LMFDB - Embedded newform 4140.2.i.b.1241.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1241,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4140.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.i (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 $ x^16 + 62x^14 + 1303x^12 + 12842x^10 + 65359x^8 + 170834x^6 + 207293x^4 + 91366*x^2 + 9604
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.7
Root			$-0.387906i$ of defining polynomial
Character	$\chi$	$=$	4140.1241
Dual form			4140.2.i.b.1241.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -0.807745i q^{7} +O(q^{10})$	$q+1.00000 q^{5} -0.807745i q^{7} -2.90866 q^{11} +2.37686 q^{13} -6.47810 q^{17} +3.20697i q^{19} +(2.82827 + 3.87309i) q^{23} +1.00000 q^{25} -7.84095i q^{29} -3.15776 q^{31} -0.807745i q^{35} -7.68653i q^{37} -0.186349i q^{41} -2.61432i q^{43} -10.5731i q^{47} +6.34755 q^{49} +5.14391 q^{53} -2.90866 q^{55} -0.0710210i q^{59} -5.76880i q^{61} +2.37686 q^{65} -2.04841i q^{67} -4.39463i q^{71} -4.19913 q^{73} +2.34945i q^{77} +0.308833i q^{79} -0.614191 q^{83} -6.47810 q^{85} -13.1731 q^{89} -1.91990i q^{91} +3.20697i q^{95} -2.32718i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5}+O(q^{10}) $ 16 * q + 16 * q^5	$ 16 q + 16 q^{5} + 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} - 4 q^{53} + 8 q^{73} - 20 q^{83} - 32 q^{89}+O(q^{100}) $ 16 * q + 16 * q^5 + 8 * q^23 + 16 * q^25 - 8 * q^31 - 40 * q^49 - 4 * q^53 + 8 * q^73 - 20 * q^83 - 32 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	0.807745i		−	0.305299i	−0.988280	−	0.152650i	$-0.951219\pi$
$7$		−	0.807745i		−	0.305299i	0.988280	−	0.152650i	$-0.0487806\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.90866			−0.876993			−0.438496	−	0.898733i	$-0.644489\pi$
$11$	−2.90866			−0.876993			−0.438496	+	0.898733i	$0.644489\pi$
$12$		0			0
$13$	2.37686			0.659221			0.329611	−	0.944117i	$-0.393082\pi$
$13$	2.37686			0.659221			0.329611	+	0.944117i	$0.393082\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.47810			−1.57117			−0.785585	−	0.618753i	$-0.787638\pi$
$17$	−6.47810			−1.57117			−0.785585	+	0.618753i	$0.787638\pi$
$18$		0			0
$19$			3.20697i			0.735729i	0.929879	+	0.367864i	$0.119911\pi$
$19$			3.20697i			0.735729i	−0.929879	+	0.367864i	$0.880089\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.82827	+	3.87309i	0.589736	+	0.807596i
$23$	2.82827	+	3.87309i	0.589736	+	0.807596i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	7.84095i		−	1.45603i	−0.685563	−	0.728014i	$-0.740444\pi$
$29$		−	7.84095i		−	1.45603i	0.685563	−	0.728014i	$-0.259556\pi$
$30$		0			0
$31$	−3.15776			−0.567150			−0.283575	−	0.958950i	$-0.591521\pi$
$31$	−3.15776			−0.567150			−0.283575	+	0.958950i	$0.591521\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	0.807745i		−	0.136534i
$36$		0			0
$37$		−	7.68653i		−	1.26366i	−0.775108	−	0.631829i	$-0.782305\pi$
$37$		−	7.68653i		−	1.26366i	0.775108	−	0.631829i	$-0.217695\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.186349i		−	0.0291029i	−0.999894	−	0.0145514i	$-0.995368\pi$
$41$		−	0.186349i		−	0.0291029i	0.999894	−	0.0145514i	$-0.00463203\pi$
$42$		0			0
$43$		−	2.61432i		−	0.398680i	−0.979930	−	0.199340i	$-0.936120\pi$
$43$		−	2.61432i		−	0.398680i	0.979930	−	0.199340i	$-0.0638797\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10.5731i		−	1.54225i	−0.636683	−	0.771126i	$-0.719694\pi$
$47$		−	10.5731i		−	1.54225i	0.636683	−	0.771126i	$-0.280306\pi$
$48$		0			0
$49$	6.34755			0.906792
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.14391			0.706570			0.353285	−	0.935516i	$-0.385065\pi$
$53$	5.14391			0.706570			0.353285	+	0.935516i	$0.385065\pi$
$54$		0			0
$55$	−2.90866			−0.392203
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	0.0710210i		−	0.00924615i	−0.999989	−	0.00462307i	$-0.998528\pi$
$59$		−	0.0710210i		−	0.00924615i	0.999989	−	0.00462307i	$-0.00147158\pi$
$60$		0			0
$61$		−	5.76880i		−	0.738619i	−0.929306	−	0.369310i	$-0.879594\pi$
$61$		−	5.76880i		−	0.738619i	0.929306	−	0.369310i	$-0.120406\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.37686			0.294813
$66$		0			0
$67$		−	2.04841i		−	0.250253i	−0.992141	−	0.125126i	$-0.960066\pi$
$67$		−	2.04841i		−	0.250253i	0.992141	−	0.125126i	$-0.0399337\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.39463i		−	0.521547i	−0.965400	−	0.260773i	$-0.916022\pi$
$71$		−	4.39463i		−	0.521547i	0.965400	−	0.260773i	$-0.0839776\pi$
$72$		0			0
$73$	−4.19913			−0.491471			−0.245736	−	0.969337i	$-0.579030\pi$
$73$	−4.19913			−0.491471			−0.245736	+	0.969337i	$0.579030\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.34945i			0.267745i
$78$		0			0
$79$			0.308833i			0.0347464i	0.999849	+	0.0173732i	$0.00553035\pi$
$79$			0.308833i			0.0347464i	−0.999849	+	0.0173732i	$0.994470\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.614191			−0.0674163			−0.0337081	−	0.999432i	$-0.510732\pi$
