LMFDB - Embedded newform 416.2.i.f.321.2

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			321.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	416.321
Dual form			416.2.i.f.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Character values

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

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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	−0.272605	−	0.962126i	$-0.587885\pi$
$3$	1.20711	−	2.09077i	0.696923	−	1.20711i	0.969528	−	0.244981i	$-0.0787816\pi$
$4$		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	−0.894674	−	0.446720i	$-0.852592\pi$
$7$	−2.20711	−	3.82282i	−0.834208	−	1.44489i	0.0604657	−	0.998170i	$-0.480741\pi$
$8$		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	$-0.995916\pi$
$11$	−1.62132	+	2.80821i	−0.488846	+	0.846707i	0.511071	+	0.859538i	$0.329249\pi$
$12$		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	0.966038	+	0.258401i	$0.0831955\pi$
$17$	2.91421	+	5.04757i	0.706801	+	1.22421i	−0.259237	+	0.965814i	$0.583471\pi$
$18$		0			0
$19$	0.621320	+	1.07616i	0.142541	+	0.246888i	0.928453	−	0.371451i	$-0.121139\pi$
$19$	0.621320	+	1.07616i	0.142541	+	0.246888i	−0.785912	+	0.618338i	$0.787806\pi$
$20$		0			0
$21$	−10.6569			−2.32552
$22$		0			0
$23$	−0.621320	+	1.07616i	−0.129554	+	0.224395i	−0.923504	−	0.383589i	$-0.874688\pi$
$23$	−0.621320	+	1.07616i	−0.129554	+	0.224395i	0.793950	+	0.607983i	$0.208021\pi$
$24$		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	0.113350	+	0.993555i	$0.463842\pi$
$29$	−4.32843	+	7.49706i	−0.803769	+	1.39217i	−0.917119	+	0.398613i	$0.869492\pi$
$30$		0			0
$31$	5.65685			1.01600			0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685			1.01600			0.508001	+	0.861357i	$0.330385\pi$
$32$		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	−0.375048	−	0.927005i	$-0.622374\pi$
$37$	3.74264	−	6.48244i	0.615286	−	1.06571i	0.990334	−	0.138702i	$-0.0442929\pi$
$38$		0			0
$39$	−8.44975	−	2.09077i	−1.35304	−	0.334791i
$40$		0			0
$41$	2.91421	−	5.04757i	0.455124	−	0.788297i	−0.543572	−	0.839363i	$-0.682928\pi$
$41$	2.91421	−	5.04757i	0.455124	−	0.788297i	0.998695	+	0.0510654i	$0.0162617\pi$
$42$		0			0
$43$	−2.03553	−	3.52565i	−0.310416	−	0.537656i	0.668036	−	0.744129i	$-0.267135\pi$
$43$	−2.03553	−	3.52565i	−0.310416	−	0.537656i	−0.978452	+	0.206472i	$0.933802\pi$
$44$		0			0
$45$	−4.00000	−	6.92820i	−0.596285	−	1.03280i
$46$		0			0
$47$	6.00000			0.875190			0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000			0.875190			0.437595	+	0.899172i	$0.355830\pi$
$48$		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	$-0.140891\pi$
$59$	0.621320	+	1.07616i	0.0808890	+	0.140104i	−0.822743	+	0.568414i	$0.807557\pi$
$60$		0			0
$61$	−3.50000	−	6.06218i	−0.448129	−	0.776182i	0.550135	−	0.835076i	$-0.314576\pi$
$61$	−3.50000	−	6.06218i	−0.448129	−	0.776182i	−0.998264	+	0.0588933i	$0.981243\pi$
$62$		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	0.104687	+	0.994505i	$0.466616\pi$
$67$	−6.62132	+	11.4685i	−0.808923	+	1.40110i	−0.913610	+	0.406591i	$0.866718\pi$
$68$		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	$-0.0252607\pi$
$71$	3.62132	+	6.27231i	0.429772	+	0.744386i	−0.567081	+	0.823662i	$0.691927\pi$
$72$		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	$-0.609590\pi$
$79$	−6.00000			−0.675053			−0.337526	+	0.941316i	$0.609590\pi$
$80$		0			0
$81$	4.74264	−	8.21449i	0.526960	−	0.912722i
