LMFDB - Embedded newform 675.2.f.h.593.3

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			593.3
Root			$-1.84278 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	675.593
Dual form			675.2.f.h.107.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.756934 + 0.756934i) q^{2} -0.854102i q^{4} +(2.16037 - 2.16037i) q^{8} +1.56231 q^{16} +(5.18810 + 5.18810i) q^{17} -8.70820i q^{19} +(6.70197 - 6.70197i) q^{23} -2.70820 q^{31} +(-3.13817 - 3.13817i) q^{32} +7.85410i q^{34} +(6.59154 - 6.59154i) q^{38} +10.1459 q^{46} +(7.34847 + 7.34847i) q^{47} +7.00000i q^{49} +(-2.16037 + 2.16037i) q^{53} -14.4164 q^{61} +(-2.04993 - 2.04993i) q^{62} -7.87539i q^{64} +(4.43117 - 4.43117i) q^{68} -7.43769 q^{76} +14.7082i q^{79} +(-9.72971 + 9.72971i) q^{83} +(-5.72417 - 5.72417i) q^{92} +11.1246i q^{94} +(-5.29854 + 5.29854i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	0.756934	+	0.756934i	0.535233	+	0.535233i	0.922125	−	0.386892i	$-0.126451\pi$
$2$	0.756934	+	0.756934i	0.535233	+	0.535233i	−0.386892	+	0.922125i	$0.626451\pi$
$3$		0			0
$3$		0			0
$4$		−	0.854102i		−	0.427051i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	2.16037	−	2.16037i	0.763805	−	0.763805i
$9$		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	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$	1.56231			0.390576
$17$	5.18810	+	5.18810i	1.25830	+	1.25830i	0.951904	+	0.306395i	$0.0991229\pi$
$17$	5.18810	+	5.18810i	1.25830	+	1.25830i	0.306395	+	0.951904i	$0.400877\pi$
$18$		0			0
$19$		−	8.70820i		−	1.99780i	−0.0469020	−	0.998899i	$-0.514935\pi$
$19$		−	8.70820i		−	1.99780i	0.0469020	−	0.998899i	$-0.485065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.70197	−	6.70197i	1.39746	−	1.39746i	0.590201	−	0.807256i	$-0.299048\pi$
$23$	6.70197	−	6.70197i	1.39746	−	1.39746i	0.807256	−	0.590201i	$-0.200952\pi$
$24$		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
$31$	−2.70820			−0.486408			−0.243204	−	0.969975i	$-0.578198\pi$
$31$	−2.70820			−0.486408			−0.243204	+	0.969975i	$0.578198\pi$
$32$	−3.13817	−	3.13817i	−0.554756	−	0.554756i
$33$		0			0
$34$			7.85410i			1.34697i
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	6.59154	−	6.59154i	1.06929	−	1.06929i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$		0			0
$46$	10.1459			1.49593
$47$	7.34847	+	7.34847i	1.07188	+	1.07188i	0.997208	+	0.0746766i	$0.0237924\pi$
$47$	7.34847	+	7.34847i	1.07188	+	1.07188i	0.0746766	+	0.997208i	$0.476208\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.16037	+	2.16037i	−0.296749	+	0.296749i	−0.839739	−	0.542990i	$-0.817292\pi$
$53$	−2.16037	+	2.16037i	−0.296749	+	0.296749i	0.542990	+	0.839739i	$0.317292\pi$
$54$		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
$61$	−14.4164			−1.84583			−0.922916	−	0.385002i	$-0.874201\pi$
$61$	−14.4164			−1.84583			−0.922916	+	0.385002i	$0.874201\pi$
$62$	−2.04993	−	2.04993i	−0.260342	−	0.260342i
$63$		0			0
$64$		−	7.87539i		−	0.984424i
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	4.43117	−	4.43117i	0.537358	−	0.537358i
$69$		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	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$		0			0
$75$		0			0
$76$	−7.43769			−0.853162
$77$		0			0
$78$		0			0
$79$			14.7082i			1.65480i	0.561611	+	0.827401i	$0.310182\pi$
$79$			14.7082i			1.65480i	−0.561611	+	0.827401i	$0.689818\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.72971	+	9.72971i	−1.06797	+	1.06797i	−0.0704594	+	0.997515i	$0.522447\pi$
$83$	−9.72971	+	9.72971i	−1.06797	+	1.06797i	−0.997515	+	0.0704594i	$0.977553\pi$
$84$		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
$92$	−5.72417	−	5.72417i	−0.596786	−	0.596786i
$93$		0			0
$94$			11.1246i			1.14742i
$95$		0			0
