LMFDB - Embedded newform 75.4.e.b.68.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(32,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.32");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	75.e (of order $4$, degree $2$, 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:	$4.42514325043$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			68.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.68
Dual form			75.4.e.b.32.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.67423 + 3.67423i) q^{2} +(3.67423 - 3.67423i) q^{3} +19.0000i q^{4} +27.0000 q^{6} +(-40.4166 + 40.4166i) q^{8} -27.0000i q^{9} +O(q^{10})$	$q+(3.67423 + 3.67423i) q^{2} +(3.67423 - 3.67423i) q^{3} +19.0000i q^{4} +27.0000 q^{6} +(-40.4166 + 40.4166i) q^{8} -27.0000i q^{9} +(69.8105 + 69.8105i) q^{12} -145.000 q^{16} +(-14.6969 - 14.6969i) q^{17} +(99.2043 - 99.2043i) q^{18} -164.000i q^{19} +(-139.621 + 139.621i) q^{23} +297.000i q^{24} +(-99.2043 - 99.2043i) q^{27} +232.000 q^{31} +(-209.431 - 209.431i) q^{32} -108.000i q^{34} +513.000 q^{36} +(602.574 - 602.574i) q^{38} -1026.00 q^{46} +(242.499 + 242.499i) q^{47} +(-532.764 + 532.764i) q^{48} +343.000i q^{49} -108.000 q^{51} +(-323.333 + 323.333i) q^{53} -729.000i q^{54} +(-602.574 - 602.574i) q^{57} -358.000 q^{61} +(852.422 + 852.422i) q^{62} -379.000i q^{64} +(279.242 - 279.242i) q^{68} +1026.00i q^{69} +(1091.25 + 1091.25i) q^{72} +3116.00 q^{76} -304.000i q^{79} -729.000 q^{81} +(-580.529 + 580.529i) q^{83} +(-2652.80 - 2652.80i) q^{92} +(852.422 - 852.422i) q^{93} +1782.00i q^{94} -1539.00 q^{96} +(-1260.26 + 1260.26i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{6}+O(q^{10}) $ 4 * q + 108 * q^6	$ 4 q + 108 q^{6} - 580 q^{16} + 928 q^{31} + 2052 q^{36} - 4104 q^{46} - 432 q^{51} - 1432 q^{61} + 12464 q^{76} - 2916 q^{81} - 6156 q^{96}+O(q^{100}) $ 4 * q + 108 * q^6 - 580 * q^16 + 928 * q^31 + 2052 * q^36 - 4104 * q^46 - 432 * q^51 - 1432 * q^61 + 12464 * q^76 - 2916 * q^81 - 6156 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	3.67423	+	3.67423i	1.29904	+	1.29904i	0.929028	+	0.370011i	$0.120646\pi$
$2$	3.67423	+	3.67423i	1.29904	+	1.29904i	0.370011	+	0.929028i	$0.379354\pi$
$3$	3.67423	−	3.67423i	0.707107	−	0.707107i
$3$	3.67423	−	3.67423i	0.707107	−	0.707107i
$4$			19.0000i			2.37500i
$5$		0			0
$5$		0			0
$6$	27.0000			1.83712
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	−40.4166	+	40.4166i	−1.78618	+	1.78618i
$9$		−	27.0000i		−	1.00000i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	69.8105	+	69.8105i	1.67938	+	1.67938i
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$	−145.000			−2.26562
$17$	−14.6969	−	14.6969i	−0.209678	−	0.209678i	0.594452	−	0.804131i	$-0.297369\pi$
$17$	−14.6969	−	14.6969i	−0.209678	−	0.209678i	−0.804131	+	0.594452i	$0.797369\pi$
$18$	99.2043	−	99.2043i	1.29904	−	1.29904i
$19$		−	164.000i		−	1.98022i	−0.140293	−	0.990110i	$-0.544805\pi$
$19$		−	164.000i		−	1.98022i	0.140293	−	0.990110i	$-0.455195\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−139.621	+	139.621i	−1.26578	+	1.26578i	−0.317535	+	0.948247i	$0.602855\pi$
$23$	−139.621	+	139.621i	−1.26578	+	1.26578i	−0.948247	+	0.317535i	$0.897145\pi$
$24$			297.000i			2.52604i
$25$		0			0
$26$		0			0
$27$	−99.2043	−	99.2043i	−0.707107	−	0.707107i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	232.000			1.34414			0.672071	−	0.740486i	$-0.265405\pi$
$31$	232.000			1.34414			0.672071	+	0.740486i	$0.265405\pi$
$32$	−209.431	−	209.431i	−1.15696	−	1.15696i
$33$		0			0
$34$		−	108.000i		−	0.544760i
$35$		0			0
$36$	513.000			2.37500
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	602.574	−	602.574i	2.57238	−	2.57238i
$39$		0			0
$40$		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
$45$		0			0
$46$	−1026.00			−3.28860
$47$	242.499	+	242.499i	0.752600	+	0.752600i	0.974964	−	0.222364i	$-0.0713773\pi$
$47$	242.499	+	242.499i	0.752600	+	0.752600i	−0.222364	+	0.974964i	$0.571377\pi$
$48$	−532.764	+	532.764i	−1.60204	+	1.60204i
$49$			343.000i			1.00000i
$50$		0			0
$51$	−108.000			−0.296530
$52$		0			0
$53$	−323.333	+	323.333i	−0.837984	+	0.837984i	−0.988593	−	0.150609i	$-0.951876\pi$
$53$	−323.333	+	323.333i	−0.837984	+	0.837984i	0.150609	+	0.988593i	$0.451876\pi$
