LMFDB - Embedded newform 4032.2.v.c.1583.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1583,4032)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 1008)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1583.3
Root			$-0.892524 + 1.09700i$ of defining polynomial
Character	$\chi$	$=$	4032.1583
Dual form			4032.2.v.c.3599.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 1.41421i) q^{5} +1.00000 q^{7} +O(q^{10})$	$q+(-1.41421 - 1.41421i) q^{5} +1.00000 q^{7} +(3.97904 - 3.97904i) q^{11} +(-1.10278 - 1.10278i) q^{13} +6.39852i q^{17} -2.97377i q^{23} -1.00000i q^{25} +(5.53860 - 5.53860i) q^{29} +2.20555i q^{31} +(-1.41421 - 1.41421i) q^{35} +(-1.00000 + 1.00000i) q^{37} -2.08676 q^{41} +(4.68111 + 4.68111i) q^{43} +8.77597 q^{47} +1.00000 q^{49} +(-3.83369 - 3.83369i) q^{53} -11.2544 q^{55} +(9.51764 - 9.51764i) q^{59} +(-5.10278 - 5.10278i) q^{61} +3.11912i q^{65} +(-6.10278 + 6.10278i) q^{67} -6.62009i q^{71} +7.04888i q^{73} +(3.97904 - 3.97904i) q^{77} -4.41110i q^{79} +(10.0448 + 10.0448i) q^{83} +(9.04888 - 9.04888i) q^{85} -13.6912 q^{89} +(-1.10278 - 1.10278i) q^{91} -15.5678 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7}+O(q^{10}) $ 12 * q + 12 * q^7	$ 12 q + 12 q^{7} + 16 q^{13} - 12 q^{37} + 20 q^{43} + 12 q^{49} - 32 q^{55} - 32 q^{61} - 44 q^{67} + 64 q^{85} + 16 q^{91} - 56 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 + 16 * q^13 - 12 * q^37 + 20 * q^43 + 12 * q^49 - 32 * q^55 - 32 * q^61 - 44 * q^67 + 64 * q^85 + 16 * q^91 - 56 * q^97

Display 100 coefficients 10 coefficients

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
$11$	3.97904	−	3.97904i	1.19973	−	1.19973i	0.225477	−	0.974248i	$-0.427606\pi$
$11$	3.97904	−	3.97904i	1.19973	−	1.19973i	0.974248	−	0.225477i	$-0.0723942\pi$
$12$		0			0
$13$	−1.10278	−	1.10278i	−0.305855	−	0.305855i	0.537444	−	0.843299i	$-0.319390\pi$
$13$	−1.10278	−	1.10278i	−0.305855	−	0.305855i	−0.843299	+	0.537444i	$0.819390\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.39852i			1.55187i	0.630813	+	0.775935i	$0.282721\pi$
$17$			6.39852i			1.55187i	−0.630813	+	0.775935i	$0.717279\pi$
$18$		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
$23$		−	2.97377i		−	0.620075i	−0.950724	−	0.310037i	$-0.899658\pi$
$23$		−	2.97377i		−	0.620075i	0.950724	−	0.310037i	$-0.100342\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.53860	−	5.53860i	1.02849	−	1.02849i	0.0289102	−	0.999582i	$-0.490796\pi$
$29$	5.53860	−	5.53860i	1.02849	−	1.02849i	0.999582	−	0.0289102i	$-0.00920367\pi$
$30$		0			0
$31$			2.20555i			0.396128i	0.980189	+	0.198064i	$0.0634655\pi$
$31$			2.20555i			0.396128i	−0.980189	+	0.198064i	$0.936535\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.41421	−	1.41421i	−0.239046	−	0.239046i
$36$		0			0
$37$	−1.00000	+	1.00000i	−0.164399	+	0.164399i	−0.784512	−	0.620113i	$-0.787087\pi$
$37$	−1.00000	+	1.00000i	−0.164399	+	0.164399i	0.620113	+	0.784512i	$0.287087\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.08676			−0.325897			−0.162949	−	0.986635i	$-0.552100\pi$
$41$	−2.08676			−0.325897			−0.162949	+	0.986635i	$0.552100\pi$
$42$		0			0
$43$	4.68111	+	4.68111i	0.713863	+	0.713863i	0.967341	−	0.253478i	$-0.0815746\pi$
$43$	4.68111	+	4.68111i	0.713863	+	0.713863i	−0.253478	+	0.967341i	$0.581575\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.77597			1.28011			0.640054	−	0.768330i	$-0.278912\pi$
$47$	8.77597			1.28011			0.640054	+	0.768330i	$0.278912\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.83369	−	3.83369i	−0.526598	−	0.526598i	0.392958	−	0.919556i	$-0.371452\pi$
$53$	−3.83369	−	3.83369i	−0.526598	−	0.526598i	−0.919556	+	0.392958i	$0.871452\pi$
$54$		0			0
$55$	−11.2544			−1.51755
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.51764	−	9.51764i	1.23909	−	1.23909i	0.278718	−	0.960373i	$-0.410090\pi$
$59$	9.51764	−	9.51764i	1.23909	−	1.23909i	0.960373	−	0.278718i	$-0.0899096\pi$
$60$		0			0
$61$	−5.10278	−	5.10278i	−0.653343	−	0.653343i	0.300453	−	0.953797i	$-0.402862\pi$
$61$	−5.10278	−	5.10278i	−0.653343	−	0.653343i	−0.953797	+	0.300453i	$0.902862\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			3.11912i			0.386879i
$66$		0			0
$67$	−6.10278	+	6.10278i	−0.745573	+	0.745573i	−0.973644	−	0.228072i	$-0.926758\pi$
$67$	−6.10278	+	6.10278i	−0.745573	+	0.745573i	0.228072	+	0.973644i	$0.426758\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.62009i		−	0.785660i	−0.919611	−	0.392830i	$-0.871496\pi$
$71$		−	6.62009i		−	0.785660i	0.919611	−	0.392830i	$-0.128504\pi$
$72$		0			0
$73$			7.04888i			0.825009i	0.910956	+	0.412504i	$0.135346\pi$
$73$			7.04888i			0.825009i	−0.910956	+	0.412504i	$0.864654\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.97904	−	3.97904i	0.453454	−	0.453454i
$78$		0			0
$79$		−	4.41110i		−	0.496288i	−0.968723	−	0.248144i	$-0.920179\pi$
$79$		−	4.41110i		−	0.496288i	0.968723	−	0.248144i	$-0.0798205\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.0448	+	10.0448i	1.10256	+	1.10256i	0.994100	+	0.108464i	$0.0345933\pi$
$83$	10.0448	+	10.0448i	1.10256	+	1.10256i	0.108464	+	0.994100i	$0.465407\pi$
$84$		0			0
$85$	9.04888	−	9.04888i	0.981488	−	0.981488i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.6912			−1.45126			−0.725630	−	0.688085i	$-0.758452\pi$
$89$	−13.6912			−1.45126			−0.725630	+	0.688085i	$0.758452\pi$