$83$	−0.614191			−0.0674163			−0.0337081	+	0.999432i	$0.510732\pi$
$84$		0			0
$85$	−6.47810			−0.702649
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−13.1731			−1.39635			−0.698175	−	0.715927i	$-0.746004\pi$
$89$	−13.1731			−1.39635			−0.698175	+	0.715927i	$0.746004\pi$
$90$		0			0
$91$		−	1.91990i		−	0.201260i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			3.20697i			0.329028i
$96$		0			0
$97$		−	2.32718i		−	0.236289i	−0.992996	−	0.118145i	$-0.962305\pi$
$97$		−	2.32718i		−	0.236289i	0.992996	−	0.118145i	$-0.0376947\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	12.5285i		−	1.24663i	−0.781971	−	0.623315i	$-0.785785\pi$
$101$		−	12.5285i		−	1.24663i	0.781971	−	0.623315i	$-0.214215\pi$
$102$		0			0
$103$		−	3.36639i		−	0.331701i	−0.986151	−	0.165850i	$-0.946963\pi$
$103$		−	3.36639i		−	0.331701i	0.986151	−	0.165850i	$-0.0530369\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.8848			−1.05228			−0.526138	−	0.850399i	$-0.676361\pi$
$107$	−10.8848			−1.05228			−0.526138	+	0.850399i	$0.676361\pi$
$108$		0			0
$109$		−	8.43356i		−	0.807789i	−0.914806	−	0.403894i	$-0.867656\pi$
$109$		−	8.43356i		−	0.807789i	0.914806	−	0.403894i	$-0.132344\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.0580			−1.32247			−0.661234	−	0.750180i	$-0.729967\pi$
$113$	−14.0580			−1.32247			−0.661234	+	0.750180i	$0.729967\pi$
$114$		0			0
$115$	2.82827	+	3.87309i	0.263738	+	0.361168i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			5.23266i			0.479677i
$120$		0			0
$121$	−2.53972			−0.230884
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	14.2897			1.26801			0.634005	−	0.773329i	$-0.281410\pi$
$127$	14.2897			1.26801			0.634005	+	0.773329i	$0.281410\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			2.46341i			0.215229i	0.994193	+	0.107615i	$0.0343212\pi$
$131$			2.46341i			0.215229i	−0.994193	+	0.107615i	$0.965679\pi$
$132$		0			0
$133$	2.59041			0.224617
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.64135			−0.738280			−0.369140	−	0.929374i	$-0.620348\pi$
$137$	−8.64135			−0.738280			−0.369140	+	0.929374i	$0.620348\pi$
$138$		0			0
$139$	15.0825			1.27928			0.639641	−	0.768673i	$-0.279083\pi$
$139$	15.0825			1.27928			0.639641	+	0.768673i	$0.279083\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.91346			−0.578132
$144$		0			0
$145$		−	7.84095i		−	0.651155i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−19.1168			−1.56611			−0.783054	−	0.621954i	$-0.786339\pi$
$149$	−19.1168			−1.56611			−0.783054	+	0.621954i	$0.786339\pi$
$150$		0			0
$151$	3.87440			0.315294			0.157647	−	0.987496i	$-0.449609\pi$
$151$	3.87440			0.315294			0.157647	+	0.987496i	$0.449609\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.15776			−0.253637
$156$		0			0
$157$			12.6141i			1.00671i	0.864079	+	0.503355i	$0.167901\pi$
$157$			12.6141i			1.00671i	−0.864079	+	0.503355i	$0.832099\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.12847	−	2.28453i	0.246558	−	0.180046i
$162$		0			0
$163$	11.9182			0.933508			0.466754	−	0.884387i	$-0.345423\pi$
$163$	11.9182			0.933508			0.466754	+	0.884387i	$0.345423\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	7.09244i		−	0.548830i	−0.961611	−	0.274415i	$-0.911516\pi$
$167$		−	7.09244i		−	0.548830i	0.961611	−	0.274415i	$-0.0884841\pi$
$168$		0			0
$169$	−7.35055			−0.565427
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	17.3090i		−	1.31598i	−0.753026	−	0.657990i	$-0.771407\pi$
$173$		−	17.3090i		−	1.31598i	0.753026	−	0.657990i	$-0.228593\pi$
$174$		0			0
$175$		−	0.807745i		−	0.0610598i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			19.8004i			1.47995i	0.672635	+	0.739974i	$0.265162\pi$
$179$			19.8004i			1.47995i	−0.672635	+	0.739974i	$0.734838\pi$
$180$		0			0
$181$		−	12.9173i		−	0.960138i	−0.877231	−	0.480069i	$-0.840612\pi$
$181$		−	12.9173i		−	0.960138i	0.877231	−	0.480069i	$-0.159388\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	7.68653i		−	0.565125i
$186$		0			0
$187$	18.8426			1.37791
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.0943			0.730398			0.365199	−	0.930929i	$-0.381001\pi$
$191$	10.0943			0.730398			0.365199	+	0.930929i	$0.381001\pi$
$192$		0			0
$193$	17.4900			1.25896			0.629478	−	0.777018i	$-0.283269\pi$
$193$	17.4900			1.25896			0.629478	+	0.777018i	$0.283269\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.22646i			0.443617i	0.975090	+	0.221808i	$0.0711959\pi$
$197$			6.22646i			0.443617i	−0.975090	+	0.221808i	$0.928804\pi$
$198$		0			0
$199$			4.36684i			0.309557i	0.987949	+	0.154779i	$0.0494664\pi$
$199$			4.36684i			0.309557i	−0.987949	+	0.154779i	$0.950534\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.33349			−0.444524
$204$		0			0
$205$		−	0.186349i		−	0.0130152i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	9.32796i		−	0.645229i
