LMFDB - Embedded newform 2240.2.b.d.1121.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1121,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2240.1121
Dual form			2240.2.b.d.1121.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -1.00000i q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.73205i q^{3} -1.00000i q^{5} +1.00000 q^{7} +1.00000i q^{11} -1.00000i q^{13} -1.73205 q^{15} -5.73205 q^{17} +2.00000i q^{19} -1.73205i q^{21} -3.46410 q^{23} -1.00000 q^{25} -5.19615i q^{27} -9.19615i q^{29} +4.00000 q^{31} +1.73205 q^{33} -1.00000i q^{35} -7.46410i q^{37} -1.73205 q^{39} -10.9282 q^{41} -3.46410i q^{43} -3.92820 q^{47} +1.00000 q^{49} +9.92820i q^{51} -5.46410i q^{53} +1.00000 q^{55} +3.46410 q^{57} +2.53590i q^{59} -5.46410i q^{61} -1.00000 q^{65} +1.46410i q^{67} +6.00000i q^{69} +0.535898 q^{71} +4.00000 q^{73} +1.73205i q^{75} +1.00000i q^{77} +5.73205 q^{79} -9.00000 q^{81} +15.4641i q^{83} +5.73205i q^{85} -15.9282 q^{87} +9.46410 q^{89} -1.00000i q^{91} -6.92820i q^{93} +2.00000 q^{95} -14.2679 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} - 16 q^{17} - 4 q^{25} + 16 q^{31} - 16 q^{41} + 12 q^{47} + 4 q^{49} + 4 q^{55} - 4 q^{65} + 16 q^{71} + 16 q^{73} + 16 q^{79} - 36 q^{81} - 36 q^{87} + 24 q^{89} + 8 q^{95} - 64 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 16 * q^17 - 4 * q^25 + 16 * q^31 - 16 * q^41 + 12 * q^47 + 4 * q^49 + 4 * q^55 - 4 * q^65 + 16 * q^71 + 16 * q^73 + 16 * q^79 - 36 * q^81 - 36 * q^87 + 24 * q^89 + 8 * q^95 - 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	− 1.73205i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	− 1.73205i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000i	0.301511i	0.988571	+	0.150756i	$0.0481707\pi$
$11$	1.00000i	0.301511i	−0.988571	+	0.150756i	$0.951829\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	0	0
$15$	−1.73205	−0.447214
$16$	0	0
$17$	−5.73205	−1.39023	−0.695113	−	0.718900i	$-0.744646\pi$
$17$	−5.73205	−1.39023	−0.695113	+	0.718900i	$0.744646\pi$
$18$	0	0
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	0	0
$21$	− 1.73205i	− 0.377964i
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 5.19615i	− 1.00000i
$28$	0	0
$29$	− 9.19615i	− 1.70768i	−0.520533	−	0.853841i	$-0.674267\pi$
$29$	− 9.19615i	− 1.70768i	0.520533	−	0.853841i	$-0.325733\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	1.73205	0.301511
$34$	0	0
$35$	− 1.00000i	− 0.169031i
$36$	0	0
$37$	− 7.46410i	− 1.22709i	−0.789659	−	0.613545i	$-0.789743\pi$
$37$	− 7.46410i	− 1.22709i	0.789659	−	0.613545i	$-0.210257\pi$
$38$	0	0
$39$	−1.73205	−0.277350
$40$	0	0
$41$	−10.9282	−1.70670	−0.853349	−	0.521340i	$-0.825432\pi$
$41$	−10.9282	−1.70670	−0.853349	+	0.521340i	$0.825432\pi$
$42$	0	0
$43$	− 3.46410i	− 0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$	− 3.46410i	− 0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.92820	−0.572987	−0.286494	−	0.958082i	$-0.592490\pi$
$47$	−3.92820	−0.572987	−0.286494	+	0.958082i	$0.592490\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.92820i	1.39023i
$52$	0	0
$53$	− 5.46410i	− 0.750552i	−0.926913	−	0.375276i	$-0.877548\pi$
$53$	− 5.46410i	− 0.750552i	0.926913	−	0.375276i	$-0.122452\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	3.46410	0.458831
$58$	0	0
$59$	2.53590i	0.330146i	0.986281	+	0.165073i	$0.0527859\pi$
$59$	2.53590i	0.330146i	−0.986281	+	0.165073i	$0.947214\pi$
$60$	0	0
$61$	− 5.46410i	− 0.699607i	−0.936823	−	0.349803i	$-0.886248\pi$
$61$	− 5.46410i	− 0.699607i	0.936823	−	0.349803i	$-0.113752\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	1.46410i	0.178868i	0.995993	+	0.0894342i	$0.0285059\pi$
$67$	1.46410i	0.178868i	−0.995993	+	0.0894342i	$0.971494\pi$
$68$	0	0
$69$	6.00000i	0.722315i
$70$	0	0
$71$	0.535898	0.0635994	0.0317997	−	0.999494i	$-0.489876\pi$
$71$	0.535898	0.0635994	0.0317997	+	0.999494i	$0.489876\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	1.73205i	0.200000i
$76$	0	0
$77$	1.00000i	0.113961i
$78$	0	0
$79$	5.73205	0.644906	0.322453	−	0.946585i	$-0.395493\pi$
$79$	5.73205	0.644906	0.322453	+	0.946585i	$0.395493\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	15.4641i	1.69741i	0.528870	+	0.848703i	$0.322616\pi$
$83$	15.4641i	1.69741i	−0.528870	+	0.848703i	$0.677384\pi$
$84$	0	0
$85$	5.73205i	0.621728i
$86$	0	0
$87$	−15.9282	−1.70768
$88$	0	0
$89$	9.46410	1.00319	0.501596	−	0.865102i	$-0.332746\pi$
$89$	9.46410	1.00319	0.501596	+	0.865102i	$0.332746\pi$
$90$	0	0
$91$	− 1.00000i	− 0.104828i
$92$	0	0
$93$	− 6.92820i	− 0.718421i
