LMFDB - Embedded newform 2240.2.h.c.671.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(671,2240)]

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([1, 1, 0, 1]))

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

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

chi := DirichletCharacter("2240.671");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.h (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:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			671.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2240.671
Dual form			2240.2.h.c.671.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +1.00000 q^{5} +(-2.44949 + 1.00000i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{3} +1.00000 q^{5} +(-2.44949 + 1.00000i) q^{7} -1.00000 q^{9} -2.00000 q^{13} -2.00000i q^{15} -4.89898i q^{17} +2.00000i q^{19} +(2.00000 + 4.89898i) q^{21} -6.00000i q^{23} +1.00000 q^{25} -4.00000i q^{27} +9.79796i q^{29} -9.79796 q^{31} +(-2.44949 + 1.00000i) q^{35} -4.89898i q^{37} +4.00000i q^{39} -4.89898 q^{43} -1.00000 q^{45} -4.89898 q^{47} +(5.00000 - 4.89898i) q^{49} -9.79796 q^{51} +4.89898i q^{53} +4.00000 q^{57} -6.00000i q^{59} -10.0000 q^{61} +(2.44949 - 1.00000i) q^{63} -2.00000 q^{65} -14.6969 q^{67} -12.0000 q^{69} +6.00000i q^{71} +4.89898i q^{73} -2.00000i q^{75} -10.0000i q^{79} -11.0000 q^{81} +6.00000i q^{83} -4.89898i q^{85} +19.5959 q^{87} -9.79796i q^{89} +(4.89898 - 2.00000i) q^{91} +19.5959i q^{93} +2.00000i q^{95} -4.89898i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 - 4 * q^9	$ 4 q + 4 q^{5} - 4 q^{9} - 8 q^{13} + 8 q^{21} + 4 q^{25} - 4 q^{45} + 20 q^{49} + 16 q^{57} - 40 q^{61} - 8 q^{65} - 48 q^{69} - 44 q^{81}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^9 - 8 * q^13 + 8 * q^21 + 4 * q^25 - 4 * q^45 + 20 * q^49 + 16 * q^57 - 40 * q^61 - 8 * q^65 - 48 * q^69 - 44 * q^81

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
$3$		−	2.00000i		−	1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$		−	2.00000i		−	1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−2.44949	+	1.00000i	−0.925820	+	0.377964i
$7$	−2.44949	+	1.00000i	−0.925820	+	0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		−	2.00000i		−	0.516398i
$16$		0			0
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$		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
$21$	2.00000	+	4.89898i	0.436436	+	1.06904i
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	4.00000i		−	0.769800i
$28$		0			0
$29$			9.79796i			1.81944i	0.415227	+	0.909718i	$0.363702\pi$
$29$			9.79796i			1.81944i	−0.415227	+	0.909718i	$0.636298\pi$
$30$		0			0
$31$	−9.79796			−1.75977			−0.879883	−	0.475191i	$-0.842379\pi$
$31$	−9.79796			−1.75977			−0.879883	+	0.475191i	$0.842379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.44949	+	1.00000i	−0.414039	+	0.169031i
$36$		0			0
$37$		−	4.89898i		−	0.805387i	−0.915335	−	0.402694i	$-0.868074\pi$
$37$		−	4.89898i		−	0.805387i	0.915335	−	0.402694i	$-0.131926\pi$
$38$		0			0
$39$			4.00000i			0.640513i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−4.89898			−0.747087			−0.373544	−	0.927613i	$-0.621857\pi$
$43$	−4.89898			−0.747087			−0.373544	+	0.927613i	$0.621857\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$	−4.89898			−0.714590			−0.357295	−	0.933992i	$-0.616301\pi$
$47$	−4.89898			−0.714590			−0.357295	+	0.933992i	$0.616301\pi$
$48$		0			0
$49$	5.00000	−	4.89898i	0.714286	−	0.699854i
$50$		0			0
$51$	−9.79796			−1.37199
$52$		0			0
$53$			4.89898i			0.672927i	0.941697	+	0.336463i	$0.109231\pi$
$53$			4.89898i			0.672927i	−0.941697	+	0.336463i	$0.890769\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.00000			0.529813
$58$		0			0
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$	2.44949	−	1.00000i	0.308607	−	0.125988i
$64$		0			0
$65$	−2.00000			−0.248069
$66$		0			0
$67$	−14.6969			−1.79552			−0.897758	−	0.440488i	$-0.854805\pi$
$67$	−14.6969			−1.79552			−0.897758	+	0.440488i	$0.854805\pi$
$68$		0			0
$69$	−12.0000			−1.44463
$70$		0			0
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		0			0
$73$			4.89898i			0.573382i	0.958023	+	0.286691i	$0.0925553\pi$
$73$			4.89898i			0.573382i	−0.958023	+	0.286691i	$0.907445\pi$
$74$		0			0
$75$		−	2.00000i		−	0.230940i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	10.0000i		−	1.12509i	−0.826767	−	0.562544i	$-0.809823\pi$
$79$		−	10.0000i		−	1.12509i	0.826767	−	0.562544i	$-0.190177\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$		0			0
$85$		−	4.89898i		−	0.531369i
$86$		0			0
$87$	19.5959			2.10090
$88$		0			0
$89$		−	9.79796i		−	1.03858i	−0.854598	−	0.519291i	$-0.826196\pi$
$89$		−	9.79796i		−	1.03858i	0.854598	−	0.519291i	$-0.173804\pi$
$90$		0			0
$91$	4.89898	−	2.00000i	0.513553	−	0.209657i
