LMFDB - Embedded newform 416.2.i.f.321.1

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			321.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	416.321
Dual form			416.2.i.f.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.207107 + 0.358719i) q^{3} -2.82843 q^{5} +(-0.792893 - 1.37333i) q^{7} +(1.41421 + 2.44949i) q^{9} +(2.62132 - 4.54026i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(0.585786 - 1.01461i) q^{15} +(0.0857864 + 0.148586i) q^{17} +(-3.62132 - 6.27231i) q^{19} +0.656854 q^{21} +(3.62132 - 6.27231i) q^{23} +3.00000 q^{25} -2.41421 q^{27} +(1.32843 - 2.30090i) q^{29} -5.65685 q^{31} +(1.08579 + 1.88064i) q^{33} +(2.24264 + 3.88437i) q^{35} +(-4.74264 + 8.21449i) q^{37} +(1.44975 + 0.358719i) q^{39} +(0.0857864 - 0.148586i) q^{41} +(5.03553 + 8.72180i) q^{43} +(-4.00000 - 6.92820i) q^{45} +6.00000 q^{47} +(2.24264 - 3.88437i) q^{49} -0.0710678 q^{51} +2.82843 q^{53} +(-7.41421 + 12.8418i) q^{55} +3.00000 q^{57} +(-3.62132 - 6.27231i) q^{59} +(-3.50000 - 6.06218i) q^{61} +(2.24264 - 3.88437i) q^{63} +(2.82843 + 9.79796i) q^{65} +(-2.37868 + 4.11999i) q^{67} +(1.50000 + 2.59808i) q^{69} +(-0.621320 - 1.07616i) q^{71} -4.48528 q^{73} +(-0.621320 + 1.07616i) q^{75} -8.31371 q^{77} -6.00000 q^{79} +(-3.74264 + 6.48244i) q^{81} +4.00000 q^{83} +(-0.242641 - 0.420266i) q^{85} +(0.550253 + 0.953065i) q^{87} +(7.32843 - 12.6932i) q^{89} +(-3.96447 + 4.11999i) q^{91} +(1.17157 - 2.02922i) q^{93} +(10.2426 + 17.7408i) q^{95} +(4.50000 + 7.79423i) q^{97} +14.8284 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.207107	+	0.358719i	−0.119573	+	0.207107i	−0.919599	−	0.392859i	$-0.871486\pi$
$3$	−0.207107	+	0.358719i	−0.119573	+	0.207107i	0.800025	+	0.599966i	$0.204819\pi$
$4$		0			0
$5$	−2.82843			−1.26491			−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843			−1.26491			−0.632456	+	0.774597i	$0.717953\pi$
$6$		0			0
$7$	−0.792893	−	1.37333i	−0.299685	−	0.519070i	0.676378	−	0.736554i	$-0.263548\pi$
$7$	−0.792893	−	1.37333i	−0.299685	−	0.519070i	−0.976064	+	0.217484i	$0.930215\pi$
$8$		0			0
$9$	1.41421	+	2.44949i	0.471405	+	0.816497i
$10$		0			0
$11$	2.62132	−	4.54026i	0.790358	−	1.36894i	−0.135388	−	0.990793i	$-0.543228\pi$
$11$	2.62132	−	4.54026i	0.790358	−	1.36894i	0.925745	−	0.378147i	$-0.123439\pi$
$12$		0			0
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$13$	−1.00000	−	3.46410i	−0.277350	−	0.960769i
$14$		0			0
$15$	0.585786	−	1.01461i	0.151249	−	0.261972i
$16$		0			0
$17$	0.0857864	+	0.148586i	0.0208063	+	0.0360375i	0.876241	−	0.481873i	$-0.160043\pi$
$17$	0.0857864	+	0.148586i	0.0208063	+	0.0360375i	−0.855435	+	0.517911i	$0.826710\pi$
$18$		0			0
$19$	−3.62132	−	6.27231i	−0.830788	−	1.43897i	−0.897414	−	0.441189i	$-0.854557\pi$
$19$	−3.62132	−	6.27231i	−0.830788	−	1.43897i	0.0666264	−	0.997778i	$-0.478776\pi$
$20$		0			0
$21$	0.656854			0.143337
$22$		0			0
$23$	3.62132	−	6.27231i	0.755097	−	1.30787i	−0.190228	−	0.981740i	$-0.560923\pi$
$23$	3.62132	−	6.27231i	0.755097	−	1.30787i	0.945326	−	0.326127i	$-0.105744\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	−2.41421			−0.464616
$28$		0			0
$29$	1.32843	−	2.30090i	0.246683	−	0.427267i	−0.715921	−	0.698182i	$-0.753993\pi$
$29$	1.32843	−	2.30090i	0.246683	−	0.427267i	0.962603	+	0.270915i	$0.0873261\pi$
$30$		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
$33$	1.08579	+	1.88064i	0.189011	+	0.327377i
$34$		0			0
$35$	2.24264	+	3.88437i	0.379075	+	0.656578i
$36$		0			0
$37$	−4.74264	+	8.21449i	−0.779685	+	1.35045i	0.152438	+	0.988313i	$0.451288\pi$
$37$	−4.74264	+	8.21449i	−0.779685	+	1.35045i	−0.932123	+	0.362142i	$0.882046\pi$
$38$		0			0
$39$	1.44975	+	0.358719i	0.232145	+	0.0574411i
$40$		0			0
$41$	0.0857864	−	0.148586i	0.0133976	−	0.0232053i	−0.859249	−	0.511558i	$-0.829069\pi$
$41$	0.0857864	−	0.148586i	0.0133976	−	0.0232053i	0.872646	+	0.488352i	$0.162402\pi$
$42$		0			0
$43$	5.03553	+	8.72180i	0.767912	+	1.33006i	0.938693	+	0.344753i	$0.112037\pi$
$43$	5.03553	+	8.72180i	0.767912	+	1.33006i	−0.170782	+	0.985309i	$0.554629\pi$
$44$		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
$49$	2.24264	−	3.88437i	0.320377	−	0.554910i
$50$		0			0
$51$	−0.0710678			−0.00995148
$52$		0			0
$53$	2.82843			0.388514			0.194257	−	0.980951i	$-0.437770\pi$
$53$	2.82843			0.388514			0.194257	+	0.980951i	$0.437770\pi$
$54$		0			0
$55$	−7.41421	+	12.8418i	−0.999732	+	1.73159i
$56$		0			0
$57$	3.00000			0.397360
$58$		0			0
$59$	−3.62132	−	6.27231i	−0.471456	−	0.816585i	0.528011	−	0.849238i	$-0.322938\pi$
$59$	−3.62132	−	6.27231i	−0.471456	−	0.816585i	−0.999467	+	0.0326522i	$0.989605\pi$
$60$		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
$63$	2.24264	−	3.88437i	0.282546	−	0.489384i
$64$		0			0
$65$	2.82843	+	9.79796i	0.350823	+	1.21529i
$66$		0			0