$54$		−	729.000i		−	1.83712i
$55$		0			0
$56$		0			0
$57$	−602.574	−	602.574i	−1.40023	−	1.40023i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−358.000			−0.751430			−0.375715	−	0.926735i	$-0.622603\pi$
$61$	−358.000			−0.751430			−0.375715	+	0.926735i	$0.622603\pi$
$62$	852.422	+	852.422i	1.74609	+	1.74609i
$63$		0			0
$64$		−	379.000i		−	0.740234i
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	279.242	−	279.242i	0.497986	−	0.497986i
$69$			1026.00i			1.79009i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	1091.25	+	1091.25i	1.78618	+	1.78618i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$		0			0
$76$	3116.00			4.70302
$77$		0			0
$78$		0			0
$79$		−	304.000i		−	0.432945i	−0.976289	−	0.216473i	$-0.930545\pi$
$79$		−	304.000i		−	0.432945i	0.976289	−	0.216473i	$-0.0694552\pi$
$80$		0			0
$81$	−729.000			−1.00000
$82$		0			0
$83$	−580.529	+	580.529i	−0.767727	+	0.767727i	−0.977706	−	0.209979i	$-0.932660\pi$
$83$	−580.529	+	580.529i	−0.767727	+	0.767727i	0.209979	+	0.977706i	$0.432660\pi$
$84$		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
$91$		0			0
$92$	−2652.80	−	2652.80i	−3.00623	−	3.00623i
$93$	852.422	−	852.422i	0.950453	−	0.950453i
$94$			1782.00i			1.95531i
$95$		0			0
$96$	−1539.00			−1.63618
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	−1260.26	+	1260.26i	−1.29904	+	1.29904i
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−396.817	−	396.817i	−0.385204	−	0.385204i
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$	−2376.00			−2.17715
$107$	1565.22	+	1565.22i	1.41417	+	1.41417i	0.712798	+	0.701370i	$0.247428\pi$
$107$	1565.22	+	1565.22i	1.41417	+	1.41417i	0.701370	+	0.712798i	$0.252572\pi$
$108$	1884.88	−	1884.88i	1.67938	−	1.67938i
$109$		−	1834.00i		−	1.61161i	−0.592182	−	0.805804i	$-0.701733\pi$
$109$		−	1834.00i		−	1.61161i	0.592182	−	0.805804i	$-0.298267\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1366.82	−	1366.82i	1.13787	−	1.13787i	0.149037	−	0.988832i	$-0.452383\pi$
$113$	1366.82	−	1366.82i	1.13787	−	1.13787i	0.988832	−	0.149037i	$-0.0476174\pi$
$114$		−	4428.00i		−	3.63790i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1331.00			1.00000
$122$	−1315.38	−	1315.38i	−0.976136	−	0.976136i
$123$		0			0
$124$			4408.00i			3.19234i
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−282.916	+	282.916i	−0.195363	+	0.195363i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1188.00			0.749045
$137$	−1778.33	−	1778.33i	−1.10900	−	1.10900i	−0.993282	−	0.115717i	$-0.963083\pi$
$137$	−1778.33	−	1778.33i	−1.10900	−	1.10900i	−0.115717	−	0.993282i	$-0.536917\pi$
$138$	−3769.76	+	3769.76i	−2.32539	+	2.32539i
$139$		−	1604.00i		−	0.978773i	−0.872067	−	0.489387i	$-0.837221\pi$
$139$		−	1604.00i		−	0.978773i	0.872067	−	0.489387i	$-0.162779\pi$
$140$		0			0
$141$	1782.00			1.06434
$142$		0			0
$143$		0			0
$144$			3915.00i			2.26562i
$145$		0			0
$146$		0			0
$147$	1260.26	+	1260.26i	0.707107	+	0.707107i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	3112.00			1.67716			0.838579	−	0.544779i	$-0.183387\pi$
$151$	3112.00			1.67716			0.838579	+	0.544779i	$0.183387\pi$
$152$	6628.32	+	6628.32i	3.53702	+	3.53702i
$153$	−396.817	+	396.817i	−0.209678	+	0.209678i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$	1116.97	−	1116.97i	0.562412	−	0.562412i
$159$			2376.00i			1.18509i
$160$		0			0
$161$		0			0
$162$	−2678.52	−	2678.52i	−1.29904	−	1.29904i
$163$		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
$166$	−4266.00			−1.99461
$167$	1491.74	+	1491.74i	0.691223	+	0.691223i	0.962501	−	0.271278i	$-0.0874463\pi$
$167$	1491.74	+	1491.74i	0.691223	+	0.691223i	−0.271278	+	0.962501i	$0.587446\pi$
$168$		0			0
$169$		−	2197.00i		−	1.00000i
$170$		0			0
$171$	−4428.00			−1.98022
$172$		0			0
$173$	3203.93	−	3203.93i	1.40804	−	1.40804i	0.638007	−	0.770031i	$-0.279759\pi$
$173$	3203.93	−	3203.93i	1.40804	−	1.40804i	0.770031	−	0.638007i	$-0.220241\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−1298.00			−0.533036			−0.266518	−	0.963830i	$-0.585873\pi$
$181$	−1298.00			−0.533036			−0.266518	+	0.963830i	$0.585873\pi$
$182$		0			0
$183$	−1315.38	+	1315.38i	−0.531341	+	0.531341i
$184$		−	11286.0i		−	4.52182i
$185$		0			0
$186$	6264.00			2.46935
