LMFDB - Embedded newform 2240.2.b.d.1121.2

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			1121.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2240.1121
Dual form			2240.2.b.d.1121.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +1.00000i q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.73205i q^{3} +1.00000i q^{5} +1.00000 q^{7} -1.00000i q^{11} +1.00000i q^{13} +1.73205 q^{15} -2.26795 q^{17} -2.00000i q^{19} -1.73205i q^{21} +3.46410 q^{23} -1.00000 q^{25} -5.19615i q^{27} -1.19615i q^{29} +4.00000 q^{31} -1.73205 q^{33} +1.00000i q^{35} +0.535898i q^{37} +1.73205 q^{39} +2.92820 q^{41} -3.46410i q^{43} +9.92820 q^{47} +1.00000 q^{49} +3.92820i q^{51} -1.46410i q^{53} +1.00000 q^{55} -3.46410 q^{57} -9.46410i q^{59} -1.46410i q^{61} -1.00000 q^{65} +5.46410i q^{67} -6.00000i q^{69} +7.46410 q^{71} +4.00000 q^{73} +1.73205i q^{75} -1.00000i q^{77} +2.26795 q^{79} -9.00000 q^{81} -8.53590i q^{83} -2.26795i q^{85} -2.07180 q^{87} +2.53590 q^{89} +1.00000i q^{91} -6.92820i q^{93} +2.00000 q^{95} -17.7321 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.951829\pi$
$11$	− 1.00000i	− 0.301511i	0.988571	−	0.150756i	$-0.0481707\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	1.73205	0.447214
$16$	0	0
$17$	−2.26795	−0.550058	−0.275029	−	0.961436i	$-0.588688\pi$
$17$	−2.26795	−0.550058	−0.275029	+	0.961436i	$0.588688\pi$
$18$	0	0
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	0	0
$21$	− 1.73205i	− 0.377964i
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 5.19615i	− 1.00000i
$28$	0	0
$29$	− 1.19615i	− 0.222120i	−0.993814	−	0.111060i	$-0.964575\pi$
$29$	− 1.19615i	− 0.222120i	0.993814	−	0.111060i	$-0.0354246\pi$
$30$	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
$37$	0.535898i	0.0881012i	0.999029	+	0.0440506i	$0.0140263\pi$
$37$	0.535898i	0.0881012i	−0.999029	+	0.0440506i	$0.985974\pi$
$38$	0	0
$39$	1.73205	0.277350
$40$	0	0
$41$	2.92820	0.457309	0.228654	−	0.973508i	$-0.426567\pi$
$41$	2.92820	0.457309	0.228654	+	0.973508i	$0.426567\pi$
$42$	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
$47$	9.92820	1.44818	0.724089	−	0.689707i	$-0.242261\pi$
$47$	9.92820	1.44818	0.724089	+	0.689707i	$0.242261\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.92820i	0.550058i
$52$	0	0
$53$	− 1.46410i	− 0.201110i	−0.994932	−	0.100555i	$-0.967938\pi$
$53$	− 1.46410i	− 0.201110i	0.994932	−	0.100555i	$-0.0320618\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−3.46410	−0.458831
$58$	0	0
$59$	− 9.46410i	− 1.23212i	−0.787699	−	0.616061i	$-0.788728\pi$
$59$	− 9.46410i	− 1.23212i	0.787699	−	0.616061i	$-0.211272\pi$
$60$	0	0
$61$	− 1.46410i	− 0.187459i	−0.995598	−	0.0937295i	$-0.970121\pi$
$61$	− 1.46410i	− 0.187459i	0.995598	−	0.0937295i	$-0.0298789\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	5.46410i	0.667546i	0.942653	+	0.333773i	$0.108322\pi$
$67$	5.46410i	0.667546i	−0.942653	+	0.333773i	$0.891678\pi$
$68$	0	0
$69$	− 6.00000i	− 0.722315i
$70$	0	0
$71$	7.46410	0.885826	0.442913	−	0.896565i	$-0.353945\pi$
$71$	7.46410	0.885826	0.442913	+	0.896565i	$0.353945\pi$
$72$	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
$79$	2.26795	0.255164	0.127582	−	0.991828i	$-0.459278\pi$
$79$	2.26795	0.255164	0.127582	+	0.991828i	$0.459278\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	− 8.53590i	− 0.936937i	−0.883480	−	0.468468i	$-0.844806\pi$
$83$	− 8.53590i	− 0.936937i	0.883480	−	0.468468i	$-0.155194\pi$
$84$	0	0
$85$	− 2.26795i	− 0.245994i
$86$	0	0
$87$	−2.07180	−0.222120
$88$	0	0
$89$	2.53590	0.268805	0.134402	−	0.990927i	$-0.457089\pi$
$89$	2.53590	0.268805	0.134402	+	0.990927i	$0.457089\pi$
$90$	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
$97$	−17.7321	−1.80042	−0.900208	−	0.435459i	$-0.856586\pi$
$97$	−17.7321	−1.80042	−0.900208	+	0.435459i	$0.856586\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 14.3923i	− 1.43209i	−0.698055	−	0.716044i	$-0.745951\pi$
$101$	− 14.3923i	− 1.43209i	0.698055	−	0.716044i	$-0.254049\pi$
$102$	0	0
$103$	7.92820	0.781189	0.390595	−	0.920563i	$-0.372269\pi$
$103$	7.92820	0.781189	0.390595	+	0.920563i	$0.372269\pi$
$104$	0	0
$105$	1.73205	0.169031
$106$	0	0
