LMFDB - Embedded newform 2240.2.b.h.1121.6

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 362 x^{8} - 794 x^{7} + 1463 x^{6} - 2048 x^{5} + 2258 x^{4} + \cdots + 61 $ x^12 - 6x^11 + 35x^10 - 120x^9 + 362x^8 - 794x^7 + 1463x^6 - 2048x^5 + 2258x^4 - 1822x^3 + 1035x^2 - 364*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.6
Root			$0.500000 + 2.51441i$ of defining polynomial
Character	$\chi$	$=$	2240.1121
Dual form			2240.2.b.h.1121.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.191804i q^{3} +1.00000i q^{5} +1.00000 q^{7} +2.96321 q^{9} +O(q^{10})$	$q-0.191804i q^{3} +1.00000i q^{5} +1.00000 q^{7} +2.96321 q^{9} -2.28284i q^{11} -7.09308i q^{13} +0.191804 q^{15} -4.56906 q^{17} -0.962523i q^{19} -0.191804i q^{21} -7.09510 q^{23} -1.00000 q^{25} -1.14377i q^{27} -8.27096i q^{29} -5.79834 q^{31} -0.437858 q^{33} +1.00000i q^{35} +4.01460i q^{37} -1.36048 q^{39} +4.14580 q^{41} +8.77547i q^{43} +2.96321i q^{45} -2.18259 q^{47} +1.00000 q^{49} +0.876363i q^{51} -4.79700i q^{53} +2.28284 q^{55} -0.184616 q^{57} -13.2877i q^{59} +0.486527i q^{61} +2.96321 q^{63} +7.09308 q^{65} +7.20902i q^{67} +1.36087i q^{69} +1.30311 q^{71} -8.33844 q^{73} +0.191804i q^{75} -2.28284i q^{77} -12.6482 q^{79} +8.67025 q^{81} -12.6474i q^{83} -4.56906i q^{85} -1.58640 q^{87} +9.09510 q^{89} -7.09308i q^{91} +1.11214i q^{93} +0.962523 q^{95} +1.64085 q^{97} -6.76455i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q + 12 * q^7 - 20 * q^9	$ 12 q + 12 q^{7} - 20 q^{9} + 16 q^{17} + 8 q^{23} - 12 q^{25} - 8 q^{31} + 72 q^{33} - 32 q^{39} - 32 q^{47} + 12 q^{49} + 8 q^{55} + 8 q^{57} - 20 q^{63} + 8 q^{65} + 48 q^{71} + 8 q^{73} - 16 q^{79} + 92 q^{81} - 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 - 20 * q^9 + 16 * q^17 + 8 * q^23 - 12 * q^25 - 8 * q^31 + 72 * q^33 - 32 * q^39 - 32 * q^47 + 12 * q^49 + 8 * q^55 + 8 * q^57 - 20 * q^63 + 8 * q^65 + 48 * q^71 + 8 * q^73 - 16 * q^79 + 92 * q^81 - 32 * q^87 + 16 * q^89 + 32 * 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
$3$	− 0.191804i	− 0.110738i	−0.998466	−	0.0553690i	$-0.982366\pi$
$3$	− 0.191804i	− 0.110738i	0.998466	−	0.0553690i	$-0.0176335\pi$
$4$	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
$9$	2.96321	0.987737
$10$	0	0
$11$	− 2.28284i	− 0.688303i	−0.938914	−	0.344152i	$-0.888167\pi$
$11$	− 2.28284i	− 0.688303i	0.938914	−	0.344152i	$-0.111833\pi$
$12$	0	0
$13$	− 7.09308i	− 1.96727i	−0.180185	−	0.983633i	$-0.557670\pi$
$13$	− 7.09308i	− 1.96727i	0.180185	−	0.983633i	$-0.442330\pi$
$14$	0	0
$15$	0.191804	0.0495236
$16$	0	0
$17$	−4.56906	−1.10816	−0.554080	−	0.832464i	$-0.686930\pi$
$17$	−4.56906	−1.10816	−0.554080	+	0.832464i	$0.686930\pi$
$18$	0	0
$19$	− 0.962523i	− 0.220818i	−0.993886	−	0.110409i	$-0.964784\pi$
$19$	− 0.962523i	− 0.220818i	0.993886	−	0.110409i	$-0.0352161\pi$
$20$	0	0
$21$	− 0.191804i	− 0.0418551i
$22$	0	0
$23$	−7.09510	−1.47943	−0.739715	−	0.672920i	$-0.765040\pi$
$23$	−7.09510	−1.47943	−0.739715	+	0.672920i	$0.765040\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.14377i	− 0.220118i
$28$	0	0
$29$	− 8.27096i	− 1.53588i	−0.640522	−	0.767940i	$-0.721282\pi$
$29$	− 8.27096i	− 1.53588i	0.640522	−	0.767940i	$-0.278718\pi$
$30$	0	0
$31$	−5.79834	−1.04141	−0.520706	−	0.853736i	$-0.674331\pi$
$31$	−5.79834	−1.04141	−0.520706	+	0.853736i	$0.674331\pi$
$32$	0	0
$33$	−0.437858	−0.0762214
$34$	0	0
$35$	1.00000i	0.169031i
$36$	0	0
$37$	4.01460i	0.659997i	0.943981	+	0.329998i	$0.107048\pi$
$37$	4.01460i	0.659997i	−0.943981	+	0.329998i	$0.892952\pi$
$38$	0	0
$39$	−1.36048	−0.217851
$40$	0	0
$41$	4.14580	0.647466	0.323733	−	0.946149i	$-0.395062\pi$
$41$	4.14580	0.647466	0.323733	+	0.946149i	$0.395062\pi$
$42$	0	0
$43$	8.77547i	1.33825i	0.743152	+	0.669123i	$0.233330\pi$
$43$	8.77547i	1.33825i	−0.743152	+	0.669123i	$0.766670\pi$
$44$	0	0
$45$	2.96321i	0.441729i
$46$	0	0
$47$	−2.18259	−0.318364	−0.159182	−	0.987249i	$-0.550886\pi$
$47$	−2.18259	−0.318364	−0.159182	+	0.987249i	$0.550886\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.876363i	0.122715i
$52$	0	0
$53$	− 4.79700i	− 0.658919i	−0.944170	−	0.329460i	$-0.893133\pi$
$53$	− 4.79700i	− 0.658919i	0.944170	−	0.329460i	$-0.106867\pi$
$54$	0	0
$55$	2.28284	0.307819
$56$	0	0
$57$	−0.184616	−0.0244530
$58$	0	0
$59$	− 13.2877i	− 1.72992i	−0.501844	−	0.864958i	$-0.667345\pi$
$59$	− 13.2877i	− 1.72992i	0.501844	−	0.864958i	$-0.332655\pi$
$60$	0	0
$61$	0.486527i	0.0622934i	0.999515	+	0.0311467i	$0.00991591\pi$
$61$	0.486527i	0.0622934i	−0.999515	+	0.0311467i	$0.990084\pi$
$62$	0	0
$63$	2.96321	0.373330
$64$	0	0
$65$	7.09308	0.879788
$66$	0	0
$67$	7.20902i	0.880722i	0.897821	+	0.440361i	$0.145150\pi$
$67$	7.20902i	0.880722i	−0.897821	+	0.440361i	$0.854850\pi$
$68$	0	0
$69$	1.36087i	0.163829i
$70$	0	0
$71$	1.30311	0.154651	0.0773256	−	0.997006i	$-0.475362\pi$