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		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	−0.763729	−	0.645537i	$-0.776634\pi$
$89$	1.67157	−	2.89525i	0.177186	−	0.306896i	0.940916	+	0.338641i	$0.109967\pi$
$90$		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	0.998796	−	0.0490655i	$-0.0156243\pi$
$97$	4.50000	+	7.79423i	0.456906	+	0.791384i	−0.541890	+	0.840450i	$0.682291\pi$
$98$		0			0
$99$	9.17157			0.921778
$100$		0			0
$101$	−5.91421	+	10.2437i	−0.588486	+	1.01929i	0.405945	+	0.913898i	$0.366943\pi$
$101$	−5.91421	+	10.2437i	−0.588486	+	1.01929i	−0.994431	+	0.105390i	$0.966391\pi$
$102$		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	−0.147029	+	0.989132i	$0.546971\pi$
$107$	−9.62132	+	16.6646i	−0.930128	+	1.61103i	−0.783099	+	0.621897i	$0.786362\pi$
$108$		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	0.870032	−	0.492995i	$-0.164098\pi$
$113$	0.0857864	+	0.148586i	0.00807011	+	0.0139778i	−0.861962	+	0.506973i	$0.830765\pi$
$114$		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	$-0.855747\pi$
$127$	−0.792893	+	1.37333i	−0.0703579	+	0.121863i	0.828700	+	0.559693i	$0.189081\pi$
$128$		0			0
$129$	−9.82843			−0.865345
$130$		0			0
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		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	0.594498	−	0.804097i	$-0.297351\pi$
$137$	−4.67157	−	8.09140i	−0.399119	−	0.691295i	−0.993617	+	0.112802i	$0.964017\pi$
$138$		0			0
$139$	0.621320	+	1.07616i	0.0526997	+	0.0912786i	0.891172	−	0.453666i	$-0.149884\pi$
$139$	0.621320	+	1.07616i	0.0526997	+	0.0912786i	−0.838472	+	0.544944i	$0.816551\pi$
$140$		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	0.926337	−	0.376696i	$-0.122940\pi$
$149$	1.67157	+	2.89525i	0.136941	+	0.237188i	−0.789397	+	0.613884i	$0.789606\pi$
$150$		0			0
$151$	−16.9706			−1.38104			−0.690522	−	0.723311i	$-0.742619\pi$
$151$	−16.9706			−1.38104			−0.690522	+	0.723311i	$0.742619\pi$
$152$		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	−0.981574	−	0.191082i	$-0.938800\pi$
$163$	−8.37868	−	14.5123i	−0.656269	−	1.13669i	0.325305	−	0.945609i	$-0.394533\pi$
$164$		0			0
$165$	11.0711	+	19.1757i	0.861881	+	1.49282i
$166$		0			0
$167$	7.86396	−	13.6208i	0.608532	−	1.05401i	−0.382951	−	0.923769i	$-0.625092\pi$
$167$	7.86396	−	13.6208i	0.608532	−	1.05401i	0.991483	−	0.130239i	$-0.0415745\pi$
$168$		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	0.984828	−	0.173534i	$-0.0555188\pi$
$173$	4.50000	+	7.79423i	0.342129	+	0.592584i	−0.642699	+	0.766119i	$0.722185\pi$
$174$		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	$-0.675143\pi$
$179$	6.37868	−	11.0482i	0.476765	−	0.825781i	0.999645	+	0.0266249i	$0.00847597\pi$
$180$		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	$-0.245282\pi$
$191$	−3.37868	−	5.85204i	−0.244473	−	0.423439i	−0.961983	+	0.273109i	$0.911948\pi$
$192$		0			0
$193$	9.74264	−	16.8747i	0.701291	−	1.21467i	−0.266723	−	0.963773i	$-0.585941\pi$
$193$	9.74264	−	16.8747i	0.701291	−	1.21467i	0.968014	−	0.250898i	$-0.0807258\pi$
$194$		0			0
$195$	−23.8995	−	5.91359i	−1.71148	−	0.423481i
$196$		0			0
$197$	−1.50000	+	2.59808i	−0.106871	+	0.185105i	−0.914501	−	0.404584i	$-0.867416\pi$
$197$	−1.50000	+	2.59808i	−0.106871	+	0.185105i	0.807630	+	0.589689i	$0.200750\pi$
$198$		0			0
$199$	12.1066	+	20.9692i	0.858214	+	1.48647i	0.873631	+	0.486590i	$0.161759\pi$
