LMFDB - Embedded newform 252.2.b.c.55.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,2,Mod(55,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("252.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	252.b (of order $2$, degree $1$, 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:	$2.01223013094$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			55.2
Root			$-1.32288 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	252.55
Dual form			252.2.b.c.55.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32288 + 0.500000i) q^{2} +(1.50000 - 1.32288i) q^{4} -2.64575i q^{7} +(-1.32288 + 2.50000i) q^{8} +O(q^{10})$	$q+(-1.32288 + 0.500000i) q^{2} +(1.50000 - 1.32288i) q^{4} -2.64575i q^{7} +(-1.32288 + 2.50000i) q^{8} -4.00000i q^{11} +(1.32288 + 3.50000i) q^{14} +(0.500000 - 3.96863i) q^{16} +(2.00000 + 5.29150i) q^{22} -8.00000i q^{23} +5.00000 q^{25} +(-3.50000 - 3.96863i) q^{28} +10.5830 q^{29} +(1.32288 + 5.50000i) q^{32} -6.00000 q^{37} +5.29150i q^{43} +(-5.29150 - 6.00000i) q^{44} +(4.00000 + 10.5830i) q^{46} -7.00000 q^{49} +(-6.61438 + 2.50000i) q^{50} -10.5830 q^{53} +(6.61438 + 3.50000i) q^{56} +(-14.0000 + 5.29150i) q^{58} +(-4.50000 - 6.61438i) q^{64} +15.8745i q^{67} +16.0000i q^{71} +(7.93725 - 3.00000i) q^{74} -10.5830 q^{77} -15.8745i q^{79} +(-2.64575 - 7.00000i) q^{86} +(10.0000 + 5.29150i) q^{88} +(-10.5830 - 12.0000i) q^{92} +(9.26013 - 3.50000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4}+O(q^{10}) $ 4 * q + 6 * q^4	$ 4 q + 6 q^{4} + 2 q^{16} + 8 q^{22} + 20 q^{25} - 14 q^{28} - 24 q^{37} + 16 q^{46} - 28 q^{49} - 56 q^{58} - 18 q^{64} + 40 q^{88}+O(q^{100}) $ 4 * q + 6 * q^4 + 2 * q^16 + 8 * q^22 + 20 * q^25 - 14 * q^28 - 24 * q^37 + 16 * q^46 - 28 * q^49 - 56 * q^58 - 18 * q^64 + 40 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.32288	+	0.500000i	−0.935414	+	0.353553i
$2$	−1.32288	+	0.500000i	−0.935414	+	0.353553i
$3$		0			0
$3$		0			0
$4$	1.50000	−	1.32288i	0.750000	−	0.661438i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		−	2.64575i		−	1.00000i
$7$		−	2.64575i		−	1.00000i
$8$	−1.32288	+	2.50000i	−0.467707	+	0.883883i
$9$		0			0
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	1.32288	+	3.50000i	0.353553	+	0.935414i
$15$		0			0
$16$	0.500000	−	3.96863i	0.125000	−	0.992157i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	2.00000	+	5.29150i	0.426401	+	1.12815i
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$	−3.50000	−	3.96863i	−0.661438	−	0.750000i
$29$	10.5830			1.96521			0.982607	−	0.185695i	$-0.0594537\pi$
$29$	10.5830			1.96521			0.982607	+	0.185695i	$0.0594537\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	1.32288	+	5.50000i	0.233854	+	0.972272i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$			5.29150i			0.806947i	0.914991	+	0.403473i	$0.132197\pi$
$43$			5.29150i			0.806947i	−0.914991	+	0.403473i	$0.867803\pi$
$44$	−5.29150	−	6.00000i	−0.797724	−	0.904534i
$45$		0			0
$46$	4.00000	+	10.5830i	0.589768	+	1.56038i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$	−6.61438	+	2.50000i	−0.935414	+	0.353553i
$51$		0			0
$52$		0			0
$53$	−10.5830			−1.45369			−0.726844	−	0.686803i	$-0.759014\pi$
$53$	−10.5830			−1.45369			−0.726844	+	0.686803i	$0.759014\pi$
$54$		0			0
$55$		0			0
$56$	6.61438	+	3.50000i	0.883883	+	0.467707i
$57$		0			0
$58$	−14.0000	+	5.29150i	−1.83829	+	0.694808i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−4.50000	−	6.61438i	−0.562500	−	0.826797i
$65$		0			0
$66$		0			0
$67$			15.8745i			1.93938i	0.244339	+	0.969690i	$0.421429\pi$
$67$			15.8745i			1.93938i	−0.244339	+	0.969690i	$0.578571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			16.0000i			1.89885i	0.313993	+	0.949425i	$0.398333\pi$
