LMFDB - Embedded newform 2420.3.f.a.241.3

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} - 3 x^{15} + 33 x^{14} - 111 x^{13} + 735 x^{12} - 1436 x^{11} + 10633 x^{10} - 25103 x^{9} + \cdots + 75625 $ x^16 - 3x^15 + 33x^14 - 111x^13 + 735x^12 - 1436x^11 + 10633x^10 - 25103x^9 + 232616x^8 - 600030x^7 + 1738315x^6 - 2296850x^5 + 2290475x^4 - 1918750x^3 + 1679000x^2 - 543125*x + 75625
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 5 $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.3
Root			$1.27755 + 3.93190i$ of defining polynomial
Character	$\chi$	$=$	2420.241
Dual form			2420.3.f.a.241.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.51621 q^{3} -2.23607 q^{5} -6.88461i q^{7} +3.36371 q^{9} +10.5628i q^{13} +7.86248 q^{15} -24.6915i q^{17} +3.05760i q^{19} +24.2077i q^{21} -28.7662 q^{23} +5.00000 q^{25} +19.8184 q^{27} -15.2201i q^{29} -28.2944 q^{31} +15.3945i q^{35} -2.74224 q^{37} -37.1409i q^{39} -61.6737i q^{41} -40.5154i q^{43} -7.52149 q^{45} +22.6838 q^{47} +1.60211 q^{49} +86.8204i q^{51} -86.4387 q^{53} -10.7512i q^{57} +73.7352 q^{59} -58.3068i q^{61} -23.1578i q^{63} -23.6191i q^{65} +64.6875 q^{67} +101.148 q^{69} -61.0928 q^{71} -84.3417i q^{73} -17.5810 q^{75} +76.3135i q^{79} -99.9588 q^{81} -18.9043i q^{83} +55.2119i q^{85} +53.5169i q^{87} +127.459 q^{89} +72.7206 q^{91} +99.4890 q^{93} -6.83701i q^{95} -88.6031 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{3} + 46 q^{9} - 30 q^{15} - 168 q^{23} + 80 q^{25} + 30 q^{27} - 190 q^{31} + 104 q^{37} - 30 q^{45} - 268 q^{47} - 228 q^{49} - 368 q^{53} + 78 q^{59} - 68 q^{67} - 212 q^{69} + 270 q^{71}+ \cdots - 726 q^{97}+O(q^{100}) $ 16 * q - 6 * q^3 + 46 * q^9 - 30 * q^15 - 168 * q^23 + 80 * q^25 + 30 * q^27 - 190 * q^31 + 104 * q^37 - 30 * q^45 - 268 * q^47 - 228 * q^49 - 368 * q^53 + 78 * q^59 - 68 * q^67 - 212 * q^69 + 270 * q^71 - 30 * q^75 + 1052 * q^81 - 160 * q^89 - 844 * q^91 - 78 * q^93 - 726 * 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
$3$	−3.51621	−1.17207	−0.586034	−	0.810286i	$-0.699312\pi$
$3$	−3.51621	−1.17207	−0.586034	+	0.810286i	$0.699312\pi$
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	− 6.88461i	− 0.983516i	−0.870732	−	0.491758i	$-0.836354\pi$
$7$	− 6.88461i	− 0.983516i	0.870732	−	0.491758i	$-0.163646\pi$
$8$	0	0
$9$	3.36371	0.373746
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	10.5628i	0.812521i	0.913757	+	0.406260i	$0.133167\pi$
$13$	10.5628i	0.812521i	−0.913757	+	0.406260i	$0.866833\pi$
$14$	0	0
$15$	7.86248	0.524165
$16$	0	0
$17$	− 24.6915i	− 1.45244i	−0.687462	−	0.726221i	$-0.741275\pi$
$17$	− 24.6915i	− 1.45244i	0.687462	−	0.726221i	$-0.258725\pi$
$18$	0	0
$19$	3.05760i	0.160926i	0.996758	+	0.0804632i	$0.0256400\pi$
$19$	3.05760i	0.160926i	−0.996758	+	0.0804632i	$0.974360\pi$
$20$	0	0
$21$	24.2077i	1.15275i
$22$	0	0
$23$	−28.7662	−1.25071	−0.625353	−	0.780342i	$-0.715045\pi$
$23$	−28.7662	−1.25071	−0.625353	+	0.780342i	$0.715045\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	19.8184	0.734013
$28$	0	0
$29$	− 15.2201i	− 0.524830i	−0.964955	−	0.262415i	$-0.915481\pi$
$29$	− 15.2201i	− 0.524830i	0.964955	−	0.262415i	$-0.0845190\pi$
$30$	0	0
$31$	−28.2944	−0.912723	−0.456361	−	0.889795i	$-0.650848\pi$
$31$	−28.2944	−0.912723	−0.456361	+	0.889795i	$0.650848\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	15.3945i	0.439842i
$36$	0	0
$37$	−2.74224	−0.0741147	−0.0370573	−	0.999313i	$-0.511798\pi$
$37$	−2.74224	−0.0741147	−0.0370573	+	0.999313i	$0.511798\pi$
$38$	0	0
$39$	− 37.1409i	− 0.952330i
$40$	0	0
$41$	− 61.6737i	− 1.50424i	−0.659028	−	0.752118i	$-0.729032\pi$
$41$	− 61.6737i	− 1.50424i	0.659028	−	0.752118i	$-0.270968\pi$
$42$	0	0
$43$	− 40.5154i	− 0.942220i	−0.882075	−	0.471110i	$-0.843854\pi$
$43$	− 40.5154i	− 0.942220i	0.882075	−	0.471110i	$-0.156146\pi$
$44$	0	0
$45$	−7.52149	−0.167144
$46$	0	0
$47$	22.6838	0.482634	0.241317	−	0.970446i	$-0.422421\pi$
$47$	22.6838	0.482634	0.241317	+	0.970446i	$0.422421\pi$
$48$	0	0
$49$	1.60211	0.0326961
$50$	0	0
$51$	86.8204i	1.70236i
$52$	0	0
$53$	−86.4387	−1.63092	−0.815460	−	0.578814i	$-0.803516\pi$
$53$	−86.4387	−1.63092	−0.815460	+	0.578814i	$0.803516\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 10.7512i	− 0.188617i
$58$	0	0
$59$	73.7352	1.24975	0.624874	−	0.780725i	$-0.285150\pi$
$59$	73.7352	1.24975	0.624874	+	0.780725i	$0.285150\pi$
$60$	0	0
$61$	− 58.3068i	− 0.955850i	−0.878401	−	0.477925i	$-0.841389\pi$
$61$	− 58.3068i	− 0.955850i	0.878401	−	0.477925i	$-0.158611\pi$
$62$	0	0
$63$	− 23.1578i	− 0.367585i
$64$	0	0
$65$	− 23.6191i	− 0.363370i
$66$	0	0
$67$	64.6875	0.965485	0.482742	−	0.875762i	$-0.339641\pi$
$67$	64.6875	0.965485	0.482742	+	0.875762i	$0.339641\pi$
