LMFDB - Embedded newform 1575.2.d.g.1324.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1324,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1575.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.d (of order $2$, degree $1$, 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:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1324.3
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	1575.1324
Dual form			1575.2.d.g.1324.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30278i q^{2} +0.302776 q^{4} +1.00000i q^{7} +3.00000i q^{8} +O(q^{10})$	$q+1.30278i q^{2} +0.302776 q^{4} +1.00000i q^{7} +3.00000i q^{8} +3.00000 q^{11} -4.60555i q^{13} -1.30278 q^{14} -3.30278 q^{16} +2.60555i q^{17} +0.605551 q^{19} +3.90833i q^{22} +8.21110i q^{23} +6.00000 q^{26} +0.302776i q^{28} -0.394449 q^{29} +7.21110 q^{31} +1.69722i q^{32} -3.39445 q^{34} +10.2111i q^{37} +0.788897i q^{38} -2.39445i q^{43} +0.908327 q^{44} -10.6972 q^{46} -3.39445i q^{47} -1.00000 q^{49} -1.39445i q^{52} -11.2111i q^{53} -3.00000 q^{56} -0.513878i q^{58} -3.39445 q^{59} +13.2111 q^{61} +9.39445i q^{62} -8.81665 q^{64} +8.39445i q^{67} +0.788897i q^{68} +3.00000 q^{71} +6.60555i q^{73} -13.3028 q^{74} +0.183346 q^{76} +3.00000i q^{77} -6.81665 q^{79} +11.2111i q^{83} +3.11943 q^{86} +9.00000i q^{88} -13.8167 q^{89} +4.60555 q^{91} +2.48612i q^{92} +4.42221 q^{94} -15.2111i q^{97} -1.30278i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{4}+O(q^{10}) $ 4 * q - 6 * q^4	$ 4 q - 6 q^{4} + 12 q^{11} + 2 q^{14} - 6 q^{16} - 12 q^{19} + 24 q^{26} - 16 q^{29} - 28 q^{34} - 18 q^{44} - 50 q^{46} - 4 q^{49} - 12 q^{56} - 28 q^{59} + 24 q^{61} + 8 q^{64} + 12 q^{71} - 46 q^{74} + 44 q^{76} + 16 q^{79} - 38 q^{86} - 12 q^{89} + 4 q^{91} - 40 q^{94}+O(q^{100}) $ 4 * q - 6 * q^4 + 12 * q^11 + 2 * q^14 - 6 * q^16 - 12 * q^19 + 24 * q^26 - 16 * q^29 - 28 * q^34 - 18 * q^44 - 50 * q^46 - 4 * q^49 - 12 * q^56 - 28 * q^59 + 24 * q^61 + 8 * q^64 + 12 * q^71 - 46 * q^74 + 44 * q^76 + 16 * q^79 - 38 * q^86 - 12 * q^89 + 4 * q^91 - 40 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.30278i	0.921201i	0.887607	+	0.460601i	$0.152366\pi$
$2$	1.30278i	0.921201i	−0.887607	+	0.460601i	$0.847634\pi$
$3$	0	0
$3$	0	0
$4$	0.302776	0.151388
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	3.00000i	1.06066i
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	− 4.60555i	− 1.27735i	−0.769477	−	0.638675i	$-0.779483\pi$
$13$	− 4.60555i	− 1.27735i	0.769477	−	0.638675i	$-0.220517\pi$
$14$	−1.30278	−0.348181
$15$	0	0
$16$	−3.30278	−0.825694
$17$	2.60555i	0.631939i	0.948769	+	0.315970i	$0.102330\pi$
$17$	2.60555i	0.631939i	−0.948769	+	0.315970i	$0.897670\pi$
$18$	0	0
$19$	0.605551	0.138923	0.0694615	−	0.997585i	$-0.477872\pi$
$19$	0.605551	0.138923	0.0694615	+	0.997585i	$0.477872\pi$
$20$	0	0
$21$	0	0
$22$	3.90833i	0.833258i
$23$	8.21110i	1.71213i	0.516865	+	0.856067i	$0.327099\pi$
$23$	8.21110i	1.71213i	−0.516865	+	0.856067i	$0.672901\pi$
$24$	0	0
$25$	0	0
$26$	6.00000	1.17670
$27$	0	0
$28$	0.302776i	0.0572192i
$29$	−0.394449	−0.0732473	−0.0366236	−	0.999329i	$-0.511660\pi$
$29$	−0.394449	−0.0732473	−0.0366236	+	0.999329i	$0.511660\pi$
$30$	0	0
$31$	7.21110	1.29515	0.647576	−	0.762001i	$-0.275783\pi$
$31$	7.21110	1.29515	0.647576	+	0.762001i	$0.275783\pi$
$32$	1.69722i	0.300030i
$33$	0	0
$34$	−3.39445	−0.582143
$35$	0	0
$36$	0	0
$37$	10.2111i	1.67869i	0.543595	+	0.839347i	$0.317063\pi$
$37$	10.2111i	1.67869i	−0.543595	+	0.839347i	$0.682937\pi$
$38$	0.788897i	0.127976i
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 2.39445i	− 0.365150i	−0.983192	−	0.182575i	$-0.941557\pi$
$43$	− 2.39445i	− 0.365150i	0.983192	−	0.182575i	$-0.0584432\pi$
$44$	0.908327	0.136935
$45$	0	0
$46$	−10.6972	−1.57722
$47$	− 3.39445i	− 0.495131i	−0.968871	−	0.247566i	$-0.920369\pi$
$47$	− 3.39445i	− 0.495131i	0.968871	−	0.247566i	$-0.0796306\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 1.39445i	− 0.193375i
$53$	− 11.2111i	− 1.53996i	−0.638066	−	0.769982i	$-0.720265\pi$
$53$	− 11.2111i	− 1.53996i	0.638066	−	0.769982i	$-0.279735\pi$
$54$	0	0
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	− 0.513878i	− 0.0674755i
$59$	−3.39445	−0.441920	−0.220960	−	0.975283i	$-0.570919\pi$
$59$	−3.39445	−0.441920	−0.220960	+	0.975283i	$0.570919\pi$
$60$	0	0
$61$	13.2111	1.69151	0.845754	−	0.533573i	$-0.179151\pi$
$61$	13.2111	1.69151	0.845754	+	0.533573i	$0.179151\pi$
$62$	9.39445i	1.19310i
$63$	0	0
$64$	−8.81665	−1.10208
$65$	0	0
$66$	0	0
$67$	8.39445i	1.02555i	0.858524	+	0.512773i	$0.171382\pi$
$67$	8.39445i	1.02555i	−0.858524	+	0.512773i	$0.828618\pi$
$68$	0.788897i	0.0956679i
