LMFDB - Embedded newform 4140.2.i.b.1241.5

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			1241.5
Root			$-2.31631i$ of defining polynomial
Character	$\chi$	$=$	4140.1241
Dual form			4140.2.i.b.1241.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.73994i q^{7} +O(q^{10})$	$q+1.00000 q^{5} -1.73994i q^{7} +1.48546 q^{11} -2.14610 q^{13} -3.22657 q^{17} +5.06099i q^{19} +(-4.42186 - 1.85665i) q^{23} +1.00000 q^{25} +5.21742i q^{29} -5.45116 q^{31} -1.73994i q^{35} -2.87861i q^{37} +10.9887i q^{41} +4.50476i q^{43} +4.76294i q^{47} +3.97260 q^{49} -0.239954 q^{53} +1.48546 q^{55} +5.20247i q^{59} +6.30502i q^{61} -2.14610 q^{65} +2.61074i q^{67} +3.17216i q^{71} -15.7765 q^{73} -2.58462i q^{77} -16.1921i q^{79} -0.583836 q^{83} -3.22657 q^{85} +14.7009 q^{89} +3.73410i q^{91} +5.06099i q^{95} -1.66138i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	1.73994i		−	0.657637i	−0.944393	−	0.328818i	$-0.893350\pi$
$7$		−	1.73994i		−	0.657637i	0.944393	−	0.328818i	$-0.106650\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.48546			0.447884			0.223942	−	0.974603i	$-0.428107\pi$
$11$	1.48546			0.447884			0.223942	+	0.974603i	$0.428107\pi$
$12$		0			0
$13$	−2.14610			−0.595222			−0.297611	−	0.954687i	$-0.596190\pi$
$13$	−2.14610			−0.595222			−0.297611	+	0.954687i	$0.596190\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.22657			−0.782558			−0.391279	−	0.920272i	$-0.627967\pi$
$17$	−3.22657			−0.782558			−0.391279	+	0.920272i	$0.627967\pi$
$18$		0			0
$19$			5.06099i			1.16107i	0.814235	+	0.580535i	$0.197157\pi$
$19$			5.06099i			1.16107i	−0.814235	+	0.580535i	$0.802843\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.42186	−	1.85665i	−0.922022	−	0.387137i
$23$	−4.42186	−	1.85665i	−0.922022	−	0.387137i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.21742i			0.968851i	0.874832	+	0.484426i	$0.160971\pi$
$29$			5.21742i			0.968851i	−0.874832	+	0.484426i	$0.839029\pi$
$30$		0			0
$31$	−5.45116			−0.979057			−0.489528	−	0.871987i	$-0.662831\pi$
$31$	−5.45116			−0.979057			−0.489528	+	0.871987i	$0.662831\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1.73994i		−	0.294104i
$36$		0			0
$37$		−	2.87861i		−	0.473241i	−0.971602	−	0.236620i	$-0.923960\pi$
$37$		−	2.87861i		−	0.473241i	0.971602	−	0.236620i	$-0.0760398\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.9887i			1.71615i	0.513525	+	0.858074i	$0.328339\pi$
$41$			10.9887i			1.71615i	−0.513525	+	0.858074i	$0.671661\pi$
$42$		0			0
$43$			4.50476i			0.686969i	0.939158	+	0.343485i	$0.111607\pi$
$43$			4.50476i			0.686969i	−0.939158	+	0.343485i	$0.888393\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			4.76294i			0.694746i	0.937727	+	0.347373i	$0.112926\pi$
$47$			4.76294i			0.694746i	−0.937727	+	0.347373i	$0.887074\pi$
$48$		0			0
$49$	3.97260			0.567514
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.239954			−0.0329601			−0.0164801	−	0.999864i	$-0.505246\pi$
$53$	−0.239954			−0.0329601			−0.0164801	+	0.999864i	$0.505246\pi$
$54$		0			0
$55$	1.48546			0.200300
$56$		0			0
$57$		0			0
$58$		0			0
$59$			5.20247i			0.677304i	0.940912	+	0.338652i	$0.109971\pi$
$59$			5.20247i			0.677304i	−0.940912	+	0.338652i	$0.890029\pi$
$60$		0			0
$61$			6.30502i			0.807275i	0.914919	+	0.403638i	$0.132254\pi$
$61$			6.30502i			0.807275i	−0.914919	+	0.403638i	$0.867746\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.14610			−0.266191
$66$		0			0
$67$			2.61074i			0.318952i	0.987202	+	0.159476i	$0.0509805\pi$
$67$			2.61074i			0.318952i	−0.987202	+	0.159476i	$0.949019\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.17216i			0.376466i	0.982124	+	0.188233i	$0.0602760\pi$
$71$			3.17216i			0.376466i	−0.982124	+	0.188233i	$0.939724\pi$
$72$		0			0
$73$	−15.7765			−1.84649			−0.923247	−	0.384207i	$-0.874475\pi$
$73$	−15.7765			−1.84649			−0.923247	+	0.384207i	$0.874475\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.58462i		−	0.294545i
$78$		0			0
$79$		−	16.1921i		−	1.82175i	−0.412682	−	0.910875i	$-0.635408\pi$
$79$		−	16.1921i		−	1.82175i	0.412682	−	0.910875i	$-0.364592\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.583836			−0.0640843			−0.0320421	−	0.999487i	$-0.510201\pi$
$83$	−0.583836			−0.0640843			−0.0320421	+	0.999487i	$0.510201\pi$
$84$		0			0
$85$	−3.22657			−0.349971
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.7009			1.55829			0.779147	−	0.626842i	$-0.215653\pi$
$89$	14.7009			1.55829			0.779147	+	0.626842i	$0.215653\pi$
$90$		0			0
$91$			3.73410i			0.391440i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			5.06099i			0.519246i
$96$		0			0
$97$		−	1.66138i		−	0.168687i	−0.996437	−	0.0843436i	$-0.973121\pi$
