LMFDB - Embedded newform 225.3.g.c.118.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,3,Mod(82,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("225.82");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	225.g (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:	$6.13080594811$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{30})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 225 $ x^4 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			118.1
Root			$-2.73861 + 2.73861i$ of defining polynomial
Character	$\chi$	$=$	225.118
Dual form			225.3.g.c.82.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.73861 + 2.73861i) q^{2} -11.0000i q^{4} +(19.1703 + 19.1703i) q^{8} -61.0000 q^{16} +(-21.9089 + 21.9089i) q^{17} +22.0000i q^{19} +(-21.9089 - 21.9089i) q^{23} -2.00000 q^{31} +(90.3742 - 90.3742i) q^{32} -120.000i q^{34} +(-60.2495 - 60.2495i) q^{38} +120.000 q^{46} +(-65.7267 + 65.7267i) q^{47} +49.0000i q^{49} +(-43.8178 - 43.8178i) q^{53} -118.000 q^{61} +(5.47723 - 5.47723i) q^{62} +251.000i q^{64} +(240.998 + 240.998i) q^{68} +242.000 q^{76} +98.0000i q^{79} +(43.8178 + 43.8178i) q^{83} +(-240.998 + 240.998i) q^{92} -360.000i q^{94} +(-134.192 - 134.192i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 244 q^{16} - 8 q^{31} + 480 q^{46} - 472 q^{61} + 968 q^{76}+O(q^{100}) $ 4 * q - 244 * q^16 - 8 * q^31 + 480 * q^46 - 472 * q^61 + 968 * q^76

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−2.73861	+	2.73861i	−1.36931	+	1.36931i	−0.507877	+	0.861430i	$0.669569\pi$
$2$	−2.73861	+	2.73861i	−1.36931	+	1.36931i	−0.861430	+	0.507877i	$0.830431\pi$
$3$		0			0
$3$		0			0
$4$		−	11.0000i		−	2.75000i
$5$		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	$0.250000\pi$
$8$	19.1703	+	19.1703i	2.39629	+	2.39629i
$9$		0			0
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		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
$16$	−61.0000			−3.81250
$17$	−21.9089	+	21.9089i	−1.28876	+	1.28876i	−0.353218	+	0.935541i	$0.614913\pi$
$17$	−21.9089	+	21.9089i	−1.28876	+	1.28876i	−0.935541	+	0.353218i	$0.885087\pi$
$18$		0			0
$19$			22.0000i			1.15789i	0.815365	+	0.578947i	$0.196536\pi$
$19$			22.0000i			1.15789i	−0.815365	+	0.578947i	$0.803464\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−21.9089	−	21.9089i	−0.952561	−	0.952561i	0.0463637	−	0.998925i	$-0.485237\pi$
$23$	−21.9089	−	21.9089i	−0.952561	−	0.952561i	−0.998925	+	0.0463637i	$0.985237\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−2.00000			−0.0645161			−0.0322581	−	0.999480i	$-0.510270\pi$
$31$	−2.00000			−0.0645161			−0.0322581	+	0.999480i	$0.510270\pi$
$32$	90.3742	−	90.3742i	2.82419	−	2.82419i
$33$		0			0
$34$		−	120.000i		−	3.52941i
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$	−60.2495	−	60.2495i	−1.58551	−	1.58551i
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		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
$46$	120.000			2.60870
$47$	−65.7267	+	65.7267i	−1.39844	+	1.39844i	−0.593907	+	0.804534i	$0.702415\pi$
$47$	−65.7267	+	65.7267i	−1.39844	+	1.39844i	−0.804534	+	0.593907i	$0.797585\pi$
$48$		0			0
$49$			49.0000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−43.8178	−	43.8178i	−0.826751	−	0.826751i	0.160315	−	0.987066i	$-0.448749\pi$
$53$	−43.8178	−	43.8178i	−0.826751	−	0.826751i	−0.987066	+	0.160315i	$0.948749\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−118.000			−1.93443			−0.967213	−	0.253966i	$-0.918265\pi$
$61$	−118.000			−1.93443			−0.967213	+	0.253966i	$0.918265\pi$
