LMFDB - Embedded newform 416.2.ba.a.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(17,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("416.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.ba (of order $6$, degree $2$, not minimal)

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:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	416.17
Dual form			416.2.ba.a.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.633975 - 0.366025i) q^{3} +3.73205 q^{5} +(3.00000 - 1.73205i) q^{7} +(-1.23205 - 2.13397i) q^{9} +O(q^{10})$	\(q+(-0.633975 - 0.366025i) q^{3} +3.73205 q^{5} +(3.00000 - 1.73205i) q^{7} +(-1.23205 - 2.13397i) q^{9} +(-1.00000 + 1.73205i) q^{11} +(-2.59808 + 2.50000i) q^{13} +(-2.36603 - 1.36603i) q^{15} +(0.232051 + 0.401924i) q^{17} +(-0.633975 - 1.09808i) q^{19} -2.53590 q^{21} +(4.09808 - 7.09808i) q^{23} +8.92820 q^{25} +4.00000i q^{27} +(2.59808 + 1.50000i) q^{29} +4.73205i q^{31} +(1.26795 - 0.732051i) q^{33} +(11.1962 - 6.46410i) q^{35} +(2.13397 - 3.69615i) q^{37} +(2.56218 - 0.633975i) q^{39} +(-7.96410 - 4.59808i) q^{41} +(-2.19615 + 1.26795i) q^{43} +(-4.59808 - 7.96410i) q^{45} +6.73205i q^{47} +(2.50000 - 4.33013i) q^{49} -0.339746i q^{51} +3.92820i q^{53} +(-3.73205 + 6.46410i) q^{55} +0.928203i q^{57} +(-0.267949 - 0.464102i) q^{59} +(0.866025 - 0.500000i) q^{61} +(-7.39230 - 4.26795i) q^{63} +(-9.69615 + 9.33013i) q^{65} +(-3.63397 + 6.29423i) q^{67} +(-5.19615 + 3.00000i) q^{69} +(-8.02628 + 4.63397i) q^{71} +1.73205i q^{73} +(-5.66025 - 3.26795i) q^{75} +6.92820i q^{77} +10.3923 q^{79} +(-2.23205 + 3.86603i) q^{81} -1.46410 q^{83} +(0.866025 + 1.50000i) q^{85} +(-1.09808 - 1.90192i) q^{87} +(6.46410 + 3.73205i) q^{89} +(-3.46410 + 12.0000i) q^{91} +(1.73205 - 3.00000i) q^{93} +(-2.36603 - 4.09808i) q^{95} +(-5.19615 + 3.00000i) q^{97} +4.92820 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{3} + 8 q^{5} + 12 q^{7} + 2 q^{9}+O(q^{10}) $ 4 * q - 6 * q^3 + 8 * q^5 + 12 * q^7 + 2 * q^9	$ 4 q - 6 q^{3} + 8 q^{5} + 12 q^{7} + 2 q^{9} - 4 q^{11} - 6 q^{15} - 6 q^{17} - 6 q^{19} - 24 q^{21} + 6 q^{23} + 8 q^{25} + 12 q^{33} + 24 q^{35} + 12 q^{37} - 14 q^{39} - 18 q^{41} + 12 q^{43} - 8 q^{45} + 10 q^{49} - 8 q^{55} - 8 q^{59} + 12 q^{63} - 18 q^{65} - 18 q^{67} + 6 q^{71} + 12 q^{75} - 2 q^{81} + 8 q^{83} + 6 q^{87} + 12 q^{89} - 6 q^{95} - 8 q^{99}+O(q^{100}) $ 4 * q - 6 * q^3 + 8 * q^5 + 12 * q^7 + 2 * q^9 - 4 * q^11 - 6 * q^15 - 6 * q^17 - 6 * q^19 - 24 * q^21 + 6 * q^23 + 8 * q^25 + 12 * q^33 + 24 * q^35 + 12 * q^37 - 14 * q^39 - 18 * q^41 + 12 * q^43 - 8 * q^45 + 10 * q^49 - 8 * q^55 - 8 * q^59 + 12 * q^63 - 18 * q^65 - 18 * q^67 + 6 * q^71 + 12 * q^75 - 2 * q^81 + 8 * q^83 + 6 * q^87 + 12 * q^89 - 6 * q^95 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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
$3$	−0.633975	−	0.366025i	−0.366025	−	0.211325i	0.305695	−	0.952129i	$-0.401111\pi$
$3$	−0.633975	−	0.366025i	−0.366025	−	0.211325i	−0.671721	+	0.740805i	$0.734444\pi$
$4$		0			0
$5$	3.73205			1.66902			0.834512	−	0.550990i	$-0.185750\pi$
$5$	3.73205			1.66902			0.834512	+	0.550990i	$0.185750\pi$
$6$		0			0
$7$	3.00000	−	1.73205i	1.13389	−	0.654654i	0.188982	−	0.981981i	$-0.439481\pi$
$7$	3.00000	−	1.73205i	1.13389	−	0.654654i	0.944911	+	0.327327i	$0.106148\pi$
$8$		0			0
$9$	−1.23205	−	2.13397i	−0.410684	−	0.711325i
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	−2.59808	+	2.50000i	−0.720577	+	0.693375i
$13$	−2.59808	+	2.50000i	−0.720577	+	0.693375i
$14$		0			0
$15$	−2.36603	−	1.36603i	−0.610905	−	0.352706i
$16$		0			0
$17$	0.232051	+	0.401924i	0.0562806	+	0.0974808i	0.892793	−	0.450467i	$-0.148743\pi$
$17$	0.232051	+	0.401924i	0.0562806	+	0.0974808i	−0.836512	+	0.547948i	$0.815409\pi$
$18$		0			0
$19$	−0.633975	−	1.09808i	−0.145444	−	0.251916i	0.784095	−	0.620641i	$-0.213128\pi$
$19$	−0.633975	−	1.09808i	−0.145444	−	0.251916i	−0.929538	+	0.368725i	$0.879794\pi$
$20$		0			0
$21$	−2.53590			−0.553378
$22$		0			0
$23$	4.09808	−	7.09808i	0.854508	−	1.48005i	−0.0225928	−	0.999745i	$-0.507192\pi$
$23$	4.09808	−	7.09808i	0.854508	−	1.48005i	0.877101	−	0.480306i	$-0.159475\pi$
$24$		0			0
$25$	8.92820			1.78564
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	2.59808	+	1.50000i	0.482451	+	0.278543i	0.721437	−	0.692480i	$-0.243482\pi$
$29$	2.59808	+	1.50000i	0.482451	+	0.278543i	−0.238987	+	0.971023i	$0.576815\pi$
$30$		0			0
$31$			4.73205i			0.849901i	0.905216	+	0.424951i	$0.139709\pi$
$31$			4.73205i			0.849901i	−0.905216	+	0.424951i	$0.860291\pi$
$32$		0			0
$33$	1.26795	−	0.732051i	0.220722	−	0.127434i
$34$		0			0
$35$	11.1962	−	6.46410i	1.89250	−	1.09263i
$36$		0			0
$37$	2.13397	−	3.69615i	0.350823	−	0.607644i	−0.635571	−	0.772043i	$-0.719235\pi$
$37$	2.13397	−	3.69615i	0.350823	−	0.607644i	0.986394	+	0.164399i	$0.0525685\pi$
$38$		0			0
$39$	2.56218	−	0.633975i	0.410277	−	0.101517i
$40$		0			0
$41$	−7.96410	−	4.59808i	−1.24378	−	0.718099i	−0.273921	−	0.961752i	$-0.588321\pi$
$41$	−7.96410	−	4.59808i	−1.24378	−	0.718099i	−0.969862	+	0.243653i	$0.921654\pi$
$42$		0			0
$43$	−2.19615	+	1.26795i	−0.334910	+	0.193360i	−0.658019	−	0.753001i	$-0.728605\pi$
$43$	−2.19615	+	1.26795i	−0.334910	+	0.193360i	0.323109	+	0.946362i	$0.395272\pi$
$44$		0			0