$97$		−	1.66138i		−	0.168687i	0.996437	−	0.0843436i	$-0.0268793\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	6.21263i		−	0.618180i	−0.951033	−	0.309090i	$-0.899976\pi$
$101$		−	6.21263i		−	0.618180i	0.951033	−	0.309090i	$-0.100024\pi$
$102$		0			0
$103$			3.57047i			0.351809i	0.984407	+	0.175904i	$0.0562850\pi$
$103$			3.57047i			0.351809i	−0.984407	+	0.175904i	$0.943715\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.4060			1.77938			0.889689	−	0.456566i	$-0.150921\pi$
$107$	18.4060			1.77938			0.889689	+	0.456566i	$0.150921\pi$
$108$		0			0
$109$			17.6728i			1.69275i	0.532590	+	0.846374i	$0.321219\pi$
$109$			17.6728i			1.69275i	−0.532590	+	0.846374i	$0.678781\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	11.9644			1.12551			0.562757	−	0.826622i	$-0.309741\pi$
$113$	11.9644			1.12551			0.562757	+	0.826622i	$0.309741\pi$
$114$		0			0
$115$	−4.42186	−	1.85665i	−0.412341	−	0.173133i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			5.61405i			0.514639i
$120$		0			0
$121$	−8.79340			−0.799400
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−15.2155			−1.35016			−0.675081	−	0.737744i	$-0.735891\pi$
$127$	−15.2155			−1.35016			−0.675081	+	0.737744i	$0.735891\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			21.1936i			1.85170i	0.377897	+	0.925848i	$0.376647\pi$
$131$			21.1936i			1.85170i	−0.377897	+	0.925848i	$0.623353\pi$
$132$		0			0
$133$	8.80583			0.763562
$134$		0			0
$135$		0			0
$136$		0			0
$137$	19.3615			1.65417			0.827084	−	0.562078i	$-0.189998\pi$
$137$	19.3615			1.65417			0.827084	+	0.562078i	$0.189998\pi$
$138$		0			0
$139$	−12.3341			−1.04617			−0.523083	−	0.852282i	$-0.675218\pi$
$139$	−12.3341			−1.04617			−0.523083	+	0.852282i	$0.675218\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.18795			−0.266590
$144$		0			0
$145$			5.21742i			0.433283i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.5487			−1.02803			−0.514015	−	0.857781i	$-0.671842\pi$
$149$	−12.5487			−1.02803			−0.514015	+	0.857781i	$0.671842\pi$
$150$		0			0
$151$	17.3650			1.41314			0.706570	−	0.707643i	$-0.250242\pi$
$151$	17.3650			1.41314			0.706570	+	0.707643i	$0.250242\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.45116			−0.437847
$156$		0			0
$157$		−	17.5785i		−	1.40291i	−0.712711	−	0.701457i	$-0.752533\pi$
$157$		−	17.5785i		−	1.40291i	0.712711	−	0.701457i	$-0.247467\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.23046	+	7.69379i	−0.254596	+	0.606356i
$162$		0			0
$163$	15.0296			1.17721			0.588606	−	0.808420i	$-0.299677\pi$
$163$	15.0296			1.17721			0.588606	+	0.808420i	$0.299677\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	11.4559i		−	0.886485i	−0.896402	−	0.443242i	$-0.853828\pi$
$167$		−	11.4559i		−	0.886485i	0.896402	−	0.443242i	$-0.146172\pi$
$168$		0			0
$169$	−8.39424			−0.645711
$170$		0			0
$171$		0			0
$172$		0			0
$173$			7.36507i			0.559956i	0.960006	+	0.279978i	$0.0903272\pi$
$173$			7.36507i			0.559956i	−0.960006	+	0.279978i	$0.909673\pi$
$174$		0			0
$175$		−	1.73994i		−	0.131527i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	13.0710i		−	0.976975i	−0.872571	−	0.488487i	$-0.837549\pi$
$179$		−	13.0710i		−	0.976975i	0.872571	−	0.488487i	$-0.162451\pi$
$180$		0			0
$181$			19.0486i			1.41587i	0.706276	+	0.707937i	$0.250374\pi$
$181$			19.0486i			1.41587i	−0.706276	+	0.707937i	$0.749626\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	2.87861i		−	0.211640i
$186$		0			0
$187$	−4.79295			−0.350495
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.27526			−0.309347			−0.154673	−	0.987966i	$-0.549433\pi$
$191$	−4.27526			−0.309347			−0.154673	+	0.987966i	$0.549433\pi$
$192$		0			0
$193$	−18.8399			−1.35613			−0.678063	−	0.735004i	$-0.737180\pi$
$193$	−18.8399			−1.35613			−0.678063	+	0.735004i	$0.737180\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			26.8545i			1.91330i	0.291233	+	0.956652i	$0.405935\pi$
$197$			26.8545i			1.91330i	−0.291233	+	0.956652i	$0.594065\pi$
$198$		0			0
$199$		−	10.8944i		−	0.772282i	−0.922440	−	0.386141i	$-0.873808\pi$
$199$		−	10.8944i		−	0.772282i	0.922440	−	0.386141i	$-0.126192\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.07802			0.637152
$204$		0			0
$205$			10.9887i			0.767485i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			7.51790i			0.520024i
$210$		0			0
$211$	13.8858			0.955941			0.477970	−	0.878376i	$-0.341373\pi$
$211$	13.8858			0.955941			0.477970	+	0.878376i	$0.341373\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4.50476i			0.307222i
$216$		0			0
$217$			9.48470i			0.643864i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.92455			0.465796
$222$		0			0
$223$	−20.1599			−1.35001			−0.675003	−	0.737815i	$-0.735858\pi$
$223$	−20.1599			−1.35001			−0.675003	+	0.737815i	$0.735858\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.4260			−1.35572			−0.677861	−	0.735190i	$-0.737093\pi$