$62$	5.47723	−	5.47723i	0.0883423	−	0.0883423i
$63$		0			0
$64$			251.000i			3.92188i
$65$		0			0
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	240.998	+	240.998i	3.54409	+	3.54409i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		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
$76$	242.000			3.18421
$77$		0			0
$78$		0			0
$79$			98.0000i			1.24051i	0.784402	+	0.620253i	$0.212970\pi$
$79$			98.0000i			1.24051i	−0.784402	+	0.620253i	$0.787030\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	43.8178	+	43.8178i	0.527925	+	0.527925i	0.919953	−	0.392028i	$-0.128226\pi$
$83$	43.8178	+	43.8178i	0.527925	+	0.527925i	−0.392028	+	0.919953i	$0.628226\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−240.998	+	240.998i	−2.61954	+	2.61954i
$93$		0			0
$94$		−	360.000i		−	3.82979i
$95$		0			0
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$	−134.192	−	134.192i	−1.36931	−	1.36931i
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		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
$106$	240.000			2.26415
$107$	131.453	−	131.453i	1.22854	−	1.22854i	0.264019	−	0.964517i	$-0.414952\pi$
$107$	131.453	−	131.453i	1.22854	−	1.22854i	0.964517	−	0.264019i	$-0.0850482\pi$
$108$		0			0
$109$		−	22.0000i		−	0.201835i	−0.994895	−	0.100917i	$-0.967822\pi$
$109$		−	22.0000i		−	0.201835i	0.994895	−	0.100917i	$-0.0321778\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	65.7267	+	65.7267i	0.581652	+	0.581652i	0.935357	−	0.353705i	$-0.115078\pi$
$113$	65.7267	+	65.7267i	0.581652	+	0.581652i	−0.353705	+	0.935357i	$0.615078\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−121.000			−1.00000
$122$	323.156	−	323.156i	2.64882	−	2.64882i
$123$		0			0
$124$			22.0000i			0.177419i
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	−325.895	−	325.895i	−2.54605	−	2.54605i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−840.000			−6.17647
$137$	109.545	−	109.545i	0.799595	−	0.799595i	−0.183437	−	0.983032i	$-0.558722\pi$
$137$	109.545	−	109.545i	0.799595	−	0.799595i	0.983032	+	0.183437i	$0.0587222\pi$
$138$		0			0
$139$			262.000i			1.88489i	0.334358	+	0.942446i	$0.391480\pi$
$139$			262.000i			1.88489i	−0.334358	+	0.942446i	$0.608520\pi$
$140$		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		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	238.000			1.57616			0.788079	−	0.615574i	$-0.211076\pi$
$151$	238.000			1.57616			0.788079	+	0.615574i	$0.211076\pi$
$152$	−421.746	+	421.746i	−2.77465	+	2.77465i
$153$		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	$0.250000\pi$
$158$	−268.384	−	268.384i	−1.69863	−	1.69863i
$159$		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
$166$	−240.000			−1.44578
$167$	153.362	−	153.362i	0.918337	−	0.918337i	−0.0785713	−	0.996908i	$-0.525036\pi$
$167$	153.362	−	153.362i	0.918337	−	0.918337i	0.996908	+	0.0785713i	$0.0250358\pi$
$168$		0			0
$169$		−	169.000i		−	1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	219.089	+	219.089i	1.26641	+	1.26641i	0.947929	+	0.318481i	$0.103173\pi$
$173$	219.089	+	219.089i	1.26641	+	1.26641i	0.318481	+	0.947929i	$0.396827\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−122.000			−0.674033			−0.337017	−	0.941499i	$-0.609418\pi$
$181$	−122.000			−0.674033			−0.337017	+	0.941499i	$0.609418\pi$
$182$		0			0
$183$		0			0
$184$		−	840.000i		−	4.56522i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	722.994	+	722.994i	3.84571	+	3.84571i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		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
