LMFDB - Embedded newform 2160.2.q.g.721.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,2,Mod(721,2160)] 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("2160.721"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.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:	$17.2476868366$
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			721.2
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2160.721
Dual form			2160.2.q.g.1441.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(1.72474 - 2.98735i) q^{7} +2.00000 q^{17} +6.89898 q^{19} +(3.72474 + 6.45145i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.949490 + 1.64456i) q^{29} +(0.550510 + 0.953512i) q^{31} -3.44949 q^{35} -6.00000 q^{37} +(-4.94949 - 8.57277i) q^{41} +(5.89898 - 10.2173i) q^{43} +(4.72474 - 8.18350i) q^{47} +(-2.44949 - 4.24264i) q^{49} -7.79796 q^{53} +(0.550510 + 0.953512i) q^{59} +(1.50000 - 2.59808i) q^{61} +(-6.62372 - 11.4726i) q^{67} +9.79796 q^{71} +13.7980 q^{73} +(-3.44949 + 5.97469i) q^{79} +(2.72474 - 4.71940i) q^{83} +(-1.00000 - 1.73205i) q^{85} -2.79796 q^{89} +(-3.44949 - 5.97469i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$		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
$7$	1.72474	−	2.98735i	0.651892	−	1.12911i	−0.330771	−	0.943711i	$-0.607309\pi$
$7$	1.72474	−	2.98735i	0.651892	−	1.12911i	0.982663	−	0.185399i	$-0.0593579\pi$
$8$		0			0
$9$		0			0
$10$		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			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
$19$	6.89898			1.58273			0.791367	−	0.611341i	$-0.209370\pi$
$19$	6.89898			1.58273			0.791367	+	0.611341i	$0.209370\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.72474	+	6.45145i	0.776663	+	1.34522i	0.933855	+	0.357652i	$0.116422\pi$
$23$	3.72474	+	6.45145i	0.776663	+	1.34522i	−0.157192	+	0.987568i	$0.550244\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.949490	+	1.64456i	−0.176316	+	0.305388i	−0.940616	−	0.339473i	$-0.889751\pi$
$29$	−0.949490	+	1.64456i	−0.176316	+	0.305388i	0.764300	+	0.644861i	$0.223085\pi$
$30$		0			0
$31$	0.550510	+	0.953512i	0.0988746	+	0.171256i	0.911219	−	0.411922i	$-0.135142\pi$
$31$	0.550510	+	0.953512i	0.0988746	+	0.171256i	−0.812345	+	0.583178i	$0.801809\pi$
$32$		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	−0.935923	−	0.352206i	$-0.885432\pi$
$41$	−4.94949	−	8.57277i	−0.772980	−	1.33884i	0.162942	−	0.986636i	$-0.447902\pi$
$42$		0			0
$43$	5.89898	−	10.2173i	0.899586	−	1.55813i	0.0715617	−	0.997436i	$-0.477202\pi$
$43$	5.89898	−	10.2173i	0.899586	−	1.55813i	0.828024	−	0.560692i	$-0.189465\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.72474	−	8.18350i	0.689175	−	1.19369i	−0.282930	−	0.959140i	$-0.591307\pi$
$47$	4.72474	−	8.18350i	0.689175	−	1.19369i	0.972105	−	0.234545i	$-0.0753602\pi$
$48$		0			0
$49$	−2.44949	−	4.24264i	−0.349927	−	0.606092i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.79796			−1.07113			−0.535566	−	0.844493i	$-0.679902\pi$
$53$	−7.79796			−1.07113			−0.535566	+	0.844493i	$0.679902\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.550510	+	0.953512i	0.0716703	+	0.124137i	0.899633	−	0.436646i	$-0.143834\pi$
$59$	0.550510	+	0.953512i	0.0716703	+	0.124137i	−0.827963	+	0.560783i	$0.810500\pi$
$60$		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
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.62372	−	11.4726i	−0.809217	−	1.40160i	−0.913407	−	0.407047i	$-0.866558\pi$
$67$	−6.62372	−	11.4726i	−0.809217	−	1.40160i	0.104190	−	0.994557i	$-0.466775\pi$
$68$		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
$73$	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
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−3.44949	+	5.97469i	−0.388098	+	0.672205i	−0.992194	−	0.124706i	$-0.960201\pi$
$79$	−3.44949	+	5.97469i	−0.388098	+	0.672205i	0.604096	+	0.796912i	$0.293534\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.72474	−	4.71940i	0.299080	−	0.518021i	−0.676846	−	0.736125i	$-0.736654\pi$
$83$	2.72474	−	4.71940i	0.299080	−	0.518021i	0.975926	+	0.218104i	$0.0699871\pi$
$84$		0			0
$85$	−1.00000	−	1.73205i	−0.108465	−	0.187867i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.79796			−0.296583			−0.148292	−	0.988944i	$-0.547377\pi$
$89$	−2.79796			−0.296583			−0.148292	+	0.988944i	$0.547377\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.44949	−	5.97469i	−0.353910	−	0.612990i
