LMFDB - Embedded newform 1120.2.w.a.657.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1120,2,Mod(433,1120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1120, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1120.433"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.w (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.40282095616.8
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 8x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 8x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			657.4
Root			$-1.71331 - 0.254137i$ of defining polynomial
Character	$\chi$	$=$	1120.657
Dual form			1120.2.w.a.433.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.96744 - 1.96744i) q^{3} +(0.254137 - 2.22158i) q^{5} +(-1.87083 - 1.87083i) q^{7} -4.74166i q^{9} +(-4.88578 + 4.88578i) q^{13} +(-3.87083 - 4.87083i) q^{15} -2.41006i q^{19} -7.36149 q^{21} +(6.74166 - 6.74166i) q^{23} +(-4.87083 - 1.12917i) q^{25} +(-3.42661 - 3.42661i) q^{27} +(-4.63164 + 3.68075i) q^{35} +19.2250i q^{39} +(-10.5340 - 1.20503i) q^{45} +7.00000i q^{49} +(-4.74166 - 4.74166i) q^{57} -10.4111i q^{59} +6.47626 q^{61} +(-8.87083 + 8.87083i) q^{63} +(9.61249 + 12.0958i) q^{65} -26.5276i q^{69} +15.2250 q^{71} +(-11.8047 + 7.36149i) q^{75} -8.25834i q^{79} +0.741657 q^{81} +(-12.2473 + 12.2473i) q^{83} +18.2809 q^{91} +(-5.35414 - 0.612486i) q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{15} + 24 q^{23} - 24 q^{25} - 8 q^{57} - 56 q^{63} + 32 q^{65} + 32 q^{71} - 24 q^{81} + 32 q^{95}+O(q^{100}) $ 8 * q - 16 * q^15 + 24 * q^23 - 24 * q^25 - 8 * q^57 - 56 * q^63 + 32 * q^65 + 32 * q^71 - 24 * q^81 + 32 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.96744	−	1.96744i	1.13590	−	1.13590i	0.146726	−	0.989177i	$-0.453126\pi$
$3$	1.96744	−	1.96744i	1.13590	−	1.13590i	0.989177	−	0.146726i	$-0.0468736\pi$
$4$		0			0
$5$	0.254137	−	2.22158i	0.113654	−	0.993520i
$5$	0.254137	−	2.22158i	0.113654	−	0.993520i
$6$		0			0
$7$	−1.87083	−	1.87083i	−0.707107	−	0.707107i
$7$	−1.87083	−	1.87083i	−0.707107	−	0.707107i
$8$		0			0
$9$		−	4.74166i		−	1.58055i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−4.88578	+	4.88578i	−1.35507	+	1.35507i	−0.475185	+	0.879886i	$0.657619\pi$
$13$	−4.88578	+	4.88578i	−1.35507	+	1.35507i	−0.879886	+	0.475185i	$0.842381\pi$
$14$		0			0
$15$	−3.87083	−	4.87083i	−0.999444	−	1.25764i
$16$		0			0
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$		−	2.41006i		−	0.552906i	−0.961027	−	0.276453i	$-0.910841\pi$
$19$		−	2.41006i		−	0.552906i	0.961027	−	0.276453i	$-0.0891590\pi$
$20$		0			0
$21$	−7.36149			−1.60641
$22$		0			0
$23$	6.74166	−	6.74166i	1.40573	−	1.40573i	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.74166	−	6.74166i	1.40573	−	1.40573i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−4.87083	−	1.12917i	−0.974166	−	0.225834i
$26$		0			0
$27$	−3.42661	−	3.42661i	−0.659451	−	0.659451i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.63164	+	3.68075i	−0.782890	+	0.622160i
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$			19.2250i			3.07846i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	−10.5340	−	1.20503i	−1.57031	−	0.179635i
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.74166	−	4.74166i	−0.628048	−	0.628048i
$58$		0			0
$59$		−	10.4111i		−	1.35541i	−0.735332	−	0.677707i	$-0.762974\pi$
$59$		−	10.4111i		−	1.35541i	0.735332	−	0.677707i	$-0.237026\pi$
$60$		0			0
$61$	6.47626			0.829199			0.414600	−	0.910004i	$-0.363922\pi$
$61$	6.47626			0.829199			0.414600	+	0.910004i	$0.363922\pi$
$62$		0			0
$63$	−8.87083	+	8.87083i	−1.11762	+	1.11762i
$64$		0			0
$65$	9.61249	+	12.0958i	1.19228	+	1.50030i
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		−	26.5276i		−	3.19355i
