LMFDB - Embedded newform 9306.2.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9306,2,Mod(1,9306)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9306, 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("9306.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9306 = 2 \cdot 3^{2} \cdot 11 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9306.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:	$74.3087841210$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2949696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 10x^{3} + 16x^{2} + 10x - 16 $ x^5 - 2x^4 - 10x^3 + 16x^2 + 10x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3102)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.16009$ of defining polynomial
Character	$\chi$	$=$	9306.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.16009 q^{5} -3.24650 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.16009 q^{5} -3.24650 q^{7} -1.00000 q^{8} +1.16009 q^{10} -1.00000 q^{11} +1.43222 q^{13} +3.24650 q^{14} +1.00000 q^{16} +2.02921 q^{17} +5.69987 q^{19} -1.16009 q^{20} +1.00000 q^{22} -5.30371 q^{23} -3.65419 q^{25} -1.43222 q^{26} -3.24650 q^{28} +5.78161 q^{29} +2.00000 q^{31} -1.00000 q^{32} -2.02921 q^{34} +3.76623 q^{35} -2.13894 q^{37} -5.69987 q^{38} +1.16009 q^{40} -7.01274 q^{41} +4.41107 q^{43} -1.00000 q^{44} +5.30371 q^{46} -1.00000 q^{47} +3.53978 q^{49} +3.65419 q^{50} +1.43222 q^{52} -12.5222 q^{53} +1.16009 q^{55} +3.24650 q^{56} -5.78161 q^{58} +4.38775 q^{59} +1.73703 q^{61} -2.00000 q^{62} +1.00000 q^{64} -1.66150 q^{65} -0.535105 q^{67} +2.02921 q^{68} -3.76623 q^{70} -3.63753 q^{71} +0.844586 q^{73} +2.13894 q^{74} +5.69987 q^{76} +3.24650 q^{77} +9.26655 q^{79} -1.16009 q^{80} +7.01274 q^{82} -13.0992 q^{83} -2.35406 q^{85} -4.41107 q^{86} +1.00000 q^{88} +7.90070 q^{89} -4.64971 q^{91} -5.30371 q^{92} +1.00000 q^{94} -6.61236 q^{95} +12.2300 q^{97} -3.53978 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8	$ 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8} + 2 q^{10} - 5 q^{11} + 6 q^{13} - 6 q^{14} + 5 q^{16} - 6 q^{17} + 12 q^{19} - 2 q^{20} + 5 q^{22} - 2 q^{23} - q^{25} - 6 q^{26} + 6 q^{28} + 4 q^{29} + 10 q^{31} - 5 q^{32} + 6 q^{34} + 8 q^{35} - 12 q^{38} + 2 q^{40} - 2 q^{41} + 14 q^{43} - 5 q^{44} + 2 q^{46} - 5 q^{47} + 5 q^{49} + q^{50} + 6 q^{52} - 2 q^{53} + 2 q^{55} - 6 q^{56} - 4 q^{58} - 10 q^{59} + 14 q^{61} - 10 q^{62} + 5 q^{64} - 6 q^{68} - 8 q^{70} - 12 q^{71} - 2 q^{73} + 12 q^{76} - 6 q^{77} - 2 q^{80} + 2 q^{82} - 14 q^{83} + 22 q^{85} - 14 q^{86} + 5 q^{88} + 12 q^{91} - 2 q^{92} + 5 q^{94} - 4 q^{95} + 22 q^{97} - 5 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8 + 2 * q^10 - 5 * q^11 + 6 * q^13 - 6 * q^14 + 5 * q^16 - 6 * q^17 + 12 * q^19 - 2 * q^20 + 5 * q^22 - 2 * q^23 - q^25 - 6 * q^26 + 6 * q^28 + 4 * q^29 + 10 * q^31 - 5 * q^32 + 6 * q^34 + 8 * q^35 - 12 * q^38 + 2 * q^40 - 2 * q^41 + 14 * q^43 - 5 * q^44 + 2 * q^46 - 5 * q^47 + 5 * q^49 + q^50 + 6 * q^52 - 2 * q^53 + 2 * q^55 - 6 * q^56 - 4 * q^58 - 10 * q^59 + 14 * q^61 - 10 * q^62 + 5 * q^64 - 6 * q^68 - 8 * q^70 - 12 * q^71 - 2 * q^73 + 12 * q^76 - 6 * q^77 - 2 * q^80 + 2 * q^82 - 14 * q^83 + 22 * q^85 - 14 * q^86 + 5 * q^88 + 12 * q^91 - 2 * q^92 + 5 * q^94 - 4 * q^95 + 22 * q^97 - 5 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.16009	−0.518808	−0.259404	−	0.965769i	$-0.583526\pi$
$5$	−1.16009	−0.518808	−0.259404	+	0.965769i	$0.583526\pi$
$6$	0	0
$7$	−3.24650	−1.22706	−0.613531	−	0.789670i	$-0.710252\pi$
$7$	−3.24650	−1.22706	−0.613531	+	0.789670i	$0.710252\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.16009	0.366853
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.43222	0.397227	0.198613	−	0.980078i	$-0.436356\pi$
$13$	1.43222	0.397227	0.198613	+	0.980078i	$0.436356\pi$
$14$	3.24650	0.867664
$15$	0	0
$16$	1.00000	0.250000
$17$	2.02921	0.492155	0.246078	−	0.969250i	$-0.420858\pi$
$17$	2.02921	0.492155	0.246078	+	0.969250i	$0.420858\pi$
$18$	0	0
$19$	5.69987	1.30764	0.653820	−	0.756650i	$-0.273165\pi$
$19$	5.69987	1.30764	0.653820	+	0.756650i	$0.273165\pi$
$20$	−1.16009	−0.259404
$21$	0	0
$22$	1.00000	0.213201
$23$	−5.30371	−1.10590	−0.552950	−	0.833215i	$-0.686498\pi$
$23$	−5.30371	−1.10590	−0.552950	+	0.833215i	$0.686498\pi$
$24$	0	0
$25$	−3.65419	−0.730838
$26$	−1.43222	−0.280882
$27$	0	0
$28$	−3.24650	−0.613531
$29$	5.78161	1.07362	0.536809	−	0.843704i	$-0.319630\pi$
$29$	5.78161	1.07362	0.536809	+	0.843704i	$0.319630\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.02921	−0.348006
$35$	3.76623	0.636610
$36$	0	0
$37$	−2.13894	−0.351640	−0.175820	−	0.984422i	$-0.556258\pi$
$37$	−2.13894	−0.351640	−0.175820	+	0.984422i	$0.556258\pi$
$38$	−5.69987	−0.924641
$39$	0	0
$40$	1.16009	0.183426
$41$	−7.01274	−1.09521	−0.547603	−	0.836738i	$-0.684459\pi$
$41$	−7.01274	−1.09521	−0.547603	+	0.836738i	$0.684459\pi$
$42$	0	0
$43$	4.41107	0.672683	0.336341	−	0.941740i	$-0.390810\pi$
$43$	4.41107	0.672683	0.336341	+	0.941740i	$0.390810\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	5.30371	0.781989
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	3.53978	0.505683
$50$	3.65419	0.516781