$107$	3.07180i	0.296962i	0.988915	+	0.148481i	$0.0474383\pi$
$107$	3.07180i	0.296962i	−0.988915	+	0.148481i	$0.952562\pi$
$108$	0	0
$109$	2.80385i	0.268560i	0.990943	+	0.134280i	$0.0428721\pi$
$109$	2.80385i	0.268560i	−0.990943	+	0.134280i	$0.957128\pi$
$110$	0	0
$111$	0.928203	0.0881012
$112$	0	0
$113$	4.92820	0.463606	0.231803	−	0.972763i	$-0.425537\pi$
$113$	4.92820	0.463606	0.231803	+	0.972763i	$0.425537\pi$
$114$	0	0
$115$	3.46410i	0.323029i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.26795	−0.207903
$120$	0	0
$121$	10.0000	0.909091
$122$	0	0
$123$	− 5.07180i	− 0.457309i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	0.392305	0.0348114	0.0174057	−	0.999849i	$-0.494459\pi$
$127$	0.392305	0.0348114	0.0174057	+	0.999849i	$0.494459\pi$
$128$	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
$137$	−16.3923	−1.40049	−0.700245	−	0.713903i	$-0.746926\pi$
$137$	−16.3923	−1.40049	−0.700245	+	0.713903i	$0.746926\pi$
$138$	0	0
$139$	− 20.9282i	− 1.77511i	−0.460705	−	0.887554i	$-0.652403\pi$
$139$	− 20.9282i	− 1.77511i	0.460705	−	0.887554i	$-0.347597\pi$
$140$	0	0
$141$	− 17.1962i	− 1.44818i
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	1.19615	0.0993351
$146$	0	0
$147$	− 1.73205i	− 0.142857i
$148$	0	0
$149$	− 21.8564i	− 1.79055i	−0.445517	−	0.895273i	$-0.646980\pi$
$149$	− 21.8564i	− 1.79055i	0.445517	−	0.895273i	$-0.353020\pi$
$150$	0	0
$151$	−0.660254	−0.0537307	−0.0268654	−	0.999639i	$-0.508553\pi$
$151$	−0.660254	−0.0537307	−0.0268654	+	0.999639i	$0.508553\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.00000i	0.321288i
$156$	0	0
$157$	4.92820i	0.393313i	0.980472	+	0.196657i	$0.0630084\pi$
$157$	4.92820i	0.393313i	−0.980472	+	0.196657i	$0.936992\pi$
$158$	0	0
$159$	−2.53590	−0.201110
$160$	0	0
$161$	3.46410	0.273009
$162$	0	0
$163$	11.4641i	0.897938i	0.893547	+	0.448969i	$0.148209\pi$
$163$	11.4641i	0.897938i	−0.893547	+	0.448969i	$0.851791\pi$
$164$	0	0
$165$	− 1.73205i	− 0.134840i
$166$	0	0
$167$	−7.92820	−0.613503	−0.306751	−	0.951790i	$-0.599242\pi$
$167$	−7.92820	−0.613503	−0.306751	+	0.951790i	$0.599242\pi$
$168$	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	$-0.939128\pi$
$173$	− 5.00000i	− 0.380143i	0.981770	−	0.190071i	$-0.0608720\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−16.3923	−1.23212
$178$	0	0
$179$	− 1.07180i	− 0.0801099i	−0.999197	−	0.0400549i	$-0.987247\pi$
$179$	− 1.07180i	− 0.0801099i	0.999197	−	0.0400549i	$-0.0127533\pi$
$180$	0	0
$181$	4.39230i	0.326477i	0.986587	+	0.163239i	$0.0521941\pi$
$181$	4.39230i	0.326477i	−0.986587	+	0.163239i	$0.947806\pi$
$182$	0	0
$183$	−2.53590	−0.187459
$184$	0	0
$185$	−0.535898	−0.0394000
$186$	0	0
$187$	2.26795i	0.165849i
$188$	0	0
$189$	− 5.19615i	− 0.377964i
$190$	0	0
$191$	8.12436	0.587858	0.293929	−	0.955827i	$-0.405037\pi$
$191$	8.12436	0.587858	0.293929	+	0.955827i	$0.405037\pi$
$192$	0	0
$193$	−23.3205	−1.67865	−0.839323	−	0.543632i	$-0.817049\pi$
$193$	−23.3205	−1.67865	−0.839323	+	0.543632i	$0.817049\pi$
$194$	0	0
$195$	1.73205i	0.124035i
$196$	0	0
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	0	0
$199$	−17.4641	−1.23800	−0.618999	−	0.785392i	$-0.712461\pi$
$199$	−17.4641	−1.23800	−0.618999	+	0.785392i	$0.712461\pi$
$200$	0	0
$201$	9.46410	0.667546
$202$	0	0
$203$	− 1.19615i	− 0.0839534i
$204$	0	0
$205$	2.92820i	0.204515i
$206$	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	$-0.989041\pi$
$211$	− 1.00000i	− 0.0688428i	0.999407	−	0.0344214i	$-0.0109588\pi$
$212$	0	0
$213$	− 12.9282i	− 0.885826i
$214$	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
$221$	− 2.26795i	− 0.152559i
$222$	0	0
$223$	−18.8564	−1.26272	−0.631359	−	0.775491i	$-0.717503\pi$
$223$	−18.8564	−1.26272	−0.631359	+	0.775491i	$0.717503\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.12436i	− 0.539232i	−0.962968	−	0.269616i	$-0.913103\pi$
$227$	− 8.12436i	− 0.539232i	0.962968	−	0.269616i	$-0.0868968\pi$
$228$	0	0
$229$	8.00000i	0.528655i	0.964433	+	0.264327i	$0.0851500\pi$
$229$	8.00000i	0.528655i	−0.964433	+	0.264327i	$0.914850\pi$
$230$	0	0
$231$	−1.73205	−0.113961
$232$	0	0
$233$	10.9282	0.715930	0.357965	−	0.933735i	$-0.383471\pi$
$233$	10.9282	0.715930	0.357965	+	0.933735i	$0.383471\pi$