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	6.60555i	0.773121i	0.922264	+	0.386561i	$0.126337\pi$
$73$	6.60555i	0.773121i	−0.922264	+	0.386561i	$0.873663\pi$
$74$	−13.3028	−1.54642
$75$	0	0
$76$	0.183346	0.0210312
$77$	3.00000i	0.341882i
$78$	0	0
$79$	−6.81665	−0.766933	−0.383467	−	0.923555i	$-0.625270\pi$
$79$	−6.81665	−0.766933	−0.383467	+	0.923555i	$0.625270\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.2111i	1.23058i	0.788301	+	0.615289i	$0.210961\pi$
$83$	11.2111i	1.23058i	−0.788301	+	0.615289i	$0.789039\pi$
$84$	0	0
$85$	0	0
$86$	3.11943	0.336377
$87$	0	0
$88$	9.00000i	0.959403i
$89$	−13.8167	−1.46456	−0.732281	−	0.681002i	$-0.761544\pi$
$89$	−13.8167	−1.46456	−0.732281	+	0.681002i	$0.761544\pi$
$90$	0	0
$91$	4.60555	0.482793
$92$	2.48612i	0.259196i
$93$	0	0
$94$	4.42221	0.456116
$95$	0	0
$96$	0	0
$97$	− 15.2111i	− 1.54445i	−0.635347	−	0.772227i	$-0.719143\pi$
$97$	− 15.2111i	− 1.54445i	0.635347	−	0.772227i	$-0.280857\pi$
$98$	− 1.30278i	− 0.131600i
$99$	0	0
$100$	0	0
$101$	−9.39445	−0.934783	−0.467391	−	0.884051i	$-0.654806\pi$
$101$	−9.39445	−0.934783	−0.467391	+	0.884051i	$0.654806\pi$
$102$	0	0
$103$	17.8167i	1.75553i	0.479094	+	0.877764i	$0.340965\pi$
$103$	17.8167i	1.75553i	−0.479094	+	0.877764i	$0.659035\pi$
$104$	13.8167	1.35483
$105$	0	0
$106$	14.6056	1.41862
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	12.2111	1.16961	0.584806	−	0.811173i	$-0.301171\pi$
$109$	12.2111	1.16961	0.584806	+	0.811173i	$0.301171\pi$
$110$	0	0
$111$	0	0
$112$	− 3.30278i	− 0.312083i
$113$	10.8167i	1.01755i	0.860901	+	0.508773i	$0.169901\pi$
$113$	10.8167i	1.01755i	−0.860901	+	0.508773i	$0.830099\pi$
$114$	0	0
$115$	0	0
$116$	−0.119429	−0.0110887
$117$	0	0
$118$	− 4.42221i	− 0.407097i
$119$	−2.60555	−0.238850
$120$	0	0
$121$	−2.00000	−0.181818
$122$	17.2111i	1.55822i
$123$	0	0
$124$	2.18335	0.196070
$125$	0	0
$126$	0	0
$127$	− 8.81665i	− 0.782352i	−0.920316	−	0.391176i	$-0.872068\pi$
$127$	− 8.81665i	− 0.782352i	0.920316	−	0.391176i	$-0.127932\pi$
$128$	− 8.09167i	− 0.715210i
$129$	0	0
$130$	0	0
$131$	14.6056	1.27609	0.638046	−	0.769998i	$-0.279743\pi$
$131$	14.6056	1.27609	0.638046	+	0.769998i	$0.279743\pi$
$132$	0	0
$133$	0.605551i	0.0525080i
$134$	−10.9361	−0.944734
$135$	0	0
$136$	−7.81665	−0.670273
$137$	− 11.2111i	− 0.957829i	−0.877862	−	0.478915i	$-0.841030\pi$
$137$	− 11.2111i	− 0.957829i	0.877862	−	0.478915i	$-0.158970\pi$
$138$	0	0
$139$	17.0278	1.44428	0.722138	−	0.691749i	$-0.243160\pi$
$139$	17.0278	1.44428	0.722138	+	0.691749i	$0.243160\pi$
$140$	0	0
$141$	0	0
$142$	3.90833i	0.327980i
$143$	− 13.8167i	− 1.15541i
$144$	0	0
$145$	0	0
$146$	−8.60555	−0.712200
$147$	0	0
$148$	3.09167i	0.254134i
$149$	−23.6056	−1.93384	−0.966921	−	0.255076i	$-0.917900\pi$
$149$	−23.6056	−1.93384	−0.966921	+	0.255076i	$0.917900\pi$
$150$	0	0
$151$	−14.8167	−1.20576	−0.602881	−	0.797831i	$-0.705981\pi$
$151$	−14.8167	−1.20576	−0.602881	+	0.797831i	$0.705981\pi$
$152$	1.81665i	0.147350i
$153$	0	0
$154$	−3.90833	−0.314942
$155$	0	0
$156$	0	0
$157$	− 14.4222i	− 1.15102i	−0.817796	−	0.575509i	$-0.804804\pi$
$157$	− 14.4222i	− 1.15102i	0.817796	−	0.575509i	$-0.195196\pi$
$158$	− 8.88057i	− 0.706500i
$159$	0	0
$160$	0	0
$161$	−8.21110	−0.647126
$162$	0	0
$163$	− 2.78890i	− 0.218443i	−0.994017	−	0.109222i	$-0.965164\pi$
$163$	− 2.78890i	− 0.218443i	0.994017	−	0.109222i	$-0.0348358\pi$
$164$	0	0
$165$	0	0
$166$	−14.6056	−1.13361
$167$	− 19.0278i	− 1.47241i	−0.676758	−	0.736206i	$-0.736615\pi$
$167$	− 19.0278i	− 1.47241i	0.676758	−	0.736206i	$-0.263385\pi$
$168$	0	0
$169$	−8.21110	−0.631623
$170$	0	0
$171$	0	0
$172$	− 0.724981i	− 0.0552793i
$173$	13.8167i	1.05046i	0.850960	+	0.525230i	$0.176021\pi$
$173$	13.8167i	1.05046i	−0.850960	+	0.525230i	$0.823979\pi$
$174$	0	0
$175$	0	0
$176$	−9.90833	−0.746868
$177$	0	0
$178$	− 18.0000i	− 1.34916i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	5.39445	0.400966	0.200483	−	0.979697i	$-0.435749\pi$
$181$	5.39445	0.400966	0.200483	+	0.979697i	$0.435749\pi$
$182$	6.00000i	0.444750i
$183$	0	0
$184$	−24.6333	−1.81599
$185$	0	0
$186$	0	0
$187$	7.81665i	0.571610i
$188$	− 1.02776i	− 0.0749568i
$189$	0	0
$190$	0	0
$191$	17.2111	1.24535	0.622676	−	0.782480i	$-0.286046\pi$
$191$	17.2111	1.24535	0.622676	+	0.782480i	$0.286046\pi$
$192$	0	0
$193$	6.21110i	0.447085i	0.974694	+	0.223542i	$0.0717621\pi$
$193$	6.21110i	0.447085i	−0.974694	+	0.223542i	$0.928238\pi$
$194$	19.8167	1.42275
$195$	0	0
$196$	−0.302776	−0.0216268
$197$	4.81665i	0.343172i	0.985169	+	0.171586i	$0.0548892\pi$
$197$	4.81665i	0.343172i	−0.985169	+	0.171586i	$0.945111\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	− 12.2389i	− 0.861123i