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	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	−0.911479	−	0.411346i	$-0.865059\pi$
$101$	−1.00000	+	1.73205i	−0.0995037	+	0.172345i	0.811976	+	0.583691i	$0.198392\pi$
$102$		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
$105$		0			0
$106$		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
$111$		0			0
$112$		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
$115$	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
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−0.348469			−0.0309216			−0.0154608	−	0.999880i	$-0.504922\pi$
$127$	−0.348469			−0.0309216			−0.0154608	+	0.999880i	$0.504922\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.44949	−	2.51059i	−0.126643	−	0.219351i	0.795731	−	0.605650i	$-0.207087\pi$
$131$	−1.44949	−	2.51059i	−0.126643	−	0.219351i	−0.922374	+	0.386299i	$0.873753\pi$
$132$		0			0
$133$	11.8990	−	20.6096i	1.03177	−	1.78708i
$134$		0			0
$135$		0			0
$136$		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	$0.145594\pi$
$139$	9.79796	+	16.9706i	0.831052	+	1.43942i	−0.0661527	+	0.997810i	$0.521072\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	1.89898			0.157702
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.5000	−	18.1865i	−0.860194	−	1.48990i	−0.871742	−	0.489966i	$-0.837009\pi$
$149$	−10.5000	−	18.1865i	−0.860194	−	1.48990i	0.0115483	−	0.999933i	$-0.496324\pi$
$150$		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
$153$		0			0
$154$		0			0
$155$	0.550510	−	0.953512i	0.0442180	−	0.0765879i
$156$		0			0
$157$	11.8990	+	20.6096i	0.949642	+	1.64483i	0.746179	+	0.665745i	$0.231886\pi$
$157$	11.8990	+	20.6096i	0.949642	+	1.64483i	0.203463	+	0.979083i	$0.434780\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	25.6969			2.02520
$162$		0			0
$163$	7.79796			0.610783			0.305392	−	0.952227i	$-0.401213\pi$
$163$	7.79796			0.610783			0.305392	+	0.952227i	$0.401213\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.17423	−	7.22999i	−0.323012	−	0.559473i	0.658096	−	0.752934i	$-0.271362\pi$
$167$	−4.17423	−	7.22999i	−0.323012	−	0.559473i	−0.981108	+	0.193461i	$0.938029\pi$
$168$		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	−0.998755	−	0.0498898i	$-0.984113\pi$
$173$	−6.00000	+	10.3923i	−0.456172	+	0.790112i	0.542583	+	0.840002i	$0.317446\pi$
$174$		0			0
$175$	1.72474	+	2.98735i	0.130378	+	0.225822i
$176$		0			0
$177$		0			0
$178$		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
$183$		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
$191$	7.44949	−	12.9029i	0.539026	−	0.933621i	−0.459931	−	0.887955i	$-0.652126\pi$
$191$	7.44949	−	12.9029i	0.539026	−	0.933621i	0.998957	−	0.0456658i	$-0.0145409\pi$
$192$		0			0
$193$	1.10102	+	1.90702i	0.0792532	+	0.137271i	0.902928	−	0.429792i	$-0.141413\pi$
$193$	1.10102	+	1.90702i	0.0792532	+	0.137271i	−0.823675	+	0.567063i	$0.808080\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.5959			1.11116			0.555582	−	0.831462i	$-0.312496\pi$
$197$	15.5959			1.11116			0.555582	+	0.831462i	$0.312496\pi$
$198$		0			0
$199$	−22.8990			−1.62327			−0.811633	−	0.584168i	$-0.801421\pi$
$199$	−22.8990			−1.62327			−0.811633	+	0.584168i	$0.801421\pi$
$200$		0			0
$201$		0			0
$202$		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
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	0.995222	−	0.0976347i	$-0.0311277\pi$
$211$	6.00000	+	10.3923i	0.413057	+	0.715436i	−0.582165	+	0.813070i	$0.697794\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−11.7980			−0.804614
$216$		0			0
$217$	3.79796			0.257822
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	2.72474	−	4.71940i	0.182462	−	0.316034i	−0.760256	−	0.649623i	$-0.774927\pi$
$223$	2.72474	−	4.71940i	0.182462	−	0.316034i	0.942718	+	0.333589i	$0.108260\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.10102	−	7.10318i	0.272194	−	0.471454i	−0.697229	−	0.716848i	$-0.745584\pi$
$227$	4.10102	−	7.10318i	0.272194	−	0.471454i	0.969423	+	0.245394i	$0.0789173\pi$
$228$		0			0
$229$	2.05051	+	3.55159i	0.135502	+	0.234696i	0.925789	−	0.378041i	$-0.123402\pi$
$229$	2.05051	+	3.55159i	0.135502	+	0.234696i	−0.790287	+	0.612736i	$0.790069\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	19.7980			1.29701			0.648504	−	0.761211i	$-0.275395\pi$
$233$	19.7980			1.29701			0.648504	+	0.761211i	$0.275395\pi$
$234$		0			0