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−14.2679	−1.44869	−0.724345	−	0.689437i	$-0.757858\pi$
$97$	−14.2679	−1.44869	−0.724345	+	0.689437i	$0.757858\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 6.39230i	− 0.636058i	−0.948081	−	0.318029i	$-0.896979\pi$
$101$	− 6.39230i	− 0.636058i	0.948081	−	0.318029i	$-0.103021\pi$
$102$	0	0
$103$	−5.92820	−0.584123	−0.292062	−	0.956400i	$-0.594341\pi$
$103$	−5.92820	−0.584123	−0.292062	+	0.956400i	$0.594341\pi$
$104$	0	0
$105$	−1.73205	−0.169031
$106$	0	0
$107$	− 16.9282i	− 1.63651i	−0.574855	−	0.818256i	$-0.694941\pi$
$107$	− 16.9282i	− 1.63651i	0.574855	−	0.818256i	$-0.305059\pi$
$108$	0	0
$109$	− 13.1962i	− 1.26396i	−0.774984	−	0.631981i	$-0.782242\pi$
$109$	− 13.1962i	− 1.26396i	0.774984	−	0.631981i	$-0.217758\pi$
$110$	0	0
$111$	−12.9282	−1.22709
$112$	0	0
$113$	−8.92820	−0.839895	−0.419947	−	0.907548i	$-0.637951\pi$
$113$	−8.92820	−0.839895	−0.419947	+	0.907548i	$0.637951\pi$
$114$	0	0
$115$	3.46410i	0.323029i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.73205	−0.525456
$120$	0	0
$121$	10.0000	0.909091
$122$	0	0
$123$	18.9282i	1.70670i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−20.3923	−1.80952	−0.904762	−	0.425917i	$-0.859952\pi$
$127$	−20.3923	−1.80952	−0.904762	+	0.425917i	$0.859952\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	10.3923i	0.907980i	0.891007	+	0.453990i	$0.150000\pi$
$131$	10.3923i	0.907980i	−0.891007	+	0.453990i	$0.850000\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	0	0
$135$	−5.19615	−0.447214
$136$	0	0
$137$	4.39230	0.375260	0.187630	−	0.982240i	$-0.439919\pi$
$137$	4.39230	0.375260	0.187630	+	0.982240i	$0.439919\pi$
$138$	0	0
$139$	7.07180i	0.599822i	0.953967	+	0.299911i	$0.0969570\pi$
$139$	7.07180i	0.599822i	−0.953967	+	0.299911i	$0.903043\pi$
$140$	0	0
$141$	6.80385i	0.572987i
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−9.19615	−0.763699
$146$	0	0
$147$	− 1.73205i	− 0.142857i
$148$	0	0
$149$	− 5.85641i	− 0.479776i	−0.970801	−	0.239888i	$-0.922889\pi$
$149$	− 5.85641i	− 0.479776i	0.970801	−	0.239888i	$-0.0771106\pi$
$150$	0	0
$151$	16.6603	1.35579	0.677896	−	0.735158i	$-0.262892\pi$
$151$	16.6603	1.35579	0.677896	+	0.735158i	$0.262892\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4.00000i	− 0.321288i
$156$	0	0
$157$	8.92820i	0.712548i	0.934381	+	0.356274i	$0.115953\pi$
$157$	8.92820i	0.712548i	−0.934381	+	0.356274i	$0.884047\pi$
$158$	0	0
$159$	−9.46410	−0.750552
$160$	0	0
$161$	−3.46410	−0.273009
$162$	0	0
$163$	− 4.53590i	− 0.355279i	−0.984096	−	0.177639i	$-0.943154\pi$
$163$	− 4.53590i	− 0.355279i	0.984096	−	0.177639i	$-0.0568461\pi$
$164$	0	0
$165$	− 1.73205i	− 0.134840i
$166$	0	0
$167$	5.92820	0.458738	0.229369	−	0.973340i	$-0.426334\pi$
$167$	5.92820	0.458738	0.229369	+	0.973340i	$0.426334\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.00000i	0.380143i	0.981770	+	0.190071i	$0.0608720\pi$
$173$	5.00000i	0.380143i	−0.981770	+	0.190071i	$0.939128\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	4.39230	0.330146
$178$	0	0
$179$	14.9282i	1.11579i	0.829913	+	0.557893i	$0.188390\pi$
$179$	14.9282i	1.11579i	−0.829913	+	0.557893i	$0.811610\pi$
$180$	0	0
$181$	16.3923i	1.21843i	0.793005	+	0.609215i	$0.208515\pi$
$181$	16.3923i	1.21843i	−0.793005	+	0.609215i	$0.791485\pi$
$182$	0	0
$183$	−9.46410	−0.699607
$184$	0	0
$185$	−7.46410	−0.548772
$186$	0	0
$187$	− 5.73205i	− 0.419169i
$188$	0	0
$189$	− 5.19615i	− 0.377964i
$190$	0	0
$191$	−16.1244	−1.16672	−0.583359	−	0.812215i	$-0.698262\pi$
$191$	−16.1244	−1.16672	−0.583359	+	0.812215i	$0.698262\pi$
$192$	0	0
$193$	11.3205	0.814868	0.407434	−	0.913235i	$-0.366424\pi$
$193$	11.3205	0.814868	0.407434	+	0.913235i	$0.366424\pi$
$194$	0	0
$195$	1.73205i	0.124035i
$196$	0	0
$197$	− 8.00000i	− 0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$	− 8.00000i	− 0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$	0	0
$199$	−10.5359	−0.746870	−0.373435	−	0.927656i	$-0.621820\pi$
$199$	−10.5359	−0.746870	−0.373435	+	0.927656i	$0.621820\pi$
$200$	0	0
$201$	2.53590	0.178868
$202$	0	0
$203$	− 9.19615i	− 0.645443i
$204$	0	0
$205$	10.9282i	0.763259i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	1.00000i	0.0688428i	0.999407	+	0.0344214i	$0.0109588\pi$
$211$	1.00000i	0.0688428i	−0.999407	+	0.0344214i	$0.989041\pi$
$212$	0	0
$213$	− 0.928203i	− 0.0635994i
$214$	0	0
$215$	−3.46410	−0.236250
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	− 6.92820i	− 0.468165i
$220$	0	0
$221$	5.73205i	0.385579i
$222$	0	0
$223$	8.85641	0.593069	0.296534	−	0.955022i	$-0.404169\pi$
