LMFDB - Embedded newform 720.2.q.g.241.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,2,Mod(241,720)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))

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

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

chi := DirichletCharacter("720.241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	720.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.74922894553$
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:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			241.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	720.241
Dual form			720.2.q.g.481.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.72474 - 0.158919i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-0.724745 + 1.25529i) q^{7} +(2.94949 + 0.548188i) q^{9} +O(q^{10})$	\(q+(-1.72474 - 0.158919i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-0.724745 + 1.25529i) q^{7} +(2.94949 + 0.548188i) q^{9} +(-0.724745 - 1.57313i) q^{15} -2.00000 q^{17} -2.89898 q^{19} +(1.44949 - 2.04989i) q^{21} +(-1.27526 - 2.20881i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-5.00000 - 1.41421i) q^{27} +(-3.94949 + 6.84072i) q^{29} +(5.44949 + 9.43879i) q^{31} -1.44949 q^{35} -6.00000 q^{37} +(0.0505103 + 0.0874863i) q^{41} +(-3.89898 + 6.75323i) q^{43} +(1.00000 + 2.82843i) q^{45} +(-2.27526 + 3.94086i) q^{47} +(2.44949 + 4.24264i) q^{49} +(3.44949 + 0.317837i) q^{51} -11.7980 q^{53} +(5.00000 + 0.460702i) q^{57} +(-5.44949 - 9.43879i) q^{59} +(1.50000 - 2.59808i) q^{61} +(-2.82577 + 3.30518i) q^{63} +(5.62372 + 9.74058i) q^{67} +(1.84847 + 4.01229i) q^{69} +9.79796 q^{71} -5.79796 q^{73} +(1.00000 - 1.41421i) q^{75} +(1.44949 - 2.51059i) q^{79} +(8.39898 + 3.23375i) q^{81} +(-0.275255 + 0.476756i) q^{83} +(-1.00000 - 1.73205i) q^{85} +(7.89898 - 11.1708i) q^{87} -16.7980 q^{89} +(-7.89898 - 17.1455i) q^{93} +(-1.44949 - 2.51059i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	$ 4 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{15} - 8 q^{17} + 8 q^{19} - 4 q^{21} - 10 q^{23} - 2 q^{25} - 20 q^{27} - 6 q^{29} + 12 q^{31} + 4 q^{35} - 24 q^{37} + 10 q^{41} + 4 q^{43} + 4 q^{45} - 14 q^{47} + 4 q^{51} - 8 q^{53} + 20 q^{57} - 12 q^{59} + 6 q^{61} - 26 q^{63} - 2 q^{67} - 22 q^{69} + 16 q^{73} + 4 q^{75} - 4 q^{79} + 14 q^{81} - 6 q^{83} - 4 q^{85} + 12 q^{87} - 28 q^{89} - 12 q^{93} + 4 q^{95} - 4 q^{97}+O(q^{100}) $ 4 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^15 - 8 * q^17 + 8 * q^19 - 4 * q^21 - 10 * q^23 - 2 * q^25 - 20 * q^27 - 6 * q^29 + 12 * q^31 + 4 * q^35 - 24 * q^37 + 10 * q^41 + 4 * q^43 + 4 * q^45 - 14 * q^47 + 4 * q^51 - 8 * q^53 + 20 * q^57 - 12 * q^59 + 6 * q^61 - 26 * q^63 - 2 * q^67 - 22 * q^69 + 16 * q^73 + 4 * q^75 - 4 * q^79 + 14 * q^81 - 6 * q^83 - 4 * q^85 + 12 * q^87 - 28 * q^89 - 12 * q^93 + 4 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	−1.72474	−	0.158919i	−0.995782	−	0.0917517i
$3$	−1.72474	−	0.158919i	−0.995782	−	0.0917517i
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	−0.724745	+	1.25529i	−0.273928	+	0.474457i	−0.969864	−	0.243647i	$-0.921656\pi$
$7$	−0.724745	+	1.25529i	−0.273928	+	0.474457i	0.695936	+	0.718104i	$0.254990\pi$
$8$		0			0
$9$	2.94949	+	0.548188i	0.983163	+	0.182729i
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$13$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$	−0.724745	−	1.57313i	−0.187128	−	0.406181i
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	−2.89898			−0.665072			−0.332536	−	0.943091i	$-0.607904\pi$
$19$	−2.89898			−0.665072			−0.332536	+	0.943091i	$0.607904\pi$
$20$		0			0
$21$	1.44949	−	2.04989i	0.316305	−	0.447322i
$22$		0			0
$23$	−1.27526	−	2.20881i	−0.265909	−	0.460568i	0.701892	−	0.712283i	$-0.252339\pi$
$23$	−1.27526	−	2.20881i	−0.265909	−	0.460568i	−0.967801	+	0.251715i	$0.919005\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$	−5.00000	−	1.41421i	−0.962250	−	0.272166i
$28$		0			0
$29$	−3.94949	+	6.84072i	−0.733402	+	1.27029i	0.222019	+	0.975042i	$0.428735\pi$
$29$	−3.94949	+	6.84072i	−0.733402	+	1.27029i	−0.955421	+	0.295247i	$0.904598\pi$
$30$		0			0
$31$	5.44949	+	9.43879i	0.978757	+	1.69526i	0.666933	+	0.745117i	$0.267607\pi$
$31$	5.44949	+	9.43879i	0.978757	+	1.69526i	0.311824	+	0.950140i	$0.399060\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.44949			−0.245008
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.0505103	+	0.0874863i	0.00788838	+	0.0136631i	0.869943	−	0.493153i	$-0.164156\pi$
$41$	0.0505103	+	0.0874863i	0.00788838	+	0.0136631i	−0.862054	+	0.506816i	$0.830822\pi$
$42$		0			0
$43$	−3.89898	+	6.75323i	−0.594589	+	1.02986i	0.399016	+	0.916944i	$0.369352\pi$
$43$	−3.89898	+	6.75323i	−0.594589	+	1.02986i	−0.993605	+	0.112914i	$0.963982\pi$
$44$		0			0
$45$	1.00000	+	2.82843i	0.149071	+	0.421637i
$46$		0			0
$47$	−2.27526	+	3.94086i	−0.331880	+	0.574833i	−0.982880	−	0.184244i	$-0.941016\pi$
$47$	−2.27526	+	3.94086i	−0.331880	+	0.574833i	0.651000	+	0.759077i	$0.274350\pi$
$48$		0			0