$235$	−9.44949			−0.616417
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.10102	−	1.90702i	−0.0712191	−	0.123355i	0.828217	−	0.560408i	$-0.189356\pi$
$239$	−1.10102	−	1.90702i	−0.0712191	−	0.123355i	−0.899436	+	0.437053i	$0.856022\pi$
$240$		0			0
$241$	1.84847	−	3.20164i	0.119070	−	0.206236i	−0.800329	−	0.599561i	$-0.795342\pi$
$241$	1.84847	−	3.20164i	0.119070	−	0.206236i	0.919400	+	0.393325i	$0.128675\pi$
$242$		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
$251$	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
$255$		0			0
$256$		0			0
$257$	13.8990	+	24.0737i	0.866995	+	1.50168i	0.865053	+	0.501680i	$0.167285\pi$
$257$	13.8990	+	24.0737i	0.866995	+	1.50168i	0.00194150	+	0.999998i	$0.499382\pi$
$258$		0			0
$259$	−10.3485	+	17.9241i	−0.643023	+	1.11375i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.00000	+	15.5885i	−0.554964	+	0.961225i	0.442943	+	0.896550i	$0.353935\pi$
$263$	−9.00000	+	15.5885i	−0.554964	+	0.961225i	−0.997906	+	0.0646755i	$0.979399\pi$
$264$		0			0
$265$	3.89898	+	6.75323i	0.239512	+	0.414848i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.5959			−1.37770			−0.688849	−	0.724905i	$-0.741884\pi$
$269$	−22.5959			−1.37770			−0.688849	+	0.724905i	$0.741884\pi$
$270$		0			0
$271$	−30.8990			−1.87698			−0.938490	−	0.345307i	$-0.887775\pi$
$271$	−30.8990			−1.87698			−0.938490	+	0.345307i	$0.887775\pi$
$272$		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
$279$		0			0
$280$		0			0
$281$	−4.05051	+	7.01569i	−0.241633	+	0.418521i	−0.961180	−	0.275923i	$-0.911016\pi$
$281$	−4.05051	+	7.01569i	−0.241633	+	0.418521i	0.719546	+	0.694444i	$0.244350\pi$
$282$		0			0
$283$	−1.82577	−	3.16232i	−0.108530	−	0.187980i	0.806645	−	0.591037i	$-0.201281\pi$
$283$	−1.82577	−	3.16232i	−0.108530	−	0.187980i	−0.915175	+	0.403056i	$0.867948\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−34.1464			−2.01560
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8.79796	+	15.2385i	0.513982	+	0.890243i	0.999868	+	0.0162213i	$0.00516362\pi$
$293$	8.79796	+	15.2385i	0.513982	+	0.890243i	−0.485886	+	0.874022i	$0.661503\pi$
$294$		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
$303$		0			0
$304$		0			0
$305$	−3.00000			−0.171780
$306$		0			0
$307$	4.75255			0.271242			0.135621	−	0.990761i	$-0.456697\pi$
$307$	4.75255			0.271242			0.135621	+	0.990761i	$0.456697\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.44949	+	5.97469i	0.195603	+	0.338794i	0.947098	−	0.320945i	$-0.104000\pi$
$311$	3.44949	+	5.97469i	0.195603	+	0.338794i	−0.751495	+	0.659738i	$0.770667\pi$
$312$		0			0
$313$	7.79796	−	13.5065i	0.440767	−	0.763430i	−0.556980	−	0.830526i	$-0.688040\pi$
$313$	7.79796	−	13.5065i	0.440767	−	0.763430i	0.997747	+	0.0670957i	$0.0213733\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.8990	+	22.3417i	−0.724479	+	1.25483i	0.234709	+	0.972066i	$0.424586\pi$
$317$	−12.8990	+	22.3417i	−0.724479	+	1.25483i	−0.959188	+	0.282769i	$0.908747\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	13.7980			0.767739
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		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	$-0.894053\pi$
$331$	−3.44949	+	5.97469i	−0.189601	+	0.328399i	0.755516	+	0.655130i	$0.227386\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.62372	+	11.4726i	−0.361893	+	0.626817i
$336$		0			0
$337$	14.8990	+	25.8058i	0.811599	+	1.40573i	0.911744	+	0.410758i	$0.134736\pi$
$337$	14.8990	+	25.8058i	0.811599	+	1.40573i	−0.100145	+	0.994973i	$0.531931\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	7.24745			0.391325
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.7980	+	22.1667i	0.687030	+	1.18997i	0.972794	+	0.231671i	$0.0744194\pi$
$347$	12.7980	+	22.1667i	0.687030	+	1.18997i	−0.285764	+	0.958300i	$0.592247\pi$
$348$		0			0
$349$	1.15153	−	1.99451i	0.0616400	−	0.106764i	−0.833559	−	0.552431i	$-0.813700\pi$
$349$	1.15153	−	1.99451i	0.0616400	−	0.106764i	0.895199	+	0.445667i	$0.147034\pi$
$350$		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
$357$		0			0
$358$		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
$363$		0			0
$364$		0			0
$365$	−6.89898	−	11.9494i	−0.361109	−	0.625459i
$366$		0			0
$367$	11.0000	−	19.0526i	0.574195	−	0.994535i	−0.421933	−	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			0
$370$		0			0
$371$	−13.4495	+	23.2952i	−0.698263	+	1.20943i
$372$		0			0