$67$	−2.37868	+	4.11999i	−0.290602	+	0.503337i	−0.973952	−	0.226753i	$-0.927189\pi$
$67$	−2.37868	+	4.11999i	−0.290602	+	0.503337i	0.683350	+	0.730091i	$0.260522\pi$
$68$		0			0
$69$	1.50000	+	2.59808i	0.180579	+	0.312772i
$70$		0			0
$71$	−0.621320	−	1.07616i	−0.0737372	−	0.127717i	0.826799	−	0.562497i	$-0.190159\pi$
$71$	−0.621320	−	1.07616i	−0.0737372	−	0.127717i	−0.900536	+	0.434781i	$0.856826\pi$
$72$		0			0
$73$	−4.48528			−0.524962			−0.262481	−	0.964937i	$-0.584541\pi$
$73$	−4.48528			−0.524962			−0.262481	+	0.964937i	$0.584541\pi$
$74$		0			0
$75$	−0.621320	+	1.07616i	−0.0717439	+	0.124264i
$76$		0			0
$77$	−8.31371			−0.947435
$78$		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
$81$	−3.74264	+	6.48244i	−0.415849	+	0.720272i
$82$		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
$85$	−0.242641	−	0.420266i	−0.0263181	−	0.0455842i
$86$		0			0
$87$	0.550253	+	0.953065i	0.0589933	+	0.102179i
$88$		0			0
$89$	7.32843	−	12.6932i	0.776812	−	1.34548i	−0.156959	−	0.987605i	$-0.550169\pi$
$89$	7.32843	−	12.6932i	0.776812	−	1.34548i	0.933771	−	0.357872i	$-0.116498\pi$
$90$		0			0
$91$	−3.96447	+	4.11999i	−0.415589	+	0.431893i
$92$		0			0
$93$	1.17157	−	2.02922i	0.121486	−	0.210421i
$94$		0			0
$95$	10.2426	+	17.7408i	1.05087	+	1.82017i
$96$		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
$99$	14.8284			1.49031
$100$		0			0
$101$	−3.08579	+	5.34474i	−0.307047	+	0.531821i	−0.977715	−	0.209936i	$-0.932674\pi$
$101$	−3.08579	+	5.34474i	−0.307047	+	0.531821i	0.670668	+	0.741758i	$0.266008\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$	−1.85786			−0.181309
$106$		0			0
$107$	−5.37868	+	9.31615i	−0.519977	+	0.900626i	0.479754	+	0.877403i	$0.340726\pi$
$107$	−5.37868	+	9.31615i	−0.519977	+	0.900626i	−0.999730	+	0.0232227i	$0.992607\pi$
$108$		0			0
$109$	8.48528			0.812743			0.406371	−	0.913708i	$-0.366794\pi$
$109$	8.48528			0.812743			0.406371	+	0.913708i	$0.366794\pi$
$110$		0			0
$111$	−1.96447	−	3.40256i	−0.186459	−	0.322956i
$112$		0			0
$113$	2.91421	+	5.04757i	0.274146	+	0.474835i	0.969919	−	0.243426i	$-0.0782715\pi$
$113$	2.91421	+	5.04757i	0.274146	+	0.474835i	−0.695773	+	0.718262i	$0.744938\pi$
$114$		0			0
$115$	−10.2426	+	17.7408i	−0.955131	+	1.65434i
$116$		0			0
$117$	7.07107	−	7.34847i	0.653720	−	0.679366i
$118$		0			0
$119$	0.136039	−	0.235626i	0.0124707	−	0.0215998i
$120$		0			0
$121$	−8.24264	−	14.2767i	−0.749331	−	1.29788i
$122$		0			0
$123$	0.0355339	+	0.0615465i	0.00320398	+	0.00554946i
$124$		0			0
$125$	5.65685			0.505964
$126$		0			0
$127$	−2.20711	+	3.82282i	−0.195849	+	0.339221i	−0.947179	−	0.320707i	$-0.896080\pi$
$127$	−2.20711	+	3.82282i	−0.195849	+	0.339221i	0.751329	+	0.659927i	$0.229413\pi$
$128$		0			0
$129$	−4.17157			−0.367287
$130$		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
$133$	−5.74264	+	9.94655i	−0.497950	+	0.862475i
$134$		0			0
$135$	6.82843			0.587697
$136$		0			0
$137$	−10.3284	−	17.8894i	−0.882417	−	1.52839i	−0.848646	−	0.528962i	$-0.822582\pi$
$137$	−10.3284	−	17.8894i	−0.882417	−	1.52839i	−0.0337713	−	0.999430i	$-0.510752\pi$
$138$		0			0
$139$	−3.62132	−	6.27231i	−0.307156	−	0.532010i	0.670583	−	0.741835i	$-0.266044\pi$
$139$	−3.62132	−	6.27231i	−0.307156	−	0.532010i	−0.977739	+	0.209824i	$0.932711\pi$
$140$		0			0
$141$	−1.24264	+	2.15232i	−0.104649	+	0.181258i
$142$		0			0
$143$	−18.3492	−	4.54026i	−1.53444	−	0.379676i
$144$		0			0
$145$	−3.75736	+	6.50794i	−0.312032	+	0.540455i
$146$		0			0
$147$	0.928932	+	1.60896i	0.0766170	+	0.132705i
$148$		0			0
$149$	7.32843	+	12.6932i	0.600368	+	1.03987i	0.992765	+	0.120072i	$0.0383126\pi$
$149$	7.32843	+	12.6932i	0.600368	+	1.03987i	−0.392397	+	0.919796i	$0.628354\pi$
$150$		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
$153$	−0.242641	+	0.420266i	−0.0196163	+	0.0339765i
$154$		0			0
$155$	16.0000			1.28515
$156$		0			0
$157$	16.4853			1.31567			0.657834	−	0.753163i	$-0.271473\pi$
$157$	16.4853			1.31567			0.657834	+	0.753163i	$0.271473\pi$
$158$		0			0
$159$	−0.585786	+	1.01461i	−0.0464559	+	0.0804640i
$160$		0			0
$161$	−11.4853			−0.905167
$162$		0			0
$163$	−12.6213	−	21.8608i	−0.988578	−	1.71227i	−0.624807	−	0.780779i	$-0.714822\pi$
$163$	−12.6213	−	21.8608i	−0.988578	−	1.71227i	−0.363771	−	0.931488i	$-0.618511\pi$
$164$		0			0
$165$	−3.07107	−	5.31925i	−0.239082	−	0.414103i
$166$		0			0
$167$	−4.86396	+	8.42463i	−0.376385	+	0.651917i	−0.990533	−	0.137273i	$-0.956166\pi$
$167$	−4.86396	+	8.42463i	−0.376385	+	0.651917i	0.614149	+	0.789190i	$0.289500\pi$
$168$		0			0
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$		0			0
$171$	10.2426	−	17.7408i	0.783274	−	1.35667i
$172$		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
$175$	−2.37868	−	4.11999i	−0.179811	−	0.311442i
$176$		0			0
$177$	3.00000			0.225494
$178$		0			0