$51$	0	0
$52$	1.43222	0.198613
$53$	−12.5222	−1.72006	−0.860029	−	0.510245i	$-0.829555\pi$
$53$	−12.5222	−1.72006	−0.860029	+	0.510245i	$0.829555\pi$
$54$	0	0
$55$	1.16009	0.156426
$56$	3.24650	0.433832
$57$	0	0
$58$	−5.78161	−0.759162
$59$	4.38775	0.571237	0.285618	−	0.958343i	$-0.407801\pi$
$59$	4.38775	0.571237	0.285618	+	0.958343i	$0.407801\pi$
$60$	0	0
$61$	1.73703	0.222403	0.111202	−	0.993798i	$-0.464530\pi$
$61$	1.73703	0.222403	0.111202	+	0.993798i	$0.464530\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.66150	−0.206084
$66$	0	0
$67$	−0.535105	−0.0653735	−0.0326867	−	0.999466i	$-0.510406\pi$
$67$	−0.535105	−0.0653735	−0.0326867	+	0.999466i	$0.510406\pi$
$68$	2.02921	0.246078
$69$	0	0
$70$	−3.76623	−0.450151
$71$	−3.63753	−0.431695	−0.215848	−	0.976427i	$-0.569251\pi$
$71$	−3.63753	−0.431695	−0.215848	+	0.976427i	$0.569251\pi$
$72$	0	0
$73$	0.844586	0.0988513	0.0494257	−	0.998778i	$-0.484261\pi$
$73$	0.844586	0.0988513	0.0494257	+	0.998778i	$0.484261\pi$
$74$	2.13894	0.248647
$75$	0	0
$76$	5.69987	0.653820
$77$	3.24650	0.369973
$78$	0	0
$79$	9.26655	1.04257	0.521284	−	0.853383i	$-0.325453\pi$
$79$	9.26655	1.04257	0.521284	+	0.853383i	$0.325453\pi$
$80$	−1.16009	−0.129702
$81$	0	0
$82$	7.01274	0.774427
$83$	−13.0992	−1.43782	−0.718909	−	0.695104i	$-0.755358\pi$
$83$	−13.0992	−1.43782	−0.718909	+	0.695104i	$0.755358\pi$
$84$	0	0
$85$	−2.35406	−0.255334
$86$	−4.41107	−0.475658
$87$	0	0
$88$	1.00000	0.106600
$89$	7.90070	0.837472	0.418736	−	0.908108i	$-0.362473\pi$
$89$	7.90070	0.837472	0.418736	+	0.908108i	$0.362473\pi$
$90$	0	0
$91$	−4.64971	−0.487422
$92$	−5.30371	−0.552950
$93$	0	0
$94$	1.00000	0.103142
$95$	−6.61236	−0.678414
$96$	0	0
$97$	12.2300	1.24177	0.620886	−	0.783901i	$-0.286773\pi$
$97$	12.2300	1.24177	0.620886	+	0.783901i	$0.286773\pi$
$98$	−3.53978	−0.357572
$99$	0	0
$100$	−3.65419	−0.365419
$101$	−11.3913	−1.13348	−0.566740	−	0.823897i	$-0.691796\pi$
$101$	−11.3913	−1.13348	−0.566740	+	0.823897i	$0.691796\pi$
$102$	0	0
$103$	−5.48148	−0.540106	−0.270053	−	0.962845i	$-0.587041\pi$
$103$	−5.48148	−0.540106	−0.270053	+	0.962845i	$0.587041\pi$
$104$	−1.43222	−0.140441
$105$	0	0
$106$	12.5222	1.21626
$107$	−16.0448	−1.55111	−0.775553	−	0.631282i	$-0.782529\pi$
$107$	−16.0448	−1.55111	−0.775553	+	0.631282i	$0.782529\pi$
$108$	0	0
$109$	1.97547	0.189216	0.0946078	−	0.995515i	$-0.469840\pi$
$109$	1.97547	0.189216	0.0946078	+	0.995515i	$0.469840\pi$
$110$	−1.16009	−0.110610
$111$	0	0
$112$	−3.24650	−0.306766
$113$	−3.08404	−0.290122	−0.145061	−	0.989423i	$-0.546338\pi$
$113$	−3.08404	−0.290122	−0.145061	+	0.989423i	$0.546338\pi$
$114$	0	0
$115$	6.15278	0.573749
$116$	5.78161	0.536809
$117$	0	0
$118$	−4.38775	−0.403925
$119$	−6.58783	−0.603905
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.73703	−0.157263
$123$	0	0
$124$	2.00000	0.179605
$125$	10.0396	0.897973
$126$	0	0
$127$	5.77118	0.512109	0.256055	−	0.966662i	$-0.417577\pi$
$127$	5.77118	0.512109	0.256055	+	0.966662i	$0.417577\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.66150	0.145724
$131$	−0.958804	−0.0837711	−0.0418855	−	0.999122i	$-0.513336\pi$
$131$	−0.958804	−0.0837711	−0.0418855	+	0.999122i	$0.513336\pi$
$132$	0	0
$133$	−18.5046	−1.60456
$134$	0.535105	0.0462260
$135$	0	0
$136$	−2.02921	−0.174003
$137$	−16.9711	−1.44994	−0.724969	−	0.688781i	$-0.758146\pi$
$137$	−16.9711	−1.44994	−0.724969	+	0.688781i	$0.758146\pi$
$138$	0	0
$139$	−14.0985	−1.19582	−0.597910	−	0.801563i	$-0.704002\pi$
$139$	−14.0985	−1.19582	−0.597910	+	0.801563i	$0.704002\pi$
$140$	3.76623	0.318305
$141$	0	0
$142$	3.63753	0.305255
$143$	−1.43222	−0.119768
$144$	0	0
$145$	−6.70718	−0.557001
$146$	−0.844586	−0.0698985
$147$	0	0
$148$	−2.13894	−0.175820
$149$	16.3696	1.34105	0.670526	−	0.741886i	$-0.266068\pi$
$149$	16.3696	1.34105	0.670526	+	0.741886i	$0.266068\pi$
$150$	0	0
$151$	2.19661	0.178758	0.0893788	−	0.995998i	$-0.471512\pi$
$151$	2.19661	0.178758	0.0893788	+	0.995998i	$0.471512\pi$
$152$	−5.69987	−0.462321
$153$	0	0
$154$	−3.24650	−0.261611
$155$	−2.32018	−0.186361
$156$	0	0
$157$	14.0014	1.11743	0.558717	−	0.829358i	$-0.311294\pi$
$157$	14.0014	1.11743	0.558717	+	0.829358i	$0.311294\pi$
$158$	−9.26655	−0.737207
$159$	0	0
$160$	1.16009	0.0917131
$161$	17.2185	1.35701
$162$	0	0
$163$	−1.15168	−0.0902065	−0.0451033	−	0.998982i	$-0.514362\pi$
$163$	−1.15168	−0.0902065	−0.0451033	+	0.998982i	$0.514362\pi$
$164$	−7.01274	−0.547603
$165$	0	0
$166$	13.0992	1.01669
$167$	3.27898	0.253735	0.126868	−	0.991920i	$-0.459508\pi$
$167$	3.27898	0.253735	0.126868	+	0.991920i	$0.459508\pi$
$168$	0	0
$169$	−10.9487	−0.842211
$170$	2.35406	0.180548
$171$	0	0
$172$	4.41107	0.336341
$173$	1.16990	0.0889460	0.0444730	−	0.999011i	$-0.485839\pi$
$173$	1.16990	0.0889460	0.0444730	+	0.999011i	$0.485839\pi$
$174$	0	0
$175$	11.8633	0.896785
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−7.90070	−0.592182
$179$	21.6745	1.62003	0.810016	−	0.586408i	$-0.199458\pi$
$179$	21.6745	1.62003	0.810016	+	0.586408i	$0.199458\pi$
$180$	0	0
$181$	−14.2604	−1.05997	−0.529985	−	0.848007i	$-0.677803\pi$
$181$	−14.2604	−1.05997	−0.529985	+	0.848007i	$0.677803\pi$
$182$	4.64971	0.344659
$183$	0	0