$196$	539.000			2.75000
$197$	87.6356	−	87.6356i	0.444851	−	0.444851i	−0.448788	−	0.893638i	$-0.648144\pi$
$197$	87.6356	−	87.6356i	0.444851	−	0.444851i	0.893638	+	0.448788i	$0.148144\pi$
$198$		0			0
$199$		−	142.000i		−	0.713568i	−0.934187	−	0.356784i	$-0.883873\pi$
$199$		−	142.000i		−	0.713568i	0.934187	−	0.356784i	$-0.116127\pi$
$200$		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
$211$	362.000			1.71564			0.857820	−	0.513950i	$-0.171818\pi$
$211$	362.000			1.71564			0.857820	+	0.513950i	$0.171818\pi$
$212$	−481.996	+	481.996i	−2.27357	+	2.27357i
$213$		0			0
$214$			720.000i			3.36449i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	60.2495	+	60.2495i	0.276374	+	0.276374i
$219$		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
$226$	−360.000			−1.59292
$227$	−306.725	+	306.725i	−1.35121	+	1.35121i	−0.466899	+	0.884310i	$0.654629\pi$
$227$	−306.725	+	306.725i	−1.35121	+	1.35121i	−0.884310	+	0.466899i	$0.845371\pi$
$228$		0			0
$229$		−	218.000i		−	0.951965i	−0.879455	−	0.475983i	$-0.842093\pi$
$229$		−	218.000i		−	0.951965i	0.879455	−	0.475983i	$-0.157907\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−328.634	−	328.634i	−1.41044	−	1.41044i	−0.756814	−	0.653631i	$-0.773245\pi$
$233$	−328.634	−	328.634i	−1.41044	−	1.41044i	−0.653631	−	0.756814i	$-0.726755\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−478.000			−1.98340			−0.991701	−	0.128564i	$-0.958963\pi$
$241$	−478.000			−1.98340			−0.991701	+	0.128564i	$0.958963\pi$
$242$	331.372	−	331.372i	1.36931	−	1.36931i
$243$		0			0
$244$			1298.00i			5.31967i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−38.3406	−	38.3406i	−0.154599	−	0.154599i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	781.000			3.05078
$257$	−153.362	+	153.362i	−0.596741	+	0.596741i	−0.939444	−	0.342703i	$-0.888657\pi$
$257$	−153.362	+	153.362i	−0.596741	+	0.596741i	0.342703	+	0.939444i	$0.388657\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	197.180	+	197.180i	0.749734	+	0.749734i	0.974429	−	0.224695i	$-0.0721385\pi$
$263$	197.180	+	197.180i	0.749734	+	0.749734i	−0.224695	+	0.974429i	$0.572139\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	482.000			1.77860			0.889299	−	0.457326i	$-0.151193\pi$
$271$	482.000			1.77860			0.889299	+	0.457326i	$0.151193\pi$
$272$	1336.44	−	1336.44i	4.91339	−	4.91339i
$273$		0			0
$274$			600.000i			2.18978i
$275$		0			0
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$	−717.517	−	717.517i	−2.58099	−	2.58099i
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		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
$289$		−	671.000i		−	2.32180i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−306.725	−	306.725i	−1.04684	−	1.04684i	−0.998848	−	0.0479941i	$-0.984717\pi$
$293$	−306.725	−	306.725i	−1.04684	−	1.04684i	−0.0479941	−	0.998848i	$-0.515283\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−651.790	+	651.790i	−2.15824	+	2.15824i
$303$		0			0
$304$		−	1342.00i		−	4.41447i
$305$		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			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		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
$316$	1078.00			3.41139
$317$	−438.178	+	438.178i	−1.38227	+	1.38227i	−0.541681	+	0.840584i	$0.682212\pi$
$317$	−438.178	+	438.178i	−1.38227	+	1.38227i	−0.840584	+	0.541681i	$0.817788\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−481.996	−	481.996i	−1.49225	−	1.49225i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	122.000			0.368580			0.184290	−	0.982872i	$-0.441001\pi$