$71$	1.30311	0.154651	0.0773256	+	0.997006i	$0.475362\pi$
$72$	0	0
$73$	−8.33844	−0.975941	−0.487970	−	0.872860i	$-0.662263\pi$
$73$	−8.33844	−0.975941	−0.487970	+	0.872860i	$0.662263\pi$
$74$	0	0
$75$	0.191804i	0.0221476i
$76$	0	0
$77$	− 2.28284i	− 0.260154i
$78$	0	0
$79$	−12.6482	−1.42304	−0.711518	−	0.702668i	$-0.751992\pi$
$79$	−12.6482	−1.42304	−0.711518	+	0.702668i	$0.751992\pi$
$80$	0	0
$81$	8.67025	0.963362
$82$	0	0
$83$	− 12.6474i	− 1.38823i	−0.719864	−	0.694115i	$-0.755796\pi$
$83$	− 12.6474i	− 1.38823i	0.719864	−	0.694115i	$-0.244204\pi$
$84$	0	0
$85$	− 4.56906i	− 0.495584i
$86$	0	0
$87$	−1.58640	−0.170080
$88$	0	0
$89$	9.09510	0.964079	0.482039	−	0.876150i	$-0.339896\pi$
$89$	9.09510	0.964079	0.482039	+	0.876150i	$0.339896\pi$
$90$	0	0
$91$	− 7.09308i	− 0.743556i
$92$	0	0
$93$	1.11214i	0.115324i
$94$	0	0
$95$	0.962523	0.0987528
$96$	0	0
$97$	1.64085	0.166604	0.0833018	−	0.996524i	$-0.473453\pi$
$97$	1.64085	0.166604	0.0833018	+	0.996524i	$0.473453\pi$
$98$	0	0
$99$	− 6.76455i	− 0.679863i
$100$	0	0
$101$	− 2.67268i	− 0.265942i	−0.991120	−	0.132971i	$-0.957548\pi$
$101$	− 2.67268i	− 0.265942i	0.991120	−	0.132971i	$-0.0424516\pi$
$102$	0	0
$103$	2.18259	0.215057	0.107529	−	0.994202i	$-0.465706\pi$
$103$	2.18259	0.215057	0.107529	+	0.994202i	$0.465706\pi$
$104$	0	0
$105$	0.191804	0.0187182
$106$	0	0
$107$	− 2.66710i	− 0.257838i	−0.991655	−	0.128919i	$-0.958849\pi$
$107$	− 2.66710i	− 0.257838i	0.991655	−	0.128919i	$-0.0411507\pi$
$108$	0	0
$109$	− 5.63437i	− 0.539675i	−0.962906	−	0.269837i	$-0.913030\pi$
$109$	− 5.63437i	− 0.539675i	0.962906	−	0.269837i	$-0.0869699\pi$
$110$	0	0
$111$	0.770017	0.0730868
$112$	0	0
$113$	20.3892	1.91805	0.959027	−	0.283315i	$-0.0914343\pi$
$113$	20.3892	1.91805	0.959027	+	0.283315i	$0.0914343\pi$
$114$	0	0
$115$	− 7.09510i	− 0.661621i
$116$	0	0
$117$	− 21.0183i	− 1.94314i
$118$	0	0
$119$	−4.56906	−0.418845
$120$	0	0
$121$	5.78863	0.526239
$122$	0	0
$123$	− 0.795182i	− 0.0716991i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−8.31003	−0.737396	−0.368698	−	0.929549i	$-0.620196\pi$
$127$	−8.31003	−0.737396	−0.368698	+	0.929549i	$0.620196\pi$
$128$	0	0
$129$	1.68317	0.148195
$130$	0	0
$131$	− 9.60724i	− 0.839388i	−0.907666	−	0.419694i	$-0.862137\pi$
$131$	− 9.60724i	− 0.839388i	0.907666	−	0.419694i	$-0.137863\pi$
$132$	0	0
$133$	− 0.962523i	− 0.0834613i
$134$	0	0
$135$	1.14377	0.0984398
$136$	0	0
$137$	−0.486527	−0.0415668	−0.0207834	−	0.999784i	$-0.506616\pi$
$137$	−0.486527	−0.0415668	−0.0207834	+	0.999784i	$0.506616\pi$
$138$	0	0
$139$	17.6713i	1.49886i	0.662081	+	0.749432i	$0.269673\pi$
$139$	17.6713i	1.49886i	−0.662081	+	0.749432i	$0.730327\pi$
$140$	0	0
$141$	0.418630i	0.0352550i
$142$	0	0
$143$	−16.1924	−1.35407
$144$	0	0
$145$	8.27096	0.686866
$146$	0	0
$147$	− 0.191804i	− 0.0158197i
$148$	0	0
$149$	4.66847i	0.382456i	0.981546	+	0.191228i	$0.0612470\pi$
$149$	4.66847i	0.382456i	−0.981546	+	0.191228i	$0.938753\pi$
$150$	0	0
$151$	−14.5733	−1.18596	−0.592978	−	0.805219i	$-0.702048\pi$
$151$	−14.5733	−1.18596	−0.592978	+	0.805219i	$0.702048\pi$
$152$	0	0
$153$	−13.5391	−1.09457
$154$	0	0
$155$	− 5.79834i	− 0.465734i
$156$	0	0
$157$	− 15.2593i	− 1.21783i	−0.793237	−	0.608913i	$-0.791606\pi$
$157$	− 15.2593i	− 1.21783i	0.793237	−	0.608913i	$-0.208394\pi$
$158$	0	0
$159$	−0.920084	−0.0729674
$160$	0	0
$161$	−7.09510	−0.559172
$162$	0	0
$163$	5.31448i	0.416262i	0.978101	+	0.208131i	$0.0667381\pi$
$163$	5.31448i	0.416262i	−0.978101	+	0.208131i	$0.933262\pi$
$164$	0	0
$165$	− 0.437858i	− 0.0340872i
$166$	0	0
$167$	−18.3701	−1.42152	−0.710761	−	0.703433i	$-0.751649\pi$
$167$	−18.3701	−1.42152	−0.710761	+	0.703433i	$0.751649\pi$
$168$	0	0
$169$	−37.3117	−2.87013
$170$	0	0
$171$	− 2.85216i	− 0.218110i
$172$	0	0
$173$	− 1.83246i	− 0.139319i	−0.997571	−	0.0696596i	$-0.977809\pi$
$173$	− 1.83246i	− 0.139319i	0.997571	−	0.0696596i	$-0.0221913\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−2.54864	−0.191568
$178$	0	0
$179$	12.4499i	0.930551i	0.885166	+	0.465276i	$0.154045\pi$
$179$	12.4499i	0.930551i	−0.885166	+	0.465276i	$0.845955\pi$
$180$	0	0
$181$	− 3.81930i	− 0.283886i	−0.989875	−	0.141943i	$-0.954665\pi$
$181$	− 3.81930i	− 0.283886i	0.989875	−	0.141943i	$-0.0453350\pi$
$182$	0	0
$183$	0.0933179	0.00689826
$184$	0	0
$185$	−4.01460	−0.295160
$186$	0	0
$187$	10.4304i	0.762749i
$188$	0	0
$189$	− 1.14377i	− 0.0831969i
$190$	0	0
$191$	22.5957	1.63497	0.817485	−	0.575950i	$-0.195368\pi$
$191$	22.5957	1.63497	0.817485	+	0.575950i	$0.195368\pi$
$192$	0	0
$193$	23.5300	1.69372	0.846862	−	0.531812i	$-0.178489\pi$
$193$	23.5300	1.69372	0.846862	+	0.531812i	$0.178489\pi$
$194$	0	0
$195$	− 1.36048i	− 0.0974260i
$196$	0	0
$197$	− 6.06709i	− 0.432262i	−0.976364	−	0.216131i	$-0.930656\pi$
$197$	− 6.06709i	− 0.432262i	0.976364	−	0.216131i	$-0.0693439\pi$
