LMFDB - Embedded newform 1080.2.q.c.361.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,2,Mod(361,1080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1080.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$8.62384341830$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1080.361
Dual form			1080.2.q.c.721.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(0.724745 + 1.25529i) q^{7} +2.00000 q^{17} +2.89898 q^{19} +(-1.27526 + 2.20881i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(3.94949 + 6.84072i) q^{29} +(-5.44949 + 9.43879i) q^{31} -1.44949 q^{35} -6.00000 q^{37} +(-0.0505103 + 0.0874863i) q^{41} +(3.89898 + 6.75323i) q^{43} +(-2.27526 - 3.94086i) q^{47} +(2.44949 - 4.24264i) q^{49} +11.7980 q^{53} +(-5.44949 + 9.43879i) q^{59} +(1.50000 + 2.59808i) q^{61} +(-5.62372 + 9.74058i) q^{67} +9.79796 q^{71} -5.79796 q^{73} +(-1.44949 - 2.51059i) q^{79} +(-0.275255 - 0.476756i) q^{83} +(-1.00000 + 1.73205i) q^{85} +16.7980 q^{89} +(-1.44949 + 2.51059i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} - 2 q^{7} + 8 q^{17} - 8 q^{19} - 10 q^{23} - 2 q^{25} + 6 q^{29} - 12 q^{31} + 4 q^{35} - 24 q^{37} - 10 q^{41} - 4 q^{43} - 14 q^{47} + 8 q^{53} - 12 q^{59} + 6 q^{61} + 2 q^{67} + 16 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 - 2 * q^7 + 8 * q^17 - 8 * q^19 - 10 * q^23 - 2 * q^25 + 6 * q^29 - 12 * q^31 + 4 * q^35 - 24 * q^37 - 10 * q^41 - 4 * q^43 - 14 * q^47 + 8 * q^53 - 12 * q^59 + 6 * q^61 + 2 * q^67 + 16 * q^73 + 4 * q^79 - 6 * q^83 - 4 * q^85 + 28 * q^89 + 4 * q^95 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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			0
$3$		0			0
$4$		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	$-0.0783437\pi$
$7$	0.724745	+	1.25529i	0.273928	+	0.474457i	−0.695936	+	0.718104i	$0.745010\pi$
$8$		0			0
$9$		0			0
$10$		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
$15$		0			0
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	2.89898			0.665072			0.332536	−	0.943091i	$-0.392096\pi$
$19$	2.89898			0.665072			0.332536	+	0.943091i	$0.392096\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.27526	+	2.20881i	−0.265909	+	0.460568i	−0.967801	−	0.251715i	$-0.919005\pi$
$23$	−1.27526	+	2.20881i	−0.265909	+	0.460568i	0.701892	+	0.712283i	$0.252339\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.94949	+	6.84072i	0.733402	+	1.27029i	0.955421	+	0.295247i	$0.0954019\pi$
$29$	3.94949	+	6.84072i	0.733402	+	1.27029i	−0.222019	+	0.975042i	$0.571265\pi$
$30$		0			0
$31$	−5.44949	+	9.43879i	−0.978757	+	1.69526i	−0.311824	+	0.950140i	$0.600940\pi$
$31$	−5.44949	+	9.43879i	−0.978757	+	1.69526i	−0.666933	+	0.745117i	$0.732393\pi$
$32$		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	$-0.835844\pi$
$41$	−0.0505103	+	0.0874863i	−0.00788838	+	0.0136631i	0.862054	+	0.506816i	$0.169178\pi$
$42$		0			0
$43$	3.89898	+	6.75323i	0.594589	+	1.02986i	0.993605	+	0.112914i	$0.0360185\pi$
$43$	3.89898	+	6.75323i	0.594589	+	1.02986i	−0.399016	+	0.916944i	$0.630648\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.27526	−	3.94086i	−0.331880	−	0.574833i	0.651000	−	0.759077i	$-0.274350\pi$
$47$	−2.27526	−	3.94086i	−0.331880	−	0.574833i	−0.982880	+	0.184244i	$0.941016\pi$
$48$		0			0
$49$	2.44949	−	4.24264i	0.349927	−	0.606092i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	11.7980			1.62057			0.810287	−	0.586033i	$-0.199311\pi$
$53$	11.7980			1.62057			0.810287	+	0.586033i	$0.199311\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.44949	+	9.43879i	−0.709463	+	1.22883i	0.255593	+	0.966784i	$0.417729\pi$
$59$	−5.44949	+	9.43879i	−0.709463	+	1.22883i	−0.965057	+	0.262042i	$0.915604\pi$
$60$		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
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.62372	+	9.74058i	−0.687047	+	1.19000i	0.285741	+	0.958307i	$0.407760\pi$
$67$	−5.62372	+	9.74058i	−0.687047	+	1.19000i	−0.972789	+	0.231694i	$0.925573\pi$
$68$		0			0
$69$		0			0
$70$		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
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.44949	−	2.51059i	−0.163080	−	0.282463i	0.772892	−	0.634538i	$-0.218810\pi$
$79$	−1.44949	−	2.51059i	−0.163080	−	0.282463i	−0.935972	+	0.352075i	$0.885476\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.275255	−	0.476756i	−0.0302132	−	0.0523308i	0.850523	−	0.525937i	$-0.176285\pi$
$83$	−0.275255	−	0.476756i	−0.0302132	−	0.0523308i	−0.880737	+	0.473606i	$0.842952\pi$
$84$		0			0