$234$	0	0
$235$	9.92820i	0.647645i
$236$	0	0
$237$	− 3.92820i	− 0.255164i
$238$	0	0
$239$	−1.73205	−0.112037	−0.0560185	−	0.998430i	$-0.517841\pi$
$239$	−1.73205	−0.112037	−0.0560185	+	0.998430i	$0.517841\pi$
$240$	0	0
$241$	9.32051	0.600387	0.300193	−	0.953878i	$-0.402949\pi$
$241$	9.32051	0.600387	0.300193	+	0.953878i	$0.402949\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000i	0.0638877i
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	−14.7846	−0.936937
$250$	0	0
$251$	− 11.0718i	− 0.698846i	−0.936965	−	0.349423i	$-0.886378\pi$
$251$	− 11.0718i	− 0.698846i	0.936965	−	0.349423i	$-0.113622\pi$
$252$	0	0
$253$	− 3.46410i	− 0.217786i
$254$	0	0
$255$	−3.92820	−0.245994
$256$	0	0
$257$	−20.7846	−1.29651	−0.648254	−	0.761424i	$-0.724501\pi$
$257$	−20.7846	−1.29651	−0.648254	+	0.761424i	$0.724501\pi$
$258$	0	0
$259$	0.535898i	0.0332991i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.4641	0.953557	0.476779	−	0.879023i	$-0.341804\pi$
$263$	15.4641	0.953557	0.476779	+	0.879023i	$0.341804\pi$
$264$	0	0
$265$	1.46410	0.0899390
$266$	0	0
$267$	− 4.39230i	− 0.268805i
$268$	0	0
$269$	15.4641i	0.942863i	0.881903	+	0.471431i	$0.156263\pi$
$269$	15.4641i	0.942863i	−0.881903	+	0.471431i	$0.843737\pi$
$270$	0	0
$271$	14.5359	0.882993	0.441496	−	0.897263i	$-0.354448\pi$
$271$	14.5359	0.882993	0.441496	+	0.897263i	$0.354448\pi$
$272$	0	0
$273$	1.73205	0.104828
$274$	0	0
$275$	1.00000i	0.0603023i
$276$	0	0
$277$	27.8564i	1.67373i	0.547410	+	0.836865i	$0.315614\pi$
$277$	27.8564i	1.67373i	−0.547410	+	0.836865i	$0.684386\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	27.7846	1.65749	0.828745	−	0.559626i	$-0.189055\pi$
$281$	27.7846	1.65749	0.828745	+	0.559626i	$0.189055\pi$
$282$	0	0
$283$	− 16.1244i	− 0.958493i	−0.877680	−	0.479247i	$-0.840910\pi$
$283$	− 16.1244i	− 0.958493i	0.877680	−	0.479247i	$-0.159090\pi$
$284$	0	0
$285$	− 3.46410i	− 0.205196i
$286$	0	0
$287$	2.92820	0.172846
$288$	0	0
$289$	−11.8564	−0.697436
$290$	0	0
$291$	30.7128i	1.80042i
$292$	0	0
$293$	20.8564i	1.21844i	0.793000	+	0.609222i	$0.208518\pi$
$293$	20.8564i	1.21844i	−0.793000	+	0.609222i	$0.791482\pi$
$294$	0	0
$295$	9.46410	0.551021
$296$	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
$303$	−24.9282	−1.43209
$304$	0	0
$305$	1.46410	0.0838342
$306$	0	0
$307$	− 11.5885i	− 0.661388i	−0.943738	−	0.330694i	$-0.892717\pi$
$307$	− 11.5885i	− 0.661388i	0.943738	−	0.330694i	$-0.107283\pi$
$308$	0	0
$309$	− 13.7321i	− 0.781189i
$310$	0	0
$311$	−6.39230	−0.362474	−0.181237	−	0.983439i	$-0.558010\pi$
$311$	−6.39230	−0.362474	−0.181237	+	0.983439i	$0.558010\pi$
$312$	0	0
$313$	−3.58846	−0.202832	−0.101416	−	0.994844i	$-0.532337\pi$
$313$	−3.58846	−0.202832	−0.101416	+	0.994844i	$0.532337\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.4641i	0.643888i	0.946759	+	0.321944i	$0.104336\pi$
$317$	11.4641i	0.643888i	−0.946759	+	0.321944i	$0.895664\pi$
$318$	0	0
$319$	−1.19615	−0.0669717
$320$	0	0
$321$	5.32051	0.296962
$322$	0	0
$323$	4.53590i	0.252384i
$324$	0	0
$325$	− 1.00000i	− 0.0554700i
$326$	0	0
$327$	4.85641	0.268560
$328$	0	0
$329$	9.92820	0.547360
$330$	0	0
$331$	28.7846i	1.58215i	0.611722	+	0.791073i	$0.290477\pi$
$331$	28.7846i	1.58215i	−0.611722	+	0.791073i	$0.709523\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.46410	−0.298536
$336$	0	0
$337$	10.3923	0.566105	0.283052	−	0.959104i	$-0.408653\pi$
$337$	10.3923	0.566105	0.283052	+	0.959104i	$0.408653\pi$
$338$	0	0
$339$	− 8.53590i	− 0.463606i
$340$	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	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	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
$353$	1.19615	0.0636648	0.0318324	−	0.999493i	$-0.489866\pi$
$353$	1.19615	0.0636648	0.0318324	+	0.999493i	$0.489866\pi$
$354$	0	0
$355$	7.46410i	0.396153i
$356$	0	0
$357$	3.92820i	0.207903i
$358$	0	0
$359$	13.6077	0.718187	0.359093	−	0.933302i	$-0.383086\pi$
$359$	13.6077	0.718187	0.359093	+	0.933302i	$0.383086\pi$
$360$	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
$367$	−4.85641	−0.253502	−0.126751	−	0.991935i	$-0.540455\pi$
$367$	−4.85641	−0.253502	−0.126751	+	0.991935i	$0.540455\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.46410i	− 0.0760124i