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	−5.29854	+	5.29854i	−0.535233	+	0.535233i
$99$		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	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$	−3.27051			−0.317660
$107$	7.34847	+	7.34847i	0.710403	+	0.710403i	0.966620	−	0.256216i	$-0.0824759\pi$
$107$	7.34847	+	7.34847i	0.710403	+	0.710403i	−0.256216	+	0.966620i	$0.582476\pi$
$108$		0			0
$109$		−	20.4164i		−	1.95554i	−0.209687	−	0.977769i	$-0.567244\pi$
$109$		−	20.4164i		−	1.95554i	0.209687	−	0.977769i	$-0.432756\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.6969	+	14.6969i	−1.38257	+	1.38257i	−0.542545	+	0.840027i	$0.682539\pi$
$113$	−14.6969	+	14.6969i	−1.38257	+	1.38257i	−0.840027	+	0.542545i	$0.817461\pi$
$114$		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
$122$	−10.9123	−	10.9123i	−0.987950	−	0.987950i
$123$		0			0
$124$			2.31308i			0.207721i
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−0.315193	+	0.315193i	−0.0278594	+	0.0278594i
$129$		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
$136$	22.4164			1.92219
$137$	−11.2436	−	11.2436i	−0.960603	−	0.960603i	0.0386495	−	0.999253i	$-0.487694\pi$
$137$	−11.2436	−	11.2436i	−0.960603	−	0.960603i	−0.999253	+	0.0386495i	$0.987694\pi$
$138$		0			0
$139$			4.00000i			0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$			4.00000i			0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$		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	$0.605538\pi$
$152$	−18.8129	−	18.8129i	−1.52593	−	1.52593i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$157$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$158$	−11.1331	+	11.1331i	−0.885705	+	0.885705i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$	−14.7295			−1.14323
$167$	14.0504	+	14.0504i	1.08726	+	1.08726i	0.995810	+	0.0914456i	$0.0291488\pi$
$167$	14.0504	+	14.0504i	1.08726	+	1.08726i	0.0914456	+	0.995810i	$0.470851\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−18.5920	+	18.5920i	−1.41353	+	1.41353i	−0.684731	+	0.728796i	$0.740080\pi$
$173$	−18.5920	+	18.5920i	−1.41353	+	1.41353i	−0.728796	+	0.684731i	$0.759920\pi$
$174$		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
$181$	2.41641			0.179610			0.0898051	−	0.995959i	$-0.471376\pi$
$181$	2.41641			0.179610			0.0898051	+	0.995959i	$0.471376\pi$
$182$		0			0
$183$		0			0
$184$		−	28.9574i		−	2.13477i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	6.27634	−	6.27634i	0.457749	−	0.457749i
$189$		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	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$	5.97871			0.427051
$197$	−12.5366	−	12.5366i	−0.893194	−	0.893194i	0.101629	−	0.994822i	$-0.467595\pi$
$197$	−12.5366	−	12.5366i	−0.893194	−	0.893194i	−0.994822	+	0.101629i	$0.967595\pi$
$198$		0			0
$199$		−	16.0000i		−	1.13421i	−0.823646	−	0.567105i	$-0.808063\pi$
$199$		−	16.0000i		−	1.13421i	0.823646	−	0.567105i	$-0.191937\pi$
$200$		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
$211$	20.7082			1.42561			0.712806	−	0.701361i	$-0.247424\pi$
$211$	20.7082			1.42561			0.712806	+	0.701361i	$0.247424\pi$
$212$	1.84517	+	1.84517i	0.126727	+	0.126727i
$213$		0			0
$214$			11.1246i			0.760463i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	15.4539	−	15.4539i	1.04667	−	1.04667i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		0			0
$226$	−22.2492			−1.48000
$227$	−2.38124	−	2.38124i	−0.158048	−	0.158048i	0.623653	−	0.781701i	$-0.285648\pi$
$227$	−2.38124	−	2.38124i	−0.158048	−	0.158048i	−0.781701	+	0.623653i	$0.785648\pi$
$228$		0			0
$229$			26.4164i			1.74565i	0.488037	+	0.872823i	$0.337713\pi$
$229$			26.4164i			1.74565i	−0.488037	+	0.872823i	$0.662287\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.6969	+	14.6969i	−0.962828	+	0.962828i	−0.999333	−	0.0365050i	$-0.988378\pi$