$70$		0			0
$71$	15.2250			1.80687			0.903436	−	0.428723i	$-0.141036\pi$
$71$	15.2250			1.80687			0.903436	+	0.428723i	$0.141036\pi$
$72$		0			0
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$	−11.8047	+	7.36149i	−1.36308	+	0.850032i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	8.25834i		−	0.929136i	−0.885537	−	0.464568i	$-0.846210\pi$
$79$		−	8.25834i		−	0.929136i	0.885537	−	0.464568i	$-0.153790\pi$
$80$		0			0
$81$	0.741657			0.0824064
$82$		0			0
$83$	−12.2473	+	12.2473i	−1.34431	+	1.34431i	−0.452598	+	0.891715i	$0.649503\pi$
$83$	−12.2473	+	12.2473i	−1.34431	+	1.34431i	−0.891715	+	0.452598i	$0.850497\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	18.2809			1.91636
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.35414	−	0.612486i	−0.549324	−	0.0628397i
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.39351			0.138660			0.0693299	−	0.997594i	$-0.477914\pi$
$101$	1.39351			0.138660			0.0693299	+	0.997594i	$0.477914\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$	−1.87083	+	16.3541i	−0.182574	+	1.59600i
$106$		0			0
$107$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$107$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	11.2250	−	11.2250i	1.05596	−	1.05596i	0.0576178	−	0.998339i	$-0.481650\pi$
$113$	11.2250	−	11.2250i	1.05596	−	1.05596i	0.998339	−	0.0576178i	$-0.0183505\pi$
$114$		0			0
$115$	−13.2638	−	16.6904i	−1.23686	−	1.55639i
$116$		0			0
$117$	23.1667	+	23.1667i	2.14176	+	2.14176i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.74640	+	10.5340i	−0.335088	+	0.942187i
$126$		0			0
$127$	−10.2250	−	10.2250i	−0.907320	−	0.907320i	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	−10.2250	−	10.2250i	−0.907320	−	0.907320i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.3129			−1.07579			−0.537893	−	0.843013i	$-0.680779\pi$
$131$	−12.3129			−1.07579			−0.537893	+	0.843013i	$0.680779\pi$
$132$		0			0
$133$	−4.50881	+	4.50881i	−0.390964	+	0.390964i
$134$		0			0
$135$	−8.48331	+	6.74166i	−0.730127	+	0.580229i
$136$		0			0
$137$	1.51669	+	1.51669i	0.129579	+	0.129579i	0.768922	−	0.639343i	$-0.220793\pi$
$137$	1.51669	+	1.51669i	0.129579	+	0.129579i	−0.639343	+	0.768922i	$0.720793\pi$
$138$		0			0
$139$		−	5.32840i		−	0.451949i	−0.974133	−	0.225974i	$-0.927443\pi$
$139$		−	5.32840i		−	0.451949i	0.974133	−	0.225974i	$-0.0725566\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	13.7721	+	13.7721i	1.13590	+	1.13590i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	22.4499			1.82695			0.913475	−	0.406894i	$-0.133388\pi$
$151$	22.4499			1.82695			0.913475	+	0.406894i	$0.133388\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−16.3135	−	16.3135i	−1.30196	−	1.30196i	−0.927071	−	0.374885i	$-0.877682\pi$
$157$	−16.3135	−	16.3135i	−1.30196	−	1.30196i	−0.374885	−	0.927071i	$-0.622318\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−25.2250			−1.98801
$162$		0			0
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$		−	34.7417i		−	2.67244i
$170$		0			0
$171$	−11.4277			−0.873897
$172$		0			0
$173$	0.573929	−	0.573929i	0.0436350	−	0.0436350i	−0.684953	−	0.728588i	$-0.740177\pi$
$173$	0.573929	−	0.573929i	0.0436350	−	0.0436350i	0.728588	+	0.684953i	$0.240177\pi$
$174$		0			0
$175$	7.00000	+	11.2250i	0.529150	+	0.848528i
$176$		0			0
$177$	−20.4833	−	20.4833i	−1.53962	−	1.53962i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	26.9046			1.99980			0.999902	−	0.0140098i	$-0.00445961\pi$
$181$	26.9046			1.99980			0.999902	+	0.0140098i	$0.00445961\pi$
$182$		0			0
$183$	12.7417	−	12.7417i	0.941890	−	0.941890i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			12.8212i			0.932605i
$190$		0			0
$191$	−4.77503			−0.345509			−0.172754	−	0.984965i	$-0.555267\pi$