$85$	−1.00000	+	1.73205i	−0.108465	+	0.187867i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	16.7980			1.78058			0.890290	−	0.455394i	$-0.150502\pi$
$89$	16.7980			1.78058			0.890290	+	0.455394i	$0.150502\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.44949	+	2.51059i	−0.148715	+	0.257581i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	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	0.811976	−	0.583691i	$-0.198392\pi$
$101$	−1.00000	−	1.73205i	−0.0995037	−	0.172345i	−0.911479	+	0.411346i	$0.865059\pi$
$102$		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
$105$		0			0
$106$		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
$111$		0			0
$112$		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
$115$	−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			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−14.3485			−1.27322			−0.636610	−	0.771186i	$-0.719664\pi$
$127$	−14.3485			−1.27322			−0.636610	+	0.771186i	$0.719664\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.44949	+	5.97469i	−0.301383	+	0.522011i	−0.976450	−	0.215746i	$-0.930782\pi$
$131$	−3.44949	+	5.97469i	−0.301383	+	0.522011i	0.675066	+	0.737757i	$0.264115\pi$
$132$		0			0
$133$	2.10102	+	3.63907i	0.182182	+	0.315548i
$134$		0			0
$135$		0			0
$136$		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	$-0.521072\pi$
$139$	9.79796	−	16.9706i	0.831052	−	1.43942i	0.897205	−	0.441615i	$-0.145594\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−7.89898			−0.655975
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.5000	+	18.1865i	−0.860194	+	1.48990i	0.0115483	+	0.999933i	$0.496324\pi$
$149$	−10.5000	+	18.1865i	−0.860194	+	1.48990i	−0.871742	+	0.489966i	$0.837009\pi$
$150$		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
$153$		0			0
$154$		0			0
$155$	−5.44949	−	9.43879i	−0.437714	−	0.758142i
$156$		0			0
$157$	2.10102	−	3.63907i	0.167680	−	0.290430i	−0.769924	−	0.638136i	$-0.779706\pi$
$157$	2.10102	−	3.63907i	0.167680	−	0.290430i	0.937604	+	0.347706i	$0.113039\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.69694			−0.291360
$162$		0			0
$163$	11.7980			0.924087			0.462044	−	0.886857i	$-0.347116\pi$
$163$	11.7980			0.924087			0.462044	+	0.886857i	$0.347116\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.17423	+	5.49794i	−0.245630	+	0.425443i	−0.962308	−	0.271960i	$-0.912328\pi$
$167$	−3.17423	+	5.49794i	−0.245630	+	0.425443i	0.716679	+	0.697403i	$0.245661\pi$
$168$		0			0
$169$	6.50000	+	11.2583i	0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.00000	−	10.3923i	−0.456172	−	0.790112i	0.542583	−	0.840002i	$-0.317446\pi$
$173$	−6.00000	−	10.3923i	−0.456172	−	0.790112i	−0.998755	+	0.0498898i	$0.984113\pi$
$174$		0			0
$175$	0.724745	−	1.25529i	0.0547856	−	0.0948914i
$176$		0			0
$177$		0			0
$178$		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
$183$		0			0
$184$		0			0
$185$	3.00000	−	5.19615i	0.220564	−	0.382029i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.55051	−	4.41761i	−0.184548	−	0.319647i	0.758876	−	0.651235i	$-0.225749\pi$
$191$	−2.55051	−	4.41761i	−0.184548	−	0.319647i	−0.943424	+	0.331588i	$0.892416\pi$
$192$		0			0
$193$	10.8990	−	18.8776i	0.784526	−	1.35884i	−0.144756	−	0.989467i	$-0.546240\pi$
$193$	10.8990	−	18.8776i	0.784526	−	1.35884i	0.929282	−	0.369371i	$-0.120427\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−23.5959			−1.68114			−0.840570	−	0.541703i	$-0.817780\pi$
$197$	−23.5959			−1.68114			−0.840570	+	0.541703i	$0.817780\pi$
$198$		0			0
$199$	13.1010			0.928707			0.464353	−	0.885650i	$-0.346287\pi$
$199$	13.1010			0.928707			0.464353	+	0.885650i	$0.346287\pi$
$200$		0			0
$201$		0			0
$202$		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
$207$		0			0
$208$		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
$213$		0			0
$214$		0			0
$215$	−7.79796			−0.531816
$216$		0			0
$217$	−15.7980			−1.07244
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.275255	−	0.476756i	−0.0184324	−	0.0319259i	0.856662	−	0.515878i	$-0.172534\pi$
$223$	−0.275255	−	0.476756i	−0.0184324	−	0.0319259i	−0.875094	+	0.483952i	$0.839201\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.8990	−	24.0737i	−0.922508	−	1.59783i	−0.795521	−	0.605926i	$-0.792803\pi$
$227$	−13.8990	−	24.0737i	−0.922508	−	1.59783i	−0.126986	−	0.991904i	$-0.540530\pi$
$228$		0			0
$229$	6.94949	−	12.0369i	0.459235	−	0.795419i	−0.539686	−	0.841867i	$-0.681457\pi$
$229$	6.94949	−	12.0369i	0.459235	−	0.795419i	0.998921	+	0.0464480i	$0.0147902\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.202041			0.0132361			0.00661807	−	0.999978i	$-0.497893\pi$