$203$	− 0.394449i	− 0.0276849i
$204$	0	0
$205$	0	0
$206$	−23.2111	−1.61719
$207$	0	0
$208$	15.2111i	1.05470i
$209$	1.81665	0.125661
$210$	0	0
$211$	−2.42221	−0.166751	−0.0833757	−	0.996518i	$-0.526570\pi$
$211$	−2.42221	−0.166751	−0.0833757	+	0.996518i	$0.526570\pi$
$212$	− 3.39445i	− 0.233132i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.21110i	0.489522i
$218$	15.9083i	1.07745i
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	− 22.6056i	− 1.51378i	−0.653542	−	0.756890i	$-0.726718\pi$
$223$	− 22.6056i	− 1.51378i	0.653542	−	0.756890i	$-0.273282\pi$
$224$	−1.69722	−0.113401
$225$	0	0
$226$	−14.0917	−0.937364
$227$	− 15.3944i	− 1.02177i	−0.859650	−	0.510883i	$-0.829319\pi$
$227$	− 15.3944i	− 1.02177i	0.859650	−	0.510883i	$-0.170681\pi$
$228$	0	0
$229$	−7.21110	−0.476523	−0.238262	−	0.971201i	$-0.576578\pi$
$229$	−7.21110	−0.476523	−0.238262	+	0.971201i	$0.576578\pi$
$230$	0	0
$231$	0	0
$232$	− 1.18335i	− 0.0776905i
$233$	− 22.8167i	− 1.49477i	−0.664392	−	0.747384i	$-0.731309\pi$
$233$	− 22.8167i	− 1.49477i	0.664392	−	0.747384i	$-0.268691\pi$
$234$	0	0
$235$	0	0
$236$	−1.02776	−0.0669012
$237$	0	0
$238$	− 3.39445i	− 0.220029i
$239$	29.2111	1.88951	0.944755	−	0.327779i	$-0.106300\pi$
$239$	29.2111	1.88951	0.944755	+	0.327779i	$0.106300\pi$
$240$	0	0
$241$	16.6056	1.06966	0.534829	−	0.844960i	$-0.320376\pi$
$241$	16.6056	1.06966	0.534829	+	0.844960i	$0.320376\pi$
$242$	− 2.60555i	− 0.167491i
$243$	0	0
$244$	4.00000	0.256074
$245$	0	0
$246$	0	0
$247$	− 2.78890i	− 0.177453i
$248$	21.6333i	1.37372i
$249$	0	0
$250$	0	0
$251$	−7.81665	−0.493383	−0.246691	−	0.969094i	$-0.579343\pi$
$251$	−7.81665	−0.493383	−0.246691	+	0.969094i	$0.579343\pi$
$252$	0	0
$253$	24.6333i	1.54868i
$254$	11.4861	0.720703
$255$	0	0
$256$	−7.09167	−0.443230
$257$	− 21.6333i	− 1.34945i	−0.738070	−	0.674724i	$-0.764263\pi$
$257$	− 21.6333i	− 1.34945i	0.738070	−	0.674724i	$-0.235737\pi$
$258$	0	0
$259$	−10.2111	−0.634487
$260$	0	0
$261$	0	0
$262$	19.0278i	1.17554i
$263$	− 20.2111i	− 1.24627i	−0.782114	−	0.623135i	$-0.785859\pi$
$263$	− 20.2111i	− 1.24627i	0.782114	−	0.623135i	$-0.214141\pi$
$264$	0	0
$265$	0	0
$266$	−0.788897	−0.0483704
$267$	0	0
$268$	2.54163i	0.155255i
$269$	11.2111	0.683553	0.341776	−	0.939781i	$-0.388971\pi$
$269$	11.2111	0.683553	0.341776	+	0.939781i	$0.388971\pi$
$270$	0	0
$271$	−19.3944	−1.17813	−0.589064	−	0.808086i	$-0.700504\pi$
$271$	−19.3944	−1.17813	−0.589064	+	0.808086i	$0.700504\pi$
$272$	− 8.60555i	− 0.521788i
$273$	0	0
$274$	14.6056	0.882354
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	22.1833i	1.33047i
$279$	0	0
$280$	0	0
$281$	22.8167	1.36113	0.680564	−	0.732689i	$-0.261735\pi$
$281$	22.8167	1.36113	0.680564	+	0.732689i	$0.261735\pi$
$282$	0	0
$283$	− 29.6333i	− 1.76152i	−0.473565	−	0.880759i	$-0.657033\pi$
$283$	− 29.6333i	− 1.76152i	0.473565	−	0.880759i	$-0.342967\pi$
$284$	0.908327	0.0538993
$285$	0	0
$286$	18.0000	1.06436
$287$	0	0
$288$	0	0
$289$	10.2111	0.600653
$290$	0	0
$291$	0	0
$292$	2.00000i	0.117041i
$293$	3.39445i	0.198306i	0.995072	+	0.0991529i	$0.0316133\pi$
$293$	3.39445i	0.198306i	−0.995072	+	0.0991529i	$0.968387\pi$
$294$	0	0
$295$	0	0
$296$	−30.6333	−1.78052
$297$	0	0
$298$	− 30.7527i	− 1.78146i
$299$	37.8167	2.18699
$300$	0	0
$301$	2.39445	0.138014
$302$	− 19.3028i	− 1.11075i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	− 1.39445i	− 0.0795854i	−0.999208	−	0.0397927i	$-0.987330\pi$
$307$	− 1.39445i	− 0.0795854i	0.999208	−	0.0397927i	$-0.0126698\pi$
$308$	0.908327i	0.0517567i
$309$	0	0
$310$	0	0
$311$	7.81665	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$311$	7.81665	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$312$	0	0
$313$	2.18335i	0.123410i	0.998094	+	0.0617050i	$0.0196538\pi$
$313$	2.18335i	0.123410i	−0.998094	+	0.0617050i	$0.980346\pi$
$314$	18.7889	1.06032
$315$	0	0
$316$	−2.06392	−0.116104
$317$	23.6056i	1.32582i	0.748699	+	0.662910i	$0.230679\pi$
$317$	23.6056i	1.32582i	−0.748699	+	0.662910i	$0.769321\pi$
$318$	0	0
$319$	−1.18335	−0.0662547
$320$	0	0
$321$	0	0
$322$	− 10.6972i	− 0.596133i
$323$	1.57779i	0.0877909i
$324$	0	0
$325$	0	0
$326$	3.63331	0.201230
$327$	0	0
$328$	0	0
$329$	3.39445	0.187142
$330$	0	0
$331$	29.2389	1.60711	0.803557	−	0.595228i	$-0.202938\pi$
$331$	29.2389	1.60711	0.803557	+	0.595228i	$0.202938\pi$
$332$	3.39445i	0.186295i
$333$	0	0
$334$	24.7889	1.35639
$335$	0	0
$336$	0	0
$337$	7.21110i	0.392814i	0.980522	+	0.196407i	$0.0629273\pi$
$337$	7.21110i	0.392814i	−0.980522	+	0.196407i	$0.937073\pi$
$338$	− 10.6972i	− 0.581852i
$339$	0	0
$340$	0	0
$341$	21.6333	1.17151
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	7.18335	0.387300
