LMFDB - Embedded newform 4032.2.v.a.1583.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4032,2,Mod(1583,4032)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4032, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4032.1583"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.v (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-4,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1583.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.a.3599.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 1.41421i) q^{5} -1.00000 q^{7} +(-1.41421 + 1.41421i) q^{11} +(2.00000 + 2.00000i) q^{13} +7.07107i q^{23} -1.00000i q^{25} +(1.41421 - 1.41421i) q^{29} -8.00000i q^{31} +(1.41421 + 1.41421i) q^{35} +(5.00000 - 5.00000i) q^{37} -2.82843 q^{41} +(3.00000 + 3.00000i) q^{43} -5.65685 q^{47} +1.00000 q^{49} +4.00000 q^{55} +(2.82843 - 2.82843i) q^{59} +(-2.00000 - 2.00000i) q^{61} -5.65685i q^{65} +(5.00000 - 5.00000i) q^{67} -9.89949i q^{71} +14.0000i q^{73} +(1.41421 - 1.41421i) q^{77} -16.0000i q^{79} +(2.82843 + 2.82843i) q^{83} -11.3137 q^{89} +(-2.00000 - 2.00000i) q^{91} -6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} + 8 q^{13} + 20 q^{37} + 12 q^{43} + 4 q^{49} + 16 q^{55} - 8 q^{61} + 20 q^{67} - 8 q^{91} - 24 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 8 * q^13 + 20 * q^37 + 12 * q^43 + 4 * q^49 + 16 * q^55 - 8 * q^61 + 20 * q^67 - 8 * q^91 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.41421	−	1.41421i	−0.632456	−	0.632456i	0.316228	−	0.948683i	$-0.397584\pi$
$5$	−1.41421	−	1.41421i	−0.632456	−	0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.41421	+	1.41421i	−0.426401	+	0.426401i	−0.887401	−	0.460999i	$-0.847491\pi$
$11$	−1.41421	+	1.41421i	−0.426401	+	0.426401i	0.460999	+	0.887401i	$0.347491\pi$
$12$		0			0
$13$	2.00000	+	2.00000i	0.554700	+	0.554700i	0.927794	−	0.373094i	$-0.121703\pi$
$13$	2.00000	+	2.00000i	0.554700	+	0.554700i	−0.373094	+	0.927794i	$0.621703\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			7.07107i			1.47442i	0.675664	+	0.737210i	$0.263857\pi$
$23$			7.07107i			1.47442i	−0.675664	+	0.737210i	$0.736143\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.41421	−	1.41421i	0.262613	−	0.262613i	−0.563502	−	0.826115i	$-0.690546\pi$
$29$	1.41421	−	1.41421i	0.262613	−	0.262613i	0.826115	+	0.563502i	$0.190546\pi$
$30$		0			0
$31$		−	8.00000i		−	1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$		−	8.00000i		−	1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.41421	+	1.41421i	0.239046	+	0.239046i
$36$		0			0
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	0.986394	+	0.164399i	$0.0525685\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.82843			−0.441726			−0.220863	−	0.975305i	$-0.570887\pi$
$41$	−2.82843			−0.441726			−0.220863	+	0.975305i	$0.570887\pi$
$42$		0			0
$43$	3.00000	+	3.00000i	0.457496	+	0.457496i	0.897833	−	0.440337i	$-0.145141\pi$
$43$	3.00000	+	3.00000i	0.457496	+	0.457496i	−0.440337	+	0.897833i	$0.645141\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.65685			−0.825137			−0.412568	−	0.910927i	$-0.635368\pi$
$47$	−5.65685			−0.825137			−0.412568	+	0.910927i	$0.635368\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		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
$55$	4.00000			0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.82843	−	2.82843i	0.368230	−	0.368230i	−0.498601	−	0.866831i	$-0.666153\pi$
$59$	2.82843	−	2.82843i	0.368230	−	0.368230i	0.866831	+	0.498601i	$0.166153\pi$
$60$		0			0
$61$	−2.00000	−	2.00000i	−0.256074	−	0.256074i	0.567381	−	0.823455i	$-0.307957\pi$
$61$	−2.00000	−	2.00000i	−0.256074	−	0.256074i	−0.823455	+	0.567381i	$0.807957\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	5.65685i		−	0.701646i
$66$		0			0
$67$	5.00000	−	5.00000i	0.610847	−	0.610847i	−0.332320	−	0.943167i	$-0.607831\pi$
$67$	5.00000	−	5.00000i	0.610847	−	0.610847i	0.943167	+	0.332320i	$0.107831\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.89949i		−	1.17485i	−0.809277	−	0.587427i	$-0.800141\pi$
$71$		−	9.89949i		−	1.17485i	0.809277	−	0.587427i	$-0.199859\pi$
$72$		0			0
$73$			14.0000i			1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$			14.0000i			1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.41421	−	1.41421i	0.161165	−	0.161165i
$78$		0			0
$79$		−	16.0000i		−	1.80014i	−0.435745	−	0.900070i	$-0.643515\pi$
$79$		−	16.0000i		−	1.80014i	0.435745	−	0.900070i	$-0.356485\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.82843	+	2.82843i	0.310460	+	0.310460i	0.845088	−	0.534628i	$-0.179548\pi$
$83$	2.82843	+	2.82843i	0.310460	+	0.310460i	−0.534628	+	0.845088i	$0.679548\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.3137			−1.19925			−0.599625	−	0.800281i	$-0.704684\pi$
$89$	−11.3137			−1.19925			−0.599625	+	0.800281i	$0.704684\pi$
$90$		0			0