$199$	12.1066	+	20.9692i	0.858214	+	1.48647i	−0.0154165	+	0.999881i	$0.504907\pi$
$200$		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	$-0.988230\pi$
$211$	−6.79289	+	11.7656i	−0.467642	+	0.809980i	0.531674	+	0.846949i	$0.321563\pi$
$212$		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	$-0.907537\pi$
$223$	−3.44975	+	5.97514i	−0.231012	+	0.400125i	0.727094	+	0.686538i	$0.240870\pi$
$224$		0			0
$225$	−4.24264	−	7.34847i	−0.282843	−	0.489898i
$226$		0			0
$227$	−7.86396	−	13.6208i	−0.521949	−	0.904043i	−0.999674	−	0.0255332i	$-0.991872\pi$
$227$	−7.86396	−	13.6208i	−0.521949	−	0.904043i	0.477725	−	0.878510i	$-0.341462\pi$
$228$		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	$-0.593698\pi$
$239$	−8.97056			−0.580257			−0.290129	+	0.956988i	$0.593698\pi$
$240$		0			0
$241$	−15.2279	−	26.3755i	−0.980917	−	1.69900i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	−15.2279	−	26.3755i	−0.980917	−	1.69900i	−0.322078	−	0.946713i	$-0.604381\pi$
$242$		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	−0.997539	−	0.0701119i	$-0.977664\pi$
$251$	−8.86396	−	15.3528i	−0.559488	−	0.969062i	0.438051	−	0.898950i	$-0.355669\pi$
$252$		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	0.475665	+	0.879626i	$0.342207\pi$
$257$	−8.39949	+	14.5484i	−0.523946	+	0.907501i	−0.999611	+	0.0278749i	$0.991126\pi$
$258$		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	$-0.925315\pi$
$263$	−4.62132	+	8.00436i	−0.284963	+	0.493570i	0.687637	+	0.726054i	$0.258648\pi$
$264$		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	0.770220	−	0.637778i	$-0.220146\pi$
$269$	−2.74264	−	4.75039i	−0.167222	−	0.289637i	−0.937442	+	0.348141i	$0.886813\pi$
$270$		0			0
$271$	−12.6213	+	21.8608i	−0.766691	+	1.32795i	0.172658	+	0.984982i	$0.444765\pi$
$271$	−12.6213	+	21.8608i	−0.766691	+	1.32795i	−0.939348	+	0.342965i	$0.888569\pi$
$272$		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	0.589210	−	0.807980i	$-0.299439\pi$
$277$	−6.74264	−	11.6786i	−0.405126	−	0.701699i	−0.994336	+	0.106281i	$0.966106\pi$
$278$		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	$-0.733161\pi$
$283$	5.20711	−	9.01897i	0.309530	−	0.536122i	0.978260	+	0.207384i	$0.0664948\pi$
$284$		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	0.830253	−	0.557386i	$-0.188196\pi$
$293$	−1.15685	−	2.00373i	−0.0675841	−	0.117059i	−0.897837	+	0.440327i	$0.854862\pi$
$294$		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	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		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	$-0.102882\pi$
$331$	3.62132	+	6.27231i	0.199046	+	0.344757i	−0.749174	+	0.662374i	$0.769549\pi$
$332$		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	0.891068	+	0.453869i	$0.149957\pi$
$347$	15.6213	+	27.0569i	0.838596	+	1.45249i	−0.0524719	+	0.998622i	$0.516710\pi$
$348$		0			0
$349$	−17.2279	+	29.8396i	−0.922190	+	1.59728i	−0.126170	+	0.992009i	$0.540268\pi$
$349$	−17.2279	+	29.8396i	−0.922190	+	1.59728i	−0.796020	+	0.605270i	$0.793065\pi$
$350$		0			0
$351$	−0.414214	−	1.43488i	−0.0221091	−	0.0765881i
$352$		0			0
$353$	7.32843	−	12.6932i	0.390053	−	0.675591i	−0.602403	−	0.798192i	$-0.705790\pi$
$353$	7.32843	−	12.6932i	0.390053	−	0.675591i	0.992456	+	0.122601i	$0.0391234\pi$
$354$		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	$-0.567708\pi$
$359$	−8.00000			−0.422224			−0.211112	+	0.977462i	$0.567708\pi$
$360$		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	$-0.902602\pi$