$233$	0.202041			0.0132361			0.00661807	+	0.999978i	$0.497893\pi$
$234$		0			0
$235$	4.55051			0.296843
$236$		0			0
$237$		0			0
$238$		0			0
$239$	10.8990	−	18.8776i	0.704996	−	1.22109i	−0.261696	−	0.965150i	$-0.584282\pi$
$239$	10.8990	−	18.8776i	0.704996	−	1.22109i	0.966693	−	0.255939i	$-0.0823847\pi$
$240$		0			0
$241$	−12.8485	−	22.2542i	−0.827643	−	1.43352i	−0.899883	−	0.436132i	$-0.856348\pi$
$241$	−12.8485	−	22.2542i	−0.827643	−	1.43352i	0.0722401	−	0.997387i	$-0.476985\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.44949	+	4.24264i	0.156492	+	0.271052i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		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
$255$		0			0
$256$		0			0
$257$	4.10102	−	7.10318i	0.255815	−	0.443084i	−0.709302	−	0.704905i	$-0.750990\pi$
$257$	4.10102	−	7.10318i	0.255815	−	0.443084i	0.965116	+	0.261821i	$0.0843230\pi$
$258$		0			0
$259$	−4.34847	−	7.53177i	−0.270201	−	0.468001i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.00000	+	15.5885i	0.554964	+	0.961225i	0.997906	+	0.0646755i	$0.0206012\pi$
$263$	9.00000	+	15.5885i	0.554964	+	0.961225i	−0.442943	+	0.896550i	$0.646065\pi$
$264$		0			0
$265$	−5.89898	+	10.2173i	−0.362371	+	0.627646i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.5959			1.01187			0.505935	−	0.862571i	$-0.331147\pi$
$269$	16.5959			1.01187			0.505935	+	0.862571i	$0.331147\pi$
$270$		0			0
$271$	21.1010			1.28180			0.640898	−	0.767626i	$-0.278562\pi$
$271$	21.1010			1.28180			0.640898	+	0.767626i	$0.278562\pi$
$272$		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
$279$		0			0
$280$		0			0
$281$	−8.94949	−	15.5010i	−0.533882	−	0.924710i	−0.999217	−	0.0395756i	$-0.987399\pi$
$281$	−8.94949	−	15.5010i	−0.533882	−	0.924710i	0.465335	−	0.885135i	$-0.345934\pi$
$282$		0			0
$283$	9.17423	−	15.8902i	0.545352	−	0.944577i	−0.453233	−	0.891392i	$-0.649729\pi$
$283$	9.17423	−	15.8902i	0.545352	−	0.944577i	0.998585	−	0.0531847i	$-0.0169372\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.146428			−0.00864338
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.7980	+	18.7026i	−0.630823	+	1.09262i	0.356560	+	0.934272i	$0.383949\pi$
$293$	−10.7980	+	18.7026i	−0.630823	+	1.09262i	−0.987384	+	0.158346i	$0.949384\pi$
$294$		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
$303$		0			0
$304$		0			0
$305$	−3.00000			−0.171780
$306$		0			0
$307$	−29.2474			−1.66924			−0.834620	−	0.550826i	$-0.814313\pi$
$307$	−29.2474			−1.66924			−0.834620	+	0.550826i	$0.814313\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.44949	−	2.51059i	0.0821930	−	0.142362i	−0.821999	−	0.569489i	$-0.807141\pi$
$311$	1.44949	−	2.51059i	0.0821930	−	0.142362i	0.904192	+	0.427127i	$0.140474\pi$
$312$		0			0
$313$	−11.7980	−	20.4347i	−0.666860	−	1.15504i	−0.978777	−	0.204926i	$-0.934305\pi$
$313$	−11.7980	−	20.4347i	−0.666860	−	1.15504i	0.311917	−	0.950109i	$-0.399029\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.10102	−	5.37113i	−0.174171	−	0.301672i	0.765703	−	0.643194i	$-0.222391\pi$
$317$	−3.10102	−	5.37113i	−0.174171	−	0.301672i	−0.939874	+	0.341522i	$0.889058\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.79796			0.322607
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		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	$-0.192054\pi$
$331$	−1.44949	−	2.51059i	−0.0796712	−	0.137994i	−0.903108	+	0.429413i	$0.858720\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.62372	−	9.74058i	−0.307257	−	0.532185i
$336$		0			0
$337$	5.10102	−	8.83523i	0.277870	−	0.481285i	−0.692985	−	0.720952i	$-0.743705\pi$
$337$	5.10102	−	8.83523i	0.277870	−	0.481285i	0.970855	+	0.239667i	$0.0770381\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	17.2474			0.931275
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.79796	−	11.7744i	0.364934	−	0.632083i	−0.623832	−	0.781559i	$-0.714425\pi$
$347$	6.79796	−	11.7744i	0.364934	−	0.632083i	0.988765	+	0.149475i	$0.0477584\pi$
$348$		0			0
$349$	15.8485	+	27.4504i	0.848349	+	1.46938i	0.882681	+	0.469973i	$0.155736\pi$
$349$	15.8485	+	27.4504i	0.848349	+	1.46938i	−0.0343315	+	0.999410i	$0.510930\pi$
$350$		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
$357$		0			0
$358$		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
$363$		0			0
$364$		0			0
$365$	2.89898	−	5.02118i	0.151740	−	0.262821i
$366$		0			0
$367$	−11.0000	−	19.0526i	−0.574195	−	0.994535i	−0.996129	−	0.0879086i	$-0.971982\pi$