$91$	−2.00000	−	2.00000i	−0.209657	−	0.209657i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−6.00000			−0.609208			−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000			−0.609208			−0.304604	+	0.952479i	$0.598524\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.7279	−	12.7279i	−1.26648	−	1.26648i	−0.947896	−	0.318579i	$-0.896794\pi$
$101$	−12.7279	−	12.7279i	−1.26648	−	1.26648i	−0.318579	−	0.947896i	$-0.603206\pi$
$102$		0			0
$103$	−8.00000			−0.788263			−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.82843	+	2.82843i	−0.273434	+	0.273434i	−0.830481	−	0.557047i	$-0.811934\pi$
$107$	−2.82843	+	2.82843i	−0.273434	+	0.273434i	0.557047	+	0.830481i	$0.311934\pi$
$108$		0			0
$109$	−1.00000	−	1.00000i	−0.0957826	−	0.0957826i	0.657592	−	0.753374i	$-0.271575\pi$
$109$	−1.00000	−	1.00000i	−0.0957826	−	0.0957826i	−0.753374	+	0.657592i	$0.771575\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.24264i			0.399114i	0.979886	+	0.199557i	$0.0639503\pi$
$113$			4.24264i			0.399114i	−0.979886	+	0.199557i	$0.936050\pi$
$114$		0			0
$115$	10.0000	−	10.0000i	0.932505	−	0.932505i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$			7.00000i			0.636364i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.48528	+	8.48528i	−0.758947	+	0.758947i
$126$		0			0
$127$		−	10.0000i		−	0.887357i	−0.896186	−	0.443678i	$-0.853673\pi$
$127$		−	10.0000i		−	0.887357i	0.896186	−	0.443678i	$-0.146327\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.82843	−	2.82843i	−0.247121	−	0.247121i	0.572667	−	0.819788i	$-0.305909\pi$
$131$	−2.82843	−	2.82843i	−0.247121	−	0.247121i	−0.819788	+	0.572667i	$0.805909\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−15.5563			−1.32907			−0.664534	−	0.747258i	$-0.731370\pi$
$137$	−15.5563			−1.32907			−0.664534	+	0.747258i	$0.731370\pi$
$138$		0			0
$139$	6.00000	+	6.00000i	0.508913	+	0.508913i	0.914193	−	0.405279i	$-0.132826\pi$
$139$	6.00000	+	6.00000i	0.508913	+	0.508913i	−0.405279	+	0.914193i	$0.632826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.65685			−0.473050
$144$		0			0
$145$	−4.00000			−0.332182
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.07107	−	7.07107i	−0.579284	−	0.579284i	0.355422	−	0.934706i	$-0.384337\pi$
$149$	−7.07107	−	7.07107i	−0.579284	−	0.579284i	−0.934706	+	0.355422i	$0.884337\pi$
$150$		0			0
$151$	−10.0000			−0.813788			−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000			−0.813788			−0.406894	+	0.913475i	$0.633388\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−11.3137	+	11.3137i	−0.908739	+	0.908739i
$156$		0			0
$157$	−8.00000	−	8.00000i	−0.638470	−	0.638470i	0.311708	−	0.950178i	$-0.399099\pi$
$157$	−8.00000	−	8.00000i	−0.638470	−	0.638470i	−0.950178	+	0.311708i	$0.899099\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	7.07107i		−	0.557278i
$162$		0			0
$163$	15.0000	−	15.0000i	1.17489	−	1.17489i	0.193862	−	0.981029i	$-0.437899\pi$
$163$	15.0000	−	15.0000i	1.17489	−	1.17489i	0.981029	−	0.193862i	$-0.0621013\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	19.7990i		−	1.53209i	−0.642786	−	0.766046i	$-0.722221\pi$
$167$		−	19.7990i		−	1.53209i	0.642786	−	0.766046i	$-0.277779\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.89949	−	9.89949i	0.752645	−	0.752645i	−0.222327	−	0.974972i	$-0.571365\pi$
$173$	9.89949	−	9.89949i	0.752645	−	0.752645i	0.974972	+	0.222327i	$0.0713654\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.48528	−	8.48528i	−0.634220	−	0.634220i	0.314904	−	0.949124i	$-0.398028\pi$
$179$	−8.48528	−	8.48528i	−0.634220	−	0.634220i	−0.949124	+	0.314904i	$0.898028\pi$
$180$		0			0
$181$	14.0000	−	14.0000i	1.04061	−	1.04061i	0.0414721	−	0.999140i	$-0.486795\pi$
$181$	14.0000	−	14.0000i	1.04061	−	1.04061i	0.999140	−	0.0414721i	$-0.0132048\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−14.1421			−1.03975
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.24264			0.306987			0.153493	−	0.988150i	$-0.450948\pi$
$191$	4.24264			0.306987			0.153493	+	0.988150i	$0.450948\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.89949	+	9.89949i	0.705310	+	0.705310i	0.965545	−	0.260235i	$-0.0838002\pi$
$197$	9.89949	+	9.89949i	0.705310	+	0.705310i	−0.260235	+	0.965545i	$0.583800\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.41421	+	1.41421i	−0.0992583	+	0.0992583i
$204$		0			0
$205$	4.00000	+	4.00000i	0.279372	+	0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−9.00000	+	9.00000i	−0.619586	+	0.619586i	−0.945425	−	0.325840i	$-0.894353\pi$
$211$	−9.00000	+	9.00000i	−0.619586	+	0.619586i	0.325840	+	0.945425i	$0.394353\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	8.48528i		−	0.578691i
$216$		0			0
$217$			8.00000i			0.543075i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.48528	−	8.48528i	−0.563188	−	0.563188i	0.367024	−	0.930212i	$-0.380377\pi$
$227$	−8.48528	−	8.48528i	−0.563188	−	0.563188i	−0.930212	+	0.367024i	$0.880377\pi$
$228$		0			0
$229$	18.0000	−	18.0000i	1.18947	−	1.18947i	0.212260	−	0.977213i	$-0.431918\pi$
