LMFDB - Embedded newform 75.4.e.a.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.a.32.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.67423 + 3.67423i) q^{3} -8.00000i q^{4} +(22.0454 - 22.0454i) q^{7} +27.0000i q^{9} +O(q^{10})$	$q+(3.67423 + 3.67423i) q^{3} -8.00000i q^{4} +(22.0454 - 22.0454i) q^{7} +27.0000i q^{9} +(29.3939 - 29.3939i) q^{12} +(44.0908 + 44.0908i) q^{13} -64.0000 q^{16} -56.0000i q^{19} +162.000 q^{21} +(-99.2043 + 99.2043i) q^{27} +(-176.363 - 176.363i) q^{28} -308.000 q^{31} +216.000 q^{36} +(-308.636 + 308.636i) q^{37} +324.000i q^{39} +(154.318 + 154.318i) q^{43} +(-235.151 - 235.151i) q^{48} -629.000i q^{49} +(352.727 - 352.727i) q^{52} +(205.757 - 205.757i) q^{57} +182.000 q^{61} +(595.226 + 595.226i) q^{63} +512.000i q^{64} +(462.954 - 462.954i) q^{67} +(264.545 + 264.545i) q^{73} -448.000 q^{76} +884.000i q^{79} -729.000 q^{81} -1296.00i q^{84} +1944.00 q^{91} +(-1131.66 - 1131.66i) q^{93} +(-969.998 + 969.998i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 256 q^{16} + 648 q^{21} - 1232 q^{31} + 864 q^{36} + 728 q^{61} - 1792 q^{76} - 2916 q^{81} + 7776 q^{91}+O(q^{100}) $ 4 * q - 256 * q^16 + 648 * q^21 - 1232 * q^31 + 864 * q^36 + 728 * q^61 - 1792 * q^76 - 2916 * q^81 + 7776 * q^91

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 $

$2$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$2$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$3$	3.67423	+	3.67423i	0.707107	+	0.707107i
$3$	3.67423	+	3.67423i	0.707107	+	0.707107i
$4$		−	8.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	22.0454	−	22.0454i	1.19034	−	1.19034i	0.213368	−	0.976972i	$-0.431557\pi$
$7$	22.0454	−	22.0454i	1.19034	−	1.19034i	0.976972	−	0.213368i	$-0.0684434\pi$
$8$		0			0
$9$			27.0000i			1.00000i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	29.3939	−	29.3939i	0.707107	−	0.707107i
$13$	44.0908	+	44.0908i	0.940661	+	0.940661i	0.998335	−	0.0576745i	$-0.0183686\pi$
$13$	44.0908	+	44.0908i	0.940661	+	0.940661i	−0.0576745	+	0.998335i	$0.518369\pi$
$14$		0			0
$15$		0			0
$16$	−64.0000			−1.00000
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$		−	56.0000i		−	0.676173i	−0.941115	−	0.338086i	$-0.890220\pi$
$19$		−	56.0000i		−	0.676173i	0.941115	−	0.338086i	$-0.109780\pi$
$20$		0			0
$21$	162.000			1.68340
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−99.2043	+	99.2043i	−0.707107	+	0.707107i
$28$	−176.363	−	176.363i	−1.19034	−	1.19034i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−308.000			−1.78447			−0.892233	−	0.451576i	$-0.850862\pi$
$31$	−308.000			−1.78447			−0.892233	+	0.451576i	$0.850862\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	216.000			1.00000
$37$	−308.636	+	308.636i	−1.37134	+	1.37134i	−0.512867	+	0.858468i	$0.671417\pi$
$37$	−308.636	+	308.636i	−1.37134	+	1.37134i	−0.858468	+	0.512867i	$0.828583\pi$
$38$		0			0
$39$			324.000i			1.33030i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	154.318	+	154.318i	0.547285	+	0.547285i	0.925655	−	0.378370i	$-0.123515\pi$
$43$	154.318	+	154.318i	0.547285	+	0.547285i	−0.378370	+	0.925655i	$0.623515\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	−235.151	−	235.151i	−0.707107	−	0.707107i
$49$		−	629.000i		−	1.83382i
$50$		0			0
$51$		0			0
$52$	352.727	−	352.727i	0.940661	−	0.940661i
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	205.757	−	205.757i	0.478126	−	0.478126i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	182.000			0.382012			0.191006	−	0.981589i	$-0.438825\pi$
$61$	182.000			0.382012			0.191006	+	0.981589i	$0.438825\pi$
$62$		0			0
$63$	595.226	+	595.226i	1.19034	+	1.19034i
$64$			512.000i			1.00000i
$65$		0			0