$372$	0	0
$373$	30.5359i	1.58109i	0.612405	+	0.790544i	$0.290202\pi$
$373$	30.5359i	1.58109i	−0.612405	+	0.790544i	$0.709798\pi$
$374$	0	0
$375$	−1.73205	−0.0894427
$376$	0	0
$377$	1.19615	0.0616050
$378$	0	0
$379$	6.14359i	0.315575i	0.987473	+	0.157788i	$0.0504361\pi$
$379$	6.14359i	0.315575i	−0.987473	+	0.157788i	$0.949564\pi$
$380$	0	0
$381$	− 0.679492i	− 0.0348114i
$382$	0	0
$383$	6.14359	0.313923	0.156961	−	0.987605i	$-0.449830\pi$
$383$	6.14359	0.313923	0.156961	+	0.987605i	$0.449830\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	0	0
$388$	0	0
$389$	31.0526i	1.57443i	0.616680	+	0.787214i	$0.288477\pi$
$389$	31.0526i	1.57443i	−0.616680	+	0.787214i	$0.711523\pi$
$390$	0	0
$391$	−7.85641	−0.397316
$392$	0	0
$393$	18.0000	0.907980
$394$	0	0
$395$	2.26795i	0.114113i
$396$	0	0
$397$	29.7846i	1.49485i	0.664348	+	0.747423i	$0.268709\pi$
$397$	29.7846i	1.49485i	−0.664348	+	0.747423i	$0.731291\pi$
$398$	0	0
$399$	−3.46410	−0.173422
$400$	0	0
$401$	23.7846	1.18775	0.593873	−	0.804559i	$-0.297598\pi$
$401$	23.7846	1.18775	0.593873	+	0.804559i	$0.297598\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	0	0
$405$	− 9.00000i	− 0.447214i
$406$	0	0
$407$	0.535898	0.0265635
$408$	0	0
$409$	33.3205	1.64759	0.823797	−	0.566886i	$-0.191852\pi$
$409$	33.3205	1.64759	0.823797	+	0.566886i	$0.191852\pi$
$410$	0	0
$411$	28.3923i	1.40049i
$412$	0	0
$413$	− 9.46410i	− 0.465698i
$414$	0	0
$415$	8.53590	0.419011
$416$	0	0
$417$	−36.2487	−1.77511
$418$	0	0
$419$	− 25.1769i	− 1.22997i	−0.788538	−	0.614986i	$-0.789161\pi$
$419$	− 25.1769i	− 1.22997i	0.788538	−	0.614986i	$-0.210839\pi$
$420$	0	0
$421$	19.5885i	0.954683i	0.878718	+	0.477341i	$0.158400\pi$
$421$	19.5885i	0.954683i	−0.878718	+	0.477341i	$0.841600\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.26795	0.110012
$426$	0	0
$427$	− 1.46410i	− 0.0708528i
$428$	0	0
$429$	− 1.73205i	− 0.0836242i
$430$	0	0
$431$	33.1962	1.59900	0.799501	−	0.600664i	$-0.205097\pi$
$431$	33.1962	1.59900	0.799501	+	0.600664i	$0.205097\pi$
$432$	0	0
$433$	12.7846	0.614389	0.307195	−	0.951647i	$-0.400610\pi$
$433$	12.7846	0.614389	0.307195	+	0.951647i	$0.400610\pi$
$434$	0	0
$435$	− 2.07180i	− 0.0993351i
$436$	0	0
$437$	− 6.92820i	− 0.331421i
$438$	0	0
$439$	−28.2487	−1.34824	−0.674119	−	0.738623i	$-0.735476\pi$
$439$	−28.2487	−1.34824	−0.674119	+	0.738623i	$0.735476\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.3205i	1.01297i	0.862249	+	0.506484i	$0.169055\pi$
$443$	21.3205i	1.01297i	−0.862249	+	0.506484i	$0.830945\pi$
$444$	0	0
$445$	2.53590i	0.120213i
$446$	0	0
$447$	−37.8564	−1.79055
$448$	0	0
$449$	−22.8564	−1.07866	−0.539330	−	0.842094i	$-0.681323\pi$
$449$	−22.8564	−1.07866	−0.539330	+	0.842094i	$0.681323\pi$
$450$	0	0
$451$	− 2.92820i	− 0.137884i
$452$	0	0
$453$	1.14359i	0.0537307i
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	21.0718	0.985697	0.492849	−	0.870115i	$-0.335956\pi$
$457$	21.0718	0.985697	0.492849	+	0.870115i	$0.335956\pi$
$458$	0	0
$459$	11.7846i	0.550058i
$460$	0	0
$461$	− 12.0000i	− 0.558896i	−0.960161	−	0.279448i	$-0.909849\pi$
$461$	− 12.0000i	− 0.558896i	0.960161	−	0.279448i	$-0.0901514\pi$
$462$	0	0
$463$	−34.2487	−1.59167	−0.795836	−	0.605512i	$-0.792968\pi$
$463$	−34.2487	−1.59167	−0.795836	+	0.605512i	$0.792968\pi$
$464$	0	0
$465$	6.92820	0.321288
$466$	0	0
$467$	− 17.7321i	− 0.820541i	−0.911964	−	0.410271i	$-0.865434\pi$
$467$	− 17.7321i	− 0.820541i	0.911964	−	0.410271i	$-0.134566\pi$
$468$	0	0
$469$	5.46410i	0.252309i
$470$	0	0
$471$	8.53590	0.393313
$472$	0	0
$473$	−3.46410	−0.159280
$474$	0	0
$475$	2.00000i	0.0917663i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.14359	−0.189326	−0.0946628	−	0.995509i	$-0.530177\pi$
$479$	−4.14359	−0.189326	−0.0946628	+	0.995509i	$0.530177\pi$
$480$	0	0
$481$	−0.535898	−0.0244349
$482$	0	0
$483$	− 6.00000i	− 0.273009i
$484$	0	0
$485$	− 17.7321i	− 0.805171i
$486$	0	0
$487$	−21.7128	−0.983901	−0.491951	−	0.870623i	$-0.663716\pi$
$487$	−21.7128	−0.983901	−0.491951	+	0.870623i	$0.663716\pi$
$488$	0	0
$489$	19.8564	0.897938
$490$	0	0
$491$	19.7846i	0.892867i	0.894817	+	0.446433i	$0.147306\pi$