$229$	18.0000	−	18.0000i	1.18947	−	1.18947i	0.977213	−	0.212260i	$-0.0680825\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.2132			−1.38972			−0.694862	−	0.719144i	$-0.744534\pi$
$233$	−21.2132			−1.38972			−0.694862	+	0.719144i	$0.744534\pi$
$234$		0			0
$235$	8.00000	+	8.00000i	0.521862	+	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.41421			0.0914779			0.0457389	−	0.998953i	$-0.485436\pi$
$239$	1.41421			0.0914779			0.0457389	+	0.998953i	$0.485436\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.41421	−	1.41421i	−0.0903508	−	0.0903508i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$251$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$252$		0			0
$253$	−10.0000	−	10.0000i	−0.628695	−	0.628695i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	11.3137i		−	0.705730i	−0.935674	−	0.352865i	$-0.885208\pi$
$257$		−	11.3137i		−	0.705730i	0.935674	−	0.352865i	$-0.114792\pi$
$258$		0			0
$259$	−5.00000	+	5.00000i	−0.310685	+	0.310685i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	4.24264i		−	0.261612i	−0.991408	−	0.130806i	$-0.958243\pi$
$263$		−	4.24264i		−	0.261612i	0.991408	−	0.130806i	$-0.0417566\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.2132	+	21.2132i	−1.29339	+	1.29339i	−0.360716	+	0.932676i	$0.617468\pi$
$269$	−21.2132	+	21.2132i	−1.29339	+	1.29339i	−0.932676	+	0.360716i	$0.882532\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.41421	+	1.41421i	0.0852803	+	0.0852803i
$276$		0			0
$277$	−7.00000	+	7.00000i	−0.420589	+	0.420589i	−0.885407	−	0.464817i	$-0.846120\pi$
$277$	−7.00000	+	7.00000i	−0.420589	+	0.420589i	0.464817	+	0.885407i	$0.346120\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.89949			0.590554			0.295277	−	0.955412i	$-0.404588\pi$
$281$	9.89949			0.590554			0.295277	+	0.955412i	$0.404588\pi$
$282$		0			0
$283$	6.00000	+	6.00000i	0.356663	+	0.356663i	0.862581	−	0.505918i	$-0.168846\pi$
$283$	6.00000	+	6.00000i	0.356663	+	0.356663i	−0.505918	+	0.862581i	$0.668846\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.82843			0.166957
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	21.2132	+	21.2132i	1.23929	+	1.23929i	0.960292	+	0.278996i	$0.0900018\pi$
$293$	21.2132	+	21.2132i	1.23929	+	1.23929i	0.278996	+	0.960292i	$0.409998\pi$
$294$		0			0
$295$	−8.00000			−0.465778
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−14.1421	+	14.1421i	−0.817861	+	0.817861i
$300$		0			0
$301$	−3.00000	−	3.00000i	−0.172917	−	0.172917i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			5.65685i			0.323911i
$306$		0			0
$307$	12.0000	−	12.0000i	0.684876	−	0.684876i	−0.276219	−	0.961095i	$-0.589081\pi$
$307$	12.0000	−	12.0000i	0.684876	−	0.684876i	0.961095	+	0.276219i	$0.0890814\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	33.9411i		−	1.92462i	−0.271947	−	0.962312i	$-0.587667\pi$
$311$		−	33.9411i		−	1.92462i	0.271947	−	0.962312i	$-0.412333\pi$
$312$		0			0
$313$		−	22.0000i		−	1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$		−	22.0000i		−	1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$		0			0
$315$		0			0
$316$		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
$319$			4.00000i			0.223957i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.00000	−	2.00000i	0.110940	−	0.110940i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.65685			0.311872
$330$		0			0
$331$	1.00000	+	1.00000i	0.0549650	+	0.0549650i	0.734055	−	0.679090i	$-0.237625\pi$
$331$	1.00000	+	1.00000i	0.0549650	+	0.0549650i	−0.679090	+	0.734055i	$0.737625\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.1421			−0.772667
$336$		0			0
$337$	30.0000			1.63420			0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000			1.63420			0.817102	+	0.576493i	$0.195579\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	11.3137	+	11.3137i	0.612672	+	0.612672i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.1421	−	14.1421i	0.759190	−	0.759190i	−0.216985	−	0.976175i	$-0.569622\pi$
$347$	14.1421	−	14.1421i	0.759190	−	0.759190i	0.976175	+	0.216985i	$0.0696224\pi$
$348$		0			0
$349$	−16.0000	−	16.0000i	−0.856460	−	0.856460i	0.134459	−	0.990919i	$-0.457070\pi$
$349$	−16.0000	−	16.0000i	−0.856460	−	0.856460i	−0.990919	+	0.134459i	$0.957070\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	31.1127i		−	1.65596i	−0.560756	−	0.827981i	$-0.689490\pi$
$353$		−	31.1127i		−	1.65596i	0.560756	−	0.827981i	$-0.310510\pi$
$354$		0			0
$355$	−14.0000	+	14.0000i	−0.743043	+	0.743043i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			7.07107i			0.373197i	0.982436	+	0.186598i	$0.0597463\pi$
$359$			7.07107i			0.373197i	−0.982436	+	0.186598i	$0.940254\pi$
$360$		0			0
$361$			19.0000i			1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	19.7990	−	19.7990i	1.03633	−	1.03633i
$366$		0			0