$331$	122.000			0.368580			0.184290	+	0.982872i	$0.441001\pi$
$332$	481.996	−	481.996i	1.45179	−	1.45179i
$333$		0			0
$334$			840.000i			2.51497i
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	462.826	+	462.826i	1.36931	+	1.36931i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1200.00			−3.46821
$347$	−262.907	+	262.907i	−0.757657	+	0.757657i	−0.975895	−	0.218239i	$-0.929969\pi$
$347$	−262.907	+	262.907i	−0.757657	+	0.757657i	0.218239	+	0.975895i	$0.429969\pi$
$348$		0			0
$349$		−	458.000i		−	1.31232i	−0.754621	−	0.656160i	$-0.772179\pi$
$349$		−	458.000i		−	1.31232i	0.754621	−	0.656160i	$-0.227821\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	460.087	+	460.087i	1.30336	+	1.30336i	0.926111	+	0.377252i	$0.123131\pi$
$353$	460.087	+	460.087i	1.30336	+	1.30336i	0.377252	+	0.926111i	$0.376869\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−123.000			−0.340720
$362$	334.111	−	334.111i	0.922958	−	0.922958i
$363$		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	$0.250000\pi$
$368$	1336.44	+	1336.44i	3.63164	+	3.63164i
$369$		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
$376$	−2520.00			−6.70213
$377$		0			0
$378$		0			0
$379$			742.000i			1.95778i	0.204379	+	0.978892i	$0.434482\pi$
$379$			742.000i			1.95778i	−0.204379	+	0.978892i	$0.565518\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−240.998	−	240.998i	−0.629237	−	0.629237i	0.318639	−	0.947876i	$-0.396774\pi$
$383$	−240.998	−	240.998i	−0.629237	−	0.629237i	−0.947876	+	0.318639i	$0.896774\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	960.000			2.45524
$392$	−939.344	+	939.344i	−2.39629	+	2.39629i
$393$		0			0
$394$			480.000i			1.21827i
$395$		0			0
$396$		0			0
$397$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$397$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$398$	388.883	+	388.883i	0.977093	+	0.977093i
$399$		0			0
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			142.000i			0.347188i	0.984817	+	0.173594i	$0.0555381\pi$
$409$			142.000i			0.347188i	−0.984817	+	0.173594i	$0.944462\pi$
$410$		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		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−602.000			−1.42993			−0.714964	−	0.699161i	$-0.753557\pi$
$421$	−602.000			−1.42993			−0.714964	+	0.699161i	$0.753557\pi$
$422$	−991.378	+	991.378i	−2.34924	+	2.34924i
$423$		0			0
$424$		−	1680.00i		−	3.96226i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1445.99	−	1445.99i	−3.37848	−	3.37848i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		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
$436$	−242.000			−0.555046
$437$	481.996	−	481.996i	1.10297	−	1.10297i
$438$		0			0
$439$		−	622.000i		−	1.41686i	−0.705783	−	0.708428i	$-0.749405\pi$
$439$		−	622.000i		−	1.41686i	0.705783	−	0.708428i	$-0.250595\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	481.996	+	481.996i	1.08803	+	1.08803i	0.995732	+	0.0922950i	$0.0294203\pi$
$443$	481.996	+	481.996i	1.08803	+	1.08803i	0.0922950	+	0.995732i	$0.470580\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$	722.994	−	722.994i	1.59954	−	1.59954i
$453$		0			0
$454$		−	1680.00i		−	3.70044i
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$	597.018	+	597.018i	1.30353	+	1.30353i
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		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
$466$	1800.00			3.86266