$90$		0			0
$91$	−1.10278	−	1.10278i	−0.115602	−	0.115602i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−15.5678			−1.58067			−0.790334	−	0.612676i	$-0.790093\pi$
$97$	−15.5678			−1.58067			−0.790334	+	0.612676i	$0.790093\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.97377	+	2.97377i	0.295901	+	0.295901i	0.839406	−	0.543505i	$-0.182903\pi$
$101$	2.97377	+	2.97377i	0.295901	+	0.295901i	−0.543505	+	0.839406i	$0.682903\pi$
$102$		0			0
$103$	−2.84333			−0.280161			−0.140081	−	0.990140i	$-0.544736\pi$
$103$	−2.84333			−0.280161			−0.140081	+	0.990140i	$0.544736\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.00527	−	1.00527i	0.0971829	−	0.0971829i	−0.656844	−	0.754027i	$-0.728109\pi$
$107$	1.00527	−	1.00527i	0.0971829	−	0.0971829i	0.754027	+	0.656844i	$0.228109\pi$
$108$		0			0
$109$	1.00000	+	1.00000i	0.0957826	+	0.0957826i	0.753374	−	0.657592i	$-0.228425\pi$
$109$	1.00000	+	1.00000i	0.0957826	+	0.0957826i	−0.657592	+	0.753374i	$0.728425\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.4914i		−	1.17509i	−0.809190	−	0.587547i	$-0.800094\pi$
$113$		−	12.4914i		−	1.17509i	0.809190	−	0.587547i	$-0.199906\pi$
$114$		0			0
$115$	−4.20555	+	4.20555i	−0.392170	+	0.392170i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			6.39852i			0.586552i
$120$		0			0
$121$		−	20.6655i		−	1.87868i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.48528	+	8.48528i	−0.758947	+	0.758947i
$126$		0			0
$127$		−	3.45998i		−	0.307023i	−0.988147	−	0.153512i	$-0.950942\pi$
$127$		−	3.45998i		−	0.307023i	0.988147	−	0.153512i	$-0.0490583\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.38799	+	4.38799i	0.383380	+	0.383380i	0.872318	−	0.488938i	$-0.162616\pi$
$131$	4.38799	+	4.38799i	0.383380	+	0.383380i	−0.488938	+	0.872318i	$0.662616\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	19.6316			1.67724			0.838620	−	0.544716i	$-0.183363\pi$
$137$	19.6316			1.67724			0.838620	+	0.544716i	$0.183363\pi$
$138$		0			0
$139$	−10.4111	−	10.4111i	−0.883058	−	0.883058i	0.110786	−	0.993844i	$-0.464663\pi$
$139$	−10.4111	−	10.4111i	−0.883058	−	0.883058i	−0.993844	+	0.110786i	$0.964663\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.77597			−0.733884
$144$		0			0
$145$	−15.6655			−1.30095
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−16.6158	−	16.6158i	−1.36122	−	1.36122i	−0.872366	−	0.488853i	$-0.837415\pi$
$149$	−16.6158	−	16.6158i	−1.36122	−	1.36122i	−0.488853	−	0.872366i	$-0.662585\pi$
$150$		0			0
$151$	−14.3033			−1.16399			−0.581993	−	0.813194i	$-0.697727\pi$
$151$	−14.3033			−1.16399			−0.581993	+	0.813194i	$0.697727\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.11912	−	3.11912i	0.250534	−	0.250534i
$156$		0			0
$157$	−1.94610	−	1.94610i	−0.155316	−	0.155316i	0.625172	−	0.780487i	$-0.285029\pi$
$157$	−1.94610	−	1.94610i	−0.155316	−	0.155316i	−0.780487	+	0.625172i	$0.785029\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.97377i		−	0.234366i
$162$		0			0
$163$	12.2005	−	12.2005i	0.955619	−	0.955619i	−0.0434371	−	0.999056i	$-0.513831\pi$
$163$	12.2005	−	12.2005i	0.955619	−	0.955619i	0.999056	+	0.0434371i	$0.0138308\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	9.06666i		−	0.701600i	−0.936450	−	0.350800i	$-0.885910\pi$
$167$		−	9.06666i		−	0.701600i	0.936450	−	0.350800i	$-0.114090\pi$
$168$		0			0
$169$		−	10.5678i		−	0.812906i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	15.0291	−	15.0291i	1.14265	−	1.14265i	0.154680	−	0.987965i	$-0.450565\pi$
$173$	15.0291	−	15.0291i	1.14265	−	1.14265i	0.987965	−	0.154680i	$-0.0494348\pi$
$174$		0			0
$175$		−	1.00000i		−	0.0755929i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.478067	+	0.478067i	0.0357324	+	0.0357324i	0.724747	−	0.689015i	$-0.241957\pi$
$179$	0.478067	+	0.478067i	0.0357324	+	0.0357324i	−0.689015	+	0.724747i	$0.741957\pi$
$180$		0			0
$181$	8.35720	−	8.35720i	0.621186	−	0.621186i	−0.324649	−	0.945835i	$-0.605246\pi$
$181$	8.35720	−	8.35720i	0.621186	−	0.621186i	0.945835	+	0.324649i	$0.105246\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.82843			0.207950
$186$		0			0
$187$	25.4600	+	25.4600i	1.86182	+	1.86182i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−24.2561			−1.75511			−0.877555	−	0.479476i	$-0.840827\pi$
$191$	−24.2561			−1.75511			−0.877555	+	0.479476i	$0.840827\pi$
$192$		0			0
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.22684	+	1.22684i	0.0874086	+	0.0874086i	0.749459	−	0.662051i	$-0.230314\pi$
$197$	1.22684	+	1.22684i	0.0874086	+	0.0874086i	−0.662051	+	0.749459i	$0.730314\pi$
$198$		0			0
$199$	21.4600			1.52126			0.760629	−	0.649187i	$-0.224891\pi$
$199$	21.4600			1.52126			0.760629	+	0.649187i	$0.224891\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.53860	−	5.53860i	0.388734	−	0.388734i
$204$		0			0
$205$	2.95112	+	2.95112i	0.206115	+	0.206115i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−8.47556	+	8.47556i	−0.583482	+	0.583482i	−0.935858	−	0.352377i	$-0.885374\pi$
$211$	−8.47556	+	8.47556i	−0.583482	+	0.583482i	0.352377	+	0.935858i	$0.385374\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	13.2402i		−	0.902973i
$216$		0			0
$217$			2.20555i			0.149722i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.05613	−	7.05613i	0.474647	−	0.474647i
$222$		0			0
$223$		−	19.5577i		−	1.30968i	−0.755767	−	0.654841i	$-0.772736\pi$
