LMFDB - Embedded newform 2160.4.h.a.431.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,4,Mod(431,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2160.431"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,-188] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$127.444125612$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2160.431
Dual form			2160.4.h.a.431.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000i q^{5} +20.7846i q^{7} -36.3731 q^{11} -47.0000 q^{13} +21.0000i q^{17} -62.3538i q^{19} -36.3731 q^{23} -25.0000 q^{25} -123.000i q^{29} +25.9808i q^{31} +103.923 q^{35} -178.000 q^{37} -342.000i q^{41} +233.827i q^{43} -306.573 q^{47} -89.0000 q^{49} -414.000i q^{53} +181.865i q^{55} +446.869 q^{59} +542.000 q^{61} +235.000i q^{65} +155.885i q^{67} +852.169 q^{71} +232.000 q^{73} -756.000i q^{77} +348.142i q^{79} +405.300 q^{83} +105.000 q^{85} +1356.00i q^{89} -976.877i q^{91} -311.769 q^{95} -1046.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 188 q^{13} - 100 q^{25} - 712 q^{37} - 356 q^{49} + 2168 q^{61} + 928 q^{73} + 420 q^{85} - 4184 q^{97}+O(q^{100}) $ 4 * q - 188 * q^13 - 100 * q^25 - 712 * q^37 - 356 * q^49 + 2168 * q^61 + 928 * q^73 + 420 * q^85 - 4184 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 5.00000i	− 0.447214i
$5$	− 5.00000i	− 0.447214i
$6$	0	0
$7$	20.7846i	1.12226i	0.827727	+	0.561132i	$0.189634\pi$
$7$	20.7846i	1.12226i	−0.827727	+	0.561132i	$0.810366\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−36.3731	−0.996990	−0.498495	−	0.866893i	$-0.666114\pi$
$11$	−36.3731	−0.996990	−0.498495	+	0.866893i	$0.666114\pi$
$12$	0	0
$13$	−47.0000	−1.00273	−0.501364	−	0.865237i	$-0.667168\pi$
$13$	−47.0000	−1.00273	−0.501364	+	0.865237i	$0.667168\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	21.0000i	0.299603i	0.988716	+	0.149801i	$0.0478634\pi$
$17$	21.0000i	0.299603i	−0.988716	+	0.149801i	$0.952137\pi$
$18$	0	0
$19$	− 62.3538i	− 0.752892i	−0.926439	−	0.376446i	$-0.877146\pi$
$19$	− 62.3538i	− 0.752892i	0.926439	−	0.376446i	$-0.122854\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−36.3731	−0.329753	−0.164876	−	0.986314i	$-0.552722\pi$
$23$	−36.3731	−0.329753	−0.164876	+	0.986314i	$0.552722\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 123.000i	− 0.787604i	−0.919195	−	0.393802i	$-0.871159\pi$
$29$	− 123.000i	− 0.787604i	0.919195	−	0.393802i	$-0.128841\pi$
$30$	0	0
$31$	25.9808i	0.150525i	0.997164	+	0.0752626i	$0.0239795\pi$
$31$	25.9808i	0.150525i	−0.997164	+	0.0752626i	$0.976020\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	103.923	0.501891
$36$	0	0
$37$	−178.000	−0.790892	−0.395446	−	0.918489i	$-0.629410\pi$
$37$	−178.000	−0.790892	−0.395446	+	0.918489i	$0.629410\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 342.000i	− 1.30272i	−0.758770	−	0.651359i	$-0.774199\pi$
$41$	− 342.000i	− 1.30272i	0.758770	−	0.651359i	$-0.225801\pi$
$42$	0	0
$43$	233.827i	0.829262i	0.909990	+	0.414631i	$0.136089\pi$
$43$	233.827i	0.829262i	−0.909990	+	0.414631i	$0.863911\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−306.573	−0.951452	−0.475726	−	0.879593i	$-0.657815\pi$
$47$	−306.573	−0.951452	−0.475726	+	0.879593i	$0.657815\pi$
$48$	0	0
$49$	−89.0000	−0.259475
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 414.000i	− 1.07297i	−0.843911	−	0.536484i	$-0.819752\pi$
$53$	− 414.000i	− 1.07297i	0.843911	−	0.536484i	$-0.180248\pi$
$54$	0	0
$55$	181.865i	0.445868i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	446.869	0.986058	0.493029	−	0.870013i	$-0.335890\pi$
$59$	446.869	0.986058	0.493029	+	0.870013i	$0.335890\pi$
$60$	0	0
$61$	542.000	1.13764	0.568820	−	0.822462i	$-0.307400\pi$
$61$	542.000	1.13764	0.568820	+	0.822462i	$0.307400\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	235.000i	0.448433i
$66$	0	0
$67$	155.885i	0.284244i	0.989849	+	0.142122i	$0.0453925\pi$
$67$	155.885i	0.284244i	−0.989849	+	0.142122i	$0.954608\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	852.169	1.42442	0.712210	−	0.701966i	$-0.247694\pi$
$71$	852.169	1.42442	0.712210	+	0.701966i	$0.247694\pi$
$72$	0	0
$73$	232.000	0.371966	0.185983	−	0.982553i	$-0.440453\pi$
$73$	232.000	0.371966	0.185983	+	0.982553i	$0.440453\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 756.000i	− 1.11889i
$78$	0	0
$79$	348.142i	0.495811i	0.968784	+	0.247905i	$0.0797422\pi$
$79$	348.142i	0.495811i	−0.968784	+	0.247905i	$0.920258\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	405.300	0.535993	0.267997	−	0.963420i	$-0.413638\pi$