$184$	5.30371	0.390995
$185$	2.48137	0.182434
$186$	0	0
$187$	−2.02921	−0.148390
$188$	−1.00000	−0.0729325
$189$	0	0
$190$	6.61236	0.479711
$191$	−20.2855	−1.46781	−0.733904	−	0.679253i	$-0.762304\pi$
$191$	−20.2855	−1.46781	−0.733904	+	0.679253i	$0.762304\pi$
$192$	0	0
$193$	−8.30261	−0.597635	−0.298818	−	0.954310i	$-0.596592\pi$
$193$	−8.30261	−0.597635	−0.298818	+	0.954310i	$0.596592\pi$
$194$	−12.2300	−0.878065
$195$	0	0
$196$	3.53978	0.252841
$197$	−2.02352	−0.144170	−0.0720848	−	0.997399i	$-0.522965\pi$
$197$	−2.02352	−0.144170	−0.0720848	+	0.997399i	$0.522965\pi$
$198$	0	0
$199$	23.0667	1.63515	0.817576	−	0.575820i	$-0.195317\pi$
$199$	23.0667	1.63515	0.817576	+	0.575820i	$0.195317\pi$
$200$	3.65419	0.258390
$201$	0	0
$202$	11.3913	0.801491
$203$	−18.7700	−1.31740
$204$	0	0
$205$	8.13540	0.568201
$206$	5.48148	0.381913
$207$	0	0
$208$	1.43222	0.0993067
$209$	−5.69987	−0.394268
$210$	0	0
$211$	15.2537	1.05011	0.525053	−	0.851070i	$-0.324046\pi$
$211$	15.2537	1.05011	0.525053	+	0.851070i	$0.324046\pi$
$212$	−12.5222	−0.860029
$213$	0	0
$214$	16.0448	1.09680
$215$	−5.11724	−0.348993
$216$	0	0
$217$	−6.49301	−0.440774
$218$	−1.97547	−0.133796
$219$	0	0
$220$	1.16009	0.0782132
$221$	2.90627	0.195497
$222$	0	0
$223$	6.85502	0.459046	0.229523	−	0.973303i	$-0.426283\pi$
$223$	6.85502	0.459046	0.229523	+	0.973303i	$0.426283\pi$
$224$	3.24650	0.216916
$225$	0	0
$226$	3.08404	0.205148
$227$	2.26886	0.150589	0.0752947	−	0.997161i	$-0.476010\pi$
$227$	2.26886	0.150589	0.0752947	+	0.997161i	$0.476010\pi$
$228$	0	0
$229$	−18.8342	−1.24460	−0.622299	−	0.782780i	$-0.713801\pi$
$229$	−18.8342	−1.24460	−0.622299	+	0.782780i	$0.713801\pi$
$230$	−6.15278	−0.405702
$231$	0	0
$232$	−5.78161	−0.379581
$233$	10.2721	0.672950	0.336475	−	0.941692i	$-0.390765\pi$
$233$	10.2721	0.672950	0.336475	+	0.941692i	$0.390765\pi$
$234$	0	0
$235$	1.16009	0.0756759
$236$	4.38775	0.285618
$237$	0	0
$238$	6.58783	0.427025
$239$	16.6126	1.07458	0.537288	−	0.843399i	$-0.319449\pi$
$239$	16.6126	1.07458	0.537288	+	0.843399i	$0.319449\pi$
$240$	0	0
$241$	5.04107	0.324724	0.162362	−	0.986731i	$-0.448089\pi$
$241$	5.04107	0.324724	0.162362	+	0.986731i	$0.448089\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	1.73703	0.111202
$245$	−4.10646	−0.262352
$246$	0	0
$247$	8.16347	0.519429
$248$	−2.00000	−0.127000
$249$	0	0
$250$	−10.0396	−0.634962
$251$	−0.343632	−0.0216899	−0.0108449	−	0.999941i	$-0.503452\pi$
$251$	−0.343632	−0.0216899	−0.0108449	+	0.999941i	$0.503452\pi$
$252$	0	0
$253$	5.30371	0.333441
$254$	−5.77118	−0.362116
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.2068	1.69711	0.848557	−	0.529103i	$-0.177472\pi$
$257$	27.2068	1.69711	0.848557	+	0.529103i	$0.177472\pi$
$258$	0	0
$259$	6.94408	0.431484
$260$	−1.66150	−0.103042
$261$	0	0
$262$	0.958804	0.0592351
$263$	−27.3393	−1.68582	−0.842908	−	0.538058i	$-0.819158\pi$
$263$	−27.3393	−1.68582	−0.842908	+	0.538058i	$0.819158\pi$
$264$	0	0
$265$	14.5269	0.892380
$266$	18.5046	1.13459
$267$	0	0
$268$	−0.535105	−0.0326867
$269$	26.9194	1.64130	0.820652	−	0.571429i	$-0.193611\pi$
$269$	26.9194	1.64130	0.820652	+	0.571429i	$0.193611\pi$
$270$	0	0
$271$	6.94637	0.421962	0.210981	−	0.977490i	$-0.432334\pi$
$271$	6.94637	0.421962	0.210981	+	0.977490i	$0.432334\pi$
$272$	2.02921	0.123039
$273$	0	0
$274$	16.9711	1.02526
$275$	3.65419	0.220356
$276$	0	0
$277$	−18.4073	−1.10599	−0.552995	−	0.833185i	$-0.686515\pi$
$277$	−18.4073	−1.10599	−0.552995	+	0.833185i	$0.686515\pi$
$278$	14.0985	0.845573
$279$	0	0
$280$	−3.76623	−0.225076
$281$	−7.50620	−0.447783	−0.223891	−	0.974614i	$-0.571876\pi$
$281$	−7.50620	−0.447783	−0.223891	+	0.974614i	$0.571876\pi$
$282$	0	0
$283$	31.4277	1.86818	0.934091	−	0.357034i	$-0.116212\pi$
$283$	31.4277	1.86818	0.934091	+	0.357034i	$0.116212\pi$
$284$	−3.63753	−0.215848
$285$	0	0
$286$	1.43222	0.0846890
$287$	22.7669	1.34389
$288$	0	0
$289$	−12.8823	−0.757783
$290$	6.70718	0.393859
$291$	0	0
$292$	0.844586	0.0494257
$293$	−2.72157	−0.158996	−0.0794980	−	0.996835i	$-0.525332\pi$
$293$	−2.72157	−0.158996	−0.0794980	+	0.996835i	$0.525332\pi$
$294$	0	0
$295$	−5.09019	−0.296362
$296$	2.13894	0.124324
$297$	0	0
$298$	−16.3696	−0.948267
$299$	−7.59608	−0.439293
$300$	0	0
$301$	−14.3206	−0.825424
$302$	−2.19661	−0.126401
$303$	0	0
$304$	5.69987	0.326910
$305$	−2.01511	−0.115385
$306$	0	0
$307$	19.5679	1.11680	0.558399	−	0.829572i	$-0.311416\pi$
$307$	19.5679	1.11680	0.558399	+	0.829572i	$0.311416\pi$
$308$	3.24650	0.184987
$309$	0	0
$310$	2.32018	0.131777
$311$	18.7709	1.06440	0.532200	−	0.846619i	$-0.321366\pi$
$311$	18.7709	1.06440	0.532200	+	0.846619i	$0.321366\pi$
$312$	0	0
$313$	6.23919	0.352660	0.176330	−	0.984331i	$-0.443577\pi$
$313$	6.23919	0.352660	0.176330	+	0.984331i	$0.443577\pi$
$314$	−14.0014	−0.790145
$315$	0	0
$316$	9.26655	0.521284
$317$	14.1968	0.797372	0.398686	−	0.917088i	$-0.369466\pi$
$317$	14.1968	0.797372	0.398686	+	0.917088i	$0.369466\pi$
$318$	0	0
$319$	−5.78161	−0.323708
$320$	−1.16009	−0.0648510
$321$	0	0
$322$	−17.2185	−0.959550
$323$	11.5662	0.643562
$324$	0	0
$325$	−5.23361	−0.290309
$326$	1.15168	0.0637857
$327$	0	0
$328$	7.01274	0.387214
$329$	3.24650	0.178985
$330$	0	0