$71$			16.0000i			1.89885i	−0.313993	+	0.949425i	$0.601667\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	7.93725	−	3.00000i	0.922687	−	0.348743i
$75$		0			0
$76$		0			0
$77$	−10.5830			−1.20605
$78$		0			0
$79$		−	15.8745i		−	1.78602i	−0.450035	−	0.893011i	$-0.648589\pi$
$79$		−	15.8745i		−	1.78602i	0.450035	−	0.893011i	$-0.351411\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	−2.64575	−	7.00000i	−0.285299	−	0.754829i
$87$		0			0
$88$	10.0000	+	5.29150i	1.06600	+	0.564076i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−10.5830	−	12.0000i	−1.10335	−	1.25109i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	9.26013	−	3.50000i	0.935414	−	0.353553i
$99$		0			0
$100$	7.50000	−	6.61438i	0.750000	−	0.661438i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	14.0000	−	5.29150i	1.35980	−	0.513956i
$107$			20.0000i			1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$			20.0000i			1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$		0			0
$109$	18.0000			1.72409			0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000			1.72409			0.862044	+	0.506834i	$0.169184\pi$
$110$		0			0
$111$		0			0
$112$	−10.5000	−	1.32288i	−0.992157	−	0.125000i
$113$	21.1660			1.99113			0.995565	−	0.0940721i	$-0.0299884\pi$
$113$	21.1660			1.99113			0.995565	+	0.0940721i	$0.0299884\pi$
$114$		0			0
$115$		0			0
$116$	15.8745	−	14.0000i	1.47391	−	1.29987i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	15.8745i		−	1.40863i	−0.709885	−	0.704317i	$-0.751253\pi$
$127$		−	15.8745i		−	1.40863i	0.709885	−	0.704317i	$-0.248747\pi$
$128$	9.26013	+	6.50000i	0.818488	+	0.574524i
$129$		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
$134$	−7.93725	−	21.0000i	−0.685674	−	1.81412i
$135$		0			0
$136$		0			0
$137$	−21.1660			−1.80833			−0.904167	−	0.427179i	$-0.859507\pi$
$137$	−21.1660			−1.80833			−0.904167	+	0.427179i	$0.859507\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−8.00000	−	21.1660i	−0.671345	−	1.77621i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−9.00000	+	7.93725i	−0.739795	+	0.652438i
$149$	10.5830			0.866994			0.433497	−	0.901155i	$-0.357280\pi$
$149$	10.5830			0.866994			0.433497	+	0.901155i	$0.357280\pi$
$150$		0			0
$151$		−	5.29150i		−	0.430616i	−0.976546	−	0.215308i	$-0.930924\pi$
$151$		−	5.29150i		−	0.430616i	0.976546	−	0.215308i	$-0.0690756\pi$
$152$		0			0
$153$		0			0
$154$	14.0000	−	5.29150i	1.12815	−	0.426401i
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	7.93725	+	21.0000i	0.631454	+	1.67067i
$159$		0			0
$160$		0			0
$161$	−21.1660			−1.66812
$162$		0			0
$163$			15.8745i			1.24339i	0.783260	+	0.621694i	$0.213555\pi$
$163$			15.8745i			1.24339i	−0.783260	+	0.621694i	$0.786445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$	7.00000	+	7.93725i	0.533745	+	0.605210i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		−	13.2288i		−	1.00000i
$176$	−15.8745	−	2.00000i	−1.19659	−	0.150756i
$177$		0			0
$178$		0			0
$179$			4.00000i			0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$			4.00000i			0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	20.0000	+	10.5830i	1.47442	+	0.780189i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			8.00000i			0.578860i	0.957199	+	0.289430i	$0.0934657\pi$
$191$			8.00000i			0.578860i	−0.957199	+	0.289430i	$0.906534\pi$
$192$		0			0
$193$	−18.0000			−1.29567			−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000			−1.29567			−0.647834	+	0.761781i	$0.724325\pi$
$194$		0			0
$195$		0			0
$196$	−10.5000	+	9.26013i	−0.750000	+	0.661438i
$197$	10.5830			0.754008			0.377004	−	0.926212i	$-0.376954\pi$