$223$	8.85641	0.593069	0.296534	+	0.955022i	$0.404169\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.1244i	− 1.07021i	−0.844785	−	0.535106i	$-0.820272\pi$
$227$	− 16.1244i	− 1.07021i	0.844785	−	0.535106i	$-0.179728\pi$
$228$	0	0
$229$	− 8.00000i	− 0.528655i	−0.964433	−	0.264327i	$-0.914850\pi$
$229$	− 8.00000i	− 0.528655i	0.964433	−	0.264327i	$-0.0851500\pi$
$230$	0	0
$231$	1.73205	0.113961
$232$	0	0
$233$	−2.92820	−0.191833	−0.0959165	−	0.995389i	$-0.530578\pi$
$233$	−2.92820	−0.191833	−0.0959165	+	0.995389i	$0.530578\pi$
$234$	0	0
$235$	3.92820i	0.256248i
$236$	0	0
$237$	− 9.92820i	− 0.644906i
$238$	0	0
$239$	1.73205	0.112037	0.0560185	−	0.998430i	$-0.482159\pi$
$239$	1.73205	0.112037	0.0560185	+	0.998430i	$0.482159\pi$
$240$	0	0
$241$	−25.3205	−1.63104	−0.815519	−	0.578731i	$-0.803548\pi$
$241$	−25.3205	−1.63104	−0.815519	+	0.578731i	$0.803548\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.00000i	− 0.0638877i
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	26.7846	1.69741
$250$	0	0
$251$	24.9282i	1.57345i	0.617301	+	0.786727i	$0.288226\pi$
$251$	24.9282i	1.57345i	−0.617301	+	0.786727i	$0.711774\pi$
$252$	0	0
$253$	− 3.46410i	− 0.217786i
$254$	0	0
$255$	9.92820	0.621728
$256$	0	0
$257$	20.7846	1.29651	0.648254	−	0.761424i	$-0.275499\pi$
$257$	20.7846	1.29651	0.648254	+	0.761424i	$0.275499\pi$
$258$	0	0
$259$	− 7.46410i	− 0.463797i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.53590	0.526346	0.263173	−	0.964749i	$-0.415231\pi$
$263$	8.53590	0.526346	0.263173	+	0.964749i	$0.415231\pi$
$264$	0	0
$265$	−5.46410	−0.335657
$266$	0	0
$267$	− 16.3923i	− 1.00319i
$268$	0	0
$269$	− 8.53590i	− 0.520443i	−0.965549	−	0.260221i	$-0.916204\pi$
$269$	− 8.53590i	− 0.520443i	0.965549	−	0.260221i	$-0.0837956\pi$
$270$	0	0
$271$	21.4641	1.30385	0.651926	−	0.758283i	$-0.273961\pi$
$271$	21.4641	1.30385	0.651926	+	0.758283i	$0.273961\pi$
$272$	0	0
$273$	−1.73205	−0.104828
$274$	0	0
$275$	− 1.00000i	− 0.0603023i
$276$	0	0
$277$	− 0.143594i	− 0.00862770i	−0.999991	−	0.00431385i	$-0.998627\pi$
$277$	− 0.143594i	− 0.00862770i	0.999991	−	0.00431385i	$-0.00137315\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.7846	−0.822321	−0.411160	−	0.911563i	$-0.634876\pi$
$281$	−13.7846	−0.822321	−0.411160	+	0.911563i	$0.634876\pi$
$282$	0	0
$283$	− 8.12436i	− 0.482943i	−0.970408	−	0.241471i	$-0.922370\pi$
$283$	− 8.12436i	− 0.482943i	0.970408	−	0.241471i	$-0.0776300\pi$
$284$	0	0
$285$	− 3.46410i	− 0.205196i
$286$	0	0
$287$	−10.9282	−0.645071
$288$	0	0
$289$	15.8564	0.932730
$290$	0	0
$291$	24.7128i	1.44869i
$292$	0	0
$293$	6.85641i	0.400556i	0.979739	+	0.200278i	$0.0641845\pi$
$293$	6.85641i	0.400556i	−0.979739	+	0.200278i	$0.935816\pi$
$294$	0	0
$295$	2.53590	0.147646
$296$	0	0
$297$	5.19615	0.301511
$298$	0	0
$299$	3.46410i	0.200334i
$300$	0	0
$301$	− 3.46410i	− 0.199667i
$302$	0	0
$303$	−11.0718	−0.636058
$304$	0	0
$305$	−5.46410	−0.312874
$306$	0	0
$307$	− 19.5885i	− 1.11797i	−0.829177	−	0.558986i	$-0.811190\pi$
$307$	− 19.5885i	− 1.11797i	0.829177	−	0.558986i	$-0.188810\pi$
$308$	0	0
$309$	10.2679i	0.584123i
$310$	0	0
$311$	14.3923	0.816113	0.408056	−	0.912957i	$-0.366207\pi$
$311$	14.3923	0.816113	0.408056	+	0.912957i	$0.366207\pi$
$312$	0	0
$313$	27.5885	1.55939	0.779696	−	0.626158i	$-0.215374\pi$
$313$	27.5885	1.55939	0.779696	+	0.626158i	$0.215374\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4.53590i	− 0.254761i	−0.991854	−	0.127381i	$-0.959343\pi$
$317$	− 4.53590i	− 0.254761i	0.991854	−	0.127381i	$-0.0406570\pi$
$318$	0	0
$319$	9.19615	0.514886
$320$	0	0
$321$	−29.3205	−1.63651
$322$	0	0
$323$	− 11.4641i	− 0.637880i
$324$	0	0
$325$	1.00000i	0.0554700i
$326$	0	0
$327$	−22.8564	−1.26396
$328$	0	0
$329$	−3.92820	−0.216569
$330$	0	0
$331$	12.7846i	0.702706i	0.936243	+	0.351353i	$0.114278\pi$
$331$	12.7846i	0.702706i	−0.936243	+	0.351353i	$0.885722\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.46410	0.0799924
$336$	0	0
$337$	−10.3923	−0.566105	−0.283052	−	0.959104i	$-0.591347\pi$
$337$	−10.3923	−0.566105	−0.283052	+	0.959104i	$0.591347\pi$
$338$	0	0
$339$	15.4641i	0.839895i
$340$	0	0
$341$	4.00000i	0.216612i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	− 2.00000i	− 0.107366i	−0.998558	−	0.0536828i	$-0.982904\pi$
$347$	− 2.00000i	− 0.107366i	0.998558	−	0.0536828i	$-0.0170960\pi$
$348$	0	0
$349$	17.3205i	0.927146i	0.886059	+	0.463573i	$0.153433\pi$
$349$	17.3205i	0.927146i	−0.886059	+	0.463573i	$0.846567\pi$
$350$	0	0
$351$	−5.19615	−0.277350
$352$	0	0
$353$	−9.19615	−0.489462	−0.244731	−	0.969591i	$-0.578700\pi$
$353$	−9.19615	−0.489462	−0.244731	+	0.969591i	$0.578700\pi$
$354$	0	0
