LMFDB - Embedded newform 2420.3.f.c.241.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2420,3,Mod(241,2420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2420, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2420.241");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2420 = 2^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2420.f (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:	$65.9402239752$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 5 x^{14} + 36 x^{13} + 396 x^{12} + 1918 x^{11} + 8573 x^{10} + 28624 x^{9} + \cdots + 121 $ x^16 - 2x^15 + 5x^14 + 36x^13 + 396x^12 + 1918x^11 + 8573x^10 + 28624x^9 + 80803x^8 + 170874x^7 + 284613x^6 + 345888x^5 + 305926x^4 + 174966x^3 + 69685x^2 + 4598*x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 5\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.16
Root			$-0.0370387 + 0.0269102i$ of defining polynomial
Character	$\chi$	$=$	2420.241
Dual form			2420.3.f.c.241.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.15878 q^{3} +2.23607 q^{5} +10.1572i q^{7} +17.6130 q^{9} -3.34439i q^{13} +11.5354 q^{15} +20.1864i q^{17} -5.53109i q^{19} +52.3990i q^{21} +34.6484 q^{23} +5.00000 q^{25} +44.4328 q^{27} +44.8670i q^{29} -43.3440 q^{31} +22.7123i q^{35} -59.4002 q^{37} -17.2530i q^{39} +33.0912i q^{41} -65.3574i q^{43} +39.3839 q^{45} +68.8573 q^{47} -54.1697 q^{49} +104.137i q^{51} -3.21911 q^{53} -28.5337i q^{57} +66.0207 q^{59} +47.3889i q^{61} +178.900i q^{63} -7.47829i q^{65} -31.6113 q^{67} +178.744 q^{69} +3.76017 q^{71} -22.4126i q^{73} +25.7939 q^{75} -78.2156i q^{79} +70.7017 q^{81} -22.1936i q^{83} +45.1381i q^{85} +231.459i q^{87} +68.2014 q^{89} +33.9698 q^{91} -223.602 q^{93} -12.3679i q^{95} +61.4375 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 14 q^{3} + 46 q^{9} + 10 q^{15} + 72 q^{23} + 80 q^{25} + 50 q^{27} + 50 q^{31} - 16 q^{37} + 30 q^{45} + 232 q^{47} - 188 q^{49} + 132 q^{53} + 78 q^{59} + 152 q^{67} + 748 q^{69} + 230 q^{71} + 70 q^{75}+ \cdots + 854 q^{97}+O(q^{100}) $ 16 * q + 14 * q^3 + 46 * q^9 + 10 * q^15 + 72 * q^23 + 80 * q^25 + 50 * q^27 + 50 * q^31 - 16 * q^37 + 30 * q^45 + 232 * q^47 - 188 * q^49 + 132 * q^53 + 78 * q^59 + 152 * q^67 + 748 * q^69 + 230 * q^71 + 70 * q^75 - 148 * q^81 + 160 * q^89 + 836 * q^91 - 698 * q^93 + 854 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2420\mathbb{Z}\right)^\times$.

$n$	$1211$	$1937$	$2301$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	5.15878	1.71959	0.859797	−	0.510636i	$-0.170590\pi$
$3$	5.15878	1.71959	0.859797	+	0.510636i	$0.170590\pi$
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	10.1572i	1.45104i	0.688204	+	0.725518i	$0.258400\pi$
$7$	10.1572i	1.45104i	−0.688204	+	0.725518i	$0.741600\pi$
$8$	0	0
$9$	17.6130	1.95700
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 3.34439i	− 0.257261i	−0.991693	−	0.128630i	$-0.958942\pi$
$13$	− 3.34439i	− 0.257261i	0.991693	−	0.128630i	$-0.0410581\pi$
$14$	0	0
$15$	11.5354	0.769026
$16$	0	0
$17$	20.1864i	1.18743i	0.804674	+	0.593717i	$0.202340\pi$
$17$	20.1864i	1.18743i	−0.804674	+	0.593717i	$0.797660\pi$
$18$	0	0
$19$	− 5.53109i	− 0.291110i	−0.989350	−	0.145555i	$-0.953503\pi$
$19$	− 5.53109i	− 0.291110i	0.989350	−	0.145555i	$-0.0464968\pi$
$20$	0	0
$21$	52.3990i	2.49519i
$22$	0	0
$23$	34.6484	1.50645	0.753227	−	0.657761i	$-0.228496\pi$
$23$	34.6484	1.50645	0.753227	+	0.657761i	$0.228496\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	44.4328	1.64566
$28$	0	0
$29$	44.8670i	1.54714i	0.633712	+	0.773569i	$0.281530\pi$
$29$	44.8670i	1.54714i	−0.633712	+	0.773569i	$0.718470\pi$
$30$	0	0
$31$	−43.3440	−1.39819	−0.699097	−	0.715027i	$-0.746415\pi$
$31$	−43.3440	−1.39819	−0.699097	+	0.715027i	$0.746415\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	22.7123i	0.648923i
$36$	0	0
$37$	−59.4002	−1.60541	−0.802706	−	0.596375i	$-0.796607\pi$
$37$	−59.4002	−1.60541	−0.802706	+	0.596375i	$0.796607\pi$
$38$	0	0
$39$	− 17.2530i	− 0.442384i
$40$	0	0
$41$	33.0912i	0.807102i	0.914957	+	0.403551i	$0.132224\pi$
$41$	33.0912i	0.807102i	−0.914957	+	0.403551i	$0.867776\pi$
$42$	0	0
$43$	− 65.3574i	− 1.51994i	−0.649958	−	0.759970i	$-0.725213\pi$
$43$	− 65.3574i	− 1.51994i	0.649958	−	0.759970i	$-0.274787\pi$
$44$	0	0
$45$	39.3839	0.875199
$46$	0	0
$47$	68.8573	1.46505	0.732524	−	0.680741i	$-0.238342\pi$
$47$	68.8573	1.46505	0.732524	+	0.680741i	$0.238342\pi$
$48$	0	0
$49$	−54.1697	−1.10550
$50$	0	0
$51$	104.137i	2.04190i
$52$	0	0
$53$	−3.21911	−0.0607379	−0.0303689	−	0.999539i	$-0.509668\pi$
$53$	−3.21911	−0.0607379	−0.0303689	+	0.999539i	$0.509668\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 28.5337i	− 0.500591i
$58$	0	0
$59$	66.0207	1.11899	0.559497	−	0.828832i	$-0.310994\pi$