$179$	10.6213	−	18.3967i	0.793875	−	1.37503i	−0.129676	−	0.991556i	$-0.541394\pi$
$179$	10.6213	−	18.3967i	0.793875	−	1.37503i	0.923551	−	0.383475i	$-0.125273\pi$
$180$		0			0
$181$	16.4853			1.22534			0.612671	−	0.790338i	$-0.290095\pi$
$181$	16.4853			1.22534			0.612671	+	0.790338i	$0.290095\pi$
$182$		0			0
$183$	2.89949			0.214337
$184$		0			0
$185$	13.4142	−	23.2341i	0.986233	−	1.70820i
$186$		0			0
$187$	0.899495			0.0657776
$188$		0			0
$189$	1.91421	+	3.31552i	0.139239	+	0.241168i
$190$		0			0
$191$	−7.62132	−	13.2005i	−0.551459	−	0.955156i	−0.998170	−	0.0604770i	$-0.980738\pi$
$191$	−7.62132	−	13.2005i	−0.551459	−	0.955156i	0.446710	−	0.894679i	$-0.352596\pi$
$192$		0			0
$193$	1.25736	−	2.17781i	0.0905067	−	0.156762i	−0.817218	−	0.576329i	$-0.804485\pi$
$193$	1.25736	−	2.17781i	0.0905067	−	0.156762i	0.907724	+	0.419567i	$0.137818\pi$
$194$		0			0
$195$	−4.10051	−	1.01461i	−0.293643	−	0.0726579i
$196$		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
$199$	−9.10660	−	15.7731i	−0.645550	−	1.11813i	−0.984174	−	0.177204i	$-0.943295\pi$
$199$	−9.10660	−	15.7731i	−0.645550	−	1.11813i	0.338624	−	0.940922i	$-0.390038\pi$
$200$		0			0
$201$	−0.985281	−	1.70656i	−0.0694964	−	0.120371i
$202$		0			0
$203$	−4.21320			−0.295709
$204$		0			0
$205$	−0.242641	+	0.420266i	−0.0169468	+	0.0293527i
$206$		0			0
$207$	20.4853			1.42383
$208$		0			0
$209$	−37.9706			−2.62648
$210$		0			0
$211$	−8.20711	+	14.2151i	−0.565001	+	0.978610i	0.432049	+	0.901850i	$0.357791\pi$
$211$	−8.20711	+	14.2151i	−0.565001	+	0.978610i	−0.997050	+	0.0767595i	$0.975543\pi$
$212$		0			0
$213$	0.514719			0.0352679
$214$		0			0
$215$	−14.2426	−	24.6690i	−0.971340	−	1.68241i
$216$		0			0
$217$	4.48528	+	7.76874i	0.304481	+	0.527376i
$218$		0			0
$219$	0.928932	−	1.60896i	0.0627714	−	0.108723i
$220$		0			0
$221$	0.428932	−	0.445759i	0.0288531	−	0.0299850i
$222$		0			0
$223$	6.44975	−	11.1713i	0.431907	−	0.748085i	−0.565130	−	0.825002i	$-0.691174\pi$
$223$	6.44975	−	11.1713i	0.431907	−	0.748085i	0.997038	+	0.0769166i	$0.0245075\pi$
$224$		0			0
$225$	4.24264	+	7.34847i	0.282843	+	0.489898i
$226$		0			0
$227$	4.86396	+	8.42463i	0.322832	+	0.559162i	0.981071	−	0.193647i	$-0.0620316\pi$
$227$	4.86396	+	8.42463i	0.322832	+	0.559162i	−0.658239	+	0.752809i	$0.728698\pi$
$228$		0			0
$229$	−8.48528			−0.560723			−0.280362	−	0.959894i	$-0.590454\pi$
$229$	−8.48528			−0.560723			−0.280362	+	0.959894i	$0.590454\pi$
$230$		0			0
$231$	1.72183	−	2.98229i	0.113288	−	0.196220i
$232$		0			0
$233$	−3.51472			−0.230257			−0.115128	−	0.993351i	$-0.536728\pi$
$233$	−3.51472			−0.230257			−0.115128	+	0.993351i	$0.536728\pi$
$234$		0			0
$235$	−16.9706			−1.10704
$236$		0			0
$237$	1.24264	−	2.15232i	0.0807182	−	0.139808i
$238$		0			0
$239$	24.9706			1.61521			0.807606	−	0.589723i	$-0.200763\pi$
$239$	24.9706			1.61521			0.807606	+	0.589723i	$0.200763\pi$
$240$		0			0
$241$	10.2279	+	17.7153i	0.658838	+	1.14114i	0.980917	+	0.194429i	$0.0622852\pi$
$241$	10.2279	+	17.7153i	0.658838	+	1.14114i	−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$	−5.17157	−	8.95743i	−0.331757	−	0.574619i
$244$		0			0
$245$	−6.34315	+	10.9867i	−0.405249	+	0.701911i
$246$		0			0
$247$	−18.1066	+	18.8169i	−1.15210	+	1.19729i
$248$		0			0
$249$	−0.828427	+	1.43488i	−0.0524994	+	0.0909317i
$250$		0			0
$251$	3.86396	+	6.69258i	0.243891	+	0.422432i	0.961819	−	0.273685i	$-0.0882427\pi$
$251$	3.86396	+	6.69258i	0.243891	+	0.422432i	−0.717928	+	0.696117i	$0.754909\pi$
$252$		0			0
$253$	−18.9853	−	32.8835i	−1.19359	−	2.06737i
$254$		0			0
$255$	0.201010			0.0125877
$256$		0			0
$257$	11.3995	−	19.7445i	0.711081	−	1.23163i	−0.253371	−	0.967369i	$-0.581539\pi$
$257$	11.3995	−	19.7445i	0.711081	−	1.23163i	0.964452	−	0.264259i	$-0.0851274\pi$
$258$		0			0
$259$	15.0416			0.934641
$260$		0			0
$261$	7.51472			0.465149
$262$		0			0
$263$	−0.378680	+	0.655892i	−0.0233504	+	0.0404441i	−0.877464	−	0.479642i	$-0.840767\pi$
$263$	−0.378680	+	0.655892i	−0.0233504	+	0.0404441i	0.854114	+	0.520086i	$0.174100\pi$
$264$		0			0
$265$	−8.00000			−0.491436
$266$		0			0
$267$	3.03553	+	5.25770i	0.185772	+	0.321766i
$268$		0			0
$269$	5.74264	+	9.94655i	0.350135	+	0.606452i	0.986273	−	0.165124i	$-0.0528024\pi$
$269$	5.74264	+	9.94655i	0.350135	+	0.606452i	−0.636138	+	0.771575i	$0.719469\pi$
$270$		0			0
$271$	−8.37868	+	14.5123i	−0.508969	+	0.881559i	0.490978	+	0.871172i	$0.336640\pi$
$271$	−8.37868	+	14.5123i	−0.508969	+	0.881559i	−0.999946	+	0.0103871i	$0.996694\pi$
$272$		0			0
$273$	−0.656854	−	2.27541i	−0.0397546	−	0.137714i
$274$		0			0
$275$	7.86396	−	13.6208i	0.474215	−	0.821364i
$276$		0			0
$277$	1.74264	+	3.01834i	0.104705	+	0.181355i	0.913618	−	0.406574i	$-0.133277\pi$
$277$	1.74264	+	3.01834i	0.104705	+	0.181355i	−0.808913	+	0.587929i	$0.799943\pi$
$278$		0			0