$49$	2.44949	+	4.24264i	0.349927	+	0.606092i
$50$		0			0
$51$	3.44949	+	0.317837i	0.483025	+	0.0445061i
$52$		0			0
$53$	−11.7980			−1.62057			−0.810287	−	0.586033i	$-0.800689\pi$
$53$	−11.7980			−1.62057			−0.810287	+	0.586033i	$0.800689\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	5.00000	+	0.460702i	0.662266	+	0.0610214i
$58$		0			0
$59$	−5.44949	−	9.43879i	−0.709463	−	1.22883i	−0.965057	−	0.262042i	$-0.915604\pi$
$59$	−5.44949	−	9.43879i	−0.709463	−	1.22883i	0.255593	−	0.966784i	$-0.417729\pi$
$60$		0			0
$61$	1.50000	−	2.59808i	0.192055	−	0.332650i	−0.753876	−	0.657017i	$-0.771818\pi$
$61$	1.50000	−	2.59808i	0.192055	−	0.332650i	0.945931	+	0.324367i	$0.105151\pi$
$62$		0			0
$63$	−2.82577	+	3.30518i	−0.356013	+	0.416414i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.62372	+	9.74058i	0.687047	+	1.19000i	0.972789	+	0.231694i	$0.0744268\pi$
$67$	5.62372	+	9.74058i	0.687047	+	1.19000i	−0.285741	+	0.958307i	$0.592240\pi$
$68$		0			0
$69$	1.84847	+	4.01229i	0.222530	+	0.483023i
$70$		0			0
$71$	9.79796			1.16280			0.581402	−	0.813617i	$-0.302504\pi$
$71$	9.79796			1.16280			0.581402	+	0.813617i	$0.302504\pi$
$72$		0			0
$73$	−5.79796			−0.678600			−0.339300	−	0.940678i	$-0.610190\pi$
$73$	−5.79796			−0.678600			−0.339300	+	0.940678i	$0.610190\pi$
$74$		0			0
$75$	1.00000	−	1.41421i	0.115470	−	0.163299i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.44949	−	2.51059i	0.163080	−	0.282463i	−0.772892	−	0.634538i	$-0.781190\pi$
$79$	1.44949	−	2.51059i	0.163080	−	0.282463i	0.935972	+	0.352075i	$0.114524\pi$
$80$		0			0
$81$	8.39898	+	3.23375i	0.933220	+	0.359306i
$82$		0			0
$83$	−0.275255	+	0.476756i	−0.0302132	+	0.0523308i	−0.880737	−	0.473606i	$-0.842952\pi$
$83$	−0.275255	+	0.476756i	−0.0302132	+	0.0523308i	0.850523	+	0.525937i	$0.176285\pi$
$84$		0			0
$85$	−1.00000	−	1.73205i	−0.108465	−	0.187867i
$86$		0			0
$87$	7.89898	−	11.1708i	0.846859	−	1.19764i
$88$		0			0
$89$	−16.7980			−1.78058			−0.890290	−	0.455394i	$-0.849498\pi$
$89$	−16.7980			−1.78058			−0.890290	+	0.455394i	$0.849498\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−7.89898	−	17.1455i	−0.819086	−	1.77791i
$94$		0			0
$95$	−1.44949	−	2.51059i	−0.148715	−	0.257581i
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	−0.811976	−	0.583691i	$-0.801608\pi$
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	0.911479	+	0.411346i	$0.134941\pi$
$102$		0			0
$103$	5.00000	+	8.66025i	0.492665	+	0.853320i	0.999964	−	0.00844953i	$-0.00268960\pi$
$103$	5.00000	+	8.66025i	0.492665	+	0.853320i	−0.507300	+	0.861770i	$0.669356\pi$
$104$		0			0
$105$	2.50000	+	0.230351i	0.243975	+	0.0224799i
$106$		0			0
$107$	−2.34847			−0.227035			−0.113518	−	0.993536i	$-0.536212\pi$
$107$	−2.34847			−0.227035			−0.113518	+	0.993536i	$0.536212\pi$
$108$		0			0
$109$	8.79796			0.842692			0.421346	−	0.906900i	$-0.361558\pi$
$109$	8.79796			0.842692			0.421346	+	0.906900i	$0.361558\pi$
$110$		0			0
$111$	10.3485	+	0.953512i	0.982233	+	0.0905033i
$112$		0			0
$113$	−4.89898	−	8.48528i	−0.460857	−	0.798228i	0.538147	−	0.842851i	$-0.319125\pi$
$113$	−4.89898	−	8.48528i	−0.460857	−	0.798228i	−0.999004	+	0.0446231i	$0.985791\pi$
$114$		0			0
$115$	1.27526	−	2.20881i	0.118918	−	0.205972i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.44949	−	2.51059i	0.132875	−	0.230145i
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$	−0.0732141	−	0.158919i	−0.00660149	−	0.0143292i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	14.3485			1.27322			0.636610	−	0.771186i	$-0.280336\pi$
$127$	14.3485			1.27322			0.636610	+	0.771186i	$0.280336\pi$
$128$		0			0
$129$	7.79796	−	11.0280i	0.686572	−	0.970959i
$130$		0			0
$131$	−3.44949	−	5.97469i	−0.301383	−	0.522011i	0.675066	−	0.737757i	$-0.264115\pi$
$131$	−3.44949	−	5.97469i	−0.301383	−	0.522011i	−0.976450	+	0.215746i	$0.930782\pi$
$132$		0			0
$133$	2.10102	−	3.63907i	0.182182	−	0.315548i
$134$		0			0
$135$	−1.27526	−	5.03723i	−0.109756	−	0.433536i
$136$		0			0
$137$	9.79796	−	16.9706i	0.837096	−	1.44989i	−0.0552162	−	0.998474i	$-0.517585\pi$
$137$	9.79796	−	16.9706i	0.837096	−	1.44989i	0.892312	−	0.451419i	$-0.149082\pi$
$138$		0			0
$139$	−9.79796	−	16.9706i	−0.831052	−	1.43942i	−0.897205	−	0.441615i	$-0.854406\pi$
$139$	−9.79796	−	16.9706i	−0.831052	−	1.43942i	0.0661527	−	0.997810i	$-0.478928\pi$
$140$		0			0
$141$	4.55051	−	6.43539i	0.383222	−	0.541958i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−7.89898			−0.655975
$146$		0			0
$147$	−3.55051	−	7.70674i	−0.292841	−	0.635641i
$148$		0			0
$149$	10.5000	+	18.1865i	0.860194	+	1.48990i	0.871742	+	0.489966i	$0.162991\pi$
$149$	10.5000	+	18.1865i	0.860194	+	1.48990i	−0.0115483	+	0.999933i	$0.503676\pi$
$150$		0			0
$151$	6.00000	−	10.3923i	0.488273	−	0.845714i	−0.511636	−	0.859202i	$-0.670960\pi$
$151$	6.00000	−	10.3923i	0.488273	−	0.845714i	0.999909	+	0.0134886i	$0.00429367\pi$
$152$		0			0
$153$	−5.89898	−	1.09638i	−0.476904	−	0.0886368i
$154$		0			0
$155$	−5.44949	+	9.43879i	−0.437714	+	0.758142i
$156$		0			0