$59$	66.0207	1.11899	0.559497	+	0.828832i	$0.310994\pi$
$60$	0	0
$61$	47.3889i	0.776867i	0.921477	+	0.388434i	$0.126984\pi$
$61$	47.3889i	0.776867i	−0.921477	+	0.388434i	$0.873016\pi$
$62$	0	0
$63$	178.900i	2.83968i
$64$	0	0
$65$	− 7.47829i	− 0.115051i
$66$	0	0
$67$	−31.6113	−0.471810	−0.235905	−	0.971776i	$-0.575805\pi$
$67$	−31.6113	−0.471810	−0.235905	+	0.971776i	$0.575805\pi$
$68$	0	0
$69$	178.744	2.59049
$70$	0	0
$71$	3.76017	0.0529601	0.0264801	−	0.999649i	$-0.491570\pi$
$71$	3.76017	0.0529601	0.0264801	+	0.999649i	$0.491570\pi$
$72$	0	0
$73$	− 22.4126i	− 0.307022i	−0.988147	−	0.153511i	$-0.950942\pi$
$73$	− 22.4126i	− 0.307022i	0.988147	−	0.153511i	$-0.0490580\pi$
$74$	0	0
$75$	25.7939	0.343919
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 78.2156i	− 0.990071i	−0.868873	−	0.495036i	$-0.835155\pi$
$79$	− 78.2156i	− 0.990071i	0.868873	−	0.495036i	$-0.164845\pi$
$80$	0	0
$81$	70.7017	0.872860
$82$	0	0
$83$	− 22.1936i	− 0.267392i	−0.991022	−	0.133696i	$-0.957315\pi$
$83$	− 22.1936i	− 0.267392i	0.991022	−	0.133696i	$-0.0426846\pi$
$84$	0	0
$85$	45.1381i	0.531037i
$86$	0	0
$87$	231.459i	2.66045i
$88$	0	0
$89$	68.2014	0.766308	0.383154	−	0.923684i	$-0.374838\pi$
$89$	68.2014	0.766308	0.383154	+	0.923684i	$0.374838\pi$
$90$	0	0
$91$	33.9698	0.373295
$92$	0	0
$93$	−223.602	−2.40433
$94$	0	0
$95$	− 12.3679i	− 0.130188i
$96$	0	0
$97$	61.4375	0.633376	0.316688	−	0.948530i	$-0.397429\pi$
$97$	61.4375	0.633376	0.316688	+	0.948530i	$0.397429\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 7.47252i	− 0.0739854i	−0.999316	−	0.0369927i	$-0.988222\pi$
$101$	− 7.47252i	− 0.0739854i	0.999316	−	0.0369927i	$-0.0117778\pi$
$102$	0	0
$103$	−82.6449	−0.802378	−0.401189	−	0.915995i	$-0.631403\pi$
$103$	−82.6449	−0.802378	−0.401189	+	0.915995i	$0.631403\pi$
$104$	0	0
$105$	117.168i	1.11588i
$106$	0	0
$107$	158.077i	1.47735i	0.674059	+	0.738677i	$0.264549\pi$
$107$	158.077i	1.47735i	−0.674059	+	0.738677i	$0.735451\pi$
$108$	0	0
$109$	− 79.1292i	− 0.725956i	−0.931798	−	0.362978i	$-0.881760\pi$
$109$	− 79.1292i	− 0.725956i	0.931798	−	0.362978i	$-0.118240\pi$
$110$	0	0
$111$	−306.433	−2.76066
$112$	0	0
$113$	28.2244	0.249774	0.124887	−	0.992171i	$-0.460143\pi$
$113$	28.2244	0.249774	0.124887	+	0.992171i	$0.460143\pi$
$114$	0	0
$115$	77.4763	0.673707
$116$	0	0
$117$	− 58.9049i	− 0.503461i
$118$	0	0
$119$	−205.038	−1.72301
$120$	0	0
$121$	0	0
$122$	0	0
$123$	170.710i	1.38789i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	92.9080i	0.731559i	0.930702	+	0.365779i	$0.119198\pi$
$127$	92.9080i	0.731559i	−0.930702	+	0.365779i	$0.880802\pi$
$128$	0	0
$129$	− 337.165i	− 2.61368i
$130$	0	0
$131$	− 156.944i	− 1.19804i	−0.800733	−	0.599021i	$-0.795556\pi$
$131$	− 156.944i	− 1.19804i	0.800733	−	0.599021i	$-0.204444\pi$
$132$	0	0
$133$	56.1806	0.422411
$134$	0	0
$135$	99.3547	0.735961
$136$	0	0
$137$	3.42211	0.0249789	0.0124894	−	0.999922i	$-0.496024\pi$
$137$	3.42211	0.0249789	0.0124894	+	0.999922i	$0.496024\pi$
$138$	0	0
$139$	112.664i	0.810534i	0.914198	+	0.405267i	$0.132821\pi$
$139$	112.664i	0.810534i	−0.914198	+	0.405267i	$0.867179\pi$
$140$	0	0
$141$	355.220	2.51929
$142$	0	0
$143$	0	0
$144$	0	0
$145$	100.326i	0.691901i
$146$	0	0
$147$	−279.449	−1.90102
$148$	0	0
$149$	− 133.406i	− 0.895345i	−0.894198	−	0.447673i	$-0.852253\pi$
$149$	− 133.406i	− 0.895345i	0.894198	−	0.447673i	$-0.147747\pi$
$150$	0	0
$151$	251.184i	1.66347i	0.555172	+	0.831736i	$0.312653\pi$
$151$	251.184i	1.66347i	−0.555172	+	0.831736i	$0.687347\pi$
$152$	0	0
$153$	355.543i	2.32381i
$154$	0	0
$155$	−96.9202	−0.625292
$156$	0	0
$157$	−236.581	−1.50688	−0.753442	−	0.657515i	$-0.771608\pi$
$157$	−236.581	−1.50688	−0.753442	+	0.657515i	$0.771608\pi$
$158$	0	0
$159$	−16.6067	−0.104444
$160$	0	0
$161$	351.933i	2.18592i
$162$	0	0
$163$	197.746	1.21317	0.606583	−	0.795021i	$-0.292540\pi$
$163$	197.746	1.21317	0.606583	+	0.795021i	$0.292540\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.16634i	− 0.0249481i	−0.999922	−	0.0124741i	$-0.996029\pi$
$167$	− 4.16634i	− 0.0249481i	0.999922	−	0.0124741i	$-0.00397072\pi$
$168$	0	0
$169$	157.815	0.933817
$170$	0	0
$171$	− 97.4192i	− 0.569703i
$172$	0	0
$173$	− 221.308i	− 1.27923i	−0.768694	−	0.639617i	$-0.779093\pi$
$173$	− 221.308i	− 1.27923i	0.768694	−	0.639617i	$-0.220907\pi$
$174$	0	0
$175$	50.7862i	0.290207i
$176$	0	0
$177$	340.586	1.92422
$178$	0	0
$179$	−24.0936	−0.134601	−0.0673006	−	0.997733i	$-0.521439\pi$
$179$	−24.0936	−0.134601	−0.0673006	+	0.997733i	$0.521439\pi$
$180$	0	0
$181$	−54.8750	−0.303177	−0.151589	−	0.988444i	$-0.548439\pi$
$181$	−54.8750	−0.303177	−0.151589	+	0.988444i	$0.548439\pi$
$182$	0	0
$183$	244.469i	1.33590i
$184$	0	0
$185$	−132.823	−0.717962
$186$	0	0
$187$	0	0
$188$	0	0
$189$	451.315i	2.38791i
$190$	0	0