$198$	0	0
$199$	24.4683	1.73452	0.867258	−	0.497860i	$-0.165881\pi$
$199$	24.4683	1.73452	0.867258	+	0.497860i	$0.165881\pi$
$200$	0	0
$201$	1.38272	0.0975295
$202$	0	0
$203$	− 8.27096i	− 0.580508i
$204$	0	0
$205$	4.14580i	0.289556i
$206$	0	0
$207$	−21.0243	−1.46129
$208$	0	0
$209$	−2.19729	−0.151990
$210$	0	0
$211$	− 18.0643i	− 1.24360i	−0.783177	−	0.621799i	$-0.786402\pi$
$211$	− 18.0643i	− 1.24360i	0.783177	−	0.621799i	$-0.213598\pi$
$212$	0	0
$213$	− 0.249942i	− 0.0171258i
$214$	0	0
$215$	−8.77547	−0.598482
$216$	0	0
$217$	−5.79834	−0.393617
$218$	0	0
$219$	1.59935i	0.108074i
$220$	0	0
$221$	32.4087i	2.18004i
$222$	0	0
$223$	26.8582	1.79856	0.899279	−	0.437376i	$-0.144092\pi$
$223$	26.8582	1.79856	0.899279	+	0.437376i	$0.144092\pi$
$224$	0	0
$225$	−2.96321	−0.197547
$226$	0	0
$227$	14.9436i	0.991844i	0.868367	+	0.495922i	$0.165170\pi$
$227$	14.9436i	0.991844i	−0.868367	+	0.495922i	$0.834830\pi$
$228$	0	0
$229$	16.0859i	1.06299i	0.847063	+	0.531493i	$0.178369\pi$
$229$	16.0859i	1.06299i	−0.847063	+	0.531493i	$0.821631\pi$
$230$	0	0
$231$	−0.437858	−0.0288090
$232$	0	0
$233$	−14.5949	−0.956143	−0.478072	−	0.878321i	$-0.658664\pi$
$233$	−14.5949	−0.956143	−0.478072	+	0.878321i	$0.658664\pi$
$234$	0	0
$235$	− 2.18259i	− 0.142377i
$236$	0	0
$237$	2.42598i	0.157584i
$238$	0	0
$239$	11.1332	0.720149	0.360075	−	0.932924i	$-0.382751\pi$
$239$	11.1332	0.720149	0.360075	+	0.932924i	$0.382751\pi$
$240$	0	0
$241$	17.7887	1.14587	0.572937	−	0.819600i	$-0.305804\pi$
$241$	17.7887	1.14587	0.572937	+	0.819600i	$0.305804\pi$
$242$	0	0
$243$	− 5.09429i	− 0.326799i
$244$	0	0
$245$	1.00000i	0.0638877i
$246$	0	0
$247$	−6.82725	−0.434408
$248$	0	0
$249$	−2.42582	−0.153730
$250$	0	0
$251$	− 18.1989i	− 1.14870i	−0.818608	−	0.574352i	$-0.805254\pi$
$251$	− 18.1989i	− 1.14870i	0.818608	−	0.574352i	$-0.194746\pi$
$252$	0	0
$253$	16.1970i	1.01830i
$254$	0	0
$255$	−0.876363	−0.0548800
$256$	0	0
$257$	8.71531	0.543646	0.271823	−	0.962347i	$-0.412373\pi$
$257$	8.71531	0.543646	0.271823	+	0.962347i	$0.412373\pi$
$258$	0	0
$259$	4.01460i	0.249455i
$260$	0	0
$261$	− 24.5086i	− 1.51704i
$262$	0	0
$263$	−7.22578	−0.445561	−0.222780	−	0.974869i	$-0.571513\pi$
$263$	−7.22578	−0.445561	−0.222780	+	0.974869i	$0.571513\pi$
$264$	0	0
$265$	4.79700	0.294678
$266$	0	0
$267$	− 1.74448i	− 0.106760i
$268$	0	0
$269$	− 24.0081i	− 1.46380i	−0.681412	−	0.731900i	$-0.738634\pi$
$269$	− 24.0081i	− 1.46380i	0.681412	−	0.731900i	$-0.261366\pi$
$270$	0	0
$271$	11.6009	0.704703	0.352352	−	0.935868i	$-0.385382\pi$
$271$	11.6009	0.704703	0.352352	+	0.935868i	$0.385382\pi$
$272$	0	0
$273$	−1.36048	−0.0823400
$274$	0	0
$275$	2.28284i	0.137661i
$276$	0	0
$277$	22.2267i	1.33547i	0.744397	+	0.667737i	$0.232737\pi$
$277$	22.2267i	1.33547i	−0.744397	+	0.667737i	$0.767263\pi$
$278$	0	0
$279$	−17.1817	−1.02864
$280$	0	0
$281$	19.0524	1.13657	0.568286	−	0.822831i	$-0.307607\pi$
$281$	19.0524	1.13657	0.568286	+	0.822831i	$0.307607\pi$
$282$	0	0
$283$	6.75611i	0.401609i	0.979631	+	0.200805i	$0.0643556\pi$
$283$	6.75611i	0.401609i	−0.979631	+	0.200805i	$0.935644\pi$
$284$	0	0
$285$	− 0.184616i	− 0.0109357i
$286$	0	0
$287$	4.14580	0.244719
$288$	0	0
$289$	3.87629	0.228017
$290$	0	0
$291$	− 0.314722i	− 0.0184494i
$292$	0	0
$293$	− 10.5274i	− 0.615017i	−0.951545	−	0.307508i	$-0.900505\pi$
$293$	− 10.5274i	− 0.615017i	0.951545	−	0.307508i	$-0.0994952\pi$
$294$	0	0
$295$	13.2877	0.773642
$296$	0	0
$297$	−2.61104	−0.151508
$298$	0	0
$299$	50.3261i	2.91043i
$300$	0	0
$301$	8.77547i	0.505809i
$302$	0	0
$303$	−0.512630	−0.0294499
$304$	0	0
$305$	−0.486527	−0.0278585
$306$	0	0
$307$	− 10.6866i	− 0.609915i	−0.952366	−	0.304958i	$-0.901358\pi$
$307$	− 10.6866i	− 0.609915i	0.952366	−	0.304958i	$-0.0986423\pi$
$308$	0	0
$309$	− 0.418630i	− 0.0238150i
$310$	0	0
$311$	34.2196	1.94041	0.970207	−	0.242278i	$-0.0778947\pi$
$311$	34.2196	1.94041	0.970207	+	0.242278i	$0.0778947\pi$
$312$	0	0
$313$	−15.3322	−0.866627	−0.433314	−	0.901243i	$-0.642656\pi$
$313$	−15.3322	−0.866627	−0.433314	+	0.901243i	$0.642656\pi$
$314$	0	0
$315$	2.96321i	0.166958i
$316$	0	0
$317$	− 8.05495i	− 0.452411i	−0.974080	−	0.226206i	$-0.927368\pi$
$317$	− 8.05495i	− 0.452411i	0.974080	−	0.226206i	$-0.0726321\pi$
$318$	0	0
$319$	−18.8813	−1.05715
$320$	0	0
$321$	−0.511560	−0.0285525
$322$	0	0
$323$	4.39782i	0.244701i
$324$	0	0
$325$	7.09308i	0.393453i
$326$	0	0
$327$	−1.08069	−0.0597625
$328$	0	0
$329$	−2.18259	−0.120330
$330$	0	0
$331$	− 12.7492i	− 0.700758i	−0.936608	−	0.350379i	$-0.886053\pi$
$331$	− 12.7492i	− 0.700758i	0.936608	−	0.350379i	$-0.113947\pi$
$332$	0	0
$333$	11.8961i	0.651903i
$334$	0	0
$335$	−7.20902	−0.393871
$336$	0	0
$337$	−8.80570	−0.479677	−0.239839	−	0.970813i	$-0.577095\pi$
$337$	−8.80570	−0.479677	−0.239839	+	0.970813i	$0.577095\pi$