$68$	0	0
$69$	101.148	1.46591
$70$	0	0
$71$	−61.0928	−0.860462	−0.430231	−	0.902719i	$-0.641568\pi$
$71$	−61.0928	−0.860462	−0.430231	+	0.902719i	$0.641568\pi$
$72$	0	0
$73$	− 84.3417i	− 1.15537i	−0.816261	−	0.577683i	$-0.803957\pi$
$73$	− 84.3417i	− 1.15537i	0.816261	−	0.577683i	$-0.196043\pi$
$74$	0	0
$75$	−17.5810	−0.234414
$76$	0	0
$77$	0	0
$78$	0	0
$79$	76.3135i	0.965994i	0.875622	+	0.482997i	$0.160452\pi$
$79$	76.3135i	0.965994i	−0.875622	+	0.482997i	$0.839548\pi$
$80$	0	0
$81$	−99.9588	−1.23406
$82$	0	0
$83$	− 18.9043i	− 0.227762i	−0.993494	−	0.113881i	$-0.963672\pi$
$83$	− 18.9043i	− 0.227762i	0.993494	−	0.113881i	$-0.0363283\pi$
$84$	0	0
$85$	55.2119i	0.649552i
$86$	0	0
$87$	53.5169i	0.615137i
$88$	0	0
$89$	127.459	1.43213	0.716063	−	0.698036i	$-0.245942\pi$
$89$	127.459	1.43213	0.716063	+	0.698036i	$0.245942\pi$
$90$	0	0
$91$	72.7206	0.799127
$92$	0	0
$93$	99.4890	1.06977
$94$	0	0
$95$	− 6.83701i	− 0.0719685i
$96$	0	0
$97$	−88.6031	−0.913434	−0.456717	−	0.889612i	$-0.650975\pi$
$97$	−88.6031	−0.913434	−0.456717	+	0.889612i	$0.650975\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	53.0670i	0.525416i	0.964875	+	0.262708i	$0.0846156\pi$
$101$	53.0670i	0.525416i	−0.964875	+	0.262708i	$0.915384\pi$
$102$	0	0
$103$	−30.0293	−0.291546	−0.145773	−	0.989318i	$-0.546567\pi$
$103$	−30.0293	−0.291546	−0.145773	+	0.989318i	$0.546567\pi$
$104$	0	0
$105$	− 54.1301i	− 0.515525i
$106$	0	0
$107$	− 194.317i	− 1.81605i	−0.418919	−	0.908024i	$-0.637591\pi$
$107$	− 194.317i	− 1.81605i	0.418919	−	0.908024i	$-0.362409\pi$
$108$	0	0
$109$	118.805i	1.08995i	0.838451	+	0.544977i	$0.183462\pi$
$109$	118.805i	1.08995i	−0.838451	+	0.544977i	$0.816538\pi$
$110$	0	0
$111$	9.64229	0.0868675
$112$	0	0
$113$	−179.534	−1.58880	−0.794399	−	0.607397i	$-0.792214\pi$
$113$	−179.534	−1.58880	−0.794399	+	0.607397i	$0.792214\pi$
$114$	0	0
$115$	64.3233	0.559333
$116$	0	0
$117$	35.5301i	0.303676i
$118$	0	0
$119$	−169.991	−1.42850
$120$	0	0
$121$	0	0
$122$	0	0
$123$	216.857i	1.76307i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 7.77494i	− 0.0612200i	−0.999531	−	0.0306100i	$-0.990255\pi$
$127$	− 7.77494i	− 0.0612200i	0.999531	−	0.0306100i	$-0.00974499\pi$
$128$	0	0
$129$	142.461i	1.10435i
$130$	0	0
$131$	41.8687i	0.319608i	0.987149	+	0.159804i	$0.0510863\pi$
$131$	41.8687i	0.319608i	−0.987149	+	0.159804i	$0.948914\pi$
$132$	0	0
$133$	21.0504	0.158274
$134$	0	0
$135$	−44.3152	−0.328261
$136$	0	0
$137$	126.318	0.922027	0.461013	−	0.887393i	$-0.347486\pi$
$137$	126.318	0.922027	0.461013	+	0.887393i	$0.347486\pi$
$138$	0	0
$139$	63.9915i	0.460370i	0.973147	+	0.230185i	$0.0739332\pi$
$139$	63.9915i	0.460370i	−0.973147	+	0.230185i	$0.926067\pi$
$140$	0	0
$141$	−79.7609	−0.565680
$142$	0	0
$143$	0	0
$144$	0	0
$145$	34.0331i	0.234711i
$146$	0	0
$147$	−5.63334	−0.0383221
$148$	0	0
$149$	1.82243i	0.0122310i	0.999981	+	0.00611552i	$0.00194664\pi$
$149$	1.82243i	0.0122310i	−0.999981	+	0.00611552i	$0.998053\pi$
$150$	0	0
$151$	288.221i	1.90875i	0.298616	+	0.954373i	$0.403475\pi$
$151$	288.221i	1.90875i	−0.298616	+	0.954373i	$0.596525\pi$
$152$	0	0
$153$	− 83.0551i	− 0.542844i
$154$	0	0
$155$	63.2682	0.408182
$156$	0	0
$157$	67.1103	0.427454	0.213727	−	0.976893i	$-0.431440\pi$
$157$	67.1103	0.427454	0.213727	+	0.976893i	$0.431440\pi$
$158$	0	0
$159$	303.936	1.91155
$160$	0	0
$161$	198.044i	1.23009i
$162$	0	0
$163$	162.305	0.995733	0.497867	−	0.867254i	$-0.334117\pi$
$163$	162.305	0.995733	0.497867	+	0.867254i	$0.334117\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	146.543i	0.877501i	0.898609	+	0.438751i	$0.144579\pi$
$167$	146.543i	0.877501i	−0.898609	+	0.438751i	$0.855421\pi$
$168$	0	0
$169$	57.4280	0.339810
$170$	0	0
$171$	10.2849i	0.0601456i
$172$	0	0
$173$	− 65.1838i	− 0.376785i	−0.982094	−	0.188393i	$-0.939672\pi$
$173$	− 65.1838i	− 0.376785i	0.982094	−	0.188393i	$-0.0603277\pi$
$174$	0	0
$175$	− 34.4231i	− 0.196703i
$176$	0	0
$177$	−259.268	−1.46479
$178$	0	0
$179$	−30.2729	−0.169122	−0.0845612	−	0.996418i	$-0.526949\pi$
$179$	−30.2729	−0.169122	−0.0845612	+	0.996418i	$0.526949\pi$
$180$	0	0
$181$	−164.173	−0.907035	−0.453517	−	0.891247i	$-0.649831\pi$
$181$	−164.173	−0.907035	−0.453517	+	0.891247i	$0.649831\pi$
$182$	0	0
$183$	205.019i	1.12032i
$184$	0	0
$185$	6.13184	0.0331451
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 136.442i	− 0.721914i
$190$	0	0
$191$	−262.183	−1.37268	−0.686342	−	0.727279i	$-0.740785\pi$
$191$	−262.183	−1.37268	−0.686342	+	0.727279i	$0.740785\pi$
$192$	0	0
$193$	− 135.407i	− 0.701591i	−0.936452	−	0.350796i	$-0.885911\pi$
$193$	− 135.407i	− 0.701591i	0.936452	−	0.350796i	$-0.114089\pi$
$194$	0	0
$195$	83.0495i	0.425895i
$196$	0	0