$467$	613.449	−	613.449i	1.31360	−	1.31360i	0.394850	−	0.918745i	$-0.370796\pi$
$467$	613.449	−	613.449i	1.31360	−	1.31360i	0.918745	−	0.394850i	$-0.129204\pi$
$468$		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		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$	1309.06	−	1309.06i	2.71589	−	2.71589i
$483$		0			0
$484$			1331.00i			2.75000i
$485$		0			0
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$	−2262.09	−	2262.09i	−4.63544	−	4.63544i
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	122.000			0.245968
$497$		0			0
$498$		0			0
$499$			938.000i			1.87976i	0.341506	+	0.939880i	$0.389063\pi$
$499$			938.000i			1.87976i	−0.341506	+	0.939880i	$0.610937\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	109.545	+	109.545i	0.217782	+	0.217782i	0.807563	−	0.589781i	$-0.200786\pi$
$503$	109.545	+	109.545i	0.217782	+	0.217782i	−0.589781	+	0.807563i	$0.700786\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−835.277	+	835.277i	−1.63140	+	1.63140i
$513$		0			0
$514$		−	840.000i		−	1.63424i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		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
$526$	−1080.00			−2.05323
$527$	43.8178	−	43.8178i	0.0831457	−	0.0831457i
$528$		0			0
$529$			431.000i			0.814745i
$530$		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
$541$	−1078.00			−1.99261			−0.996303	−	0.0859072i	$-0.972621\pi$
$541$	−1078.00			−1.99261			−0.996303	+	0.0859072i	$0.972621\pi$
$542$	−1320.01	+	1320.01i	−2.43545	+	2.43545i
$543$		0			0
$544$			3960.00i			7.27941i
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	−1204.99	−	1204.99i	−2.19889	−	2.19889i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	2882.00			5.18345
$557$	657.267	−	657.267i	1.18001	−	1.18001i	0.200272	−	0.979740i	$-0.435817\pi$
$557$	657.267	−	657.267i	1.18001	−	1.18001i	0.979740	−	0.200272i	$-0.0641827\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	788.720	+	788.720i	1.40092	+	1.40092i	0.797171	+	0.603753i	$0.206329\pi$
$563$	788.720	+	788.720i	1.40092	+	1.40092i	0.603753	+	0.797171i	$0.293671\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−358.000			−0.626970			−0.313485	−	0.949593i	$-0.601497\pi$
$571$	−358.000			−0.626970			−0.313485	+	0.949593i	$0.601497\pi$
$572$		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	$0.250000\pi$
$578$	1837.61	+	1837.61i	3.17925	+	3.17925i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	1680.00			2.86689
$587$	569.631	−	569.631i	0.970411	−	0.970411i	−0.0291633	−	0.999575i	$-0.509284\pi$
$587$	569.631	−	569.631i	0.970411	−	0.970411i	0.999575	+	0.0291633i	$0.00928429\pi$
$588$		0			0
$589$		−	44.0000i		−	0.0747029i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−153.362	−	153.362i	−0.258621	−	0.258621i	0.565872	−	0.824493i	$-0.308540\pi$
$593$	−153.362	−	153.362i	−0.258621	−	0.258621i	−0.824493	+	0.565872i	$0.808540\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−242.000			−0.402662			−0.201331	−	0.979523i	$-0.564527\pi$
$601$	−242.000			−0.402662			−0.201331	+	0.979523i	$0.564527\pi$
$602$		0			0
$603$		0			0
$604$		−	2618.00i		−	4.33444i
$605$		0			0
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$	1988.23	+	1988.23i	3.27012	+	3.27012i
$609$		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
$617$	240.998	−	240.998i	0.390596	−	0.390596i	−0.484304	−	0.874900i	$-0.660927\pi$
$617$	240.998	−	240.998i	0.390596	−	0.390596i	0.874900	+	0.484304i	$0.160927\pi$
$618$		0			0
$619$			698.000i			1.12763i	0.825903	+	0.563813i	$0.190666\pi$
