LMFDB - Embedded newform 2160.4.a.bj.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,4,Mod(1,2160)] 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("2160.1"); 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([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-15,0,8,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$127.444125612$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.4281.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 15 $ x^3 - x^2 - 12*x + 15
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.26757$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} -10.2514 q^{7} -44.3754 q^{11} -50.8782 q^{13} -25.2514 q^{17} +31.3754 q^{19} +76.1296 q^{23} +25.0000 q^{25} -156.008 q^{29} -134.505 q^{31} +51.2571 q^{35} +81.2423 q^{37} -326.256 q^{41} -422.526 q^{43} -452.635 q^{47} -237.909 q^{49} -98.1105 q^{53} +221.877 q^{55} +540.118 q^{59} +522.292 q^{61} +254.391 q^{65} +129.272 q^{67} +26.6034 q^{71} -147.873 q^{73} +454.910 q^{77} +1088.91 q^{79} +594.611 q^{83} +126.257 q^{85} -592.229 q^{89} +521.573 q^{91} -156.877 q^{95} -666.786 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} + 8 q^{7} - 10 q^{11} + 48 q^{13} - 37 q^{17} - 29 q^{19} - 11 q^{23} + 75 q^{25} - 28 q^{29} - 41 q^{31} - 40 q^{35} + 230 q^{37} - 370 q^{41} + 130 q^{43} - 56 q^{47} + 547 q^{49} - 805 q^{53}+ \cdots + 576 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 8 * q^7 - 10 * q^11 + 48 * q^13 - 37 * q^17 - 29 * q^19 - 11 * q^23 + 75 * q^25 - 28 * q^29 - 41 * q^31 - 40 * q^35 + 230 * q^37 - 370 * q^41 + 130 * q^43 - 56 * q^47 + 547 * q^49 - 805 * q^53 + 50 * q^55 + 576 * q^59 - 257 * q^61 - 240 * q^65 + 14 * q^67 + 1238 * q^71 - 398 * q^73 - 1296 * q^77 + 321 * q^79 + 687 * q^83 + 185 * q^85 - 2358 * q^89 + 1968 * q^91 + 145 * q^95 + 576 * q^97

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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−10.2514	−0.553524	−0.276762	−	0.960938i	$-0.589261\pi$
$7$	−10.2514	−0.553524	−0.276762	+	0.960938i	$0.589261\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−44.3754	−1.21633	−0.608167	−	0.793809i	$-0.708095\pi$
$11$	−44.3754	−1.21633	−0.608167	+	0.793809i	$0.708095\pi$
$12$	0	0
$13$	−50.8782	−1.08547	−0.542733	−	0.839905i	$-0.682611\pi$
$13$	−50.8782	−1.08547	−0.542733	+	0.839905i	$0.682611\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−25.2514	−0.360257	−0.180128	−	0.983643i	$-0.557651\pi$
$17$	−25.2514	−0.360257	−0.180128	+	0.983643i	$0.557651\pi$
$18$	0	0
$19$	31.3754	0.378842	0.189421	−	0.981896i	$-0.439339\pi$
$19$	31.3754	0.378842	0.189421	+	0.981896i	$0.439339\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	76.1296	0.690179	0.345090	−	0.938570i	$-0.387849\pi$
$23$	76.1296	0.690179	0.345090	+	0.938570i	$0.387849\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−156.008	−0.998963	−0.499481	−	0.866325i	$-0.666476\pi$
$29$	−156.008	−0.998963	−0.499481	+	0.866325i	$0.666476\pi$
$30$	0	0
$31$	−134.505	−0.779284	−0.389642	−	0.920966i	$-0.627401\pi$
$31$	−134.505	−0.779284	−0.389642	+	0.920966i	$0.627401\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	51.2571	0.247544
$36$	0	0
$37$	81.2423	0.360977	0.180488	−	0.983577i	$-0.442232\pi$
$37$	81.2423	0.360977	0.180488	+	0.983577i	$0.442232\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−326.256	−1.24275	−0.621373	−	0.783515i	$-0.713425\pi$
$41$	−326.256	−1.24275	−0.621373	+	0.783515i	$0.713425\pi$
$42$	0	0