$331$	15.3215	0.842146	0.421073	−	0.907027i	$-0.361654\pi$
$331$	15.3215	0.842146	0.421073	+	0.907027i	$0.361654\pi$
$332$	−13.0992	−0.718909
$333$	0	0
$334$	−3.27898	−0.179418
$335$	0.620770	0.0339163
$336$	0	0
$337$	5.12542	0.279199	0.139600	−	0.990208i	$-0.455418\pi$
$337$	5.12542	0.279199	0.139600	+	0.990208i	$0.455418\pi$
$338$	10.9487	0.595533
$339$	0	0
$340$	−2.35406	−0.127667
$341$	−2.00000	−0.108306
$342$	0	0
$343$	11.2336	0.606558
$344$	−4.41107	−0.237829
$345$	0	0
$346$	−1.16990	−0.0628943
$347$	11.1722	0.599755	0.299877	−	0.953978i	$-0.403054\pi$
$347$	11.1722	0.599755	0.299877	+	0.953978i	$0.403054\pi$
$348$	0	0
$349$	6.12203	0.327705	0.163852	−	0.986485i	$-0.447608\pi$
$349$	6.12203	0.327705	0.163852	+	0.986485i	$0.447608\pi$
$350$	−11.8633	−0.634122
$351$	0	0
$352$	1.00000	0.0533002
$353$	10.6355	0.566073	0.283036	−	0.959109i	$-0.408658\pi$
$353$	10.6355	0.566073	0.283036	+	0.959109i	$0.408658\pi$
$354$	0	0
$355$	4.21986	0.223967
$356$	7.90070	0.418736
$357$	0	0
$358$	−21.6745	−1.14554
$359$	22.6671	1.19632	0.598162	−	0.801375i	$-0.295898\pi$
$359$	22.6671	1.19632	0.598162	+	0.801375i	$0.295898\pi$
$360$	0	0
$361$	13.4885	0.709922
$362$	14.2604	0.749513
$363$	0	0
$364$	−4.64971	−0.243711
$365$	−0.979796	−0.0512849
$366$	0	0
$367$	22.2478	1.16133	0.580664	−	0.814144i	$-0.302793\pi$
$367$	22.2478	1.16133	0.580664	+	0.814144i	$0.302793\pi$
$368$	−5.30371	−0.276475
$369$	0	0
$370$	−2.48137	−0.129000
$371$	40.6534	2.11062
$372$	0	0
$373$	−19.9204	−1.03144	−0.515720	−	0.856757i	$-0.672476\pi$
$373$	−19.9204	−1.03144	−0.515720	+	0.856757i	$0.672476\pi$
$374$	2.02921	0.104928
$375$	0	0
$376$	1.00000	0.0515711
$377$	8.28054	0.426470
$378$	0	0
$379$	18.7218	0.961676	0.480838	−	0.876810i	$-0.340333\pi$
$379$	18.7218	0.961676	0.480838	+	0.876810i	$0.340333\pi$
$380$	−6.61236	−0.339207
$381$	0	0
$382$	20.2855	1.03790
$383$	−12.2842	−0.627695	−0.313847	−	0.949473i	$-0.601618\pi$
$383$	−12.2842	−0.627695	−0.313847	+	0.949473i	$0.601618\pi$
$384$	0	0
$385$	−3.76623	−0.191945
$386$	8.30261	0.422592
$387$	0	0
$388$	12.2300	0.620886
$389$	−10.1540	−0.514831	−0.257415	−	0.966301i	$-0.582871\pi$
$389$	−10.1540	−0.514831	−0.257415	+	0.966301i	$0.582871\pi$
$390$	0	0
$391$	−10.7623	−0.544274
$392$	−3.53978	−0.178786
$393$	0	0
$394$	2.02352	0.101943
$395$	−10.7500	−0.540893
$396$	0	0
$397$	−13.0697	−0.655952	−0.327976	−	0.944686i	$-0.606366\pi$
$397$	−13.0697	−0.655952	−0.327976	+	0.944686i	$0.606366\pi$
$398$	−23.0667	−1.15623
$399$	0	0
$400$	−3.65419	−0.182710
$401$	20.2877	1.01312	0.506560	−	0.862205i	$-0.330917\pi$
$401$	20.2877	1.01312	0.506560	+	0.862205i	$0.330917\pi$
$402$	0	0
$403$	2.86444	0.142688
$404$	−11.3913	−0.566740
$405$	0	0
$406$	18.7700	0.931540
$407$	2.13894	0.106023
$408$	0	0
$409$	5.75515	0.284574	0.142287	−	0.989825i	$-0.454554\pi$
$409$	5.75515	0.284574	0.142287	+	0.989825i	$0.454554\pi$
$410$	−8.13540	−0.401779
$411$	0	0
$412$	−5.48148	−0.270053
$413$	−14.2449	−0.700943
$414$	0	0
$415$	15.1962	0.745952
$416$	−1.43222	−0.0702204
$417$	0	0
$418$	5.69987	0.278790
$419$	35.4372	1.73122	0.865610	−	0.500719i	$-0.166931\pi$
$419$	35.4372	1.73122	0.865610	+	0.500719i	$0.166931\pi$
$420$	0	0
$421$	8.91464	0.434473	0.217237	−	0.976119i	$-0.430296\pi$
$421$	8.91464	0.434473	0.217237	+	0.976119i	$0.430296\pi$
$422$	−15.2537	−0.742537
$423$	0	0
$424$	12.5222	0.608132
$425$	−7.41511	−0.359686
$426$	0	0
$427$	−5.63926	−0.272903
$428$	−16.0448	−0.775553
$429$	0	0
$430$	5.11724	0.246775
$431$	7.63551	0.367789	0.183895	−	0.982946i	$-0.441129\pi$
$431$	7.63551	0.367789	0.183895	+	0.982946i	$0.441129\pi$
$432$	0	0
$433$	3.33393	0.160219	0.0801093	−	0.996786i	$-0.474473\pi$
$433$	3.33393	0.160219	0.0801093	+	0.996786i	$0.474473\pi$
$434$	6.49301	0.311674
$435$	0	0
$436$	1.97547	0.0946078
$437$	−30.2305	−1.44612
$438$	0	0
$439$	−12.8481	−0.613206	−0.306603	−	0.951838i	$-0.599192\pi$
$439$	−12.8481	−0.613206	−0.306603	+	0.951838i	$0.599192\pi$
$440$	−1.16009	−0.0553051
$441$	0	0
$442$	−2.90627	−0.138237
$443$	23.3886	1.11123	0.555613	−	0.831441i	$-0.312484\pi$
$443$	23.3886	1.11123	0.555613	+	0.831441i	$0.312484\pi$
$444$	0	0
$445$	−9.16551	−0.434487
$446$	−6.85502	−0.324594
$447$	0	0
$448$	−3.24650	−0.153383
$449$	31.9616	1.50836	0.754180	−	0.656667i	$-0.228034\pi$
$449$	31.9616	1.50836	0.754180	+	0.656667i	$0.228034\pi$
$450$	0	0
$451$	7.01274	0.330217
$452$	−3.08404	−0.145061
$453$	0	0
$454$	−2.26886	−0.106483
$455$	5.39408	0.252878
$456$	0	0
$457$	−7.37475	−0.344976	−0.172488	−	0.985012i	$-0.555181\pi$
$457$	−7.37475	−0.344976	−0.172488	+	0.985012i	$0.555181\pi$
$458$	18.8342	0.880063
$459$	0	0
$460$	6.15278	0.286875
$461$	−38.7302	−1.80385	−0.901923	−	0.431898i	$-0.857844\pi$
$461$	−38.7302	−1.80385	−0.901923	+	0.431898i	$0.857844\pi$
$462$	0	0
$463$	−6.29532	−0.292568	−0.146284	−	0.989243i	$-0.546731\pi$
$463$	−6.29532	−0.292568	−0.146284	+	0.989243i	$0.546731\pi$
$464$	5.78161	0.268404
$465$	0	0
$466$	−10.2721	−0.475847
$467$	12.4649	0.576806	0.288403	−	0.957509i	$-0.406876\pi$
$467$	12.4649	0.576806	0.288403	+	0.957509i	$0.406876\pi$
$468$	0	0
$469$	1.73722	0.0802174
$470$	−1.16009	−0.0535109
$471$	0	0
$472$	−4.38775	−0.201963