$338$	0	0
$339$	− 3.91073i	− 0.212402i
$340$	0	0
$341$	13.2367i	0.716807i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.36087	−0.0732667
$346$	0	0
$347$	29.8409i	1.60194i	0.598702	+	0.800972i	$0.295683\pi$
$347$	29.8409i	1.60194i	−0.598702	+	0.800972i	$0.704317\pi$
$348$	0	0
$349$	− 18.9719i	− 1.01554i	−0.861492	−	0.507772i	$-0.830469\pi$
$349$	− 18.9719i	− 1.01554i	0.861492	−	0.507772i	$-0.169531\pi$
$350$	0	0
$351$	−8.11283	−0.433031
$352$	0	0
$353$	−0.853924	−0.0454498	−0.0227249	−	0.999742i	$-0.507234\pi$
$353$	−0.853924	−0.0454498	−0.0227249	+	0.999742i	$0.507234\pi$
$354$	0	0
$355$	1.30311i	0.0691621i
$356$	0	0
$357$	0.876363i	0.0463821i
$358$	0	0
$359$	−10.1272	−0.534491	−0.267245	−	0.963629i	$-0.586113\pi$
$359$	−10.1272	−0.534491	−0.267245	+	0.963629i	$0.586113\pi$
$360$	0	0
$361$	18.0735	0.951239
$362$	0	0
$363$	− 1.11028i	− 0.0582747i
$364$	0	0
$365$	− 8.33844i	− 0.436454i
$366$	0	0
$367$	29.0927	1.51863	0.759315	−	0.650724i	$-0.225534\pi$
$367$	29.0927	1.51863	0.759315	+	0.650724i	$0.225534\pi$
$368$	0	0
$369$	12.2849	0.639526
$370$	0	0
$371$	− 4.79700i	− 0.249048i
$372$	0	0
$373$	10.6342i	0.550617i	0.961356	+	0.275309i	$0.0887800\pi$
$373$	10.6342i	0.550617i	−0.961356	+	0.275309i	$0.911220\pi$
$374$	0	0
$375$	−0.191804	−0.00990471
$376$	0	0
$377$	−58.6666	−3.02148
$378$	0	0
$379$	− 34.7389i	− 1.78442i	−0.451625	−	0.892208i	$-0.649156\pi$
$379$	− 34.7389i	− 1.78442i	0.451625	−	0.892208i	$-0.350844\pi$
$380$	0	0
$381$	1.59390i	0.0816578i
$382$	0	0
$383$	−9.23011	−0.471637	−0.235818	−	0.971797i	$-0.575777\pi$
$383$	−9.23011	−0.471637	−0.235818	+	0.971797i	$0.575777\pi$
$384$	0	0
$385$	2.28284	0.116344
$386$	0	0
$387$	26.0036i	1.32184i
$388$	0	0
$389$	− 2.25392i	− 0.114278i	−0.998366	−	0.0571391i	$-0.981802\pi$
$389$	− 2.25392i	− 0.114278i	0.998366	−	0.0571391i	$-0.0181978\pi$
$390$	0	0
$391$	32.4179	1.63944
$392$	0	0
$393$	−1.84271	−0.0929522
$394$	0	0
$395$	− 12.6482i	− 0.636401i
$396$	0	0
$397$	7.14020i	0.358356i	0.983817	+	0.179178i	$0.0573438\pi$
$397$	7.14020i	0.358356i	−0.983817	+	0.179178i	$0.942656\pi$
$398$	0	0
$399$	−0.184616	−0.00924235
$400$	0	0
$401$	25.5177	1.27429	0.637147	−	0.770743i	$-0.280115\pi$
$401$	25.5177	1.27429	0.637147	+	0.770743i	$0.280115\pi$
$402$	0	0
$403$	41.1281i	2.04873i
$404$	0	0
$405$	8.67025i	0.430828i
$406$	0	0
$407$	9.16471	0.454278
$408$	0	0
$409$	−26.9071	−1.33047	−0.665236	−	0.746634i	$-0.731669\pi$
$409$	−26.9071	−1.33047	−0.665236	+	0.746634i	$0.731669\pi$
$410$	0	0
$411$	0.0933179i	0.00460303i
$412$	0	0
$413$	− 13.2877i	− 0.653847i
$414$	0	0
$415$	12.6474	0.620836
$416$	0	0
$417$	3.38943	0.165981
$418$	0	0
$419$	− 9.57935i	− 0.467982i	−0.972239	−	0.233991i	$-0.924821\pi$
$419$	− 9.57935i	− 0.467982i	0.972239	−	0.233991i	$-0.0751786\pi$
$420$	0	0
$421$	37.7603i	1.84032i	0.391539	+	0.920161i	$0.371943\pi$
$421$	37.7603i	1.84032i	−0.391539	+	0.920161i	$0.628057\pi$
$422$	0	0
$423$	−6.46748	−0.314460
$424$	0	0
$425$	4.56906	0.221632
$426$	0	0
$427$	0.486527i	0.0235447i
$428$	0	0
$429$	3.10576i	0.149948i
$430$	0	0
$431$	14.5243	0.699613	0.349806	−	0.936822i	$-0.386247\pi$
$431$	14.5243	0.699613	0.349806	+	0.936822i	$0.386247\pi$
$432$	0	0
$433$	−21.0665	−1.01239	−0.506196	−	0.862419i	$-0.668949\pi$
$433$	−21.0665	−1.01239	−0.506196	+	0.862419i	$0.668949\pi$
$434$	0	0
$435$	− 1.58640i	− 0.0760622i
$436$	0	0
$437$	6.82920i	0.326685i
$438$	0	0
$439$	−2.15309	−0.102761	−0.0513807	−	0.998679i	$-0.516362\pi$
$439$	−2.15309	−0.102761	−0.0513807	+	0.998679i	$0.516362\pi$
$440$	0	0
$441$	2.96321	0.141105
$442$	0	0
$443$	− 33.3536i	− 1.58468i	−0.610081	−	0.792339i	$-0.708863\pi$
$443$	− 33.3536i	− 1.58468i	0.610081	−	0.792339i	$-0.291137\pi$
$444$	0	0
$445$	9.09510i	0.431149i
$446$	0	0
$447$	0.895432	0.0423525
$448$	0	0
$449$	−21.9197	−1.03446	−0.517228	−	0.855848i	$-0.673036\pi$
$449$	−21.9197	−1.03446	−0.517228	+	0.855848i	$0.673036\pi$
$450$	0	0
$451$	− 9.46422i	− 0.445653i
$452$	0	0
$453$	2.79521i	0.131330i
$454$	0	0
$455$	7.09308	0.332529
$456$	0	0
$457$	−21.2437	−0.993740	−0.496870	−	0.867825i	$-0.665517\pi$
$457$	−21.2437	−0.993740	−0.496870	+	0.867825i	$0.665517\pi$
$458$	0	0
$459$	5.22594i	0.243926i
$460$	0	0
$461$	6.02516i	0.280620i	0.990108	+	0.140310i	$0.0448099\pi$
$461$	6.02516i	0.280620i	−0.990108	+	0.140310i	$0.955190\pi$
$462$	0	0
$463$	5.26475	0.244674	0.122337	−	0.992489i	$-0.460961\pi$
$463$	5.26475	0.244674	0.122337	+	0.992489i	$0.460961\pi$
$464$	0	0
$465$	−1.11214	−0.0515745
$466$	0	0
$467$	25.5467i	1.18216i	0.806614	+	0.591079i	$0.201298\pi$
$467$	25.5467i	1.18216i	−0.806614	+	0.591079i	$0.798702\pi$
$468$	0	0
$469$	7.20902i	0.332882i
$470$	0	0
$471$	−2.92680	−0.134860
$472$	0	0
$473$	20.0330	0.921119
$474$	0	0
$475$	0.962523i	0.0441636i
$476$	0	0
$477$	− 14.2145i	− 0.650839i
$478$	0	0