$157$	2.10102	+	3.63907i	0.167680	+	0.290430i	0.937604	−	0.347706i	$-0.113039\pi$
$157$	2.10102	+	3.63907i	0.167680	+	0.290430i	−0.769924	+	0.638136i	$0.779706\pi$
$158$		0			0
$159$	20.3485	+	1.87492i	1.61374	+	0.148690i
$160$		0			0
$161$	3.69694			0.291360
$162$		0			0
$163$	−11.7980			−0.924087			−0.462044	−	0.886857i	$-0.652884\pi$
$163$	−11.7980			−0.924087			−0.462044	+	0.886857i	$0.652884\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.17423	−	5.49794i	−0.245630	−	0.425443i	0.716679	−	0.697403i	$-0.245661\pi$
$167$	−3.17423	−	5.49794i	−0.245630	−	0.425443i	−0.962308	+	0.271960i	$0.912328\pi$
$168$		0			0
$169$	6.50000	−	11.2583i	0.500000	−	0.866025i
$170$		0			0
$171$	−8.55051	−	1.58919i	−0.653874	−	0.121528i
$172$		0			0
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	−0.542583	−	0.840002i	$-0.682554\pi$
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	0.998755	+	0.0498898i	$0.0158870\pi$
$174$		0			0
$175$	−0.724745	−	1.25529i	−0.0547856	−	0.0948914i
$176$		0			0
$177$	7.89898	+	17.1455i	0.593724	+	1.28874i
$178$		0			0
$179$	−5.79796			−0.433360			−0.216680	−	0.976243i	$-0.569523\pi$
$179$	−5.79796			−0.433360			−0.216680	+	0.976243i	$0.569523\pi$
$180$		0			0
$181$	19.6969			1.46406			0.732031	−	0.681271i	$-0.238573\pi$
$181$	19.6969			1.46406			0.732031	+	0.681271i	$0.238573\pi$
$182$		0			0
$183$	−3.00000	+	4.24264i	−0.221766	+	0.313625i
$184$		0			0
$185$	−3.00000	−	5.19615i	−0.220564	−	0.382029i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	5.39898	−	5.25153i	0.392718	−	0.381993i
$190$		0			0
$191$	−2.55051	+	4.41761i	−0.184548	+	0.319647i	−0.943424	−	0.331588i	$-0.892416\pi$
$191$	−2.55051	+	4.41761i	−0.184548	+	0.319647i	0.758876	+	0.651235i	$0.225749\pi$
$192$		0			0
$193$	10.8990	+	18.8776i	0.784526	+	1.35884i	0.929282	+	0.369371i	$0.120427\pi$
$193$	10.8990	+	18.8776i	0.784526	+	1.35884i	−0.144756	+	0.989467i	$0.546240\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	23.5959			1.68114			0.840570	−	0.541703i	$-0.182220\pi$
$197$	23.5959			1.68114			0.840570	+	0.541703i	$0.182220\pi$
$198$		0			0
$199$	−13.1010			−0.928707			−0.464353	−	0.885650i	$-0.653713\pi$
$199$	−13.1010			−0.928707			−0.464353	+	0.885650i	$0.653713\pi$
$200$		0			0
$201$	−8.15153	−	17.6937i	−0.574965	−	1.24802i
$202$		0			0
$203$	−5.72474	−	9.91555i	−0.401798	−	0.695935i
$204$		0			0
$205$	−0.0505103	+	0.0874863i	−0.00352779	+	0.00611031i
$206$		0			0
$207$	−2.55051	−	7.21393i	−0.177273	−	0.501403i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	0.995222	−	0.0976347i	$-0.0311277\pi$
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	−0.582165	+	0.813070i	$0.697794\pi$
$212$		0			0
$213$	−16.8990	−	1.55708i	−1.15790	−	0.106689i
$214$		0			0
$215$	−7.79796			−0.531816
$216$		0			0
$217$	−15.7980			−1.07244
$218$		0			0
$219$	10.0000	+	0.921404i	0.675737	+	0.0622627i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.275255	−	0.476756i	0.0184324	−	0.0319259i	−0.856662	−	0.515878i	$-0.827466\pi$
$223$	0.275255	−	0.476756i	0.0184324	−	0.0319259i	0.875094	+	0.483952i	$0.160799\pi$
$224$		0			0
$225$	−1.94949	+	2.28024i	−0.129966	+	0.152016i
$226$		0			0
$227$	−13.8990	+	24.0737i	−0.922508	+	1.59783i	−0.126986	+	0.991904i	$0.540530\pi$
$227$	−13.8990	+	24.0737i	−0.922508	+	1.59783i	−0.795521	+	0.605926i	$0.792803\pi$
$228$		0			0
$229$	6.94949	+	12.0369i	0.459235	+	0.795419i	0.998921	−	0.0464480i	$-0.0147902\pi$
$229$	6.94949	+	12.0369i	0.459235	+	0.795419i	−0.539686	+	0.841867i	$0.681457\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.202041			−0.0132361			−0.00661807	−	0.999978i	$-0.502107\pi$
$233$	−0.202041			−0.0132361			−0.00661807	+	0.999978i	$0.502107\pi$
$234$		0			0
$235$	−4.55051			−0.296843
$236$		0			0
$237$	−2.89898	+	4.09978i	−0.188309	+	0.266309i
$238$		0			0
$239$	10.8990	+	18.8776i	0.704996	+	1.22109i	0.966693	+	0.255939i	$0.0823847\pi$
$239$	10.8990	+	18.8776i	0.704996	+	1.22109i	−0.261696	+	0.965150i	$0.584282\pi$
$240$		0			0
$241$	−12.8485	+	22.2542i	−0.827643	+	1.43352i	0.0722401	+	0.997387i	$0.476985\pi$
$241$	−12.8485	+	22.2542i	−0.827643	+	1.43352i	−0.899883	+	0.436132i	$0.856348\pi$
$242$		0			0
$243$	−13.9722	−	6.91215i	−0.896317	−	0.443415i
$244$		0			0
$245$	−2.44949	+	4.24264i	−0.156492	+	0.271052i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.550510	−	0.778539i	0.0348872	−	0.0493379i
$250$		0			0
$251$	6.89898			0.435460			0.217730	−	0.976009i	$-0.430135\pi$
$251$	6.89898			0.435460			0.217730	+	0.976009i	$0.430135\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	1.44949	+	3.14626i	0.0907706	+	0.197027i
$256$		0			0
$257$	−4.10102	−	7.10318i	−0.255815	−	0.443084i	0.709302	−	0.704905i	$-0.249010\pi$
$257$	−4.10102	−	7.10318i	−0.255815	−	0.443084i	−0.965116	+	0.261821i	$0.915677\pi$
$258$		0			0
$259$	4.34847	−	7.53177i	0.270201	−	0.468001i
$260$		0			0
$261$	−15.3990	+	18.0116i	−0.953173	+	1.11489i
$262$		0			0
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	−0.442943	−	0.896550i	$-0.646065\pi$
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	0.997906	−	0.0646755i	$-0.0206012\pi$