$210$		0			0
$211$	−15.3265			−1.05512			−0.527560	−	0.849517i	$-0.676893\pi$
$211$	−15.3265			−1.05512			−0.527560	+	0.849517i	$0.676893\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	2.61432i		−	0.178295i
$216$		0			0
$217$			2.55067i			0.173150i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−15.3975			−1.03575
$222$		0			0
$223$	3.81379			0.255390			0.127695	−	0.991813i	$-0.459242\pi$
$223$	3.81379			0.255390			0.127695	+	0.991813i	$0.459242\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.52259			−0.101058			−0.0505289	−	0.998723i	$-0.516091\pi$
$227$	−1.52259			−0.101058			−0.0505289	+	0.998723i	$0.516091\pi$
$228$		0			0
$229$		−	28.0932i		−	1.85645i	−0.372022	−	0.928224i	$-0.621335\pi$
$229$		−	28.0932i		−	1.85645i	0.372022	−	0.928224i	$-0.378665\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	9.86682i		−	0.646397i	−0.946331	−	0.323198i	$-0.895242\pi$
$233$		−	9.86682i		−	0.646397i	0.946331	−	0.323198i	$-0.104758\pi$
$234$		0			0
$235$		−	10.5731i		−	0.689716i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	13.1543i		−	0.850881i	−0.904986	−	0.425441i	$-0.860119\pi$
$239$		−	13.1543i		−	0.850881i	0.904986	−	0.425441i	$-0.139881\pi$
$240$		0			0
$241$			3.68770i			0.237545i	0.992921	+	0.118773i	$0.0378960\pi$
$241$			3.68770i			0.237545i	−0.992921	+	0.118773i	$0.962104\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.34755			0.405530
$246$		0			0
$247$			7.62250i			0.485008i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.26695			−0.206208			−0.103104	−	0.994671i	$-0.532878\pi$
$251$	−3.26695			−0.206208			−0.103104	+	0.994671i	$0.532878\pi$
$252$		0			0
$253$	−8.22648	−	11.2655i	−0.517194	−	0.708256i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			12.0706i			0.752946i	0.926428	+	0.376473i	$0.122863\pi$
$257$			12.0706i			0.752946i	−0.926428	+	0.376473i	$0.877137\pi$
$258$		0			0
$259$	−6.20876			−0.385794
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.46777			−0.583808			−0.291904	−	0.956448i	$-0.594289\pi$
$263$	−9.46777			−0.583808			−0.291904	+	0.956448i	$0.594289\pi$
$264$		0			0
$265$	5.14391			0.315988
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	8.25738i		−	0.503461i	−0.967797	−	0.251731i	$-0.919000\pi$
$269$		−	8.25738i		−	0.503461i	0.967797	−	0.251731i	$-0.0809997\pi$
$270$		0			0
$271$	−24.3626			−1.47992			−0.739962	−	0.672648i	$-0.765157\pi$
$271$	−24.3626			−1.47992			−0.739962	+	0.672648i	$0.765157\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.90866			−0.175399
$276$		0			0
$277$	5.72991			0.344277			0.172138	−	0.985073i	$-0.444932\pi$
$277$	5.72991			0.344277			0.172138	+	0.985073i	$0.444932\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.95139			−0.176065			−0.0880326	−	0.996118i	$-0.528058\pi$
$281$	−2.95139			−0.176065			−0.0880326	+	0.996118i	$0.528058\pi$
$282$		0			0
$283$			3.90976i			0.232411i	0.993225	+	0.116206i	$0.0370731\pi$
$283$			3.90976i			0.232411i	−0.993225	+	0.116206i	$0.962927\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.150523			−0.00888509
$288$		0			0
$289$	24.9658			1.46858
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.18409			−0.302858			−0.151429	−	0.988468i	$-0.548387\pi$
$293$	−5.18409			−0.302858			−0.151429	+	0.988468i	$0.548387\pi$
$294$		0			0
$295$		−	0.0710210i		−	0.00413500i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.72240	+	9.20579i	0.388767	+	0.532385i
$300$		0			0
$301$	−2.11170			−0.121717
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	5.76880i		−	0.330321i
$306$		0			0
$307$	−6.57119			−0.375037			−0.187519	−	0.982261i	$-0.560045\pi$
$307$	−6.57119			−0.375037			−0.187519	+	0.982261i	$0.560045\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	32.6341i		−	1.85051i	−0.379344	−	0.925256i	$-0.623850\pi$
$311$		−	32.6341i		−	1.85051i	0.379344	−	0.925256i	$-0.376150\pi$
$312$		0			0
$313$		−	12.9252i		−	0.730575i	−0.930895	−	0.365288i	$-0.880971\pi$
$313$		−	12.9252i		−	0.730575i	0.930895	−	0.365288i	$-0.119029\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	3.67373i		−	0.206337i	−0.994664	−	0.103169i	$-0.967102\pi$
$317$		−	3.67373i		−	0.206337i	0.994664	−	0.103169i	$-0.0328981\pi$
$318$		0			0
$319$			22.8066i			1.27693i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	20.7751i		−	1.15596i
$324$		0			0
$325$	2.37686			0.131844
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.54041			−0.470848
$330$		0			0
$331$	−1.62505			−0.0893207			−0.0446604	−	0.999002i	$-0.514221\pi$
$331$	−1.62505			−0.0893207			−0.0446604	+	0.999002i	$0.514221\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	2.04841i		−	0.111917i
$336$		0			0
$337$			25.5980i			1.39441i	0.716870	+	0.697207i	$0.245574\pi$
$337$			25.5980i			1.39441i	−0.716870	+	0.697207i	$0.754426\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.18483			0.497387
$342$		0			0