$473$	−4.41107	−0.202821
$474$	0	0
$475$	−20.8284	−0.955674
$476$	−6.58783	−0.301953
$477$	0	0
$478$	−16.6126	−0.759841
$479$	−27.3181	−1.24820	−0.624098	−	0.781346i	$-0.714533\pi$
$479$	−27.3181	−1.24820	−0.624098	+	0.781346i	$0.714533\pi$
$480$	0	0
$481$	−3.06344	−0.139681
$482$	−5.04107	−0.229614
$483$	0	0
$484$	1.00000	0.0454545
$485$	−14.1879	−0.644241
$486$	0	0
$487$	16.2800	0.737719	0.368859	−	0.929485i	$-0.379748\pi$
$487$	16.2800	0.737719	0.368859	+	0.929485i	$0.379748\pi$
$488$	−1.73703	−0.0786315
$489$	0	0
$490$	4.10646	0.185511
$491$	0.581943	0.0262627	0.0131314	−	0.999914i	$-0.495820\pi$
$491$	0.581943	0.0262627	0.0131314	+	0.999914i	$0.495820\pi$
$492$	0	0
$493$	11.7321	0.528386
$494$	−8.16347	−0.367292
$495$	0	0
$496$	2.00000	0.0898027
$497$	11.8092	0.529717
$498$	0	0
$499$	−29.5612	−1.32334	−0.661671	−	0.749794i	$-0.730152\pi$
$499$	−29.5612	−1.32334	−0.661671	+	0.749794i	$0.730152\pi$
$500$	10.0396	0.448986
$501$	0	0
$502$	0.343632	0.0153371
$503$	12.3320	0.549858	0.274929	−	0.961465i	$-0.411346\pi$
$503$	12.3320	0.549858	0.274929	+	0.961465i	$0.411346\pi$
$504$	0	0
$505$	13.2150	0.588058
$506$	−5.30371	−0.235779
$507$	0	0
$508$	5.77118	0.256055
$509$	−6.43966	−0.285433	−0.142717	−	0.989764i	$-0.545584\pi$
$509$	−6.43966	−0.285433	−0.142717	+	0.989764i	$0.545584\pi$
$510$	0	0
$511$	−2.74195	−0.121297
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−27.2068	−1.20004
$515$	6.35901	0.280211
$516$	0	0
$517$	1.00000	0.0439799
$518$	−6.94408	−0.305106
$519$	0	0
$520$	1.66150	0.0728618
$521$	14.0387	0.615046	0.307523	−	0.951541i	$-0.400500\pi$
$521$	14.0387	0.615046	0.307523	+	0.951541i	$0.400500\pi$
$522$	0	0
$523$	3.53994	0.154791	0.0773953	−	0.997000i	$-0.475340\pi$
$523$	3.53994	0.154791	0.0773953	+	0.997000i	$0.475340\pi$
$524$	−0.958804	−0.0418855
$525$	0	0
$526$	27.3393	1.19205
$527$	4.05842	0.176787
$528$	0	0
$529$	5.12932	0.223014
$530$	−14.5269	−0.631008
$531$	0	0
$532$	−18.5046	−0.802278
$533$	−10.0438	−0.435045
$534$	0	0
$535$	18.6134	0.804726
$536$	0.535105	0.0231130
$537$	0	0
$538$	−26.9194	−1.16058
$539$	−3.53978	−0.152469
$540$	0	0
$541$	−24.3035	−1.04489	−0.522444	−	0.852673i	$-0.674980\pi$
$541$	−24.3035	−1.04489	−0.522444	+	0.852673i	$0.674980\pi$
$542$	−6.94637	−0.298372
$543$	0	0
$544$	−2.02921	−0.0870016
$545$	−2.29172	−0.0981665
$546$	0	0
$547$	−30.8395	−1.31860	−0.659301	−	0.751879i	$-0.729148\pi$
$547$	−30.8395	−1.31860	−0.659301	+	0.751879i	$0.729148\pi$
$548$	−16.9711	−0.724969
$549$	0	0
$550$	−3.65419	−0.155815
$551$	32.9544	1.40391
$552$	0	0
$553$	−30.0839	−1.27930
$554$	18.4073	0.782053
$555$	0	0
$556$	−14.0985	−0.597910
$557$	−1.87122	−0.0792861	−0.0396431	−	0.999214i	$-0.512622\pi$
$557$	−1.87122	−0.0792861	−0.0396431	+	0.999214i	$0.512622\pi$
$558$	0	0
$559$	6.31763	0.267207
$560$	3.76623	0.159152
$561$	0	0
$562$	7.50620	0.316630
$563$	25.4635	1.07316	0.536580	−	0.843849i	$-0.319716\pi$
$563$	25.4635	1.07316	0.536580	+	0.843849i	$0.319716\pi$
$564$	0	0
$565$	3.57777	0.150518
$566$	−31.4277	−1.32100
$567$	0	0
$568$	3.63753	0.152627
$569$	6.11245	0.256247	0.128124	−	0.991758i	$-0.459105\pi$
$569$	6.11245	0.256247	0.128124	+	0.991758i	$0.459105\pi$
$570$	0	0
$571$	−35.2983	−1.47719	−0.738595	−	0.674150i	$-0.764510\pi$
$571$	−35.2983	−1.47719	−0.738595	+	0.674150i	$0.764510\pi$
$572$	−1.43222	−0.0598842
$573$	0	0
$574$	−22.7669	−0.950271
$575$	19.3808	0.808234
$576$	0	0
$577$	39.1884	1.63144	0.815718	−	0.578450i	$-0.196342\pi$
$577$	39.1884	1.63144	0.815718	+	0.578450i	$0.196342\pi$
$578$	12.8823	0.535834
$579$	0	0
$580$	−6.70718	−0.278501
$581$	42.5264	1.76429
$582$	0	0
$583$	12.5222	0.518617
$584$	−0.844586	−0.0349492
$585$	0	0
$586$	2.72157	0.112427
$587$	16.4793	0.680174	0.340087	−	0.940394i	$-0.389544\pi$
$587$	16.4793	0.680174	0.340087	+	0.940394i	$0.389544\pi$
$588$	0	0
$589$	11.3997	0.469718
$590$	5.09019	0.209560
$591$	0	0
$592$	−2.13894	−0.0879100
$593$	12.2713	0.503924	0.251962	−	0.967737i	$-0.418924\pi$
$593$	12.2713	0.503924	0.251962	+	0.967737i	$0.418924\pi$
$594$	0	0
$595$	7.64247	0.313311
$596$	16.3696	0.670526
$597$	0	0
$598$	7.59608	0.310627
$599$	15.9864	0.653185	0.326592	−	0.945165i	$-0.394100\pi$
$599$	15.9864	0.653185	0.326592	+	0.945165i	$0.394100\pi$
$600$	0	0
$601$	−4.89591	−0.199708	−0.0998541	−	0.995002i	$-0.531838\pi$
$601$	−4.89591	−0.199708	−0.0998541	+	0.995002i	$0.531838\pi$
$602$	14.3206	0.583663
$603$	0	0
$604$	2.19661	0.0893788
$605$	−1.16009	−0.0471643
$606$	0	0
$607$	1.46054	0.0592816	0.0296408	−	0.999561i	$-0.490564\pi$
$607$	1.46054	0.0592816	0.0296408	+	0.999561i	$0.490564\pi$
$608$	−5.69987	−0.231160
$609$	0	0
$610$	2.01511	0.0815893
$611$	−1.43222	−0.0579415
$612$	0	0
$613$	−26.2998	−1.06224	−0.531119	−	0.847297i	$-0.678228\pi$
$613$	−26.2998	−1.06224	−0.531119	+	0.847297i	$0.678228\pi$
$614$	−19.5679	−0.789696
$615$	0	0
$616$	−3.24650	−0.130805
$617$	−0.224471	−0.00903687	−0.00451844	−	0.999990i	$-0.501438\pi$
$617$	−0.224471	−0.00903687	−0.00451844	+	0.999990i	$0.501438\pi$
$618$	0	0
$619$	4.14035	0.166415	0.0832073	−	0.996532i	$-0.473484\pi$
$619$	4.14035	0.166415	0.0832073	+	0.996532i	$0.473484\pi$
$620$	−2.32018	−0.0931806
$621$	0	0
$622$	−18.7709	−0.752644
