LMFDB - Embedded newform 360.2.q.c.121.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,2,Mod(121,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

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

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

chi := DirichletCharacter("360.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	360.q (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:	$2.87461447277$
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			121.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	360.121
Dual form			360.2.q.c.241.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.724745 - 1.57313i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-1.72474 - 2.98735i) q^{7} +(-1.94949 + 2.28024i) q^{9} +O(q^{10})$	\(q+(-0.724745 - 1.57313i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-1.72474 - 2.98735i) q^{7} +(-1.94949 + 2.28024i) q^{9} +(-1.72474 - 0.158919i) q^{15} -2.00000 q^{17} -6.89898 q^{19} +(-3.44949 + 4.87832i) q^{21} +(3.72474 - 6.45145i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(5.00000 + 1.41421i) q^{27} +(0.949490 + 1.64456i) q^{29} +(-0.550510 + 0.953512i) q^{31} -3.44949 q^{35} -6.00000 q^{37} +(4.94949 - 8.57277i) q^{41} +(-5.89898 - 10.2173i) q^{43} +(1.00000 + 2.82843i) q^{45} +(4.72474 + 8.18350i) q^{47} +(-2.44949 + 4.24264i) q^{49} +(1.44949 + 3.14626i) q^{51} +7.79796 q^{53} +(5.00000 + 10.8530i) q^{57} +(0.550510 - 0.953512i) q^{59} +(1.50000 + 2.59808i) q^{61} +(10.1742 + 1.89097i) q^{63} +(6.62372 - 11.4726i) q^{67} +(-12.8485 - 1.18386i) q^{69} +9.79796 q^{71} +13.7980 q^{73} +(-1.00000 + 1.41421i) q^{75} +(3.44949 + 5.97469i) q^{79} +(-1.39898 - 8.89060i) q^{81} +(2.72474 + 4.71940i) q^{83} +(-1.00000 + 1.73205i) q^{85} +(1.89898 - 2.68556i) q^{87} +2.79796 q^{89} +(1.89898 + 0.174973i) q^{93} +(-3.44949 + 5.97469i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$.

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.724745	−	1.57313i	−0.418432	−	0.908248i
$3$	−0.724745	−	1.57313i	−0.418432	−	0.908248i
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−1.72474	−	2.98735i	−0.651892	−	1.12911i	−0.982663	−	0.185399i	$-0.940642\pi$
$7$	−1.72474	−	2.98735i	−0.651892	−	1.12911i	0.330771	−	0.943711i	$-0.392691\pi$
$8$		0			0
$9$	−1.94949	+	2.28024i	−0.649830	+	0.760080i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$13$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$14$		0			0
$15$	−1.72474	−	0.158919i	−0.445327	−	0.0410326i
$16$		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
$19$	−6.89898			−1.58273			−0.791367	−	0.611341i	$-0.790630\pi$
$19$	−6.89898			−1.58273			−0.791367	+	0.611341i	$0.790630\pi$
$20$		0			0
$21$	−3.44949	+	4.87832i	−0.752740	+	1.06454i
$22$		0			0
$23$	3.72474	−	6.45145i	0.776663	−	1.34522i	−0.157192	−	0.987568i	$-0.550244\pi$
$23$	3.72474	−	6.45145i	0.776663	−	1.34522i	0.933855	−	0.357652i	$-0.116422\pi$
$24$		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
$29$	0.949490	+	1.64456i	0.176316	+	0.305388i	0.940616	−	0.339473i	$-0.110249\pi$
$29$	0.949490	+	1.64456i	0.176316	+	0.305388i	−0.764300	+	0.644861i	$0.776915\pi$
$30$		0			0
$31$	−0.550510	+	0.953512i	−0.0988746	+	0.171256i	−0.911219	−	0.411922i	$-0.864858\pi$
$31$	−0.550510	+	0.953512i	−0.0988746	+	0.171256i	0.812345	+	0.583178i	$0.198191\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.44949			−0.583070
$36$		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
$41$	4.94949	−	8.57277i	0.772980	−	1.33884i	−0.162942	−	0.986636i	$-0.552098\pi$
$41$	4.94949	−	8.57277i	0.772980	−	1.33884i	0.935923	−	0.352206i	$-0.114568\pi$
$42$		0			0
$43$	−5.89898	−	10.2173i	−0.899586	−	1.55813i	−0.828024	−	0.560692i	$-0.810535\pi$
$43$	−5.89898	−	10.2173i	−0.899586	−	1.55813i	−0.0715617	−	0.997436i	$-0.522798\pi$
$44$		0			0
$45$	1.00000	+	2.82843i	0.149071	+	0.421637i
$46$		0			0
$47$	4.72474	+	8.18350i	0.689175	+	1.19369i	0.972105	+	0.234545i	$0.0753602\pi$
$47$	4.72474	+	8.18350i	0.689175	+	1.19369i	−0.282930	+	0.959140i	$0.591307\pi$
$48$		0			0
$49$	−2.44949	+	4.24264i	−0.349927	+	0.606092i
$50$		0			0
$51$	1.44949	+	3.14626i	0.202969	+	0.440565i
$52$		0			0
$53$	7.79796			1.07113			0.535566	−	0.844493i	$-0.320098\pi$
$53$	7.79796			1.07113			0.535566	+	0.844493i	$0.320098\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	5.00000	+	10.8530i	0.662266	+	1.43752i
$58$		0			0
$59$	0.550510	−	0.953512i	0.0716703	−	0.124137i	−0.827963	−	0.560783i	$-0.810500\pi$
$59$	0.550510	−	0.953512i	0.0716703	−	0.124137i	0.899633	+	0.436646i	$0.143834\pi$
$60$		0			0
$61$	1.50000	+	2.59808i	0.192055	+	0.332650i	0.945931	−	0.324367i	$-0.105151\pi$
$61$	1.50000	+	2.59808i	0.192055	+	0.332650i	−0.753876	+	0.657017i	$0.771818\pi$
