LMFDB - Embedded newform 416.2.ba.a.49.1

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			49.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	416.49
Dual form			416.2.ba.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.36603 + 1.36603i) q^{3} +0.267949 q^{5} +(3.00000 + 1.73205i) q^{7} +(2.23205 - 3.86603i) q^{9} +O(q^{10})$	\(q+(-2.36603 + 1.36603i) q^{3} +0.267949 q^{5} +(3.00000 + 1.73205i) q^{7} +(2.23205 - 3.86603i) q^{9} +(-1.00000 - 1.73205i) q^{11} +(2.59808 + 2.50000i) q^{13} +(-0.633975 + 0.366025i) q^{15} +(-3.23205 + 5.59808i) q^{17} +(-2.36603 + 4.09808i) q^{19} -9.46410 q^{21} +(-1.09808 - 1.90192i) q^{23} -4.92820 q^{25} +4.00000i q^{27} +(-2.59808 + 1.50000i) q^{29} +1.26795i q^{31} +(4.73205 + 2.73205i) q^{33} +(0.803848 + 0.464102i) q^{35} +(3.86603 + 6.69615i) q^{37} +(-9.56218 - 2.36603i) q^{39} +(-1.03590 + 0.598076i) q^{41} +(8.19615 + 4.73205i) q^{43} +(0.598076 - 1.03590i) q^{45} +3.26795i q^{47} +(2.50000 + 4.33013i) q^{49} -17.6603i q^{51} -9.92820i q^{53} +(-0.267949 - 0.464102i) q^{55} -12.9282i q^{57} +(-3.73205 + 6.46410i) q^{59} +(-0.866025 - 0.500000i) q^{61} +(13.3923 - 7.73205i) q^{63} +(0.696152 + 0.669873i) q^{65} +(-5.36603 - 9.29423i) q^{67} +(5.19615 + 3.00000i) q^{69} +(11.0263 + 6.36603i) q^{71} -1.73205i q^{73} +(11.6603 - 6.73205i) q^{75} -6.92820i q^{77} -10.3923 q^{79} +(1.23205 + 2.13397i) q^{81} +5.46410 q^{83} +(-0.866025 + 1.50000i) q^{85} +(4.09808 - 7.09808i) q^{87} +(-0.464102 + 0.267949i) q^{89} +(3.46410 + 12.0000i) q^{91} +(-1.73205 - 3.00000i) q^{93} +(-0.633975 + 1.09808i) q^{95} +(5.19615 + 3.00000i) q^{97} -8.92820 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

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
$3$	−2.36603	+	1.36603i	−1.36603	+	0.788675i	−0.990418	−	0.138104i	$-0.955899\pi$
$3$	−2.36603	+	1.36603i	−1.36603	+	0.788675i	−0.375608	+	0.926779i	$0.622566\pi$
$4$		0			0
$5$	0.267949			0.119831			0.0599153	−	0.998203i	$-0.480917\pi$
$5$	0.267949			0.119831			0.0599153	+	0.998203i	$0.480917\pi$
$6$		0			0
$7$	3.00000	+	1.73205i	1.13389	+	0.654654i	0.944911	−	0.327327i	$-0.106148\pi$
$7$	3.00000	+	1.73205i	1.13389	+	0.654654i	0.188982	+	0.981981i	$0.439481\pi$
$8$		0			0
$9$	2.23205	−	3.86603i	0.744017	−	1.28868i
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	2.59808	+	2.50000i	0.720577	+	0.693375i
$13$	2.59808	+	2.50000i	0.720577	+	0.693375i
$14$		0			0
$15$	−0.633975	+	0.366025i	−0.163692	+	0.0945074i
$16$		0			0
$17$	−3.23205	+	5.59808i	−0.783887	+	1.35773i	0.145774	+	0.989318i	$0.453433\pi$
$17$	−3.23205	+	5.59808i	−0.783887	+	1.35773i	−0.929661	+	0.368415i	$0.879901\pi$
$18$		0			0
$19$	−2.36603	+	4.09808i	−0.542803	+	0.940163i	0.455938	+	0.890011i	$0.349304\pi$
$19$	−2.36603	+	4.09808i	−0.542803	+	0.940163i	−0.998742	+	0.0501517i	$0.984030\pi$
$20$		0			0
$21$	−9.46410			−2.06524
$22$		0			0
$23$	−1.09808	−	1.90192i	−0.228965	−	0.396579i	0.728537	−	0.685007i	$-0.240201\pi$
$23$	−1.09808	−	1.90192i	−0.228965	−	0.396579i	−0.957502	+	0.288428i	$0.906867\pi$
$24$		0			0
$25$	−4.92820			−0.985641
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	−2.59808	+	1.50000i	−0.482451	+	0.278543i	−0.721437	−	0.692480i	$-0.756518\pi$
$29$	−2.59808	+	1.50000i	−0.482451	+	0.278543i	0.238987	+	0.971023i	$0.423185\pi$
$30$		0			0
$31$			1.26795i			0.227730i	0.993496	+	0.113865i	$0.0363232\pi$
$31$			1.26795i			0.227730i	−0.993496	+	0.113865i	$0.963677\pi$
$32$		0			0
$33$	4.73205	+	2.73205i	0.823744	+	0.475589i
$34$		0			0
$35$	0.803848	+	0.464102i	0.135875	+	0.0784475i
$36$		0			0
$37$	3.86603	+	6.69615i	0.635571	+	1.10084i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	3.86603	+	6.69615i	0.635571	+	1.10084i	−0.350823	+	0.936442i	$0.614098\pi$
$38$		0			0
$39$	−9.56218	−	2.36603i	−1.53117	−	0.378867i
$40$		0			0
$41$	−1.03590	+	0.598076i	−0.161780	+	0.0934038i	−0.578704	−	0.815538i	$-0.696441\pi$
$41$	−1.03590	+	0.598076i	−0.161780	+	0.0934038i	0.416924	+	0.908941i	$0.363108\pi$
$42$		0			0
$43$	8.19615	+	4.73205i	1.24990	+	0.721631i	0.971090	−	0.238715i	$-0.0767261\pi$
$43$	8.19615	+	4.73205i	1.24990	+	0.721631i	0.278812	+	0.960346i	$0.410059\pi$
$44$		0			0
$45$	0.598076	−	1.03590i	0.0891559	−	0.154423i
$46$		0			0
$47$			3.26795i			0.476679i	0.971182	+	0.238340i	$0.0766032\pi$
$47$			3.26795i			0.476679i	−0.971182	+	0.238340i	$0.923397\pi$
$48$		0			0
$49$	2.50000	+	4.33013i	0.357143	+	0.618590i
$50$		0			0
$51$		−	17.6603i		−	2.47293i
$52$		0			0
$53$		−	9.92820i		−	1.36374i	−0.731472	−	0.681872i	$-0.761166\pi$
$53$		−	9.92820i		−	1.36374i	0.731472	−	0.681872i	$-0.238834\pi$
$54$		0			0
$55$	−0.267949	−	0.464102i	−0.0361303	−	0.0625794i
$56$		0			0
$57$		−	12.9282i		−	1.71238i
$58$		0			0
$59$	−3.73205	+	6.46410i	−0.485872	+	0.841554i	−0.999868	−	0.0162379i	$-0.994831\pi$
$59$	−3.73205	+	6.46410i	−0.485872	+	0.841554i	0.513997	+	0.857792i	$0.328164\pi$
$60$		0			0
$61$	−0.866025	−	0.500000i	−0.110883	−	0.0640184i	0.443533	−	0.896258i	$-0.353725\pi$
