$
$384$	0	0
$385$	−8.79849	−0.448412
$386$	0	0
$387$	− 19.3375i	− 0.982982i
$388$	0	0
$389$	− 19.0958i	− 0.968197i	−0.875013	−	0.484099i	$-0.839148\pi$
$389$	− 19.0958i	− 0.968197i	0.875013	−	0.484099i	$-0.160852\pi$
$390$	0	0
$391$	−30.9800	−1.56673
$392$	0	0
$393$	−34.6718	−1.74896
$394$	0	0
$395$	0.957882i	0.0481963i
$396$	0	0
$397$	1.59382i	0.0799915i	0.999200	+	0.0399957i	$0.0127344\pi$
$397$	1.59382i	0.0799915i	−0.999200	+	0.0399957i	$0.987266\pi$
$398$	0	0
$399$	20.0379	1.00315
$400$	0	0
$401$	−18.6414	−0.930907	−0.465454	−	0.885072i	$-0.654109\pi$
$401$	−18.6414	−0.930907	−0.465454	+	0.885072i	$0.654109\pi$
$402$	0	0
$403$	− 7.68615i	− 0.382874i
$404$	0	0
$405$	− 90.8066i	− 4.51222i
$406$	0	0
$407$	−6.57381	−0.325852
$408$	0	0
$409$	−25.8785	−1.27961	−0.639805	−	0.768537i	$-0.720985\pi$
$409$	−25.8785	−1.27961	−0.639805	+	0.768537i	$0.720985\pi$
$410$	0	0
$411$	19.4609i	0.959938i
$412$	0	0
$413$	− 7.87028i	− 0.387271i
$414$	0	0
$415$	13.4020	0.657877
$416$	0	0
$417$	38.1846	1.86991
$418$	0	0
$419$	− 3.23952i	− 0.158261i	−0.996864	−	0.0791303i	$-0.974786\pi$
$419$	− 3.23952i	− 0.158261i	0.996864	−	0.0791303i	$-0.0252143\pi$
$420$	0	0
$421$	13.2477i	0.645656i	0.946458	+	0.322828i	$0.104633\pi$
$421$	13.2477i	0.645656i	−0.946458	+	0.322828i	$0.895367\pi$
$422$	0	0
$423$	69.2450	3.36681
$424$	0	0
$425$	20.2163	0.980633
$426$	0	0
$427$	18.2815i	0.884706i
$428$	0	0
$429$	5.46410i	0.263809i
$430$	0	0
$431$	−18.3948	−0.886048	−0.443024	−	0.896510i	$-0.646094\pi$
$431$	−18.3948	−0.886048	−0.443024	+	0.896510i	$0.646094\pi$
$432$	0	0
$433$	1.55389	0.0746753	0.0373376	−	0.999303i	$-0.488112\pi$
$433$	1.55389	0.0746753	0.0373376	+	0.999303i	$0.488112\pi$
$434$	0	0
$435$	45.5831i	2.18554i
$436$	0	0
$437$	33.4071i	1.59808i
$438$	0	0
$439$	13.7943	0.658365	0.329183	−	0.944266i	$-0.393227\pi$
$439$	13.7943	0.658365	0.329183	+	0.944266i	$0.393227\pi$
$440$	0	0
$441$	34.1180	1.62467
$442$	0	0
$443$	− 14.0980i	− 0.669817i	−0.942251	−	0.334909i	$-0.891295\pi$
$443$	− 14.0980i	− 0.669817i	0.942251	−	0.334909i	$-0.108705\pi$
$444$	0	0
$445$	− 4.00000i	− 0.189618i
$446$	0	0
$447$	2.19184	0.103671
$448$	0	0
$449$	22.8692	1.07927	0.539633	−	0.841900i	$-0.318563\pi$
$449$	22.8692	1.07927	0.539633	+	0.841900i	$0.318563\pi$
$450$	0	0
$451$	− 18.0032i	− 0.847735i
$452$	0	0
$453$	− 32.7205i	− 1.53734i
$454$	0	0
$455$	−5.27743	−0.247410
$456$	0	0
$457$	17.6267	0.824540	0.412270	−	0.911062i	$-0.364736\pi$
$457$	17.6267	0.824540	0.412270	+	0.911062i	$0.364736\pi$
$458$	0	0
$459$	− 54.7173i	− 2.55399i
$460$	0	0
$461$	− 19.2509i	− 0.896604i	−0.893882	−	0.448302i	$-0.852029\pi$
$461$	− 19.2509i	− 0.896604i	0.893882	−	0.448302i	$-0.147971\pi$
$462$	0	0
$463$	−22.3549	−1.03892	−0.519461	−	0.854494i	$-0.673867\pi$