$355$	− 0.535898i	− 0.0284425i
$356$	0	0
$357$	9.92820i	0.525456i
$358$	0	0
$359$	34.3923	1.81516	0.907578	−	0.419883i	$-0.137929\pi$
$359$	34.3923	1.81516	0.907578	+	0.419883i	$0.137929\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	− 17.3205i	− 0.909091i
$364$	0	0
$365$	− 4.00000i	− 0.209370i
$366$	0	0
$367$	22.8564	1.19309	0.596547	−	0.802578i	$-0.296539\pi$
$367$	22.8564	1.19309	0.596547	+	0.802578i	$0.296539\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 5.46410i	− 0.283682i
$372$	0	0
$373$	− 37.4641i	− 1.93982i	−0.243468	−	0.969909i	$-0.578285\pi$
$373$	− 37.4641i	− 1.93982i	0.243468	−	0.969909i	$-0.421715\pi$
$374$	0	0
$375$	1.73205	0.0894427
$376$	0	0
$377$	−9.19615	−0.473626
$378$	0	0
$379$	− 33.8564i	− 1.73909i	−0.493857	−	0.869543i	$-0.664413\pi$
$379$	− 33.8564i	− 1.73909i	0.493857	−	0.869543i	$-0.335587\pi$
$380$	0	0
$381$	35.3205i	1.80952i
$382$	0	0
$383$	33.8564	1.72998	0.864991	−	0.501788i	$-0.167324\pi$
$383$	33.8564	1.72998	0.864991	+	0.501788i	$0.167324\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.05256i	0.357579i	0.983887	+	0.178789i	$0.0572181\pi$
$389$	7.05256i	0.357579i	−0.983887	+	0.178789i	$0.942782\pi$
$390$	0	0
$391$	19.8564	1.00418
$392$	0	0
$393$	18.0000	0.907980
$394$	0	0
$395$	− 5.73205i	− 0.288411i
$396$	0	0
$397$	11.7846i	0.591453i	0.955273	+	0.295726i	$0.0955616\pi$
$397$	11.7846i	0.591453i	−0.955273	+	0.295726i	$0.904438\pi$
$398$	0	0
$399$	3.46410	0.173422
$400$	0	0
$401$	−17.7846	−0.888121	−0.444061	−	0.895997i	$-0.646462\pi$
$401$	−17.7846	−0.888121	−0.444061	+	0.895997i	$0.646462\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	0	0
$405$	9.00000i	0.447214i
$406$	0	0
$407$	7.46410	0.369982
$408$	0	0
$409$	−1.32051	−0.0652949	−0.0326475	−	0.999467i	$-0.510394\pi$
$409$	−1.32051	−0.0652949	−0.0326475	+	0.999467i	$0.510394\pi$
$410$	0	0
$411$	− 7.60770i	− 0.375260i
$412$	0	0
$413$	2.53590i	0.124783i
$414$	0	0
$415$	15.4641	0.759103
$416$	0	0
$417$	12.2487	0.599822
$418$	0	0
$419$	− 37.1769i	− 1.81621i	−0.418741	−	0.908106i	$-0.637529\pi$
$419$	− 37.1769i	− 1.81621i	0.418741	−	0.908106i	$-0.362471\pi$
$420$	0	0
$421$	11.5885i	0.564787i	0.959299	+	0.282393i	$0.0911284\pi$
$421$	11.5885i	0.564787i	−0.959299	+	0.282393i	$0.908872\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.73205	0.278045
$426$	0	0
$427$	− 5.46410i	− 0.264426i
$428$	0	0
$429$	− 1.73205i	− 0.0836242i
$430$	0	0
$431$	22.8038	1.09842	0.549211	−	0.835683i	$-0.314928\pi$
$431$	22.8038	1.09842	0.549211	+	0.835683i	$0.314928\pi$
$432$	0	0
$433$	−28.7846	−1.38330	−0.691650	−	0.722233i	$-0.743116\pi$
$433$	−28.7846	−1.38330	−0.691650	+	0.722233i	$0.743116\pi$
$434$	0	0
$435$	15.9282i	0.763699i
$436$	0	0
$437$	− 6.92820i	− 0.331421i
$438$	0	0
$439$	20.2487	0.966418	0.483209	−	0.875505i	$-0.339471\pi$
$439$	20.2487	0.966418	0.483209	+	0.875505i	$0.339471\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.3205i	0.632876i	0.948613	+	0.316438i	$0.102487\pi$
$443$	13.3205i	0.632876i	−0.948613	+	0.316438i	$0.897513\pi$
$444$	0	0
$445$	− 9.46410i	− 0.448641i
$446$	0	0
$447$	−10.1436	−0.479776
$448$	0	0
$449$	4.85641	0.229188	0.114594	−	0.993412i	$-0.463443\pi$
$449$	4.85641	0.229188	0.114594	+	0.993412i	$0.463443\pi$
$450$	0	0
$451$	− 10.9282i	− 0.514589i
$452$	0	0
$453$	− 28.8564i	− 1.35579i
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	34.9282	1.63387	0.816936	−	0.576728i	$-0.195671\pi$
$457$	34.9282	1.63387	0.816936	+	0.576728i	$0.195671\pi$
$458$	0	0
$459$	29.7846i	1.39023i
$460$	0	0
$461$	12.0000i	0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$	12.0000i	0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$	0	0
$463$	14.2487	0.662194	0.331097	−	0.943597i	$-0.392581\pi$
$463$	14.2487	0.662194	0.331097	+	0.943597i	$0.392581\pi$
$464$	0	0
$465$	−6.92820	−0.321288
$466$	0	0
$467$	14.2679i	0.660242i	0.943939	+	0.330121i	$0.107090\pi$
$467$	14.2679i	0.660242i	−0.943939	+	0.330121i	$0.892910\pi$
$468$	0	0
$469$	1.46410i	0.0676059i
$470$	0	0
$471$	15.4641	0.712548
$472$	0	0
$473$	3.46410	0.159280
$474$	0	0
$475$	− 2.00000i	− 0.0917663i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−31.8564	−1.45556	−0.727778	−	0.685813i	$-0.759447\pi$
$479$	−31.8564	−1.45556	−0.727778	+	0.685813i	$0.759447\pi$
$480$	0	0
$481$	−7.46410	−0.340334
$482$	0	0
$483$	6.00000i	0.273009i
$484$	0	0
$485$	14.2679i	0.647874i
$486$	0	0
$487$	33.7128	1.52767	0.763837	−	0.645410i	$-0.223313\pi$
$487$	33.7128	1.52767	0.763837	+	0.645410i	$0.223313\pi$
$488$	0	0
$489$	−7.85641	−0.355279
$490$	0	0
$491$	21.7846i	0.983126i	0.870842	+	0.491563i	$0.163574\pi$