$367$		−	20.0000i		−	1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$		−	20.0000i		−	1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.0000	−	11.0000i	0.569558	−	0.569558i	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	11.0000	−	11.0000i	0.569558	−	0.569558i	0.932005	+	0.362446i	$0.118058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.65685			0.291343
$378$		0			0
$379$	11.0000	+	11.0000i	0.565032	+	0.565032i	0.930733	−	0.365701i	$-0.119171\pi$
$379$	11.0000	+	11.0000i	0.565032	+	0.565032i	−0.365701	+	0.930733i	$0.619171\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−14.1421			−0.722629			−0.361315	−	0.932444i	$-0.617672\pi$
$383$	−14.1421			−0.722629			−0.361315	+	0.932444i	$0.617672\pi$
$384$		0			0
$385$	−4.00000			−0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.9706	+	16.9706i	0.860442	+	0.860442i	0.991389	−	0.130948i	$-0.0418020\pi$
$389$	16.9706	+	16.9706i	0.860442	+	0.860442i	−0.130948	+	0.991389i	$0.541802\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−22.6274	+	22.6274i	−1.13851	+	1.13851i
$396$		0			0
$397$	4.00000	+	4.00000i	0.200754	+	0.200754i	0.800323	−	0.599569i	$-0.204661\pi$
$397$	4.00000	+	4.00000i	0.200754	+	0.200754i	−0.599569	+	0.800323i	$0.704661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			1.41421i			0.0706225i	0.999376	+	0.0353112i	$0.0112422\pi$
$401$			1.41421i			0.0706225i	−0.999376	+	0.0353112i	$0.988758\pi$
$402$		0			0
$403$	16.0000	−	16.0000i	0.797017	−	0.797017i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			14.1421i			0.701000i
$408$		0			0
$409$		−	26.0000i		−	1.28562i	−0.766027	−	0.642809i	$-0.777769\pi$
$409$		−	26.0000i		−	1.28562i	0.766027	−	0.642809i	$-0.222231\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.82843	+	2.82843i	−0.139178	+	0.139178i
$414$		0			0
$415$		−	8.00000i		−	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.9706	+	16.9706i	0.829066	+	0.829066i	0.987388	−	0.158321i	$-0.0506082\pi$
$419$	16.9706	+	16.9706i	0.829066	+	0.829066i	−0.158321	+	0.987388i	$0.550608\pi$
$420$		0			0
$421$	21.0000	−	21.0000i	1.02348	−	1.02348i	0.0237597	−	0.999718i	$-0.492436\pi$
$421$	21.0000	−	21.0000i	1.02348	−	1.02348i	0.999718	−	0.0237597i	$-0.00756365\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.00000	+	2.00000i	0.0967868	+	0.0967868i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	41.0122			1.97549			0.987744	−	0.156083i	$-0.0498868\pi$
$431$	41.0122			1.97549			0.987744	+	0.156083i	$0.0498868\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−20.0000			−0.954548			−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000			−0.954548			−0.477274	+	0.878755i	$0.658375\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.3848	+	18.3848i	−0.873487	+	0.873487i	−0.992851	−	0.119364i	$-0.961915\pi$
$443$	−18.3848	+	18.3848i	−0.873487	+	0.873487i	0.119364	+	0.992851i	$0.461915\pi$
$444$		0			0
$445$	16.0000	+	16.0000i	0.758473	+	0.758473i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	4.24264i		−	0.200223i	−0.994976	−	0.100111i	$-0.968080\pi$
$449$		−	4.24264i		−	0.200223i	0.994976	−	0.100111i	$-0.0319199\pi$
$450$		0			0
$451$	4.00000	−	4.00000i	0.188353	−	0.188353i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			5.65685i			0.265197i
$456$		0			0
$457$			12.0000i			0.561336i	0.959805	+	0.280668i	$0.0905560\pi$
$457$			12.0000i			0.561336i	−0.959805	+	0.280668i	$0.909444\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.5563	+	15.5563i	−0.724531	+	0.724531i	−0.969525	−	0.244993i	$-0.921214\pi$
$461$	−15.5563	+	15.5563i	−0.724531	+	0.724531i	0.244993	+	0.969525i	$0.421214\pi$
$462$		0			0
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−25.4558	−	25.4558i	−1.17796	−	1.17796i	−0.980264	−	0.197692i	$-0.936655\pi$
$467$	−25.4558	−	25.4558i	−1.17796	−	1.17796i	−0.197692	−	0.980264i	$-0.563345\pi$
$468$		0			0
$469$	−5.00000	+	5.00000i	−0.230879	+	0.230879i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8.48528			−0.390154
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.3137			−0.516937			−0.258468	−	0.966020i	$-0.583218\pi$
$479$	−11.3137			−0.516937			−0.258468	+	0.966020i	$0.583218\pi$
$480$		0			0
$481$	20.0000			0.911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.48528	+	8.48528i	0.385297	+	0.385297i
$486$		0			0
$487$	−26.0000			−1.17817			−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000			−1.17817			−0.589086	+	0.808070i	$0.700512\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.7279	+	12.7279i	−0.574403	+	0.574403i	−0.933356	−	0.358953i	$-0.883134\pi$
$491$	−12.7279	+	12.7279i	−0.574403	+	0.574403i	0.358953	+	0.933356i	$0.383134\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			9.89949i			0.444053i
$498$		0			0
$499$	9.00000	−	9.00000i	0.402895	−	0.402895i	−0.476357	−	0.879252i	$-0.658043\pi$
$499$	9.00000	−	9.00000i	0.402895	−	0.402895i	0.879252	+	0.476357i	$0.158043\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	19.7990i		−	0.882793i	−0.897312	−	0.441397i	$-0.854483\pi$
$503$		−	19.7990i		−	0.882793i	0.897312	−	0.441397i	$-0.145517\pi$