$463$	−22.3549	−1.03892	−0.519461	+	0.854494i	$0.673867\pi$
$464$	0	0
$465$	−82.5610	−3.82867
$466$	0	0
$467$	− 1.49147i	− 0.0690170i	−0.999404	−	0.0345085i	$-0.989013\pi$
$467$	− 1.49147i	− 0.0690170i	0.999404	−	0.0345085i	$-0.0109866\pi$
$468$	0	0
$469$	− 0.143594i	− 0.00663053i
$470$	0	0
$471$	−76.7604	−3.53693
$472$	0	0
$473$	4.16447	0.191483
$474$	0	0
$475$	− 21.8001i	− 1.00026i
$476$	0	0
$477$	− 16.4873i	− 0.754900i
$478$	0	0
$479$	−16.7663	−0.766070	−0.383035	−	0.923734i	$-0.625121\pi$
$479$	−16.7663	−0.766070	−0.383035	+	0.923734i	$0.625121\pi$
$480$	0	0
$481$	−3.94304	−0.179787
$482$	0	0
$483$	− 46.4334i	− 2.11279i
$484$	0	0
$485$	45.9461i	2.08631i
$486$	0	0
$487$	19.5889	0.887657	0.443828	−	0.896112i	$-0.353620\pi$
$487$	19.5889	0.887657	0.443828	+	0.896112i	$0.353620\pi$
$488$	0	0
$489$	22.9368	1.03724
$490$	0	0
$491$	35.2509i	1.59085i	0.606051	+	0.795425i	$0.292753\pi$
$491$	35.2509i	1.59085i	−0.606051	+	0.795425i	$0.707247\pi$
$492$	0	0
$493$	14.9421i	0.672957i
$494$	0	0
$495$	42.3005	1.90127
$496$	0	0
$497$	4.91988	0.220687
$498$	0	0
$499$	7.01507i	0.314038i	0.987596	+	0.157019i	$0.0501883\pi$
$499$	7.01507i	0.314038i	−0.987596	+	0.157019i	$0.949812\pi$
$500$	0	0
$501$	− 33.0665i	− 1.47730i
$502$	0	0
$503$	11.3344	0.505375	0.252688	−	0.967548i	$-0.418686\pi$
$503$	11.3344	0.505375	0.252688	+	0.967548i	$0.418686\pi$
$504$	0	0
$505$	4.36301	0.194151
$506$	0	0
$507$	3.27743i	0.145556i
$508$	0	0
$509$	16.1383i	0.715319i	0.933852	+	0.357660i	$0.116425\pi$
$509$	16.1383i	0.715319i	−0.933852	+	0.357660i	$0.883575\pi$
$510$	0	0
$511$	−16.3163	−0.721791
$512$	0	0
$513$	−59.0040	−2.60509
$514$	0	0
$515$	− 11.8861i	− 0.523763i
$516$	0	0
$517$	14.9124i	0.655847i
$518$	0	0
$519$	−34.7170	−1.52391
$520$	0	0
$521$	−12.1056	−0.530356	−0.265178	−	0.964200i	$-0.585431\pi$
$521$	−12.1056	−0.530356	−0.265178	+	0.964200i	$0.585431\pi$
$522$	0	0
$523$	− 1.22047i	− 0.0533674i	−0.999644	−	0.0266837i	$-0.991505\pi$
$523$	− 1.22047i	− 0.0533674i	0.999644	−	0.0266837i	$-0.00849469\pi$
$524$	0	0
$525$	30.3005i	1.32242i
$526$	0	0
$527$	−27.0634	−1.17890
$528$	0	0
$529$	54.4134	2.36580
$530$	0	0
$531$	37.8380i	1.64203i
$532$	0	0
$533$	− 10.7985i	− 0.467734i
$534$	0	0
$535$	−48.5631	−2.09957
$536$	0	0
$537$	−1.03695	−0.0447477
$538$	0	0
$539$	7.34756i	0.316482i
$540$	0	0
$541$	− 27.9430i	− 1.20137i	−0.799488	−	0.600683i	$-0.794896\pi$
$541$	− 27.9430i	− 1.20137i	0.799488	−	0.600683i	$-0.205104\pi$
$542$	0	0
$543$	−30.8585	−1.32427
$544$	0	0
$545$	−8.93337	−0.382664
$546$	0	0
$547$	23.1698i	0.990670i	0.868702	+	0.495335i	$0.164955\pi$
$547$	23.1698i	0.990670i	−0.868702	+	0.495335i	$0.835045\pi$
$548$	0	0
$549$	− 87.8922i	− 3.75115i
$550$	0	0
$551$	16.1127	0.686423
$552$	0	0
$553$	−0.470617	−0.0200127
$554$	0	0
$555$	42.3543i	1.79784i