$367$	−11.0000	−	19.0526i	−0.574195	−	0.994535i	0.421933	−	0.906627i	$-0.361352\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.55051	+	14.8099i	0.443920	+	0.768893i
$372$		0			0
$373$	8.79796	−	15.2385i	0.455541	−	0.789020i	−0.543178	−	0.839618i	$-0.682779\pi$
$373$	8.79796	−	15.2385i	0.455541	−	0.789020i	0.998719	+	0.0505973i	$0.0161125\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	17.1010			0.878420			0.439210	−	0.898384i	$-0.355258\pi$
$379$	17.1010			0.878420			0.439210	+	0.898384i	$0.355258\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.00000	−	1.73205i	0.0510976	−	0.0885037i	−0.839345	−	0.543599i	$-0.817061\pi$
$383$	1.00000	−	1.73205i	0.0510976	−	0.0885037i	0.890443	+	0.455095i	$0.150395\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.2980	+	19.5686i	0.572829	+	0.992169i	0.996274	+	0.0862473i	$0.0274875\pi$
$389$	11.2980	+	19.5686i	0.572829	+	0.992169i	−0.423444	+	0.905922i	$0.639179\pi$
$390$		0			0
$391$	−2.55051	+	4.41761i	−0.128985	+	0.223408i
$392$		0			0
$393$		0			0
$394$		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
$399$		0			0
$400$		0			0
$401$	−6.79796	+	11.7744i	−0.339474	+	0.587986i	−0.984334	−	0.176315i	$-0.943582\pi$
$401$	−6.79796	+	11.7744i	−0.339474	+	0.587986i	0.644860	+	0.764301i	$0.276916\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−14.7980	+	25.6308i	−0.731712	+	1.26736i	0.224439	+	0.974488i	$0.427945\pi$
$409$	−14.7980	+	25.6308i	−0.731712	+	1.26736i	−0.956151	+	0.292874i	$0.905388\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−15.7980			−0.777367
$414$		0			0
$415$	0.550510			0.0270235
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.0000	−	17.3205i	0.488532	−	0.846162i	−0.511381	−	0.859354i	$-0.670866\pi$
$419$	10.0000	−	17.3205i	0.488532	−	0.846162i	0.999913	+	0.0131919i	$0.00419923\pi$
$420$		0			0
$421$	6.79796	+	11.7744i	0.331312	+	0.573850i	0.982769	−	0.184836i	$-0.0591753\pi$
$421$	6.79796	+	11.7744i	0.331312	+	0.573850i	−0.651457	+	0.758685i	$0.725842\pi$
$422$		0			0
$423$		0			0
$424$		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
$435$		0			0
$436$		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	$-0.180328\pi$
$439$	−0.898979	−	1.55708i	−0.0429059	−	0.0743153i	−0.886681	+	0.462382i	$0.846995\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.0732	+	22.6435i	0.621127	+	1.07582i	0.989276	+	0.146057i	$0.0466583\pi$
$443$	13.0732	+	22.6435i	0.621127	+	1.07582i	−0.368149	+	0.929767i	$0.620008\pi$
$444$		0			0
$445$	−8.39898	+	14.5475i	−0.398150	+	0.689616i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	17.5959			0.830403			0.415201	−	0.909730i	$-0.363711\pi$
$449$	17.5959			0.830403			0.415201	+	0.909730i	$0.363711\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	10.7980	+	18.7026i	0.505107	+	0.874871i	0.999983	+	0.00590738i	$0.00188039\pi$
$457$	10.7980	+	18.7026i	0.505107	+	0.874871i	−0.494875	+	0.868964i	$0.664786\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.9495	+	18.9651i	0.509969	+	0.883291i	0.999933	+	0.0115492i	$0.00367632\pi$
$461$	10.9495	+	18.9651i	0.509969	+	0.883291i	−0.489965	+	0.871742i	$0.662990\pi$
$462$		0			0
$463$	−15.8990	+	27.5378i	−0.738888	+	1.27979i	0.214108	+	0.976810i	$0.431316\pi$
$463$	−15.8990	+	27.5378i	−0.738888	+	1.27979i	−0.952996	+	0.302982i	$0.902018\pi$
$464$		0			0
$465$		0			0
$466$		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
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.44949	−	2.51059i	−0.0665072	−	0.115194i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.24745	−	16.0171i	−0.422527	−	0.731838i	0.573659	−	0.819094i	$-0.305523\pi$
$479$	−9.24745	−	16.0171i	−0.422527	−	0.731838i	−0.996186	+	0.0872564i	$0.972190\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.00000			0.0908153
$486$		0			0
$487$	25.5959			1.15986			0.579931	−	0.814666i	$-0.303080\pi$
$487$	25.5959			1.15986			0.579931	+	0.814666i	$0.303080\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.89898	−	5.02118i	0.130829	−	0.226603i	−0.793167	−	0.609004i	$-0.791569\pi$
$491$	2.89898	−	5.02118i	0.130829	−	0.226603i	0.923996	+	0.382401i	$0.124903\pi$
$492$		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	−0.361478	−	0.932381i	$-0.617728\pi$
$499$	14.0000	−	24.2487i	0.626726	−	1.08552i	0.988204	−	0.153141i	$-0.0489388\pi$
$500$		0			0
$501$		0			0
$502$		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
$507$		0			0
$508$		0			0
$509$	−6.94949	+	12.0369i	−0.308031	+	0.533525i	−0.977931	−	0.208926i	$-0.933003\pi$