$279$	−8.00000	−	13.8564i	−0.478947	−	0.829561i
$280$		0			0
$281$	26.1421			1.55951			0.779755	−	0.626085i	$-0.215344\pi$
$281$	26.1421			1.55951			0.779755	+	0.626085i	$0.215344\pi$
$282$		0			0
$283$	3.79289	−	6.56948i	0.225464	−	0.390515i	−0.730994	−	0.682383i	$-0.760943\pi$
$283$	3.79289	−	6.56948i	0.225464	−	0.390515i	0.956459	+	0.291868i	$0.0942768\pi$
$284$		0			0
$285$	−8.48528			−0.502625
$286$		0			0
$287$	−0.272078			−0.0160603
$288$		0			0
$289$	8.48528	−	14.6969i	0.499134	−	0.864526i
$290$		0			0
$291$	−3.72792			−0.218535
$292$		0			0
$293$	10.1569	+	17.5922i	0.593370	+	1.02775i	0.993775	+	0.111408i	$0.0355361\pi$
$293$	10.1569	+	17.5922i	0.593370	+	1.02775i	−0.400405	+	0.916338i	$0.631131\pi$
$294$		0			0
$295$	10.2426	+	17.7408i	0.596350	+	1.03291i
$296$		0			0
$297$	−6.32843	+	10.9612i	−0.367213	+	0.636031i
$298$		0			0
$299$	−25.3492	−	6.27231i	−1.46598	−	0.362737i
$300$		0			0
$301$	7.98528	−	13.8309i	0.460264	−	0.797201i
$302$		0			0
$303$	−1.27817	−	2.21386i	−0.0734292	−	0.127183i
$304$		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
$313$	3.51472			0.198664			0.0993318	−	0.995054i	$-0.468329\pi$
$313$	3.51472			0.198664			0.0993318	+	0.995054i	$0.468329\pi$
$314$		0			0
$315$	−6.34315	+	10.9867i	−0.357396	+	0.619028i
$316$		0			0
$317$	5.31371			0.298448			0.149224	−	0.988803i	$-0.452323\pi$
$317$	5.31371			0.298448			0.149224	+	0.988803i	$0.452323\pi$
$318$		0			0
$319$	−6.96447	−	12.0628i	−0.389935	−	0.675388i
$320$		0			0
$321$	−2.22792	−	3.85887i	−0.124350	−	0.215381i
$322$		0			0
$323$	0.621320	−	1.07616i	0.0345712	−	0.0598791i
$324$		0			0
$325$	−3.00000	−	10.3923i	−0.166410	−	0.576461i
$326$		0			0
$327$	−1.75736	+	3.04384i	−0.0971822	+	0.168324i
$328$		0			0
$329$	−4.75736	−	8.23999i	−0.262282	−	0.454285i
$330$		0			0
$331$	−0.621320	−	1.07616i	−0.0341509	−	0.0591510i	0.848445	−	0.529284i	$-0.177539\pi$
$331$	−0.621320	−	1.07616i	−0.0341509	−	0.0591510i	−0.882596	+	0.470133i	$0.844206\pi$
$332$		0			0
$333$	−26.8284			−1.47019
$334$		0			0
$335$	6.72792	−	11.6531i	0.367586	−	0.636677i
$336$		0			0
$337$	−12.4853			−0.680117			−0.340058	−	0.940404i	$-0.610447\pi$
$337$	−12.4853			−0.680117			−0.340058	+	0.940404i	$0.610447\pi$
$338$		0			0
$339$	−2.41421			−0.131122
$340$		0			0
$341$	−14.8284	+	25.6836i	−0.803004	+	1.39084i
$342$		0			0
$343$	−18.2132			−0.983421
$344$		0			0
$345$	−4.24264	−	7.34847i	−0.228416	−	0.395628i
$346$		0			0
$347$	11.3787	+	19.7085i	0.610840	+	1.05801i	0.991099	+	0.133125i	$0.0425013\pi$
$347$	11.3787	+	19.7085i	0.610840	+	1.05801i	−0.380260	+	0.924880i	$0.624165\pi$
$348$		0			0
$349$	8.22792	−	14.2512i	0.440431	−	0.762848i	−0.557291	−	0.830317i	$-0.688159\pi$
$349$	8.22792	−	14.2512i	0.440431	−	0.762848i	0.997721	+	0.0674693i	$0.0214925\pi$
$350$		0			0
$351$	2.41421	+	8.36308i	0.128861	+	0.446388i
$352$		0			0
$353$	1.67157	−	2.89525i	0.0889688	−	0.154099i	−0.818107	−	0.575066i	$-0.804976\pi$
$353$	1.67157	−	2.89525i	0.0889688	−	0.154099i	0.907076	+	0.420968i	$0.138310\pi$
$354$		0			0
$355$	1.75736	+	3.04384i	0.0932709	+	0.161550i
$356$		0			0
$357$	0.0563492	+	0.0975997i	0.00298232	+	0.00516552i
$358$		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
$361$	−16.7279	+	28.9736i	−0.880417	+	1.52493i
$362$		0			0
$363$	6.82843			0.358399
$364$		0			0
$365$	12.6863			0.664031
$366$		0			0
$367$	−16.8640	+	29.2092i	−0.880292	+	1.52471i	−0.0292747	+	0.999571i	$0.509320\pi$
$367$	−16.8640	+	29.2092i	−0.880292	+	1.52471i	−0.851017	+	0.525138i	$0.824014\pi$
$368$		0			0
$369$	0.485281			0.0252627
$370$		0			0
$371$	−2.24264	−	3.88437i	−0.116432	−	0.201666i
$372$		0			0
$373$	−10.2574	−	17.7663i	−0.531106	−	0.919902i	−0.999341	−	0.0362985i	$-0.988443\pi$
$373$	−10.2574	−	17.7663i	−0.531106	−	0.919902i	0.468235	−	0.883604i	$-0.344890\pi$
$374$		0			0
$375$	−1.17157	+	2.02922i	−0.0604998	+	0.104789i
$376$		0			0
$377$	−9.29899	−	2.30090i	−0.478922	−	0.118503i
$378$		0			0
$379$	−4.69239	+	8.12745i	−0.241032	+	0.417479i	−0.961008	−	0.276519i	$-0.910819\pi$
$379$	−4.69239	+	8.12745i	−0.241032	+	0.417479i	0.719977	+	0.693998i	$0.244152\pi$
$380$		0			0
$381$	−0.914214	−	1.58346i	−0.0468366	−	0.0811233i
$382$		0			0
$383$	4.86396	+	8.42463i	0.248537	+	0.430478i	0.963120	−	0.269072i	$-0.0867170\pi$
$383$	4.86396	+	8.42463i	0.248537	+	0.430478i	−0.714583	+	0.699550i	$0.753384\pi$
$384$		0			0
$385$	23.5147			1.19842
$386$		0			0
$387$	−14.2426	+	24.6690i	−0.723994	+	1.25399i
$388$		0			0
$389$	28.6274			1.45147			0.725734	−	0.687976i	$-0.241500\pi$
$389$	28.6274			1.45147			0.725734	+	0.687976i	$0.241500\pi$
$390$		0			0
$391$	1.24264			0.0628430
$392$		0			0
$393$	0.828427	−	1.43488i	0.0417886	−	0.0723800i
$394$		0			0
$395$	16.9706			0.853882
$396$		0			0
$397$	−2.74264	−	4.75039i	−0.137649	−	0.238415i	0.788957	−	0.614448i	$-0.210621\pi$
