LMFDB - Embedded newform 5780.2.c.g.5201.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5780,2,Mod(5201,5780)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5780, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5780.5201");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5780 = 2^{2} \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5780.c (of order $2$, degree $1$, not 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:	$46.1535323683$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.851059918206111744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 33x^{10} + 360x^{8} + 1423x^{6} + 1269x^{4} + 234x^{2} + 9 $ x^12 + 33x^10 + 360x^8 + 1423x^6 + 1269x^4 + 234*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5201.3
Root			$-0.423390i$ of defining polynomial
Character	$\chi$	$=$	5780.5201
Dual form			5780.2.c.g.5201.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278i q^{3} +1.00000i q^{5} +1.53209i q^{7} -2.30278 q^{9} +2.44842i q^{11} -6.18216 q^{13} +2.30278 q^{15} +0.345896 q^{19} +3.52806 q^{21} -9.04065i q^{23} -1.00000 q^{25} -1.60555i q^{27} +5.05397i q^{29} +2.71070i q^{31} +5.63816 q^{33} -1.53209 q^{35} +4.08706i q^{37} +14.2361i q^{39} +1.14564i q^{41} -0.730943 q^{43} -2.30278i q^{45} +0.594059 q^{47} +4.65270 q^{49} +9.59749 q^{53} -2.44842 q^{55} -0.796521i q^{57} -6.74937 q^{59} -10.5856i q^{61} -3.52806i q^{63} -6.18216i q^{65} +12.3030 q^{67} -20.8186 q^{69} -1.08582i q^{71} -9.34420i q^{73} +2.30278i q^{75} -3.75119 q^{77} -11.0401i q^{79} -10.6056 q^{81} +12.2524 q^{83} +11.6382 q^{87} -10.8303 q^{89} -9.47162i q^{91} +6.24214 q^{93} +0.345896i q^{95} -6.97153i q^{97} -5.63816i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{9} - 30 q^{13} + 6 q^{15} + 6 q^{19} - 12 q^{25} - 6 q^{43} - 30 q^{47} + 60 q^{49} + 24 q^{53} + 12 q^{59} - 12 q^{67} + 6 q^{77} - 84 q^{81} + 30 q^{83} + 72 q^{87} - 12 q^{89} + 72 q^{93}+O(q^{100}) $ 12 * q - 6 * q^9 - 30 * q^13 + 6 * q^15 + 6 * q^19 - 12 * q^25 - 6 * q^43 - 30 * q^47 + 60 * q^49 + 24 * q^53 + 12 * q^59 - 12 * q^67 + 6 * q^77 - 84 * q^81 + 30 * q^83 + 72 * q^87 - 12 * q^89 + 72 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5780\mathbb{Z}\right)^\times$.

$n$	$581$	$1157$	$2891$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 2.30278i	− 1.32951i	−0.747062	−	0.664754i	$-0.768536\pi$
$3$	− 2.30278i	− 1.32951i	0.747062	−	0.664754i	$-0.231464\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	1.53209i	0.579075i	0.957167	+	0.289538i	$0.0935015\pi$
$7$	1.53209i	0.579075i	−0.957167	+	0.289538i	$0.906498\pi$
$8$	0	0
$9$	−2.30278	−0.767592
$10$	0	0
$11$	2.44842i	0.738226i	0.929385	+	0.369113i	$0.120338\pi$
$11$	2.44842i	0.738226i	−0.929385	+	0.369113i	$0.879662\pi$
$12$	0	0
$13$	−6.18216	−1.71462	−0.857311	−	0.514798i	$-0.827867\pi$
$13$	−6.18216	−1.71462	−0.857311	+	0.514798i	$0.827867\pi$
$14$	0	0
$15$	2.30278	0.594574
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	0.345896	0.0793540	0.0396770	−	0.999213i	$-0.487367\pi$
$19$	0.345896	0.0793540	0.0396770	+	0.999213i	$0.487367\pi$
$20$	0	0
$21$	3.52806	0.769885
$22$	0	0
$23$	− 9.04065i	− 1.88511i	−0.334056	−	0.942553i	$-0.608418\pi$
$23$	− 9.04065i	− 1.88511i	0.334056	−	0.942553i	$-0.391582\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.60555i	− 0.308988i
$28$	0	0
$29$	5.05397i	0.938498i	0.883066	+	0.469249i	$0.155475\pi$
$29$	5.05397i	0.938498i	−0.883066	+	0.469249i	$0.844525\pi$
$30$	0	0
$31$	2.71070i	0.486857i	0.969919	+	0.243428i	$0.0782721\pi$
$31$	2.71070i	0.486857i	−0.969919	+	0.243428i	$0.921728\pi$
$32$	0	0
$33$	5.63816	0.981477
$34$	0	0
$35$	−1.53209	−0.258970
$36$	0	0
$37$	4.08706i	0.671908i	0.941878	+	0.335954i	$0.109059\pi$
$37$	4.08706i	0.671908i	−0.941878	+	0.335954i	$0.890941\pi$
$38$	0	0
$39$	14.2361i	2.27961i
$40$	0	0
$41$	1.14564i	0.178919i	0.995990	+	0.0894596i	$0.0285140\pi$
$41$	1.14564i	0.178919i	−0.995990	+	0.0894596i	$0.971486\pi$
$42$	0	0
$43$	−0.730943	−0.111468	−0.0557339	−	0.998446i	$-0.517750\pi$
$43$	−0.730943	−0.111468	−0.0557339	+	0.998446i	$0.517750\pi$
$44$	0	0
$45$	− 2.30278i	− 0.343278i
$46$	0	0
$47$	0.594059	0.0866524	0.0433262	−	0.999061i	$-0.486205\pi$
$47$	0.594059	0.0866524	0.0433262	+	0.999061i	$0.486205\pi$
$48$	0	0
$49$	4.65270	0.664672
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.59749	1.31832	0.659158	−	0.752004i	$-0.270913\pi$
$53$	9.59749	1.31832	0.659158	+	0.752004i	$0.270913\pi$
$54$	0	0
$55$	−2.44842	−0.330145
$56$	0	0
$57$	− 0.796521i	− 0.105502i
$58$	0	0
$59$	−6.74937	−0.878693	−0.439346	−	0.898318i	$-0.644790\pi$
$59$	−6.74937	−0.878693	−0.439346	+	0.898318i	$0.644790\pi$
$60$	0	0
$61$	− 10.5856i	− 1.35534i	−0.735365	−	0.677672i	$-0.762989\pi$
$61$	− 10.5856i	− 1.35534i	0.735365	−	0.677672i	$-0.237011\pi$