$66$		0			0
$67$	462.954	−	462.954i	0.844161	−	0.844161i	−0.145236	−	0.989397i	$-0.546394\pi$
$67$	462.954	−	462.954i	0.844161	−	0.844161i	0.989397	+	0.145236i	$0.0463942\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	264.545	+	264.545i	0.424146	+	0.424146i	0.886628	−	0.462483i	$-0.153041\pi$
$73$	264.545	+	264.545i	0.424146	+	0.424146i	−0.462483	+	0.886628i	$0.653041\pi$
$74$		0			0
$75$		0			0
$76$	−448.000			−0.676173
$77$		0			0
$78$		0			0
$79$			884.000i			1.25896i	0.777017	+	0.629480i	$0.216732\pi$
$79$			884.000i			1.25896i	−0.777017	+	0.629480i	$0.783268\pi$
$80$		0			0
$81$	−729.000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		−	1296.00i		−	1.68340i
$85$		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
$91$	1944.00			2.23941
$92$		0			0
$93$	−1131.66	−	1131.66i	−1.26181	−	1.26181i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−969.998	+	969.998i	−1.01534	+	1.01534i	−0.0154636	+	0.999880i	$0.504922\pi$
$97$	−969.998	+	969.998i	−1.01534	+	1.01534i	−0.999880	+	0.0154636i	$0.995078\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−727.498	−	727.498i	−0.695947	−	0.695947i	0.267587	−	0.963534i	$-0.413774\pi$
$103$	−727.498	−	727.498i	−0.695947	−	0.695947i	−0.963534	+	0.267587i	$0.913774\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$	793.635	+	793.635i	0.707107	+	0.707107i
$109$		−	646.000i		−	0.567666i	−0.958874	−	0.283833i	$-0.908394\pi$
$109$		−	646.000i		−	0.567666i	0.958874	−	0.283833i	$-0.0916061\pi$
$110$		0			0
$111$	−2268.00			−1.93936
$112$	−1410.91	+	1410.91i	−1.19034	+	1.19034i
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1190.45	+	1190.45i	−0.940661	+	0.940661i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1331.00			1.00000
$122$		0			0
$123$		0			0
$124$			2464.00i			1.78447i
$125$		0			0
$126$		0			0
$127$	2006.13	−	2006.13i	1.40170	−	1.40170i	0.606977	−	0.794720i	$-0.292382\pi$
$127$	2006.13	−	2006.13i	1.40170	−	1.40170i	0.794720	−	0.606977i	$-0.207618\pi$
$128$		0			0
$129$			1134.00i			0.773978i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	−1234.54	−	1234.54i	−0.804875	−	0.804875i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		−	2576.00i		−	1.57190i	−0.618293	−	0.785948i	$-0.712175\pi$
$139$		−	2576.00i		−	1.57190i	0.618293	−	0.785948i	$-0.287825\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		−	1728.00i		−	1.00000i
$145$		0			0
$146$		0			0
$147$	2311.09	−	2311.09i	1.29671	−	1.29671i
$148$	2469.09	+	2469.09i	1.37134	+	1.37134i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−1748.00			−0.942054			−0.471027	−	0.882119i	$-0.656117\pi$
$151$	−1748.00			−0.942054			−0.471027	+	0.882119i	$0.656117\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	2592.00			1.33030
$157$	573.181	−	573.181i	0.291368	−	0.291368i	−0.546252	−	0.837621i	$-0.683946\pi$
$157$	573.181	−	573.181i	0.291368	−	0.291368i	0.837621	+	0.546252i	$0.183946\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1697.50	+	1697.50i	0.815694	+	0.815694i	0.985481	−	0.169787i	$-0.0543078\pi$
$163$	1697.50	+	1697.50i	0.815694	+	0.815694i	−0.169787	+	0.985481i	$0.554308\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			1691.00i			0.769686i
$170$		0			0
$171$	1512.00			0.676173
$172$	1234.54	−	1234.54i	0.547285	−	0.547285i
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		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
$181$	−3458.00			−1.42006			−0.710031	−	0.704171i	$-0.751319\pi$
$181$	−3458.00			−1.42006			−0.710031	+	0.704171i	$0.751319\pi$
$182$		0			0
$183$	668.711	+	668.711i	0.270123	+	0.270123i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			4374.00i			1.68340i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−1881.21	+	1881.21i	−0.707107	+	0.707107i