$479$	−22.4709	−1.02672	−0.513360	−	0.858173i	$-0.671599\pi$
$479$	−22.4709	−1.02672	−0.513360	+	0.858173i	$0.671599\pi$
$480$	0	0
$481$	28.4759	1.29839
$482$	0	0
$483$	1.36087i	0.0619216i
$484$	0	0
$485$	1.64085i	0.0745074i
$486$	0	0
$487$	31.4033	1.42302	0.711510	−	0.702676i	$-0.248012\pi$
$487$	31.4033	1.42302	0.711510	+	0.702676i	$0.248012\pi$
$488$	0	0
$489$	1.01934	0.0460961
$490$	0	0
$491$	4.61526i	0.208284i	0.994562	+	0.104142i	$0.0332096\pi$
$491$	4.61526i	0.208284i	−0.994562	+	0.104142i	$0.966790\pi$
$492$	0	0
$493$	37.7905i	1.70200i
$494$	0	0
$495$	6.76455	0.304044
$496$	0	0
$497$	1.30311	0.0584527
$498$	0	0
$499$	13.2110i	0.591408i	0.955280	+	0.295704i	$0.0955541\pi$
$499$	13.2110i	0.591408i	−0.955280	+	0.295704i	$0.904446\pi$
$500$	0	0
$501$	3.52346i	0.157417i
$502$	0	0
$503$	12.0644	0.537926	0.268963	−	0.963150i	$-0.413319\pi$
$503$	12.0644	0.537926	0.268963	+	0.963150i	$0.413319\pi$
$504$	0	0
$505$	2.67268	0.118933
$506$	0	0
$507$	7.15654i	0.317833i
$508$	0	0
$509$	24.8433i	1.10116i	0.834782	+	0.550580i	$0.185594\pi$
$509$	24.8433i	1.10116i	−0.834782	+	0.550580i	$0.814406\pi$
$510$	0	0
$511$	−8.33844	−0.368871
$512$	0	0
$513$	−1.10090	−0.0486060
$514$	0	0
$515$	2.18259i	0.0961765i
$516$	0	0
$517$	4.98252i	0.219131i
$518$	0	0
$519$	−0.351473	−0.0154279
$520$	0	0
$521$	−16.8102	−0.736470	−0.368235	−	0.929733i	$-0.620038\pi$
$521$	−16.8102	−0.736470	−0.368235	+	0.929733i	$0.620038\pi$
$522$	0	0
$523$	0.150199i	0.00656775i	0.999995	+	0.00328388i	$0.00104529\pi$
$523$	0.150199i	0.00656775i	−0.999995	+	0.00328388i	$0.998955\pi$
$524$	0	0
$525$	0.191804i	0.00837101i
$526$	0	0
$527$	26.4929	1.15405
$528$	0	0
$529$	27.3404	1.18871
$530$	0	0
$531$	− 39.3744i	− 1.70870i
$532$	0	0
$533$	− 29.4065i	− 1.27374i
$534$	0	0
$535$	2.66710	0.115309
$536$	0	0
$537$	2.38795	0.103047
$538$	0	0
$539$	− 2.28284i	− 0.0983290i
$540$	0	0
$541$	37.0978i	1.59496i	0.603347	+	0.797479i	$0.293833\pi$
$541$	37.0978i	1.59496i	−0.603347	+	0.797479i	$0.706167\pi$
$542$	0	0
$543$	−0.732556	−0.0314370
$544$	0	0
$545$	5.63437	0.241350
$546$	0	0
$547$	24.2098i	1.03514i	0.855642	+	0.517568i	$0.173162\pi$
$547$	24.2098i	1.03514i	−0.855642	+	0.517568i	$0.826838\pi$
$548$	0	0
$549$	1.44168i	0.0615295i
$550$	0	0
$551$	−7.96099	−0.339150
$552$	0	0
$553$	−12.6482	−0.537857
$554$	0	0
$555$	0.770017i	0.0326854i
$556$	0	0
$557$	7.17924i	0.304194i	0.988366	+	0.152097i	$0.0486026\pi$
$557$	7.17924i	0.304194i	−0.988366	+	0.152097i	$0.951397\pi$
$558$	0	0
$559$	62.2451	2.63269
$560$	0	0
$561$	2.00060	0.0844654
$562$	0	0
$563$	− 33.9927i	− 1.43262i	−0.697781	−	0.716312i	$-0.745829\pi$
$563$	− 33.9927i	− 1.43262i	0.697781	−	0.716312i	$-0.254171\pi$
$564$	0	0
$565$	20.3892i	0.857780i
$566$	0	0
$567$	8.67025	0.364116
$568$	0	0
$569$	−1.38269	−0.0579654	−0.0289827	−	0.999580i	$-0.509227\pi$
$569$	−1.38269	−0.0579654	−0.0289827	+	0.999580i	$0.509227\pi$
$570$	0	0
$571$	27.5840i	1.15435i	0.816619	+	0.577177i	$0.195846\pi$
$571$	27.5840i	1.15435i	−0.816619	+	0.577177i	$0.804154\pi$
$572$	0	0
$573$	− 4.33395i	− 0.181053i
$574$	0	0
$575$	7.09510	0.295886
$576$	0	0
$577$	−25.1968	−1.04895	−0.524477	−	0.851424i	$-0.675739\pi$
$577$	−25.1968	−1.04895	−0.524477	+	0.851424i	$0.675739\pi$
$578$	0	0
$579$	− 4.51314i	− 0.187560i
$580$	0	0
$581$	− 12.6474i	− 0.524702i
$582$	0	0
$583$	−10.9508	−0.453536
$584$	0	0
$585$	21.0183	0.868999
$586$	0	0
$587$	− 17.3370i	− 0.715574i	−0.933803	−	0.357787i	$-0.883531\pi$
$587$	− 17.3370i	− 0.715574i	0.933803	−	0.357787i	$-0.116469\pi$
$588$	0	0
$589$	5.58104i	0.229963i
$590$	0	0
$591$	−1.16369	−0.0478679
$592$	0	0
$593$	3.02762	0.124329	0.0621647	−	0.998066i	$-0.480200\pi$
$593$	3.02762	0.124329	0.0621647	+	0.998066i	$0.480200\pi$
$594$	0	0
$595$	− 4.56906i	− 0.187313i
$596$	0	0
$597$	− 4.69313i	− 0.192077i
$598$	0	0
$599$	20.0391	0.818774	0.409387	−	0.912361i	$-0.365743\pi$
$599$	20.0391	0.818774	0.409387	+	0.912361i	$0.365743\pi$
$600$	0	0
$601$	−41.5248	−1.69383	−0.846915	−	0.531728i	$-0.821543\pi$
$601$	−41.5248	−1.69383	−0.846915	+	0.531728i	$0.821543\pi$
$602$	0	0
$603$	21.3619i	0.869922i
$604$	0	0
$605$	5.78863i	0.235341i
$606$	0	0
$607$	−28.6125	−1.16134	−0.580672	−	0.814138i	$-0.697210\pi$
$607$	−28.6125	−1.16134	−0.580672	+	0.814138i	$0.697210\pi$
$608$	0	0
$609$	−1.58640	−0.0642843
$610$	0	0
$611$	15.4813i	0.626306i
$612$	0	0
$613$	− 14.5762i	− 0.588727i	−0.955694	−	0.294363i	$-0.904892\pi$
$613$	− 14.5762i	− 0.588727i	0.955694	−	0.294363i	$-0.0951076\pi$
$614$	0	0
$615$	0.795182	0.0320648
$616$	0	0
$617$	−17.9847	−0.724035	−0.362017	−	0.932171i	$-0.617912\pi$
$617$	−17.9847	−0.724035	−0.362017	+	0.932171i	$0.617912\pi$
$618$	0	0
$619$	11.7216i	0.471132i	0.971858	+	0.235566i	$0.0756943\pi$
$619$	11.7216i	0.471132i	−0.971858	+	0.235566i	$0.924306\pi$
$620$	0	0
$621$	8.11514i	0.325650i
$622$	0	0
