LMFDB - Embedded newform 416.2.i.c.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.c.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.207107 - 0.358719i) q^{3} -2.82843 q^{5} +(0.792893 + 1.37333i) q^{7} +(1.41421 + 2.44949i) q^{9} +(-2.62132 + 4.54026i) q^{11} +(-1.00000 - 3.46410i) q^{13} +(-0.585786 + 1.01461i) q^{15} +(0.0857864 + 0.148586i) q^{17} +(3.62132 + 6.27231i) q^{19} +0.656854 q^{21} +(-3.62132 + 6.27231i) q^{23} +3.00000 q^{25} +2.41421 q^{27} +(1.32843 - 2.30090i) q^{29} +5.65685 q^{31} +(1.08579 + 1.88064i) q^{33} +(-2.24264 - 3.88437i) q^{35} +(-4.74264 + 8.21449i) q^{37} +(-1.44975 - 0.358719i) q^{39} +(0.0857864 - 0.148586i) q^{41} +(-5.03553 - 8.72180i) q^{43} +(-4.00000 - 6.92820i) q^{45} -6.00000 q^{47} +(2.24264 - 3.88437i) q^{49} +0.0710678 q^{51} +2.82843 q^{53} +(7.41421 - 12.8418i) q^{55} +3.00000 q^{57} +(3.62132 + 6.27231i) q^{59} +(-3.50000 - 6.06218i) q^{61} +(-2.24264 + 3.88437i) q^{63} +(2.82843 + 9.79796i) q^{65} +(2.37868 - 4.11999i) q^{67} +(1.50000 + 2.59808i) q^{69} +(0.621320 + 1.07616i) q^{71} -4.48528 q^{73} +(0.621320 - 1.07616i) q^{75} -8.31371 q^{77} +6.00000 q^{79} +(-3.74264 + 6.48244i) q^{81} -4.00000 q^{83} +(-0.242641 - 0.420266i) q^{85} +(-0.550253 - 0.953065i) q^{87} +(7.32843 - 12.6932i) q^{89} +(3.96447 - 4.11999i) q^{91} +(1.17157 - 2.02922i) q^{93} +(-10.2426 - 17.7408i) q^{95} +(4.50000 + 7.79423i) q^{97} -14.8284 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	−0.800025	−	0.599966i	$-0.795181\pi$
$3$	0.207107	−	0.358719i	0.119573	−	0.207107i	0.919599	+	0.392859i	$0.128514\pi$
$4$		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	0.976064	−	0.217484i	$-0.0697849\pi$
$7$	0.792893	+	1.37333i	0.299685	+	0.519070i	−0.676378	+	0.736554i	$0.736452\pi$
$8$		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	$0.456772\pi$
$11$	−2.62132	+	4.54026i	−0.790358	+	1.36894i	−0.925745	+	0.378147i	$0.876561\pi$
$12$		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	$0.145443\pi$
$19$	3.62132	+	6.27231i	0.830788	+	1.43897i	−0.0666264	+	0.997778i	$0.521224\pi$
$20$		0			0
$21$	0.656854			0.143337
$22$		0			0
$23$	−3.62132	+	6.27231i	−0.755097	+	1.30787i	0.190228	+	0.981740i	$0.439077\pi$
$23$	−3.62132	+	6.27231i	−0.755097	+	1.30787i	−0.945326	+	0.326127i	$0.894256\pi$
$24$		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	$-0.330385\pi$
$31$	5.65685			1.01600			0.508001	+	0.861357i	$0.330385\pi$
$32$		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	$-0.887963\pi$
$43$	−5.03553	−	8.72180i	−0.767912	−	1.33006i	0.170782	−	0.985309i	$-0.445371\pi$
$44$		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
$49$	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	0.999467	−	0.0326522i	$-0.0103954\pi$
$59$	3.62132	+	6.27231i	0.471456	+	0.816585i	−0.528011	+	0.849238i	$0.677062\pi$
$60$		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	−0.683350	−	0.730091i	$-0.739478\pi$
$67$	2.37868	−	4.11999i	0.290602	−	0.503337i	0.973952	+	0.226753i	$0.0728111\pi$
$68$		0			0
$69$	1.50000	+	2.59808i	0.180579	+	0.312772i
$70$		0			0
$71$	0.621320	+	1.07616i	0.0737372	+	0.127717i	0.900536	−	0.434781i	$-0.143174\pi$
$71$	0.621320	+	1.07616i	0.0737372	+	0.127717i	−0.826799	+	0.562497i	$0.809841\pi$
$72$		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	$-0.390410\pi$