$343$		−	10.7814i		−	0.582142i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			31.0915i			1.66908i	0.550949	+	0.834539i	$0.314266\pi$
$347$			31.0915i			1.66908i	−0.550949	+	0.834539i	$0.685734\pi$
$348$		0			0
$349$	−21.2598			−1.13801			−0.569006	−	0.822333i	$-0.692672\pi$
$349$	−21.2598			−1.13801			−0.569006	+	0.822333i	$0.692672\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			28.2173i			1.50185i	0.660385	+	0.750927i	$0.270393\pi$
$353$			28.2173i			1.50185i	−0.660385	+	0.750927i	$0.729607\pi$
$354$		0			0
$355$		−	4.39463i		−	0.233243i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.45276			0.498898			0.249449	−	0.968388i	$-0.419751\pi$
$359$	9.45276			0.498898			0.249449	+	0.968388i	$0.419751\pi$
$360$		0			0
$361$	8.71536			0.458703
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.19913			−0.219793
$366$		0			0
$367$		−	2.53118i		−	0.132126i	−0.997815	−	0.0660632i	$-0.978956\pi$
$367$		−	2.53118i		−	0.132126i	0.997815	−	0.0660632i	$-0.0210439\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	4.15497i		−	0.215715i
$372$		0			0
$373$			3.91984i			0.202961i	0.994838	+	0.101481i	$0.0323580\pi$
$373$			3.91984i			0.202961i	−0.994838	+	0.101481i	$0.967642\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	18.6368i		−	0.959844i
$378$		0			0
$379$			10.1284i			0.520261i	0.965573	+	0.260131i	$0.0837657\pi$
$379$			10.1284i			0.520261i	−0.965573	+	0.260131i	$0.916234\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.8286			−0.859903			−0.429951	−	0.902852i	$-0.641469\pi$
$383$	−16.8286			−0.859903			−0.429951	+	0.902852i	$0.641469\pi$
$384$		0			0
$385$			2.34945i			0.119739i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.18442			0.110754			0.0553771	−	0.998466i	$-0.482364\pi$
$389$	2.18442			0.110754			0.0553771	+	0.998466i	$0.482364\pi$
$390$		0			0
$391$	−18.3219	−	25.0903i	−0.926576	−	1.26887i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			0.308833i			0.0155391i
$396$		0			0
$397$	31.4132			1.57658			0.788290	−	0.615303i	$-0.210966\pi$
$397$	31.4132			1.57658			0.788290	+	0.615303i	$0.210966\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.64192			−0.181869			−0.0909345	−	0.995857i	$-0.528985\pi$
$401$	−3.64192			−0.181869			−0.0909345	+	0.995857i	$0.528985\pi$
$402$		0			0
$403$	−7.50554			−0.373878
$404$		0			0
$405$		0			0
$406$		0			0
$407$			22.3575i			1.10822i
$408$		0			0
$409$	37.6984			1.86407			0.932034	−	0.362371i	$-0.118033\pi$
$409$	37.6984			1.86407			0.932034	+	0.362371i	$0.118033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.0573669			−0.00282284
$414$		0			0
$415$	−0.614191			−0.0301495
$416$		0			0
$417$		0			0
$418$		0			0
$419$	38.6455			1.88795			0.943977	−	0.330011i	$-0.107053\pi$
$419$	38.6455			1.88795			0.943977	+	0.330011i	$0.107053\pi$
$420$		0			0
$421$		−	27.1993i		−	1.32561i	−0.748791	−	0.662806i	$-0.769365\pi$
$421$		−	27.1993i		−	1.32561i	0.748791	−	0.662806i	$-0.230635\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.47810			−0.314234
$426$		0			0
$427$	−4.65972			−0.225500
$428$		0			0
$429$		0			0
$430$		0			0
$431$	22.8403			1.10018			0.550090	−	0.835105i	$-0.314593\pi$
$431$	22.8403			1.10018			0.550090	+	0.835105i	$0.314593\pi$
$432$		0			0
$433$		−	32.1300i		−	1.54407i	−0.635581	−	0.772034i	$-0.719239\pi$
$433$		−	32.1300i		−	1.54407i	0.635581	−	0.772034i	$-0.280761\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.4209	+	9.07018i	−0.594172	+	0.433886i
$438$		0			0
$439$	−1.23387			−0.0588893			−0.0294446	−	0.999566i	$-0.509374\pi$
$439$	−1.23387			−0.0588893			−0.0294446	+	0.999566i	$0.509374\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			7.21380i			0.342738i	0.985207	+	0.171369i	$0.0548190\pi$
$443$			7.21380i			0.342738i	−0.985207	+	0.171369i	$0.945181\pi$
$444$		0			0
$445$	−13.1731			−0.624467
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	13.6458i		−	0.643987i	−0.946742	−	0.321993i	$-0.895647\pi$
$449$		−	13.6458i		−	0.643987i	0.946742	−	0.321993i	$-0.104353\pi$
$450$		0			0
$451$			0.542026i			0.0255230i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1.91990i		−	0.0900061i
$456$		0			0
$457$		−	6.85297i		−	0.320568i	−0.987071	−	0.160284i	$-0.948759\pi$
$457$		−	6.85297i		−	0.320568i	0.987071	−	0.160284i	$-0.0512411\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	22.8823i		−	1.06574i	−0.846198	−	0.532868i	$-0.821114\pi$
$461$		−	22.8823i		−	1.06574i	0.846198	−	0.532868i	$-0.178886\pi$
$462$		0			0
$463$	−23.1173			−1.07435			−0.537176	−	0.843470i	$-0.680509\pi$
$463$	−23.1173			−1.07435			−0.537176	+	0.843470i	$0.680509\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.1739			−0.470793			−0.235397	−	0.971899i	$-0.575639\pi$
$467$	−10.1739			−0.470793			−0.235397	+	0.971899i	$0.575639\pi$
$468$		0			0
$469$	−1.65459			−0.0764020
$470$		0			0
$471$		0			0
$472$		0			0
$473$			7.60415i			0.349639i
$474$		0			0