$623$	−25.6496	−1.02763
$624$	0	0
$625$	6.62408	0.264963
$626$	−6.23919	−0.249368
$627$	0	0
$628$	14.0014	0.558717
$629$	−4.34036	−0.173061
$630$	0	0
$631$	−18.1584	−0.722876	−0.361438	−	0.932396i	$-0.617714\pi$
$631$	−18.1584	−0.722876	−0.361438	+	0.932396i	$0.617714\pi$
$632$	−9.26655	−0.368604
$633$	0	0
$634$	−14.1968	−0.563827
$635$	−6.69508	−0.265686
$636$	0	0
$637$	5.06975	0.200871
$638$	5.78161	0.228896
$639$	0	0
$640$	1.16009	0.0458566
$641$	20.0116	0.790412	0.395206	−	0.918592i	$-0.370673\pi$
$641$	20.0116	0.790412	0.395206	+	0.918592i	$0.370673\pi$
$642$	0	0
$643$	−29.9515	−1.18117	−0.590586	−	0.806975i	$-0.701103\pi$
$643$	−29.9515	−1.18117	−0.590586	+	0.806975i	$0.701103\pi$
$644$	17.2185	0.678504
$645$	0	0
$646$	−11.5662	−0.455067
$647$	8.92933	0.351048	0.175524	−	0.984475i	$-0.443838\pi$
$647$	8.92933	0.351048	0.175524	+	0.984475i	$0.443838\pi$
$648$	0	0
$649$	−4.38775	−0.172234
$650$	5.23361	0.205279
$651$	0	0
$652$	−1.15168	−0.0451033
$653$	28.1013	1.09969	0.549845	−	0.835267i	$-0.314687\pi$
$653$	28.1013	1.09969	0.549845	+	0.835267i	$0.314687\pi$
$654$	0	0
$655$	1.11230	0.0434611
$656$	−7.01274	−0.273801
$657$	0	0
$658$	−3.24650	−0.126562
$659$	−29.0752	−1.13261	−0.566304	−	0.824196i	$-0.691627\pi$
$659$	−29.0752	−1.13261	−0.566304	+	0.824196i	$0.691627\pi$
$660$	0	0
$661$	−6.78486	−0.263900	−0.131950	−	0.991256i	$-0.542124\pi$
$661$	−6.78486	−0.263900	−0.131950	+	0.991256i	$0.542124\pi$
$662$	−15.3215	−0.595487
$663$	0	0
$664$	13.0992	0.508346
$665$	21.4670	0.832456
$666$	0	0
$667$	−30.6640	−1.18731
$668$	3.27898	0.126868
$669$	0	0
$670$	−0.620770	−0.0239824
$671$	−1.73703	−0.0670572
$672$	0	0
$673$	20.3719	0.785280	0.392640	−	0.919692i	$-0.371562\pi$
$673$	20.3719	0.785280	0.392640	+	0.919692i	$0.371562\pi$
$674$	−5.12542	−0.197424
$675$	0	0
$676$	−10.9487	−0.421105
$677$	−51.0642	−1.96256	−0.981278	−	0.192594i	$-0.938310\pi$
$677$	−51.0642	−1.96256	−0.981278	+	0.192594i	$0.938310\pi$
$678$	0	0
$679$	−39.7048	−1.52373
$680$	2.35406	0.0902742
$681$	0	0
$682$	2.00000	0.0765840
$683$	−21.9274	−0.839028	−0.419514	−	0.907749i	$-0.637799\pi$
$683$	−21.9274	−0.839028	−0.419514	+	0.907749i	$0.637799\pi$
$684$	0	0
$685$	19.6880	0.752240
$686$	−11.2336	−0.428901
$687$	0	0
$688$	4.41107	0.168171
$689$	−17.9346	−0.683253
$690$	0	0
$691$	17.1209	0.651311	0.325655	−	0.945489i	$-0.394415\pi$
$691$	17.1209	0.651311	0.325655	+	0.945489i	$0.394415\pi$
$692$	1.16990	0.0444730
$693$	0	0
$694$	−11.1722	−0.424091
$695$	16.3555	0.620401
$696$	0	0
$697$	−14.2303	−0.539011
$698$	−6.12203	−0.231722
$699$	0	0
$700$	11.8633	0.448392
$701$	29.9913	1.13276	0.566378	−	0.824145i	$-0.308344\pi$
$701$	29.9913	1.13276	0.566378	+	0.824145i	$0.308344\pi$
$702$	0	0
$703$	−12.1917	−0.459819
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−10.6355	−0.400274
$707$	36.9820	1.39085
$708$	0	0
$709$	−41.2852	−1.55050	−0.775249	−	0.631655i	$-0.782376\pi$
$709$	−41.2852	−1.55050	−0.775249	+	0.631655i	$0.782376\pi$
$710$	−4.21986	−0.158368
$711$	0	0
$712$	−7.90070	−0.296091
$713$	−10.6074	−0.397251
$714$	0	0
$715$	1.66150	0.0621368
$716$	21.6745	0.810016
$717$	0	0
$718$	−22.6671	−0.845929
$719$	26.6671	0.994517	0.497258	−	0.867603i	$-0.334340\pi$
$719$	26.6671	0.994517	0.497258	+	0.867603i	$0.334340\pi$
$720$	0	0
$721$	17.7956	0.662744
$722$	−13.4885	−0.501991
$723$	0	0
$724$	−14.2604	−0.529985
$725$	−21.1271	−0.784641
$726$	0	0
$727$	30.0989	1.11631	0.558153	−	0.829738i	$-0.311510\pi$
$727$	30.0989	1.11631	0.558153	+	0.829738i	$0.311510\pi$
$728$	4.64971	0.172330
$729$	0	0
$730$	0.979796	0.0362639
$731$	8.95099	0.331064
$732$	0	0
$733$	−4.74937	−0.175422	−0.0877109	−	0.996146i	$-0.527955\pi$
$733$	−4.74937	−0.175422	−0.0877109	+	0.996146i	$0.527955\pi$
$734$	−22.2478	−0.821182
$735$	0	0
$736$	5.30371	0.195497
$737$	0.535105	0.0197108
$738$	0	0
$739$	38.3923	1.41228	0.706141	−	0.708071i	$-0.250434\pi$
$739$	38.3923	1.41228	0.706141	+	0.708071i	$0.250434\pi$
$740$	2.48137	0.0912168
$741$	0	0
$742$	−40.6534	−1.49243
$743$	−26.6933	−0.979281	−0.489641	−	0.871924i	$-0.662872\pi$
$743$	−26.6933	−0.979281	−0.489641	+	0.871924i	$0.662872\pi$
$744$	0	0
$745$	−18.9902	−0.695748
$746$	19.9204	0.729338
$747$	0	0
$748$	−2.02921	−0.0741952
$749$	52.0894	1.90331
$750$	0	0
$751$	−4.24976	−0.155076	−0.0775379	−	0.996989i	$-0.524706\pi$
$751$	−4.24976	−0.155076	−0.0775379	+	0.996989i	$0.524706\pi$
$752$	−1.00000	−0.0364662
$753$	0	0
$754$	−8.28054	−0.301560
$755$	−2.54826	−0.0927408
$756$	0	0
$757$	13.0642	0.474826	0.237413	−	0.971409i	$-0.423701\pi$
$757$	13.0642	0.474826	0.237413	+	0.971409i	$0.423701\pi$
$758$	−18.7218	−0.680007
$759$	0	0
$760$	6.61236	0.239856
$761$	41.2953	1.49695	0.748477	−	0.663161i	$-0.230786\pi$
$761$	41.2953	1.49695	0.748477	+	0.663161i	$0.230786\pi$
$762$	0	0
$763$	−6.41336	−0.232179
$764$	−20.2855	−0.733904
$765$	0	0
$766$	12.2842	0.443847
$767$	6.28423	0.226910
$768$	0	0
$769$	25.4994	0.919532	0.459766	−	0.888040i	$-0.347933\pi$
$769$	25.4994	0.919532	0.459766	+	0.888040i	$0.347933\pi$
$770$	3.76623	0.135726
$771$	0	0
$772$	−8.30261	−0.298818
$773$	17.6692	0.635515	0.317758	−	0.948172i	$-0.397070\pi$
$773$	17.6692	0.635515	0.317758	+	0.948172i	$0.397070\pi$