$504$		0			0
$505$			36.0000i			1.60198i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	21.2132	−	21.2132i	0.940259	−	0.940259i	−0.0580547	−	0.998313i	$-0.518490\pi$
$509$	21.2132	−	21.2132i	0.940259	−	0.940259i	0.998313	+	0.0580547i	$0.0184898\pi$
$510$		0			0
$511$		−	14.0000i		−	0.619324i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	11.3137	+	11.3137i	0.498542	+	0.498542i
$516$		0			0
$517$	8.00000	−	8.00000i	0.351840	−	0.351840i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.2843			−1.23916			−0.619578	−	0.784935i	$-0.712696\pi$
$521$	−28.2843			−1.23916			−0.619578	+	0.784935i	$0.712696\pi$
$522$		0			0
$523$	12.0000	+	12.0000i	0.524723	+	0.524723i	0.918994	−	0.394271i	$-0.129003\pi$
$523$	12.0000	+	12.0000i	0.524723	+	0.524723i	−0.394271	+	0.918994i	$0.629003\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−27.0000			−1.17391
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5.65685	−	5.65685i	−0.245026	−	0.245026i
$534$		0			0
$535$	8.00000			0.345870
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.41421	+	1.41421i	−0.0609145	+	0.0609145i
$540$		0			0
$541$	−21.0000	−	21.0000i	−0.902861	−	0.902861i	0.0928222	−	0.995683i	$-0.470411\pi$
$541$	−21.0000	−	21.0000i	−0.902861	−	0.902861i	−0.995683	+	0.0928222i	$0.970411\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			2.82843i			0.121157i
$546$		0			0
$547$	−27.0000	+	27.0000i	−1.15444	+	1.15444i	−0.168783	+	0.985653i	$0.553984\pi$
$547$	−27.0000	+	27.0000i	−1.15444	+	1.15444i	−0.985653	+	0.168783i	$0.946016\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12.7279	−	12.7279i	0.539299	−	0.539299i	−0.384024	−	0.923323i	$-0.625462\pi$
$557$	12.7279	−	12.7279i	0.539299	−	0.539299i	0.923323	+	0.384024i	$0.125462\pi$
$558$		0			0
$559$			12.0000i			0.507546i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.48528	+	8.48528i	0.357612	+	0.357612i	0.862932	−	0.505320i	$-0.168626\pi$
$563$	8.48528	+	8.48528i	0.357612	+	0.357612i	−0.505320	+	0.862932i	$0.668626\pi$
$564$		0			0
$565$	6.00000	−	6.00000i	0.252422	−	0.252422i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.07107			−0.296435			−0.148217	−	0.988955i	$-0.547354\pi$
$569$	−7.07107			−0.296435			−0.148217	+	0.988955i	$0.547354\pi$
$570$		0			0
$571$	15.0000	+	15.0000i	0.627730	+	0.627730i	0.947497	−	0.319766i	$-0.103604\pi$
$571$	15.0000	+	15.0000i	0.627730	+	0.627730i	−0.319766	+	0.947497i	$0.603604\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7.07107			0.294884
$576$		0			0
$577$	18.0000			0.749350			0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000			0.749350			0.374675	+	0.927156i	$0.377754\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.82843	−	2.82843i	−0.117343	−	0.117343i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	11.3137	−	11.3137i	0.466967	−	0.466967i	−0.433964	−	0.900930i	$-0.642885\pi$
$587$	11.3137	−	11.3137i	0.466967	−	0.466967i	0.900930	+	0.433964i	$0.142885\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			19.7990i			0.813047i	0.913640	+	0.406524i	$0.133259\pi$
$593$			19.7990i			0.813047i	−0.913640	+	0.406524i	$0.866741\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			38.1838i			1.56015i	0.625688	+	0.780073i	$0.284818\pi$
$599$			38.1838i			1.56015i	−0.625688	+	0.780073i	$0.715182\pi$
$600$		0			0
$601$			2.00000i			0.0815817i	0.999168	+	0.0407909i	$0.0129877\pi$
$601$			2.00000i			0.0815817i	−0.999168	+	0.0407909i	$0.987012\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.89949	−	9.89949i	0.402472	−	0.402472i
$606$		0			0
$607$			32.0000i			1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$			32.0000i			1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−11.3137	−	11.3137i	−0.457704	−	0.457704i
$612$		0			0
$613$	−3.00000	+	3.00000i	−0.121169	+	0.121169i	−0.765091	−	0.643922i	$-0.777306\pi$
$613$	−3.00000	+	3.00000i	−0.121169	+	0.121169i	0.643922	+	0.765091i	$0.277306\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.0416			0.967880			0.483940	−	0.875101i	$-0.339205\pi$
$617$	24.0416			0.967880			0.483940	+	0.875101i	$0.339205\pi$
$618$		0			0
$619$	−24.0000	−	24.0000i	−0.964641	−	0.964641i	0.0347544	−	0.999396i	$-0.488935\pi$
$619$	−24.0000	−	24.0000i	−0.964641	−	0.964641i	−0.999396	+	0.0347544i	$0.988935\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	11.3137			0.453274
$624$		0			0
$625$	19.0000			0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−30.0000			−1.19428			−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000			−1.19428			−0.597141	+	0.802137i	$0.703697\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−14.1421	+	14.1421i	−0.561214	+	0.561214i
$636$		0			0
$637$	2.00000	+	2.00000i	0.0792429	+	0.0792429i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	18.3848i		−	0.726155i	−0.931759	−	0.363078i	$-0.881726\pi$
$641$		−	18.3848i		−	0.726155i	0.931759	−	0.363078i	$-0.118274\pi$
$642$		0			0