$373$	−10.7980	−	18.7026i	−0.559097	−	0.968385i	−0.997572	−	0.0696412i	$-0.977815\pi$
$373$	−10.7980	−	18.7026i	−0.559097	−	0.968385i	0.438475	−	0.898743i	$-0.355519\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−26.8990			−1.38171			−0.690854	−	0.722994i	$-0.742765\pi$
$379$	−26.8990			−1.38171			−0.690854	+	0.722994i	$0.742765\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.00000	−	1.73205i	−0.0510976	−	0.0885037i	0.839345	−	0.543599i	$-0.182939\pi$
$383$	−1.00000	−	1.73205i	−0.0510976	−	0.0885037i	−0.890443	+	0.455095i	$0.849605\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.29796	+	14.3725i	−0.420723	+	0.728714i	−0.996010	−	0.0892375i	$-0.971557\pi$
$389$	−8.29796	+	14.3725i	−0.420723	+	0.728714i	0.575287	+	0.817952i	$0.304890\pi$
$390$		0			0
$391$	7.44949	+	12.9029i	0.376737	+	0.652527i
$392$		0			0
$393$		0			0
$394$		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
$399$		0			0
$400$		0			0
$401$	12.7980	+	22.1667i	0.639100	+	1.10695i	0.985631	+	0.168914i	$0.0540262\pi$
$401$	12.7980	+	22.1667i	0.639100	+	1.10695i	−0.346531	+	0.938038i	$0.612641\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	4.79796	+	8.31031i	0.237244	+	0.410918i	0.959922	−	0.280266i	$-0.0904226\pi$
$409$	4.79796	+	8.31031i	0.237244	+	0.410918i	−0.722679	+	0.691184i	$0.757089\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.79796			0.186885
$414$		0			0
$415$	−5.44949			−0.267505
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.0000	−	17.3205i	−0.488532	−	0.846162i	0.511381	−	0.859354i	$-0.329134\pi$
$419$	−10.0000	−	17.3205i	−0.488532	−	0.846162i	−0.999913	+	0.0131919i	$0.995801\pi$
$420$		0			0
$421$	−12.7980	+	22.1667i	−0.623734	+	1.08034i	0.365050	+	0.930988i	$0.381052\pi$
$421$	−12.7980	+	22.1667i	−0.623734	+	1.08034i	−0.988784	+	0.149352i	$0.952281\pi$
$422$		0			0
$423$		0			0
$424$		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
$435$		0			0
$436$		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	$-0.972963\pi$
$439$	−8.89898	+	15.4135i	−0.424725	+	0.735645i	0.571670	+	0.820484i	$0.306296\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.07321	−	7.05501i	0.193524	−	0.335194i	−0.752892	−	0.658145i	$-0.771342\pi$
$443$	4.07321	−	7.05501i	0.193524	−	0.335194i	0.946416	+	0.322951i	$0.104675\pi$
$444$		0			0
$445$	1.39898	+	2.42310i	0.0663180	+	0.114866i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−21.5959			−1.01917			−0.509587	−	0.860419i	$-0.670202\pi$
$449$	−21.5959			−1.01917			−0.509587	+	0.860419i	$0.670202\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.79796	+	15.2385i	−0.411551	+	0.712828i	−0.995060	−	0.0992796i	$-0.968346\pi$
$457$	−8.79796	+	15.2385i	−0.411551	+	0.712828i	0.583508	+	0.812107i	$0.301680\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.05051	−	10.4798i	0.281800	−	0.488093i	−0.690028	−	0.723783i	$-0.742402\pi$
$461$	6.05051	−	10.4798i	0.281800	−	0.488093i	0.971828	+	0.235690i	$0.0757351\pi$
$462$		0			0
$463$	6.10102	+	10.5673i	0.283538	+	0.491103i	0.972254	−	0.233929i	$-0.0751583\pi$
$463$	6.10102	+	10.5673i	0.283538	+	0.491103i	−0.688715	+	0.725032i	$0.741825\pi$
$464$		0			0
$465$		0			0
$466$		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
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−3.44949	+	5.97469i	−0.158273	+	0.274138i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.2474	+	26.4094i	−0.696674	+	1.20667i	0.272939	+	0.962031i	$0.412004\pi$
$479$	−15.2474	+	26.4094i	−0.696674	+	1.20667i	−0.969613	+	0.244643i	$0.921329\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.00000			0.0908153
$486$		0			0
$487$	13.5959			0.616090			0.308045	−	0.951372i	$-0.400325\pi$
$487$	13.5959			0.616090			0.308045	+	0.951372i	$0.400325\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.89898	+	11.9494i	0.311347	+	0.539268i	0.978654	−	0.205514i	$-0.0658867\pi$
$491$	6.89898	+	11.9494i	0.311347	+	0.539268i	−0.667308	+	0.744782i	$0.732553\pi$
$492$		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	−0.988204	−	0.153141i	$-0.951061\pi$
$499$	−14.0000	−	24.2487i	−0.626726	−	1.08552i	0.361478	−	0.932381i	$-0.382272\pi$
$500$		0			0
$501$		0			0
$502$		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
$507$		0			0
$508$		0			0
$509$	−2.05051	−	3.55159i	−0.0908873	−	0.157421i	0.816997	−	0.576641i	$-0.195637\pi$
$509$	−2.05051	−	3.55159i	−0.0908873	−	0.157421i	−0.907885	+	0.419220i	$0.862304\pi$