$264$		0			0
$265$	−5.89898	−	10.2173i	−0.362371	−	0.627646i
$266$		0			0
$267$	28.9722	+	2.66951i	1.77307	+	0.163371i
$268$		0			0
$269$	−16.5959			−1.01187			−0.505935	−	0.862571i	$-0.668853\pi$
$269$	−16.5959			−1.01187			−0.505935	+	0.862571i	$0.668853\pi$
$270$		0			0
$271$	−21.1010			−1.28180			−0.640898	−	0.767626i	$-0.721438\pi$
$271$	−21.1010			−1.28180			−0.640898	+	0.767626i	$0.721438\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7.00000	−	12.1244i	0.420589	−	0.728482i	−0.575408	−	0.817867i	$-0.695157\pi$
$277$	7.00000	−	12.1244i	0.420589	−	0.728482i	0.995997	+	0.0893846i	$0.0284900\pi$
$278$		0			0
$279$	10.8990	+	30.8270i	0.652505	+	1.84556i
$280$		0			0
$281$	8.94949	−	15.5010i	0.533882	−	0.924710i	−0.465335	−	0.885135i	$-0.654066\pi$
$281$	8.94949	−	15.5010i	0.533882	−	0.924710i	0.999217	−	0.0395756i	$-0.0126006\pi$
$282$		0			0
$283$	−9.17423	−	15.8902i	−0.545352	−	0.944577i	−0.998585	−	0.0531847i	$-0.983063\pi$
$283$	−9.17423	−	15.8902i	−0.545352	−	0.944577i	0.453233	−	0.891392i	$-0.350271\pi$
$284$		0			0
$285$	2.10102	+	4.56048i	0.124454	+	0.270139i
$286$		0			0
$287$	−0.146428			−0.00864338
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$	2.00000	−	2.82843i	0.117242	−	0.165805i
$292$		0			0
$293$	10.7980	+	18.7026i	0.630823	+	1.09262i	0.987384	+	0.158346i	$0.0506161\pi$
$293$	10.7980	+	18.7026i	0.630823	+	1.09262i	−0.356560	+	0.934272i	$0.616051\pi$
$294$		0			0
$295$	5.44949	−	9.43879i	0.317282	−	0.549548i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−5.65153	−	9.78874i	−0.325749	−	0.564214i
$302$		0			0
$303$	−2.00000	+	2.82843i	−0.114897	+	0.162489i
$304$		0			0
$305$	3.00000			0.171780
$306$		0			0
$307$	29.2474			1.66924			0.834620	−	0.550826i	$-0.185687\pi$
$307$	29.2474			1.66924			0.834620	+	0.550826i	$0.185687\pi$
$308$		0			0
$309$	−7.24745	−	15.7313i	−0.412293	−	0.894924i
$310$		0			0
$311$	1.44949	+	2.51059i	0.0821930	+	0.142362i	0.904192	−	0.427127i	$-0.140474\pi$
$311$	1.44949	+	2.51059i	0.0821930	+	0.142362i	−0.821999	+	0.569489i	$0.807141\pi$
$312$		0			0
$313$	−11.7980	+	20.4347i	−0.666860	+	1.15504i	0.311917	+	0.950109i	$0.399029\pi$
$313$	−11.7980	+	20.4347i	−0.666860	+	1.15504i	−0.978777	+	0.204926i	$0.934305\pi$
$314$		0			0
$315$	−4.27526	−	0.794593i	−0.240883	−	0.0447703i
$316$		0			0
$317$	3.10102	−	5.37113i	0.174171	−	0.301672i	−0.765703	−	0.643194i	$-0.777609\pi$
$317$	3.10102	−	5.37113i	0.174171	−	0.301672i	0.939874	+	0.341522i	$0.110942\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	4.05051	+	0.373215i	0.226077	+	0.0208309i
$322$		0			0
$323$	5.79796			0.322607
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−15.1742	−	1.39816i	−0.839137	−	0.0773184i
$328$		0			0
$329$	−3.29796	−	5.71223i	−0.181822	−	0.314926i
$330$		0			0
$331$	1.44949	−	2.51059i	0.0796712	−	0.137994i	−0.823437	−	0.567408i	$-0.807946\pi$
$331$	1.44949	−	2.51059i	0.0796712	−	0.137994i	0.903108	+	0.429413i	$0.141280\pi$
$332$		0			0
$333$	−17.6969	−	3.28913i	−0.969786	−	0.180243i
$334$		0			0
$335$	−5.62372	+	9.74058i	−0.307257	+	0.532185i
$336$		0			0
$337$	5.10102	+	8.83523i	0.277870	+	0.481285i	0.970855	−	0.239667i	$-0.0770381\pi$
$337$	5.10102	+	8.83523i	0.277870	+	0.481285i	−0.692985	+	0.720952i	$0.743705\pi$
$338$		0			0
$339$	7.10102	+	15.4135i	0.385674	+	0.837146i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−17.2474			−0.931275
$344$		0			0
$345$	−2.55051	+	3.60697i	−0.137315	+	0.194193i
$346$		0			0
$347$	6.79796	+	11.7744i	0.364934	+	0.632083i	0.988765	−	0.149475i	$-0.0477584\pi$
$347$	6.79796	+	11.7744i	0.364934	+	0.632083i	−0.623832	+	0.781559i	$0.714425\pi$
$348$		0			0
$349$	15.8485	−	27.4504i	0.848349	−	1.46938i	−0.0343315	−	0.999410i	$-0.510930\pi$
$349$	15.8485	−	27.4504i	0.848349	−	1.46938i	0.882681	−	0.469973i	$-0.155736\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$	4.89898	+	8.48528i	0.260011	+	0.450352i
$356$		0			0
$357$	−2.89898	+	4.09978i	−0.153430	+	0.216983i
$358$		0			0
$359$	0.696938			0.0367830			0.0183915	−	0.999831i	$-0.494145\pi$
$359$	0.696938			0.0367830			0.0183915	+	0.999831i	$0.494145\pi$
$360$		0			0
$361$	−10.5959			−0.557680
$362$		0			0
$363$	−7.97219	−	17.3045i	−0.418432	−	0.908248i
$364$		0			0
$365$	−2.89898	−	5.02118i	−0.151740	−	0.262821i
$366$		0			0
$367$	11.0000	−	19.0526i	0.574195	−	0.994535i	−0.421933	−	0.906627i	$-0.638648\pi$
$367$	11.0000	−	19.0526i	0.574195	−	0.994535i	0.996129	−	0.0879086i	$-0.0280183\pi$
$368$		0			0
$369$	0.101021	+	0.285729i	0.00525892	+	0.0148745i
$370$		0			0
$371$	8.55051	−	14.8099i	0.443920	−	0.768893i
$372$		0			0
$373$	8.79796	+	15.2385i	0.455541	+	0.789020i	0.998719	−	0.0505973i	$-0.0161125\pi$
$373$	8.79796	+	15.2385i	0.455541	+	0.789020i	−0.543178	+	0.839618i	$0.682779\pi$
$374$		0			0
$375$	1.72474	+	0.158919i	0.0890654	+	0.00820652i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−17.1010			−0.878420			−0.439210	−	0.898384i	$-0.644742\pi$
$379$	−17.1010			−0.878420			−0.439210	+	0.898384i	$0.644742\pi$