$491$	19.7846i	0.892867i	−0.894817	+	0.446433i	$0.852694\pi$
$492$	0	0
$493$	2.71281i	0.122179i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.46410	0.334811
$498$	0	0
$499$	− 3.78461i	− 0.169422i	−0.996406	−	0.0847112i	$-0.973003\pi$
$499$	− 3.78461i	− 0.169422i	0.996406	−	0.0847112i	$-0.0269968\pi$
$500$	0	0
$501$	13.7321i	0.613503i
$502$	0	0
$503$	3.78461	0.168747	0.0843737	−	0.996434i	$-0.473111\pi$
$503$	3.78461	0.168747	0.0843737	+	0.996434i	$0.473111\pi$
$504$	0	0
$505$	14.3923	0.640449
$506$	0	0
$507$	− 20.7846i	− 0.923077i
$508$	0	0
$509$	31.4641i	1.39462i	0.716769	+	0.697311i	$0.245620\pi$
$509$	31.4641i	1.39462i	−0.716769	+	0.697311i	$0.754380\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	0	0
$513$	−10.3923	−0.458831
$514$	0	0
$515$	7.92820i	0.349358i
$516$	0	0
$517$	− 9.92820i	− 0.436642i
$518$	0	0
$519$	−8.66025	−0.380143
$520$	0	0
$521$	26.7846	1.17346	0.586728	−	0.809784i	$-0.300416\pi$
$521$	26.7846	1.17346	0.586728	+	0.809784i	$0.300416\pi$
$522$	0	0
$523$	22.3923i	0.979147i	0.871962	+	0.489574i	$0.162848\pi$
$523$	22.3923i	0.979147i	−0.871962	+	0.489574i	$0.837152\pi$
$524$	0	0
$525$	1.73205i	0.0755929i
$526$	0	0
$527$	−9.07180	−0.395174
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.92820i	0.126835i
$534$	0	0
$535$	−3.07180	−0.132805
$536$	0	0
$537$	−1.85641	−0.0801099
$538$	0	0
$539$	− 1.00000i	− 0.0430730i
$540$	0	0
$541$	15.0526i	0.647160i	0.946201	+	0.323580i	$0.104886\pi$
$541$	15.0526i	0.647160i	−0.946201	+	0.323580i	$0.895114\pi$
$542$	0	0
$543$	7.60770	0.326477
$544$	0	0
$545$	−2.80385	−0.120104
$546$	0	0
$547$	− 29.1769i	− 1.24751i	−0.781618	−	0.623757i	$-0.785605\pi$
$547$	− 29.1769i	− 1.24751i	0.781618	−	0.623757i	$-0.214395\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.39230	−0.101916
$552$	0	0
$553$	2.26795	0.0964430
$554$	0	0
$555$	0.928203i	0.0394000i
$556$	0	0
$557$	− 4.78461i	− 0.202730i	−0.994849	−	0.101365i	$-0.967679\pi$
$557$	− 4.78461i	− 0.202730i	0.994849	−	0.101365i	$-0.0323211\pi$
$558$	0	0
$559$	3.46410	0.146516
$560$	0	0
$561$	3.92820	0.165849
$562$	0	0
$563$	− 5.32051i	− 0.224233i	−0.993695	−	0.112116i	$-0.964237\pi$
$563$	− 5.32051i	− 0.224233i	0.993695	−	0.112116i	$-0.0357629\pi$
$564$	0	0
$565$	4.92820i	0.207331i
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−36.6410	−1.53607	−0.768036	−	0.640407i	$-0.778766\pi$
$569$	−36.6410	−1.53607	−0.768036	+	0.640407i	$0.778766\pi$
$570$	0	0
$571$	23.7128i	0.992350i	0.868222	+	0.496175i	$0.165263\pi$
$571$	23.7128i	0.992350i	−0.868222	+	0.496175i	$0.834737\pi$
$572$	0	0
$573$	− 14.0718i	− 0.587858i
$574$	0	0
$575$	−3.46410	−0.144463
$576$	0	0
$577$	3.33975	0.139035	0.0695177	−	0.997581i	$-0.477854\pi$
$577$	3.33975	0.139035	0.0695177	+	0.997581i	$0.477854\pi$
$578$	0	0
$579$	40.3923i	1.67865i
$580$	0	0
$581$	− 8.53590i	− 0.354129i
$582$	0	0
$583$	−1.46410	−0.0606369
$584$	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
$593$	−28.1244	−1.15493	−0.577464	−	0.816416i	$-0.695958\pi$
$593$	−28.1244	−1.15493	−0.577464	+	0.816416i	$0.695958\pi$
$594$	0	0
$595$	− 2.26795i	− 0.0929769i
$596$	0	0
$597$	30.2487i	1.23800i
$598$	0	0
$599$	2.51666	0.102828	0.0514140	−	0.998677i	$-0.483627\pi$
$599$	2.51666	0.102828	0.0514140	+	0.998677i	$0.483627\pi$
$600$	0	0
$601$	35.4641	1.44661	0.723305	−	0.690528i	$-0.242622\pi$
$601$	35.4641	1.44661	0.723305	+	0.690528i	$0.242622\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.0000i	0.406558i
$606$	0	0
$607$	−7.92820	−0.321796	−0.160898	−	0.986971i	$-0.551439\pi$
$607$	−7.92820	−0.321796	−0.160898	+	0.986971i	$0.551439\pi$
$608$	0	0
$609$	−2.07180	−0.0839534
$610$	0	0
$611$	9.92820i	0.401652i
$612$	0	0
$613$	39.1769i	1.58234i	0.611596	+	0.791170i	$0.290528\pi$
$613$	39.1769i	1.58234i	−0.611596	+	0.791170i	$0.709472\pi$
$614$	0	0
$615$	5.07180	0.204515
$616$	0	0
$617$	−1.32051	−0.0531617	−0.0265808	−	0.999647i	$-0.508462\pi$
$617$	−1.32051	−0.0531617	−0.0265808	+	0.999647i	$0.508462\pi$
$618$	0	0
$619$	20.5359i	0.825407i	0.910865	+	0.412704i	$0.135416\pi$
$619$	20.5359i	0.825407i	−0.910865	+	0.412704i	$0.864584\pi$
$620$	0	0
$621$	− 18.0000i	− 0.722315i
$622$	0	0
$623$	2.53590	0.101599