$61$	−0.866025	−	0.500000i	−0.110883	−	0.0640184i	−0.554416	+	0.832240i	$0.687058\pi$
$62$		0			0
$63$	13.3923	−	7.73205i	1.68727	−	0.974147i
$64$		0			0
$65$	0.696152	+	0.669873i	0.0863471	+	0.0830875i
$66$		0			0
$67$	−5.36603	−	9.29423i	−0.655564	−	1.13547i	−0.981752	−	0.190166i	$-0.939097\pi$
$67$	−5.36603	−	9.29423i	−0.655564	−	1.13547i	0.326188	−	0.945305i	$-0.394236\pi$
$68$		0			0
$69$	5.19615	+	3.00000i	0.625543	+	0.361158i
$70$		0			0
$71$	11.0263	+	6.36603i	1.30858	+	0.755508i	0.981859	−	0.189613i	$-0.0607234\pi$
$71$	11.0263	+	6.36603i	1.30858	+	0.755508i	0.326720	+	0.945121i	$0.394057\pi$
$72$		0			0
$73$		−	1.73205i		−	0.202721i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$		−	1.73205i		−	0.202721i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$		0			0
$75$	11.6603	−	6.73205i	1.34641	−	0.777350i
$76$		0			0
$77$		−	6.92820i		−	0.789542i
$78$		0			0
$79$	−10.3923			−1.16923			−0.584613	−	0.811312i	$-0.698754\pi$
$79$	−10.3923			−1.16923			−0.584613	+	0.811312i	$0.698754\pi$
$80$		0			0
$81$	1.23205	+	2.13397i	0.136895	+	0.237108i
$82$		0			0
$83$	5.46410			0.599763			0.299882	−	0.953976i	$-0.403053\pi$
$83$	5.46410			0.599763			0.299882	+	0.953976i	$0.403053\pi$
$84$		0			0
$85$	−0.866025	+	1.50000i	−0.0939336	+	0.162698i
$86$		0			0
$87$	4.09808	−	7.09808i	0.439360	−	0.760994i
$88$		0			0
$89$	−0.464102	+	0.267949i	−0.0491947	+	0.0284026i	−0.524396	−	0.851475i	$-0.675709\pi$
$89$	−0.464102	+	0.267949i	−0.0491947	+	0.0284026i	0.475201	+	0.879877i	$0.342375\pi$
$90$		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
$95$	−0.633975	+	1.09808i	−0.0650444	+	0.112660i
$96$		0			0
$97$	5.19615	+	3.00000i	0.527589	+	0.304604i	0.740034	−	0.672569i	$-0.234809\pi$
$97$	5.19615	+	3.00000i	0.527589	+	0.304604i	−0.212445	+	0.977173i	$0.568143\pi$
$98$		0			0
$99$	−8.92820			−0.897318
$100$		0			0
$101$	15.9904	−	9.23205i	1.59110	−	0.918623i	0.597984	−	0.801508i	$-0.295969\pi$
$101$	15.9904	−	9.23205i	1.59110	−	0.918623i	0.993118	−	0.117115i	$-0.0373647\pi$
$102$		0			0
$103$	4.19615			0.413459			0.206730	−	0.978398i	$-0.433718\pi$
$103$	4.19615			0.413459			0.206730	+	0.978398i	$0.433718\pi$
$104$		0			0
$105$	−2.53590			−0.247478
$106$		0			0
$107$	0.294229	−	0.169873i	0.0284442	−	0.0164222i	−0.485710	−	0.874120i	$-0.661439\pi$
$107$	0.294229	−	0.169873i	0.0284442	−	0.0164222i	0.514155	+	0.857697i	$0.328106\pi$
$108$		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
$111$	−18.2942	−	10.5622i	−1.73641	−	1.00252i
$112$		0			0
$113$	1.50000	−	2.59808i	0.141108	−	0.244406i	−0.786806	−	0.617200i	$-0.788267\pi$
$113$	1.50000	−	2.59808i	0.141108	−	0.244406i	0.927914	+	0.372794i	$0.121600\pi$
$114$		0			0
$115$	−0.294229	−	0.509619i	−0.0274370	−	0.0475222i
$116$		0			0
$117$	15.4641	−	4.46410i	1.42966	−	0.412706i
$118$		0			0
$119$	−19.3923	+	11.1962i	−1.77769	+	1.02635i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$	1.63397	−	2.83013i	0.147331	−	0.255184i
$124$		0			0
$125$	−2.66025			−0.237940
$126$		0			0
$127$	8.83013	+	15.2942i	0.783547	+	1.35714i	0.929863	+	0.367905i	$0.119925\pi$
$127$	8.83013	+	15.2942i	0.783547	+	1.35714i	−0.146316	+	0.989238i	$0.546742\pi$
$128$		0			0
$129$	−25.8564			−2.27653
$130$		0			0
$131$		−	7.26795i		−	0.635004i	−0.948258	−	0.317502i	$-0.897156\pi$
$131$		−	7.26795i		−	0.635004i	0.948258	−	0.317502i	$-0.102844\pi$
$132$		0			0
$133$	−14.1962	+	8.19615i	−1.23096	+	0.710697i
$134$		0			0
$135$			1.07180i			0.0922456i
$136$		0			0
$137$	−15.2321	−	8.79423i	−1.30136	−	0.751342i	−0.320724	−	0.947173i	$-0.603926\pi$
$137$	−15.2321	−	8.79423i	−1.30136	−	0.751342i	−0.980638	+	0.195831i	$0.937260\pi$
$138$		0			0
$139$	2.19615	+	1.26795i	0.186275	+	0.107546i	0.590238	−	0.807230i	$-0.299034\pi$
$139$	2.19615	+	1.26795i	0.186275	+	0.107546i	−0.403962	+	0.914776i	$0.632367\pi$
$140$		0			0
$141$	−4.46410	−	7.73205i	−0.375945	−	0.651156i
$142$		0			0
$143$	1.73205	−	7.00000i	0.144841	−	0.585369i
$144$		0			0
$145$	−0.696152	+	0.401924i	−0.0578123	+	0.0333780i
$146$		0			0
$147$	−11.8301	−	6.83013i	−0.975732	−	0.563339i
$148$		0			0
$149$	−4.13397	+	7.16025i	−0.338668	+	0.586591i	−0.984183	−	0.177157i	$-0.943310\pi$
$149$	−4.13397	+	7.16025i	−0.338668	+	0.586591i	0.645514	+	0.763748i	$0.276643\pi$
$150$		0			0
$151$		−	8.19615i		−	0.666993i	−0.942751	−	0.333497i	$-0.891771\pi$
$151$		−	8.19615i		−	0.666993i	0.942751	−	0.333497i	$-0.108229\pi$
$152$		0			0
$153$	14.4282	+	24.9904i	1.16645	+	2.02035i
$154$		0			0
$155$			0.339746i			0.0272891i
$156$		0			0
$157$			9.92820i			0.792357i	0.918174	+	0.396178i	$0.129664\pi$
$157$			9.92820i			0.792357i	−0.918174	+	0.396178i	$0.870336\pi$
$158$		0			0
$159$	13.5622	+	23.4904i	1.07555	+	1.86291i
$160$		0			0
$161$		−	7.60770i		−	0.599570i
$162$		0			0
$163$	−2.19615	+	3.80385i	−0.172016	+	0.297940i	−0.939125	−	0.343577i	$-0.888361\pi$
$163$	−2.19615	+	3.80385i	−0.172016	+	0.297940i	0.767109	+	0.641517i	$0.221695\pi$
$164$		0			0
$165$	1.26795	+	0.732051i	0.0987097	+	0.0569901i