$197$	149.591i	0.759344i	0.925121	+	0.379672i	$0.123963\pi$
$197$	149.591i	0.759344i	−0.925121	+	0.379672i	$0.876037\pi$
$198$	0	0
$199$	−367.773	−1.84811	−0.924053	−	0.382264i	$-0.875145\pi$
$199$	−367.773	−1.84811	−0.924053	+	0.382264i	$0.875145\pi$
$200$	0	0
$201$	−227.455	−1.13161
$202$	0	0
$203$	−104.784	−0.516179
$204$	0	0
$205$	137.907i	0.672715i
$206$	0	0
$207$	−96.7613	−0.467446
$208$	0	0
$209$	0	0
$210$	0	0
$211$	344.013i	1.63039i	0.579185	+	0.815196i	$0.303371\pi$
$211$	344.013i	1.63039i	−0.579185	+	0.815196i	$0.696629\pi$
$212$	0	0
$213$	214.815	1.00852
$214$	0	0
$215$	90.5953i	0.421373i
$216$	0	0
$217$	194.796i	0.897677i
$218$	0	0
$219$	296.563i	1.35417i
$220$	0	0
$221$	260.811	1.18014
$222$	0	0
$223$	−212.858	−0.954520	−0.477260	−	0.878762i	$-0.658370\pi$
$223$	−212.858	−0.954520	−0.477260	+	0.878762i	$0.658370\pi$
$224$	0	0
$225$	16.8186	0.0747491
$226$	0	0
$227$	115.597i	0.509237i	0.967042	+	0.254619i	$0.0819500\pi$
$227$	115.597i	0.509237i	−0.967042	+	0.254619i	$0.918050\pi$
$228$	0	0
$229$	−444.472	−1.94093	−0.970463	−	0.241248i	$-0.922443\pi$
$229$	−444.472	−1.94093	−0.970463	+	0.241248i	$0.922443\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 90.3930i	− 0.387953i	−0.981006	−	0.193976i	$-0.937861\pi$
$233$	− 90.3930i	− 0.387953i	0.981006	−	0.193976i	$-0.0621385\pi$
$234$	0	0
$235$	−50.7225	−0.215840
$236$	0	0
$237$	− 268.334i	− 1.13221i
$238$	0	0
$239$	− 5.74820i	− 0.0240510i	−0.999928	−	0.0120255i	$-0.996172\pi$
$239$	− 5.74820i	− 0.0240510i	0.999928	−	0.0120255i	$-0.00382793\pi$
$240$	0	0
$241$	− 227.149i	− 0.942527i	−0.881992	−	0.471264i	$-0.843798\pi$
$241$	− 227.149i	− 0.942527i	0.881992	−	0.471264i	$-0.156202\pi$
$242$	0	0
$243$	173.111	0.712390
$244$	0	0
$245$	−3.58242	−0.0146221
$246$	0	0
$247$	−32.2968	−0.130756
$248$	0	0
$249$	66.4714i	0.266953i
$250$	0	0
$251$	−137.485	−0.547750	−0.273875	−	0.961765i	$-0.588305\pi$
$251$	−137.485	−0.547750	−0.273875	+	0.961765i	$0.588305\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	− 194.136i	− 0.761319i
$256$	0	0
$257$	412.814	1.60628	0.803141	−	0.595789i	$-0.203161\pi$
$257$	412.814	1.60628	0.803141	+	0.595789i	$0.203161\pi$
$258$	0	0
$259$	18.8793i	0.0728930i
$260$	0	0
$261$	− 51.1959i	− 0.196153i
$262$	0	0
$263$	378.661i	1.43978i	0.694090	+	0.719889i	$0.255807\pi$
$263$	378.661i	1.43978i	−0.694090	+	0.719889i	$0.744193\pi$
$264$	0	0
$265$	193.283	0.729369
$266$	0	0
$267$	−448.173	−1.67855
$268$	0	0
$269$	−65.1141	−0.242060	−0.121030	−	0.992649i	$-0.538620\pi$
$269$	−65.1141	−0.242060	−0.121030	+	0.992649i	$0.538620\pi$
$270$	0	0
$271$	530.380i	1.95712i	0.205956	+	0.978561i	$0.433970\pi$
$271$	530.380i	1.95712i	−0.205956	+	0.978561i	$0.566030\pi$
$272$	0	0
$273$	−255.701	−0.936632
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 138.890i	− 0.501407i	−0.968064	−	0.250704i	$-0.919338\pi$
$277$	− 138.890i	− 0.501407i	0.968064	−	0.250704i	$-0.0806620\pi$
$278$	0	0
$279$	−95.1742	−0.341126
$280$	0	0
$281$	311.037i	1.10689i	0.832885	+	0.553446i	$0.186688\pi$
$281$	311.037i	1.10689i	−0.832885	+	0.553446i	$0.813312\pi$
$282$	0	0
$283$	391.642i	1.38389i	0.721948	+	0.691947i	$0.243247\pi$
$283$	391.642i	1.38389i	−0.721948	+	0.691947i	$0.756753\pi$
$284$	0	0
$285$	24.0403i	0.0843521i
$286$	0	0
$287$	−424.599	−1.47944
$288$	0	0
$289$	−320.670	−1.10959
$290$	0	0
$291$	311.547	1.07061
$292$	0	0
$293$	214.392i	0.731714i	0.930671	+	0.365857i	$0.119224\pi$
$293$	214.392i	0.731714i	−0.930671	+	0.365857i	$0.880776\pi$
$294$	0	0
$295$	−164.877	−0.558905
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 303.851i	− 1.01622i
$300$	0	0
$301$	−278.933	−0.926688
$302$	0	0
$303$	− 186.595i	− 0.615824i
$304$	0	0
$305$	130.378i	0.427469i
$306$	0	0
$307$	471.912i	1.53717i	0.639745	+	0.768587i	$0.279040\pi$
$307$	471.912i	1.53717i	−0.639745	+	0.768587i	$0.720960\pi$
$308$	0	0
$309$	105.589	0.341712
$310$	0	0
$311$	−4.91728	−0.0158112	−0.00790560	−	0.999969i	$-0.502516\pi$
$311$	−4.91728	−0.0158112	−0.00790560	+	0.999969i	$0.502516\pi$
$312$	0	0
$313$	−203.179	−0.649135	−0.324568	−	0.945863i	$-0.605219\pi$
$313$	−203.179	−0.649135	−0.324568	+	0.945863i	$0.605219\pi$
$314$	0	0
$315$	51.7825i	0.164389i
$316$	0	0
$317$	−0.493990	−0.00155833	−0.000779164	−	1.00000i	$-0.500248\pi$
$317$	−0.493990	−0.00155833	−0.000779164		1.00000i	$0.500248\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	683.259i	2.12853i
$322$	0	0
$323$	75.4968	0.233736
$324$	0	0
$325$	52.8138i	0.162504i
$326$	0	0
$327$	− 417.743i	− 1.27750i
$328$	0	0
$329$	− 156.169i	− 0.474678i
$330$	0	0
$331$	−490.502	−1.48188	−0.740940	−	0.671571i	$-0.765620\pi$
$331$	−490.502	−1.48188	−0.740940	+	0.671571i	$0.765620\pi$
$332$	0	0
$333$	−9.22411	−0.0277000
$334$	0	0
