LMFDB - Embedded newform 416.2.i.c.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(289,416)]

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, 0, 2]))

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

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

chi := DirichletCharacter("416.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.i (of order $3$, degree $2$, 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 $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	416.289
Dual form			416.2.i.c.321.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.20711 - 2.09077i) q^{3} +2.82843 q^{5} +(2.20711 - 3.82282i) q^{7} +(-1.41421 + 2.44949i) q^{9} +O(q^{10})$	\(q+(-1.20711 - 2.09077i) q^{3} +2.82843 q^{5} +(2.20711 - 3.82282i) q^{7} +(-1.41421 + 2.44949i) q^{9} +(1.62132 + 2.80821i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(-3.41421 - 5.91359i) q^{15} +(2.91421 - 5.04757i) q^{17} +(-0.621320 + 1.07616i) q^{19} -10.6569 q^{21} +(0.621320 + 1.07616i) q^{23} +3.00000 q^{25} -0.414214 q^{27} +(-4.32843 - 7.49706i) q^{29} -5.65685 q^{31} +(3.91421 - 6.77962i) q^{33} +(6.24264 - 10.8126i) q^{35} +(3.74264 + 6.48244i) q^{37} +(8.44975 - 2.09077i) q^{39} +(2.91421 + 5.04757i) q^{41} +(2.03553 - 3.52565i) q^{43} +(-4.00000 + 6.92820i) q^{45} -6.00000 q^{47} +(-6.24264 - 10.8126i) q^{49} -14.0711 q^{51} -2.82843 q^{53} +(4.58579 + 7.94282i) q^{55} +3.00000 q^{57} +(-0.621320 + 1.07616i) q^{59} +(-3.50000 + 6.06218i) q^{61} +(6.24264 + 10.8126i) q^{63} +(-2.82843 + 9.79796i) q^{65} +(6.62132 + 11.4685i) q^{67} +(1.50000 - 2.59808i) q^{69} +(-3.62132 + 6.27231i) q^{71} +12.4853 q^{73} +(-3.62132 - 6.27231i) q^{75} +14.3137 q^{77} +6.00000 q^{79} +(4.74264 + 8.21449i) q^{81} -4.00000 q^{83} +(8.24264 - 14.2767i) q^{85} +(-10.4497 + 18.0995i) q^{87} +(1.67157 + 2.89525i) q^{89} +(11.0355 + 11.4685i) q^{91} +(6.82843 + 11.8272i) q^{93} +(-1.75736 + 3.04384i) q^{95} +(4.50000 - 7.79423i) q^{97} -9.17157 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 6 q^{7}+O(q^{10}) $ 4 * q - 2 * q^3 + 6 * q^7	$ 4 q - 2 q^{3} + 6 q^{7} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 6 q^{19} - 20 q^{21} - 6 q^{23} + 12 q^{25} + 4 q^{27} - 6 q^{29} + 10 q^{33} + 8 q^{35} - 2 q^{37} + 14 q^{39} + 6 q^{41} - 6 q^{43} - 16 q^{45} - 24 q^{47} - 8 q^{49} - 28 q^{51} + 24 q^{55} + 12 q^{57} + 6 q^{59} - 14 q^{61} + 8 q^{63} + 18 q^{67} + 6 q^{69} - 6 q^{71} + 16 q^{73} - 6 q^{75} + 12 q^{77} + 24 q^{79} + 2 q^{81} - 16 q^{83} + 16 q^{85} - 22 q^{87} + 18 q^{89} + 30 q^{91} + 16 q^{93} - 24 q^{95} + 18 q^{97} - 48 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 6 * q^7 - 2 * q^11 - 4 * q^13 - 8 * q^15 + 6 * q^17 + 6 * q^19 - 20 * q^21 - 6 * q^23 + 12 * q^25 + 4 * q^27 - 6 * q^29 + 10 * q^33 + 8 * q^35 - 2 * q^37 + 14 * q^39 + 6 * q^41 - 6 * q^43 - 16 * q^45 - 24 * q^47 - 8 * q^49 - 28 * q^51 + 24 * q^55 + 12 * q^57 + 6 * q^59 - 14 * q^61 + 8 * q^63 + 18 * q^67 + 6 * q^69 - 6 * q^71 + 16 * q^73 - 6 * q^75 + 12 * q^77 + 24 * q^79 + 2 * q^81 - 16 * q^83 + 16 * q^85 - 22 * q^87 + 18 * q^89 + 30 * q^91 + 16 * q^93 - 24 * q^95 + 18 * q^97 - 48 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	−1.20711	−	2.09077i	−0.696923	−	1.20711i	−0.969528	−	0.244981i	$-0.921218\pi$
$3$	−1.20711	−	2.09077i	−0.696923	−	1.20711i	0.272605	−	0.962126i	$-0.412115\pi$
$4$		0			0
$5$	2.82843			1.26491			0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843			1.26491			0.632456	+	0.774597i	$0.282047\pi$
$6$		0			0
$7$	2.20711	−	3.82282i	0.834208	−	1.44489i	−0.0604657	−	0.998170i	$-0.519259\pi$
$7$	2.20711	−	3.82282i	0.834208	−	1.44489i	0.894674	−	0.446720i	$-0.147408\pi$
$8$		0			0
$9$	−1.41421	+	2.44949i	−0.471405	+	0.816497i
$10$		0			0
$11$	1.62132	+	2.80821i	0.488846	+	0.846707i	0.999918	−	0.0128314i	$-0.00408449\pi$
$11$	1.62132	+	2.80821i	0.488846	+	0.846707i	−0.511071	+	0.859538i	$0.670751\pi$
$12$		0			0
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$14$		0			0
$15$	−3.41421	−	5.91359i	−0.881546	−	1.52688i
$16$		0			0
$17$	2.91421	−	5.04757i	0.706801	−	1.22421i	−0.259237	−	0.965814i	$-0.583471\pi$
$17$	2.91421	−	5.04757i	0.706801	−	1.22421i	0.966038	−	0.258401i	$-0.0831955\pi$
$18$		0			0
$19$	−0.621320	+	1.07616i	−0.142541	+	0.246888i	−0.928453	−	0.371451i	$-0.878861\pi$
$19$	−0.621320	+	1.07616i	−0.142541	+	0.246888i	0.785912	+	0.618338i	$0.212194\pi$
$20$		0			0
$21$	−10.6569			−2.32552
$22$		0			0
$23$	0.621320	+	1.07616i	0.129554	+	0.224395i	0.923504	−	0.383589i	$-0.125312\pi$
$23$	0.621320	+	1.07616i	0.129554	+	0.224395i	−0.793950	+	0.607983i	$0.791979\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	−0.414214			−0.0797154
$28$		0			0
$29$	−4.32843	−	7.49706i	−0.803769	−	1.39217i	−0.917119	−	0.398613i	$-0.869492\pi$
$29$	−4.32843	−	7.49706i	−0.803769	−	1.39217i	0.113350	−	0.993555i	$-0.463842\pi$
$30$		0			0
$31$	−5.65685			−1.01600			−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685			−1.01600			−0.508001	+	0.861357i	$0.669615\pi$
$32$		0			0
$33$	3.91421	−	6.77962i	0.681377	−	1.18018i
$34$		0			0
$35$	6.24264	−	10.8126i	1.05520	−	1.82766i
$36$		0			0
$37$	3.74264	+	6.48244i	0.615286	+	1.06571i	0.990334	+	0.138702i	$0.0442929\pi$
$37$	3.74264	+	6.48244i	0.615286	+	1.06571i	−0.375048	+	0.927005i	$0.622374\pi$