$79$	6.00000			0.675053			0.337526	+	0.941316i	$0.390410\pi$
$80$		0			0
$81$	−3.74264	+	6.48244i	−0.415849	+	0.720272i
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		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	$-0.659274\pi$
$107$	5.37868	−	9.31615i	0.519977	−	0.900626i	0.999730	−	0.0232227i	$-0.00739267\pi$
$108$		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	−0.751329	−	0.659927i	$-0.770587\pi$
$127$	2.20711	−	3.82282i	0.195849	−	0.339221i	0.947179	+	0.320707i	$0.103920\pi$
$128$		0			0
$129$	−4.17157			−0.367287
$130$		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
$133$	−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	0.977739	−	0.209824i	$-0.0672892\pi$
$139$	3.62132	+	6.27231i	0.307156	+	0.532010i	−0.670583	+	0.741835i	$0.733956\pi$
$140$		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	$-0.742619\pi$
$151$	−16.9706			−1.38104			−0.690522	+	0.723311i	$0.742619\pi$
$152$		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	$0.285178\pi$
$163$	12.6213	+	21.8608i	0.988578	+	1.71227i	0.363771	+	0.931488i	$0.381489\pi$
$164$		0			0
$165$	−3.07107	−	5.31925i	−0.239082	−	0.414103i
$166$		0			0
$167$	4.86396	−	8.42463i	0.376385	−	0.651917i	−0.614149	−	0.789190i	$-0.710500\pi$
$167$	4.86396	−	8.42463i	0.376385	−	0.651917i	0.990533	+	0.137273i	$0.0438338\pi$
$168$		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	$0.458606\pi$
$179$	−10.6213	+	18.3967i	−0.793875	+	1.37503i	−0.923551	+	0.383475i	$0.874727\pi$
$180$		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	$0.0192622\pi$
$191$	7.62132	+	13.2005i	0.551459	+	0.955156i	−0.446710	+	0.894679i	$0.647404\pi$
$192$		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	$0.0567051\pi$
$199$	9.10660	+	15.7731i	0.645550	+	1.11813i	−0.338624	+	0.940922i	$0.609962\pi$
$200$		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	$-0.642209\pi$
$211$	8.20711	−	14.2151i	0.565001	−	0.978610i	0.997050	−	0.0767595i	$-0.0244574\pi$
$212$		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	−0.997038	−	0.0769166i	$-0.975492\pi$
$223$	−6.44975	+	11.1713i	−0.431907	+	0.748085i	0.565130	+	0.825002i	$0.308826\pi$
$224$		0			0
$225$	4.24264	+	7.34847i	0.282843	+	0.489898i
$226$		0			0
$227$	−4.86396	−	8.42463i	−0.322832	−	0.559162i	0.658239	−	0.752809i	$-0.271302\pi$
$227$	−4.86396	−	8.42463i	−0.322832	−	0.559162i	−0.981071	+	0.193647i	$0.937968\pi$
$228$		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	$-0.799237\pi$
$239$	−24.9706			−1.61521			−0.807606	+	0.589723i	$0.799237\pi$
$240$		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	0.717928	−	0.696117i	$-0.245091\pi$
$251$	−3.86396	−	6.69258i	−0.243891	−	0.422432i	−0.961819	+	0.273685i	$0.911757\pi$
$252$		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	−0.854114	−	0.520086i	$-0.825900\pi$
$263$	0.378680	−	0.655892i	0.0233504	−	0.0404441i	0.877464	+	0.479642i	$0.159233\pi$
$264$		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	$-0.663360\pi$
$271$	8.37868	−	14.5123i	0.508969	−	0.881559i	0.999946	−	0.0103871i	$-0.00330638\pi$
$272$		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	−0.956459	−	0.291868i	$-0.905723\pi$
$283$	−3.79289	+	6.56948i	−0.225464	+	0.390515i	0.730994	+	0.682383i	$0.239057\pi$
$284$		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	$-0.611253\pi$