$45$	−4.59808	−	7.96410i	−0.685441	−	1.18722i
$46$		0			0
$47$			6.73205i			0.981971i	0.871168	+	0.490985i	$0.163363\pi$
$47$			6.73205i			0.981971i	−0.871168	+	0.490985i	$0.836637\pi$
$48$		0			0
$49$	2.50000	−	4.33013i	0.357143	−	0.618590i
$50$		0			0
$51$		−	0.339746i		−	0.0475740i
$52$		0			0
$53$			3.92820i			0.539580i	0.962919	+	0.269790i	$0.0869543\pi$
$53$			3.92820i			0.539580i	−0.962919	+	0.269790i	$0.913046\pi$
$54$		0			0
$55$	−3.73205	+	6.46410i	−0.503230	+	0.871619i
$56$		0			0
$57$			0.928203i			0.122944i
$58$		0			0
$59$	−0.267949	−	0.464102i	−0.0348840	−	0.0604209i	0.848056	−	0.529906i	$-0.177773\pi$
$59$	−0.267949	−	0.464102i	−0.0348840	−	0.0604209i	−0.882940	+	0.469485i	$0.844440\pi$
$60$		0			0
$61$	0.866025	−	0.500000i	0.110883	−	0.0640184i	−0.443533	−	0.896258i	$-0.646275\pi$
$61$	0.866025	−	0.500000i	0.110883	−	0.0640184i	0.554416	+	0.832240i	$0.312942\pi$
$62$		0			0
$63$	−7.39230	−	4.26795i	−0.931343	−	0.537711i
$64$		0			0
$65$	−9.69615	+	9.33013i	−1.20266	+	1.15726i
$66$		0			0
$67$	−3.63397	+	6.29423i	−0.443961	+	0.768962i	−0.997979	−	0.0635419i	$-0.979760\pi$
$67$	−3.63397	+	6.29423i	−0.443961	+	0.768962i	0.554019	+	0.832504i	$0.313094\pi$
$68$		0			0
$69$	−5.19615	+	3.00000i	−0.625543	+	0.361158i
$70$		0			0
$71$	−8.02628	+	4.63397i	−0.952544	+	0.549952i	−0.893870	−	0.448326i	$-0.852021\pi$
$71$	−8.02628	+	4.63397i	−0.952544	+	0.549952i	−0.0586738	+	0.998277i	$0.518687\pi$
$72$		0			0
$73$			1.73205i			0.202721i	0.994850	+	0.101361i	$0.0323196\pi$
$73$			1.73205i			0.202721i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	−5.66025	−	3.26795i	−0.653590	−	0.377350i
$76$		0			0
$77$			6.92820i			0.789542i
$78$		0			0
$79$	10.3923			1.16923			0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923			1.16923			0.584613	+	0.811312i	$0.301246\pi$
$80$		0			0
$81$	−2.23205	+	3.86603i	−0.248006	+	0.429558i
$82$		0			0
$83$	−1.46410			−0.160706			−0.0803530	−	0.996766i	$-0.525605\pi$
$83$	−1.46410			−0.160706			−0.0803530	+	0.996766i	$0.525605\pi$
$84$		0			0
$85$	0.866025	+	1.50000i	0.0939336	+	0.162698i
$86$		0			0
$87$	−1.09808	−	1.90192i	−0.117726	−	0.203908i
$88$		0			0
$89$	6.46410	+	3.73205i	0.685193	+	0.395597i	0.801809	−	0.597581i	$-0.203871\pi$
$89$	6.46410	+	3.73205i	0.685193	+	0.395597i	−0.116615	+	0.993177i	$0.537205\pi$
$90$		0			0
$91$	−3.46410	+	12.0000i	−0.363137	+	1.25794i
$92$		0			0
$93$	1.73205	−	3.00000i	0.179605	−	0.311086i
$94$		0			0
$95$	−2.36603	−	4.09808i	−0.242749	−	0.420454i
$96$		0			0
$97$	−5.19615	+	3.00000i	−0.527589	+	0.304604i	−0.740034	−	0.672569i	$-0.765191\pi$
$97$	−5.19615	+	3.00000i	−0.527589	+	0.304604i	0.212445	+	0.977173i	$0.431857\pi$
$98$		0			0
$99$	4.92820			0.495303
$100$		0			0
$101$	−9.99038	−	5.76795i	−0.994080	−	0.573932i	−0.0875887	−	0.996157i	$-0.527916\pi$
$101$	−9.99038	−	5.76795i	−0.994080	−	0.573932i	−0.906491	+	0.422224i	$0.861249\pi$
$102$		0			0
$103$	−6.19615			−0.610525			−0.305263	−	0.952268i	$-0.598744\pi$
$103$	−6.19615			−0.610525			−0.305263	+	0.952268i	$0.598744\pi$
$104$		0			0
$105$	−9.46410			−0.923602
$106$		0			0
$107$	−15.2942	−	8.83013i	−1.47855	−	0.853641i	−0.478843	−	0.877900i	$-0.658944\pi$
$107$	−15.2942	−	8.83013i	−1.47855	−	0.853641i	−0.999706	+	0.0242598i	$0.992277\pi$
$108$		0			0
$109$	6.00000			0.574696			0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000			0.574696			0.287348	+	0.957826i	$0.407226\pi$
$110$		0			0
$111$	−2.70577	+	1.56218i	−0.256820	+	0.148275i
$112$		0			0
$113$	1.50000	+	2.59808i	0.141108	+	0.244406i	0.927914	−	0.372794i	$-0.121600\pi$
$113$	1.50000	+	2.59808i	0.141108	+	0.244406i	−0.786806	+	0.617200i	$0.788267\pi$
$114$		0			0
$115$	15.2942	−	26.4904i	1.42619	−	2.47024i
$116$		0			0
$117$	8.53590	+	2.46410i	0.789144	+	0.227806i
$118$		0			0
$119$	1.39230	+	0.803848i	0.127632	+	0.0736886i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$	3.36603	+	5.83013i	0.303504	+	0.525685i
$124$		0			0
$125$	14.6603			1.31125
$126$		0			0
$127$	0.169873	−	0.294229i	0.0150738	−	0.0261086i	−0.858390	−	0.512997i	$-0.828535\pi$
$127$	0.169873	−	0.294229i	0.0150738	−	0.0261086i	0.873464	+	0.486889i	$0.161868\pi$
$128$		0			0
$129$	1.85641			0.163447
$130$		0			0
$131$		−	10.7321i		−	0.937664i	−0.883287	−	0.468832i	$-0.844675\pi$
$131$		−	10.7321i		−	0.937664i	0.883287	−	0.468832i	$-0.155325\pi$
$132$		0			0
$133$	−3.80385	−	2.19615i	−0.329835	−	0.190431i
$134$		0			0
$135$			14.9282i			1.28482i
$136$		0			0
$137$	−11.7679	+	6.79423i	−1.00540	+	0.580470i	−0.909843	−	0.414953i	$-0.863798\pi$
$137$	−11.7679	+	6.79423i	−1.00540	+	0.580470i	−0.0955611	+	0.995424i	$0.530465\pi$
$138$		0			0
$139$	−8.19615	+	4.73205i	−0.695189	+	0.401367i	−0.805553	−	0.592524i	$-0.798132\pi$
$139$	−8.19615	+	4.73205i	−0.695189	+	0.401367i	0.110364	+	0.993891i	$0.464798\pi$
$140$		0			0
$141$	2.46410	−	4.26795i	0.207515	−	0.359426i
$142$		0			0
$143$	−1.73205	−	7.00000i	−0.144841	−	0.585369i
$144$		0			0
$145$	9.69615	+	5.59808i	0.805222	+	0.464895i
$146$		0			0
$147$	−3.16987	+	1.83013i	−0.261447	+	0.150946i
$148$		0			0
$149$	−5.86603	−	10.1603i	−0.480564	−	0.832360i	0.519188	−	0.854660i	$-0.326235\pi$
$149$	−5.86603	−	10.1603i	−0.480564	−	0.832360i	−0.999751	+	0.0222997i	$0.992901\pi$
$150$		0			0