$193$	−3703.63	−	3703.63i	−1.38131	−	1.38131i	−0.842297	−	0.539014i	$-0.818797\pi$
$193$	−3703.63	−	3703.63i	−1.38131	−	1.38131i	−0.539014	−	0.842297i	$-0.681203\pi$
$194$		0			0
$195$		0			0
$196$	−5032.00			−1.83382
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		−	5236.00i		−	1.86518i	−0.360942	−	0.932588i	$-0.617545\pi$
$199$		−	5236.00i		−	1.86518i	0.360942	−	0.932588i	$-0.382455\pi$
$200$		0			0
$201$	3402.00			1.19382
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−2821.81	−	2821.81i	−0.940661	−	0.940661i
$209$		0			0
$210$		0			0
$211$	6032.00			1.96806			0.984028	−	0.178011i	$-0.0569664\pi$
$211$	6032.00			1.96806			0.984028	+	0.178011i	$0.0569664\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−6789.99	+	6789.99i	−2.12412	+	2.12412i
$218$		0			0
$219$			1944.00i			0.599833i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	4122.49	+	4122.49i	1.23795	+	1.23795i	0.960838	+	0.277110i	$0.0893766\pi$
$223$	4122.49	+	4122.49i	1.23795	+	1.23795i	0.277110	+	0.960838i	$0.410623\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$	−1646.06	−	1646.06i	−0.478126	−	0.478126i
$229$		−	4466.00i		−	1.28874i	−0.764714	−	0.644370i	$-0.777120\pi$
$229$		−	4466.00i		−	1.28874i	0.764714	−	0.644370i	$-0.222880\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−3248.02	+	3248.02i	−0.890219	+	0.890219i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−7378.00			−1.97203			−0.986014	−	0.166662i	$-0.946701\pi$
$241$	−7378.00			−1.97203			−0.986014	+	0.166662i	$0.946701\pi$
$242$		0			0
$243$	−2678.52	−	2678.52i	−0.707107	−	0.707107i
$244$		−	1456.00i		−	0.382012i
$245$		0			0
$246$		0			0
$247$	2469.09	−	2469.09i	0.636049	−	0.636049i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	4761.81	−	4761.81i	1.19034	−	1.19034i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4096.00			1.00000
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$			13608.0i			3.26471i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−3703.63	−	3703.63i	−0.844161	−	0.844161i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	812.000			0.182013			0.0910064	−	0.995850i	$-0.470992\pi$
$271$	812.000			0.182013			0.0910064	+	0.995850i	$0.470992\pi$
$272$		0			0
$273$	7142.71	+	7142.71i	1.58350	+	1.58350i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5864.08	−	5864.08i	1.27198	−	1.27198i	0.326931	−	0.945048i	$-0.393985\pi$
$277$	5864.08	−	5864.08i	1.27198	−	1.27198i	0.945048	−	0.326931i	$-0.106015\pi$
$278$		0			0
$279$		−	8316.00i		−	1.78447i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	5445.22	+	5445.22i	1.14376	+	1.14376i	0.987756	+	0.156005i	$0.0498616\pi$
$283$	5445.22	+	5445.22i	1.14376	+	1.14376i	0.156005	+	0.987756i	$0.450138\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	4913.00i		−	1.00000i
$290$		0			0
$291$	−7128.00			−1.43591
$292$	2116.36	−	2116.36i	0.424146	−	0.424146i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	6804.00			1.30291
$302$		0			0
$303$		0			0
$304$			3584.00i			0.676173i
$305$		0			0
$306$		0			0
$307$	1124.32	−	1124.32i	0.209017	−	0.209017i	−0.594833	−	0.803849i	$-0.702782\pi$
$307$	1124.32	−	1124.32i	0.209017	−	0.209017i	0.803849	+	0.594833i	$0.202782\pi$
$308$		0			0
$309$		−	5346.00i		−	0.984218i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	3350.90	+	3350.90i	0.605125	+	0.605125i	0.941668	−	0.336543i	$-0.109258\pi$
$313$	3350.90	+	3350.90i	0.605125	+	0.605125i	−0.336543	+	0.941668i	$0.609258\pi$
$314$		0			0
$315$		0			0
$316$	7072.00			1.25896
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$			5832.00i			1.00000i
$325$		0			0
$326$		0			0