$191$	228.706	1.19741	0.598706	−	0.800969i	$-0.295682\pi$
$191$	228.706	1.19741	0.598706	+	0.800969i	$0.295682\pi$
$192$	0	0
$193$	− 140.392i	− 0.727421i	−0.931512	−	0.363711i	$-0.881510\pi$
$193$	− 140.392i	− 0.727421i	0.931512	−	0.363711i	$-0.118490\pi$
$194$	0	0
$195$	− 38.5789i	− 0.197840i
$196$	0	0
$197$	301.867i	1.53232i	0.642651	+	0.766159i	$0.277834\pi$
$197$	301.867i	1.53232i	−0.642651	+	0.766159i	$0.722166\pi$
$198$	0	0
$199$	169.860	0.853570	0.426785	−	0.904353i	$-0.359646\pi$
$199$	169.860	0.853570	0.426785	+	0.904353i	$0.359646\pi$
$200$	0	0
$201$	−163.076	−0.811322
$202$	0	0
$203$	−455.725	−2.24495
$204$	0	0
$205$	73.9941i	0.360947i
$206$	0	0
$207$	610.264	2.94814
$208$	0	0
$209$	0	0
$210$	0	0
$211$	87.0263i	0.412447i	0.978505	+	0.206224i	$0.0661174\pi$
$211$	87.0263i	0.412447i	−0.978505	+	0.206224i	$0.933883\pi$
$212$	0	0
$213$	19.3979	0.0910699
$214$	0	0
$215$	− 146.144i	− 0.679738i
$216$	0	0
$217$	− 440.256i	− 2.02883i
$218$	0	0
$219$	− 115.622i	− 0.527952i
$220$	0	0
$221$	67.5112	0.305480
$222$	0	0
$223$	−23.3059	−0.104511	−0.0522553	−	0.998634i	$-0.516641\pi$
$223$	−23.3059	−0.104511	−0.0522553	+	0.998634i	$0.516641\pi$
$224$	0	0
$225$	88.0652	0.391401
$226$	0	0
$227$	− 394.768i	− 1.73907i	−0.493875	−	0.869533i	$-0.664420\pi$
$227$	− 394.768i	− 1.73907i	0.493875	−	0.869533i	$-0.335580\pi$
$228$	0	0
$229$	429.708	1.87645	0.938227	−	0.346022i	$-0.112468\pi$
$229$	429.708	1.87645	0.938227	+	0.346022i	$0.112468\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 119.958i	− 0.514841i	−0.966300	−	0.257420i	$-0.917127\pi$
$233$	− 119.958i	− 0.514841i	0.966300	−	0.257420i	$-0.0828725\pi$
$234$	0	0
$235$	153.969	0.655189
$236$	0	0
$237$	− 403.497i	− 1.70252i
$238$	0	0
$239$	− 206.753i	− 0.865076i	−0.901616	−	0.432538i	$-0.857618\pi$
$239$	− 206.753i	− 0.865076i	0.901616	−	0.432538i	$-0.142382\pi$
$240$	0	0
$241$	− 314.829i	− 1.30634i	−0.757209	−	0.653172i	$-0.773438\pi$
$241$	− 314.829i	− 1.30634i	0.757209	−	0.653172i	$-0.226562\pi$
$242$	0	0
$243$	−35.1603	−0.144693
$244$	0	0
$245$	−121.127	−0.494396
$246$	0	0
$247$	−18.4981	−0.0748912
$248$	0	0
$249$	− 114.492i	− 0.459806i
$250$	0	0
$251$	−85.3576	−0.340070	−0.170035	−	0.985438i	$-0.554388\pi$
$251$	−85.3576	−0.340070	−0.170035	+	0.985438i	$0.554388\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	232.858i	0.913167i
$256$	0	0
$257$	−295.270	−1.14891	−0.574455	−	0.818536i	$-0.694786\pi$
$257$	−295.270	−1.14891	−0.574455	+	0.818536i	$0.694786\pi$
$258$	0	0
$259$	− 603.343i	− 2.32951i
$260$	0	0
$261$	790.244i	3.02775i
$262$	0	0
$263$	264.881i	1.00715i	0.863951	+	0.503577i	$0.167983\pi$
$263$	264.881i	1.00715i	−0.863951	+	0.503577i	$0.832017\pi$
$264$	0	0
$265$	−7.19814	−0.0271628
$266$	0	0
$267$	351.836	1.31774
$268$	0	0
$269$	89.8558	0.334036	0.167018	−	0.985954i	$-0.446586\pi$
$269$	89.8558	0.334036	0.167018	+	0.985954i	$0.446586\pi$
$270$	0	0
$271$	− 101.229i	− 0.373538i	−0.982404	−	0.186769i	$-0.940198\pi$
$271$	− 101.229i	− 0.373538i	0.982404	−	0.186769i	$-0.0598016\pi$
$272$	0	0
$273$	175.243	0.641915
$274$	0	0
$275$	0	0
$276$	0	0
$277$	258.444i	0.933010i	0.884518	+	0.466505i	$0.154487\pi$
$277$	258.444i	0.933010i	−0.884518	+	0.466505i	$0.845513\pi$
$278$	0	0
$279$	−763.420	−2.73627
$280$	0	0
$281$	− 383.628i	− 1.36523i	−0.730780	−	0.682613i	$-0.760844\pi$
$281$	− 383.628i	− 1.36523i	0.730780	−	0.682613i	$-0.239156\pi$
$282$	0	0
$283$	94.3964i	0.333556i	0.985994	+	0.166778i	$0.0533364\pi$
$283$	94.3964i	0.333556i	−0.985994	+	0.166778i	$0.946664\pi$
$284$	0	0
$285$	− 63.8032i	− 0.223871i
$286$	0	0
$287$	−336.115	−1.17113
$288$	0	0
$289$	−118.490	−0.410000
$290$	0	0
$291$	316.943	1.08915
$292$	0	0
$293$	6.58908i	0.0224883i	0.999937	+	0.0112442i	$0.00357921\pi$
$293$	6.58908i	0.0224883i	−0.999937	+	0.0112442i	$0.996421\pi$
$294$	0	0
$295$	147.627	0.500430
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 115.878i	− 0.387552i
$300$	0	0
$301$	663.852	2.20549
$302$	0	0
$303$	− 38.5491i	− 0.127225i
$304$	0	0
$305$	105.965i	0.347426i
$306$	0	0
$307$	− 598.880i	− 1.95075i	−0.220557	−	0.975374i	$-0.570787\pi$
$307$	− 598.880i	− 1.95075i	0.220557	−	0.975374i	$-0.429213\pi$
$308$	0	0
$309$	−426.347	−1.37976
$310$	0	0
$311$	367.259	1.18090	0.590449	−	0.807075i	$-0.298951\pi$
$311$	367.259	1.18090	0.590449	+	0.807075i	$0.298951\pi$
$312$	0	0
$313$	372.762	1.19093	0.595466	−	0.803380i	$-0.296967\pi$
$313$	372.762	1.19093	0.595466	+	0.803380i	$0.296967\pi$
$314$	0	0
$315$	400.032i	1.26994i
$316$	0	0
$317$	−385.788	−1.21700	−0.608499	−	0.793555i	$-0.708228\pi$
$317$	−385.788	−1.21700	−0.608499	+	0.793555i	$0.708228\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	815.485i	2.54045i
$322$	0	0
$323$	111.653	0.345674
$324$	0	0
$325$	− 16.7220i	− 0.0514522i