$92$		0			0
$93$			19.5959i			2.03200i
$94$		0			0
$95$			2.00000i			0.205196i
$96$		0			0
$97$		−	4.89898i		−	0.497416i	−0.968579	−	0.248708i	$-0.919994\pi$
$97$		−	4.89898i		−	0.497416i	0.968579	−	0.248708i	$-0.0800060\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	4.89898			0.482711			0.241355	−	0.970437i	$-0.422408\pi$
$103$	4.89898			0.482711			0.241355	+	0.970437i	$0.422408\pi$
$104$		0			0
$105$	2.00000	+	4.89898i	0.195180	+	0.478091i
$106$		0			0
$107$	−14.6969			−1.42081			−0.710403	−	0.703795i	$-0.751487\pi$
$107$	−14.6969			−1.42081			−0.710403	+	0.703795i	$0.751487\pi$
$108$		0			0
$109$			9.79796i			0.938474i	0.883072	+	0.469237i	$0.155471\pi$
$109$			9.79796i			0.938474i	−0.883072	+	0.469237i	$0.844529\pi$
$110$		0			0
$111$	−9.79796			−0.929981
$112$		0			0
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$		0			0
$115$		−	6.00000i		−	0.559503i
$116$		0			0
$117$	2.00000			0.184900
$118$		0			0
$119$	4.89898	+	12.0000i	0.449089	+	1.10004i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$			10.0000i			0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$			10.0000i			0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$		0			0
$129$			9.79796i			0.862662i
$130$		0			0
$131$		−	6.00000i		−	0.524222i	−0.965038	−	0.262111i	$-0.915581\pi$
$131$		−	6.00000i		−	0.524222i	0.965038	−	0.262111i	$-0.0844187\pi$
$132$		0			0
$133$	−2.00000	−	4.89898i	−0.173422	−	0.424795i
$134$		0			0
$135$		−	4.00000i		−	0.344265i
$136$		0			0
$137$	−6.00000			−0.512615			−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000			−0.512615			−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$			10.0000i			0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$			10.0000i			0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$		0			0
$141$			9.79796i			0.825137i
$142$		0			0
$143$		0			0
$144$		0			0
$145$			9.79796i			0.813676i
$146$		0			0
$147$	−9.79796	−	10.0000i	−0.808122	−	0.824786i
$148$		0			0
$149$			9.79796i			0.802680i	0.915929	+	0.401340i	$0.131455\pi$
$149$			9.79796i			0.802680i	−0.915929	+	0.401340i	$0.868545\pi$
$150$		0			0
$151$		−	10.0000i		−	0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$		−	10.0000i		−	0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$		0			0
$153$			4.89898i			0.396059i
$154$		0			0
$155$	−9.79796			−0.786991
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$	9.79796			0.777029
$160$		0			0
$161$	6.00000	+	14.6969i	0.472866	+	1.15828i
$162$		0			0
$163$	14.6969			1.15115			0.575577	−	0.817748i	$-0.304778\pi$
$163$	14.6969			1.15115			0.575577	+	0.817748i	$0.304778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	14.6969			1.13728			0.568642	−	0.822585i	$-0.307469\pi$
$167$	14.6969			1.13728			0.568642	+	0.822585i	$0.307469\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		−	2.00000i		−	0.152944i
$172$		0			0
$173$	−18.0000			−1.36851			−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000			−1.36851			−0.684257	+	0.729241i	$0.739873\pi$
$174$		0			0
$175$	−2.44949	+	1.00000i	−0.185164	+	0.0755929i
$176$		0			0
$177$	−12.0000			−0.901975
$178$		0			0
$179$	9.79796			0.732334			0.366167	−	0.930549i	$-0.380670\pi$
$179$	9.79796			0.732334			0.366167	+	0.930549i	$0.380670\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$			20.0000i			1.47844i
$184$		0			0
$185$		−	4.89898i		−	0.360180i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.00000	+	9.79796i	0.290957	+	0.712697i
$190$		0			0
$191$		−	18.0000i		−	1.30243i	−0.758891	−	0.651217i	$-0.774259\pi$
$191$		−	18.0000i		−	1.30243i	0.758891	−	0.651217i	$-0.225741\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$			4.00000i			0.286446i
$196$		0			0
$197$		−	4.89898i		−	0.349038i	−0.984654	−	0.174519i	$-0.944163\pi$
$197$		−	4.89898i		−	0.349038i	0.984654	−	0.174519i	$-0.0558370\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$			29.3939i			2.07328i
$202$		0			0
$203$	−9.79796	−	24.0000i	−0.687682	−	1.68447i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			6.00000i			0.417029i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$	12.0000			0.822226
$214$		0			0
$215$	−4.89898			−0.334108
$216$		0			0
$217$	24.0000	−	9.79796i	1.62923	−	0.665129i
$218$		0			0
$219$	9.79796			0.662085
$220$		0			0
$221$			9.79796i			0.659082i
$222$		0			0
$223$	4.89898			0.328060			0.164030	−	0.986455i	$-0.447551\pi$
$223$	4.89898			0.328060			0.164030	+	0.986455i	$0.447551\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$		−	18.0000i		−	1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$		−	18.0000i		−	1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$		0			0
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$	−4.89898			−0.319574
$236$		0			0
$237$	−20.0000			−1.29914
$238$		0			0
