LMFDB - Embedded newform 1764.2.e.f.1079.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(1079,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1764.1079");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.e (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:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1079.2
Root			$-0.581861 - 1.28897i$ of defining polynomial
Character	$\chi$	$=$	1764.1079
Dual form			1764.2.e.f.1079.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.28897 + 0.581861i) q^{2} +(1.32288 - 1.50000i) q^{4} +(-0.832353 + 2.70318i) q^{8} +O(q^{10})$	$q+(-1.28897 + 0.581861i) q^{2} +(1.32288 - 1.50000i) q^{4} +(-0.832353 + 2.70318i) q^{8} -0.913230 q^{11} +(-0.500000 - 3.96863i) q^{16} +(1.17712 - 0.531373i) q^{22} -9.39851 q^{23} +5.00000 q^{25} +8.89753i q^{29} +(2.95367 + 4.82450i) q^{32} -10.5830 q^{37} -12.0000i q^{43} +(-1.20809 + 1.36985i) q^{44} +(12.1144 - 5.46863i) q^{46} +(-6.44484 + 2.90930i) q^{50} -0.412247i q^{53} +(-5.17712 - 11.4686i) q^{58} +(-6.61438 - 4.50000i) q^{64} +15.8745i q^{67} -7.57205 q^{71} +(13.6412 - 6.15784i) q^{74} +15.8745i q^{79} +(6.98233 + 15.4676i) q^{86} +(0.760130 - 2.46863i) q^{88} +(-12.4331 + 14.0978i) q^{92} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 4 q^{16} + 20 q^{22} + 40 q^{25} + 44 q^{46} - 52 q^{58} - 68 q^{88}+O(q^{100}) $ 8 * q - 4 * q^16 + 20 * q^22 + 40 * q^25 + 44 * q^46 - 52 * q^58 - 68 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	−1.28897	+	0.581861i	−0.911438	+	0.411438i
$2$	−1.28897	+	0.581861i	−0.911438	+	0.411438i
$3$		0			0
$3$		0			0
$4$	1.32288	−	1.50000i	0.661438	−	0.750000i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−0.832353	+	2.70318i	−0.294281	+	0.955719i
$9$		0			0
$10$		0			0
$11$	−0.913230			−0.275349			−0.137675	−	0.990478i	$-0.543963\pi$
$11$	−0.913230			−0.275349			−0.137675	+	0.990478i	$0.543963\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		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		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$	1.17712	−	0.531373i	0.250964	−	0.113289i
$23$	−9.39851			−1.95973			−0.979863	−	0.199673i	$-0.936012\pi$
$23$	−9.39851			−1.95973			−0.979863	+	0.199673i	$0.936012\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.89753i			1.65223i	0.563502	+	0.826115i	$0.309454\pi$
$29$			8.89753i			1.65223i	−0.563502	+	0.826115i	$0.690546\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	2.95367	+	4.82450i	0.522141	+	0.852859i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−10.5830			−1.73984			−0.869918	−	0.493197i	$-0.835828\pi$
$37$	−10.5830			−1.73984			−0.869918	+	0.493197i	$0.835828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	12.0000i		−	1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$		−	12.0000i		−	1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	−1.20809	+	1.36985i	−0.182126	+	0.206512i
$45$		0			0
$46$	12.1144	−	5.46863i	1.78617	−	0.806305i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$		0			0
$50$	−6.44484	+	2.90930i	−0.911438	+	0.411438i
$51$		0			0
$52$		0			0
$53$		−	0.412247i		−	0.0566265i	−0.999599	−	0.0283132i	$-0.990986\pi$
$53$		−	0.412247i		−	0.0566265i	0.999599	−	0.0283132i	$-0.00901359\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−5.17712	−	11.4686i	−0.679790	−	1.50590i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	−6.61438	−	4.50000i	−0.826797	−	0.562500i
$65$		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
$71$	−7.57205			−0.898637			−0.449319	−	0.893372i	$-0.648333\pi$
$71$	−7.57205			−0.898637			−0.449319	+	0.893372i	$0.648333\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	13.6412	−	6.15784i	1.58575	−	0.715834i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			15.8745i			1.78602i	0.450035	+	0.893011i	$0.351411\pi$