$509$	−6.94949	+	12.0369i	−0.308031	+	0.533525i	0.669901	+	0.742451i	$0.266337\pi$
$510$		0			0
$511$	−4.20204	−	7.27815i	−0.185887	−	0.321966i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.00000	−	8.66025i	−0.220326	−	0.381616i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.7980			−1.26166			−0.630831	−	0.775920i	$-0.717286\pi$
$521$	−28.7980			−1.26166			−0.630831	+	0.775920i	$0.717286\pi$
$522$		0			0
$523$	−19.6515			−0.859301			−0.429651	−	0.902995i	$-0.641363\pi$
$523$	−19.6515			−0.859301			−0.429651	+	0.902995i	$0.641363\pi$
$524$		0			0
$525$		0			0
$526$		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
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	1.17423	−	2.03383i	0.0507666	−	0.0879303i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		0			0
$545$	−4.39898	+	7.61926i	−0.188432	+	0.326373i
$546$		0			0
$547$	20.6237	+	35.7213i	0.881807	+	1.52733i	0.849330	+	0.527862i	$0.177006\pi$
$547$	20.6237	+	35.7213i	0.881807	+	1.52733i	0.0324764	+	0.999473i	$0.489661\pi$
$548$		0			0
$549$		0			0
$550$		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
$555$		0			0
$556$		0			0
$557$	−24.2020			−1.02547			−0.512737	−	0.858546i	$-0.671368\pi$
$557$	−24.2020			−1.02547			−0.512737	+	0.858546i	$0.671368\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.17423	−	3.76588i	0.0916331	−	0.158713i	−0.816565	−	0.577253i	$-0.804125\pi$
$563$	2.17423	−	3.76588i	0.0916331	−	0.158713i	0.908198	+	0.418540i	$0.137458\pi$
$564$		0			0
$565$	4.89898	+	8.48528i	0.206102	+	0.356978i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.00000	+	8.66025i	0.209611	+	0.363057i	0.951592	−	0.307364i	$-0.0994469\pi$
$569$	5.00000	+	8.66025i	0.209611	+	0.363057i	−0.741981	+	0.670421i	$0.766114\pi$
$570$		0			0
$571$	−3.10102	+	5.37113i	−0.129774	+	0.224775i	−0.923589	−	0.383385i	$-0.874758\pi$
$571$	−3.10102	+	5.37113i	−0.129774	+	0.224775i	0.793815	+	0.608159i	$0.208092\pi$
$572$		0			0
$573$		0			0
$574$		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
$579$		0			0
$580$		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	0.999022	−	0.0442259i	$-0.0140821\pi$
$587$	11.1742	+	19.3543i	0.461210	+	0.798839i	−0.537812	+	0.843065i	$0.680749\pi$
$588$		0			0
$589$	−15.7980	+	27.3629i	−0.650944	+	1.12747i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−27.3939			−1.12493			−0.562466	−	0.826821i	$-0.690147\pi$
$593$	−27.3939			−1.12493			−0.562466	+	0.826821i	$0.690147\pi$
$594$		0			0
$595$	−2.89898			−0.118847
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.34847	−	4.06767i	0.0959559	−	0.166200i	−0.814051	−	0.580793i	$-0.802743\pi$
$599$	2.34847	−	4.06767i	0.0959559	−	0.166200i	0.910007	+	0.414593i	$0.136076\pi$
$600$		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
$603$		0			0
$604$		0			0
$605$	5.50000	+	9.52628i	0.223607	+	0.387298i
$606$		0			0
$607$	16.0732	−	27.8396i	0.652392	−	1.12998i	−0.330149	−	0.943929i	$-0.607099\pi$
$607$	16.0732	−	27.8396i	0.652392	−	1.12998i	0.982541	−	0.186047i	$-0.0595676\pi$
$608$		0			0
$609$		0			0
$610$		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			0
$616$		0			0
$617$	21.5959	−	37.4052i	0.869419	−	1.50588i	0.00682740	−	0.999977i	$-0.497827\pi$
$617$	21.5959	−	37.4052i	0.869419	−	1.50588i	0.862592	−	0.505901i	$-0.168840\pi$
$618$		0			0
$619$	−2.34847	−	4.06767i	−0.0943929	−	0.163493i	0.814962	−	0.579514i	$-0.196758\pi$
$619$	−2.34847	−	4.06767i	−0.0943929	−	0.163493i	−0.909355	+	0.416021i	$0.863424\pi$
$620$		0			0
$621$		0			0
$622$		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	$-0.344373\pi$
$631$	23.5959			0.939339			0.469669	+	0.882842i	$0.344373\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.17423	−	12.4261i	0.284701	−	0.493116i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.05051	−	3.55159i	−0.0809903	−	0.140279i	0.822685	−	0.568497i	$-0.192475\pi$
$641$	−2.05051	−	3.55159i	−0.0809903	−	0.140279i	−0.903676	+	0.428218i	$0.859142\pi$
$642$		0			0
$643$	−7.07321	+	12.2512i	−0.278940	+	0.483139i	−0.971122	−	0.238585i	$-0.923317\pi$
$643$	−7.07321	+	12.2512i	−0.278940	+	0.483139i	0.692181	+	0.721724i	$0.256650\pi$
$644$		0			0
$645$		0			0
$646$		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
$651$		0			0
$652$		0			0
$653$	4.10102	−	7.10318i	0.160485	−	0.277969i	−0.774558	−	0.632503i	$-0.782027\pi$
$653$	4.10102	−	7.10318i	0.160485	−	0.277969i	0.935043	+	0.354535i	$0.115361\pi$
$654$		0			0