$223$		−	19.5577i		−	1.30968i	0.755767	−	0.654841i	$-0.227264\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.83896	−	4.83896i	−0.321173	−	0.321173i	0.528044	−	0.849217i	$-0.322926\pi$
$227$	−4.83896	−	4.83896i	−0.321173	−	0.321173i	−0.849217	+	0.528044i	$0.822926\pi$
$228$		0			0
$229$	20.3572	−	20.3572i	1.34524	−	1.34524i	0.454490	−	0.890752i	$-0.349822\pi$
$229$	20.3572	−	20.3572i	1.34524	−	1.34524i	0.890752	−	0.454490i	$-0.150178\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	26.3429			1.72578			0.862889	−	0.505394i	$-0.168653\pi$
$233$	26.3429			1.72578			0.862889	+	0.505394i	$0.168653\pi$
$234$		0			0
$235$	−12.4111	−	12.4111i	−0.809611	−	0.809611i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.01764			−0.130510			−0.0652551	−	0.997869i	$-0.520786\pi$
$239$	−2.01764			−0.130510			−0.0652551	+	0.997869i	$0.520786\pi$
$240$		0			0
$241$	−5.25443			−0.338467			−0.169234	−	0.985576i	$-0.554129\pi$
$241$	−5.25443			−0.338467			−0.169234	+	0.985576i	$0.554129\pi$
$242$		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
$251$	17.9267	−	17.9267i	1.13152	−	1.13152i	0.141599	−	0.989924i	$-0.454776\pi$
$251$	17.9267	−	17.9267i	1.13152	−	1.13152i	0.989924	−	0.141599i	$-0.0452243\pi$
$252$		0			0
$253$	−11.8328	−	11.8328i	−0.743919	−	0.743919i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	7.88186i		−	0.491657i	−0.969313	−	0.245828i	$-0.920940\pi$
$257$		−	7.88186i		−	0.491657i	0.969313	−	0.245828i	$-0.0790600\pi$
$258$		0			0
$259$	−1.00000	+	1.00000i	−0.0621370	+	0.0621370i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	15.2436i		−	0.939962i	−0.882677	−	0.469981i	$-0.844261\pi$
$263$		−	15.2436i		−	0.939962i	0.882677	−	0.469981i	$-0.155739\pi$
$264$		0			0
$265$			10.8433i			0.666100i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.672546	−	0.672546i	0.0410059	−	0.0410059i	−0.686307	−	0.727312i	$-0.740769\pi$
$269$	0.672546	−	0.672546i	0.0410059	−	0.0410059i	0.727312	+	0.686307i	$0.240769\pi$
$270$		0			0
$271$			5.15667i			0.313246i	0.987658	+	0.156623i	$0.0500607\pi$
$271$			5.15667i			0.313246i	−0.987658	+	0.156623i	$0.949939\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.97904	−	3.97904i	−0.239945	−	0.239945i
$276$		0			0
$277$	−13.7839	+	13.7839i	−0.828194	+	0.828194i	−0.987267	−	0.159073i	$-0.949149\pi$
$277$	−13.7839	+	13.7839i	−0.828194	+	0.828194i	0.159073	+	0.987267i	$0.449149\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.748771			−0.0446679			−0.0223340	−	0.999751i	$-0.507110\pi$
$281$	−0.748771			−0.0446679			−0.0223340	+	0.999751i	$0.507110\pi$
$282$		0			0
$283$	6.30330	+	6.30330i	0.374692	+	0.374692i	0.869183	−	0.494491i	$-0.164645\pi$
$283$	6.30330	+	6.30330i	0.374692	+	0.374692i	−0.494491	+	0.869183i	$0.664645\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.08676			−0.123178
$288$		0			0
$289$	−23.9411			−1.40830
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.31887	−	4.31887i	−0.252311	−	0.252311i	0.569607	−	0.821917i	$-0.307096\pi$
$293$	−4.31887	−	4.31887i	−0.252311	−	0.252311i	−0.821917	+	0.569607i	$0.807096\pi$
$294$		0			0
$295$	−26.9200			−1.56734
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.27940	+	3.27940i	−0.189653	+	0.189653i
$300$		0			0
$301$	4.68111	+	4.68111i	0.269815	+	0.269815i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			14.4328i			0.826421i
$306$		0			0
$307$	−18.5089	+	18.5089i	−1.05636	+	1.05636i	−0.0580418	+	0.998314i	$0.518486\pi$
$307$	−18.5089	+	18.5089i	−1.05636	+	1.05636i	−0.998314	+	0.0580418i	$0.981514\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			27.8113i			1.57703i	0.615015	+	0.788516i	$0.289150\pi$
$311$			27.8113i			1.57703i	−0.615015	+	0.788516i	$0.710850\pi$
$312$		0			0
$313$			2.30330i			0.130190i	0.997879	+	0.0650952i	$0.0207351\pi$
$313$			2.30330i			0.130190i	−0.997879	+	0.0650952i	$0.979265\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.82316	−	1.82316i	0.102399	−	0.102399i	−0.654051	−	0.756450i	$-0.726932\pi$
$317$	1.82316	−	1.82316i	0.102399	−	0.102399i	0.756450	+	0.654051i	$0.226932\pi$
$318$		0			0
$319$		−	44.0766i		−	2.46782i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.10278	+	1.10278i	−0.0611710	+	0.0611710i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.77597			0.483835
$330$		0			0
$331$	9.41664	+	9.41664i	0.517585	+	0.517585i	0.916840	−	0.399255i	$-0.130731\pi$
$331$	9.41664	+	9.41664i	0.517585	+	0.517585i	−0.399255	+	0.916840i	$0.630731\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	17.2613			0.943083
$336$		0			0
$337$	−32.9200			−1.79326			−0.896632	−	0.442776i	$-0.853994\pi$
$337$	−32.9200			−1.79326			−0.896632	+	0.442776i	$0.853994\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.77597	+	8.77597i	0.475246	+	0.475246i
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.60772	−	5.60772i	0.301038	−	0.301038i	−0.540382	−	0.841420i	$-0.681720\pi$
$347$	5.60772	−	5.60772i	0.301038	−	0.301038i	0.841420	+	0.540382i	$0.181720\pi$
$348$		0			0
$349$	−2.99498	−	2.99498i	−0.160317	−	0.160317i	0.622390	−	0.782707i	$-0.286162\pi$
$349$	−2.99498	−	2.99498i	−0.160317	−	0.160317i	−0.782707	+	0.622390i	$0.786162\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			13.8294i			0.736065i	0.929813	+	0.368032i	$0.119968\pi$
$353$			13.8294i			0.736065i	−0.929813	+	0.368032i	$0.880032\pi$