$326$	0	0
$327$	− 408.210i	− 1.24835i
$328$	0	0
$329$	699.400i	2.12584i
$330$	0	0
$331$	−617.568	−1.86576	−0.932882	−	0.360182i	$-0.882715\pi$
$331$	−617.568	−1.86576	−0.932882	+	0.360182i	$0.882715\pi$
$332$	0	0
$333$	−1046.22	−3.14180
$334$	0	0
$335$	−70.6850	−0.211000
$336$	0	0
$337$	170.873i	0.507042i	0.967330	+	0.253521i	$0.0815887\pi$
$337$	170.873i	0.507042i	−0.967330	+	0.253521i	$0.918411\pi$
$338$	0	0
$339$	145.604	0.429509
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 52.5095i	− 0.153089i
$344$	0	0
$345$	399.683	1.15850
$346$	0	0
$347$	− 423.730i	− 1.22112i	−0.791969	−	0.610562i	$-0.790944\pi$
$347$	− 423.730i	− 1.22112i	0.791969	−	0.610562i	$-0.209056\pi$
$348$	0	0
$349$	286.057i	0.819647i	0.912165	+	0.409823i	$0.134410\pi$
$349$	286.057i	0.819647i	−0.912165	+	0.409823i	$0.865590\pi$
$350$	0	0
$351$	− 148.601i	− 0.423364i
$352$	0	0
$353$	124.306	0.352142	0.176071	−	0.984377i	$-0.443661\pi$
$353$	124.306	0.352142	0.176071	+	0.984377i	$0.443661\pi$
$354$	0	0
$355$	8.40799	0.0236845
$356$	0	0
$357$	−1057.75	−2.96288
$358$	0	0
$359$	− 284.135i	− 0.791463i	−0.918366	−	0.395732i	$-0.870491\pi$
$359$	− 284.135i	− 0.791463i	0.918366	−	0.395732i	$-0.129509\pi$
$360$	0	0
$361$	330.407	0.915255
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 50.1160i	− 0.137304i
$366$	0	0
$367$	−268.901	−0.732700	−0.366350	−	0.930477i	$-0.619393\pi$
$367$	−268.901	−0.732700	−0.366350	+	0.930477i	$0.619393\pi$
$368$	0	0
$369$	582.836i	1.57950i
$370$	0	0
$371$	− 32.6973i	− 0.0881328i
$372$	0	0
$373$	− 556.864i	− 1.49293i	−0.665423	−	0.746466i	$-0.731749\pi$
$373$	− 556.864i	− 1.49293i	0.665423	−	0.746466i	$-0.268251\pi$
$374$	0	0
$375$	57.6769	0.153805
$376$	0	0
$377$	150.053	0.398018
$378$	0	0
$379$	−606.719	−1.60084	−0.800421	−	0.599438i	$-0.795391\pi$
$379$	−606.719	−1.60084	−0.800421	+	0.599438i	$0.795391\pi$
$380$	0	0
$381$	479.292i	1.25798i
$382$	0	0
$383$	681.707	1.77991	0.889957	−	0.456044i	$-0.150734\pi$
$383$	681.707	1.77991	0.889957	+	0.456044i	$0.150734\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1151.14i	− 2.97453i
$388$	0	0
$389$	−130.409	−0.335241	−0.167620	−	0.985852i	$-0.553608\pi$
$389$	−130.409	−0.335241	−0.167620	+	0.985852i	$0.553608\pi$
$390$	0	0
$391$	699.427i	1.78882i
$392$	0	0
$393$	− 809.638i	− 2.06015i
$394$	0	0
$395$	− 174.895i	− 0.442773i
$396$	0	0
$397$	47.4096	0.119420	0.0597098	−	0.998216i	$-0.480982\pi$
$397$	47.4096	0.119420	0.0597098	+	0.998216i	$0.480982\pi$
$398$	0	0
$399$	289.824	0.726375
$400$	0	0
$401$	−378.577	−0.944082	−0.472041	−	0.881577i	$-0.656483\pi$
$401$	−378.577	−0.944082	−0.472041	+	0.881577i	$0.656483\pi$
$402$	0	0
$403$	144.959i	0.359701i
$404$	0	0
$405$	158.094	0.390355
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 1.66384i	− 0.00406807i	−0.999998	−	0.00203403i	$-0.999353\pi$
$409$	− 1.66384i	− 0.00406807i	0.999998	−	0.00203403i	$-0.000647453\pi$
$410$	0	0
$411$	17.6539	0.0429536
$412$	0	0
$413$	670.589i	1.62370i
$414$	0	0
$415$	− 49.6263i	− 0.119581i
$416$	0	0
$417$	581.210i	1.39379i
$418$	0	0
$419$	230.024	0.548983	0.274491	−	0.961590i	$-0.411491\pi$
$419$	230.024	0.548983	0.274491	+	0.961590i	$0.411491\pi$
$420$	0	0
$421$	184.393	0.437987	0.218994	−	0.975726i	$-0.429723\pi$
$421$	184.393	0.437987	0.218994	+	0.975726i	$0.429723\pi$
$422$	0	0
$423$	1212.79	2.86710
$424$	0	0
$425$	100.932i	0.237487i
$426$	0	0
$427$	−481.341	−1.12726
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 97.4064i	− 0.226001i	−0.993595	−	0.113000i	$-0.963954\pi$
$431$	− 97.4064i	− 0.226001i	0.993595	−	0.113000i	$-0.0360462\pi$
$432$	0	0
$433$	310.412	0.716887	0.358444	−	0.933551i	$-0.383308\pi$
$433$	310.412	0.716887	0.358444	+	0.933551i	$0.383308\pi$
$434$	0	0
$435$	517.558i	1.18979i
$436$	0	0
$437$	− 191.644i	− 0.438544i
$438$	0	0
$439$	124.229i	0.282982i	0.989940	+	0.141491i	$0.0451896\pi$
$439$	124.229i	0.282982i	−0.989940	+	0.141491i	$0.954810\pi$
$440$	0	0
$441$	−954.092	−2.16347
$442$	0	0
$443$	336.800	0.760270	0.380135	−	0.924931i	$-0.375878\pi$
$443$	336.800	0.760270	0.380135	+	0.924931i	$0.375878\pi$
$444$	0	0
$445$	152.503	0.342703
$446$	0	0
$447$	− 688.215i	− 1.53963i
$448$	0	0
$449$	202.686	0.451417	0.225709	−	0.974195i	$-0.427530\pi$
$449$	202.686	0.451417	0.225709	+	0.974195i	$0.427530\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	1295.80i	2.86050i
$454$	0	0
$455$	75.9588	0.166942
$456$	0	0
$457$	− 817.158i	− 1.78809i	−0.447976	−	0.894046i	$-0.647855\pi$
$457$	− 817.158i	− 1.78809i	0.447976	−	0.894046i	$-0.352145\pi$
$458$	0	0
$459$	896.937i	1.95411i
$460$	0	0
$461$	− 572.857i	− 1.24264i	−0.783557	−	0.621320i	$-0.786597\pi$
$461$	− 572.857i	− 1.24264i	0.783557	−	0.621320i	$-0.213403\pi$
$462$	0	0
$463$	809.225	1.74779	0.873893	−	0.486119i	$-0.161588\pi$