$166$		0			0
$167$	7.73205	−	4.46410i	0.598324	−	0.345443i	−0.170058	−	0.985434i	$-0.554395\pi$
$167$	7.73205	−	4.46410i	0.598324	−	0.345443i	0.768382	+	0.639992i	$0.221062\pi$
$168$		0			0
$169$	0.500000	+	12.9904i	0.0384615	+	0.999260i
$170$		0			0
$171$	10.5622	+	18.2942i	0.807710	+	1.39899i
$172$		0			0
$173$	4.39230	+	2.53590i	0.333941	+	0.192801i	0.657589	−	0.753377i	$-0.271576\pi$
$173$	4.39230	+	2.53590i	0.333941	+	0.192801i	−0.323649	+	0.946177i	$0.604910\pi$
$174$		0			0
$175$	−14.7846	−	8.53590i	−1.11761	−	0.645253i
$176$		0			0
$177$		−	20.3923i		−	1.53278i
$178$		0			0
$179$	−12.0000	+	6.92820i	−0.896922	+	0.517838i	−0.876200	−	0.481947i	$-0.839930\pi$
$179$	−12.0000	+	6.92820i	−0.896922	+	0.517838i	−0.0207218	+	0.999785i	$0.506596\pi$
$180$		0			0
$181$			11.5359i			0.857457i	0.903433	+	0.428728i	$0.141038\pi$
$181$			11.5359i			0.857457i	−0.903433	+	0.428728i	$0.858962\pi$
$182$		0			0
$183$	2.73205			0.201959
$184$		0			0
$185$	1.03590	+	1.79423i	0.0761608	+	0.131914i
$186$		0			0
$187$	12.9282			0.945404
$188$		0			0
$189$	−6.92820	+	12.0000i	−0.503953	+	0.872872i
$190$		0			0
$191$	7.09808	−	12.2942i	0.513599	−	0.889579i	−0.486277	−	0.873805i	$-0.661645\pi$
$191$	7.09808	−	12.2942i	0.513599	−	0.889579i	0.999876	−	0.0157743i	$-0.00502134\pi$
$192$		0			0
$193$	21.6962	−	12.5263i	1.56172	−	0.901661i	0.564640	−	0.825337i	$-0.309015\pi$
$193$	21.6962	−	12.5263i	1.56172	−	0.901661i	0.997083	−	0.0763241i	$-0.0243184\pi$
$194$		0			0
$195$	−2.56218	−	0.633975i	−0.183481	−	0.0453999i
$196$		0			0
$197$	−9.73205	−	16.8564i	−0.693380	−	1.20097i	−0.970724	−	0.240199i	$-0.922787\pi$
$197$	−9.73205	−	16.8564i	−0.693380	−	1.20097i	0.277344	−	0.960771i	$-0.410546\pi$
$198$		0			0
$199$	4.56218	−	7.90192i	0.323404	−	0.560153i	−0.657784	−	0.753207i	$-0.728506\pi$
$199$	4.56218	−	7.90192i	0.323404	−	0.560153i	0.981188	+	0.193054i	$0.0618393\pi$
$200$		0			0
$201$	25.3923	+	14.6603i	1.79104	+	1.03405i
$202$		0			0
$203$	−10.3923			−0.729397
$204$		0			0
$205$	−0.277568	+	0.160254i	−0.0193862	+	0.0111926i
$206$		0			0
$207$	−9.80385			−0.681415
$208$		0			0
$209$	9.46410			0.654646
$210$		0			0
$211$	2.24167	−	1.29423i	0.154323	−	0.0890984i	−0.420850	−	0.907130i	$-0.638268\pi$
$211$	2.24167	−	1.29423i	0.154323	−	0.0890984i	0.575173	+	0.818032i	$0.304935\pi$
$212$		0			0
$213$	−34.7846			−2.38340
$214$		0			0
$215$	2.19615	+	1.26795i	0.149776	+	0.0864734i
$216$		0			0
$217$	−2.19615	+	3.80385i	−0.149085	+	0.258222i
$218$		0			0
$219$	2.36603	+	4.09808i	0.159881	+	0.276922i
$220$		0			0
$221$	−22.3923	+	6.46410i	−1.50627	+	0.434823i
$222$		0			0
$223$	16.9019	−	9.75833i	1.13184	−	0.653466i	0.187440	−	0.982276i	$-0.439981\pi$
$223$	16.9019	−	9.75833i	1.13184	−	0.653466i	0.944396	+	0.328810i	$0.106648\pi$
$224$		0			0
$225$	−11.0000	+	19.0526i	−0.733333	+	1.27017i
$226$		0			0
$227$	6.02628	−	10.4378i	0.399978	−	0.692783i	−0.593745	−	0.804654i	$-0.702351\pi$
$227$	6.02628	−	10.4378i	0.399978	−	0.692783i	0.993723	+	0.111871i	$0.0356844\pi$
$228$		0			0
$229$	−0.928203			−0.0613374			−0.0306687	−	0.999530i	$-0.509764\pi$
$229$	−0.928203			−0.0613374			−0.0306687	+	0.999530i	$0.509764\pi$
$230$		0			0
$231$	9.46410	+	16.3923i	0.622692	+	1.07853i
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$			0.875644i			0.0571207i
$236$		0			0
$237$	24.5885	−	14.1962i	1.59719	−	0.922139i
$238$		0			0
$239$			2.39230i			0.154745i	0.997002	+	0.0773727i	$0.0246531\pi$
$239$			2.39230i			0.154745i	−0.997002	+	0.0773727i	$0.975347\pi$
$240$		0			0
$241$	−0.696152	−	0.401924i	−0.0448431	−	0.0258902i	0.477411	−	0.878680i	$-0.341575\pi$
$241$	−0.696152	−	0.401924i	−0.0448431	−	0.0258902i	−0.522254	+	0.852790i	$0.674909\pi$
$242$		0			0
$243$	−16.2224	−	9.36603i	−1.04067	−	0.600831i
$244$		0			0
$245$	0.669873	+	1.16025i	0.0427966	+	0.0741259i
$246$		0			0
$247$	−16.3923	+	4.73205i	−1.04302	+	0.301093i
$248$		0			0
$249$	−12.9282	+	7.46410i	−0.819292	+	0.473018i
$250$		0			0
$251$	1.90192	+	1.09808i	0.120048	+	0.0693100i	0.558822	−	0.829288i	$-0.311254\pi$
$251$	1.90192	+	1.09808i	0.120048	+	0.0693100i	−0.438773	+	0.898598i	$0.644587\pi$
$252$		0			0
$253$	−2.19615	+	3.80385i	−0.138071	+	0.239146i
$254$		0			0
$255$		−	4.73205i		−	0.296333i
$256$		0			0
$257$	2.42820	+	4.20577i	0.151467	+	0.262349i	0.931767	−	0.363057i	$-0.118267\pi$
$257$	2.42820	+	4.20577i	0.151467	+	0.262349i	−0.780300	+	0.625406i	$0.784934\pi$
$258$		0			0
$259$			26.7846i			1.66431i
$260$		0			0
$261$			13.3923i			0.828963i
$262$		0			0
$263$	−2.19615	−	3.80385i	−0.135421	−	0.234555i	0.790337	−	0.612672i	$-0.209905\pi$
$263$	−2.19615	−	3.80385i	−0.135421	−	0.234555i	−0.925758	+	0.378116i	$0.876572\pi$
$264$		0			0
$265$		−	2.66025i		−	0.163418i
$266$		0			0
$267$	0.732051	−	1.26795i	0.0448008	−	0.0775972i
$268$		0			0
$269$	−2.19615	−	1.26795i	−0.133902	−	0.0773082i	0.431553	−	0.902088i	$-0.357966\pi$
$269$	−2.19615	−	1.26795i	−0.133902	−	0.0773082i	−0.565455	+	0.824779i	$0.691299\pi$
$270$		0			0