$655$	−3.44949	−	5.97469i	−0.134783	−	0.233451i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.8990	+	25.8058i	0.580382	+	1.00525i	0.995434	+	0.0954532i	$0.0304300\pi$
$659$	14.8990	+	25.8058i	0.580382	+	1.00525i	−0.415052	+	0.909798i	$0.636237\pi$
$660$		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
$665$	−4.20204			−0.162948
$666$		0			0
$667$	−20.1464			−0.780073
$668$		0			0
$669$		0			0
$670$		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
$675$		0			0
$676$		0			0
$677$	16.8990	+	29.2699i	0.649481	+	1.12493i	0.983247	+	0.182278i	$0.0583469\pi$
$677$	16.8990	+	29.2699i	0.649481	+	1.12493i	−0.333767	+	0.942656i	$0.608320\pi$
$678$		0			0
$679$	1.44949	−	2.51059i	0.0556263	−	0.0963476i
$680$		0			0
$681$		0			0
$682$		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
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−14.5505	−	25.2022i	−0.553527	−	0.958738i	−0.998016	−	0.0629534i	$-0.979948\pi$
$691$	−14.5505	−	25.2022i	−0.553527	−	0.958738i	0.444489	−	0.895784i	$-0.353385\pi$
$692$		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			0
$700$		0			0
$701$	28.3939			1.07242			0.536211	−	0.844084i	$-0.319855\pi$
$701$	28.3939			1.07242			0.536211	+	0.844084i	$0.319855\pi$
$702$		0			0
$703$	−17.3939			−0.656022
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.44949	−	2.51059i	0.0545137	−	0.0944205i
$708$		0			0
$709$	9.84847	+	17.0580i	0.369867	+	0.640628i	0.989544	−	0.144228i	$-0.0460699\pi$
$709$	9.84847	+	17.0580i	0.369867	+	0.640628i	−0.619677	+	0.784857i	$0.712737\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.8990	−	24.0737i	−0.520521	−	0.901569i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		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
$723$		0			0
$724$		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	$-0.126703\pi$
$727$	3.37628	+	5.84788i	0.125219	+	0.216886i	−0.796599	+	0.604508i	$0.793370\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7.79796	+	13.5065i	0.288418	+	0.499555i
$732$		0			0
$733$	−4.79796	+	8.31031i	−0.177217	+	0.306948i	−0.940926	−	0.338612i	$-0.890043\pi$
$733$	−4.79796	+	8.31031i	−0.177217	+	0.306948i	0.763710	+	0.645560i	$0.223376\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−42.8990			−1.57806			−0.789032	−	0.614352i	$-0.789418\pi$
$739$	−42.8990			−1.57806			−0.789032	+	0.614352i	$0.789418\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.5227	−	32.0823i	0.679532	−	1.17698i	−0.295590	−	0.955315i	$-0.595516\pi$
$743$	18.5227	−	32.0823i	0.679532	−	1.17698i	0.975122	−	0.221669i	$-0.0711505\pi$
$744$		0			0
$745$	−10.5000	−	18.1865i	−0.384690	−	0.666303i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.70204	−	2.94802i	−0.0621912	−	0.107718i
$750$		0			0
$751$	−22.8990	+	39.6622i	−0.835596	+	1.44729i	0.0579489	+	0.998320i	$0.481544\pi$
$751$	−22.8990	+	39.6622i	−0.835596	+	1.44729i	−0.893545	+	0.448975i	$0.851789\pi$
$752$		0			0
$753$		0			0
$754$		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	−0.352273	−	0.935897i	$-0.614591\pi$
$761$	17.5000	−	30.3109i	0.634375	−	1.09877i	0.986647	−	0.162872i	$-0.0520756\pi$
$762$		0			0
$763$	6.37628	+	11.0440i	0.230837	+	0.399821i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	12.3990	−	21.4757i	0.447119	−	0.774432i	−0.551078	−	0.834453i	$-0.685784\pi$
$769$	12.3990	−	21.4757i	0.447119	−	0.774432i	0.998197	+	0.0600212i	$0.0191168\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−25.7980			−0.927888			−0.463944	−	0.885865i	$-0.653566\pi$
$773$	−25.7980			−0.927888			−0.463944	+	0.885865i	$0.653566\pi$
$774$		0			0
$775$	10.8990			0.391503
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.146428	+	0.253621i	−0.00524633	+	0.00908692i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		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	$-0.773965\pi$
$787$	5.20204	−	9.01020i	0.185433	−	0.321179i	0.943722	+	0.330739i	$0.107298\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	14.2020			0.504966
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.00000	+	6.92820i	−0.141687	+	0.245410i	−0.928132	−	0.372251i	$-0.878586\pi$
$797$	−4.00000	+	6.92820i	−0.141687	+	0.245410i	0.786445	+	0.617661i	$0.211919\pi$
$798$		0			0
$799$	−4.55051	−	7.88171i	−0.160985	−	0.278835i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	1.84847	−	3.20164i	0.0651500	−	0.112843i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$	22.8990			0.804092			0.402046	−	0.915619i	$-0.368299\pi$
$811$	22.8990			0.804092			0.402046	+	0.915619i	$0.368299\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.89898	+	10.2173i	−0.206632	+	0.357898i