$233$	−14.6969	+	14.6969i	−0.962828	+	0.962828i	0.0365050	+	0.999333i	$0.488378\pi$
$234$		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
$241$	25.8328			1.66404			0.832019	−	0.554747i	$-0.187185\pi$
$241$	25.8328			1.66404			0.832019	+	0.554747i	$0.187185\pi$
$242$	8.32627	+	8.32627i	0.535233	+	0.535233i
$243$		0			0
$244$			12.3131i			0.788264i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−5.85071	+	5.85071i	−0.371521	+	0.371521i
$249$		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
$256$	−16.2279			−1.01425
$257$	21.6198	+	21.6198i	1.34860	+	1.34860i	0.887179	+	0.461426i	$0.152662\pi$
$257$	21.6198	+	21.6198i	1.34860	+	1.34860i	0.461426	+	0.887179i	$0.347338\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	22.0454	−	22.0454i	1.35938	−	1.35938i	0.484695	−	0.874683i	$-0.338931\pi$
$263$	22.0454	−	22.0454i	1.35938	−	1.35938i	0.874683	−	0.484695i	$-0.161069\pi$
$264$		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
$271$	−9.29180			−0.564436			−0.282218	−	0.959350i	$-0.591070\pi$
$271$	−9.29180			−0.564436			−0.282218	+	0.959350i	$0.591070\pi$
$272$	8.10540	+	8.10540i	0.491462	+	0.491462i
$273$		0			0
$274$		−	17.0213i		−	1.02829i
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	−3.02774	+	3.02774i	−0.181592	+	0.181592i
$279$		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	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			36.8328i			2.16664i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.2713	−	14.2713i	0.833739	−	0.833739i	−0.154287	−	0.988026i	$-0.549308\pi$
$293$	14.2713	−	14.2713i	0.833739	−	0.833739i	0.988026	+	0.154287i	$0.0493081\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−6.05547	−	6.05547i	−0.348453	−	0.348453i
$303$		0			0
$304$		−	13.6049i		−	0.780293i
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$	12.5623			0.706685
$317$	3.89510	+	3.89510i	0.218771	+	0.218771i	0.807980	−	0.589209i	$-0.200561\pi$
$317$	3.89510	+	3.89510i	0.218771	+	0.218771i	−0.589209	+	0.807980i	$0.700561\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	45.1791	−	45.1791i	2.51383	−	2.51383i
$324$		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	$0.779499\pi$
$332$	8.31016	+	8.31016i	0.456079	+	0.456079i
$333$		0			0
$334$			21.2705i			1.16387i
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	9.84014	−	9.84014i	0.535233	−	0.535233i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−28.1459			−1.51313
$347$	7.34847	+	7.34847i	0.394486	+	0.394486i	0.876283	−	0.481797i	$-0.160016\pi$
$347$	7.34847	+	7.34847i	0.394486	+	0.394486i	−0.481797	+	0.876283i	$0.660016\pi$
$348$		0			0
$349$		−	3.58359i		−	0.191825i	−0.995390	−	0.0959126i	$-0.969423\pi$
$349$		−	3.58359i		−	0.191825i	0.995390	−	0.0959126i	$-0.0305769\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.6969	+	14.6969i	−0.782239	+	0.782239i	−0.980208	−	0.197969i	$-0.936565\pi$
$353$	−14.6969	+	14.6969i	−0.782239	+	0.782239i	0.197969	+	0.980208i	$0.436565\pi$
$354$		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
$361$	−56.8328			−2.99120
$362$	1.82906	+	1.82906i	0.0961333	+	0.0961333i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$	10.4705	−	10.4705i	0.545814	−	0.545814i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$	31.7508			1.63742
$377$		0			0
$378$		0			0
$379$			31.5410i			1.62015i	0.586324	+	0.810077i	$0.300575\pi$
$379$			31.5410i			1.62015i	−0.586324	+	0.810077i	$0.699425\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.99497	−	7.99497i	0.408524	−	0.408524i	−0.472700	−	0.881224i	$-0.656720\pi$
$383$	7.99497	−	7.99497i	0.408524	−	0.408524i	0.881224	+	0.472700i	$0.156720\pi$
$384$		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
$391$	69.5410			3.51684