$475$			3.20697i			0.147146i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.5581			−0.893635			−0.446817	−	0.894625i	$-0.647443\pi$
$479$	−19.5581			−0.893635			−0.446817	+	0.894625i	$0.647443\pi$
$480$		0			0
$481$		−	18.2698i		−	0.833030i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	2.32718i		−	0.105672i
$486$		0			0
$487$	26.1609			1.18546			0.592732	−	0.805400i	$-0.298049\pi$
$487$	26.1609			1.18546			0.592732	+	0.805400i	$0.298049\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	11.4128i		−	0.515052i	−0.966271	−	0.257526i	$-0.917093\pi$
$491$		−	11.4128i		−	0.515052i	0.966271	−	0.257526i	$-0.0829073\pi$
$492$		0			0
$493$			50.7945i			2.28767i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.54974			−0.159228
$498$		0			0
$499$	−12.5811			−0.563207			−0.281603	−	0.959531i	$-0.590866\pi$
$499$	−12.5811			−0.563207			−0.281603	+	0.959531i	$0.590866\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	28.5487			1.27292			0.636462	−	0.771308i	$-0.280397\pi$
$503$	28.5487			1.27292			0.636462	+	0.771308i	$0.280397\pi$
$504$		0			0
$505$		−	12.5285i		−	0.557510i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	12.3543i		−	0.547593i	−0.961788	−	0.273796i	$-0.911721\pi$
$509$		−	12.3543i		−	0.547593i	0.961788	−	0.273796i	$-0.0882794\pi$
$510$		0			0
$511$			3.39183i			0.150046i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	3.36639i		−	0.148341i
$516$		0			0
$517$			30.7536i			1.35254i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−14.7874			−0.647848			−0.323924	−	0.946083i	$-0.605002\pi$
$521$	−14.7874			−0.647848			−0.323924	+	0.946083i	$0.605002\pi$
$522$		0			0
$523$			14.9607i			0.654184i	0.944992	+	0.327092i	$0.106069\pi$
$523$			14.9607i			0.654184i	−0.944992	+	0.327092i	$0.893931\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	20.4563			0.891090
$528$		0			0
$529$	−7.00173	+	21.9084i	−0.304423	+	0.952537i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	0.442926i		−	0.0191853i
$534$		0			0
$535$	−10.8848			−0.470592
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18.4628			−0.795250
$540$		0			0
$541$	0.260952			0.0112192			0.00560961	−	0.999984i	$-0.498214\pi$
$541$	0.260952			0.0112192			0.00560961	+	0.999984i	$0.498214\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	8.43356i		−	0.361254i
$546$		0			0
$547$	−44.5929			−1.90666			−0.953328	−	0.301936i	$-0.902367\pi$
$547$	−44.5929			−1.90666			−0.953328	+	0.301936i	$0.902367\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	25.1457			1.07124
$552$		0			0
$553$	0.249459			0.0106081
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−33.3290			−1.41219			−0.706097	−	0.708115i	$-0.749546\pi$
$557$	−33.3290			−1.41219			−0.706097	+	0.708115i	$0.749546\pi$
$558$		0			0
$559$		−	6.21386i		−	0.262818i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	36.3300			1.53113			0.765564	−	0.643359i	$-0.222460\pi$
$563$	36.3300			1.53113			0.765564	+	0.643359i	$0.222460\pi$
$564$		0			0
$565$	−14.0580			−0.591425
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.82194			−0.369835			−0.184917	−	0.982754i	$-0.559202\pi$
$569$	−8.82194			−0.369835			−0.184917	+	0.982754i	$0.559202\pi$
$570$		0			0
$571$		−	33.0248i		−	1.38204i	−0.722833	−	0.691022i	$-0.757161\pi$
$571$		−	33.0248i		−	1.38204i	0.722833	−	0.691022i	$-0.242839\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.82827	+	3.87309i	0.117947	+	0.161519i
$576$		0			0
$577$	17.7753			0.739994			0.369997	−	0.929033i	$-0.379359\pi$
$577$	17.7753			0.739994			0.369997	+	0.929033i	$0.379359\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			0.496110i			0.0205821i
$582$		0			0
$583$	−14.9619			−0.619657
$584$		0			0
$585$		0			0
$586$		0			0
$587$			14.5579i			0.600867i	0.953803	+	0.300434i	$0.0971314\pi$
$587$			14.5579i			0.600867i	−0.953803	+	0.300434i	$0.902869\pi$
$588$		0			0
$589$		−	10.1268i		−	0.417269i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	23.5386i		−	0.966613i	−0.875451	−	0.483307i	$-0.839436\pi$
$593$		−	23.5386i		−	0.966613i	0.875451	−	0.483307i	$-0.160564\pi$
$594$		0			0
$595$			5.23266i			0.214518i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			40.0975i			1.63834i	0.573551	+	0.819170i	$0.305566\pi$
$599$			40.0975i			1.63834i	−0.573551	+	0.819170i	$0.694434\pi$
$600$		0			0
$601$	−36.5540			−1.49107			−0.745534	−	0.666468i	$-0.767805\pi$
$601$	−36.5540			−1.49107			−0.745534	+	0.666468i	$0.767805\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.53972			−0.103254
$606$		0			0
$607$	21.6574			0.879045			0.439522	−	0.898232i	$-0.355148\pi$
$607$	21.6574			0.879045			0.439522	+	0.898232i	$0.355148\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	25.1309i		−	1.01669i
$612$		0			0
$613$			26.1752i			1.05721i	0.848868	+	0.528604i	$0.177284\pi$
