LMFDB - Embedded newform 675.2.f.h.107.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(107,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("675.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.f (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:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.3317760000.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 2x^{6} + 8x^{5} + 13x^{4} - 44x^{3} + 164x^{2} - 140x + 145 $ x^8 - 4x^7 + 2x^6 + 8x^5 + 13x^4 - 44x^3 + 164x^2 - 140*x + 145
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			107.4
Root			$2.84278 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	675.107
Dual form			675.2.f.h.593.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.98168 - 1.98168i) q^{2} -5.85410i q^{4} +(-7.63759 - 7.63759i) q^{8} -18.5623 q^{16} +(0.289123 - 0.289123i) q^{17} -4.70820i q^{19} +(4.25248 + 4.25248i) q^{23} +10.7082 q^{31} +(-21.5093 + 21.5093i) q^{32} -1.14590i q^{34} +(-9.33015 - 9.33015i) q^{38} +16.8541 q^{46} +(-7.34847 + 7.34847i) q^{47} -7.00000i q^{49} +(7.63759 + 7.63759i) q^{53} +12.4164 q^{61} +(21.2202 - 21.2202i) q^{62} +48.1246i q^{64} +(-1.69256 - 1.69256i) q^{68} -27.5623 q^{76} -1.29180i q^{79} +(-12.1792 - 12.1792i) q^{83} +(24.8945 - 24.8945i) q^{92} +29.1246i q^{94} +(-13.8718 - 13.8718i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 68 q^{16} + 32 q^{31} + 108 q^{46} - 8 q^{61} - 140 q^{76}+O(q^{100}) $ 8 * q - 68 * q^16 + 32 * q^31 + 108 * q^46 - 8 * q^61 - 140 * q^76

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	1.98168	−	1.98168i	1.40126	−	1.40126i	0.605138	−	0.796121i	$-0.293118\pi$
$2$	1.98168	−	1.98168i	1.40126	−	1.40126i	0.796121	−	0.605138i	$-0.206882\pi$
$3$		0			0
$3$		0			0
$4$		−	5.85410i		−	2.92705i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−7.63759	−	7.63759i	−2.70030	−	2.70030i
$9$		0			0
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	−18.5623			−4.64058
$17$	0.289123	−	0.289123i	0.0701226	−	0.0701226i	−0.671176	−	0.741298i	$-0.734210\pi$
$17$	0.289123	−	0.289123i	0.0701226	−	0.0701226i	0.741298	+	0.671176i	$0.234210\pi$
$18$		0			0
$19$		−	4.70820i		−	1.08014i	−0.841621	−	0.540068i	$-0.818398\pi$
$19$		−	4.70820i		−	1.08014i	0.841621	−	0.540068i	$-0.181602\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.25248	+	4.25248i	0.886704	+	0.886704i	0.994205	−	0.107501i	$-0.0342850\pi$
$23$	4.25248	+	4.25248i	0.886704	+	0.886704i	−0.107501	+	0.994205i	$0.534285\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	10.7082			1.92325			0.961625	−	0.274367i	$-0.0884683\pi$
$31$	10.7082			1.92325			0.961625	+	0.274367i	$0.0884683\pi$
$32$	−21.5093	+	21.5093i	−3.80235	+	3.80235i
$33$		0			0
$34$		−	1.14590i		−	0.196520i
$35$		0			0
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	−9.33015	−	9.33015i	−1.51355	−	1.51355i
$39$		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	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$	16.8541			2.48500
$47$	−7.34847	+	7.34847i	−1.07188	+	1.07188i	−0.0746766	+	0.997208i	$0.523792\pi$
$47$	−7.34847	+	7.34847i	−1.07188	+	1.07188i	−0.997208	+	0.0746766i	$0.976208\pi$
$48$		0			0
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.63759	+	7.63759i	1.04910	+	1.04910i	0.998730	+	0.0503735i	$0.0160412\pi$