$62$		0			0
$63$	10.1742	+	1.89097i	1.28183	+	0.238240i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.62372	−	11.4726i	0.809217	−	1.40160i	−0.104190	−	0.994557i	$-0.533225\pi$
$67$	6.62372	−	11.4726i	0.809217	−	1.40160i	0.913407	−	0.407047i	$-0.133442\pi$
$68$		0			0
$69$	−12.8485	−	1.18386i	−1.54677	−	0.142520i
$70$		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
$73$	13.7980			1.61493			0.807464	−	0.589916i	$-0.200839\pi$
$73$	13.7980			1.61493			0.807464	+	0.589916i	$0.200839\pi$
$74$		0			0
$75$	−1.00000	+	1.41421i	−0.115470	+	0.163299i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	3.44949	+	5.97469i	0.388098	+	0.672205i	0.992194	−	0.124706i	$-0.0397989\pi$
$79$	3.44949	+	5.97469i	0.388098	+	0.672205i	−0.604096	+	0.796912i	$0.706466\pi$
$80$		0			0
$81$	−1.39898	−	8.89060i	−0.155442	−	0.987845i
$82$		0			0
$83$	2.72474	+	4.71940i	0.299080	+	0.518021i	0.975926	−	0.218104i	$-0.0699871\pi$
$83$	2.72474	+	4.71940i	0.299080	+	0.518021i	−0.676846	+	0.736125i	$0.736654\pi$
$84$		0			0
$85$	−1.00000	+	1.73205i	−0.108465	+	0.187867i
$86$		0			0
$87$	1.89898	−	2.68556i	0.203592	−	0.287923i
$88$		0			0
$89$	2.79796			0.296583			0.148292	−	0.988944i	$-0.452623\pi$
$89$	2.79796			0.296583			0.148292	+	0.988944i	$0.452623\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.89898	+	0.174973i	0.196915	+	0.0181438i
$94$		0			0
$95$	−3.44949	+	5.97469i	−0.353910	+	0.612990i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	0.911479	−	0.411346i	$-0.134941\pi$
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	−0.811976	+	0.583691i	$0.801608\pi$
$102$		0			0
$103$	−5.00000	+	8.66025i	−0.492665	+	0.853320i	−0.999964	−	0.00844953i	$-0.997310\pi$
$103$	−5.00000	+	8.66025i	−0.492665	+	0.853320i	0.507300	+	0.861770i	$0.330644\pi$
$104$		0			0
$105$	2.50000	+	5.42650i	0.243975	+	0.529573i
$106$		0			0
$107$	−12.3485			−1.19377			−0.596886	−	0.802326i	$-0.703595\pi$
$107$	−12.3485			−1.19377			−0.596886	+	0.802326i	$0.703595\pi$
$108$		0			0
$109$	−10.7980			−1.03426			−0.517128	−	0.855908i	$-0.672999\pi$
$109$	−10.7980			−1.03426			−0.517128	+	0.855908i	$0.672999\pi$
$110$		0			0
$111$	4.34847	+	9.43879i	0.412738	+	0.895891i
$112$		0			0
$113$	4.89898	−	8.48528i	0.460857	−	0.798228i	−0.538147	−	0.842851i	$-0.680875\pi$
$113$	4.89898	−	8.48528i	0.460857	−	0.798228i	0.999004	+	0.0446231i	$0.0142087\pi$
$114$		0			0
$115$	−3.72474	−	6.45145i	−0.347334	−	0.601601i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.44949	+	5.97469i	0.316214	+	0.547699i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$	−17.0732	−	1.57313i	−1.53944	−	0.141845i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	0.348469			0.0309216			0.0154608	−	0.999880i	$-0.495078\pi$
$127$	0.348469			0.0309216			0.0154608	+	0.999880i	$0.495078\pi$
$128$		0			0
$129$	−11.7980	+	16.6848i	−1.03875	+	1.46902i
$130$		0			0
$131$	−1.44949	+	2.51059i	−0.126643	+	0.219351i	−0.922374	−	0.386299i	$-0.873753\pi$
$131$	−1.44949	+	2.51059i	−0.126643	+	0.219351i	0.795731	+	0.605650i	$0.207087\pi$
$132$		0			0
$133$	11.8990	+	20.6096i	1.03177	+	1.78708i
$134$		0			0
$135$	3.72474	−	3.62302i	0.320575	−	0.311820i
$136$		0			0
$137$	−9.79796	−	16.9706i	−0.837096	−	1.44989i	−0.892312	−	0.451419i	$-0.850918\pi$
$137$	−9.79796	−	16.9706i	−0.837096	−	1.44989i	0.0552162	−	0.998474i	$-0.482415\pi$
$138$		0			0
$139$	−9.79796	+	16.9706i	−0.831052	+	1.43942i	0.0661527	+	0.997810i	$0.478928\pi$
$139$	−9.79796	+	16.9706i	−0.831052	+	1.43942i	−0.897205	+	0.441615i	$0.854406\pi$
$140$		0			0
$141$	9.44949	−	13.3636i	0.795791	−	1.12542i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	1.89898			0.157702
$146$		0			0
$147$	8.44949	+	0.778539i	0.696902	+	0.0642128i
$148$		0			0
$149$	10.5000	−	18.1865i	0.860194	−	1.48990i	−0.0115483	−	0.999933i	$-0.503676\pi$
$149$	10.5000	−	18.1865i	0.860194	−	1.48990i	0.871742	−	0.489966i	$-0.162991\pi$
$150$		0			0
$151$	−6.00000	−	10.3923i	−0.488273	−	0.845714i	0.511636	−	0.859202i	$-0.329040\pi$
$151$	−6.00000	−	10.3923i	−0.488273	−	0.845714i	−0.999909	+	0.0134886i	$0.995706\pi$
$152$		0			0
$153$	3.89898	−	4.56048i	0.315214	−	0.368693i
$154$		0			0
$155$	0.550510	+	0.953512i	0.0442180	+	0.0765879i
$156$		0			0
$157$	11.8990	−	20.6096i	0.949642	−	1.64483i	0.203463	−	0.979083i	$-0.434780\pi$
$157$	11.8990	−	20.6096i	0.949642	−	1.64483i	0.746179	−	0.665745i	$-0.231886\pi$
$158$		0			0
$159$	−5.65153	−	12.2672i	−0.448196	−	0.972854i
$160$		0			0
$161$	−25.6969			−2.02520
$162$		0			0
$163$	−7.79796			−0.610783			−0.305392	−	0.952227i	$-0.598787\pi$
$163$	−7.79796			−0.610783			−0.305392	+	0.952227i	$0.598787\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.17423	+	7.22999i	−0.323012	+	0.559473i	−0.981108	−	0.193461i	$-0.938029\pi$