$491$	21.7846i	0.983126i	−0.870842	+	0.491563i	$0.836426\pi$
$492$	0	0
$493$	52.7128i	2.37407i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.535898	0.0240383
$498$	0	0
$499$	− 37.7846i	− 1.69147i	−0.533602	−	0.845736i	$-0.679162\pi$
$499$	− 37.7846i	− 1.69147i	0.533602	−	0.845736i	$-0.320838\pi$
$500$	0	0
$501$	− 10.2679i	− 0.458738i
$502$	0	0
$503$	−37.7846	−1.68473	−0.842366	−	0.538905i	$-0.818838\pi$
$503$	−37.7846	−1.68473	−0.842366	+	0.538905i	$0.818838\pi$
$504$	0	0
$505$	−6.39230	−0.284454
$506$	0	0
$507$	− 20.7846i	− 0.923077i
$508$	0	0
$509$	− 24.5359i	− 1.08753i	−0.839236	−	0.543767i	$-0.816997\pi$
$509$	− 24.5359i	− 1.08753i	0.839236	−	0.543767i	$-0.183003\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	10.3923	0.458831
$514$	0	0
$515$	5.92820i	0.261228i
$516$	0	0
$517$	− 3.92820i	− 0.172762i
$518$	0	0
$519$	8.66025	0.380143
$520$	0	0
$521$	−14.7846	−0.647726	−0.323863	−	0.946104i	$-0.604982\pi$
$521$	−14.7846	−0.647726	−0.323863	+	0.946104i	$0.604982\pi$
$522$	0	0
$523$	− 1.60770i	− 0.0702996i	−0.999382	−	0.0351498i	$-0.988809\pi$
$523$	− 1.60770i	− 0.0702996i	0.999382	−	0.0351498i	$-0.0111908\pi$
$524$	0	0
$525$	1.73205i	0.0755929i
$526$	0	0
$527$	−22.9282	−0.998768
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.9282i	0.473353i
$534$	0	0
$535$	−16.9282	−0.731870
$536$	0	0
$537$	25.8564	1.11579
$538$	0	0
$539$	1.00000i	0.0430730i
$540$	0	0
$541$	23.0526i	0.991107i	0.868577	+	0.495553i	$0.165035\pi$
$541$	23.0526i	0.991107i	−0.868577	+	0.495553i	$0.834965\pi$
$542$	0	0
$543$	28.3923	1.21843
$544$	0	0
$545$	−13.1962	−0.565261
$546$	0	0
$547$	− 33.1769i	− 1.41854i	−0.704936	−	0.709271i	$-0.749024\pi$
$547$	− 33.1769i	− 1.41854i	0.704936	−	0.709271i	$-0.250976\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	18.3923	0.783539
$552$	0	0
$553$	5.73205	0.243752
$554$	0	0
$555$	12.9282i	0.548772i
$556$	0	0
$557$	− 36.7846i	− 1.55861i	−0.626642	−	0.779307i	$-0.715571\pi$
$557$	− 36.7846i	− 1.55861i	0.626642	−	0.779307i	$-0.284429\pi$
$558$	0	0
$559$	−3.46410	−0.146516
$560$	0	0
$561$	−9.92820	−0.419169
$562$	0	0
$563$	− 29.3205i	− 1.23571i	−0.786291	−	0.617856i	$-0.788001\pi$
$563$	− 29.3205i	− 1.23571i	0.786291	−	0.617856i	$-0.211999\pi$
$564$	0	0
$565$	8.92820i	0.375612i
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	32.6410	1.36838	0.684191	−	0.729303i	$-0.260155\pi$
$569$	32.6410	1.36838	0.684191	+	0.729303i	$0.260155\pi$
$570$	0	0
$571$	31.7128i	1.32714i	0.748114	+	0.663570i	$0.230959\pi$
$571$	31.7128i	1.32714i	−0.748114	+	0.663570i	$0.769041\pi$
$572$	0	0
$573$	27.9282i	1.16672i
$574$	0	0
$575$	3.46410	0.144463
$576$	0	0
$577$	20.6603	0.860098	0.430049	−	0.902806i	$-0.358496\pi$
$577$	20.6603	0.860098	0.430049	+	0.902806i	$0.358496\pi$
$578$	0	0
$579$	− 19.6077i	− 0.814868i
$580$	0	0
$581$	15.4641i	0.641559i
$582$	0	0
$583$	5.46410	0.226300
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 10.3923i	− 0.428936i	−0.976731	−	0.214468i	$-0.931198\pi$
$587$	− 10.3923i	− 0.428936i	0.976731	−	0.214468i	$-0.0688018\pi$
$588$	0	0
$589$	8.00000i	0.329634i
$590$	0	0
$591$	−13.8564	−0.569976
$592$	0	0
$593$	−3.87564	−0.159154	−0.0795768	−	0.996829i	$-0.525357\pi$
$593$	−3.87564	−0.159154	−0.0795768	+	0.996829i	$0.525357\pi$
$594$	0	0
$595$	5.73205i	0.234991i
$596$	0	0
$597$	18.2487i	0.746870i
$598$	0	0
$599$	−42.5167	−1.73718	−0.868592	−	0.495528i	$-0.834975\pi$
$599$	−42.5167	−1.73718	−0.868592	+	0.495528i	$0.834975\pi$
$600$	0	0
$601$	28.5359	1.16400	0.582002	−	0.813188i	$-0.302270\pi$
$601$	28.5359	1.16400	0.582002	+	0.813188i	$0.302270\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 10.0000i	− 0.406558i
$606$	0	0
$607$	5.92820	0.240618	0.120309	−	0.992736i	$-0.461611\pi$
$607$	5.92820	0.240618	0.120309	+	0.992736i	$0.461611\pi$
$608$	0	0
$609$	−15.9282	−0.645443
$610$	0	0
$611$	3.92820i	0.158918i
$612$	0	0
$613$	23.1769i	0.936107i	0.883700	+	0.468053i	$0.155045\pi$
$613$	23.1769i	0.936107i	−0.883700	+	0.468053i	$0.844955\pi$
$614$	0	0
$615$	18.9282	0.763259
$616$	0	0
$617$	33.3205	1.34143	0.670717	−	0.741714i	$-0.265987\pi$
$617$	33.3205	1.34143	0.670717	+	0.741714i	$0.265987\pi$
$618$	0	0
$619$	− 27.4641i	− 1.10388i	−0.833885	−	0.551938i	$-0.813889\pi$
$619$	− 27.4641i	− 1.10388i	0.833885	−	0.551938i	$-0.186111\pi$
$620$	0	0
$621$	18.0000i	0.722315i
$622$	0	0
$623$	9.46410	0.379171
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.46410i	0.138343i
$628$	0	0
$629$	42.7846i	1.70593i
$630$	0	0
$631$	9.98076	0.397328	0.198664	−	0.980068i	$-0.436340\pi$
$631$	9.98076	0.397328	0.198664	+	0.980068i	$0.436340\pi$