$623$	9.09510	0.364387
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.421449i	0.0168310i
$628$	0	0
$629$	− 18.3430i	− 0.731382i
$630$	0	0
$631$	24.9398	0.992839	0.496419	−	0.868083i	$-0.334648\pi$
$631$	24.9398	0.992839	0.496419	+	0.868083i	$0.334648\pi$
$632$	0	0
$633$	−3.46480	−0.137714
$634$	0	0
$635$	− 8.31003i	− 0.329773i
$636$	0	0
$637$	− 7.09308i	− 0.281038i
$638$	0	0
$639$	3.86140	0.152755
$640$	0	0
$641$	33.5072	1.32346	0.661728	−	0.749744i	$-0.269823\pi$
$641$	33.5072	1.32346	0.661728	+	0.749744i	$0.269823\pi$
$642$	0	0
$643$	30.4807i	1.20204i	0.799233	+	0.601022i	$0.205239\pi$
$643$	30.4807i	1.20204i	−0.799233	+	0.601022i	$0.794761\pi$
$644$	0	0
$645$	1.68317i	0.0662747i
$646$	0	0
$647$	−36.4027	−1.43114	−0.715568	−	0.698543i	$-0.753832\pi$
$647$	−36.4027	−1.43114	−0.715568	+	0.698543i	$0.753832\pi$
$648$	0	0
$649$	−30.3338	−1.19071
$650$	0	0
$651$	1.11214i	0.0435884i
$652$	0	0
$653$	11.4937i	0.449782i	0.974384	+	0.224891i	$0.0722026\pi$
$653$	11.4937i	0.449782i	−0.974384	+	0.224891i	$0.927797\pi$
$654$	0	0
$655$	9.60724	0.375386
$656$	0	0
$657$	−24.7086	−0.963973
$658$	0	0
$659$	5.83329i	0.227233i	0.993525	+	0.113616i	$0.0362435\pi$
$659$	5.83329i	0.227233i	−0.993525	+	0.113616i	$0.963757\pi$
$660$	0	0
$661$	− 20.9106i	− 0.813326i	−0.913578	−	0.406663i	$-0.866692\pi$
$661$	− 20.9106i	− 0.813326i	0.913578	−	0.406663i	$-0.133308\pi$
$662$	0	0
$663$	6.21611	0.241414
$664$	0	0
$665$	0.962523	0.0373250
$666$	0	0
$667$	58.6833i	2.27223i
$668$	0	0
$669$	− 5.15151i	− 0.199169i
$670$	0	0
$671$	1.11067	0.0428768
$672$	0	0
$673$	−37.0222	−1.42710	−0.713551	−	0.700603i	$-0.752914\pi$
$673$	−37.0222	−1.42710	−0.713551	+	0.700603i	$0.752914\pi$
$674$	0	0
$675$	1.14377i	0.0440236i
$676$	0	0
$677$	− 9.39877i	− 0.361224i	−0.983554	−	0.180612i	$-0.942192\pi$
$677$	− 9.39877i	− 0.361224i	0.983554	−	0.180612i	$-0.0578078\pi$
$678$	0	0
$679$	1.64085	0.0629702
$680$	0	0
$681$	2.86625	0.109835
$682$	0	0
$683$	46.4388i	1.77693i	0.458945	+	0.888465i	$0.348228\pi$
$683$	46.4388i	1.77693i	−0.458945	+	0.888465i	$0.651772\pi$
$684$	0	0
$685$	− 0.486527i	− 0.0185893i
$686$	0	0
$687$	3.08534	0.117713
$688$	0	0
$689$	−34.0255	−1.29627
$690$	0	0
$691$	12.7719i	0.485865i	0.970043	+	0.242932i	$0.0781093\pi$
$691$	12.7719i	0.485865i	−0.970043	+	0.242932i	$0.921891\pi$
$692$	0	0
$693$	− 6.76455i	− 0.256964i
$694$	0	0
$695$	−17.6713	−0.670312
$696$	0	0
$697$	−18.9424	−0.717495
$698$	0	0
$699$	2.79936i	0.105881i
$700$	0	0
$701$	− 3.07006i	− 0.115954i	−0.998318	−	0.0579772i	$-0.981535\pi$
$701$	− 3.07006i	− 0.115954i	0.998318	−	0.0579772i	$-0.0184651\pi$
$702$	0	0
$703$	3.86415	0.145739
$704$	0	0
$705$	−0.418630	−0.0157665
$706$	0	0
$707$	− 2.67268i	− 0.100516i
$708$	0	0
$709$	33.3877i	1.25390i	0.779059	+	0.626951i	$0.215697\pi$
$709$	33.3877i	1.25390i	−0.779059	+	0.626951i	$0.784303\pi$
$710$	0	0
$711$	−37.4793	−1.40558
$712$	0	0
$713$	41.1398	1.54070
$714$	0	0
$715$	− 16.1924i	− 0.605561i
$716$	0	0
$717$	− 2.13540i	− 0.0797479i
$718$	0	0
$719$	−33.8589	−1.26273	−0.631363	−	0.775488i	$-0.717504\pi$
$719$	−33.8589	−1.26273	−0.631363	+	0.775488i	$0.717504\pi$
$720$	0	0
$721$	2.18259	0.0812840
$722$	0	0
$723$	− 3.41195i	− 0.126892i
$724$	0	0
$725$	8.27096i	0.307176i
$726$	0	0
$727$	−29.0976	−1.07917	−0.539585	−	0.841931i	$-0.681419\pi$
$727$	−29.0976	−1.07917	−0.539585	+	0.841931i	$0.681419\pi$
$728$	0	0
$729$	25.0337	0.927173
$730$	0	0
$731$	− 40.0956i	− 1.48299i
$732$	0	0
$733$	37.5192i	1.38580i	0.721032	+	0.692902i	$0.243668\pi$
$733$	37.5192i	1.38580i	−0.721032	+	0.692902i	$0.756332\pi$
$734$	0	0
$735$	0.191804	0.00707480
$736$	0	0
$737$	16.4571	0.606204
$738$	0	0
$739$	34.3673i	1.26422i	0.774878	+	0.632111i	$0.217811\pi$
$739$	34.3673i	1.26422i	−0.774878	+	0.632111i	$0.782189\pi$
$740$	0	0
$741$	1.30949i	0.0481055i
$742$	0	0
$743$	4.67144	0.171378	0.0856892	−	0.996322i	$-0.472691\pi$
$743$	4.67144	0.171378	0.0856892	+	0.996322i	$0.472691\pi$
$744$	0	0
$745$	−4.66847	−0.171040
$746$	0	0
$747$	− 37.4769i	− 1.37121i
$748$	0	0
$749$	− 2.66710i	− 0.0974536i
$750$	0	0
$751$	13.1081	0.478321	0.239161	−	0.970980i	$-0.423128\pi$
$751$	13.1081	0.478321	0.239161	+	0.970980i	$0.423128\pi$
$752$	0	0
$753$	−3.49062	−0.127205
$754$	0	0
$755$	− 14.5733i	− 0.530375i
$756$	0	0
$757$	− 34.7064i	− 1.26142i	−0.776017	−	0.630712i	$-0.782763\pi$
$757$	− 34.7064i	− 1.26142i	0.776017	−	0.630712i	$-0.217237\pi$
$758$	0	0
$759$	3.10665	0.112764
$760$	0	0
$761$	19.0543	0.690718	0.345359	−	0.938471i	$-0.387757\pi$
$761$	19.0543	0.690718	0.345359	+	0.938471i	$0.387757\pi$
$762$	0	0
$763$	− 5.63437i	− 0.203978i
$764$	0	0
$765$	− 13.5391i	− 0.489507i
$766$	0	0
$767$	−94.2509	−3.40320
$768$	0	0
$769$	23.3346	0.841466	0.420733	−	0.907184i	$-0.361773\pi$
$769$	23.3346	0.841466	0.420733	+	0.907184i	$0.361773\pi$
$770$	0	0