$510$		0			0
$511$	23.7980	−	41.2193i	1.05276	−	1.82343i
$512$		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
$521$	−9.20204			−0.403149			−0.201574	−	0.979473i	$-0.564606\pi$
$521$	−9.20204			−0.403149			−0.201574	+	0.979473i	$0.564606\pi$
$522$		0			0
$523$	34.3485			1.50195			0.750977	−	0.660329i	$-0.229583\pi$
$523$	34.3485			1.50195			0.750977	+	0.660329i	$0.229583\pi$
$524$		0			0
$525$		0			0
$526$		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
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	6.17423	+	10.6941i	0.266935	+	0.462346i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		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	$-0.949924\pi$
$547$	−8.37628	+	14.5081i	−0.358144	+	0.620323i	0.629507	+	0.776995i	$0.283257\pi$
$548$		0			0
$549$		0			0
$550$		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
$555$		0			0
$556$		0			0
$557$	−43.7980			−1.85578			−0.927890	−	0.372855i	$-0.878379\pi$
$557$	−43.7980			−1.85578			−0.927890	+	0.372855i	$0.878379\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.17423	+	8.96204i	0.218068	+	0.377705i	0.954217	−	0.299114i	$-0.0966912\pi$
$563$	5.17423	+	8.96204i	0.218068	+	0.377705i	−0.736149	+	0.676819i	$0.763358\pi$
$564$		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	−0.741981	−	0.670421i	$-0.766114\pi$
$569$	5.00000	−	8.66025i	0.209611	−	0.363057i	0.951592	+	0.307364i	$0.0994469\pi$
$570$		0			0
$571$	12.8990	+	22.3417i	0.539805	+	0.934971i	0.998914	+	0.0465904i	$0.0148355\pi$
$571$	12.8990	+	22.3417i	0.539805	+	0.934971i	−0.459109	+	0.888380i	$0.651831\pi$
$572$		0			0
$573$		0			0
$574$		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
$579$		0			0
$580$		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	−0.934113	−	0.356976i	$-0.883808\pi$
$587$	−3.82577	+	6.62642i	−0.157906	+	0.273502i	0.776207	+	0.630478i	$0.217141\pi$
$588$		0			0
$589$	3.79796	+	6.57826i	0.156492	+	0.271052i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	31.3939			1.28919			0.644596	−	0.764523i	$-0.277026\pi$
$593$	31.3939			1.28919			0.644596	+	0.764523i	$0.277026\pi$
$594$		0			0
$595$	−6.89898			−0.282831
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.3485	+	21.3882i	0.504545	+	0.873897i	0.999986	+	0.00525583i	$0.00167299\pi$
$599$	12.3485	+	21.3882i	0.504545	+	0.873897i	−0.495441	+	0.868641i	$0.664994\pi$
$600$		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
$603$		0			0
$604$		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	$-0.152797\pi$
$607$	1.07321	+	1.85886i	0.0435604	+	0.0754489i	−0.843423	+	0.537250i	$0.819463\pi$
$608$		0			0
$609$		0			0
$610$		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
$615$		0			0
$616$		0			0
$617$	−17.5959	−	30.4770i	−0.708385	−	1.22696i	−0.965456	−	0.260566i	$-0.916091\pi$
$617$	−17.5959	−	30.4770i	−0.708385	−	1.22696i	0.257071	−	0.966393i	$-0.417243\pi$
$618$		0			0
$619$	−12.3485	+	21.3882i	−0.496327	+	0.859663i	−0.999991	−	0.00423617i	$-0.998652\pi$
$619$	−12.3485	+	21.3882i	−0.496327	+	0.859663i	0.503664	+	0.863900i	$0.331985\pi$
$620$		0			0
$621$		0			0
$622$		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	$-0.399526\pi$
$631$	15.5959			0.620864			0.310432	+	0.950596i	$0.399526\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.174235	+	0.301783i	0.00691429	+	0.0119759i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.94949	+	12.0369i	−0.274488	+	0.475428i	−0.970006	−	0.243081i	$-0.921842\pi$
$641$	−6.94949	+	12.0369i	−0.274488	+	0.475428i	0.695518	+	0.718509i	$0.255175\pi$
$642$		0			0
$643$	−10.0732	−	17.4473i	−0.397249	−	0.688055i	0.596137	−	0.802883i	$-0.296702\pi$
$643$	−10.0732	−	17.4473i	−0.397249	−	0.688055i	−0.993385	+	0.114828i	$0.963368\pi$
$644$		0			0
$645$		0			0
$646$		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
$651$		0			0
$652$		0			0
$653$	13.8990	+	24.0737i	0.543909	+	0.942078i	0.998675	+	0.0514670i	$0.0163897\pi$
$653$	13.8990	+	24.0737i	0.543909	+	0.942078i	−0.454766	+	0.890611i	$0.650277\pi$
$654$		0			0
$655$	−1.44949	+	2.51059i	−0.0566363	+	0.0980969i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.10102	+	8.83523i	−0.198708	+	0.344172i	−0.948110	−	0.317944i	$-0.897008\pi$
$659$	−5.10102	+	8.83523i	−0.198708	+	0.344172i	0.749402	+	0.662115i	$0.230341\pi$
$660$		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
$665$	−23.7980			−0.922845
$666$		0			0