$197$	10.5830			0.754008			0.377004	+	0.926212i	$0.376954\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−6.61438	+	12.5000i	−0.467707	+	0.883883i
$201$		0			0
$202$		0			0
$203$		−	28.0000i		−	1.96521i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			26.4575i			1.82141i	0.413057	+	0.910705i	$0.364461\pi$
$211$			26.4575i			1.82141i	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−15.8745	+	14.0000i	−1.09027	+	0.961524i
$213$		0			0
$214$	−10.0000	−	26.4575i	−0.683586	−	1.80860i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−23.8118	+	9.00000i	−1.61274	+	0.609557i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	14.5516	−	3.50000i	0.972272	−	0.233854i
$225$		0			0
$226$	−28.0000	+	10.5830i	−1.86253	+	0.703971i
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−14.0000	+	26.4575i	−0.919145	+	1.73702i
$233$	21.1660			1.38663			0.693316	−	0.720634i	$-0.256149\pi$
$233$	21.1660			1.38663			0.693316	+	0.720634i	$0.256149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	16.0000i		−	1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$		−	16.0000i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	6.61438	−	2.50000i	0.425188	−	0.160706i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−32.0000			−2.01182
$254$	7.93725	+	21.0000i	0.498028	+	1.31766i
$255$		0			0
$256$	−15.5000	−	3.96863i	−0.968750	−	0.248039i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$			15.8745i			0.986394i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	32.0000i		−	1.97320i	−0.163144	−	0.986602i	$-0.552164\pi$
$263$		−	32.0000i		−	1.97320i	0.163144	−	0.986602i	$-0.447836\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	21.0000	+	23.8118i	1.28278	+	1.45453i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	28.0000	−	10.5830i	1.69154	−	0.639343i
$275$		−	20.0000i		−	1.20605i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	21.1660			1.26266			0.631329	−	0.775515i	$-0.282510\pi$
$281$	21.1660			1.26266			0.631329	+	0.775515i	$0.282510\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	21.1660	+	24.0000i	1.25597	+	1.42414i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	7.93725	−	15.0000i	0.461344	−	0.871857i
$297$		0			0
$298$	−14.0000	+	5.29150i	−0.810998	+	0.306529i
$299$		0			0
$300$		0			0
$301$	14.0000			0.806947
$302$	2.64575	+	7.00000i	0.152246	+	0.402805i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−15.8745	+	14.0000i	−0.904534	+	0.797724i
$309$		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		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$	−21.0000	−	23.8118i	−1.18134	−	1.33952i
$317$	−10.5830			−0.594401			−0.297200	−	0.954815i	$-0.596053\pi$
$317$	−10.5830			−0.594401			−0.297200	+	0.954815i	$0.596053\pi$
$318$		0			0
$319$		−	42.3320i		−	2.37014i
$320$		0			0
$321$		0			0
$322$	28.0000	−	10.5830i	1.56038	−	0.589768i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−7.93725	−	21.0000i	−0.439604	−	1.16308i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			5.29150i			0.290847i	0.989369	+	0.145424i	$0.0464545\pi$
$331$			5.29150i			0.290847i	−0.989369	+	0.145424i	$0.953545\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	30.0000			1.63420			0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000			1.63420			0.817102	+	0.576493i	$0.195579\pi$
$338$	−17.1974	+	6.50000i	−0.935414	+	0.353553i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			18.5203i			1.00000i
$344$	−13.2288	−	7.00000i	−0.713247	−	0.377415i
$345$		0			0
$346$		0			0
$347$			4.00000i			0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$			4.00000i			0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	6.61438	+	17.5000i	0.353553	+	0.935414i