$43$	−422.526	−1.49848	−0.749240	−	0.662299i	$-0.769581\pi$
$43$	−422.526	−1.49848	−0.749240	+	0.662299i	$0.769581\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−452.635	−1.40476	−0.702379	−	0.711803i	$-0.747879\pi$
$47$	−452.635	−1.40476	−0.702379	+	0.711803i	$0.747879\pi$
$48$	0	0
$49$	−237.909	−0.693611
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−98.1105	−0.254274	−0.127137	−	0.991885i	$-0.540579\pi$
$53$	−98.1105	−0.254274	−0.127137	+	0.991885i	$0.540579\pi$
$54$	0	0
$55$	221.877	0.543961
$56$	0	0
$57$	0	0
$58$	0	0
$59$	540.118	1.19182	0.595910	−	0.803051i	$-0.296792\pi$
$59$	540.118	1.19182	0.595910	+	0.803051i	$0.296792\pi$
$60$	0	0
$61$	522.292	1.09627	0.548136	−	0.836389i	$-0.315338\pi$
$61$	522.292	1.09627	0.548136	+	0.836389i	$0.315338\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	254.391	0.485436
$66$	0	0
$67$	129.272	0.235717	0.117859	−	0.993030i	$-0.462397\pi$
$67$	129.272	0.235717	0.117859	+	0.993030i	$0.462397\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	26.6034	0.0444682	0.0222341	−	0.999753i	$-0.492922\pi$
$71$	26.6034	0.0444682	0.0222341	+	0.999753i	$0.492922\pi$
$72$	0	0
$73$	−147.873	−0.237085	−0.118542	−	0.992949i	$-0.537822\pi$
$73$	−147.873	−0.237085	−0.118542	+	0.992949i	$0.537822\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	454.910	0.673270
$78$	0	0
$79$	1088.91	1.55079	0.775394	−	0.631478i	$-0.217551\pi$
$79$	1088.91	1.55079	0.775394	+	0.631478i	$0.217551\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	594.611	0.786350	0.393175	−	0.919464i	$-0.371377\pi$
$83$	594.611	0.786350	0.393175	+	0.919464i	$0.371377\pi$
$84$	0	0
$85$	126.257	0.161112
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−592.229	−0.705350	−0.352675	−	0.935746i	$-0.614728\pi$
$89$	−592.229	−0.705350	−0.352675	+	0.935746i	$0.614728\pi$
$90$	0	0
$91$	521.573	0.600832
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−156.877	−0.169423
$96$	0	0
$97$	−666.786	−0.697958	−0.348979	−	0.937131i	$-0.613471\pi$
$97$	−666.786	−0.697958	−0.348979	+	0.937131i	$0.613471\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1218.56	−1.20051	−0.600254	−	0.799809i	$-0.704934\pi$
$101$	−1218.56	−1.20051	−0.600254	+	0.799809i	$0.704934\pi$
$102$	0	0
$103$	231.789	0.221736	0.110868	−	0.993835i	$-0.464637\pi$
$103$	231.789	0.221736	0.110868	+	0.993835i	$0.464637\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1979.96	1.78888	0.894439	−	0.447189i	$-0.147575\pi$
$107$	1979.96	1.78888	0.894439	+	0.447189i	$0.147575\pi$
$108$	0	0
$109$	1075.41	0.945005	0.472502	−	0.881329i	$-0.343351\pi$
$109$	1075.41	0.945005	0.472502	+	0.881329i	$0.343351\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−693.410	−0.577261	−0.288631	−	0.957441i	$-0.593200\pi$
$113$	−693.410	−0.577261	−0.288631	+	0.957441i	$0.593200\pi$
$114$	0	0
$115$	−380.648	−0.308657
$116$	0	0
$117$	0	0
$118$	0	0
$119$	258.863	0.199411
$120$	0	0
$121$	638.173	0.479469
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	833.579	0.582427	0.291213	−	0.956658i	$-0.405941\pi$
$127$	833.579	0.582427	0.291213	+	0.956658i	$0.405941\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	353.721	0.235914	0.117957	−	0.993019i	$-0.462365\pi$
$131$	353.721	0.235914	0.117957	+	0.993019i	$0.462365\pi$
$132$	0	0
$133$	−321.642	−0.209698
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1509.90	−0.941601	−0.470800	−	0.882240i	$-0.656035\pi$
$137$	−1509.90	−0.941601	−0.470800	+	0.882240i	$0.656035\pi$
$138$	0	0