$327$	2373.56	−	2373.56i	0.401400	−	0.401400i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	992.000			0.164729			0.0823644	−	0.996602i	$-0.473753\pi$
$331$	992.000			0.164729			0.0823644	+	0.996602i	$0.473753\pi$
$332$		0			0
$333$	−8333.16	−	8333.16i	−1.37134	−	1.37134i
$334$		0			0
$335$		0			0
$336$	−10368.0			−1.68340
$337$	−8024.53	+	8024.53i	−1.29710	+	1.29710i	−0.366806	+	0.930297i	$0.619549\pi$
$337$	−8024.53	+	8024.53i	−1.29710	+	1.29710i	−0.930297	+	0.366806i	$0.880451\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6304.99	−	6304.99i	−0.992529	−	0.992529i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$			11914.0i			1.82734i	0.406456	+	0.913670i	$0.366764\pi$
$349$			11914.0i			1.82734i	−0.406456	+	0.913670i	$0.633236\pi$
$350$		0			0
$351$	−8748.00			−1.33030
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		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
$361$	3723.00			0.542790
$362$		0			0
$363$	4890.41	+	4890.41i	0.707107	+	0.707107i
$364$		−	15552.0i		−	2.23941i
$365$		0			0
$366$		0			0
$367$	−9457.48	+	9457.48i	−1.34517	+	1.34517i	−0.454338	+	0.890829i	$0.650124\pi$
$367$	−9457.48	+	9457.48i	−1.34517	+	1.34517i	−0.890829	+	0.454338i	$0.849876\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−9053.31	+	9053.31i	−1.26181	+	1.26181i
$373$	−5246.81	−	5246.81i	−0.728336	−	0.728336i	0.241952	−	0.970288i	$-0.422212\pi$
$373$	−5246.81	−	5246.81i	−0.728336	−	0.728336i	−0.970288	+	0.241952i	$0.922212\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			8584.00i			1.16340i	0.813402	+	0.581702i	$0.197613\pi$
$379$			8584.00i			1.16340i	−0.813402	+	0.581702i	$0.802387\pi$
$380$		0			0
$381$	14742.0			1.98230
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4166.58	+	4166.58i	−0.547285	+	0.547285i
$388$	7759.98	+	7759.98i	1.01534	+	1.01534i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11155.0	−	11155.0i	1.41021	−	1.41021i	0.651915	−	0.758292i	$-0.273966\pi$
$397$	11155.0	−	11155.0i	1.41021	−	1.41021i	0.758292	−	0.651915i	$-0.226034\pi$
$398$		0			0
$399$		−	9072.00i		−	1.13827i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−13580.0	−	13580.0i	−1.67858	−	1.67858i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	8246.00i		−	0.996916i	−0.866914	−	0.498458i	$-0.833900\pi$
$409$		−	8246.00i		−	0.996916i	0.866914	−	0.498458i	$-0.166100\pi$
$410$		0			0
$411$		0			0
$412$	−5819.99	+	5819.99i	−0.695947	+	0.695947i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	9464.83	−	9464.83i	1.11150	−	1.11150i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−17138.0			−1.98398			−0.991989	−	0.126322i	$-0.959683\pi$
$421$	−17138.0			−1.98398			−0.991989	+	0.126322i	$0.959683\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4012.26	−	4012.26i	0.454724	−	0.454724i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	6349.08	−	6349.08i	0.707107	−	0.707107i
$433$	12610.0	+	12610.0i	1.39953	+	1.39953i	0.801395	+	0.598135i	$0.204091\pi$
$433$	12610.0	+	12610.0i	1.39953	+	1.39953i	0.598135	+	0.801395i	$0.295909\pi$
$434$		0			0
$435$		0			0
$436$	−5168.00			−0.567666
$437$		0			0
$438$		0			0
$439$			14924.0i			1.62251i	0.584690	+	0.811257i	$0.301216\pi$
$439$			14924.0i			1.62251i	−0.584690	+	0.811257i	$0.698784\pi$
$440$		0			0
$441$	16983.0			1.83382
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$			18144.0i			1.93936i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	11287.2	+	11287.2i	1.19034	+	1.19034i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−6422.56	−	6422.56i	−0.666133	−	0.666133i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	10493.6	−	10493.6i	1.07412	−	1.07412i	0.0770909	−	0.997024i	$-0.475437\pi$