$38$		0			0
$39$	8.44975	−	2.09077i	1.35304	−	0.334791i
$40$		0			0
$41$	2.91421	+	5.04757i	0.455124	+	0.788297i	0.998695	−	0.0510654i	$-0.0162617\pi$
$41$	2.91421	+	5.04757i	0.455124	+	0.788297i	−0.543572	+	0.839363i	$0.682928\pi$
$42$		0			0
$43$	2.03553	−	3.52565i	0.310416	−	0.537656i	−0.668036	−	0.744129i	$-0.732865\pi$
$43$	2.03553	−	3.52565i	0.310416	−	0.537656i	0.978452	+	0.206472i	$0.0661983\pi$
$44$		0			0
$45$	−4.00000	+	6.92820i	−0.596285	+	1.03280i
$46$		0			0
$47$	−6.00000			−0.875190			−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000			−0.875190			−0.437595	+	0.899172i	$0.644170\pi$
$48$		0			0
$49$	−6.24264	−	10.8126i	−0.891806	−	1.54465i
$50$		0			0
$51$	−14.0711			−1.97034
$52$		0			0
$53$	−2.82843			−0.388514			−0.194257	−	0.980951i	$-0.562230\pi$
$53$	−2.82843			−0.388514			−0.194257	+	0.980951i	$0.562230\pi$
$54$		0			0
$55$	4.58579	+	7.94282i	0.618347	+	1.07101i
$56$		0			0
$57$	3.00000			0.397360
$58$		0			0
$59$	−0.621320	+	1.07616i	−0.0808890	+	0.140104i	−0.903632	−	0.428310i	$-0.859109\pi$
$59$	−0.621320	+	1.07616i	−0.0808890	+	0.140104i	0.822743	+	0.568414i	$0.192443\pi$
$60$		0			0
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	−0.998264	−	0.0588933i	$-0.981243\pi$
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	0.550135	+	0.835076i	$0.314576\pi$
$62$		0			0
$63$	6.24264	+	10.8126i	0.786499	+	1.36226i
$64$		0			0
$65$	−2.82843	+	9.79796i	−0.350823	+	1.21529i
$66$		0			0
$67$	6.62132	+	11.4685i	0.808923	+	1.40110i	0.913610	+	0.406591i	$0.133282\pi$
$67$	6.62132	+	11.4685i	0.808923	+	1.40110i	−0.104687	+	0.994505i	$0.533384\pi$
$68$		0			0
$69$	1.50000	−	2.59808i	0.180579	−	0.312772i
$70$		0			0
$71$	−3.62132	+	6.27231i	−0.429772	+	0.744386i	−0.996853	−	0.0792756i	$-0.974739\pi$
$71$	−3.62132	+	6.27231i	−0.429772	+	0.744386i	0.567081	+	0.823662i	$0.308073\pi$
$72$		0			0
$73$	12.4853			1.46129			0.730646	−	0.682757i	$-0.239219\pi$
$73$	12.4853			1.46129			0.730646	+	0.682757i	$0.239219\pi$
$74$		0			0
$75$	−3.62132	−	6.27231i	−0.418154	−	0.724264i
$76$		0			0
$77$	14.3137			1.63120
$78$		0			0
$79$	6.00000			0.675053			0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000			0.675053			0.337526	+	0.941316i	$0.390410\pi$
$80$		0			0
$81$	4.74264	+	8.21449i	0.526960	+	0.912722i
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$	8.24264	−	14.2767i	0.894040	−	1.54852i
$86$		0			0
$87$	−10.4497	+	18.0995i	−1.12033	+	1.94047i
$88$		0			0
$89$	1.67157	+	2.89525i	0.177186	+	0.306896i	0.940916	−	0.338641i	$-0.109967\pi$
$89$	1.67157	+	2.89525i	0.177186	+	0.306896i	−0.763729	+	0.645537i	$0.776634\pi$
$90$		0			0
$91$	11.0355	+	11.4685i	1.15684	+	1.20222i
$92$		0			0
$93$	6.82843	+	11.8272i	0.708075	+	1.22642i
$94$		0			0
$95$	−1.75736	+	3.04384i	−0.180301	+	0.312291i
$96$		0			0
$97$	4.50000	−	7.79423i	0.456906	−	0.791384i	−0.541890	−	0.840450i	$-0.682291\pi$
$97$	4.50000	−	7.79423i	0.456906	−	0.791384i	0.998796	+	0.0490655i	$0.0156243\pi$
$98$		0			0
$99$	−9.17157			−0.921778
$100$		0			0
$101$	−5.91421	−	10.2437i	−0.588486	−	1.01929i	−0.994431	−	0.105390i	$-0.966391\pi$
$101$	−5.91421	−	10.2437i	−0.588486	−	1.01929i	0.405945	−	0.913898i	$-0.366943\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$	−30.1421			−2.94157
$106$		0			0
$107$	9.62132	+	16.6646i	0.930128	+	1.61103i	0.783099	+	0.621897i	$0.213638\pi$
$107$	9.62132	+	16.6646i	0.930128	+	1.61103i	0.147029	+	0.989132i	$0.453029\pi$
$108$		0			0
$109$	−8.48528			−0.812743			−0.406371	−	0.913708i	$-0.633206\pi$
$109$	−8.48528			−0.812743			−0.406371	+	0.913708i	$0.633206\pi$
$110$		0			0
$111$	9.03553	−	15.6500i	0.857615	−	1.48543i
$112$		0			0
$113$	0.0857864	−	0.148586i	0.00807011	−	0.0139778i	−0.861962	−	0.506973i	$-0.830765\pi$
$113$	0.0857864	−	0.148586i	0.00807011	−	0.0139778i	0.870032	+	0.492995i	$0.164098\pi$
$114$		0			0
$115$	1.75736	+	3.04384i	0.163875	+	0.283839i
$116$		0			0
$117$	−7.07107	−	7.34847i	−0.653720	−	0.679366i
$118$		0			0
$119$	−12.8640	−	22.2810i	−1.17924	−	2.04250i
$120$		0			0
$121$	0.242641	−	0.420266i	0.0220582	−	0.0382060i
$122$		0			0
$123$	7.03553	−	12.1859i	0.634373	−	1.09877i
$124$		0			0
$125$	−5.65685			−0.505964
$126$		0			0
$127$	0.792893	+	1.37333i	0.0703579	+	0.121863i	0.899058	−	0.437829i	$-0.144253\pi$
$127$	0.792893	+	1.37333i	0.0703579	+	0.121863i	−0.828700	+	0.559693i	$0.810919\pi$
$128$		0			0
$129$	−9.82843			−0.865345
$130$		0			0
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$		0			0
$133$	2.74264	+	4.75039i	0.237817	+	0.411911i
$134$		0			0
$135$	−1.17157			−0.100833
$136$		0			0
$137$	−4.67157	+	8.09140i	−0.399119	+	0.691295i	−0.993617	−	0.112802i	$-0.964017\pi$
$137$	−4.67157	+	8.09140i	−0.399119	+	0.691295i	0.594498	+	0.804097i	$0.297351\pi$
$138$		0			0
$139$	−0.621320	+	1.07616i	−0.0526997	+	0.0912786i	−0.891172	−	0.453666i	$-0.850116\pi$
$139$	−0.621320	+	1.07616i	−0.0526997	+	0.0912786i	0.838472	+	0.544944i	$0.183449\pi$
$140$		0			0
$141$	7.24264	+	12.5446i	0.609940	+	1.05645i
$142$		0			0
$143$	−11.3492	+	2.80821i	−0.949071	+	0.234834i
$144$		0			0
$145$	−12.2426	−	21.2049i	−1.01670	−	1.76097i
$146$		0			0
$147$	−15.0711	+	26.1039i	−1.24304	+	2.15301i
$148$		0			0
$149$	1.67157	−	2.89525i	0.136941	−	0.237188i	−0.789397	−	0.613884i	$-0.789606\pi$