$191$	−4.77503			−0.345509			−0.172754	+	0.984965i	$0.555267\pi$
$192$		0			0
$193$	−6.00000	+	6.00000i	−0.431889	+	0.431889i	−0.889271	−	0.457381i	$-0.848787\pi$
$193$	−6.00000	+	6.00000i	−0.431889	+	0.431889i	0.457381	+	0.889271i	$0.348787\pi$
$194$		0			0
$195$	42.7098	+	4.88578i	3.05851	+	0.349878i
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−31.9666	−	31.9666i	−2.22183	−	2.22183i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	29.9543	−	29.9543i	2.05243	−	2.05243i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$	−5.35414	+	23.0958i	−0.356943	+	1.53972i
$226$		0			0
$227$	17.0674	+	17.0674i	1.13280	+	1.13280i	0.989709	+	0.143094i	$0.0457050\pi$
$227$	17.0674	+	17.0674i	1.13280	+	1.13280i	0.143094	+	0.989709i	$0.454295\pi$
$228$		0			0
$229$			30.0856i			1.98811i	0.108880	+	0.994055i	$0.465274\pi$
$229$			30.0856i			1.98811i	−0.108880	+	0.994055i	$0.534726\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.9666	−	11.9666i	0.783960	−	0.783960i	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	11.9666	−	11.9666i	0.783960	−	0.783960i	0.980497	+	0.196537i	$0.0629694\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−16.2478	−	16.2478i	−1.05541	−	1.05541i
$238$		0			0
$239$			7.48331i			0.484055i	0.970269	+	0.242028i	$0.0778125\pi$
$239$			7.48331i			0.484055i	−0.970269	+	0.242028i	$0.922188\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	11.7390	−	11.7390i	0.753057	−	0.753057i
$244$		0			0
$245$	15.5511	+	1.77896i	0.993520	+	0.113654i
$246$		0			0
$247$	11.7750	+	11.7750i	0.749227	+	0.749227i
$248$		0			0
$249$			48.1916i			3.05402i
$250$		0			0
$251$	13.1982			0.833061			0.416530	−	0.909122i	$-0.363246\pi$
$251$	13.1982			0.833061			0.416530	+	0.909122i	$0.363246\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.2250	+	11.2250i	−0.692161	+	0.692161i	−0.962707	−	0.270546i	$-0.912796\pi$
$263$	−11.2250	+	11.2250i	−0.692161	+	0.692161i	0.270546	+	0.962707i	$0.412796\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			17.0017i			1.03661i	0.855194	+	0.518307i	$0.173438\pi$
$269$			17.0017i			1.03661i	−0.855194	+	0.518307i	$0.826562\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	35.9666	−	35.9666i	2.17680	−	2.17680i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.9666			0.892834			0.446417	−	0.894825i	$-0.352700\pi$
$281$	14.9666			0.892834			0.446417	+	0.894825i	$0.352700\pi$
$282$		0			0
$283$	2.34441	−	2.34441i	0.139361	−	0.139361i	−0.633985	−	0.773345i	$-0.718582\pi$
$283$	2.34441	−	2.34441i	0.139361	−	0.139361i	0.773345	+	0.633985i	$0.218582\pi$
$284$		0			0
$285$	−11.7390	+	9.32894i	−0.695358	+	0.552599i
$286$		0			0
$287$		0			0
$288$		0			0
$289$			17.0000i			1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.1832	+	24.1832i	−1.41280	+	1.41280i	−0.674788	+	0.738011i	$0.735765\pi$
$293$	−24.1832	+	24.1832i	−1.41280	+	1.41280i	−0.738011	+	0.674788i	$0.764235\pi$
$294$		0			0
$295$	−23.1292	−	2.64586i	−1.34663	−	0.154048i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			65.8765i			3.80974i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	2.74166	−	2.74166i	0.157504	−	0.157504i
$304$		0			0
$305$	1.64586	−	14.3875i	0.0942415	−	0.823827i
$306$		0			0
$307$	−17.1987	−	17.1987i	−0.981582	−	0.981582i	0.0182515	−	0.999833i	$-0.494190\pi$
$307$	−17.1987	−	17.1987i	−0.981582	−	0.981582i	−0.999833	+	0.0182515i	$0.994190\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$	17.4528	+	21.9617i	0.983356	+	1.23740i
$316$		0			0
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	29.3147	−	18.2809i	1.62609	−	1.01404i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	18.0000	+	18.0000i	0.980522	+	0.980522i	0.999814	−	0.0192914i	$-0.00614103\pi$