$457$	10493.6	−	10493.6i	1.07412	−	1.07412i	0.997024	−	0.0770909i	$-0.0245632\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	1697.50	+	1697.50i	0.170387	+	0.170387i	0.787150	−	0.616762i	$-0.211556\pi$
$463$	1697.50	+	1697.50i	0.170387	+	0.170387i	−0.616762	+	0.787150i	$0.711556\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	9523.62	+	9523.62i	0.940661	+	0.940661i
$469$		−	20412.0i		−	2.00968i
$470$		0			0
$471$	4212.00			0.412057
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−27216.0			−2.57992
$482$		0			0
$483$		0			0
$484$		−	10648.0i		−	1.00000i
$485$		0			0
$486$		0			0
$487$	3549.31	−	3549.31i	0.330256	−	0.330256i	−0.522428	−	0.852684i	$-0.674974\pi$
$487$	3549.31	−	3549.31i	0.330256	−	0.330256i	0.852684	+	0.522428i	$0.174974\pi$
$488$		0			0
$489$			12474.0i			1.15357i
$490$		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
$496$	19712.0			1.78447
$497$		0			0
$498$		0			0
$499$		−	15136.0i		−	1.35788i	−0.734195	−	0.678938i	$-0.762440\pi$
$499$		−	15136.0i		−	1.35788i	0.734195	−	0.678938i	$-0.237560\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−6213.13	+	6213.13i	−0.544250	+	0.544250i
$508$	−16049.1	−	16049.1i	−1.40170	−	1.40170i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	11664.0			1.00976
$512$		0			0
$513$	5555.44	+	5555.44i	0.478126	+	0.478126i
$514$		0			0
$515$		0			0
$516$	9072.00			0.773978
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−14616.1	−	14616.1i	−1.22202	−	1.22202i	−0.966912	−	0.255110i	$-0.917888\pi$
$523$	−14616.1	−	14616.1i	−1.22202	−	1.22202i	−0.255110	−	0.966912i	$-0.582112\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			12167.0i			1.00000i
$530$		0			0
$531$		0			0
$532$	−9876.34	+	9876.34i	−0.804875	+	0.804875i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−22678.0			−1.80222			−0.901112	−	0.433586i	$-0.857248\pi$
$541$	−22678.0			−1.80222			−0.901112	+	0.433586i	$0.857248\pi$
$542$		0			0
$543$	−12705.5	−	12705.5i	−1.00414	−	1.00414i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−18055.2	+	18055.2i	−1.41131	+	1.41131i	−0.660330	+	0.750976i	$0.729584\pi$
$547$	−18055.2	+	18055.2i	−1.41131	+	1.41131i	−0.750976	+	0.660330i	$0.770416\pi$
$548$		0			0
$549$			4914.00i			0.382012i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	19488.1	+	19488.1i	1.49859	+	1.49859i
$554$		0			0
$555$		0			0
$556$	−20608.0			−1.57190
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$			13608.0i			1.02962i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−16071.1	+	16071.1i	−1.19034	+	1.19034i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	23312.0			1.70854			0.854270	−	0.519829i	$-0.174004\pi$
$571$	23312.0			1.70854			0.854270	+	0.519829i	$0.174004\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−13824.0			−1.00000
$577$	−15079.1	+	15079.1i	−1.08795	+	1.08795i	−0.0922148	+	0.995739i	$0.529395\pi$
$577$	−15079.1	+	15079.1i	−1.08795	+	1.08795i	−0.995739	+	0.0922148i	$0.970605\pi$
$578$		0			0
$579$		−	27216.0i		−	1.95347i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	−18488.7	−	18488.7i	−1.29671	−	1.29671i
$589$			17248.0i			1.20661i
$590$		0			0
$591$		0			0
$592$	19752.7	−	19752.7i	1.37134	−	1.37134i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	19238.3	−	19238.3i	1.31888	−	1.31888i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	29302.0			1.98877			0.994387	−	0.105801i	$-0.0337408\pi$
$601$	29302.0			1.98877			0.994387	+	0.105801i	$0.0337408\pi$
$602$		0			0
$603$	12499.7	+	12499.7i	0.844161	+	0.844161i
$604$			13984.0i			0.942054i
$605$		0			0
$606$		0			0
$607$	−6591.58	+	6591.58i	−0.440764	+	0.440764i	−0.892269	−	0.451505i	$-0.850887\pi$