$239$			6.00000i			0.388108i	0.980991	+	0.194054i	$0.0621637\pi$
$239$			6.00000i			0.388108i	−0.980991	+	0.194054i	$0.937836\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$			10.0000i			0.641500i
$244$		0			0
$245$	5.00000	−	4.89898i	0.319438	−	0.312984i
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$		0			0
$249$	12.0000			0.760469
$250$		0			0
$251$			18.0000i			1.13615i	0.822977	+	0.568075i	$0.192312\pi$
$251$			18.0000i			1.13615i	−0.822977	+	0.568075i	$0.807688\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−9.79796			−0.613572
$256$		0			0
$257$		−	24.4949i		−	1.52795i	−0.645246	−	0.763975i	$-0.723245\pi$
$257$		−	24.4949i		−	1.52795i	0.645246	−	0.763975i	$-0.276755\pi$
$258$		0			0
$259$	4.89898	+	12.0000i	0.304408	+	0.745644i
$260$		0			0
$261$		−	9.79796i		−	0.606478i
$262$		0			0
$263$		−	6.00000i		−	0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$		−	6.00000i		−	0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$		0			0
$265$			4.89898i			0.300942i
$266$		0			0
$267$	−19.5959			−1.19925
$268$		0			0
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$		0			0
$271$	−9.79796			−0.595184			−0.297592	−	0.954693i	$-0.596183\pi$
$271$	−9.79796			−0.595184			−0.297592	+	0.954693i	$0.596183\pi$
$272$		0			0
$273$	−4.00000	−	9.79796i	−0.242091	−	0.592999i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	24.4949i		−	1.47176i	−0.677114	−	0.735878i	$-0.736770\pi$
$277$		−	24.4949i		−	1.47176i	0.677114	−	0.735878i	$-0.263230\pi$
$278$		0			0
$279$	9.79796			0.586588
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$			14.0000i			0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$			14.0000i			0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$		0			0
$285$	4.00000			0.236940
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$	−9.79796			−0.574367
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$		−	6.00000i		−	0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			12.0000i			0.693978i
$300$		0			0
$301$	12.0000	−	4.89898i	0.691669	−	0.282372i
$302$		0			0
$303$		−	12.0000i		−	0.689382i
$304$		0			0
$305$	−10.0000			−0.572598
$306$		0			0
$307$			22.0000i			1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$			22.0000i			1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$		0			0
$309$		−	9.79796i		−	0.557386i
$310$		0			0
$311$	−29.3939			−1.66677			−0.833387	−	0.552690i	$-0.813601\pi$
$311$	−29.3939			−1.66677			−0.833387	+	0.552690i	$0.813601\pi$
$312$		0			0
$313$		−	14.6969i		−	0.830720i	−0.909657	−	0.415360i	$-0.863656\pi$
$313$		−	14.6969i		−	0.830720i	0.909657	−	0.415360i	$-0.136344\pi$
$314$		0			0
$315$	2.44949	−	1.00000i	0.138013	−	0.0563436i
$316$		0			0
$317$		−	4.89898i		−	0.275154i	−0.990491	−	0.137577i	$-0.956069\pi$
$317$		−	4.89898i		−	0.275154i	0.990491	−	0.137577i	$-0.0439315\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$			29.3939i			1.64061i
$322$		0			0
$323$	9.79796			0.545173
$324$		0			0
$325$	−2.00000			−0.110940
$326$		0			0
$327$	19.5959			1.08366
$328$		0			0
$329$	12.0000	−	4.89898i	0.661581	−	0.270089i
$330$		0			0
$331$	19.5959			1.07709			0.538545	−	0.842597i	$-0.318974\pi$
$331$	19.5959			1.07709			0.538545	+	0.842597i	$0.318974\pi$
$332$		0			0
$333$			4.89898i			0.268462i
$334$		0			0
$335$	−14.6969			−0.802980
$336$		0			0
$337$	34.0000			1.85210			0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000			1.85210			0.926049	+	0.377403i	$0.123183\pi$
$338$		0			0
$339$		−	36.0000i		−	1.95525i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−7.34847	+	17.0000i	−0.396780	+	0.917914i
$344$		0			0
$345$	−12.0000			−0.646058
$346$		0			0
$347$	24.4949			1.31495			0.657477	−	0.753474i	$-0.271623\pi$
$347$	24.4949			1.31495			0.657477	+	0.753474i	$0.271623\pi$
$348$		0			0
$349$	−26.0000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$350$		0			0
$351$			8.00000i			0.427008i
$352$		0			0
$353$		−	14.6969i		−	0.782239i	−0.920340	−	0.391120i	$-0.872088\pi$
$353$		−	14.6969i		−	0.782239i	0.920340	−	0.391120i	$-0.127912\pi$
$354$		0			0
$355$			6.00000i			0.318447i
$356$		0			0
$357$	24.0000	−	9.79796i	1.27021	−	0.518563i
$358$		0			0
$359$		−	18.0000i		−	0.950004i	−0.879985	−	0.475002i	$-0.842447\pi$
$359$		−	18.0000i		−	0.950004i	0.879985	−	0.475002i	$-0.157553\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$			22.0000i			1.15470i
$364$		0			0
$365$			4.89898i			0.256424i
$366$		0			0
$367$	14.6969			0.767174			0.383587	−	0.923505i	$-0.374689\pi$
$367$	14.6969			0.767174			0.383587	+	0.923505i	$0.374689\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.89898	−	12.0000i	−0.254342	−	0.623009i
$372$		0			0
$373$		−	34.2929i		−	1.77562i	−0.460213	−	0.887808i	$-0.652227\pi$
$373$		−	34.2929i		−	1.77562i	0.460213	−	0.887808i	$-0.347773\pi$