$337$	18.0000	+	18.0000i	0.980522	+	0.980522i	−0.0192914	+	0.999814i	$0.506141\pi$
$338$		0			0
$339$		−	44.1690i		−	2.39893i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	13.0958	−	13.0958i	0.707107	−	0.707107i
$344$		0			0
$345$	−58.9333	−	6.74166i	−3.17286	−	0.362959i
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		−	17.2644i		−	0.924140i	−0.886843	−	0.462070i	$-0.847107\pi$
$349$		−	17.2644i		−	0.924140i	0.886843	−	0.462070i	$-0.152893\pi$
$350$		0			0
$351$	33.4833			1.78721
$352$		0			0
$353$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$353$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$	3.86923	−	33.8235i	0.205357	−	1.79516i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6.00000i			0.316668i	0.987386	+	0.158334i	$0.0506123\pi$
$359$			6.00000i			0.316668i	−0.987386	+	0.158334i	$0.949388\pi$
$360$		0			0
$361$	13.1916			0.694295
$362$		0			0
$363$	21.6419	−	21.6419i	1.13590	−	1.13590i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$	13.3541	+	28.0958i	0.689605	+	1.45086i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−40.2341			−2.06125
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−24.2250	+	24.2250i	−1.22199	+	1.22199i
$394$		0			0
$395$	−18.3466	−	2.09875i	−0.923116	−	0.105600i
$396$		0			0
$397$	28.1181	+	28.1181i	1.41121	+	1.41121i	0.751680	+	0.659528i	$0.229244\pi$
$397$	28.1181	+	28.1181i	1.41121	+	1.41121i	0.659528	+	0.751680i	$0.270756\pi$
$398$		0			0
$399$			17.7417i			0.888194i
$400$		0			0
$401$	14.7750			0.737830			0.368915	−	0.929463i	$-0.379729\pi$
$401$	14.7750			0.737830			0.368915	+	0.929463i	$0.379729\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.188483	−	1.64765i	0.00936578	−	0.0818724i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	5.96798			0.294379
$412$		0			0
$413$	−19.4775	+	19.4775i	−0.958423	+	0.958423i
$414$		0			0
$415$	24.0958	+	30.3208i	1.18282	+	1.48839i
$416$		0			0
$417$	−10.4833	−	10.4833i	−0.513370	−	0.513370i
$418$		0			0
$419$			26.7733i			1.30796i	0.756511	+	0.653981i	$0.226902\pi$
$419$			26.7733i			1.30796i	−0.756511	+	0.653981i	$0.773098\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−12.1160	−	12.1160i	−0.586333	−	0.586333i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	18.0000			0.867029			0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000			0.867029			0.433515	+	0.901146i	$0.357273\pi$
$432$		0			0
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−16.2478	−	16.2478i	−0.777238	−	0.777238i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	33.1916			1.58055
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			29.9333i			1.41264i	0.707894	+	0.706319i	$0.249646\pi$
$449$			29.9333i			1.41264i	−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	44.1690	−	44.1690i	2.07524	−	2.07524i
$454$		0			0
$455$	4.64586	−	40.6125i	0.217801	−	1.90394i
$456$		0			0
$457$	3.74166	+	3.74166i	0.175027	+	0.175027i	0.789184	−	0.614157i	$-0.210504\pi$
$457$	3.74166	+	3.74166i	0.175027	+	0.175027i	−0.614157	+	0.789184i	$0.710504\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−41.6276			−1.93879			−0.969395	−	0.245505i	$-0.921046\pi$
$461$	−41.6276			−1.93879			−0.969395	+	0.245505i	$0.921046\pi$
$462$		0			0
$463$	−24.0000	+	24.0000i	−1.11537	+	1.11537i	−0.122963	+	0.992411i	$0.539240\pi$
$463$	−24.0000	+	24.0000i	−1.11537	+	1.11537i	−0.992411	+	0.122963i	$0.960760\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.8548	+	18.8548i	0.872498	+	0.872498i	0.992744	−	0.120246i	$-0.0383684\pi$
$467$	18.8548	+	18.8548i	0.872498	+	0.872498i	−0.120246	+	0.992744i	$0.538368\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−64.1916			−2.95779