$463$	809.225	1.74779	0.873893	+	0.486119i	$0.161588\pi$
$464$	0	0
$465$	−499.990	−1.07525
$466$	0	0
$467$	−800.542	−1.71422	−0.857111	−	0.515132i	$-0.827743\pi$
$467$	−800.542	−1.71422	−0.857111	+	0.515132i	$0.827743\pi$
$468$	0	0
$469$	− 321.084i	− 0.684613i
$470$	0	0
$471$	−1220.47	−2.59123
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 27.6554i	− 0.0582220i
$476$	0	0
$477$	−56.6983	−0.118864
$478$	0	0
$479$	663.266i	1.38469i	0.721567	+	0.692345i	$0.243422\pi$
$479$	663.266i	1.38469i	−0.721567	+	0.692345i	$0.756578\pi$
$480$	0	0
$481$	198.658i	0.413010i
$482$	0	0
$483$	1815.54i	3.75889i
$484$	0	0
$485$	137.378	0.283255
$486$	0	0
$487$	−247.807	−0.508843	−0.254422	−	0.967093i	$-0.581885\pi$
$487$	−247.807	−0.508843	−0.254422	+	0.967093i	$0.581885\pi$
$488$	0	0
$489$	1020.13	2.08615
$490$	0	0
$491$	834.509i	1.69961i	0.527096	+	0.849806i	$0.323281\pi$
$491$	834.509i	1.69961i	−0.527096	+	0.849806i	$0.676719\pi$
$492$	0	0
$493$	−905.702	−1.83712
$494$	0	0
$495$	0	0
$496$	0	0
$497$	38.1929i	0.0768470i
$498$	0	0
$499$	525.408	1.05292	0.526461	−	0.850199i	$-0.323518\pi$
$499$	525.408	1.05292	0.526461	+	0.850199i	$0.323518\pi$
$500$	0	0
$501$	− 21.4932i	− 0.0429006i
$502$	0	0
$503$	327.969i	0.652026i	0.945365	+	0.326013i	$0.105705\pi$
$503$	327.969i	0.652026i	−0.945365	+	0.326013i	$0.894295\pi$
$504$	0	0
$505$	− 16.7091i	− 0.0330873i
$506$	0	0
$507$	814.133	1.60579
$508$	0	0
$509$	−392.787	−0.771683	−0.385842	−	0.922565i	$-0.626089\pi$
$509$	−392.787	−0.771683	−0.385842	+	0.922565i	$0.626089\pi$
$510$	0	0
$511$	227.650	0.445499
$512$	0	0
$513$	− 245.762i	− 0.479067i
$514$	0	0
$515$	−184.800	−0.358834
$516$	0	0
$517$	0	0
$518$	0	0
$519$	− 1141.68i	− 2.19976i
$520$	0	0
$521$	−92.8535	−0.178222	−0.0891109	−	0.996022i	$-0.528403\pi$
$521$	−92.8535	−0.178222	−0.0891109	+	0.996022i	$0.528403\pi$
$522$	0	0
$523$	340.960i	0.651932i	0.945382	+	0.325966i	$0.105689\pi$
$523$	340.960i	0.651932i	−0.945382	+	0.325966i	$0.894311\pi$
$524$	0	0
$525$	261.995i	0.499038i
$526$	0	0
$527$	− 874.959i	− 1.66026i
$528$	0	0
$529$	671.515	1.26940
$530$	0	0
$531$	1162.82	2.18988
$532$	0	0
$533$	110.670	0.207636
$534$	0	0
$535$	353.471i	0.660693i
$536$	0	0
$537$	−124.294	−0.231459
$538$	0	0
$539$	0	0
$540$	0	0
$541$	186.702i	0.345104i	0.985000	+	0.172552i	$0.0552014\pi$
$541$	186.702i	0.345104i	−0.985000	+	0.172552i	$0.944799\pi$
$542$	0	0
$543$	−283.088	−0.521341
$544$	0	0
$545$	− 176.938i	− 0.324657i
$546$	0	0
$547$	80.5995i	0.147348i	0.997282	+	0.0736741i	$0.0234725\pi$
$547$	80.5995i	0.147348i	−0.997282	+	0.0736741i	$0.976528\pi$
$548$	0	0
$549$	834.663i	1.52033i
$550$	0	0
$551$	248.163	0.450387
$552$	0	0
$553$	794.455	1.43663
$554$	0	0
$555$	−685.205	−1.23460
$556$	0	0
$557$	24.3519i	0.0437198i	0.999761	+	0.0218599i	$0.00695878\pi$
$557$	24.3519i	0.0437198i	−0.999761	+	0.0218599i	$0.993041\pi$
$558$	0	0
$559$	−218.581	−0.391021
$560$	0	0
$561$	0	0
$562$	0	0
$563$	319.673i	0.567803i	0.958853	+	0.283902i	$0.0916289\pi$
$563$	319.673i	0.567803i	−0.958853	+	0.283902i	$0.908371\pi$
$564$	0	0
$565$	63.1117	0.111702
$566$	0	0
$567$	718.135i	1.26655i
$568$	0	0
$569$	− 744.166i	− 1.30785i	−0.756560	−	0.653924i	$-0.773121\pi$
$569$	− 744.166i	− 1.30785i	0.756560	−	0.653924i	$-0.226879\pi$
$570$	0	0
$571$	− 265.828i	− 0.465549i	−0.972531	−	0.232774i	$-0.925220\pi$
$571$	− 265.828i	− 0.465549i	0.972531	−	0.232774i	$-0.0747804\pi$
$572$	0	0
$573$	1179.84	2.05906
$574$	0	0
$575$	173.242	0.301291
$576$	0	0
$577$	220.270	0.381750	0.190875	−	0.981614i	$-0.438867\pi$
$577$	220.270	0.381750	0.190875	+	0.981614i	$0.438867\pi$
$578$	0	0
$579$	− 724.253i	− 1.25087i
$580$	0	0
$581$	225.425	0.387996
$582$	0	0
$583$	0	0
$584$	0	0
$585$	− 131.715i	− 0.225154i
$586$	0	0
$587$	−451.754	−0.769598	−0.384799	−	0.923000i	$-0.625729\pi$
$587$	−451.754	−0.769598	−0.384799	+	0.923000i	$0.625729\pi$
$588$	0	0
$589$	239.740i	0.407028i
$590$	0	0
$591$	1557.26i	2.63496i
$592$	0	0
$593$	− 337.627i	− 0.569353i	−0.958624	−	0.284677i	$-0.908114\pi$
$593$	− 337.627i	− 0.569353i	0.958624	−	0.284677i	$-0.0918863\pi$
$594$	0	0
$595$	−458.479	−0.770553
$596$	0	0
$597$	876.273	1.46779
$598$	0	0
$599$	−659.439	−1.10090	−0.550450	−	0.834868i	$-0.685544\pi$
$599$	−659.439	−1.10090	−0.550450	+	0.834868i	$0.685544\pi$
$600$	0	0
$601$	− 592.806i	− 0.986367i	−0.869925	−	0.493183i	$-0.835833\pi$
$601$	− 592.806i	− 0.986367i	0.869925	−	0.493183i	$-0.164167\pi$
$602$	0	0
$603$	−556.771	−0.923335
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 615.618i	− 1.01420i	−0.861888	−	0.507099i	$-0.830718\pi$
$607$	− 615.618i	− 1.01420i	0.861888	−	0.507099i	$-0.169282\pi$
$608$	0	0
$609$	−2350.99	−3.86040
$610$	0	0
$611$	− 230.286i	− 0.376900i
$612$	0	0
$613$	− 468.595i	− 0.764428i	−0.924074	−	0.382214i	$-0.875162\pi$