$227$	−20.4260			−1.35572			−0.677861	+	0.735190i	$0.737093\pi$
$228$		0			0
$229$		−	12.6822i		−	0.838064i	−0.907971	−	0.419032i	$-0.862369\pi$
$229$		−	12.6822i		−	0.838064i	0.907971	−	0.419032i	$-0.137631\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			18.4814i			1.21076i	0.795938	+	0.605378i	$0.206978\pi$
$233$			18.4814i			1.21076i	−0.795938	+	0.605378i	$0.793022\pi$
$234$		0			0
$235$			4.76294i			0.310700i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	9.96320i		−	0.644466i	−0.946660	−	0.322233i	$-0.895567\pi$
$239$		−	9.96320i		−	0.644466i	0.946660	−	0.322233i	$-0.104433\pi$
$240$		0			0
$241$			4.57276i			0.294557i	0.989095	+	0.147279i	$0.0470514\pi$
$241$			4.57276i			0.294557i	−0.989095	+	0.147279i	$0.952949\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.97260			0.253800
$246$		0			0
$247$		−	10.8614i		−	0.691094i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.15628			−0.136103			−0.0680515	−	0.997682i	$-0.521678\pi$
$251$	−2.15628			−0.136103			−0.0680515	+	0.997682i	$0.521678\pi$
$252$		0			0
$253$	−6.56851	−	2.75798i	−0.412959	−	0.173392i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			18.9962i			1.18495i	0.805589	+	0.592474i	$0.201849\pi$
$257$			18.9962i			1.18495i	−0.805589	+	0.592474i	$0.798151\pi$
$258$		0			0
$259$	−5.00862			−0.311221
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.44986			−0.0894022			−0.0447011	−	0.999000i	$-0.514234\pi$
$263$	−1.44986			−0.0894022			−0.0447011	+	0.999000i	$0.514234\pi$
$264$		0			0
$265$	−0.239954			−0.0147402
$266$		0			0
$267$		0			0
$268$		0			0
$269$			10.9723i			0.668992i	0.942397	+	0.334496i	$0.108566\pi$
$269$			10.9723i			0.668992i	−0.942397	+	0.334496i	$0.891434\pi$
$270$		0			0
$271$	−18.9514			−1.15122			−0.575608	−	0.817726i	$-0.695235\pi$
$271$	−18.9514			−1.15122			−0.575608	+	0.817726i	$0.695235\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.48546			0.0895767
$276$		0			0
$277$	4.76990			0.286595			0.143298	−	0.989680i	$-0.454229\pi$
$277$	4.76990			0.286595			0.143298	+	0.989680i	$0.454229\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.47200			−0.505397			−0.252699	−	0.967545i	$-0.581318\pi$
$281$	−8.47200			−0.505397			−0.252699	+	0.967545i	$0.581318\pi$
$282$		0			0
$283$		−	3.48520i		−	0.207174i	−0.994620	−	0.103587i	$-0.966968\pi$
$283$		−	3.48520i		−	0.207174i	0.994620	−	0.103587i	$-0.0330320\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	19.1197			1.12860
$288$		0			0
$289$	−6.58925			−0.387603
$290$		0			0
$291$		0			0
$292$		0			0
$293$	16.5320			0.965811			0.482905	−	0.875672i	$-0.339581\pi$
$293$	16.5320			0.965811			0.482905	+	0.875672i	$0.339581\pi$
$294$		0			0
$295$			5.20247i			0.302899i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.48977	+	3.98455i	0.548808	+	0.230433i
$300$		0			0
$301$	7.83802			0.451776
$302$		0			0
$303$		0			0
$304$		0			0
$305$			6.30502i			0.361024i
$306$		0			0
$307$	−12.9569			−0.739491			−0.369746	−	0.929133i	$-0.620555\pi$
$307$	−12.9569			−0.739491			−0.369746	+	0.929133i	$0.620555\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	2.07291i		−	0.117544i	−0.998271	−	0.0587720i	$-0.981282\pi$
$311$		−	2.07291i		−	0.117544i	0.998271	−	0.0587720i	$-0.0187185\pi$
$312$		0			0
$313$			9.21330i			0.520767i	0.965505	+	0.260383i	$0.0838489\pi$
$313$			9.21330i			0.520767i	−0.965505	+	0.260383i	$0.916151\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			1.62477i			0.0912560i	0.998958	+	0.0456280i	$0.0145289\pi$
$317$			1.62477i			0.0912560i	−0.998958	+	0.0456280i	$0.985471\pi$
$318$		0			0
$319$			7.75028i			0.433933i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	16.3296i		−	0.908605i
$324$		0			0
$325$	−2.14610			−0.119044
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.28725			0.456891
$330$		0			0
$331$	15.7992			0.868402			0.434201	−	0.900816i	$-0.357031\pi$
$331$	15.7992			0.868402			0.434201	+	0.900816i	$0.357031\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			2.61074i			0.142640i
$336$		0			0
$337$			2.58536i			0.140834i	0.997518	+	0.0704169i	$0.0224330\pi$
$337$			2.58536i			0.140834i	−0.997518	+	0.0704169i	$0.977567\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.09748			−0.438503
$342$		0			0
$343$		−	19.0917i		−	1.03085i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	0.459390i		−	0.0246614i	−0.999924	−	0.0123307i	$-0.996075\pi$
$347$		−	0.459390i		−	0.0246614i	0.999924	−	0.0123307i	$-0.00392508\pi$
$348$		0			0
$349$	−20.8616			−1.11670			−0.558348	−	0.829607i	$-0.688565\pi$
$349$	−20.8616			−1.11670			−0.558348	+	0.829607i	$0.688565\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			27.4459i			1.46080i	0.683020	+	0.730400i	$0.260666\pi$