$351$		0			0
$352$	22.0000	−	5.29150i	1.17260	−	0.282038i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−2.00000	−	5.29150i	−0.105703	−	0.279665i
$359$			8.00000i			0.422224i	0.977462	+	0.211112i	$0.0677085\pi$
$359$			8.00000i			0.422224i	−0.977462	+	0.211112i	$0.932292\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−31.7490	−	4.00000i	−1.65503	−	0.208514i
$369$		0			0
$370$		0			0
$371$			28.0000i			1.45369i
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	37.0405i		−	1.90264i	−0.308199	−	0.951322i	$-0.599726\pi$
$379$		−	37.0405i		−	1.90264i	0.308199	−	0.951322i	$-0.400274\pi$
$380$		0			0
$381$		0			0
$382$	−4.00000	−	10.5830i	−0.204658	−	0.541474i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	23.8118	−	9.00000i	1.21199	−	0.458088i
$387$		0			0
$388$		0			0
$389$	−10.5830			−0.536580			−0.268290	−	0.963338i	$-0.586458\pi$
$389$	−10.5830			−0.536580			−0.268290	+	0.963338i	$0.586458\pi$
$390$		0			0
$391$		0			0
$392$	9.26013	−	17.5000i	0.467707	−	0.883883i
$393$		0			0
$394$	−14.0000	+	5.29150i	−0.705310	+	0.266582i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	2.50000	−	19.8431i	0.125000	−	0.992157i
$401$	−21.1660			−1.05698			−0.528490	−	0.848939i	$-0.677242\pi$
$401$	−21.1660			−1.05698			−0.528490	+	0.848939i	$0.677242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	14.0000	+	37.0405i	0.694808	+	1.83829i
$407$			24.0000i			1.18964i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		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
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$	−13.2288	−	35.0000i	−0.643966	−	1.70377i
$423$		0			0
$424$	14.0000	−	26.4575i	0.679900	−	1.28489i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	26.4575	+	30.0000i	1.27887	+	1.45010i
$429$		0			0
$430$		0			0
$431$			32.0000i			1.54139i	0.637207	+	0.770693i	$0.280090\pi$
$431$			32.0000i			1.54139i	−0.637207	+	0.770693i	$0.719910\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	27.0000	−	23.8118i	1.29307	−	1.14038i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	20.0000i		−	0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$		−	20.0000i		−	0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−17.5000	+	11.9059i	−0.826797	+	0.562500i
$449$	−42.3320			−1.99777			−0.998886	−	0.0471929i	$-0.984972\pi$
$449$	−42.3320			−1.99777			−0.998886	+	0.0471929i	$0.984972\pi$
$450$		0			0
$451$		0			0
$452$	31.7490	−	28.0000i	1.49335	−	1.31701i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.00000			0.280668			0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000			0.280668			0.140334	+	0.990104i	$0.455182\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		−	15.8745i		−	0.737751i	−0.929479	−	0.368875i	$-0.879743\pi$
$463$		−	15.8745i		−	0.737751i	0.929479	−	0.368875i	$-0.120257\pi$
$464$	5.29150	−	42.0000i	0.245652	−	1.94980i
$465$		0			0
$466$	−28.0000	+	10.5830i	−1.29707	+	0.490248i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	42.0000			1.93938
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.1660			0.973214
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	8.00000	+	21.1660i	0.365911	+	0.968111i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−7.50000	+	6.61438i	−0.340909	+	0.300654i
$485$		0			0
$486$		0			0
$487$			37.0405i			1.67847i	0.543772	+	0.839233i	$0.316996\pi$
$487$			37.0405i			1.67847i	−0.543772	+	0.839233i	$0.683004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			44.0000i			1.98569i	0.119401	+	0.992846i	$0.461903\pi$
$491$			44.0000i			1.98569i	−0.119401	+	0.992846i	$0.538097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	42.3320			1.89885