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 15.7889i	− 0.847592i	−0.905758	−	0.423796i	$-0.860697\pi$
$347$	− 15.7889i	− 0.847592i	0.905758	−	0.423796i	$-0.139303\pi$
$348$	0	0
$349$	33.4500	1.79054	0.895268	−	0.445529i	$-0.146984\pi$
$349$	33.4500	1.79054	0.895268	+	0.445529i	$0.146984\pi$
$350$	0	0
$351$	0	0
$352$	5.09167i	0.271387i
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	0	0
$356$	−4.18335	−0.221717
$357$	0	0
$358$	0	0
$359$	−18.6333	−0.983428	−0.491714	−	0.870757i	$-0.663630\pi$
$359$	−18.6333	−0.983428	−0.491714	+	0.870757i	$0.663630\pi$
$360$	0	0
$361$	−18.6333	−0.980700
$362$	7.02776i	0.369371i
$363$	0	0
$364$	1.39445	0.0730890
$365$	0	0
$366$	0	0
$367$	− 14.4222i	− 0.752833i	−0.926451	−	0.376416i	$-0.877156\pi$
$367$	− 14.4222i	− 0.752833i	0.926451	−	0.376416i	$-0.122844\pi$
$368$	− 27.1194i	− 1.41370i
$369$	0	0
$370$	0	0
$371$	11.2111	0.582051
$372$	0	0
$373$	1.00000i	0.0517780i	0.999665	+	0.0258890i	$0.00824165\pi$
$373$	1.00000i	0.0517780i	−0.999665	+	0.0258890i	$0.991758\pi$
$374$	−10.1833	−0.526568
$375$	0	0
$376$	10.1833	0.525166
$377$	1.81665i	0.0935624i
$378$	0	0
$379$	−6.81665	−0.350148	−0.175074	−	0.984555i	$-0.556016\pi$
$379$	−6.81665	−0.350148	−0.175074	+	0.984555i	$0.556016\pi$
$380$	0	0
$381$	0	0
$382$	22.4222i	1.14722i
$383$	24.2389i	1.23855i	0.785175	+	0.619274i	$0.212573\pi$
$383$	24.2389i	1.23855i	−0.785175	+	0.619274i	$0.787427\pi$
$384$	0	0
$385$	0	0
$386$	−8.09167	−0.411855
$387$	0	0
$388$	− 4.60555i	− 0.233811i
$389$	−7.18335	−0.364210	−0.182105	−	0.983279i	$-0.558291\pi$
$389$	−7.18335	−0.364210	−0.182105	+	0.983279i	$0.558291\pi$
$390$	0	0
$391$	−21.3944	−1.08196
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	−6.27502	−0.316131
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	− 10.4222i	− 0.522418i
$399$	0	0
$400$	0	0
$401$	−4.81665	−0.240532	−0.120266	−	0.992742i	$-0.538375\pi$
$401$	−4.81665	−0.240532	−0.120266	+	0.992742i	$0.538375\pi$
$402$	0	0
$403$	− 33.2111i	− 1.65436i
$404$	−2.84441	−0.141515
$405$	0	0
$406$	0.513878	0.0255033
$407$	30.6333i	1.51844i
$408$	0	0
$409$	−9.81665	−0.485402	−0.242701	−	0.970101i	$-0.578033\pi$
$409$	−9.81665	−0.485402	−0.242701	+	0.970101i	$0.578033\pi$
$410$	0	0
$411$	0	0
$412$	5.39445i	0.265765i
$413$	− 3.39445i	− 0.167030i
$414$	0	0
$415$	0	0
$416$	7.81665	0.383243
$417$	0	0
$418$	2.36669i	0.115759i
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	4.21110	0.205237	0.102618	−	0.994721i	$-0.467278\pi$
$421$	4.21110	0.205237	0.102618	+	0.994721i	$0.467278\pi$
$422$	− 3.15559i	− 0.153612i
$423$	0	0
$424$	33.6333	1.63338
$425$	0	0
$426$	0	0
$427$	13.2111i	0.639330i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.4222	−1.65806	−0.829030	−	0.559205i	$-0.811107\pi$
$431$	−34.4222	−1.65806	−0.829030	+	0.559205i	$0.811107\pi$
$432$	0	0
$433$	22.7889i	1.09516i	0.836752	+	0.547582i	$0.184452\pi$
$433$	22.7889i	1.09516i	−0.836752	+	0.547582i	$0.815548\pi$
$434$	−9.39445	−0.450948
$435$	0	0
$436$	3.69722	0.177065
$437$	4.97224i	0.237855i
$438$	0	0
$439$	0.605551	0.0289014	0.0144507	−	0.999896i	$-0.495400\pi$
$439$	0.605551	0.0289014	0.0144507	+	0.999896i	$0.495400\pi$
$440$	0	0
$441$	0	0
$442$	15.6333i	0.743601i
$443$	− 27.6333i	− 1.31290i	−0.754370	−	0.656449i	$-0.772058\pi$
$443$	− 27.6333i	− 1.31290i	0.754370	−	0.656449i	$-0.227942\pi$
$444$	0	0
$445$	0	0
$446$	29.4500	1.39450
$447$	0	0
$448$	− 8.81665i	− 0.416548i
$449$	−13.1833	−0.622161	−0.311080	−	0.950384i	$-0.600691\pi$
$449$	−13.1833	−0.622161	−0.311080	+	0.950384i	$0.600691\pi$
$450$	0	0
$451$	0	0
$452$	3.27502i	0.154044i
$453$	0	0
$454$	20.0555	0.941252
$455$	0	0
$456$	0	0
$457$	− 18.2111i	− 0.851879i	−0.904752	−	0.425940i	$-0.859944\pi$
$457$	− 18.2111i	− 0.851879i	0.904752	−	0.425940i	$-0.140056\pi$
$458$	− 9.39445i	− 0.438974i
$459$	0	0
$460$	0	0
$461$	−24.2389	−1.12892	−0.564458	−	0.825462i	$-0.690915\pi$
$461$	−24.2389	−1.12892	−0.564458	+	0.825462i	$0.690915\pi$
$462$	0	0
$463$	− 2.78890i	− 0.129611i	−0.997898	−	0.0648055i	$-0.979357\pi$
$463$	− 2.78890i	− 0.129611i	0.997898	−	0.0648055i	$-0.0206427\pi$
$464$	1.30278	0.0604798
$465$	0	0
$466$	29.7250	1.37698
$467$	− 28.4222i	− 1.31522i	−0.753357	−	0.657611i	$-0.771567\pi$
$467$	− 28.4222i	− 1.31522i	0.753357	−	0.657611i	$-0.228433\pi$
$468$	0	0
$469$	−8.39445	−0.387620
$470$	0	0
$471$	0	0
$472$	− 10.1833i	− 0.468727i
$473$	− 7.18335i	− 0.330291i
$474$	0	0
$475$	0	0
$476$	−0.788897	−0.0361591
$477$	0	0
$478$	38.0555i	1.74062i
$479$	27.3944	1.25168	0.625842	−	0.779950i	$-0.284755\pi$
$479$	27.3944	1.25168	0.625842	+	0.779950i	$0.284755\pi$
$480$	0	0
$481$	47.0278	2.14428
$482$	21.6333i	0.985370i