$367$	−4.13604	+	7.16383i	−0.215899	+	0.373949i	0.737651	+	0.675182i	$0.235935\pi$
$368$		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	−0.694178	−	0.719804i	$-0.744232\pi$
$373$	−18.7426	−	32.4632i	−0.970457	−	1.68088i	−0.276280	−	0.961077i	$-0.589102\pi$
$374$		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	−0.263960	−	0.964534i	$-0.585029\pi$
$379$	13.6924	−	23.7159i	0.703331	−	1.21820i	0.967291	−	0.253671i	$-0.0816380\pi$
$380$		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	$-0.298292\pi$
$383$	−7.86396	−	13.6208i	−0.401830	−	0.695989i	−0.993947	+	0.109863i	$0.964959\pi$
$384$		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	0.973384	−	0.229181i	$-0.0736049\pi$
$397$	5.74264	+	9.94655i	0.288215	+	0.499203i	−0.685169	+	0.728384i	$0.740272\pi$
$398$		0			0
$399$	−6.62132	−	11.4685i	−0.331481	−	0.574141i
$400$		0			0
$401$	−5.57107	+	9.64937i	−0.278206	+	0.481867i	−0.970939	−	0.239327i	$-0.923073\pi$
$401$	−5.57107	+	9.64937i	−0.278206	+	0.481867i	0.692733	+	0.721194i	$0.256406\pi$
$402$		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	0.966231	−	0.257678i	$-0.0829574\pi$
$409$	5.25736	+	9.10601i	0.259960	+	0.450263i	−0.706271	+	0.707941i	$0.749624\pi$
$410$		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	$-0.740850\pi$
$419$	5.86396	−	10.1567i	0.286473	−	0.496186i	0.972965	+	0.230951i	$0.0741837\pi$
$420$		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	−0.0577890	−	0.998329i	$-0.518405\pi$
$431$	17.3492	−	30.0498i	0.835684	−	1.44745i	0.893473	−	0.449118i	$-0.148262\pi$
$432$		0			0
$433$	1.25736	+	2.17781i	0.0604248	+	0.104659i	0.894655	−	0.446757i	$-0.147421\pi$
$433$	1.25736	+	2.17781i	0.0604248	+	0.104659i	−0.834231	+	0.551416i	$0.814088\pi$
$434$		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	−0.308058	−	0.951368i	$-0.599679\pi$
$439$	14.0355	−	24.3103i	0.669879	−	1.16027i	0.977937	−	0.208898i	$-0.0669876\pi$
$440$		0			0
$441$	35.3137			1.68161
$442$		0			0
$443$	−28.9706			−1.37643			−0.688216	−	0.725505i	$-0.741606\pi$
$443$	−28.9706			−1.37643			−0.688216	+	0.725505i	$0.741606\pi$
$444$		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	0.679752	−	0.733442i	$-0.262087\pi$
$449$	−6.25736	−	10.8381i	−0.295303	−	0.511480i	−0.975055	+	0.221962i	$0.928754\pi$
$450$		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	0.233890	+	0.972263i	$0.424854\pi$
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	−0.958950	+	0.283577i	$0.908479\pi$
$458$		0			0
$459$	1.20711	+	2.09077i	0.0563429	+	0.0975888i
$460$		0			0
$461$	−0.600505	−	1.04011i	−0.0279683	−	0.0484425i	0.851702	−	0.524026i	$-0.175570\pi$
$461$	−0.600505	−	1.04011i	−0.0279683	−	0.0484425i	−0.879671	+	0.475583i	$0.842237\pi$
$462$		0			0
$463$	18.3431			0.852478			0.426239	−	0.904611i	$-0.359838\pi$
$463$	18.3431			0.852478			0.426239	+	0.904611i	$0.359838\pi$
$464$		0			0
$465$	19.3137	−	33.4523i	0.895652	−	1.55131i
$466$		0			0
$467$	6.97056			0.322559			0.161280	−	0.986909i	$-0.448438\pi$
$467$	6.97056			0.322559			0.161280	+	0.986909i	$0.448438\pi$
$468$		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	$-0.769217\pi$
$479$	4.37868	−	7.58410i	0.200067	−	0.346526i	0.948550	+	0.316628i	$0.102551\pi$
$480$		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	$-0.0722574\pi$
$487$	6.44975	+	11.1713i	0.292266	+	0.506219i	−0.682079	+	0.731278i	$0.738924\pi$
$488$		0			0
$489$	−40.4558			−1.82948