$613$			26.1752i			1.05721i	−0.848868	+	0.528604i	$0.822716\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.177165			−0.00713240			−0.00356620	−	0.999994i	$-0.501135\pi$
$617$	−0.177165			−0.00713240			−0.00356620	+	0.999994i	$0.501135\pi$
$618$		0			0
$619$			39.5776i			1.59076i	0.606112	+	0.795380i	$0.292728\pi$
$619$			39.5776i			1.59076i	−0.606112	+	0.795380i	$0.707272\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			10.6405i			0.426305i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			49.7941i			1.98542i
$630$		0			0
$631$		−	29.2264i		−	1.16349i	−0.813373	−	0.581743i	$-0.802371\pi$
$631$		−	29.2264i		−	1.16349i	0.813373	−	0.581743i	$-0.197629\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.2897			0.567071
$636$		0			0
$637$	15.0872			0.597777
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.90989			0.312422			0.156211	−	0.987724i	$-0.450072\pi$
$641$	7.90989			0.312422			0.156211	+	0.987724i	$0.450072\pi$
$642$		0			0
$643$			43.8370i			1.72876i	0.502837	+	0.864381i	$0.332290\pi$
$643$			43.8370i			1.72876i	−0.502837	+	0.864381i	$0.667710\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			44.5892i			1.75298i	0.481417	+	0.876492i	$0.340122\pi$
$647$			44.5892i			1.75298i	−0.481417	+	0.876492i	$0.659878\pi$
$648$		0			0
$649$			0.206576i			0.00810880i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			29.3885i			1.15006i	0.818132	+	0.575030i	$0.195010\pi$
$653$			29.3885i			1.15006i	−0.818132	+	0.575030i	$0.804990\pi$
$654$		0			0
$655$			2.46341i			0.0962534i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.49489			0.136142			0.0680708	−	0.997680i	$-0.478316\pi$
$659$	3.49489			0.136142			0.0680708	+	0.997680i	$0.478316\pi$
$660$		0			0
$661$		−	8.00078i		−	0.311194i	−0.987821	−	0.155597i	$-0.950270\pi$
$661$		−	8.00078i		−	0.311194i	0.987821	−	0.155597i	$-0.0497302\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.59041			0.100452
$666$		0			0
$667$	30.3687	−	22.1763i	1.17588	−	0.858672i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			16.7795i			0.647764i
$672$		0			0
$673$	42.4608			1.63674			0.818372	−	0.574689i	$-0.194877\pi$
$673$	42.4608			1.63674			0.818372	+	0.574689i	$0.194877\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.8961			0.572502			0.286251	−	0.958155i	$-0.407591\pi$
$677$	14.8961			0.572502			0.286251	+	0.958155i	$0.407591\pi$
$678$		0			0
$679$	−1.87977			−0.0721390
$680$		0			0
$681$		0			0
$682$		0			0
$683$			33.2528i			1.27238i	0.771531	+	0.636191i	$0.219491\pi$
$683$			33.2528i			1.27238i	−0.771531	+	0.636191i	$0.780509\pi$
$684$		0			0
$685$	−8.64135			−0.330169
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.2263			0.465786
$690$		0			0
$691$	−22.9473			−0.872958			−0.436479	−	0.899714i	$-0.643775\pi$
$691$	−22.9473			−0.872958			−0.436479	+	0.899714i	$0.643775\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	15.0825			0.572113
$696$		0			0
$697$			1.20719i			0.0457256i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21.0334			0.794422			0.397211	−	0.917727i	$-0.369978\pi$
$701$	21.0334			0.794422			0.397211	+	0.917727i	$0.369978\pi$
$702$		0			0
$703$	24.6504			0.929709
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.1198			−0.380595
$708$		0			0
$709$			12.8536i			0.482727i	0.970435	+	0.241364i	$0.0775946\pi$
$709$			12.8536i			0.482727i	−0.970435	+	0.241364i	$0.922405\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8.93101	−	12.2303i	−0.334469	−	0.458028i
$714$		0			0
$715$	−6.91346			−0.258549
$716$		0			0
$717$		0			0
$718$		0			0
$719$			16.5125i			0.615813i	0.951417	+	0.307906i	$0.0996283\pi$
$719$			16.5125i			0.615813i	−0.951417	+	0.307906i	$0.900372\pi$
$720$		0			0
$721$	−2.71919			−0.101268
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	7.84095i		−	0.291205i
$726$		0			0
$727$			29.5648i			1.09650i	0.836316	+	0.548248i	$0.184705\pi$
$727$			29.5648i			1.09650i	−0.836316	+	0.548248i	$0.815295\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			16.9358i			0.626394i
$732$		0			0
$733$		−	1.52557i		−	0.0563481i	−0.999603	−	0.0281740i	$-0.991031\pi$
$733$		−	1.52557i		−	0.0563481i	0.999603	−	0.0281740i	$-0.00896926\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			5.95812i			0.219470i
$738$		0			0
$739$	−1.14817			−0.0422360			−0.0211180	−	0.999777i	$-0.506723\pi$
$739$	−1.14817			−0.0422360			−0.0211180	+	0.999777i	$0.506723\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	26.7137			0.980031			0.490015	−	0.871714i	$-0.336991\pi$
$743$	26.7137			0.980031			0.490015	+	0.871714i	$0.336991\pi$
$744$		0			0
$745$	−19.1168			−0.700385
$746$		0			0
$747$		0			0
$748$		0			0
$749$			8.79217i			0.321259i
$750$		0			0
$751$			30.2897i			1.10529i	0.833418	+	0.552643i	$0.186381\pi$
$751$			30.2897i			1.10529i	−0.833418	+	0.552643i	$0.813619\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.87440			0.141004
$756$		0			0
$757$			3.61798i			0.131498i	0.997836	+	0.0657489i	$0.0209436\pi$