$79$			15.8745i			1.78602i	−0.450035	+	0.893011i	$0.648589\pi$
$80$		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
$86$	6.98233	+	15.4676i	0.752924	+	1.66792i
$87$		0			0
$88$	0.760130	−	2.46863i	0.0810301	−	0.263157i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−12.4331	+	14.0978i	−1.29624	+	1.46979i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$	6.61438	−	7.50000i	0.661438	−	0.750000i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	0.239870	+	0.531373i	0.0232983	+	0.0516115i
$107$	−17.8838			−1.72889			−0.864446	−	0.502726i	$-0.832330\pi$
$107$	−17.8838			−1.72889			−0.864446	+	0.502726i	$0.832330\pi$
$108$		0			0
$109$	−10.5830			−1.01367			−0.506834	−	0.862044i	$-0.669184\pi$
$109$	−10.5830			−1.01367			−0.506834	+	0.862044i	$0.669184\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			13.5524i			1.27490i	0.770490	+	0.637452i	$0.220012\pi$
$113$			13.5524i			1.27490i	−0.770490	+	0.637452i	$0.779988\pi$
$114$		0			0
$115$		0			0
$116$	13.3463	+	11.7703i	1.23917	+	1.09285i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.1660			−0.924183
$122$		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	$-0.248747\pi$
$128$	11.1441	+	1.95171i	0.985008	+	0.172508i
$129$		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
$134$	−9.23676	−	20.4617i	−0.797934	−	1.76762i
$135$		0			0
$136$		0			0
$137$		−	22.0377i		−	1.88281i	−0.337282	−	0.941404i	$-0.609507\pi$
$137$		−	22.0377i		−	1.88281i	0.337282	−	0.941404i	$-0.390493\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	9.76013	−	4.40588i	0.819052	−	0.369733i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−14.0000	+	15.8745i	−1.15079	+	1.30488i
$149$			8.07303i			0.661369i	0.943741	+	0.330684i	$0.107280\pi$
$149$			8.07303i			0.661369i	−0.943741	+	0.330684i	$0.892720\pi$
$150$		0			0
$151$			24.0000i			1.95309i	0.215308	+	0.976546i	$0.430924\pi$
$151$			24.0000i			1.95309i	−0.215308	+	0.976546i	$0.569076\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−9.23676	−	20.4617i	−0.734837	−	1.62785i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		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
$172$	−18.0000	−	15.8745i	−1.37249	−	1.21042i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$	0.456615	+	3.62427i	0.0344187	+	0.273190i
$177$		0			0
$178$		0			0
$179$	−21.5367			−1.60973			−0.804865	−	0.593458i	$-0.797762\pi$
$179$	−21.5367			−1.60973			−0.804865	+	0.593458i	$0.797762\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	7.82288	−	25.4059i	0.576710	−	1.87295i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.0514			−0.944369			−0.472184	−	0.881500i	$-0.656534\pi$
$191$	−13.0514			−0.944369			−0.472184	+	0.881500i	$0.656534\pi$
$192$		0			0
$193$	−21.1660			−1.52356			−0.761781	−	0.647834i	$-0.775675\pi$
$193$	−21.1660			−1.52356			−0.761781	+	0.647834i	$0.775675\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	25.8681i		−	1.84303i	−0.388348	−	0.921513i	$-0.626954\pi$
$197$		−	25.8681i		−	1.84303i	0.388348	−	0.921513i	$-0.373046\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−4.16176	+	13.5159i	−0.294281	+	0.955719i
$201$		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
$211$			12.0000i			0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$			12.0000i			0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$	−0.618370	−	0.545351i	−0.0424699	−	0.0374549i
$213$		0			0
$214$	23.0516	−	10.4059i	1.57578	−	0.711331i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	13.6412	−	6.15784i	0.923895	−	0.417061i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	−7.88562	−	17.4686i	−0.524544	−	1.16200i
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	−24.0516	−	7.40588i	−1.57907	−	0.486220i
$233$		−	30.5230i		−	1.99963i	−0.0193169	−	0.999813i	$-0.506149\pi$