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−2.72137	+	11.7390i	−0.124865	+	0.538622i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−49.6287	+	49.6287i	−2.25818	+	2.25818i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−30.2250	−	30.2250i	−1.36962	−	1.36962i	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−30.2250	−	30.2250i	−1.36962	−	1.36962i	−0.508652	−	0.860972i	$-0.669856\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−28.4833	−	28.4833i	−1.27765	−	1.27765i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	0.354143	−	3.09580i	0.0157592	−	0.137761i
$506$		0			0
$507$	−68.3522	−	68.3522i	−3.03563	−	3.03563i
$508$		0			0
$509$			25.0028i			1.10823i	0.832440	+	0.554115i	$0.186943\pi$
$509$			25.0028i			1.10823i	−0.832440	+	0.554115i	$0.813057\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−8.25834	+	8.25834i	−0.364615	+	0.364615i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	2.25834i		−	0.0991302i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	26.7246	−	26.7246i	1.16859	−	1.16859i	0.186044	−	0.982541i	$-0.440433\pi$
$523$	26.7246	−	26.7246i	1.16859	−	1.16859i	0.982541	−	0.186044i	$-0.0595667\pi$
$524$		0			0
$525$	35.8566	+	8.31239i	1.56491	+	0.362782i
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	67.8999i		−	2.95217i
$530$		0			0
$531$	−49.3661			−2.14230
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	52.9333	−	52.9333i	2.27158	−	2.27158i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		−	30.7082i		−	1.31059i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−15.4499	+	15.4499i	−0.656998	+	0.656998i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.05018	−	7.05018i	0.297130	−	0.297130i	−0.542759	−	0.839889i	$-0.682620\pi$
$563$	7.05018	−	7.05018i	0.297130	−	0.297130i	0.839889	+	0.542759i	$0.182620\pi$
$564$		0			0
$565$	−22.0845	−	27.7898i	−0.929101	−	1.16913i
$566$		0			0
$567$	−1.38751	−	1.38751i	−0.0582701	−	0.0582701i
$568$		0			0
$569$			35.6749i			1.49557i	0.663941	+	0.747785i	$0.268883\pi$
$569$			35.6749i			1.49557i	−0.663941	+	0.747785i	$0.731117\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−9.39459	+	9.39459i	−0.392465	+	0.392465i
$574$		0			0
$575$	−40.4499	+	25.2250i	−1.68688	+	1.05195i
$576$		0			0
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$		0			0
$579$			23.6093i			0.981169i
$580$		0			0
$581$	45.8251			1.90115
$582$		0			0
$583$		0			0
$584$		0			0
$585$	57.3541	−	45.5791i	2.37130	−	1.88446i
$586$		0			0
$587$	−30.6595	−	30.6595i	−1.26545	−	1.26545i	−0.948412	−	0.317041i	$-0.897311\pi$
$587$	−30.6595	−	30.6595i	−1.26545	−	1.26545i	−0.317041	−	0.948412i	$-0.602689\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	25.6749i		−	1.04905i	−0.851395	−	0.524524i	$-0.824243\pi$
$599$		−	25.6749i		−	1.04905i	0.851395	−	0.524524i	$-0.175757\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.79551	−	24.4374i	0.113654	−	0.993520i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−33.6749	−	33.6749i	−1.35570	−	1.35570i	−0.879143	−	0.476558i	$-0.841884\pi$
$617$	−33.6749	−	33.6749i	−1.35570	−	1.35570i	−0.476558	−	0.879143i	$-0.658116\pi$
$618$		0			0
$619$			28.9377i			1.16310i	0.813509	+	0.581552i	$0.197554\pi$
$619$			28.9377i			1.16310i	−0.813509	+	0.581552i	$0.802446\pi$
$620$		0			0
$621$	−46.2021			−1.85402
$622$		0			0
$623$		0			0
$624$		0			0
$625$	22.4499	+	11.0000i	0.897998	+	0.440000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−2.19160			−0.0872463			−0.0436231	−	0.999048i	$-0.513890\pi$
$631$	−2.19160			−0.0872463			−0.0436231	+	0.999048i	$0.513890\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−25.3141	+	20.1170i	−1.00456	+	0.798320i
$636$		0			0
$637$	−34.2004	−	34.2004i	−1.35507	−	1.35507i
$638$		0			0
$639$		−	72.1916i		−	2.85586i
$640$		0			0