$397$	−2.74264	−	4.75039i	−0.137649	−	0.238415i	−0.926606	+	0.376033i	$0.877288\pi$
$398$		0			0
$399$	−2.37868	−	4.11999i	−0.119083	−	0.206258i
$400$		0			0
$401$	8.57107	−	14.8455i	0.428019	−	0.741350i	−0.568678	−	0.822560i	$-0.692545\pi$
$401$	8.57107	−	14.8455i	0.428019	−	0.741350i	0.996697	+	0.0812099i	$0.0258784\pi$
$402$		0			0
$403$	5.65685	+	19.5959i	0.281788	+	0.976142i
$404$		0			0
$405$	10.5858	−	18.3351i	0.526012	−	0.911079i
$406$		0			0
$407$	24.8640	+	43.0656i	1.23246	+	2.13468i
$408$		0			0
$409$	13.7426	+	23.8030i	0.679530	+	1.17698i	0.975123	+	0.221666i	$0.0711495\pi$
$409$	13.7426	+	23.8030i	0.679530	+	1.17698i	−0.295593	+	0.955314i	$0.595517\pi$
$410$		0			0
$411$	8.55635			0.422054
$412$		0			0
$413$	−5.74264	+	9.94655i	−0.282577	+	0.489438i
$414$		0			0
$415$	−11.3137			−0.555368
$416$		0			0
$417$	3.00000			0.146911
$418$		0			0
$419$	−6.86396	+	11.8887i	−0.335326	+	0.580802i	−0.983547	−	0.180650i	$-0.942180\pi$
$419$	−6.86396	+	11.8887i	−0.335326	+	0.580802i	0.648221	+	0.761452i	$0.275513\pi$
$420$		0			0
$421$	−16.4853			−0.803443			−0.401722	−	0.915762i	$-0.631588\pi$
$421$	−16.4853			−0.803443			−0.401722	+	0.915762i	$0.631588\pi$
$422$		0			0
$423$	8.48528	+	14.6969i	0.412568	+	0.714590i
$424$		0			0
$425$	0.257359	+	0.445759i	0.0124838	+	0.0216225i
$426$		0			0
$427$	−5.55025	+	9.61332i	−0.268596	+	0.465221i
$428$		0			0
$429$	5.42893	−	5.64191i	0.262111	−	0.272394i
$430$		0			0
$431$	−12.3492	+	21.3895i	−0.594842	+	1.03030i	0.398727	+	0.917070i	$0.369452\pi$
$431$	−12.3492	+	21.3895i	−0.594842	+	1.03030i	−0.993569	+	0.113227i	$0.963881\pi$
$432$		0			0
$433$	9.74264	+	16.8747i	0.468201	+	0.810949i	0.999340	−	0.0363365i	$-0.0115688\pi$
$433$	9.74264	+	16.8747i	0.468201	+	0.810949i	−0.531138	+	0.847285i	$0.678235\pi$
$434$		0			0
$435$	−1.55635	−	2.69568i	−0.0746212	−	0.129248i
$436$		0			0
$437$	−52.4558			−2.50930
$438$		0			0
$439$	6.96447	−	12.0628i	0.332396	−	0.575726i	−0.650585	−	0.759433i	$-0.725476\pi$
$439$	6.96447	−	12.0628i	0.332396	−	0.575726i	0.982981	+	0.183707i	$0.0588097\pi$
$440$		0			0
$441$	12.6863			0.604109
$442$		0			0
$443$	4.97056			0.236159			0.118079	−	0.993004i	$-0.462326\pi$
$443$	4.97056			0.236159			0.118079	+	0.993004i	$0.462326\pi$
$444$		0			0
$445$	−20.7279	+	35.9018i	−0.982598	+	1.70191i
$446$		0			0
$447$	−6.07107			−0.287152
$448$		0			0
$449$	−14.7426	−	25.5350i	−0.695748	−	1.20507i	−0.969928	−	0.243393i	$-0.921740\pi$
$449$	−14.7426	−	25.5350i	−0.695748	−	1.20507i	0.274180	−	0.961679i	$-0.411594\pi$
$450$		0			0
$451$	−0.449747	−	0.778985i	−0.0211778	−	0.0366810i
$452$		0			0
$453$	−3.51472	+	6.08767i	−0.165136	+	0.286024i
$454$		0			0
$455$	11.2132	−	11.6531i	0.525683	−	0.546306i
$456$		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
$459$	−0.207107	−	0.358719i	−0.00966692	−	0.0167436i
$460$		0			0
$461$	−20.3995	−	35.3330i	−0.950099	−	1.64562i	−0.745204	−	0.666836i	$-0.767648\pi$
$461$	−20.3995	−	35.3330i	−0.950099	−	1.64562i	−0.204895	−	0.978784i	$-0.565685\pi$
$462$		0			0
$463$	29.6569			1.37827			0.689135	−	0.724633i	$-0.257990\pi$
$463$	29.6569			1.37827			0.689135	+	0.724633i	$0.257990\pi$
$464$		0			0
$465$	−3.31371	+	5.73951i	−0.153670	+	0.266163i
$466$		0			0
$467$	−26.9706			−1.24805			−0.624024	−	0.781405i	$-0.714503\pi$
$467$	−26.9706			−1.24805			−0.624024	+	0.781405i	$0.714503\pi$
$468$		0			0
$469$	7.54416			0.348357
$470$		0			0
$471$	−3.41421	+	5.91359i	−0.157319	+	0.272484i
$472$		0			0
$473$	52.7990			2.42770
$474$		0			0
$475$	−10.8640	−	18.8169i	−0.498473	−	0.863380i
$476$		0			0
$477$	4.00000	+	6.92820i	0.183147	+	0.317221i
$478$		0			0
$479$	8.62132	−	14.9326i	0.393918	−	0.682286i	−0.599044	−	0.800716i	$-0.704453\pi$
$479$	8.62132	−	14.9326i	0.393918	−	0.682286i	0.992962	+	0.118430i	$0.0377860\pi$
$480$		0			0
$481$	33.1985	+	8.21449i	1.51372	+	0.374549i
$482$		0			0
$483$	2.37868	−	4.11999i	0.108234	−	0.187466i
$484$		0			0
$485$	−12.7279	−	22.0454i	−0.577945	−	1.00103i
$486$		0			0
$487$	−3.44975	−	5.97514i	−0.156323	−	0.270759i	0.777217	−	0.629233i	$-0.216631\pi$
$487$	−3.44975	−	5.97514i	−0.156323	−	0.270759i	−0.933540	+	0.358473i	$0.883297\pi$
$488$		0			0
$489$	10.4558			0.472830
$490$		0			0
$491$	10.1066	−	17.5051i	0.456105	−	0.789996i	−0.542646	−	0.839961i	$-0.682578\pi$
$491$	10.1066	−	17.5051i	0.456105	−	0.789996i	0.998751	+	0.0499650i	$0.0159110\pi$
$492$		0			0
$493$	0.455844			0.0205302
$494$		0			0
$495$	−41.9411			−1.88511
$496$		0			0
$497$	−0.985281	+	1.70656i	−0.0441959	+	0.0765496i
$498$		0			0
$499$	−6.34315			−0.283958			−0.141979	−	0.989870i	$-0.545347\pi$
$499$	−6.34315			−0.283958			−0.141979	+	0.989870i	$0.545347\pi$
$500$		0			0
$501$	−2.01472	−	3.48960i	−0.0900110	−	0.155904i
$502$		0			0