$151$			2.19615i			0.178720i	0.995999	+	0.0893602i	$0.0284822\pi$
$151$			2.19615i			0.178720i	−0.995999	+	0.0893602i	$0.971518\pi$
$152$		0			0
$153$	0.571797	−	0.990381i	0.0462270	−	0.0800676i
$154$		0			0
$155$			17.6603i			1.41851i
$156$		0			0
$157$		−	3.92820i		−	0.313505i	−0.987638	−	0.156752i	$-0.949898\pi$
$157$		−	3.92820i		−	0.313505i	0.987638	−	0.156752i	$-0.0501025\pi$
$158$		0			0
$159$	1.43782	−	2.49038i	0.114027	−	0.197500i
$160$		0			0
$161$		−	28.3923i		−	2.23763i
$162$		0			0
$163$	8.19615	+	14.1962i	0.641972	+	1.11193i	0.984992	+	0.172600i	$0.0552169\pi$
$163$	8.19615	+	14.1962i	0.641972	+	1.11193i	−0.343020	+	0.939328i	$0.611450\pi$
$164$		0			0
$165$	4.73205	−	2.73205i	0.368390	−	0.212690i
$166$		0			0
$167$	4.26795	+	2.46410i	0.330264	+	0.190678i	0.655958	−	0.754797i	$-0.272265\pi$
$167$	4.26795	+	2.46410i	0.330264	+	0.190678i	−0.325694	+	0.945475i	$0.605598\pi$
$168$		0			0
$169$	0.500000	−	12.9904i	0.0384615	−	0.999260i
$170$		0			0
$171$	−1.56218	+	2.70577i	−0.119463	+	0.206916i
$172$		0			0
$173$	−16.3923	+	9.46410i	−1.24628	+	0.719542i	−0.970366	−	0.241639i	$-0.922315\pi$
$173$	−16.3923	+	9.46410i	−1.24628	+	0.719542i	−0.275918	+	0.961181i	$0.588982\pi$
$174$		0			0
$175$	26.7846	−	15.4641i	2.02473	−	1.16898i
$176$		0			0
$177$			0.392305i			0.0294874i
$178$		0			0
$179$	−12.0000	−	6.92820i	−0.896922	−	0.517838i	−0.0207218	−	0.999785i	$-0.506596\pi$
$179$	−12.0000	−	6.92820i	−0.896922	−	0.517838i	−0.876200	+	0.481947i	$0.839930\pi$
$180$		0			0
$181$			18.4641i			1.37243i	0.727401	+	0.686213i	$0.240728\pi$
$181$			18.4641i			1.37243i	−0.727401	+	0.686213i	$0.759272\pi$
$182$		0			0
$183$	−0.732051			−0.0541148
$184$		0			0
$185$	7.96410	−	13.7942i	0.585532	−	1.01417i
$186$		0			0
$187$	−0.928203			−0.0678769
$188$		0			0
$189$	6.92820	+	12.0000i	0.503953	+	0.872872i
$190$		0			0
$191$	1.90192	+	3.29423i	0.137618	+	0.238362i	0.926595	−	0.376062i	$-0.122722\pi$
$191$	1.90192	+	3.29423i	0.137618	+	0.238362i	−0.788976	+	0.614424i	$0.789389\pi$
$192$		0			0
$193$	11.3038	+	6.52628i	0.813669	+	0.469772i	0.848228	−	0.529631i	$-0.177670\pi$
$193$	11.3038	+	6.52628i	0.813669	+	0.469772i	−0.0345595	+	0.999403i	$0.511003\pi$
$194$		0			0
$195$	9.56218	−	2.36603i	0.684762	−	0.169435i
$196$		0			0
$197$	−6.26795	+	10.8564i	−0.446573	+	0.773487i	−0.998160	−	0.0606302i	$-0.980689\pi$
$197$	−6.26795	+	10.8564i	−0.446573	+	0.773487i	0.551587	+	0.834117i	$0.314022\pi$
$198$		0			0
$199$	−7.56218	−	13.0981i	−0.536069	−	0.928498i	−0.999111	−	0.0421618i	$-0.986576\pi$
$199$	−7.56218	−	13.0981i	−0.536069	−	0.928498i	0.463042	−	0.886336i	$-0.346758\pi$
$200$		0			0
$201$	4.60770	−	2.66025i	0.325002	−	0.187640i
$202$		0			0
$203$	10.3923			0.729397
$204$		0			0
$205$	−29.7224	−	17.1603i	−2.07590	−	1.19852i
$206$		0			0
$207$	−20.1962			−1.40373
$208$		0			0
$209$	2.53590			0.175412
$210$		0			0
$211$	24.7583	+	14.2942i	1.70443	+	0.984055i	0.941148	+	0.337994i	$0.109748\pi$
$211$	24.7583	+	14.2942i	1.70443	+	0.984055i	0.763285	+	0.646061i	$0.223585\pi$
$212$		0			0
$213$	6.78461			0.464874
$214$		0			0
$215$	−8.19615	+	4.73205i	−0.558973	+	0.322723i
$216$		0			0
$217$	8.19615	+	14.1962i	0.556391	+	0.963698i
$218$		0			0
$219$	0.633975	−	1.09808i	0.0428400	−	0.0742011i
$220$		0			0
$221$	−1.60770	−	0.464102i	−0.108145	−	0.0312189i
$222$		0			0
$223$	22.0981	+	12.7583i	1.47980	+	0.854361i	0.999738	−	0.0228756i	$-0.00728216\pi$
$223$	22.0981	+	12.7583i	1.47980	+	0.854361i	0.480058	+	0.877237i	$0.340615\pi$
$224$		0			0
$225$	−11.0000	−	19.0526i	−0.733333	−	1.27017i
$226$		0			0
$227$	−13.0263	−	22.5622i	−0.864585	−	1.49750i	−0.867459	−	0.497508i	$-0.834248\pi$
$227$	−13.0263	−	22.5622i	−0.864585	−	1.49750i	0.00287459	−	0.999996i	$-0.499085\pi$
$228$		0			0
$229$	12.9282			0.854320			0.427160	−	0.904176i	$-0.359514\pi$
$229$	12.9282			0.854320			0.427160	+	0.904176i	$0.359514\pi$
$230$		0			0
$231$	2.53590	−	4.39230i	0.166850	−	0.288992i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$			25.1244i			1.63893i
$236$		0			0
$237$	−6.58846	−	3.80385i	−0.427966	−	0.247086i
$238$		0			0
$239$		−	18.3923i		−	1.18970i	−0.803837	−	0.594850i	$-0.797212\pi$
$239$		−	18.3923i		−	1.18970i	0.803837	−	0.594850i	$-0.202788\pi$
$240$		0			0
$241$	9.69615	−	5.59808i	0.624584	−	0.360604i	−0.154068	−	0.988060i	$-0.549237\pi$
$241$	9.69615	−	5.59808i	0.624584	−	0.360604i	0.778652	+	0.627457i	$0.215904\pi$
$242$		0			0
$243$	13.2224	−	7.63397i	0.848219	−	0.489720i
$244$		0			0
$245$	9.33013	−	16.1603i	0.596080	−	1.03244i
$246$		0			0
$247$	4.39230	+	1.26795i	0.279476	+	0.0806777i
$248$		0			0
$249$	0.928203	+	0.535898i	0.0588225	+	0.0339612i
$250$		0			0
$251$	7.09808	−	4.09808i	0.448027	−	0.258668i	−0.258970	−	0.965885i	$-0.583383\pi$
$251$	7.09808	−	4.09808i	0.448027	−	0.258668i	0.706996	+	0.707217i	$0.250050\pi$
$252$		0			0
$253$	8.19615	+	14.1962i	0.515288	+	0.892504i
$254$		0			0
$255$		−	1.26795i		−	0.0794021i
$256$		0			0
$257$	−11.4282	+	19.7942i	−0.712872	+	1.23473i	0.250903	+	0.968012i	$0.419273\pi$
$257$	−11.4282	+	19.7942i	−0.712872	+	1.23473i	−0.963775	+	0.266718i	$0.914061\pi$
$258$		0			0
$259$		−	14.7846i		−	0.918671i
$260$		0			0
$261$		−	7.39230i		−	0.457572i
$262$		0			0
$263$	8.19615	−	14.1962i	0.505396	−	0.875372i	−0.494584	−	0.869130i	$-0.664680\pi$