$187$		0			0
$188$	−4607.49	+	4607.49i	−1.78742	+	1.78742i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−1392.53	−	1392.53i	−0.523425	−	0.523425i
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$	−6517.00			−2.37500
$197$	−3468.48	−	3468.48i	−1.25441	−	1.25441i	−0.953723	−	0.300687i	$-0.902784\pi$
$197$	−3468.48	−	3468.48i	−1.25441	−	1.25441i	−0.300687	−	0.953723i	$-0.597216\pi$
$198$		0			0
$199$			5456.00i			1.94355i	0.235919	+	0.971773i	$0.424190\pi$
$199$			5456.00i			1.94355i	−0.235919	+	0.971773i	$0.575810\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		−	2052.00i		−	0.704259i
$205$		0			0
$206$		0			0
$207$	3769.76	+	3769.76i	1.26578	+	1.26578i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−4228.00			−1.37947			−0.689733	−	0.724063i	$-0.742272\pi$
$211$	−4228.00			−1.37947			−0.689733	+	0.724063i	$0.742272\pi$
$212$	−6143.32	−	6143.32i	−1.99021	−	1.99021i
$213$		0			0
$214$			11502.0i			3.67411i
$215$		0			0
$216$	8019.00			2.52604
$217$		0			0
$218$	6738.55	−	6738.55i	2.09354	−	2.09354i
$219$		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
$225$		0			0
$226$	10044.0			2.95627
$227$	−1594.62	−	1594.62i	−0.466249	−	0.466249i	0.434448	−	0.900697i	$-0.356943\pi$
$227$	−1594.62	−	1594.62i	−0.466249	−	0.466249i	−0.900697	+	0.434448i	$0.856943\pi$
$228$	11448.9	−	11448.9i	3.32554	−	3.32554i
$229$			286.000i			0.0825302i	0.999148	+	0.0412651i	$0.0131388\pi$
$229$			286.000i			0.0825302i	−0.999148	+	0.0412651i	$0.986861\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3924.08	+	3924.08i	−1.10333	+	1.10333i	−0.109320	+	0.994007i	$0.534867\pi$
$233$	−3924.08	+	3924.08i	−1.10333	+	1.10333i	−0.994007	+	0.109320i	$0.965133\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1116.97	−	1116.97i	−0.306138	−	0.306138i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−1438.00			−0.384356			−0.192178	−	0.981360i	$-0.561555\pi$
$241$	−1438.00			−0.384356			−0.192178	+	0.981360i	$0.561555\pi$
$242$	4890.41	+	4890.41i	1.29904	+	1.29904i
$243$	−2678.52	+	2678.52i	−0.707107	+	0.707107i
$244$		−	6802.00i		−	1.78465i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−9376.65	+	9376.65i	−2.40088	+	2.40088i
$249$			4266.00i			1.08573i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−5111.00			−1.24780
$257$	−3541.96	−	3541.96i	−0.859695	−	0.859695i	0.131607	−	0.991302i	$-0.457986\pi$
$257$	−3541.96	−	3541.96i	−0.859695	−	0.859695i	−0.991302	+	0.131607i	$0.957986\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4034.31	+	4034.31i	−0.945879	+	0.945879i	−0.998609	−	0.0527298i	$-0.983208\pi$
$263$	−4034.31	+	4034.31i	−0.945879	+	0.945879i	0.0527298	+	0.998609i	$0.483208\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	6752.00			1.51349			0.756743	−	0.653712i	$-0.226789\pi$
$271$	6752.00			1.51349			0.756743	+	0.653712i	$0.226789\pi$
$272$	2131.06	+	2131.06i	0.475052	+	0.475052i
$273$		0			0
$274$		−	13068.0i		−	2.88127i
$275$		0			0
$276$	−19494.0			−4.25145
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	5893.47	−	5893.47i	1.27146	−	1.27146i
$279$		−	6264.00i		−	1.34414i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	6547.49	+	6547.49i	1.38261	+	1.38261i
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−5654.65	+	5654.65i	−1.15696	+	1.15696i
$289$		−	4481.00i		−	0.912070i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6731.20	−	6731.20i	1.34212	−	1.34212i	0.448171	−	0.893948i	$-0.352076\pi$
$293$	6731.20	−	6731.20i	1.34212	−	1.34212i	0.893948	−	0.448171i	$-0.147924\pi$
$294$			9261.00i			1.83712i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	11434.2	+	11434.2i	2.17869	+	2.17869i
$303$		0			0
$304$			23780.0i			4.48644i
$305$		0			0
$306$	−2916.00			−0.544760
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		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
$315$		0			0
$316$	5776.00			1.02824
$317$	3586.05	+	3586.05i	0.635372	+	0.635372i	0.949410	−	0.314039i	$-0.101682\pi$
$317$	3586.05	+	3586.05i	0.635372	+	0.635372i	−0.314039	+	0.949410i	$0.601682\pi$
$318$	−8729.98	+	8729.98i	−1.53947	+	1.53947i
$319$		0			0
$320$		0			0
$321$	11502.0			1.99993
$322$		0			0
$323$	−2410.30	+	2410.30i	−0.415209	+	0.415209i
$324$		−	13851.0i		−	2.37500i