$53$	7.63759	+	7.63759i	1.04910	+	1.04910i	0.0503735	+	0.998730i	$0.483959\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	12.4164			1.58976			0.794879	−	0.606768i	$-0.207534\pi$
$61$	12.4164			1.58976			0.794879	+	0.606768i	$0.207534\pi$
$62$	21.2202	−	21.2202i	2.69497	−	2.69497i
$63$		0			0
$64$			48.1246i			6.01558i
$65$		0			0
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	−1.69256	−	1.69256i	−0.205253	−	0.205253i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		0			0
$75$		0			0
$76$	−27.5623			−3.16161
$77$		0			0
$78$		0			0
$79$		−	1.29180i		−	0.145338i	−0.997356	−	0.0726692i	$-0.976848\pi$
$79$		−	1.29180i		−	0.145338i	0.997356	−	0.0726692i	$-0.0231517\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.1792	−	12.1792i	−1.33684	−	1.33684i	−0.899103	−	0.437738i	$-0.855780\pi$
$83$	−12.1792	−	12.1792i	−1.33684	−	1.33684i	−0.437738	−	0.899103i	$-0.644220\pi$
$84$		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
$92$	24.8945	−	24.8945i	2.59543	−	2.59543i
$93$		0			0
$94$			29.1246i			3.00397i
$95$		0			0
$96$		0			0
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	−13.8718	−	13.8718i	−1.40126	−	1.40126i
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$	30.2705			2.94013
$107$	−7.34847	+	7.34847i	−0.710403	+	0.710403i	−0.966620	−	0.256216i	$-0.917524\pi$
$107$	−7.34847	+	7.34847i	−0.710403	+	0.710403i	0.256216	+	0.966620i	$0.417524\pi$
$108$		0			0
$109$		−	6.41641i		−	0.614580i	−0.951616	−	0.307290i	$-0.900578\pi$
$109$		−	6.41641i		−	0.614580i	0.951616	−	0.307290i	$-0.0994222\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.6969	+	14.6969i	1.38257	+	1.38257i	0.840027	+	0.542545i	$0.182539\pi$
$113$	14.6969	+	14.6969i	1.38257	+	1.38257i	0.542545	+	0.840027i	$0.317461\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	24.6053	−	24.6053i	2.22766	−	2.22766i
$123$		0			0
$124$		−	62.6869i		−	5.62945i
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	52.3488	+	52.3488i	4.62703	+	4.62703i
$129$		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
$136$	−4.41641			−0.378704
$137$	−16.1426	+	16.1426i	−1.37915	+	1.37915i	−0.533098	+	0.846054i	$0.678972\pi$
$137$	−16.1426	+	16.1426i	−1.37915	+	1.37915i	−0.846054	+	0.533098i	$0.821028\pi$
$138$		0			0
$139$		−	4.00000i		−	0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$		−	4.00000i		−	0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$	−35.9593	+	35.9593i	−2.91669	+	2.91669i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$	−2.55992	−	2.55992i	−0.203657	−	0.203657i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$	−48.2705			−3.74652
$167$	−3.09599	+	3.09599i	−0.239575	+	0.239575i	−0.816674	−	0.577099i	$-0.804185\pi$
$167$	−3.09599	+	3.09599i	−0.239575	+	0.239575i	0.577099	+	0.816674i	$0.304185\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.79408	−	8.79408i	−0.668602	−	0.668602i	0.288790	−	0.957392i	$-0.406747\pi$
$173$	−8.79408	−	8.79408i	−0.668602	−	0.668602i	−0.957392	+	0.288790i	$0.906747\pi$
$174$		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
$181$	−24.4164			−1.81486			−0.907429	−	0.420206i	$-0.861958\pi$
$181$	−24.4164			−1.81486			−0.907429	+	0.420206i	$0.861958\pi$
$182$		0			0
$183$		0			0
$184$		−	64.9574i		−	4.78873i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	43.0187	+	43.0187i	3.13746	+	3.13746i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	−40.9787			−2.92705