$774$	0	0
$775$	−7.30838	−0.262525
$776$	−12.2300	−0.439033
$777$	0	0
$778$	10.1540	0.364040
$779$	−39.9717	−1.43213
$780$	0	0
$781$	3.63753	0.130161
$782$	10.7623	0.384860
$783$	0	0
$784$	3.53978	0.126421
$785$	−16.2429	−0.579733
$786$	0	0
$787$	−17.0350	−0.607232	−0.303616	−	0.952794i	$-0.598194\pi$
$787$	−17.0350	−0.607232	−0.303616	+	0.952794i	$0.598194\pi$
$788$	−2.02352	−0.0720848
$789$	0	0
$790$	10.7500	0.382469
$791$	10.0124	0.355998
$792$	0	0
$793$	2.48781	0.0883446
$794$	13.0697	0.463828
$795$	0	0
$796$	23.0667	0.817576
$797$	16.4507	0.582714	0.291357	−	0.956614i	$-0.405893\pi$
$797$	16.4507	0.582714	0.291357	+	0.956614i	$0.405893\pi$
$798$	0	0
$799$	−2.02921	−0.0717882
$800$	3.65419	0.129195
$801$	0	0
$802$	−20.2877	−0.716384
$803$	−0.844586	−0.0298048
$804$	0	0
$805$	−19.9750	−0.704027
$806$	−2.86444	−0.100896
$807$	0	0
$808$	11.3913	0.400746
$809$	15.5145	0.545462	0.272731	−	0.962090i	$-0.412073\pi$
$809$	15.5145	0.545462	0.272731	+	0.962090i	$0.412073\pi$
$810$	0	0
$811$	47.9816	1.68486	0.842431	−	0.538805i	$-0.181124\pi$
$811$	47.9816	1.68486	0.842431	+	0.538805i	$0.181124\pi$
$812$	−18.7700	−0.658698
$813$	0	0
$814$	−2.13894	−0.0749699
$815$	1.33605	0.0467999
$816$	0	0
$817$	25.1426	0.879627
$818$	−5.75515	−0.201224
$819$	0	0
$820$	8.13540	0.284101
$821$	−12.9001	−0.450215	−0.225108	−	0.974334i	$-0.572273\pi$
$821$	−12.9001	−0.450215	−0.225108	+	0.974334i	$0.572273\pi$
$822$	0	0
$823$	35.1073	1.22376	0.611882	−	0.790949i	$-0.290413\pi$
$823$	35.1073	1.22376	0.611882	+	0.790949i	$0.290413\pi$
$824$	5.48148	0.190956
$825$	0	0
$826$	14.2449	0.495642
$827$	51.9829	1.80762	0.903812	−	0.427930i	$-0.140757\pi$
$827$	51.9829	1.80762	0.903812	+	0.427930i	$0.140757\pi$
$828$	0	0
$829$	30.3450	1.05393	0.526963	−	0.849888i	$-0.323331\pi$
$829$	30.3450	1.05393	0.526963	+	0.849888i	$0.323331\pi$
$830$	−15.1962	−0.527467
$831$	0	0
$832$	1.43222	0.0496533
$833$	7.18295	0.248874
$834$	0	0
$835$	−3.80391	−0.131640
$836$	−5.69987	−0.197134
$837$	0	0
$838$	−35.4372	−1.22416
$839$	41.2456	1.42396	0.711978	−	0.702202i	$-0.247800\pi$
$839$	41.2456	1.42396	0.711978	+	0.702202i	$0.247800\pi$
$840$	0	0
$841$	4.42699	0.152655
$842$	−8.91464	−0.307219
$843$	0	0
$844$	15.2537	0.525053
$845$	12.7015	0.436946
$846$	0	0
$847$	−3.24650	−0.111551
$848$	−12.5222	−0.430015
$849$	0	0
$850$	7.41511	0.254336
$851$	11.3443	0.388879
$852$	0	0
$853$	39.3065	1.34583	0.672914	−	0.739721i	$-0.265042\pi$
$853$	39.3065	1.34583	0.672914	+	0.739721i	$0.265042\pi$
$854$	5.63926	0.192972
$855$	0	0
$856$	16.0448	0.548399
$857$	1.74239	0.0595188	0.0297594	−	0.999557i	$-0.490526\pi$
$857$	1.74239	0.0595188	0.0297594	+	0.999557i	$0.490526\pi$
$858$	0	0
$859$	29.1665	0.995148	0.497574	−	0.867422i	$-0.334224\pi$
$859$	29.1665	0.995148	0.497574	+	0.867422i	$0.334224\pi$
$860$	−5.11724	−0.174496
$861$	0	0
$862$	−7.63551	−0.260066
$863$	−8.11418	−0.276210	−0.138105	−	0.990418i	$-0.544101\pi$
$863$	−8.11418	−0.276210	−0.138105	+	0.990418i	$0.544101\pi$
$864$	0	0
$865$	−1.35719	−0.0461459
$866$	−3.33393	−0.113292
$867$	0	0
$868$	−6.49301	−0.220387
$869$	−9.26655	−0.314346
$870$	0	0
$871$	−0.766389	−0.0259681
$872$	−1.97547	−0.0668978
$873$	0	0
$874$	30.2305	1.02256
$875$	−32.5937	−1.10187
$876$	0	0
$877$	11.1391	0.376141	0.188071	−	0.982156i	$-0.439777\pi$
$877$	11.1391	0.376141	0.188071	+	0.982156i	$0.439777\pi$
$878$	12.8481	0.433602
$879$	0	0
$880$	1.16009	0.0391066
$881$	−21.1439	−0.712357	−0.356178	−	0.934418i	$-0.615920\pi$
$881$	−21.1439	−0.712357	−0.356178	+	0.934418i	$0.615920\pi$
$882$	0	0
$883$	−43.9557	−1.47923	−0.739613	−	0.673032i	$-0.764991\pi$
$883$	−43.9557	−1.47923	−0.739613	+	0.673032i	$0.764991\pi$
$884$	2.90627	0.0977486
$885$	0	0
$886$	−23.3886	−0.785755
$887$	8.16982	0.274316	0.137158	−	0.990549i	$-0.456203\pi$
$887$	8.16982	0.274316	0.137158	+	0.990549i	$0.456203\pi$
$888$	0	0
$889$	−18.7361	−0.628390
$890$	9.16551	0.307229
$891$	0	0
$892$	6.85502	0.229523
$893$	−5.69987	−0.190739
$894$	0	0
$895$	−25.1444	−0.840485
$896$	3.24650	0.108458
$897$	0	0
$898$	−31.9616	−1.06657
$899$	11.5632	0.385655
$900$	0	0
$901$	−25.4102	−0.846536
$902$	−7.01274	−0.233499
$903$	0	0
$904$	3.08404	0.102574
$905$	16.5434	0.549921
$906$	0	0
$907$	−44.1259	−1.46518	−0.732589	−	0.680671i	$-0.761688\pi$
$907$	−44.1259	−1.46518	−0.732589	+	0.680671i	$0.761688\pi$
$908$	2.26886	0.0752947
$909$	0	0
$910$	−5.39408	−0.178812
$911$	29.8636	0.989424	0.494712	−	0.869057i	$-0.335274\pi$
$911$	29.8636	0.989424	0.494712	+	0.869057i	$0.335274\pi$
$912$	0	0
$913$	13.0992	0.433519
$914$	7.37475	0.243935
$915$	0	0
$916$	−18.8342	−0.622299
$917$	3.11276	0.102792
$918$	0	0
$919$	37.2007	1.22714	0.613570	−	0.789641i	$-0.289733\pi$
$919$	37.2007	1.22714	0.613570	+	0.789641i	$0.289733\pi$
$920$	−6.15278	−0.202851
$921$	0	0
$922$	38.7302	1.27551
$923$	−5.20974	−0.171481
$924$	0	0
$925$	7.81611	0.256992
$926$	6.29532	0.206877
$927$	0	0
$928$	−5.78161	−0.189791
$929$	−20.9254	−0.686540	−0.343270	−	0.939237i	$-0.611535\pi$
$929$	−20.9254	−0.686540	−0.343270	+	0.939237i	$0.611535\pi$
$930$	0	0
$931$	20.1763	0.661251
$932$	10.2721	0.336475
$933$	0	0
$934$	−12.4649	−0.407863