$607$	−6591.58	+	6591.58i	−0.440764	+	0.440764i	0.451505	+	0.892269i	$0.350887\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−17592.2	−	17592.2i	−1.15913	−	1.15913i	−0.984664	−	0.174461i	$-0.944182\pi$
$613$	−17592.2	−	17592.2i	−1.15913	−	1.15913i	−0.174461	−	0.984664i	$-0.555818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		−	26656.0i		−	1.73085i	−0.501040	−	0.865424i	$-0.667049\pi$
$619$		−	26656.0i		−	1.73085i	0.501040	−	0.865424i	$-0.332951\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		−	20736.0i		−	1.33030i
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−4585.44	−	4585.44i	−0.291368	−	0.291368i
$629$		0			0
$630$		0			0
$631$	1892.00			0.119365			0.0596825	−	0.998217i	$-0.480991\pi$
$631$	1892.00			0.119365			0.0596825	+	0.998217i	$0.480991\pi$
$632$		0			0
$633$	22163.0	+	22163.0i	1.39163	+	1.39163i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	27733.1	−	27733.1i	1.72500	−	1.72500i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	21097.5	+	21097.5i	1.29394	+	1.29394i	0.932330	+	0.361608i	$0.117772\pi$
$643$	21097.5	+	21097.5i	1.29394	+	1.29394i	0.361608	+	0.932330i	$0.382228\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−49896.0			−3.00396
$652$	13580.0	−	13580.0i	0.815694	−	0.815694i
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−7142.71	+	7142.71i	−0.424146	+	0.424146i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	20482.0			1.20523			0.602615	−	0.798032i	$-0.294125\pi$
$661$	20482.0			1.20523			0.602615	+	0.798032i	$0.294125\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			30294.0i			1.75072i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	17900.9	+	17900.9i	1.02530	+	1.02530i	0.999671	+	0.0256299i	$0.00815916\pi$
$673$	17900.9	+	17900.9i	1.02530	+	1.02530i	0.0256299	+	0.999671i	$0.491841\pi$
$674$		0			0
$675$		0			0
$676$	13528.0			0.769686
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$			42768.0i			2.41721i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		−	12096.0i		−	0.676173i
$685$		0			0
$686$		0			0
$687$	16409.1	−	16409.1i	0.911277	−	0.911277i
$688$	−9876.34	−	9876.34i	−0.547285	−	0.547285i
$689$		0			0
$690$		0			0
$691$	16072.0			0.884816			0.442408	−	0.896814i	$-0.354124\pi$
$691$	16072.0			0.884816			0.442408	+	0.896814i	$0.354124\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	17283.6	+	17283.6i	0.927259	+	0.927259i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	36146.0i		−	1.91466i	−0.289003	−	0.957328i	$-0.593324\pi$
$709$		−	36146.0i		−	1.91466i	0.289003	−	0.957328i	$-0.406676\pi$
$710$		0			0
$711$	−23868.0			−1.25896
$712$		0			0
$713$		0			0
$714$		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
$721$	−32076.0			−1.65683
$722$		0			0
$723$	−27108.5	−	27108.5i	−1.39443	−	1.39443i
$724$			27664.0i			1.42006i
$725$		0			0
$726$		0			0
$727$	−26652.9	+	26652.9i	−1.35970	+	1.35970i	−0.485416	+	0.874284i	$0.661332\pi$
$727$	−26652.9	+	26652.9i	−1.35970	+	1.35970i	−0.874284	+	0.485416i	$0.838668\pi$
$728$		0			0
$729$		−	19683.0i		−	1.00000i
$730$		0			0
$731$		0			0
$732$	5349.69	−	5349.69i	0.270123	−	0.270123i
$733$	−25969.5	−	25969.5i	−1.30860	−	1.30860i	−0.922425	−	0.386177i	$-0.873795\pi$
$733$	−25969.5	−	25969.5i	−1.30860	−	1.30860i	−0.386177	−	0.922425i	$-0.626205\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	31376.0i		−	1.56182i	−0.624644	−	0.780910i	$-0.714756\pi$
$739$		−	31376.0i		−	1.56182i	0.624644	−	0.780910i	$-0.285244\pi$
$740$		0			0
$741$	18144.0			0.899509
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	23452.0			1.13951			0.569757	−	0.821813i	$-0.307037\pi$