$325$		0			0
$326$		0			0
$327$	−6738.55	−	6738.55i	−1.13958	−	1.13958i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5852.00			0.971767			0.485884	−	0.874023i	$-0.338498\pi$
$331$	5852.00			0.971767			0.485884	+	0.874023i	$0.338498\pi$
$332$	−11030.1	−	11030.1i	−1.82335	−	1.82335i
$333$		0			0
$334$			10962.0i			1.79585i
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	8072.29	−	8072.29i	1.29904	−	1.29904i
$339$		−	10044.0i		−	1.60919i
$340$		0			0
$341$		0			0
$342$	−16269.5	−	16269.5i	−2.57238	−	2.57238i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	23544.0			3.65819
$347$	6856.12	+	6856.12i	1.06068	+	1.06068i	0.998036	+	0.0626439i	$0.0199532\pi$
$347$	6856.12	+	6856.12i	1.06068	+	1.06068i	0.0626439	+	0.998036i	$0.480047\pi$
$348$		0			0
$349$			3706.00i			0.568417i	0.958762	+	0.284209i	$0.0917308\pi$
$349$			3706.00i			0.568417i	−0.958762	+	0.284209i	$0.908269\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9214.98	+	9214.98i	−1.38942	+	1.38942i	−0.562872	+	0.826544i	$0.690304\pi$
$353$	−9214.98	+	9214.98i	−1.38942	+	1.38942i	−0.826544	+	0.562872i	$0.809696\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−20037.0			−2.92127
$362$	−4769.16	−	4769.16i	−0.692435	−	0.692435i
$363$	4890.41	−	4890.41i	0.707107	−	0.707107i
$364$		0			0
$365$		0			0
$366$	−9666.00			−1.38046
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$	20245.0	−	20245.0i	2.86779	−	2.86779i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	16196.0	+	16196.0i	2.25732	+	2.25732i
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$	−19602.0			−2.68855
$377$		0			0
$378$		0			0
$379$		−	4484.00i		−	0.607725i	−0.952716	−	0.303862i	$-0.901724\pi$
$379$		−	4484.00i		−	0.607725i	0.952716	−	0.303862i	$-0.0982763\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8164.15	−	8164.15i	1.08921	−	1.08921i	0.0936033	−	0.995610i	$-0.470161\pi$
$383$	8164.15	−	8164.15i	1.08921	−	1.08921i	0.995610	−	0.0936033i	$-0.0298385\pi$
$384$			2079.00i			0.276285i
$385$		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
$391$	4104.00			0.530814
$392$	−13862.9	−	13862.9i	−1.78618	−	1.78618i
$393$		0			0
$394$		−	25488.0i		−	3.25905i
$395$		0			0
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	−20046.6	+	20046.6i	−2.52474	+	2.52474i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	4364.99	−	4364.99i	0.529655	−	0.529655i
$409$			14326.0i			1.73197i	0.500071	+	0.865984i	$0.333307\pi$
$409$			14326.0i			1.73197i	−0.500071	+	0.865984i	$0.666693\pi$
$410$		0			0
$411$	−13068.0			−1.56836
$412$		0			0
$413$		0			0
$414$			27702.0i			3.28860i
$415$		0			0
$416$		0			0
$417$	−5893.47	−	5893.47i	−0.692097	−	0.692097i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−6878.00			−0.796231			−0.398115	−	0.917335i	$-0.630336\pi$
$421$	−6878.00			−0.796231			−0.398115	+	0.917335i	$0.630336\pi$
$422$	−15534.7	−	15534.7i	−1.79198	−	1.79198i
$423$	6547.49	−	6547.49i	0.752600	−	0.752600i
$424$		−	26136.0i		−	2.99358i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−29739.3	+	29739.3i	−3.35865	+	3.35865i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	14384.6	+	14384.6i	1.60204	+	1.60204i
$433$		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
$436$	34846.0			3.82757
$437$	22897.8	+	22897.8i	2.50653	+	2.50653i
$438$		0			0
$439$			16976.0i			1.84560i	0.385274	+	0.922802i	$0.374107\pi$
$439$			16976.0i			1.84560i	−0.385274	+	0.922802i	$0.625893\pi$
$440$		0			0
$441$	9261.00			1.00000
$442$		0			0
$443$	3314.16	−	3314.16i	0.355441	−	0.355441i	−0.506688	−	0.862129i	$-0.669130\pi$
$443$	3314.16	−	3314.16i	0.355441	−	0.355441i	0.862129	+	0.506688i	$0.169130\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	25969.5	+	25969.5i	2.70244	+	2.70244i
$453$	11434.2	−	11434.2i	1.18593	−	1.18593i
$454$		−	11718.0i		−	1.21135i
$455$		0			0
$456$	48708.0			5.00211
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$	−1050.83	+	1050.83i	−0.107210	+	0.107210i
$459$			2916.00i			0.296530i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	−28836.0			−2.86653
$467$	−13940.0	−	13940.0i	−1.38130	−	1.38130i	−0.842311	−	0.538992i	$-0.818805\pi$
$467$	−13940.0	−	13940.0i	−1.38130	−	1.38130i	−0.538992	−	0.842311i	$-0.681195\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		−	8208.00i		−	0.795371i