$374$		0			0
$375$		−	2.00000i		−	0.103280i
$376$		0			0
$377$		−	19.5959i		−	1.00924i
$378$		0			0
$379$	29.3939			1.50986			0.754931	−	0.655804i	$-0.227670\pi$
$379$	29.3939			1.50986			0.754931	+	0.655804i	$0.227670\pi$
$380$		0			0
$381$	20.0000			1.02463
$382$		0			0
$383$	−14.6969			−0.750978			−0.375489	−	0.926827i	$-0.622525\pi$
$383$	−14.6969			−0.750978			−0.375489	+	0.926827i	$0.622525\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.89898			0.249029
$388$		0			0
$389$		−	9.79796i		−	0.496776i	−0.968661	−	0.248388i	$-0.920099\pi$
$389$		−	9.79796i		−	0.496776i	0.968661	−	0.248388i	$-0.0799008\pi$
$390$		0			0
$391$	−29.3939			−1.48651
$392$		0			0
$393$	−12.0000			−0.605320
$394$		0			0
$395$		−	10.0000i		−	0.503155i
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$		0			0
$399$	−9.79796	+	4.00000i	−0.490511	+	0.200250i
$400$		0			0
$401$	−6.00000			−0.299626			−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000			−0.299626			−0.149813	+	0.988714i	$0.547867\pi$
$402$		0			0
$403$	19.5959			0.976142
$404$		0			0
$405$	−11.0000			−0.546594
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	29.3939i		−	1.45343i	−0.686937	−	0.726717i	$-0.741045\pi$
$409$		−	29.3939i		−	1.45343i	0.686937	−	0.726717i	$-0.258955\pi$
$410$		0			0
$411$			12.0000i			0.591916i
$412$		0			0
$413$	6.00000	+	14.6969i	0.295241	+	0.723189i
$414$		0			0
$415$			6.00000i			0.294528i
$416$		0			0
$417$	20.0000			0.979404
$418$		0			0
$419$			18.0000i			0.879358i	0.898155	+	0.439679i	$0.144908\pi$
$419$			18.0000i			0.879358i	−0.898155	+	0.439679i	$0.855092\pi$
$420$		0			0
$421$		−	19.5959i		−	0.955047i	−0.878619	−	0.477523i	$-0.841535\pi$
$421$		−	19.5959i		−	0.955047i	0.878619	−	0.477523i	$-0.158465\pi$
$422$		0			0
$423$	4.89898			0.238197
$424$		0			0
$425$		−	4.89898i		−	0.237635i
$426$		0			0
$427$	24.4949	−	10.0000i	1.18539	−	0.483934i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	18.0000i		−	0.867029i	−0.901146	−	0.433515i	$-0.857273\pi$
$431$		−	18.0000i		−	0.867029i	0.901146	−	0.433515i	$-0.142727\pi$
$432$		0			0
$433$		−	34.2929i		−	1.64801i	−0.566583	−	0.824005i	$-0.691735\pi$
$433$		−	34.2929i		−	1.64801i	0.566583	−	0.824005i	$-0.308265\pi$
$434$		0			0
$435$	19.5959			0.939552
$436$		0			0
$437$	12.0000			0.574038
$438$		0			0
$439$	−19.5959			−0.935262			−0.467631	−	0.883924i	$-0.654892\pi$
$439$	−19.5959			−0.935262			−0.467631	+	0.883924i	$0.654892\pi$
$440$		0			0
$441$	−5.00000	+	4.89898i	−0.238095	+	0.233285i
$442$		0			0
$443$	−24.4949			−1.16379			−0.581894	−	0.813265i	$-0.697688\pi$
$443$	−24.4949			−1.16379			−0.581894	+	0.813265i	$0.697688\pi$
$444$		0			0
$445$		−	9.79796i		−	0.464468i
$446$		0			0
$447$	19.5959			0.926855
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−20.0000			−0.939682
$454$		0			0
$455$	4.89898	−	2.00000i	0.229668	−	0.0937614i
$456$		0			0
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$		0			0
$459$	−19.5959			−0.914659
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$		−	14.0000i		−	0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$		−	14.0000i		−	0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$		0			0
$465$			19.5959i			0.908739i
$466$		0			0
$467$			6.00000i			0.277647i	0.990317	+	0.138823i	$0.0443321\pi$
$467$			6.00000i			0.277647i	−0.990317	+	0.138823i	$0.955668\pi$
$468$		0			0
$469$	36.0000	−	14.6969i	1.66233	−	0.678642i
$470$		0			0
$471$		−	28.0000i		−	1.29017i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			2.00000i			0.0917663i
$476$		0			0
$477$		−	4.89898i		−	0.224309i
$478$		0			0
$479$	39.1918			1.79072			0.895360	−	0.445342i	$-0.146918\pi$
$479$	39.1918			1.79072			0.895360	+	0.445342i	$0.146918\pi$
$480$		0			0
$481$			9.79796i			0.446748i
$482$		0			0
$483$	29.3939	−	12.0000i	1.33747	−	0.546019i
$484$		0			0
$485$		−	4.89898i		−	0.222451i
$486$		0			0
$487$			2.00000i			0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$			2.00000i			0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$		0			0
$489$		−	29.3939i		−	1.32924i
$490$		0			0
$491$	19.5959			0.884351			0.442176	−	0.896928i	$-0.354207\pi$
$491$	19.5959			0.884351			0.442176	+	0.896928i	$0.354207\pi$
$492$		0			0
$493$	48.0000			2.16181
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.00000	−	14.6969i	−0.269137	−	0.659248i
$498$		0			0
$499$	−9.79796			−0.438617			−0.219308	−	0.975656i	$-0.570380\pi$
$499$	−9.79796			−0.438617			−0.219308	+	0.975656i	$0.570380\pi$
$500$		0			0
$501$		−	29.3939i		−	1.31322i
$502$		0			0
$503$	−14.6969			−0.655304			−0.327652	−	0.944798i	$-0.606257\pi$
$503$	−14.6969			−0.655304			−0.327652	+	0.944798i	$0.606257\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		0			0
$507$			18.0000i			0.799408i
$508$		0			0