$197$	7.05935	−	7.05935i	0.502958	−	0.502958i	−0.409398	−	0.912356i	$-0.634261\pi$
$197$	7.05935	−	7.05935i	0.502958	−	0.502958i	0.912356	+	0.409398i	$0.134261\pi$
$198$		0			0
$199$			16.0000i			1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$			16.0000i			1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	7.29180			0.501988			0.250994	−	0.967989i	$-0.419243\pi$
$211$	7.29180			0.501988			0.250994	+	0.967989i	$0.419243\pi$
$212$	44.7112	−	44.7112i	3.07078	−	3.07078i
$213$		0			0
$214$			29.1246i			1.99092i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−12.7153	−	12.7153i	−0.861186	−	0.861186i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$		0			0
$226$	58.2492			3.87468
$227$	−19.5277	+	19.5277i	−1.29610	+	1.29610i	−0.365147	+	0.930950i	$0.618981\pi$
$227$	−19.5277	+	19.5277i	−1.29610	+	1.29610i	−0.930950	+	0.365147i	$0.881019\pi$
$228$		0			0
$229$			0.416408i			0.0275170i	0.999905	+	0.0137585i	$0.00437961\pi$
$229$			0.416408i			0.0275170i	−0.999905	+	0.0137585i	$0.995620\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.6969	+	14.6969i	0.962828	+	0.962828i	0.999333	−	0.0365050i	$-0.0116225\pi$
$233$	14.6969	+	14.6969i	0.962828	+	0.962828i	−0.0365050	+	0.999333i	$0.511622\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−27.8328			−1.79287			−0.896435	−	0.443176i	$-0.853852\pi$
$241$	−27.8328			−1.79287			−0.896435	+	0.443176i	$0.853852\pi$
$242$	21.7985	−	21.7985i	1.40126	−	1.40126i
$243$		0			0
$244$		−	72.6869i		−	4.65330i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−81.7849	−	81.7849i	−5.19335	−	5.19335i
$249$		0			0
$250$		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
$256$	111.228			6.95175
$257$	16.7208	−	16.7208i	1.04301	−	1.04301i	0.0439825	−	0.999032i	$-0.485995\pi$
$257$	16.7208	−	16.7208i	1.04301	−	1.04301i	0.999032	−	0.0439825i	$-0.0140046\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−22.0454	−	22.0454i	−1.35938	−	1.35938i	−0.874683	−	0.484695i	$-0.838931\pi$
$263$	−22.0454	−	22.0454i	−1.35938	−	1.35938i	−0.484695	−	0.874683i	$-0.661069\pi$
$264$		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
$271$	−22.7082			−1.37943			−0.689713	−	0.724083i	$-0.742263\pi$
$271$	−22.7082			−1.37943			−0.689713	+	0.724083i	$0.742263\pi$
$272$	−5.36679	+	5.36679i	−0.325409	+	0.325409i
$273$		0			0
$274$			63.9787i			3.86510i
$275$		0			0
$276$		0			0
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$	−7.92672	−	7.92672i	−0.475413	−	0.475413i
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			16.8328i			0.990166i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	24.0693	+	24.0693i	1.40614	+	1.40614i	0.778512	+	0.627630i	$0.215975\pi$
$293$	24.0693	+	24.0693i	1.40614	+	1.40614i	0.627630	+	0.778512i	$0.284025\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−15.8534	+	15.8534i	−0.912262	+	0.912262i
$303$		0			0
$304$			87.3951i			5.01245i
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		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	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$	−7.56231			−0.425413
$317$	23.4910	−	23.4910i	1.31939	−	1.31939i	0.405127	−	0.914261i	$-0.367227\pi$
$317$	23.4910	−	23.4910i	1.31939	−	1.31939i	0.914261	−	0.405127i	$-0.132773\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.36125	−	1.36125i	−0.0757420	−	0.0757420i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−28.0000			−1.53902			−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$	−71.2983	+	71.2983i	−3.91300	+	3.91300i