$771$	− 1.67163i	− 0.0602023i
$772$	0	0
$773$	− 11.9937i	− 0.431384i	−0.976461	−	0.215692i	$-0.930799\pi$
$773$	− 11.9937i	− 0.431384i	0.976461	−	0.215692i	$-0.0692007\pi$
$774$	0	0
$775$	5.79834	0.208282
$776$	0	0
$777$	0.770017	0.0276242
$778$	0	0
$779$	− 3.99043i	− 0.142972i
$780$	0	0
$781$	− 2.97481i	− 0.106447i
$782$	0	0
$783$	−9.46006	−0.338075
$784$	0	0
$785$	15.2593	0.544629
$786$	0	0
$787$	− 52.5461i	− 1.87306i	−0.350581	−	0.936532i	$-0.614016\pi$
$787$	− 52.5461i	− 1.87306i	0.350581	−	0.936532i	$-0.385984\pi$
$788$	0	0
$789$	1.38593i	0.0493405i
$790$	0	0
$791$	20.3892	0.724956
$792$	0	0
$793$	3.45098	0.122548
$794$	0	0
$795$	− 0.920084i	− 0.0326320i
$796$	0	0
$797$	10.5448i	0.373515i	0.982406	+	0.186758i	$0.0597979\pi$
$797$	10.5448i	0.373515i	−0.982406	+	0.186758i	$0.940202\pi$
$798$	0	0
$799$	9.97239	0.352798
$800$	0	0
$801$	26.9507	0.952256
$802$	0	0
$803$	19.0354i	0.671743i
$804$	0	0
$805$	− 7.09510i	− 0.250069i
$806$	0	0
$807$	−4.60485	−0.162098
$808$	0	0
$809$	28.7886	1.01215	0.506077	−	0.862488i	$-0.331095\pi$
$809$	28.7886	1.01215	0.506077	+	0.862488i	$0.331095\pi$
$810$	0	0
$811$	1.60362i	0.0563108i	0.999604	+	0.0281554i	$0.00896332\pi$
$811$	1.60362i	0.0563108i	−0.999604	+	0.0281554i	$0.991037\pi$
$812$	0	0
$813$	− 2.22510i	− 0.0780375i
$814$	0	0
$815$	−5.31448	−0.186158
$816$	0	0
$817$	8.44659	0.295509
$818$	0	0
$819$	− 21.0183i	− 0.734438i
$820$	0	0
$821$	41.8725i	1.46136i	0.682720	+	0.730680i	$0.260797\pi$
$821$	41.8725i	1.46136i	−0.682720	+	0.730680i	$0.739203\pi$
$822$	0	0
$823$	16.8752	0.588234	0.294117	−	0.955769i	$-0.404975\pi$
$823$	16.8752	0.588234	0.294117	+	0.955769i	$0.404975\pi$
$824$	0	0
$825$	0.437858	0.0152443
$826$	0	0
$827$	− 0.789683i	− 0.0274600i	−0.999906	−	0.0137300i	$-0.995629\pi$
$827$	− 0.789683i	− 0.0274600i	0.999906	−	0.0137300i	$-0.00437053\pi$
$828$	0	0
$829$	− 25.8343i	− 0.897261i	−0.893717	−	0.448631i	$-0.851912\pi$
$829$	− 25.8343i	− 0.897261i	0.893717	−	0.448631i	$-0.148088\pi$
$830$	0	0
$831$	4.26317	0.147888
$832$	0	0
$833$	−4.56906	−0.158308
$834$	0	0
$835$	− 18.3701i	− 0.635724i
$836$	0	0
$837$	6.63195i	0.229234i
$838$	0	0
$839$	18.8432	0.650538	0.325269	−	0.945621i	$-0.394545\pi$
$839$	18.8432	0.650538	0.325269	+	0.945621i	$0.394545\pi$
$840$	0	0
$841$	−39.4088	−1.35893
$842$	0	0
$843$	− 3.65433i	− 0.125862i
$844$	0	0
$845$	− 37.3117i	− 1.28356i
$846$	0	0
$847$	5.78863	0.198900
$848$	0	0
$849$	1.29585	0.0444734
$850$	0	0
$851$	− 28.4840i	− 0.976420i
$852$	0	0
$853$	− 24.8587i	− 0.851145i	−0.904924	−	0.425572i	$-0.860073\pi$
$853$	− 24.8587i	− 0.851145i	0.904924	−	0.425572i	$-0.139927\pi$
$854$	0	0
$855$	2.85216	0.0975418
$856$	0	0
$857$	−19.8448	−0.677885	−0.338943	−	0.940807i	$-0.610069\pi$
$857$	−19.8448	−0.677885	−0.338943	+	0.940807i	$0.610069\pi$
$858$	0	0
$859$	− 23.1860i	− 0.791094i	−0.918446	−	0.395547i	$-0.870555\pi$
$859$	− 23.1860i	− 0.791094i	0.918446	−	0.395547i	$-0.129445\pi$
$860$	0	0
$861$	− 0.795182i	− 0.0270997i
$862$	0	0
$863$	13.5878	0.462532	0.231266	−	0.972891i	$-0.425713\pi$
$863$	13.5878	0.462532	0.231266	+	0.972891i	$0.425713\pi$
$864$	0	0
$865$	1.83246	0.0623054
$866$	0	0
$867$	− 0.743488i	− 0.0252502i
$868$	0	0
$869$	28.8739i	0.979480i
$870$	0	0
$871$	51.1342	1.73261
$872$	0	0
$873$	4.86220	0.164560
$874$	0	0
$875$	− 1.00000i	− 0.0338062i
$876$	0	0
$877$	− 36.5476i	− 1.23412i	−0.786914	−	0.617062i	$-0.788323\pi$
$877$	− 36.5476i	− 1.23412i	0.786914	−	0.617062i	$-0.211677\pi$
$878$	0	0
$879$	−2.01920	−0.0681058
$880$	0	0
$881$	36.7800	1.23915	0.619575	−	0.784937i	$-0.287305\pi$
$881$	36.7800	1.23915	0.619575	+	0.784937i	$0.287305\pi$
$882$	0	0
$883$	− 43.3640i	− 1.45931i	−0.683813	−	0.729657i	$-0.739680\pi$
$883$	− 43.3640i	− 1.45931i	0.683813	−	0.729657i	$-0.260320\pi$
$884$	0	0
$885$	− 2.54864i	− 0.0856716i
$886$	0	0
$887$	24.6046	0.826142	0.413071	−	0.910699i	$-0.364456\pi$
$887$	24.6046	0.826142	0.413071	+	0.910699i	$0.364456\pi$
$888$	0	0
$889$	−8.31003	−0.278709
$890$	0	0
$891$	− 19.7928i	− 0.663085i
$892$	0	0
$893$	2.10080i	0.0703005i
$894$	0	0
$895$	−12.4499	−0.416155
$896$	0	0
$897$	9.65274	0.322296
$898$	0	0
$899$	47.9578i	1.59948i
$900$	0	0
$901$	21.9178i	0.730187i
$902$	0	0
$903$	1.68317	0.0560124
$904$	0	0
$905$	3.81930	0.126958
$906$	0	0
$907$	19.2254i	0.638368i	0.947693	+	0.319184i	$0.103409\pi$
$907$	19.2254i	0.638368i	−0.947693	+	0.319184i	$0.896591\pi$
$908$	0	0
$909$	− 7.91971i	− 0.262680i
$910$	0	0
$911$	−27.7858	−0.920585	−0.460293	−	0.887767i	$-0.652255\pi$
$911$	−27.7858	−0.920585	−0.460293	+	0.887767i	$0.652255\pi$
$912$	0	0
$913$	−28.8720	−0.955523
$914$	0	0
$915$	0.0933179i	0.00308499i
$916$	0	0
$917$	− 9.60724i	− 0.317259i
$918$	0	0
$919$	−49.0084	−1.61664	−0.808320	−	0.588744i	$-0.799623\pi$
$919$	−49.0084	−1.61664	−0.808320	+	0.588744i	$0.799623\pi$
$920$	0	0