$556$	0	0
$557$	− 32.5061i	− 1.37733i	−0.725080	−	0.688664i	$-0.758197\pi$
$557$	− 32.5061i	− 1.37733i	0.725080	−	0.688664i	$-0.241803\pi$
$558$	0	0
$559$	2.49790	0.105650
$560$	0	0
$561$	19.2394	0.812289
$562$	0	0
$563$	4.52757i	0.190815i	0.995438	+	0.0954073i	$0.0304153\pi$
$563$	4.52757i	0.190815i	−0.995438	+	0.0954073i	$0.969585\pi$
$564$	0	0
$565$	− 3.40513i	− 0.143255i
$566$	0	0
$567$	44.6143	1.87362
$568$	0	0
$569$	21.7057	0.909951	0.454976	−	0.890504i	$-0.349648\pi$
$569$	21.7057	0.909951	0.454976	+	0.890504i	$0.349648\pi$
$570$	0	0
$571$	37.8058i	1.58212i	0.611737	+	0.791061i	$0.290471\pi$
$571$	37.8058i	1.58212i	−0.611737	+	0.791061i	$0.709529\pi$
$572$	0	0
$573$	62.4713i	2.60978i
$574$	0	0
$575$	−50.5168	−2.10669
$576$	0	0
$577$	−1.56452	−0.0651320	−0.0325660	−	0.999470i	$-0.510368\pi$
$577$	−1.56452	−0.0651320	−0.0325660	+	0.999470i	$0.510368\pi$
$578$	0	0
$579$	81.5589i	3.38947i
$580$	0	0
$581$	6.58453i	0.273172i
$582$	0	0
$583$	3.55065	0.147053
$584$	0	0
$585$	25.3723	1.04902
$586$	0	0
$587$	− 44.3244i	− 1.82946i	−0.404062	−	0.914732i	$-0.632402\pi$
$587$	− 44.3244i	− 1.82946i	0.404062	−	0.914732i	$-0.367598\pi$
$588$	0	0
$589$	29.1836i	1.20249i
$590$	0	0
$591$	81.2583	3.34252
$592$	0	0
$593$	23.6970	0.973120	0.486560	−	0.873647i	$-0.338252\pi$
$593$	23.6970	0.973120	0.486560	+	0.873647i	$0.338252\pi$
$594$	0	0
$595$	18.5821i	0.761793i
$596$	0	0
$597$	22.3333i	0.914042i
$598$	0	0
$599$	36.8036	1.50375	0.751876	−	0.659304i	$-0.229149\pi$
$599$	36.8036	1.50375	0.751876	+	0.659304i	$0.229149\pi$
$600$	0	0
$601$	−28.6520	−1.16874	−0.584370	−	0.811487i	$-0.698658\pi$
$601$	−28.6520	−1.16874	−0.584370	+	0.811487i	$0.698658\pi$
$602$	0	0
$603$	0.690355i	0.0281134i
$604$	0	0
$605$	− 26.9420i	− 1.09535i
$606$	0	0
$607$	42.9535	1.74343	0.871714	−	0.490015i	$-0.163009\pi$
$607$	42.9535	1.74343	0.871714	+	0.490015i	$0.163009\pi$
$608$	0	0
$609$	−22.3955	−0.907510
$610$	0	0
$611$	8.94462i	0.361861i
$612$	0	0
$613$	− 31.2719i	− 1.26306i	−0.775352	−	0.631530i	$-0.782427\pi$
$613$	− 31.2719i	− 1.26306i	0.775352	−	0.631530i	$-0.217573\pi$
$614$	0	0
$615$	−115.992	−4.67726
$616$	0	0
$617$	8.99895	0.362284	0.181142	−	0.983457i	$-0.442021\pi$
$617$	8.99895	0.362284	0.181142	+	0.983457i	$0.442021\pi$
$618$	0	0
$619$	4.36249i	0.175343i	0.996149	+	0.0876716i	$0.0279426\pi$
$619$	4.36249i	0.175343i	−0.996149	+	0.0876716i	$0.972057\pi$
$620$	0	0
$621$	136.729i	5.48673i
$622$	0	0
$623$	1.96524	0.0787358
$624$	0	0
$625$	−20.7425	−0.829700
$626$	0	0
$627$	− 20.7467i	− 0.828543i
$628$	0	0
$629$	13.8837i	0.553579i
$630$	0	0
$631$	10.8718	0.432798	0.216399	−	0.976305i	$-0.430569\pi$
$631$	10.8718	0.432798	0.216399	+	0.976305i	$0.430569\pi$
$632$	0	0
$633$	−15.4779	−0.615191
$634$	0	0
$635$	− 45.5831i	− 1.80891i