$83$	405.300	0.535993	0.267997	+	0.963420i	$0.413638\pi$
$84$	0	0
$85$	105.000	0.133986
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1356.00i	1.61501i	0.589862	+	0.807504i	$0.299182\pi$
$89$	1356.00i	1.61501i	−0.589862	+	0.807504i	$0.700818\pi$
$90$	0	0
$91$	− 976.877i	− 1.12532i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−311.769	−0.336704
$96$	0	0
$97$	−1046.00	−1.09490	−0.547450	−	0.836839i	$-0.684401\pi$
$97$	−1046.00	−1.09490	−0.547450	+	0.836839i	$0.684401\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	333.000i	0.328067i	0.986455	+	0.164033i	$0.0524505\pi$
$101$	333.000i	0.328067i	−0.986455	+	0.164033i	$0.947550\pi$
$102$	0	0
$103$	1111.98i	1.06375i	0.846823	+	0.531875i	$0.178512\pi$
$103$	1111.98i	1.06375i	−0.846823	+	0.531875i	$0.821488\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1392.57	1.25817	0.629087	−	0.777335i	$-0.283429\pi$
$107$	1392.57	1.25817	0.629087	+	0.777335i	$0.283429\pi$
$108$	0	0
$109$	1474.00	1.29526	0.647631	−	0.761954i	$-0.275760\pi$
$109$	1474.00	1.29526	0.647631	+	0.761954i	$0.275760\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	255.000i	0.212287i	0.994351	+	0.106143i	$0.0338502\pi$
$113$	255.000i	0.212287i	−0.994351	+	0.106143i	$0.966150\pi$
$114$	0	0
$115$	181.865i	0.147470i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−436.477	−0.336233
$120$	0	0
$121$	−8.00000	−0.00601052
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000i	0.0894427i
$126$	0	0
$127$	− 1444.53i	− 1.00930i	−0.863323	−	0.504651i	$-0.831621\pi$
$127$	− 1444.53i	− 1.00930i	0.863323	−	0.504651i	$-0.168379\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−192.258	−0.128226	−0.0641131	−	0.997943i	$-0.520422\pi$
$131$	−192.258	−0.128226	−0.0641131	+	0.997943i	$0.520422\pi$
$132$	0	0
$133$	1296.00	0.844943
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1734.00i	1.08135i	0.841230	+	0.540677i	$0.181832\pi$
$137$	1734.00i	1.08135i	−0.841230	+	0.540677i	$0.818168\pi$
$138$	0	0
$139$	− 872.954i	− 0.532683i	−0.963879	−	0.266342i	$-0.914185\pi$
$139$	− 872.954i	− 0.532683i	0.963879	−	0.266342i	$-0.0858149\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1709.53	0.999709
$144$	0	0
$145$	−615.000	−0.352227
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1407.00i	0.773597i	0.922164	+	0.386798i	$0.126419\pi$
$149$	1407.00i	0.773597i	−0.922164	+	0.386798i	$0.873581\pi$
$150$	0	0
$151$	223.435i	0.120416i	0.998186	+	0.0602081i	$0.0191764\pi$
$151$	223.435i	0.120416i	−0.998186	+	0.0602081i	$0.980824\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	129.904	0.0673169
$156$	0	0
$157$	−3301.00	−1.67802	−0.839008	−	0.544119i	$-0.816864\pi$
$157$	−3301.00	−1.67802	−0.839008	+	0.544119i	$0.816864\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 756.000i	− 0.370069i
$162$	0	0
$163$	2114.83i	1.01624i	0.861287	+	0.508118i	$0.169659\pi$
$163$	2114.83i	1.01624i	−0.861287	+	0.508118i	$0.830341\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−72.7461	−0.0337082	−0.0168541	−	0.999858i	$-0.505365\pi$
$167$	−72.7461	−0.0337082	−0.0168541	+	0.999858i	$0.505365\pi$
$168$	0	0
$169$	12.0000	0.00546199
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2112.00i	− 0.928164i	−0.885792	−	0.464082i	$-0.846384\pi$
$173$	− 2112.00i	− 0.928164i	0.885792	−	0.464082i	$-0.153616\pi$
$174$	0	0
$175$	− 519.615i	− 0.224453i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3315.15	1.38428	0.692139	−	0.721765i	$-0.256669\pi$
$179$	3315.15	1.38428	0.692139	+	0.721765i	$0.256669\pi$
$180$	0	0
$181$	−704.000	−0.289104	−0.144552	−	0.989497i	$-0.546174\pi$
$181$	−704.000	−0.289104	−0.144552	+	0.989497i	$0.546174\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	890.000i	0.353698i
$186$	0	0
$187$	− 763.834i	− 0.298701i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1434.14	0.543302	0.271651	−	0.962396i	$-0.412430\pi$
$191$	1434.14	0.543302	0.271651	+	0.962396i	$0.412430\pi$
$192$	0	0
$193$	−1168.00	−0.435619	−0.217810	−	0.975991i	$-0.569891\pi$
$193$	−1168.00	−0.435619	−0.217810	+	0.975991i	$0.569891\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 456.000i	− 0.164917i	−0.996594	−	0.0824585i	$-0.973723\pi$
$197$	− 456.000i	− 0.164917i	0.996594	−	0.0824585i	$-0.0262772\pi$
$198$	0	0
$199$	− 743.050i	− 0.264690i	−0.991204	−	0.132345i	$-0.957749\pi$
$199$	− 743.050i	− 0.264690i	0.991204	−	0.132345i	$-0.0422508\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2556.51	0.883900
$204$	0	0
$205$	−1710.00	−0.582593