$233$		−	30.5230i		−	1.99963i	0.0193169	−	0.999813i	$-0.493851\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	30.0220			1.94196			0.970981	−	0.239158i	$-0.0768713\pi$
$239$	30.0220			1.94196			0.970981	+	0.239158i	$0.0768713\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	13.1037	−	5.91520i	0.842335	−	0.380244i
$243$		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
$253$	8.58301			0.539609
$254$	9.23676	+	20.4617i	0.579566	+	1.28388i
$255$		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
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−26.3691			−1.62599			−0.812993	−	0.582273i	$-0.802164\pi$
$263$	−26.3691			−1.62599			−0.812993	+	0.582273i	$0.802164\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	23.8118	+	21.0000i	1.45453	+	1.28278i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$	12.8229	+	28.4059i	0.774658	+	1.71606i
$275$	−4.56615			−0.275349
$276$		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
$281$			3.41815i			0.203910i	0.994789	+	0.101955i	$0.0325097\pi$
$281$			3.41815i			0.203910i	−0.994789	+	0.101955i	$0.967490\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	−10.0169	+	11.3581i	−0.594393	+	0.673978i
$285$		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
$296$	8.80879	−	28.6078i	0.512001	−	1.66279i
$297$		0			0
$298$	−4.69738	−	10.4059i	−0.272112	−	0.602796i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−13.9647	−	30.9352i	−0.803576	−	1.78012i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		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			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	23.8118	+	21.0000i	1.33952	+	1.18134i
$317$			16.5583i			0.930008i	0.885309	+	0.465004i	$0.153947\pi$
$317$			16.5583i			0.930008i	−0.885309	+	0.465004i	$0.846053\pi$
$318$		0			0
$319$		−	8.12549i		−	0.454940i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−9.23676	−	20.4617i	−0.511577	−	1.13327i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			36.0000i			1.97874i	0.145424	+	0.989369i	$0.453545\pi$
$331$			36.0000i			1.97874i	−0.145424	+	0.989369i	$0.546455\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	21.1660			1.15299			0.576493	−	0.817102i	$-0.304421\pi$
$337$	21.1660			1.15299			0.576493	+	0.817102i	$0.304421\pi$
$338$	16.7566	−	7.56419i	0.911438	−	0.411438i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	32.4382	+	9.98823i	1.74895	+	0.538529i
$345$		0			0
$346$		0			0
$347$	23.3632			1.25420			0.627100	−	0.778938i	$-0.284242\pi$
$347$	23.3632			1.25420			0.627100	+	0.778938i	$0.284242\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−2.69738	−	4.40588i	−0.143771	−	0.234834i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	27.7601	−	12.5314i	1.46717	−	0.662304i
$359$	31.8485			1.68090			0.840449	−	0.541891i	$-0.182292\pi$
$359$	31.8485			1.68090			0.840449	+	0.541891i	$0.182292\pi$
$360$		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		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	4.69926	+	37.2992i	0.244966	+	1.94435i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			12.0000i			0.616399i	0.951322	+	0.308199i	$0.0997264\pi$
$379$			12.0000i			0.616399i	−0.951322	+	0.308199i	$0.900274\pi$
$380$		0			0
$381$		0			0
$382$	16.8229	−	7.59412i	0.860733	−	0.388549i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	27.2823	−	12.3157i	1.38863	−	0.626851i
$387$		0			0
$388$		0			0
$389$			34.3534i			1.74179i	0.491473	+	0.870893i	$0.336458\pi$
$389$			34.3534i			1.74179i	−0.491473	+	0.870893i	$0.663542\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	15.0516	+	33.3431i	0.758290	+	1.67980i
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	−2.50000	−	19.8431i	−0.125000	−	0.992157i
$401$		−	39.0083i		−	1.94798i	−0.226592	−	0.973990i	$-0.572758\pi$