$333$		0			0
$334$			12.2705i			0.671412i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	25.7618	+	25.7618i	1.40126	+	1.40126i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−34.8541			−1.87377
$347$	−7.34847	+	7.34847i	−0.394486	+	0.394486i	−0.876283	−	0.481797i	$-0.839984\pi$
$347$	−7.34847	+	7.34847i	−0.394486	+	0.394486i	0.481797	+	0.876283i	$0.339984\pi$
$348$		0			0
$349$			30.4164i			1.62815i	0.580758	+	0.814076i	$0.302756\pi$
$349$			30.4164i			1.62815i	−0.580758	+	0.814076i	$0.697244\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.6969	+	14.6969i	0.782239	+	0.782239i	0.980208	−	0.197969i	$-0.0634346\pi$
$353$	14.6969	+	14.6969i	0.782239	+	0.782239i	−0.197969	+	0.980208i	$0.563435\pi$
$354$		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
$361$	−3.16718			−0.166694
$362$	−48.3855	+	48.3855i	−2.54308	+	2.54308i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$	−78.9358	−	78.9358i	−4.11482	−	4.11482i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$	112.249			5.78881
$377$		0			0
$378$		0			0
$379$			35.5410i			1.82562i	0.408385	+	0.912810i	$0.366092\pi$
$379$			35.5410i			1.82562i	−0.408385	+	0.912810i	$0.633908\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−18.9494	−	18.9494i	−0.968270	−	0.968270i	0.0312418	−	0.999512i	$-0.490054\pi$
$383$	−18.9494	−	18.9494i	−0.968270	−	0.968270i	−0.999512	+	0.0312418i	$0.990054\pi$
$384$		0			0
$385$		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
$391$	2.45898			0.124356
$392$	−53.4631	+	53.4631i	−2.70030	+	2.70030i
$393$		0			0
$394$		−	27.9787i		−	1.40955i
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	31.7069	+	31.7069i	1.58932	+	1.58932i
$399$		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
$408$		0			0
$409$		−	39.8328i		−	1.96961i	−0.173675	−	0.984803i	$-0.555564\pi$
$409$		−	39.8328i		−	1.96961i	0.173675	−	0.984803i	$-0.444436\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	5.58359			0.272128			0.136064	−	0.990700i	$-0.456555\pi$
$421$	5.58359			0.272128			0.136064	+	0.990700i	$0.456555\pi$
$422$	14.4500	−	14.4500i	0.703415	−	0.703415i
$423$		0			0
$424$		−	116.666i		−	5.66578i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	43.0187	+	43.0187i	2.07939	+	2.07939i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$	−37.5623			−1.79891
$437$	20.0215	−	20.0215i	0.957760	−	0.957760i
$438$		0			0
$439$		−	41.5410i		−	1.98264i	−0.131453	−	0.991322i	$-0.541964\pi$
$439$		−	41.5410i		−	1.98264i	0.131453	−	0.991322i	$-0.458036\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.51774	−	2.51774i	−0.119622	−	0.119622i	0.644762	−	0.764383i	$-0.276957\pi$
$443$	−2.51774	−	2.51774i	−0.119622	−	0.119622i	−0.764383	+	0.644762i	$0.776957\pi$
$444$		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
$452$	86.0374	−	86.0374i	4.04686	−	4.04686i
$453$		0			0
$454$			77.3951i			3.63233i
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$	0.825187	+	0.825187i	0.0385584	+	0.0385584i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	58.2492			2.69834
$467$	13.3357	−	13.3357i	0.617102	−	0.617102i	−0.327685	−	0.944787i	$-0.606268\pi$
$467$	13.3357	−	13.3357i	0.617102	−	0.617102i	0.944787	+	0.327685i	$0.106268\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		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