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2268.00i	0.750626i
$210$	0	0
$211$	− 2816.31i	− 0.918877i	−0.888210	−	0.459439i	$-0.848051\pi$
$211$	− 2816.31i	− 0.918877i	0.888210	−	0.459439i	$-0.151949\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1169.13	0.370857
$216$	0	0
$217$	−540.000	−0.168929
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 987.000i	− 0.300420i
$222$	0	0
$223$	− 1641.98i	− 0.493074i	−0.969133	−	0.246537i	$-0.920707\pi$
$223$	− 1641.98i	− 0.493074i	0.969133	−	0.246537i	$-0.0792926\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5591.06	1.63477	0.817383	−	0.576095i	$-0.195424\pi$
$227$	5591.06	1.63477	0.817383	+	0.576095i	$0.195424\pi$
$228$	0	0
$229$	6298.00	1.81740	0.908698	−	0.417455i	$-0.137078\pi$
$229$	6298.00	1.81740	0.908698	+	0.417455i	$0.137078\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2766.00i	− 0.777711i	−0.921299	−	0.388856i	$-0.872871\pi$
$233$	− 2766.00i	− 0.777711i	0.921299	−	0.388856i	$-0.127129\pi$
$234$	0	0
$235$	1532.86i	0.425502i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5757.34	1.55821	0.779103	−	0.626896i	$-0.215675\pi$
$239$	5757.34	1.55821	0.779103	+	0.626896i	$0.215675\pi$
$240$	0	0
$241$	1807.00	0.482984	0.241492	−	0.970403i	$-0.422363\pi$
$241$	1807.00	0.482984	0.241492	+	0.970403i	$0.422363\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	445.000i	0.116041i
$246$	0	0
$247$	2930.63i	0.754945i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2769.55	0.696464	0.348232	−	0.937408i	$-0.386782\pi$
$251$	2769.55	0.696464	0.348232	+	0.937408i	$0.386782\pi$
$252$	0	0
$253$	1323.00	0.328760
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1227.00i	− 0.297814i	−0.988851	−	0.148907i	$-0.952425\pi$
$257$	− 1227.00i	− 0.297814i	0.988851	−	0.148907i	$-0.0475755\pi$
$258$	0	0
$259$	− 3699.66i	− 0.887590i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	883.346	0.207108	0.103554	−	0.994624i	$-0.466979\pi$
$263$	883.346	0.207108	0.103554	+	0.994624i	$0.466979\pi$
$264$	0	0
$265$	−2070.00	−0.479846
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4071.00i	− 0.922726i	−0.887212	−	0.461363i	$-0.847361\pi$
$269$	− 4071.00i	− 0.922726i	0.887212	−	0.461363i	$-0.152639\pi$
$270$	0	0
$271$	3190.44i	0.715149i	0.933885	+	0.357574i	$0.116396\pi$
$271$	3190.44i	0.715149i	−0.933885	+	0.357574i	$0.883604\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	909.327	0.199398
$276$	0	0
$277$	−1370.00	−0.297167	−0.148584	−	0.988900i	$-0.547471\pi$
$277$	−1370.00	−0.297167	−0.148584	+	0.988900i	$0.547471\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1866.00i	0.396143i	0.980188	+	0.198072i	$0.0634678\pi$
$281$	1866.00i	0.396143i	−0.980188	+	0.198072i	$0.936532\pi$
$282$	0	0
$283$	− 4063.39i	− 0.853511i	−0.904367	−	0.426755i	$-0.859657\pi$
$283$	− 4063.39i	− 0.853511i	0.904367	−	0.426755i	$-0.140343\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7108.34	1.46199
$288$	0	0
$289$	4472.00	0.910238
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1632.00i	− 0.325401i	−0.986676	−	0.162700i	$-0.947980\pi$
$293$	− 1632.00i	− 0.325401i	0.986676	−	0.162700i	$-0.0520204\pi$
$294$	0	0
$295$	− 2234.35i	− 0.440978i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1709.53	0.330652
$300$	0	0
$301$	−4860.00	−0.930650
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2710.00i	− 0.508768i
$306$	0	0
$307$	1273.06i	0.236668i	0.992974	+	0.118334i	$0.0377554\pi$
$307$	1273.06i	0.236668i	−0.992974	+	0.118334i	$0.962245\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9176.41	1.67314	0.836569	−	0.547861i	$-0.184558\pi$
$311$	9176.41	1.67314	0.836569	+	0.547861i	$0.184558\pi$
$312$	0	0
$313$	8426.00	1.52162	0.760808	−	0.648977i	$-0.224803\pi$
$313$	8426.00	1.52162	0.760808	+	0.648977i	$0.224803\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5034.00i	− 0.891917i	−0.895054	−	0.445958i	$-0.852863\pi$
$317$	− 5034.00i	− 0.891917i	0.895054	−	0.445958i	$-0.147137\pi$
$318$	0	0
$319$	4473.89i	0.785234i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1309.43	0.225569
$324$	0	0
$325$	1175.00	0.200545
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 6372.00i	− 1.06778i
$330$	0	0
$331$	8396.98i	1.39438i	0.716886	+	0.697190i	$0.245567\pi$
$331$	8396.98i	1.39438i	−0.716886	+	0.697190i	$0.754433\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	779.423	0.127118
$336$	0	0
$337$	−2414.00	−0.390205	−0.195102	−	0.980783i	$-0.562504\pi$
$337$	−2414.00	−0.390205	−0.195102	+	0.980783i	$0.562504\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 945.000i	− 0.150072i