$757$			3.61798i			0.131498i	−0.997836	+	0.0657489i	$0.979056\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	2.19975i		−	0.0797410i	−0.999205	−	0.0398705i	$-0.987305\pi$
$761$		−	2.19975i		−	0.0797410i	0.999205	−	0.0398705i	$-0.0126945\pi$
$762$		0			0
$763$	−6.81217			−0.246617
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	0.168807i		−	0.00609526i
$768$		0			0
$769$		−	43.0140i		−	1.55112i	−0.631271	−	0.775562i	$-0.717467\pi$
$769$		−	43.0140i		−	1.55112i	0.631271	−	0.775562i	$-0.282533\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	33.2717			1.19670			0.598350	−	0.801235i	$-0.295823\pi$
$773$	33.2717			1.19670			0.598350	+	0.801235i	$0.295823\pi$
$774$		0			0
$775$	−3.15776			−0.113430
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.597616			0.0214118
$780$		0			0
$781$			12.7825i			0.457393i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			12.6141i			0.450215i
$786$		0			0
$787$			41.3621i			1.47440i	0.675676	+	0.737199i	$0.263852\pi$
$787$			41.3621i			1.47440i	−0.675676	+	0.737199i	$0.736148\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			11.3553i			0.403748i
$792$		0			0
$793$		−	13.7116i		−	0.486914i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.61447			0.163453			0.0817264	−	0.996655i	$-0.473957\pi$
$797$	4.61447			0.163453			0.0817264	+	0.996655i	$0.473957\pi$
$798$		0			0
$799$			68.4939i			2.42314i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.2138			0.431017
$804$		0			0
$805$	3.12847	−	2.28453i	0.110264	−	0.0805190i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	20.1201i		−	0.707386i	−0.935362	−	0.353693i	$-0.884926\pi$
$809$		−	20.1201i		−	0.707386i	0.935362	−	0.353693i	$-0.115074\pi$
$810$		0			0
$811$	−16.8303			−0.590990			−0.295495	−	0.955344i	$-0.595485\pi$
$811$	−16.8303			−0.590990			−0.295495	+	0.955344i	$0.595485\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	11.9182			0.417477
$816$		0			0
$817$	8.38403			0.293320
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	43.5480i		−	1.51984i	−0.650019	−	0.759918i	$-0.725239\pi$
$821$		−	43.5480i		−	1.51984i	0.650019	−	0.759918i	$-0.274761\pi$
$822$		0			0
$823$	−11.9759			−0.417453			−0.208727	−	0.977974i	$-0.566932\pi$
$823$	−11.9759			−0.417453			−0.208727	+	0.977974i	$0.566932\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	43.9986			1.52998			0.764991	−	0.644041i	$-0.222744\pi$
$827$	43.9986			1.52998			0.764991	+	0.644041i	$0.222744\pi$
$828$		0			0
$829$	26.5359			0.921629			0.460814	−	0.887497i	$-0.347557\pi$
$829$	26.5359			0.921629			0.460814	+	0.887497i	$0.347557\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−41.1201			−1.42473
$834$		0			0
$835$		−	7.09244i		−	0.245444i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−56.2238			−1.94106			−0.970530	−	0.240980i	$-0.922531\pi$
$839$	−56.2238			−1.94106			−0.970530	+	0.240980i	$0.922531\pi$
$840$		0			0
$841$	−32.4804			−1.12002
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7.35055			−0.252867
$846$		0			0
$847$			2.05145i			0.0704886i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	29.7707	−	21.7396i	1.02052	−	0.745224i
$852$		0			0
$853$	−15.0117			−0.513991			−0.256995	−	0.966413i	$-0.582733\pi$
$853$	−15.0117			−0.513991			−0.256995	+	0.966413i	$0.582733\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			0.515056i			0.0175940i	0.999961	+	0.00879698i	$0.00280020\pi$
$857$			0.515056i			0.0175940i	−0.999961	+	0.00879698i	$0.997200\pi$
$858$		0			0
$859$	15.8326			0.540200			0.270100	−	0.962832i	$-0.412943\pi$
$859$	15.8326			0.540200			0.270100	+	0.962832i	$0.412943\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			2.17592i			0.0740693i	0.999314	+	0.0370346i	$0.0117912\pi$
$863$			2.17592i			0.0740693i	−0.999314	+	0.0370346i	$0.988209\pi$
$864$		0			0
$865$		−	17.3090i		−	0.588524i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	0.898289i		−	0.0304724i
$870$		0			0
$871$		−	4.86877i		−	0.164972i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	0.807745i		−	0.0273068i
$876$		0			0
$877$	14.4337			0.487391			0.243696	−	0.969852i	$-0.421640\pi$
$877$	14.4337			0.487391			0.243696	+	0.969852i	$0.421640\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	6.71721			0.226308			0.113154	−	0.993577i	$-0.463905\pi$
$881$	6.71721			0.226308			0.113154	+	0.993577i	$0.463905\pi$
$882$		0			0
$883$	−38.6083			−1.29927			−0.649637	−	0.760245i	$-0.725079\pi$
$883$	−38.6083			−1.29927			−0.649637	+	0.760245i	$0.725079\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	32.9159i		−	1.10521i	−0.833444	−	0.552604i	$-0.813634\pi$
$887$		−	32.9159i		−	1.10521i	0.833444	−	0.552604i	$-0.186366\pi$
$888$		0			0
$889$		−	11.5425i		−	0.387122i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	33.9077			1.13468
$894$		0			0