$271$	15.0000	−	8.66025i	0.911185	−	0.526073i	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	15.0000	−	8.66025i	0.911185	−	0.526073i	0.880812	+	0.473466i	$0.156997\pi$
$272$		0			0
$273$	−24.5885	−	23.6603i	−1.48816	−	1.43198i
$274$		0			0
$275$	4.92820	+	8.53590i	0.297182	+	0.514734i
$276$		0			0
$277$	17.5981	+	10.1603i	1.05737	+	0.610471i	0.924702	−	0.380691i	$-0.124314\pi$
$277$	17.5981	+	10.1603i	1.05737	+	0.610471i	0.132663	+	0.991161i	$0.457647\pi$
$278$		0			0
$279$	4.90192	+	2.83013i	0.293471	+	0.169435i
$280$		0			0
$281$			6.66025i			0.397317i	0.980069	+	0.198659i	$0.0636585\pi$
$281$			6.66025i			0.397317i	−0.980069	+	0.198659i	$0.936341\pi$
$282$		0			0
$283$	−9.33975	+	5.39230i	−0.555190	+	0.320539i	−0.751213	−	0.660060i	$-0.770531\pi$
$283$	−9.33975	+	5.39230i	−0.555190	+	0.320539i	0.196022	+	0.980599i	$0.437197\pi$
$284$		0			0
$285$		−	3.46410i		−	0.205196i
$286$		0			0
$287$	−4.14359			−0.244589
$288$		0			0
$289$	−12.3923	−	21.4641i	−0.728959	−	1.26259i
$290$		0			0
$291$	−16.3923			−0.960934
$292$		0			0
$293$	6.59808	−	11.4282i	0.385464	−	0.667643i	−0.606370	−	0.795183i	$-0.707375\pi$
$293$	6.59808	−	11.4282i	0.385464	−	0.667643i	0.991833	+	0.127540i	$0.0407082\pi$
$294$		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
$299$	1.90192	−	7.68653i	0.109991	−	0.444524i
$300$		0			0
$301$	16.3923	+	28.3923i	0.944837	+	1.63651i
$302$		0			0
$303$	−25.2224	+	43.6865i	−1.44899	+	2.50973i
$304$		0			0
$305$	−0.232051	−	0.133975i	−0.0132872	−	0.00767136i
$306$		0			0
$307$	8.19615			0.467779			0.233890	−	0.972263i	$-0.424855\pi$
$307$	8.19615			0.467779			0.233890	+	0.972263i	$0.424855\pi$
$308$		0			0
$309$	−9.92820	+	5.73205i	−0.564796	+	0.326085i
$310$		0			0
$311$	−12.5885			−0.713826			−0.356913	−	0.934138i	$-0.616171\pi$
$311$	−12.5885			−0.713826			−0.356913	+	0.934138i	$0.616171\pi$
$312$		0			0
$313$	−9.46410			−0.534943			−0.267471	−	0.963566i	$-0.586188\pi$
$313$	−9.46410			−0.534943			−0.267471	+	0.963566i	$0.586188\pi$
$314$		0			0
$315$	3.58846	−	2.07180i	0.202187	−	0.116733i
$316$		0			0
$317$	14.8038			0.831467			0.415733	−	0.909486i	$-0.363525\pi$
$317$	14.8038			0.831467			0.415733	+	0.909486i	$0.363525\pi$
$318$		0			0
$319$	5.19615	+	3.00000i	0.290929	+	0.167968i
$320$		0			0
$321$	−0.464102	+	0.803848i	−0.0259036	+	0.0448664i
$322$		0			0
$323$	−15.2942	−	26.4904i	−0.850994	−	1.47396i
$324$		0			0
$325$	−12.8038	−	12.3205i	−0.710230	−	0.683419i
$326$		0			0
$327$	−14.1962	+	8.19615i	−0.785049	+	0.453248i
$328$		0			0
$329$	−5.66025	+	9.80385i	−0.312060	+	0.540504i
$330$		0			0
$331$	−6.00000	+	10.3923i	−0.329790	+	0.571213i	−0.982470	−	0.186421i	$-0.940311\pi$
$331$	−6.00000	+	10.3923i	−0.329790	+	0.571213i	0.652680	+	0.757634i	$0.273645\pi$
$332$		0			0
$333$	34.5167			1.89150
$334$		0			0
$335$	−1.43782	−	2.49038i	−0.0785566	−	0.136064i
$336$		0			0
$337$	15.2487			0.830650			0.415325	−	0.909673i	$-0.363668\pi$
$337$	15.2487			0.830650			0.415325	+	0.909673i	$0.363668\pi$
$338$		0			0
$339$			8.19615i			0.445154i
$340$		0			0
$341$	2.19615	−	1.26795i	0.118928	−	0.0686633i
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$		0			0
$345$	1.39230	+	0.803848i	0.0749592	+	0.0432777i
$346$		0			0
$347$	−4.09808	−	2.36603i	−0.219996	−	0.127015i	0.385952	−	0.922519i	$-0.373873\pi$
$347$	−4.09808	−	2.36603i	−0.219996	−	0.127015i	−0.605949	+	0.795504i	$0.707206\pi$
$348$		0			0
$349$	−14.6603	−	25.3923i	−0.784745	−	1.35922i	−0.929151	−	0.369700i	$-0.879460\pi$
$349$	−14.6603	−	25.3923i	−0.784745	−	1.35922i	0.144406	−	0.989519i	$-0.453873\pi$
$350$		0			0
$351$	−10.0000	+	10.3923i	−0.533761	+	0.554700i
$352$		0			0
$353$	18.8205	−	10.8660i	1.00171	−	0.578340i	0.0929594	−	0.995670i	$-0.470367\pi$
$353$	18.8205	−	10.8660i	1.00171	−	0.578340i	0.908755	+	0.417330i	$0.137034\pi$
$354$		0			0
$355$	2.95448	+	1.70577i	0.156808	+	0.0905329i
$356$		0			0
$357$	30.5885	−	52.9808i	1.61891	−	2.80404i
$358$		0			0
$359$		−	26.9282i		−	1.42122i	−0.703588	−	0.710608i	$-0.748420\pi$
$359$		−	26.9282i		−	1.42122i	0.703588	−	0.710608i	$-0.251580\pi$
$360$		0			0
$361$	−1.69615	−	2.93782i	−0.0892712	−	0.154622i
$362$		0			0
$363$			19.1244i			1.00377i
$364$		0			0
$365$		−	0.464102i		−	0.0242922i
$366$		0			0
$367$	−8.90192	−	15.4186i	−0.464677	−	0.804844i	0.534510	−	0.845162i	$-0.320496\pi$
$367$	−8.90192	−	15.4186i	−0.464677	−	0.804844i	−0.999187	+	0.0403184i	$0.987163\pi$
$368$		0			0
$369$			5.33975i			0.277976i
$370$		0			0
$371$	17.1962	−	29.7846i	0.892780	−	1.54634i
$372$		0			0
$373$	13.6699	+	7.89230i	0.707799	+	0.408648i	0.810246	−	0.586090i	$-0.199334\pi$
$373$	13.6699	+	7.89230i	0.707799	+	0.408648i	−0.102446	+	0.994739i	$0.532667\pi$
$374$		0			0
$375$	6.29423	−	3.63397i	0.325033	−	0.187658i
$376$		0			0
$377$	−10.5000	−	2.59808i	−0.540778	−	0.133808i
$378$		0			0
$379$	−8.02628	−	13.9019i	−0.412282	−	0.714094i	0.582857	−	0.812575i	$-0.301935\pi$
$379$	−8.02628	−	13.9019i	−0.412282	−	0.714094i	−0.995139	+	0.0984811i	$0.968602\pi$