$263$	8.19615	−	14.1962i	0.505396	−	0.875372i	0.999981	−	0.00624249i	$-0.00198706\pi$
$264$		0			0
$265$			14.6603i			0.900572i
$266$		0			0
$267$	−2.73205	−	4.73205i	−0.167199	−	0.289597i
$268$		0			0
$269$	8.19615	−	4.73205i	0.499728	−	0.288518i	−0.228873	−	0.973456i	$-0.573504\pi$
$269$	8.19615	−	4.73205i	0.499728	−	0.288518i	0.728601	+	0.684938i	$0.240171\pi$
$270$		0			0
$271$	15.0000	+	8.66025i	0.911185	+	0.526073i	0.880812	−	0.473466i	$-0.156997\pi$
$271$	15.0000	+	8.66025i	0.911185	+	0.526073i	0.0303728	+	0.999539i	$0.490331\pi$
$272$		0			0
$273$	6.58846	−	6.33975i	0.398752	−	0.383699i
$274$		0			0
$275$	−8.92820	+	15.4641i	−0.538391	+	0.932520i
$276$		0			0
$277$	12.4019	−	7.16025i	0.745159	−	0.430218i	−0.0787828	−	0.996892i	$-0.525103\pi$
$277$	12.4019	−	7.16025i	0.745159	−	0.430218i	0.823942	+	0.566674i	$0.191770\pi$
$278$		0			0
$279$	10.0981	−	5.83013i	0.604556	−	0.349041i
$280$		0			0
$281$		−	10.6603i		−	0.635937i	−0.948101	−	0.317969i	$-0.896999\pi$
$281$		−	10.6603i		−	0.635937i	0.948101	−	0.317969i	$-0.103001\pi$
$282$		0			0
$283$	−26.6603	−	15.3923i	−1.58479	−	0.914978i	−0.994146	−	0.108043i	$-0.965541\pi$
$283$	−26.6603	−	15.3923i	−1.58479	−	0.914978i	−0.590641	−	0.806934i	$-0.701125\pi$
$284$		0			0
$285$			3.46410i			0.205196i
$286$		0			0
$287$	−31.8564			−1.88042
$288$		0			0
$289$	8.39230	−	14.5359i	0.493665	−	0.855053i
$290$		0			0
$291$	4.39230			0.257481
$292$		0			0
$293$	1.40192	+	2.42820i	0.0819013	+	0.141857i	0.904067	−	0.427392i	$-0.140567\pi$
$293$	1.40192	+	2.42820i	0.0819013	+	0.141857i	−0.822165	+	0.569249i	$0.807234\pi$
$294$		0			0
$295$	−1.00000	−	1.73205i	−0.0582223	−	0.100844i
$296$		0			0
$297$	−6.92820	−	4.00000i	−0.402015	−	0.232104i
$298$		0			0
$299$	7.09808	+	28.6865i	0.410492	+	1.65899i
$300$		0			0
$301$	−4.39230	+	7.60770i	−0.253168	+	0.438500i
$302$		0			0
$303$	4.22243	+	7.31347i	0.242572	+	0.420148i
$304$		0			0
$305$	3.23205	−	1.86603i	0.185067	−	0.106848i
$306$		0			0
$307$	−2.19615			−0.125341			−0.0626705	−	0.998034i	$-0.519962\pi$
$307$	−2.19615			−0.125341			−0.0626705	+	0.998034i	$0.519962\pi$
$308$		0			0
$309$	3.92820	+	2.26795i	0.223468	+	0.129019i
$310$		0			0
$311$	18.5885			1.05405			0.527027	−	0.849848i	$-0.323307\pi$
$311$	18.5885			1.05405			0.527027	+	0.849848i	$0.323307\pi$
$312$		0			0
$313$	−2.53590			−0.143337			−0.0716687	−	0.997428i	$-0.522832\pi$
$313$	−2.53590			−0.143337			−0.0716687	+	0.997428i	$0.522832\pi$
$314$		0			0
$315$	−27.5885	−	15.9282i	−1.55443	−	0.897453i
$316$		0			0
$317$	25.1962			1.41516			0.707578	−	0.706635i	$-0.249788\pi$
$317$	25.1962			1.41516			0.707578	+	0.706635i	$0.249788\pi$
$318$		0			0
$319$	−5.19615	+	3.00000i	−0.290929	+	0.167968i
$320$		0			0
$321$	6.46410	+	11.1962i	0.360791	+	0.624908i
$322$		0			0
$323$	0.294229	−	0.509619i	0.0163713	−	0.0283560i
$324$		0			0
$325$	−23.1962	+	22.3205i	−1.28669	+	1.23812i
$326$		0			0
$327$	−3.80385	−	2.19615i	−0.210353	−	0.121448i
$328$		0			0
$329$	11.6603	+	20.1962i	0.642851	+	1.11345i
$330$		0			0
$331$	−6.00000	−	10.3923i	−0.329790	−	0.571213i	0.652680	−	0.757634i	$-0.273645\pi$
$331$	−6.00000	−	10.3923i	−0.329790	−	0.571213i	−0.982470	+	0.186421i	$0.940311\pi$
$332$		0			0
$333$	−10.5167			−0.576309
$334$		0			0
$335$	−13.5622	+	23.4904i	−0.740981	+	1.28342i
$336$		0			0
$337$	−33.2487			−1.81117			−0.905586	−	0.424162i	$-0.860569\pi$
$337$	−33.2487			−1.81117			−0.905586	+	0.424162i	$0.860569\pi$
$338$		0			0
$339$		−	2.19615i		−	0.119279i
$340$		0			0
$341$	−8.19615	−	4.73205i	−0.443847	−	0.256255i
$342$		0			0
$343$			6.92820i			0.374088i
$344$		0			0
$345$	−19.3923	+	11.1962i	−1.04405	+	0.602781i
$346$		0			0
$347$	1.09808	−	0.633975i	0.0589478	−	0.0340335i	−0.470236	−	0.882540i	$-0.655831\pi$
$347$	1.09808	−	0.633975i	0.0589478	−	0.0340335i	0.529184	+	0.848507i	$0.322498\pi$
$348$		0			0
$349$	2.66025	−	4.60770i	0.142400	−	0.246644i	−0.786000	−	0.618227i	$-0.787851\pi$
$349$	2.66025	−	4.60770i	0.142400	−	0.246644i	0.928400	+	0.371582i	$0.121185\pi$
$350$		0			0
$351$	−10.0000	−	10.3923i	−0.533761	−	0.554700i
$352$		0			0
$353$	−15.8205	−	9.13397i	−0.842041	−	0.486152i	0.0159167	−	0.999873i	$-0.494933\pi$
$353$	−15.8205	−	9.13397i	−0.842041	−	0.486152i	−0.857957	+	0.513721i	$0.828267\pi$
$354$		0			0
$355$	−29.9545	+	17.2942i	−1.58982	+	0.917882i
$356$		0			0
$357$	−0.588457	−	1.01924i	−0.0311445	−	0.0539438i
$358$		0			0
$359$		−	13.0718i		−	0.689903i	−0.938621	−	0.344952i	$-0.887895\pi$
$359$		−	13.0718i		−	0.689903i	0.938621	−	0.344952i	$-0.112105\pi$
$360$		0			0
$361$	8.69615	−	15.0622i	0.457692	−	0.792746i
$362$		0			0
$363$		−	5.12436i		−	0.268959i
$364$		0			0
$365$			6.46410i			0.338347i
$366$		0			0
$367$	−14.0981	+	24.4186i	−0.735914	+	1.27464i	0.218408	+	0.975858i	$0.429914\pi$
$367$	−14.0981	+	24.4186i	−0.735914	+	1.27464i	−0.954321	+	0.298782i	$0.903420\pi$
$368$		0			0
$369$			22.6603i			1.17965i
$370$		0			0
$371$	6.80385	+	11.7846i	0.353238	+	0.611826i
$372$		0			0
$373$	22.3301	−	12.8923i	1.15621	−	0.667538i	0.205817	−	0.978590i	$-0.434015\pi$
$373$	22.3301	−	12.8923i	1.15621	−	0.667538i	0.950393	+	0.311052i	$0.100681\pi$
$374$		0			0
$375$	−9.29423	−	5.36603i	−0.479952	−	0.277100i
$376$		0			0