$619$			698.000i			1.12763i	−0.825903	+	0.563813i	$0.809334\pi$
$620$		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
$631$	−238.000			−0.377179			−0.188590	−	0.982056i	$-0.560392\pi$
$631$	−238.000			−0.377179			−0.188590	+	0.982056i	$0.560392\pi$
$632$	−1878.69	+	1878.69i	−2.97261	+	2.97261i
$633$		0			0
$634$		−	2400.00i		−	3.78549i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		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
$646$	2640.00			4.08669
$647$	−766.812	+	766.812i	−1.18518	+	1.18518i	−0.206796	+	0.978384i	$0.566304\pi$
$647$	−766.812	+	766.812i	−1.18518	+	1.18518i	−0.978384	+	0.206796i	$0.933696\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	481.996	+	481.996i	0.738125	+	0.738125i	0.972215	−	0.234090i	$-0.0752109\pi$
$653$	481.996	+	481.996i	0.738125	+	0.738125i	−0.234090	+	0.972215i	$0.575211\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	838.000			1.26778			0.633888	−	0.773425i	$-0.281458\pi$
$661$	838.000			1.26778			0.633888	+	0.773425i	$0.281458\pi$
$662$	−334.111	+	334.111i	−0.504699	+	0.504699i
$663$		0			0
$664$			1680.00i			2.53012i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1686.99	−	1686.99i	−2.52543	−	2.52543i
$669$		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
$676$	−1859.00			−2.75000
$677$	−920.174	+	920.174i	−1.35919	+	1.35919i	−0.484281	+	0.874913i	$0.660919\pi$
$677$	−920.174	+	920.174i	−1.35919	+	1.35919i	−0.874913	+	0.484281i	$0.839081\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−963.992	−	963.992i	−1.41141	−	1.41141i	−0.750222	−	0.661186i	$-0.770053\pi$
$683$	−963.992	−	963.992i	−1.41141	−	1.41141i	−0.661186	−	0.750222i	$-0.729947\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1322.00			−1.91317			−0.956585	−	0.291455i	$-0.905861\pi$
$691$	−1322.00			−1.91317			−0.956585	+	0.291455i	$0.905861\pi$
$692$	2409.98	−	2409.98i	3.48263	−	3.48263i
$693$		0			0
$694$		−	1440.00i		−	2.07493i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	1254.28	+	1254.28i	1.79697	+	1.79697i
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−2520.00			−3.56941
$707$		0			0
$708$		0			0
$709$			742.000i			1.04654i	0.852166	+	0.523272i	$0.175289\pi$
$709$			742.000i			1.04654i	−0.852166	+	0.523272i	$0.824711\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	43.8178	+	43.8178i	0.0614555	+	0.0614555i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	336.849	−	336.849i	0.466550	−	0.466550i
$723$		0			0
$724$			1342.00i			1.85359i
$725$		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
$736$	−3960.00			−5.38043
$737$		0			0
$738$		0			0
$739$			1462.00i			1.97835i	0.146744	+	0.989175i	$0.453121\pi$
$739$			1462.00i			1.97835i	−0.146744	+	0.989175i	$0.546879\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−985.901	−	985.901i	−1.32692	−	1.32692i	−0.908044	−	0.418875i	$-0.862425\pi$
$743$	−985.901	−	985.901i	−1.32692	−	1.32692i	−0.418875	−	0.908044i	$-0.637575\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1438.00			1.91478			0.957390	−	0.288798i	$-0.0932555\pi$
$751$	1438.00			1.91478			0.957390	+	0.288798i	$0.0932555\pi$
$752$	4009.33	−	4009.33i	5.33155	−	5.33155i
$753$		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	$0.250000\pi$
$758$	−2032.05	−	2032.05i	−2.68081	−	2.68081i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	1320.00			1.72324
$767$		0			0
$768$		0			0
$769$			578.000i			0.751625i	0.926696	+	0.375813i	$0.122636\pi$