$353$			27.4459i			1.46080i	−0.683020	+	0.730400i	$0.739334\pi$
$354$		0			0
$355$			3.17216i			0.168361i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.9440			0.788716			0.394358	−	0.918957i	$-0.370967\pi$
$359$	14.9440			0.788716			0.394358	+	0.918957i	$0.370967\pi$
$360$		0			0
$361$	−6.61358			−0.348083
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.7765			−0.825777
$366$		0			0
$367$		−	10.6253i		−	0.554636i	−0.960778	−	0.277318i	$-0.910554\pi$
$367$		−	10.6253i		−	0.554636i	0.960778	−	0.277318i	$-0.0894456\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			0.417505i			0.0216758i
$372$		0			0
$373$			14.2372i			0.737173i	0.929593	+	0.368587i	$0.120158\pi$
$373$			14.2372i			0.737173i	−0.929593	+	0.368587i	$0.879842\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	11.1971i		−	0.576681i
$378$		0			0
$379$		−	7.81778i		−	0.401572i	−0.979635	−	0.200786i	$-0.935650\pi$
$379$		−	7.81778i		−	0.401572i	0.979635	−	0.200786i	$-0.0643496\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−33.0636			−1.68947			−0.844737	−	0.535182i	$-0.820243\pi$
$383$	−33.0636			−1.68947			−0.844737	+	0.535182i	$0.820243\pi$
$384$		0			0
$385$		−	2.58462i		−	0.131724i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.1348			−0.615259			−0.307630	−	0.951506i	$-0.599536\pi$
$389$	−12.1348			−0.615259			−0.307630	+	0.951506i	$0.599536\pi$
$390$		0			0
$391$	14.2674	+	5.99060i	0.721536	+	0.302957i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	16.1921i		−	0.814711i
$396$		0			0
$397$	−1.46588			−0.0735704			−0.0367852	−	0.999323i	$-0.511712\pi$
$397$	−1.46588			−0.0735704			−0.0367852	+	0.999323i	$0.511712\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.10416			0.155014			0.0775071	−	0.996992i	$-0.475304\pi$
$401$	3.10416			0.155014			0.0775071	+	0.996992i	$0.475304\pi$
$402$		0			0
$403$	11.6987			0.582756
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4.27607i		−	0.211957i
$408$		0			0
$409$	−24.1259			−1.19295			−0.596475	−	0.802632i	$-0.703432\pi$
$409$	−24.1259			−1.19295			−0.596475	+	0.802632i	$0.703432\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.05200			0.445420
$414$		0			0
$415$	−0.583836			−0.0286594
$416$		0			0
$417$		0			0
$418$		0			0
$419$	17.4432			0.852158			0.426079	−	0.904686i	$-0.359895\pi$
$419$	17.4432			0.852158			0.426079	+	0.904686i	$0.359895\pi$
$420$		0			0
$421$			20.3486i			0.991729i	0.868400	+	0.495864i	$0.165149\pi$
$421$			20.3486i			0.991729i	−0.868400	+	0.495864i	$0.834851\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.22657			−0.156512
$426$		0			0
$427$	10.9704			0.530894
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.4974			−1.46901			−0.734505	−	0.678603i	$-0.762586\pi$
$431$	−30.4974			−1.46901			−0.734505	+	0.678603i	$0.762586\pi$
$432$		0			0
$433$		−	19.0844i		−	0.917137i	−0.888659	−	0.458568i	$-0.848362\pi$
$433$		−	19.0844i		−	0.917137i	0.888659	−	0.458568i	$-0.151638\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.39646	−	22.3790i	0.449494	−	1.07053i
$438$		0			0
$439$	29.7420			1.41951			0.709753	−	0.704451i	$-0.248807\pi$
$439$	29.7420			1.41951			0.709753	+	0.704451i	$0.248807\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			11.9964i			0.569965i	0.958533	+	0.284983i	$0.0919878\pi$
$443$			11.9964i			0.569965i	−0.958533	+	0.284983i	$0.908012\pi$
$444$		0			0
$445$	14.7009			0.696890
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.01821i		−	0.0480525i	−0.999711	−	0.0240263i	$-0.992351\pi$
$449$		−	1.01821i		−	0.0480525i	0.999711	−	0.0240263i	$-0.00764853\pi$
$450$		0			0
$451$			16.3233i			0.768635i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			3.73410i			0.175057i
$456$		0			0
$457$		−	26.4418i		−	1.23689i	−0.785826	−	0.618447i	$-0.787762\pi$
$457$		−	26.4418i		−	1.23689i	0.785826	−	0.618447i	$-0.212238\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			35.5680i			1.65657i	0.560309	+	0.828284i	$0.310683\pi$
$461$			35.5680i			1.65657i	−0.560309	+	0.828284i	$0.689317\pi$
$462$		0			0
$463$	23.1095			1.07399			0.536995	−	0.843586i	$-0.319560\pi$
$463$	23.1095			1.07399			0.536995	+	0.843586i	$0.319560\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.60164			−0.259213			−0.129607	−	0.991566i	$-0.541371\pi$
$467$	−5.60164			−0.259213			−0.129607	+	0.991566i	$0.541371\pi$
$468$		0			0
$469$	4.54254			0.209755
$470$		0			0
$471$		0			0
$472$		0			0
$473$			6.69165i			0.307682i
$474$		0			0
$475$			5.06099i			0.232214i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.0470			0.641825			0.320912	−	0.947109i	$-0.396010\pi$
$479$	14.0470			0.641825			0.320912	+	0.947109i	$0.396010\pi$
$480$		0			0