$149$	1.67157	−	2.89525i	0.136941	−	0.237188i	0.926337	+	0.376696i	$0.122940\pi$
$150$		0			0
$151$	16.9706			1.38104			0.690522	−	0.723311i	$-0.257381\pi$
$151$	16.9706			1.38104			0.690522	+	0.723311i	$0.257381\pi$
$152$		0			0
$153$	8.24264	+	14.2767i	0.666378	+	1.15420i
$154$		0			0
$155$	−16.0000			−1.28515
$156$		0			0
$157$	−0.485281			−0.0387297			−0.0193648	−	0.999812i	$-0.506164\pi$
$157$	−0.485281			−0.0387297			−0.0193648	+	0.999812i	$0.506164\pi$
$158$		0			0
$159$	3.41421	+	5.91359i	0.270765	+	0.468978i
$160$		0			0
$161$	5.48528			0.432301
$162$		0			0
$163$	8.37868	−	14.5123i	0.656269	−	1.13669i	−0.325305	−	0.945609i	$-0.605467\pi$
$163$	8.37868	−	14.5123i	0.656269	−	1.13669i	0.981574	−	0.191082i	$-0.0611996\pi$
$164$		0			0
$165$	11.0711	−	19.1757i	0.861881	−	1.49282i
$166$		0			0
$167$	−7.86396	−	13.6208i	−0.608532	−	1.05401i	−0.991483	−	0.130239i	$-0.958426\pi$
$167$	−7.86396	−	13.6208i	−0.608532	−	1.05401i	0.382951	−	0.923769i	$-0.374908\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$		0			0
$171$	−1.75736	−	3.04384i	−0.134389	−	0.232768i
$172$		0			0
$173$	4.50000	−	7.79423i	0.342129	−	0.592584i	−0.642699	−	0.766119i	$-0.722185\pi$
$173$	4.50000	−	7.79423i	0.342129	−	0.592584i	0.984828	+	0.173534i	$0.0555188\pi$
$174$		0			0
$175$	6.62132	−	11.4685i	0.500525	−	0.866934i
$176$		0			0
$177$	3.00000			0.225494
$178$		0			0
$179$	−6.37868	−	11.0482i	−0.476765	−	0.825781i	0.522881	−	0.852406i	$-0.324857\pi$
$179$	−6.37868	−	11.0482i	−0.476765	−	0.825781i	−0.999645	+	0.0266249i	$0.991524\pi$
$180$		0			0
$181$	−0.485281			−0.0360707			−0.0180353	−	0.999837i	$-0.505741\pi$
$181$	−0.485281			−0.0360707			−0.0180353	+	0.999837i	$0.505741\pi$
$182$		0			0
$183$	16.8995			1.24925
$184$		0			0
$185$	10.5858	+	18.3351i	0.778282	+	1.34802i
$186$		0			0
$187$	18.8995			1.38207
$188$		0			0
$189$	−0.914214	+	1.58346i	−0.0664993	+	0.115180i
$190$		0			0
$191$	3.37868	−	5.85204i	0.244473	−	0.423439i	−0.717511	−	0.696548i	$-0.754718\pi$
$191$	3.37868	−	5.85204i	0.244473	−	0.423439i	0.961983	+	0.273109i	$0.0880518\pi$
$192$		0			0
$193$	9.74264	+	16.8747i	0.701291	+	1.21467i	0.968014	+	0.250898i	$0.0807258\pi$
$193$	9.74264	+	16.8747i	0.701291	+	1.21467i	−0.266723	+	0.963773i	$0.585941\pi$
$194$		0			0
$195$	23.8995	−	5.91359i	1.71148	−	0.423481i
$196$		0			0
$197$	−1.50000	−	2.59808i	−0.106871	−	0.185105i	0.807630	−	0.589689i	$-0.200750\pi$
$197$	−1.50000	−	2.59808i	−0.106871	−	0.185105i	−0.914501	+	0.404584i	$0.867416\pi$
$198$		0			0
$199$	−12.1066	+	20.9692i	−0.858214	+	1.48647i	0.0154165	+	0.999881i	$0.495093\pi$
$199$	−12.1066	+	20.9692i	−0.858214	+	1.48647i	−0.873631	+	0.486590i	$0.838241\pi$
$200$		0			0
$201$	15.9853	−	27.6873i	1.12751	−	1.95291i
$202$		0			0
$203$	−38.2132			−2.68204
$204$		0			0
$205$	8.24264	+	14.2767i	0.575691	+	0.997126i
$206$		0			0
$207$	−3.51472			−0.244290
$208$		0			0
$209$	−4.02944			−0.278722
$210$		0			0
$211$	6.79289	+	11.7656i	0.467642	+	0.809980i	0.999316	−	0.0369691i	$-0.0117703\pi$
$211$	6.79289	+	11.7656i	0.467642	+	0.809980i	−0.531674	+	0.846949i	$0.678437\pi$
$212$		0			0
$213$	17.4853			1.19807
$214$		0			0
$215$	5.75736	−	9.97204i	0.392649	−	0.680087i
$216$		0			0
$217$	−12.4853	+	21.6251i	−0.847556	+	1.46801i
$218$		0			0
$219$	−15.0711	−	26.1039i	−1.01841	−	1.76394i
$220$		0			0
$221$	14.5711	+	15.1427i	0.980156	+	1.01861i
$222$		0			0
$223$	3.44975	+	5.97514i	0.231012	+	0.400125i	0.958106	−	0.286413i	$-0.0924629\pi$
$223$	3.44975	+	5.97514i	0.231012	+	0.400125i	−0.727094	+	0.686538i	$0.759130\pi$
$224$		0			0
$225$	−4.24264	+	7.34847i	−0.282843	+	0.489898i
$226$		0			0
$227$	7.86396	−	13.6208i	0.521949	−	0.904043i	−0.477725	−	0.878510i	$-0.658538\pi$
$227$	7.86396	−	13.6208i	0.521949	−	0.904043i	0.999674	−	0.0255332i	$-0.00812837\pi$
$228$		0			0
$229$	8.48528			0.560723			0.280362	−	0.959894i	$-0.409546\pi$
$229$	8.48528			0.560723			0.280362	+	0.959894i	$0.409546\pi$
$230$		0			0
$231$	−17.2782	−	29.9267i	−1.13682	−	1.96903i
$232$		0			0
$233$	−20.4853			−1.34204			−0.671018	−	0.741441i	$-0.734143\pi$
$233$	−20.4853			−1.34204			−0.671018	+	0.741441i	$0.734143\pi$
$234$		0			0
$235$	−16.9706			−1.10704
$236$		0			0
$237$	−7.24264	−	12.5446i	−0.470460	−	0.814861i
$238$		0			0
$239$	8.97056			0.580257			0.290129	−	0.956988i	$-0.406302\pi$
$239$	8.97056			0.580257			0.290129	+	0.956988i	$0.406302\pi$
$240$		0			0
$241$	−15.2279	+	26.3755i	−0.980917	+	1.69900i	−0.322078	+	0.946713i	$0.604381\pi$
$241$	−15.2279	+	26.3755i	−0.980917	+	1.69900i	−0.658838	+	0.752285i	$0.728952\pi$
$242$		0			0
$243$	10.8284	−	18.7554i	0.694644	−	1.20316i
$244$		0			0
$245$	−17.6569	−	30.5826i	−1.12806	−	1.95385i
$246$		0			0
$247$	−3.10660	−	3.22848i	−0.197668	−	0.205423i
$248$		0			0
$249$	4.82843	+	8.36308i	0.305989	+	0.529989i
$250$		0			0
$251$	8.86396	−	15.3528i	0.559488	−	0.969062i	−0.438051	−	0.898950i	$-0.644331\pi$
$251$	8.86396	−	15.3528i	0.559488	−	0.969062i	0.997539	−	0.0701119i	$-0.0223356\pi$
$252$		0			0
$253$	−2.01472	+	3.48960i	−0.126664	+	0.219389i
$254$		0			0
$255$	−39.7990			−2.49231
$256$		0			0
$257$	−8.39949	−	14.5484i	−0.523946	−	0.907501i	−0.999611	−	0.0278749i	$-0.991126\pi$