$636$	0	0
$637$	4.40714i	0.174617i
$638$	0	0
$639$	−23.6533	−0.935711
$640$	0	0
$641$	34.4460	1.36054	0.680268	−	0.732964i	$-0.261863\pi$
$641$	34.4460	1.36054	0.680268	+	0.732964i	$0.261863\pi$
$642$	0	0
$643$	39.1589i	1.54427i	0.635455	+	0.772137i	$0.280812\pi$
$643$	39.1589i	1.54427i	−0.635455	+	0.772137i	$0.719188\pi$
$644$	0	0
$645$	− 26.8312i	− 1.05648i
$646$	0	0
$647$	14.7309	0.579131	0.289566	−	0.957158i	$-0.406489\pi$
$647$	14.7309	0.579131	0.289566	+	0.957158i	$0.406489\pi$
$648$	0	0
$649$	−8.14867	−0.319863
$650$	0	0
$651$	− 40.5631i	− 1.58979i
$652$	0	0
$653$	39.2615i	1.53642i	0.640196	+	0.768211i	$0.278853\pi$
$653$	39.2615i	1.53642i	−0.640196	+	0.768211i	$0.721147\pi$
$654$	0	0
$655$	−34.6718	−1.35474
$656$	0	0
$657$	78.4439	3.06039
$658$	0	0
$659$	41.0748i	1.60005i	0.599969	+	0.800023i	$0.295179\pi$
$659$	41.0748i	1.60005i	−0.599969	+	0.800023i	$0.704821\pi$
$660$	0	0
$661$	35.8132i	1.39297i	0.717570	+	0.696487i	$0.245254\pi$
$661$	35.8132i	1.39297i	−0.717570	+	0.696487i	$0.754746\pi$
$662$	0	0
$663$	11.5400	0.448177
$664$	0	0
$665$	20.0379	0.777037
$666$	0	0
$667$	− 37.3375i	− 1.44572i
$668$	0	0
$669$	27.0262i	1.04489i
$670$	0	0
$671$	18.9282	0.730715
$672$	0	0
$673$	28.6790	1.10549	0.552747	−	0.833349i	$-0.313579\pi$
$673$	28.6790	1.10549	0.552747	+	0.833349i	$0.313579\pi$
$674$	0	0
$675$	− 89.2234i	− 3.43421i
$676$	0	0
$677$	− 20.9958i	− 0.806934i	−0.914994	−	0.403467i	$-0.867805\pi$
$677$	− 20.9958i	− 0.806934i	0.914994	−	0.403467i	$-0.132195\pi$
$678$	0	0
$679$	−22.5738	−0.866303
$680$	0	0
$681$	−25.6159	−0.981605
$682$	0	0
$683$	38.5743i	1.47601i	0.674797	+	0.738003i	$0.264231\pi$
$683$	38.5743i	1.47601i	−0.674797	+	0.738003i	$0.735769\pi$
$684$	0	0
$685$	19.4609i	0.743565i
$686$	0	0
$687$	54.5579	2.08151
$688$	0	0
$689$	2.12972	0.0811357
$690$	0	0
$691$	29.2696i	1.11347i	0.830690	+	0.556735i	$0.187946\pi$
$691$	29.2696i	1.11347i	−0.830690	+	0.556735i	$0.812054\pi$
$692$	0	0
$693$	20.7827i	0.789469i
$694$	0	0
$695$	38.1846	1.44842
$696$	0	0
$697$	−38.0221	−1.44019
$698$	0	0
$699$	− 19.9030i	− 0.752802i
$700$	0	0
$701$	− 8.28154i	− 0.312790i	−0.987695	−	0.156395i	$-0.950013\pi$
$701$	− 8.28154i	− 0.312790i	0.987695	−	0.156395i	$-0.0499872\pi$
$702$	0	0
$703$	14.9714	0.564656
$704$	0	0
$705$	96.0789	3.61854
$706$	0	0
$707$	2.14359i	0.0806181i
$708$	0	0
$709$	36.3005i	1.36329i	0.731681	+	0.681647i	$0.238736\pi$
$709$	36.3005i	1.36329i	−0.731681	+	0.681647i	$0.761264\pi$
$710$	0	0
$711$	2.26259	0.0848537
$712$	0	0
$713$	67.6265	2.53263
$714$	0	0
$715$	5.46410i	0.204346i
$716$	0	0
$717$	− 60.6611i	− 2.26543i
$718$	0	0
$719$	−1.93872	−0.0723020	−0.0361510	−	0.999346i	$-0.511510\pi$
$719$	−1.93872	−0.0723020	−0.0361510	+	0.999346i	$0.511510\pi$
$720$	0	0
$721$	5.83976	0.217484