$167$	−4.17423	+	7.22999i	−0.323012	+	0.559473i	0.658096	+	0.752934i	$0.271362\pi$
$168$		0			0
$169$	6.50000	+	11.2583i	0.500000	+	0.866025i
$170$		0			0
$171$	13.4495	−	15.7313i	1.02851	−	1.20300i
$172$		0			0
$173$	6.00000	+	10.3923i	0.456172	+	0.790112i	0.998755	−	0.0498898i	$-0.0158870\pi$
$173$	6.00000	+	10.3923i	0.456172	+	0.790112i	−0.542583	+	0.840002i	$0.682554\pi$
$174$		0			0
$175$	−1.72474	+	2.98735i	−0.130378	+	0.225822i
$176$		0			0
$177$	−1.89898	−	0.174973i	−0.142736	−	0.0131518i
$178$		0			0
$179$	−13.7980			−1.03131			−0.515654	−	0.856797i	$-0.672451\pi$
$179$	−13.7980			−1.03131			−0.515654	+	0.856797i	$0.672451\pi$
$180$		0			0
$181$	−9.69694			−0.720768			−0.360384	−	0.932804i	$-0.617354\pi$
$181$	−9.69694			−0.720768			−0.360384	+	0.932804i	$0.617354\pi$
$182$		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
$189$	−4.39898	−	17.3759i	−0.319979	−	1.26391i
$190$		0			0
$191$	7.44949	+	12.9029i	0.539026	+	0.933621i	0.998957	+	0.0456658i	$0.0145409\pi$
$191$	7.44949	+	12.9029i	0.539026	+	0.933621i	−0.459931	+	0.887955i	$0.652126\pi$
$192$		0			0
$193$	1.10102	−	1.90702i	0.0792532	−	0.137271i	−0.823675	−	0.567063i	$-0.808080\pi$
$193$	1.10102	−	1.90702i	0.0792532	−	0.137271i	0.902928	+	0.429792i	$0.141413\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.5959			−1.11116			−0.555582	−	0.831462i	$-0.687504\pi$
$197$	−15.5959			−1.11116			−0.555582	+	0.831462i	$0.687504\pi$
$198$		0			0
$199$	22.8990			1.62327			0.811633	−	0.584168i	$-0.198579\pi$
$199$	22.8990			1.62327			0.811633	+	0.584168i	$0.198579\pi$
$200$		0			0
$201$	−22.8485	−	2.10527i	−1.61161	−	0.148494i
$202$		0			0
$203$	3.27526	−	5.67291i	0.229878	−	0.398160i
$204$		0			0
$205$	−4.94949	−	8.57277i	−0.345687	−	0.598748i
$206$		0			0
$207$	7.44949	+	21.0703i	0.517775	+	1.46449i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−6.00000	+	10.3923i	−0.413057	+	0.715436i	−0.995222	−	0.0976347i	$-0.968872\pi$
$211$	−6.00000	+	10.3923i	−0.413057	+	0.715436i	0.582165	+	0.813070i	$0.302206\pi$
$212$		0			0
$213$	−7.10102	−	15.4135i	−0.486554	−	1.05611i
$214$		0			0
$215$	−11.7980			−0.804614
$216$		0			0
$217$	3.79796			0.257822
$218$		0			0
$219$	−10.0000	−	21.7060i	−0.675737	−	1.46676i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−2.72474	−	4.71940i	−0.182462	−	0.316034i	0.760256	−	0.649623i	$-0.225073\pi$
$223$	−2.72474	−	4.71940i	−0.182462	−	0.316034i	−0.942718	+	0.333589i	$0.891740\pi$
$224$		0			0
$225$	2.94949	+	0.548188i	0.196633	+	0.0365459i
$226$		0			0
$227$	4.10102	+	7.10318i	0.272194	+	0.471454i	0.969423	−	0.245394i	$-0.0789173\pi$
$227$	4.10102	+	7.10318i	0.272194	+	0.471454i	−0.697229	+	0.716848i	$0.745584\pi$
$228$		0			0
$229$	2.05051	−	3.55159i	0.135502	−	0.234696i	−0.790287	−	0.612736i	$-0.790069\pi$
$229$	2.05051	−	3.55159i	0.135502	−	0.234696i	0.925789	+	0.378041i	$0.123402\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.7980			−1.29701			−0.648504	−	0.761211i	$-0.724605\pi$
$233$	−19.7980			−1.29701			−0.648504	+	0.761211i	$0.724605\pi$
$234$		0			0
$235$	9.44949			0.616417
$236$		0			0
$237$	6.89898	−	9.75663i	0.448137	−	0.633761i
$238$		0			0
$239$	−1.10102	+	1.90702i	−0.0712191	+	0.123355i	−0.899436	−	0.437053i	$-0.856022\pi$
$239$	−1.10102	+	1.90702i	−0.0712191	+	0.123355i	0.828217	+	0.560408i	$0.189356\pi$
$240$		0			0
$241$	1.84847	+	3.20164i	0.119070	+	0.206236i	0.919400	−	0.393325i	$-0.128675\pi$
$241$	1.84847	+	3.20164i	0.119070	+	0.206236i	−0.800329	+	0.599561i	$0.795342\pi$
$242$		0			0
$243$	−12.9722	+	8.64420i	−0.832167	+	0.554526i
$244$		0			0
$245$	2.44949	+	4.24264i	0.156492	+	0.271052i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	5.44949	−	7.70674i	0.345347	−	0.488395i
$250$		0			0
$251$	2.89898			0.182982			0.0914910	−	0.995806i	$-0.470837\pi$
$251$	2.89898			0.182982			0.0914910	+	0.995806i	$0.470837\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	3.44949	+	0.317837i	0.216015	+	0.0199037i
$256$		0			0
$257$	−13.8990	+	24.0737i	−0.866995	+	1.50168i	−0.00194150	+	0.999998i	$0.500618\pi$
$257$	−13.8990	+	24.0737i	−0.866995	+	1.50168i	−0.865053	+	0.501680i	$0.832715\pi$
$258$		0			0
$259$	10.3485	+	17.9241i	0.643023	+	1.11375i
$260$		0			0
$261$	−5.60102	−	1.04100i	−0.346694	−	0.0644362i
$262$		0			0
$263$	−9.00000	−	15.5885i	−0.554964	−	0.961225i	−0.997906	−	0.0646755i	$-0.979399\pi$
$263$	−9.00000	−	15.5885i	−0.554964	−	0.961225i	0.442943	−	0.896550i	$-0.353935\pi$
$264$		0			0
$265$	3.89898	−	6.75323i	0.239512	−	0.414848i
$266$		0			0
$267$	−2.02781	−	4.40156i	−0.124100	−	0.269371i
$268$		0			0
$269$	22.5959			1.37770			0.688849	−	0.724905i	$-0.258116\pi$
$269$	22.5959			1.37770			0.688849	+	0.724905i	$0.258116\pi$