$392$	15.1226	+	15.1226i	0.763805	+	0.763805i
$393$		0			0
$394$		−	18.9787i		−	0.956134i
$395$		0			0
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	12.1109	−	12.1109i	0.607067	−	0.607067i
$399$		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
$409$		−	13.8328i		−	0.683989i	−0.939702	−	0.341994i	$-0.888898\pi$
$409$		−	13.8328i		−	0.683989i	0.939702	−	0.341994i	$-0.111102\pi$
$410$		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
$421$	32.4164			1.57988			0.789940	−	0.613185i	$-0.210112\pi$
$421$	32.4164			1.57988			0.789940	+	0.613185i	$0.210112\pi$
$422$	15.6747	+	15.6747i	0.763035	+	0.763035i
$423$		0			0
$424$			9.33437i			0.453317i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	6.27634	−	6.27634i	0.303378	−	0.303378i
$429$		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	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$	−17.4377			−0.835114
$437$	−58.3621	−	58.3621i	−2.79184	−	2.79184i
$438$		0			0
$439$		−	25.5410i		−	1.21901i	−0.792784	−	0.609503i	$-0.791369\pi$
$439$		−	25.5410i		−	1.21901i	0.792784	−	0.609503i	$-0.208631\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	24.4266	−	24.4266i	1.16054	−	1.16054i	0.176188	−	0.984356i	$-0.443623\pi$
$443$	24.4266	−	24.4266i	1.16054	−	1.16054i	0.984356	−	0.176188i	$-0.0563768\pi$
$444$		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
$452$	12.5527	+	12.5527i	0.590429	+	0.590429i
$453$		0			0
$454$		−	3.60488i		−	0.169185i
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$	−19.9955	+	19.9955i	−0.934327	+	0.934327i
$459$		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	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	−22.2492			−1.03068
$467$	30.4821	+	30.4821i	1.41054	+	1.41054i	0.756177	+	0.654367i	$0.227065\pi$
$467$	30.4821	+	30.4821i	1.41054	+	1.41054i	0.654367	+	0.756177i	$0.272935\pi$
$468$		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
$482$	19.5537	+	19.5537i	0.890648	+	0.890648i
$483$		0			0
$484$		−	9.39512i		−	0.427051i
$485$		0			0
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	−31.1447	+	31.1447i	−1.40986	+	1.40986i
$489$		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
$496$	−4.23104			−0.189979
$497$		0			0
$498$		0			0
$499$		−	15.2918i		−	0.684555i	−0.939599	−	0.342277i	$-0.888802\pi$
$499$		−	15.2918i		−	0.684555i	0.939599	−	0.342277i	$-0.111198\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−26.1614	+	26.1614i	−1.16648	+	1.16648i	−0.183449	+	0.983029i	$0.558726\pi$
$503$	−26.1614	+	26.1614i	−1.16648	+	1.16648i	−0.983029	+	0.183449i	$0.941274\pi$
$504$		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
$512$	−11.6531	−	11.6531i	−0.514999	−	0.514999i
$513$		0			0
$514$			32.7295i			1.44364i
$515$		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	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$	33.3738			1.45517
$527$	−14.0504	−	14.0504i	−0.612047	−	0.612047i
$528$		0			0
$529$		−	66.8328i		−	2.90577i
$530$		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	$0.486311\pi$
$542$	−7.03328	−	7.03328i	−0.302105	−	0.302105i
$543$		0			0
$544$		−	32.5623i		−	1.39610i
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	−9.60316	+	9.60316i	−0.410227	+	0.410227i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	3.41641			0.144888
$557$	−29.3939	−	29.3939i	−1.24546	−	1.24546i	−0.957704	−	0.287754i	$-0.907091\pi$
$557$	−29.3939	−	29.3939i	−1.24546	−	1.24546i	−0.287754	−	0.957704i	$-0.592909\pi$
$558$		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	$-0.521836\pi$
$563$	22.0454	−	22.0454i	0.929103	−	0.929103i	0.997648	+	0.0685449i	$0.0218356\pi$
$564$		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
$571$	−19.5410			−0.817766			−0.408883	−	0.912587i	$-0.634082\pi$