$481$			6.17780i			0.281683i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	1.66138i		−	0.0754392i
$486$		0			0
$487$	−34.3972			−1.55868			−0.779342	−	0.626598i	$-0.784447\pi$
$487$	−34.3972			−1.55868			−0.779342	+	0.626598i	$0.784447\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	39.4728i		−	1.78138i	−0.454610	−	0.890691i	$-0.650221\pi$
$491$		−	39.4728i		−	1.78138i	0.454610	−	0.890691i	$-0.349779\pi$
$492$		0			0
$493$		−	16.8344i		−	0.758182i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.51938			0.247578
$498$		0			0
$499$	19.1017			0.855108			0.427554	−	0.903990i	$-0.359375\pi$
$499$	19.1017			0.855108			0.427554	+	0.903990i	$0.359375\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.1989			0.454749			0.227374	−	0.973807i	$-0.426986\pi$
$503$	10.1989			0.454749			0.227374	+	0.973807i	$0.426986\pi$
$504$		0			0
$505$		−	6.21263i		−	0.276458i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			0.706053i			0.0312953i	0.999878	+	0.0156476i	$0.00498100\pi$
$509$			0.706053i			0.0312953i	−0.999878	+	0.0156476i	$0.995019\pi$
$510$		0			0
$511$			27.4501i			1.21432i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			3.57047i			0.157334i
$516$		0			0
$517$			7.07517i			0.311166i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−38.3159			−1.67865			−0.839325	−	0.543630i	$-0.817049\pi$
$521$	−38.3159			−1.67865			−0.839325	+	0.543630i	$0.817049\pi$
$522$		0			0
$523$			28.1352i			1.23027i	0.788423	+	0.615133i	$0.210898\pi$
$523$			28.1352i			1.23027i	−0.788423	+	0.615133i	$0.789102\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	17.5885			0.766169
$528$		0			0
$529$	16.1057	+	16.4197i	0.700249	+	0.713898i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	23.5829i		−	1.02149i
$534$		0			0
$535$	18.4060			0.795762
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.90114			0.254180
$540$		0			0
$541$	−21.9400			−0.943272			−0.471636	−	0.881793i	$-0.656336\pi$
$541$	−21.9400			−0.943272			−0.471636	+	0.881793i	$0.656336\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			17.6728i			0.757020i
$546$		0			0
$547$	−0.611160			−0.0261313			−0.0130657	−	0.999915i	$-0.504159\pi$
$547$	−0.611160			−0.0261313			−0.0130657	+	0.999915i	$0.504159\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−26.4053			−1.12490
$552$		0			0
$553$	−28.1733			−1.19805
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−24.5939			−1.04208			−0.521039	−	0.853533i	$-0.674455\pi$
$557$	−24.5939			−1.04208			−0.521039	+	0.853533i	$0.674455\pi$
$558$		0			0
$559$		−	9.66767i		−	0.408899i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5.35384			0.225637			0.112819	−	0.993616i	$-0.464012\pi$
$563$	5.35384			0.225637			0.112819	+	0.993616i	$0.464012\pi$
$564$		0			0
$565$	11.9644			0.503345
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.18277			0.133429			0.0667144	−	0.997772i	$-0.478748\pi$
$569$	3.18277			0.133429			0.0667144	+	0.997772i	$0.478748\pi$
$570$		0			0
$571$		−	20.9849i		−	0.878191i	−0.898440	−	0.439096i	$-0.855299\pi$
$571$		−	20.9849i		−	0.878191i	0.898440	−	0.439096i	$-0.144701\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.42186	−	1.85665i	−0.184404	−	0.0774275i
$576$		0			0
$577$	−20.1387			−0.838384			−0.419192	−	0.907897i	$-0.637687\pi$
$577$	−20.1387			−0.838384			−0.419192	+	0.907897i	$0.637687\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			1.01584i			0.0421442i
$582$		0			0
$583$	−0.356442			−0.0147623
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	45.4769i		−	1.87703i	−0.345237	−	0.938516i	$-0.612201\pi$
$587$		−	45.4769i		−	1.87703i	0.345237	−	0.938516i	$-0.387799\pi$
$588$		0			0
$589$		−	27.5882i		−	1.13675i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	24.0282i		−	0.986722i	−0.869825	−	0.493361i	$-0.835768\pi$
$593$		−	24.0282i		−	0.986722i	0.869825	−	0.493361i	$-0.164232\pi$
$594$		0			0
$595$			5.61405i			0.230154i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	2.54733i		−	0.104081i	−0.998645	−	0.0520406i	$-0.983427\pi$
$599$		−	2.54733i		−	0.104081i	0.998645	−	0.0520406i	$-0.0165725\pi$
$600$		0			0
$601$	12.7927			0.521825			0.260913	−	0.965362i	$-0.415977\pi$
$601$	12.7927			0.521825			0.260913	+	0.965362i	$0.415977\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8.79340			−0.357503
$606$		0			0
$607$	22.4815			0.912497			0.456248	−	0.889852i	$-0.349193\pi$
$607$	22.4815			0.912497			0.456248	+	0.889852i	$0.349193\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	10.2218i		−	0.413528i
$612$		0			0
$613$		−	1.99075i		−	0.0804055i	−0.999192	−	0.0402027i	$-0.987200\pi$
$613$		−	1.99075i		−	0.0804055i	0.999192	−	0.0402027i	$-0.0128004\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	46.5579			1.87435			0.937175	−	0.348860i	$-0.113431\pi$