$270$		0			0
$271$	30.8990			1.87698			0.938490	−	0.345307i	$-0.112225\pi$
$271$	30.8990			1.87698			0.938490	+	0.345307i	$0.112225\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7.00000	+	12.1244i	0.420589	+	0.728482i	0.995997	−	0.0893846i	$-0.0284900\pi$
$277$	7.00000	+	12.1244i	0.420589	+	0.728482i	−0.575408	+	0.817867i	$0.695157\pi$
$278$		0			0
$279$	−1.10102	−	3.11416i	−0.0659164	−	0.186440i
$280$		0			0
$281$	4.05051	+	7.01569i	0.241633	+	0.418521i	0.961180	−	0.275923i	$-0.0889836\pi$
$281$	4.05051	+	7.01569i	0.241633	+	0.418521i	−0.719546	+	0.694444i	$0.755650\pi$
$282$		0			0
$283$	1.82577	−	3.16232i	0.108530	−	0.187980i	−0.806645	−	0.591037i	$-0.798719\pi$
$283$	1.82577	−	3.16232i	0.108530	−	0.187980i	0.915175	+	0.403056i	$0.132052\pi$
$284$		0			0
$285$	11.8990	+	1.09638i	0.704835	+	0.0649437i
$286$		0			0
$287$	−34.1464			−2.01560
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$	−2.00000	+	2.82843i	−0.117242	+	0.165805i
$292$		0			0
$293$	−8.79796	+	15.2385i	−0.513982	+	0.890243i	0.485886	+	0.874022i	$0.338497\pi$
$293$	−8.79796	+	15.2385i	−0.513982	+	0.890243i	−0.999868	+	0.0162213i	$0.994836\pi$
$294$		0			0
$295$	−0.550510	−	0.953512i	−0.0320519	−	0.0555156i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−20.3485	+	35.2446i	−1.17287	+	2.03146i
$302$		0			0
$303$	2.00000	−	2.82843i	0.114897	−	0.162489i
$304$		0			0
$305$	3.00000			0.171780
$306$		0			0
$307$	−4.75255			−0.271242			−0.135621	−	0.990761i	$-0.543303\pi$
$307$	−4.75255			−0.271242			−0.135621	+	0.990761i	$0.543303\pi$
$308$		0			0
$309$	17.2474	+	1.58919i	0.981173	+	0.0904056i
$310$		0			0
$311$	3.44949	−	5.97469i	0.195603	−	0.338794i	−0.751495	−	0.659738i	$-0.770667\pi$
$311$	3.44949	−	5.97469i	0.195603	−	0.338794i	0.947098	+	0.320945i	$0.104000\pi$
$312$		0			0
$313$	7.79796	+	13.5065i	0.440767	+	0.763430i	0.997747	−	0.0670957i	$-0.0213733\pi$
$313$	7.79796	+	13.5065i	0.440767	+	0.763430i	−0.556980	+	0.830526i	$0.688040\pi$
$314$		0			0
$315$	6.72474	−	7.86566i	0.378896	−	0.443180i
$316$		0			0
$317$	12.8990	+	22.3417i	0.724479	+	1.25483i	0.959188	+	0.282769i	$0.0912528\pi$
$317$	12.8990	+	22.3417i	0.724479	+	1.25483i	−0.234709	+	0.972066i	$0.575414\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	8.94949	+	19.4258i	0.499512	+	1.08424i
$322$		0			0
$323$	13.7980			0.767739
$324$		0			0
$325$		0			0
$326$		0			0
$327$	7.82577	+	16.9866i	0.432766	+	0.939362i
$328$		0			0
$329$	16.2980	−	28.2289i	0.898536	−	1.55631i
$330$		0			0
$331$	3.44949	+	5.97469i	0.189601	+	0.328399i	0.945117	−	0.326731i	$-0.105947\pi$
$331$	3.44949	+	5.97469i	0.189601	+	0.328399i	−0.755516	+	0.655130i	$0.772614\pi$
$332$		0			0
$333$	11.6969	−	13.6814i	0.640988	−	0.749738i
$334$		0			0
$335$	−6.62372	−	11.4726i	−0.361893	−	0.626817i
$336$		0			0
$337$	14.8990	−	25.8058i	0.811599	−	1.40573i	−0.100145	−	0.994973i	$-0.531931\pi$
$337$	14.8990	−	25.8058i	0.811599	−	1.40573i	0.911744	−	0.410758i	$-0.134736\pi$
$338$		0			0
$339$	−16.8990	−	1.55708i	−0.917827	−	0.0845689i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−7.24745			−0.391325
$344$		0			0
$345$	−7.44949	+	10.5352i	−0.401067	+	0.567194i
$346$		0			0
$347$	12.7980	−	22.1667i	0.687030	−	1.18997i	−0.285764	−	0.958300i	$-0.592247\pi$
$347$	12.7980	−	22.1667i	0.687030	−	1.18997i	0.972794	−	0.231671i	$-0.0744194\pi$
$348$		0			0
$349$	1.15153	+	1.99451i	0.0616400	+	0.106764i	0.895199	−	0.445667i	$-0.147034\pi$
$349$	1.15153	+	1.99451i	0.0616400	+	0.106764i	−0.833559	+	0.552431i	$0.813700\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$	4.89898	−	8.48528i	0.260011	−	0.450352i
$356$		0			0
$357$	6.89898	−	9.75663i	0.365133	−	0.516376i
$358$		0			0
$359$	28.6969			1.51457			0.757283	−	0.653087i	$-0.226526\pi$
$359$	28.6969			1.51457			0.757283	+	0.653087i	$0.226526\pi$
$360$		0			0
$361$	28.5959			1.50505
$362$		0			0
$363$	−18.9722	−	1.74810i	−0.995782	−	0.0917517i
$364$		0			0
$365$	6.89898	−	11.9494i	0.361109	−	0.625459i
$366$		0			0
$367$	−11.0000	−	19.0526i	−0.574195	−	0.994535i	−0.996129	−	0.0879086i	$-0.971982\pi$
$367$	−11.0000	−	19.0526i	−0.574195	−	0.994535i	0.421933	−	0.906627i	$-0.361352\pi$
$368$		0			0
$369$	9.89898	+	27.9985i	0.515320	+	1.45755i
$370$		0			0
$371$	−13.4495	−	23.2952i	−0.698263	−	1.20943i
$372$		0			0
$373$	−10.7980	+	18.7026i	−0.559097	+	0.968385i	0.438475	+	0.898743i	$0.355519\pi$
$373$	−10.7980	+	18.7026i	−0.559097	+	0.968385i	−0.997572	+	0.0696412i	$0.977815\pi$
$374$		0			0
$375$	0.724745	+	1.57313i	0.0374257	+	0.0812362i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	26.8990			1.38171			0.690854	−	0.722994i	$-0.257235\pi$
$379$	26.8990			1.38171			0.690854	+	0.722994i	$0.257235\pi$