$62$	0	0
$63$	− 3.52806i	− 0.444493i
$64$	0	0
$65$	− 6.18216i	− 0.766803i
$66$	0	0
$67$	12.3030	1.50306	0.751528	−	0.659701i	$-0.229317\pi$
$67$	12.3030	1.50306	0.751528	+	0.659701i	$0.229317\pi$
$68$	0	0
$69$	−20.8186	−2.50626
$70$	0	0
$71$	− 1.08582i	− 0.128863i	−0.997922	−	0.0644314i	$-0.979477\pi$
$71$	− 1.08582i	− 0.128863i	0.997922	−	0.0644314i	$-0.0205234\pi$
$72$	0	0
$73$	− 9.34420i	− 1.09366i	−0.837245	−	0.546828i	$-0.815835\pi$
$73$	− 9.34420i	− 1.09366i	0.837245	−	0.546828i	$-0.184165\pi$
$74$	0	0
$75$	2.30278i	0.265902i
$76$	0	0
$77$	−3.75119	−0.427488
$78$	0	0
$79$	− 11.0401i	− 1.24210i	−0.783769	−	0.621052i	$-0.786705\pi$
$79$	− 11.0401i	− 1.24210i	0.783769	−	0.621052i	$-0.213295\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	12.2524	1.34487	0.672436	−	0.740156i	$-0.265248\pi$
$83$	12.2524	1.34487	0.672436	+	0.740156i	$0.265248\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.6382	1.24774
$88$	0	0
$89$	−10.8303	−1.14801	−0.574005	−	0.818852i	$-0.694611\pi$
$89$	−10.8303	−1.14801	−0.574005	+	0.818852i	$0.694611\pi$
$90$	0	0
$91$	− 9.47162i	− 0.992896i
$92$	0	0
$93$	6.24214	0.647280
$94$	0	0
$95$	0.345896i	0.0354882i
$96$	0	0
$97$	− 6.97153i	− 0.707851i	−0.935274	−	0.353926i	$-0.884847\pi$
$97$	− 6.97153i	− 0.707851i	0.935274	−	0.353926i	$-0.115153\pi$
$98$	0	0
$99$	− 5.63816i	− 0.566656i
$100$	0	0
$101$	16.9191	1.68351	0.841756	−	0.539858i	$-0.181522\pi$
$101$	16.9191	1.68351	0.841756	+	0.539858i	$0.181522\pi$
$102$	0	0
$103$	9.06890	0.893586	0.446793	−	0.894637i	$-0.352566\pi$
$103$	9.06890	0.893586	0.446793	+	0.894637i	$0.352566\pi$
$104$	0	0
$105$	3.52806i	0.344303i
$106$	0	0
$107$	− 5.38006i	− 0.520110i	−0.965594	−	0.260055i	$-0.916259\pi$
$107$	− 5.38006i	− 0.520110i	0.965594	−	0.260055i	$-0.0837408\pi$
$108$	0	0
$109$	7.14640i	0.684501i	0.939609	+	0.342250i	$0.111189\pi$
$109$	7.14640i	0.684501i	−0.939609	+	0.342250i	$0.888811\pi$
$110$	0	0
$111$	9.41158	0.893308
$112$	0	0
$113$	18.8990i	1.77787i	0.458033	+	0.888935i	$0.348554\pi$
$113$	18.8990i	1.77787i	−0.458033	+	0.888935i	$0.651446\pi$
$114$	0	0
$115$	9.04065	0.843045
$116$	0	0
$117$	14.2361	1.31613
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00525	0.455023
$122$	0	0
$123$	2.63816	0.237874
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	18.6094	1.65131	0.825657	−	0.564172i	$-0.190805\pi$
$127$	18.6094	1.65131	0.825657	+	0.564172i	$0.190805\pi$
$128$	0	0
$129$	1.68320i	0.148197i
$130$	0	0
$131$	3.69739i	0.323043i	0.986869	+	0.161521i	$0.0516401\pi$
$131$	3.69739i	0.323043i	−0.986869	+	0.161521i	$0.948360\pi$
$132$	0	0
$133$	0.529944i	0.0459519i
$134$	0	0
$135$	1.60555	0.138184
$136$	0	0
$137$	−6.28824	−0.537240	−0.268620	−	0.963246i	$-0.586568\pi$
$137$	−6.28824	−0.537240	−0.268620	+	0.963246i	$0.586568\pi$
$138$	0	0
$139$	10.0556i	0.852903i	0.904510	+	0.426452i	$0.140237\pi$
$139$	10.0556i	0.852903i	−0.904510	+	0.426452i	$0.859763\pi$
$140$	0	0
$141$	− 1.36798i	− 0.115205i
$142$	0	0
$143$	− 15.1365i	− 1.26578i
$144$	0	0
$145$	−5.05397	−0.419709
$146$	0	0
$147$	− 10.7141i	− 0.883687i
$148$	0	0
$149$	1.27054	0.104086	0.0520432	−	0.998645i	$-0.483427\pi$
$149$	1.27054	0.104086	0.0520432	+	0.998645i	$0.483427\pi$
$150$	0	0
$151$	22.4948	1.83060	0.915300	−	0.402773i	$-0.131954\pi$
$151$	22.4948	1.83060	0.915300	+	0.402773i	$0.131954\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.71070	−0.217729
$156$	0	0
$157$	17.0193	1.35829	0.679143	−	0.734006i	$-0.262352\pi$
$157$	17.0193	1.35829	0.679143	+	0.734006i	$0.262352\pi$
$158$	0	0
$159$	− 22.1009i	− 1.75271i
$160$	0	0
$161$	13.8511	1.09162
$162$	0	0
$163$	0.799925i	0.0626550i	0.999509	+	0.0313275i	$0.00997348\pi$
$163$	0.799925i	0.0626550i	−0.999509	+	0.0313275i	$0.990027\pi$
$164$	0	0
$165$	5.63816i	0.438930i
$166$	0	0
$167$	− 17.8105i	− 1.37822i	−0.724656	−	0.689111i	$-0.758001\pi$
$167$	− 17.8105i	− 1.37822i	0.724656	−	0.689111i	$-0.241999\pi$
$168$	0	0
$169$	25.2191	1.93993
$170$	0	0
$171$	−0.796521	−0.0609115
$172$	0	0
$173$	− 6.13517i	− 0.466448i	−0.972423	−	0.233224i	$-0.925072\pi$
$173$	− 6.13517i	− 0.466448i	0.972423	−	0.233224i	$-0.0749275\pi$
$174$	0	0
$175$	− 1.53209i	− 0.115815i
$176$	0	0
$177$	15.5423i	1.16823i
$178$	0	0
$179$	7.55270	0.564516	0.282258	−	0.959339i	$-0.408917\pi$
$179$	7.55270	0.564516	0.282258	+	0.959339i	$0.408917\pi$
$180$	0	0
$181$	− 16.2037i	− 1.20442i	−0.798340	−	0.602208i	$-0.794288\pi$
$181$	− 16.2037i	− 1.20442i	0.798340	−	0.602208i	$-0.205712\pi$
$182$	0	0
$183$	−24.3762	−1.80194
$184$	0	0
$185$	−4.08706	−0.300487
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.45985	0.178928
$190$	0	0
$191$	21.8640	1.58203	0.791014	−	0.611799i	$-0.209554\pi$
$191$	21.8640	1.58203	0.791014	+	0.611799i	$0.209554\pi$