$380$		0			0
$381$	−41.7846	−	24.1244i	−2.14069	−	1.23593i
$382$		0			0
$383$	17.3205	+	10.0000i	0.885037	+	0.510976i	0.872316	−	0.488943i	$-0.162617\pi$
$383$	17.3205	+	10.0000i	0.885037	+	0.510976i	0.0127209	+	0.999919i	$0.495951\pi$
$384$		0			0
$385$		−	1.85641i		−	0.0946112i
$386$		0			0
$387$	36.5885	−	21.1244i	1.85990	−	1.07381i
$388$		0			0
$389$			36.7128i			1.86141i	0.365766	+	0.930707i	$0.380807\pi$
$389$			36.7128i			1.86141i	−0.365766	+	0.930707i	$0.619193\pi$
$390$		0			0
$391$	14.1962			0.717930
$392$		0			0
$393$	9.92820	+	17.1962i	0.500812	+	0.867431i
$394$		0			0
$395$	−2.78461			−0.140109
$396$		0			0
$397$	12.1244	−	21.0000i	0.608504	−	1.05396i	−0.382983	−	0.923755i	$-0.625103\pi$
$397$	12.1244	−	21.0000i	0.608504	−	1.05396i	0.991487	−	0.130204i	$-0.0415634\pi$
$398$		0			0
$399$	22.3923	−	38.7846i	1.12102	−	1.94166i
$400$		0			0
$401$	−5.42820	+	3.13397i	−0.271072	+	0.156503i	−0.629374	−	0.777102i	$-0.716689\pi$
$401$	−5.42820	+	3.13397i	−0.271072	+	0.156503i	0.358303	+	0.933605i	$0.383355\pi$
$402$		0			0
$403$	−3.16987	+	3.29423i	−0.157903	+	0.164097i
$404$		0			0
$405$	0.330127	+	0.571797i	0.0164041	+	0.0284128i
$406$		0			0
$407$	7.73205	−	13.3923i	0.383264	−	0.663832i
$408$		0			0
$409$	−9.69615	−	5.59808i	−0.479444	−	0.276807i	0.240741	−	0.970589i	$-0.422610\pi$
$409$	−9.69615	−	5.59808i	−0.479444	−	0.276807i	−0.720185	+	0.693782i	$0.755943\pi$
$410$		0			0
$411$	48.0526			2.37026
$412$		0			0
$413$	−22.3923	+	12.9282i	−1.10185	+	0.636155i
$414$		0			0
$415$	1.46410			0.0718699
$416$		0			0
$417$	−6.92820			−0.339276
$418$		0			0
$419$	22.0981	−	12.7583i	1.07956	−	0.623285i	0.148784	−	0.988870i	$-0.452464\pi$
$419$	22.0981	−	12.7583i	1.07956	−	0.623285i	0.930778	+	0.365585i	$0.119131\pi$
$420$		0			0
$421$	−5.87564			−0.286361			−0.143181	−	0.989697i	$-0.545733\pi$
$421$	−5.87564			−0.286361			−0.143181	+	0.989697i	$0.545733\pi$
$422$		0			0
$423$	12.6340	+	7.29423i	0.614285	+	0.354658i
$424$		0			0
$425$	15.9282	−	27.5885i	0.772631	−	1.33824i
$426$		0			0
$427$	−1.73205	−	3.00000i	−0.0838198	−	0.145180i
$428$		0			0
$429$	5.46410	+	18.9282i	0.263809	+	0.913862i
$430$		0			0
$431$	−8.53590	+	4.92820i	−0.411160	+	0.237383i	−0.691288	−	0.722579i	$-0.742956\pi$
$431$	−8.53590	+	4.92820i	−0.411160	+	0.237383i	0.280128	+	0.959963i	$0.409623\pi$
$432$		0			0
$433$	−10.5000	+	18.1865i	−0.504598	+	0.873989i	0.495388	+	0.868672i	$0.335026\pi$
$433$	−10.5000	+	18.1865i	−0.504598	+	0.873989i	−0.999986	+	0.00531724i	$0.998307\pi$
$434$		0			0
$435$	1.09808	−	1.90192i	0.0526487	−	0.0911903i
$436$		0			0
$437$	10.3923			0.497131
$438$		0			0
$439$	18.1962	+	31.5167i	0.868455	+	1.50421i	0.863575	+	0.504220i	$0.168220\pi$
$439$	18.1962	+	31.5167i	0.868455	+	1.50421i	0.00487976	+	0.999988i	$0.498447\pi$
$440$		0			0
$441$	22.3205			1.06288
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−0.124356	+	0.0717968i	−0.00589502	+	0.00340349i
$446$		0			0
$447$		−	22.5885i		−	1.06840i
$448$		0			0
$449$	11.5359	+	6.66025i	0.544413	+	0.314317i	0.746865	−	0.664975i	$-0.231558\pi$
$449$	11.5359	+	6.66025i	0.544413	+	0.314317i	−0.202453	+	0.979292i	$0.564891\pi$
$450$		0			0
$451$	2.07180	+	1.19615i	0.0975571	+	0.0563246i
$452$		0			0
$453$	11.1962	+	19.3923i	0.526041	+	0.911130i
$454$		0			0
$455$	0.928203	+	3.21539i	0.0435148	+	0.150740i
$456$		0			0
$457$	−3.69615	+	2.13397i	−0.172899	+	0.0998231i	−0.583952	−	0.811788i	$-0.698494\pi$
$457$	−3.69615	+	2.13397i	−0.172899	+	0.0998231i	0.411053	+	0.911611i	$0.365161\pi$
$458$		0			0
$459$	−22.3923	−	12.9282i	−1.04518	−	0.603437i
$460$		0			0
$461$	−5.66987	+	9.82051i	−0.264072	+	0.457387i	−0.967320	−	0.253558i	$-0.918399\pi$
$461$	−5.66987	+	9.82051i	−0.264072	+	0.457387i	0.703248	+	0.710945i	$0.251732\pi$
$462$		0			0
$463$			4.39230i			0.204128i	0.994778	+	0.102064i	$0.0325446\pi$
$463$			4.39230i			0.204128i	−0.994778	+	0.102064i	$0.967455\pi$
$464$		0			0
$465$	−0.464102	−	0.803848i	−0.0215222	−	0.0372775i
$466$		0			0
$467$			27.4641i			1.27089i	0.772147	+	0.635444i	$0.219183\pi$
$467$			27.4641i			1.27089i	−0.772147	+	0.635444i	$0.780817\pi$
$468$		0			0
$469$		−	37.1769i		−	1.71667i
$470$		0			0
$471$	−13.5622	−	23.4904i	−0.624912	−	1.08238i
$472$		0			0
$473$		−	18.9282i		−	0.870320i
$474$		0			0
$475$	11.6603	−	20.1962i	0.535009	−	0.926663i
$476$		0			0
$477$	−38.3827	−	22.1603i	−1.75742	−	1.01465i
$478$		0			0
$479$	−28.3468	+	16.3660i	−1.29520	+	0.747783i	−0.979571	−	0.201100i	$-0.935548\pi$
$479$	−28.3468	+	16.3660i	−1.29520	+	0.747783i	−0.315627	+	0.948883i	$0.602215\pi$
$480$		0			0
$481$	−6.69615	+	27.0622i	−0.305318	+	1.23393i
$482$		0			0
$483$	10.3923	+	18.0000i	0.472866	+	0.819028i
$484$		0			0
$485$	1.39230	+	0.803848i	0.0632213	+	0.0365008i
$486$		0			0
$487$	−9.80385	−	5.66025i	−0.444255	−	0.256491i	0.261146	−	0.965299i	$-0.415900\pi$
$487$	−9.80385	−	5.66025i	−0.444255	−	0.256491i	−0.705401	+	0.708809i	$0.749233\pi$
$488$		0			0
$489$		−	12.0000i		−	0.542659i