$257$	−8.39949	−	14.5484i	−0.523946	−	0.907501i	0.475665	−	0.879626i	$-0.342207\pi$
$258$		0			0
$259$	33.0416			2.05311
$260$		0			0
$261$	24.4853			1.51560
$262$		0			0
$263$	4.62132	+	8.00436i	0.284963	+	0.493570i	0.972600	−	0.232484i	$-0.0746854\pi$
$263$	4.62132	+	8.00436i	0.284963	+	0.493570i	−0.687637	+	0.726054i	$0.741352\pi$
$264$		0			0
$265$	−8.00000			−0.491436
$266$		0			0
$267$	4.03553	−	6.98975i	0.246971	−	0.427766i
$268$		0			0
$269$	−2.74264	+	4.75039i	−0.167222	+	0.289637i	−0.937442	−	0.348141i	$-0.886813\pi$
$269$	−2.74264	+	4.75039i	−0.167222	+	0.289637i	0.770220	+	0.637778i	$0.220146\pi$
$270$		0			0
$271$	12.6213	+	21.8608i	0.766691	+	1.32795i	0.939348	+	0.342965i	$0.111431\pi$
$271$	12.6213	+	21.8608i	0.766691	+	1.32795i	−0.172658	+	0.984982i	$0.555235\pi$
$272$		0			0
$273$	10.6569	−	36.9164i	0.644982	−	2.23428i
$274$		0			0
$275$	4.86396	+	8.42463i	0.293308	+	0.508024i
$276$		0			0
$277$	−6.74264	+	11.6786i	−0.405126	+	0.701699i	−0.994336	−	0.106281i	$-0.966106\pi$
$277$	−6.74264	+	11.6786i	−0.405126	+	0.701699i	0.589210	+	0.807980i	$0.299439\pi$
$278$		0			0
$279$	8.00000	−	13.8564i	0.478947	−	0.829561i
$280$		0			0
$281$	−2.14214			−0.127789			−0.0638945	−	0.997957i	$-0.520352\pi$
$281$	−2.14214			−0.127789			−0.0638945	+	0.997957i	$0.520352\pi$
$282$		0			0
$283$	−5.20711	−	9.01897i	−0.309530	−	0.536122i	0.668729	−	0.743506i	$-0.266839\pi$
$283$	−5.20711	−	9.01897i	−0.309530	−	0.536122i	−0.978260	+	0.207384i	$0.933505\pi$
$284$		0			0
$285$	8.48528			0.502625
$286$		0			0
$287$	25.7279			1.51867
$288$		0			0
$289$	−8.48528	−	14.6969i	−0.499134	−	0.864526i
$290$		0			0
$291$	−21.7279			−1.27371
$292$		0			0
$293$	−1.15685	+	2.00373i	−0.0675841	+	0.117059i	−0.897837	−	0.440327i	$-0.854862\pi$
$293$	−1.15685	+	2.00373i	−0.0675841	+	0.117059i	0.830253	+	0.557386i	$0.188196\pi$
$294$		0			0
$295$	−1.75736	+	3.04384i	−0.102317	+	0.177219i
$296$		0			0
$297$	−0.671573	−	1.16320i	−0.0389686	−	0.0674956i
$298$		0			0
$299$	−4.34924	+	1.07616i	−0.251523	+	0.0622358i
$300$		0			0
$301$	−8.98528	−	15.5630i	−0.517903	−	0.897034i
$302$		0			0
$303$	−14.2782	+	24.7305i	−0.820260	+	1.42073i
$304$		0			0
$305$	−9.89949	+	17.1464i	−0.566843	+	0.981802i
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$	20.4853			1.15790			0.578948	−	0.815364i	$-0.303463\pi$
$313$	20.4853			1.15790			0.578948	+	0.815364i	$0.303463\pi$
$314$		0			0
$315$	17.6569	+	30.5826i	0.994851	+	1.72313i
$316$		0			0
$317$	−17.3137			−0.972435			−0.486217	−	0.873838i	$-0.661624\pi$
$317$	−17.3137			−0.972435			−0.486217	+	0.873838i	$0.661624\pi$
$318$		0			0
$319$	14.0355	−	24.3103i	0.785839	−	1.36111i
$320$		0			0
$321$	23.2279	−	40.2319i	1.29646	−	2.24553i
$322$		0			0
$323$	3.62132	+	6.27231i	0.201496	+	0.349001i
$324$		0			0
$325$	−3.00000	+	10.3923i	−0.166410	+	0.576461i
$326$		0			0
$327$	10.2426	+	17.7408i	0.566419	+	0.981067i
$328$		0			0
$329$	−13.2426	+	22.9369i	−0.730090	+	1.26455i
$330$		0			0
$331$	−3.62132	+	6.27231i	−0.199046	+	0.344757i	−0.948219	−	0.317616i	$-0.897118\pi$
$331$	−3.62132	+	6.27231i	−0.199046	+	0.344757i	0.749174	+	0.662374i	$0.230451\pi$
$332$		0			0
$333$	−21.1716			−1.16020
$334$		0			0
$335$	18.7279	+	32.4377i	1.02322	+	1.77226i
$336$		0			0
$337$	4.48528			0.244329			0.122164	−	0.992510i	$-0.461016\pi$
$337$	4.48528			0.244329			0.122164	+	0.992510i	$0.461016\pi$
$338$		0			0
$339$	−0.414214			−0.0224970
$340$		0			0
$341$	−9.17157	−	15.8856i	−0.496669	−	0.860255i
$342$		0			0
$343$	−24.2132			−1.30739
$344$		0			0
$345$	4.24264	−	7.34847i	0.228416	−	0.395628i
$346$		0			0
$347$	−15.6213	+	27.0569i	−0.838596	+	1.45249i	0.0524719	+	0.998622i	$0.483290\pi$
$347$	−15.6213	+	27.0569i	−0.838596	+	1.45249i	−0.891068	+	0.453869i	$0.850043\pi$
$348$		0			0
$349$	−17.2279	−	29.8396i	−0.922190	−	1.59728i	−0.796020	−	0.605270i	$-0.793065\pi$
$349$	−17.2279	−	29.8396i	−0.922190	−	1.59728i	−0.126170	−	0.992009i	$-0.540268\pi$
$350$		0			0
$351$	0.414214	−	1.43488i	0.0221091	−	0.0765881i
$352$		0			0
$353$	7.32843	+	12.6932i	0.390053	+	0.675591i	0.992456	−	0.122601i	$-0.0391234\pi$
$353$	7.32843	+	12.6932i	0.390053	+	0.675591i	−0.602403	+	0.798192i	$0.705790\pi$
$354$		0			0
$355$	−10.2426	+	17.7408i	−0.543623	+	0.941583i
$356$		0			0
$357$	−31.0563	+	53.7912i	−1.64368	+	2.84693i
$358$		0			0
$359$	8.00000			0.422224			0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000			0.422224			0.211112	+	0.977462i	$0.432292\pi$
$360$		0			0
$361$	8.72792	+	15.1172i	0.459364	+	0.795642i
$362$		0			0
$363$	−1.17157			−0.0614916
$364$		0			0
$365$	35.3137			1.84840
$366$		0			0
$367$	4.13604	+	7.16383i	0.215899	+	0.373949i	0.953550	−	0.301233i	$-0.0973983\pi$
$367$	4.13604	+	7.16383i	0.215899	+	0.373949i	−0.737651	+	0.675182i	$0.764065\pi$
$368$		0			0
$369$	−16.4853			−0.858189
$370$		0			0
$371$	−6.24264	+	10.8126i	−0.324102	+	0.561361i
$372$		0			0
$373$	−18.7426	+	32.4632i	−0.970457	+	1.68088i	−0.276280	+	0.961077i	$0.589102\pi$
$373$	−18.7426	+	32.4632i	−0.970457	+	1.68088i	−0.694178	+	0.719804i	$0.744232\pi$
$374$		0			0
$375$	6.82843	+	11.8272i	0.352618	+	0.610753i
$376$		0			0
$377$	30.2990	−	7.49706i	1.56048	−	0.386118i
$378$		0			0
$379$	−13.6924	−	23.7159i	−0.703331	−	1.21820i	−0.967291	−	0.253671i	$-0.918362\pi$