$139$	618.485	0.377404	0.188702	−	0.982034i	$-0.439572\pi$
$139$	618.485	0.377404	0.188702	+	0.982034i	$0.439572\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2257.74	1.32029
$144$	0	0
$145$	780.039	0.446750
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2405.41	−1.32254	−0.661271	−	0.750147i	$-0.729983\pi$
$149$	−2405.41	−1.32254	−0.661271	+	0.750147i	$0.729983\pi$
$150$	0	0
$151$	−1601.13	−0.862900	−0.431450	−	0.902137i	$-0.641998\pi$
$151$	−1601.13	−0.862900	−0.431450	+	0.902137i	$0.641998\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	672.525	0.348506
$156$	0	0
$157$	3208.39	1.63094	0.815470	−	0.578799i	$-0.196479\pi$
$157$	3208.39	1.63094	0.815470	+	0.578799i	$0.196479\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−780.436	−0.382031
$162$	0	0
$163$	392.481	0.188598	0.0942991	−	0.995544i	$-0.469939\pi$
$163$	392.481	0.188598	0.0942991	+	0.995544i	$0.469939\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−260.818	−0.120854	−0.0604272	−	0.998173i	$-0.519246\pi$
$167$	−260.818	−0.120854	−0.0604272	+	0.998173i	$0.519246\pi$
$168$	0	0
$169$	391.590	0.178239
$170$	0	0
$171$	0	0
$172$	0	0
$173$	204.870	0.0900346	0.0450173	−	0.998986i	$-0.485666\pi$
$173$	204.870	0.0900346	0.0450173	+	0.998986i	$0.485666\pi$
$174$	0	0
$175$	−256.285	−0.110705
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1836.92	−0.767029	−0.383514	−	0.923535i	$-0.625286\pi$
$179$	−1836.92	−0.767029	−0.383514	+	0.923535i	$0.625286\pi$
$180$	0	0
$181$	−2880.67	−1.18298	−0.591488	−	0.806314i	$-0.701459\pi$
$181$	−2880.67	−1.18298	−0.591488	+	0.806314i	$0.701459\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−406.211	−0.161434
$186$	0	0
$187$	1120.54	0.438193
$188$	0	0
$189$	0	0
$190$	0	0
$191$	748.242	0.283460	0.141730	−	0.989905i	$-0.454733\pi$
$191$	748.242	0.283460	0.141730	+	0.989905i	$0.454733\pi$
$192$	0	0
$193$	621.620	0.231840	0.115920	−	0.993259i	$-0.463018\pi$
$193$	621.620	0.231840	0.115920	+	0.993259i	$0.463018\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−588.243	−0.212744	−0.106372	−	0.994326i	$-0.533923\pi$
$197$	−588.243	−0.212744	−0.106372	+	0.994326i	$0.533923\pi$
$198$	0	0
$199$	4015.31	1.43034	0.715171	−	0.698950i	$-0.246349\pi$
$199$	4015.31	1.43034	0.715171	+	0.698950i	$0.246349\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1599.30	0.552950
$204$	0	0
$205$	1631.28	0.555773
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1392.29	−0.460799
$210$	0	0
$211$	640.632	0.209019	0.104509	−	0.994524i	$-0.466673\pi$
$211$	640.632	0.209019	0.104509	+	0.994524i	$0.466673\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2112.63	0.670141
$216$	0	0
$217$	1378.87	0.431353
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1284.75	0.391047
$222$	0	0
$223$	4147.71	1.24552	0.622761	−	0.782412i	$-0.286011\pi$
$223$	4147.71	1.24552	0.622761	+	0.782412i	$0.286011\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5012.97	1.46574	0.732869	−	0.680370i	$-0.238181\pi$
$227$	5012.97	1.46574	0.732869	+	0.680370i	$0.238181\pi$
$228$	0	0
$229$	6763.07	1.95160	0.975800	−	0.218664i	$-0.0701700\pi$
$229$	6763.07	1.95160	0.975800	+	0.218664i	$0.0701700\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1995.20	0.560987	0.280493	−	0.959856i	$-0.409502\pi$
$233$	1995.20	0.560987	0.280493	+	0.959856i	$0.409502\pi$
$234$	0	0
$235$	2263.18	0.628227
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1681.95	0.455213	0.227607	−	0.973753i	$-0.426910\pi$
$239$	1681.95	0.455213	0.227607	+	0.973753i	$0.426910\pi$