$335$	−144.646	−0.431778
$336$	0	0
$337$	− 377.893i	− 1.12134i	−0.828038	−	0.560672i	$-0.810543\pi$
$337$	− 377.893i	− 1.12134i	0.828038	−	0.560672i	$-0.189457\pi$
$338$	0	0
$339$	631.279	1.86218
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 348.376i	− 1.01567i
$344$	0	0
$345$	−226.174	−0.655577
$346$	0	0
$347$	76.8713i	0.221531i	0.993847	+	0.110766i	$0.0353303\pi$
$347$	76.8713i	0.221531i	−0.993847	+	0.110766i	$0.964670\pi$
$348$	0	0
$349$	− 109.877i	− 0.314833i	−0.987532	−	0.157417i	$-0.949683\pi$
$349$	− 109.877i	− 0.314833i	0.987532	−	0.157417i	$-0.0503165\pi$
$350$	0	0
$351$	209.337i	0.596401i
$352$	0	0
$353$	497.456	1.40922	0.704612	−	0.709593i	$-0.251121\pi$
$353$	497.456	1.40922	0.704612	+	0.709593i	$0.251121\pi$
$354$	0	0
$355$	136.608	0.384810
$356$	0	0
$357$	597.725	1.67430
$358$	0	0
$359$	21.7532i	0.0605938i	0.999541	+	0.0302969i	$0.00964529\pi$
$359$	21.7532i	0.0605938i	−0.999541	+	0.0302969i	$0.990355\pi$
$360$	0	0
$361$	351.651	0.974103
$362$	0	0
$363$	0	0
$364$	0	0
$365$	188.594i	0.516695i
$366$	0	0
$367$	536.421	1.46164	0.730819	−	0.682571i	$-0.239138\pi$
$367$	536.421	1.46164	0.730819	+	0.682571i	$0.239138\pi$
$368$	0	0
$369$	− 207.452i	− 0.562202i
$370$	0	0
$371$	595.097i	1.60404i
$372$	0	0
$373$	− 332.971i	− 0.892685i	−0.894862	−	0.446342i	$-0.852726\pi$
$373$	− 332.971i	− 0.892685i	0.894862	−	0.446342i	$-0.147274\pi$
$374$	0	0
$375$	39.3124	0.104833
$376$	0	0
$377$	160.766	0.426435
$378$	0	0
$379$	378.033	0.997449	0.498724	−	0.866761i	$-0.333802\pi$
$379$	378.033	0.997449	0.498724	+	0.866761i	$0.333802\pi$
$380$	0	0
$381$	27.3383i	0.0717541i
$382$	0	0
$383$	−123.477	−0.322394	−0.161197	−	0.986922i	$-0.551535\pi$
$383$	−123.477	−0.322394	−0.161197	+	0.986922i	$0.551535\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 136.282i	− 0.352150i
$388$	0	0
$389$	731.435	1.88029	0.940147	−	0.340768i	$-0.110687\pi$
$389$	731.435	1.88029	0.940147	+	0.340768i	$0.110687\pi$
$390$	0	0
$391$	710.282i	1.81658i
$392$	0	0
$393$	− 147.219i	− 0.374603i
$394$	0	0
$395$	− 170.642i	− 0.432006i
$396$	0	0
$397$	123.206	0.310344	0.155172	−	0.987887i	$-0.450407\pi$
$397$	123.206	0.310344	0.155172	+	0.987887i	$0.450407\pi$
$398$	0	0
$399$	−74.0176	−0.185508
$400$	0	0
$401$	506.436	1.26293	0.631466	−	0.775404i	$-0.282454\pi$
$401$	506.436	1.26293	0.631466	+	0.775404i	$0.282454\pi$
$402$	0	0
$403$	− 298.867i	− 0.741606i
$404$	0	0
$405$	223.515	0.551888
$406$	0	0
$407$	0	0
$408$	0	0
$409$	699.526i	1.71033i	0.518354	+	0.855166i	$0.326545\pi$
$409$	699.526i	1.71033i	−0.518354	+	0.855166i	$0.673455\pi$
$410$	0	0
$411$	−444.159	−1.08068
$412$	0	0
$413$	− 507.638i	− 1.22915i
$414$	0	0
$415$	42.2713i	0.101858i
$416$	0	0
$417$	− 225.007i	− 0.539586i
$418$	0	0
$419$	−240.430	−0.573818	−0.286909	−	0.957958i	$-0.592628\pi$
$419$	−240.430	−0.573818	−0.286909	+	0.957958i	$0.592628\pi$
$420$	0	0
$421$	312.565	0.742435	0.371217	−	0.928546i	$-0.378940\pi$
$421$	312.565	0.742435	0.371217	+	0.928546i	$0.378940\pi$
$422$	0	0
$423$	76.3017	0.180382
$424$	0	0
$425$	− 123.458i	− 0.290488i
$426$	0	0
$427$	−401.420	−0.940093
$428$	0	0
$429$	0	0
$430$	0	0
$431$	730.105i	1.69398i	0.531609	+	0.846990i	$0.321588\pi$
$431$	730.105i	1.69398i	−0.531609	+	0.846990i	$0.678412\pi$
$432$	0	0
$433$	−431.087	−0.995583	−0.497791	−	0.867297i	$-0.665855\pi$
$433$	−431.087	−0.995583	−0.497791	+	0.867297i	$0.665855\pi$
$434$	0	0
$435$	− 119.668i	− 0.275098i
$436$	0	0
$437$	− 87.9558i	− 0.201272i
$438$	0	0
$439$	41.1901i	0.0938270i	0.998899	+	0.0469135i	$0.0149385\pi$
$439$	41.1901i	0.0938270i	−0.998899	+	0.0469135i	$0.985061\pi$
$440$	0	0
$441$	5.38903	0.0122200
$442$	0	0
$443$	210.738	0.475707	0.237853	−	0.971301i	$-0.423556\pi$
$443$	210.738	0.475707	0.237853	+	0.971301i	$0.423556\pi$
$444$	0	0
$445$	−285.008	−0.640466
$446$	0	0
$447$	− 6.40803i	− 0.0143356i
$448$	0	0
$449$	−357.921	−0.797152	−0.398576	−	0.917135i	$-0.630496\pi$
$449$	−357.921	−0.797152	−0.398576	+	0.917135i	$0.630496\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 1013.44i	− 2.23718i
$454$	0	0
$455$	−162.608	−0.357380
$456$	0	0
$457$	0.950272i	0.00207937i	0.999999	+	0.00103968i	$0.000330942\pi$
$457$	0.950272i	0.00207937i	−0.999999	+	0.00103968i	$0.999669\pi$
$458$	0	0
$459$	− 489.345i	− 1.06611i
$460$	0	0
$461$	511.859i	1.11032i	0.831743	+	0.555161i	$0.187343\pi$
$461$	511.859i	1.11032i	−0.831743	+	0.555161i	$0.812657\pi$
$462$	0	0
$463$	32.0919	0.0693130	0.0346565	−	0.999399i	$-0.488966\pi$
$463$	32.0919	0.0693130	0.0346565	+	0.999399i	$0.488966\pi$
$464$	0	0
$465$	−222.464	−0.478417
$466$	0	0
$467$	−215.116	−0.460634	−0.230317	−	0.973116i	$-0.573976\pi$
$467$	−215.116	−0.460634	−0.230317	+	0.973116i	$0.573976\pi$
$468$	0	0
$469$	− 445.348i	− 0.949570i