$380$		0			0
$381$	−24.7474	−	2.28024i	−1.26785	−	0.116820i
$382$		0			0
$383$	1.00000	+	1.73205i	0.0510976	+	0.0885037i	0.890443	−	0.455095i	$-0.150395\pi$
$383$	1.00000	+	1.73205i	0.0510976	+	0.0885037i	−0.839345	+	0.543599i	$0.817061\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−15.2020	+	17.7812i	−0.772763	+	0.903870i
$388$		0			0
$389$	−11.2980	+	19.5686i	−0.572829	+	0.992169i	0.423444	+	0.905922i	$0.360821\pi$
$389$	−11.2980	+	19.5686i	−0.572829	+	0.992169i	−0.996274	+	0.0862473i	$0.972512\pi$
$390$		0			0
$391$	2.55051	+	4.41761i	0.128985	+	0.223408i
$392$		0			0
$393$	5.00000	+	10.8530i	0.252217	+	0.547462i
$394$		0			0
$395$	2.89898			0.145863
$396$		0			0
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$		0			0
$399$	−4.20204	+	5.94258i	−0.210365	+	0.297501i
$400$		0			0
$401$	6.79796	+	11.7744i	0.339474	+	0.587986i	0.984334	−	0.176315i	$-0.0564177\pi$
$401$	6.79796	+	11.7744i	0.339474	+	0.587986i	−0.644860	+	0.764301i	$0.723084\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.39898	+	8.89060i	0.0695158	+	0.441778i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−14.7980	−	25.6308i	−0.731712	−	1.26736i	−0.956151	−	0.292874i	$-0.905388\pi$
$409$	−14.7980	−	25.6308i	−0.731712	−	1.26736i	0.224439	−	0.974488i	$-0.427945\pi$
$410$		0			0
$411$	−19.5959	+	27.7128i	−0.966595	+	1.36697i
$412$		0			0
$413$	15.7980			0.777367
$414$		0			0
$415$	−0.550510			−0.0270235
$416$		0			0
$417$	14.2020	+	30.8270i	0.695477	+	1.50960i
$418$		0			0
$419$	10.0000	+	17.3205i	0.488532	+	0.846162i	0.999913	−	0.0131919i	$-0.00419923\pi$
$419$	10.0000	+	17.3205i	0.488532	+	0.846162i	−0.511381	+	0.859354i	$0.670866\pi$
$420$		0			0
$421$	6.79796	−	11.7744i	0.331312	−	0.573850i	−0.651457	−	0.758685i	$-0.725842\pi$
$421$	6.79796	−	11.7744i	0.331312	−	0.573850i	0.982769	+	0.184836i	$0.0591753\pi$
$422$		0			0
$423$	−8.87117	+	10.3763i	−0.431331	+	0.504511i
$424$		0			0
$425$	1.00000	−	1.73205i	0.0485071	−	0.0840168i
$426$		0			0
$427$	2.17423	+	3.76588i	0.105219	+	0.182244i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.10102			−0.245708			−0.122854	−	0.992425i	$-0.539205\pi$
$431$	−5.10102			−0.245708			−0.122854	+	0.992425i	$0.539205\pi$
$432$		0			0
$433$	7.79796			0.374746			0.187373	−	0.982289i	$-0.440003\pi$
$433$	7.79796			0.374746			0.187373	+	0.982289i	$0.440003\pi$
$434$		0			0
$435$	13.6237	+	1.25529i	0.653208	+	0.0601868i
$436$		0			0
$437$	3.69694	+	6.40329i	0.176849	+	0.306311i
$438$		0			0
$439$	0.898979	−	1.55708i	0.0429059	−	0.0743153i	−0.843775	−	0.536697i	$-0.819672\pi$
$439$	0.898979	−	1.55708i	0.0429059	−	0.0743153i	0.886681	+	0.462382i	$0.153005\pi$
$440$		0			0
$441$	4.89898	+	13.8564i	0.233285	+	0.659829i
$442$		0			0
$443$	13.0732	−	22.6435i	0.621127	−	1.07582i	−0.368149	−	0.929767i	$-0.620008\pi$
$443$	13.0732	−	22.6435i	0.621127	−	1.07582i	0.989276	−	0.146057i	$-0.0466583\pi$
$444$		0			0
$445$	−8.39898	−	14.5475i	−0.398150	−	0.689616i
$446$		0			0
$447$	−15.2196	−	33.0358i	−0.719864	−	1.56254i
$448$		0			0
$449$	−17.5959			−0.830403			−0.415201	−	0.909730i	$-0.636289\pi$
$449$	−17.5959			−0.830403			−0.415201	+	0.909730i	$0.636289\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−12.0000	+	16.9706i	−0.563809	+	0.797347i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	10.7980	−	18.7026i	0.505107	−	0.874871i	−0.494875	−	0.868964i	$-0.664786\pi$
$457$	10.7980	−	18.7026i	0.505107	−	0.874871i	0.999983	−	0.00590738i	$-0.00188039\pi$
$458$		0			0
$459$	10.0000	+	2.82843i	0.466760	+	0.132020i
$460$		0			0
$461$	−10.9495	+	18.9651i	−0.509969	+	0.883291i	0.489965	+	0.871742i	$0.337010\pi$
$461$	−10.9495	+	18.9651i	−0.509969	+	0.883291i	−0.999933	+	0.0115492i	$0.996324\pi$
$462$		0			0
$463$	15.8990	+	27.5378i	0.738888	+	1.27979i	0.952996	+	0.302982i	$0.0979822\pi$
$463$	15.8990	+	27.5378i	0.738888	+	1.27979i	−0.214108	+	0.976810i	$0.568684\pi$
$464$		0			0
$465$	10.8990	−	15.4135i	0.505428	−	0.714783i
$466$		0			0
$467$	11.7980			0.545944			0.272972	−	0.962022i	$-0.411993\pi$
$467$	11.7980			0.545944			0.272972	+	0.962022i	$0.411993\pi$
$468$		0			0
$469$	−16.3031			−0.752805
$470$		0			0
$471$	−3.04541	−	6.61037i	−0.140325	−	0.304590i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.44949	−	2.51059i	0.0665072	−	0.115194i
$476$		0			0
$477$	−34.7980	−	6.46750i	−1.59329	−	0.296127i
$478$		0			0
$479$	−9.24745	+	16.0171i	−0.422527	+	0.731838i	−0.996186	−	0.0872564i	$-0.972190\pi$
$479$	−9.24745	+	16.0171i	−0.422527	+	0.731838i	0.573659	+	0.819094i	$0.305523\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−6.37628	−	0.587512i	−0.290131	−	0.0267327i
$484$		0			0
$485$	−2.00000			−0.0908153
$486$		0			0
$487$	−25.5959			−1.15986			−0.579931	−	0.814666i	$-0.696920\pi$
$487$	−25.5959			−1.15986			−0.579931	+	0.814666i	$0.696920\pi$
$488$		0			0
$489$	20.3485	+	1.87492i	0.920190	+	0.0847866i
$490$		0			0
$491$	2.89898	+	5.02118i	0.130829	+	0.226603i	0.923996	−	0.382401i	$-0.124903\pi$
$491$	2.89898	+	5.02118i	0.130829	+	0.226603i	−0.793167	+	0.609004i	$0.791569\pi$