$380$		0			0
$381$	−0.252551	−	0.548188i	−0.0129386	−	0.0280845i
$382$		0			0
$383$	−1.00000	+	1.73205i	−0.0510976	+	0.0885037i	−0.890443	−	0.455095i	$-0.849605\pi$
$383$	−1.00000	+	1.73205i	−0.0510976	+	0.0885037i	0.839345	+	0.543599i	$0.182939\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	34.7980	+	6.46750i	1.76888	+	0.328762i
$388$		0			0
$389$	8.29796	+	14.3725i	0.420723	+	0.728714i	0.996010	−	0.0892375i	$-0.0284430\pi$
$389$	8.29796	+	14.3725i	0.420723	+	0.728714i	−0.575287	+	0.817952i	$0.695110\pi$
$390$		0			0
$391$	−7.44949	+	12.9029i	−0.376737	+	0.652527i
$392$		0			0
$393$	5.00000	+	0.460702i	0.252217	+	0.0232393i
$394$		0			0
$395$	6.89898			0.347125
$396$		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
$399$	23.7980	−	33.6554i	1.19139	−	1.68488i
$400$		0			0
$401$	−12.7980	+	22.1667i	−0.639100	+	1.10695i	0.346531	+	0.938038i	$0.387359\pi$
$401$	−12.7980	+	22.1667i	−0.639100	+	1.10695i	−0.985631	+	0.168914i	$0.945974\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−8.39898	−	3.23375i	−0.417349	−	0.160686i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	4.79796	−	8.31031i	0.237244	−	0.410918i	−0.722679	−	0.691184i	$-0.757089\pi$
$409$	4.79796	−	8.31031i	0.237244	−	0.410918i	0.959922	+	0.280266i	$0.0904226\pi$
$410$		0			0
$411$	−19.5959	+	27.7128i	−0.966595	+	1.36697i
$412$		0			0
$413$	−3.79796			−0.186885
$414$		0			0
$415$	5.44949			0.267505
$416$		0			0
$417$	33.7980	+	3.11416i	1.65509	+	0.152501i
$418$		0			0
$419$	−10.0000	+	17.3205i	−0.488532	+	0.846162i	−0.999913	−	0.0131919i	$-0.995801\pi$
$419$	−10.0000	+	17.3205i	−0.488532	+	0.846162i	0.511381	+	0.859354i	$0.329134\pi$
$420$		0			0
$421$	−12.7980	−	22.1667i	−0.623734	−	1.08034i	−0.988784	−	0.149352i	$-0.952281\pi$
$421$	−12.7980	−	22.1667i	−0.623734	−	1.08034i	0.365050	−	0.930988i	$-0.381052\pi$
$422$		0			0
$423$	−27.8712	−	5.18010i	−1.35514	−	0.251865i
$424$		0			0
$425$	1.00000	+	1.73205i	0.0485071	+	0.0840168i
$426$		0			0
$427$	5.17423	−	8.96204i	0.250399	−	0.433703i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14.8990			0.717659			0.358829	−	0.933403i	$-0.383176\pi$
$431$	14.8990			0.717659			0.358829	+	0.933403i	$0.383176\pi$
$432$		0			0
$433$	−11.7980			−0.566974			−0.283487	−	0.958976i	$-0.591491\pi$
$433$	−11.7980			−0.566974			−0.283487	+	0.958976i	$0.591491\pi$
$434$		0			0
$435$	−1.37628	−	2.98735i	−0.0659874	−	0.143232i
$436$		0			0
$437$	−25.6969	+	44.5084i	−1.22925	+	2.12913i
$438$		0			0
$439$	8.89898	+	15.4135i	0.424725	+	0.735645i	0.996395	−	0.0848384i	$-0.0270374\pi$
$439$	8.89898	+	15.4135i	0.424725	+	0.735645i	−0.571670	+	0.820484i	$0.693704\pi$
$440$		0			0
$441$	−4.89898	−	13.8564i	−0.233285	−	0.659829i
$442$		0			0
$443$	4.07321	+	7.05501i	0.193524	+	0.335194i	0.946416	−	0.322951i	$-0.104675\pi$
$443$	4.07321	+	7.05501i	0.193524	+	0.335194i	−0.752892	+	0.658145i	$0.771342\pi$
$444$		0			0
$445$	1.39898	−	2.42310i	0.0663180	−	0.114866i
$446$		0			0
$447$	−36.2196	−	3.33729i	−1.71313	−	0.157848i
$448$		0			0
$449$	21.5959			1.01917			0.509587	−	0.860419i	$-0.329798\pi$
$449$	21.5959			1.01917			0.509587	+	0.860419i	$0.329798\pi$
$450$		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
$457$	−8.79796	−	15.2385i	−0.411551	−	0.712828i	0.583508	−	0.812107i	$-0.301680\pi$
$457$	−8.79796	−	15.2385i	−0.411551	−	0.712828i	−0.995060	+	0.0992796i	$0.968346\pi$
$458$		0			0
$459$	−10.0000	−	2.82843i	−0.466760	−	0.132020i
$460$		0			0
$461$	−6.05051	−	10.4798i	−0.281800	−	0.488093i	0.690028	−	0.723783i	$-0.257598\pi$
$461$	−6.05051	−	10.4798i	−0.281800	−	0.488093i	−0.971828	+	0.235690i	$0.924265\pi$
$462$		0			0
$463$	−6.10102	+	10.5673i	−0.283538	+	0.491103i	−0.972254	−	0.233929i	$-0.924842\pi$
$463$	−6.10102	+	10.5673i	−0.283538	+	0.491103i	0.688715	+	0.725032i	$0.258175\pi$
$464$		0			0
$465$	1.10102	−	1.55708i	0.0510586	−	0.0722078i
$466$		0			0
$467$	7.79796			0.360847			0.180423	−	0.983589i	$-0.442253\pi$
$467$	7.79796			0.360847			0.180423	+	0.983589i	$0.442253\pi$
$468$		0			0
$469$	−45.6969			−2.11009
$470$		0			0
$471$	−41.0454	−	3.78194i	−1.89127	−	0.174263i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	3.44949	+	5.97469i	0.158273	+	0.274138i
$476$		0			0
$477$	−15.2020	+	17.7812i	−0.696054	+	0.814146i
$478$		0			0
$479$	−15.2474	−	26.4094i	−0.696674	−	1.20667i	−0.969613	−	0.244643i	$-0.921329\pi$
$479$	−15.2474	−	26.4094i	−0.696674	−	1.20667i	0.272939	−	0.962031i	$-0.412004\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	18.6237	+	40.4247i	0.847409	+	1.83939i
$484$		0			0
$485$	−2.00000			−0.0908153
$486$		0			0
$487$	−13.5959			−0.616090			−0.308045	−	0.951372i	$-0.599675\pi$
$487$	−13.5959			−0.616090			−0.308045	+	0.951372i	$0.599675\pi$
$488$		0			0