$192$	0	0
$193$	2.35506i	0.169521i	0.996401	+	0.0847606i	$0.0270125\pi$
$193$	2.35506i	0.169521i	−0.996401	+	0.0847606i	$0.972987\pi$
$194$	0	0
$195$	−14.2361	−1.01947
$196$	0	0
$197$	9.36359i	0.667128i	0.942727	+	0.333564i	$0.108251\pi$
$197$	9.36359i	0.667128i	−0.942727	+	0.333564i	$0.891749\pi$
$198$	0	0
$199$	12.3432i	0.874987i	0.899222	+	0.437494i	$0.144134\pi$
$199$	12.3432i	0.874987i	−0.899222	+	0.437494i	$0.855866\pi$
$200$	0	0
$201$	− 28.3312i	− 1.99833i
$202$	0	0
$203$	−7.74313	−0.543461
$204$	0	0
$205$	−1.14564	−0.0800151
$206$	0	0
$207$	20.8186i	1.44699i
$208$	0	0
$209$	0.846898i	0.0585812i
$210$	0	0
$211$	5.16250i	0.355401i	0.984085	+	0.177701i	$0.0568659\pi$
$211$	5.16250i	0.355401i	−0.984085	+	0.177701i	$0.943134\pi$
$212$	0	0
$213$	−2.50039	−0.171324
$214$	0	0
$215$	− 0.730943i	− 0.0498499i
$216$	0	0
$217$	−4.15304	−0.281927
$218$	0	0
$219$	−21.5176	−1.45402
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.9115	−0.998547	−0.499274	−	0.866444i	$-0.666400\pi$
$223$	−14.9115	−0.998547	−0.499274	+	0.866444i	$0.666400\pi$
$224$	0	0
$225$	2.30278	0.153518
$226$	0	0
$227$	− 19.9006i	− 1.32085i	−0.750892	−	0.660425i	$-0.770376\pi$
$227$	− 19.9006i	− 1.32085i	0.750892	−	0.660425i	$-0.229624\pi$
$228$	0	0
$229$	−25.4356	−1.68083	−0.840417	−	0.541941i	$-0.817690\pi$
$229$	−25.4356	−1.68083	−0.840417	+	0.541941i	$0.817690\pi$
$230$	0	0
$231$	8.63816i	0.568349i
$232$	0	0
$233$	− 18.0696i	− 1.18378i	−0.806019	−	0.591890i	$-0.798382\pi$
$233$	− 18.0696i	− 1.18378i	0.806019	−	0.591890i	$-0.201618\pi$
$234$	0	0
$235$	0.594059i	0.0387521i
$236$	0	0
$237$	−25.4228	−1.65139
$238$	0	0
$239$	−3.46127	−0.223891	−0.111945	−	0.993714i	$-0.535708\pi$
$239$	−3.46127	−0.223891	−0.111945	+	0.993714i	$0.535708\pi$
$240$	0	0
$241$	− 14.0380i	− 0.904268i	−0.891950	−	0.452134i	$-0.850663\pi$
$241$	− 14.0380i	− 0.904268i	0.891950	−	0.452134i	$-0.149337\pi$
$242$	0	0
$243$	19.6056i	1.25770i
$244$	0	0
$245$	4.65270i	0.297250i
$246$	0	0
$247$	−2.13839	−0.136062
$248$	0	0
$249$	− 28.2144i	− 1.78802i
$250$	0	0
$251$	−3.61602	−0.228241	−0.114121	−	0.993467i	$-0.536405\pi$
$251$	−3.61602	−0.228241	−0.114121	+	0.993467i	$0.536405\pi$
$252$	0	0
$253$	22.1353	1.39163
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.91274	0.618340	0.309170	−	0.951007i	$-0.399949\pi$
$257$	9.91274	0.618340	0.309170	+	0.951007i	$0.399949\pi$
$258$	0	0
$259$	−6.26174	−0.389085
$260$	0	0
$261$	− 11.6382i	− 0.720384i
$262$	0	0
$263$	22.8485	1.40890	0.704449	−	0.709755i	$-0.251194\pi$
$263$	22.8485	1.40890	0.704449	+	0.709755i	$0.251194\pi$
$264$	0	0
$265$	9.59749i	0.589569i
$266$	0	0
$267$	24.9398i	1.52629i
$268$	0	0
$269$	16.3450i	0.996571i	0.867013	+	0.498286i	$0.166037\pi$
$269$	16.3450i	0.996571i	−0.867013	+	0.498286i	$0.833963\pi$
$270$	0	0
$271$	−24.3015	−1.47621	−0.738105	−	0.674686i	$-0.764279\pi$
$271$	−24.3015	−1.47621	−0.738105	+	0.674686i	$0.764279\pi$
$272$	0	0
$273$	−21.8110	−1.32006
$274$	0	0
$275$	− 2.44842i	− 0.147645i
$276$	0	0
$277$	− 22.9348i	− 1.37802i	−0.724752	−	0.689010i	$-0.758046\pi$
$277$	− 22.9348i	− 1.37802i	0.724752	−	0.689010i	$-0.241954\pi$
$278$	0	0
$279$	− 6.24214i	− 0.373707i
$280$	0	0
$281$	16.4468	0.981132	0.490566	−	0.871404i	$-0.336790\pi$
$281$	16.4468	0.981132	0.490566	+	0.871404i	$0.336790\pi$
$282$	0	0
$283$	− 4.05544i	− 0.241071i	−0.992709	−	0.120535i	$-0.961539\pi$
$283$	− 4.05544i	− 0.241071i	0.992709	−	0.120535i	$-0.0384611\pi$
$284$	0	0
$285$	0.796521	0.0471818
$286$	0	0
$287$	−1.75522	−0.103608
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−16.0539	−0.941094
$292$	0	0
$293$	3.85188	0.225029	0.112515	−	0.993650i	$-0.464109\pi$
$293$	3.85188	0.225029	0.112515	+	0.993650i	$0.464109\pi$
$294$	0	0
$295$	− 6.74937i	− 0.392963i
$296$	0	0
$297$	3.93106	0.228103
$298$	0	0
$299$	55.8908i	3.23225i
$300$	0	0
$301$	− 1.11987i	− 0.0645482i
$302$	0	0
$303$	− 38.9609i	− 2.23824i
$304$	0	0
$305$	10.5856	0.606128
$306$	0	0
$307$	−11.6595	−0.665444	−0.332722	−	0.943025i	$-0.607967\pi$
$307$	−11.6595	−0.665444	−0.332722	+	0.943025i	$0.607967\pi$
$308$	0	0
$309$	− 20.8836i	− 1.18803i
$310$	0	0
$311$	12.6065i	0.714850i	0.933942	+	0.357425i	$0.116345\pi$
$311$	12.6065i	0.714850i	−0.933942	+	0.357425i	$0.883655\pi$
$312$	0	0
$313$	− 17.1453i	− 0.969112i	−0.874760	−	0.484556i	$-0.838981\pi$
$313$	− 17.1453i	− 0.969112i	0.874760	−	0.484556i	$-0.161019\pi$
$314$	0	0
$315$	3.52806	0.198783
$316$	0	0
$317$	2.62353i	0.147352i	0.997282	+	0.0736762i	$0.0234731\pi$
$317$	2.62353i	0.147352i	−0.997282	+	0.0736762i	$0.976527\pi$
$318$	0	0
$319$	−12.3742	−0.692824
$320$	0	0
$321$	−12.3891	−0.691491
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.18216	0.342925
$326$	0	0
$327$	16.4566	0.910049
$328$	0	0
$329$	0.910151i	0.0501783i