$492$		0			0
$493$	7.89898	−	13.6814i	0.355752	−	0.616181i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.10102	+	12.2993i	−0.318524	+	0.551700i
$498$		0			0
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	−0.988204	−	0.153141i	$-0.951061\pi$
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	0.361478	−	0.932381i	$-0.382272\pi$
$500$		0			0
$501$	4.60102	+	9.98698i	0.205558	+	0.446185i
$502$		0			0
$503$	19.0454			0.849193			0.424596	−	0.905383i	$-0.360416\pi$
$503$	19.0454			0.849193			0.424596	+	0.905383i	$0.360416\pi$
$504$		0			0
$505$	2.00000			0.0889988
$506$		0			0
$507$	−13.0000	+	18.3848i	−0.577350	+	0.816497i
$508$		0			0
$509$	6.94949	+	12.0369i	0.308031	+	0.533525i	0.977931	−	0.208926i	$-0.0669967\pi$
$509$	6.94949	+	12.0369i	0.308031	+	0.533525i	−0.669901	+	0.742451i	$0.733663\pi$
$510$		0			0
$511$	4.20204	−	7.27815i	0.185887	−	0.321966i
$512$		0			0
$513$	14.4949	+	4.09978i	0.639965	+	0.181010i
$514$		0			0
$515$	−5.00000	+	8.66025i	−0.220326	+	0.381616i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−12.0000	+	16.9706i	−0.526742	+	0.744925i
$520$		0			0
$521$	28.7980			1.26166			0.630831	−	0.775920i	$-0.282714\pi$
$521$	28.7980			1.26166			0.630831	+	0.775920i	$0.282714\pi$
$522$		0			0
$523$	19.6515			0.859301			0.429651	−	0.902995i	$-0.358637\pi$
$523$	19.6515			0.859301			0.429651	+	0.902995i	$0.358637\pi$
$524$		0			0
$525$	1.05051	+	2.28024i	0.0458480	+	0.0995178i
$526$		0			0
$527$	−10.8990	−	18.8776i	−0.474767	−	0.822321i
$528$		0			0
$529$	8.24745	−	14.2850i	0.358585	−	0.621087i
$530$		0			0
$531$	−10.8990	−	30.8270i	−0.472975	−	1.33778i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.17423	−	2.03383i	−0.0507666	−	0.0879303i
$536$		0			0
$537$	10.0000	+	0.921404i	0.431532	+	0.0397615i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	21.8990			0.941511			0.470755	−	0.882264i	$-0.343981\pi$
$541$	21.8990			0.941511			0.470755	+	0.882264i	$0.343981\pi$
$542$		0			0
$543$	−33.9722	−	3.13021i	−1.45789	−	0.134330i
$544$		0			0
$545$	4.39898	+	7.61926i	0.188432	+	0.326373i
$546$		0			0
$547$	−20.6237	+	35.7213i	−0.881807	+	1.52733i	−0.0324764	+	0.999473i	$0.510339\pi$
$547$	−20.6237	+	35.7213i	−0.881807	+	1.52733i	−0.849330	+	0.527862i	$0.822994\pi$
$548$		0			0
$549$	5.84847	−	6.84072i	0.249607	−	0.291955i
$550$		0			0
$551$	11.4495	−	19.8311i	0.487765	−	0.844833i
$552$		0			0
$553$	2.10102	+	3.63907i	0.0893445	+	0.154749i
$554$		0			0
$555$	4.34847	+	9.43879i	0.184582	+	0.400654i
$556$		0			0
$557$	24.2020			1.02547			0.512737	−	0.858546i	$-0.328632\pi$
$557$	24.2020			1.02547			0.512737	+	0.858546i	$0.328632\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.17423	+	3.76588i	0.0916331	+	0.158713i	0.908198	−	0.418540i	$-0.137458\pi$
$563$	2.17423	+	3.76588i	0.0916331	+	0.158713i	−0.816565	+	0.577253i	$0.804125\pi$
$564$		0			0
$565$	4.89898	−	8.48528i	0.206102	−	0.356978i
$566$		0			0
$567$	−10.1464	+	8.19955i	−0.426110	+	0.344349i
$568$		0			0
$569$	−5.00000	+	8.66025i	−0.209611	+	0.363057i	−0.951592	−	0.307364i	$-0.900553\pi$
$569$	−5.00000	+	8.66025i	−0.209611	+	0.363057i	0.741981	+	0.670421i	$0.233886\pi$
$570$		0			0
$571$	3.10102	+	5.37113i	0.129774	+	0.224775i	0.923589	−	0.383385i	$-0.125242\pi$
$571$	3.10102	+	5.37113i	0.129774	+	0.224775i	−0.793815	+	0.608159i	$0.791908\pi$
$572$		0			0
$573$	5.10102	−	7.21393i	0.213098	−	0.301366i
$574$		0			0
$575$	2.55051			0.106364
$576$		0			0
$577$	−26.0000			−1.08239			−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000			−1.08239			−0.541197	+	0.840896i	$0.682029\pi$
$578$		0			0
$579$	−15.7980	−	34.2911i	−0.656541	−	1.42509i
$580$		0			0
$581$	−0.398979	−	0.691053i	−0.0165525	−	0.0286697i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	11.1742	−	19.3543i	0.461210	−	0.798839i	−0.537812	−	0.843065i	$-0.680749\pi$
$587$	11.1742	−	19.3543i	0.461210	−	0.798839i	0.999022	+	0.0442259i	$0.0140821\pi$
$588$		0			0
$589$	−15.7980	−	27.3629i	−0.650944	−	1.12747i
$590$		0			0
$591$	−40.6969	−	3.74983i	−1.67405	−	0.154247i
$592$		0			0
$593$	27.3939			1.12493			0.562466	−	0.826821i	$-0.309853\pi$
$593$	27.3939			1.12493			0.562466	+	0.826821i	$0.309853\pi$
$594$		0			0
$595$	2.89898			0.118847
$596$		0			0
$597$	22.5959	+	2.08200i	0.924789	+	0.0852104i
$598$		0			0
$599$	2.34847	+	4.06767i	0.0959559	+	0.166200i	0.910007	−	0.414593i	$-0.136076\pi$
$599$	2.34847	+	4.06767i	0.0959559	+	0.166200i	−0.814051	+	0.580793i	$0.802743\pi$
$600$		0			0
$601$	−7.00000	+	12.1244i	−0.285536	+	0.494563i	−0.972739	−	0.231903i	$-0.925505\pi$
$601$	−7.00000	+	12.1244i	−0.285536	+	0.494563i	0.687203	+	0.726465i	$0.258838\pi$
$602$		0			0
$603$	11.2474	+	31.8126i	0.458032	+	1.29551i
$604$		0			0
$605$	−5.50000	+	9.52628i	−0.223607	+	0.387298i
$606$		0			0
$607$	−16.0732	−	27.8396i	−0.652392	−	1.12998i	−0.982541	−	0.186047i	$-0.940432\pi$
$607$	−16.0732	−	27.8396i	−0.652392	−	1.12998i	0.330149	−	0.943929i	$-0.392901\pi$
$608$		0			0