$470$	0	0
$471$	−235.974	−0.501006
$472$	0	0
$473$	0	0
$474$	0	0
$475$	15.2880i	0.0321853i
$476$	0	0
$477$	−290.755	−0.609549
$478$	0	0
$479$	− 231.124i	− 0.482514i	−0.970461	−	0.241257i	$-0.922440\pi$
$479$	− 231.124i	− 0.482514i	0.970461	−	0.241257i	$-0.0775597\pi$
$480$	0	0
$481$	− 28.9657i	− 0.0602197i
$482$	0	0
$483$	− 696.365i	− 1.44175i
$484$	0	0
$485$	198.123	0.408500
$486$	0	0
$487$	−0.548101	−0.00112546	−0.000562732	−	1.00000i	$-0.500179\pi$
$487$	−0.548101	−0.00112546	−0.000562732		1.00000i	$0.500179\pi$
$488$	0	0
$489$	−570.696	−1.16707
$490$	0	0
$491$	− 311.536i	− 0.634494i	−0.948343	−	0.317247i	$-0.897242\pi$
$491$	− 311.536i	− 0.634494i	0.948343	−	0.317247i	$-0.102758\pi$
$492$	0	0
$493$	−375.807	−0.762285
$494$	0	0
$495$	0	0
$496$	0	0
$497$	420.600i	0.846278i
$498$	0	0
$499$	−642.572	−1.28772	−0.643860	−	0.765143i	$-0.722668\pi$
$499$	−642.572	−1.28772	−0.643860	+	0.765143i	$0.722668\pi$
$500$	0	0
$501$	− 515.275i	− 1.02849i
$502$	0	0
$503$	247.519i	0.492086i	0.969259	+	0.246043i	$0.0791305\pi$
$503$	247.519i	0.492086i	−0.969259	+	0.246043i	$0.920870\pi$
$504$	0	0
$505$	− 118.661i	− 0.234973i
$506$	0	0
$507$	−201.929	−0.398281
$508$	0	0
$509$	751.728	1.47687	0.738437	−	0.674323i	$-0.235564\pi$
$509$	751.728	1.47687	0.738437	+	0.674323i	$0.235564\pi$
$510$	0	0
$511$	−580.660	−1.13632
$512$	0	0
$513$	60.5967i	0.118122i
$514$	0	0
$515$	67.1475	0.130383
$516$	0	0
$517$	0	0
$518$	0	0
$519$	229.200i	0.441618i
$520$	0	0
$521$	−610.473	−1.17173	−0.585866	−	0.810408i	$-0.699246\pi$
$521$	−610.473	−1.17173	−0.585866	+	0.810408i	$0.699246\pi$
$522$	0	0
$523$	620.110i	1.18568i	0.805320	+	0.592840i	$0.201993\pi$
$523$	620.110i	1.18568i	−0.805320	+	0.592840i	$0.798007\pi$
$524$	0	0
$525$	121.039i	0.230550i
$526$	0	0
$527$	698.631i	1.32568i
$528$	0	0
$529$	298.497	0.564266
$530$	0	0
$531$	248.024	0.467088
$532$	0	0
$533$	651.445	1.22222
$534$	0	0
$535$	434.506i	0.812161i
$536$	0	0
$537$	106.446	0.198223
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 302.908i	− 0.559904i	−0.960014	−	0.279952i	$-0.909681\pi$
$541$	− 302.908i	− 0.559904i	0.960014	−	0.279952i	$-0.0903186\pi$
$542$	0	0
$543$	577.267	1.06311
$544$	0	0
$545$	− 265.656i	− 0.487442i
$546$	0	0
$547$	− 781.304i	− 1.42834i	−0.699970	−	0.714172i	$-0.746803\pi$
$547$	− 781.304i	− 1.42834i	0.699970	−	0.714172i	$-0.253197\pi$
$548$	0	0
$549$	− 196.127i	− 0.357245i
$550$	0	0
$551$	46.5370	0.0844591
$552$	0	0
$553$	525.389	0.950071
$554$	0	0
$555$	−21.5608	−0.0388483
$556$	0	0
$557$	− 1016.07i	− 1.82418i	−0.409988	−	0.912091i	$-0.634467\pi$
$557$	− 1016.07i	− 1.82418i	0.409988	−	0.912091i	$-0.365533\pi$
$558$	0	0
$559$	427.955	0.765573
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 804.378i	− 1.42874i	−0.699770	−	0.714368i	$-0.746714\pi$
$563$	− 804.378i	− 1.42874i	0.699770	−	0.714368i	$-0.253286\pi$
$564$	0	0
$565$	401.450	0.710532
$566$	0	0
$567$	688.178i	1.21372i
$568$	0	0
$569$	− 916.303i	− 1.61037i	−0.593021	−	0.805187i	$-0.702065\pi$
$569$	− 916.303i	− 1.61037i	0.593021	−	0.805187i	$-0.297935\pi$
$570$	0	0
$571$	− 629.066i	− 1.10169i	−0.834607	−	0.550846i	$-0.814305\pi$
$571$	− 629.066i	− 1.10169i	0.834607	−	0.550846i	$-0.185695\pi$
$572$	0	0
$573$	921.888	1.60888
$574$	0	0
$575$	−143.831	−0.250141
$576$	0	0
$577$	−1013.71	−1.75686	−0.878432	−	0.477867i	$-0.841410\pi$
$577$	−1013.71	−1.75686	−0.878432	+	0.477867i	$0.841410\pi$
$578$	0	0
$579$	476.119i	0.822313i
$580$	0	0
$581$	−130.149	−0.224008
$582$	0	0
$583$	0	0
$584$	0	0
$585$	− 79.4477i	− 0.135808i
$586$	0	0
$587$	658.827	1.12236	0.561182	−	0.827693i	$-0.310347\pi$
$587$	658.827	1.12236	0.561182	+	0.827693i	$0.310347\pi$
$588$	0	0
$589$	− 86.5130i	− 0.146881i
$590$	0	0
$591$	− 525.992i	− 0.890003i
$592$	0	0
$593$	− 293.233i	− 0.494490i	−0.968953	−	0.247245i	$-0.920475\pi$
$593$	− 293.233i	− 0.494490i	0.968953	−	0.247245i	$-0.0795253\pi$
$594$	0	0
$595$	380.112	0.638844
$596$	0	0
$597$	1293.17	2.16611
$598$	0	0
$599$	−510.619	−0.852452	−0.426226	−	0.904617i	$-0.640157\pi$
$599$	−510.619	−0.852452	−0.426226	+	0.904617i	$0.640157\pi$
$600$	0	0
$601$	1008.32i	1.67773i	0.544339	+	0.838865i	$0.316780\pi$
$601$	1008.32i	1.67773i	−0.544339	+	0.838865i	$0.683220\pi$
$602$	0	0
$603$	217.590	0.360846
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13.9092i	0.0229147i	0.999934	+	0.0114573i	$0.00364706\pi$
$607$	13.9092i	0.0229147i	−0.999934	+	0.0114573i	$0.996353\pi$
$608$	0	0
$609$	368.443	0.604997
$610$	0	0
$611$	239.604i	0.392150i
$612$	0	0
$613$	418.153i	0.682142i	0.940038	+	0.341071i	$0.110790\pi$
$613$	418.153i	0.682142i	−0.940038	+	0.341071i	$0.889210\pi$
$614$	0	0
$615$	− 484.908i	− 0.788468i
$616$	0	0
$617$	−1045.51	−1.69451	−0.847255	−	0.531187i	$-0.821746\pi$