$498$		0			0
$499$			26.4575i			1.18440i	0.805791	+	0.592200i	$0.201741\pi$
$499$			26.4575i			1.18440i	−0.805791	+	0.592200i	$0.798259\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$	42.3320	−	16.0000i	1.88189	−	0.711287i
$507$		0			0
$508$	−21.0000	−	23.8118i	−0.931724	−	1.05648i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	22.4889	−	2.50000i	0.993878	−	0.110485i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−7.93725	−	21.0000i	−0.348743	−	0.922687i
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	16.0000	+	42.3320i	0.697633	+	1.84576i
$527$		0			0
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−39.6863	−	21.0000i	−1.71419	−	0.907062i
$537$		0			0
$538$		0			0
$539$			28.0000i			1.20605i
$540$		0			0
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			15.8745i			0.678745i	0.940652	+	0.339372i	$0.110215\pi$
$547$			15.8745i			0.678745i	−0.940652	+	0.339372i	$0.889785\pi$
$548$	−31.7490	+	28.0000i	−1.35625	+	1.19610i
$549$		0			0
$550$	10.0000	+	26.4575i	0.426401	+	1.12815i
$551$		0			0
$552$		0			0
$553$	−42.0000			−1.78602
$554$	13.2288	−	5.00000i	0.562036	−	0.212430i
$555$		0			0
$556$		0			0
$557$	10.5830			0.448416			0.224208	−	0.974541i	$-0.428020\pi$
$557$	10.5830			0.448416			0.224208	+	0.974541i	$0.428020\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−28.0000	+	10.5830i	−1.18111	+	0.446417i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−40.0000	−	21.1660i	−1.67836	−	0.888106i
$569$	−42.3320			−1.77465			−0.887325	−	0.461144i	$-0.847439\pi$
$569$	−42.3320			−1.77465			−0.887325	+	0.461144i	$0.847439\pi$
$570$		0			0
$571$		−	47.6235i		−	1.99298i	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$		−	47.6235i		−	1.99298i	0.0836974	−	0.996491i	$-0.473327\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	40.0000i		−	1.66812i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−22.4889	+	8.50000i	−0.935414	+	0.353553i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			42.3320i			1.75321i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−3.00000	+	23.8118i	−0.123299	+	0.978657i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	15.8745	−	14.0000i	0.650245	−	0.573462i
$597$		0			0
$598$		0			0
$599$			32.0000i			1.30748i	0.756717	+	0.653742i	$0.226802\pi$
$599$			32.0000i			1.30748i	−0.756717	+	0.653742i	$0.773198\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−18.5203	+	7.00000i	−0.754829	+	0.285299i
$603$		0			0
$604$	−7.00000	−	7.93725i	−0.284826	−	0.322962i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$		0			0
$615$		0			0
$616$	14.0000	−	26.4575i	0.564076	−	1.06600i
$617$	−42.3320			−1.70422			−0.852111	−	0.523360i	$-0.824678\pi$
$617$	−42.3320			−1.70422			−0.852111	+	0.523360i	$0.824678\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			47.6235i			1.89586i	0.318475	+	0.947931i	$0.396829\pi$
$631$			47.6235i			1.89586i	−0.318475	+	0.947931i	$0.603171\pi$
$632$	39.6863	+	21.0000i	1.57864	+	0.835335i
$633$		0			0
$634$	14.0000	−	5.29150i	0.556011	−	0.210152i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	21.1660	+	56.0000i	0.837970	+	2.21706i
$639$		0			0
$640$		0			0
$641$	21.1660			0.836007			0.418004	−	0.908445i	$-0.362730\pi$
$641$	21.1660			0.836007			0.418004	+	0.908445i	$0.362730\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	−31.7490	+	28.0000i	−1.25109	+	1.10335i
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	21.0000	+	23.8118i	0.822423	+	0.932541i
$653$	10.5830			0.414145			0.207072	−	0.978326i	$-0.433606\pi$
$653$	10.5830			0.414145			0.207072	+	0.978326i	$0.433606\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	44.0000i		−	1.71400i	−0.515319	−	0.856998i	$-0.672327\pi$