$342$	0	0
$343$	5279.29i	0.831064i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5580.67	−0.863360	−0.431680	−	0.902027i	$-0.642079\pi$
$347$	−5580.67	−0.863360	−0.431680	+	0.902027i	$0.642079\pi$
$348$	0	0
$349$	−4300.00	−0.659524	−0.329762	−	0.944064i	$-0.606968\pi$
$349$	−4300.00	−0.659524	−0.329762	+	0.944064i	$0.606968\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11619.0i	1.75189i	0.482411	+	0.875945i	$0.339761\pi$
$353$	11619.0i	1.75189i	−0.482411	+	0.875945i	$0.660239\pi$
$354$	0	0
$355$	− 4260.84i	− 0.637020i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8802.28	−1.29406	−0.647029	−	0.762466i	$-0.723989\pi$
$359$	−8802.28	−1.29406	−0.647029	+	0.762466i	$0.723989\pi$
$360$	0	0
$361$	2971.00	0.433154
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1160.00i	− 0.166348i
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8604.83	1.20415
$372$	0	0
$373$	−2149.00	−0.298314	−0.149157	−	0.988814i	$-0.547656\pi$
$373$	−2149.00	−0.298314	−0.149157	+	0.988814i	$0.547656\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5781.00i	0.789752i
$378$	0	0
$379$	− 7711.09i	− 1.04510i	−0.852609	−	0.522549i	$-0.824981\pi$
$379$	− 7711.09i	− 1.04510i	0.852609	−	0.522549i	$-0.175019\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10262.4	1.36915	0.684575	−	0.728943i	$-0.259988\pi$
$383$	10262.4	1.36915	0.684575	+	0.728943i	$0.259988\pi$
$384$	0	0
$385$	−3780.00	−0.500381
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6399.00i	0.834042i	0.908897	+	0.417021i	$0.136926\pi$
$389$	6399.00i	0.834042i	−0.908897	+	0.417021i	$0.863074\pi$
$390$	0	0
$391$	− 763.834i	− 0.0987948i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1740.71	0.221733
$396$	0	0
$397$	−7873.00	−0.995301	−0.497651	−	0.867378i	$-0.665804\pi$
$397$	−7873.00	−0.995301	−0.497651	+	0.867378i	$0.665804\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4098.00i	− 0.510335i	−0.966897	−	0.255168i	$-0.917869\pi$
$401$	− 4098.00i	− 0.510335i	0.966897	−	0.255168i	$-0.0821306\pi$
$402$	0	0
$403$	− 1221.10i	− 0.150936i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6474.41	0.788512
$408$	0	0
$409$	1519.00	0.183642	0.0918212	−	0.995776i	$-0.470731\pi$
$409$	1519.00	0.183642	0.0918212	+	0.995776i	$0.470731\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9288.00i	1.10662i
$414$	0	0
$415$	− 2026.50i	− 0.239703i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14014.0	1.63396	0.816981	−	0.576665i	$-0.195646\pi$
$419$	14014.0	1.63396	0.816981	+	0.576665i	$0.195646\pi$
$420$	0	0
$421$	−2162.00	−0.250284	−0.125142	−	0.992139i	$-0.539939\pi$
$421$	−2162.00	−0.250284	−0.125142	+	0.992139i	$0.539939\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 525.000i	− 0.0599206i
$426$	0	0
$427$	11265.3i	1.27673i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6495.19	−0.725899	−0.362949	−	0.931809i	$-0.618230\pi$
$431$	−6495.19	−0.725899	−0.362949	+	0.931809i	$0.618230\pi$
$432$	0	0
$433$	−9574.00	−1.06258	−0.531290	−	0.847190i	$-0.678293\pi$
$433$	−9574.00	−1.06258	−0.531290	+	0.847190i	$0.678293\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2268.00i	0.248268i
$438$	0	0
$439$	− 197.454i	− 0.0214669i	−0.999942	−	0.0107334i	$-0.996583\pi$
$439$	− 197.454i	− 0.0214669i	0.999942	−	0.0107334i	$-0.00341662\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16409.4	1.75990	0.879951	−	0.475065i	$-0.157575\pi$
$443$	16409.4	1.75990	0.879951	+	0.475065i	$0.157575\pi$
$444$	0	0
$445$	6780.00	0.722254
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 2160.00i	− 0.227031i	−0.993536	−	0.113515i	$-0.963789\pi$
$449$	− 2160.00i	− 0.227031i	0.993536	−	0.113515i	$-0.0362111\pi$
$450$	0	0
$451$	12439.6i	1.29880i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4884.38	−0.503260
$456$	0	0
$457$	1186.00	0.121398	0.0606988	−	0.998156i	$-0.480667\pi$
$457$	1186.00	0.121398	0.0606988	+	0.998156i	$0.480667\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19026.0i	1.92219i	0.276220	+	0.961095i	$0.410918\pi$
$461$	19026.0i	1.92219i	−0.276220	+	0.961095i	$0.589082\pi$
$462$	0	0
$463$	6183.42i	0.620665i	0.950628	+	0.310333i	$0.100440\pi$
$463$	6183.42i	0.620665i	−0.950628	+	0.310333i	$0.899560\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10922.3	1.08228	0.541140	−	0.840933i	$-0.317993\pi$
$467$	10922.3	1.08228	0.541140	+	0.840933i	$0.317993\pi$
$468$	0	0
$469$	−3240.00	−0.318996
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 8505.00i	− 0.826766i
$474$	0	0