$617$	46.5579			1.87435			0.937175	+	0.348860i	$0.113431\pi$
$618$		0			0
$619$		−	25.2164i		−	1.01353i	−0.862084	−	0.506766i	$-0.830841\pi$
$619$		−	25.2164i		−	1.01353i	0.862084	−	0.506766i	$-0.169159\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	25.5787i		−	1.02479i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			9.28804i			0.370338i
$630$		0			0
$631$			49.3547i			1.96478i	0.186840	+	0.982390i	$0.440175\pi$
$631$			49.3547i			1.96478i	−0.186840	+	0.982390i	$0.559825\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−15.2155			−0.603810
$636$		0			0
$637$	−8.52560			−0.337797
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.85956			0.310434			0.155217	−	0.987880i	$-0.450392\pi$
$641$	7.85956			0.310434			0.155217	+	0.987880i	$0.450392\pi$
$642$		0			0
$643$			8.93166i			0.352230i	0.984370	+	0.176115i	$0.0563531\pi$
$643$			8.93166i			0.352230i	−0.984370	+	0.176115i	$0.943647\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			3.29757i			0.129641i	0.997897	+	0.0648204i	$0.0206474\pi$
$647$			3.29757i			0.129641i	−0.997897	+	0.0648204i	$0.979353\pi$
$648$		0			0
$649$			7.72807i			0.303353i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	33.1565i		−	1.29752i	−0.760995	−	0.648758i	$-0.775289\pi$
$653$		−	33.1565i		−	1.29752i	0.760995	−	0.648758i	$-0.224711\pi$
$654$		0			0
$655$			21.1936i			0.828103i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.27163			0.127444			0.0637222	−	0.997968i	$-0.479703\pi$
$659$	3.27163			0.127444			0.0637222	+	0.997968i	$0.479703\pi$
$660$		0			0
$661$			46.2040i			1.79713i	0.438844	+	0.898563i	$0.355388\pi$
$661$			46.2040i			1.79713i	−0.438844	+	0.898563i	$0.644612\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.80583			0.341475
$666$		0			0
$667$	9.68690	−	23.0707i	0.375078	−	0.893302i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			9.36587i			0.361565i
$672$		0			0
$673$	42.4260			1.63540			0.817701	−	0.575643i	$-0.195248\pi$
$673$	42.4260			1.63540			0.817701	+	0.575643i	$0.195248\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	30.8444			1.18545			0.592723	−	0.805407i	$-0.298053\pi$
$677$	30.8444			1.18545			0.592723	+	0.805407i	$0.298053\pi$
$678$		0			0
$679$	−2.89070			−0.110935
$680$		0			0
$681$		0			0
$682$		0			0
$683$			12.4892i			0.477887i	0.971033	+	0.238944i	$0.0768011\pi$
$683$			12.4892i			0.477887i	−0.971033	+	0.238944i	$0.923199\pi$
$684$		0			0
$685$	19.3615			0.739767
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.514965			0.0196186
$690$		0			0
$691$	28.2420			1.07438			0.537189	−	0.843462i	$-0.319486\pi$
$691$	28.2420			1.07438			0.537189	+	0.843462i	$0.319486\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.3341			−0.467859
$696$		0			0
$697$		−	35.4558i		−	1.34299i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−14.8720			−0.561708			−0.280854	−	0.959751i	$-0.590618\pi$
$701$	−14.8720			−0.561708			−0.280854	+	0.959751i	$0.590618\pi$
$702$		0			0
$703$	14.5686			0.549466
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−10.8096			−0.406538
$708$		0			0
$709$		−	24.4780i		−	0.919290i	−0.888103	−	0.459645i	$-0.847977\pi$
$709$		−	24.4780i		−	0.919290i	0.888103	−	0.459645i	$-0.152023\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	24.1043	+	10.1209i	0.902712	+	0.379029i
$714$		0			0
$715$	−3.18795			−0.119223
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	14.4680i		−	0.539566i	−0.962921	−	0.269783i	$-0.913048\pi$
$719$		−	14.4680i		−	0.539566i	0.962921	−	0.269783i	$-0.0869520\pi$
$720$		0			0
$721$	6.21242			0.231362
$722$		0			0
$723$		0			0
$724$		0			0
$725$			5.21742i			0.193770i
$726$		0			0
$727$		−	36.4058i		−	1.35022i	−0.737718	−	0.675109i	$-0.764097\pi$
$727$		−	36.4058i		−	1.35022i	0.737718	−	0.675109i	$-0.235903\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	14.5349i		−	0.537593i
$732$		0			0
$733$		−	39.6907i		−	1.46601i	−0.680223	−	0.733005i	$-0.738117\pi$
$733$		−	39.6907i		−	1.46601i	0.680223	−	0.733005i	$-0.261883\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			3.87815i			0.142854i
$738$		0			0
$739$	−26.6534			−0.980462			−0.490231	−	0.871592i	$-0.663088\pi$
$739$	−26.6534			−0.980462			−0.490231	+	0.871592i	$0.663088\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4.33994			0.159217			0.0796085	−	0.996826i	$-0.474633\pi$
$743$	4.33994			0.159217			0.0796085	+	0.996826i	$0.474633\pi$
$744$		0			0
$745$	−12.5487			−0.459749
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	32.0255i		−	1.17018i
$750$		0			0
$751$			22.0717i			0.805408i	0.915330	+	0.402704i	$0.131930\pi$
$751$			22.0717i			0.805408i	−0.915330	+	0.402704i	$0.868070\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	17.3650			0.631975
$756$		0			0