$503$	−11.6213	−	20.1287i	−0.518169	−	0.897495i	−0.999777	−	0.0211085i	$-0.993280\pi$
$503$	−11.6213	−	20.1287i	−0.518169	−	0.897495i	0.481608	−	0.876387i	$-0.340053\pi$
$504$		0			0
$505$	8.72792	−	15.1172i	0.388387	−	0.672707i
$506$		0			0
$507$	−0.207107	−	5.38079i	−0.00919794	−	0.238969i
$508$		0			0
$509$	−11.5711	+	20.0417i	−0.512879	+	0.888332i	0.487010	+	0.873396i	$0.338088\pi$
$509$	−11.5711	+	20.0417i	−0.512879	+	0.888332i	−0.999888	+	0.0149353i	$0.995246\pi$
$510$		0			0
$511$	3.55635	+	6.15978i	0.157324	+	0.272493i
$512$		0			0
$513$	8.74264	+	15.1427i	0.385997	+	0.668566i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	15.7279	−	27.2416i	0.691713	−	1.19808i
$518$		0			0
$519$	−3.72792			−0.163638
$520$		0			0
$521$	−14.8284			−0.649645			−0.324823	−	0.945775i	$-0.605305\pi$
$521$	−14.8284			−0.649645			−0.324823	+	0.945775i	$0.605305\pi$
$522$		0			0
$523$	6.79289	−	11.7656i	0.297032	−	0.514475i	−0.678423	−	0.734671i	$-0.737336\pi$
$523$	6.79289	−	11.7656i	0.297032	−	0.514475i	0.975456	+	0.220196i	$0.0706697\pi$
$524$		0			0
$525$	1.97056			0.0860024
$526$		0			0
$527$	−0.485281	−	0.840532i	−0.0211392	−	0.0366141i
$528$		0			0
$529$	−14.7279	−	25.5095i	−0.640344	−	1.10911i
$530$		0			0
$531$	10.2426	−	17.7408i	0.444493	−	0.769884i
$532$		0			0
$533$	−0.600505	−	0.148586i	−0.0260108	−	0.00643599i
$534$		0			0
$535$	15.2132	−	26.3500i	0.657724	−	1.13921i
$536$		0			0
$537$	4.39949	+	7.62015i	0.189852	+	0.328834i
$538$		0			0
$539$	−11.7574	−	20.3643i	−0.506425	−	0.877154i
$540$		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
$543$	−3.41421	+	5.91359i	−0.146518	+	0.253776i
$544$		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
$551$	−19.2426			−0.819764
$552$		0			0
$553$	4.75736	+	8.23999i	0.202303	+	0.350400i
$554$		0			0
$555$	5.55635	+	9.62388i	0.235854	+	0.408511i
$556$		0			0
$557$	1.32843	−	2.30090i	0.0562873	−	0.0974924i	−0.836509	−	0.547954i	$-0.815407\pi$
$557$	1.32843	−	2.30090i	0.0562873	−	0.0974924i	0.892796	+	0.450461i	$0.148740\pi$
$558$		0			0
$559$	25.1777	−	26.1654i	1.06490	−	1.10668i
$560$		0			0
$561$	−0.186292	+	0.322666i	−0.00786523	+	0.0136230i
$562$		0			0
$563$	13.3492	+	23.1216i	0.562603	+	0.974458i	0.997268	+	0.0738655i	$0.0235335\pi$
$563$	13.3492	+	23.1216i	0.562603	+	0.974458i	−0.434665	+	0.900592i	$0.643133\pi$
$564$		0			0
$565$	−8.24264	−	14.2767i	−0.346770	−	0.600624i
$566$		0			0
$567$	11.8701			0.498496
$568$		0			0
$569$	−3.98528	+	6.90271i	−0.167072	+	0.289377i	−0.937389	−	0.348284i	$-0.886764\pi$
$569$	−3.98528	+	6.90271i	−0.167072	+	0.289377i	0.770317	+	0.637661i	$0.220098\pi$
$570$		0			0
$571$	−16.2843			−0.681476			−0.340738	−	0.940158i	$-0.610677\pi$
$571$	−16.2843			−0.681476			−0.340738	+	0.940158i	$0.610677\pi$
$572$		0			0
$573$	6.31371			0.263759
$574$		0			0
$575$	10.8640	−	18.8169i	0.453058	−	0.784720i
$576$		0			0
$577$	−12.4853			−0.519769			−0.259885	−	0.965640i	$-0.583685\pi$
$577$	−12.4853			−0.519769			−0.259885	+	0.965640i	$0.583685\pi$
$578$		0			0
$579$	0.520815	+	0.902079i	0.0216443	+	0.0374891i
$580$		0			0
$581$	−3.17157	−	5.49333i	−0.131579	−	0.227902i
$582$		0			0
$583$	7.41421	−	12.8418i	0.307065	−	0.531853i
$584$		0			0
$585$	−20.0000	+	20.7846i	−0.826898	+	0.859338i
$586$		0			0
$587$	−19.3492	+	33.5139i	−0.798629	+	1.38327i	0.121881	+	0.992545i	$0.461108\pi$
$587$	−19.3492	+	33.5139i	−0.798629	+	1.38327i	−0.920509	+	0.390721i	$0.872226\pi$
$588$		0			0
$589$	20.4853	+	35.4815i	0.844081	+	1.46199i
$590$		0			0
$591$	−0.621320	−	1.07616i	−0.0255577	−	0.0442672i
$592$		0			0
$593$	9.17157			0.376631			0.188316	−	0.982109i	$-0.439697\pi$
$593$	9.17157			0.376631			0.188316	+	0.982109i	$0.439697\pi$
$594$		0			0
$595$	−0.384776	+	0.666452i	−0.0157743	+	0.0273219i
$596$		0			0
$597$	7.54416			0.308762
$598$		0			0
$599$	−19.9411			−0.814772			−0.407386	−	0.913256i	$-0.633560\pi$
$599$	−19.9411			−0.814772			−0.407386	+	0.913256i	$0.633560\pi$
$600$		0			0
$601$	13.7426	−	23.8030i	0.560574	−	0.970943i	−0.436872	−	0.899523i	$-0.643914\pi$
$601$	13.7426	−	23.8030i	0.560574	−	0.970943i	0.997446	−	0.0714192i	$-0.0227528\pi$
$602$		0			0
$603$	−13.4558			−0.547964
$604$		0			0
$605$	23.3137	+	40.3805i	0.947837	+	1.64170i
$606$		0			0
$607$	2.20711	+	3.82282i	0.0895837	+	0.155164i	0.907335	−	0.420408i	$-0.138113\pi$
$607$	2.20711	+	3.82282i	0.0895837	+	0.155164i	−0.817752	+	0.575571i	$0.804780\pi$
$608$		0			0
$609$	0.872583	−	1.51136i	0.0353588	−	0.0612433i
$610$		0			0
$611$	−6.00000	−	20.7846i	−0.242734	−	0.840855i
$612$		0			0
$613$	15.7426	−	27.2671i	0.635839	−	1.10131i	−0.350497	−	0.936564i	$-0.613987\pi$
$613$	15.7426	−	27.2671i	0.635839	−	1.10131i	0.986337	−	0.164742i	$-0.0526792\pi$
$614$		0			0
$615$	−0.100505	−	0.174080i	−0.00405276	−	0.00701958i
$616$		0			0