$483$	0	0
$484$	−0.605551	−0.0275251
$485$	0	0
$486$	0	0
$487$	0.816654i	0.0370061i	0.999829	+	0.0185031i	$0.00589004\pi$
$487$	0.816654i	0.0370061i	−0.999829	+	0.0185031i	$0.994110\pi$
$488$	39.6333i	1.79412i
$489$	0	0
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	− 1.02776i	− 0.0462878i
$494$	3.63331	0.163470
$495$	0	0
$496$	−23.8167	−1.06940
$497$	3.00000i	0.134568i
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	− 10.1833i	− 0.454505i
$503$	− 23.2111i	− 1.03493i	−0.855704	−	0.517466i	$-0.826875\pi$
$503$	− 23.2111i	− 1.03493i	0.855704	−	0.517466i	$-0.173125\pi$
$504$	0	0
$505$	0	0
$506$	−32.0917	−1.42665
$507$	0	0
$508$	− 2.66947i	− 0.118438i
$509$	8.60555	0.381434	0.190717	−	0.981645i	$-0.438919\pi$
$509$	8.60555	0.381434	0.190717	+	0.981645i	$0.438919\pi$
$510$	0	0
$511$	−6.60555	−0.292212
$512$	− 25.4222i	− 1.12351i
$513$	0	0
$514$	28.1833	1.24311
$515$	0	0
$516$	0	0
$517$	− 10.1833i	− 0.447863i
$518$	− 13.3028i	− 0.584490i
$519$	0	0
$520$	0	0
$521$	7.57779	0.331989	0.165995	−	0.986127i	$-0.446917\pi$
$521$	7.57779	0.331989	0.165995	+	0.986127i	$0.446917\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	4.42221	0.193185
$525$	0	0
$526$	26.3305	1.14807
$527$	18.7889i	0.818457i
$528$	0	0
$529$	−44.4222	−1.93140
$530$	0	0
$531$	0	0
$532$	0.183346i	0.00794906i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−25.1833	−1.08775
$537$	0	0
$538$	14.6056i	0.629690i
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−2.57779	−0.110828	−0.0554140	−	0.998463i	$-0.517648\pi$
$541$	−2.57779	−0.110828	−0.0554140	+	0.998463i	$0.517648\pi$
$542$	− 25.2666i	− 1.08529i
$543$	0	0
$544$	−4.42221	−0.189600
$545$	0	0
$546$	0	0
$547$	− 22.3944i	− 0.957517i	−0.877947	−	0.478759i	$-0.841087\pi$
$547$	− 22.3944i	− 0.957517i	0.877947	−	0.478759i	$-0.158913\pi$
$548$	− 3.39445i	− 0.145004i
$549$	0	0
$550$	0	0
$551$	−0.238859	−0.0101757
$552$	0	0
$553$	− 6.81665i	− 0.289874i
$554$	13.0278	0.553496
$555$	0	0
$556$	5.15559	0.218646
$557$	39.2389i	1.66260i	0.555821	+	0.831302i	$0.312404\pi$
$557$	39.2389i	1.66260i	−0.555821	+	0.831302i	$0.687596\pi$
$558$	0	0
$559$	−11.0278	−0.466424
$560$	0	0
$561$	0	0
$562$	29.7250i	1.25387i
$563$	0.788897i	0.0332481i	0.999862	+	0.0166240i	$0.00529184\pi$
$563$	0.788897i	0.0332481i	−0.999862	+	0.0166240i	$0.994708\pi$
$564$	0	0
$565$	0	0
$566$	38.6056	1.62271
$567$	0	0
$568$	9.00000i	0.377632i
$569$	−22.8167	−0.956524	−0.478262	−	0.878217i	$-0.658733\pi$
$569$	−22.8167	−0.956524	−0.478262	+	0.878217i	$0.658733\pi$
$570$	0	0
$571$	1.60555	0.0671902	0.0335951	−	0.999436i	$-0.489304\pi$
$571$	1.60555	0.0671902	0.0335951	+	0.999436i	$0.489304\pi$
$572$	− 4.18335i	− 0.174914i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.3944i	0.973924i	0.873423	+	0.486962i	$0.161895\pi$
$577$	23.3944i	0.973924i	−0.873423	+	0.486962i	$0.838105\pi$
$578$	13.3028i	0.553323i
$579$	0	0
$580$	0	0
$581$	−11.2111	−0.465115
$582$	0	0
$583$	− 33.6333i	− 1.39295i
$584$	−19.8167	−0.820019
$585$	0	0
$586$	−4.42221	−0.182680
$587$	− 16.1833i	− 0.667958i	−0.942580	−	0.333979i	$-0.891609\pi$
$587$	− 16.1833i	− 0.667958i	0.942580	−	0.333979i	$-0.108391\pi$
$588$	0	0
$589$	4.36669	0.179926
$590$	0	0
$591$	0	0
$592$	− 33.7250i	− 1.38609i
$593$	− 15.6333i	− 0.641983i	−0.947082	−	0.320991i	$-0.895984\pi$
$593$	− 15.6333i	− 0.641983i	0.947082	−	0.320991i	$-0.104016\pi$
$594$	0	0
$595$	0	0
$596$	−7.14719	−0.292760
$597$	0	0
$598$	49.2666i	2.01466i
$599$	20.2111	0.825803	0.412902	−	0.910776i	$-0.364515\pi$
$599$	20.2111	0.825803	0.412902	+	0.910776i	$0.364515\pi$
$600$	0	0
$601$	2.78890	0.113761	0.0568807	−	0.998381i	$-0.481885\pi$
$601$	2.78890	0.113761	0.0568807	+	0.998381i	$0.481885\pi$
$602$	3.11943i	0.127138i
$603$	0	0
$604$	−4.48612	−0.182538
$605$	0	0
$606$	0	0
$607$	− 2.18335i	− 0.0886193i	−0.999018	−	0.0443096i	$-0.985891\pi$
$607$	− 2.18335i	− 0.0886193i	0.999018	−	0.0443096i	$-0.0141088\pi$
$608$	1.02776i	0.0416810i
$609$	0	0
$610$	0	0
$611$	−15.6333	−0.632456
$612$	0	0
$613$	20.5778i	0.831129i	0.909564	+	0.415565i	$0.136416\pi$
$613$	20.5778i	0.831129i	−0.909564	+	0.415565i	$0.863584\pi$
$614$	1.81665	0.0733142
$615$	0	0
$616$	−9.00000	−0.362620
$617$	− 1.18335i	− 0.0476397i	−0.999716	−	0.0238199i	$-0.992417\pi$
$617$	− 1.18335i	− 0.0476397i	0.999716	−	0.0238199i	$-0.00758281\pi$
$618$	0	0
$619$	−4.60555	−0.185113	−0.0925564	−	0.995707i	$-0.529504\pi$
$619$	−4.60555	−0.185113	−0.0925564	+	0.995707i	$0.529504\pi$
$620$	0	0
$621$	0	0
$622$	10.1833i	0.408315i
$623$	− 13.8167i	− 0.553553i
$624$	0	0
$625$	0	0
$626$	−2.84441	−0.113685
$627$	0	0
$628$	− 4.36669i	− 0.174250i
$629$	−26.6056	−1.06083
$630$	0	0