$481$		0			0
$482$	−55.1557	+	55.1557i	−2.51227	+	2.51227i
$483$		0			0
$484$		−	64.3951i		−	2.92705i
$485$		0			0
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	−94.8315	−	94.8315i	−4.29282	−	4.29282i
$489$		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
$496$	−198.769			−8.92499
$497$		0			0
$498$		0			0
$499$			28.7082i			1.28516i	0.766220	+	0.642578i	$0.222135\pi$
$499$			28.7082i			1.28516i	−0.766220	+	0.642578i	$0.777865\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−28.6109	−	28.6109i	−1.27570	−	1.27570i	−0.943053	−	0.332643i	$-0.892060\pi$
$503$	−28.6109	−	28.6109i	−1.27570	−	1.27570i	−0.332643	−	0.943053i	$-0.607940\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	115.720	−	115.720i	5.11417	−	5.11417i
$513$		0			0
$514$		−	66.2705i		−	2.92307i
$515$		0			0
$516$		0			0
$517$		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
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$	−87.3738			−3.80968
$527$	3.09599	−	3.09599i	0.134863	−	0.134863i
$528$		0			0
$529$			13.1672i			0.572486i
$530$		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
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$	−45.0004	+	45.0004i	−1.93293	+	1.93293i
$543$		0			0
$544$			12.4377i			0.533262i
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	94.5002	+	94.5002i	4.03685	+	4.03685i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−23.4164			−0.993077
$557$	29.3939	−	29.3939i	1.24546	−	1.24546i	0.287754	−	0.957704i	$-0.407091\pi$
$557$	29.3939	−	29.3939i	1.24546	−	1.24546i	0.957704	−	0.287754i	$-0.0929086\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−22.0454	−	22.0454i	−0.929103	−	0.929103i	0.0685449	−	0.997648i	$-0.478164\pi$
$563$	−22.0454	−	22.0454i	−0.929103	−	0.929103i	−0.997648	+	0.0685449i	$0.978164\pi$
$564$		0			0
$565$		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
$571$	47.5410			1.98953			0.994765	−	0.102190i	$-0.0325850\pi$
$571$	47.5410			1.98953			0.994765	+	0.102190i	$0.0325850\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	33.3572	+	33.3572i	1.38748	+	1.38748i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	95.3951			3.94074
$587$	26.8761	−	26.8761i	1.10930	−	1.10930i	0.116054	−	0.993243i	$-0.462975\pi$
$587$	26.8761	−	26.8761i	1.10930	−	1.10930i	0.993243	−	0.116054i	$-0.0370245\pi$
$588$		0			0
$589$		−	50.4164i		−	2.07737i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.4078	+	14.4078i	0.591658	+	0.591658i	0.938079	−	0.346421i	$-0.112603\pi$
$593$	14.4078	+	14.4078i	0.591658	+	0.591658i	−0.346421	+	0.938079i	$0.612603\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	45.8328			1.86956			0.934780	−	0.355228i	$-0.115597\pi$
$601$	45.8328			1.86956			0.934780	+	0.355228i	$0.115597\pi$
$602$		0			0
$603$		0			0
$604$			46.8328i			1.90560i
$605$		0			0
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$	101.270	+	101.270i	4.10706	+	4.10706i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$613$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−32.5742	+	32.5742i	−1.31139	+	1.31139i	−0.390997	+	0.920392i	$0.627870\pi$
$617$	−32.5742	+	32.5742i	−1.31139	+	1.31139i	−0.920392	+	0.390997i	$0.872130\pi$
$618$		0			0
$619$		−	44.0000i		−	1.76851i	−0.467005	−	0.884255i	$-0.654667\pi$
$619$		−	44.0000i		−	1.76851i	0.467005	−	0.884255i	$-0.345333\pi$