$489$	5.65153	+	12.2672i	0.255571	+	0.554743i
$490$		0			0
$491$	6.89898	−	11.9494i	0.311347	−	0.539268i	−0.667308	−	0.744782i	$-0.732553\pi$
$491$	6.89898	−	11.9494i	0.311347	−	0.539268i	0.978654	+	0.205514i	$0.0658867\pi$
$492$		0			0
$493$	−1.89898	−	3.28913i	−0.0855257	−	0.148135i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.8990	−	29.2699i	−0.758023	−	1.31293i
$498$		0			0
$499$	14.0000	−	24.2487i	0.626726	−	1.08552i	−0.361478	−	0.932381i	$-0.617728\pi$
$499$	14.0000	−	24.2487i	0.626726	−	1.08552i	0.988204	−	0.153141i	$-0.0489388\pi$
$500$		0			0
$501$	14.3990	+	1.32673i	0.643299	+	0.0592738i
$502$		0			0
$503$	25.0454			1.11672			0.558360	−	0.829599i	$-0.311431\pi$
$503$	25.0454			1.11672			0.558360	+	0.829599i	$0.311431\pi$
$504$		0			0
$505$	2.00000			0.0889988
$506$		0			0
$507$	13.0000	−	18.3848i	0.577350	−	0.816497i
$508$		0			0
$509$	2.05051	−	3.55159i	0.0908873	−	0.157421i	−0.816997	−	0.576641i	$-0.804363\pi$
$509$	2.05051	−	3.55159i	0.0908873	−	0.157421i	0.907885	+	0.419220i	$0.137696\pi$
$510$		0			0
$511$	−23.7980	−	41.2193i	−1.05276	−	1.82343i
$512$		0			0
$513$	−34.4949	−	9.75663i	−1.52299	−	0.430766i
$514$		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
$521$	9.20204			0.403149			0.201574	−	0.979473i	$-0.435394\pi$
$521$	9.20204			0.403149			0.201574	+	0.979473i	$0.435394\pi$
$522$		0			0
$523$	−34.3485			−1.50195			−0.750977	−	0.660329i	$-0.770417\pi$
$523$	−34.3485			−1.50195			−0.750977	+	0.660329i	$0.770417\pi$
$524$		0			0
$525$	5.94949	+	0.548188i	0.259657	+	0.0239249i
$526$		0			0
$527$	1.10102	−	1.90702i	0.0479612	−	0.0830712i
$528$		0			0
$529$	−16.2474	−	28.1414i	−0.706411	−	1.22354i
$530$		0			0
$531$	1.10102	+	3.11416i	0.0477802	+	0.135143i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−6.17423	+	10.6941i	−0.266935	+	0.462346i
$536$		0			0
$537$	10.0000	+	21.7060i	0.431532	+	0.936684i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	12.1010			0.520264			0.260132	−	0.965573i	$-0.416234\pi$
$541$	12.1010			0.520264			0.260132	+	0.965573i	$0.416234\pi$
$542$		0			0
$543$	7.02781	+	15.2546i	0.301592	+	0.654636i
$544$		0			0
$545$	−5.39898	+	9.35131i	−0.231267	+	0.400566i
$546$		0			0
$547$	8.37628	+	14.5081i	0.358144	+	0.620323i	0.987651	−	0.156672i	$-0.0500765\pi$
$547$	8.37628	+	14.5081i	0.358144	+	0.620323i	−0.629507	+	0.776995i	$0.716743\pi$
$548$		0			0
$549$	−8.84847	−	1.64456i	−0.377643	−	0.0701883i
$550$		0			0
$551$	−6.55051	−	11.3458i	−0.279061	−	0.483348i
$552$		0			0
$553$	11.8990	−	20.6096i	0.505996	−	0.876411i
$554$		0			0
$555$	10.3485	+	0.953512i	0.439268	+	0.0404743i
$556$		0			0
$557$	43.7980			1.85578			0.927890	−	0.372855i	$-0.121621\pi$
$557$	43.7980			1.85578			0.927890	+	0.372855i	$0.121621\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.17423	−	8.96204i	0.218068	−	0.377705i	−0.736149	−	0.676819i	$-0.763358\pi$
$563$	5.17423	−	8.96204i	0.218068	−	0.377705i	0.954217	+	0.299114i	$0.0966912\pi$
$564$		0			0
$565$	−4.89898	−	8.48528i	−0.206102	−	0.356978i
$566$		0			0
$567$	−24.1464	+	19.5133i	−1.01405	+	0.819480i
$568$		0			0
$569$	−5.00000	−	8.66025i	−0.209611	−	0.363057i	0.741981	−	0.670421i	$-0.233886\pi$
$569$	−5.00000	−	8.66025i	−0.209611	−	0.363057i	−0.951592	+	0.307364i	$0.900553\pi$
$570$		0			0
$571$	−12.8990	+	22.3417i	−0.539805	+	0.934971i	0.459109	+	0.888380i	$0.348169\pi$
$571$	−12.8990	+	22.3417i	−0.539805	+	0.934971i	−0.998914	+	0.0465904i	$0.985164\pi$
$572$		0			0
$573$	14.8990	−	21.0703i	0.622414	−	0.880226i
$574$		0			0
$575$	−7.44949			−0.310665
$576$		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
$579$	−3.79796	−	0.349945i	−0.157838	−	0.0145432i
$580$		0			0
$581$	9.39898	−	16.2795i	0.389935	−	0.675388i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.82577	−	6.62642i	−0.157906	−	0.273502i	0.776207	−	0.630478i	$-0.217141\pi$
$587$	−3.82577	−	6.62642i	−0.157906	−	0.273502i	−0.934113	+	0.356976i	$0.883808\pi$
$588$		0			0
$589$	3.79796	−	6.57826i	0.156492	−	0.271052i
$590$		0			0
$591$	11.3031	+	24.5344i	0.464946	+	1.00921i
$592$		0			0
$593$	−31.3939			−1.28919			−0.644596	−	0.764523i	$-0.722974\pi$
$593$	−31.3939			−1.28919			−0.644596	+	0.764523i	$0.722974\pi$
$594$		0			0
$595$	6.89898			0.282831
$596$		0			0
$597$	−16.5959	−	36.0231i	−0.679226	−	1.47433i
$598$		0			0
$599$	12.3485	−	21.3882i	0.504545	−	0.873897i	−0.495441	−	0.868641i	$-0.664994\pi$
$599$	12.3485	−	21.3882i	0.504545	−	0.873897i	0.999986	−	0.00525583i	$-0.00167299\pi$
$600$		0			0
$601$	−7.00000	−	12.1244i	−0.285536	−	0.494563i	0.687203	−	0.726465i	$-0.258838\pi$
$601$	−7.00000	−	12.1244i	−0.285536	−	0.494563i	−0.972739	+	0.231903i	$0.925505\pi$
$602$		0			0
$603$	13.2474	+	37.4694i	0.539478	+	1.52587i