$659$		−	44.0000i		−	1.71400i	0.515319	−	0.856998i	$-0.327673\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−2.64575	−	7.00000i	−0.102830	−	0.272063i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	84.6640i		−	3.27820i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	30.0000			1.15642			0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000			1.15642			0.578208	+	0.815890i	$0.303752\pi$
$674$	−39.6863	+	15.0000i	−1.52866	+	0.577778i
$675$		0			0
$676$	19.5000	−	17.1974i	0.750000	−	0.661438i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	52.0000i		−	1.98972i	−0.101237	−	0.994862i	$-0.532280\pi$
$683$		−	52.0000i		−	1.98972i	0.101237	−	0.994862i	$-0.467720\pi$
$684$		0			0
$685$		0			0
$686$	−9.26013	−	24.5000i	−0.353553	−	0.935414i
$687$		0			0
$688$	21.0000	+	2.64575i	0.800617	+	0.100868i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$	−2.00000	−	5.29150i	−0.0759190	−	0.200863i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−17.5000	−	19.8431i	−0.661438	−	0.750000i
$701$	−52.9150			−1.99857			−0.999286	−	0.0377695i	$-0.987975\pi$
$701$	−52.9150			−1.99857			−0.999286	+	0.0377695i	$0.987975\pi$
$702$		0			0
$703$		0			0
$704$	−26.4575	+	18.0000i	−0.997155	+	0.678401i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−6.00000			−0.225335			−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000			−0.225335			−0.112667	+	0.993633i	$0.535939\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	5.29150	+	6.00000i	0.197753	+	0.224231i
$717$		0			0
$718$	−4.00000	−	10.5830i	−0.149279	−	0.394954i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	25.1346	−	9.50000i	0.935414	−	0.353553i
$723$		0			0
$724$		0			0
$725$	52.9150			1.96521
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	44.0000	−	10.5830i	1.62186	−	0.390095i
$737$	63.4980			2.33898
$738$		0			0
$739$			15.8745i			0.583953i	0.956425	+	0.291977i	$0.0943129\pi$
$739$			15.8745i			0.583953i	−0.956425	+	0.291977i	$0.905687\pi$
$740$		0			0
$741$		0			0
$742$	−14.0000	−	37.0405i	−0.513956	−	1.35980i
$743$		−	40.0000i		−	1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$		−	40.0000i		−	1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$		0			0
$745$		0			0
$746$	−29.1033	+	11.0000i	−1.06555	+	0.402739i
$747$		0			0
$748$		0			0
$749$	52.9150			1.93347
$750$		0			0
$751$		−	26.4575i		−	0.965448i	−0.875772	−	0.482724i	$-0.839647\pi$
$751$		−	26.4575i		−	0.965448i	0.875772	−	0.482724i	$-0.160353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−54.0000			−1.96266			−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−54.0000			−1.96266			−0.981332	+	0.192323i	$0.938398\pi$
$758$	18.5203	+	49.0000i	0.672686	+	1.77976i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		−	47.6235i		−	1.72409i
$764$	10.5830	+	12.0000i	0.382880	+	0.434145i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−27.0000	+	23.8118i	−0.971751	+	0.857004i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	14.0000	−	5.29150i	0.501924	−	0.189710i
$779$		0			0
$780$		0			0
$781$	64.0000			2.29010
$782$		0			0
$783$		0			0
$784$	−3.50000	+	27.7804i	−0.125000	+	0.992157i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	15.8745	−	14.0000i	0.565506	−	0.498729i
$789$		0			0
$790$		0			0
$791$		−	56.0000i		−	1.99113i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	6.61438	+	27.5000i	0.233854	+	0.972272i
$801$		0			0
$802$	28.0000	−	10.5830i	0.988714	−	0.373699i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	42.3320			1.48831			0.744157	−	0.668004i	$-0.232851\pi$