$240$	0	0
$241$	3573.58	0.955164	0.477582	−	0.878587i	$-0.341513\pi$
$241$	3573.58	0.955164	0.477582	+	0.878587i	$0.341513\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1189.54	0.310192
$246$	0	0
$247$	−1596.32	−0.411221
$248$	0	0
$249$	0	0
$250$	0	0
$251$	808.137	0.203224	0.101612	−	0.994824i	$-0.467600\pi$
$251$	808.137	0.203224	0.101612	+	0.994824i	$0.467600\pi$
$252$	0	0
$253$	−3378.28	−0.839488
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4824.76	−1.17105	−0.585526	−	0.810654i	$-0.699112\pi$
$257$	−4824.76	−1.17105	−0.585526	+	0.810654i	$0.699112\pi$
$258$	0	0
$259$	−832.848	−0.199809
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4252.92	−0.997134	−0.498567	−	0.866851i	$-0.666140\pi$
$263$	−4252.92	−0.997134	−0.498567	+	0.866851i	$0.666140\pi$
$264$	0	0
$265$	490.553	0.113715
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2198.92	0.498403	0.249202	−	0.968452i	$-0.419832\pi$
$269$	2198.92	0.498403	0.249202	+	0.968452i	$0.419832\pi$
$270$	0	0
$271$	4904.13	1.09928	0.549639	−	0.835402i	$-0.314765\pi$
$271$	4904.13	1.09928	0.549639	+	0.835402i	$0.314765\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1109.38	−0.243267
$276$	0	0
$277$	933.019	0.202382	0.101191	−	0.994867i	$-0.467735\pi$
$277$	933.019	0.202382	0.101191	+	0.994867i	$0.467735\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4412.79	−0.936814	−0.468407	−	0.883513i	$-0.655172\pi$
$281$	−4412.79	−0.936814	−0.468407	+	0.883513i	$0.655172\pi$
$282$	0	0
$283$	−748.931	−0.157312	−0.0786561	−	0.996902i	$-0.525063\pi$
$283$	−748.931	−0.157312	−0.0786561	+	0.996902i	$0.525063\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3344.58	0.687890
$288$	0	0
$289$	−4275.37	−0.870215
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1706.45	−0.340246	−0.170123	−	0.985423i	$-0.554416\pi$
$293$	−1706.45	−0.340246	−0.170123	+	0.985423i	$0.554416\pi$
$294$	0	0
$295$	−2700.59	−0.532998
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3873.34	−0.749167
$300$	0	0
$301$	4331.49	0.829445
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2611.46	−0.490268
$306$	0	0
$307$	−5773.97	−1.07341	−0.536706	−	0.843769i	$-0.680332\pi$
$307$	−5773.97	−1.07341	−0.536706	+	0.843769i	$0.680332\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3777.25	0.688709	0.344354	−	0.938840i	$-0.388098\pi$
$311$	3777.25	0.688709	0.344354	+	0.938840i	$0.388098\pi$
$312$	0	0
$313$	6966.88	1.25812	0.629060	−	0.777357i	$-0.283440\pi$
$313$	6966.88	1.25812	0.629060	+	0.777357i	$0.283440\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8482.29	1.50288	0.751440	−	0.659801i	$-0.229360\pi$
$317$	8482.29	1.50288	0.751440	+	0.659801i	$0.229360\pi$
$318$	0	0
$319$	6922.90	1.21507
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−792.272	−0.136481
$324$	0	0
$325$	−1271.95	−0.217093
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4640.15	0.777568
$330$	0	0
$331$	7673.68	1.27427	0.637136	−	0.770752i	$-0.280119\pi$
$331$	7673.68	1.27427	0.637136	+	0.770752i	$0.280119\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−646.359	−0.105416
$336$	0	0
$337$	7135.56	1.15341	0.576704	−	0.816953i	$-0.304339\pi$
$337$	7135.56	1.15341	0.576704	+	0.816953i	$0.304339\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5968.71	0.947870
$342$	0	0
$343$	5955.13	0.937455
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6809.59	−1.05348	−0.526741	−	0.850026i	$-0.676586\pi$
$347$	−6809.59	−1.05348	−0.526741	+	0.850026i	$0.676586\pi$