$475$	1558.85i	0.150578i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10194.9	0.972473	0.486237	−	0.873827i	$-0.338369\pi$
$479$	10194.9	0.972473	0.486237	+	0.873827i	$0.338369\pi$
$480$	0	0
$481$	8366.00	0.793049
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5230.00i	0.489654i
$486$	0	0
$487$	− 18373.6i	− 1.70962i	−0.518938	−	0.854812i	$-0.673672\pi$
$487$	− 18373.6i	− 1.70962i	0.518938	−	0.854812i	$-0.326328\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3211.22	0.295154	0.147577	−	0.989051i	$-0.452853\pi$
$491$	3211.22	0.295154	0.147577	+	0.989051i	$0.452853\pi$
$492$	0	0
$493$	2583.00	0.235968
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17712.0i	1.59858i
$498$	0	0
$499$	− 2047.28i	− 0.183665i	−0.995774	−	0.0918327i	$-0.970728\pi$
$499$	− 2047.28i	− 0.183665i	0.995774	−	0.0918327i	$-0.0292725\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12174.6	−1.07920	−0.539600	−	0.841921i	$-0.681425\pi$
$503$	−12174.6	−1.07920	−0.539600	+	0.841921i	$0.681425\pi$
$504$	0	0
$505$	1665.00	0.146716
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 18063.0i	− 1.57294i	−0.617626	−	0.786472i	$-0.711905\pi$
$509$	− 18063.0i	− 1.57294i	0.617626	−	0.786472i	$-0.288095\pi$
$510$	0	0
$511$	4822.03i	0.417444i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5559.88	0.475724
$516$	0	0
$517$	11151.0	0.948589
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 9270.00i	− 0.779513i	−0.920918	−	0.389756i	$-0.872559\pi$
$521$	− 9270.00i	− 0.779513i	0.920918	−	0.389756i	$-0.127441\pi$
$522$	0	0
$523$	− 8599.63i	− 0.718997i	−0.933146	−	0.359499i	$-0.882948\pi$
$523$	− 8599.63i	− 0.718997i	0.933146	−	0.359499i	$-0.117052\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−545.596	−0.0450978
$528$	0	0
$529$	−10844.0	−0.891263
$530$	0	0
$531$	0	0
$532$	0	0
$533$	16074.0i	1.30627i
$534$	0	0
$535$	− 6962.84i	− 0.562673i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3237.20	0.258694
$540$	0	0
$541$	−416.000	−0.0330596	−0.0165298	−	0.999863i	$-0.505262\pi$
$541$	−416.000	−0.0330596	−0.0165298	+	0.999863i	$0.505262\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 7370.00i	− 0.579259i
$546$	0	0
$547$	4245.26i	0.331836i	0.986140	+	0.165918i	$0.0530586\pi$
$547$	4245.26i	0.331836i	−0.986140	+	0.165918i	$0.946941\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7669.52	−0.592981
$552$	0	0
$553$	−7236.00	−0.556430
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 19050.0i	− 1.44915i	−0.689198	−	0.724573i	$-0.742037\pi$
$557$	− 19050.0i	− 1.44915i	0.689198	−	0.724573i	$-0.257963\pi$
$558$	0	0
$559$	− 10989.9i	− 0.831524i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1101.58	−0.0824622	−0.0412311	−	0.999150i	$-0.513128\pi$
$563$	−1101.58	−0.0824622	−0.0412311	+	0.999150i	$0.513128\pi$
$564$	0	0
$565$	1275.00	0.0949374
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 3156.00i	− 0.232524i	−0.993219	−	0.116262i	$-0.962909\pi$
$569$	− 3156.00i	− 0.232524i	0.993219	−	0.116262i	$-0.0370913\pi$
$570$	0	0
$571$	− 23122.9i	− 1.69468i	−0.531051	−	0.847340i	$-0.678203\pi$
$571$	− 23122.9i	− 1.69468i	0.531051	−	0.847340i	$-0.321797\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	909.327	0.0659505
$576$	0	0
$577$	14348.0	1.03521	0.517604	−	0.855620i	$-0.326824\pi$
$577$	14348.0	1.03521	0.517604	+	0.855620i	$0.326824\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8424.00i	0.601526i
$582$	0	0
$583$	15058.4i	1.06974i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4770.07	0.335403	0.167702	−	0.985838i	$-0.446365\pi$
$587$	4770.07	0.335403	0.167702	+	0.985838i	$0.446365\pi$
$588$	0	0
$589$	1620.00	0.113329
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 7617.00i	− 0.527475i	−0.964594	−	0.263738i	$-0.915045\pi$
$593$	− 7617.00i	− 0.527475i	0.964594	−	0.263738i	$-0.0849553\pi$
$594$	0	0
$595$	2182.38i	0.150368i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4053.00	0.276463	0.138231	−	0.990400i	$-0.455858\pi$
$599$	4053.00	0.276463	0.138231	+	0.990400i	$0.455858\pi$
$600$	0	0
$601$	15365.0	1.04285	0.521424	−	0.853298i	$-0.325401\pi$
$601$	15365.0	1.04285	0.521424	+	0.853298i	$0.325401\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	40.0000i	0.00268799i
$606$	0	0
$607$	15494.9i	1.03611i	0.855347	+	0.518056i	$0.173344\pi$
$607$	15494.9i	1.03611i	−0.855347	+	0.518056i	$0.826656\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14408.9	0.954047
$612$	0	0
$613$	6131.00	0.403962	0.201981	−	0.979389i	$-0.435262\pi$