$571$	−19.5410			−0.817766			−0.408883	+	0.912587i	$0.634082\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$577$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$578$	−27.8800	+	27.8800i	−1.15966	+	1.15966i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	21.6049			0.892489
$587$	−4.96723	−	4.96723i	−0.205020	−	0.205020i	0.597127	−	0.802147i	$-0.296309\pi$
$587$	−4.96723	−	4.96723i	−0.205020	−	0.205020i	−0.802147	+	0.597127i	$0.796309\pi$
$588$		0			0
$589$			23.5836i			0.971745i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−19.8850	+	19.8850i	−0.816581	+	0.816581i	−0.985611	−	0.169030i	$-0.945936\pi$
$593$	−19.8850	+	19.8850i	−0.816581	+	0.816581i	0.169030	+	0.985611i	$0.445936\pi$
$594$		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
$601$	−7.83282			−0.319507			−0.159754	−	0.987157i	$-0.551070\pi$
$601$	−7.83282			−0.319507			−0.159754	+	0.987157i	$0.551070\pi$
$602$		0			0
$603$		0			0
$604$			6.83282i			0.278023i
$605$		0			0
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$	−27.3278	+	27.3278i	−1.10829	+	1.10829i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−27.6753	−	27.6753i	−1.11416	−	1.11416i	−0.992581	−	0.121582i	$-0.961203\pi$
$617$	−27.6753	−	27.6753i	−1.11416	−	1.11416i	−0.121582	−	0.992581i	$-0.538797\pi$
$618$		0			0
$619$			44.0000i			1.76851i	0.467005	+	0.884255i	$0.345333\pi$
$619$			44.0000i			1.76851i	−0.467005	+	0.884255i	$0.654667\pi$
$620$		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
$631$	−49.5410			−1.97220			−0.986098	−	0.166162i	$-0.946862\pi$
$631$	−49.5410			−1.97220			−0.986098	+	0.166162i	$0.946862\pi$
$632$	31.7751	+	31.7751i	1.26395	+	1.26395i
$633$		0			0
$634$			5.89667i			0.234187i
$635$		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	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$	68.3951			2.69097
$647$	−18.8129	−	18.8129i	−0.739612	−	0.739612i	0.232891	−	0.972503i	$-0.425181\pi$
$647$	−18.8129	−	18.8129i	−0.739612	−	0.739612i	−0.972503	+	0.232891i	$0.925181\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−35.0237	+	35.0237i	−1.37058	+	1.37058i	−0.511008	+	0.859576i	$0.670728\pi$
$653$	−35.0237	+	35.0237i	−1.37058	+	1.37058i	−0.859576	+	0.511008i	$0.829272\pi$
$654$		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	$0.359271\pi$
$662$	−21.1942	−	21.1942i	−0.823734	−	0.823734i
$663$		0			0
$664$			42.0395i			1.63145i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	12.0005	−	12.0005i	0.464314	−	0.464314i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$673$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$674$		0			0
$675$		0			0
$676$	−11.1033			−0.427051
$677$	−29.3939	−	29.3939i	−1.12970	−	1.12970i	−0.990226	−	0.139473i	$-0.955459\pi$
$677$	−29.3939	−	29.3939i	−1.12970	−	1.12970i	−0.139473	−	0.990226i	$-0.544541\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.43671	+	8.43671i	−0.322822	+	0.322822i	−0.849849	−	0.527027i	$-0.823307\pi$
$683$	−8.43671	+	8.43671i	−0.322822	+	0.322822i	0.527027	+	0.849849i	$0.323307\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−32.7082			−1.24428			−0.622139	−	0.782907i	$-0.713736\pi$
$691$	−32.7082			−1.24428			−0.622139	+	0.782907i	$0.713736\pi$
$692$	15.8795	+	15.8795i	0.603648	+	0.603648i
$693$		0			0
$694$			11.1246i			0.422284i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	2.71254	−	2.71254i	0.102671	−	0.102671i
$699$		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
$706$	−22.2492			−0.837361
$707$		0			0
$708$		0			0
$709$		−	26.0000i		−	0.976450i	−0.872718	−	0.488225i	$-0.837644\pi$
$709$		−	26.0000i		−	0.976450i	0.872718	−	0.488225i	$-0.162356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−18.1503	+	18.1503i	−0.679734	+	0.679734i
$714$		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