$617$	10.8431	+	18.7809i	0.436529	+	0.756090i	0.997419	−	0.0718003i	$-0.0228744\pi$
$617$	10.8431	+	18.7809i	0.436529	+	0.756090i	−0.560890	+	0.827890i	$0.689541\pi$
$618$		0			0
$619$	28.9706			1.16443			0.582213	−	0.813037i	$-0.302187\pi$
$619$	28.9706			1.16443			0.582213	+	0.813037i	$0.302187\pi$
$620$		0			0
$621$	−8.74264	+	15.1427i	−0.350830	+	0.607656i
$622$		0			0
$623$	−23.2426			−0.931197
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	7.86396	−	13.6208i	0.314056	−	0.543962i
$628$		0			0
$629$	−1.62742			−0.0648894
$630$		0			0
$631$	−0.621320	−	1.07616i	−0.0247344	−	0.0428412i	0.853393	−	0.521268i	$-0.174541\pi$
$631$	−0.621320	−	1.07616i	−0.0247344	−	0.0428412i	−0.878128	+	0.478426i	$0.841207\pi$
$632$		0			0
$633$	−3.39949	−	5.88810i	−0.135118	−	0.234031i
$634$		0			0
$635$	6.24264	−	10.8126i	0.247732	−	0.429084i
$636$		0			0
$637$	−15.6985	−	3.88437i	−0.621997	−	0.153904i
$638$		0			0
$639$	1.75736	−	3.04384i	0.0695201	−	0.120412i
$640$		0			0
$641$	−8.39949	−	14.5484i	−0.331760	−	0.574625i	0.651097	−	0.758995i	$-0.274309\pi$
$641$	−8.39949	−	14.5484i	−0.331760	−	0.574625i	−0.982857	+	0.184369i	$0.940976\pi$
$642$		0			0
$643$	−21.1066	−	36.5577i	−0.832363	−	1.44170i	−0.896159	−	0.443732i	$-0.853654\pi$
$643$	−21.1066	−	36.5577i	−0.832363	−	1.44170i	0.0637963	−	0.997963i	$-0.479679\pi$
$644$		0			0
$645$	11.7990			0.464585
$646$		0			0
$647$	−7.37868	+	12.7802i	−0.290086	+	0.502443i	−0.973830	−	0.227279i	$-0.927017\pi$
$647$	−7.37868	+	12.7802i	−0.290086	+	0.502443i	0.683744	+	0.729722i	$0.260351\pi$
$648$		0			0
$649$	−37.9706			−1.49047
$650$		0			0
$651$	−3.71573			−0.145631
$652$		0			0
$653$	2.57107	−	4.45322i	0.100614	−	0.174268i	−0.811324	−	0.584597i	$-0.801253\pi$
$653$	2.57107	−	4.45322i	0.100614	−	0.174268i	0.911938	+	0.410329i	$0.134586\pi$
$654$		0			0
$655$	11.3137			0.442063
$656$		0			0
$657$	−6.34315	−	10.9867i	−0.247470	−	0.428630i
$658$		0			0
$659$	−9.10660	−	15.7731i	−0.354743	−	0.614433i	0.632331	−	0.774698i	$-0.282098\pi$
$659$	−9.10660	−	15.7731i	−0.354743	−	0.614433i	−0.987074	+	0.160266i	$0.948765\pi$
$660$		0			0
$661$	−13.2279	+	22.9114i	−0.514507	+	0.891151i	0.485352	+	0.874319i	$0.338692\pi$
$661$	−13.2279	+	22.9114i	−0.514507	+	0.891151i	−0.999858	+	0.0168325i	$0.994642\pi$
$662$		0			0
$663$	0.0710678	+	0.246186i	0.00276005	+	0.00956108i
$664$		0			0
$665$	16.2426	−	28.1331i	0.629863	−	1.09095i
$666$		0			0
$667$	−9.62132	−	16.6646i	−0.372539	−	0.645256i
$668$		0			0
$669$	2.67157	+	4.62730i	0.103289	+	0.178902i
$670$		0			0
$671$	−36.6985			−1.41673
$672$		0			0
$673$	20.9853	−	36.3476i	0.808923	−	1.40110i	−0.104687	−	0.994505i	$-0.533384\pi$
$673$	20.9853	−	36.3476i	0.808923	−	1.40110i	0.913610	−	0.406591i	$-0.133283\pi$
$674$		0			0
$675$	−7.24264			−0.278769
$676$		0			0
$677$	25.4558			0.978348			0.489174	−	0.872186i	$-0.337298\pi$
$677$	25.4558			0.978348			0.489174	+	0.872186i	$0.337298\pi$
$678$		0			0
$679$	7.13604	−	12.3600i	0.273856	−	0.474333i
$680$		0			0
$681$	−4.02944			−0.154408
$682$		0			0
$683$	0.378680	+	0.655892i	0.0144898	+	0.0250970i	0.873179	−	0.487399i	$-0.162054\pi$
$683$	0.378680	+	0.655892i	0.0144898	+	0.0250970i	−0.858690	+	0.512496i	$0.828721\pi$
$684$		0			0
$685$	29.2132	+	50.5988i	1.11618	+	1.93328i
$686$		0			0
$687$	1.75736	−	3.04384i	0.0670474	−	0.116130i
$688$		0			0
$689$	−2.82843	−	9.79796i	−0.107754	−	0.373273i
$690$		0			0
$691$	0.964466	−	1.67050i	0.0366900	−	0.0635490i	−0.847097	−	0.531438i	$-0.821652\pi$
$691$	0.964466	−	1.67050i	0.0366900	−	0.0635490i	0.883787	+	0.467889i	$0.154985\pi$
$692$		0			0
$693$	−11.7574	−	20.3643i	−0.446625	−	0.773577i
$694$		0			0
$695$	10.2426	+	17.7408i	0.388526	+	0.672946i
$696$		0			0
$697$	0.0294373			0.00111502
$698$		0			0
$699$	0.727922	−	1.26080i	0.0275325	−	0.0476878i
$700$		0			0
$701$	8.48528			0.320485			0.160242	−	0.987078i	$-0.448772\pi$
$701$	8.48528			0.320485			0.160242	+	0.987078i	$0.448772\pi$
$702$		0			0
$703$	68.6985			2.59101
$704$		0			0
$705$	3.51472	−	6.08767i	0.132372	−	0.229275i
$706$		0			0
$707$	9.78680			0.368070
$708$		0			0
$709$	1.25736	+	2.17781i	0.0472211	+	0.0817894i	0.888670	−	0.458548i	$-0.151630\pi$
$709$	1.25736	+	2.17781i	0.0472211	+	0.0817894i	−0.841449	+	0.540337i	$0.818297\pi$
$710$		0			0
$711$	−8.48528	−	14.6969i	−0.318223	−	0.551178i
$712$		0			0
$713$	−20.4853	+	35.4815i	−0.767180	+	1.32879i
$714$		0			0
$715$	51.8995	+	12.8418i	1.94093	+	0.480256i
$716$		0			0
$717$	−5.17157	+	8.95743i	−0.193136	+	0.334521i
$718$		0			0
$719$	−15.6213	−	27.0569i	−0.582577	−	1.00905i	−0.995173	−	0.0981387i	$-0.968711\pi$
$719$	−15.6213	−	27.0569i	−0.582577	−	1.00905i	0.412596	−	0.910914i	$-0.364622\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−8.47309			−0.315118
$724$		0			0
$725$	3.98528	−	6.90271i	0.148010	−	0.256360i