$377$	−10.5000	+	2.59808i	−0.540778	+	0.133808i
$378$		0			0
$379$	11.0263	−	19.0981i	0.566382	−	0.981002i	−0.430538	−	0.902573i	$-0.641676\pi$
$379$	11.0263	−	19.0981i	0.566382	−	0.981002i	0.996920	−	0.0784297i	$-0.0249906\pi$
$380$		0			0
$381$	−0.215390	+	0.124356i	−0.0110348	+	0.00637093i
$382$		0			0
$383$	−17.3205	+	10.0000i	−0.885037	+	0.510976i	−0.872316	−	0.488943i	$-0.837383\pi$
$383$	−17.3205	+	10.0000i	−0.885037	+	0.510976i	−0.0127209	+	0.999919i	$0.504049\pi$
$384$		0			0
$385$			25.8564i			1.31776i
$386$		0			0
$387$	5.41154	+	3.12436i	0.275084	+	0.158820i
$388$		0			0
$389$		−	18.7128i		−	0.948777i	−0.880316	−	0.474389i	$-0.842669\pi$
$389$		−	18.7128i		−	0.948777i	0.880316	−	0.474389i	$-0.157331\pi$
$390$		0			0
$391$	3.80385			0.192369
$392$		0			0
$393$	−3.92820	+	6.80385i	−0.198152	+	0.343209i
$394$		0			0
$395$	38.7846			1.95147
$396$		0			0
$397$	−12.1244	−	21.0000i	−0.608504	−	1.05396i	−0.991487	−	0.130204i	$-0.958437\pi$
$397$	−12.1244	−	21.0000i	−0.608504	−	1.05396i	0.382983	−	0.923755i	$-0.374897\pi$
$398$		0			0
$399$	1.60770	+	2.78461i	0.0804854	+	0.139405i
$400$		0			0
$401$	8.42820	+	4.86603i	0.420884	+	0.242998i	0.695456	−	0.718569i	$-0.255203\pi$
$401$	8.42820	+	4.86603i	0.420884	+	0.242998i	−0.274571	+	0.961567i	$0.588536\pi$
$402$		0			0
$403$	−11.8301	−	12.2942i	−0.589301	−	0.612419i
$404$		0			0
$405$	−8.33013	+	14.4282i	−0.413927	+	0.716943i
$406$		0			0
$407$	4.26795	+	7.39230i	0.211554	+	0.366423i
$408$		0			0
$409$	0.696152	−	0.401924i	0.0344225	−	0.0198739i	−0.482690	−	0.875791i	$-0.660340\pi$
$409$	0.696152	−	0.401924i	0.0344225	−	0.0198739i	0.517113	+	0.855917i	$0.327007\pi$
$410$		0			0
$411$	9.94744			0.490671
$412$		0			0
$413$	−1.60770	−	0.928203i	−0.0791095	−	0.0456739i
$414$		0			0
$415$	−5.46410			−0.268222
$416$		0			0
$417$	6.92820			0.339276
$418$		0			0
$419$	16.9019	+	9.75833i	0.825713	+	0.476726i	0.852383	−	0.522919i	$-0.175157\pi$
$419$	16.9019	+	9.75833i	0.825713	+	0.476726i	−0.0266696	+	0.999644i	$0.508490\pi$
$420$		0			0
$421$	−30.1244			−1.46817			−0.734086	−	0.679057i	$-0.762389\pi$
$421$	−30.1244			−1.46817			−0.734086	+	0.679057i	$0.762389\pi$
$422$		0			0
$423$	14.3660	−	8.29423i	0.698500	−	0.403279i
$424$		0			0
$425$	2.07180	+	3.58846i	0.100497	+	0.174066i
$426$		0			0
$427$	1.73205	−	3.00000i	0.0838198	−	0.145180i
$428$		0			0
$429$	−1.46410	+	5.07180i	−0.0706875	+	0.244869i
$430$		0			0
$431$	−15.4641	−	8.92820i	−0.744880	−	0.430056i	0.0789612	−	0.996878i	$-0.474840\pi$
$431$	−15.4641	−	8.92820i	−0.744880	−	0.430056i	−0.823841	+	0.566821i	$0.808173\pi$
$432$		0			0
$433$	−10.5000	−	18.1865i	−0.504598	−	0.873989i	−0.999986	−	0.00531724i	$-0.998307\pi$
$433$	−10.5000	−	18.1865i	−0.504598	−	0.873989i	0.495388	−	0.868672i	$-0.335026\pi$
$434$		0			0
$435$	−4.09808	−	7.09808i	−0.196488	−	0.340327i
$436$		0			0
$437$	−10.3923			−0.497131
$438$		0			0
$439$	7.80385	−	13.5167i	0.372457	−	0.645115i	−0.617486	−	0.786582i	$-0.711849\pi$
$439$	7.80385	−	13.5167i	0.372457	−	0.645115i	0.989943	+	0.141467i	$0.0451820\pi$
$440$		0			0
$441$	−12.3205			−0.586691
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	24.1244	+	13.9282i	1.14360	+	0.660260i
$446$		0			0
$447$			8.58846i			0.406220i
$448$		0			0
$449$	18.4641	−	10.6603i	0.871375	−	0.503088i	0.00356996	−	0.999994i	$-0.498864\pi$
$449$	18.4641	−	10.6603i	0.871375	−	0.503088i	0.867805	+	0.496905i	$0.165530\pi$
$450$		0			0
$451$	15.9282	−	9.19615i	0.750030	−	0.433030i
$452$		0			0
$453$	0.803848	−	1.39230i	0.0377681	−	0.0654162i
$454$		0			0
$455$	−12.9282	+	44.7846i	−0.606084	+	2.09953i
$456$		0			0
$457$	6.69615	+	3.86603i	0.313233	+	0.180845i	0.648372	−	0.761324i	$-0.275450\pi$
$457$	6.69615	+	3.86603i	0.313233	+	0.180845i	−0.335139	+	0.942169i	$0.608783\pi$
$458$		0			0
$459$	−1.60770	+	0.928203i	−0.0750408	+	0.0433248i
$460$		0			0
$461$	−14.3301	−	24.8205i	−0.667421	−	1.15601i	−0.978623	−	0.205663i	$-0.934065\pi$
$461$	−14.3301	−	24.8205i	−0.667421	−	1.15601i	0.311202	−	0.950344i	$-0.399268\pi$
$462$		0			0
$463$		−	16.3923i		−	0.761815i	−0.924613	−	0.380908i	$-0.875612\pi$
$463$		−	16.3923i		−	0.761815i	0.924613	−	0.380908i	$-0.124388\pi$
$464$		0			0
$465$	6.46410	−	11.1962i	0.299766	−	0.519209i
$466$		0			0
$467$			20.5359i			0.950288i	0.879908	+	0.475144i	$0.157604\pi$
$467$			20.5359i			0.950288i	−0.879908	+	0.475144i	$0.842396\pi$
$468$		0			0
$469$			25.1769i			1.16256i
$470$		0			0
$471$	−1.43782	+	2.49038i	−0.0662513	+	0.114751i
$472$		0			0
$473$		−	5.07180i		−	0.233201i
$474$		0			0
$475$	−5.66025	−	9.80385i	−0.259710	−	0.449831i
$476$		0			0
$477$	8.38269	−	4.83975i	0.383817	−	0.221597i
$478$		0			0
$479$	25.3468	+	14.6340i	1.15812	+	0.668643i	0.950854	−	0.309640i	$-0.100209\pi$
$479$	25.3468	+	14.6340i	1.15812	+	0.668643i	0.207271	+	0.978284i	$0.433542\pi$
$480$		0			0
$481$	3.69615	+	14.9378i	0.168530	+	0.681106i
$482$		0			0
$483$	−10.3923	+	18.0000i	−0.472866	+	0.819028i
$484$		0			0
$485$	−19.3923	+	11.1962i	−0.880559	+	0.508391i
$486$		0			0
$487$	−20.1962	+	11.6603i	−0.915175	+	0.528377i	−0.882093	−	0.471076i	$-0.843866\pi$
$487$	−20.1962	+	11.6603i	−0.915175	+	0.528377i	−0.0330824	+	0.999453i	$0.510532\pi$
$488$		0			0
$489$		−	12.0000i		−	0.542659i
$490$		0			0
$491$	6.58846	+	3.80385i	0.297333	+	0.171665i	0.641244	−	0.767337i	$-0.278419\pi$