$921$	−2.04973	−0.0675409
$922$	0	0
$923$	− 9.24309i	− 0.304240i
$924$	0	0
$925$	− 4.01460i	− 0.131999i
$926$	0	0
$927$	6.46748	0.212420
$928$	0	0
$929$	−17.9803	−0.589914	−0.294957	−	0.955511i	$-0.595305\pi$
$929$	−17.9803	−0.589914	−0.294957	+	0.955511i	$0.595305\pi$
$930$	0	0
$931$	− 0.962523i	− 0.0315454i
$932$	0	0
$933$	− 6.56345i	− 0.214878i
$934$	0	0
$935$	−10.4304	−0.341112
$936$	0	0
$937$	16.8426	0.550224	0.275112	−	0.961412i	$-0.411285\pi$
$937$	16.8426	0.550224	0.275112	+	0.961412i	$0.411285\pi$
$938$	0	0
$939$	2.94078i	0.0959686i
$940$	0	0
$941$	− 9.51080i	− 0.310043i	−0.987911	−	0.155022i	$-0.950455\pi$
$941$	− 9.51080i	− 0.310043i	0.987911	−	0.155022i	$-0.0495447\pi$
$942$	0	0
$943$	−29.4149	−0.957881
$944$	0	0
$945$	1.14377	0.0372068
$946$	0	0
$947$	− 21.3596i	− 0.694093i	−0.937848	−	0.347046i	$-0.887185\pi$
$947$	− 21.3596i	− 0.694093i	0.937848	−	0.347046i	$-0.112815\pi$
$948$	0	0
$949$	59.1452i	1.91993i
$950$	0	0
$951$	−1.54497	−0.0500991
$952$	0	0
$953$	−28.7813	−0.932317	−0.466158	−	0.884701i	$-0.654362\pi$
$953$	−28.7813	−0.932317	−0.466158	+	0.884701i	$0.654362\pi$
$954$	0	0
$955$	22.5957i	0.731181i
$956$	0	0
$957$	3.62151i	0.117067i
$958$	0	0
$959$	−0.486527	−0.0157108
$960$	0	0
$961$	2.62073	0.0845397
$962$	0	0
$963$	− 7.90317i	− 0.254676i
$964$	0	0
$965$	23.5300i	0.757457i
$966$	0	0
$967$	−6.14801	−0.197707	−0.0988534	−	0.995102i	$-0.531517\pi$
$967$	−6.14801	−0.197707	−0.0988534	+	0.995102i	$0.531517\pi$
$968$	0	0
$969$	0.843520	0.0270978
$970$	0	0
$971$	− 36.7772i	− 1.18024i	−0.807316	−	0.590119i	$-0.799081\pi$
$971$	− 36.7772i	− 1.18024i	0.807316	−	0.590119i	$-0.200919\pi$
$972$	0	0
$973$	17.6713i	0.566517i
$974$	0	0
$975$	1.36048	0.0435702
$976$	0	0
$977$	34.4277	1.10144	0.550719	−	0.834690i	$-0.314353\pi$
$977$	34.4277	1.10144	0.550719	+	0.834690i	$0.314353\pi$
$978$	0	0
$979$	− 20.7627i	− 0.663578i
$980$	0	0
$981$	− 16.6958i	− 0.533057i
$982$	0	0
$983$	30.4682	0.971786	0.485893	−	0.874018i	$-0.338495\pi$
$983$	30.4682	0.971786	0.485893	+	0.874018i	$0.338495\pi$
$984$	0	0
$985$	6.06709	0.193314
$986$	0	0
$987$	0.418630i	0.0133251i
$988$	0	0
$989$	− 62.2628i	− 1.97984i
$990$	0	0
$991$	44.7242	1.42071	0.710356	−	0.703843i	$-0.248534\pi$
$991$	44.7242	1.42071	0.710356	+	0.703843i	$0.248534\pi$
$992$	0	0
$993$	−2.44534	−0.0776006
$994$	0	0
$995$	24.4683i	0.775699i
$996$	0	0
$997$	− 10.5274i	− 0.333406i	−0.986007	−	0.166703i	$-0.946688\pi$
$997$	− 10.5274i	− 0.333406i	0.986007	−	0.166703i	$-0.0533121\pi$
$998$	0	0
$999$	4.59178	0.145277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.b.h.1121.6	yes	12
4.3	odd	2		2240.2.b.g.1121.7	yes	12
8.3	odd	2		2240.2.b.g.1121.6	✓	12
8.5	even	2	inner	2240.2.b.h.1121.7	yes	12
16.3	odd	4		8960.2.a.ch.1.4		6
16.5	even	4		8960.2.a.cb.1.4		6
16.11	odd	4		8960.2.a.cc.1.3		6
16.13	even	4		8960.2.a.ce.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.g.1121.6	✓	12	8.3	odd	2
2240.2.b.g.1121.7	yes	12	4.3	odd	2
2240.2.b.h.1121.6	yes	12	1.1	even	1	trivial
2240.2.b.h.1121.7	yes	12	8.5	even	2	inner
8960.2.a.cb.1.4		6	16.5	even	4
8960.2.a.cc.1.3		6	16.11	odd	4
8960.2.a.ce.1.3		6	16.13	even	4
8960.2.a.ch.1.4		6	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)

\(f(q)\)	\(=\)	\(q-0.191804i q^{3} +1.00000i q^{5} +1.00000 q^{7} +2.96321 q^{9} +O(q^{10})\)	\(q-0.191804i q^{3} +1.00000i q^{5} +1.00000 q^{7} +2.96321 q^{9} -2.28284i q^{11} -7.09308i q^{13} +0.191804 q^{15} -4.56906 q^{17} -0.962523i q^{19} -0.191804i q^{21} -7.09510 q^{23} -1.00000 q^{25} -1.14377i q^{27} -8.27096i q^{29} -5.79834 q^{31} -0.437858 q^{33} +1.00000i q^{35} +4.01460i q^{37} -1.36048 q^{39} +4.14580 q^{41} +8.77547i q^{43} +2.96321i q^{45} -2.18259 q^{47} +1.00000 q^{49} +0.876363i q^{51} -4.79700i q^{53} +2.28284 q^{55} -0.184616 q^{57} -13.2877i q^{59} +0.486527i q^{61} +2.96321 q^{63} +7.09308 q^{65} +7.20902i q^{67} +1.36087i q^{69} +1.30311 q^{71} -8.33844 q^{73} +0.191804i q^{75} -2.28284i q^{77} -12.6482 q^{79} +8.67025 q^{81} -12.6474i q^{83} -4.56906i q^{85} -1.58640 q^{87} +9.09510 q^{89} -7.09308i q^{91} +1.11214i q^{93} +0.962523 q^{95} +1.64085 q^{97} -6.76455i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q + 12 * q^7 - 20 * q^9	\( 12 q + 12 q^{7} - 20 q^{9} + 16 q^{17} + 8 q^{23} - 12 q^{25} - 8 q^{31} + 72 q^{33} - 32 q^{39} - 32 q^{47} + 12 q^{49} + 8 q^{55} + 8 q^{57} - 20 q^{63} + 8 q^{65} + 48 q^{71} + 8 q^{73} - 16 q^{79} + 92 q^{81} - 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 - 20 * q^9 + 16 * q^17 + 8 * q^23 - 12 * q^25 - 8 * q^31 + 72 * q^33 - 32 * q^39 - 32 * q^47 + 12 * q^49 + 8 * q^55 + 8 * q^57 - 20 * q^63 + 8 * q^65 + 48 * q^71 + 8 * q^73 - 16 * q^79 + 92 * q^81 - 32 * q^87 + 16 * q^89 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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