$895$			19.8004i			0.661853i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			24.7598i			0.825786i
$900$		0			0
$901$	−33.3228			−1.11014
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	12.9173i		−	0.429387i
$906$		0			0
$907$			49.3816i			1.63969i	0.572585	+	0.819845i	$0.305941\pi$
$907$			49.3816i			1.63969i	−0.572585	+	0.819845i	$0.694059\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	46.5813			1.54331			0.771654	−	0.636042i	$-0.219430\pi$
$911$	46.5813			1.54331			0.771654	+	0.636042i	$0.219430\pi$
$912$		0			0
$913$	1.78647			0.0591236
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.98981			0.0657093
$918$		0			0
$919$		−	42.0253i		−	1.38629i	−0.720799	−	0.693144i	$-0.756225\pi$
$919$		−	42.0253i		−	1.38629i	0.720799	−	0.693144i	$-0.243775\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	10.4454i		−	0.343815i
$924$		0			0
$925$		−	7.68653i		−	0.252732i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			0.747452i			0.0245231i	0.999925	+	0.0122615i	$0.00390307\pi$
$929$			0.747452i			0.0245231i	−0.999925	+	0.0122615i	$0.996097\pi$
$930$		0			0
$931$			20.3564i			0.667153i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	18.8426			0.616218
$936$		0			0
$937$		−	20.4521i		−	0.668142i	−0.942548	−	0.334071i	$-0.891578\pi$
$937$		−	20.4521i		−	0.668142i	0.942548	−	0.334071i	$-0.108422\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	44.5440			1.45209			0.726046	−	0.687646i	$-0.241356\pi$
$941$	44.5440			1.45209			0.726046	+	0.687646i	$0.241356\pi$
$942$		0			0
$943$	0.721749	−	0.527047i	0.0235034	−	0.0171630i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			8.36166i			0.271717i	0.990728	+	0.135859i	$0.0433793\pi$
$947$			8.36166i			0.271717i	−0.990728	+	0.135859i	$0.956621\pi$
$948$		0			0
$949$	−9.98074			−0.323988
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−46.0089			−1.49037			−0.745187	−	0.666855i	$-0.767640\pi$
$953$	−46.0089			−1.49037			−0.745187	+	0.666855i	$0.767640\pi$
$954$		0			0
$955$	10.0943			0.326644
$956$		0			0
$957$		0			0
$958$		0			0
$959$			6.98001i			0.225396i
$960$		0			0
$961$	−21.0286			−0.678341
$962$		0			0
$963$		0			0
$964$		0			0
$965$	17.4900			0.563022
$966$		0			0
$967$	−42.9649			−1.38166			−0.690830	−	0.723018i	$-0.742754\pi$
$967$	−42.9649			−1.38166			−0.690830	+	0.723018i	$0.742754\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.0564			−0.900373			−0.450187	−	0.892934i	$-0.648643\pi$
$971$	−28.0564			−0.900373			−0.450187	+	0.892934i	$0.648643\pi$
$972$		0			0
$973$		−	12.1828i		−	0.390564i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.3810			−0.620052			−0.310026	−	0.950728i	$-0.600338\pi$
$977$	−19.3810			−0.620052			−0.310026	+	0.950728i	$0.600338\pi$
$978$		0			0
$979$	38.3161			1.22459
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−11.8728			−0.378684			−0.189342	−	0.981911i	$-0.560636\pi$
$983$	−11.8728			−0.378684			−0.189342	+	0.981911i	$0.560636\pi$
$984$		0			0
$985$			6.22646i			0.198391i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	10.1255	−	7.39401i	0.321972	−	0.235116i
$990$		0			0
$991$	−20.1005			−0.638512			−0.319256	−	0.947668i	$-0.603433\pi$
$991$	−20.1005			−0.638512			−0.319256	+	0.947668i	$0.603433\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			4.36684i			0.138438i
$996$		0			0
$997$	2.52792			0.0800602			0.0400301	−	0.999198i	$-0.487255\pi$
$997$	2.52792			0.0800602			0.0400301	+	0.999198i	$0.487255\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.i.b.1241.7	yes	16
3.2	odd	2		4140.2.i.a.1241.7	✓	16
23.22	odd	2		4140.2.i.a.1241.10	yes	16
69.68	even	2	inner	4140.2.i.b.1241.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.i.a.1241.7	✓	16	3.2	odd	2
4140.2.i.a.1241.10	yes	16	23.22	odd	2
4140.2.i.b.1241.7	yes	16	1.1	even	1	trivial
4140.2.i.b.1241.10	yes	16	69.68	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.807745i q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -0.807745i q^{7} -2.90866 q^{11} +2.37686 q^{13} -6.47810 q^{17} +3.20697i q^{19} +(2.82827 + 3.87309i) q^{23} +1.00000 q^{25} -7.84095i q^{29} -3.15776 q^{31} -0.807745i q^{35} -7.68653i q^{37} -0.186349i q^{41} -2.61432i q^{43} -10.5731i q^{47} +6.34755 q^{49} +5.14391 q^{53} -2.90866 q^{55} -0.0710210i q^{59} -5.76880i q^{61} +2.37686 q^{65} -2.04841i q^{67} -4.39463i q^{71} -4.19913 q^{73} +2.34945i q^{77} +0.308833i q^{79} -0.614191 q^{83} -6.47810 q^{85} -13.1731 q^{89} -1.91990i q^{91} +3.20697i q^{95} -2.32718i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5}+O(q^{10}) \) 16 * q + 16 * q^5	\( 16 q + 16 q^{5} + 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} - 4 q^{53} + 8 q^{73} - 20 q^{83} - 32 q^{89}+O(q^{100}) \) 16 * q + 16 * q^5 + 8 * q^23 + 16 * q^25 - 8 * q^31 - 40 * q^49 - 4 * q^53 + 8 * q^73 - 20 * q^83 - 32 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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