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$	−4.89898	−	12.0000i	−0.216718	−	0.530849i
$512$		0			0
$513$	8.00000			0.353209
$514$		0			0
$515$	4.89898			0.215875
$516$		0			0
$517$		0			0
$518$		0			0
$519$			36.0000i			1.58022i
$520$		0			0
$521$			39.1918i			1.71703i	0.512792	+	0.858513i	$0.328611\pi$
$521$			39.1918i			1.71703i	−0.512792	+	0.858513i	$0.671389\pi$
$522$		0			0
$523$		−	34.0000i		−	1.48672i	−0.668894	−	0.743358i	$-0.733232\pi$
$523$		−	34.0000i		−	1.48672i	0.668894	−	0.743358i	$-0.266768\pi$
$524$		0			0
$525$	2.00000	+	4.89898i	0.0872872	+	0.213809i
$526$		0			0
$527$			48.0000i			2.09091i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$			6.00000i			0.260378i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−14.6969			−0.635404
$536$		0			0
$537$		−	19.5959i		−	0.845626i
$538$		0			0
$539$		0			0
$540$		0			0
$541$			19.5959i			0.842494i	0.906946	+	0.421247i	$0.138408\pi$
$541$			19.5959i			0.842494i	−0.906946	+	0.421247i	$0.861592\pi$
$542$		0			0
$543$			20.0000i			0.858282i
$544$		0			0
$545$			9.79796i			0.419698i
$546$		0			0
$547$	24.4949			1.04733			0.523663	−	0.851925i	$-0.324565\pi$
$547$	24.4949			1.04733			0.523663	+	0.851925i	$0.324565\pi$
$548$		0			0
$549$	10.0000			0.426790
$550$		0			0
$551$	−19.5959			−0.834814
$552$		0			0
$553$	10.0000	+	24.4949i	0.425243	+	1.04163i
$554$		0			0
$555$	−9.79796			−0.415900
$556$		0			0
$557$			34.2929i			1.45303i	0.687148	+	0.726517i	$0.258862\pi$
$557$			34.2929i			1.45303i	−0.687148	+	0.726517i	$0.741138\pi$
$558$		0			0
$559$	9.79796			0.414410
$560$		0			0
$561$		0			0
$562$		0			0
$563$			6.00000i			0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$			6.00000i			0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$		0			0
$565$	18.0000			0.757266
$566$		0			0
$567$	26.9444	−	11.0000i	1.13156	−	0.461957i
$568$		0			0
$569$	42.0000			1.76073			0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000			1.76073			0.880366	+	0.474295i	$0.157297\pi$
$570$		0			0
$571$	39.1918			1.64013			0.820064	−	0.572272i	$-0.193938\pi$
$571$	39.1918			1.64013			0.820064	+	0.572272i	$0.193938\pi$
$572$		0			0
$573$	−36.0000			−1.50392
$574$		0			0
$575$		−	6.00000i		−	0.250217i
$576$		0			0
$577$		−	4.89898i		−	0.203947i	−0.994787	−	0.101974i	$-0.967484\pi$
$577$		−	4.89898i		−	0.203947i	0.994787	−	0.101974i	$-0.0325157\pi$
$578$		0			0
$579$			28.0000i			1.16364i
$580$		0			0
$581$	−6.00000	−	14.6969i	−0.248922	−	0.609732i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	2.00000			0.0826898
$586$		0			0
$587$			30.0000i			1.23823i	0.785299	+	0.619116i	$0.212509\pi$
$587$			30.0000i			1.23823i	−0.785299	+	0.619116i	$0.787491\pi$
$588$		0			0
$589$		−	19.5959i		−	0.807436i
$590$		0			0
$591$	−9.79796			−0.403034
$592$		0			0
$593$			24.4949i			1.00588i	0.864320	+	0.502942i	$0.167749\pi$
$593$			24.4949i			1.00588i	−0.864320	+	0.502942i	$0.832251\pi$
$594$		0			0
$595$	4.89898	+	12.0000i	0.200839	+	0.491952i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	18.0000i		−	0.735460i	−0.929933	−	0.367730i	$-0.880135\pi$
$599$		−	18.0000i		−	0.735460i	0.929933	−	0.367730i	$-0.119865\pi$
$600$		0			0
$601$			19.5959i			0.799334i	0.916660	+	0.399667i	$0.130874\pi$
$601$			19.5959i			0.799334i	−0.916660	+	0.399667i	$0.869126\pi$
$602$		0			0
$603$	14.6969			0.598506
$604$		0			0
$605$	−11.0000			−0.447214
$606$		0			0
$607$	34.2929			1.39190			0.695952	−	0.718088i	$-0.254983\pi$
$607$	34.2929			1.39190			0.695952	+	0.718088i	$0.254983\pi$
$608$		0			0
$609$	−48.0000	+	19.5959i	−1.94506	+	0.794067i
$610$		0			0
$611$	9.79796			0.396383
$612$		0			0
$613$		−	34.2929i		−	1.38508i	−0.721382	−	0.692538i	$-0.756493\pi$
$613$		−	34.2929i		−	1.38508i	0.721382	−	0.692538i	$-0.243507\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$		0			0
$619$		−	38.0000i		−	1.52735i	−0.645601	−	0.763674i	$-0.723393\pi$
$619$		−	38.0000i		−	1.52735i	0.645601	−	0.763674i	$-0.276607\pi$
$620$		0			0
$621$	−24.0000			−0.963087
$622$		0			0
$623$	9.79796	+	24.0000i	0.392547	+	0.961540i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−24.0000			−0.956943
$630$		0			0
$631$		−	26.0000i		−	1.03504i	−0.855670	−	0.517522i	$-0.826855\pi$
$631$		−	26.0000i		−	1.03504i	0.855670	−	0.517522i	$-0.173145\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			10.0000i			0.396838i
$636$		0			0
$637$	−10.0000	+	9.79796i	−0.396214	+	0.388209i
$638$		0			0
$639$		−	6.00000i		−	0.237356i
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$			38.0000i			1.49857i	0.662246	+	0.749287i	$0.269604\pi$
$643$			38.0000i			1.49857i	−0.662246	+	0.749287i	$0.730396\pi$
$644$		0			0
$645$			9.79796i			0.385794i
$646$		0			0
$647$	−24.4949			−0.962994			−0.481497	−	0.876448i	$-0.659907\pi$