$613$	− 468.595i	− 0.764428i	0.924074	−	0.382214i	$-0.124838\pi$
$614$	0	0
$615$	381.719i	0.620682i
$616$	0	0
$617$	−94.7516	−0.153568	−0.0767841	−	0.997048i	$-0.524465\pi$
$617$	−94.7516	−0.153568	−0.0767841	+	0.997048i	$0.524465\pi$
$618$	0	0
$619$	−15.2679	−0.0246655	−0.0123327	−	0.999924i	$-0.503926\pi$
$619$	−15.2679	−0.0246655	−0.0123327	+	0.999924i	$0.503926\pi$
$620$	0	0
$621$	1539.53	2.47911
$622$	0	0
$623$	692.739i	1.11194i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1199.08i	− 1.90632i
$630$	0	0
$631$	961.254	1.52338	0.761691	−	0.647940i	$-0.224369\pi$
$631$	961.254	1.52338	0.761691	+	0.647940i	$0.224369\pi$
$632$	0	0
$633$	448.950i	0.709241i
$634$	0	0
$635$	207.749i	0.327163i
$636$	0	0
$637$	181.165i	0.284403i
$638$	0	0
$639$	66.2280	0.103643
$640$	0	0
$641$	157.742	0.246087	0.123044	−	0.992401i	$-0.460734\pi$
$641$	157.742	0.246087	0.123044	+	0.992401i	$0.460734\pi$
$642$	0	0
$643$	−895.842	−1.39322	−0.696611	−	0.717449i	$-0.745310\pi$
$643$	−895.842	−1.39322	−0.696611	+	0.717449i	$0.745310\pi$
$644$	0	0
$645$	− 753.923i	− 1.16887i
$646$	0	0
$647$	−169.006	−0.261215	−0.130608	−	0.991434i	$-0.541693\pi$
$647$	−169.006	−0.261215	−0.130608	+	0.991434i	$0.541693\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 2271.18i	− 3.48876i
$652$	0	0
$653$	131.764	0.201783	0.100891	−	0.994897i	$-0.467831\pi$
$653$	131.764	0.201783	0.100891	+	0.994897i	$0.467831\pi$
$654$	0	0
$655$	− 350.937i	− 0.535781i
$656$	0	0
$657$	− 394.753i	− 0.600842i
$658$	0	0
$659$	− 956.323i	− 1.45117i	−0.688131	−	0.725586i	$-0.741569\pi$
$659$	− 956.323i	− 1.45117i	0.688131	−	0.725586i	$-0.258431\pi$
$660$	0	0
$661$	−649.628	−0.982795	−0.491398	−	0.870935i	$-0.663514\pi$
$661$	−649.628	−0.982795	−0.491398	+	0.870935i	$0.663514\pi$
$662$	0	0
$663$	348.275	0.525302
$664$	0	0
$665$	125.624	0.188908
$666$	0	0
$667$	1554.57i	2.33069i
$668$	0	0
$669$	−120.230	−0.179716
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 758.633i	− 1.12724i	−0.826034	−	0.563621i	$-0.809408\pi$
$673$	− 758.633i	− 1.12724i	0.826034	−	0.563621i	$-0.190592\pi$
$674$	0	0
$675$	222.164	0.329132
$676$	0	0
$677$	− 141.672i	− 0.209265i	−0.994511	−	0.104632i	$-0.966633\pi$
$677$	− 141.672i	− 0.209265i	0.994511	−	0.104632i	$-0.0333666\pi$
$678$	0	0
$679$	624.036i	0.919052i
$680$	0	0
$681$	− 2036.52i	− 2.99049i
$682$	0	0
$683$	101.169	0.148124	0.0740621	−	0.997254i	$-0.476404\pi$
$683$	101.169	0.148124	0.0740621	+	0.997254i	$0.476404\pi$
$684$	0	0
$685$	7.65207	0.0111709
$686$	0	0
$687$	2216.77	3.22674
$688$	0	0
$689$	10.7660i	0.0156255i
$690$	0	0
$691$	−630.444	−0.912364	−0.456182	−	0.889886i	$-0.650783\pi$
$691$	−630.444	−0.912364	−0.456182	+	0.889886i	$0.650783\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	251.925i	0.362482i
$696$	0	0
$697$	−667.991	−0.958380
$698$	0	0
$699$	− 618.837i	− 0.885317i
$700$	0	0
$701$	260.870i	0.372140i	0.982536	+	0.186070i	$0.0595752\pi$
$701$	260.870i	0.372140i	−0.982536	+	0.186070i	$0.940425\pi$
$702$	0	0
$703$	328.548i	0.467351i
$704$	0	0
$705$	794.295	1.12666
$706$	0	0
$707$	75.9003	0.107355
$708$	0	0
$709$	−638.185	−0.900120	−0.450060	−	0.892998i	$-0.648597\pi$
$709$	−638.185	−0.900120	−0.450060	+	0.892998i	$0.648597\pi$
$710$	0	0
$711$	− 1377.61i	− 1.93757i
$712$	0	0
$713$	−1501.80	−2.10632
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1066.59i	− 1.48758i
$718$	0	0
$719$	900.685	1.25269	0.626345	−	0.779546i	$-0.284550\pi$
$719$	900.685	1.25269	0.626345	+	0.779546i	$0.284550\pi$
$720$	0	0
$721$	− 839.445i	− 1.16428i
$722$	0	0
$723$	− 1624.13i	− 2.24638i
$724$	0	0
$725$	224.335i	0.309428i
$726$	0	0
$727$	−356.679	−0.490618	−0.245309	−	0.969445i	$-0.578889\pi$
$727$	−356.679	−0.490618	−0.245309	+	0.969445i	$0.578889\pi$
$728$	0	0
$729$	−817.700	−1.12167
$730$	0	0
$731$	1319.33	1.80483
$732$	0	0
$733$	183.795i	0.250743i	0.992110	+	0.125371i	$0.0400123\pi$
$733$	183.795i	0.250743i	−0.992110	+	0.125371i	$0.959988\pi$
$734$	0	0
$735$	−624.868	−0.850161
$736$	0	0
$737$	0	0
$738$	0	0
$739$	625.309i	0.846156i	0.906093	+	0.423078i	$0.139050\pi$
$739$	625.309i	0.846156i	−0.906093	+	0.423078i	$0.860950\pi$
$740$	0	0
$741$	−95.4278	−0.128782
$742$	0	0
$743$	478.632i	0.644188i	0.946708	+	0.322094i	$0.104387\pi$
$743$	478.632i	0.644188i	−0.946708	+	0.322094i	$0.895613\pi$
$744$	0	0
$745$	− 298.306i	− 0.400411i
$746$	0	0
$747$	− 390.896i	− 0.523288i
$748$	0	0
$749$	−1605.63	−2.14369
$750$	0	0
$751$	−725.136	−0.965560	−0.482780	−	0.875742i	$-0.660373\pi$
$751$	−725.136	−0.965560	−0.482780	+	0.875742i	$0.660373\pi$
$752$	0	0
$753$	−440.341	−0.584782
$754$	0	0
$755$	561.665i	0.743927i
$756$	0	0
$757$	555.723	0.734112	0.367056	−	0.930199i	$-0.380366\pi$