$354$		0			0
$355$	−9.36222	+	9.36222i	−0.496895	+	0.496895i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			30.4943i			1.60943i	0.593662	+	0.804715i	$0.297682\pi$
$359$			30.4943i			1.60943i	−0.593662	+	0.804715i	$0.702318\pi$
$360$		0			0
$361$			19.0000i			1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.96862	−	9.96862i	0.521781	−	0.521781i
$366$		0			0
$367$		−	14.9511i		−	0.780442i	−0.920721	−	0.390221i	$-0.872399\pi$
$367$		−	14.9511i		−	0.780442i	0.920721	−	0.390221i	$-0.127601\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.83369	−	3.83369i	−0.199036	−	0.199036i
$372$		0			0
$373$	−4.52946	+	4.52946i	−0.234527	+	0.234527i	−0.814579	−	0.580052i	$-0.803032\pi$
$373$	−4.52946	+	4.52946i	−0.234527	+	0.234527i	0.580052	+	0.814579i	$0.303032\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.2157			−0.629138
$378$		0			0
$379$	−0.886662	−	0.886662i	−0.0455448	−	0.0455448i	0.683968	−	0.729512i	$-0.260253\pi$
$379$	−0.886662	−	0.886662i	−0.0455448	−	0.0455448i	−0.729512	+	0.683968i	$0.760253\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.2069			0.828132			0.414066	−	0.910247i	$-0.364108\pi$
$383$	16.2069			0.828132			0.414066	+	0.910247i	$0.364108\pi$
$384$		0			0
$385$	−11.2544			−0.573579
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−25.3227	−	25.3227i	−1.28391	−	1.28391i	−0.938424	−	0.345485i	$-0.887714\pi$
$389$	−25.3227	−	25.3227i	−1.28391	−	1.28391i	−0.345485	−	0.938424i	$-0.612286\pi$
$390$		0			0
$391$	19.0278			0.962275
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.23824	+	6.23824i	−0.313880	+	0.313880i
$396$		0			0
$397$	11.4061	+	11.4061i	0.572455	+	0.572455i	0.932814	−	0.360359i	$-0.117346\pi$
$397$	11.4061	+	11.4061i	0.572455	+	0.572455i	−0.360359	+	0.932814i	$0.617346\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3.13406i			0.156507i	0.996933	+	0.0782537i	$0.0249344\pi$
$401$			3.13406i			0.156507i	−0.996933	+	0.0782537i	$0.975066\pi$
$402$		0			0
$403$	2.43223	−	2.43223i	0.121158	−	0.121158i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			7.95808i			0.394467i
$408$		0			0
$409$			27.7633i			1.37281i	0.727221	+	0.686403i	$0.240811\pi$
$409$			27.7633i			1.37281i	−0.727221	+	0.686403i	$0.759189\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.51764	−	9.51764i	0.468332	−	0.468332i
$414$		0			0
$415$		−	28.4111i		−	1.39465i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−19.4863	−	19.4863i	−0.951966	−	0.951966i	0.0469322	−	0.998898i	$-0.485056\pi$
$419$	−19.4863	−	19.4863i	−0.951966	−	0.951966i	−0.998898	+	0.0469322i	$0.985056\pi$
$420$		0			0
$421$	9.47054	−	9.47054i	0.461566	−	0.461566i	−0.437603	−	0.899168i	$-0.644172\pi$
$421$	9.47054	−	9.47054i	0.461566	−	0.461566i	0.899168	+	0.437603i	$0.144172\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.39852			0.310374
$426$		0			0
$427$	−5.10278	−	5.10278i	−0.246941	−	0.246941i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.91078			0.332881			0.166440	−	0.986052i	$-0.446773\pi$
$431$	6.91078			0.332881			0.166440	+	0.986052i	$0.446773\pi$
$432$		0			0
$433$	26.3033			1.26406			0.632028	−	0.774946i	$-0.282223\pi$
$433$	26.3033			1.26406			0.632028	+	0.774946i	$0.282223\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−24.1955			−1.15479			−0.577394	−	0.816466i	$-0.695930\pi$
$439$	−24.1955			−1.15479			−0.577394	+	0.816466i	$0.695930\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.7131	+	20.7131i	−0.984109	+	0.984109i	−0.999876	−	0.0157669i	$-0.994981\pi$
$443$	−20.7131	+	20.7131i	−0.984109	+	0.984109i	0.0157669	+	0.999876i	$0.494981\pi$
$444$		0			0
$445$	19.3622	+	19.3622i	0.917857	+	0.917857i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.41421i		−	0.0667409i	−0.999443	−	0.0333704i	$-0.989376\pi$
$449$		−	1.41421i		−	0.0667409i	0.999443	−	0.0333704i	$-0.0106241\pi$
$450$		0			0
$451$	−8.30330	+	8.30330i	−0.390987	+	0.390987i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			3.11912i			0.146227i
$456$		0			0
$457$			19.6655i			0.919915i	0.887941	+	0.459957i	$0.152135\pi$
$457$			19.6655i			0.919915i	−0.887941	+	0.459957i	$0.847865\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.8365	−	13.8365i	0.644430	−	0.644430i	−0.307211	−	0.951641i	$-0.599396\pi$
$461$	13.8365	−	13.8365i	0.644430	−	0.644430i	0.951641	+	0.307211i	$0.0993958\pi$
$462$		0			0
$463$		−	16.8222i		−	0.781794i	−0.920434	−	0.390897i	$-0.872165\pi$
$463$		−	16.8222i		−	0.781794i	0.920434	−	0.390897i	$-0.127835\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.83896	+	4.83896i	0.223920	+	0.223920i	0.810147	−	0.586227i	$-0.199387\pi$
$467$	4.83896	+	4.83896i	0.223920	+	0.223920i	−0.586227	+	0.810147i	$0.699387\pi$
$468$		0			0
$469$	−6.10278	+	6.10278i	−0.281800	+	0.281800i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	37.2527			1.71288
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.6167			−0.896308			−0.448154	−	0.893956i	$-0.647918\pi$
$479$	−19.6167			−0.896308			−0.448154	+	0.893956i	$0.647918\pi$
$480$		0			0
$481$	2.20555			0.100564
$482$		0			0
$483$		0			0
$484$		0			0
$485$	22.0162	+	22.0162i	0.999702	+	0.999702i
$486$		0			0
$487$	19.7944			0.896972			0.448486	−	0.893790i	$-0.351963\pi$
$487$	19.7944			0.896972			0.448486	+	0.893790i	$0.351963\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	15.8741	−	15.8741i	0.716390	−	0.716390i	−0.251474	−	0.967864i	$-0.580915\pi$