$647$	−24.4949			−0.962994			−0.481497	+	0.876448i	$0.659907\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−19.5959	−	48.0000i	−0.768025	−	1.88127i
$652$		0			0
$653$		−	34.2929i		−	1.34198i	−0.741465	−	0.670992i	$-0.765869\pi$
$653$		−	34.2929i		−	1.34198i	0.741465	−	0.670992i	$-0.234131\pi$
$654$		0			0
$655$		−	6.00000i		−	0.234439i
$656$		0			0
$657$		−	4.89898i		−	0.191127i
$658$		0			0
$659$	−48.9898			−1.90837			−0.954186	−	0.299215i	$-0.903275\pi$
$659$	−48.9898			−1.90837			−0.954186	+	0.299215i	$0.903275\pi$
$660$		0			0
$661$	46.0000			1.78919			0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000			1.78919			0.894596	+	0.446875i	$0.147463\pi$
$662$		0			0
$663$	19.5959			0.761042
$664$		0			0
$665$	−2.00000	−	4.89898i	−0.0775567	−	0.189974i
$666$		0			0
$667$	58.7878			2.27627
$668$		0			0
$669$		−	9.79796i		−	0.378811i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	2.00000			0.0770943			0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000			0.0770943			0.0385472	+	0.999257i	$0.487727\pi$
$674$		0			0
$675$		−	4.00000i		−	0.153960i
$676$		0			0
$677$	−42.0000			−1.61419			−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000			−1.61419			−0.807096	+	0.590421i	$0.798962\pi$
$678$		0			0
$679$	4.89898	+	12.0000i	0.188006	+	0.460518i
$680$		0			0
$681$	−36.0000			−1.37952
$682$		0			0
$683$	−44.0908			−1.68709			−0.843544	−	0.537060i	$-0.819535\pi$
$683$	−44.0908			−1.68709			−0.843544	+	0.537060i	$0.819535\pi$
$684$		0			0
$685$	−6.00000			−0.229248
$686$		0			0
$687$		−	28.0000i		−	1.06827i
$688$		0			0
$689$		−	9.79796i		−	0.373273i
$690$		0			0
$691$			10.0000i			0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$			10.0000i			0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			10.0000i			0.379322i
$696$		0			0
$697$		0			0
$698$		0			0
$699$			12.0000i			0.453882i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	9.79796			0.369537
$704$		0			0
$705$			9.79796i			0.369012i
$706$		0			0
$707$	−14.6969	+	6.00000i	−0.552735	+	0.225653i
$708$		0			0
$709$		−	48.9898i		−	1.83985i	−0.392094	−	0.919925i	$-0.628249\pi$
$709$		−	48.9898i		−	1.83985i	0.392094	−	0.919925i	$-0.371751\pi$
$710$		0			0
$711$			10.0000i			0.375029i
$712$		0			0
$713$			58.7878i			2.20162i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	12.0000			0.448148
$718$		0			0
$719$	39.1918			1.46161			0.730804	−	0.682587i	$-0.239145\pi$
$719$	39.1918			1.46161			0.730804	+	0.682587i	$0.239145\pi$
$720$		0			0
$721$	−12.0000	+	4.89898i	−0.446903	+	0.182448i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			9.79796i			0.363887i
$726$		0			0
$727$	−4.89898			−0.181693			−0.0908465	−	0.995865i	$-0.528957\pi$
$727$	−4.89898			−0.181693			−0.0908465	+	0.995865i	$0.528957\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$			24.0000i			0.887672i
$732$		0			0
$733$	−2.00000			−0.0738717			−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000			−0.0738717			−0.0369358	+	0.999318i	$0.511760\pi$
$734$		0			0
$735$	−9.79796	−	10.0000i	−0.361403	−	0.368856i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	29.3939			1.08127			0.540636	−	0.841257i	$-0.318184\pi$
$739$	29.3939			1.08127			0.540636	+	0.841257i	$0.318184\pi$
$740$		0			0
$741$	−8.00000			−0.293887
$742$		0			0
$743$		−	30.0000i		−	1.10059i	−0.834969	−	0.550297i	$-0.814515\pi$
$743$		−	30.0000i		−	1.10059i	0.834969	−	0.550297i	$-0.185485\pi$
$744$		0			0
$745$			9.79796i			0.358969i
$746$		0			0
$747$		−	6.00000i		−	0.219529i
$748$		0			0
$749$	36.0000	−	14.6969i	1.31541	−	0.537014i
$750$		0			0
$751$			46.0000i			1.67856i	0.543696	+	0.839282i	$0.317024\pi$
$751$			46.0000i			1.67856i	−0.543696	+	0.839282i	$0.682976\pi$
$752$		0			0
$753$	36.0000			1.31191
$754$		0			0
$755$		−	10.0000i		−	0.363937i
$756$		0			0
$757$		−	4.89898i		−	0.178056i	−0.996029	−	0.0890282i	$-0.971624\pi$
$757$		−	4.89898i		−	0.178056i	0.996029	−	0.0890282i	$-0.0283761\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			19.5959i			0.710351i	0.934800	+	0.355176i	$0.115579\pi$
$761$			19.5959i			0.710351i	−0.934800	+	0.355176i	$0.884421\pi$
$762$		0			0
$763$	−9.79796	−	24.0000i	−0.354710	−	0.868858i
$764$		0			0
$765$			4.89898i			0.177123i
$766$		0			0
$767$			12.0000i			0.433295i
$768$		0			0
$769$			29.3939i			1.05997i	0.848007	+	0.529985i	$0.177803\pi$
$769$			29.3939i			1.05997i	−0.848007	+	0.529985i	$0.822197\pi$
$770$		0			0
$771$	−48.9898			−1.76432
$772$		0			0
$773$	54.0000			1.94225			0.971123	−	0.238581i	$-0.0766824\pi$
$773$	54.0000			1.94225			0.971123	+	0.238581i	$0.0766824\pi$
$774$		0			0
$775$	−9.79796			−0.351953
$776$		0			0
$777$	24.0000	−	9.79796i	0.860995	−	0.351500i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	39.1918			1.40060
$784$		0			0
$785$	14.0000			0.499681
$786$		0			0