$722$	0	0
$723$	− 99.5292i	− 3.70153i
$724$	0	0
$725$	24.3649i	0.904891i
$726$	0	0
$727$	33.0642	1.22628	0.613142	−	0.789973i	$-0.289905\pi$
$727$	33.0642	1.22628	0.613142	+	0.789973i	$0.289905\pi$
$728$	0	0
$729$	−61.6983	−2.28512
$730$	0	0
$731$	− 8.79524i	− 0.325304i
$732$	0	0
$733$	20.5390i	0.758624i	0.925269	+	0.379312i	$0.123839\pi$
$733$	20.5390i	0.758624i	−0.925269	+	0.379312i	$0.876161\pi$
$734$	0	0
$735$	47.3395	1.74614
$736$	0	0
$737$	−0.148673	−0.00547643
$738$	0	0
$739$	32.6977i	1.20281i	0.798946	+	0.601403i	$0.205391\pi$
$739$	32.6977i	1.20281i	−0.798946	+	0.601403i	$0.794609\pi$
$740$	0	0
$741$	− 12.4441i	− 0.457145i
$742$	0	0
$743$	−14.6871	−0.538818	−0.269409	−	0.963026i	$-0.586828\pi$
$743$	−14.6871	−0.538818	−0.269409	+	0.963026i	$0.586828\pi$
$744$	0	0
$745$	2.19184	0.0803029
$746$	0	0
$747$	− 31.6565i	− 1.15825i
$748$	0	0
$749$	− 23.8596i	− 0.871809i
$750$	0	0
$751$	43.9758	1.60470	0.802350	−	0.596854i	$-0.203583\pi$
$751$	43.9758	1.60470	0.802350	+	0.596854i	$0.203583\pi$
$752$	0	0
$753$	10.5652	0.385017
$754$	0	0
$755$	− 32.7205i	− 1.19082i
$756$	0	0
$757$	11.6024i	0.421698i	0.977519	+	0.210849i	$0.0676228\pi$
$757$	11.6024i	0.421698i	−0.977519	+	0.210849i	$0.932377\pi$
$758$	0	0
$759$	−48.0758	−1.74504
$760$	0	0
$761$	19.8046	0.717917	0.358958	−	0.933354i	$-0.383132\pi$
$761$	19.8046	0.717917	0.358958	+	0.933354i	$0.383132\pi$
$762$	0	0
$763$	− 4.38906i	− 0.158895i
$764$	0	0
$765$	− 89.3374i	− 3.23000i
$766$	0	0
$767$	−4.88766	−0.176483
$768$	0	0
$769$	−32.8090	−1.18312	−0.591562	−	0.806260i	$-0.701488\pi$
$769$	−32.8090	−1.18312	−0.591562	+	0.806260i	$0.701488\pi$
$770$	0	0
$771$	− 16.7636i	− 0.603728i
$772$	0	0
$773$	− 46.5472i	− 1.67419i	−0.547060	−	0.837093i	$-0.684253\pi$
$773$	− 46.5472i	− 1.67419i	0.547060	−	0.837093i	$-0.315747\pi$
$774$	0	0
$775$	−44.1302	−1.58521
$776$	0	0
$777$	−20.8091	−0.746523
$778$	0	0
$779$	41.0009i	1.46901i
$780$	0	0
$781$	− 5.09391i	− 0.182274i
$782$	0	0
$783$	65.9461	2.35672
$784$	0	0
$785$	−76.7604	−2.73970
$786$	0	0
$787$	− 30.5605i	− 1.08936i	−0.838643	−	0.544681i	$-0.816651\pi$
$787$	− 30.5605i	− 1.08936i	0.838643	−	0.544681i	$-0.183349\pi$
$788$	0	0
$789$	− 61.9737i	− 2.20632i
$790$	0	0
$791$	1.67298	0.0594842
$792$	0	0
$793$	11.3533	0.403169
$794$	0	0
$795$	− 22.8764i	− 0.811343i
$796$	0	0
$797$	− 17.3988i	− 0.616298i	−0.951338	−	0.308149i	$-0.900290\pi$
$797$	− 17.3988i	− 0.616298i	0.951338	−	0.308149i	$-0.0997095\pi$
$798$	0	0
$799$	31.4945	1.11420
$800$	0	0
$801$	−9.44830	−0.333839
$802$	0	0
$803$	16.8934i	0.596157i
$804$	0	0
$805$	− 46.4334i	− 1.63656i
$806$	0	0
$807$	−30.2815	−1.06596
$808$	0	0
$809$	−15.0717	−0.529893	−0.264946	−	0.964263i	$-0.585354\pi$
$809$	−15.0717	−0.529893	−0.264946	+	0.964263i	$0.585354\pi$
$810$	0	0