$722$	−43.0187	−	43.0187i	−1.60099	−	1.60099i
$723$		0			0
$724$		−	2.06386i		−	0.0767027i
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$	−42.0639			−1.55049
$737$		0			0
$738$		0			0
$739$		−	48.9574i		−	1.80093i	−0.434930	−	0.900464i	$-0.643227\pi$
$739$		−	48.9574i		−	1.80093i	0.434930	−	0.900464i	$-0.356773\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	22.0454	−	22.0454i	0.808768	−	0.808768i	−0.175680	−	0.984447i	$-0.556212\pi$
$743$	22.0454	−	22.0454i	0.808768	−	0.808768i	0.984447	+	0.175680i	$0.0562123\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−42.9574			−1.56754			−0.783769	−	0.621052i	$-0.786706\pi$
$751$	−42.9574			−1.56754			−0.783769	+	0.621052i	$0.786706\pi$
$752$	11.4806	+	11.4806i	0.418653	+	0.418653i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	−23.8745	+	23.8745i	−0.867160	+	0.867160i
$759$		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
$766$	12.1033			0.437311
$767$		0			0
$768$		0			0
$769$			49.8328i			1.79702i	0.438956	+	0.898509i	$0.355348\pi$
$769$			49.8328i			1.79702i	−0.438956	+	0.898509i	$0.644652\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16.8573	−	16.8573i	0.606315	−	0.606315i	−0.335666	−	0.941981i	$-0.608961\pi$
$773$	16.8573	−	16.8573i	0.606315	−	0.606315i	0.941981	+	0.335666i	$0.108961\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	52.6380	+	52.6380i	1.88233	+	1.88233i
$783$		0			0
$784$			10.9361i			0.390576i
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	−10.7075	+	10.7075i	−0.381439	+	0.381439i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−13.6656			−0.484365
$797$	20.3268	+	20.3268i	0.720012	+	0.720012i	0.968607	−	0.248596i	$-0.0799691\pi$
$797$	20.3268	+	20.3268i	0.720012	+	0.720012i	−0.248596	+	0.968607i	$0.579969\pi$
$798$		0			0
$799$			76.2492i			2.69750i
$800$		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
$818$	10.4705	−	10.4705i	0.366093	−	0.366093i
$819$		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	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−37.8306	−	37.8306i	−1.31550	−	1.31550i	−0.917299	−	0.398200i	$-0.869635\pi$
$827$	−37.8306	−	37.8306i	−1.31550	−	1.31550i	−0.398200	−	0.917299i	$-0.630365\pi$
$828$		0			0
$829$			34.0000i			1.18087i	0.807086	+	0.590434i	$0.201044\pi$
$829$			34.0000i			1.18087i	−0.807086	+	0.590434i	$0.798956\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−36.3167	+	36.3167i	−1.25830	+	1.25830i
$834$		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
$842$	24.5371	+	24.5371i	0.845604	+	0.845604i
$843$		0			0
$844$		−	17.6869i		−	0.608809i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−3.37515	+	3.37515i	−0.115903	+	0.115903i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$	31.7508			1.08522
$857$	38.0515	+	38.0515i	1.29981	+	1.29981i	0.928513	+	0.371300i	$0.121088\pi$
$857$	38.0515	+	38.0515i	1.29981	+	1.29981i	0.371300	+	0.928513i	$0.378912\pi$
$858$		0			0
$859$		−	55.5410i		−	1.89504i	−0.319704	−	0.947518i	$-0.603583\pi$
$859$		−	55.5410i		−	1.89504i	0.319704	−	0.947518i	$-0.396417\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	40.8583	−	40.8583i	1.39083	−	1.39083i	0.567371	−	0.823462i	$-0.307960\pi$
$863$	40.8583	−	40.8583i	1.39083	−	1.39083i	0.823462	−	0.567371i	$-0.192040\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−44.1069	−	44.1069i	−1.49365	−	1.49365i
$873$		0			0
$874$		−	88.3525i		−	2.98857i
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$	19.3329	−	19.3329i	0.652453	−	0.652453i
$879$		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	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$	36.9787			1.24232
$887$	15.3434	+	15.3434i	0.515182	+	0.515182i	0.916110	−	0.400928i	$-0.131312\pi$