$787$		−	26.0000i		−	0.926800i	−0.886149	−	0.463400i	$-0.846629\pi$
$787$		−	26.0000i		−	0.926800i	0.886149	−	0.463400i	$-0.153371\pi$
$788$		0			0
$789$	−12.0000			−0.427211
$790$		0			0
$791$	−44.0908	+	18.0000i	−1.56769	+	0.640006i
$792$		0			0
$793$	20.0000			0.710221
$794$		0			0
$795$	9.79796			0.347498
$796$		0			0
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$			24.0000i			0.849059i
$800$		0			0
$801$			9.79796i			0.346194i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	6.00000	+	14.6969i	0.211472	+	0.517999i
$806$		0			0
$807$			36.0000i			1.26726i
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$			50.0000i			1.75574i	0.478901	+	0.877869i	$0.341035\pi$
$811$			50.0000i			1.75574i	−0.478901	+	0.877869i	$0.658965\pi$
$812$		0			0
$813$			19.5959i			0.687259i
$814$		0			0
$815$	14.6969			0.514811
$816$		0			0
$817$		−	9.79796i		−	0.342787i
$818$		0			0
$819$	−4.89898	+	2.00000i	−0.171184	+	0.0698857i
$820$		0			0
$821$			19.5959i			0.683902i	0.939718	+	0.341951i	$0.111088\pi$
$821$			19.5959i			0.683902i	−0.939718	+	0.341951i	$0.888912\pi$
$822$		0			0
$823$		−	14.0000i		−	0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$		−	14.0000i		−	0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	24.4949			0.851771			0.425886	−	0.904777i	$-0.359963\pi$
$827$	24.4949			0.851771			0.425886	+	0.904777i	$0.359963\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$	−48.9898			−1.69944
$832$		0			0
$833$	−24.0000	−	24.4949i	−0.831551	−	0.848698i
$834$		0			0
$835$	14.6969			0.508609
$836$		0			0
$837$			39.1918i			1.35467i
$838$		0			0
$839$	19.5959			0.676526			0.338263	−	0.941052i	$-0.390161\pi$
$839$	19.5959			0.676526			0.338263	+	0.941052i	$0.390161\pi$
$840$		0			0
$841$	−67.0000			−2.31034
$842$		0			0
$843$			12.0000i			0.413302i
$844$		0			0
$845$	−9.00000			−0.309609
$846$		0			0
$847$	26.9444	−	11.0000i	0.925820	−	0.377964i
$848$		0			0
$849$	28.0000			0.960958
$850$		0			0
$851$	−29.3939			−1.00761
$852$		0			0
$853$	−10.0000			−0.342393			−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000			−0.342393			−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$		−	2.00000i		−	0.0683986i
$856$		0			0
$857$		−	24.4949i		−	0.836730i	−0.908279	−	0.418365i	$-0.862603\pi$
$857$		−	24.4949i		−	0.836730i	0.908279	−	0.418365i	$-0.137397\pi$
$858$		0			0
$859$			10.0000i			0.341196i	0.985341	+	0.170598i	$0.0545699\pi$
$859$			10.0000i			0.341196i	−0.985341	+	0.170598i	$0.945430\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	6.00000i		−	0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$		−	6.00000i		−	0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$		0			0
$865$	−18.0000			−0.612018
$866$		0			0
$867$			14.0000i			0.475465i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	29.3939			0.995974
$872$		0			0
$873$			4.89898i			0.165805i
$874$		0			0
$875$	−2.44949	+	1.00000i	−0.0828079	+	0.0338062i
$876$		0			0
$877$			53.8888i			1.81969i	0.414943	+	0.909847i	$0.363801\pi$
$877$			53.8888i			1.81969i	−0.414943	+	0.909847i	$0.636199\pi$
$878$		0			0
$879$		−	12.0000i		−	0.404750i
$880$		0			0
$881$		−	19.5959i		−	0.660203i	−0.943945	−	0.330102i	$-0.892917\pi$
$881$		−	19.5959i		−	0.660203i	0.943945	−	0.330102i	$-0.107083\pi$
$882$		0			0
$883$	−4.89898			−0.164864			−0.0824319	−	0.996597i	$-0.526269\pi$
$883$	−4.89898			−0.164864			−0.0824319	+	0.996597i	$0.526269\pi$
$884$		0			0
$885$	−12.0000			−0.403376
$886$		0			0
$887$	14.6969			0.493475			0.246737	−	0.969082i	$-0.420641\pi$
$887$	14.6969			0.493475			0.246737	+	0.969082i	$0.420641\pi$
$888$		0			0
$889$	−10.0000	−	24.4949i	−0.335389	−	0.821532i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	9.79796i		−	0.327876i
$894$		0			0
$895$	9.79796			0.327510
$896$		0			0
$897$	24.0000			0.801337
$898$		0			0
$899$		−	96.0000i		−	3.20178i
$900$		0			0
$901$	24.0000			0.799556
$902$		0			0
$903$	−9.79796	−	24.0000i	−0.326056	−	0.798670i
$904$		0			0
$905$	−10.0000			−0.332411
$906$		0			0
$907$	4.89898			0.162668			0.0813340	−	0.996687i	$-0.474082\pi$
$907$	4.89898			0.162668			0.0813340	+	0.996687i	$0.474082\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$		−	18.0000i		−	0.596367i	−0.954509	−	0.298183i	$-0.903619\pi$
$911$		−	18.0000i		−	0.596367i	0.954509	−	0.298183i	$-0.0963807\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$			20.0000i			0.661180i
$916$		0			0
$917$	6.00000	+	14.6969i	0.198137	+	0.485336i
$918$		0			0
$919$			14.0000i			0.461817i	0.972975	+	0.230909i	$0.0741699\pi$
$919$			14.0000i			0.461817i	−0.972975	+	0.230909i	$0.925830\pi$
$920$		0			0
$921$	44.0000			1.44985
$922$		0			0
$923$		−	12.0000i		−	0.394985i
$924$		0			0
$925$		−	4.89898i		−	0.161077i
$926$		0			0
$927$	−4.89898			−0.160904
$928$		0			0