$617$	−1045.51	−1.69451	−0.847255	+	0.531187i	$0.821746\pi$
$618$	0	0
$619$	−614.916	−0.993403	−0.496701	−	0.867922i	$-0.665456\pi$
$619$	−614.916	−0.993403	−0.496701	+	0.867922i	$0.665456\pi$
$620$	0	0
$621$	−570.100	−0.918035
$622$	0	0
$623$	− 877.508i	− 1.40852i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	67.7101i	0.107647i
$630$	0	0
$631$	22.4531	0.0355834	0.0177917	−	0.999842i	$-0.494336\pi$
$631$	22.4531	0.0355834	0.0177917	+	0.999842i	$0.494336\pi$
$632$	0	0
$633$	− 1209.62i	− 1.91093i
$634$	0	0
$635$	17.3853i	0.0273784i
$636$	0	0
$637$	16.9227i	0.0265662i
$638$	0	0
$639$	−205.499	−0.321594
$640$	0	0
$641$	737.445	1.15046	0.575230	−	0.817992i	$-0.304913\pi$
$641$	737.445	1.15046	0.575230	+	0.817992i	$0.304913\pi$
$642$	0	0
$643$	−287.361	−0.446907	−0.223453	−	0.974715i	$-0.571733\pi$
$643$	−287.361	−0.446907	−0.223453	+	0.974715i	$0.571733\pi$
$644$	0	0
$645$	− 318.552i	− 0.493879i
$646$	0	0
$647$	851.654	1.31631	0.658156	−	0.752882i	$-0.271337\pi$
$647$	851.654	1.31631	0.658156	+	0.752882i	$0.271337\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 684.943i	− 1.05214i
$652$	0	0
$653$	854.829	1.30908	0.654540	−	0.756027i	$-0.272862\pi$
$653$	854.829	1.30908	0.654540	+	0.756027i	$0.272862\pi$
$654$	0	0
$655$	− 93.6213i	− 0.142933i
$656$	0	0
$657$	− 283.701i	− 0.431813i
$658$	0	0
$659$	− 328.629i	− 0.498679i	−0.968416	−	0.249339i	$-0.919787\pi$
$659$	− 328.629i	− 0.498679i	0.968416	−	0.249339i	$-0.0802135\pi$
$660$	0	0
$661$	−329.541	−0.498550	−0.249275	−	0.968433i	$-0.580192\pi$
$661$	−329.541	−0.498550	−0.249275	+	0.968433i	$0.580192\pi$
$662$	0	0
$663$	−917.064	−1.38320
$664$	0	0
$665$	−47.0702	−0.0707822
$666$	0	0
$667$	437.824i	0.656409i
$668$	0	0
$669$	748.453	1.11876
$670$	0	0
$671$	0	0
$672$	0	0
$673$	222.239i	0.330222i	0.986275	+	0.165111i	$0.0527982\pi$
$673$	222.239i	0.330222i	−0.986275	+	0.165111i	$0.947202\pi$
$674$	0	0
$675$	99.0918	0.146803
$676$	0	0
$677$	− 831.844i	− 1.22872i	−0.789025	−	0.614361i	$-0.789414\pi$
$677$	− 831.844i	− 1.22872i	0.789025	−	0.614361i	$-0.210586\pi$
$678$	0	0
$679$	609.998i	0.898377i
$680$	0	0
$681$	− 406.463i	− 0.596861i
$682$	0	0
$683$	623.160	0.912387	0.456194	−	0.889881i	$-0.349212\pi$
$683$	623.160	0.912387	0.456194	+	0.889881i	$0.349212\pi$
$684$	0	0
$685$	−282.455	−0.412343
$686$	0	0
$687$	1562.86	2.27490
$688$	0	0
$689$	− 913.032i	− 1.32516i
$690$	0	0
$691$	−156.947	−0.227130	−0.113565	−	0.993531i	$-0.536227\pi$
$691$	−156.947	−0.227130	−0.113565	+	0.993531i	$0.536227\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 143.089i	− 0.205884i
$696$	0	0
$697$	−1522.82	−2.18482
$698$	0	0
$699$	317.841i	0.454708i
$700$	0	0
$701$	− 620.329i	− 0.884920i	−0.896788	−	0.442460i	$-0.854106\pi$
$701$	− 620.329i	− 0.884920i	0.896788	−	0.442460i	$-0.145894\pi$
$702$	0	0
$703$	− 8.38469i	− 0.0119270i
$704$	0	0
$705$	178.351	0.252980
$706$	0	0
$707$	365.346	0.516755
$708$	0	0
$709$	−18.2861	−0.0257914	−0.0128957	−	0.999917i	$-0.504105\pi$
$709$	−18.2861	−0.0257914	−0.0128957	+	0.999917i	$0.504105\pi$
$710$	0	0
$711$	256.697i	0.361036i
$712$	0	0
$713$	813.924	1.14155
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20.2118i	0.0281895i
$718$	0	0
$719$	−1078.37	−1.49982	−0.749909	−	0.661541i	$-0.769903\pi$
$719$	−1078.37	−1.49982	−0.749909	+	0.661541i	$0.769903\pi$
$720$	0	0
$721$	206.740i	0.286740i
$722$	0	0
$723$	798.703i	1.10471i
$724$	0	0
$725$	− 76.1004i	− 0.104966i
$726$	0	0
$727$	739.254	1.01686	0.508428	−	0.861104i	$-0.330227\pi$
$727$	739.254	1.01686	0.508428	+	0.861104i	$0.330227\pi$
$728$	0	0
$729$	290.936	0.399090
$730$	0	0
$731$	−1000.39	−1.36852
$732$	0	0
$733$	− 1204.04i	− 1.64263i	−0.570478	−	0.821313i	$-0.693242\pi$
$733$	− 1204.04i	− 1.64263i	0.570478	−	0.821313i	$-0.306758\pi$
$734$	0	0
$735$	12.5965	0.0171381
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 303.552i	− 0.410761i	−0.978682	−	0.205380i	$-0.934157\pi$
$739$	− 303.552i	− 0.410761i	0.978682	−	0.205380i	$-0.0658431\pi$
$740$	0	0
$741$	113.562	0.153255
$742$	0	0
$743$	194.593i	0.261902i	0.991389	+	0.130951i	$0.0418031\pi$
$743$	194.593i	0.261902i	−0.991389	+	0.130951i	$0.958197\pi$
$744$	0	0
$745$	− 4.07507i	− 0.00546989i
$746$	0	0
$747$	− 63.5886i	− 0.0851252i
$748$	0	0
$749$	−1337.80	−1.78611
$750$	0	0
$751$	706.270	0.940440	0.470220	−	0.882549i	$-0.344175\pi$
$751$	706.270	0.940440	0.470220	+	0.882549i	$0.344175\pi$
$752$	0	0
$753$	483.426	0.642001
$754$	0	0
$755$	− 644.481i	− 0.853617i
$756$	0	0
$757$	−236.493	−0.312408	−0.156204	−	0.987725i	$-0.549926\pi$
$757$	−236.493	−0.312408	−0.156204	+	0.987725i	$0.549926\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	204.376i	0.268562i	0.990943	+	0.134281i	$0.0428725\pi$