$307$	−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
$313$	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	0.882596	−	0.470133i	$-0.155794\pi$
$331$	0.621320	+	1.07616i	0.0341509	+	0.0591510i	−0.848445	+	0.529284i	$0.822461\pi$
$332$		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	$-0.957499\pi$
$347$	−11.3787	−	19.7085i	−0.610840	−	1.05801i	0.380260	−	0.924880i	$-0.375835\pi$
$348$		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	$-0.432292\pi$
$359$	8.00000			0.422224			0.211112	+	0.977462i	$0.432292\pi$
$360$		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	$-0.490680\pi$
$367$	16.8640	−	29.2092i	0.880292	−	1.52471i	0.851017	−	0.525138i	$-0.175986\pi$
$368$		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	−0.719977	−	0.693998i	$-0.755848\pi$
$379$	4.69239	−	8.12745i	0.241032	−	0.417479i	0.961008	+	0.276519i	$0.0891809\pi$
$380$		0			0
$381$	−0.914214	−	1.58346i	−0.0468366	−	0.0811233i
$382$		0			0
$383$	−4.86396	−	8.42463i	−0.248537	−	0.430478i	0.714583	−	0.699550i	$-0.246616\pi$
$383$	−4.86396	−	8.42463i	−0.248537	−	0.430478i	−0.963120	+	0.269072i	$0.913283\pi$
$384$		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	−0.648221	−	0.761452i	$-0.724487\pi$
$419$	6.86396	−	11.8887i	0.335326	−	0.580802i	0.983547	+	0.180650i	$0.0578201\pi$
$420$		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	$-0.630548\pi$
$431$	12.3492	−	21.3895i	0.594842	−	1.03030i	0.993569	−	0.113227i	$-0.0361187\pi$
$432$		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	−0.982981	−	0.183707i	$-0.941190\pi$
$439$	−6.96447	+	12.0628i	−0.332396	+	0.575726i	0.650585	+	0.759433i	$0.274524\pi$
$440$		0			0
$441$	12.6863			0.604109
$442$		0			0
$443$	−4.97056			−0.236159			−0.118079	−	0.993004i	$-0.537674\pi$
$443$	−4.97056			−0.236159			−0.118079	+	0.993004i	$0.537674\pi$
$444$		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	$-0.742010\pi$
$463$	−29.6569			−1.37827			−0.689135	+	0.724633i	$0.742010\pi$
$464$		0			0
$465$	−3.31371	+	5.73951i	−0.153670	+	0.266163i
$466$		0			0
$467$	26.9706			1.24805			0.624024	−	0.781405i	$-0.285497\pi$
$467$	26.9706			1.24805			0.624024	+	0.781405i	$0.285497\pi$
$468$		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	−0.992962	−	0.118430i	$-0.962214\pi$
$479$	−8.62132	+	14.9326i	−0.393918	+	0.682286i	0.599044	+	0.800716i	$0.295547\pi$
$480$		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	0.933540	−	0.358473i	$-0.116703\pi$
$487$	3.44975	+	5.97514i	0.156323	+	0.270759i	−0.777217	+	0.629233i	$0.783369\pi$
$488$		0			0
$489$	10.4558			0.472830
$490$		0			0
$491$	−10.1066	+	17.5051i	−0.456105	+	0.789996i	−0.998751	−	0.0499650i	$-0.984089\pi$
$491$	−10.1066	+	17.5051i	−0.456105	+	0.789996i	0.542646	+	0.839961i	$0.317422\pi$
$492$		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	$-0.454653\pi$
$499$	6.34315			0.283958			0.141979	+	0.989870i	$0.454653\pi$
$500$		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	$0.00671953\pi$
$503$	11.6213	+	20.1287i	0.518169	+	0.897495i	−0.481608	+	0.876387i	$0.659947\pi$
$504$		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	−0.975456	−	0.220196i	$-0.929330\pi$
$523$	−6.79289	+	11.7656i	−0.297032	+	0.514475i	0.678423	+	0.734671i	$0.262664\pi$