$769$			578.000i			0.751625i	−0.926696	+	0.375813i	$0.877364\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	175.271	+	175.271i	0.226742	+	0.226742i	0.811330	−	0.584588i	$-0.198744\pi$
$773$	175.271	+	175.271i	0.226742	+	0.226742i	−0.584588	+	0.811330i	$0.698744\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−2629.07	+	2629.07i	−3.36198	+	3.36198i
$783$		0			0
$784$		−	2989.00i		−	3.81250i
$785$		0			0
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$	−963.992	−	963.992i	−1.22334	−	1.22334i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−1562.00			−1.96231
$797$	−963.992	+	963.992i	−1.20953	+	1.20953i	−0.238345	+	0.971181i	$0.576605\pi$
$797$	−963.992	+	963.992i	−1.20953	+	1.20953i	−0.971181	+	0.238345i	$0.923395\pi$
$798$		0			0
$799$		−	2880.00i		−	3.60451i
$800$		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		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−1082.00			−1.33416			−0.667078	−	0.744988i	$-0.732455\pi$
$811$	−1082.00			−1.33416			−0.667078	+	0.744988i	$0.732455\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−388.883	−	388.883i	−0.475407	−	0.475407i
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		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
$827$	−1139.26	+	1139.26i	−1.37759	+	1.37759i	−0.528903	+	0.848683i	$0.677396\pi$
$827$	−1139.26	+	1139.26i	−1.37759	+	1.37759i	−0.848683	+	0.528903i	$0.822604\pi$
$828$		0			0
$829$			502.000i			0.605549i	0.953062	+	0.302774i	$0.0979129\pi$
$829$			502.000i			0.605549i	−0.953062	+	0.302774i	$0.902087\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1073.54	−	1073.54i	−1.28876	−	1.28876i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	841.000			1.00000
$842$	1648.64	−	1648.64i	1.95801	−	1.95801i
$843$		0			0
$844$		−	3982.00i		−	4.71801i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	2672.89	+	2672.89i	3.15199	+	3.15199i
$849$		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
$856$	5040.00			5.88785
$857$	−284.816	+	284.816i	−0.332340	+	0.332340i	−0.853475	−	0.521134i	$-0.825509\pi$
$857$	−284.816	+	284.816i	−0.332340	+	0.332340i	0.521134	+	0.853475i	$0.325509\pi$
$858$		0			0
$859$			218.000i			0.253783i	0.991917	+	0.126892i	$0.0405001\pi$
$859$			218.000i			0.253783i	−0.991917	+	0.126892i	$0.959500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1204.99	+	1204.99i	1.39628	+	1.39628i	0.810392	+	0.585888i	$0.199254\pi$
$863$	1204.99	+	1204.99i	1.39628	+	1.39628i	0.585888	+	0.810392i	$0.300746\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	421.746	−	421.746i	0.483654	−	0.483654i
$873$		0			0
$874$			2640.00i			3.02059i
$875$		0			0
$876$		0			0
$877$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$877$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$878$	1703.42	+	1703.42i	1.94011	+	1.94011i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		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
$886$	−2640.00			−2.97968
$887$	372.451	−	372.451i	0.419900	−	0.419900i	−0.465269	−	0.885169i	$-0.654042\pi$
$887$	372.451	−	372.451i	0.419900	−	0.419900i	0.885169	+	0.465269i	$0.154042\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1445.99	−	1445.99i	−1.61925	−	1.61925i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	1920.00			2.13097
$902$		0			0
$903$		0			0
$904$			2520.00i			2.78761i
$905$		0			0
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$	3373.97	+	3373.97i	3.71583	+	3.71583i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−2398.00			−2.61790
$917$		0			0
$918$		0			0
$919$		−	1298.00i		−	1.41240i	−0.708010	−	0.706202i	$-0.750407\pi$