$475$		0			0
$476$		0			0
$477$	8729.98	+	8729.98i	0.837984	+	0.837984i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	−5283.55	−	5283.55i	−0.499293	−	0.499293i
$483$		0			0
$484$			25289.0i			2.37500i
$485$		0			0
$486$	−19683.0			−1.83712
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	14469.1	−	14469.1i	1.34219	−	1.34219i
$489$		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
$496$	−33640.0			−3.04532
$497$		0			0
$498$	−15674.3	+	15674.3i	−1.41040	+	1.41040i
$499$			19316.0i			1.73287i	0.499289	+	0.866436i	$0.333595\pi$
$499$			19316.0i			1.73287i	−0.499289	+	0.866436i	$0.666405\pi$
$500$		0			0
$501$	10962.0			0.977537
$502$		0			0
$503$	−3666.89	+	3666.89i	−0.325046	+	0.325046i	−0.850699	−	0.525653i	$-0.823821\pi$
$503$	−3666.89	+	3666.89i	−0.325046	+	0.325046i	0.525653	+	0.850699i	$0.323821\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−8072.29	−	8072.29i	−0.707107	−	0.707107i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−16515.7	−	16515.7i	−1.42558	−	1.42558i
$513$	−16269.5	+	16269.5i	−1.40023	+	1.40023i
$514$		−	26028.0i		−	2.23355i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	23544.0i		−	1.99127i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$	−29646.0			−2.45747
$527$	−3409.69	−	3409.69i	−0.281838	−	0.281838i
$528$		0			0
$529$		−	26821.0i		−	2.20441i
$530$		0			0
$531$		0			0
$532$		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
$541$	−3238.00			−0.257324			−0.128662	−	0.991688i	$-0.541068\pi$
$541$	−3238.00			−0.257324			−0.128662	+	0.991688i	$0.541068\pi$
$542$	24808.4	+	24808.4i	1.96608	+	1.96608i
$543$	−4769.16	+	4769.16i	−0.376914	+	0.376914i
$544$			6156.00i			0.485177i
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	33788.3	−	33788.3i	2.63387	−	2.63387i
$549$			9666.00i			0.751430i
$550$		0			0
$551$		0			0
$552$	−41467.4	−	41467.4i	−3.19741	−	3.19741i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	30476.0			2.32459
$557$	1675.45	+	1675.45i	0.127453	+	0.127453i	0.767956	−	0.640503i	$-0.221274\pi$
$557$	1675.45	+	1675.45i	0.127453	+	0.127453i	−0.640503	+	0.767956i	$0.721274\pi$
$558$	23015.4	−	23015.4i	1.74609	−	1.74609i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15806.6	−	15806.6i	1.18325	−	1.18325i	0.204346	−	0.978899i	$-0.434493\pi$
$563$	15806.6	−	15806.6i	1.18325	−	1.18325i	0.978899	−	0.204346i	$-0.0655069\pi$
$564$			33858.0i			2.52780i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	26012.0			1.90642			0.953212	−	0.302302i	$-0.0977551\pi$
$571$	26012.0			1.90642			0.953212	+	0.302302i	$0.0977551\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−10233.0			−0.740234
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	16464.2	−	16464.2i	1.18481	−	1.18481i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	49464.0			3.48693
$587$	−8502.18	−	8502.18i	−0.597824	−	0.597824i	0.341909	−	0.939733i	$-0.388926\pi$
$587$	−8502.18	−	8502.18i	−0.597824	−	0.597824i	−0.939733	+	0.341909i	$0.888926\pi$
$588$	−23945.0	+	23945.0i	−1.67938	+	1.67938i
$589$		−	38048.0i		−	2.66170i
$590$		0			0
$591$	−25488.0			−1.77400
$592$		0			0
$593$	−19649.8	+	19649.8i	−1.36074	+	1.36074i	−0.487773	+	0.872971i	$0.662191\pi$
$593$	−19649.8	+	19649.8i	−1.36074	+	1.36074i	−0.872971	+	0.487773i	$0.837809\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	20046.6	+	20046.6i	1.37429	+	1.37429i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	13642.0			0.925905			0.462952	−	0.886383i	$-0.346790\pi$
$601$	13642.0			0.925905			0.462952	+	0.886383i	$0.346790\pi$
$602$		0			0
$603$		0			0
$604$			59128.0i			3.98325i
$605$		0			0
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$	−34346.7	+	34346.7i	−2.29103	+	2.29103i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−7539.53	−	7539.53i	−0.497986	−	0.497986i
$613$		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
$617$	−8832.86	−	8832.86i	−0.576333	−	0.576333i	0.357558	−	0.933891i	$-0.383609\pi$
$617$	−8832.86	−	8832.86i	−0.576333	−	0.576333i	−0.933891	+	0.357558i	$0.883609\pi$
$618$		0			0
$619$			3476.00i			0.225706i	0.993612	+	0.112853i	$0.0359990\pi$
$619$			3476.00i			0.225706i	−0.993612	+	0.112853i	$0.964001\pi$
$620$		0			0