$613$	6131.00	0.403962	0.201981	+	0.979389i	$0.435262\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 22305.0i	− 1.45537i	−0.685910	−	0.727687i	$-0.740595\pi$
$617$	− 22305.0i	− 1.45537i	0.685910	−	0.727687i	$-0.259405\pi$
$618$	0	0
$619$	− 26074.3i	− 1.69308i	−0.532328	−	0.846538i	$-0.678683\pi$
$619$	− 26074.3i	− 1.69308i	0.532328	−	0.846538i	$-0.321317\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−28183.9	−1.81246
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3738.00i	− 0.236954i
$630$	0	0
$631$	9779.16i	0.616961i	0.951231	+	0.308480i	$0.0998204\pi$
$631$	9779.16i	0.616961i	−0.951231	+	0.308480i	$0.900180\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7222.65	−0.451374
$636$	0	0
$637$	4183.00	0.260183
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 10884.0i	− 0.670659i	−0.942101	−	0.335329i	$-0.891152\pi$
$641$	− 10884.0i	− 0.670659i	0.942101	−	0.335329i	$-0.108848\pi$
$642$	0	0
$643$	29696.0i	1.82130i	0.413178	+	0.910650i	$0.364419\pi$
$643$	29696.0i	1.82130i	−0.413178	+	0.910650i	$0.635581\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23891.9	−1.45176	−0.725879	−	0.687822i	$-0.758567\pi$
$647$	−23891.9	−1.45176	−0.725879	+	0.687822i	$0.758567\pi$
$648$	0	0
$649$	−16254.0	−0.983090
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 11790.0i	− 0.706552i	−0.935519	−	0.353276i	$-0.885068\pi$
$653$	− 11790.0i	− 0.706552i	0.935519	−	0.353276i	$-0.114932\pi$
$654$	0	0
$655$	961.288i	0.0573445i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13063.1	−0.772181	−0.386090	−	0.922461i	$-0.626175\pi$
$659$	−13063.1	−0.772181	−0.386090	+	0.922461i	$0.626175\pi$
$660$	0	0
$661$	−8998.00	−0.529473	−0.264736	−	0.964321i	$-0.585285\pi$
$661$	−8998.00	−0.529473	−0.264736	+	0.964321i	$0.585285\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 6480.00i	− 0.377870i
$666$	0	0
$667$	4473.89i	0.259715i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−19714.2	−1.13422
$672$	0	0
$673$	−18898.0	−1.08241	−0.541207	−	0.840890i	$-0.682032\pi$
$673$	−18898.0	−1.08241	−0.541207	+	0.840890i	$0.682032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24948.0i	1.41629i	0.706066	+	0.708146i	$0.250468\pi$
$677$	24948.0i	1.41629i	−0.706066	+	0.708146i	$0.749532\pi$
$678$	0	0
$679$	− 21740.7i	− 1.22877i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21688.7	1.21508	0.607538	−	0.794291i	$-0.292157\pi$
$683$	21688.7	1.21508	0.607538	+	0.794291i	$0.292157\pi$
$684$	0	0
$685$	8670.00	0.483597
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19458.0i	1.07589i
$690$	0	0
$691$	5258.51i	0.289498i	0.989468	+	0.144749i	$0.0462374\pi$
$691$	5258.51i	0.289498i	−0.989468	+	0.144749i	$0.953763\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4364.77	−0.238223
$696$	0	0
$697$	7182.00	0.390298
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4413.00i	− 0.237770i	−0.992908	−	0.118885i	$-0.962068\pi$
$701$	− 4413.00i	− 0.237770i	0.992908	−	0.118885i	$-0.0379320\pi$
$702$	0	0
$703$	11099.0i	0.595457i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6921.28	−0.368177
$708$	0	0
$709$	−9830.00	−0.520696	−0.260348	−	0.965515i	$-0.583837\pi$
$709$	−9830.00	−0.520696	−0.260348	+	0.965515i	$0.583837\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 945.000i	− 0.0496361i
$714$	0	0
$715$	− 8547.67i	− 0.447084i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9675.24	−0.501843	−0.250922	−	0.968007i	$-0.580734\pi$
$719$	−9675.24	−0.501843	−0.250922	+	0.968007i	$0.580734\pi$
$720$	0	0
$721$	−23112.0	−1.19381
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3075.00i	0.157521i
$726$	0	0
$727$	− 31374.4i	− 1.60057i	−0.599623	−	0.800283i	$-0.704683\pi$
$727$	− 31374.4i	− 1.60057i	0.599623	−	0.800283i	$-0.295317\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4910.36	−0.248449
$732$	0	0
$733$	17350.0	0.874266	0.437133	−	0.899397i	$-0.355994\pi$
$733$	17350.0	0.874266	0.437133	+	0.899397i	$0.355994\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5670.00i	− 0.283388i
$738$	0	0
$739$	− 21553.6i	− 1.07289i	−0.843936	−	0.536443i	$-0.819767\pi$
$739$	− 21553.6i	− 1.07289i	0.843936	−	0.536443i	$-0.180233\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6386.07	0.315319	0.157660	−	0.987494i	$-0.449605\pi$
$743$	6386.07	0.315319	0.157660	+	0.987494i	$0.449605\pi$
$744$	0	0
$745$	7035.00	0.345963
$746$	0	0
$747$	0	0
$748$	0	0
$749$	28944.0i	1.41200i
$750$	0	0
$751$	− 24957.1i	− 1.21265i	−0.795218	−	0.606324i	$-0.792644\pi$
$751$	− 24957.1i	− 1.21265i	0.795218	−	0.606324i	$-0.207356\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1117.17	0.0538518