$643$	−6.00000	+	6.00000i	−0.236617	+	0.236617i	−0.815448	−	0.578831i	$-0.803509\pi$
$643$	−6.00000	+	6.00000i	−0.236617	+	0.236617i	0.578831	+	0.815448i	$0.303509\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	19.7990i		−	0.778379i	−0.921158	−	0.389189i	$-0.872755\pi$
$647$		−	19.7990i		−	0.778379i	0.921158	−	0.389189i	$-0.127245\pi$
$648$		0			0
$649$			8.00000i			0.314027i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	29.6985	−	29.6985i	1.16219	−	1.16219i	0.178197	−	0.983995i	$-0.442974\pi$
$653$	29.6985	−	29.6985i	1.16219	−	1.16219i	0.983995	−	0.178197i	$-0.0570263\pi$
$654$		0			0
$655$			8.00000i			0.312586i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	25.4558	+	25.4558i	0.991619	+	0.991619i	0.999965	−	0.00834627i	$-0.00265673\pi$
$659$	25.4558	+	25.4558i	0.991619	+	0.991619i	−0.00834627	+	0.999965i	$0.502657\pi$
$660$		0			0
$661$	34.0000	−	34.0000i	1.32245	−	1.32245i	0.410657	−	0.911790i	$-0.365299\pi$
$661$	34.0000	−	34.0000i	1.32245	−	1.32245i	0.911790	−	0.410657i	$-0.134701\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10.0000	+	10.0000i	0.387202	+	0.387202i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.65685			0.218380
$672$		0			0
$673$	−20.0000			−0.770943			−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000			−0.770943			−0.385472	+	0.922720i	$0.625961\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.2132	+	21.2132i	0.815290	+	0.815290i	0.985421	−	0.170132i	$-0.0544193\pi$
$677$	21.2132	+	21.2132i	0.815290	+	0.815290i	−0.170132	+	0.985421i	$0.554419\pi$
$678$		0			0
$679$	6.00000			0.230259
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21.2132	−	21.2132i	0.811701	−	0.811701i	−0.173188	−	0.984889i	$-0.555407\pi$
$683$	21.2132	−	21.2132i	0.811701	−	0.811701i	0.984889	+	0.173188i	$0.0554069\pi$
$684$		0			0
$685$	22.0000	+	22.0000i	0.840577	+	0.840577i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−26.0000	+	26.0000i	−0.989087	+	0.989087i	−0.999941	−	0.0108545i	$-0.996545\pi$
$691$	−26.0000	+	26.0000i	−0.989087	+	0.989087i	0.0108545	+	0.999941i	$0.496545\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	16.9706i		−	0.643730i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−8.48528	+	8.48528i	−0.320485	+	0.320485i	−0.848953	−	0.528468i	$-0.822767\pi$
$701$	−8.48528	+	8.48528i	−0.320485	+	0.320485i	0.528468	+	0.848953i	$0.322767\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.7279	+	12.7279i	0.478683	+	0.478683i
$708$		0			0
$709$	−27.0000	+	27.0000i	−1.01401	+	1.01401i	−0.0141058	+	0.999901i	$0.504490\pi$
$709$	−27.0000	+	27.0000i	−1.01401	+	1.01401i	−0.999901	+	0.0141058i	$0.995510\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	56.5685			2.11851
$714$		0			0
$715$	8.00000	+	8.00000i	0.299183	+	0.299183i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.82843			−0.105483			−0.0527413	−	0.998608i	$-0.516796\pi$
$719$	−2.82843			−0.105483			−0.0527413	+	0.998608i	$0.516796\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.41421	−	1.41421i	−0.0525226	−	0.0525226i
$726$		0			0
$727$	−32.0000			−1.18681			−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000			−1.18681			−0.593407	+	0.804902i	$0.702218\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−28.0000	−	28.0000i	−1.03420	−	1.03420i	−0.999394	−	0.0348096i	$-0.988918\pi$
$733$	−28.0000	−	28.0000i	−1.03420	−	1.03420i	−0.0348096	−	0.999394i	$-0.511082\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			14.1421i			0.520932i
$738$		0			0
$739$	17.0000	−	17.0000i	0.625355	−	0.625355i	−0.321541	−	0.946896i	$-0.604201\pi$
$739$	17.0000	−	17.0000i	0.625355	−	0.625355i	0.946896	+	0.321541i	$0.104201\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			9.89949i			0.363177i	0.983375	+	0.181589i	$0.0581239\pi$
$743$			9.89949i			0.363177i	−0.983375	+	0.181589i	$0.941876\pi$
$744$		0			0
$745$			20.0000i			0.732743i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.82843	−	2.82843i	0.103348	−	0.103348i
$750$		0			0
$751$		−	32.0000i		−	1.16770i	−0.811863	−	0.583848i	$-0.801546\pi$
$751$		−	32.0000i		−	1.16770i	0.811863	−	0.583848i	$-0.198454\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14.1421	+	14.1421i	0.514685	+	0.514685i
$756$		0			0
$757$	−17.0000	+	17.0000i	−0.617876	+	0.617876i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−17.0000	+	17.0000i	−0.617876	+	0.617876i	0.327111	+	0.944986i	$0.393925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.82843			0.102530			0.0512652	−	0.998685i	$-0.483675\pi$
$761$	2.82843			0.102530			0.0512652	+	0.998685i	$0.483675\pi$
$762$		0			0
$763$	1.00000	+	1.00000i	0.0362024	+	0.0362024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11.3137			0.408514
$768$		0			0
$769$	−54.0000			−1.94729			−0.973645	−	0.228069i	$-0.926759\pi$
$769$	−54.0000			−1.94729			−0.973645	+	0.228069i	$0.926759\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.89949	−	9.89949i	−0.356060	−	0.356060i	0.506298	−	0.862358i	$-0.331013\pi$