$330$	0	0
$331$	12.6609	0.695906	0.347953	−	0.937512i	$-0.386877\pi$
$331$	12.6609	0.695906	0.347953	+	0.937512i	$0.386877\pi$
$332$	0	0
$333$	− 9.41158i	− 0.515751i
$334$	0	0
$335$	12.3030i	0.672187i
$336$	0	0
$337$	− 4.77135i	− 0.259912i	−0.991520	−	0.129956i	$-0.958516\pi$
$337$	− 4.77135i	− 0.259912i	0.991520	−	0.129956i	$-0.0414836\pi$
$338$	0	0
$339$	43.5202	2.36369
$340$	0	0
$341$	−6.63693	−0.359410
$342$	0	0
$343$	17.8530i	0.963970i
$344$	0	0
$345$	− 20.8186i	− 1.12084i
$346$	0	0
$347$	− 8.54464i	− 0.458700i	−0.973344	−	0.229350i	$-0.926340\pi$
$347$	− 8.54464i	− 0.458700i	0.973344	−	0.229350i	$-0.0736601\pi$
$348$	0	0
$349$	17.7864	0.952083	0.476041	−	0.879423i	$-0.342071\pi$
$349$	17.7864	0.952083	0.476041	+	0.879423i	$0.342071\pi$
$350$	0	0
$351$	9.92578i	0.529799i
$352$	0	0
$353$	−32.1311	−1.71016	−0.855082	−	0.518492i	$-0.826493\pi$
$353$	−32.1311	−1.71016	−0.855082	+	0.518492i	$0.826493\pi$
$354$	0	0
$355$	1.08582	0.0576292
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.0091	−1.63660	−0.818299	−	0.574793i	$-0.805082\pi$
$359$	−31.0091	−1.63660	−0.818299	+	0.574793i	$0.805082\pi$
$360$	0	0
$361$	−18.8804	−0.993703
$362$	0	0
$363$	− 11.5260i	− 0.604957i
$364$	0	0
$365$	9.34420	0.489098
$366$	0	0
$367$	23.4378i	1.22344i	0.791073	+	0.611722i	$0.209523\pi$
$367$	23.4378i	1.22344i	−0.791073	+	0.611722i	$0.790477\pi$
$368$	0	0
$369$	− 2.63816i	− 0.137337i
$370$	0	0
$371$	14.7042i	0.763404i
$372$	0	0
$373$	−22.4192	−1.16082	−0.580412	−	0.814323i	$-0.697108\pi$
$373$	−22.4192	−1.16082	−0.580412	+	0.814323i	$0.697108\pi$
$374$	0	0
$375$	−2.30278	−0.118915
$376$	0	0
$377$	− 31.2444i	− 1.60917i
$378$	0	0
$379$	− 29.9129i	− 1.53652i	−0.640136	−	0.768262i	$-0.721122\pi$
$379$	− 29.9129i	− 1.53652i	0.640136	−	0.768262i	$-0.278878\pi$
$380$	0	0
$381$	− 42.8532i	− 2.19544i
$382$	0	0
$383$	28.1941	1.44065	0.720325	−	0.693637i	$-0.243993\pi$
$383$	28.1941	1.44065	0.720325	+	0.693637i	$0.243993\pi$
$384$	0	0
$385$	− 3.75119i	− 0.191178i
$386$	0	0
$387$	1.68320	0.0855618
$388$	0	0
$389$	5.87615	0.297933	0.148966	−	0.988842i	$-0.452405\pi$
$389$	5.87615	0.297933	0.148966	+	0.988842i	$0.452405\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	8.51427	0.429488
$394$	0	0
$395$	11.0401	0.555486
$396$	0	0
$397$	39.1164i	1.96319i	0.190966	+	0.981597i	$0.438838\pi$
$397$	39.1164i	1.96319i	−0.190966	+	0.981597i	$0.561162\pi$
$398$	0	0
$399$	1.22034	0.0610935
$400$	0	0
$401$	− 27.9328i	− 1.39490i	−0.716635	−	0.697448i	$-0.754319\pi$
$401$	− 27.9328i	− 1.39490i	0.716635	−	0.697448i	$-0.245681\pi$
$402$	0	0
$403$	− 16.7580i	− 0.834776i
$404$	0	0
$405$	− 10.6056i	− 0.526994i
$406$	0	0
$407$	−10.0068	−0.496020
$408$	0	0
$409$	−3.35372	−0.165831	−0.0829154	−	0.996557i	$-0.526423\pi$
$409$	−3.35372	−0.165831	−0.0829154	+	0.996557i	$0.526423\pi$
$410$	0	0
$411$	14.4804i	0.714266i
$412$	0	0
$413$	− 10.3406i	− 0.508829i
$414$	0	0
$415$	12.2524i	0.601445i
$416$	0	0
$417$	23.1558	1.13394
$418$	0	0
$419$	− 17.7106i	− 0.865219i	−0.901581	−	0.432610i	$-0.857593\pi$
$419$	− 17.7106i	− 0.865219i	0.901581	−	0.432610i	$-0.142407\pi$
$420$	0	0
$421$	−39.1191	−1.90655	−0.953274	−	0.302106i	$-0.902310\pi$
$421$	−39.1191	−1.90655	−0.953274	+	0.302106i	$0.902310\pi$
$422$	0	0
$423$	−1.36798	−0.0665137
$424$	0	0
$425$	0	0
$426$	0	0
$427$	16.2180	0.784846
$428$	0	0
$429$	−34.8560	−1.68286
$430$	0	0
$431$	34.4019i	1.65708i	0.559927	+	0.828542i	$0.310829\pi$
$431$	34.4019i	1.65708i	−0.559927	+	0.828542i	$0.689171\pi$
$432$	0	0
$433$	6.72002	0.322943	0.161472	−	0.986877i	$-0.448376\pi$
$433$	6.72002	0.322943	0.161472	+	0.986877i	$0.448376\pi$
$434$	0	0
$435$	11.6382i	0.558007i
$436$	0	0
$437$	− 3.12713i	− 0.149591i
$438$	0	0
$439$	28.4568i	1.35817i	0.734060	+	0.679085i	$0.237623\pi$
$439$	28.4568i	1.35817i	−0.734060	+	0.679085i	$0.762377\pi$
$440$	0	0
$441$	−10.7141	−0.510197
$442$	0	0
$443$	−6.39254	−0.303719	−0.151859	−	0.988402i	$-0.548526\pi$
$443$	−6.39254	−0.303719	−0.151859	+	0.988402i	$0.548526\pi$
$444$	0	0
$445$	− 10.8303i	− 0.513406i
$446$	0	0
$447$	− 2.92576i	− 0.138384i
$448$	0	0
$449$	24.8875i	1.17451i	0.809400	+	0.587257i	$0.199792\pi$
$449$	24.8875i	1.17451i	−0.809400	+	0.587257i	$0.800208\pi$
$450$	0	0
$451$	−2.80501	−0.132083
$452$	0	0
$453$	− 51.8004i	− 2.43380i
$454$	0	0
$455$	9.47162	0.444036
$456$	0	0
$457$	−15.0624	−0.704589	−0.352294	−	0.935889i	$-0.614598\pi$
$457$	−15.0624	−0.704589	−0.352294	+	0.935889i	$0.614598\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.44077	−0.160253	−0.0801264	−	0.996785i	$-0.525532\pi$
$461$	−3.44077	−0.160253	−0.0801264	+	0.996785i	$0.525532\pi$
$462$	0	0
$463$	13.7212	0.637680	0.318840	−	0.947809i	$-0.396707\pi$
$463$	13.7212	0.637680	0.318840	+	0.947809i	$0.396707\pi$
$464$	0	0