$929$			9.79796i			0.321461i	0.986998	+	0.160730i	$0.0513849\pi$
$929$			9.79796i			0.321461i	−0.986998	+	0.160730i	$0.948615\pi$
$930$		0			0
$931$	9.79796	+	10.0000i	0.321115	+	0.327737i
$932$		0			0
$933$			58.7878i			1.92462i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	4.89898i		−	0.160043i	−0.996793	−	0.0800213i	$-0.974501\pi$
$937$		−	4.89898i		−	0.160043i	0.996793	−	0.0800213i	$-0.0254988\pi$
$938$		0			0
$939$	−29.3939			−0.959233
$940$		0			0
$941$	6.00000			0.195594			0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000			0.195594			0.0977972	+	0.995206i	$0.468820\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	4.00000	+	9.79796i	0.130120	+	0.318728i
$946$		0			0
$947$	−14.6969			−0.477586			−0.238793	−	0.971070i	$-0.576752\pi$
$947$	−14.6969			−0.477586			−0.238793	+	0.971070i	$0.576752\pi$
$948$		0			0
$949$		−	9.79796i		−	0.318055i
$950$		0			0
$951$	−9.79796			−0.317721
$952$		0			0
$953$	−54.0000			−1.74923			−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000			−1.74923			−0.874616	+	0.484817i	$0.838886\pi$
$954$		0			0
$955$		−	18.0000i		−	0.582466i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.6969	−	6.00000i	0.474589	−	0.193750i
$960$		0			0
$961$	65.0000			2.09677
$962$		0			0
$963$	14.6969			0.473602
$964$		0			0
$965$	−14.0000			−0.450676
$966$		0			0
$967$		−	22.0000i		−	0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$		−	22.0000i		−	0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$		0			0
$969$		−	19.5959i		−	0.629512i
$970$		0			0
$971$		−	30.0000i		−	0.962746i	−0.876516	−	0.481373i	$-0.840138\pi$
$971$		−	30.0000i		−	0.962746i	0.876516	−	0.481373i	$-0.159862\pi$
$972$		0			0
$973$	−10.0000	−	24.4949i	−0.320585	−	0.785270i
$974$		0			0
$975$			4.00000i			0.128103i
$976$		0			0
$977$	−30.0000			−0.959785			−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000			−0.959785			−0.479893	+	0.877327i	$0.659324\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	9.79796i		−	0.312825i
$982$		0			0
$983$	4.89898			0.156253			0.0781266	−	0.996943i	$-0.475106\pi$
$983$	4.89898			0.156253			0.0781266	+	0.996943i	$0.475106\pi$
$984$		0			0
$985$		−	4.89898i		−	0.156094i
$986$		0			0
$987$	−9.79796	−	24.0000i	−0.311872	−	0.763928i
$988$		0			0
$989$			29.3939i			0.934671i
$990$		0			0
$991$		−	2.00000i		−	0.0635321i	−0.999495	−	0.0317660i	$-0.989887\pi$
$991$		−	2.00000i		−	0.0635321i	0.999495	−	0.0317660i	$-0.0101131\pi$
$992$		0			0
$993$		−	39.1918i		−	1.24372i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−10.0000			−0.316703			−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000			−0.316703			−0.158352	+	0.987383i	$0.550618\pi$
$998$		0			0
$999$	−19.5959			−0.619987

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.h.c.671.1	yes	4
4.3	odd	2	inner	2240.2.h.c.671.4	yes	4
7.6	odd	2		2240.2.h.a.671.4	yes	4
8.3	odd	2		2240.2.h.a.671.2	yes	4
8.5	even	2		2240.2.h.a.671.3	yes	4
28.27	even	2		2240.2.h.a.671.1	✓	4
56.13	odd	2	inner	2240.2.h.c.671.2	yes	4
56.27	even	2	inner	2240.2.h.c.671.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.h.a.671.1	✓	4	28.27	even	2
2240.2.h.a.671.2	yes	4	8.3	odd	2
2240.2.h.a.671.3	yes	4	8.5	even	2
2240.2.h.a.671.4	yes	4	7.6	odd	2
2240.2.h.c.671.1	yes	4	1.1	even	1	trivial
2240.2.h.c.671.2	yes	4	56.13	odd	2	inner
2240.2.h.c.671.3	yes	4	56.27	even	2	inner
2240.2.h.c.671.4	yes	4	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +1.00000 q^{5} +(-2.44949 + 1.00000i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} +1.00000 q^{5} +(-2.44949 + 1.00000i) q^{7} -1.00000 q^{9} -2.00000 q^{13} -2.00000i q^{15} -4.89898i q^{17} +2.00000i q^{19} +(2.00000 + 4.89898i) q^{21} -6.00000i q^{23} +1.00000 q^{25} -4.00000i q^{27} +9.79796i q^{29} -9.79796 q^{31} +(-2.44949 + 1.00000i) q^{35} -4.89898i q^{37} +4.00000i q^{39} -4.89898 q^{43} -1.00000 q^{45} -4.89898 q^{47} +(5.00000 - 4.89898i) q^{49} -9.79796 q^{51} +4.89898i q^{53} +4.00000 q^{57} -6.00000i q^{59} -10.0000 q^{61} +(2.44949 - 1.00000i) q^{63} -2.00000 q^{65} -14.6969 q^{67} -12.0000 q^{69} +6.00000i q^{71} +4.89898i q^{73} -2.00000i q^{75} -10.0000i q^{79} -11.0000 q^{81} +6.00000i q^{83} -4.89898i q^{85} +19.5959 q^{87} -9.79796i q^{89} +(4.89898 - 2.00000i) q^{91} +19.5959i q^{93} +2.00000i q^{95} -4.89898i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 - 4 * q^9	\( 4 q + 4 q^{5} - 4 q^{9} - 8 q^{13} + 8 q^{21} + 4 q^{25} - 4 q^{45} + 20 q^{49} + 16 q^{57} - 40 q^{61} - 8 q^{65} - 48 q^{69} - 44 q^{81}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^9 - 8 * q^13 + 8 * q^21 + 4 * q^25 - 4 * q^45 + 20 * q^49 + 16 * q^57 - 40 * q^61 - 8 * q^65 - 48 * q^69 - 44 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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