$379$	−13.6924	−	23.7159i	−0.703331	−	1.21820i	0.263960	−	0.964534i	$-0.414971\pi$
$380$		0			0
$381$	1.91421	−	3.31552i	0.0980681	−	0.169859i
$382$		0			0
$383$	7.86396	−	13.6208i	0.401830	−	0.695989i	−0.592117	−	0.805852i	$-0.701708\pi$
$383$	7.86396	−	13.6208i	0.401830	−	0.695989i	0.993947	+	0.109863i	$0.0350411\pi$
$384$		0			0
$385$	40.4853			2.06332
$386$		0			0
$387$	5.75736	+	9.97204i	0.292663	+	0.506907i
$388$		0			0
$389$	−16.6274			−0.843044			−0.421522	−	0.906818i	$-0.638504\pi$
$389$	−16.6274			−0.843044			−0.421522	+	0.906818i	$0.638504\pi$
$390$		0			0
$391$	7.24264			0.366276
$392$		0			0
$393$	−4.82843	−	8.36308i	−0.243562	−	0.421862i
$394$		0			0
$395$	16.9706			0.853882
$396$		0			0
$397$	5.74264	−	9.94655i	0.288215	−	0.499203i	−0.685169	−	0.728384i	$-0.740272\pi$
$397$	5.74264	−	9.94655i	0.288215	−	0.499203i	0.973384	+	0.229181i	$0.0736049\pi$
$398$		0			0
$399$	6.62132	−	11.4685i	0.331481	−	0.574141i
$400$		0			0
$401$	−5.57107	−	9.64937i	−0.278206	−	0.481867i	0.692733	−	0.721194i	$-0.256406\pi$
$401$	−5.57107	−	9.64937i	−0.278206	−	0.481867i	−0.970939	+	0.239327i	$0.923073\pi$
$402$		0			0
$403$	5.65685	−	19.5959i	0.281788	−	0.976142i
$404$		0			0
$405$	13.4142	+	23.2341i	0.666558	+	1.15451i
$406$		0			0
$407$	−12.1360	+	21.0202i	−0.601561	+	1.04193i
$408$		0			0
$409$	5.25736	−	9.10601i	0.259960	−	0.450263i	−0.706271	−	0.707941i	$-0.749624\pi$
$409$	5.25736	−	9.10601i	0.259960	−	0.450263i	0.966231	+	0.257678i	$0.0829574\pi$
$410$		0			0
$411$	22.5563			1.11262
$412$		0			0
$413$	2.74264	+	4.75039i	0.134957	+	0.233752i
$414$		0			0
$415$	−11.3137			−0.555368
$416$		0			0
$417$	3.00000			0.146911
$418$		0			0
$419$	−5.86396	−	10.1567i	−0.286473	−	0.496186i	0.686492	−	0.727137i	$-0.259150\pi$
$419$	−5.86396	−	10.1567i	−0.286473	−	0.496186i	−0.972965	+	0.230951i	$0.925816\pi$
$420$		0			0
$421$	0.485281			0.0236512			0.0118256	−	0.999930i	$-0.496236\pi$
$421$	0.485281			0.0236512			0.0118256	+	0.999930i	$0.496236\pi$
$422$		0			0
$423$	8.48528	−	14.6969i	0.412568	−	0.714590i
$424$		0			0
$425$	8.74264	−	15.1427i	0.424080	−	0.734529i
$426$		0			0
$427$	15.4497	+	26.7597i	0.747666	+	1.29499i
$428$		0			0
$429$	19.5711	+	20.3389i	0.944900	+	0.981969i
$430$		0			0
$431$	−17.3492	−	30.0498i	−0.835684	−	1.44745i	−0.893473	−	0.449118i	$-0.851738\pi$
$431$	−17.3492	−	30.0498i	−0.835684	−	1.44745i	0.0577890	−	0.998329i	$-0.481595\pi$
$432$		0			0
$433$	1.25736	−	2.17781i	0.0604248	−	0.104659i	−0.834231	−	0.551416i	$-0.814088\pi$
$433$	1.25736	−	2.17781i	0.0604248	−	0.104659i	0.894655	+	0.446757i	$0.147421\pi$
$434$		0			0
$435$	−29.5563	+	51.1931i	−1.41712	+	2.45452i
$436$		0			0
$437$	−1.54416			−0.0738670
$438$		0			0
$439$	−14.0355	−	24.3103i	−0.669879	−	1.16027i	−0.977937	−	0.208898i	$-0.933012\pi$
$439$	−14.0355	−	24.3103i	−0.669879	−	1.16027i	0.308058	−	0.951368i	$-0.400321\pi$
$440$		0			0
$441$	35.3137			1.68161
$442$		0			0
$443$	28.9706			1.37643			0.688216	−	0.725505i	$-0.258394\pi$
$443$	28.9706			1.37643			0.688216	+	0.725505i	$0.258394\pi$
$444$		0			0
$445$	4.72792	+	8.18900i	0.224125	+	0.388196i
$446$		0			0
$447$	−8.07107			−0.381748
$448$		0			0
$449$	−6.25736	+	10.8381i	−0.295303	+	0.511480i	−0.975055	−	0.221962i	$-0.928754\pi$
$449$	−6.25736	+	10.8381i	−0.295303	+	0.511480i	0.679752	+	0.733442i	$0.262087\pi$
$450$		0			0
$451$	−9.44975	+	16.3674i	−0.444971	+	0.770713i
$452$		0			0
$453$	−20.4853	−	35.4815i	−0.962482	−	1.66707i
$454$		0			0
$455$	31.2132	+	32.4377i	1.46330	+	1.52070i
$456$		0			0
$457$	−15.5000	−	26.8468i	−0.725059	−	1.25584i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−15.5000	−	26.8468i	−0.725059	−	1.25584i	0.233890	−	0.972263i	$-0.424854\pi$
$458$		0			0
$459$	−1.20711	+	2.09077i	−0.0563429	+	0.0975888i
$460$		0			0
$461$	−0.600505	+	1.04011i	−0.0279683	+	0.0484425i	−0.879671	−	0.475583i	$-0.842237\pi$
$461$	−0.600505	+	1.04011i	−0.0279683	+	0.0484425i	0.851702	+	0.524026i	$0.175570\pi$
$462$		0			0
$463$	−18.3431			−0.852478			−0.426239	−	0.904611i	$-0.640162\pi$
$463$	−18.3431			−0.852478			−0.426239	+	0.904611i	$0.640162\pi$
$464$		0			0
$465$	19.3137	+	33.4523i	0.895652	+	1.55131i
$466$		0			0
$467$	−6.97056			−0.322559			−0.161280	−	0.986909i	$-0.551562\pi$
$467$	−6.97056			−0.322559			−0.161280	+	0.986909i	$0.551562\pi$
$468$		0			0
$469$	58.4558			2.69924
$470$		0			0
$471$	0.585786	+	1.01461i	0.0269916	+	0.0467508i
$472$		0			0
$473$	13.2010			0.606983
$474$		0			0
$475$	−1.86396	+	3.22848i	−0.0855244	+	0.148133i
$476$		0			0
$477$	4.00000	−	6.92820i	0.183147	−	0.317221i
$478$		0			0
$479$	−4.37868	−	7.58410i	−0.200067	−	0.346526i	0.748483	−	0.663154i	$-0.230783\pi$
$479$	−4.37868	−	7.58410i	−0.200067	−	0.346526i	−0.948550	+	0.316628i	$0.897449\pi$
$480$		0			0
$481$	−26.1985	+	6.48244i	−1.19455	+	0.295574i
$482$		0			0
$483$	−6.62132	−	11.4685i	−0.301281	−	0.521833i
$484$		0			0
$485$	12.7279	−	22.0454i	0.577945	−	1.00103i
$486$		0			0
$487$	−6.44975	+	11.1713i	−0.292266	+	0.506219i	−0.974345	−	0.225059i	$-0.927743\pi$
$487$	−6.44975	+	11.1713i	−0.292266	+	0.506219i	0.682079	+	0.731278i	$0.261076\pi$
$488$		0			0
$489$	−40.4558			−1.82948
$490$		0			0
$491$	11.1066	+	19.2372i	0.501234	+	0.868163i	0.999999	+	0.00142539i	$0.000453717\pi$