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.46410i	0.138343i
$628$	0	0
$629$	− 1.21539i	− 0.0484608i
$630$	0	0
$631$	−41.9808	−1.67123	−0.835614	−	0.549317i	$-0.814888\pi$
$631$	−41.9808	−1.67123	−0.835614	+	0.549317i	$0.814888\pi$
$632$	0	0
$633$	−1.73205	−0.0688428
$634$	0	0
$635$	0.392305i	0.0155681i
$636$	0	0
$637$	1.00000i	0.0396214i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−21.7128	−0.857605	−0.428802	−	0.903398i	$-0.641064\pi$
$641$	−21.7128	−0.857605	−0.428802	+	0.903398i	$0.641064\pi$
$642$	0	0
$643$	14.8038i	0.583807i	0.956448	+	0.291903i	$0.0942886\pi$
$643$	14.8038i	0.583807i	−0.956448	+	0.291903i	$0.905711\pi$
$644$	0	0
$645$	− 6.00000i	− 0.236250i
$646$	0	0
$647$	−37.8564	−1.48829	−0.744144	−	0.668019i	$-0.767143\pi$
$647$	−37.8564	−1.48829	−0.744144	+	0.668019i	$0.767143\pi$
$648$	0	0
$649$	−9.46410	−0.371498
$650$	0	0
$651$	− 6.92820i	− 0.271538i
$652$	0	0
$653$	16.9282i	0.662452i	0.943551	+	0.331226i	$0.107462\pi$
$653$	16.9282i	0.662452i	−0.943551	+	0.331226i	$0.892538\pi$
$654$	0	0
$655$	−10.3923	−0.406061
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 34.8564i	− 1.35781i	−0.734225	−	0.678906i	$-0.762454\pi$
$659$	− 34.8564i	− 1.35781i	0.734225	−	0.678906i	$-0.237546\pi$
$660$	0	0
$661$	− 14.9282i	− 0.580640i	−0.956930	−	0.290320i	$-0.906238\pi$
$661$	− 14.9282i	− 0.580640i	0.956930	−	0.290320i	$-0.0937617\pi$
$662$	0	0
$663$	−3.92820	−0.152559
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	− 4.14359i	− 0.160441i
$668$	0	0
$669$	32.6603i	1.26272i
$670$	0	0
$671$	−1.46410	−0.0565210
$672$	0	0
$673$	−3.60770	−0.139066	−0.0695332	−	0.997580i	$-0.522151\pi$
$673$	−3.60770	−0.139066	−0.0695332	+	0.997580i	$0.522151\pi$
$674$	0	0
$675$	5.19615i	0.200000i
$676$	0	0
$677$	− 12.7128i	− 0.488593i	−0.969701	−	0.244297i	$-0.921443\pi$
$677$	− 12.7128i	− 0.488593i	0.969701	−	0.244297i	$-0.0785570\pi$
$678$	0	0
$679$	−17.7321	−0.680494
$680$	0	0
$681$	−14.0718	−0.539232
$682$	0	0
$683$	42.2487i	1.61660i	0.588769	+	0.808301i	$0.299613\pi$
$683$	42.2487i	1.61660i	−0.588769	+	0.808301i	$0.700387\pi$
$684$	0	0
$685$	− 16.3923i	− 0.626318i
$686$	0	0
$687$	13.8564	0.528655
$688$	0	0
$689$	1.46410	0.0557778
$690$	0	0
$691$	− 2.14359i	− 0.0815461i	−0.999168	−	0.0407731i	$-0.987018\pi$
$691$	− 2.14359i	− 0.0815461i	0.999168	−	0.0407731i	$-0.0129821\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.9282	0.793852
$696$	0	0
$697$	−6.64102	−0.251546
$698$	0	0
$699$	− 18.9282i	− 0.715930i
$700$	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
$703$	1.07180	0.0404236
$704$	0	0
$705$	17.1962	0.647645
$706$	0	0
$707$	− 14.3923i	− 0.541278i
$708$	0	0
$709$	− 38.2679i	− 1.43718i	−0.695432	−	0.718591i	$-0.744787\pi$
$709$	− 38.2679i	− 1.43718i	0.695432	−	0.718591i	$-0.255213\pi$
$710$	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
$721$	7.92820	0.295262
$722$	0	0
$723$	− 16.1436i	− 0.600387i
$724$	0	0
$725$	1.19615i	0.0444240i
$726$	0	0
$727$	−49.5692	−1.83842	−0.919210	−	0.393767i	$-0.871172\pi$
$727$	−49.5692	−1.83842	−0.919210	+	0.393767i	$0.871172\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	7.85641i	0.290580i
$732$	0	0
$733$	− 4.21539i	− 0.155699i	−0.996965	−	0.0778495i	$-0.975195\pi$
$733$	− 4.21539i	− 0.155699i	0.996965	−	0.0778495i	$-0.0248054\pi$
$734$	0	0
$735$	1.73205	0.0638877
$736$	0	0
$737$	5.46410	0.201273
$738$	0	0
$739$	39.7846i	1.46350i	0.681573	+	0.731750i	$0.261296\pi$
$739$	39.7846i	1.46350i	−0.681573	+	0.731750i	$0.738704\pi$
$740$	0	0
$741$	− 3.46410i	− 0.127257i
$742$	0	0
$743$	18.2487	0.669480	0.334740	−	0.942310i	$-0.391351\pi$
$743$	18.2487	0.669480	0.334740	+	0.942310i	$0.391351\pi$
$744$	0	0
$745$	21.8564	0.800757
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.07180i	0.112241i
$750$	0	0
$751$	−32.1244	−1.17223	−0.586117	−	0.810226i	$-0.699344\pi$
$751$	−32.1244	−1.17223	−0.586117	+	0.810226i	$0.699344\pi$
$752$	0	0
$753$	−19.1769	−0.698846
$754$	0	0
$755$	− 0.660254i	− 0.0240291i
$756$	0	0
$757$	7.71281i	0.280327i	0.990128	+	0.140163i	$0.0447628\pi$
$757$	7.71281i	0.280327i	−0.990128	+	0.140163i	$0.955237\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−26.3923	−0.956720	−0.478360	−	0.878164i	$-0.658769\pi$