$935$	2.35406	0.0769861
$936$	0	0
$937$	54.2932	1.77368	0.886841	−	0.462075i	$-0.152895\pi$
$937$	54.2932	1.77368	0.886841	+	0.462075i	$0.152895\pi$
$938$	−1.73722	−0.0567222
$939$	0	0
$940$	1.16009	0.0378379
$941$	−32.2508	−1.05135	−0.525674	−	0.850686i	$-0.676187\pi$
$941$	−32.2508	−1.05135	−0.525674	+	0.850686i	$0.676187\pi$
$942$	0	0
$943$	37.1935	1.21119
$944$	4.38775	0.142809
$945$	0	0
$946$	4.41107	0.143416
$947$	9.64960	0.313570	0.156785	−	0.987633i	$-0.449887\pi$
$947$	9.64960	0.313570	0.156785	+	0.987633i	$0.449887\pi$
$948$	0	0
$949$	1.20963	0.0392664
$950$	20.8284	0.675763
$951$	0	0
$952$	6.58783	0.213513
$953$	38.6335	1.25146	0.625731	−	0.780039i	$-0.284801\pi$
$953$	38.6335	1.25146	0.625731	+	0.780039i	$0.284801\pi$
$954$	0	0
$955$	23.5330	0.761510
$956$	16.6126	0.537288
$957$	0	0
$958$	27.3181	0.882608
$959$	55.0967	1.77917
$960$	0	0
$961$	−27.0000	−0.870968
$962$	3.06344	0.0987693
$963$	0	0
$964$	5.04107	0.162362
$965$	9.63177	0.310058
$966$	0	0
$967$	−48.0568	−1.54540	−0.772702	−	0.634769i	$-0.781095\pi$
$967$	−48.0568	−1.54540	−0.772702	+	0.634769i	$0.781095\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	14.1879	0.455547
$971$	−2.93372	−0.0941475	−0.0470737	−	0.998891i	$-0.514990\pi$
$971$	−2.93372	−0.0941475	−0.0470737	+	0.998891i	$0.514990\pi$
$972$	0	0
$973$	45.7709	1.46735
$974$	−16.2800	−0.521646
$975$	0	0
$976$	1.73703	0.0556009
$977$	−17.4540	−0.558403	−0.279202	−	0.960233i	$-0.590070\pi$
$977$	−17.4540	−0.558403	−0.279202	+	0.960233i	$0.590070\pi$
$978$	0	0
$979$	−7.90070	−0.252507
$980$	−4.10646	−0.131176
$981$	0	0
$982$	−0.581943	−0.0185705
$983$	−9.20575	−0.293618	−0.146809	−	0.989165i	$-0.546900\pi$
$983$	−9.20575	−0.293618	−0.146809	+	0.989165i	$0.546900\pi$
$984$	0	0
$985$	2.34746	0.0747963
$986$	−11.7321	−0.373626
$987$	0	0
$988$	8.16347	0.259715
$989$	−23.3951	−0.743919
$990$	0	0
$991$	11.0727	0.351736	0.175868	−	0.984414i	$-0.443727\pi$
$991$	11.0727	0.351736	0.175868	+	0.984414i	$0.443727\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	−11.8092	−0.374566
$995$	−26.7594	−0.848330
$996$	0	0
$997$	−28.7812	−0.911509	−0.455754	−	0.890106i	$-0.650630\pi$
$997$	−28.7812	−0.911509	−0.455754	+	0.890106i	$0.650630\pi$
$998$	29.5612	0.935744
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9306.2.a.be.1.3		5
3.2	odd	2		3102.2.a.v.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3102.2.a.v.1.3	✓	5	3.2	odd	2
9306.2.a.be.1.3		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9306 = 2 \cdot 3^{2} \cdot 11 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9306.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.16009 q^{5} -3.24650 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.16009 q^{5} -3.24650 q^{7} -1.00000 q^{8} +1.16009 q^{10} -1.00000 q^{11} +1.43222 q^{13} +3.24650 q^{14} +1.00000 q^{16} +2.02921 q^{17} +5.69987 q^{19} -1.16009 q^{20} +1.00000 q^{22} -5.30371 q^{23} -3.65419 q^{25} -1.43222 q^{26} -3.24650 q^{28} +5.78161 q^{29} +2.00000 q^{31} -1.00000 q^{32} -2.02921 q^{34} +3.76623 q^{35} -2.13894 q^{37} -5.69987 q^{38} +1.16009 q^{40} -7.01274 q^{41} +4.41107 q^{43} -1.00000 q^{44} +5.30371 q^{46} -1.00000 q^{47} +3.53978 q^{49} +3.65419 q^{50} +1.43222 q^{52} -12.5222 q^{53} +1.16009 q^{55} +3.24650 q^{56} -5.78161 q^{58} +4.38775 q^{59} +1.73703 q^{61} -2.00000 q^{62} +1.00000 q^{64} -1.66150 q^{65} -0.535105 q^{67} +2.02921 q^{68} -3.76623 q^{70} -3.63753 q^{71} +0.844586 q^{73} +2.13894 q^{74} +5.69987 q^{76} +3.24650 q^{77} +9.26655 q^{79} -1.16009 q^{80} +7.01274 q^{82} -13.0992 q^{83} -2.35406 q^{85} -4.41107 q^{86} +1.00000 q^{88} +7.90070 q^{89} -4.64971 q^{91} -5.30371 q^{92} +1.00000 q^{94} -6.61236 q^{95} +12.2300 q^{97} -3.53978 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8	\( 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8} + 2 q^{10} - 5 q^{11} + 6 q^{13} - 6 q^{14} + 5 q^{16} - 6 q^{17} + 12 q^{19} - 2 q^{20} + 5 q^{22} - 2 q^{23} - q^{25} - 6 q^{26} + 6 q^{28} + 4 q^{29} + 10 q^{31} - 5 q^{32} + 6 q^{34} + 8 q^{35} - 12 q^{38} + 2 q^{40} - 2 q^{41} + 14 q^{43} - 5 q^{44} + 2 q^{46} - 5 q^{47} + 5 q^{49} + q^{50} + 6 q^{52} - 2 q^{53} + 2 q^{55} - 6 q^{56} - 4 q^{58} - 10 q^{59} + 14 q^{61} - 10 q^{62} + 5 q^{64} - 6 q^{68} - 8 q^{70} - 12 q^{71} - 2 q^{73} + 12 q^{76} - 6 q^{77} - 2 q^{80} + 2 q^{82} - 14 q^{83} + 22 q^{85} - 14 q^{86} + 5 q^{88} + 12 q^{91} - 2 q^{92} + 5 q^{94} - 4 q^{95} + 22 q^{97} - 5 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8 + 2 * q^10 - 5 * q^11 + 6 * q^13 - 6 * q^14 + 5 * q^16 - 6 * q^17 + 12 * q^19 - 2 * q^20 + 5 * q^22 - 2 * q^23 - q^25 - 6 * q^26 + 6 * q^28 + 4 * q^29 + 10 * q^31 - 5 * q^32 + 6 * q^34 + 8 * q^35 - 12 * q^38 + 2 * q^40 - 2 * q^41 + 14 * q^43 - 5 * q^44 + 2 * q^46 - 5 * q^47 + 5 * q^49 + q^50 + 6 * q^52 - 2 * q^53 + 2 * q^55 - 6 * q^56 - 4 * q^58 - 10 * q^59 + 14 * q^61 - 10 * q^62 + 5 * q^64 - 6 * q^68 - 8 * q^70 - 12 * q^71 - 2 * q^73 + 12 * q^76 - 6 * q^77 - 2 * q^80 + 2 * q^82 - 14 * q^83 + 22 * q^85 - 14 * q^86 + 5 * q^88 + 12 * q^91 - 2 * q^92 + 5 * q^94 - 4 * q^95 + 22 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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