$632$	0	0
$633$	1.73205	0.0688428
$634$	0	0
$635$	20.3923i	0.809244i
$636$	0	0
$637$	− 1.00000i	− 0.0396214i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.7128	1.33158	0.665788	−	0.746141i	$-0.268096\pi$
$641$	33.7128	1.33158	0.665788	+	0.746141i	$0.268096\pi$
$642$	0	0
$643$	− 25.1962i	− 0.993639i	−0.867854	−	0.496820i	$-0.834501\pi$
$643$	− 25.1962i	− 0.993639i	0.867854	−	0.496820i	$-0.165499\pi$
$644$	0	0
$645$	6.00000i	0.236250i
$646$	0	0
$647$	−10.1436	−0.398786	−0.199393	−	0.979920i	$-0.563897\pi$
$647$	−10.1436	−0.398786	−0.199393	+	0.979920i	$0.563897\pi$
$648$	0	0
$649$	−2.53590	−0.0995427
$650$	0	0
$651$	− 6.92820i	− 0.271538i
$652$	0	0
$653$	− 3.07180i	− 0.120209i	−0.998192	−	0.0601043i	$-0.980857\pi$
$653$	− 3.07180i	− 0.120209i	0.998192	−	0.0601043i	$-0.0191433\pi$
$654$	0	0
$655$	10.3923	0.406061
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.14359i	0.278275i	0.990273	+	0.139137i	$0.0444330\pi$
$659$	7.14359i	0.278275i	−0.990273	+	0.139137i	$0.955567\pi$
$660$	0	0
$661$	1.07180i	0.0416881i	0.999783	+	0.0208440i	$0.00663534\pi$
$661$	1.07180i	0.0416881i	−0.999783	+	0.0208440i	$0.993365\pi$
$662$	0	0
$663$	9.92820	0.385579
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	31.8564i	1.23348i
$668$	0	0
$669$	− 15.3397i	− 0.593069i
$670$	0	0
$671$	5.46410	0.210939
$672$	0	0
$673$	−24.3923	−0.940254	−0.470127	−	0.882599i	$-0.655792\pi$
$673$	−24.3923	−0.940254	−0.470127	+	0.882599i	$0.655792\pi$
$674$	0	0
$675$	5.19615i	0.200000i
$676$	0	0
$677$	− 42.7128i	− 1.64159i	−0.571225	−	0.820793i	$-0.693532\pi$
$677$	− 42.7128i	− 1.64159i	0.571225	−	0.820793i	$-0.306468\pi$
$678$	0	0
$679$	−14.2679	−0.547554
$680$	0	0
$681$	−27.9282	−1.07021
$682$	0	0
$683$	6.24871i	0.239100i	0.992828	+	0.119550i	$0.0381452\pi$
$683$	6.24871i	0.239100i	−0.992828	+	0.119550i	$0.961855\pi$
$684$	0	0
$685$	− 4.39230i	− 0.167821i
$686$	0	0
$687$	−13.8564	−0.528655
$688$	0	0
$689$	−5.46410	−0.208166
$690$	0	0
$691$	29.8564i	1.13579i	0.823101	+	0.567896i	$0.192242\pi$
$691$	29.8564i	1.13579i	−0.823101	+	0.567896i	$0.807758\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.07180	0.268249
$696$	0	0
$697$	62.6410	2.37270
$698$	0	0
$699$	5.07180i	0.191833i
$700$	0	0
$701$	− 19.0526i	− 0.719605i	−0.933028	−	0.359803i	$-0.882844\pi$
$701$	− 19.0526i	− 0.719605i	0.933028	−	0.359803i	$-0.117156\pi$
$702$	0	0
$703$	14.9282	0.563028
$704$	0	0
$705$	6.80385	0.256248
$706$	0	0
$707$	− 6.39230i	− 0.240407i
$708$	0	0
$709$	41.7321i	1.56728i	0.621215	+	0.783640i	$0.286639\pi$
$709$	41.7321i	1.56728i	−0.621215	+	0.783640i	$0.713361\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−13.8564	−0.518927
$714$	0	0
$715$	− 1.00000i	− 0.0373979i
$716$	0	0
$717$	− 3.00000i	− 0.112037i
$718$	0	0
$719$	−14.0000	−0.522112	−0.261056	−	0.965324i	$-0.584071\pi$
$719$	−14.0000	−0.522112	−0.261056	+	0.965324i	$0.584071\pi$
$720$	0	0
$721$	−5.92820	−0.220778
$722$	0	0
$723$	43.8564i	1.63104i
$724$	0	0
$725$	9.19615i	0.341537i
$726$	0	0
$727$	33.5692	1.24501	0.622507	−	0.782614i	$-0.286114\pi$
$727$	33.5692	1.24501	0.622507	+	0.782614i	$0.286114\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	19.8564i	0.734416i
$732$	0	0
$733$	45.7846i	1.69109i	0.533902	+	0.845547i	$0.320725\pi$
$733$	45.7846i	1.69109i	−0.533902	+	0.845547i	$0.679275\pi$
$734$	0	0
$735$	−1.73205	−0.0638877
$736$	0	0
$737$	−1.46410	−0.0539309
$738$	0	0
$739$	1.78461i	0.0656479i	0.999461	+	0.0328240i	$0.0104501\pi$
$739$	1.78461i	0.0656479i	−0.999461	+	0.0328240i	$0.989550\pi$
$740$	0	0
$741$	− 3.46410i	− 0.127257i
$742$	0	0
$743$	−30.2487	−1.10972	−0.554859	−	0.831945i	$-0.687228\pi$
$743$	−30.2487	−1.10972	−0.554859	+	0.831945i	$0.687228\pi$
$744$	0	0
$745$	−5.85641	−0.214562
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 16.9282i	− 0.618543i
$750$	0	0
$751$	−7.87564	−0.287386	−0.143693	−	0.989622i	$-0.545898\pi$
$751$	−7.87564	−0.287386	−0.143693	+	0.989622i	$0.545898\pi$
$752$	0	0
$753$	43.1769	1.57345
$754$	0	0
$755$	− 16.6603i	− 0.606329i
$756$	0	0
$757$	47.7128i	1.73415i	0.498176	+	0.867076i	$0.334003\pi$
$757$	47.7128i	1.73415i	−0.498176	+	0.867076i	$0.665997\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−5.60770	−0.203279	−0.101639	−	0.994821i	$-0.532409\pi$
$761$	−5.60770	−0.203279	−0.101639	+	0.994821i	$0.532409\pi$
$762$	0	0
$763$	− 13.1962i	− 0.477733i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.53590	0.0915660
$768$	0	0
$769$	36.2487	1.30716	0.653581	−	0.756857i	$-0.273266\pi$
$769$	36.2487	1.30716	0.653581	+	0.756857i	$0.273266\pi$
$770$	0	0
$771$	− 36.0000i	− 1.29651i
$772$	0	0