$773$	−9.89949	−	9.89949i	−0.356060	−	0.356060i	−0.862358	+	0.506298i	$0.831013\pi$
$774$		0			0
$775$	−8.00000			−0.287368
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	14.0000	+	14.0000i	0.500959	+	0.500959i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			22.6274i			0.807607i
$786$		0			0
$787$	2.00000	−	2.00000i	0.0712923	−	0.0712923i	−0.670562	−	0.741854i	$-0.733947\pi$
$787$	2.00000	−	2.00000i	0.0712923	−	0.0712923i	0.741854	+	0.670562i	$0.233947\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	4.24264i		−	0.150851i
$792$		0			0
$793$		−	8.00000i		−	0.284088i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12.7279	+	12.7279i	−0.450846	+	0.450846i	−0.895635	−	0.444789i	$-0.853279\pi$
$797$	−12.7279	+	12.7279i	−0.450846	+	0.450846i	0.444789	+	0.895635i	$0.353279\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.7990	−	19.7990i	−0.698691	−	0.698691i
$804$		0			0
$805$	−10.0000	+	10.0000i	−0.352454	+	0.352454i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	35.3553			1.24303			0.621514	−	0.783403i	$-0.286518\pi$
$809$	35.3553			1.24303			0.621514	+	0.783403i	$0.286518\pi$
$810$		0			0
$811$	12.0000	+	12.0000i	0.421377	+	0.421377i	0.885678	−	0.464301i	$-0.153694\pi$
$811$	12.0000	+	12.0000i	0.421377	+	0.421377i	−0.464301	+	0.885678i	$0.653694\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−42.4264			−1.48613
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22.6274	−	22.6274i	−0.789702	−	0.789702i	0.191743	−	0.981445i	$-0.438586\pi$
$821$	−22.6274	−	22.6274i	−0.789702	−	0.789702i	−0.981445	+	0.191743i	$0.938586\pi$
$822$		0			0
$823$	2.00000			0.0697156			0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000			0.0697156			0.0348578	+	0.999392i	$0.488902\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.4558	−	25.4558i	0.885186	−	0.885186i	−0.108870	−	0.994056i	$-0.534723\pi$
$827$	25.4558	−	25.4558i	0.885186	−	0.885186i	0.994056	+	0.108870i	$0.0347231\pi$
$828$		0			0
$829$	−12.0000	−	12.0000i	−0.416777	−	0.416777i	0.467314	−	0.884091i	$-0.345222\pi$
$829$	−12.0000	−	12.0000i	−0.416777	−	0.416777i	−0.884091	+	0.467314i	$0.845222\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−28.0000	+	28.0000i	−0.968980	+	0.968980i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	2.82843i		−	0.0976481i	−0.998807	−	0.0488241i	$-0.984453\pi$
$839$		−	2.82843i		−	0.0976481i	0.998807	−	0.0488241i	$-0.0155474\pi$
$840$		0			0
$841$			25.0000i			0.862069i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.07107	+	7.07107i	−0.243252	+	0.243252i
$846$		0			0
$847$		−	7.00000i		−	0.240523i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	35.3553	+	35.3553i	1.21197	+	1.21197i
$852$		0			0
$853$	−16.0000	+	16.0000i	−0.547830	+	0.547830i	−0.925813	−	0.377983i	$-0.876618\pi$
$853$	−16.0000	+	16.0000i	−0.547830	+	0.547830i	0.377983	+	0.925813i	$0.376618\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.82843			−0.0966172			−0.0483086	−	0.998832i	$-0.515383\pi$
$857$	−2.82843			−0.0966172			−0.0483086	+	0.998832i	$0.515383\pi$
$858$		0			0
$859$	32.0000	+	32.0000i	1.09183	+	1.09183i	0.995334	+	0.0964922i	$0.0307623\pi$
$859$	32.0000	+	32.0000i	1.09183	+	1.09183i	0.0964922	+	0.995334i	$0.469238\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.0416			0.818387			0.409193	−	0.912448i	$-0.365810\pi$
$863$	24.0416			0.818387			0.409193	+	0.912448i	$0.365810\pi$
$864$		0			0
$865$	−28.0000			−0.952029
$866$		0			0
$867$		0			0
$868$		0			0
$869$	22.6274	+	22.6274i	0.767583	+	0.767583i
$870$		0			0
$871$	20.0000			0.677674
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.48528	−	8.48528i	0.286855	−	0.286855i
$876$		0			0
$877$	21.0000	+	21.0000i	0.709120	+	0.709120i	0.966350	−	0.257230i	$-0.0828100\pi$
$877$	21.0000	+	21.0000i	0.709120	+	0.709120i	−0.257230	+	0.966350i	$0.582810\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	14.1421i		−	0.476461i	−0.971209	−	0.238230i	$-0.923433\pi$
$881$		−	14.1421i		−	0.476461i	0.971209	−	0.238230i	$-0.0765673\pi$
$882$		0			0
$883$	−15.0000	+	15.0000i	−0.504790	+	0.504790i	−0.912923	−	0.408133i	$-0.866180\pi$
$883$	−15.0000	+	15.0000i	−0.504790	+	0.504790i	0.408133	+	0.912923i	$0.366180\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	2.82843i		−	0.0949693i	−0.998872	−	0.0474846i	$-0.984879\pi$
$887$		−	2.82843i		−	0.0949693i	0.998872	−	0.0474846i	$-0.0151205\pi$
$888$		0			0
$889$			10.0000i			0.335389i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$			24.0000i			0.802232i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.3137	−	11.3137i	−0.377333	−	0.377333i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−39.5980			−1.31628
$906$		0			0
$907$	−29.0000	−	29.0000i	−0.962929	−	0.962929i	0.0364078	−	0.999337i	$-0.488408\pi$
$907$	−29.0000	−	29.0000i	−0.962929	−	0.962929i	−0.999337	+	0.0364078i	$0.988408\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	52.3259			1.73363			0.866817	−	0.498626i	$-0.166162\pi$