$491$	6.58846	+	3.80385i	0.297333	+	0.171665i	−0.343911	+	0.939002i	$0.611752\pi$
$492$		0			0
$493$			1.39230i			0.0627063i
$494$		0			0
$495$	18.3923			0.826673
$496$		0			0
$497$	−16.0526	+	27.8038i	−0.720056	+	1.24717i
$498$		0			0
$499$	−8.87564			−0.397328			−0.198664	−	0.980068i	$-0.563660\pi$
$499$	−8.87564			−0.397328			−0.198664	+	0.980068i	$0.563660\pi$
$500$		0			0
$501$	−1.80385	−	3.12436i	−0.0805900	−	0.139586i
$502$		0			0
$503$	4.39230	+	7.60770i	0.195843	+	0.339210i	0.947177	−	0.320712i	$-0.103922\pi$
$503$	4.39230	+	7.60770i	0.195843	+	0.339210i	−0.751333	+	0.659923i	$0.770589\pi$
$504$		0			0
$505$	−37.2846	−	21.5263i	−1.65914	−	0.957907i
$506$		0			0
$507$	−5.07180	+	8.05256i	−0.225246	+	0.357627i
$508$		0			0
$509$	3.93782	−	6.82051i	0.174541	−	0.302314i	−0.765461	−	0.643482i	$-0.777489\pi$
$509$	3.93782	−	6.82051i	0.174541	−	0.302314i	0.940002	+	0.341168i	$0.110823\pi$
$510$		0			0
$511$	3.00000	+	5.19615i	0.132712	+	0.229864i
$512$		0			0
$513$	4.39230	−	2.53590i	0.193925	−	0.111963i
$514$		0			0
$515$	−23.1244			−1.01898
$516$		0			0
$517$	−11.6603	−	6.73205i	−0.512817	−	0.296075i
$518$		0			0
$519$	13.8564			0.608229
$520$		0			0
$521$	9.24871			0.405193			0.202597	−	0.979262i	$-0.435062\pi$
$521$	9.24871			0.405193			0.202597	+	0.979262i	$0.435062\pi$
$522$		0			0
$523$	0.803848	+	0.464102i	0.0351498	+	0.0202937i	0.517472	−	0.855700i	$-0.326873\pi$
$523$	0.803848	+	0.464102i	0.0351498	+	0.0202937i	−0.482322	+	0.875994i	$0.660207\pi$
$524$		0			0
$525$	−22.6410			−0.988135
$526$		0			0
$527$	−1.90192	+	1.09808i	−0.0828491	+	0.0478330i
$528$		0			0
$529$	−22.0885	−	38.2583i	−0.960368	−	1.66341i
$530$		0			0
$531$	−0.660254	+	1.14359i	−0.0286526	+	0.0496277i
$532$		0			0
$533$	32.1865	−	7.96410i	1.39415	−	0.344964i
$534$		0			0
$535$	−57.0788	−	32.9545i	−2.46773	−	1.42475i
$536$		0			0
$537$	5.07180	+	8.78461i	0.218864	+	0.379084i
$538$		0			0
$539$	5.00000	+	8.66025i	0.215365	+	0.373024i
$540$		0			0
$541$	35.4449			1.52389			0.761947	−	0.647640i	$-0.224244\pi$
$541$	35.4449			1.52389			0.761947	+	0.647640i	$0.224244\pi$
$542$		0			0
$543$	6.75833	−	11.7058i	0.290028	−	0.502343i
$544$		0			0
$545$	22.3923			0.959181
$546$		0			0
$547$		−	18.5885i		−	0.794785i	−0.917649	−	0.397393i	$-0.869915\pi$
$547$		−	18.5885i		−	0.794785i	0.917649	−	0.397393i	$-0.130085\pi$
$548$		0			0
$549$	−2.13397	−	1.23205i	−0.0910758	−	0.0525826i
$550$		0			0
$551$		−	3.80385i		−	0.162049i
$552$		0			0
$553$	31.1769	−	18.0000i	1.32578	−	0.765438i
$554$		0			0
$555$	−10.0981	+	5.83013i	−0.428639	+	0.247475i
$556$		0			0
$557$	−2.40192	+	4.16025i	−0.101773	+	0.176276i	−0.912415	−	0.409266i	$-0.865785\pi$
$557$	−2.40192	+	4.16025i	−0.101773	+	0.176276i	0.810642	+	0.585542i	$0.199118\pi$
$558$		0			0
$559$	2.53590	−	8.78461i	0.107257	−	0.371549i
$560$		0			0
$561$	0.588457	+	0.339746i	0.0248447	+	0.0143441i
$562$		0			0
$563$	−13.9808	+	8.07180i	−0.589219	+	0.340186i	−0.764789	−	0.644281i	$-0.777157\pi$
$563$	−13.9808	+	8.07180i	−0.589219	+	0.340186i	0.175570	+	0.984467i	$0.443823\pi$
$564$		0			0
$565$	5.59808	+	9.69615i	0.235513	+	0.407920i
$566$		0			0
$567$			15.4641i			0.649431i
$568$		0			0
$569$	12.0000	−	20.7846i	0.503066	−	0.871336i	−0.496928	−	0.867792i	$-0.665539\pi$
$569$	12.0000	−	20.7846i	0.503066	−	0.871336i	0.999994	−	0.00354413i	$-0.00112814\pi$
$570$		0			0
$571$			2.39230i			0.100115i	0.998746	+	0.0500574i	$0.0159404\pi$
$571$			2.39230i			0.100115i	−0.998746	+	0.0500574i	$0.984060\pi$
$572$		0			0
$573$		−	2.78461i		−	0.116329i
$574$		0			0
$575$	36.5885	−	63.3731i	1.52584	−	2.64284i
$576$		0			0
$577$			34.2679i			1.42659i	0.700862	+	0.713297i	$0.252799\pi$
$577$			34.2679i			1.42659i	−0.700862	+	0.713297i	$0.747201\pi$
$578$		0			0
$579$	−4.77757	−	8.27499i	−0.198549	−	0.343897i
$580$		0			0
$581$	−4.39230	+	2.53590i	−0.182224	+	0.105207i
$582$		0			0
$583$	−6.80385	−	3.92820i	−0.281787	−	0.162690i
$584$		0			0
$585$	31.8564	+	9.19615i	1.31710	+	0.380214i
$586$		0			0
$587$	22.5885	−	39.1244i	0.932325	−	1.61483i	0.152990	−	0.988228i	$-0.451110\pi$
$587$	22.5885	−	39.1244i	0.932325	−	1.61483i	0.779335	−	0.626607i	$-0.215557\pi$
$588$		0			0
$589$	5.19615	−	3.00000i	0.214104	−	0.123613i
$590$		0			0
$591$	7.94744	−	4.58846i	0.326914	−	0.188744i
$592$		0			0
$593$		−	1.58846i		−	0.0652301i	−0.999468	−	0.0326151i	$-0.989616\pi$
$593$		−	1.58846i		−	0.0652301i	0.999468	−	0.0326151i	$-0.0103835\pi$
$594$		0			0
$595$	5.19615	+	3.00000i	0.213021	+	0.122988i
$596$		0			0
$597$			11.0718i			0.453138i
$598$		0			0
$599$	13.2679			0.542114			0.271057	−	0.962563i	$-0.412627\pi$
$599$	13.2679			0.542114			0.271057	+	0.962563i	$0.412627\pi$
$600$		0			0
$601$	−13.0359	+	22.5788i	−0.531745	+	0.921010i	0.467568	+	0.883957i	$0.345130\pi$
$601$	−13.0359	+	22.5788i	−0.531745	+	0.921010i	−0.999313	+	0.0370529i	$0.988203\pi$
$602$		0			0
$603$	17.9090			0.729309
$604$		0			0
$605$	13.0622	+	22.6244i	0.531053	+	0.919811i
$606$		0			0
$607$	1.29423	+	2.24167i	0.0525311	+	0.0909866i	0.891095	−	0.453816i	$-0.149938\pi$
$607$	1.29423	+	2.24167i	0.0525311	+	0.0909866i	−0.838564	+	0.544803i	$0.816604\pi$
$608$		0			0
$609$	−6.58846	−	3.80385i	−0.266978	−	0.154140i
$610$		0			0