$919$		−	1298.00i		−	1.41240i	0.708010	−	0.706202i	$-0.249593\pi$
$920$		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		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	−1078.00			−1.15789
$932$	−3614.97	+	3614.97i	−3.87872	+	3.87872i
$933$		0			0
$934$			3360.00i			3.59743i
$935$		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			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−744.903	+	744.903i	−0.786592	+	0.786592i	−0.980934	−	0.194342i	$-0.937743\pi$
$947$	−744.903	+	744.903i	−0.786592	+	0.786592i	0.194342	+	0.980934i	$0.437743\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	854.447	+	854.447i	0.896587	+	0.896587i	0.995133	−	0.0985458i	$-0.0314191\pi$
$953$	854.447	+	854.447i	0.896587	+	0.896587i	−0.0985458	+	0.995133i	$0.531419\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−957.000			−0.995838
$962$		0			0
$963$		0			0
$964$			5258.00i			5.45436i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−2319.61	−	2319.61i	−2.39629	−	2.39629i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	7198.00			7.37500
$977$	197.180	−	197.180i	0.201822	−	0.201822i	−0.598958	−	0.800780i	$-0.704418\pi$
$977$	197.180	−	197.180i	0.201822	−	0.201822i	0.800780	+	0.598958i	$0.204418\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−153.362	−	153.362i	−0.156015	−	0.156015i	0.624783	−	0.780798i	$-0.285187\pi$
$983$	−153.362	−	153.362i	−0.156015	−	0.156015i	−0.780798	+	0.624783i	$0.785187\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−958.000			−0.966700			−0.483350	−	0.875427i	$-0.660580\pi$
$991$	−958.000			−0.966700			−0.483350	+	0.875427i	$0.660580\pi$
$992$	−180.748	+	180.748i	−0.182206	+	0.182206i
$993$		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	$0.250000\pi$
$998$	−2568.82	−	2568.82i	−2.57397	−	2.57397i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.3.g.c.118.1	yes	4
3.2	odd	2	inner	225.3.g.c.118.2	yes	4
5.2	odd	4	inner	225.3.g.c.82.1	✓	4
5.3	odd	4	inner	225.3.g.c.82.2	yes	4
5.4	even	2	inner	225.3.g.c.118.2	yes	4
15.2	even	4	inner	225.3.g.c.82.2	yes	4
15.8	even	4	inner	225.3.g.c.82.1	✓	4
15.14	odd	2	CM	225.3.g.c.118.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.3.g.c.82.1	✓	4	5.2	odd	4	inner
225.3.g.c.82.1	✓	4	15.8	even	4	inner
225.3.g.c.82.2	yes	4	5.3	odd	4	inner
225.3.g.c.82.2	yes	4	15.2	even	4	inner
225.3.g.c.118.1	yes	4	1.1	even	1	trivial
225.3.g.c.118.1	yes	4	15.14	odd	2	CM
225.3.g.c.118.2	yes	4	3.2	odd	2	inner
225.3.g.c.118.2	yes	4	5.4	even	2	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	225.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.73861 + 2.73861i) q^{2} -11.0000i q^{4} +(19.1703 + 19.1703i) q^{8} -61.0000 q^{16} +(-21.9089 + 21.9089i) q^{17} +22.0000i q^{19} +(-21.9089 - 21.9089i) q^{23} -2.00000 q^{31} +(90.3742 - 90.3742i) q^{32} -120.000i q^{34} +(-60.2495 - 60.2495i) q^{38} +120.000 q^{46} +(-65.7267 + 65.7267i) q^{47} +49.0000i q^{49} +(-43.8178 - 43.8178i) q^{53} -118.000 q^{61} +(5.47723 - 5.47723i) q^{62} +251.000i q^{64} +(240.998 + 240.998i) q^{68} +242.000 q^{76} +98.0000i q^{79} +(43.8178 + 43.8178i) q^{83} +(-240.998 + 240.998i) q^{92} -360.000i q^{94} +(-134.192 - 134.192i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 244 q^{16} - 8 q^{31} + 480 q^{46} - 472 q^{61} + 968 q^{76}+O(q^{100}) \) 4 * q - 244 * q^16 - 8 * q^31 + 480 * q^46 - 472 * q^61 + 968 * q^76

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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