$401$		−	39.0083i		−	1.94798i	0.226592	−	0.973990i	$-0.427242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.66472			0.479062
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		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	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$	−6.98233	−	15.4676i	−0.339895	−	0.752952i
$423$		0			0
$424$	1.11438	+	0.343135i	0.0541190	+	0.0166641i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−23.6580	+	26.8257i	−1.14355	+	1.29667i
$429$		0			0
$430$		0			0
$431$	3.91913			0.188778			0.0943889	−	0.995535i	$-0.469910\pi$
$431$	3.91913			0.188778			0.0943889	+	0.995535i	$0.469910\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−14.0000	+	15.8745i	−0.670478	+	0.760251i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	40.3337			1.91631			0.958157	−	0.286244i	$-0.0924067\pi$
$443$	40.3337			1.91631			0.958157	+	0.286244i	$0.0924067\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	31.3475i		−	1.47938i	−0.672948	−	0.739689i	$-0.734972\pi$
$449$		−	31.3475i		−	1.47938i	0.672948	−	0.739689i	$-0.265028\pi$
$450$		0			0
$451$		0			0
$452$	20.3286	+	17.9282i	0.956178	+	0.843270i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	42.3320			1.98021			0.990104	−	0.140334i	$-0.0448177\pi$
$457$	42.3320			1.98021			0.990104	+	0.140334i	$0.0448177\pi$
$458$		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	$0.120257\pi$
$463$			15.8745i			0.737751i	−0.929479	+	0.368875i	$0.879743\pi$
$464$	35.3110	−	4.44876i	1.63927	−	0.206529i
$465$		0			0
$466$	17.7601	+	39.3431i	0.822722	+	1.82254i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			10.9588i			0.503884i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−38.6974	+	17.4686i	−1.76998	+	0.798996i
$479$		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
$484$	−13.4484	+	15.2490i	−0.611289	+	0.693137i
$485$		0			0
$486$		0			0
$487$		−	24.0000i		−	1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$		−	24.0000i		−	1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	34.8544			1.57296			0.786478	−	0.617619i	$-0.211903\pi$
$491$	34.8544			1.57296			0.786478	+	0.617619i	$0.211903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	36.0000i		−	1.61158i	−0.592200	−	0.805791i	$-0.701741\pi$
$499$		−	36.0000i		−	1.61158i	0.592200	−	0.805791i	$-0.298259\pi$
$500$		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
$506$	−11.0632	+	4.99412i	−0.491820	+	0.222016i
$507$		0			0
$508$	−23.8118	−	21.0000i	−1.05648	−	0.931724i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	17.6698	−	14.1343i	0.780903	−	0.624653i
$513$		0			0
$514$		0			0
$515$		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		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	33.9889	−	15.3431i	1.48199	−	0.668992i
$527$		0			0
$528$		0			0
$529$	65.3320			2.84052
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−42.9117	−	13.2132i	−1.85350	−	0.570723i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		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	$-0.889785\pi$
$547$		−	15.8745i		−	0.678745i	0.940652	−	0.339372i	$-0.110215\pi$
$548$	−33.0565	−	29.1531i	−1.41211	−	1.24536i
$549$		0			0
$550$	5.88562	−	2.65687i	0.250964	−	0.113289i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	12.8897	−	5.81861i	0.547630	−	0.247209i
$555$		0			0
$556$		0			0
$557$			25.0436i			1.06113i	0.847644	+	0.530566i	$0.178020\pi$
$557$			25.0436i			1.06113i	−0.847644	+	0.530566i	$0.821980\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.98889	−	4.40588i	−0.0838961	−	0.185851i
$563$		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
$568$	6.30262	−	20.4686i	0.264452	−	0.858845i
$569$			14.3769i			0.602711i	0.953512	+	0.301356i	$0.0974392\pi$
$569$			14.3769i			0.602711i	−0.953512	+	0.301356i	$0.902561\pi$