$611$	−16.8301	−	17.4904i	−0.680874	−	0.707585i
$612$		0			0
$613$	20.2583	−	35.0885i	0.818226	−	1.41721i	−0.0887617	−	0.996053i	$-0.528291\pi$
$613$	20.2583	−	35.0885i	0.818226	−	1.41721i	0.906988	−	0.421157i	$-0.138376\pi$
$614$		0			0
$615$	12.5622	+	21.7583i	0.506556	+	0.877381i
$616$		0			0
$617$	15.3564	−	8.86603i	0.618226	−	0.356933i	−0.157952	−	0.987447i	$-0.550489\pi$
$617$	15.3564	−	8.86603i	0.618226	−	0.356933i	0.776178	+	0.630514i	$0.217156\pi$
$618$		0			0
$619$	24.2487			0.974638			0.487319	−	0.873224i	$-0.337975\pi$
$619$	24.2487			0.974638			0.487319	+	0.873224i	$0.337975\pi$
$620$		0			0
$621$	28.3923	+	16.3923i	1.13934	+	0.657801i
$622$		0			0
$623$	25.8564			1.03592
$624$		0			0
$625$	10.0718			0.402872
$626$		0			0
$627$	−1.60770	−	0.928203i	−0.0642052	−	0.0370689i
$628$		0			0
$629$	1.98076			0.0789782
$630$		0			0
$631$	14.7846	−	8.53590i	0.588566	−	0.339809i	−0.175964	−	0.984397i	$-0.556304\pi$
$631$	14.7846	−	8.53590i	0.588566	−	0.339809i	0.764530	+	0.644588i	$0.222971\pi$
$632$		0			0
$633$	−10.4641	−	18.1244i	−0.415911	−	0.720378i
$634$		0			0
$635$	0.633975	−	1.09808i	0.0251585	−	0.0435758i
$636$		0			0
$637$	4.33013	+	17.5000i	0.171566	+	0.693375i
$638$		0			0
$639$	19.7776	+	11.4186i	0.782389	+	0.451712i
$640$		0			0
$641$	7.03590	+	12.1865i	0.277901	+	0.481339i	0.970863	−	0.239635i	$-0.0770279\pi$
$641$	7.03590	+	12.1865i	0.277901	+	0.481339i	−0.692962	+	0.720974i	$0.743695\pi$
$642$		0			0
$643$	19.3923	+	33.5885i	0.764758	+	1.32460i	0.940375	+	0.340141i	$0.110475\pi$
$643$	19.3923	+	33.5885i	0.764758	+	1.32460i	−0.175617	+	0.984459i	$0.556192\pi$
$644$		0			0
$645$	6.92820			0.272798
$646$		0			0
$647$	14.8301	−	25.6865i	0.583032	−	1.00984i	−0.412085	−	0.911145i	$-0.635199\pi$
$647$	14.8301	−	25.6865i	0.583032	−	1.00984i	0.995118	−	0.0986965i	$-0.0314673\pi$
$648$		0			0
$649$	1.07180			0.0420717
$650$		0			0
$651$		−	12.0000i		−	0.470317i
$652$		0			0
$653$	−31.1769	−	18.0000i	−1.22005	−	0.704394i	−0.255119	−	0.966910i	$-0.582115\pi$
$653$	−31.1769	−	18.0000i	−1.22005	−	0.704394i	−0.964928	+	0.262515i	$0.915448\pi$
$654$		0			0
$655$		−	40.0526i		−	1.56498i
$656$		0			0
$657$	3.69615	−	2.13397i	0.144201	−	0.0832543i
$658$		0			0
$659$	−26.7846	+	15.4641i	−1.04338	+	0.602396i	−0.920789	−	0.390061i	$-0.872454\pi$
$659$	−26.7846	+	15.4641i	−1.04338	+	0.602396i	−0.122591	+	0.992457i	$0.539120\pi$
$660$		0			0
$661$	6.06218	−	10.5000i	0.235791	−	0.408403i	−0.723711	−	0.690103i	$-0.757565\pi$
$661$	6.06218	−	10.5000i	0.235791	−	0.408403i	0.959502	+	0.281701i	$0.0908985\pi$
$662$		0			0
$663$	0.849365	+	0.882686i	0.0329866	+	0.0342807i
$664$		0			0
$665$	−14.1962	−	8.19615i	−0.550503	−	0.317833i
$666$		0			0
$667$	21.2942	−	12.2942i	0.824516	−	0.476034i
$668$		0			0
$669$	−9.33975	−	16.1769i	−0.361095	−	0.625436i
$670$		0			0
$671$			2.00000i			0.0772091i
$672$		0			0
$673$	5.50000	−	9.52628i	0.212009	−	0.367211i	−0.740334	−	0.672239i	$-0.765333\pi$
$673$	5.50000	−	9.52628i	0.212009	−	0.367211i	0.952343	+	0.305028i	$0.0986659\pi$
$674$		0			0
$675$			35.7128i			1.37459i
$676$		0			0
$677$		−	7.85641i		−	0.301946i	−0.988538	−	0.150973i	$-0.951759\pi$
$677$		−	7.85641i		−	0.301946i	0.988538	−	0.150973i	$-0.0482407\pi$
$678$		0			0
$679$	−10.3923	+	18.0000i	−0.398820	+	0.690777i
$680$		0			0
$681$			19.0718i			0.730833i
$682$		0			0
$683$	−5.63397	−	9.75833i	−0.215578	−	0.373392i	0.737873	−	0.674939i	$-0.235830\pi$
$683$	−5.63397	−	9.75833i	−0.215578	−	0.373392i	−0.953451	+	0.301547i	$0.902497\pi$
$684$		0			0
$685$	−43.9186	+	25.3564i	−1.67804	+	0.968818i
$686$		0			0
$687$	−8.19615	−	4.73205i	−0.312703	−	0.180539i
$688$		0			0
$689$	−9.82051	−	10.2058i	−0.374132	−	0.388809i
$690$		0			0
$691$	−9.00000	+	15.5885i	−0.342376	+	0.593013i	−0.984873	−	0.173275i	$-0.944565\pi$
$691$	−9.00000	+	15.5885i	−0.342376	+	0.593013i	0.642497	+	0.766288i	$0.277898\pi$
$692$		0			0
$693$	14.7846	−	8.53590i	0.561621	−	0.324252i
$694$		0			0
$695$	−30.5885	+	17.6603i	−1.16029	+	0.669892i
$696$		0			0
$697$		−	4.26795i		−	0.161660i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			8.78461i			0.331790i	0.986143	+	0.165895i	$0.0530513\pi$
$701$			8.78461i			0.331790i	−0.986143	+	0.165895i	$0.946949\pi$
$702$		0			0
$703$	−5.41154			−0.204100
$704$		0			0
$705$	9.19615	−	15.9282i	0.346347	−	0.599891i
$706$		0			0
$707$	−39.9615			−1.50291
$708$		0			0
$709$	20.5981	+	35.6769i	0.773577	+	1.33987i	0.935591	+	0.353086i	$0.114868\pi$
$709$	20.5981	+	35.6769i	0.773577	+	1.33987i	−0.162014	+	0.986788i	$0.551799\pi$
$710$		0			0
$711$	−12.8038	−	22.1769i	−0.480182	−	0.831699i
$712$		0			0
$713$	33.5885	+	19.3923i	1.25790	+	0.726248i
$714$		0			0
$715$	−6.46410	−	26.1244i	−0.241744	−	0.976996i
$716$		0			0
$717$	−6.73205	+	11.6603i	−0.251413	+	0.435460i
$718$		0			0
$719$	−16.8564	−	29.1962i	−0.628638	−	1.08883i	−0.987825	−	0.155567i	$-0.950279\pi$
$719$	−16.8564	−	29.1962i	−0.628638	−	1.08883i	0.359187	−	0.933265i	$-0.383054\pi$
$720$		0			0
$721$	−18.5885	+	10.7321i	−0.692270	+	0.399682i
$722$		0			0
$723$	−8.19615			−0.304818
$724$		0			0
$725$	23.1962	+	13.3923i	0.861483	+	0.497378i
$726$		0			0
$727$	13.6077			0.504681			0.252341	−	0.967638i	$-0.418800\pi$
$727$	13.6077			0.504681			0.252341	+	0.967638i	$0.418800\pi$
$728$		0			0