$604$		0			0
$605$	−5.50000	−	9.52628i	−0.223607	−	0.387298i
$606$		0			0
$607$	−1.07321	+	1.85886i	−0.0435604	+	0.0754489i	−0.886984	−	0.461801i	$-0.847203\pi$
$607$	−1.07321	+	1.85886i	−0.0435604	+	0.0754489i	0.843423	+	0.537250i	$0.180537\pi$
$608$		0			0
$609$	−11.2980	−	1.04100i	−0.457816	−	0.0421834i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−29.5959			−1.19537			−0.597684	−	0.801732i	$-0.703912\pi$
$613$	−29.5959			−1.19537			−0.597684	+	0.801732i	$0.703912\pi$
$614$		0			0
$615$	−9.89898	+	13.9993i	−0.399165	+	0.564505i
$616$		0			0
$617$	17.5959	−	30.4770i	0.708385	−	1.22696i	−0.257071	−	0.966393i	$-0.582757\pi$
$617$	17.5959	−	30.4770i	0.708385	−	1.22696i	0.965456	−	0.260566i	$-0.0839092\pi$
$618$		0			0
$619$	12.3485	+	21.3882i	0.496327	+	0.859663i	0.999991	−	0.00423617i	$-0.00134842\pi$
$619$	12.3485	+	21.3882i	0.496327	+	0.859663i	−0.503664	+	0.863900i	$0.668015\pi$
$620$		0			0
$621$	27.7474	−	26.9897i	1.11347	−	1.08306i
$622$		0			0
$623$	−4.82577	−	8.35847i	−0.193340	−	0.334875i
$624$		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
$631$	−15.5959			−0.620864			−0.310432	−	0.950596i	$-0.600474\pi$
$631$	−15.5959			−0.620864			−0.310432	+	0.950596i	$0.600474\pi$
$632$		0			0
$633$	20.6969	+	1.90702i	0.822629	+	0.0757974i
$634$		0			0
$635$	0.174235	−	0.301783i	0.00691429	−	0.0119759i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−19.1010	+	22.3417i	−0.755625	+	0.883824i
$640$		0			0
$641$	6.94949	+	12.0369i	0.274488	+	0.475428i	0.970006	−	0.243081i	$-0.0781582\pi$
$641$	6.94949	+	12.0369i	0.274488	+	0.475428i	−0.695518	+	0.718509i	$0.744825\pi$
$642$		0			0
$643$	10.0732	−	17.4473i	0.397249	−	0.688055i	−0.596137	−	0.802883i	$-0.703298\pi$
$643$	10.0732	−	17.4473i	0.397249	−	0.688055i	0.993385	+	0.114828i	$0.0366317\pi$
$644$		0			0
$645$	8.55051	+	18.5597i	0.336676	+	0.730789i
$646$		0			0
$647$	6.55051			0.257527			0.128764	−	0.991675i	$-0.458899\pi$
$647$	6.55051			0.257527			0.128764	+	0.991675i	$0.458899\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−2.75255	−	5.97469i	−0.107881	−	0.234167i
$652$		0			0
$653$	−13.8990	+	24.0737i	−0.543909	+	0.942078i	0.454766	+	0.890611i	$0.349723\pi$
$653$	−13.8990	+	24.0737i	−0.543909	+	0.942078i	−0.998675	+	0.0514670i	$0.983610\pi$
$654$		0			0
$655$	1.44949	+	2.51059i	0.0566363	+	0.0980969i
$656$		0			0
$657$	−26.8990	+	31.4626i	−1.04943	+	1.22747i
$658$		0			0
$659$	−5.10102	−	8.83523i	−0.198708	−	0.344172i	0.749402	−	0.662115i	$-0.230341\pi$
$659$	−5.10102	−	8.83523i	−0.198708	−	0.344172i	−0.948110	+	0.317944i	$0.897008\pi$
$660$		0			0
$661$	−5.00000	+	8.66025i	−0.194477	+	0.336845i	−0.946729	−	0.322031i	$-0.895634\pi$
$661$	−5.00000	+	8.66025i	−0.194477	+	0.336845i	0.752252	+	0.658876i	$0.228968\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	23.7980			0.922845
$666$		0			0
$667$	14.1464			0.547752
$668$		0			0
$669$	−5.44949	+	7.70674i	−0.210689	+	0.297960i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	8.00000	+	13.8564i	0.308377	+	0.534125i	0.978008	−	0.208569i	$-0.0668807\pi$
$673$	8.00000	+	13.8564i	0.308377	+	0.534125i	−0.669630	+	0.742695i	$0.733547\pi$
$674$		0			0
$675$	−1.27526	−	5.03723i	−0.0490846	−	0.193883i
$676$		0			0
$677$	−7.10102	−	12.2993i	−0.272914	−	0.472702i	0.696692	−	0.717370i	$-0.254654\pi$
$677$	−7.10102	−	12.2993i	−0.272914	−	0.472702i	−0.969607	+	0.244668i	$0.921321\pi$
$678$		0			0
$679$	−3.44949	+	5.97469i	−0.132379	+	0.229288i
$680$		0			0
$681$	8.20204	−	11.5994i	0.314303	−	0.444491i
$682$		0			0
$683$	23.3939			0.895142			0.447571	−	0.894248i	$-0.352289\pi$
$683$	23.3939			0.895142			0.447571	+	0.894248i	$0.352289\pi$
$684$		0			0
$685$	−19.5959			−0.748722
$686$		0			0
$687$	−7.07321	−	0.651729i	−0.269860	−	0.0248650i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−19.4495	−	33.6875i	−0.739893	−	1.28153i	−0.952543	−	0.304404i	$-0.901543\pi$
$691$	−19.4495	−	33.6875i	−0.739893	−	1.28153i	0.212649	−	0.977129i	$-0.431791\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.79796	+	16.9706i	0.371658	+	0.643730i
$696$		0			0
$697$	−9.89898	+	17.1455i	−0.374951	+	0.649433i
$698$		0			0
$699$	14.3485	+	31.1448i	0.542709	+	1.17800i
$700$		0			0
$701$	30.3939			1.14796			0.573980	−	0.818869i	$-0.305399\pi$
$701$	30.3939			1.14796			0.573980	+	0.818869i	$0.305399\pi$
$702$		0			0
$703$	41.3939			1.56120
$704$		0			0
$705$	−6.84847	−	14.8653i	−0.257928	−	0.559859i
$706$		0			0
$707$	3.44949	−	5.97469i	0.129731	−	0.224701i
$708$		0			0
$709$	−4.84847	−	8.39780i	−0.182088	−	0.315386i	0.760503	−	0.649334i	$-0.224952\pi$
$709$	−4.84847	−	8.39780i	−0.182088	−	0.315386i	−0.942591	+	0.333948i	$0.891619\pi$
$710$		0			0
$711$	−20.3485	−	3.78194i	−0.763127	−	0.141834i