$491$	11.1066	+	19.2372i	0.501234	+	0.868163i	−0.498765	+	0.866737i	$0.666213\pi$
$492$		0			0
$493$	−50.4558			−2.27242
$494$		0			0
$495$	−25.9411			−1.16597
$496$		0			0
$497$	15.9853	+	27.6873i	0.717038	+	1.24195i
$498$		0			0
$499$	17.6569			0.790429			0.395215	−	0.918589i	$-0.370670\pi$
$499$	17.6569			0.790429			0.395215	+	0.918589i	$0.370670\pi$
$500$		0			0
$501$	−18.9853	+	32.8835i	−0.848200	+	1.46913i
$502$		0			0
$503$	7.37868	−	12.7802i	0.328999	−	0.569843i	−0.653314	−	0.757087i	$-0.726622\pi$
$503$	7.37868	−	12.7802i	0.328999	−	0.569843i	0.982314	+	0.187244i	$0.0599554\pi$
$504$		0			0
$505$	−16.7279	−	28.9736i	−0.744383	−	1.28931i
$506$		0			0
$507$	−1.20711	+	31.3616i	−0.0536095	+	1.39282i
$508$		0			0
$509$	2.57107	+	4.45322i	0.113961	+	0.197386i	0.917364	−	0.398050i	$-0.130313\pi$
$509$	2.57107	+	4.45322i	0.113961	+	0.197386i	−0.803403	+	0.595435i	$0.796980\pi$
$510$		0			0
$511$	27.5563	−	47.7290i	1.21902	−	2.11141i
$512$		0			0
$513$	0.257359	−	0.445759i	0.0113627	−	0.0196808i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−9.72792	−	16.8493i	−0.427834	−	0.741029i
$518$		0			0
$519$	−21.7279			−0.953750
$520$		0			0
$521$	−9.17157			−0.401814			−0.200907	−	0.979610i	$-0.564389\pi$
$521$	−9.17157			−0.401814			−0.200907	+	0.979610i	$0.564389\pi$
$522$		0			0
$523$	−8.20711	−	14.2151i	−0.358872	−	0.621584i	0.628901	−	0.777485i	$-0.283505\pi$
$523$	−8.20711	−	14.2151i	−0.358872	−	0.621584i	−0.987773	+	0.155901i	$0.950172\pi$
$524$		0			0
$525$	−31.9706			−1.39531
$526$		0			0
$527$	−16.4853	+	28.5533i	−0.718110	+	1.24380i
$528$		0			0
$529$	10.7279	−	18.5813i	0.466431	−	0.807883i
$530$		0			0
$531$	−1.75736	−	3.04384i	−0.0762629	−	0.132091i
$532$		0			0
$533$	−20.3995	+	5.04757i	−0.883600	+	0.218634i
$534$		0			0
$535$	27.2132	+	47.1347i	1.17653	+	2.03781i
$536$		0			0
$537$	−15.3995	+	26.6727i	−0.664537	+	1.15101i
$538$		0			0
$539$	20.2426	−	35.0613i	0.871912	−	1.51020i
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$	0.585786	+	1.01461i	0.0251385	+	0.0435412i
$544$		0			0
$545$	−24.0000			−1.02805
$546$		0			0
$547$	−36.0000			−1.53925			−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000			−1.53925			−0.769624	+	0.638497i	$0.779557\pi$
$548$		0			0
$549$	−9.89949	−	17.1464i	−0.422500	−	0.731792i
$550$		0			0
$551$	10.7574			0.458279
$552$		0			0
$553$	13.2426	−	22.9369i	0.563134	−	0.975377i
$554$		0			0
$555$	25.5563	−	44.2649i	1.08481	−	1.87894i
$556$		0			0
$557$	−4.32843	−	7.49706i	−0.183401	−	0.317660i	0.759635	−	0.650349i	$-0.225377\pi$
$557$	−4.32843	−	7.49706i	−0.183401	−	0.317660i	−0.943037	+	0.332689i	$0.892044\pi$
$558$		0			0
$559$	10.1777	+	10.5769i	0.430470	+	0.447357i
$560$		0			0
$561$	−22.8137	−	39.5145i	−0.963196	−	1.66830i
$562$		0			0
$563$	16.3492	−	28.3177i	0.689038	−	1.19345i	−0.283111	−	0.959087i	$-0.591366\pi$
$563$	16.3492	−	28.3177i	0.689038	−	1.19345i	0.972149	−	0.234362i	$-0.0753002\pi$
$564$		0			0
$565$	0.242641	−	0.420266i	0.0102080	−	0.0176807i
$566$		0			0
$567$	41.8701			1.75838
$568$		0			0
$569$	12.9853	+	22.4912i	0.544371	+	0.942879i	0.998646	+	0.0520172i	$0.0165651\pi$
$569$	12.9853	+	22.4912i	0.544371	+	0.942879i	−0.454275	+	0.890862i	$0.650102\pi$
$570$		0			0
$571$	−40.2843			−1.68584			−0.842922	−	0.538036i	$-0.819167\pi$
$571$	−40.2843			−1.68584			−0.842922	+	0.538036i	$0.819167\pi$
$572$		0			0
$573$	−16.3137			−0.681515
$574$		0			0
$575$	1.86396	+	3.22848i	0.0777325	+	0.134637i
$576$		0			0
$577$	4.48528			0.186725			0.0933624	−	0.995632i	$-0.470238\pi$
$577$	4.48528			0.186725			0.0933624	+	0.995632i	$0.470238\pi$
$578$		0			0
$579$	23.5208	−	40.7392i	0.977492	−	1.69307i
$580$		0			0
$581$	−8.82843	+	15.2913i	−0.366265	+	0.634389i
$582$		0			0
$583$	−4.58579	−	7.94282i	−0.189924	−	0.328958i
$584$		0			0
$585$	−20.0000	−	20.7846i	−0.826898	−	0.859338i
$586$		0			0
$587$	−10.3492	−	17.9254i	−0.427159	−	0.739861i	0.569460	−	0.822019i	$-0.307152\pi$
$587$	−10.3492	−	17.9254i	−0.427159	−	0.739861i	−0.996619	+	0.0821578i	$0.973819\pi$
$588$		0			0
$589$	3.51472	−	6.08767i	0.144821	−	0.250838i
$590$		0			0
$591$	−3.62132	+	6.27231i	−0.148961	+	0.258008i
$592$		0			0
$593$	14.8284			0.608931			0.304465	−	0.952523i	$-0.401522\pi$
$593$	14.8284			0.608931			0.304465	+	0.952523i	$0.401522\pi$
$594$		0			0
$595$	−36.3848	−	63.0203i	−1.49163	−	2.58358i
$596$		0			0
$597$	58.4558			2.39244
$598$		0			0
$599$	−47.9411			−1.95882			−0.979411	−	0.201878i	$-0.935295\pi$
$599$	−47.9411			−1.95882			−0.979411	+	0.201878i	$0.935295\pi$
$600$		0			0
$601$	5.25736	+	9.10601i	0.214452	+	0.371442i	0.953103	−	0.302646i	$-0.0978701\pi$
$601$	5.25736	+	9.10601i	0.214452	+	0.371442i	−0.738651	+	0.674088i	$0.764537\pi$
$602$		0			0
$603$	−37.4558			−1.52532
$604$		0			0
$605$	0.686292	−	1.18869i	0.0279017	−	0.0483272i
$606$		0			0
$607$	−0.792893	+	1.37333i	−0.0321825	+	0.0557418i	−0.881668	−	0.471870i	$-0.843579\pi$
$607$	−0.792893	+	1.37333i	−0.0321825	+	0.0557418i	0.849486	+	0.527612i	$0.176912\pi$
$608$		0			0
$609$	46.1274	+	79.8950i	1.86918	+	3.23751i
$610$		0			0
$611$	6.00000	−	20.7846i	0.242734	−	0.840855i
$612$		0			0
$613$	7.25736	+	12.5701i	0.293122	+	0.507702i	0.974546	−	0.224186i	$-0.0719725\pi$
$613$	7.25736	+	12.5701i	0.293122	+	0.507702i	−0.681424	+	0.731889i	$0.738639\pi$