$570$		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
$575$	−46.9926			−1.95973
$576$		0			0
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−21.9125	+	9.89164i	−0.911438	+	0.411438i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			0.376476i			0.0155921i
$584$		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
$592$	5.29150	+	42.0000i	0.217479	+	1.72619i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	12.1096	+	10.6796i	0.496027	+	0.437454i
$597$		0			0
$598$		0			0
$599$	48.8190			1.99469			0.997346	−	0.0728143i	$-0.0231980\pi$
$599$	48.8190			1.99469			0.997346	+	0.0728143i	$0.0231980\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$	36.0000	+	31.7490i	1.46482	+	1.29185i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−38.0000			−1.53481			−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000			−1.53481			−0.767403	+	0.641165i	$0.778451\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			48.3180i			1.94521i	0.232462	+	0.972605i	$0.425322\pi$
$617$			48.3180i			1.94521i	−0.232462	+	0.972605i	$0.574678\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		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	$0.603171\pi$
$632$	−42.9117	−	13.2132i	−1.70693	−	0.525592i
$633$		0			0
$634$	−9.63464	−	21.3431i	−0.382640	−	0.847644i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	4.72791	+	10.4735i	0.187180	+	0.414650i
$639$		0			0
$640$		0			0
$641$		−	47.4935i		−	1.87588i	−0.346795	−	0.937941i	$-0.612730\pi$
$641$		−	47.4935i		−	1.87588i	0.346795	−	0.937941i	$-0.387270\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		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
$652$	23.8118	+	21.0000i	0.932541	+	0.822423i
$653$			42.8387i			1.67641i	0.545358	+	0.838203i	$0.316394\pi$
$653$			42.8387i			1.67641i	−0.545358	+	0.838203i	$0.683606\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.4044			−0.483207			−0.241604	−	0.970375i	$-0.577673\pi$
$659$	−12.4044			−0.483207			−0.241604	+	0.970375i	$0.577673\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−20.9470	−	46.4028i	−0.814128	−	1.80350i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	83.6235i		−	3.23792i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−42.3320			−1.63178			−0.815890	−	0.578208i	$-0.803752\pi$
$673$	−42.3320			−1.63178			−0.815890	+	0.578208i	$0.803752\pi$
$674$	−27.2823	+	12.3157i	−1.05088	+	0.474382i
$675$		0			0
$676$	−17.1974	+	19.5000i	−0.661438	+	0.750000i
$677$		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
$683$	−33.0279			−1.26378			−0.631889	−	0.775059i	$-0.717720\pi$
$683$	−33.0279			−1.26378			−0.631889	+	0.775059i	$0.717720\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−47.6235	+	6.00000i	−1.81563	+	0.228748i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	−30.1144	+	13.5941i	−1.14313	+	0.516026i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	36.0024i		−	1.35979i	−0.733309	−	0.679895i	$-0.762025\pi$
$701$		−	36.0024i		−	1.35979i	0.733309	−	0.679895i	$-0.237975\pi$
$702$		0			0
$703$		0			0
$704$	6.04045	+	4.10954i	0.227658	+	0.154884i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	52.9150			1.98727			0.993633	−	0.112667i	$-0.0359394\pi$
$709$	52.9150			1.98727			0.993633	+	0.112667i	$0.0359394\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−28.4904	+	32.3051i	−1.06474	+	1.20730i
$717$		0			0
$718$	−41.0516	+	18.5314i	−1.53203	+	0.691585i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−24.4904	+	11.0554i	−0.911438	+	0.411438i
$723$		0			0
$724$		0			0
$725$			44.4876i			1.65223i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	−27.7601	−	45.3431i	−1.02325	−	1.67137i
$737$		−	14.4971i		−	0.534007i
$738$		0			0
$739$		−	15.8745i		−	0.583953i	−0.956425	−	0.291977i	$-0.905687\pi$