$729$	2.21539			0.0820515
$730$		0			0
$731$	−1.01924	−	0.588457i	−0.0376979	−	0.0217649i
$732$		0			0
$733$	4.94744			0.182738			0.0913690	−	0.995817i	$-0.470876\pi$
$733$	4.94744			0.182738			0.0913690	+	0.995817i	$0.470876\pi$
$734$		0			0
$735$	−11.8301	+	6.83013i	−0.436361	+	0.251933i
$736$		0			0
$737$	−7.26795	−	12.5885i	−0.267718	−	0.463702i
$738$		0			0
$739$	−9.46410	+	16.3923i	−0.348143	+	0.603001i	−0.985920	−	0.167220i	$-0.946521\pi$
$739$	−9.46410	+	16.3923i	−0.348143	+	0.603001i	0.637777	+	0.770221i	$0.279854\pi$
$740$		0			0
$741$	−2.32051	−	2.41154i	−0.0852460	−	0.0885902i
$742$		0			0
$743$	−4.73205	−	2.73205i	−0.173602	−	0.100229i	0.410681	−	0.911779i	$-0.365291\pi$
$743$	−4.73205	−	2.73205i	−0.173602	−	0.100229i	−0.584283	+	0.811550i	$0.698624\pi$
$744$		0			0
$745$	−21.8923	−	37.9186i	−0.802072	−	1.38923i
$746$		0			0
$747$	1.80385	+	3.12436i	0.0659993	+	0.114314i
$748$		0			0
$749$	−61.1769			−2.23536
$750$		0			0
$751$	−18.1962	+	31.5167i	−0.663987	+	1.15006i	0.315572	+	0.948902i	$0.397804\pi$
$751$	−18.1962	+	31.5167i	−0.663987	+	1.15006i	−0.979559	+	0.201158i	$0.935530\pi$
$752$		0			0
$753$	−6.00000			−0.218652
$754$		0			0
$755$			8.19615i			0.298289i
$756$		0			0
$757$	−18.3397	−	10.5885i	−0.666569	−	0.384844i	0.128206	−	0.991748i	$-0.459078\pi$
$757$	−18.3397	−	10.5885i	−0.666569	−	0.384844i	−0.794776	+	0.606904i	$0.792411\pi$
$758$		0			0
$759$		−	12.0000i		−	0.435572i
$760$		0			0
$761$	−10.8564	+	6.26795i	−0.393544	+	0.227213i	−0.683695	−	0.729768i	$-0.739628\pi$
$761$	−10.8564	+	6.26795i	−0.393544	+	0.227213i	0.290150	+	0.956981i	$0.406295\pi$
$762$		0			0
$763$	18.0000	−	10.3923i	0.651644	−	0.376227i
$764$		0			0
$765$	2.13397	−	3.69615i	0.0771540	−	0.133635i
$766$		0			0
$767$	1.85641	+	0.535898i	0.0670310	+	0.0193502i
$768$		0			0
$769$	19.3923	+	11.1962i	0.699304	+	0.403744i	0.807088	−	0.590431i	$-0.201042\pi$
$769$	19.3923	+	11.1962i	0.699304	+	0.403744i	−0.107784	+	0.994174i	$0.534375\pi$
$770$		0			0
$771$	14.4904	−	8.36603i	0.521858	−	0.301295i
$772$		0			0
$773$	26.2487	+	45.4641i	0.944101	+	1.63523i	0.757542	+	0.652786i	$0.226400\pi$
$773$	26.2487	+	45.4641i	0.944101	+	1.63523i	0.186558	+	0.982444i	$0.440267\pi$
$774$		0			0
$775$			42.2487i			1.51762i
$776$		0			0
$777$	−5.41154	+	9.37307i	−0.194138	+	0.336257i
$778$		0			0
$779$			11.6603i			0.417772i
$780$		0			0
$781$		−	18.5359i		−	0.663267i
$782$		0			0
$783$	−6.00000	+	10.3923i	−0.214423	+	0.371391i
$784$		0			0
$785$		−	14.6603i		−	0.523247i
$786$		0			0
$787$	20.3660	+	35.2750i	0.725970	+	1.25742i	0.958573	+	0.284846i	$0.0919424\pi$
$787$	20.3660	+	35.2750i	0.725970	+	1.25742i	−0.232603	+	0.972572i	$0.574724\pi$

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.ba (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.633975 - 0.366025i) q^{3} +3.73205 q^{5} +(3.00000 - 1.73205i) q^{7} +(-1.23205 - 2.13397i) q^{9} +O(q^{10})\)	\(q+(-0.633975 - 0.366025i) q^{3} +3.73205 q^{5} +(3.00000 - 1.73205i) q^{7} +(-1.23205 - 2.13397i) q^{9} +(-1.00000 + 1.73205i) q^{11} +(-2.59808 + 2.50000i) q^{13} +(-2.36603 - 1.36603i) q^{15} +(0.232051 + 0.401924i) q^{17} +(-0.633975 - 1.09808i) q^{19} -2.53590 q^{21} +(4.09808 - 7.09808i) q^{23} +8.92820 q^{25} +4.00000i q^{27} +(2.59808 + 1.50000i) q^{29} +4.73205i q^{31} +(1.26795 - 0.732051i) q^{33} +(11.1962 - 6.46410i) q^{35} +(2.13397 - 3.69615i) q^{37} +(2.56218 - 0.633975i) q^{39} +(-7.96410 - 4.59808i) q^{41} +(-2.19615 + 1.26795i) q^{43} +(-4.59808 - 7.96410i) q^{45} +6.73205i q^{47} +(2.50000 - 4.33013i) q^{49} -0.339746i q^{51} +3.92820i q^{53} +(-3.73205 + 6.46410i) q^{55} +0.928203i q^{57} +(-0.267949 - 0.464102i) q^{59} +(0.866025 - 0.500000i) q^{61} +(-7.39230 - 4.26795i) q^{63} +(-9.69615 + 9.33013i) q^{65} +(-3.63397 + 6.29423i) q^{67} +(-5.19615 + 3.00000i) q^{69} +(-8.02628 + 4.63397i) q^{71} +1.73205i q^{73} +(-5.66025 - 3.26795i) q^{75} +6.92820i q^{77} +10.3923 q^{79} +(-2.23205 + 3.86603i) q^{81} -1.46410 q^{83} +(0.866025 + 1.50000i) q^{85} +(-1.09808 - 1.90192i) q^{87} +(6.46410 + 3.73205i) q^{89} +(-3.46410 + 12.0000i) q^{91} +(1.73205 - 3.00000i) q^{93} +(-2.36603 - 4.09808i) q^{95} +(-5.19615 + 3.00000i) q^{97} +4.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} + 8 q^{5} + 12 q^{7} + 2 q^{9}+O(q^{10}) \) 4 * q - 6 * q^3 + 8 * q^5 + 12 * q^7 + 2 * q^9	\( 4 q - 6 q^{3} + 8 q^{5} + 12 q^{7} + 2 q^{9} - 4 q^{11} - 6 q^{15} - 6 q^{17} - 6 q^{19} - 24 q^{21} + 6 q^{23} + 8 q^{25} + 12 q^{33} + 24 q^{35} + 12 q^{37} - 14 q^{39} - 18 q^{41} + 12 q^{43} - 8 q^{45} + 10 q^{49} - 8 q^{55} - 8 q^{59} + 12 q^{63} - 18 q^{65} - 18 q^{67} + 6 q^{71} + 12 q^{75} - 2 q^{81} + 8 q^{83} + 6 q^{87} + 12 q^{89} - 6 q^{95} - 8 q^{99}+O(q^{100}) \) 4 * q - 6 * q^3 + 8 * q^5 + 12 * q^7 + 2 * q^9 - 4 * q^11 - 6 * q^15 - 6 * q^17 - 6 * q^19 - 24 * q^21 + 6 * q^23 + 8 * q^25 + 12 * q^33 + 24 * q^35 + 12 * q^37 - 14 * q^39 - 18 * q^41 + 12 * q^43 - 8 * q^45 + 10 * q^49 - 8 * q^55 - 8 * q^59 + 12 * q^63 - 18 * q^65 - 18 * q^67 + 6 * q^71 + 12 * q^75 - 2 * q^81 + 8 * q^83 + 6 * q^87 + 12 * q^89 - 6 * q^95 - 8 * q^99

Introduction

L-functions

Modular forms

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Database

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