$614$		0			0
$615$	19.8995	−	34.4669i	0.802425	−	1.38984i
$616$		0			0
$617$	22.1569	−	38.3768i	0.892001	−	1.54499i	0.0545289	−	0.998512i	$-0.482634\pi$
$617$	22.1569	−	38.3768i	0.892001	−	1.54499i	0.837472	−	0.546479i	$-0.184032\pi$
$618$		0			0
$619$	4.97056			0.199784			0.0998919	−	0.994998i	$-0.468150\pi$
$619$	4.97056			0.199784			0.0998919	+	0.994998i	$0.468150\pi$
$620$		0			0
$621$	−0.257359	−	0.445759i	−0.0103275	−	0.0178877i
$622$		0			0
$623$	14.7574			0.591241
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	4.86396	+	8.42463i	0.194248	+	0.336447i
$628$		0			0
$629$	43.6274			1.73954
$630$		0			0
$631$	−3.62132	+	6.27231i	−0.144162	+	0.249697i	−0.929060	−	0.369929i	$-0.879382\pi$
$631$	−3.62132	+	6.27231i	−0.144162	+	0.249697i	0.784898	+	0.619625i	$0.212715\pi$
$632$		0			0
$633$	16.3995	−	28.4048i	0.651821	−	1.12899i
$634$		0			0
$635$	2.24264	+	3.88437i	0.0889965	+	0.154146i
$636$		0			0
$637$	43.6985	−	10.8126i	1.73140	−	0.428410i
$638$		0			0
$639$	−10.2426	−	17.7408i	−0.405193	−	0.701814i
$640$		0			0
$641$	11.3995	−	19.7445i	0.450253	−	0.779861i	−0.548148	−	0.836381i	$-0.684667\pi$
$641$	11.3995	−	19.7445i	0.450253	−	0.779861i	0.998401	+	0.0565200i	$0.0180005\pi$
$642$		0			0
$643$	−0.106602	+	0.184640i	−0.00420396	+	0.00728147i	−0.868120	−	0.496355i	$-0.834671\pi$
$643$	−0.106602	+	0.184640i	−0.00420396	+	0.00728147i	0.863916	+	0.503636i	$0.168005\pi$
$644$		0			0
$645$	−27.7990			−1.09458
$646$		0			0
$647$	11.6213	+	20.1287i	0.456881	+	0.791342i	0.998794	−	0.0490931i	$-0.0156331\pi$
$647$	11.6213	+	20.1287i	0.456881	+	0.791342i	−0.541913	+	0.840435i	$0.682300\pi$
$648$		0			0
$649$	−4.02944			−0.158169
$650$		0			0
$651$	60.2843			2.36273
$652$		0			0
$653$	−11.5711	−	20.0417i	−0.452811	−	0.784291i	0.545749	−	0.837949i	$-0.316245\pi$
$653$	−11.5711	−	20.0417i	−0.452811	−	0.784291i	−0.998559	+	0.0536576i	$0.982912\pi$
$654$		0			0
$655$	11.3137			0.442063
$656$		0			0
$657$	−17.6569	+	30.5826i	−0.688859	+	1.19314i
$658$		0			0
$659$	−12.1066	+	20.9692i	−0.471606	+	0.816846i	−0.999472	−	0.0324816i	$-0.989659\pi$
$659$	−12.1066	+	20.9692i	−0.471606	+	0.816846i	0.527866	+	0.849328i	$0.322992\pi$
$660$		0			0
$661$	12.2279	+	21.1794i	0.475611	+	0.823782i	0.999610	−	0.0279366i	$-0.00889365\pi$
$661$	12.2279	+	21.1794i	0.475611	+	0.823782i	−0.523999	+	0.851719i	$0.675560\pi$
$662$		0			0
$663$	14.0711	−	48.7436i	0.546475	−	1.89304i
$664$		0			0
$665$	7.75736	+	13.4361i	0.300817	+	0.521031i
$666$		0			0
$667$	5.37868	−	9.31615i	0.208263	−	0.360723i
$668$		0			0
$669$	8.32843	−	14.4253i	0.321996	−	0.557713i
$670$		0			0
$671$	−22.6985			−0.876265
$672$		0			0
$673$	4.01472	+	6.95370i	0.154756	+	0.268045i	0.932970	−	0.359954i	$-0.117208\pi$
$673$	4.01472	+	6.95370i	0.154756	+	0.268045i	−0.778214	+	0.627999i	$0.783874\pi$
$674$		0			0
$675$	−1.24264			−0.0478293
$676$		0			0
$677$	−25.4558			−0.978348			−0.489174	−	0.872186i	$-0.662702\pi$
$677$	−25.4558			−0.978348			−0.489174	+	0.872186i	$0.662702\pi$
$678$		0			0
$679$	−19.8640	−	34.4054i	−0.762309	−	1.32036i
$680$		0			0
$681$	−37.9706			−1.45504
$682$		0			0
$683$	−4.62132	+	8.00436i	−0.176830	+	0.306278i	−0.940793	−	0.338982i	$-0.889918\pi$
$683$	−4.62132	+	8.00436i	−0.176830	+	0.306278i	0.763963	+	0.645260i	$0.223251\pi$
$684$		0			0
$685$	−13.2132	+	22.8859i	−0.504851	+	0.874427i
$686$		0			0
$687$	−10.2426	−	17.7408i	−0.390781	−	0.676853i
$688$		0			0
$689$	2.82843	−	9.79796i	0.107754	−	0.373273i
$690$		0			0
$691$	−8.03553	−	13.9180i	−0.305686	−	0.529464i	0.671728	−	0.740798i	$-0.265552\pi$
$691$	−8.03553	−	13.9180i	−0.305686	−	0.529464i	−0.977414	+	0.211334i	$0.932219\pi$
$692$		0			0
$693$	−20.2426	+	35.0613i	−0.768954	+	1.33187i
$694$		0			0
$695$	−1.75736	+	3.04384i	−0.0666604	+	0.115459i
$696$		0			0
$697$	33.9706			1.28673
$698$		0			0
$699$	24.7279	+	42.8300i	0.935296	+	1.61998i
$700$		0			0
$701$	−8.48528			−0.320485			−0.160242	−	0.987078i	$-0.551228\pi$
$701$	−8.48528			−0.320485			−0.160242	+	0.987078i	$0.551228\pi$
$702$		0			0
$703$	−9.30152			−0.350813
$704$		0			0
$705$	20.4853	+	35.4815i	0.771520	+	1.33631i
$706$		0			0
$707$	−52.2132			−1.96368
$708$		0			0
$709$	9.74264	−	16.8747i	0.365893	−	0.633744i	−0.623026	−	0.782201i	$-0.714097\pi$
$709$	9.74264	−	16.8747i	0.365893	−	0.633744i	0.988919	+	0.148456i	$0.0474304\pi$
$710$		0			0
$711$	−8.48528	+	14.6969i	−0.318223	+	0.551178i
$712$		0			0
$713$	−3.51472	−	6.08767i	−0.131627	−	0.227985i
$714$		0			0
$715$	−32.1005	+	7.94282i	−1.20049	+	0.297044i
$716$		0			0
$717$	−10.8284	−	18.7554i	−0.404395	−	0.700433i
$718$		0			0
$719$	11.3787	−	19.7085i	0.424353	−	0.735001i	−0.572007	−	0.820249i	$-0.693835\pi$
$719$	11.3787	−	19.7085i	0.424353	−	0.735001i	0.996360	+	0.0852478i	$0.0271682\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	73.5269			2.73450
$724$		0			0
$725$	−12.9853	−	22.4912i	−0.482261	−	0.835301i
$726$		0			0
$727$	35.3137			1.30971			0.654856	−	0.755753i	$-0.272729\pi$
$727$	35.3137			1.30971			0.654856	+	0.755753i	$0.272729\pi$
$728$		0			0
$729$	−23.8284			−0.882534
$730$		0			0
$731$	−11.8640	−	20.5490i	−0.438804	−	0.760032i
$732$		0			0
$733$	43.9411			1.62300			0.811501	−	0.584351i	$-0.198651\pi$