$491$	15.8741	−	15.8741i	0.716390	−	0.716390i	0.967864	+	0.251474i	$0.0809154\pi$
$492$		0			0
$493$	35.4389	+	35.4389i	1.59609	+	1.59609i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	6.62009i		−	0.296952i
$498$		0			0
$499$	3.72999	−	3.72999i	0.166977	−	0.166977i	−0.618672	−	0.785649i	$-0.712329\pi$
$499$	3.72999	−	3.72999i	0.166977	−	0.166977i	0.785649	+	0.618672i	$0.212329\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	31.6941i		−	1.41317i	−0.707629	−	0.706585i	$-0.750235\pi$
$503$		−	31.6941i		−	1.41317i	0.707629	−	0.706585i	$-0.249765\pi$
$504$		0			0
$505$		−	8.41110i		−	0.374289i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−29.3858	+	29.3858i	−1.30250	+	1.30250i	−0.375800	+	0.926701i	$0.622632\pi$
$509$	−29.3858	+	29.3858i	−1.30250	+	1.30250i	−0.926701	+	0.375800i	$0.877368\pi$
$510$		0			0
$511$			7.04888i			0.311824i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.02107	+	4.02107i	0.177190	+	0.177190i
$516$		0			0
$517$	34.9200	−	34.9200i	1.53578	−	1.53578i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8.46324			−0.370781			−0.185391	−	0.982665i	$-0.559355\pi$
$521$	−8.46324			−0.370781			−0.185391	+	0.982665i	$0.559355\pi$
$522$		0			0
$523$	23.0278	+	23.0278i	1.00693	+	1.00693i	0.999976	+	0.00695743i	$0.00221464\pi$
$523$	23.0278	+	23.0278i	1.00693	+	1.00693i	0.00695743	+	0.999976i	$0.497785\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.1123			−0.614740
$528$		0			0
$529$	14.1567			0.615508
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.30123	+	2.30123i	0.0996772	+	0.0996772i
$534$		0			0
$535$	−2.84333			−0.122928
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.97904	−	3.97904i	0.171389	−	0.171389i
$540$		0			0
$541$	27.5472	+	27.5472i	1.18435	+	1.18435i	0.978607	+	0.205738i	$0.0659595\pi$
$541$	27.5472	+	27.5472i	1.18435	+	1.18435i	0.205738	+	0.978607i	$0.434041\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	2.82843i		−	0.121157i
$546$		0			0
$547$	8.74055	−	8.74055i	0.373719	−	0.373719i	−0.495111	−	0.868830i	$-0.664873\pi$
$547$	8.74055	−	8.74055i	0.373719	−	0.373719i	0.868830	+	0.495111i	$0.164873\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	4.41110i		−	0.187579i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.26703	+	2.26703i	−0.0960572	+	0.0960572i	−0.753502	−	0.657445i	$-0.771637\pi$
$557$	−2.26703	+	2.26703i	−0.0960572	+	0.0960572i	0.657445	+	0.753502i	$0.271637\pi$
$558$		0			0
$559$		−	10.3244i		−	0.436677i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15.3269	+	15.3269i	0.645954	+	0.645954i	0.952013	−	0.306059i	$-0.0990104\pi$
$563$	15.3269	+	15.3269i	0.645954	+	0.645954i	−0.306059	+	0.952013i	$0.599010\pi$
$564$		0			0
$565$	−17.6655	+	17.6655i	−0.743194	+	0.743194i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−35.1730			−1.47453			−0.737265	−	0.675604i	$-0.763883\pi$
$569$	−35.1730			−1.47453			−0.737265	+	0.675604i	$0.763883\pi$
$570$		0			0
$571$	11.0439	+	11.0439i	0.462171	+	0.462171i	0.899366	−	0.437196i	$-0.144028\pi$
$571$	11.0439	+	11.0439i	0.462171	+	0.462171i	−0.437196	+	0.899366i	$0.644028\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.97377			−0.124015
$576$		0			0
$577$	4.91995			0.204820			0.102410	−	0.994742i	$-0.467345\pi$
$577$	4.91995			0.204820			0.102410	+	0.994742i	$0.467345\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10.0448	+	10.0448i	0.416730	+	0.416730i
$582$		0			0
$583$	−30.5089			−1.26355
$584$		0			0
$585$		0			0
$586$		0			0
$587$	17.9267	−	17.9267i	0.739914	−	0.739914i	−0.232647	−	0.972561i	$-0.574739\pi$
$587$	17.9267	−	17.9267i	0.739914	−	0.739914i	0.972561	+	0.232647i	$0.0747388\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			43.2764i			1.77715i	0.458731	+	0.888575i	$0.348304\pi$
$593$			43.2764i			1.77715i	−0.458731	+	0.888575i	$0.651696\pi$
$594$		0			0
$595$	9.04888	−	9.04888i	0.370968	−	0.370968i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	21.1912i		−	0.865847i	−0.901431	−	0.432924i	$-0.857482\pi$
$599$		−	21.1912i		−	0.865847i	0.901431	−	0.432924i	$-0.142518\pi$
$600$		0			0
$601$			42.6066i			1.73796i	0.494848	+	0.868980i	$0.335224\pi$
$601$			42.6066i			1.73796i	−0.494848	+	0.868980i	$0.664776\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−29.2255	+	29.2255i	−1.18818	+	1.18818i
$606$		0			0
$607$			31.1466i			1.26420i	0.774886	+	0.632101i	$0.217807\pi$
$607$			31.1466i			1.26420i	−0.774886	+	0.632101i	$0.782193\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.67792	−	9.67792i	−0.391527	−	0.391527i
$612$		0			0
$613$	1.57834	−	1.57834i	0.0637484	−	0.0637484i	−0.674514	−	0.738262i	$-0.735647\pi$
$613$	1.57834	−	1.57834i	0.0637484	−	0.0637484i	0.738262	+	0.674514i	$0.235647\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−20.4495			−0.823266			−0.411633	−	0.911350i	$-0.635041\pi$
$617$	−20.4495			−0.823266			−0.411633	+	0.911350i	$0.635041\pi$
$618$		0			0
$619$	−5.79445	−	5.79445i	−0.232899	−	0.232899i	0.581003	−	0.813901i	$-0.302660\pi$
$619$	−5.79445	−	5.79445i	−0.232899	−	0.232899i	−0.813901	+	0.581003i	$0.802660\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−13.6912			−0.548525
$624$		0			0
$625$	19.0000			0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.39852	−	6.39852i	−0.255126	−	0.255126i