$465$	6.24214i	0.289472i
$466$	0	0
$467$	3.18218	0.147254	0.0736269	−	0.997286i	$-0.476543\pi$
$467$	3.18218	0.147254	0.0736269	+	0.997286i	$0.476543\pi$
$468$	0	0
$469$	18.8494i	0.870383i
$470$	0	0
$471$	− 39.1916i	− 1.80585i
$472$	0	0
$473$	− 1.78965i	− 0.0822884i
$474$	0	0
$475$	−0.345896	−0.0158708
$476$	0	0
$477$	−22.1009	−1.01193
$478$	0	0
$479$	− 34.1428i	− 1.56002i	−0.625765	−	0.780012i	$-0.715213\pi$
$479$	− 34.1428i	− 1.56002i	0.625765	−	0.780012i	$-0.284787\pi$
$480$	0	0
$481$	− 25.2669i	− 1.15207i
$482$	0	0
$483$	− 31.8959i	− 1.45132i
$484$	0	0
$485$	6.97153	0.316561
$486$	0	0
$487$	10.0187i	0.453989i	0.973896	+	0.226995i	$0.0728900\pi$
$487$	10.0187i	0.453989i	−0.973896	+	0.226995i	$0.927110\pi$
$488$	0	0
$489$	1.84205	0.0833003
$490$	0	0
$491$	−0.623364	−0.0281320	−0.0140660	−	0.999901i	$-0.504478\pi$
$491$	−0.623364	−0.0281320	−0.0140660	+	0.999901i	$0.504478\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	5.63816	0.253416
$496$	0	0
$497$	1.66357	0.0746212
$498$	0	0
$499$	1.53671i	0.0687925i	0.999408	+	0.0343962i	$0.0109508\pi$
$499$	1.53671i	0.0687925i	−0.999408	+	0.0343962i	$0.989049\pi$
$500$	0	0
$501$	−41.0137	−1.83236
$502$	0	0
$503$	− 11.6041i	− 0.517401i	−0.965958	−	0.258700i	$-0.916706\pi$
$503$	− 11.6041i	− 0.517401i	0.965958	−	0.258700i	$-0.0832943\pi$
$504$	0	0
$505$	16.9191i	0.752889i
$506$	0	0
$507$	− 58.0740i	− 2.57916i
$508$	0	0
$509$	38.2148	1.69384	0.846921	−	0.531719i	$-0.178454\pi$
$509$	38.2148	1.69384	0.846921	+	0.531719i	$0.178454\pi$
$510$	0	0
$511$	14.3161	0.633309
$512$	0	0
$513$	− 0.555354i	− 0.0245195i
$514$	0	0
$515$	9.06890i	0.399624i
$516$	0	0
$517$	1.45450i	0.0639690i
$518$	0	0
$519$	−14.1279	−0.620146
$520$	0	0
$521$	− 27.4152i	− 1.20108i	−0.799595	−	0.600540i	$-0.794952\pi$
$521$	− 27.4152i	− 1.20108i	0.799595	−	0.600540i	$-0.205048\pi$
$522$	0	0
$523$	−12.8711	−0.562815	−0.281408	−	0.959588i	$-0.590801\pi$
$523$	−12.8711	−0.562815	−0.281408	+	0.959588i	$0.590801\pi$
$524$	0	0
$525$	−3.52806	−0.153977
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−58.7334	−2.55363
$530$	0	0
$531$	15.5423	0.674478
$532$	0	0
$533$	− 7.08254i	− 0.306779i
$534$	0	0
$535$	5.38006	0.232600
$536$	0	0
$537$	− 17.3922i	− 0.750528i
$538$	0	0
$539$	11.3918i	0.490678i
$540$	0	0
$541$	− 38.7809i	− 1.66732i	−0.552278	−	0.833660i	$-0.686241\pi$
$541$	− 38.7809i	− 1.66732i	0.552278	−	0.833660i	$-0.313759\pi$
$542$	0	0
$543$	−37.3136	−1.60128
$544$	0	0
$545$	−7.14640	−0.306118
$546$	0	0
$547$	34.1248i	1.45907i	0.683942	+	0.729536i	$0.260264\pi$
$547$	34.1248i	1.45907i	−0.683942	+	0.729536i	$0.739736\pi$
$548$	0	0
$549$	24.3762i	1.04035i
$550$	0	0
$551$	1.74815i	0.0744736i
$552$	0	0
$553$	16.9144	0.719272
$554$	0	0
$555$	9.41158i	0.399499i
$556$	0	0
$557$	33.8036	1.43231	0.716153	−	0.697943i	$-0.245901\pi$
$557$	33.8036	1.43231	0.716153	+	0.697943i	$0.245901\pi$
$558$	0	0
$559$	4.51881	0.191125
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.6208	−0.869063	−0.434532	−	0.900657i	$-0.643086\pi$
$563$	−20.6208	−0.869063	−0.434532	+	0.900657i	$0.643086\pi$
$564$	0	0
$565$	−18.8990	−0.795088
$566$	0	0
$567$	− 16.2486i	− 0.682379i
$568$	0	0
$569$	15.7173	0.658906	0.329453	−	0.944172i	$-0.393136\pi$
$569$	15.7173	0.658906	0.329453	+	0.944172i	$0.393136\pi$
$570$	0	0
$571$	39.4858i	1.65243i	0.563357	+	0.826214i	$0.309510\pi$
$571$	39.4858i	1.65243i	−0.563357	+	0.826214i	$0.690490\pi$
$572$	0	0
$573$	− 50.3480i	− 2.10332i
$574$	0	0
$575$	9.04065i	0.377021i
$576$	0	0
$577$	−10.9057	−0.454008	−0.227004	−	0.973894i	$-0.572893\pi$
$577$	−10.9057	−0.454008	−0.227004	+	0.973894i	$0.572893\pi$
$578$	0	0
$579$	5.42318	0.225380
$580$	0	0
$581$	18.7717i	0.778781i
$582$	0	0
$583$	23.4987i	0.973215i
$584$	0	0
$585$	14.2361i	0.588592i
$586$	0	0
$587$	−15.2969	−0.631371	−0.315686	−	0.948864i	$-0.602234\pi$
$587$	−15.2969	−0.631371	−0.315686	+	0.948864i	$0.602234\pi$
$588$	0	0
$589$	0.937622i	0.0386340i
$590$	0	0
$591$	21.5622	0.886952
$592$	0	0
$593$	−25.4750	−1.04613	−0.523067	−	0.852292i	$-0.675212\pi$
$593$	−25.4750	−1.04613	−0.523067	+	0.852292i	$0.675212\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	28.4237	1.16330
$598$	0	0
$599$	−19.8734	−0.812006	−0.406003	−	0.913872i	$-0.633078\pi$
$599$	−19.8734	−0.812006	−0.406003	+	0.913872i	$0.633078\pi$
$600$	0	0
$601$	− 36.9027i	− 1.50529i	−0.658424	−	0.752647i	$-0.728777\pi$
$601$	− 36.9027i	− 1.50529i	0.658424	−	0.752647i	$-0.271223\pi$
$602$	0	0
$603$	−28.3312	−1.15373
$604$	0	0
$605$	5.00525i	0.203492i
$606$	0	0
$607$	27.0762i	1.09899i	0.835497	+	0.549494i	$0.185180\pi$
$607$	27.0762i	1.09899i	−0.835497	+	0.549494i	$0.814820\pi$
$608$	0	0
$609$	17.8307i	0.722536i
$610$	0	0
$611$	−3.67257	−0.148576
$612$	0	0
$613$	34.9934	1.41337	0.706685	−	0.707529i	$-0.250190\pi$