$816$		0			0
$817$	11.3031	+	19.5775i	0.395444	+	0.684929i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−8.39898	−	14.5475i	−0.293126	−	0.507710i	0.681421	−	0.731892i	$-0.261362\pi$
$821$	−8.39898	−	14.5475i	−0.293126	−	0.507710i	−0.974547	+	0.224182i	$0.928029\pi$
$822$		0			0
$823$	−3.17423	+	5.49794i	−0.110647	+	0.191646i	−0.916031	−	0.401107i	$-0.868626\pi$
$823$	−3.17423	+	5.49794i	−0.110647	+	0.191646i	0.805384	+	0.592753i	$0.201959\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	39.5403			1.37495			0.687476	−	0.726208i	$-0.258719\pi$
$827$	39.5403			1.37495			0.687476	+	0.726208i	$0.258719\pi$
$828$		0			0
$829$	−28.1918			−0.979143			−0.489571	−	0.871963i	$-0.662847\pi$
$829$	−28.1918			−0.979143			−0.489571	+	0.871963i	$0.662847\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.89898	−	8.48528i	0.169740	−	0.293998i
$834$		0			0
$835$	−3.17423	−	5.49794i	−0.109849	−	0.190264i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.3485	−	21.3882i	−0.426317	−	0.738402i	0.570226	−	0.821488i	$-0.306856\pi$
$839$	−12.3485	−	21.3882i	−0.426317	−	0.738402i	−0.996542	+	0.0830861i	$0.973522\pi$
$840$		0			0
$841$	−16.6969	+	28.9199i	−0.575756	+	0.997240i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.0000			−0.447214
$846$		0			0
$847$	15.9444			0.547856
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.65153	−	13.2528i	0.262291	−	0.454302i
$852$		0			0
$853$	−18.8990	−	32.7340i	−0.647089	−	1.12079i	−0.983815	−	0.179188i	$-0.942653\pi$
$853$	−18.8990	−	32.7340i	−0.647089	−	1.12079i	0.336726	−	0.941603i	$-0.390680\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19.7980	+	34.2911i	0.676285	+	1.17136i	0.976091	+	0.217360i	$0.0697447\pi$
$857$	19.7980	+	34.2911i	0.676285	+	1.17136i	−0.299806	+	0.954000i	$0.596922\pi$
$858$		0			0
$859$	19.5959	−	33.9411i	0.668604	−	1.15806i	−0.309691	−	0.950837i	$-0.600225\pi$
$859$	19.5959	−	33.9411i	0.668604	−	1.15806i	0.978295	−	0.207219i	$-0.0664412\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−52.5505			−1.78884			−0.894420	−	0.447228i	$-0.852411\pi$
$863$	−52.5505			−1.78884			−0.894420	+	0.447228i	$0.852411\pi$
$864$		0			0
$865$	12.0000			0.408012
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.724745	+	1.25529i	0.0245008	+	0.0424367i
$876$		0			0
$877$	19.7980	−	34.2911i	0.668530	−	1.15793i	−0.309786	−	0.950806i	$-0.600257\pi$
$877$	19.7980	−	34.2911i	0.668530	−	1.15793i	0.978315	−	0.207121i	$-0.0664093\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−15.8990			−0.535650			−0.267825	−	0.963468i	$-0.586305\pi$
$881$	−15.8990			−0.535650			−0.267825	+	0.963468i	$0.586305\pi$
$882$		0			0
$883$	−45.0454			−1.51590			−0.757949	−	0.652313i	$-0.773799\pi$
$883$	−45.0454			−1.51590			−0.757949	+	0.652313i	$0.773799\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.7980	+	32.5590i	−0.631174	+	1.09322i	0.356138	+	0.934433i	$0.384093\pi$
$887$	−18.7980	+	32.5590i	−0.631174	+	1.09322i	−0.987312	+	0.158792i	$0.949240\pi$
$888$		0			0
$889$	−10.3990	−	18.0116i	−0.348771	−	0.604088i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.59592	−	11.4245i	−0.220724	−	0.382305i
$894$		0			0
$895$	2.89898	−	5.02118i	0.0969022	−	0.167840i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−86.0908			−2.87129
$900$		0			0
$901$	23.5959			0.786094
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9.84847	+	17.0580i	−0.327374	+	0.567029i
$906$		0			0
$907$	18.1742	+	31.4787i	0.603466	+	1.04523i	0.992292	+	0.123922i	$0.0395473\pi$
$907$	18.1742	+	31.4787i	0.603466	+	1.04523i	−0.388826	+	0.921311i	$0.627119\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−19.2474	−	33.3376i	−0.637696	−	1.10452i	−0.985937	−	0.167117i	$-0.946554\pi$
$911$	−19.2474	−	33.3376i	−0.637696	−	1.10452i	0.348241	−	0.937405i	$-0.386779\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.0000			−0.330229
$918$		0			0
$919$	−4.69694			−0.154938			−0.0774689	−	0.996995i	$-0.524684\pi$
$919$	−4.69694			−0.154938			−0.0774689	+	0.996995i	$0.524684\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.00000	+	5.19615i	0.0986394	+	0.170848i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.797959	−	1.38211i	−0.0261802	−	0.0453454i	0.852638	−	0.522501i	$-0.175001\pi$
$929$	−0.797959	−	1.38211i	−0.0261802	−	0.0453454i	−0.878819	+	0.477156i	$0.841668\pi$
$930$		0			0
$931$	7.10102	−	12.2993i	0.232727	−	0.403094i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	56.9898			1.86178			0.930888	−	0.365305i	$-0.119035\pi$