$348$	0	0
$349$	−10666.0	−1.63593	−0.817963	−	0.575271i	$-0.804897\pi$
$349$	−10666.0	−1.63593	−0.817963	+	0.575271i	$0.804897\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6399.50	−0.964904	−0.482452	−	0.875922i	$-0.660254\pi$
$353$	−6399.50	−0.964904	−0.482452	+	0.875922i	$0.660254\pi$
$354$	0	0
$355$	−133.017	−0.0198868
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1340.16	−0.197022	−0.0985112	−	0.995136i	$-0.531408\pi$
$359$	−1340.16	−0.197022	−0.0985112	+	0.995136i	$0.531408\pi$
$360$	0	0
$361$	−5874.59	−0.856479
$362$	0	0
$363$	0	0
$364$	0	0
$365$	739.363	0.106027
$366$	0	0
$367$	−2856.96	−0.406354	−0.203177	−	0.979142i	$-0.565127\pi$
$367$	−2856.96	−0.406354	−0.203177	+	0.979142i	$0.565127\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1005.77	0.140747
$372$	0	0
$373$	9057.97	1.25738	0.628691	−	0.777655i	$-0.283591\pi$
$373$	9057.97	1.25738	0.628691	+	0.777655i	$0.283591\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7937.39	1.08434
$378$	0	0
$379$	10770.0	1.45968	0.729839	−	0.683619i	$-0.239595\pi$
$379$	10770.0	1.45968	0.729839	+	0.683619i	$0.239595\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1249.75	0.166734	0.0833671	−	0.996519i	$-0.473433\pi$
$383$	1249.75	0.166734	0.0833671	+	0.996519i	$0.473433\pi$
$384$	0	0
$385$	−2274.55	−0.301096
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14409.5	−1.87812	−0.939060	−	0.343753i	$-0.888302\pi$
$389$	−14409.5	−1.87812	−0.939060	+	0.343753i	$0.888302\pi$
$390$	0	0
$391$	−1922.38	−0.248642
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5444.56	−0.693533
$396$	0	0
$397$	8504.39	1.07512	0.537561	−	0.843225i	$-0.319346\pi$
$397$	8504.39	1.07512	0.537561	+	0.843225i	$0.319346\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11812.6	1.47106	0.735528	−	0.677494i	$-0.236934\pi$
$401$	11812.6	1.47106	0.735528	+	0.677494i	$0.236934\pi$
$402$	0	0
$403$	6843.37	0.845887
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3605.16	−0.439069
$408$	0	0
$409$	−3009.71	−0.363864	−0.181932	−	0.983311i	$-0.558235\pi$
$409$	−3009.71	−0.363864	−0.181932	+	0.983311i	$0.558235\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5536.97	−0.659701
$414$	0	0
$415$	−2973.06	−0.351667
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1354.28	−0.157902	−0.0789510	−	0.996878i	$-0.525157\pi$
$419$	−1354.28	−0.157902	−0.0789510	+	0.996878i	$0.525157\pi$
$420$	0	0
$421$	14411.5	1.66835	0.834175	−	0.551500i	$-0.185944\pi$
$421$	14411.5	1.66835	0.834175	+	0.551500i	$0.185944\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−631.285	−0.0720514
$426$	0	0
$427$	−5354.23	−0.606813
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6762.24	−0.755744	−0.377872	−	0.925858i	$-0.623344\pi$
$431$	−6762.24	−0.755744	−0.377872	+	0.925858i	$0.623344\pi$
$432$	0	0
$433$	−3323.67	−0.368881	−0.184440	−	0.982844i	$-0.559047\pi$
$433$	−3323.67	−0.368881	−0.184440	+	0.982844i	$0.559047\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2388.59	0.261469
$438$	0	0
$439$	−4649.90	−0.505530	−0.252765	−	0.967528i	$-0.581340\pi$
$439$	−4649.90	−0.505530	−0.252765	+	0.967528i	$0.581340\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14434.0	−1.54804	−0.774019	−	0.633162i	$-0.781757\pi$
$443$	−14434.0	−1.54804	−0.774019	+	0.633162i	$0.781757\pi$
$444$	0	0
$445$	2961.14	0.315442
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3428.29	−0.360336	−0.180168	−	0.983636i	$-0.557664\pi$
$449$	−3428.29	−0.360336	−0.180168	+	0.983636i	$0.557664\pi$