$757$		−	28.4839i		−	1.03527i	−0.855603	−	0.517633i	$-0.826813\pi$
$757$		−	28.4839i		−	1.03527i	0.855603	−	0.517633i	$-0.173187\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			29.8896i			1.08350i	0.840540	+	0.541749i	$0.182238\pi$
$761$			29.8896i			1.08350i	−0.840540	+	0.541749i	$0.817762\pi$
$762$		0			0
$763$	30.7497			1.11321
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	11.1650i		−	0.403146i
$768$		0			0
$769$		−	1.87428i		−	0.0675881i	−0.999429	−	0.0337941i	$-0.989241\pi$
$769$		−	1.87428i		−	0.0675881i	0.999429	−	0.0337941i	$-0.0107590\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.80634			0.244807			0.122404	−	0.992480i	$-0.460940\pi$
$773$	6.80634			0.244807			0.122404	+	0.992480i	$0.460940\pi$
$774$		0			0
$775$	−5.45116			−0.195811
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−55.6137			−1.99257
$780$		0			0
$781$			4.71212i			0.168613i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	17.5785i		−	0.627403i
$786$		0			0
$787$		−	33.9079i		−	1.20869i	−0.796724	−	0.604344i	$-0.793435\pi$
$787$		−	33.9079i		−	1.20869i	0.796724	−	0.604344i	$-0.206565\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	20.8173i		−	0.740179i
$792$		0			0
$793$		−	13.5312i		−	0.480508i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.345368			−0.0122336			−0.00611678	−	0.999981i	$-0.501947\pi$
$797$	−0.345368			−0.0122336			−0.00611678	+	0.999981i	$0.501947\pi$
$798$		0			0
$799$		−	15.3680i		−	0.543679i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−23.4353			−0.827014
$804$		0			0
$805$	−3.23046	+	7.69379i	−0.113859	+	0.271170i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			8.81115i			0.309784i	0.987931	+	0.154892i	$0.0495029\pi$
$809$			8.81115i			0.309784i	−0.987931	+	0.154892i	$0.950497\pi$
$810$		0			0
$811$	9.96662			0.349975			0.174988	−	0.984571i	$-0.444011\pi$
$811$	9.96662			0.349975			0.174988	+	0.984571i	$0.444011\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	15.0296			0.526466
$816$		0			0
$817$	−22.7985			−0.797619
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	28.9308i		−	1.00969i	−0.863210	−	0.504845i	$-0.831550\pi$
$821$		−	28.9308i		−	1.00969i	0.863210	−	0.504845i	$-0.168450\pi$
$822$		0			0
$823$	33.4278			1.16522			0.582610	−	0.812752i	$-0.302032\pi$
$823$	33.4278			1.16522			0.582610	+	0.812752i	$0.302032\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11.3409			−0.394363			−0.197182	−	0.980367i	$-0.563179\pi$
$827$	−11.3409			−0.394363			−0.197182	+	0.980367i	$0.563179\pi$
$828$		0			0
$829$	0.900479			0.0312749			0.0156375	−	0.999878i	$-0.495022\pi$
$829$	0.900479			0.0312749			0.0156375	+	0.999878i	$0.495022\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.8179			−0.444113
$834$		0			0
$835$		−	11.4559i		−	0.396448i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−36.8397			−1.27185			−0.635924	−	0.771751i	$-0.719381\pi$
$839$	−36.8397			−1.27185			−0.635924	+	0.771751i	$0.719381\pi$
$840$		0			0
$841$	1.77850			0.0613275
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.39424			−0.288771
$846$		0			0
$847$			15.3000i			0.525715i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.34456	+	12.7288i	−0.183209	+	0.436338i
$852$		0			0
$853$	−16.7815			−0.574588			−0.287294	−	0.957843i	$-0.592756\pi$
$853$	−16.7815			−0.574588			−0.287294	+	0.957843i	$0.592756\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			22.6280i			0.772957i	0.922298	+	0.386479i	$0.126309\pi$
$857$			22.6280i			0.772957i	−0.922298	+	0.386479i	$0.873691\pi$
$858$		0			0
$859$	−0.122682			−0.00418587			−0.00209293	−	0.999998i	$-0.500666\pi$
$859$	−0.122682			−0.00418587			−0.00209293	+	0.999998i	$0.500666\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	12.3766i		−	0.421306i	−0.977561	−	0.210653i	$-0.932441\pi$
$863$		−	12.3766i		−	0.421306i	0.977561	−	0.210653i	$-0.0675590\pi$
$864$		0			0
$865$			7.36507i			0.250420i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	24.0527i		−	0.815932i
$870$		0			0
$871$		−	5.60291i		−	0.189847i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	1.73994i		−	0.0588208i
$876$		0			0
$877$	−49.1426			−1.65943			−0.829714	−	0.558188i	$-0.811497\pi$
$877$	−49.1426			−1.65943			−0.829714	+	0.558188i	$0.811497\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	10.8206			0.364554			0.182277	−	0.983247i	$-0.441653\pi$
$881$	10.8206			0.364554			0.182277	+	0.983247i	$0.441653\pi$
$882$		0			0
$883$	−40.8514			−1.37476			−0.687380	−	0.726298i	$-0.741239\pi$
$883$	−40.8514			−1.37476			−0.687380	+	0.726298i	$0.741239\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	45.5013i		−	1.52779i	−0.645343	−	0.763893i	$-0.723286\pi$
$887$		−	45.5013i		−	1.52779i	0.645343	−	0.763893i	$-0.276714\pi$
$888$		0			0