$911$	52.3259			1.73363			0.866817	+	0.498626i	$0.166162\pi$
$912$		0			0
$913$	−8.00000			−0.264761
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.82843	+	2.82843i	0.0934029	+	0.0934029i
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	19.7990	−	19.7990i	0.651692	−	0.651692i
$924$		0			0
$925$	−5.00000	−	5.00000i	−0.164399	−	0.164399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			36.7696i			1.20637i	0.797601	+	0.603185i	$0.206102\pi$
$929$			36.7696i			1.20637i	−0.797601	+	0.603185i	$0.793898\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			14.0000i			0.457360i	0.973502	+	0.228680i	$0.0734410\pi$
$937$			14.0000i			0.457360i	−0.973502	+	0.228680i	$0.926559\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.7279	+	12.7279i	−0.414918	+	0.414918i	−0.883448	−	0.468529i	$-0.844784\pi$
$941$	−12.7279	+	12.7279i	−0.414918	+	0.414918i	0.468529	+	0.883448i	$0.344784\pi$
$942$		0			0
$943$		−	20.0000i		−	0.651290i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−19.7990	−	19.7990i	−0.643381	−	0.643381i	0.308004	−	0.951385i	$-0.400339\pi$
$947$	−19.7990	−	19.7990i	−0.643381	−	0.643381i	−0.951385	+	0.308004i	$0.900339\pi$
$948$		0			0
$949$	−28.0000	+	28.0000i	−0.908918	+	0.908918i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.41421			0.0458109			0.0229054	−	0.999738i	$-0.492708\pi$
$953$	1.41421			0.0458109			0.0229054	+	0.999738i	$0.492708\pi$
$954$		0			0
$955$	−6.00000	−	6.00000i	−0.194155	−	0.194155i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	15.5563			0.502341
$960$		0			0
$961$	−33.0000			−1.06452
$962$		0			0
$963$		0			0
$964$		0			0
$965$	19.7990	+	19.7990i	0.637352	+	0.637352i
$966$		0			0
$967$	40.0000			1.28631			0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000			1.28631			0.643157	+	0.765735i	$0.277624\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.7696	−	36.7696i	1.17999	−	1.17999i	0.200245	−	0.979746i	$-0.435826\pi$
$971$	36.7696	−	36.7696i	1.17999	−	1.17999i	0.979746	−	0.200245i	$-0.0641739\pi$
$972$		0			0
$973$	−6.00000	−	6.00000i	−0.192351	−	0.192351i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			41.0122i			1.31210i	0.754719	+	0.656048i	$0.227773\pi$
$977$			41.0122i			1.31210i	−0.754719	+	0.656048i	$0.772227\pi$
$978$		0			0
$979$	16.0000	−	16.0000i	0.511362	−	0.511362i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			33.9411i			1.08255i	0.840844	+	0.541277i	$0.182059\pi$
$983$			33.9411i			1.08255i	−0.840844	+	0.541277i	$0.817941\pi$
$984$		0			0
$985$		−	28.0000i		−	0.892154i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−21.2132	+	21.2132i	−0.674541	+	0.674541i
$990$		0			0
$991$			26.0000i			0.825917i	0.910750	+	0.412959i	$0.135505\pi$
$991$			26.0000i			0.825917i	−0.910750	+	0.412959i	$0.864495\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.65685	+	5.65685i	0.179334	+	0.179334i
$996$		0			0
$997$	−2.00000	+	2.00000i	−0.0633406	+	0.0633406i	−0.738068	−	0.674727i	$-0.764261\pi$
$997$	−2.00000	+	2.00000i	−0.0633406	+	0.0633406i	0.674727	+	0.738068i	$0.264261\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.a.1583.1		4
3.2	odd	2	inner	4032.2.v.a.1583.2		4
4.3	odd	2		1008.2.v.b.323.2	yes	4
12.11	even	2		1008.2.v.b.323.1	✓	4
16.5	even	4		1008.2.v.b.827.1	yes	4
16.11	odd	4	inner	4032.2.v.a.3599.2		4
48.5	odd	4		1008.2.v.b.827.2	yes	4
48.11	even	4	inner	4032.2.v.a.3599.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.b.323.1	✓	4	12.11	even	2
1008.2.v.b.323.2	yes	4	4.3	odd	2
1008.2.v.b.827.1	yes	4	16.5	even	4
1008.2.v.b.827.2	yes	4	48.5	odd	4
4032.2.v.a.1583.1		4	1.1	even	1	trivial
4032.2.v.a.1583.2		4	3.2	odd	2	inner
4032.2.v.a.3599.1		4	48.11	even	4	inner
4032.2.v.a.3599.2		4	16.11	odd	4	inner

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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.v (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 1.41421i) q^{5} -1.00000 q^{7} +(-1.41421 + 1.41421i) q^{11} +(2.00000 + 2.00000i) q^{13} +7.07107i q^{23} -1.00000i q^{25} +(1.41421 - 1.41421i) q^{29} -8.00000i q^{31} +(1.41421 + 1.41421i) q^{35} +(5.00000 - 5.00000i) q^{37} -2.82843 q^{41} +(3.00000 + 3.00000i) q^{43} -5.65685 q^{47} +1.00000 q^{49} +4.00000 q^{55} +(2.82843 - 2.82843i) q^{59} +(-2.00000 - 2.00000i) q^{61} -5.65685i q^{65} +(5.00000 - 5.00000i) q^{67} -9.89949i q^{71} +14.0000i q^{73} +(1.41421 - 1.41421i) q^{77} -16.0000i q^{79} +(2.82843 + 2.82843i) q^{83} -11.3137 q^{89} +(-2.00000 - 2.00000i) q^{91} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} + 8 q^{13} + 20 q^{37} + 12 q^{43} + 4 q^{49} + 16 q^{55} - 8 q^{61} + 20 q^{67} - 8 q^{91} - 24 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 8 * q^13 + 20 * q^37 + 12 * q^43 + 4 * q^49 + 16 * q^55 - 8 * q^61 + 20 * q^67 - 8 * q^91 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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