$773$	− 4.85641i	− 0.174673i	−0.996179	−	0.0873364i	$-0.972164\pi$
$773$	− 4.85641i	− 0.174673i	0.996179	−	0.0873364i	$-0.0278355\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−12.9282	−0.463797
$778$	0	0
$779$	− 21.8564i	− 0.783087i
$780$	0	0
$781$	0.535898i	0.0191760i
$782$	0	0
$783$	−47.7846	−1.70768
$784$	0	0
$785$	8.92820	0.318661
$786$	0	0
$787$	− 6.51666i	− 0.232294i	−0.993232	−	0.116147i	$-0.962946\pi$
$787$	− 6.51666i	− 0.232294i	0.993232	−	0.116147i	$-0.0370543\pi$
$788$	0	0
$789$	− 14.7846i	− 0.526346i
$790$	0	0
$791$	−8.92820	−0.317450
$792$	0	0
$793$	−5.46410	−0.194036
$794$	0	0
$795$	9.46410i	0.335657i
$796$	0	0
$797$	− 38.5692i	− 1.36619i	−0.730329	−	0.683096i	$-0.760633\pi$
$797$	− 38.5692i	− 1.36619i	0.730329	−	0.683096i	$-0.239367\pi$
$798$	0	0
$799$	22.5167	0.796582
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.00000i	0.141157i
$804$	0	0
$805$	3.46410i	0.122094i
$806$	0	0
$807$	−14.7846	−0.520443
$808$	0	0
$809$	36.7128	1.29075	0.645377	−	0.763864i	$-0.276700\pi$
$809$	36.7128	1.29075	0.645377	+	0.763864i	$0.276700\pi$
$810$	0	0
$811$	− 19.0718i	− 0.669701i	−0.942271	−	0.334851i	$-0.891314\pi$
$811$	− 19.0718i	− 0.669701i	0.942271	−	0.334851i	$-0.108686\pi$
$812$	0	0
$813$	− 37.1769i	− 1.30385i
$814$	0	0
$815$	−4.53590	−0.158886
$816$	0	0
$817$	6.92820	0.242387
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 45.4449i	− 1.58604i	−0.609198	−	0.793018i	$-0.708508\pi$
$821$	− 45.4449i	− 1.58604i	0.609198	−	0.793018i	$-0.291492\pi$
$822$	0	0
$823$	−38.6410	−1.34694	−0.673471	−	0.739214i	$-0.735197\pi$
$823$	−38.6410	−1.34694	−0.673471	+	0.739214i	$0.735197\pi$
$824$	0	0
$825$	−1.73205	−0.0603023
$826$	0	0
$827$	32.5359i	1.13138i	0.824617	+	0.565692i	$0.191391\pi$
$827$	32.5359i	1.13138i	−0.824617	+	0.565692i	$0.808609\pi$
$828$	0	0
$829$	11.7128i	0.406803i	0.979095	+	0.203401i	$0.0651996\pi$
$829$	11.7128i	0.406803i	−0.979095	+	0.203401i	$0.934800\pi$
$830$	0	0
$831$	−0.248711	−0.00862770
$832$	0	0
$833$	−5.73205	−0.198604
$834$	0	0
$835$	− 5.92820i	− 0.205154i
$836$	0	0
$837$	− 20.7846i	− 0.718421i
$838$	0	0
$839$	−35.5692	−1.22799	−0.613993	−	0.789312i	$-0.710438\pi$
$839$	−35.5692	−1.22799	−0.613993	+	0.789312i	$0.710438\pi$
$840$	0	0
$841$	−55.5692	−1.91618
$842$	0	0
$843$	23.8756i	0.822321i
$844$	0	0
$845$	− 12.0000i	− 0.412813i
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	−14.0718	−0.482943
$850$	0	0
$851$	25.8564i	0.886346i
$852$	0	0
$853$	− 34.0000i	− 1.16414i	−0.813139	−	0.582069i	$-0.802243\pi$
$853$	− 34.0000i	− 1.16414i	0.813139	−	0.582069i	$-0.197757\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.00000	−0.136637	−0.0683187	−	0.997664i	$-0.521763\pi$
$857$	−4.00000	−0.136637	−0.0683187	+	0.997664i	$0.521763\pi$
$858$	0	0
$859$	46.6410i	1.59137i	0.605710	+	0.795685i	$0.292889\pi$
$859$	46.6410i	1.59137i	−0.605710	+	0.795685i	$0.707111\pi$
$860$	0	0
$861$	18.9282i	0.645071i
$862$	0	0
$863$	−33.0718	−1.12578	−0.562889	−	0.826533i	$-0.690310\pi$
$863$	−33.0718	−1.12578	−0.562889	+	0.826533i	$0.690310\pi$
$864$	0	0
$865$	5.00000	0.170005
$866$	0	0
$867$	− 27.4641i	− 0.932730i
$868$	0	0
$869$	5.73205i	0.194447i
$870$	0	0
$871$	1.46410	0.0496092
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.00000i	0.0338062i
$876$	0	0
$877$	17.7128i	0.598119i	0.954234	+	0.299060i	$0.0966729\pi$
$877$	17.7128i	0.598119i	−0.954234	+	0.299060i	$0.903327\pi$
$878$	0	0
$879$	11.8756	0.400556
$880$	0	0
$881$	−33.5692	−1.13098	−0.565488	−	0.824757i	$-0.691312\pi$
$881$	−33.5692	−1.13098	−0.565488	+	0.824757i	$0.691312\pi$
$882$	0	0
$883$	− 19.3205i	− 0.650187i	−0.945682	−	0.325093i	$-0.894604\pi$
$883$	− 19.3205i	− 0.650187i	0.945682	−	0.325093i	$-0.105396\pi$
$884$	0	0
$885$	− 4.39230i	− 0.147646i
$886$	0	0
$887$	24.7846	0.832186	0.416093	−	0.909322i	$-0.363399\pi$
$887$	24.7846	0.832186	0.416093	+	0.909322i	$0.363399\pi$
$888$	0	0
$889$	−20.3923	−0.683936
$890$	0	0
$891$	− 9.00000i	− 0.301511i
$892$	0	0
$893$	− 7.85641i	− 0.262905i
$894$	0	0
$895$	14.9282	0.498995
$896$	0	0
$897$	6.00000	0.200334
$898$	0	0
$899$	− 36.7846i	− 1.22684i
$900$	0	0
$901$	31.3205i	1.04344i
$902$	0	0
$903$	−6.00000	−0.199667
$904$	0	0
$905$	16.3923	0.544899
$906$	0	0
$907$	18.7846i	0.623733i	0.950126	+	0.311866i	$0.100954\pi$
$907$	18.7846i	0.623733i	−0.950126	+	0.311866i	$0.899046\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22.3923	−0.741890	−0.370945	−	0.928655i	$-0.620966\pi$
$911$	−22.3923	−0.741890	−0.370945	+	0.928655i	$0.620966\pi$
$912$	0	0
$913$	−15.4641	−0.511787
$914$	0	0
$915$	9.46410i	0.312874i
$916$	0	0
$917$	10.3923i	0.343184i