$739$		−	15.8745i		−	0.583953i	0.956425	−	0.291977i	$-0.0943129\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.09267			−0.0767726			−0.0383863	−	0.999263i	$-0.512222\pi$
$743$	−2.09267			−0.0767726			−0.0383863	+	0.999263i	$0.512222\pi$
$744$		0			0
$745$		0			0
$746$	28.3573	−	12.8009i	1.03823	−	0.468676i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	48.0000i		−	1.75154i	−0.482724	−	0.875772i	$-0.660353\pi$
$751$		−	48.0000i		−	1.75154i	0.482724	−	0.875772i	$-0.339647\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−10.5830			−0.384646			−0.192323	−	0.981332i	$-0.561602\pi$
$757$	−10.5830			−0.384646			−0.192323	+	0.981332i	$0.561602\pi$
$758$	−6.98233	−	15.4676i	−0.253610	−	0.561809i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$	−17.2654	+	19.5771i	−0.624641	+	0.708276i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	−28.0000	+	31.7490i	−1.00774	+	1.14267i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−19.9889	−	44.2804i	−0.716636	−	1.58753i
$779$		0			0
$780$		0			0
$781$	6.91503			0.247439
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−38.8021	−	34.2203i	−1.38227	−	1.21905i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		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
$800$	14.7684	+	24.1225i	0.522141	+	0.852859i
$801$		0			0
$802$	22.6974	+	50.2804i	0.801472	+	1.77546i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	56.8033i		−	1.99710i	−0.0538482	−	0.998549i	$-0.517149\pi$
$809$		−	56.8033i		−	1.99710i	0.0538482	−	0.998549i	$-0.482851\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	−12.4575	+	5.62352i	−0.436636	+	0.197104i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			52.9729i			1.84877i	0.381464	+	0.924384i	$0.375420\pi$
$821$			52.9729i			1.84877i	−0.381464	+	0.924384i	$0.624580\pi$
$822$		0			0
$823$		−	47.6235i		−	1.66005i	−0.557725	−	0.830026i	$-0.688326\pi$
$823$		−	47.6235i		−	1.66005i	0.557725	−	0.830026i	$-0.311674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−57.3043			−1.99267			−0.996333	−	0.0855616i	$-0.972732\pi$
$827$	−57.3043			−1.99267			−0.996333	+	0.0855616i	$0.972732\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		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
$841$	−50.1660			−1.72986
$842$	−33.5132	+	15.1284i	−1.15494	+	0.521359i
$843$		0			0
$844$	18.0000	+	15.8745i	0.619586	+	0.546423i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−1.63605	+	0.206123i	−0.0561823	+	0.00707831i
$849$		0			0
$850$		0			0
$851$	99.4645			3.40960
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	14.8856	−	48.3431i	0.508780	−	1.65233i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	−5.05163	+	2.28039i	−0.172059	+	0.0776703i
$863$	35.5014			1.20848			0.604240	−	0.796802i	$-0.293477\pi$
$863$	35.5014			1.20848			0.604240	+	0.796802i	$0.293477\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	14.4971i		−	0.491780i
$870$		0			0
$871$		0			0
$872$	8.80879	−	28.6078i	0.298303	−	0.968782i
$873$		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
$883$		−	12.0000i		−	0.403832i	−0.979403	−	0.201916i	$-0.935283\pi$
$883$		−	12.0000i		−	0.403832i	0.979403	−	0.201916i	$-0.0647168\pi$
$884$		0			0
$885$		0			0
$886$	−51.9889	+	23.4686i	−1.74660	+	0.788444i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	18.2399	+	40.4059i	0.608672	+	1.34836i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−36.6346	−	11.2804i	−1.21845	−	0.375180i
$905$		0			0
$906$		0			0
$907$			60.0000i			1.99227i	0.0878507	+	0.996134i	$0.472000\pi$
$907$			60.0000i			1.99227i	−0.0878507	+	0.996134i	$0.528000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−52.4719			−1.73847			−0.869236	−	0.494397i	$-0.835389\pi$
$911$	−52.4719			−1.73847			−0.869236	+	0.494397i	$0.835389\pi$
$912$		0			0
$913$		0			0