$726$		0			0
$727$	−12.6863			−0.470509			−0.235254	−	0.971934i	$-0.575592\pi$
$727$	−12.6863			−0.470509			−0.235254	+	0.971934i	$0.575592\pi$
$728$		0			0
$729$	−18.1716			−0.673021
$730$		0			0
$731$	−0.863961	+	1.49642i	−0.0319548	+	0.0553473i
$732$		0			0
$733$	−23.9411			−0.884286			−0.442143	−	0.896945i	$-0.645782\pi$
$733$	−23.9411			−0.884286			−0.442143	+	0.896945i	$0.645782\pi$
$734$		0			0
$735$	−2.62742	−	4.55082i	−0.0969137	−	0.167860i
$736$		0			0
$737$	12.4706	+	21.5996i	0.459359	+	0.795633i
$738$		0			0
$739$	6.10660	−	10.5769i	0.224635	−	0.389079i	−0.731575	−	0.681761i	$-0.761214\pi$
$739$	6.10660	−	10.5769i	0.224635	−	0.389079i	0.956210	+	0.292682i	$0.0945477\pi$
$740$		0			0
$741$	−3.00000	−	10.3923i	−0.110208	−	0.381771i
$742$		0			0
$743$	0.621320	−	1.07616i	0.0227940	−	0.0394804i	−0.854403	−	0.519610i	$-0.826077\pi$
$743$	0.621320	−	1.07616i	0.0227940	−	0.0394804i	0.877197	+	0.480130i	$0.159410\pi$
$744$		0			0
$745$	−20.7279	−	35.9018i	−0.759412	−	1.31534i
$746$		0			0
$747$	5.65685	+	9.79796i	0.206973	+	0.358489i
$748$		0			0
$749$	17.0589			0.623318
$750$		0			0
$751$	18.4497	−	31.9559i	0.673241	−	1.16609i	−0.303739	−	0.952755i	$-0.598235\pi$
$751$	18.4497	−	31.9559i	0.673241	−	1.16609i	0.976980	−	0.213332i	$-0.0684316\pi$
$752$		0			0
$753$	−3.20101			−0.116651
$754$		0			0
$755$	−48.0000			−1.74690
$756$		0			0
$757$	3.25736	−	5.64191i	0.118391	−	0.205059i	−0.800739	−	0.599013i	$-0.795560\pi$
$757$	3.25736	−	5.64191i	0.118391	−	0.205059i	0.919130	+	0.393954i	$0.128893\pi$
$758$		0			0
$759$	15.7279			0.570887
$760$		0			0
$761$	−6.25736	−	10.8381i	−0.226829	−	0.392880i	0.730038	−	0.683407i	$-0.239503\pi$
$761$	−6.25736	−	10.8381i	−0.226829	−	0.392880i	−0.956867	+	0.290528i	$0.906169\pi$
$762$		0			0
$763$	−6.72792	−	11.6531i	−0.243567	−	0.421871i
$764$		0			0
$765$	0.686292	−	1.18869i	0.0248129	−	0.0429772i
$766$		0			0
$767$	−18.1066	+	18.8169i	−0.653791	+	0.679440i
$768$		0			0
$769$	−19.7132	+	34.1443i	−0.710876	+	1.23127i	0.253652	+	0.967295i	$0.418368\pi$
$769$	−19.7132	+	34.1443i	−0.710876	+	1.23127i	−0.964529	+	0.263978i	$0.914965\pi$
$770$		0			0
$771$	4.72183	+	8.17844i	0.170052	+	0.294539i
$772$		0			0
$773$	−2.74264	−	4.75039i	−0.0986459	−	0.170860i	0.812478	−	0.582991i	$-0.198118\pi$
$773$	−2.74264	−	4.75039i	−0.0986459	−	0.170860i	−0.911124	+	0.412131i	$0.864784\pi$
$774$		0			0
$775$	−16.9706			−0.609601
$776$		0			0
$777$	−3.11522	+	5.39573i	−0.111758	+	0.193571i
$778$		0			0
$779$	−1.24264			−0.0445222
$780$		0			0
$781$	−6.51472			−0.233115
$782$		0			0
$783$	−3.20711	+	5.55487i	−0.114613	+	0.198515i
$784$		0			0
$785$	−46.6274			−1.66420
$786$		0			0
$787$	22.0061	+	38.1157i	0.784433	+	1.35868i	0.929337	+	0.369232i	$0.120379\pi$
$787$	22.0061	+	38.1157i	0.784433	+	1.35868i	−0.144905	+	0.989446i	$0.546288\pi$
$788$		0			0
$789$	−0.156854	−	0.271680i	−0.00558416	−	0.00967205i
$790$		0			0
$791$	4.62132	−	8.00436i	0.164315	−	0.284602i

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)

\(f(q)\)	\(=\)	\(q+(-0.207107 + 0.358719i) q^{3} -2.82843 q^{5} +(-0.792893 - 1.37333i) q^{7} +(1.41421 + 2.44949i) q^{9} +(2.62132 - 4.54026i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(0.585786 - 1.01461i) q^{15} +(0.0857864 + 0.148586i) q^{17} +(-3.62132 - 6.27231i) q^{19} +0.656854 q^{21} +(3.62132 - 6.27231i) q^{23} +3.00000 q^{25} -2.41421 q^{27} +(1.32843 - 2.30090i) q^{29} -5.65685 q^{31} +(1.08579 + 1.88064i) q^{33} +(2.24264 + 3.88437i) q^{35} +(-4.74264 + 8.21449i) q^{37} +(1.44975 + 0.358719i) q^{39} +(0.0857864 - 0.148586i) q^{41} +(5.03553 + 8.72180i) q^{43} +(-4.00000 - 6.92820i) q^{45} +6.00000 q^{47} +(2.24264 - 3.88437i) q^{49} -0.0710678 q^{51} +2.82843 q^{53} +(-7.41421 + 12.8418i) q^{55} +3.00000 q^{57} +(-3.62132 - 6.27231i) q^{59} +(-3.50000 - 6.06218i) q^{61} +(2.24264 - 3.88437i) q^{63} +(2.82843 + 9.79796i) q^{65} +(-2.37868 + 4.11999i) q^{67} +(1.50000 + 2.59808i) q^{69} +(-0.621320 - 1.07616i) q^{71} -4.48528 q^{73} +(-0.621320 + 1.07616i) q^{75} -8.31371 q^{77} -6.00000 q^{79} +(-3.74264 + 6.48244i) q^{81} +4.00000 q^{83} +(-0.242641 - 0.420266i) q^{85} +(0.550253 + 0.953065i) q^{87} +(7.32843 - 12.6932i) q^{89} +(-3.96447 + 4.11999i) q^{91} +(1.17157 - 2.02922i) q^{93} +(10.2426 + 17.7408i) q^{95} +(4.50000 + 7.79423i) q^{97} +14.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 6 q^{7} + 2 q^{11} - 4 q^{13} + 8 q^{15} + 6 q^{17} - 6 q^{19} - 20 q^{21} + 6 q^{23} + 12 q^{25} - 4 q^{27} - 6 q^{29} + 10 q^{33} - 8 q^{35} - 2 q^{37} - 14 q^{39} + 6 q^{41} + 6 q^{43}+ \cdots + 48 q^{99}+O(q^{100}) \) 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

Modular forms

Varieties

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Database

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