$809$	42.3320			1.48831			0.744157	+	0.668004i	$0.232851\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−37.0405	−	42.0000i	−1.29987	−	1.47391i
$813$		0			0
$814$	−12.0000	−	31.7490i	−0.420600	−	1.11280i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−52.9150			−1.84675			−0.923374	−	0.383903i	$-0.874580\pi$
$821$	−52.9150			−1.84675			−0.923374	+	0.383903i	$0.874580\pi$
$822$		0			0
$823$			47.6235i			1.66005i	0.557725	+	0.830026i	$0.311674\pi$
$823$			47.6235i			1.66005i	−0.557725	+	0.830026i	$0.688326\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	44.0000i		−	1.53003i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$		−	44.0000i		−	1.53003i	0.644013	−	0.765015i	$-0.277268\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		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
$841$	83.0000			2.86207
$842$	34.3948	−	13.0000i	1.18532	−	0.448010i
$843$		0			0
$844$	35.0000	+	39.6863i	1.20475	+	1.36606i
$845$		0			0
$846$		0			0
$847$			13.2288i			0.454545i
$848$	−5.29150	+	42.0000i	−0.181711	+	1.44229i
$849$		0			0
$850$		0			0
$851$			48.0000i			1.64542i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−50.0000	−	26.4575i	−1.70896	−	0.904299i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	−16.0000	−	42.3320i	−0.544962	−	1.44183i
$863$		−	8.00000i		−	0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$		−	8.00000i		−	0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−63.4980			−2.15402
$870$		0			0
$871$		0			0
$872$	−23.8118	+	45.0000i	−0.806368	+	1.52389i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−50.0000			−1.68838			−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000			−1.68838			−0.844190	+	0.536044i	$0.819918\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		−	58.2065i		−	1.95881i	−0.201916	−	0.979403i	$-0.564717\pi$
$883$		−	58.2065i		−	1.95881i	0.201916	−	0.979403i	$-0.435283\pi$
$884$		0			0
$885$		0			0
$886$	10.0000	+	26.4575i	0.335957	+	0.888858i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−42.0000			−1.40863
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	17.1974	−	24.5000i	0.574524	−	0.818488i
$897$		0			0
$898$	56.0000	−	21.1660i	1.86874	−	0.706319i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−28.0000	+	52.9150i	−0.931266	+	1.75993i
$905$		0			0
$906$		0			0
$907$			5.29150i			0.175701i	0.996134	+	0.0878507i	$0.0279999\pi$
$907$			5.29150i			0.175701i	−0.996134	+	0.0878507i	$0.972000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			16.0000i			0.530104i	0.964234	+	0.265052i	$0.0853891\pi$
$911$			16.0000i			0.530104i	−0.964234	+	0.265052i	$0.914611\pi$
$912$		0			0
$913$		0			0
$914$	−7.93725	+	3.00000i	−0.262541	+	0.0992312i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			37.0405i			1.22185i	0.791687	+	0.610927i	$0.209203\pi$
$919$			37.0405i			1.22185i	−0.791687	+	0.610927i	$0.790797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−30.0000			−0.986394
$926$	7.93725	+	21.0000i	0.260834	+	0.690103i
$927$		0			0
$928$	14.0000	+	58.2065i	0.459573	+	1.91072i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	31.7490	−	28.0000i	1.03997	−	0.917170i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−55.5608	+	21.0000i	−1.81412	+	0.685674i
$939$		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
$946$	−28.0000	+	10.5830i	−0.910359	+	0.344083i
$947$			20.0000i			0.649913i	0.945729	+	0.324956i	$0.105350\pi$
$947$			20.0000i			0.649913i	−0.945729	+	0.324956i	$0.894650\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21.1660			−0.685634			−0.342817	−	0.939402i	$-0.611381\pi$
$953$	−21.1660			−0.685634			−0.342817	+	0.939402i	$0.611381\pi$
$954$		0			0
$955$		0			0
$956$	−21.1660	−	24.0000i	−0.684558	−	0.776215i