$811$	− 6.69351i	− 0.235041i	−0.993070	−	0.117520i	$-0.962505\pi$
$811$	− 6.69351i	− 0.235041i	0.993070	−	0.117520i	$-0.0374946\pi$
$812$	0	0
$813$	52.4651i	1.84003i
$814$	0	0
$815$	22.9368	0.803442
$816$	0	0
$817$	−9.48429	−0.331813
$818$	0	0
$819$	12.4657i	0.435586i
$820$	0	0
$821$	− 8.15931i	− 0.284762i	−0.989812	−	0.142381i	$-0.954524\pi$
$821$	− 8.15931i	− 0.284762i	0.989812	−	0.142381i	$-0.0454758\pi$
$822$	0	0
$823$	24.8198	0.865162	0.432581	−	0.901595i	$-0.357603\pi$
$823$	24.8198	0.865162	0.432581	+	0.901595i	$0.357603\pi$
$824$	0	0
$825$	31.3723	1.09224
$826$	0	0
$827$	22.6872i	0.788911i	0.918915	+	0.394456i	$0.129067\pi$
$827$	22.6872i	0.788911i	−0.918915	+	0.394456i	$0.870933\pi$
$828$	0	0
$829$	− 3.88608i	− 0.134969i	−0.997720	−	0.0674847i	$-0.978503\pi$
$829$	− 3.88608i	− 0.134969i	0.997720	−	0.0674847i	$-0.0214974\pi$
$830$	0	0
$831$	−31.6570	−1.09817
$832$	0	0
$833$	15.5178	0.537660
$834$	0	0
$835$	− 33.0665i	− 1.14431i
$836$	0	0
$837$	119.443i	4.12855i
$838$	0	0
$839$	−44.2113	−1.52634	−0.763172	−	0.646195i	$-0.776359\pi$
$839$	−44.2113	−1.52634	−0.763172	+	0.646195i	$0.776359\pi$
$840$	0	0
$841$	10.9916	0.379020
$842$	0	0
$843$	− 58.9068i	− 2.02886i
$844$	0	0
$845$	3.27743i	0.112747i
$846$	0	0
$847$	13.2369	0.454825
$848$	0	0
$849$	−74.1980	−2.54647
$850$	0	0
$851$	− 34.6928i	− 1.18925i
$852$	0	0
$853$	42.9586i	1.47088i	0.677592	+	0.735438i	$0.263023\pi$
$853$	42.9586i	1.47088i	−0.677592	+	0.735438i	$0.736977\pi$
$854$	0	0
$855$	−96.3363	−3.29463
$856$	0	0
$857$	0.662461	0.0226292	0.0113146	−	0.999936i	$-0.496398\pi$
$857$	0.662461	0.0226292	0.0113146	+	0.999936i	$0.496398\pi$
$858$	0	0
$859$	− 6.32387i	− 0.215768i	−0.994164	−	0.107884i	$-0.965593\pi$
$859$	− 6.32387i	− 0.215768i	0.994164	−	0.107884i	$-0.0344075\pi$
$860$	0	0
$861$	− 56.9882i	− 1.94215i
$862$	0	0
$863$	−29.6165	−1.00816	−0.504079	−	0.863657i	$-0.668168\pi$
$863$	−29.6165	−1.00816	−0.504079	+	0.863657i	$0.668168\pi$
$864$	0	0
$865$	−34.7170	−1.18041
$866$	0	0
$867$	15.0832i	0.512252i
$868$	0	0
$869$	0.487264i	0.0165293i
$870$	0	0
$871$	−0.0891755	−0.00302160
$872$	0	0
$873$	108.528	3.67312
$874$	0	0
$875$	3.91336i	0.132296i
$876$	0	0
$877$	− 46.1572i	− 1.55862i	−0.626640	−	0.779309i	$-0.715570\pi$
$877$	− 46.1572i	− 1.55862i	0.626640	−	0.779309i	$-0.284430\pi$
$878$	0	0
$879$	57.1128	1.92637
$880$	0	0
$881$	−34.2459	−1.15377	−0.576886	−	0.816825i	$-0.695732\pi$
$881$	−34.2459	−1.15377	−0.576886	+	0.816825i	$0.695732\pi$
$882$	0	0
$883$	16.4579i	0.553852i	0.960891	+	0.276926i	$0.0893157\pi$
$883$	16.4579i	0.553852i	−0.960891	+	0.276926i	$0.910684\pi$
$884$	0	0
$885$	52.5010i	1.76480i
$886$	0	0
$887$	−2.55255	−0.0857061	−0.0428530	−	0.999081i	$-0.513645\pi$
$887$	−2.55255	−0.0857061	−0.0428530	+	0.999081i	$0.513645\pi$