$621$	27702.0			1.79009
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−27808.0			−1.75439			−0.877194	−	0.480136i	$-0.840587\pi$
$631$	−27808.0			−1.75439			−0.877194	+	0.480136i	$0.840587\pi$
$632$	12286.6	+	12286.6i	0.773317	+	0.773317i
$633$	−15534.7	+	15534.7i	−0.975430	+	0.975430i
$634$			26352.0i			1.65074i
$635$		0			0
$636$	−45144.0			−2.81458
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	42261.0	+	42261.0i	2.59799	+	2.59799i
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$	−17712.0			−1.07875
$647$	−23199.1	−	23199.1i	−1.40966	−	1.40966i	−0.761516	−	0.648146i	$-0.775544\pi$
$647$	−23199.1	−	23199.1i	−1.40966	−	1.40966i	−0.648146	−	0.761516i	$-0.724456\pi$
$648$	29463.7	−	29463.7i	1.78618	−	1.78618i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	17313.0	−	17313.0i	1.03753	−	1.03753i	0.0382666	−	0.999268i	$-0.487816\pi$
$653$	17313.0	−	17313.0i	1.03753	−	1.03753i	0.999268	−	0.0382666i	$-0.0121836\pi$
$654$		−	49518.0i		−	2.96071i
$655$		0			0
$656$		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
$661$	−32978.0			−1.94054			−0.970269	−	0.242029i	$-0.922187\pi$
$661$	−32978.0			−1.94054			−0.970269	+	0.242029i	$0.922187\pi$
$662$	21501.6	+	21501.6i	1.26236	+	1.26236i
$663$		0			0
$664$		−	46926.0i		−	2.74259i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−28343.0	+	28343.0i	−1.64165	+	1.64165i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$	41743.0			2.37500
$677$	−8906.34	−	8906.34i	−0.505611	−	0.505611i	0.407565	−	0.913176i	$-0.366378\pi$
$677$	−8906.34	−	8906.34i	−0.505611	−	0.505611i	−0.913176	+	0.407565i	$0.866378\pi$
$678$	36904.0	−	36904.0i	2.09040	−	2.09040i
$679$		0			0
$680$		0			0
$681$	−11718.0			−0.659376
$682$		0			0
$683$	−16085.8	+	16085.8i	−0.901180	+	0.901180i	−0.995538	−	0.0943582i	$-0.969920\pi$
$683$	−16085.8	+	16085.8i	−0.901180	+	0.901180i	0.0943582	+	0.995538i	$0.469920\pi$
$684$		−	84132.0i		−	4.70302i
$685$		0			0
$686$		0			0
$687$	1050.83	+	1050.83i	0.0583577	+	0.0583577i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	32812.0			1.80641			0.903204	−	0.429212i	$-0.141209\pi$
$691$	32812.0			1.80641			0.903204	+	0.429212i	$0.141209\pi$
$692$	60874.7	+	60874.7i	3.34409	+	3.34409i
$693$		0			0
$694$			50382.0i			2.75573i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−13616.7	+	13616.7i	−0.738396	+	0.738396i
$699$			28836.0i			1.56034i
$700$		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
$706$	−67716.0			−3.60981
$707$		0			0
$708$		0			0
$709$			37726.0i			1.99835i	0.0406201	+	0.999175i	$0.487067\pi$
$709$			37726.0i			1.99835i	−0.0406201	+	0.999175i	$0.512933\pi$
$710$		0			0
$711$	−8208.00			−0.432945
$712$		0			0
$713$	−32392.1	+	32392.1i	−1.70139	+	1.70139i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−73620.6	−	73620.6i	−3.79484	−	3.79484i
$723$	−5283.55	+	5283.55i	−0.271781	+	0.271781i
$724$		−	24662.0i		−	1.26596i
$725$		0			0
$726$	35937.0			1.83712
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$			19683.0i			1.00000i
$730$		0			0
$731$		0			0
$732$	−24992.1	−	24992.1i	−1.26193	−	1.26193i
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$	58482.0			2.92891
$737$		0			0
$738$		0			0
$739$		−	8804.00i		−	0.438241i	−0.975698	−	0.219121i	$-0.929681\pi$
$739$		−	8804.00i		−	0.438241i	0.975698	−	0.219121i	$-0.0703188\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	27711.1	−	27711.1i	1.36826	−	1.36826i	0.505351	−	0.862914i	$-0.331363\pi$
$743$	27711.1	−	27711.1i	1.36826	−	1.36826i	0.862914	−	0.505351i	$-0.168637\pi$
$744$			68904.0i			3.39535i
$745$		0			0
$746$		0			0
$747$	15674.3	+	15674.3i	0.767727	+	0.767727i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	17512.0			0.850895			0.425447	−	0.904983i	$-0.360117\pi$
$751$	17512.0			0.850895			0.425447	+	0.904983i	$0.360117\pi$
$752$	−35162.4	−	35162.4i	−1.70511	−	1.70511i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	16475.3	−	16475.3i	0.789457	−	0.789457i
$759$		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
$766$	59994.0			2.82986
$767$		0			0
$768$	−18779.0	+	18779.0i	−0.882330	+	0.882330i
$769$			8786.00i			0.412004i	0.978552	+	0.206002i	$0.0660454\pi$
$769$			8786.00i			0.412004i	−0.978552	+	0.206002i	$0.933955\pi$
$770$		0			0