$631$	−14.0278	−0.558436	−0.279218	−	0.960228i	$-0.590075\pi$
$631$	−14.0278	−0.558436	−0.279218	+	0.960228i	$0.590075\pi$
$632$	− 20.4500i	− 0.813456i
$633$	0	0
$634$	−30.7527	−1.22135
$635$	0	0
$636$	0	0
$637$	4.60555i	0.182479i
$638$	− 1.54163i	− 0.0610339i
$639$	0	0
$640$	0	0
$641$	−1.18335	−0.0467394	−0.0233697	−	0.999727i	$-0.507439\pi$
$641$	−1.18335	−0.0467394	−0.0233697	+	0.999727i	$0.507439\pi$
$642$	0	0
$643$	− 9.57779i	− 0.377711i	−0.982005	−	0.188856i	$-0.939522\pi$
$643$	− 9.57779i	− 0.377711i	0.982005	−	0.188856i	$-0.0604778\pi$
$644$	−2.48612	−0.0979669
$645$	0	0
$646$	−2.05551	−0.0808731
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	0	0
$649$	−10.1833	−0.399731
$650$	0	0
$651$	0	0
$652$	− 0.844410i	− 0.0330697i
$653$	11.2111i	0.438724i	0.975643	+	0.219362i	$0.0703976\pi$
$653$	11.2111i	0.438724i	−0.975643	+	0.219362i	$0.929602\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	4.42221i	0.172396i
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−41.8167	−1.62648	−0.813240	−	0.581929i	$-0.802298\pi$
$661$	−41.8167	−1.62648	−0.813240	+	0.581929i	$0.802298\pi$
$662$	38.0917i	1.48047i
$663$	0	0
$664$	−33.6333	−1.30523
$665$	0	0
$666$	0	0
$667$	− 3.23886i	− 0.125409i
$668$	− 5.76114i	− 0.222905i
$669$	0	0
$670$	0	0
$671$	39.6333	1.53003
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	−9.39445	−0.361861
$675$	0	0
$676$	−2.48612	−0.0956201
$677$	23.2111i	0.892075i	0.895014	+	0.446038i	$0.147165\pi$
$677$	23.2111i	0.892075i	−0.895014	+	0.446038i	$0.852835\pi$
$678$	0	0
$679$	15.2111	0.583749
$680$	0	0
$681$	0	0
$682$	28.1833i	1.07920i
$683$	24.6333i	0.942567i	0.881982	+	0.471284i	$0.156209\pi$
$683$	24.6333i	0.942567i	−0.881982	+	0.471284i	$0.843791\pi$
$684$	0	0
$685$	0	0
$686$	1.30278	0.0497402
$687$	0	0
$688$	7.90833i	0.301502i
$689$	−51.6333	−1.96707
$690$	0	0
$691$	26.7889	1.01910	0.509549	−	0.860442i	$-0.329812\pi$
$691$	26.7889	1.01910	0.509549	+	0.860442i	$0.329812\pi$
$692$	4.18335i	0.159027i
$693$	0	0
$694$	20.5694	0.780803
$695$	0	0
$696$	0	0
$697$	0	0
$698$	43.5778i	1.64944i
$699$	0	0
$700$	0	0
$701$	35.2111	1.32990	0.664952	−	0.746886i	$-0.268452\pi$
$701$	35.2111	1.32990	0.664952	+	0.746886i	$0.268452\pi$
$702$	0	0
$703$	6.18335i	0.233209i
$704$	−26.4500	−0.996870
$705$	0	0
$706$	39.0833	1.47092
$707$	− 9.39445i	− 0.353315i
$708$	0	0
$709$	−22.8444	−0.857940	−0.428970	−	0.903319i	$-0.641123\pi$
$709$	−22.8444	−0.857940	−0.428970	+	0.903319i	$0.641123\pi$
$710$	0	0
$711$	0	0
$712$	− 41.4500i	− 1.55340i
$713$	59.2111i	2.21747i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	− 24.2750i	− 0.905936i
$719$	41.2111	1.53691	0.768457	−	0.639901i	$-0.221025\pi$
$719$	41.2111	1.53691	0.768457	+	0.639901i	$0.221025\pi$
$720$	0	0
$721$	−17.8167	−0.663527
$722$	− 24.2750i	− 0.903423i
$723$	0	0
$724$	1.63331	0.0607014
$725$	0	0
$726$	0	0
$727$	− 30.6056i	− 1.13510i	−0.823340	−	0.567549i	$-0.807892\pi$
$727$	− 30.6056i	− 1.13510i	0.823340	−	0.567549i	$-0.192108\pi$
$728$	13.8167i	0.512079i
$729$	0	0
$730$	0	0
$731$	6.23886	0.230753
$732$	0	0
$733$	− 20.0000i	− 0.738717i	−0.929287	−	0.369358i	$-0.879577\pi$
$733$	− 20.0000i	− 0.738717i	0.929287	−	0.369358i	$-0.120423\pi$
$734$	18.7889	0.693511
$735$	0	0
$736$	−13.9361	−0.513691
$737$	25.1833i	0.927640i
$738$	0	0
$739$	−24.8167	−0.912895	−0.456448	−	0.889750i	$-0.650878\pi$
$739$	−24.8167	−0.912895	−0.456448	+	0.889750i	$0.650878\pi$
$740$	0	0
$741$	0	0
$742$	14.6056i	0.536187i
$743$	15.6333i	0.573530i	0.958001	+	0.286765i	$0.0925800\pi$
$743$	15.6333i	0.573530i	−0.958001	+	0.286765i	$0.907420\pi$
$744$	0	0
$745$	0	0
$746$	−1.30278	−0.0476980
$747$	0	0
$748$	2.36669i	0.0865348i
$749$	0	0
$750$	0	0
$751$	−43.6333	−1.59220	−0.796101	−	0.605164i	$-0.793108\pi$
$751$	−43.6333	−1.59220	−0.796101	+	0.605164i	$0.793108\pi$
$752$	11.2111i	0.408827i
$753$	0	0
$754$	−2.36669	−0.0861899
$755$	0	0
$756$	0	0
$757$	41.0000i	1.49017i	0.666969	+	0.745085i	$0.267591\pi$
$757$	41.0000i	1.49017i	−0.666969	+	0.745085i	$0.732409\pi$
$758$	− 8.88057i	− 0.322557i
$759$	0	0
$760$	0	0
$761$	−15.6333	−0.566707	−0.283353	−	0.959016i	$-0.591447\pi$
$761$	−15.6333	−0.566707	−0.283353	+	0.959016i	$0.591447\pi$
$762$	0	0
$763$	12.2111i	0.442072i
$764$	5.21110	0.188531
$765$	0	0
$766$	−31.5778	−1.14095
$767$	15.6333i	0.564486i
$768$	0	0
$769$	−11.6333	−0.419508	−0.209754	−	0.977754i	$-0.567266\pi$
$769$	−11.6333	−0.419508	−0.209754	+	0.977754i	$0.567266\pi$
$770$	0	0
$771$	0	0
$772$	1.88057i	0.0676832i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	45.6333	1.63814
$777$	0	0
$778$	− 9.35829i	− 0.335511i
$779$	0	0
$780$	0	0
$781$	9.00000	0.322045
$782$	− 27.8722i	− 0.996707i