$761$	204.376i	0.268562i	−0.990943	+	0.134281i	$0.957127\pi$
$762$	0	0
$763$	817.926	1.07199
$764$	0	0
$765$	185.717i	0.242767i
$766$	0	0
$767$	778.847i	1.01545i
$768$	0	0
$769$	404.940i	0.526580i	0.964717	+	0.263290i	$0.0848076\pi$
$769$	404.940i	0.526580i	−0.964717	+	0.263290i	$0.915192\pi$
$770$	0	0
$771$	−1451.54	−1.88267
$772$	0	0
$773$	−1286.73	−1.66460	−0.832298	−	0.554329i	$-0.812975\pi$
$773$	−1286.73	−1.66460	−0.832298	+	0.554329i	$0.812975\pi$
$774$	0	0
$775$	−141.472	−0.182545
$776$	0	0
$777$	− 66.3835i	− 0.0854356i
$778$	0	0
$779$	188.574	0.242071
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 301.637i	− 0.385232i
$784$	0	0
$785$	−150.063	−0.191163
$786$	0	0
$787$	1178.17i	1.49703i	0.663115	+	0.748517i	$0.269234\pi$
$787$	1178.17i	1.49703i	−0.663115	+	0.748517i	$0.730766\pi$
$788$	0	0
$789$	− 1331.45i	− 1.68752i
$790$	0	0
$791$	1236.02i	1.56261i
$792$	0	0
$793$	615.881	0.776647
$794$	0	0
$795$	−679.623	−0.854871
$796$	0	0
$797$	−306.811	−0.384958	−0.192479	−	0.981301i	$-0.561653\pi$
$797$	−306.811	−0.384958	−0.192479	+	0.981301i	$0.561653\pi$
$798$	0	0
$799$	− 560.097i	− 0.700997i
$800$	0	0
$801$	428.736	0.535251
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 442.841i	− 0.550113i
$806$	0	0
$807$	228.955	0.283711
$808$	0	0
$809$	1270.14i	1.57002i	0.619485	+	0.785009i	$0.287342\pi$
$809$	1270.14i	1.57002i	−0.619485	+	0.785009i	$0.712658\pi$
$810$	0	0
$811$	777.502i	0.958696i	0.877625	+	0.479348i	$0.159127\pi$
$811$	777.502i	0.958696i	−0.877625	+	0.479348i	$0.840873\pi$
$812$	0	0
$813$	− 1864.93i	− 2.29388i
$814$	0	0
$815$	−362.924	−0.445305
$816$	0	0
$817$	123.880	0.151628
$818$	0	0
$819$	244.611	0.298670
$820$	0	0
$821$	− 896.003i	− 1.09136i	−0.837995	−	0.545678i	$-0.816272\pi$
$821$	− 896.003i	− 1.09136i	0.837995	−	0.545678i	$-0.183728\pi$
$822$	0	0
$823$	957.496	1.16342	0.581711	−	0.813396i	$-0.302384\pi$
$823$	957.496	1.16342	0.581711	+	0.813396i	$0.302384\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1329.17i	− 1.60721i	−0.595160	−	0.803607i	$-0.702911\pi$
$827$	− 1329.17i	− 1.60721i	0.595160	−	0.803607i	$-0.297089\pi$
$828$	0	0
$829$	−87.7710	−0.105876	−0.0529379	−	0.998598i	$-0.516859\pi$
$829$	−87.7710	−0.105876	−0.0529379	+	0.998598i	$0.516859\pi$
$830$	0	0
$831$	488.365i	0.587684i
$832$	0	0
$833$	− 39.5585i	− 0.0474892i
$834$	0	0
$835$	− 327.679i	− 0.392431i
$836$	0	0
$837$	−560.749	−0.669951
$838$	0	0
$839$	−53.7571	−0.0640728	−0.0320364	−	0.999487i	$-0.510199\pi$
$839$	−53.7571	−0.0640728	−0.0320364	+	0.999487i	$0.510199\pi$
$840$	0	0
$841$	609.349	0.724553
$842$	0	0
$843$	− 1093.67i	− 1.29735i
$844$	0	0
$845$	−128.413	−0.151968
$846$	0	0
$847$	0	0
$848$	0	0
$849$	− 1377.10i	− 1.62202i
$850$	0	0
$851$	78.8840	0.0926957
$852$	0	0
$853$	381.990i	0.447819i	0.974610	+	0.223910i	$0.0718821\pi$
$853$	381.990i	0.447819i	−0.974610	+	0.223910i	$0.928118\pi$
$854$	0	0
$855$	− 22.9977i	− 0.0268979i
$856$	0	0
$857$	− 735.176i	− 0.857849i	−0.903340	−	0.428924i	$-0.858893\pi$
$857$	− 735.176i	− 0.857849i	0.903340	−	0.428924i	$-0.141107\pi$
$858$	0	0
$859$	1221.00	1.42142	0.710710	−	0.703485i	$-0.248374\pi$
$859$	1221.00	1.42142	0.710710	+	0.703485i	$0.248374\pi$
$860$	0	0
$861$	1492.98	1.73401
$862$	0	0
$863$	−1089.51	−1.26247	−0.631234	−	0.775593i	$-0.717451\pi$
$863$	−1089.51	−1.26247	−0.631234	+	0.775593i	$0.717451\pi$
$864$	0	0
$865$	145.756i	0.168503i
$866$	0	0
$867$	1127.54	1.30051
$868$	0	0
$869$	0	0
$870$	0	0
$871$	683.279i	0.784476i
$872$	0	0
$873$	−298.035	−0.341392
$874$	0	0
$875$	76.9723i	0.0879684i
$876$	0	0
$877$	− 1002.19i	− 1.14275i	−0.820690	−	0.571374i	$-0.806411\pi$
$877$	− 1002.19i	− 1.14275i	0.820690	−	0.571374i	$-0.193589\pi$
$878$	0	0
$879$	− 753.847i	− 0.857619i
$880$	0	0
$881$	−319.501	−0.362657	−0.181329	−	0.983423i	$-0.558040\pi$
$881$	−319.501	−0.362657	−0.181329	+	0.983423i	$0.558040\pi$
$882$	0	0
$883$	−1282.67	−1.45263	−0.726313	−	0.687364i	$-0.758768\pi$
$883$	−1282.67	−1.45263	−0.726313	+	0.687364i	$0.758768\pi$
$884$	0	0
$885$	579.741	0.655075
$886$	0	0
$887$	213.370i	0.240553i	0.992740	+	0.120276i	$0.0383781\pi$
$887$	213.370i	0.240553i	−0.992740	+	0.120276i	$0.961622\pi$
$888$	0	0
$889$	−53.5275	−0.0602109
$890$	0	0
$891$	0	0
$892$	0	0
$893$	69.3580i	0.0776686i
$894$	0	0
$895$	67.6923	0.0756338
$896$	0	0
$897$	1068.40i	1.19109i
$898$	0	0
$899$	430.643i	0.479024i
$900$	0	0
$901$	2134.30i	2.36881i
$902$	0	0
$903$	980.787	1.08614
$904$	0	0
$905$	367.103	0.405638
$906$	0	0
$907$	−1433.33	−1.58030	−0.790149	−	0.612915i	$-0.789997\pi$
$907$	−1433.33	−1.58030	−0.790149	+	0.612915i	$0.789997\pi$
$908$	0	0
$909$	178.502i	0.196372i
$910$	0	0
$911$	1648.12	1.80913	0.904565	−	0.426335i	$-0.140195\pi$