$620$		0			0
$621$		0			0
$622$		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
$631$	17.5410			0.698297			0.349148	−	0.937067i	$-0.386471\pi$
$631$	17.5410			0.698297			0.349148	+	0.937067i	$0.386471\pi$
$632$	−9.86621	+	9.86621i	−0.392457	+	0.392457i
$633$		0			0
$634$		−	93.1033i		−	3.69761i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$	−5.39512			−0.212268
$647$	−35.9593	+	35.9593i	−1.41371	+	1.41371i	−0.687941	+	0.725767i	$0.741485\pi$
$647$	−35.9593	+	35.9593i	−1.41371	+	1.41371i	−0.725767	+	0.687941i	$0.758515\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−25.2258	−	25.2258i	−0.987160	−	0.987160i	0.0127583	−	0.999919i	$-0.495939\pi$
$653$	−25.2258	−	25.2258i	−0.987160	−	0.987160i	−0.999919	+	0.0127583i	$0.995939\pi$
$654$		0			0
$655$		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
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$	−55.4870	+	55.4870i	−2.15656	+	2.15656i
$663$		0			0
$664$			186.039i			7.21973i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	18.1242	+	18.1242i	0.701248	+	0.701248i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$		0			0
$676$	76.1033			2.92705
$677$	29.3939	−	29.3939i	1.12970	−	1.12970i	0.139473	−	0.990226i	$-0.455459\pi$
$677$	29.3939	−	29.3939i	1.12970	−	1.12970i	0.990226	−	0.139473i	$-0.0445407\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−35.3811	−	35.3811i	−1.35382	−	1.35382i	−0.881343	−	0.472477i	$-0.843360\pi$
$683$	−35.3811	−	35.3811i	−1.35382	−	1.35382i	−0.472477	−	0.881343i	$-0.656640\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−19.2918			−0.733895			−0.366947	−	0.930242i	$-0.619597\pi$
$691$	−19.2918			−0.733895			−0.366947	+	0.930242i	$0.619597\pi$
$692$	−51.4815	+	51.4815i	−1.95703	+	1.95703i
$693$		0			0
$694$			29.1246i			1.10556i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	60.2756	+	60.2756i	2.28146	+	2.28146i
$699$		0			0
$700$		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
$706$	58.2492			2.19224
$707$		0			0
$708$		0			0
$709$			26.0000i			0.976450i	0.872718	+	0.488225i	$0.162356\pi$
$709$			26.0000i			0.976450i	−0.872718	+	0.488225i	$0.837644\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	45.5364	+	45.5364i	1.70535	+	1.70535i
$714$		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
$722$	−6.27634	+	6.27634i	−0.233581	+	0.233581i
$723$		0			0
$724$			142.936i			5.31218i
$725$		0			0
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$733$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$		0			0
$736$	−182.936			−6.74312
$737$		0			0
$738$		0			0
$739$		−	44.9574i		−	1.65379i	−0.562360	−	0.826893i	$-0.690106\pi$
$739$		−	44.9574i		−	1.65379i	0.562360	−	0.826893i	$-0.309894\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−22.0454	−	22.0454i	−0.808768	−	0.808768i	0.175680	−	0.984447i	$-0.443788\pi$
$743$	−22.0454	−	22.0454i	−0.808768	−	0.808768i	−0.984447	+	0.175680i	$0.943788\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	50.9574			1.85946			0.929731	−	0.368238i	$-0.120039\pi$
$751$	50.9574			1.85946			0.929731	+	0.368238i	$0.120039\pi$
$752$	136.405	−	136.405i	4.97416	−	4.97416i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$	70.4309	+	70.4309i	2.55816	+	2.55816i
$759$		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
$766$	−75.1033			−2.71359
$767$		0			0
$768$		0			0