$889$			26.4742i			0.887916i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−24.1052			−0.806649
$894$		0			0
$895$		−	13.0710i		−	0.436916i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	28.4410i		−	0.948560i
$900$		0			0
$901$	0.774227			0.0257932
$902$		0			0
$903$		0			0
$904$		0			0
$905$			19.0486i			0.633198i
$906$		0			0
$907$		−	40.4240i		−	1.34226i	−0.741341	−	0.671129i	$-0.765810\pi$
$907$		−	40.4240i		−	1.34226i	0.741341	−	0.671129i	$-0.234190\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−14.5112			−0.480776			−0.240388	−	0.970677i	$-0.577275\pi$
$911$	−14.5112			−0.480776			−0.240388	+	0.970677i	$0.577275\pi$
$912$		0			0
$913$	−0.867265			−0.0287023
$914$		0			0
$915$		0			0
$916$		0			0
$917$	36.8757			1.21774
$918$		0			0
$919$		−	57.0808i		−	1.88292i	−0.337124	−	0.941460i	$-0.609454\pi$
$919$		−	57.0808i		−	1.88292i	0.337124	−	0.941460i	$-0.390546\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	6.80778i		−	0.224081i
$924$		0			0
$925$		−	2.87861i		−	0.0946482i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	7.48787i		−	0.245669i	−0.992427	−	0.122835i	$-0.960802\pi$
$929$		−	7.48787i		−	0.245669i	0.992427	−	0.122835i	$-0.0391984\pi$
$930$		0			0
$931$			20.1053i			0.658923i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.79295			−0.156746
$936$		0			0
$937$			5.76692i			0.188397i	0.995553	+	0.0941986i	$0.0300289\pi$
$937$			5.76692i			0.188397i	−0.995553	+	0.0941986i	$0.969971\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−22.0222			−0.717902			−0.358951	−	0.933356i	$-0.616866\pi$
$941$	−22.0222			−0.717902			−0.358951	+	0.933356i	$0.616866\pi$
$942$		0			0
$943$	20.4021	−	48.5906i	0.664385	−	1.58233i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	40.0318i		−	1.30086i	−0.759567	−	0.650429i	$-0.774589\pi$
$947$		−	40.0318i		−	1.30086i	0.759567	−	0.650429i	$-0.225411\pi$
$948$		0			0
$949$	33.8579			1.09907
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−28.9291			−0.937105			−0.468553	−	0.883436i	$-0.655224\pi$
$953$	−28.9291			−0.937105			−0.468553	+	0.883436i	$0.655224\pi$
$954$		0			0
$955$	−4.27526			−0.138344
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	33.6880i		−	1.08784i
$960$		0			0
$961$	−1.28490			−0.0414484
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.8399			−0.606478
$966$		0			0
$967$	47.3299			1.52203			0.761014	−	0.648736i	$-0.224702\pi$
$967$	47.3299			1.52203			0.761014	+	0.648736i	$0.224702\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−38.0244			−1.22026			−0.610131	−	0.792301i	$-0.708883\pi$
$971$	−38.0244			−1.22026			−0.610131	+	0.792301i	$0.708883\pi$
$972$		0			0
$973$			21.4606i			0.687997i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.4601			0.654575			0.327288	−	0.944925i	$-0.393865\pi$
$977$	20.4601			0.654575			0.327288	+	0.944925i	$0.393865\pi$
$978$		0			0
$979$	21.8376			0.697934
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−20.7804			−0.662792			−0.331396	−	0.943492i	$-0.607520\pi$
$983$	−20.7804			−0.662792			−0.331396	+	0.943492i	$0.607520\pi$
$984$		0			0
$985$			26.8545i			0.855656i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.36374	−	19.9194i	0.265951	−	0.633401i
$990$		0			0
$991$	9.91895			0.315086			0.157543	−	0.987512i	$-0.449643\pi$
$991$	9.91895			0.315086			0.157543	+	0.987512i	$0.449643\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	10.8944i		−	0.345375i
$996$		0			0
$997$	34.8855			1.10484			0.552418	−	0.833568i	$-0.313705\pi$
$997$	34.8855			1.10484			0.552418	+	0.833568i	$0.313705\pi$
$998$		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.1241.5	yes	16
3.2	odd	2		4140.2.i.a.1241.5	✓	16
23.22	odd	2		4140.2.i.a.1241.12	yes	16
69.68	even	2	inner	4140.2.i.b.1241.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.i.a.1241.5	✓	16	3.2	odd	2
4140.2.i.a.1241.12	yes	16	23.22	odd	2
4140.2.i.b.1241.5	yes	16	1.1	even	1	trivial
4140.2.i.b.1241.12	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.73994i q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -1.73994i q^{7} +1.48546 q^{11} -2.14610 q^{13} -3.22657 q^{17} +5.06099i q^{19} +(-4.42186 - 1.85665i) q^{23} +1.00000 q^{25} +5.21742i q^{29} -5.45116 q^{31} -1.73994i q^{35} -2.87861i q^{37} +10.9887i q^{41} +4.50476i q^{43} +4.76294i q^{47} +3.97260 q^{49} -0.239954 q^{53} +1.48546 q^{55} +5.20247i q^{59} +6.30502i q^{61} -2.14610 q^{65} +2.61074i q^{67} +3.17216i q^{71} -15.7765 q^{73} -2.58462i q^{77} -16.1921i q^{79} -0.583836 q^{83} -3.22657 q^{85} +14.7009 q^{89} +3.73410i q^{91} +5.06099i q^{95} -1.66138i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5}+O(q^{10}) \) 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

Related objects

Downloads

Learn more