$771$	−26028.0			−1.21579
$772$		0			0
$773$	29511.5	−	29511.5i	1.37316	−	1.37316i	0.517443	−	0.855718i	$-0.326884\pi$
$773$	29511.5	−	29511.5i	1.37316	−	1.37316i	0.855718	−	0.517443i	$-0.173116\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	15079.1	+	15079.1i	0.689547	+	0.689547i
$783$		0			0
$784$		−	49735.0i		−	2.26562i
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	65901.1	−	65901.1i	2.97922	−	2.97922i
$789$			29646.0i			1.33767i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−103664.			−4.61592
$797$	31804.2	+	31804.2i	1.41350	+	1.41350i	0.729168	+	0.684335i	$0.239907\pi$
$797$	31804.2	+	31804.2i	1.41350	+	1.41350i	0.684335	+	0.729168i	$0.260093\pi$
$798$		0			0
$799$		−	7128.00i		−	0.315608i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	14092.0			0.610157			0.305078	−	0.952327i	$-0.401317\pi$
$811$	14092.0			0.610157			0.305078	+	0.952327i	$0.401317\pi$
$812$		0			0
$813$	24808.4	−	24808.4i	1.07020	−	1.07020i
$814$		0			0
$815$		0			0
$816$	15660.0			0.671826
$817$		0			0
$818$	−52637.1	+	52637.1i	−2.24989	+	2.24989i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−48014.9	−	48014.9i	−2.03736	−	2.03736i
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14425.0	+	14425.0i	0.606539	+	0.606539i	0.942040	−	0.335501i	$-0.108905\pi$
$827$	14425.0	+	14425.0i	0.606539	+	0.606539i	−0.335501	+	0.942040i	$0.608905\pi$
$828$	−71625.5	+	71625.5i	−3.00623	+	3.00623i
$829$		−	45254.0i		−	1.89594i	−0.318356	−	0.947971i	$-0.603131\pi$
$829$		−	45254.0i		−	1.89594i	0.318356	−	0.947971i	$-0.396869\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5041.05	−	5041.05i	0.209678	−	0.209678i
$834$		−	43308.0i		−	1.79812i
$835$		0			0
$836$		0			0
$837$	−23015.4	−	23015.4i	−0.950453	−	0.950453i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−24389.0			−1.00000
$842$	−25271.4	−	25271.4i	−1.03433	−	1.03433i
$843$		0			0
$844$		−	80332.0i		−	3.27623i
$845$		0			0
$846$	48114.0			1.95531
$847$		0			0
$848$	46883.2	−	46883.2i	1.89856	−	1.89856i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$	−126522.			−5.05191
$857$	−12360.1	−	12360.1i	−0.492665	−	0.492665i	0.416480	−	0.909145i	$-0.363264\pi$
$857$	−12360.1	−	12360.1i	−0.492665	−	0.492665i	−0.909145	+	0.416480i	$0.863264\pi$
$858$		0			0
$859$		−	28204.0i		−	1.12027i	−0.828403	−	0.560133i	$-0.810750\pi$
$859$		−	28204.0i		−	1.12027i	0.828403	−	0.560133i	$-0.189250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−30635.8	+	30635.8i	−1.20841	+	1.20841i	−0.236862	+	0.971543i	$0.576119\pi$
$863$	−30635.8	+	30635.8i	−1.20841	+	1.20841i	−0.971543	+	0.236862i	$0.923881\pi$
$864$			41553.0i			1.63618i
$865$		0			0
$866$		0			0
$867$	−16464.2	−	16464.2i	−0.644931	−	0.644931i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	74124.0	+	74124.0i	2.87862	+	2.87862i
$873$		0			0
$874$			168264.i			6.51214i
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$	−62373.8	+	62373.8i	−2.39751	+	2.39751i
$879$		−	49464.0i		−	1.89804i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	34027.1	+	34027.1i	1.29904	+	1.29904i
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$	24354.0			0.923463
$887$	33604.5	+	33604.5i	1.27207	+	1.27207i	0.944998	+	0.327077i	$0.106064\pi$
$887$	33604.5	+	33604.5i	1.27207	+	1.27207i	0.327077	+	0.944998i	$0.393936\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	39769.9	−	39769.9i	1.49031	−	1.49031i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	9504.00			0.351414
$902$		0			0
$903$		0			0
$904$			110484.i			4.06487i
$905$		0			0
$906$	84024.0			3.08114
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$	30297.7	−	30297.7i	1.10734	−	1.10734i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	87373.3	+	87373.3i	3.17239	+	3.17239i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−5434.00			−0.196009
$917$		0			0
$918$	−10714.1	+	10714.1i	−0.385204	+	0.385204i
$919$		−	21224.0i		−	0.761823i	−0.924612	−	0.380911i	$-0.875610\pi$
$919$		−	21224.0i		−	0.761823i	0.924612	−	0.380911i	$-0.124390\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		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
$931$	56252.0			1.98022
$932$	−74557.6	−	74557.6i	−2.62040	−	2.62040i
$933$		0			0
$934$		−	102438.i		−	3.58873i