$733$	43.9411			1.62300			0.811501	+	0.584351i	$0.198651\pi$
$734$		0			0
$735$	−42.6274	+	73.8329i	−1.57234	+	2.72337i
$736$		0			0
$737$	−21.4706	+	37.1881i	−0.790878	+	1.36984i
$738$		0			0
$739$	15.1066	+	26.1654i	0.555705	+	0.962510i	0.997848	+	0.0655653i	$0.0208851\pi$
$739$	15.1066	+	26.1654i	0.555705	+	0.962510i	−0.442143	+	0.896945i	$0.645782\pi$
$740$		0			0
$741$	−3.00000	+	10.3923i	−0.110208	+	0.381771i
$742$		0			0
$743$	3.62132	+	6.27231i	0.132853	+	0.230109i	0.924775	−	0.380513i	$-0.124253\pi$
$743$	3.62132	+	6.27231i	0.132853	+	0.230109i	−0.791922	+	0.610622i	$0.790919\pi$
$744$		0			0
$745$	4.72792	−	8.18900i	0.173218	−	0.300022i
$746$		0			0
$747$	5.65685	−	9.79796i	0.206973	−	0.358489i
$748$		0			0
$749$	84.9411			3.10368
$750$		0			0
$751$	−8.55025	−	14.8095i	−0.312003	−	0.540405i	0.666793	−	0.745243i	$-0.267667\pi$
$751$	−8.55025	−	14.8095i	−0.312003	−	0.540405i	−0.978796	+	0.204838i	$0.934333\pi$
$752$		0			0
$753$	−42.7990			−1.55968
$754$		0			0
$755$	48.0000			1.74690
$756$		0			0
$757$	11.7426	+	20.3389i	0.426794	+	0.739228i	0.996586	−	0.0825605i	$-0.0263098\pi$
$757$	11.7426	+	20.3389i	0.426794	+	0.739228i	−0.569793	+	0.821789i	$0.692976\pi$
$758$		0			0
$759$	9.72792			0.353101
$760$		0			0
$761$	−14.7426	+	25.5350i	−0.534420	+	0.925643i	0.464771	+	0.885431i	$0.346137\pi$
$761$	−14.7426	+	25.5350i	−0.534420	+	0.925643i	−0.999191	+	0.0402121i	$0.987197\pi$
$762$		0			0
$763$	−18.7279	+	32.4377i	−0.677996	+	1.17432i
$764$		0			0
$765$	23.3137	+	40.3805i	0.842909	+	1.45996i
$766$		0			0
$767$	−3.10660	−	3.22848i	−0.112173	−	0.116573i
$768$		0			0
$769$	22.7132	+	39.3404i	0.819059	+	1.41865i	0.906376	+	0.422471i	$0.138837\pi$
$769$	22.7132	+	39.3404i	0.819059	+	1.41865i	−0.0873172	+	0.996181i	$0.527829\pi$
$770$		0			0
$771$	−20.2782	+	35.1228i	−0.730301	+	1.26492i
$772$		0			0
$773$	5.74264	−	9.94655i	0.206548	−	0.357752i	−0.744077	−	0.668094i	$-0.767110\pi$
$773$	5.74264	−	9.94655i	0.206548	−	0.357752i	0.950625	+	0.310342i	$0.100444\pi$
$774$		0			0
$775$	−16.9706			−0.609601
$776$		0			0
$777$	−39.8848	−	69.0825i	−1.43086	−	2.47832i
$778$		0			0
$779$	−7.24264			−0.259495
$780$		0			0
$781$	−23.4853			−0.840369
$782$		0			0
$783$	1.79289	+	3.10538i	0.0640728	+	0.110977i
$784$		0			0
$785$	−1.37258			−0.0489896
$786$		0			0
$787$	19.0061	−	32.9195i	0.677494	−	1.17345i	−0.298239	−	0.954491i	$-0.596399\pi$
$787$	19.0061	−	32.9195i	0.677494	−	1.17345i	0.975733	−	0.218963i	$-0.0702674\pi$
$788$		0			0
$789$	11.1569	−	19.3242i	0.397195	−	0.687961i
$790$		0

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.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.20711 - 2.09077i) q^{3} +2.82843 q^{5} +(2.20711 - 3.82282i) q^{7} +(-1.41421 + 2.44949i) q^{9} +O(q^{10})\)	\(q+(-1.20711 - 2.09077i) q^{3} +2.82843 q^{5} +(2.20711 - 3.82282i) q^{7} +(-1.41421 + 2.44949i) q^{9} +(1.62132 + 2.80821i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(-3.41421 - 5.91359i) q^{15} +(2.91421 - 5.04757i) q^{17} +(-0.621320 + 1.07616i) q^{19} -10.6569 q^{21} +(0.621320 + 1.07616i) q^{23} +3.00000 q^{25} -0.414214 q^{27} +(-4.32843 - 7.49706i) q^{29} -5.65685 q^{31} +(3.91421 - 6.77962i) q^{33} +(6.24264 - 10.8126i) q^{35} +(3.74264 + 6.48244i) q^{37} +(8.44975 - 2.09077i) q^{39} +(2.91421 + 5.04757i) q^{41} +(2.03553 - 3.52565i) q^{43} +(-4.00000 + 6.92820i) q^{45} -6.00000 q^{47} +(-6.24264 - 10.8126i) q^{49} -14.0711 q^{51} -2.82843 q^{53} +(4.58579 + 7.94282i) q^{55} +3.00000 q^{57} +(-0.621320 + 1.07616i) q^{59} +(-3.50000 + 6.06218i) q^{61} +(6.24264 + 10.8126i) q^{63} +(-2.82843 + 9.79796i) q^{65} +(6.62132 + 11.4685i) q^{67} +(1.50000 - 2.59808i) q^{69} +(-3.62132 + 6.27231i) q^{71} +12.4853 q^{73} +(-3.62132 - 6.27231i) q^{75} +14.3137 q^{77} +6.00000 q^{79} +(4.74264 + 8.21449i) q^{81} -4.00000 q^{83} +(8.24264 - 14.2767i) q^{85} +(-10.4497 + 18.0995i) q^{87} +(1.67157 + 2.89525i) q^{89} +(11.0355 + 11.4685i) q^{91} +(6.82843 + 11.8272i) q^{93} +(-1.75736 + 3.04384i) q^{95} +(4.50000 - 7.79423i) q^{97} -9.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 6 q^{7}+O(q^{10}) \) 4 * q - 2 * q^3 + 6 * q^7	\( 4 q - 2 q^{3} + 6 q^{7} - 2 q^{11} - 4 q^{13} - 8 q^{15} + 6 q^{17} + 6 q^{19} - 20 q^{21} - 6 q^{23} + 12 q^{25} + 4 q^{27} - 6 q^{29} + 10 q^{33} + 8 q^{35} - 2 q^{37} + 14 q^{39} + 6 q^{41} - 6 q^{43} - 16 q^{45} - 24 q^{47} - 8 q^{49} - 28 q^{51} + 24 q^{55} + 12 q^{57} + 6 q^{59} - 14 q^{61} + 8 q^{63} + 18 q^{67} + 6 q^{69} - 6 q^{71} + 16 q^{73} - 6 q^{75} + 12 q^{77} + 24 q^{79} + 2 q^{81} - 16 q^{83} + 16 q^{85} - 22 q^{87} + 18 q^{89} + 30 q^{91} + 16 q^{93} - 24 q^{95} + 18 q^{97} - 48 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 6 * q^7 - 2 * q^11 - 4 * q^13 - 8 * q^15 + 6 * q^17 + 6 * q^19 - 20 * q^21 - 6 * q^23 + 12 * q^25 + 4 * q^27 - 6 * q^29 + 10 * q^33 + 8 * q^35 - 2 * q^37 + 14 * q^39 + 6 * q^41 - 6 * q^43 - 16 * q^45 - 24 * q^47 - 8 * q^49 - 28 * q^51 + 24 * q^55 + 12 * q^57 + 6 * q^59 - 14 * q^61 + 8 * q^63 + 18 * q^67 + 6 * q^69 - 6 * q^71 + 16 * q^73 - 6 * q^75 + 12 * q^77 + 24 * q^79 + 2 * q^81 - 16 * q^83 + 16 * q^85 - 22 * q^87 + 18 * q^89 + 30 * q^91 + 16 * q^93 - 24 * q^95 + 18 * q^97 - 48 * q^99

Introduction

L-functions

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