$490$		0			0
$491$	−24.5885	+	14.1962i	−1.10966	+	0.640663i	−0.938742	−	0.344622i	$-0.888007\pi$
$491$	−24.5885	+	14.1962i	−1.10966	+	0.640663i	−0.170920	+	0.985285i	$0.554674\pi$
$492$		0			0
$493$		−	19.3923i		−	0.873385i
$494$		0			0
$495$	−2.39230			−0.107526
$496$		0			0
$497$	22.0526	+	38.1962i	0.989192	+	1.71333i
$498$		0			0
$499$	−33.1244			−1.48285			−0.741425	−	0.671036i	$-0.765850\pi$
$499$	−33.1244			−1.48285			−0.741425	+	0.671036i	$0.765850\pi$
$500$		0			0
$501$	−12.1962	+	21.1244i	−0.544884	+	0.943767i
$502$		0			0
$503$	−16.3923	+	28.3923i	−0.730897	+	1.26595i	0.225604	+	0.974219i	$0.427565\pi$
$503$	−16.3923	+	28.3923i	−0.730897	+	1.26595i	−0.956500	+	0.291731i	$0.905769\pi$
$504$		0			0
$505$	4.28461	−	2.47372i	0.190663	−	0.110079i
$506$		0			0
$507$	−18.9282	−	30.0526i	−0.840631	−	1.33468i
$508$		0			0
$509$	16.0622	+	27.8205i	0.711944	+	1.23312i	0.964127	+	0.265443i	$0.0855181\pi$
$509$	16.0622	+	27.8205i	0.711944	+	1.23312i	−0.252183	+	0.967680i	$0.581149\pi$
$510$		0			0
$511$	3.00000	−	5.19615i	0.132712	−	0.229864i
$512$		0			0
$513$	−16.3923	−	9.46410i	−0.723738	−	0.417850i
$514$		0			0
$515$	1.12436			0.0495450
$516$		0			0
$517$	5.66025	−	3.26795i	0.248938	−	0.143724i
$518$		0			0
$519$	−13.8564			−0.608229
$520$		0			0
$521$	−39.2487			−1.71952			−0.859759	−	0.510701i	$-0.829386\pi$
$521$	−39.2487			−1.71952			−0.859759	+	0.510701i	$0.829386\pi$
$522$		0			0
$523$	11.1962	−	6.46410i	0.489574	−	0.282655i	−0.234824	−	0.972038i	$-0.575451\pi$
$523$	11.1962	−	6.46410i	0.489574	−	0.282655i	0.724398	+	0.689382i	$0.242118\pi$
$524$		0			0
$525$	46.6410			2.03558
$526$		0			0
$527$	−7.09808	−	4.09808i	−0.309197	−	0.178515i
$528$		0			0
$529$	9.08846	−	15.7417i	0.395150	−	0.684420i
$530$		0			0
$531$	16.6603	+	28.8564i	0.722993	+	1.25226i
$532$		0			0
$533$	−4.18653	−	1.03590i	−0.181339	−	0.0448697i
$534$		0			0
$535$	0.0788383	−	0.0455173i	0.00340848	−	0.00196789i
$536$		0			0
$537$	18.9282	−	32.7846i	0.816812	−	1.41476i
$538$		0			0
$539$	5.00000	−	8.66025i	0.215365	−	0.373024i
$540$		0			0
$541$	−23.4449			−1.00797			−0.503987	−	0.863711i	$-0.668134\pi$
$541$	−23.4449			−1.00797			−0.503987	+	0.863711i	$0.668134\pi$
$542$		0			0
$543$	−15.7583	−	27.2942i	−0.676255	−	1.17131i
$544$		0			0
$545$	1.60770			0.0688661
$546$		0			0
$547$			12.5885i			0.538244i	0.963106	+	0.269122i	$0.0867334\pi$
$547$			12.5885i			0.538244i	−0.963106	+	0.269122i	$0.913267\pi$
$548$		0			0
$549$	−3.86603	+	2.23205i	−0.164998	+	0.0952616i
$550$		0			0
$551$		−	14.1962i		−	0.604776i
$552$		0			0
$553$	−31.1769	−	18.0000i	−1.32578	−	0.765438i
$554$		0			0
$555$	−4.90192	−	2.83013i	−0.208075	−	0.120132i
$556$		0			0
$557$	−7.59808	−	13.1603i	−0.321941	−	0.557618i	0.658948	−	0.752189i	$-0.271002\pi$
$557$	−7.59808	−	13.1603i	−0.321941	−	0.557618i	−0.980888	+	0.194571i	$0.937669\pi$
$558$		0			0
$559$	9.46410	+	32.7846i	0.400289	+	1.38664i
$560$		0			0
$561$	−30.5885	+	17.6603i	−1.29145	+	0.745617i
$562$		0			0
$563$	37.9808	+	21.9282i	1.60070	+	0.924164i	0.991348	+	0.131263i	$0.0419031\pi$
$563$	37.9808	+	21.9282i	1.60070	+	0.924164i	0.609351	+	0.792901i	$0.291430\pi$
$564$		0			0
$565$	0.401924	−	0.696152i	0.0169091	−	0.0292874i
$566$		0			0
$567$			8.53590i			0.358474i
$568$		0			0
$569$	12.0000	+	20.7846i	0.503066	+	0.871336i	0.999994	+	0.00354413i	$0.00112814\pi$
$569$	12.0000	+	20.7846i	0.503066	+	0.871336i	−0.496928	+	0.867792i	$0.665539\pi$
$570$		0			0
$571$		−	18.3923i		−	0.769694i	−0.922980	−	0.384847i	$-0.874254\pi$
$571$		−	18.3923i		−	0.769694i	0.922980	−	0.384847i	$-0.125746\pi$
$572$		0			0
$573$			38.7846i			1.62025i
$574$		0			0
$575$	5.41154	+	9.37307i	0.225677	+	0.390884i
$576$		0			0
$577$			37.7321i			1.57081i	0.618985	+	0.785403i	$0.287544\pi$
$577$			37.7321i			1.57081i	−0.618985	+	0.785403i	$0.712456\pi$
$578$		0			0
$579$	−34.2224	+	59.2750i	−1.42224	+	2.46338i
$580$		0			0
$581$	16.3923	+	9.46410i	0.680067	+	0.392637i
$582$		0			0
$583$	−17.1962	+	9.92820i	−0.712192	+	0.411184i
$584$		0			0
$585$	4.14359	−	1.19615i	0.171317	−	0.0494548i
$586$		0			0
$587$	−8.58846	−	14.8756i	−0.354484	−	0.613984i	0.632546	−	0.774523i	$-0.282010\pi$
$587$	−8.58846	−	14.8756i	−0.354484	−	0.613984i	−0.987029	+	0.160539i	$0.948677\pi$
$588$		0			0
$589$	−5.19615	−	3.00000i	−0.214104	−	0.123613i
$590$		0			0
$591$	46.0526	+	26.5885i	1.89435	+	1.09370i
$592$		0			0
$593$			29.5885i			1.21505i	0.794300	+	0.607526i	$0.207838\pi$
$593$			29.5885i			1.21505i	−0.794300	+	0.607526i	$0.792162\pi$
$594$		0			0
$595$	−5.19615	+	3.00000i	−0.213021	+	0.122988i
$596$		0			0
$597$			24.9282i			1.02024i
$598$		0			0
$599$	16.7321			0.683653			0.341827	−	0.939763i	$-0.388954\pi$
$599$	16.7321			0.683653			0.341827	+	0.939763i	$0.388954\pi$
$600$		0			0
$601$	−19.9641	−	34.5788i	−0.814353	−	1.41050i	−0.909792	−	0.415065i	$-0.863759\pi$
$601$	−19.9641	−	34.5788i	−0.814353	−	1.41050i	0.0954391	−	0.995435i	$-0.469574\pi$
$602$		0			0
$603$	−47.9090			−1.95100
$604$		0			0