$524$		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	$-0.779557\pi$
$547$	−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
$551$	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	$-0.976466\pi$
$563$	−13.3492	−	23.1216i	−0.562603	−	0.974458i	0.434665	−	0.900592i	$-0.356867\pi$
$564$		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	$-0.389323\pi$
$571$	16.2843			0.681476			0.340738	+	0.940158i	$0.389323\pi$
$572$		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	$-0.538892\pi$
$587$	19.3492	−	33.5139i	0.798629	−	1.38327i	0.920509	−	0.390721i	$-0.127774\pi$
$588$		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	$-0.366440\pi$
$599$	19.9411			0.814772			0.407386	+	0.913256i	$0.366440\pi$
$600$		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	0.817752	−	0.575571i	$-0.195220\pi$
$607$	−2.20711	−	3.82282i	−0.0895837	−	0.155164i	−0.907335	+	0.420408i	$0.861887\pi$
$608$		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	$-0.697813\pi$
$619$	−28.9706			−1.16443			−0.582213	+	0.813037i	$0.697813\pi$
$620$		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	0.878128	−	0.478426i	$-0.158793\pi$
$631$	0.621320	+	1.07616i	0.0247344	+	0.0428412i	−0.853393	+	0.521268i	$0.825459\pi$
$632$		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	$0.146346\pi$
$643$	21.1066	+	36.5577i	0.832363	+	1.44170i	−0.0637963	+	0.997963i	$0.520321\pi$
$644$		0			0
$645$	11.7990			0.464585
$646$		0			0
$647$	7.37868	−	12.7802i	0.290086	−	0.502443i	−0.683744	−	0.729722i	$-0.739649\pi$
$647$	7.37868	−	12.7802i	0.290086	−	0.502443i	0.973830	+	0.227279i	$0.0729828\pi$
$648$		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	0.987074	−	0.160266i	$-0.0512351\pi$
$659$	9.10660	+	15.7731i	0.354743	+	0.614433i	−0.632331	+	0.774698i	$0.717902\pi$
$660$		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	0.858690	−	0.512496i	$-0.171279\pi$
$683$	−0.378680	−	0.655892i	−0.0144898	−	0.0250970i	−0.873179	+	0.487399i	$0.837946\pi$
$684$		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	−0.883787	−	0.467889i	$-0.845015\pi$
$691$	−0.964466	+	1.67050i	−0.0366900	+	0.0635490i	0.847097	+	0.531438i	$0.178348\pi$
$692$		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	$0.0312889\pi$
$719$	15.6213	+	27.0569i	0.582577	+	1.00905i	−0.412596	+	0.910914i	$0.635378\pi$
$720$		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	$-0.424408\pi$
$727$	12.6863			0.470509			0.235254	+	0.971934i	$0.424408\pi$
$728$		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	−0.956210	−	0.292682i	$-0.905452\pi$
$739$	−6.10660	+	10.5769i	−0.224635	+	0.389079i	0.731575	+	0.681761i	$0.238786\pi$
$740$		0			0
$741$	−3.00000	−	10.3923i	−0.110208	−	0.381771i
$742$		0			0
$743$	−0.621320	+	1.07616i	−0.0227940	+	0.0394804i	−0.877197	−	0.480130i	$-0.840590\pi$
$743$	−0.621320	+	1.07616i	−0.0227940	+	0.0394804i	0.854403	+	0.519610i	$0.173923\pi$
$744$		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	$0.401765\pi$
$751$	−18.4497	+	31.9559i	−0.673241	+	1.16609i	−0.976980	+	0.213332i	$0.931568\pi$
$752$		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	$-0.879621\pi$
$787$	−22.0061	−	38.1157i	−0.784433	−	1.35868i	0.144905	−	0.989446i	$-0.453712\pi$
$788$		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

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