$761$	−26.3923	−0.956720	−0.478360	+	0.878164i	$0.658769\pi$
$762$	0	0
$763$	2.80385i	0.101506i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.46410	0.341729
$768$	0	0
$769$	−12.2487	−0.441700	−0.220850	−	0.975308i	$-0.570883\pi$
$769$	−12.2487	−0.441700	−0.220850	+	0.975308i	$0.570883\pi$
$770$	0	0
$771$	36.0000i	1.29651i
$772$	0	0
$773$	− 22.8564i	− 0.822088i	−0.911616	−	0.411044i	$-0.865164\pi$
$773$	− 22.8564i	− 0.822088i	0.911616	−	0.411044i	$-0.134836\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	0.928203	0.0332991
$778$	0	0
$779$	− 5.85641i	− 0.209828i
$780$	0	0
$781$	− 7.46410i	− 0.267087i
$782$	0	0
$783$	−6.21539	−0.222120
$784$	0	0
$785$	−4.92820	−0.175895
$786$	0	0
$787$	− 38.5167i	− 1.37297i	−0.727144	−	0.686485i	$-0.759153\pi$
$787$	− 38.5167i	− 1.37297i	0.727144	−	0.686485i	$-0.240847\pi$
$788$	0	0
$789$	− 26.7846i	− 0.953557i
$790$	0	0
$791$	4.92820	0.175227
$792$	0	0
$793$	1.46410	0.0519918
$794$	0	0
$795$	− 2.53590i	− 0.0899390i
$796$	0	0
$797$	− 44.5692i	− 1.57872i	−0.613929	−	0.789361i	$-0.710412\pi$
$797$	− 44.5692i	− 1.57872i	0.613929	−	0.789361i	$-0.289588\pi$
$798$	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
$807$	26.7846	0.942863
$808$	0	0
$809$	−18.7128	−0.657907	−0.328954	−	0.944346i	$-0.606696\pi$
$809$	−18.7128	−0.657907	−0.328954	+	0.944346i	$0.606696\pi$
$810$	0	0
$811$	32.9282i	1.15627i	0.815943	+	0.578133i	$0.196218\pi$
$811$	32.9282i	1.15627i	−0.815943	+	0.578133i	$0.803782\pi$
$812$	0	0
$813$	− 25.1769i	− 0.882993i
$814$	0	0
$815$	−11.4641	−0.401570
$816$	0	0
$817$	−6.92820	−0.242387
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 13.4449i	− 0.469229i	−0.972088	−	0.234614i	$-0.924617\pi$
$821$	− 13.4449i	− 0.469229i	0.972088	−	0.234614i	$-0.0753828\pi$
$822$	0	0
$823$	30.6410	1.06808	0.534039	−	0.845460i	$-0.320673\pi$
$823$	30.6410	1.06808	0.534039	+	0.845460i	$0.320673\pi$
$824$	0	0
$825$	1.73205	0.0603023
$826$	0	0
$827$	− 39.4641i	− 1.37230i	−0.727459	−	0.686151i	$-0.759299\pi$
$827$	− 39.4641i	− 1.37230i	0.727459	−	0.686151i	$-0.240701\pi$
$828$	0	0
$829$	43.7128i	1.51821i	0.650969	+	0.759104i	$0.274363\pi$
$829$	43.7128i	1.51821i	−0.650969	+	0.759104i	$0.725637\pi$
$830$	0	0
$831$	48.2487	1.67373
$832$	0	0
$833$	−2.26795	−0.0785798
$834$	0	0
$835$	− 7.92820i	− 0.274367i
$836$	0	0
$837$	− 20.7846i	− 0.718421i
$838$	0	0
$839$	47.5692	1.64227	0.821136	−	0.570733i	$-0.193341\pi$
$839$	47.5692	1.64227	0.821136	+	0.570733i	$0.193341\pi$
$840$	0	0
$841$	27.5692	0.950663
$842$	0	0
$843$	− 48.1244i	− 1.65749i
$844$	0	0
$845$	12.0000i	0.412813i
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	−27.9282	−0.958493
$850$	0	0
$851$	1.85641i	0.0636368i
$852$	0	0
$853$	34.0000i	1.16414i	0.813139	+	0.582069i	$0.197757\pi$
$853$	34.0000i	1.16414i	−0.813139	+	0.582069i	$0.802243\pi$
$854$	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
$859$	22.6410i	0.772502i	0.922394	+	0.386251i	$0.126230\pi$
$859$	22.6410i	0.772502i	−0.922394	+	0.386251i	$0.873770\pi$
$860$	0	0
$861$	− 5.07180i	− 0.172846i
$862$	0	0
$863$	−46.9282	−1.59745	−0.798727	−	0.601693i	$-0.794493\pi$
$863$	−46.9282	−1.59745	−0.798727	+	0.601693i	$0.794493\pi$
$864$	0	0
$865$	5.00000	0.170005
$866$	0	0
$867$	20.5359i	0.697436i
$868$	0	0
$869$	− 2.26795i	− 0.0769349i
$870$	0	0
$871$	−5.46410	−0.185144
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.00000i	− 0.0338062i
$876$	0	0
$877$	37.7128i	1.27347i	0.771082	+	0.636736i	$0.219716\pi$
$877$	37.7128i	1.27347i	−0.771082	+	0.636736i	$0.780284\pi$
$878$	0	0
$879$	36.1244	1.21844
$880$	0	0
$881$	49.5692	1.67003	0.835015	−	0.550228i	$-0.185459\pi$
$881$	49.5692	1.67003	0.835015	+	0.550228i	$0.185459\pi$
$882$	0	0
$883$	− 15.3205i	− 0.515576i	−0.966201	−	0.257788i	$-0.917006\pi$
$883$	− 15.3205i	− 0.515576i	0.966201	−	0.257788i	$-0.0829936\pi$
$884$	0	0
$885$	− 16.3923i	− 0.551021i
$886$	0	0
$887$	−16.7846	−0.563572	−0.281786	−	0.959477i	$-0.590927\pi$
$887$	−16.7846	−0.563572	−0.281786	+	0.959477i	$0.590927\pi$
$888$	0	0
$889$	0.392305	0.0131575
$890$	0	0
$891$	9.00000i	0.301511i
$892$	0	0
$893$	− 19.8564i	− 0.664469i
$894$	0	0
$895$	1.07180	0.0358262
$896$	0	0
$897$	6.00000	0.200334
$898$	0	0