$712$		0			0
$713$	4.10102	+	7.10318i	0.153584	+	0.266016i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	3.79796	+	0.349945i	0.141837	+	0.0130689i
$718$		0			0
$719$	−28.2929			−1.05515			−0.527573	−	0.849510i	$-0.676898\pi$
$719$	−28.2929			−1.05515			−0.527573	+	0.849510i	$0.676898\pi$
$720$		0			0
$721$	34.4949			1.28466
$722$		0			0
$723$	3.69694	−	5.22826i	0.137491	−	0.194441i
$724$		0			0
$725$	0.949490	−	1.64456i	0.0352632	−	0.0610776i
$726$		0			0
$727$	15.6237	+	27.0611i	0.579452	+	1.00364i	0.995542	+	0.0943168i	$0.0300667\pi$
$727$	15.6237	+	27.0611i	0.579452	+	1.00364i	−0.416090	+	0.909323i	$0.636600\pi$
$728$		0			0
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$	11.7980	+	20.4347i	0.436363	+	0.755803i
$732$		0			0
$733$	14.7980	−	25.6308i	0.546575	−	0.946696i	−0.451931	−	0.892053i	$-0.649265\pi$
$733$	14.7980	−	25.6308i	0.546575	−	0.946696i	0.998506	−	0.0546429i	$-0.0174020\pi$
$734$		0			0
$735$	4.89898	−	6.92820i	0.180702	−	0.255551i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−33.1010			−1.21764			−0.608820	−	0.793308i	$-0.708357\pi$
$739$	−33.1010			−1.21764			−0.608820	+	0.793308i	$0.708357\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.52270	−	6.10150i	0.129235	−	0.223842i	−0.794145	−	0.607728i	$-0.792081\pi$
$743$	3.52270	−	6.10150i	0.129235	−	0.223842i	0.923381	+	0.383886i	$0.125414\pi$
$744$		0			0
$745$	−10.5000	−	18.1865i	−0.384690	−	0.666303i
$746$		0			0
$747$	−16.0732	−	2.98735i	−0.588088	−	0.109301i
$748$		0			0
$749$	21.2980	+	36.8891i	0.778210	+	1.34790i
$750$		0			0
$751$	−13.1010	+	22.6916i	−0.478063	+	0.828029i	−0.999684	−	0.0251480i	$-0.991994\pi$
$751$	−13.1010	+	22.6916i	−0.478063	+	0.828029i	0.521621	+	0.853177i	$0.325328\pi$
$752$		0			0
$753$	−2.10102	−	4.56048i	−0.0765654	−	0.166193i
$754$		0			0
$755$	−12.0000			−0.436725
$756$		0			0
$757$	31.5959			1.14837			0.574187	−	0.818724i	$-0.305318\pi$
$757$	31.5959			1.14837			0.574187	+	0.818724i	$0.305318\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17.5000	+	30.3109i	−0.634375	+	1.09877i	0.352273	+	0.935897i	$0.385409\pi$
$761$	−17.5000	+	30.3109i	−0.634375	+	1.09877i	−0.986647	+	0.162872i	$0.947924\pi$
$762$		0			0
$763$	18.6237	+	32.2572i	0.674224	+	1.16779i
$764$		0			0
$765$	−2.00000	−	5.65685i	−0.0723102	−	0.204524i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.60102	−	4.50510i	0.0937952	−	0.162458i	−0.815310	−	0.579025i	$-0.803433\pi$
$769$	2.60102	−	4.50510i	0.0937952	−	0.162458i	0.909105	+	0.416567i	$0.136767\pi$
$770$		0			0
$771$	47.9444	+	4.41761i	1.72667	+	0.159096i
$772$		0			0
$773$	6.20204			0.223072			0.111536	−	0.993760i	$-0.464423\pi$
$773$	6.20204			0.223072			0.111536	+	0.993760i	$0.464423\pi$
$774$		0			0
$775$	1.10102			0.0395498
$776$		0			0
$777$	20.6969	−	29.2699i	0.742499	−	1.05005i
$778$		0			0
$779$	−34.1464	+	59.1433i	−1.22342	+	2.11903i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	2.42168	+	9.56560i	0.0865439	+	0.341847i
$784$		0			0
$785$	−11.8990	−	20.6096i	−0.424693	−	0.735589i
$786$		0			0
$787$	24.7980	−	42.9513i	0.883952	−	1.53105i	0.0370414	−	0.999314i	$-0.488207\pi$
$787$	24.7980	−	42.9513i	0.883952	−	1.53105i	0.846910	−	0.531736i	$-0.178460\pi$
$788$		0			0
$789$	−18.0000	+	25.4558i	−0.640817	+	0.906252i
$790$		0			0
$791$	−33.7980			−1.20172
$792$		0			0
$793$

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:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.724745 - 1.57313i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-1.72474 - 2.98735i) q^{7} +(-1.94949 + 2.28024i) q^{9} +O(q^{10})\)	\(q+(-0.724745 - 1.57313i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-1.72474 - 2.98735i) q^{7} +(-1.94949 + 2.28024i) q^{9} +(-1.72474 - 0.158919i) q^{15} -2.00000 q^{17} -6.89898 q^{19} +(-3.44949 + 4.87832i) q^{21} +(3.72474 - 6.45145i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(5.00000 + 1.41421i) q^{27} +(0.949490 + 1.64456i) q^{29} +(-0.550510 + 0.953512i) q^{31} -3.44949 q^{35} -6.00000 q^{37} +(4.94949 - 8.57277i) q^{41} +(-5.89898 - 10.2173i) q^{43} +(1.00000 + 2.82843i) q^{45} +(4.72474 + 8.18350i) q^{47} +(-2.44949 + 4.24264i) q^{49} +(1.44949 + 3.14626i) q^{51} +7.79796 q^{53} +(5.00000 + 10.8530i) q^{57} +(0.550510 - 0.953512i) q^{59} +(1.50000 + 2.59808i) q^{61} +(10.1742 + 1.89097i) q^{63} +(6.62372 - 11.4726i) q^{67} +(-12.8485 - 1.18386i) q^{69} +9.79796 q^{71} +13.7980 q^{73} +(-1.00000 + 1.41421i) q^{75} +(3.44949 + 5.97469i) q^{79} +(-1.39898 - 8.89060i) q^{81} +(2.72474 + 4.71940i) q^{83} +(-1.00000 + 1.73205i) q^{85} +(1.89898 - 2.68556i) q^{87} +2.79796 q^{89} +(1.89898 + 0.174973i) q^{93} +(-3.44949 + 5.97469i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more