$957$		0			0
$958$		0			0
$959$			56.0000i			1.80833i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			47.6235i			1.53147i	0.643157	+	0.765735i	$0.277624\pi$
$967$			47.6235i			1.53147i	−0.643157	+	0.765735i	$0.722376\pi$
$968$	6.61438	−	12.5000i	0.212594	−	0.401765i
$969$		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
$974$	−18.5203	−	49.0000i	−0.593427	−	1.57006i
$975$		0			0
$976$		0			0
$977$	−42.3320			−1.35432			−0.677161	−	0.735835i	$-0.736790\pi$
$977$	−42.3320			−1.35432			−0.677161	+	0.735835i	$0.736790\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−22.0000	−	58.2065i	−0.702048	−	1.85744i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	42.3320			1.34608
$990$		0			0
$991$			58.2065i			1.84899i	0.381193	+	0.924496i	$0.375513\pi$
$991$			58.2065i			1.84899i	−0.381193	+	0.924496i	$0.624487\pi$
$992$		0			0
$993$		0			0
$994$	−56.0000	+	21.1660i	−1.77621	+	0.671345i
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	−13.2288	−	35.0000i	−0.418749	−	1.10791i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.b.c.55.2	yes	4
3.2	odd	2	inner	252.2.b.c.55.3	yes	4
4.3	odd	2	inner	252.2.b.c.55.1	✓	4
7.6	odd	2	CM	252.2.b.c.55.2	yes	4
8.3	odd	2		4032.2.b.m.3583.3		4
8.5	even	2		4032.2.b.m.3583.2		4
12.11	even	2	inner	252.2.b.c.55.4	yes	4
21.20	even	2	inner	252.2.b.c.55.3	yes	4
24.5	odd	2		4032.2.b.m.3583.1		4
24.11	even	2		4032.2.b.m.3583.4		4
28.27	even	2	inner	252.2.b.c.55.1	✓	4
56.13	odd	2		4032.2.b.m.3583.2		4
56.27	even	2		4032.2.b.m.3583.3		4
84.83	odd	2	inner	252.2.b.c.55.4	yes	4
168.83	odd	2		4032.2.b.m.3583.4		4
168.125	even	2		4032.2.b.m.3583.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.b.c.55.1	✓	4	4.3	odd	2	inner
252.2.b.c.55.1	✓	4	28.27	even	2	inner
252.2.b.c.55.2	yes	4	1.1	even	1	trivial
252.2.b.c.55.2	yes	4	7.6	odd	2	CM
252.2.b.c.55.3	yes	4	3.2	odd	2	inner
252.2.b.c.55.3	yes	4	21.20	even	2	inner
252.2.b.c.55.4	yes	4	12.11	even	2	inner
252.2.b.c.55.4	yes	4	84.83	odd	2	inner
4032.2.b.m.3583.1		4	24.5	odd	2
4032.2.b.m.3583.1		4	168.125	even	2
4032.2.b.m.3583.2		4	8.5	even	2
4032.2.b.m.3583.2		4	56.13	odd	2
4032.2.b.m.3583.3		4	8.3	odd	2
4032.2.b.m.3583.3		4	56.27	even	2
4032.2.b.m.3583.4		4	24.11	even	2
4032.2.b.m.3583.4		4	168.83	odd	2

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	252.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.32288 + 0.500000i) q^{2} +(1.50000 - 1.32288i) q^{4} -2.64575i q^{7} +(-1.32288 + 2.50000i) q^{8} +O(q^{10})\)	\(q+(-1.32288 + 0.500000i) q^{2} +(1.50000 - 1.32288i) q^{4} -2.64575i q^{7} +(-1.32288 + 2.50000i) q^{8} -4.00000i q^{11} +(1.32288 + 3.50000i) q^{14} +(0.500000 - 3.96863i) q^{16} +(2.00000 + 5.29150i) q^{22} -8.00000i q^{23} +5.00000 q^{25} +(-3.50000 - 3.96863i) q^{28} +10.5830 q^{29} +(1.32288 + 5.50000i) q^{32} -6.00000 q^{37} +5.29150i q^{43} +(-5.29150 - 6.00000i) q^{44} +(4.00000 + 10.5830i) q^{46} -7.00000 q^{49} +(-6.61438 + 2.50000i) q^{50} -10.5830 q^{53} +(6.61438 + 3.50000i) q^{56} +(-14.0000 + 5.29150i) q^{58} +(-4.50000 - 6.61438i) q^{64} +15.8745i q^{67} +16.0000i q^{71} +(7.93725 - 3.00000i) q^{74} -10.5830 q^{77} -15.8745i q^{79} +(-2.64575 - 7.00000i) q^{86} +(10.0000 + 5.29150i) q^{88} +(-10.5830 - 12.0000i) q^{92} +(9.26013 - 3.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4}+O(q^{10}) \) 4 * q + 6 * q^4	\( 4 q + 6 q^{4} + 2 q^{16} + 8 q^{22} + 20 q^{25} - 14 q^{28} - 24 q^{37} + 16 q^{46} - 28 q^{49} - 56 q^{58} - 18 q^{64} + 40 q^{88}+O(q^{100}) \) 4 * q + 6 * q^4 + 2 * q^16 + 8 * q^22 + 20 * q^25 - 14 * q^28 - 24 * q^37 + 16 * q^46 - 28 * q^49 - 56 * q^58 - 18 * q^64 + 40 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more