$490$		0			0
$491$	−11.1066	+	19.2372i	−0.501234	+	0.868163i	0.498765	+	0.866737i	$0.333787\pi$
$491$	−11.1066	+	19.2372i	−0.501234	+	0.868163i	−0.999999	+	0.00142539i	$0.999546\pi$
$492$		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	$-0.629330\pi$
$499$	−17.6569			−0.790429			−0.395215	+	0.918589i	$0.629330\pi$
$500$		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	$-0.273378\pi$
$503$	−7.37868	−	12.7802i	−0.328999	−	0.569843i	−0.982314	+	0.187244i	$0.940045\pi$
$504$		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	−0.803403	−	0.595435i	$-0.796980\pi$
$509$	2.57107	−	4.45322i	0.113961	−	0.197386i	0.917364	+	0.398050i	$0.130313\pi$
$510$		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	$-0.716495\pi$
$523$	8.20711	−	14.2151i	0.358872	−	0.621584i	0.987773	+	0.155901i	$0.0498282\pi$
$524$		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	$-0.220443\pi$
$547$	36.0000			1.53925			0.769624	+	0.638497i	$0.220443\pi$
$548$		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	−0.943037	−	0.332689i	$-0.892044\pi$
$557$	−4.32843	+	7.49706i	−0.183401	+	0.317660i	0.759635	+	0.650349i	$0.225377\pi$
$558$		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	−0.972149	−	0.234362i	$-0.924700\pi$
$563$	−16.3492	−	28.3177i	−0.689038	−	1.19345i	0.283111	−	0.959087i	$-0.408634\pi$
$564$		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	−0.454275	−	0.890862i	$-0.650102\pi$
$569$	12.9853	−	22.4912i	0.544371	−	0.942879i	0.998646	−	0.0520172i	$-0.0165651\pi$
$570$		0			0
$571$	40.2843			1.68584			0.842922	−	0.538036i	$-0.180833\pi$
$571$	40.2843			1.68584			0.842922	+	0.538036i	$0.180833\pi$
$572$		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	$-0.692848\pi$
$587$	10.3492	−	17.9254i	0.427159	−	0.739861i	0.996619	+	0.0821578i	$0.0261811\pi$
$588$		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	$-0.0647045\pi$
$599$	47.9411			1.95882			0.979411	+	0.201878i	$0.0647045\pi$
$600$		0			0
$601$	5.25736	−	9.10601i	0.214452	−	0.371442i	−0.738651	−	0.674088i	$-0.764537\pi$
$601$	5.25736	−	9.10601i	0.214452	−	0.371442i	0.953103	+	0.302646i	$0.0978701\pi$
$602$		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	$-0.156421\pi$
$607$	0.792893	+	1.37333i	0.0321825	+	0.0557418i	−0.849486	+	0.527612i	$0.823088\pi$
$608$		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	−0.681424	−	0.731889i	$-0.738639\pi$
$613$	7.25736	−	12.5701i	0.293122	−	0.507702i	0.974546	+	0.224186i	$0.0719725\pi$
$614$		0			0
$615$	−19.8995	−	34.4669i	−0.802425	−	1.38984i
$616$		0			0
$617$	22.1569	+	38.3768i	0.892001	+	1.54499i	0.837472	+	0.546479i	$0.184032\pi$
$617$	22.1569	+	38.3768i	0.892001	+	1.54499i	0.0545289	+	0.998512i	$0.482634\pi$
$618$		0			0
$619$	−4.97056			−0.199784			−0.0998919	−	0.994998i	$-0.531850\pi$
$619$	−4.97056			−0.199784			−0.0998919	+	0.994998i	$0.531850\pi$
$620$		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	$-0.120618\pi$
$631$	3.62132	+	6.27231i	0.144162	+	0.249697i	−0.784898	+	0.619625i	$0.787285\pi$
$632$		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	0.998401	−	0.0565200i	$-0.0180005\pi$
$641$	11.3995	+	19.7445i	0.450253	+	0.779861i	−0.548148	+	0.836381i	$0.684667\pi$
$642$		0			0
$643$	0.106602	+	0.184640i	0.00420396	+	0.00728147i	0.868120	−	0.496355i	$-0.165329\pi$
$643$	0.106602	+	0.184640i	0.00420396	+	0.00728147i	−0.863916	+	0.503636i	$0.831995\pi$
$644$		0			0
$645$	−27.7990			−1.09458
$646$		0			0