$914$	−54.5646	+	24.6314i	−1.80484	+	0.814733i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	48.0000i		−	1.58337i	−0.610927	−	0.791687i	$-0.709203\pi$
$919$		−	48.0000i		−	1.58337i	0.610927	−	0.791687i	$-0.290797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−52.9150			−1.73984
$926$	−9.23676	−	20.4617i	−0.303539	−	0.672414i
$927$		0			0
$928$	−42.9261	+	26.2804i	−1.40912	+	0.862696i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−45.7845	−	40.3781i	−1.49972	−	1.32263i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		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
$946$	−6.37648	−	14.1255i	−0.207317	−	0.459259i
$947$	27.0161			0.877905			0.438953	−	0.898510i	$-0.355350\pi$
$947$	27.0161			0.877905			0.438953	+	0.898510i	$0.355350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			55.9788i			1.81333i	0.421849	+	0.906666i	$0.361381\pi$
$953$			55.9788i			1.81333i	−0.421849	+	0.906666i	$0.638619\pi$
$954$		0			0
$955$		0			0
$956$	39.7154	−	45.0330i	1.28449	−	1.45647i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		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	$-0.722376\pi$
$967$		−	47.6235i		−	1.53147i	0.643157	−	0.765735i	$-0.277624\pi$
$968$	8.46171	−	27.4806i	0.271970	−	0.883259i
$969$		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
$974$	13.9647	+	30.9352i	0.447457	+	0.991229i
$975$		0			0
$976$		0			0
$977$		−	2.59365i		−	0.0829783i	−0.999139	−	0.0414892i	$-0.986790\pi$
$977$		−	2.59365i		−	0.0829783i	0.999139	−	0.0414892i	$-0.0132102\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−44.9261	+	20.2804i	−1.43365	+	0.647173i
$983$		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
$989$			112.782i			3.58626i
$990$		0			0
$991$			24.0000i			0.762385i	0.924496	+	0.381193i	$0.124487\pi$
$991$			24.0000i			0.762385i	−0.924496	+	0.381193i	$0.875513\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	20.9470	+	46.4028i	0.663066	+	1.46886i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.e.f.1079.2	yes	8
3.2	odd	2	inner	1764.2.e.f.1079.7	yes	8
4.3	odd	2	inner	1764.2.e.f.1079.8	yes	8
7.6	odd	2	CM	1764.2.e.f.1079.2	yes	8
12.11	even	2	inner	1764.2.e.f.1079.1	✓	8
21.20	even	2	inner	1764.2.e.f.1079.7	yes	8
28.27	even	2	inner	1764.2.e.f.1079.8	yes	8
84.83	odd	2	inner	1764.2.e.f.1079.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.e.f.1079.1	✓	8	12.11	even	2	inner
1764.2.e.f.1079.1	✓	8	84.83	odd	2	inner
1764.2.e.f.1079.2	yes	8	1.1	even	1	trivial
1764.2.e.f.1079.2	yes	8	7.6	odd	2	CM
1764.2.e.f.1079.7	yes	8	3.2	odd	2	inner
1764.2.e.f.1079.7	yes	8	21.20	even	2	inner
1764.2.e.f.1079.8	yes	8	4.3	odd	2	inner
1764.2.e.f.1079.8	yes	8	28.27	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.28897 + 0.581861i) q^{2} +(1.32288 - 1.50000i) q^{4} +(-0.832353 + 2.70318i) q^{8} +O(q^{10})\)	\(q+(-1.28897 + 0.581861i) q^{2} +(1.32288 - 1.50000i) q^{4} +(-0.832353 + 2.70318i) q^{8} -0.913230 q^{11} +(-0.500000 - 3.96863i) q^{16} +(1.17712 - 0.531373i) q^{22} -9.39851 q^{23} +5.00000 q^{25} +8.89753i q^{29} +(2.95367 + 4.82450i) q^{32} -10.5830 q^{37} -12.0000i q^{43} +(-1.20809 + 1.36985i) q^{44} +(12.1144 - 5.46863i) q^{46} +(-6.44484 + 2.90930i) q^{50} -0.412247i q^{53} +(-5.17712 - 11.4686i) q^{58} +(-6.61438 - 4.50000i) q^{64} +15.8745i q^{67} -7.57205 q^{71} +(13.6412 - 6.15784i) q^{74} +15.8745i q^{79} +(6.98233 + 15.4676i) q^{86} +(0.760130 - 2.46863i) q^{88} +(-12.4331 + 14.0978i) q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 4 q^{16} + 20 q^{22} + 40 q^{25} + 44 q^{46} - 52 q^{58} - 68 q^{88}+O(q^{100}) \) 8 * q - 4 * q^16 + 20 * q^22 + 40 * q^25 + 44 * q^46 - 52 * q^58 - 68 * q^88

Introduction

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Modular forms

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Database

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