$899$	− 4.78461i	− 0.159576i
$900$	0	0
$901$	3.32051i	0.110622i
$902$	0	0
$903$	−6.00000	−0.199667
$904$	0	0
$905$	−4.39230	−0.146005
$906$	0	0
$907$	22.7846i	0.756551i	0.925693	+	0.378275i	$0.123483\pi$
$907$	22.7846i	0.756551i	−0.925693	+	0.378275i	$0.876517\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.60770	−0.0532653	−0.0266327	−	0.999645i	$-0.508478\pi$
$911$	−1.60770	−0.0532653	−0.0266327	+	0.999645i	$0.508478\pi$
$912$	0	0
$913$	−8.53590	−0.282497
$914$	0	0
$915$	− 2.53590i	− 0.0838342i
$916$	0	0
$917$	10.3923i	0.343184i
$918$	0	0
$919$	49.7321	1.64051	0.820254	−	0.571999i	$-0.193832\pi$
$919$	49.7321	1.64051	0.820254	+	0.571999i	$0.193832\pi$
$920$	0	0
$921$	−20.0718	−0.661388
$922$	0	0
$923$	7.46410i	0.245684i
$924$	0	0
$925$	− 0.535898i	− 0.0176202i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−33.4641	−1.09792	−0.548961	−	0.835848i	$-0.684976\pi$
$929$	−33.4641	−1.09792	−0.548961	+	0.835848i	$0.684976\pi$
$930$	0	0
$931$	− 2.00000i	− 0.0655474i
$932$	0	0
$933$	11.0718i	0.362474i
$934$	0	0
$935$	−2.26795	−0.0741699
$936$	0	0
$937$	17.1962	0.561774	0.280887	−	0.959741i	$-0.409371\pi$
$937$	17.1962	0.561774	0.280887	+	0.959741i	$0.409371\pi$
$938$	0	0
$939$	6.21539i	0.202832i
$940$	0	0
$941$	− 4.67949i	− 0.152547i	−0.997087	−	0.0762735i	$-0.975698\pi$
$941$	− 4.67949i	− 0.152547i	0.997087	−	0.0762735i	$-0.0243022\pi$
$942$	0	0
$943$	10.1436	0.330321
$944$	0	0
$945$	5.19615	0.169031
$946$	0	0
$947$	− 44.0000i	− 1.42981i	−0.699223	−	0.714904i	$-0.746470\pi$
$947$	− 44.0000i	− 1.42981i	0.699223	−	0.714904i	$-0.253530\pi$
$948$	0	0
$949$	4.00000i	0.129845i
$950$	0	0
$951$	19.8564	0.643888
$952$	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
$955$	8.12436i	0.262898i
$956$	0	0
$957$	2.07180i	0.0669717i
$958$	0	0
$959$	−16.3923	−0.529335
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 23.3205i	− 0.750714i
$966$	0	0
$967$	−16.5359	−0.531759	−0.265879	−	0.964006i	$-0.585662\pi$
$967$	−16.5359	−0.531759	−0.265879	+	0.964006i	$0.585662\pi$
$968$	0	0
$969$	7.85641	0.252384
$970$	0	0
$971$	− 12.0000i	− 0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$	− 12.0000i	− 0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$	0	0
$973$	− 20.9282i	− 0.670927i
$974$	0	0
$975$	−1.73205	−0.0554700
$976$	0	0
$977$	1.60770	0.0514347	0.0257174	−	0.999669i	$-0.491813\pi$
$977$	1.60770	0.0514347	0.0257174	+	0.999669i	$0.491813\pi$
$978$	0	0
$979$	− 2.53590i	− 0.0810477i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.7128	1.29854	0.649269	−	0.760559i	$-0.275075\pi$
$983$	40.7128	1.29854	0.649269	+	0.760559i	$0.275075\pi$
$984$	0	0
$985$	−8.00000	−0.254901
$986$	0	0
$987$	− 17.1962i	− 0.547360i
$988$	0	0
$989$	− 12.0000i	− 0.381578i
$990$	0	0
$991$	48.2487	1.53267	0.766335	−	0.642441i	$-0.222078\pi$
$991$	48.2487	1.53267	0.766335	+	0.642441i	$0.222078\pi$
$992$	0	0
$993$	49.8564	1.58215
$994$	0	0
$995$	− 17.4641i	− 0.553649i
$996$	0	0
$997$	− 42.5692i	− 1.34818i	−0.738649	−	0.674090i	$-0.764536\pi$
$997$	− 42.5692i	− 1.34818i	0.738649	−	0.674090i	$-0.235464\pi$
$998$	0	0
$999$	2.78461	0.0881012

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +1.00000i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.73205i q^{3} +1.00000i q^{5} +1.00000 q^{7} -1.00000i q^{11} +1.00000i q^{13} +1.73205 q^{15} -2.26795 q^{17} -2.00000i q^{19} -1.73205i q^{21} +3.46410 q^{23} -1.00000 q^{25} -5.19615i q^{27} -1.19615i q^{29} +4.00000 q^{31} -1.73205 q^{33} +1.00000i q^{35} +0.535898i q^{37} +1.73205 q^{39} +2.92820 q^{41} -3.46410i q^{43} +9.92820 q^{47} +1.00000 q^{49} +3.92820i q^{51} -1.46410i q^{53} +1.00000 q^{55} -3.46410 q^{57} -9.46410i q^{59} -1.46410i q^{61} -1.00000 q^{65} +5.46410i q^{67} -6.00000i q^{69} +7.46410 q^{71} +4.00000 q^{73} +1.73205i q^{75} -1.00000i q^{77} +2.26795 q^{79} -9.00000 q^{81} -8.53590i q^{83} -2.26795i q^{85} -2.07180 q^{87} +2.53590 q^{89} +1.00000i q^{91} -6.92820i q^{93} +2.00000 q^{95} -17.7321 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 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

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Database

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