$667$	−14.1464			−0.547752
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	8.00000	−	13.8564i	0.308377	−	0.534125i	−0.669630	−	0.742695i	$-0.733547\pi$
$673$	8.00000	−	13.8564i	0.308377	−	0.534125i	0.978008	+	0.208569i	$0.0668807\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.10102	−	12.2993i	0.272914	−	0.472702i	−0.696692	−	0.717370i	$-0.745346\pi$
$677$	7.10102	−	12.2993i	0.272914	−	0.472702i	0.969607	+	0.244668i	$0.0786791\pi$
$678$		0			0
$679$	3.44949	+	5.97469i	0.132379	+	0.229288i
$680$		0			0
$681$		0			0
$682$		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
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	19.4495	−	33.6875i	0.739893	−	1.28153i	−0.212649	−	0.977129i	$-0.568209\pi$
$691$	19.4495	−	33.6875i	0.739893	−	1.28153i	0.952543	−	0.304404i	$-0.0984574\pi$
$692$		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
$699$		0			0
$700$		0			0
$701$	−30.3939			−1.14796			−0.573980	−	0.818869i	$-0.694601\pi$
$701$	−30.3939			−1.14796			−0.573980	+	0.818869i	$0.694601\pi$
$702$		0			0
$703$	−41.3939			−1.56120
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.44949	+	5.97469i	0.129731	+	0.224701i
$708$		0			0
$709$	−4.84847	+	8.39780i	−0.182088	+	0.315386i	−0.942591	−	0.333948i	$-0.891619\pi$
$709$	−4.84847	+	8.39780i	−0.182088	+	0.315386i	0.760503	+	0.649334i	$0.224952\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.10102	+	7.10318i	−0.153584	+	0.266016i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		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
$723$		0			0
$724$		0			0
$725$	−0.949490	−	1.64456i	−0.0352632	−	0.0610776i
$726$		0			0
$727$	−15.6237	+	27.0611i	−0.579452	+	1.00364i	0.416090	+	0.909323i	$0.363400\pi$
$727$	−15.6237	+	27.0611i	−0.579452	+	1.00364i	−0.995542	+	0.0943168i	$0.969933\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	11.7980	−	20.4347i	0.436363	−	0.755803i
$732$		0			0
$733$	14.7980	+	25.6308i	0.546575	+	0.946696i	0.998506	+	0.0546429i	$0.0174020\pi$
$733$	14.7980	+	25.6308i	0.546575	+	0.946696i	−0.451931	+	0.892053i	$0.649265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	33.1010			1.21764			0.608820	−	0.793308i	$-0.291643\pi$
$739$	33.1010			1.21764			0.608820	+	0.793308i	$0.291643\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.52270	+	6.10150i	0.129235	+	0.223842i	0.923381	−	0.383886i	$-0.125414\pi$
$743$	3.52270	+	6.10150i	0.129235	+	0.223842i	−0.794145	+	0.607728i	$0.792081\pi$
$744$		0			0
$745$	−10.5000	+	18.1865i	−0.384690	+	0.666303i
$746$		0			0
$747$		0			0
$748$		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	$-0.00800571\pi$
$751$	13.1010	+	22.6916i	0.478063	+	0.828029i	−0.521621	+	0.853177i	$0.674672\pi$
$752$		0			0
$753$		0			0
$754$		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	0.986647	+	0.162872i	$0.0520756\pi$
$761$	17.5000	+	30.3109i	0.634375	+	1.09877i	−0.352273	+	0.935897i	$0.614591\pi$
$762$		0			0
$763$	−18.6237	+	32.2572i	−0.674224	+	1.16779i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.60102	+	4.50510i	0.0937952	+	0.162458i	0.909105	−	0.416567i	$-0.136767\pi$
$769$	2.60102	+	4.50510i	0.0937952	+	0.162458i	−0.815310	+	0.579025i	$0.803433\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.20204			−0.223072			−0.111536	−	0.993760i	$-0.535577\pi$
$773$	−6.20204			−0.223072			−0.111536	+	0.993760i	$0.535577\pi$
$774$		0			0
$775$	−1.10102			−0.0395498
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−34.1464	−	59.1433i	−1.22342	−	2.11903i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.8990	−	20.6096i	0.424693	−	0.735589i
$786$		0			0
$787$	−24.7980	−	42.9513i	−0.883952	−	1.53105i	−0.846910	−	0.531736i	$-0.821540\pi$
$787$	−24.7980	−	42.9513i	−0.883952	−	1.53105i	−0.0370414	−	0.999314i	$-0.511793\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−33.7980			−1.20172
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.00000	−	6.92820i	−0.141687	−	0.245410i	0.786445	−	0.617661i	$-0.211919\pi$
$797$	−4.00000	−	6.92820i	−0.141687	−	0.245410i	−0.928132	+	0.372251i	$0.878586\pi$
$798$		0			0
$799$	9.44949	−	16.3670i	0.334299	−	0.579023i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−12.8485	−	22.2542i	−0.452849	−	0.784358i
$806$		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
$811$	−13.1010			−0.460039			−0.230020	−	0.973186i	$-0.573879\pi$