$605$	0.937822	−	1.62436i	0.0381279	−	0.0660394i
$606$		0			0
$607$	−14.2942	+	24.7583i	−0.580185	+	1.00491i	0.415272	+	0.909697i	$0.363686\pi$
$607$	−14.2942	+	24.7583i	−0.580185	+	1.00491i	−0.995457	+	0.0952124i	$0.969647\pi$
$608$		0			0
$609$	24.5885	−	14.1962i	0.996375	−	0.575257i
$610$		0			0
$611$	−8.16987	+	8.49038i	−0.330518	+	0.343484i
$612$		0			0
$613$	−2.25833	−	3.91154i	−0.0912131	−	0.157986i	0.816809	−	0.576909i	$-0.195741\pi$
$613$	−2.25833	−	3.91154i	−0.0912131	−	0.157986i	−0.908022	+	0.418923i	$0.862408\pi$
$614$		0			0
$615$	0.437822	−	0.758330i	0.0176547	−	0.0305788i
$616$		0			0
$617$	−12.3564	−	7.13397i	−0.497450	−	0.287203i	0.230210	−	0.973141i	$-0.426059\pi$
$617$	−12.3564	−	7.13397i	−0.497450	−	0.287203i	−0.727660	+	0.685938i	$0.759392\pi$
$618$		0			0
$619$	−24.2487			−0.974638			−0.487319	−	0.873224i	$-0.662025\pi$
$619$	−24.2487			−0.974638			−0.487319	+	0.873224i	$0.662025\pi$
$620$		0			0
$621$	7.60770	−	4.39230i	0.305286	−	0.176257i
$622$		0			0
$623$	−1.85641			−0.0743754
$624$		0			0
$625$	23.9282			0.957128
$626$		0			0
$627$	−22.3923	+	12.9282i	−0.894263	+	0.516303i
$628$		0			0
$629$	−49.9808			−1.99286
$630$		0			0
$631$	−26.7846	−	15.4641i	−1.06628	−	0.615616i	−0.139116	−	0.990276i	$-0.544426\pi$
$631$	−26.7846	−	15.4641i	−1.06628	−	0.615616i	−0.927162	+	0.374660i	$0.877759\pi$
$632$		0			0
$633$	−3.53590	+	6.12436i	−0.140539	+	0.243421i
$634$		0			0
$635$	2.36603	+	4.09808i	0.0938929	+	0.162627i
$636$		0			0
$637$	−4.33013	+	17.5000i	−0.171566	+	0.693375i
$638$		0			0
$639$	49.2224	−	28.4186i	1.94721	−	1.12422i
$640$		0			0
$641$	13.9641	−	24.1865i	0.551549	−	0.955311i	−0.446614	−	0.894727i	$-0.647370\pi$
$641$	13.9641	−	24.1865i	0.551549	−	0.955311i	0.998163	−	0.0605840i	$-0.0192963\pi$
$642$		0			0
$643$	−1.39230	+	2.41154i	−0.0549071	+	0.0951020i	−0.892173	−	0.451695i	$-0.850820\pi$
$643$	−1.39230	+	2.41154i	−0.0549071	+	0.0951020i	0.837265	+	0.546797i	$0.184153\pi$
$644$		0			0
$645$	−6.92820			−0.272798
$646$		0			0
$647$	6.16987	+	10.6865i	0.242563	+	0.420131i	0.961444	−	0.275002i	$-0.0886786\pi$
$647$	6.16987	+	10.6865i	0.242563	+	0.420131i	−0.718881	+	0.695133i	$0.755345\pi$
$648$		0			0
$649$	14.9282			0.585983
$650$		0			0
$651$		−	12.0000i		−	0.470317i
$652$		0			0
$653$	31.1769	−	18.0000i	1.22005	−	0.704394i	0.255119	−	0.966910i	$-0.417885\pi$
$653$	31.1769	−	18.0000i	1.22005	−	0.704394i	0.964928	+	0.262515i	$0.0845520\pi$
$654$		0			0
$655$		−	1.94744i		−	0.0760928i
$656$		0			0
$657$	−6.69615	−	3.86603i	−0.261242	−	0.150828i
$658$		0			0
$659$	14.7846	+	8.53590i	0.575927	+	0.332511i	0.759513	−	0.650492i	$-0.225437\pi$
$659$	14.7846	+	8.53590i	0.575927	+	0.332511i	−0.183586	+	0.983004i	$0.558771\pi$
$660$		0			0
$661$	−6.06218	−	10.5000i	−0.235791	−	0.408403i	0.723711	−	0.690103i	$-0.242435\pi$
$661$	−6.06218	−	10.5000i	−0.235791	−	0.408403i	−0.959502	+	0.281701i	$0.909102\pi$
$662$		0			0
$663$	44.1506	−	45.8827i	1.71467	−	1.78194i
$664$		0			0
$665$	−3.80385	+	2.19615i	−0.147507	+	0.0851631i
$666$		0			0
$667$	5.70577	+	3.29423i	0.220928	+	0.127553i
$668$		0			0
$669$	−26.6603	+	46.1769i	−1.03074	+	1.78530i
$670$		0			0
$671$			2.00000i			0.0772091i
$672$		0			0
$673$	5.50000	+	9.52628i	0.212009	+	0.367211i	0.952343	−	0.305028i	$-0.0986659\pi$
$673$	5.50000	+	9.52628i	0.212009	+	0.367211i	−0.740334	+	0.672239i	$0.765333\pi$
$674$		0			0
$675$		−	19.7128i		−	0.758747i
$676$		0			0
$677$			19.8564i			0.763144i	0.924339	+	0.381572i	$0.124617\pi$
$677$			19.8564i			0.763144i	−0.924339	+	0.381572i	$0.875383\pi$
$678$		0			0
$679$	10.3923	+	18.0000i	0.398820	+	0.690777i
$680$		0			0
$681$			32.9282i			1.26181i
$682$		0			0
$683$	−7.36603	+	12.7583i	−0.281853	+	0.488184i	−0.971841	−	0.235637i	$-0.924282\pi$
$683$	−7.36603	+	12.7583i	−0.281853	+	0.488184i	0.689988	+	0.723821i	$0.257616\pi$
$684$		0			0
$685$	−4.08142	−	2.35641i	−0.155943	−	0.0900337i
$686$		0			0
$687$	2.19615	−	1.26795i	0.0837884	−	0.0483753i
$688$		0			0
$689$	24.8205	−	25.7942i	0.945586	−	0.982682i
$690$		0			0
$691$	−9.00000	−	15.5885i	−0.342376	−	0.593013i	0.642497	−	0.766288i	$-0.277898\pi$
$691$	−9.00000	−	15.5885i	−0.342376	−	0.593013i	−0.984873	+	0.173275i	$0.944565\pi$
$692$		0			0
$693$	−26.7846	−	15.4641i	−1.01746	−	0.587433i
$694$		0			0
$695$	0.588457	+	0.339746i	0.0223215	+	0.0128873i
$696$		0			0
$697$		−	7.73205i		−	0.292872i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	32.7846i		−	1.23826i	−0.785289	−	0.619129i	$-0.787486\pi$
$701$		−	32.7846i		−	1.23826i	0.785289	−	0.619129i	$-0.212514\pi$
$702$		0			0
$703$	−36.5885			−1.37996
$704$		0			0
$705$	−1.19615	−	2.07180i	−0.0450497	−	0.0780284i
$706$		0			0
$707$	63.9615			2.40552
$708$		0			0
$709$	15.4019	−	26.6769i	0.578431	−	1.00187i	−0.417228	−	0.908802i	$-0.636998\pi$
$709$	15.4019	−	26.6769i	0.578431	−	1.00187i	0.995659	−	0.0930708i	$-0.0296683\pi$
$710$		0			0
$711$	−23.1962	+	40.1769i	−0.869924	+	1.50675i
$712$		0			0
$713$	2.41154	−	1.39230i	0.0903130	−	0.0521422i