$888$	0	0
$889$	22.3955	0.751120
$890$	0	0
$891$	− 46.1924i	− 1.54750i
$892$	0	0
$893$	− 33.9619i	− 1.13649i
$894$	0	0
$895$	−1.03695	−0.0346614
$896$	0	0
$897$	−28.8364	−0.962819
$898$	0	0
$899$	− 32.6172i	− 1.08784i
$900$	0	0
$901$	− 7.49886i	− 0.249823i
$902$	0	0
$903$	13.1825	0.438685
$904$	0	0
$905$	−30.8585	−1.02577
$906$	0	0
$907$	− 44.9934i	− 1.49398i	−0.664835	−	0.746991i	$-0.731498\pi$
$907$	− 44.9934i	− 1.49398i	0.664835	−	0.746991i	$-0.268502\pi$
$908$	0	0
$909$	− 10.3058i	− 0.341820i
$910$	0	0
$911$	−46.3523	−1.53572	−0.767860	−	0.640618i	$-0.778678\pi$
$911$	−46.3523	−1.53572	−0.767860	+	0.640618i	$0.778678\pi$
$912$	0	0
$913$	6.81744	0.225624
$914$	0	0
$915$	− 121.952i	− 4.03162i
$916$	0	0
$917$	− 17.0346i	− 0.562533i
$918$	0	0
$919$	44.5578	1.46983	0.734914	−	0.678161i	$-0.237223\pi$
$919$	44.5578	1.46983	0.734914	+	0.678161i	$0.237223\pi$
$920$	0	0
$921$	29.3944	0.968579
$922$	0	0
$923$	− 3.05538i	− 0.100569i
$924$	0	0
$925$	22.6391i	0.744369i
$926$	0	0
$927$	−28.0758	−0.922131
$928$	0	0
$929$	27.6677	0.907748	0.453874	−	0.891066i	$-0.350042\pi$
$929$	27.6677	0.907748	0.453874	+	0.891066i	$0.350042\pi$
$930$	0	0
$931$	− 16.7335i	− 0.548419i
$932$	0	0
$933$	− 53.6286i	− 1.75572i
$934$	0	0
$935$	19.2394	0.629196
$936$	0	0
$937$	−34.9261	−1.14099	−0.570493	−	0.821302i	$-0.693248\pi$
$937$	−34.9261	−1.14099	−0.570493	+	0.821302i	$0.693248\pi$
$938$	0	0
$939$	44.9092i	1.46555i
$940$	0	0
$941$	47.8192i	1.55886i	0.626487	+	0.779431i	$0.284492\pi$
$941$	47.8192i	1.55886i	−0.626487	+	0.779431i	$0.715508\pi$
$942$	0	0
$943$	95.0103	3.09396
$944$	0	0
$945$	82.0113	2.66783
$946$	0	0
$947$	35.9993i	1.16982i	0.811098	+	0.584910i	$0.198870\pi$
$947$	35.9993i	1.16982i	−0.811098	+	0.584910i	$0.801130\pi$
$948$	0	0
$949$	10.1329i	0.328927i
$950$	0	0
$951$	50.6370	1.64202
$952$	0	0
$953$	−34.3385	−1.11233	−0.556167	−	0.831071i	$-0.687728\pi$
$953$	−34.3385	−1.11233	−0.556167	+	0.831071i	$0.687728\pi$
$954$	0	0
$955$	62.4713i	2.02152i
$956$	0	0
$957$	23.1876i	0.749550i
$958$	0	0
$959$	−9.56137	−0.308753
$960$	0	0
$961$	28.0769	0.905706
$962$	0	0
$963$	114.710i	3.69647i
$964$	0	0
$965$	81.5589i	2.62547i
$966$	0	0
$967$	−3.38556	−0.108872	−0.0544361	−	0.998517i	$-0.517336\pi$
$967$	−3.38556	−0.108872	−0.0544361	+	0.998517i	$0.517336\pi$
$968$	0	0
$969$	−43.8164	−1.40759
$970$	0	0
$971$	− 49.1884i	− 1.57853i	−0.614052	−	0.789265i	$-0.710462\pi$
$971$	− 49.1884i	− 1.57853i	0.614052	−	0.789265i	$-0.289538\pi$
$972$	0	0
$973$	18.7605i	0.601434i
$974$	0	0
$975$	18.8174	0.602640
$976$	0	0
$977$	−17.9884	−0.575501	−0.287750	−	0.957705i	$-0.592907\pi$
$977$	−17.9884	−0.575501	−0.287750	+	0.957705i	$0.592907\pi$
$978$	0	0
$979$	− 2.03476i	− 0.0650311i
$980$	0	0
$981$	21.1013i	0.673713i
$982$	0	0