$751$	23452.0			1.13951			0.569757	+	0.821813i	$0.307037\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	34992.0			1.68340
$757$	2777.72	−	2777.72i	0.133366	−	0.133366i	−0.637273	−	0.770638i	$-0.719937\pi$
$757$	2777.72	−	2777.72i	0.133366	−	0.133366i	0.770638	+	0.637273i	$0.219937\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−14241.3	−	14241.3i	−0.675715	−	0.675715i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	15049.7	+	15049.7i	0.707107	+	0.707107i
$769$		−	4606.00i		−	0.215990i	−0.994151	−	0.107995i	$-0.965557\pi$
$769$		−	4606.00i		−	0.215990i	0.994151	−	0.107995i	$-0.0344431\pi$
$770$		0			0
$771$		0			0
$772$	−29629.0	+	29629.0i	−1.38131	+	1.38131i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−49999.0	+	49999.0i	−2.30850	+	2.30850i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			40256.0i			1.83382i
$785$		0			0
$786$		0			0
$787$	5753.85	−	5753.85i	0.260613	−	0.260613i	−0.564690	−	0.825303i	$-0.691004\pi$
$787$	5753.85	−	5753.85i	0.260613	−	0.260613i	0.825303	+	0.564690i	$0.191004\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	8024.53	+	8024.53i	0.359343	+	0.359343i
$794$		0			0
$795$		0			0
$796$	−41888.0			−1.86518
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	27216.0i		−	1.19382i
$805$		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
$811$	−39368.0			−1.70456			−0.852280	−	0.523087i	$-0.824780\pi$
$811$	−39368.0			−1.70456			−0.852280	+	0.523087i	$0.824780\pi$
$812$		0			0
$813$	2983.48	+	2983.48i	0.128703	+	0.128703i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	8641.80	−	8641.80i	0.370059	−	0.370059i
$818$		0			0
$819$			52488.0i			2.23941i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−32252.4	−	32252.4i	−1.36604	−	1.36604i	−0.866009	−	0.500029i	$-0.833323\pi$
$823$	−32252.4	−	32252.4i	−1.36604	−	1.36604i	−0.500029	−	0.866009i	$-0.666677\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		−	17066.0i		−	0.714990i	−0.933915	−	0.357495i	$-0.883631\pi$
$829$		−	17066.0i		−	0.714990i	0.933915	−	0.357495i	$-0.116369\pi$
$830$		0			0
$831$	43092.0			1.79885
$832$	−22574.5	+	22574.5i	−0.940661	+	0.940661i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	30554.9	−	30554.9i	1.26181	−	1.26181i
$838$		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
$842$		0			0
$843$		0			0
$844$		−	48256.0i		−	1.96806i
$845$		0			0
$846$		0			0
$847$	29342.4	−	29342.4i	1.19034	−	1.19034i
$848$		0			0
$849$			40014.0i			1.61752i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−12301.3	−	12301.3i	−0.493775	−	0.493775i	0.415719	−	0.909493i	$-0.363530\pi$
$853$	−12301.3	−	12301.3i	−0.493775	−	0.493775i	−0.909493	+	0.415719i	$0.863530\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$			31304.0i			1.24340i	0.783256	+	0.621699i	$0.213557\pi$
$859$			31304.0i			1.24340i	−0.783256	+	0.621699i	$0.786443\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	18051.5	−	18051.5i	0.707107	−	0.707107i
$868$	54319.9	+	54319.9i	2.12412	+	2.12412i
$869$		0			0
$870$		0			0
$871$	40824.0			1.58814
$872$		0			0
$873$	−26189.9	−	26189.9i	−1.01534	−	1.01534i
$874$		0			0
$875$		0			0
$876$	15552.0			0.599833
$877$	−9567.71	+	9567.71i	−0.368391	+	0.368391i	−0.866890	−	0.498499i	$-0.833885\pi$
$877$	−9567.71	+	9567.71i	−0.368391	+	0.368391i	0.498499	+	0.866890i	$0.333885\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	34104.2	+	34104.2i	1.29977	+	1.29977i	0.928540	+	0.371233i	$0.121065\pi$
$883$	34104.2	+	34104.2i	1.29977	+	1.29977i	0.371233	+	0.928540i	$0.378935\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		−	88452.0i		−	3.33699i
$890$		0			0
$891$		0			0