$887$	15.3434	+	15.3434i	0.515182	+	0.515182i	−0.400928	+	0.916110i	$0.631312\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	63.9920	−	63.9920i	2.14141	−	2.14141i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−22.4164			−0.746799
$902$		0			0
$903$		0			0
$904$			63.5016i			2.11203i
$905$		0			0
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$	−2.03382	+	2.03382i	−0.0674947	+	0.0674947i
$909$		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
$916$	22.5623			0.745480
$917$		0			0
$918$		0			0
$919$		−	56.0000i		−	1.84727i	−0.383274	−	0.923635i	$-0.625203\pi$
$919$		−	56.0000i		−	1.84727i	0.383274	−	0.923635i	$-0.374797\pi$
$920$		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
$931$	60.9574			1.99780
$932$	12.5527	+	12.5527i	0.411177	+	0.411177i
$933$		0			0
$934$			46.1459i			1.50994i
$935$		0			0
$936$		0			0
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	31.7751	+	31.7751i	1.03255	+	1.03255i	0.999452	+	0.0331004i	$0.0105381\pi$
$947$	31.7751	+	31.7751i	1.03255	+	1.03255i	0.0331004	+	0.999452i	$0.489462\pi$
$948$		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	$-0.859290\pi$
$953$	−14.6969	+	14.6969i	−0.476081	+	0.476081i	0.427795	+	0.903876i	$0.359290\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−23.6656			−0.763407
$962$		0			0
$963$		0			0
$964$		−	22.0639i		−	0.710629i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	23.7640	−	23.7640i	0.763805	−	0.763805i
$969$		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
$976$	−22.5228			−0.720939
$977$	44.0908	+	44.0908i	1.41059	+	1.41059i	0.755880	+	0.654710i	$0.227209\pi$
$977$	44.0908	+	44.0908i	1.41059	+	1.41059i	0.654710	+	0.755880i	$0.272791\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	39.5653	−	39.5653i	1.26194	−	1.26194i	0.311785	−	0.950153i	$-0.399073\pi$
$983$	39.5653	−	39.5653i	1.26194	−	1.26194i	0.950153	−	0.311785i	$-0.100927\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	30.9574			0.983395			0.491698	−	0.870766i	$-0.336377\pi$
$991$	30.9574			0.983395			0.491698	+	0.870766i	$0.336377\pi$
$992$	8.49881	+	8.49881i	0.269837	+	0.269837i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$	11.5749	−	11.5749i	0.366396	−	0.366396i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.f.h.593.3	yes	8
3.2	odd	2	inner	675.2.f.h.593.2	yes	8
5.2	odd	4	inner	675.2.f.h.107.2	✓	8
5.3	odd	4	inner	675.2.f.h.107.3	yes	8
5.4	even	2	inner	675.2.f.h.593.2	yes	8
15.2	even	4	inner	675.2.f.h.107.3	yes	8
15.8	even	4	inner	675.2.f.h.107.2	✓	8
15.14	odd	2	CM	675.2.f.h.593.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.2.f.h.107.2	✓	8	5.2	odd	4	inner
675.2.f.h.107.2	✓	8	15.8	even	4	inner
675.2.f.h.107.3	yes	8	5.3	odd	4	inner
675.2.f.h.107.3	yes	8	15.2	even	4	inner
675.2.f.h.593.2	yes	8	3.2	odd	2	inner
675.2.f.h.593.2	yes	8	5.4	even	2	inner
675.2.f.h.593.3	yes	8	1.1	even	1	trivial
675.2.f.h.593.3	yes	8	15.14	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.756934 + 0.756934i) q^{2} -0.854102i q^{4} +(2.16037 - 2.16037i) q^{8} +1.56231 q^{16} +(5.18810 + 5.18810i) q^{17} -8.70820i q^{19} +(6.70197 - 6.70197i) q^{23} -2.70820 q^{31} +(-3.13817 - 3.13817i) q^{32} +7.85410i q^{34} +(6.59154 - 6.59154i) q^{38} +10.1459 q^{46} +(7.34847 + 7.34847i) q^{47} +7.00000i q^{49} +(-2.16037 + 2.16037i) q^{53} -14.4164 q^{61} +(-2.04993 - 2.04993i) q^{62} -7.87539i q^{64} +(4.43117 - 4.43117i) q^{68} -7.43769 q^{76} +14.7082i q^{79} +(-9.72971 + 9.72971i) q^{83} +(-5.72417 - 5.72417i) q^{92} +11.1246i q^{94} +(-5.29854 + 5.29854i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 68 q^{16} + 32 q^{31} + 108 q^{46} - 8 q^{61} - 140 q^{76}+O(q^{100}) \) 8 * q - 68 * q^16 + 32 * q^31 + 108 * q^46 - 8 * q^61 - 140 * q^76

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