$811$	−13.1010			−0.460039			−0.230020	+	0.973186i	$0.573879\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.89898	−	6.75323i	−0.136575	−	0.236555i
$816$		0			0
$817$	40.6969	−	70.4892i	1.42381	−	2.46610i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.39898	−	2.42310i	0.0488247	−	0.0845669i	−0.840580	−	0.541687i	$-0.817786\pi$
$821$	1.39898	−	2.42310i	0.0488247	−	0.0845669i	0.889405	+	0.457120i	$0.151119\pi$
$822$		0			0
$823$	−4.17423	−	7.22999i	−0.145505	−	0.252021i	0.784056	−	0.620690i	$-0.213147\pi$
$823$	−4.17423	−	7.22999i	−0.145505	−	0.252021i	−0.929561	+	0.368668i	$0.879814\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	53.5403			1.86178			0.930889	−	0.365301i	$-0.119034\pi$
$827$	53.5403			1.86178			0.930889	+	0.365301i	$0.119034\pi$
$828$		0			0
$829$	50.1918			1.74323			0.871617	−	0.490187i	$-0.163072\pi$
$829$	50.1918			1.74323			0.871617	+	0.490187i	$0.163072\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.89898	−	8.48528i	−0.169740	−	0.293998i
$834$		0			0
$835$	−4.17423	+	7.22999i	−0.144455	+	0.250204i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.34847	+	4.06767i	−0.0810782	+	0.140431i	−0.903713	−	0.428138i	$-0.859170\pi$
$839$	−2.34847	+	4.06767i	−0.0810782	+	0.140431i	0.822635	+	0.568570i	$0.192503\pi$
$840$		0			0
$841$	12.6969	+	21.9917i	0.437825	+	0.758336i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.0000			−0.447214
$846$		0			0
$847$	37.9444			1.30378
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−22.3485	−	38.7087i	−0.766096	−	1.32692i
$852$		0			0
$853$	−9.10102	+	15.7634i	−0.311613	+	0.539730i	−0.978712	−	0.205240i	$-0.934203\pi$
$853$	−9.10102	+	15.7634i	−0.311613	+	0.539730i	0.667099	+	0.744969i	$0.267536\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.202041	−	0.349945i	0.00690159	−	0.0119539i	−0.862554	−	0.505965i	$-0.831136\pi$
$857$	0.202041	−	0.349945i	0.00690159	−	0.0119539i	0.869456	+	0.494011i	$0.164470\pi$
$858$		0			0
$859$	19.5959	+	33.9411i	0.668604	+	1.15806i	0.978295	+	0.207219i	$0.0664412\pi$
$859$	19.5959	+	33.9411i	0.668604	+	1.15806i	−0.309691	+	0.950837i	$0.600225\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	57.4495			1.95560			0.977802	−	0.209532i	$-0.0671942\pi$
$863$	57.4495			1.95560			0.977802	+	0.209532i	$0.0671942\pi$
$864$		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
$875$	1.72474	−	2.98735i	0.0583070	−	0.100991i
$876$		0			0
$877$	0.202041	+	0.349945i	0.00682244	+	0.0118168i	0.869416	−	0.494080i	$-0.164495\pi$
$877$	0.202041	+	0.349945i	0.00682244	+	0.0118168i	−0.862594	+	0.505897i	$0.831162\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−6.10102			−0.205549			−0.102774	−	0.994705i	$-0.532772\pi$
$881$	−6.10102			−0.205549			−0.102774	+	0.994705i	$0.532772\pi$
$882$		0			0
$883$	0.954592			0.0321246			0.0160623	−	0.999871i	$-0.494887\pi$
$883$	0.954592			0.0321246			0.0160623	+	0.999871i	$0.494887\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.797959	−	1.38211i	−0.0267928	−	0.0464066i	0.852318	−	0.523024i	$-0.175196\pi$
$887$	−0.797959	−	1.38211i	−0.0267928	−	0.0464066i	−0.879111	+	0.476617i	$0.841863\pi$
$888$		0			0
$889$	−0.601021	+	1.04100i	−0.0201576	+	0.0349140i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	32.5959	−	56.4578i	1.09078	−	1.88929i
$894$		0			0
$895$	6.89898	+	11.9494i	0.230607	+	0.399424i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.09082			−0.0697326
$900$		0			0
$901$	−15.5959			−0.519575
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.84847	+	8.39780i	0.161169	+	0.279152i
$906$		0			0
$907$	−10.8258	+	18.7508i	−0.359464	+	0.622609i	−0.987871	−	0.155275i	$-0.950374\pi$
$907$	−10.8258	+	18.7508i	−0.359464	+	0.622609i	0.628408	+	0.777884i	$0.283707\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5.24745	+	9.08885i	−0.173856	+	0.301127i	−0.939765	−	0.341822i	$-0.888956\pi$
$911$	−5.24745	+	9.08885i	−0.173856	+	0.301127i	0.765909	+	0.642949i	$0.222289\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.0000			−0.330229
$918$		0			0
$919$	−24.6969			−0.814677			−0.407338	−	0.913277i	$-0.633543\pi$
$919$	−24.6969			−0.814677			−0.407338	+	0.913277i	$0.633543\pi$
$920$		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
$929$	18.7980	−	32.5590i	0.616741	−	1.06823i	−0.373335	−	0.927696i	$-0.621786\pi$
$929$	18.7980	−	32.5590i	0.616741	−	1.06823i	0.990076	−	0.140530i	$-0.0448808\pi$
$930$		0			0