$937$	56.9898			1.86178			0.930888	+	0.365305i	$0.119035\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.8485	−	23.9863i	0.451447	−	0.781929i	−0.547029	−	0.837114i	$-0.684241\pi$
$941$	13.8485	−	23.9863i	0.451447	−	0.781929i	0.998476	+	0.0551842i	$0.0175746\pi$
$942$		0			0
$943$	−0.128827	−	0.223135i	−0.00419518	−	0.00726627i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−23.0732	−	39.9640i	−0.749779	−	1.29865i	−0.947929	−	0.318483i	$-0.896827\pi$
$947$	−23.0732	−	39.9640i	−0.749779	−	1.29865i	0.198150	−	0.980172i	$-0.436507\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	33.7980			1.09482			0.547412	−	0.836863i	$-0.315613\pi$
$953$	33.7980			1.09482			0.547412	+	0.836863i	$0.315613\pi$
$954$		0			0
$955$	5.10102			0.165065
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.2020	−	24.5987i	0.458608	−	0.794332i
$960$		0			0
$961$	−43.8939	−	76.0264i	−1.41593	−	2.45247i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10.8990	+	18.8776i	0.350851	+	0.607691i
$966$		0			0
$967$	7.42168	−	12.8547i	0.238665	−	0.413380i	−0.721666	−	0.692241i	$-0.756623\pi$
$967$	7.42168	−	12.8547i	0.238665	−	0.413380i	0.960332	+	0.278861i	$0.0899568\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−60.6969			−1.94786			−0.973929	−	0.226854i	$-0.927156\pi$
$971$	−60.6969			−1.94786			−0.973929	+	0.226854i	$0.927156\pi$
$972$		0			0
$973$	28.4041			0.910593
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.6969	−	39.3123i	0.726139	−	1.25771i	−0.232364	−	0.972629i	$-0.574646\pi$
$977$	22.6969	−	39.3123i	0.726139	−	1.25771i	0.958503	−	0.285081i	$-0.0920206\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.27526	−	5.67291i	−0.104464	−	0.180938i	0.809055	−	0.587733i	$-0.199980\pi$
$983$	−3.27526	−	5.67291i	−0.104464	−	0.180938i	−0.913519	+	0.406795i	$0.866646\pi$
$984$		0			0
$985$	11.7980	−	20.4347i	0.375914	−	0.651103i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.8888			−0.632426
$990$		0			0
$991$	−0.696938			−0.0221390			−0.0110695	−	0.999939i	$-0.503524\pi$
$991$	−0.696938			−0.0221390			−0.0110695	+	0.999939i	$0.503524\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.55051	+	11.3458i	−0.207665	+	0.359687i
$996$		0			0
$997$	−17.6969	−	30.6520i	−0.560468	−	0.970758i	−0.997456	−	0.0712911i	$-0.977288\pi$
$997$	−17.6969	−	30.6520i	−0.560468	−	0.970758i	0.436988	−	0.899467i	$-0.356045\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.2.q.c.361.2		4
3.2	odd	2		360.2.q.c.121.2	✓	4
4.3	odd	2		2160.2.q.g.1441.1		4
9.2	odd	6		360.2.q.c.241.2	yes	4
9.4	even	3		3240.2.a.o.1.1		2
9.5	odd	6		3240.2.a.j.1.1		2
9.7	even	3	inner	1080.2.q.c.721.2		4
12.11	even	2		720.2.q.g.481.1		4
36.7	odd	6		2160.2.q.g.721.1		4
36.11	even	6		720.2.q.g.241.1		4
36.23	even	6		6480.2.a.bc.1.2		2
36.31	odd	6		6480.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.c.121.2	✓	4	3.2	odd	2
360.2.q.c.241.2	yes	4	9.2	odd	6
720.2.q.g.241.1		4	36.11	even	6
720.2.q.g.481.1		4	12.11	even	2
1080.2.q.c.361.2		4	1.1	even	1	trivial
1080.2.q.c.721.2		4	9.7	even	3	inner
2160.2.q.g.721.1		4	36.7	odd	6
2160.2.q.g.1441.1		4	4.3	odd	2
3240.2.a.j.1.1		2	9.5	odd	6
3240.2.a.o.1.1		2	9.4	even	3
6480.2.a.bc.1.2		2	36.23	even	6
6480.2.a.bl.1.2		2	36.31	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} +(0.724745 + 1.25529i) q^{7} +2.00000 q^{17} +2.89898 q^{19} +(-1.27526 + 2.20881i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(3.94949 + 6.84072i) q^{29} +(-5.44949 + 9.43879i) q^{31} -1.44949 q^{35} -6.00000 q^{37} +(-0.0505103 + 0.0874863i) q^{41} +(3.89898 + 6.75323i) q^{43} +(-2.27526 - 3.94086i) q^{47} +(2.44949 - 4.24264i) q^{49} +11.7980 q^{53} +(-5.44949 + 9.43879i) q^{59} +(1.50000 + 2.59808i) q^{61} +(-5.62372 + 9.74058i) q^{67} +9.79796 q^{71} -5.79796 q^{73} +(-1.44949 - 2.51059i) q^{79} +(-0.275255 - 0.476756i) q^{83} +(-1.00000 + 1.73205i) q^{85} +16.7980 q^{89} +(-1.44949 + 2.51059i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} - 2 q^{7} + 8 q^{17} - 8 q^{19} - 10 q^{23} - 2 q^{25} + 6 q^{29} - 12 q^{31} + 4 q^{35} - 24 q^{37} - 10 q^{41} - 4 q^{43} - 14 q^{47} + 8 q^{53} - 12 q^{59} + 6 q^{61} + 2 q^{67} + 16 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 - 2 * q^7 + 8 * q^17 - 8 * q^19 - 10 * q^23 - 2 * q^25 + 6 * q^29 - 12 * q^31 + 4 * q^35 - 24 * q^37 - 10 * q^41 - 4 * q^43 - 14 * q^47 + 8 * q^53 - 12 * q^59 + 6 * q^61 + 2 * q^67 + 16 * q^73 + 4 * q^79 - 6 * q^83 - 4 * q^85 + 28 * q^89 + 4 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more