$769$			3.83282i			0.138215i	0.997609	+	0.0691074i	$0.0220151\pi$
$769$			3.83282i			0.138215i	−0.997609	+	0.0691074i	$0.977985\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−22.3345	−	22.3345i	−0.803317	−	0.803317i	0.180295	−	0.983613i	$-0.442295\pi$
$773$	−22.3345	−	22.3345i	−0.803317	−	0.803317i	−0.983613	+	0.180295i	$0.942295\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	4.87291	−	4.87291i	0.174255	−	0.174255i
$783$		0			0
$784$			129.936i			4.64058i
$785$		0			0
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−41.3261	−	41.3261i	−1.47218	−	1.47218i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	93.6656			3.31989
$797$	39.9227	−	39.9227i	1.41413	−	1.41413i	0.699594	−	0.714541i	$-0.253364\pi$
$797$	39.9227	−	39.9227i	1.41413	−	1.41413i	0.714541	−	0.699594i	$-0.246636\pi$
$798$		0			0
$799$			4.24922i			0.150327i
$800$		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
$811$	52.0000			1.82597			0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000			1.82597			0.912983	+	0.407997i	$0.133772\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−78.9358	−	78.9358i	−2.75993	−	2.75993i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5.98722	+	5.98722i	−0.208196	+	0.208196i	−0.803500	−	0.595304i	$-0.797032\pi$
$827$	−5.98722	+	5.98722i	−0.208196	+	0.208196i	0.595304	+	0.803500i	$0.297032\pi$
$828$		0			0
$829$		−	34.0000i		−	1.18087i	−0.807086	−	0.590434i	$-0.798956\pi$
$829$		−	34.0000i		−	1.18087i	0.807086	−	0.590434i	$-0.201044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.02386	−	2.02386i	−0.0701226	−	0.0701226i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	11.0649	−	11.0649i	0.381321	−	0.381321i
$843$		0			0
$844$		−	42.6869i		−	1.46934i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−141.771	−	141.771i	−4.86845	−	4.86845i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$	112.249			3.83660
$857$	33.1525	−	33.1525i	1.13247	−	1.13247i	0.142701	−	0.989766i	$-0.454421\pi$
$857$	33.1525	−	33.1525i	1.13247	−	1.13247i	0.989766	−	0.142701i	$-0.0455788\pi$
$858$		0			0
$859$		−	11.5410i		−	0.393775i	−0.980426	−	0.196887i	$-0.936917\pi$
$859$		−	11.5410i		−	0.393775i	0.980426	−	0.196887i	$-0.0630833\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	13.9139	+	13.9139i	0.473636	+	0.473636i	0.903089	−	0.429453i	$-0.141294\pi$
$863$	13.9139	+	13.9139i	0.473636	+	0.473636i	−0.429453	+	0.903089i	$0.641294\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−49.0059	+	49.0059i	−1.65955	+	1.65955i
$873$		0			0
$874$		−	79.3525i		−	2.68414i
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$	−82.3210	−	82.3210i	−2.77820	−	2.77820i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$	−9.97871			−0.335241
$887$	−26.2979	+	26.2979i	−0.882997	+	0.882997i	−0.993838	−	0.110841i	$-0.964645\pi$
$887$	−26.2979	+	26.2979i	−0.882997	+	0.882997i	0.110841	+	0.993838i	$0.464645\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	34.5981	+	34.5981i	1.15778	+	1.15778i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	4.41641			0.147132
$902$		0			0
$903$		0			0
$904$		−	224.498i		−	7.46671i
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	114.317	+	114.317i	3.79374	+	3.79374i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	2.43769			0.0805437
$917$		0			0
$918$		0			0
$919$			56.0000i			1.84727i	0.383274	+	0.923635i	$0.374797\pi$