$641$	−45.2250			−1.78628			−0.893140	−	0.449780i	$-0.851502\pi$
$641$	−45.2250			−1.78628			−0.893140	+	0.449780i	$0.851502\pi$
$642$		0			0
$643$	−3.11530	+	3.11530i	−0.122855	+	0.122855i	−0.765861	−	0.643006i	$-0.777687\pi$
$643$	−3.11530	+	3.11530i	−0.122855	+	0.122855i	0.643006	+	0.765861i	$0.277687\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$	−3.12917	+	27.3541i	−0.122267	+	1.06881i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−12.4442			−0.484025			−0.242012	−	0.970273i	$-0.577807\pi$
$661$	−12.4442			−0.484025			−0.242012	+	0.970273i	$0.577807\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.87083	+	11.1625i	0.343996	+	0.432865i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−35.4499	+	35.4499i	−1.36649	+	1.36649i	−0.501113	+	0.865382i	$0.667076\pi$
$673$	−35.4499	+	35.4499i	−1.36649	+	1.36649i	−0.865382	+	0.501113i	$0.832924\pi$
$674$		0			0
$675$	12.8212	+	20.5597i	0.493488	+	0.791342i
$676$		0			0
$677$	−35.9879	−	35.9879i	−1.38313	−	1.38313i	−0.839008	−	0.544119i	$-0.816864\pi$
$677$	−35.9879	−	35.9879i	−1.38313	−	1.38313i	−0.544119	−	0.839008i	$-0.683136\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	67.1582			2.57351
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	3.75488	−	2.98399i	0.143467	−	0.114012i
$686$		0			0
$687$	59.1916	+	59.1916i	2.25830	+	2.25830i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−46.5790			−1.77195			−0.885975	−	0.463733i	$-0.846510\pi$
$691$	−46.5790			−1.77195			−0.885975	+	0.463733i	$0.846510\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−11.8375	−	1.35414i	−0.449020	−	0.0513656i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		−	47.0873i		−	1.78101i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.60703	−	2.60703i	−0.0980473	−	0.0980473i
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$	−39.1582			−1.46855
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	14.7230	+	14.7230i	0.549840	+	0.549840i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$		−	43.9666i		−	1.62839i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−19.1005	+	19.1005i	−0.705493	+	0.705493i	−0.965584	−	0.260091i	$-0.916247\pi$
$733$	−19.1005	+	19.1005i	−0.705493	+	0.705493i	0.260091	+	0.965584i	$0.416247\pi$
$734$		0			0
$735$	34.0958	−	27.0958i	1.25764	−	0.999444i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$	46.3334			1.70210
$742$		0			0
$743$	−23.2583	+	23.2583i	−0.853266	+	0.853266i	−0.990534	−	0.137268i	$-0.956168\pi$
$743$	−23.2583	+	23.2583i	−0.853266	+	0.853266i	0.137268	+	0.990534i	$0.456168\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	58.0724	+	58.0724i	2.12476	+	2.12476i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	22.4499			0.819210			0.409605	−	0.912263i	$-0.365667\pi$
$751$	22.4499			0.819210			0.409605	+	0.912263i	$0.365667\pi$
$752$		0			0
$753$	25.9666	−	25.9666i	0.946277	−	0.946277i
$754$		0			0
$755$	5.70536	−	49.8743i	0.207639	−	1.81511i
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	50.8665	+	50.8665i	1.83668	+	1.83668i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−39.1519	+	39.1519i	−1.40820	+	1.40820i	−0.638932	+	0.769263i	$0.720624\pi$
$773$	−39.1519	+	39.1519i	−1.40820	+	1.40820i	−0.769263	+	0.638932i	$0.779376\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−40.3875	+	32.0958i	−1.44149	+	1.14555i
$786$		0			0
$787$	38.5293	+	38.5293i	1.37342	+	1.37342i	0.855321	+	0.518099i	$0.173360\pi$
$787$	38.5293	+	38.5293i	1.37342	+	1.37342i	0.518099	+	0.855321i	$0.326640\pi$
$788$		0			0
$789$			44.1690i			1.57246i
$790$		0			0
$791$	−42.0000			−1.49335
$792$		0			0
$793$	−31.6415	+	31.6415i	−1.12362	+	1.12362i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	34.3318	+	34.3318i	1.21609	+	1.21609i	0.968989	+	0.247104i	$0.0794790\pi$