$613$	34.9934	1.41337	0.706685	+	0.707529i	$0.250190\pi$
$614$	0	0
$615$	2.63816i	0.106381i
$616$	0	0
$617$	− 34.4944i	− 1.38869i	−0.719641	−	0.694346i	$-0.755694\pi$
$617$	− 34.4944i	− 1.38869i	0.719641	−	0.694346i	$-0.244306\pi$
$618$	0	0
$619$	13.2602i	0.532971i	0.963839	+	0.266485i	$0.0858624\pi$
$619$	13.2602i	0.532971i	−0.963839	+	0.266485i	$0.914138\pi$
$620$	0	0
$621$	−14.5152	−0.582476
$622$	0	0
$623$	− 16.5930i	− 0.664784i
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.95022	0.0778841
$628$	0	0
$629$	0	0
$630$	0	0
$631$	10.2715	0.408900	0.204450	−	0.978877i	$-0.434459\pi$
$631$	10.2715	0.408900	0.204450	+	0.978877i	$0.434459\pi$
$632$	0	0
$633$	11.8881	0.472509
$634$	0	0
$635$	18.6094i	0.738490i
$636$	0	0
$637$	−28.7638	−1.13966
$638$	0	0
$639$	2.50039i	0.0989140i
$640$	0	0
$641$	− 1.97743i	− 0.0781037i	−0.999237	−	0.0390519i	$-0.987566\pi$
$641$	− 1.97743i	− 0.0781037i	0.999237	−	0.0390519i	$-0.0124338\pi$
$642$	0	0
$643$	43.7321i	1.72462i	0.506377	+	0.862312i	$0.330984\pi$
$643$	43.7321i	1.72462i	−0.506377	+	0.862312i	$0.669016\pi$
$644$	0	0
$645$	−1.68320	−0.0662758
$646$	0	0
$647$	44.5243	1.75043	0.875216	−	0.483732i	$-0.160719\pi$
$647$	44.5243	1.75043	0.875216	+	0.483732i	$0.160719\pi$
$648$	0	0
$649$	− 16.5253i	− 0.648674i
$650$	0	0
$651$	9.56352i	0.374824i
$652$	0	0
$653$	− 10.5300i	− 0.412071i	−0.978545	−	0.206035i	$-0.933944\pi$
$653$	− 10.5300i	− 0.412071i	0.978545	−	0.206035i	$-0.0660562\pi$
$654$	0	0
$655$	−3.69739	−0.144469
$656$	0	0
$657$	21.5176i	0.839481i
$658$	0	0
$659$	−19.1763	−0.747004	−0.373502	−	0.927629i	$-0.621843\pi$
$659$	−19.1763	−0.747004	−0.373502	+	0.927629i	$0.621843\pi$
$660$	0	0
$661$	38.9860	1.51638	0.758190	−	0.652034i	$-0.226084\pi$
$661$	38.9860	1.51638	0.758190	+	0.652034i	$0.226084\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.529944	−0.0205503
$666$	0	0
$667$	45.6912	1.76917
$668$	0	0
$669$	34.3378i	1.32758i
$670$	0	0
$671$	25.9179	1.00055
$672$	0	0
$673$	− 45.3288i	− 1.74730i	−0.486558	−	0.873648i	$-0.661748\pi$
$673$	− 45.3288i	− 1.74730i	0.486558	−	0.873648i	$-0.338252\pi$
$674$	0	0
$675$	1.60555i	0.0617977i
$676$	0	0
$677$	− 33.3408i	− 1.28139i	−0.767794	−	0.640696i	$-0.778646\pi$
$677$	− 33.3408i	− 1.28139i	0.767794	−	0.640696i	$-0.221354\pi$
$678$	0	0
$679$	10.6810	0.409899
$680$	0	0
$681$	−45.8266	−1.75608
$682$	0	0
$683$	− 10.9924i	− 0.420613i	−0.977636	−	0.210306i	$-0.932554\pi$
$683$	− 10.9924i	− 0.420613i	0.977636	−	0.210306i	$-0.0674461\pi$
$684$	0	0
$685$	− 6.28824i	− 0.240261i
$686$	0	0
$687$	58.5725i	2.23468i
$688$	0	0
$689$	−59.3332	−2.26042
$690$	0	0
$691$	37.6221i	1.43121i	0.698503	+	0.715607i	$0.253850\pi$
$691$	37.6221i	1.43121i	−0.698503	+	0.715607i	$0.746150\pi$
$692$	0	0
$693$	8.63816	0.328136
$694$	0	0
$695$	−10.0556	−0.381430
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−41.6103	−1.57384
$700$	0	0
$701$	21.0184	0.793855	0.396927	−	0.917850i	$-0.370076\pi$
$701$	21.0184	0.793855	0.396927	+	0.917850i	$0.370076\pi$
$702$	0	0
$703$	1.41370i	0.0533186i
$704$	0	0
$705$	1.36798	0.0515213
$706$	0	0
$707$	25.9215i	0.974880i
$708$	0	0
$709$	35.0263i	1.31544i	0.753263	+	0.657720i	$0.228479\pi$
$709$	35.0263i	1.31544i	−0.753263	+	0.657720i	$0.771521\pi$
$710$	0	0
$711$	25.4228i	0.953429i
$712$	0	0
$713$	24.5065	0.917777
$714$	0	0
$715$	15.1365	0.566073
$716$	0	0
$717$	7.97053i	0.297665i
$718$	0	0
$719$	12.7653i	0.476066i	0.971257	+	0.238033i	$0.0765026\pi$
$719$	12.7653i	0.476066i	−0.971257	+	0.238033i	$0.923497\pi$
$720$	0	0
$721$	13.8944i	0.517453i
$722$	0	0
$723$	−32.3264	−1.20223
$724$	0	0
$725$	− 5.05397i	− 0.187700i
$726$	0	0
$727$	−40.6908	−1.50914	−0.754569	−	0.656221i	$-0.772154\pi$
$727$	−40.6908	−1.50914	−0.754569	+	0.656221i	$0.772154\pi$
$728$	0	0
$729$	13.3305	0.493723
$730$	0	0
$731$	0	0
$732$	0	0
$733$	15.1041	0.557881	0.278941	−	0.960308i	$-0.410017\pi$
$733$	15.1041	0.557881	0.278941	+	0.960308i	$0.410017\pi$
$734$	0	0
$735$	10.7141	0.395197
$736$	0	0
$737$	30.1230i	1.10959i
$738$	0	0
$739$	−10.4340	−0.383822	−0.191911	−	0.981412i	$-0.561468\pi$
$739$	−10.4340	−0.383822	−0.191911	+	0.981412i	$0.561468\pi$
$740$	0	0
$741$	4.92422i	0.180896i
$742$	0	0
$743$	− 21.8006i	− 0.799786i	−0.916562	−	0.399893i	$-0.869047\pi$
$743$	− 21.8006i	− 0.799786i	0.916562	−	0.399893i	$-0.130953\pi$
$744$	0	0
$745$	1.27054i	0.0465489i
$746$	0	0
$747$	−28.2144	−1.03231
$748$	0	0
$749$	8.24274	0.301183
$750$	0	0
$751$	− 26.5462i	− 0.968683i	−0.874879	−	0.484342i	$-0.839059\pi$
$751$	− 26.5462i	− 0.968683i	0.874879	−	0.484342i	$-0.160941\pi$
$752$	0	0
$753$	8.32688i	0.303448i
$754$	0	0
$755$	22.4948i	0.818669i
$756$	0	0
$757$	25.5323	0.927988	0.463994	−	0.885838i	$-0.346416\pi$
$757$	25.5323	0.927988	0.463994	+	0.885838i	$0.346416\pi$