$931$	−16.8990	−	29.2699i	−0.553842	−	0.959282i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−40.9898			−1.33908			−0.669539	−	0.742777i	$-0.733508\pi$
$937$	−40.9898			−1.33908			−0.669539	+	0.742777i	$0.733508\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.848469	−	1.46959i	−0.0276593	−	0.0479073i	0.851864	−	0.523762i	$-0.175472\pi$
$941$	−0.848469	−	1.46959i	−0.0276593	−	0.0479073i	−0.879524	+	0.475855i	$0.842139\pi$
$942$		0			0
$943$	36.8712	−	63.8627i	1.20069	−	2.07966i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.92679	−	10.2655i	0.192595	−	0.333584i	−0.753515	−	0.657431i	$-0.771643\pi$
$947$	5.92679	−	10.2655i	0.192595	−	0.333584i	0.946109	+	0.323847i	$0.104976\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14.2020			0.460049			0.230025	−	0.973185i	$-0.426119\pi$
$953$	14.2020			0.460049			0.230025	+	0.973185i	$0.426119\pi$
$954$		0			0
$955$	−14.8990			−0.482120
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−33.7980	−	58.5398i	−1.09139	−	1.89035i
$960$		0			0
$961$	14.8939	−	25.7970i	0.480448	−	0.832160i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.10102	−	1.90702i	0.0354431	−	0.0613893i
$966$		0			0
$967$	24.4217	+	42.2996i	0.785348	+	1.36026i	0.928791	+	0.370605i	$0.120850\pi$
$967$	24.4217	+	42.2996i	0.785348	+	1.36026i	−0.143442	+	0.989659i	$0.545817\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	31.3031			1.00456			0.502282	−	0.864704i	$-0.332494\pi$
$971$	31.3031			1.00456			0.502282	+	0.864704i	$0.332494\pi$
$972$		0			0
$973$	67.5959			2.16703
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.69694	−	11.5994i	−0.214254	−	0.371099i	0.738788	−	0.673938i	$-0.235399\pi$
$977$	−6.69694	−	11.5994i	−0.214254	−	0.371099i	−0.953042	+	0.302840i	$0.902065\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	5.72474	−	9.91555i	0.182591	−	0.316257i	−0.760171	−	0.649723i	$-0.774885\pi$
$983$	5.72474	−	9.91555i	0.182591	−	0.316257i	0.942762	+	0.333466i	$0.108218\pi$
$984$		0			0
$985$	−7.79796	−	13.5065i	−0.248464	−	0.430352i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	87.8888			2.79470
$990$		0			0
$991$	−28.6969			−0.911588			−0.455794	−	0.890085i	$-0.650645\pi$
$991$	−28.6969			−0.911588			−0.455794	+	0.890085i	$0.650645\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	11.4495	+	19.8311i	0.362973	+	0.628688i
$996$		0			0
$997$	11.6969	−	20.2597i	0.370446	−	0.641631i	−0.619188	−	0.785242i	$-0.712538\pi$
$997$	11.6969	−	20.2597i	0.370446	−	0.641631i	0.989634	+	0.143612i	$0.0458716\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.q.g.721.2		4
3.2	odd	2		720.2.q.g.241.2		4
4.3	odd	2		1080.2.q.c.721.1		4
9.2	odd	6		6480.2.a.bc.1.1		2
9.4	even	3	inner	2160.2.q.g.1441.2		4
9.5	odd	6		720.2.q.g.481.2		4
9.7	even	3		6480.2.a.bl.1.1		2
12.11	even	2		360.2.q.c.241.1	yes	4
36.7	odd	6		3240.2.a.o.1.2		2
36.11	even	6		3240.2.a.j.1.2		2
36.23	even	6		360.2.q.c.121.1	✓	4
36.31	odd	6		1080.2.q.c.361.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.c.121.1	✓	4	36.23	even	6
360.2.q.c.241.1	yes	4	12.11	even	2
720.2.q.g.241.2		4	3.2	odd	2
720.2.q.g.481.2		4	9.5	odd	6
1080.2.q.c.361.1		4	36.31	odd	6
1080.2.q.c.721.1		4	4.3	odd	2
2160.2.q.g.721.2		4	1.1	even	1	trivial
2160.2.q.g.1441.2		4	9.4	even	3	inner
3240.2.a.j.1.2		2	36.11	even	6
3240.2.a.o.1.2		2	36.7	odd	6
6480.2.a.bc.1.1		2	9.2	odd	6
6480.2.a.bl.1.1		2	9.7	even	3

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(1.72474 - 2.98735i) q^{7} +2.00000 q^{17} +6.89898 q^{19} +(3.72474 + 6.45145i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.949490 + 1.64456i) q^{29} +(0.550510 + 0.953512i) q^{31} -3.44949 q^{35} -6.00000 q^{37} +(-4.94949 - 8.57277i) q^{41} +(5.89898 - 10.2173i) q^{43} +(4.72474 - 8.18350i) q^{47} +(-2.44949 - 4.24264i) q^{49} -7.79796 q^{53} +(0.550510 + 0.953512i) q^{59} +(1.50000 - 2.59808i) q^{61} +(-6.62372 - 11.4726i) q^{67} +9.79796 q^{71} +13.7980 q^{73} +(-3.44949 + 5.97469i) q^{79} +(2.72474 - 4.71940i) q^{83} +(-1.00000 - 1.73205i) q^{85} -2.79796 q^{89} +(-3.44949 - 5.97469i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 2 q^{7} + 8 q^{17} + 8 q^{19} + 10 q^{23} - 2 q^{25} + 6 q^{29} + 12 q^{31} - 4 q^{35} - 24 q^{37} - 10 q^{41} + 4 q^{43} + 14 q^{47} + 8 q^{53} + 12 q^{59} + 6 q^{61} - 2 q^{67} + 16 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) \) 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

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