$797$	34.3318	+	34.3318i	1.21609	+	1.21609i	0.247104	+	0.968989i	$0.420521\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−6.41060	+	56.0393i	−0.225944	+	1.97512i
$806$		0			0
$807$	33.4499	+	33.4499i	1.17749	+	1.17749i
$808$		0			0
$809$			55.6749i			1.95743i	0.205234	+	0.978713i	$0.434204\pi$
$809$			55.6749i			1.95743i	−0.205234	+	0.978713i	$0.565796\pi$
$810$		0			0
$811$	56.2193			1.97413			0.987063	−	0.160333i	$-0.0512570\pi$
$811$	56.2193			1.97413			0.987063	+	0.160333i	$0.0512570\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		−	86.6818i		−	3.02891i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	36.0000	−	36.0000i	1.25488	−	1.25488i	0.301376	−	0.953506i	$-0.402554\pi$
$823$	36.0000	−	36.0000i	1.25488	−	1.25488i	0.953506	−	0.301376i	$-0.0974458\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$		−	53.6949i		−	1.86490i	−0.361299	−	0.932450i	$-0.617667\pi$
$829$		−	53.6949i		−	1.86490i	0.361299	−	0.932450i	$-0.382333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	29.4460	−	29.4460i	1.01417	−	1.01417i
$844$		0			0
$845$	−77.1813	−	8.82914i	−2.65512	−	0.303732i
$846$		0			0
$847$	−20.5791	−	20.5791i	−0.707107	−	0.707107i
$848$		0			0
$849$		−	9.22497i		−	0.316600i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	39.9228	−	39.9228i	1.36693	−	1.36693i	0.502148	−	0.864782i	$-0.332543\pi$
$853$	39.9228	−	39.9228i	1.36693	−	1.36693i	0.864782	−	0.502148i	$-0.167457\pi$
$854$		0			0
$855$	−2.90420	+	25.3875i	−0.0993215	+	0.868235i
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		−	41.7589i		−	1.42480i	−0.701776	−	0.712398i	$-0.747609\pi$
$859$		−	41.7589i		−	1.42480i	0.701776	−	0.712398i	$-0.252391\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−11.2250	+	11.2250i	−0.382102	+	0.382102i	−0.871859	−	0.489757i	$-0.837086\pi$
$863$	−11.2250	+	11.2250i	−0.382102	+	0.382102i	0.489757	+	0.871859i	$0.337086\pi$
$864$		0			0
$865$	−1.12917	−	1.42088i	−0.0383930	−	0.0483115i
$866$		0			0
$867$	33.4465	+	33.4465i	1.13590	+	1.13590i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	26.7161	−	12.6984i	0.903170	−	0.429283i
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$		0			0
$879$			95.1582i			3.20961i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$	−50.7109	+	40.2997i	−1.70463	+	1.35466i
$886$		0			0
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		0			0
$889$			38.2583i			1.28314i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	129.608	+	129.608i	4.32749	+	4.32749i
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.83746	−	59.7707i	0.227285	−	1.98685i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$		−	6.60756i		−	0.219159i
$910$		0			0
$911$	−52.3832			−1.73553			−0.867766	−	0.496972i	$-0.834445\pi$
$911$	−52.3832			−1.73553			−0.867766	+	0.496972i	$0.834445\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−25.0685	−	31.5447i	−0.828738	−	1.04284i
$916$		0			0
$917$	23.0354	+	23.0354i	0.760695	+	0.760695i
$918$		0			0
$919$			29.1582i			0.961841i	0.876764	+	0.480921i	$0.159697\pi$
$919$			29.1582i			0.961841i	−0.876764	+	0.480921i	$0.840303\pi$
$920$		0			0
$921$	−67.6749			−2.22996
$922$		0			0
$923$	−74.3858	+	74.3858i	−2.44844	+	2.44844i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	16.8704			0.552906
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	61.1707			1.99411			0.997054	−	0.0767020i	$-0.0244390\pi$
$941$	61.1707			1.99411			0.997054	+	0.0767020i	$0.0244390\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	28.4833	+	3.25834i	0.926562	+	0.105994i
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	41.9666	−	41.9666i	1.35943	−	1.35943i	0.484817	−	0.874616i	$-0.338886\pi$