$758$	0	0
$759$	− 50.9726i	− 1.85019i
$760$	0	0
$761$	−13.4659	−0.488139	−0.244070	−	0.969758i	$-0.578483\pi$
$761$	−13.4659	−0.488139	−0.244070	+	0.969758i	$0.578483\pi$
$762$	0	0
$763$	−10.9489	−0.396377
$764$	0	0
$765$	0	0
$766$	0	0
$767$	41.7257	1.50663
$768$	0	0
$769$	−30.5735	−1.10251	−0.551255	−	0.834337i	$-0.685851\pi$
$769$	−30.5735	−1.10251	−0.551255	+	0.834337i	$0.685851\pi$
$770$	0	0
$771$	− 22.8268i	− 0.822088i
$772$	0	0
$773$	19.6308	0.706073	0.353036	−	0.935610i	$-0.385149\pi$
$773$	19.6308	0.706073	0.353036	+	0.935610i	$0.385149\pi$
$774$	0	0
$775$	− 2.71070i	− 0.0973714i
$776$	0	0
$777$	14.4194i	0.517292i
$778$	0	0
$779$	0.396273i	0.0141980i
$780$	0	0
$781$	2.65853	0.0951298
$782$	0	0
$783$	8.11441	0.289985
$784$	0	0
$785$	17.0193i	0.607444i
$786$	0	0
$787$	− 12.7767i	− 0.455439i	−0.973727	−	0.227720i	$-0.926873\pi$
$787$	− 12.7767i	− 0.455439i	0.973727	−	0.227720i	$-0.0731269\pi$
$788$	0	0
$789$	− 52.6149i	− 1.87314i
$790$	0	0
$791$	−28.9550	−1.02952
$792$	0	0
$793$	65.4417i	2.32390i
$794$	0	0
$795$	22.1009	0.783837
$796$	0	0
$797$	9.76822	0.346008	0.173004	−	0.984921i	$-0.444653\pi$
$797$	9.76822	0.346008	0.173004	+	0.984921i	$0.444653\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	24.9398	0.881203
$802$	0	0
$803$	22.8785	0.807365
$804$	0	0
$805$	13.8511i	0.488187i
$806$	0	0
$807$	37.6388	1.32495
$808$	0	0
$809$	− 39.9796i	− 1.40561i	−0.711384	−	0.702803i	$-0.751931\pi$
$809$	− 39.9796i	− 1.40561i	0.711384	−	0.702803i	$-0.248069\pi$
$810$	0	0
$811$	− 6.22771i	− 0.218684i	−0.994004	−	0.109342i	$-0.965126\pi$
$811$	− 6.22771i	− 0.218684i	0.994004	−	0.109342i	$-0.0348744\pi$
$812$	0	0
$813$	55.9609i	1.96263i
$814$	0	0
$815$	−0.799925	−0.0280202
$816$	0	0
$817$	−0.252830	−0.00884541
$818$	0	0
$819$	21.8110i	0.762139i
$820$	0	0
$821$	− 3.43848i	− 0.120004i	−0.998198	−	0.0600019i	$-0.980889\pi$
$821$	− 3.43848i	− 0.120004i	0.998198	−	0.0600019i	$-0.0191107\pi$
$822$	0	0
$823$	29.2112i	1.01824i	0.860696	+	0.509119i	$0.170029\pi$
$823$	29.2112i	1.01824i	−0.860696	+	0.509119i	$0.829971\pi$
$824$	0	0
$825$	−5.63816	−0.196295
$826$	0	0
$827$	54.9557i	1.91100i	0.294999	+	0.955498i	$0.404681\pi$
$827$	54.9557i	1.91100i	−0.294999	+	0.955498i	$0.595319\pi$
$828$	0	0
$829$	11.5572	0.401397	0.200698	−	0.979653i	$-0.435679\pi$
$829$	11.5572	0.401397	0.200698	+	0.979653i	$0.435679\pi$
$830$	0	0
$831$	−52.8137	−1.83209
$832$	0	0
$833$	0	0
$834$	0	0
$835$	17.8105	0.616359
$836$	0	0
$837$	4.35217	0.150433
$838$	0	0
$839$	− 49.5915i	− 1.71209i	−0.516901	−	0.856045i	$-0.672915\pi$
$839$	− 49.5915i	− 1.71209i	0.516901	−	0.856045i	$-0.327085\pi$
$840$	0	0
$841$	3.45740	0.119221
$842$	0	0
$843$	− 37.8732i	− 1.30442i
$844$	0	0
$845$	25.2191i	0.867564i
$846$	0	0
$847$	7.66849i	0.263493i
$848$	0	0
$849$	−9.33877	−0.320506
$850$	0	0
$851$	36.9497	1.26662
$852$	0	0
$853$	− 1.12515i	− 0.0385242i	−0.999814	−	0.0192621i	$-0.993868\pi$
$853$	− 1.12515i	− 0.0385242i	0.999814	−	0.0192621i	$-0.00613170\pi$
$854$	0	0
$855$	− 0.796521i	− 0.0272404i
$856$	0	0
$857$	− 16.8681i	− 0.576202i	−0.957600	−	0.288101i	$-0.906976\pi$
$857$	− 16.8681i	− 0.576202i	0.957600	−	0.288101i	$-0.0930240\pi$
$858$	0	0
$859$	26.9983	0.921168	0.460584	−	0.887616i	$-0.347640\pi$
$859$	26.9983	0.921168	0.460584	+	0.887616i	$0.347640\pi$
$860$	0	0
$861$	4.04189i	0.137747i
$862$	0	0
$863$	27.1761	0.925086	0.462543	−	0.886597i	$-0.346937\pi$
$863$	27.1761	0.925086	0.462543	+	0.886597i	$0.346937\pi$
$864$	0	0
$865$	6.13517	0.208602
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27.0307	0.916953
$870$	0	0
$871$	−76.0594	−2.57717
$872$	0	0
$873$	16.0539i	0.543341i
$874$	0	0
$875$	1.53209	0.0517941
$876$	0	0
$877$	− 30.5465i	− 1.03148i	−0.856745	−	0.515741i	$-0.827517\pi$
$877$	− 30.5465i	− 1.03148i	0.856745	−	0.515741i	$-0.172483\pi$
$878$	0	0
$879$	− 8.87002i	− 0.299178i
$880$	0	0
$881$	31.8061i	1.07157i	0.844353	+	0.535787i	$0.179985\pi$
$881$	31.8061i	1.07157i	−0.844353	+	0.535787i	$0.820015\pi$
$882$	0	0
$883$	−4.30075	−0.144732	−0.0723658	−	0.997378i	$-0.523055\pi$
$883$	−4.30075	−0.144732	−0.0723658	+	0.997378i	$0.523055\pi$
$884$	0	0
$885$	−15.5423	−0.522448
$886$	0	0
$887$	− 19.7877i	− 0.664405i	−0.943208	−	0.332202i	$-0.892208\pi$
$887$	− 19.7877i	− 0.664405i	0.943208	−	0.332202i	$-0.107792\pi$
$888$	0	0
$889$	28.5112i	0.956235i
$890$	0	0
$891$	− 25.9668i	− 0.869921i
$892$	0	0
$893$	0.205483	0.00687622
$894$	0	0
$895$	7.55270i	0.252459i
$896$	0	0
$897$	128.704	4.29730
$898$	0	0
$899$	−13.6998	−0.456914
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−2.57881	−0.0858174
$904$	0	0
$905$	16.2037	0.538631
$906$	0	0
$907$	− 36.4938i	− 1.21176i	−0.795558	−	0.605878i	$-0.792822\pi$
$907$	− 36.4938i	− 1.21176i	0.795558	−	0.605878i	$-0.207178\pi$