$609$	8.29796	+	18.0116i	0.336250	+	0.729865i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	9.59592			0.387575			0.193788	−	0.981043i	$-0.437923\pi$
$613$	9.59592			0.387575			0.193788	+	0.981043i	$0.437923\pi$
$614$		0			0
$615$	0.101021	−	0.142865i	0.00407354	−	0.00576086i
$616$		0			0
$617$	−21.5959	−	37.4052i	−0.869419	−	1.50588i	−0.862592	−	0.505901i	$-0.831160\pi$
$617$	−21.5959	−	37.4052i	−0.869419	−	1.50588i	−0.00682740	−	0.999977i	$-0.502173\pi$
$618$		0			0
$619$	2.34847	−	4.06767i	0.0943929	−	0.163493i	−0.814962	−	0.579514i	$-0.803242\pi$
$619$	2.34847	−	4.06767i	0.0943929	−	0.163493i	0.909355	+	0.416021i	$0.136576\pi$
$620$		0			0
$621$	3.25255	+	12.8475i	0.130520	+	0.515553i
$622$		0			0
$623$	12.1742	−	21.0864i	0.487750	−	0.844808i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	−23.5959			−0.939339			−0.469669	−	0.882842i	$-0.655627\pi$
$631$	−23.5959			−0.939339			−0.469669	+	0.882842i	$0.655627\pi$
$632$		0			0
$633$	−8.69694	−	18.8776i	−0.345672	−	0.750317i
$634$		0			0
$635$	7.17423	+	12.4261i	0.284701	+	0.493116i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	28.8990	+	5.37113i	1.14323	+	0.212478i
$640$		0			0
$641$	2.05051	−	3.55159i	0.0809903	−	0.140279i	−0.822685	−	0.568497i	$-0.807525\pi$
$641$	2.05051	−	3.55159i	0.0809903	−	0.140279i	0.903676	+	0.428218i	$0.140858\pi$
$642$		0			0
$643$	7.07321	+	12.2512i	0.278940	+	0.483139i	0.971122	−	0.238585i	$-0.0766835\pi$
$643$	7.07321	+	12.2512i	0.278940	+	0.483139i	−0.692181	+	0.721724i	$0.743350\pi$
$644$		0			0
$645$	13.4495	+	1.23924i	0.529573	+	0.0487951i
$646$		0			0
$647$	−11.4495			−0.450126			−0.225063	−	0.974344i	$-0.572259\pi$
$647$	−11.4495			−0.450126			−0.225063	+	0.974344i	$0.572259\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	27.2474	+	2.51059i	1.06791	+	0.0983978i
$652$		0			0
$653$	−4.10102	−	7.10318i	−0.160485	−	0.277969i	0.774558	−	0.632503i	$-0.217973\pi$
$653$	−4.10102	−	7.10318i	−0.160485	−	0.277969i	−0.935043	+	0.354535i	$0.884639\pi$
$654$		0			0
$655$	3.44949	−	5.97469i	0.134783	−	0.233451i
$656$		0			0
$657$	−17.1010	−	3.17837i	−0.667174	−	0.124000i
$658$		0			0
$659$	14.8990	−	25.8058i	0.580382	−	1.00525i	−0.415052	−	0.909798i	$-0.636237\pi$
$659$	14.8990	−	25.8058i	0.580382	−	1.00525i	0.995434	−	0.0954532i	$-0.0304300\pi$
$660$		0			0
$661$	−5.00000	−	8.66025i	−0.194477	−	0.336845i	0.752252	−	0.658876i	$-0.228968\pi$
$661$	−5.00000	−	8.66025i	−0.194477	−	0.336845i	−0.946729	+	0.322031i	$0.895634\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.20204			0.162948
$666$		0			0
$667$	20.1464			0.780073
$668$		0			0
$669$	−0.550510	+	0.778539i	−0.0212840	+	0.0301001i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	8.00000	−	13.8564i	0.308377	−	0.534125i	−0.669630	−	0.742695i	$-0.733547\pi$
$673$	8.00000	−	13.8564i	0.308377	−	0.534125i	0.978008	+	0.208569i	$0.0668807\pi$
$674$		0			0
$675$	3.72474	−	3.62302i	0.143365	−	0.139450i
$676$		0			0
$677$	−16.8990	+	29.2699i	−0.649481	+	1.12493i	0.333767	+	0.942656i	$0.391680\pi$
$677$	−16.8990	+	29.2699i	−0.649481	+	1.12493i	−0.983247	+	0.182278i	$0.941653\pi$
$678$		0			0
$679$	−1.44949	−	2.51059i	−0.0556263	−	0.0963476i
$680$		0			0
$681$	27.7980	−	39.3123i	1.06522	−	1.50645i
$682$		0			0
$683$	35.3939			1.35431			0.677155	−	0.735841i	$-0.263213\pi$
$683$	35.3939			1.35431			0.677155	+	0.735841i	$0.263213\pi$
$684$		0			0
$685$	19.5959			0.748722
$686$		0			0
$687$	−10.0732	−	21.8649i	−0.384317	−	0.834199i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	14.5505	−	25.2022i	0.553527	−	0.958738i	−0.444489	−	0.895784i	$-0.646615\pi$
$691$	14.5505	−	25.2022i	0.553527	−	0.958738i	0.998016	−	0.0629534i	$-0.0200520\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.79796	−	16.9706i	0.371658	−	0.643730i
$696$		0			0
$697$	−0.101021	−	0.174973i	−0.00382642	−	0.00662756i
$698$		0			0
$699$	0.348469	+	0.0321081i	0.0131803	+	0.00121444i
$700$		0			0
$701$	−28.3939			−1.07242			−0.536211	−	0.844084i	$-0.680145\pi$
$701$	−28.3939			−1.07242			−0.536211	+	0.844084i	$0.680145\pi$
$702$		0			0
$703$	17.3939			0.656022
$704$		0			0
$705$	7.84847	+	0.723161i	0.295590	+	0.0272358i
$706$		0			0
$707$	1.44949	+	2.51059i	0.0545137	+	0.0944205i
$708$		0			0
$709$	9.84847	−	17.0580i	0.369867	−	0.640628i	−0.619677	−	0.784857i	$-0.712737\pi$
$709$	9.84847	−	17.0580i	0.369867	−	0.640628i	0.989544	+	0.144228i	$0.0460699\pi$
$710$		0			0
$711$	5.65153	−	6.61037i	0.211949	−	0.247908i
$712$		0			0
$713$	13.8990	−	24.0737i	0.520521	−	0.901569i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−15.7980	−	34.2911i	−0.589986	−	1.28062i
$718$		0			0
$719$	−40.2929			−1.50267			−0.751335	−	0.659921i	$-0.770590\pi$
$719$	−40.2929			−1.50267			−0.751335	+	0.659921i	$0.770590\pi$
$720$		0			0
$721$	−14.4949			−0.539818
$722$		0			0
$723$	25.6969	−	36.3410i	0.955679	−	1.35153i
$724$		0			0
$725$	−3.94949	−	6.84072i	−0.146680	−	0.254058i
$726$		0			0
$727$	−3.37628	+	5.84788i	−0.125219	+	0.216886i	−0.921819	−	0.387622i	$-0.873297\pi$
$727$	−3.37628	+	5.84788i	−0.125219	+	0.216886i	0.796599	+	0.604508i	$0.206630\pi$