$953$	41.9666	−	41.9666i	1.35943	−	1.35943i	0.874616	−	0.484817i	$-0.161114\pi$
$954$		0			0
$955$	−1.21351	+	10.6081i	−0.0392683	+	0.343270i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	5.67492i		−	0.183253i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.8047	+	14.8543i	0.380005	+	0.478177i
$966$		0			0
$967$	−40.2250	−	40.2250i	−1.29355	−	1.29355i	−0.932577	−	0.360971i	$-0.882445\pi$
$967$	−40.2250	−	40.2250i	−1.29355	−	1.29355i	−0.360971	−	0.932577i	$-0.617555\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	57.6298			1.84943			0.924713	−	0.380664i	$-0.124305\pi$
$971$	57.6298			1.84943			0.924713	+	0.380664i	$0.124305\pi$
$972$		0			0
$973$	−9.96852	+	9.96852i	−0.319576	+	0.319576i
$974$		0			0
$975$	21.7083	−	93.6415i	0.695222	−	2.99893i
$976$		0			0
$977$	38.9333	+	38.9333i	1.24559	+	1.24559i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	38.9333	+	38.9333i	1.24559	+	1.24559i	0.287936	+	0.957650i	$0.407031\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−62.1916			−1.97558			−0.987791	−	0.155787i	$-0.950209\pi$
$991$	−62.1916			−1.97558			−0.987791	+	0.155787i	$0.950209\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.7401	+	19.7401i	0.625174	+	0.625174i	0.946850	−	0.321675i	$-0.104246\pi$
$997$	19.7401	+	19.7401i	0.625174	+	0.625174i	−0.321675	+	0.946850i	$0.604246\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.w.a.657.4		8
4.3	odd	2		280.2.s.a.237.1	yes	8
5.3	odd	4	inner	1120.2.w.a.433.4		8
7.6	odd	2	inner	1120.2.w.a.657.1		8
8.3	odd	2		280.2.s.a.237.4	yes	8
8.5	even	2	inner	1120.2.w.a.657.1		8
20.3	even	4		280.2.s.a.13.1	✓	8
28.27	even	2		280.2.s.a.237.4	yes	8
35.13	even	4	inner	1120.2.w.a.433.1		8
40.3	even	4		280.2.s.a.13.4	yes	8
40.13	odd	4	inner	1120.2.w.a.433.1		8
56.13	odd	2	CM	1120.2.w.a.657.4		8
56.27	even	2		280.2.s.a.237.1	yes	8
140.83	odd	4		280.2.s.a.13.4	yes	8
280.13	even	4	inner	1120.2.w.a.433.4		8
280.83	odd	4		280.2.s.a.13.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.s.a.13.1	✓	8	20.3	even	4
280.2.s.a.13.1	✓	8	280.83	odd	4
280.2.s.a.13.4	yes	8	40.3	even	4
280.2.s.a.13.4	yes	8	140.83	odd	4
280.2.s.a.237.1	yes	8	4.3	odd	2
280.2.s.a.237.1	yes	8	56.27	even	2
280.2.s.a.237.4	yes	8	8.3	odd	2
280.2.s.a.237.4	yes	8	28.27	even	2
1120.2.w.a.433.1		8	35.13	even	4	inner
1120.2.w.a.433.1		8	40.13	odd	4	inner
1120.2.w.a.433.4		8	5.3	odd	4	inner
1120.2.w.a.433.4		8	280.13	even	4	inner
1120.2.w.a.657.1		8	7.6	odd	2	inner
1120.2.w.a.657.1		8	8.5	even	2	inner
1120.2.w.a.657.4		8	1.1	even	1	trivial
1120.2.w.a.657.4		8	56.13	odd	2	CM

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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.w (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.96744 - 1.96744i) q^{3} +(0.254137 - 2.22158i) q^{5} +(-1.87083 - 1.87083i) q^{7} -4.74166i q^{9} +(-4.88578 + 4.88578i) q^{13} +(-3.87083 - 4.87083i) q^{15} -2.41006i q^{19} -7.36149 q^{21} +(6.74166 - 6.74166i) q^{23} +(-4.87083 - 1.12917i) q^{25} +(-3.42661 - 3.42661i) q^{27} +(-4.63164 + 3.68075i) q^{35} +19.2250i q^{39} +(-10.5340 - 1.20503i) q^{45} +7.00000i q^{49} +(-4.74166 - 4.74166i) q^{57} -10.4111i q^{59} +6.47626 q^{61} +(-8.87083 + 8.87083i) q^{63} +(9.61249 + 12.0958i) q^{65} -26.5276i q^{69} +15.2250 q^{71} +(-11.8047 + 7.36149i) q^{75} -8.25834i q^{79} +0.741657 q^{81} +(-12.2473 + 12.2473i) q^{83} +18.2809 q^{91} +(-5.35414 - 0.612486i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{15} + 24 q^{23} - 24 q^{25} - 8 q^{57} - 56 q^{63} + 32 q^{65} + 32 q^{71} - 24 q^{81} + 32 q^{95}+O(q^{100}) \) 8 * q - 16 * q^15 + 24 * q^23 - 24 * q^25 - 8 * q^57 - 56 * q^63 + 32 * q^65 + 32 * q^71 - 24 * q^81 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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