$918$	0	0
$919$	46.2679	1.52624	0.763119	−	0.646258i	$-0.223667\pi$
$919$	46.2679	1.52624	0.763119	+	0.646258i	$0.223667\pi$
$920$	0	0
$921$	−33.9282	−1.11797
$922$	0	0
$923$	− 0.535898i	− 0.0176393i
$924$	0	0
$925$	7.46410i	0.245418i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−26.5359	−0.870615	−0.435307	−	0.900282i	$-0.643360\pi$
$929$	−26.5359	−0.870615	−0.435307	+	0.900282i	$0.643360\pi$
$930$	0	0
$931$	2.00000i	0.0655474i
$932$	0	0
$933$	− 24.9282i	− 0.816113i
$934$	0	0
$935$	−5.73205	−0.187458
$936$	0	0
$937$	6.80385	0.222272	0.111136	−	0.993805i	$-0.464551\pi$
$937$	6.80385	0.222272	0.111136	+	0.993805i	$0.464551\pi$
$938$	0	0
$939$	− 47.7846i	− 1.55939i
$940$	0	0
$941$	39.3205i	1.28181i	0.767619	+	0.640906i	$0.221441\pi$
$941$	39.3205i	1.28181i	−0.767619	+	0.640906i	$0.778559\pi$
$942$	0	0
$943$	37.8564	1.23277
$944$	0	0
$945$	−5.19615	−0.169031
$946$	0	0
$947$	44.0000i	1.42981i	0.699223	+	0.714904i	$0.253530\pi$
$947$	44.0000i	1.42981i	−0.699223	+	0.714904i	$0.746470\pi$
$948$	0	0
$949$	− 4.00000i	− 0.129845i
$950$	0	0
$951$	−7.85641	−0.254761
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	16.1244i	0.521772i
$956$	0	0
$957$	− 15.9282i	− 0.514886i
$958$	0	0
$959$	4.39230	0.141835
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 11.3205i	− 0.364420i
$966$	0	0
$967$	−23.4641	−0.754555	−0.377277	−	0.926100i	$-0.623140\pi$
$967$	−23.4641	−0.754555	−0.377277	+	0.926100i	$0.623140\pi$
$968$	0	0
$969$	−19.8564	−0.637880
$970$	0	0
$971$	12.0000i	0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$	12.0000i	0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$	0	0
$973$	7.07180i	0.226711i
$974$	0	0
$975$	1.73205	0.0554700
$976$	0	0
$977$	22.3923	0.716393	0.358197	−	0.933646i	$-0.383392\pi$
$977$	22.3923	0.716393	0.358197	+	0.933646i	$0.383392\pi$
$978$	0	0
$979$	9.46410i	0.302474i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−14.7128	−0.469266	−0.234633	−	0.972084i	$-0.575389\pi$
$983$	−14.7128	−0.469266	−0.234633	+	0.972084i	$0.575389\pi$
$984$	0	0
$985$	−8.00000	−0.254901
$986$	0	0
$987$	6.80385i	0.216569i
$988$	0	0
$989$	12.0000i	0.381578i
$990$	0	0
$991$	−0.248711	−0.00790058	−0.00395029	−	0.999992i	$-0.501257\pi$
$991$	−0.248711	−0.00790058	−0.00395029	+	0.999992i	$0.501257\pi$
$992$	0	0
$993$	22.1436	0.702706
$994$	0	0
$995$	10.5359i	0.334010i
$996$	0	0
$997$	− 40.5692i	− 1.28484i	−0.766353	−	0.642420i	$-0.777930\pi$
$997$	− 40.5692i	− 1.28484i	0.766353	−	0.642420i	$-0.222070\pi$
$998$	0	0
$999$	−38.7846	−1.22709

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.b.d.1121.1	yes	4
4.3	odd	2		2240.2.b.c.1121.3	yes	4
8.3	odd	2		2240.2.b.c.1121.2	✓	4
8.5	even	2	inner	2240.2.b.d.1121.4	yes	4
16.3	odd	4		8960.2.a.z.1.1		2
16.5	even	4		8960.2.a.ba.1.1		2
16.11	odd	4		8960.2.a.bb.1.2		2
16.13	even	4		8960.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.c.1121.2	✓	4	8.3	odd	2
2240.2.b.c.1121.3	yes	4	4.3	odd	2
2240.2.b.d.1121.1	yes	4	1.1	even	1	trivial
2240.2.b.d.1121.4	yes	4	8.5	even	2	inner
8960.2.a.y.1.2		2	16.13	even	4
8960.2.a.z.1.1		2	16.3	odd	4
8960.2.a.ba.1.1		2	16.5	even	4
8960.2.a.bb.1.2		2	16.11	odd	4

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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -1.00000i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.73205i q^{3} -1.00000i q^{5} +1.00000 q^{7} +1.00000i q^{11} -1.00000i q^{13} -1.73205 q^{15} -5.73205 q^{17} +2.00000i q^{19} -1.73205i q^{21} -3.46410 q^{23} -1.00000 q^{25} -5.19615i q^{27} -9.19615i q^{29} +4.00000 q^{31} +1.73205 q^{33} -1.00000i q^{35} -7.46410i q^{37} -1.73205 q^{39} -10.9282 q^{41} -3.46410i q^{43} -3.92820 q^{47} +1.00000 q^{49} +9.92820i q^{51} -5.46410i q^{53} +1.00000 q^{55} +3.46410 q^{57} +2.53590i q^{59} -5.46410i q^{61} -1.00000 q^{65} +1.46410i q^{67} +6.00000i q^{69} +0.535898 q^{71} +4.00000 q^{73} +1.73205i q^{75} +1.00000i q^{77} +5.73205 q^{79} -9.00000 q^{81} +15.4641i q^{83} +5.73205i q^{85} -15.9282 q^{87} +9.46410 q^{89} -1.00000i q^{91} -6.92820i q^{93} +2.00000 q^{95} -14.2679 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} - 16 q^{17} - 4 q^{25} + 16 q^{31} - 16 q^{41} + 12 q^{47} + 4 q^{49} + 4 q^{55} - 4 q^{65} + 16 q^{71} + 16 q^{73} + 16 q^{79} - 36 q^{81} - 36 q^{87} + 24 q^{89} + 8 q^{95} - 64 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 16 * q^17 - 4 * q^25 + 16 * q^31 - 16 * q^41 + 12 * q^47 + 4 * q^49 + 4 * q^55 - 4 * q^65 + 16 * q^71 + 16 * q^73 + 16 * q^79 - 36 * q^81 - 36 * q^87 + 24 * q^89 + 8 * q^95 - 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more