$919$			56.0000i			1.84727i	−0.383274	+	0.923635i	$0.625203\pi$
$920$		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
$931$	−32.9574			−1.08014
$932$	86.0374	−	86.0374i	2.81825	−	2.81825i
$933$		0			0
$934$		−	52.8541i		−	1.72944i
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		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
$947$	−9.86621	+	9.86621i	−0.320609	+	0.320609i	−0.849001	−	0.528392i	$-0.822795\pi$
$947$	−9.86621	+	9.86621i	−0.320609	+	0.320609i	0.528392	+	0.849001i	$0.322795\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14.6969	+	14.6969i	0.476081	+	0.476081i	0.903876	−	0.427795i	$-0.140710\pi$
$953$	14.6969	+	14.6969i	0.476081	+	0.476081i	−0.427795	+	0.903876i	$0.640710\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	83.6656			2.69889
$962$		0			0
$963$		0			0
$964$			162.936i			5.24782i
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−84.0135	−	84.0135i	−2.70030	−	2.70030i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−230.477			−7.37739
$977$	−44.0908	+	44.0908i	−1.41059	+	1.41059i	−0.654710	+	0.755880i	$0.727209\pi$
$977$	−44.0908	+	44.0908i	−1.41059	+	1.41059i	−0.755880	+	0.654710i	$0.772791\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	37.1158	+	37.1158i	1.18381	+	1.18381i	0.978749	+	0.205062i	$0.0657397\pi$
$983$	37.1158	+	37.1158i	1.18381	+	1.18381i	0.205062	+	0.978749i	$0.434260\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−62.9574			−1.99991			−0.999954	−	0.00956046i	$-0.996957\pi$
$991$	−62.9574			−1.99991			−0.999954	+	0.00956046i	$0.996957\pi$
$992$	−230.326	+	230.326i	−7.31287	+	7.31287i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$	56.8904	+	56.8904i	1.80084	+	1.80084i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.f.h.107.4	yes	8
3.2	odd	2	inner	675.2.f.h.107.1	✓	8
5.2	odd	4	inner	675.2.f.h.593.4	yes	8
5.3	odd	4	inner	675.2.f.h.593.1	yes	8
5.4	even	2	inner	675.2.f.h.107.1	✓	8
15.2	even	4	inner	675.2.f.h.593.1	yes	8
15.8	even	4	inner	675.2.f.h.593.4	yes	8
15.14	odd	2	CM	675.2.f.h.107.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.2.f.h.107.1	✓	8	3.2	odd	2	inner
675.2.f.h.107.1	✓	8	5.4	even	2	inner
675.2.f.h.107.4	yes	8	1.1	even	1	trivial
675.2.f.h.107.4	yes	8	15.14	odd	2	CM
675.2.f.h.593.1	yes	8	5.3	odd	4	inner
675.2.f.h.593.1	yes	8	15.2	even	4	inner
675.2.f.h.593.4	yes	8	5.2	odd	4	inner
675.2.f.h.593.4	yes	8	15.8	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.98168 - 1.98168i) q^{2} -5.85410i q^{4} +(-7.63759 - 7.63759i) q^{8} -18.5623 q^{16} +(0.289123 - 0.289123i) q^{17} -4.70820i q^{19} +(4.25248 + 4.25248i) q^{23} +10.7082 q^{31} +(-21.5093 + 21.5093i) q^{32} -1.14590i q^{34} +(-9.33015 - 9.33015i) q^{38} +16.8541 q^{46} +(-7.34847 + 7.34847i) q^{47} -7.00000i q^{49} +(7.63759 + 7.63759i) q^{53} +12.4164 q^{61} +(21.2202 - 21.2202i) q^{62} +48.1246i q^{64} +(-1.69256 - 1.69256i) q^{68} -27.5623 q^{76} -1.29180i q^{79} +(-12.1792 - 12.1792i) q^{83} +(24.8945 - 24.8945i) q^{92} +29.1246i q^{94} +(-13.8718 - 13.8718i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 68 q^{16} + 32 q^{31} + 108 q^{46} - 8 q^{61} - 140 q^{76}+O(q^{100}) \) 8 * q - 68 * q^16 + 32 * q^31 + 108 * q^46 - 8 * q^61 - 140 * q^76

Introduction

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Modular forms

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