$647$	−11.6213	+	20.1287i	−0.456881	+	0.791342i	−0.998794	−	0.0490931i	$-0.984367\pi$
$647$	−11.6213	+	20.1287i	−0.456881	+	0.791342i	0.541913	+	0.840435i	$0.317700\pi$
$648$		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	−0.998559	−	0.0536576i	$-0.982912\pi$
$653$	−11.5711	+	20.0417i	−0.452811	+	0.784291i	0.545749	+	0.837949i	$0.316245\pi$
$654$		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	$-0.0103410\pi$
$659$	12.1066	+	20.9692i	0.471606	+	0.816846i	−0.527866	+	0.849328i	$0.677008\pi$
$660$		0			0
$661$	12.2279	−	21.1794i	0.475611	−	0.823782i	−0.523999	−	0.851719i	$-0.675560\pi$
$661$	12.2279	−	21.1794i	0.475611	−	0.823782i	0.999610	+	0.0279366i	$0.00889365\pi$
$662$		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	−0.778214	−	0.627999i	$-0.783874\pi$
$673$	4.01472	−	6.95370i	0.154756	−	0.268045i	0.932970	+	0.359954i	$0.117208\pi$
$674$		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	$-0.110082\pi$
$683$	4.62132	+	8.00436i	0.176830	+	0.306278i	−0.763963	+	0.645260i	$0.776749\pi$
$684$		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	$-0.734448\pi$
$691$	8.03553	−	13.9180i	0.305686	−	0.529464i	0.977414	+	0.211334i	$0.0677808\pi$
$692$		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	0.988919	−	0.148456i	$-0.0474304\pi$
$709$	9.74264	+	16.8747i	0.365893	+	0.633744i	−0.623026	+	0.782201i	$0.714097\pi$
$710$		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	$-0.306165\pi$
$719$	−11.3787	−	19.7085i	−0.424353	−	0.735001i	−0.996360	+	0.0852478i	$0.972832\pi$
$720$		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	$-0.727271\pi$
$727$	−35.3137			−1.30971			−0.654856	+	0.755753i	$0.727271\pi$
$728$		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	0.442143	+	0.896945i	$0.354218\pi$
$739$	−15.1066	+	26.1654i	−0.555705	+	0.962510i	−0.997848	+	0.0655653i	$0.979115\pi$
$740$		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	$-0.875747\pi$
$743$	−3.62132	+	6.27231i	−0.132853	+	0.230109i	0.791922	+	0.610622i	$0.209081\pi$
$744$		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	$-0.732333\pi$
$751$	8.55025	−	14.8095i	0.312003	−	0.540405i	0.978796	+	0.204838i	$0.0656667\pi$
$752$		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	−0.569793	−	0.821789i	$-0.692976\pi$
$757$	11.7426	−	20.3389i	0.426794	−	0.739228i	0.996586	+	0.0825605i	$0.0263098\pi$
$758$		0			0
$759$	−9.72792			−0.353101
$760$		0			0
$761$	−14.7426	−	25.5350i	−0.534420	−	0.925643i	−0.999191	−	0.0402121i	$-0.987197\pi$
$761$	−14.7426	−	25.5350i	−0.534420	−	0.925643i	0.464771	−	0.885431i	$-0.346137\pi$
$762$		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	−0.0873172	−	0.996181i	$-0.527829\pi$
$769$	22.7132	−	39.3404i	0.819059	−	1.41865i	0.906376	−	0.422471i	$-0.138837\pi$
$770$		0			0
$771$	20.2782	+	35.1228i	0.730301	+	1.26492i
$772$		0			0
$773$	5.74264	+	9.94655i	0.206548	+	0.357752i	0.950625	−	0.310342i	$-0.100444\pi$
$773$	5.74264	+	9.94655i	0.206548	+	0.357752i	−0.744077	+	0.668094i	$0.767110\pi$
$774$		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	−0.975733	−	0.218963i	$-0.929733\pi$
$787$	−19.0061	−	32.9195i	−0.677494	−	1.17345i	0.298239	−	0.954491i	$-0.403601\pi$
$788$		0			0
$789$	11.1569	+	19.3242i	0.397195	+	0.687961i
$790$		0			0
$791$	0.378680	−	0.655892i	0.0134643	−	0.0233208i
$792$

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)

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