$783$	0	0
$784$	3.30278	0.117956
$785$	0	0
$786$	0	0
$787$	− 22.2389i	− 0.792730i	−0.918093	−	0.396365i	$-0.870272\pi$
$787$	− 22.2389i	− 0.792730i	0.918093	−	0.396365i	$-0.129728\pi$
$788$	1.45837i	0.0519521i
$789$	0	0
$790$	0	0
$791$	−10.8167	−0.384596
$792$	0	0
$793$	− 60.8444i	− 2.16065i
$794$	−2.60555	−0.0924676
$795$	0	0
$796$	−2.42221	−0.0858528
$797$	− 35.4500i	− 1.25570i	−0.778334	−	0.627851i	$-0.783935\pi$
$797$	− 35.4500i	− 1.25570i	0.778334	−	0.627851i	$-0.216065\pi$
$798$	0	0
$799$	8.84441	0.312893
$800$	0	0
$801$	0	0
$802$	− 6.27502i	− 0.221579i
$803$	19.8167i	0.699315i
$804$	0	0
$805$	0	0
$806$	43.2666	1.52400
$807$	0	0
$808$	− 28.1833i	− 0.991487i
$809$	−32.4500	−1.14088	−0.570440	−	0.821339i	$-0.693227\pi$
$809$	−32.4500	−1.14088	−0.570440	+	0.821339i	$0.693227\pi$
$810$	0	0
$811$	−42.8444	−1.50447	−0.752235	−	0.658894i	$-0.771024\pi$
$811$	−42.8444	−1.50447	−0.752235	+	0.658894i	$0.771024\pi$
$812$	− 0.119429i	− 0.00419115i
$813$	0	0
$814$	−39.9083	−1.39879
$815$	0	0
$816$	0	0
$817$	− 1.44996i	− 0.0507277i
$818$	− 12.7889i	− 0.447153i
$819$	0	0
$820$	0	0
$821$	37.2666	1.30061	0.650307	−	0.759672i	$-0.274640\pi$
$821$	37.2666	1.30061	0.650307	+	0.759672i	$0.274640\pi$
$822$	0	0
$823$	− 3.18335i	− 0.110964i	−0.998460	−	0.0554822i	$-0.982330\pi$
$823$	− 3.18335i	− 0.110964i	0.998460	−	0.0554822i	$-0.0176696\pi$
$824$	−53.4500	−1.86202
$825$	0	0
$826$	4.42221	0.153868
$827$	− 12.6333i	− 0.439303i	−0.975578	−	0.219652i	$-0.929508\pi$
$827$	− 12.6333i	− 0.439303i	0.975578	−	0.219652i	$-0.0704921\pi$
$828$	0	0
$829$	−50.2389	−1.74487	−0.872434	−	0.488732i	$-0.837460\pi$
$829$	−50.2389	−1.74487	−0.872434	+	0.488732i	$0.837460\pi$
$830$	0	0
$831$	0	0
$832$	40.6056i	1.40774i
$833$	− 2.60555i	− 0.0902770i
$834$	0	0
$835$	0	0
$836$	0.550039	0.0190235
$837$	0	0
$838$	7.81665i	0.270022i
$839$	−43.8167	−1.51272	−0.756359	−	0.654156i	$-0.773024\pi$
$839$	−43.8167	−1.51272	−0.756359	+	0.654156i	$0.773024\pi$
$840$	0	0
$841$	−28.8444	−0.994635
$842$	5.48612i	0.189064i
$843$	0	0
$844$	−0.733385	−0.0252441
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	37.0278i	1.27154i
$849$	0	0
$850$	0	0
$851$	−83.8444	−2.87415
$852$	0	0
$853$	21.2111i	0.726254i	0.931740	+	0.363127i	$0.118291\pi$
$853$	21.2111i	0.726254i	−0.931740	+	0.363127i	$0.881709\pi$
$854$	−17.2111	−0.588952
$855$	0	0
$856$	0	0
$857$	15.6333i	0.534024i	0.963693	+	0.267012i	$0.0860363\pi$
$857$	15.6333i	0.534024i	−0.963693	+	0.267012i	$0.913964\pi$
$858$	0	0
$859$	6.36669	0.217229	0.108614	−	0.994084i	$-0.465359\pi$
$859$	6.36669	0.217229	0.108614	+	0.994084i	$0.465359\pi$
$860$	0	0
$861$	0	0
$862$	− 44.8444i	− 1.52741i
$863$	48.6333i	1.65550i	0.561099	+	0.827749i	$0.310379\pi$
$863$	48.6333i	1.65550i	−0.561099	+	0.827749i	$0.689621\pi$
$864$	0	0
$865$	0	0
$866$	−29.6888	−1.00887
$867$	0	0
$868$	2.18335i	0.0741076i
$869$	−20.4500	−0.693717
$870$	0	0
$871$	38.6611	1.30998
$872$	36.6333i	1.24056i
$873$	0	0
$874$	−6.47772	−0.219112
$875$	0	0
$876$	0	0
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	0.788897i	0.0266240i
$879$	0	0
$880$	0	0
$881$	9.63331	0.324554	0.162277	−	0.986745i	$-0.448116\pi$
$881$	9.63331	0.324554	0.162277	+	0.986745i	$0.448116\pi$
$882$	0	0
$883$	− 8.39445i	− 0.282496i	−0.989974	−	0.141248i	$-0.954889\pi$
$883$	− 8.39445i	− 0.282496i	0.989974	−	0.141248i	$-0.0451114\pi$
$884$	3.63331	0.122201
$885$	0	0
$886$	36.0000	1.20944
$887$	29.2111i	0.980813i	0.871494	+	0.490406i	$0.163152\pi$
$887$	29.2111i	0.980813i	−0.871494	+	0.490406i	$0.836848\pi$
$888$	0	0
$889$	8.81665	0.295701
$890$	0	0
$891$	0	0
$892$	− 6.84441i	− 0.229168i
$893$	− 2.05551i	− 0.0687851i
$894$	0	0
$895$	0	0
$896$	8.09167	0.270324
$897$	0	0
$898$	− 17.1749i	− 0.573135i
$899$	−2.84441	−0.0948664
$900$	0	0
$901$	29.2111	0.973163
$902$	0	0
$903$	0	0
$904$	−32.4500	−1.07927
$905$	0	0
$906$	0	0
$907$	1.21110i	0.0402140i	0.999798	+	0.0201070i	$0.00640069\pi$
$907$	1.21110i	0.0402140i	−0.999798	+	0.0201070i	$0.993599\pi$
$908$	− 4.66106i	− 0.154683i
$909$	0	0
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	0	0
$913$	33.6333i	1.11310i
$914$	23.7250	0.784753
$915$	0	0
$916$	−2.18335	−0.0721398
$917$	14.6056i	0.482318i
$918$	0	0
$919$	10.3944	0.342881	0.171441	−	0.985194i	$-0.445158\pi$
$919$	10.3944	0.342881	0.171441	+	0.985194i	$0.445158\pi$
$920$	0	0
$921$	0	0
$922$	− 31.5778i	− 1.03996i
$923$	− 13.8167i	− 0.454781i
$924$	0	0
$925$	0	0
$926$	3.63331	0.119398
$927$	0	0
$928$	− 0.669468i	− 0.0219764i
$929$	30.2389	0.992105	0.496052	−	0.868293i	$-0.334782\pi$
$929$	30.2389	0.992105	0.496052	+	0.868293i	$0.334782\pi$
$930$	0	0
$931$	−0.605551	−0.0198461
$932$	− 6.90833i	− 0.226290i