$935$		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
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26109.1	+	26109.1i	0.895916	+	0.895916i	0.995072	−	0.0991562i	$-0.0316144\pi$
$947$	26109.1	+	26109.1i	0.895916	+	0.895916i	−0.0991562	+	0.995072i	$0.531614\pi$
$948$	21222.4	−	21222.4i	0.727079	−	0.727079i
$949$		0			0
$950$		0			0
$951$	26352.0			0.898551
$952$		0			0
$953$	−35669.5	+	35669.5i	−1.21243	+	1.21243i	−0.242207	+	0.970225i	$0.577871\pi$
$953$	−35669.5	+	35669.5i	−1.21243	+	1.21243i	−0.970225	+	0.242207i	$0.922129\pi$
$954$			64152.0i			2.17715i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	24033.0			0.806720
$962$		0			0
$963$	42261.0	−	42261.0i	1.41417	−	1.41417i
$964$		−	27322.0i		−	0.912845i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−53794.5	+	53794.5i	−1.78618	+	1.78618i
$969$			17712.0i			0.587194i
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−50891.8	−	50891.8i	−1.67938	−	1.67938i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	51910.0			1.70246
$977$	−42194.9	−	42194.9i	−1.38171	−	1.38171i	−0.841577	−	0.540137i	$-0.818372\pi$
$977$	−42194.9	−	42194.9i	−1.38171	−	1.38171i	−0.540137	−	0.841577i	$-0.681628\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−49518.0			−1.61161
$982$		0			0
$983$	−7194.15	+	7194.15i	−0.233426	+	0.233426i	−0.814121	−	0.580695i	$-0.802781\pi$
$983$	−7194.15	+	7194.15i	−0.233426	+	0.233426i	0.580695	+	0.814121i	$0.302781\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−62368.0			−1.99918			−0.999589	−	0.0286779i	$-0.990870\pi$
$991$	−62368.0			−1.99918			−0.999589	+	0.0286779i	$0.990870\pi$
$992$	−48588.1	−	48588.1i	−1.55511	−	1.55511i
$993$	21501.6	−	21501.6i	0.687143	−	0.687143i
$994$		0			0
$995$		0			0
$996$	−81054.0			−2.57861
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$	−70971.5	+	70971.5i	−2.25107	+	2.25107i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.e.b.68.2	yes	4
3.2	odd	2	inner	75.4.e.b.68.1	yes	4
5.2	odd	4	inner	75.4.e.b.32.1	✓	4
5.3	odd	4	inner	75.4.e.b.32.2	yes	4
5.4	even	2	inner	75.4.e.b.68.1	yes	4
15.2	even	4	inner	75.4.e.b.32.2	yes	4
15.8	even	4	inner	75.4.e.b.32.1	✓	4
15.14	odd	2	CM	75.4.e.b.68.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.e.b.32.1	✓	4	5.2	odd	4	inner
75.4.e.b.32.1	✓	4	15.8	even	4	inner
75.4.e.b.32.2	yes	4	5.3	odd	4	inner
75.4.e.b.32.2	yes	4	15.2	even	4	inner
75.4.e.b.68.1	yes	4	3.2	odd	2	inner
75.4.e.b.68.1	yes	4	5.4	even	2	inner
75.4.e.b.68.2	yes	4	1.1	even	1	trivial
75.4.e.b.68.2	yes	4	15.14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.67423 + 3.67423i) q^{2} +(3.67423 - 3.67423i) q^{3} +19.0000i q^{4} +27.0000 q^{6} +(-40.4166 + 40.4166i) q^{8} -27.0000i q^{9} +O(q^{10})\)	\(q+(3.67423 + 3.67423i) q^{2} +(3.67423 - 3.67423i) q^{3} +19.0000i q^{4} +27.0000 q^{6} +(-40.4166 + 40.4166i) q^{8} -27.0000i q^{9} +(69.8105 + 69.8105i) q^{12} -145.000 q^{16} +(-14.6969 - 14.6969i) q^{17} +(99.2043 - 99.2043i) q^{18} -164.000i q^{19} +(-139.621 + 139.621i) q^{23} +297.000i q^{24} +(-99.2043 - 99.2043i) q^{27} +232.000 q^{31} +(-209.431 - 209.431i) q^{32} -108.000i q^{34} +513.000 q^{36} +(602.574 - 602.574i) q^{38} -1026.00 q^{46} +(242.499 + 242.499i) q^{47} +(-532.764 + 532.764i) q^{48} +343.000i q^{49} -108.000 q^{51} +(-323.333 + 323.333i) q^{53} -729.000i q^{54} +(-602.574 - 602.574i) q^{57} -358.000 q^{61} +(852.422 + 852.422i) q^{62} -379.000i q^{64} +(279.242 - 279.242i) q^{68} +1026.00i q^{69} +(1091.25 + 1091.25i) q^{72} +3116.00 q^{76} -304.000i q^{79} -729.000 q^{81} +(-580.529 + 580.529i) q^{83} +(-2652.80 - 2652.80i) q^{92} +(852.422 - 852.422i) q^{93} +1782.00i q^{94} -1539.00 q^{96} +(-1260.26 + 1260.26i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{6}+O(q^{10}) \) 4 * q + 108 * q^6	\( 4 q + 108 q^{6} - 580 q^{16} + 928 q^{31} + 2052 q^{36} - 4104 q^{46} - 432 q^{51} - 1432 q^{61} + 12464 q^{76} - 2916 q^{81} - 6156 q^{96}+O(q^{100}) \) 4 * q + 108 * q^6 - 580 * q^16 + 928 * q^31 + 2052 * q^36 - 4104 * q^46 - 432 * q^51 - 1432 * q^61 + 12464 * q^76 - 2916 * q^81 - 6156 * q^96

Introduction

L-functions

Modular forms

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Database

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