$630$		0			0
$631$	7.87108			0.313343			0.156671	−	0.987651i	$-0.449924\pi$
$631$	7.87108			0.313343			0.156671	+	0.987651i	$0.449924\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.89315	+	4.89315i	−0.194179	+	0.194179i
$636$		0			0
$637$	−1.10278	−	1.10278i	−0.0436935	−	0.0436935i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	5.35122i		−	0.211361i	−0.994400	−	0.105680i	$-0.966298\pi$
$641$		−	5.35122i		−	0.211361i	0.994400	−	0.105680i	$-0.0337020\pi$
$642$		0			0
$643$	4.61665	−	4.61665i	0.182063	−	0.182063i	−0.610191	−	0.792254i	$-0.708907\pi$
$643$	4.61665	−	4.61665i	0.182063	−	0.182063i	0.792254	+	0.610191i	$0.208907\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			4.31176i			0.169513i	0.996402	+	0.0847564i	$0.0270112\pi$
$647$			4.31176i			0.169513i	−0.996402	+	0.0847564i	$0.972989\pi$
$648$		0			0
$649$		−	75.7422i		−	2.97314i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.42143	−	9.42143i	0.368689	−	0.368689i	−0.498310	−	0.866999i	$-0.666046\pi$
$653$	9.42143	−	9.42143i	0.368689	−	0.368689i	0.866999	+	0.498310i	$0.166046\pi$
$654$		0			0
$655$		−	12.4111i		−	0.484942i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.40649	+	9.40649i	0.366425	+	0.366425i	0.866172	−	0.499747i	$-0.166574\pi$
$659$	9.40649	+	9.40649i	0.366425	+	0.366425i	−0.499747	+	0.866172i	$0.666574\pi$
$660$		0			0
$661$	22.9739	−	22.9739i	0.893579	−	0.893579i	−0.101279	−	0.994858i	$-0.532293\pi$
$661$	22.9739	−	22.9739i	0.893579	−	0.893579i	0.994858	+	0.101279i	$0.0322934\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−16.4705	−	16.4705i	−0.637742	−	0.637742i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−40.6083			−1.56767
$672$		0			0
$673$	−38.9200			−1.50025			−0.750127	−	0.661294i	$-0.770008\pi$
$673$	−38.9200			−1.50025			−0.750127	+	0.661294i	$0.770008\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−23.9434	−	23.9434i	−0.920218	−	0.920218i	0.0768263	−	0.997044i	$-0.475521\pi$
$677$	−23.9434	−	23.9434i	−0.920218	−	0.920218i	−0.997044	+	0.0768263i	$0.975521\pi$
$678$		0			0
$679$	−15.5678			−0.597436
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.0352	+	11.0352i	−0.422249	+	0.422249i	−0.885977	−	0.463728i	$-0.846511\pi$
$683$	−11.0352	+	11.0352i	−0.422249	+	0.422249i	0.463728	+	0.885977i	$0.346511\pi$
$684$		0			0
$685$	−27.7633	−	27.7633i	−1.06078	−	1.06078i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			8.45541i			0.322125i
$690$		0			0
$691$	−3.15667	+	3.15667i	−0.120086	+	0.120086i	−0.764596	−	0.644510i	$-0.777061\pi$
$691$	−3.15667	+	3.15667i	−0.120086	+	0.120086i	0.644510	+	0.764596i	$0.277061\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			29.4470i			1.11699i
$696$		0			0
$697$		−	13.3522i		−	0.505750i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12.7479	−	12.7479i	0.481482	−	0.481482i	−0.424123	−	0.905605i	$-0.639418\pi$
$701$	12.7479	−	12.7479i	0.481482	−	0.481482i	0.905605	+	0.424123i	$0.139418\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.97377	+	2.97377i	0.111840	+	0.111840i
$708$		0			0
$709$	−0.254426	+	0.254426i	−0.00955517	+	0.00955517i	−0.711868	−	0.702313i	$-0.752151\pi$
$709$	−0.254426	+	0.254426i	−0.00955517	+	0.00955517i	0.702313	+	0.711868i	$0.252151\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.55881			0.245629
$714$		0			0
$715$	12.4111	+	12.4111i	0.464149	+	0.464149i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−31.0686			−1.15866			−0.579332	−	0.815092i	$-0.696686\pi$
$719$	−31.0686			−1.15866			−0.579332	+	0.815092i	$0.696686\pi$
$720$		0			0
$721$	−2.84333			−0.105891
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.53860	−	5.53860i	−0.205698	−	0.205698i
$726$		0			0
$727$	47.2544			1.75257			0.876285	−	0.481793i	$-0.160014\pi$
$727$	47.2544			1.75257			0.876285	+	0.481793i	$0.160014\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−29.9522	+	29.9522i	−1.10782	+	1.10782i
$732$		0			0
$733$	−11.6217	−	11.6217i	−0.429256	−	0.429256i	0.459119	−	0.888375i	$-0.348165\pi$
$733$	−11.6217	−	11.6217i	−0.429256	−	0.429256i	−0.888375	+	0.459119i	$0.848165\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			48.5664i			1.78897i
$738$		0			0
$739$	13.5628	−	13.5628i	0.498914	−	0.498914i	−0.412186	−	0.911100i	$-0.635235\pi$
$739$	13.5628	−	13.5628i	0.498914	−	0.498914i	0.911100	+	0.412186i	$0.135235\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			32.5889i			1.19557i	0.801656	+	0.597786i	$0.203953\pi$
$743$			32.5889i			1.19557i	−0.801656	+	0.597786i	$0.796047\pi$
$744$		0			0
$745$			46.9966i			1.72182i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.00527	−	1.00527i	0.0367317	−	0.0367317i
$750$		0			0
$751$			30.2933i			1.10542i	0.833375	+	0.552708i	$0.186406\pi$
$751$			30.2933i			1.10542i	−0.833375	+	0.552708i	$0.813594\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	20.2279	+	20.2279i	0.736170	+	0.736170i
$756$		0			0
$757$	−34.2544	+	34.2544i	−1.24500	+	1.24500i	−0.287097	+	0.957902i	$0.592690\pi$
$757$	−34.2544	+	34.2544i	−1.24500	+	1.24500i	−0.957902	+	0.287097i	$0.907310\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.96078			−0.361078			−0.180539	−	0.983568i	$-0.557784\pi$
$761$	−9.96078			−0.361078			−0.180539	+	0.983568i	$0.557784\pi$
$762$		0			0