$757$	555.723	0.734112	0.367056	+	0.930199i	$0.380366\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1023.24i	− 1.34460i	−0.740280	−	0.672299i	$-0.765307\pi$
$761$	− 1023.24i	− 1.34460i	0.740280	−	0.672299i	$-0.234693\pi$
$762$	0	0
$763$	803.735	1.05339
$764$	0	0
$765$	795.019i	1.03924i
$766$	0	0
$767$	− 220.799i	− 0.287874i
$768$	0	0
$769$	− 500.191i	− 0.650443i	−0.945638	−	0.325222i	$-0.894561\pi$
$769$	− 500.191i	− 0.650443i	0.945638	−	0.325222i	$-0.105439\pi$
$770$	0	0
$771$	−1523.23	−1.97566
$772$	0	0
$773$	−755.149	−0.976907	−0.488453	−	0.872590i	$-0.662439\pi$
$773$	−755.149	−0.976907	−0.488453	+	0.872590i	$0.662439\pi$
$774$	0	0
$775$	−216.720	−0.279639
$776$	0	0
$777$	− 3112.52i	− 4.00581i
$778$	0	0
$779$	183.030	0.234955
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1993.56i	2.54606i
$784$	0	0
$785$	−529.010	−0.673899
$786$	0	0
$787$	1401.42i	1.78071i	0.455271	+	0.890353i	$0.349543\pi$
$787$	1401.42i	1.78071i	−0.455271	+	0.890353i	$0.650457\pi$
$788$	0	0
$789$	1366.47i	1.73190i
$790$	0	0
$791$	286.682i	0.362430i
$792$	0	0
$793$	158.487	0.199858
$794$	0	0
$795$	−37.1337	−0.0467090
$796$	0	0
$797$	66.4088	0.0833235	0.0416617	−	0.999132i	$-0.486735\pi$
$797$	66.4088	0.0833235	0.0416617	+	0.999132i	$0.486735\pi$
$798$	0	0
$799$	1389.98i	1.73965i
$800$	0	0
$801$	1201.23	1.49967
$802$	0	0
$803$	0	0
$804$	0	0
$805$	786.946i	0.977572i
$806$	0	0
$807$	463.546	0.574407
$808$	0	0
$809$	6.37591i	0.00788122i	0.999992	+	0.00394061i	$0.00125434\pi$
$809$	6.37591i	0.00788122i	−0.999992	+	0.00394061i	$0.998746\pi$
$810$	0	0
$811$	− 672.405i	− 0.829107i	−0.910025	−	0.414553i	$-0.863938\pi$
$811$	− 672.405i	− 0.829107i	0.910025	−	0.414553i	$-0.136062\pi$
$812$	0	0
$813$	− 522.217i	− 0.642334i
$814$	0	0
$815$	442.173	0.542544
$816$	0	0
$817$	−361.498	−0.442470
$818$	0	0
$819$	598.312	0.730539
$820$	0	0
$821$	79.6541i	0.0970208i	0.998823	+	0.0485104i	$0.0154474\pi$
$821$	79.6541i	0.0970208i	−0.998823	+	0.0485104i	$0.984553\pi$
$822$	0	0
$823$	−691.951	−0.840766	−0.420383	−	0.907347i	$-0.638104\pi$
$823$	−691.951	−0.840766	−0.420383	+	0.907347i	$0.638104\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1050.43i	1.27017i	0.772440	+	0.635087i	$0.219036\pi$
$827$	1050.43i	1.27017i	−0.772440	+	0.635087i	$0.780964\pi$
$828$	0	0
$829$	333.117	0.401830	0.200915	−	0.979609i	$-0.435608\pi$
$829$	333.117	0.401830	0.200915	+	0.979609i	$0.435608\pi$
$830$	0	0
$831$	1333.26i	1.60440i
$832$	0	0
$833$	− 1093.49i	− 1.31271i
$834$	0	0
$835$	− 9.31621i	− 0.0111571i
$836$	0	0
$837$	−1925.90	−2.30095
$838$	0	0
$839$	29.8091	0.0355294	0.0177647	−	0.999842i	$-0.494345\pi$
$839$	29.8091	0.0355294	0.0177647	+	0.999842i	$0.494345\pi$
$840$	0	0
$841$	−1172.05	−1.39363
$842$	0	0
$843$	− 1979.05i	− 2.34763i
$844$	0	0
$845$	352.885	0.417616
$846$	0	0
$847$	0	0
$848$	0	0
$849$	486.971i	0.573581i
$850$	0	0
$851$	−2058.13	−2.41848
$852$	0	0
$853$	− 540.502i	− 0.633648i	−0.948484	−	0.316824i	$-0.897383\pi$
$853$	− 540.502i	− 0.633648i	0.948484	−	0.316824i	$-0.102617\pi$
$854$	0	0
$855$	− 217.836i	− 0.254779i
$856$	0	0
$857$	273.436i	0.319062i	0.987193	+	0.159531i	$0.0509981\pi$
$857$	273.436i	0.319062i	−0.987193	+	0.159531i	$0.949002\pi$
$858$	0	0
$859$	−275.966	−0.321264	−0.160632	−	0.987014i	$-0.551353\pi$
$859$	−275.966	−0.321264	−0.160632	+	0.987014i	$0.551353\pi$
$860$	0	0
$861$	−1733.95	−2.01387
$862$	0	0
$863$	−36.8569	−0.0427079	−0.0213539	−	0.999772i	$-0.506798\pi$
$863$	−36.8569	−0.0427079	−0.0213539	+	0.999772i	$0.506798\pi$
$864$	0	0
$865$	− 494.859i	− 0.572091i
$866$	0	0
$867$	−611.264	−0.705033
$868$	0	0
$869$	0	0
$870$	0	0
$871$	105.721i	0.121378i
$872$	0	0
$873$	1082.10	1.23952
$874$	0	0
$875$	113.561i	0.129785i
$876$	0	0
$877$	448.626i	0.511546i	0.966737	+	0.255773i	$0.0823300\pi$
$877$	448.626i	0.511546i	−0.966737	+	0.255773i	$0.917670\pi$
$878$	0	0
$879$	33.9916i	0.0386708i
$880$	0	0
$881$	710.818	0.806831	0.403415	−	0.915017i	$-0.367823\pi$
$881$	710.818	0.806831	0.403415	+	0.915017i	$0.367823\pi$
$882$	0	0
$883$	−1282.97	−1.45297	−0.726486	−	0.687182i	$-0.758848\pi$
$883$	−1282.97	−1.45297	−0.726486	+	0.687182i	$0.758848\pi$
$884$	0	0
$885$	761.574	0.860536
$886$	0	0
$887$	− 1023.50i	− 1.15389i	−0.816785	−	0.576943i	$-0.804246\pi$
$887$	− 1023.50i	− 1.15389i	0.816785	−	0.576943i	$-0.195754\pi$
$888$	0	0
$889$	−943.689	−1.06152
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 380.856i	− 0.426490i
$894$	0	0
$895$	−53.8750	−0.0601955
$896$	0	0
$897$	− 597.789i	− 0.666432i
$898$	0	0
$899$	− 1944.72i	− 2.16320i
$900$	0	0
$901$	− 64.9821i	− 0.0721222i
$902$	0	0
$903$	3424.67	3.79254
$904$	0	0
$905$	−122.704	−0.135585
$906$	0	0
$907$	1544.86	1.70327	0.851633	−	0.524138i	$-0.175613\pi$
$907$	1544.86	1.70327	0.851633	+	0.524138i	$0.175613\pi$
$908$	0	0
$909$	− 131.614i	− 0.144790i