$983$	−14.8718	−0.474336	−0.237168	−	0.971469i	$-0.576219\pi$
$983$	−14.8718	−0.474336	−0.237168	+	0.971469i	$0.576219\pi$
$984$	0	0
$985$	81.2583	2.58910
$986$	0	0
$987$	47.2046i	1.50254i
$988$	0	0
$989$	21.9777i	0.698851i
$990$	0	0
$991$	−4.70353	−0.149412	−0.0747062	−	0.997206i	$-0.523802\pi$
$991$	−4.70353	−0.149412	−0.0747062	+	0.997206i	$0.523802\pi$
$992$	0	0
$993$	74.1272	2.35236
$994$	0	0
$995$	22.3333i	0.708014i
$996$	0	0
$997$	12.9337i	0.409613i	0.978802	+	0.204807i	$0.0656566\pi$
$997$	12.9337i	0.409613i	−0.978802	+	0.204807i	$0.934343\pi$
$998$	0	0
$999$	61.2749	1.93865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.2.b.c.417.1	✓	8
4.3	odd	2		832.2.b.d.417.8	yes	8
8.3	odd	2		832.2.b.d.417.1	yes	8
8.5	even	2	inner	832.2.b.c.417.8	yes	8
16.3	odd	4		3328.2.a.bi.1.1		4
16.5	even	4		3328.2.a.bj.1.1		4
16.11	odd	4		3328.2.a.bn.1.4		4
16.13	even	4		3328.2.a.bm.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.b.c.417.1	✓	8	1.1	even	1	trivial
832.2.b.c.417.8	yes	8	8.5	even	2	inner
832.2.b.d.417.1	yes	8	8.3	odd	2
832.2.b.d.417.8	yes	8	4.3	odd	2
3328.2.a.bi.1.1		4	16.3	odd	4
3328.2.a.bj.1.1		4	16.5	even	4
3328.2.a.bm.1.4		4	16.13	even	4
3328.2.a.bn.1.4		4	16.11	odd	4

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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.27743i q^{3} -3.27743i q^{5} +1.61023 q^{7} -7.74153 q^{9} -1.66719i q^{11} -1.00000i q^{13} -10.7415 q^{15} -3.52106 q^{17} +3.79691i q^{19} -5.27743i q^{21} +8.79849 q^{23} -5.74153 q^{25} +15.5400i q^{27} -4.24363i q^{29} +7.68615 q^{31} -5.46410 q^{33} -5.27743i q^{35} -3.94304i q^{37} -3.27743 q^{39} +10.7985 q^{41} +2.49790i q^{43} +25.3723i q^{45} -8.94462 q^{47} -4.40714 q^{49} +11.5400i q^{51} +2.12972i q^{53} -5.46410 q^{55} +12.4441 q^{57} -4.88766i q^{59} +11.3533i q^{61} -12.4657 q^{63} -3.27743 q^{65} -0.0891755i q^{67} -28.8364i q^{69} +3.05538 q^{71} -10.1329 q^{73} +18.8174i q^{75} -2.68457i q^{77} -0.292266 q^{79} +27.7067 q^{81} +4.08918i q^{83} +11.5400i q^{85} -13.9082 q^{87} +1.22047 q^{89} -1.61023i q^{91} -25.1908i q^{93} +12.4441 q^{95} -14.0190 q^{97} +12.9066i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{9} - 36 q^{15} - 4 q^{17} + 24 q^{23} + 4 q^{25} - 20 q^{31} - 16 q^{33} - 4 q^{39} + 40 q^{41} - 40 q^{47} - 4 q^{49} - 16 q^{55} - 8 q^{57} - 44 q^{63} - 4 q^{65} + 56 q^{71} - 16 q^{73} - 32 q^{79}+ \cdots - 40 q^{97}+O(q^{100}) \) 8 * q - 12 * q^9 - 36 * q^15 - 4 * q^17 + 24 * q^23 + 4 * q^25 - 20 * q^31 - 16 * q^33 - 4 * q^39 + 40 * q^41 - 40 * q^47 - 4 * q^49 - 16 * q^55 - 8 * q^57 - 44 * q^63 - 4 * q^65 + 56 * q^71 - 16 * q^73 - 32 * q^79 + 40 * q^81 + 24 * q^87 - 16 * q^89 - 8 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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