$763$	1.00000	+	1.00000i	0.0362024	+	0.0362024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−20.9916			−0.757964
$768$		0			0
$769$	−12.4211			−0.447918			−0.223959	−	0.974599i	$-0.571898\pi$
$769$	−12.4211			−0.447918			−0.223959	+	0.974599i	$0.571898\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.85660	+	6.85660i	0.246615	+	0.246615i	0.819580	−	0.572965i	$-0.194207\pi$
$773$	6.85660	+	6.85660i	0.246615	+	0.246615i	−0.572965	+	0.819580i	$0.694207\pi$
$774$		0			0
$775$	2.20555			0.0792257
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−26.3416	−	26.3416i	−0.942577	−	0.942577i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			5.50440i			0.196461i
$786$		0			0
$787$	30.8222	−	30.8222i	1.09869	−	1.09869i	0.104129	−	0.994564i	$-0.466795\pi$
$787$	30.8222	−	30.8222i	1.09869	−	1.09869i	0.994564	−	0.104129i	$-0.0332055\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	12.4914i		−	0.444144i
$792$		0			0
$793$			11.2544i			0.399656i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−24.0958	+	24.0958i	−0.853518	+	0.853518i	−0.990565	−	0.137047i	$-0.956239\pi$
$797$	−24.0958	+	24.0958i	−0.853518	+	0.853518i	0.137047	+	0.990565i	$0.456239\pi$
$798$		0			0
$799$			56.1533i			1.98656i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	28.0478	+	28.0478i	0.989784	+	0.989784i
$804$		0			0
$805$	−4.20555	+	4.20555i	−0.148226	+	0.148226i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	31.4725			1.10651			0.553257	−	0.833010i	$-0.313385\pi$
$809$	31.4725			1.10651			0.553257	+	0.833010i	$0.313385\pi$
$810$		0			0
$811$	−13.5678	−	13.5678i	−0.476429	−	0.476429i	0.427559	−	0.903988i	$-0.359374\pi$
$811$	−13.5678	−	13.5678i	−0.476429	−	0.476429i	−0.903988	+	0.427559i	$0.859374\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−34.5083			−1.20877
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.7183	+	13.7183i	0.478770	+	0.478770i	0.904738	−	0.425968i	$-0.140066\pi$
$821$	13.7183	+	13.7183i	0.478770	+	0.478770i	−0.425968	+	0.904738i	$0.640066\pi$
$822$		0			0
$823$	−32.9511			−1.14860			−0.574302	−	0.818644i	$-0.694726\pi$
$823$	−32.9511			−1.14860			−0.574302	+	0.818644i	$0.694726\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	31.9356	−	31.9356i	1.11051	−	1.11051i	0.117430	−	0.993081i	$-0.462534\pi$
$827$	31.9356	−	31.9356i	1.11051	−	1.11051i	0.993081	−	0.117430i	$-0.0374655\pi$
$828$		0			0
$829$	26.6605	+	26.6605i	0.925958	+	0.925958i	0.997442	−	0.0714842i	$-0.0227735\pi$
$829$	26.6605	+	26.6605i	0.925958	+	0.925958i	−0.0714842	+	0.997442i	$0.522774\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			6.39852i			0.221696i
$834$		0			0
$835$	−12.8222	+	12.8222i	−0.443731	+	0.443731i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			17.2613i			0.595925i	0.954578	+	0.297962i	$0.0963070\pi$
$839$			17.2613i			0.595925i	−0.954578	+	0.297962i	$0.903693\pi$
$840$		0			0
$841$		−	32.3522i		−	1.11559i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−14.9451	+	14.9451i	−0.514127	+	0.514127i
$846$		0			0
$847$		−	20.6655i		−	0.710076i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.97377	+	2.97377i	0.101940	+	0.101940i
$852$		0			0
$853$	−6.88718	+	6.88718i	−0.235813	+	0.235813i	−0.815114	−	0.579301i	$-0.803326\pi$
$853$	−6.88718	+	6.88718i	−0.235813	+	0.235813i	0.579301	+	0.815114i	$0.303326\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−43.1240			−1.47309			−0.736544	−	0.676390i	$-0.763543\pi$
$857$	−43.1240			−1.47309			−0.736544	+	0.676390i	$0.763543\pi$
$858$		0			0
$859$	−3.48110	−	3.48110i	−0.118774	−	0.118774i	0.645222	−	0.763995i	$-0.276765\pi$
$859$	−3.48110	−	3.48110i	−0.118774	−	0.118774i	−0.763995	+	0.645222i	$0.776765\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	13.9968			0.476456			0.238228	−	0.971209i	$-0.423433\pi$
$863$	13.9968			0.476456			0.238228	+	0.971209i	$0.423433\pi$
$864$		0			0
$865$	−42.5089			−1.44534
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−17.5519	−	17.5519i	−0.595409	−	0.595409i
$870$		0			0
$871$	13.4600			0.456074
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.48528	+	8.48528i	−0.286855	+	0.286855i
$876$		0			0
$877$	0.0871891	+	0.0871891i	0.00294417	+	0.00294417i	0.708577	−	0.705633i	$-0.249337\pi$
$877$	0.0871891	+	0.0871891i	0.00294417	+	0.00294417i	−0.705633	+	0.708577i	$0.749337\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			0.0220418i			0.000742608i	1.00000		0.000371304i	$0.000118190\pi$
$881$			0.0220418i			0.000742608i	−1.00000		0.000371304i	$0.999882\pi$
$882$		0			0
$883$	11.3955	−	11.3955i	0.383490	−	0.383490i	−0.488868	−	0.872358i	$-0.662590\pi$
$883$	11.3955	−	11.3955i	0.383490	−	0.383490i	0.872358	+	0.488868i	$0.162590\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			56.3862i			1.89326i	0.322317	+	0.946632i	$0.395538\pi$
$887$			56.3862i			1.89326i	−0.322317	+	0.946632i	$0.604462\pi$
$888$		0			0
$889$		−	3.45998i		−	0.116044i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	1.35218i		−	0.0451983i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.2157	+	12.2157i	0.407415	+	0.407415i
$900$		0			0
$901$	24.5300	−	24.5300i	0.817212	−	0.817212i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−23.6377			−0.785745
$906$		0			0
$907$	11.5244	+	11.5244i	0.382663	+	0.382663i	0.872061	−	0.489398i	$-0.162783\pi$