$756$	0	0
$757$	27133.0	1.30273	0.651364	−	0.758765i	$-0.274197\pi$
$757$	27133.0	1.30273	0.651364	+	0.758765i	$0.274197\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 30630.0i	− 1.45905i	−0.683954	−	0.729525i	$-0.739741\pi$
$761$	− 30630.0i	− 1.45905i	0.683954	−	0.729525i	$-0.260259\pi$
$762$	0	0
$763$	30636.5i	1.45363i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21002.8	−0.988747
$768$	0	0
$769$	−9727.00	−0.456131	−0.228065	−	0.973646i	$-0.573240\pi$
$769$	−9727.00	−0.456131	−0.228065	+	0.973646i	$0.573240\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 34260.0i	− 1.59411i	−0.603907	−	0.797055i	$-0.706390\pi$
$773$	− 34260.0i	− 1.59411i	0.603907	−	0.797055i	$-0.293610\pi$
$774$	0	0
$775$	− 649.519i	− 0.0301050i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−21325.0	−0.980806
$780$	0	0
$781$	−30996.0	−1.42013
$782$	0	0
$783$	0	0
$784$	0	0
$785$	16505.0i	0.750431i
$786$	0	0
$787$	5554.69i	0.251592i	0.992056	+	0.125796i	$0.0401485\pi$
$787$	5554.69i	0.251592i	−0.992056	+	0.125796i	$0.959851\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5300.08	−0.238241
$792$	0	0
$793$	−25474.0	−1.14074
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 10680.0i	− 0.474661i	−0.971429	−	0.237331i	$-0.923728\pi$
$797$	− 10680.0i	− 0.474661i	0.971429	−	0.237331i	$-0.0762725\pi$
$798$	0	0
$799$	− 6438.03i	− 0.285058i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8438.55	−0.370847
$804$	0	0
$805$	−3780.00	−0.165500
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4062.00i	− 0.176529i	−0.996097	−	0.0882647i	$-0.971868\pi$
$809$	− 4062.00i	− 0.176529i	0.996097	−	0.0882647i	$-0.0281321\pi$
$810$	0	0
$811$	34564.8i	1.49659i	0.663366	+	0.748295i	$0.269127\pi$
$811$	34564.8i	1.49659i	−0.663366	+	0.748295i	$0.730873\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	10574.2	0.454475
$816$	0	0
$817$	14580.0	0.624345
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35430.0i	1.50611i	0.657958	+	0.753055i	$0.271420\pi$
$821$	35430.0i	1.50611i	−0.657958	+	0.753055i	$0.728580\pi$
$822$	0	0
$823$	25107.8i	1.06343i	0.846923	+	0.531715i	$0.178452\pi$
$823$	25107.8i	1.06343i	−0.846923	+	0.531715i	$0.821548\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33650.3	1.41492	0.707458	−	0.706756i	$-0.249842\pi$
$827$	33650.3	1.41492	0.707458	+	0.706756i	$0.249842\pi$
$828$	0	0
$829$	−24032.0	−1.00683	−0.503417	−	0.864043i	$-0.667924\pi$
$829$	−24032.0	−1.00683	−0.503417	+	0.864043i	$0.667924\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1869.00i	− 0.0777395i
$834$	0	0
$835$	363.731i	0.0150748i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36580.9	−1.50526	−0.752630	−	0.658444i	$-0.771215\pi$
$839$	−36580.9	−1.50526	−0.752630	+	0.658444i	$0.771215\pi$
$840$	0	0
$841$	9260.00	0.379679
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 60.0000i	− 0.00244268i
$846$	0	0
$847$	− 166.277i	− 0.00674539i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6474.41	0.260799
$852$	0	0
$853$	18947.0	0.760531	0.380265	−	0.924877i	$-0.375833\pi$
$853$	18947.0	0.760531	0.380265	+	0.924877i	$0.375833\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23406.0i	0.932945i	0.884536	+	0.466472i	$0.154475\pi$
$857$	23406.0i	0.932945i	−0.884536	+	0.466472i	$0.845525\pi$
$858$	0	0
$859$	6391.27i	0.253862i	0.991912	+	0.126931i	$0.0405126\pi$
$859$	6391.27i	0.253862i	−0.991912	+	0.126931i	$0.959487\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16258.8	−0.641315	−0.320657	−	0.947195i	$-0.603904\pi$
$863$	−16258.8	−0.641315	−0.320657	+	0.947195i	$0.603904\pi$
$864$	0	0
$865$	−10560.0	−0.415088
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 12663.0i	− 0.494319i
$870$	0	0
$871$	− 7326.57i	− 0.285019i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2598.08	−0.100378
$876$	0	0
$877$	−6001.00	−0.231060	−0.115530	−	0.993304i	$-0.536857\pi$
$877$	−6001.00	−0.231060	−0.115530	+	0.993304i	$0.536857\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20214.0i	0.773016i	0.922286	+	0.386508i	$0.126319\pi$
$881$	20214.0i	0.773016i	−0.922286	+	0.386508i	$0.873681\pi$
$882$	0	0
$883$	26843.3i	1.02305i	0.859270	+	0.511523i	$0.170919\pi$
$883$	26843.3i	1.02305i	−0.859270	+	0.511523i	$0.829081\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40659.9	−1.53915	−0.769575	−	0.638557i	$-0.779532\pi$
$887$	−40659.9	−1.53915	−0.769575	+	0.638557i	$0.779532\pi$
$888$	0	0
$889$	30024.0	1.13270
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19116.0i	0.716341i