$728$		0			0
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$	7.79796	−	13.5065i	0.288418	−	0.499555i
$732$		0			0
$733$	−4.79796	−	8.31031i	−0.177217	−	0.306948i	0.763710	−	0.645560i	$-0.223376\pi$
$733$	−4.79796	−	8.31031i	−0.177217	−	0.306948i	−0.940926	+	0.338612i	$0.890043\pi$
$734$		0			0
$735$	4.89898	−	6.92820i	0.180702	−	0.255551i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	42.8990			1.57806			0.789032	−	0.614352i	$-0.210582\pi$
$739$	42.8990			1.57806			0.789032	+	0.614352i	$0.210582\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.5227	+	32.0823i	0.679532	+	1.17698i	0.975122	+	0.221669i	$0.0711505\pi$
$743$	18.5227	+	32.0823i	0.679532	+	1.17698i	−0.295590	+	0.955315i	$0.595516\pi$
$744$		0			0
$745$	−10.5000	+	18.1865i	−0.384690	+	0.666303i
$746$		0			0
$747$	−1.07321	+	1.25529i	−0.0392669	+	0.0459288i
$748$		0			0
$749$	1.70204	−	2.94802i	0.0621912	−	0.107718i
$750$		0			0
$751$	22.8990	+	39.6622i	0.835596	+	1.44729i	0.893545	+	0.448975i	$0.148211\pi$
$751$	22.8990	+	39.6622i	0.835596	+	1.44729i	−0.0579489	+	0.998320i	$0.518456\pi$
$752$		0			0
$753$	−11.8990	−	1.09638i	−0.433623	−	0.0399542i
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	−7.59592			−0.276078			−0.138039	−	0.990427i	$-0.544080\pi$
$757$	−7.59592			−0.276078			−0.138039	+	0.990427i	$0.544080\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.5000	−	30.3109i	−0.634375	−	1.09877i	−0.986647	−	0.162872i	$-0.947924\pi$
$761$	−17.5000	−	30.3109i	−0.634375	−	1.09877i	0.352273	−	0.935897i	$-0.385409\pi$
$762$		0			0
$763$	−6.37628	+	11.0440i	−0.230837	+	0.399821i
$764$		0			0
$765$	−2.00000	−	5.65685i	−0.0723102	−	0.204524i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	12.3990	+	21.4757i	0.447119	+	0.774432i	0.998197	−	0.0600212i	$-0.0191168\pi$
$769$	12.3990	+	21.4757i	0.447119	+	0.774432i	−0.551078	+	0.834453i	$0.685784\pi$
$770$		0			0
$771$	5.94439	+	12.9029i	0.214082	+	0.464686i
$772$		0			0
$773$	25.7980			0.927888			0.463944	−	0.885865i	$-0.346434\pi$
$773$	25.7980			0.927888			0.463944	+	0.885865i	$0.346434\pi$
$774$		0			0
$775$	−10.8990			−0.391503
$776$		0			0
$777$	−8.69694	+	12.2993i	−0.312001	+	0.441236i
$778$		0			0
$779$	−0.146428	−	0.253621i	−0.00524633	−	0.00908692i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	29.4217	−	28.6182i	1.05145	−	1.02273i
$784$		0			0
$785$	−2.10102	+	3.63907i	−0.0749886	+	0.129884i
$786$		0			0
$787$	−5.20204	−	9.01020i	−0.185433	−	0.321179i	0.758290	−	0.651918i	$-0.226035\pi$
$787$	−5.20204	−	9.01020i	−0.185433	−	0.321179i	−0.943722	+	0.330739i	$0.892702\pi$
$788$		0			0
$789$	−18.0000	+	25.4558i	−0.640817	+	0.906252i
$790$		0			0
$791$	14.2020			0.504966
$792$		0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.72474 - 0.158919i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-0.724745 + 1.25529i) q^{7} +(2.94949 + 0.548188i) q^{9} +O(q^{10})\)	\(q+(-1.72474 - 0.158919i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-0.724745 + 1.25529i) q^{7} +(2.94949 + 0.548188i) q^{9} +(-0.724745 - 1.57313i) q^{15} -2.00000 q^{17} -2.89898 q^{19} +(1.44949 - 2.04989i) q^{21} +(-1.27526 - 2.20881i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-5.00000 - 1.41421i) q^{27} +(-3.94949 + 6.84072i) q^{29} +(5.44949 + 9.43879i) q^{31} -1.44949 q^{35} -6.00000 q^{37} +(0.0505103 + 0.0874863i) q^{41} +(-3.89898 + 6.75323i) q^{43} +(1.00000 + 2.82843i) q^{45} +(-2.27526 + 3.94086i) q^{47} +(2.44949 + 4.24264i) q^{49} +(3.44949 + 0.317837i) q^{51} -11.7980 q^{53} +(5.00000 + 0.460702i) q^{57} +(-5.44949 - 9.43879i) q^{59} +(1.50000 - 2.59808i) q^{61} +(-2.82577 + 3.30518i) q^{63} +(5.62372 + 9.74058i) q^{67} +(1.84847 + 4.01229i) q^{69} +9.79796 q^{71} -5.79796 q^{73} +(1.00000 - 1.41421i) q^{75} +(1.44949 - 2.51059i) q^{79} +(8.39898 + 3.23375i) q^{81} +(-0.275255 + 0.476756i) q^{83} +(-1.00000 - 1.73205i) q^{85} +(7.89898 - 11.1708i) q^{87} -16.7980 q^{89} +(-7.89898 - 17.1455i) q^{93} +(-1.44949 - 2.51059i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 4 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{15} - 8 q^{17} + 8 q^{19} - 4 q^{21} - 10 q^{23} - 2 q^{25} - 20 q^{27} - 6 q^{29} + 12 q^{31} + 4 q^{35} - 24 q^{37} + 10 q^{41} + 4 q^{43} + 4 q^{45} - 14 q^{47} + 4 q^{51} - 8 q^{53} + 20 q^{57} - 12 q^{59} + 6 q^{61} - 26 q^{63} - 2 q^{67} - 22 q^{69} + 16 q^{73} + 4 q^{75} - 4 q^{79} + 14 q^{81} - 6 q^{83} - 4 q^{85} + 12 q^{87} - 28 q^{89} - 12 q^{93} + 4 q^{95} - 4 q^{97}+O(q^{100}) \) 4 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^15 - 8 * q^17 + 8 * q^19 - 4 * q^21 - 10 * q^23 - 2 * q^25 - 20 * q^27 - 6 * q^29 + 12 * q^31 + 4 * q^35 - 24 * q^37 + 10 * q^41 + 4 * q^43 + 4 * q^45 - 14 * q^47 + 4 * q^51 - 8 * q^53 + 20 * q^57 - 12 * q^59 + 6 * q^61 - 26 * q^63 - 2 * q^67 - 22 * q^69 + 16 * q^73 + 4 * q^75 - 4 * q^79 + 14 * q^81 - 6 * q^83 - 4 * q^85 + 12 * q^87 - 28 * q^89 - 12 * q^93 + 4 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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