$910$	0	0
$911$	−376.609	−0.413401	−0.206701	−	0.978404i	$-0.566273\pi$
$911$	−376.609	−0.413401	−0.206701	+	0.978404i	$0.566273\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	546.649i	0.597431i
$916$	0	0
$917$	1594.11	1.73840
$918$	0	0
$919$	− 1416.00i	− 1.54081i	−0.637556	−	0.770404i	$-0.720054\pi$
$919$	− 1416.00i	− 1.54081i	0.637556	−	0.770404i	$-0.279946\pi$
$920$	0	0
$921$	− 3089.49i	− 3.35450i
$922$	0	0
$923$	− 12.5755i	− 0.0136246i
$924$	0	0
$925$	−297.001	−0.321082
$926$	0	0
$927$	−1455.63	−1.57026
$928$	0	0
$929$	−800.241	−0.861400	−0.430700	−	0.902495i	$-0.641733\pi$
$929$	−800.241	−0.861400	−0.430700	+	0.902495i	$0.641733\pi$
$930$	0	0
$931$	299.617i	0.321823i
$932$	0	0
$933$	1894.61	2.03067
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 954.407i	− 1.01858i	−0.860596	−	0.509288i	$-0.829909\pi$
$937$	− 954.407i	− 1.01858i	0.860596	−	0.509288i	$-0.170091\pi$
$938$	0	0
$939$	1923.00	2.04792
$940$	0	0
$941$	− 874.736i	− 0.929581i	−0.885421	−	0.464791i	$-0.846130\pi$
$941$	− 874.736i	− 0.929581i	0.885421	−	0.464791i	$-0.153870\pi$
$942$	0	0
$943$	1146.56i	1.21586i
$944$	0	0
$945$	1009.17i	1.06790i
$946$	0	0
$947$	614.074	0.648441	0.324221	−	0.945982i	$-0.394898\pi$
$947$	614.074	0.648441	0.324221	+	0.945982i	$0.394898\pi$
$948$	0	0
$949$	−74.9564	−0.0789846
$950$	0	0
$951$	−1990.20	−2.09274
$952$	0	0
$953$	− 612.752i	− 0.642971i	−0.946914	−	0.321486i	$-0.895818\pi$
$953$	− 612.752i	− 0.642971i	0.946914	−	0.321486i	$-0.104182\pi$
$954$	0	0
$955$	511.401	0.535499
$956$	0	0
$957$	0	0
$958$	0	0
$959$	34.7592i	0.0362453i
$960$	0	0
$961$	917.705	0.954948
$962$	0	0
$963$	2784.22i	2.89119i
$964$	0	0
$965$	− 313.927i	− 0.325313i
$966$	0	0
$967$	− 200.077i	− 0.206905i	−0.994634	−	0.103452i	$-0.967011\pi$
$967$	− 200.077i	− 0.206905i	0.994634	−	0.103452i	$-0.0329890\pi$
$968$	0	0
$969$	575.992	0.594419
$970$	0	0
$971$	946.128	0.974385	0.487193	−	0.873295i	$-0.338021\pi$
$971$	946.128	0.974385	0.487193	+	0.873295i	$0.338021\pi$
$972$	0	0
$973$	−1144.36	−1.17611
$974$	0	0
$975$	− 86.2650i	− 0.0884769i
$976$	0	0
$977$	−343.304	−0.351386	−0.175693	−	0.984445i	$-0.556217\pi$
$977$	−343.304	−0.351386	−0.175693	+	0.984445i	$0.556217\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 1393.71i	− 1.42070i
$982$	0	0
$983$	−195.765	−0.199151	−0.0995753	−	0.995030i	$-0.531748\pi$
$983$	−195.765	−0.199151	−0.0995753	+	0.995030i	$0.531748\pi$
$984$	0	0
$985$	674.994i	0.685273i
$986$	0	0
$987$	3608.05i	3.65558i
$988$	0	0
$989$	− 2264.53i	− 2.28972i
$990$	0	0
$991$	−1157.74	−1.16826	−0.584130	−	0.811660i	$-0.698564\pi$
$991$	−1157.74	−1.16826	−0.584130	+	0.811660i	$0.698564\pi$
$992$	0	0
$993$	−3185.90	−3.20836
$994$	0	0
$995$	379.819	0.381728
$996$	0	0
$997$	1469.14i	1.47356i	0.676133	+	0.736779i	$0.263654\pi$
$997$	1469.14i	1.47356i	−0.676133	+	0.736779i	$0.736346\pi$
$998$	0	0
$999$	−2639.32	−2.64196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2420.3.f.c.241.16		16
11.6	odd	10		220.3.p.a.41.4	✓	16
11.9	even	5		220.3.p.a.161.4	yes	16
11.10	odd	2	inner	2420.3.f.c.241.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.3.p.a.41.4	✓	16	11.6	odd	10
220.3.p.a.161.4	yes	16	11.9	even	5
2420.3.f.c.241.15		16	11.10	odd	2	inner
2420.3.f.c.241.16		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2420.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.15878 q^{3} +2.23607 q^{5} +10.1572i q^{7} +17.6130 q^{9} -3.34439i q^{13} +11.5354 q^{15} +20.1864i q^{17} -5.53109i q^{19} +52.3990i q^{21} +34.6484 q^{23} +5.00000 q^{25} +44.4328 q^{27} +44.8670i q^{29} -43.3440 q^{31} +22.7123i q^{35} -59.4002 q^{37} -17.2530i q^{39} +33.0912i q^{41} -65.3574i q^{43} +39.3839 q^{45} +68.8573 q^{47} -54.1697 q^{49} +104.137i q^{51} -3.21911 q^{53} -28.5337i q^{57} +66.0207 q^{59} +47.3889i q^{61} +178.900i q^{63} -7.47829i q^{65} -31.6113 q^{67} +178.744 q^{69} +3.76017 q^{71} -22.4126i q^{73} +25.7939 q^{75} -78.2156i q^{79} +70.7017 q^{81} -22.1936i q^{83} +45.1381i q^{85} +231.459i q^{87} +68.2014 q^{89} +33.9698 q^{91} -223.602 q^{93} -12.3679i q^{95} +61.4375 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 14 q^{3} + 46 q^{9} + 10 q^{15} + 72 q^{23} + 80 q^{25} + 50 q^{27} + 50 q^{31} - 16 q^{37} + 30 q^{45} + 232 q^{47} - 188 q^{49} + 132 q^{53} + 78 q^{59} + 152 q^{67} + 748 q^{69} + 230 q^{71} + 70 q^{75}+ \cdots + 854 q^{97}+O(q^{100}) \) 16 * q + 14 * q^3 + 46 * q^9 + 10 * q^15 + 72 * q^23 + 80 * q^25 + 50 * q^27 + 50 * q^31 - 16 * q^37 + 30 * q^45 + 232 * q^47 - 188 * q^49 + 132 * q^53 + 78 * q^59 + 152 * q^67 + 748 * q^69 + 230 * q^71 + 70 * q^75 - 148 * q^81 + 160 * q^89 + 836 * q^91 - 698 * q^93 + 854 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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