$892$	32979.9	−	32979.9i	1.23795	−	1.23795i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	24999.5	+	24999.5i	0.921297	+	0.921297i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	22067.5	−	22067.5i	0.807870	−	0.807870i	−0.176441	−	0.984311i	$-0.556459\pi$
$907$	22067.5	−	22067.5i	0.807870	−	0.807870i	0.984311	+	0.176441i	$0.0564586\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−13168.5	+	13168.5i	−0.478126	+	0.478126i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−35728.0			−1.28874
$917$		0			0
$918$		0			0
$919$		−	2756.00i		−	0.0989250i	−0.998776	−	0.0494625i	$-0.984249\pi$
$919$		−	2756.00i		−	0.0989250i	0.998776	−	0.0494625i	$-0.0157508\pi$
$920$		0			0
$921$	8262.00			0.295594
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	19642.5	−	19642.5i	0.695947	−	0.695947i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−35224.0			−1.23998
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−10229.1	+	10229.1i	−0.356637	+	0.356637i	−0.862572	−	0.505935i	$-0.831148\pi$
$937$	−10229.1	+	10229.1i	−0.356637	+	0.356637i	0.505935	+	0.862572i	$0.331148\pi$
$938$		0			0
$939$			24624.0i			0.855776i
$940$		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
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$	25984.2	+	25984.2i	0.890219	+	0.890219i
$949$			23328.0i			0.797955i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	65073.0			2.18432
$962$		0			0
$963$		0			0
$964$			59024.0i			1.97203i
$965$		0			0
$966$		0			0
$967$	23610.6	−	23610.6i	0.785178	−	0.785178i	−0.195522	−	0.980699i	$-0.562640\pi$
$967$	23610.6	−	23610.6i	0.785178	−	0.785178i	0.980699	+	0.195522i	$0.0626400\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−21428.1	+	21428.1i	−0.707107	+	0.707107i
$973$	−56789.0	−	56789.0i	−1.87109	−	1.87109i
$974$		0			0
$975$		0			0
$976$	−11648.0			−0.382012
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	17442.0			0.567666
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−19752.7	−	19752.7i	−0.636049	−	0.636049i
$989$		0			0
$990$		0			0
$991$	−45628.0			−1.46258			−0.731292	−	0.682064i	$-0.761082\pi$
$991$	−45628.0			−1.46258			−0.731292	+	0.682064i	$0.761082\pi$
$992$		0			0
$993$	3644.84	+	3644.84i	0.116481	+	0.116481i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−39549.5	+	39549.5i	−1.25631	+	1.25631i	−0.303473	+	0.952840i	$0.598146\pi$
$997$	−39549.5	+	39549.5i	−1.25631	+	1.25631i	−0.952840	+	0.303473i	$0.901854\pi$
$998$		0			0
$999$		−	61236.0i		−	1.93936i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(3.67423 + 3.67423i) q^{3} -8.00000i q^{4} +(22.0454 - 22.0454i) q^{7} +27.0000i q^{9} +O(q^{10})\)	\(q+(3.67423 + 3.67423i) q^{3} -8.00000i q^{4} +(22.0454 - 22.0454i) q^{7} +27.0000i q^{9} +(29.3939 - 29.3939i) q^{12} +(44.0908 + 44.0908i) q^{13} -64.0000 q^{16} -56.0000i q^{19} +162.000 q^{21} +(-99.2043 + 99.2043i) q^{27} +(-176.363 - 176.363i) q^{28} -308.000 q^{31} +216.000 q^{36} +(-308.636 + 308.636i) q^{37} +324.000i q^{39} +(154.318 + 154.318i) q^{43} +(-235.151 - 235.151i) q^{48} -629.000i q^{49} +(352.727 - 352.727i) q^{52} +(205.757 - 205.757i) q^{57} +182.000 q^{61} +(595.226 + 595.226i) q^{63} +512.000i q^{64} +(462.954 - 462.954i) q^{67} +(264.545 + 264.545i) q^{73} -448.000 q^{76} +884.000i q^{79} -729.000 q^{81} -1296.00i q^{84} +1944.00 q^{91} +(-1131.66 - 1131.66i) q^{93} +(-969.998 + 969.998i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 256 q^{16} + 648 q^{21} - 1232 q^{31} + 864 q^{36} + 728 q^{61} - 1792 q^{76} - 2916 q^{81} + 7776 q^{91}+O(q^{100}) \) 4 * q - 256 * q^16 + 648 * q^21 - 1232 * q^31 + 864 * q^36 + 728 * q^61 - 1792 * q^76 - 2916 * q^81 + 7776 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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