$907$	11.5244	+	11.5244i	0.382663	+	0.382663i	−0.489398	+	0.872061i	$0.662783\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	42.0147			1.39201			0.696004	−	0.718038i	$-0.254959\pi$
$911$	42.0147			1.39201			0.696004	+	0.718038i	$0.254959\pi$
$912$		0			0
$913$	79.9377			2.64555
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.38799	+	4.38799i	0.144904	+	0.144904i
$918$		0			0
$919$	36.1955			1.19398			0.596990	−	0.802249i	$-0.296363\pi$
$919$	36.1955			1.19398			0.596990	+	0.802249i	$0.296363\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7.30047	+	7.30047i	−0.240298	+	0.240298i
$924$		0			0
$925$	1.00000	+	1.00000i	0.0328798	+	0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1.03236i			0.0338706i	0.999857	+	0.0169353i	$0.00539093\pi$
$929$			1.03236i			0.0338706i	−0.999857	+	0.0169353i	$0.994609\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	72.0117i		−	2.35503i
$936$		0			0
$937$		−	1.17780i		−	0.0384770i	−0.999815	−	0.0192385i	$-0.993876\pi$
$937$		−	1.17780i		−	0.0384770i	0.999815	−	0.0192385i	$-0.00612419\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−16.8574	+	16.8574i	−0.549534	+	0.549534i	−0.926306	−	0.376772i	$-0.877034\pi$
$941$	−16.8574	+	16.8574i	−0.549534	+	0.549534i	0.376772	+	0.926306i	$0.377034\pi$
$942$		0			0
$943$			6.20555i			0.202081i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−10.8898	−	10.8898i	−0.353872	−	0.353872i	0.507676	−	0.861548i	$-0.330505\pi$
$947$	−10.8898	−	10.8898i	−0.353872	−	0.353872i	−0.861548	+	0.507676i	$0.830505\pi$
$948$		0			0
$949$	7.77332	−	7.77332i	0.252333	−	0.252333i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−19.8139			−0.641836			−0.320918	−	0.947107i	$-0.603991\pi$
$953$	−19.8139			−0.641836			−0.320918	+	0.947107i	$0.603991\pi$
$954$		0			0
$955$	34.3033	+	34.3033i	1.11003	+	1.11003i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	19.6316			0.633937
$960$		0			0
$961$	26.1355			0.843082
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.82843	+	2.82843i	0.0910503	+	0.0910503i
$966$		0			0
$967$	10.3133			0.331655			0.165827	−	0.986155i	$-0.446971\pi$
$967$	10.3133			0.331655			0.165827	+	0.986155i	$0.446971\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	7.00195	−	7.00195i	0.224703	−	0.224703i	−0.585772	−	0.810476i	$-0.699209\pi$
$971$	7.00195	−	7.00195i	0.224703	−	0.224703i	0.810476	+	0.585772i	$0.199209\pi$
$972$		0			0
$973$	−10.4111	−	10.4111i	−0.333765	−	0.333765i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	26.8558i		−	0.859195i	−0.903021	−	0.429597i	$-0.858655\pi$
$977$		−	26.8558i		−	0.859195i	0.903021	−	0.429597i	$-0.141345\pi$
$978$		0			0
$979$	−54.4777	+	54.4777i	−1.74111	+	1.74111i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			16.3451i			0.521328i	0.965430	+	0.260664i	$0.0839414\pi$
$983$			16.3451i			0.521328i	−0.965430	+	0.260664i	$0.916059\pi$
$984$		0			0
$985$		−	3.47002i		−	0.110564i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	13.9206	−	13.9206i	0.442648	−	0.442648i
$990$		0			0
$991$			22.5189i			0.715336i	0.933849	+	0.357668i	$0.116428\pi$
$991$			22.5189i			0.715336i	−0.933849	+	0.357668i	$0.883572\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−30.3490	−	30.3490i	−0.962128	−	0.962128i
$996$		0			0
$997$	13.4372	−	13.4372i	0.425562	−	0.425562i	−0.461551	−	0.887113i	$-0.652707\pi$
$997$	13.4372	−	13.4372i	0.425562	−	0.425562i	0.887113	+	0.461551i	$0.152707\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.v.c.1583.3		12
3.2	odd	2	inner	4032.2.v.c.1583.4		12
4.3	odd	2		1008.2.v.c.323.4	yes	12
12.11	even	2		1008.2.v.c.323.3	✓	12
16.5	even	4		1008.2.v.c.827.3	yes	12
16.11	odd	4	inner	4032.2.v.c.3599.4		12
48.5	odd	4		1008.2.v.c.827.4	yes	12
48.11	even	4	inner	4032.2.v.c.3599.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.v.c.323.3	✓	12	12.11	even	2
1008.2.v.c.323.4	yes	12	4.3	odd	2
1008.2.v.c.827.3	yes	12	16.5	even	4
1008.2.v.c.827.4	yes	12	48.5	odd	4
4032.2.v.c.1583.3		12	1.1	even	1	trivial
4032.2.v.c.1583.4		12	3.2	odd	2	inner
4032.2.v.c.3599.3		12	48.11	even	4	inner
4032.2.v.c.3599.4		12	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 1.41421i) q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+(-1.41421 - 1.41421i) q^{5} +1.00000 q^{7} +(3.97904 - 3.97904i) q^{11} +(-1.10278 - 1.10278i) q^{13} +6.39852i q^{17} -2.97377i q^{23} -1.00000i q^{25} +(5.53860 - 5.53860i) q^{29} +2.20555i q^{31} +(-1.41421 - 1.41421i) q^{35} +(-1.00000 + 1.00000i) q^{37} -2.08676 q^{41} +(4.68111 + 4.68111i) q^{43} +8.77597 q^{47} +1.00000 q^{49} +(-3.83369 - 3.83369i) q^{53} -11.2544 q^{55} +(9.51764 - 9.51764i) q^{59} +(-5.10278 - 5.10278i) q^{61} +3.11912i q^{65} +(-6.10278 + 6.10278i) q^{67} -6.62009i q^{71} +7.04888i q^{73} +(3.97904 - 3.97904i) q^{77} -4.41110i q^{79} +(10.0448 + 10.0448i) q^{83} +(9.04888 - 9.04888i) q^{85} -13.6912 q^{89} +(-1.10278 - 1.10278i) q^{91} -15.5678 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7}+O(q^{10}) \) 12 * q + 12 * q^7	\( 12 q + 12 q^{7} + 16 q^{13} - 12 q^{37} + 20 q^{43} + 12 q^{49} - 32 q^{55} - 32 q^{61} - 44 q^{67} + 64 q^{85} + 16 q^{91} - 56 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 + 16 * q^13 - 12 * q^37 + 20 * q^43 + 12 * q^49 - 32 * q^55 - 32 * q^61 - 44 * q^67 + 64 * q^85 + 16 * q^91 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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