$714$		0			0
$715$	0.464102	−	1.87564i	0.0173564	−	0.0701451i
$716$		0			0
$717$	−3.26795	−	5.66025i	−0.122044	−	0.211386i
$718$		0			0
$719$	10.8564	−	18.8038i	0.404876	−	0.701265i	−0.589431	−	0.807818i	$-0.700648\pi$
$719$	10.8564	−	18.8038i	0.404876	−	0.701265i	0.994307	+	0.106553i	$0.0339815\pi$
$720$		0			0
$721$	12.5885	+	7.26795i	0.468819	+	0.270673i
$722$		0			0
$723$	2.19615			0.0816758
$724$		0			0
$725$	12.8038	−	7.39230i	0.475523	−	0.274543i
$726$		0			0
$727$	34.3923			1.27554			0.637770	−	0.770227i	$-0.279857\pi$
$727$	34.3923			1.27554			0.637770	+	0.770227i	$0.279857\pi$
$728$		0			0
$729$	43.7846			1.62165
$730$		0			0
$731$	−52.9808	+	30.5885i	−1.95956	+	1.13135i
$732$		0			0
$733$	43.0526			1.59018			0.795091	−	0.606490i	$-0.207423\pi$
$733$	43.0526			1.59018			0.795091	+	0.606490i	$0.207423\pi$
$734$		0			0
$735$	−3.16987	−	1.83013i	−0.116923	−	0.0675053i
$736$		0			0
$737$	−10.7321	+	18.5885i	−0.395320	+	0.684715i
$738$		0			0
$739$	−2.53590	−	4.39230i	−0.0932845	−	0.161574i	0.815607	−	0.578607i	$-0.196403\pi$
$739$	−2.53590	−	4.39230i	−0.0932845	−	0.161574i	−0.908891	+	0.417033i	$0.863070\pi$
$740$		0			0
$741$	32.3205	−	33.5885i	1.18732	−	1.23390i
$742$		0			0
$743$	−1.26795	+	0.732051i	−0.0465165	+	0.0268563i	−0.523078	−	0.852285i	$-0.675216\pi$
$743$	−1.26795	+	0.732051i	−0.0465165	+	0.0268563i	0.476561	+	0.879141i	$0.341883\pi$
$744$		0			0
$745$	−1.10770	+	1.91858i	−0.0405828	+	0.0702915i
$746$		0			0
$747$	12.1962	−	21.1244i	0.446234	−	0.772900i
$748$		0			0
$749$	1.17691			0.0430035
$750$		0			0
$751$	−7.80385	−	13.5167i	−0.284766	−	0.493230i	0.687786	−	0.725913i	$-0.258583\pi$
$751$	−7.80385	−	13.5167i	−0.284766	−	0.493230i	−0.972553	+	0.232683i	$0.925249\pi$
$752$		0			0
$753$	−6.00000			−0.218652
$754$		0			0
$755$		−	2.19615i		−	0.0799262i
$756$		0			0
$757$	−35.6603	+	20.5885i	−1.29609	+	0.748300i	−0.979727	−	0.200337i	$-0.935796\pi$
$757$	−35.6603	+	20.5885i	−1.29609	+	0.748300i	−0.316367	+	0.948637i	$0.602463\pi$
$758$		0			0
$759$		−	12.0000i		−	0.435572i
$760$		0			0
$761$	16.8564	+	9.73205i	0.611044	+	0.352787i	0.773374	−	0.633950i	$-0.218568\pi$
$761$	16.8564	+	9.73205i	0.611044	+	0.352787i	−0.162330	+	0.986737i	$0.551901\pi$
$762$		0			0
$763$	18.0000	+	10.3923i	0.651644	+	0.376227i
$764$		0			0
$765$	3.86603	+	6.69615i	0.139776	+	0.242100i
$766$		0			0
$767$	−25.8564	+	7.46410i	−0.933621	+	0.269513i
$768$		0			0
$769$	−1.39230	+	0.803848i	−0.0502078	+	0.0289875i	−0.524894	−	0.851168i	$-0.675895\pi$
$769$	−1.39230	+	0.803848i	−0.0502078	+	0.0289875i	0.474686	+	0.880155i	$0.342562\pi$
$770$		0			0
$771$	−11.4904	−	6.63397i	−0.413816	−	0.238917i
$772$		0			0
$773$	−22.2487	+	38.5359i	−0.800231	+	1.38604i	0.119234	+	0.992866i	$0.461956\pi$
$773$	−22.2487	+	38.5359i	−0.800231	+	1.38604i	−0.919464	+	0.393174i	$0.871377\pi$
$774$		0			0
$775$		−	6.24871i		−	0.224460i
$776$		0			0
$777$	−36.5885	−	63.3731i	−1.31260	−	2.27350i
$778$		0			0
$779$		−	5.66025i		−	0.202800i
$780$		0			0
$781$		−	25.4641i		−	0.911177i
$782$		0			0
$783$	−6.00000	−	10.3923i	−0.214423	−	0.371391i
$784$		0			0
$785$			2.66025i			0.0949485i
$786$		0			0
$787$	18.6340	−	32.2750i	0.664229	−	1.15048i	−0.315264	−	0.949004i	$-0.602093\pi$
$787$	18.6340	−	32.2750i	0.664229	−	1.15048i	0.979494	−	0.201475i	$-0.0645735\pi$

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)

\(f(q)\)	\(=\)	\(q+(-2.36603 + 1.36603i) q^{3} +0.267949 q^{5} +(3.00000 + 1.73205i) q^{7} +(2.23205 - 3.86603i) q^{9} +O(q^{10})\)	\(q+(-2.36603 + 1.36603i) q^{3} +0.267949 q^{5} +(3.00000 + 1.73205i) q^{7} +(2.23205 - 3.86603i) q^{9} +(-1.00000 - 1.73205i) q^{11} +(2.59808 + 2.50000i) q^{13} +(-0.633975 + 0.366025i) q^{15} +(-3.23205 + 5.59808i) q^{17} +(-2.36603 + 4.09808i) q^{19} -9.46410 q^{21} +(-1.09808 - 1.90192i) q^{23} -4.92820 q^{25} +4.00000i q^{27} +(-2.59808 + 1.50000i) q^{29} +1.26795i q^{31} +(4.73205 + 2.73205i) q^{33} +(0.803848 + 0.464102i) q^{35} +(3.86603 + 6.69615i) q^{37} +(-9.56218 - 2.36603i) q^{39} +(-1.03590 + 0.598076i) q^{41} +(8.19615 + 4.73205i) q^{43} +(0.598076 - 1.03590i) q^{45} +3.26795i q^{47} +(2.50000 + 4.33013i) q^{49} -17.6603i q^{51} -9.92820i q^{53} +(-0.267949 - 0.464102i) q^{55} -12.9282i q^{57} +(-3.73205 + 6.46410i) q^{59} +(-0.866025 - 0.500000i) q^{61} +(13.3923 - 7.73205i) q^{63} +(0.696152 + 0.669873i) q^{65} +(-5.36603 - 9.29423i) q^{67} +(5.19615 + 3.00000i) q^{69} +(11.0263 + 6.36603i) q^{71} -1.73205i q^{73} +(11.6603 - 6.73205i) q^{75} -6.92820i q^{77} -10.3923 q^{79} +(1.23205 + 2.13397i) q^{81} +5.46410 q^{83} +(-0.866025 + 1.50000i) q^{85} +(4.09808 - 7.09808i) q^{87} +(-0.464102 + 0.267949i) q^{89} +(3.46410 + 12.0000i) q^{91} +(-1.73205 - 3.00000i) q^{93} +(-0.633975 + 1.09808i) q^{95} +(5.19615 + 3.00000i) q^{97} -8.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} + 8 q^{5} + 12 q^{7} + 2 q^{9}+O(q^{10}) \) 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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