$894$	0	0
$895$	− 16575.7i	− 0.619068i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3195.63	0.118554
$900$	0	0
$901$	8694.00	0.321464
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3520.00i	0.129291i
$906$	0	0
$907$	− 3393.09i	− 0.124218i	−0.998069	−	0.0621089i	$-0.980217\pi$
$907$	− 3393.09i	− 0.124218i	0.998069	−	0.0621089i	$-0.0197826\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12668.2	−0.460721	−0.230361	−	0.973105i	$-0.573991\pi$
$911$	−12668.2	−0.460721	−0.230361	+	0.973105i	$0.573991\pi$
$912$	0	0
$913$	−14742.0	−0.534380
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3996.00i	− 0.143904i
$918$	0	0
$919$	− 12694.2i	− 0.455651i	−0.973702	−	0.227825i	$-0.926838\pi$
$919$	− 12694.2i	− 0.455651i	0.973702	−	0.227825i	$-0.0731615\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−40051.9	−1.42831
$924$	0	0
$925$	4450.00	0.158178
$926$	0	0
$927$	0	0
$928$	0	0
$929$	51366.0i	1.81406i	0.421064	+	0.907031i	$0.361657\pi$
$929$	51366.0i	1.81406i	−0.421064	+	0.907031i	$0.638343\pi$
$930$	0	0
$931$	5549.49i	0.195357i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3819.17	−0.133583
$936$	0	0
$937$	40700.0	1.41901	0.709504	−	0.704701i	$-0.248919\pi$
$937$	40700.0	1.41901	0.709504	+	0.704701i	$0.248919\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 41649.0i	− 1.44285i	−0.692494	−	0.721423i	$-0.743488\pi$
$941$	− 41649.0i	− 1.44285i	0.692494	−	0.721423i	$-0.256512\pi$
$942$	0	0
$943$	12439.6i	0.429574i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27092.7	−0.929668	−0.464834	−	0.885398i	$-0.653886\pi$
$947$	−27092.7	−0.929668	−0.464834	+	0.885398i	$0.653886\pi$
$948$	0	0
$949$	−10904.0	−0.372981
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25593.0i	0.869925i	0.900449	+	0.434963i	$0.143238\pi$
$953$	25593.0i	0.869925i	−0.900449	+	0.434963i	$0.856762\pi$
$954$	0	0
$955$	− 7170.69i	− 0.242972i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36040.5	−1.21357
$960$	0	0
$961$	29116.0	0.977342
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5840.00i	0.194815i
$966$	0	0
$967$	39511.5i	1.31397i	0.753905	+	0.656983i	$0.228168\pi$
$967$	39511.5i	1.31397i	−0.753905	+	0.656983i	$0.771832\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−32959.2	−1.08930	−0.544650	−	0.838663i	$-0.683338\pi$
$971$	−32959.2	−1.08930	−0.544650	+	0.838663i	$0.683338\pi$
$972$	0	0
$973$	18144.0	0.597811
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1269.00i	0.0415547i	0.999784	+	0.0207773i	$0.00661411\pi$
$977$	1269.00i	0.0415547i	−0.999784	+	0.0207773i	$0.993386\pi$
$978$	0	0
$979$	− 49321.9i	− 1.61015i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−30174.1	−0.979047	−0.489523	−	0.871990i	$-0.662829\pi$
$983$	−30174.1	−0.979047	−0.489523	+	0.871990i	$0.662829\pi$
$984$	0	0
$985$	−2280.00	−0.0737531
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 8505.00i	− 0.273451i
$990$	0	0
$991$	52839.7i	1.69375i	0.531791	+	0.846876i	$0.321519\pi$
$991$	52839.7i	1.69375i	−0.531791	+	0.846876i	$0.678481\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3715.25	−0.118373
$996$	0	0
$997$	−28051.0	−0.891057	−0.445529	−	0.895268i	$-0.646984\pi$
$997$	−28051.0	−0.891057	−0.445529	+	0.895268i	$0.646984\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.h.a.431.2	yes	4
3.2	odd	2	inner	2160.4.h.a.431.4	yes	4
4.3	odd	2	inner	2160.4.h.a.431.1	✓	4
12.11	even	2	inner	2160.4.h.a.431.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.4.h.a.431.1	✓	4	4.3	odd	2	inner
2160.4.h.a.431.2	yes	4	1.1	even	1	trivial
2160.4.h.a.431.3	yes	4	12.11	even	2	inner
2160.4.h.a.431.4	yes	4	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.00000i q^{5} +20.7846i q^{7} -36.3731 q^{11} -47.0000 q^{13} +21.0000i q^{17} -62.3538i q^{19} -36.3731 q^{23} -25.0000 q^{25} -123.000i q^{29} +25.9808i q^{31} +103.923 q^{35} -178.000 q^{37} -342.000i q^{41} +233.827i q^{43} -306.573 q^{47} -89.0000 q^{49} -414.000i q^{53} +181.865i q^{55} +446.869 q^{59} +542.000 q^{61} +235.000i q^{65} +155.885i q^{67} +852.169 q^{71} +232.000 q^{73} -756.000i q^{77} +348.142i q^{79} +405.300 q^{83} +105.000 q^{85} +1356.00i q^{89} -976.877i q^{91} -311.769 q^{95} -1046.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 188 q^{13} - 100 q^{25} - 712 q^{37} - 356 q^{49} + 2168 q^{61} + 928 q^{73} + 420 q^{85} - 4184 q^{97}+O(q^{100}) \) 4 * q - 188 * q^13 - 100 * q^25 - 712 * q^37 - 356 * q^49 + 2168 * q^61 + 928 * q^73 + 420 * q^85 - 4184 * q^97

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