$911$	1648.12	1.80913	0.904565	+	0.426335i	$0.140195\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 458.436i	− 0.501023i
$916$	0	0
$917$	288.250	0.314340
$918$	0	0
$919$	398.321i	0.433429i	0.976235	+	0.216714i	$0.0695341\pi$
$919$	398.321i	0.433429i	−0.976235	+	0.216714i	$0.930466\pi$
$920$	0	0
$921$	− 1659.34i	− 1.80167i
$922$	0	0
$923$	− 645.309i	− 0.699143i
$924$	0	0
$925$	−13.7112	−0.0148229
$926$	0	0
$927$	−101.010	−0.108964
$928$	0	0
$929$	−1228.51	−1.32240	−0.661198	−	0.750212i	$-0.729952\pi$
$929$	−1228.51	−1.32240	−0.661198	+	0.750212i	$0.729952\pi$
$930$	0	0
$931$	4.89861i	0.00526167i
$932$	0	0
$933$	17.2902	0.0185318
$934$	0	0
$935$	0	0
$936$	0	0
$937$	672.767i	0.718001i	0.933337	+	0.359001i	$0.116882\pi$
$937$	672.767i	0.718001i	−0.933337	+	0.359001i	$0.883118\pi$
$938$	0	0
$939$	714.420	0.760831
$940$	0	0
$941$	1272.68i	1.35248i	0.736681	+	0.676240i	$0.236392\pi$
$941$	1272.68i	1.35248i	−0.736681	+	0.676240i	$0.763608\pi$
$942$	0	0
$943$	1774.12i	1.88136i
$944$	0	0
$945$	305.093i	0.322850i
$946$	0	0
$947$	−264.005	−0.278780	−0.139390	−	0.990238i	$-0.544514\pi$
$947$	−264.005	−0.278780	−0.139390	+	0.990238i	$0.544514\pi$
$948$	0	0
$949$	890.881	0.938758
$950$	0	0
$951$	1.73697	0.00182647
$952$	0	0
$953$	856.742i	0.898995i	0.893282	+	0.449498i	$0.148397\pi$
$953$	856.742i	0.898995i	−0.893282	+	0.449498i	$0.851603\pi$
$954$	0	0
$955$	586.258	0.613883
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 869.648i	− 0.906828i
$960$	0	0
$961$	−160.427	−0.166938
$962$	0	0
$963$	− 653.626i	− 0.678740i
$964$	0	0
$965$	302.779i	0.313761i
$966$	0	0
$967$	618.200i	0.639297i	0.947536	+	0.319648i	$0.103565\pi$
$967$	618.200i	0.639297i	−0.947536	+	0.319648i	$0.896435\pi$
$968$	0	0
$969$	−265.462	−0.273955
$970$	0	0
$971$	−752.937	−0.775424	−0.387712	−	0.921780i	$-0.626735\pi$
$971$	−752.937	−0.775424	−0.387712	+	0.921780i	$0.626735\pi$
$972$	0	0
$973$	440.556	0.452782
$974$	0	0
$975$	− 185.704i	− 0.190466i
$976$	0	0
$977$	−169.145	−0.173127	−0.0865636	−	0.996246i	$-0.527589\pi$
$977$	−169.145	−0.173127	−0.0865636	+	0.996246i	$0.527589\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	399.625i	0.407365i
$982$	0	0
$983$	1038.58	1.05654	0.528271	−	0.849076i	$-0.322841\pi$
$983$	1038.58	1.05654	0.528271	+	0.849076i	$0.322841\pi$
$984$	0	0
$985$	− 334.495i	− 0.339589i
$986$	0	0
$987$	549.123i	0.556355i
$988$	0	0
$989$	1165.48i	1.17844i
$990$	0	0
$991$	−1226.12	−1.23725	−0.618625	−	0.785686i	$-0.712310\pi$
$991$	−1226.12	−1.23725	−0.618625	+	0.785686i	$0.712310\pi$
$992$	0	0
$993$	1724.71	1.73687
$994$	0	0
$995$	822.366	0.826498
$996$	0	0
$997$	1809.11i	1.81455i	0.420536	+	0.907276i	$0.361842\pi$
$997$	1809.11i	1.81455i	−0.420536	+	0.907276i	$0.638158\pi$
$998$	0	0
$999$	−54.3468	−0.0544012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2420.3.f.a.241.3		16
11.2	odd	10		220.3.p.b.161.1	yes	16
11.5	even	5		220.3.p.b.41.1	✓	16
11.10	odd	2	inner	2420.3.f.a.241.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.3.p.b.41.1	✓	16	11.5	even	5
220.3.p.b.161.1	yes	16	11.2	odd	10
2420.3.f.a.241.3		16	1.1	even	1	trivial
2420.3.f.a.241.4		16	11.10	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:	\( N \)	\(=\)	\( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2420.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.51621 q^{3} -2.23607 q^{5} -6.88461i q^{7} +3.36371 q^{9} +10.5628i q^{13} +7.86248 q^{15} -24.6915i q^{17} +3.05760i q^{19} +24.2077i q^{21} -28.7662 q^{23} +5.00000 q^{25} +19.8184 q^{27} -15.2201i q^{29} -28.2944 q^{31} +15.3945i q^{35} -2.74224 q^{37} -37.1409i q^{39} -61.6737i q^{41} -40.5154i q^{43} -7.52149 q^{45} +22.6838 q^{47} +1.60211 q^{49} +86.8204i q^{51} -86.4387 q^{53} -10.7512i q^{57} +73.7352 q^{59} -58.3068i q^{61} -23.1578i q^{63} -23.6191i q^{65} +64.6875 q^{67} +101.148 q^{69} -61.0928 q^{71} -84.3417i q^{73} -17.5810 q^{75} +76.3135i q^{79} -99.9588 q^{81} -18.9043i q^{83} +55.2119i q^{85} +53.5169i q^{87} +127.459 q^{89} +72.7206 q^{91} +99.4890 q^{93} -6.83701i q^{95} -88.6031 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{3} + 46 q^{9} - 30 q^{15} - 168 q^{23} + 80 q^{25} + 30 q^{27} - 190 q^{31} + 104 q^{37} - 30 q^{45} - 268 q^{47} - 228 q^{49} - 368 q^{53} + 78 q^{59} - 68 q^{67} - 212 q^{69} + 270 q^{71}+ \cdots - 726 q^{97}+O(q^{100}) \) 16 * q - 6 * q^3 + 46 * q^9 - 30 * q^15 - 168 * q^23 + 80 * q^25 + 30 * q^27 - 190 * q^31 + 104 * q^37 - 30 * q^45 - 268 * q^47 - 228 * q^49 - 368 * q^53 + 78 * q^59 - 68 * q^67 - 212 * q^69 + 270 * q^71 - 30 * q^75 + 1052 * q^81 - 160 * q^89 - 844 * q^91 - 78 * q^93 - 726 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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