$908$	0	0
$909$	−38.9609	−1.29225
$910$	0	0
$911$	− 29.9058i	− 0.990825i	−0.868658	−	0.495412i	$-0.835017\pi$
$911$	− 29.9058i	− 0.990825i	0.868658	−	0.495412i	$-0.164983\pi$
$912$	0	0
$913$	29.9989i	0.992818i
$914$	0	0
$915$	− 24.3762i	− 0.805852i
$916$	0	0
$917$	−5.66474	−0.187066
$918$	0	0
$919$	−20.3390	−0.670923	−0.335462	−	0.942054i	$-0.608892\pi$
$919$	−20.3390	−0.670923	−0.335462	+	0.942054i	$0.608892\pi$
$920$	0	0
$921$	26.8493i	0.884713i
$922$	0	0
$923$	6.71269i	0.220951i
$924$	0	0
$925$	− 4.08706i	− 0.134382i
$926$	0	0
$927$	−20.8836	−0.685909
$928$	0	0
$929$	30.0855i	0.987074i	0.869725	+	0.493537i	$0.164296\pi$
$929$	30.0855i	0.987074i	−0.869725	+	0.493537i	$0.835704\pi$
$930$	0	0
$931$	1.60935	0.0527444
$932$	0	0
$933$	29.0300	0.950399
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.4044	0.666583	0.333292	−	0.942824i	$-0.391841\pi$
$937$	20.4044	0.666583	0.333292	+	0.942824i	$0.391841\pi$
$938$	0	0
$939$	−39.4819	−1.28844
$940$	0	0
$941$	34.5644i	1.12677i	0.826196	+	0.563383i	$0.190501\pi$
$941$	34.5644i	1.12677i	−0.826196	+	0.563383i	$0.809499\pi$
$942$	0	0
$943$	10.3573	0.337282
$944$	0	0
$945$	2.45985i	0.0800188i
$946$	0	0
$947$	0.166858i	0.00542215i	0.999996	+	0.00271107i	$0.000862963\pi$
$947$	0.166858i	0.00542215i	−0.999996	+	0.00271107i	$0.999137\pi$
$948$	0	0
$949$	57.7673i	1.87521i
$950$	0	0
$951$	6.04141	0.195906
$952$	0	0
$953$	−26.0938	−0.845261	−0.422631	−	0.906302i	$-0.638893\pi$
$953$	−26.0938	−0.845261	−0.422631	+	0.906302i	$0.638893\pi$
$954$	0	0
$955$	21.8640i	0.707504i
$956$	0	0
$957$	28.4951i	0.921115i
$958$	0	0
$959$	− 9.63414i	− 0.311103i
$960$	0	0
$961$	23.6521	0.762970
$962$	0	0
$963$	12.3891i	0.399233i
$964$	0	0
$965$	−2.35506	−0.0758122
$966$	0	0
$967$	28.5476	0.918030	0.459015	−	0.888428i	$-0.348202\pi$
$967$	28.5476	0.918030	0.459015	+	0.888428i	$0.348202\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.54834	0.145963	0.0729815	−	0.997333i	$-0.476749\pi$
$971$	4.54834	0.145963	0.0729815	+	0.997333i	$0.476749\pi$
$972$	0	0
$973$	−15.4060	−0.493895
$974$	0	0
$975$	− 14.2361i	− 0.455921i
$976$	0	0
$977$	−25.7740	−0.824583	−0.412292	−	0.911052i	$-0.635272\pi$
$977$	−25.7740	−0.824583	−0.412292	+	0.911052i	$0.635272\pi$
$978$	0	0
$979$	− 26.5171i	− 0.847490i
$980$	0	0
$981$	− 16.4566i	− 0.525417i
$982$	0	0
$983$	− 18.1671i	− 0.579441i	−0.957111	−	0.289720i	$-0.906438\pi$
$983$	− 18.1671i	− 0.579441i	0.957111	−	0.289720i	$-0.0935623\pi$
$984$	0	0
$985$	−9.36359	−0.298349
$986$	0	0
$987$	2.09587	0.0667124
$988$	0	0
$989$	6.60820i	0.210129i
$990$	0	0
$991$	30.3752i	0.964900i	0.875923	+	0.482450i	$0.160253\pi$
$991$	30.3752i	0.964900i	−0.875923	+	0.482450i	$0.839747\pi$
$992$	0	0
$993$	− 29.1552i	− 0.925212i
$994$	0	0
$995$	−12.3432	−0.391306
$996$	0	0
$997$	29.8884i	0.946574i	0.880908	+	0.473287i	$0.156933\pi$
$997$	29.8884i	0.946574i	−0.880908	+	0.473287i	$0.843067\pi$
$998$	0	0
$999$	6.56198	0.207612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5780.2.c.g.5201.3		12
17.4	even	4		5780.2.a.l.1.1	✓	6
17.13	even	4		5780.2.a.o.1.6	yes	6
17.16	even	2	inner	5780.2.c.g.5201.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5780.2.a.l.1.1	✓	6	17.4	even	4
5780.2.a.o.1.6	yes	6	17.13	even	4
5780.2.c.g.5201.3		12	1.1	even	1	trivial
5780.2.c.g.5201.10		12	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5780.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.30278i q^{3} +1.00000i q^{5} +1.53209i q^{7} -2.30278 q^{9} +2.44842i q^{11} -6.18216 q^{13} +2.30278 q^{15} +0.345896 q^{19} +3.52806 q^{21} -9.04065i q^{23} -1.00000 q^{25} -1.60555i q^{27} +5.05397i q^{29} +2.71070i q^{31} +5.63816 q^{33} -1.53209 q^{35} +4.08706i q^{37} +14.2361i q^{39} +1.14564i q^{41} -0.730943 q^{43} -2.30278i q^{45} +0.594059 q^{47} +4.65270 q^{49} +9.59749 q^{53} -2.44842 q^{55} -0.796521i q^{57} -6.74937 q^{59} -10.5856i q^{61} -3.52806i q^{63} -6.18216i q^{65} +12.3030 q^{67} -20.8186 q^{69} -1.08582i q^{71} -9.34420i q^{73} +2.30278i q^{75} -3.75119 q^{77} -11.0401i q^{79} -10.6056 q^{81} +12.2524 q^{83} +11.6382 q^{87} -10.8303 q^{89} -9.47162i q^{91} +6.24214 q^{93} +0.345896i q^{95} -6.97153i q^{97} -5.63816i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{9} - 30 q^{13} + 6 q^{15} + 6 q^{19} - 12 q^{25} - 6 q^{43} - 30 q^{47} + 60 q^{49} + 24 q^{53} + 12 q^{59} - 12 q^{67} + 6 q^{77} - 84 q^{81} + 30 q^{83} + 72 q^{87} - 12 q^{89} + 72 q^{93}+O(q^{100}) \) 12 * q - 6 * q^9 - 30 * q^13 + 6 * q^15 + 6 * q^19 - 12 * q^25 - 6 * q^43 - 30 * q^47 + 60 * q^49 + 24 * q^53 + 12 * q^59 - 12 * q^67 + 6 * q^77 - 84 * q^81 + 30 * q^83 + 72 * q^87 - 12 * q^89 + 72 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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