$933$	0	0
$934$	37.0278	1.21159
$935$	0	0
$936$	0	0
$937$	44.4777i	1.45302i	0.687154	+	0.726512i	$0.258860\pi$
$937$	44.4777i	1.45302i	−0.687154	+	0.726512i	$0.741140\pi$
$938$	− 10.9361i	− 0.357076i
$939$	0	0
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0	0
$944$	11.2111	0.364890
$945$	0	0
$946$	9.35829	0.304264
$947$	− 56.8444i	− 1.84720i	−0.383363	−	0.923598i	$-0.625234\pi$
$947$	− 56.8444i	− 1.84720i	0.383363	−	0.923598i	$-0.374766\pi$
$948$	0	0
$949$	30.4222	0.987547
$950$	0	0
$951$	0	0
$952$	− 7.81665i	− 0.253339i
$953$	− 6.39445i	− 0.207137i	−0.994622	−	0.103568i	$-0.966974\pi$
$953$	− 6.39445i	− 0.207137i	0.994622	−	0.103568i	$-0.0330260\pi$
$954$	0	0
$955$	0	0
$956$	8.84441	0.286049
$957$	0	0
$958$	35.6888i	1.15305i
$959$	11.2111	0.362025
$960$	0	0
$961$	21.0000	0.677419
$962$	61.2666i	1.97531i
$963$	0	0
$964$	5.02776	0.161933
$965$	0	0
$966$	0	0
$967$	16.8444i	0.541680i	0.962624	+	0.270840i	$0.0873014\pi$
$967$	16.8444i	0.541680i	−0.962624	+	0.270840i	$0.912699\pi$
$968$	− 6.00000i	− 0.192847i
$969$	0	0
$970$	0	0
$971$	39.8722	1.27956	0.639779	−	0.768559i	$-0.279026\pi$
$971$	39.8722	1.27956	0.639779	+	0.768559i	$0.279026\pi$
$972$	0	0
$973$	17.0278i	0.545885i
$974$	−1.06392	−0.0340901
$975$	0	0
$976$	−43.6333	−1.39667
$977$	− 34.0278i	− 1.08864i	−0.838876	−	0.544322i	$-0.816787\pi$
$977$	− 34.0278i	− 1.08864i	0.838876	−	0.544322i	$-0.183213\pi$
$978$	0	0
$979$	−41.4500	−1.32475
$980$	0	0
$981$	0	0
$982$	3.90833i	0.124720i
$983$	− 32.0555i	− 1.02241i	−0.859458	−	0.511206i	$-0.829199\pi$
$983$	− 32.0555i	− 1.02241i	0.859458	−	0.511206i	$-0.170801\pi$
$984$	0	0
$985$	0	0
$986$	1.33894	0.0426404
$987$	0	0
$988$	− 0.844410i	− 0.0268643i
$989$	19.6611	0.625185
$990$	0	0
$991$	9.97224	0.316779	0.158389	−	0.987377i	$-0.449370\pi$
$991$	9.97224	0.316779	0.158389	+	0.987377i	$0.449370\pi$
$992$	12.2389i	0.388584i
$993$	0	0
$994$	−3.90833	−0.123965
$995$	0	0
$996$	0	0
$997$	49.4500i	1.56610i	0.621961	+	0.783048i	$0.286336\pi$
$997$	49.4500i	1.56610i	−0.621961	+	0.783048i	$0.713664\pi$
$998$	36.4777i	1.15468i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.d.g.1324.3		4
3.2	odd	2		525.2.d.d.274.2		4
5.2	odd	4		1575.2.a.t.1.1		2
5.3	odd	4		1575.2.a.o.1.2		2
5.4	even	2	inner	1575.2.d.g.1324.2		4
15.2	even	4		525.2.a.f.1.2	✓	2
15.8	even	4		525.2.a.h.1.1	yes	2
15.14	odd	2		525.2.d.d.274.3		4
60.23	odd	4		8400.2.a.cw.1.2		2
60.47	odd	4		8400.2.a.df.1.1		2
105.62	odd	4		3675.2.a.w.1.2		2
105.83	odd	4		3675.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.a.f.1.2	✓	2	15.2	even	4
525.2.a.h.1.1	yes	2	15.8	even	4
525.2.d.d.274.2		4	3.2	odd	2
525.2.d.d.274.3		4	15.14	odd	2
1575.2.a.o.1.2		2	5.3	odd	4
1575.2.a.t.1.1		2	5.2	odd	4
1575.2.d.g.1324.2		4	5.4	even	2	inner
1575.2.d.g.1324.3		4	1.1	even	1	trivial
3675.2.a.w.1.2		2	105.62	odd	4
3675.2.a.bb.1.1		2	105.83	odd	4
8400.2.a.cw.1.2		2	60.23	odd	4
8400.2.a.df.1.1		2	60.47	odd	4

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.30278i q^{2} +0.302776 q^{4} +1.00000i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+1.30278i q^{2} +0.302776 q^{4} +1.00000i q^{7} +3.00000i q^{8} +3.00000 q^{11} -4.60555i q^{13} -1.30278 q^{14} -3.30278 q^{16} +2.60555i q^{17} +0.605551 q^{19} +3.90833i q^{22} +8.21110i q^{23} +6.00000 q^{26} +0.302776i q^{28} -0.394449 q^{29} +7.21110 q^{31} +1.69722i q^{32} -3.39445 q^{34} +10.2111i q^{37} +0.788897i q^{38} -2.39445i q^{43} +0.908327 q^{44} -10.6972 q^{46} -3.39445i q^{47} -1.00000 q^{49} -1.39445i q^{52} -11.2111i q^{53} -3.00000 q^{56} -0.513878i q^{58} -3.39445 q^{59} +13.2111 q^{61} +9.39445i q^{62} -8.81665 q^{64} +8.39445i q^{67} +0.788897i q^{68} +3.00000 q^{71} +6.60555i q^{73} -13.3028 q^{74} +0.183346 q^{76} +3.00000i q^{77} -6.81665 q^{79} +11.2111i q^{83} +3.11943 q^{86} +9.00000i q^{88} -13.8167 q^{89} +4.60555 q^{91} +2.48612i q^{92} +4.42221 q^{94} -15.2111i q^{97} -1.30278i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4}+O(q^{10}) \) 4 * q - 6 * q^4	\( 4 q - 6 q^{4} + 12 q^{11} + 2 q^{14} - 6 q^{16} - 12 q^{19} + 24 q^{26} - 16 q^{29} - 28 q^{34} - 18 q^{44} - 50 q^{46} - 4 q^{49} - 12 q^{56} - 28 q^{59} + 24 q^{61} + 8 q^{64} + 12 q^{71} - 46 q^{74} + 44 q^{76} + 16 q^{79} - 38 q^{86} - 12 q^{89} + 4 q^{91} - 40 q^{94}+O(q^{100}) \) 4 * q - 6 * q^4 + 12 * q^11 + 2 * q^14 - 6 * q^16 - 12 * q^19 + 24 * q^26 - 16 * q^29 - 28 * q^34 - 18 * q^44 - 50 * q^46 - 4 * q^49 - 12 * q^56 - 28 * q^59 + 24 * q^61 + 8 * q^64 + 12 * q^71 - 46 * q^74 + 44 * q^76 + 16 * q^79 - 38 * q^86 - 12 * q^89 + 4 * q^91 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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