$450$	0	0
$451$	14477.7	1.51159
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2607.87	−0.268700
$456$	0	0
$457$	18647.8	1.90877	0.954384	−	0.298583i	$-0.0965141\pi$
$457$	18647.8	1.90877	0.954384	+	0.298583i	$0.0965141\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−144.345	−0.0145831	−0.00729157	−	0.999973i	$-0.502321\pi$
$461$	−144.345	−0.0145831	−0.00729157	+	0.999973i	$0.502321\pi$
$462$	0	0
$463$	−3418.69	−0.343154	−0.171577	−	0.985171i	$-0.554886\pi$
$463$	−3418.69	−0.343154	−0.171577	+	0.985171i	$0.554886\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9571.86	−0.948465	−0.474232	−	0.880400i	$-0.657274\pi$
$467$	−9571.86	−0.948465	−0.474232	+	0.880400i	$0.657274\pi$
$468$	0	0
$469$	−1325.22	−0.130475
$470$	0	0
$471$	0	0
$472$	0	0
$473$	18749.8	1.82265
$474$	0	0
$475$	784.384	0.0757685
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17144.0	−1.63534	−0.817670	−	0.575687i	$-0.804735\pi$
$479$	−17144.0	−1.63534	−0.817670	+	0.575687i	$0.804735\pi$
$480$	0	0
$481$	−4133.46	−0.391829
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3333.93	0.312136
$486$	0	0
$487$	16588.4	1.54351	0.771757	−	0.635917i	$-0.219378\pi$
$487$	16588.4	1.54351	0.771757	+	0.635917i	$0.219378\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5835.61	−0.536369	−0.268185	−	0.963368i	$-0.586424\pi$
$491$	−5835.61	−0.536369	−0.268185	+	0.963368i	$0.586424\pi$
$492$	0	0
$493$	3939.42	0.359883
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−272.722	−0.0246142
$498$	0	0
$499$	−9028.41	−0.809954	−0.404977	−	0.914327i	$-0.632720\pi$
$499$	−9028.41	−0.809954	−0.404977	+	0.914327i	$0.632720\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8169.36	0.724163	0.362081	−	0.932147i	$-0.382066\pi$
$503$	8169.36	0.724163	0.362081	+	0.932147i	$0.382066\pi$
$504$	0	0
$505$	6092.80	0.536884
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7955.70	0.692791	0.346395	−	0.938089i	$-0.387406\pi$
$509$	7955.70	0.692791	0.346395	+	0.938089i	$0.387406\pi$
$510$	0	0
$511$	1515.90	0.131232
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1158.94	−0.0991634
$516$	0	0
$517$	20085.9	1.70866
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5276.52	0.443701	0.221851	−	0.975081i	$-0.428790\pi$
$521$	5276.52	0.443701	0.221851	+	0.975081i	$0.428790\pi$
$522$	0	0
$523$	17227.2	1.44033	0.720167	−	0.693801i	$-0.244065\pi$
$523$	17227.2	1.44033	0.720167	+	0.693801i	$0.244065\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3396.44	0.280742
$528$	0	0
$529$	−6371.28	−0.523653
$530$	0	0
$531$	0	0
$532$	0	0
$533$	16599.3	1.34896
$534$	0	0
$535$	−9899.80	−0.800011
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10557.3	0.843663
$540$	0	0
$541$	4830.69	0.383895	0.191948	−	0.981405i	$-0.438520\pi$
$541$	4830.69	0.383895	0.191948	+	0.981405i	$0.438520\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5377.04	−0.422619
$546$	0	0
$547$	−16766.0	−1.31053	−0.655267	−	0.755398i	$-0.727444\pi$
$547$	−16766.0	−1.31053	−0.655267	+	0.755398i	$0.727444\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4894.80	−0.378449
$552$	0	0
$553$	−11162.9	−0.858398
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17819.2	1.35552	0.677758	−	0.735285i	$-0.262952\pi$
$557$	17819.2	1.35552	0.677758	+	0.735285i	$0.262952\pi$
$558$	0	0
$559$	21497.4	1.62655
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13031.8	0.975534	0.487767	−	0.872974i	$-0.337811\pi$
$563$	13031.8	0.975534	0.487767	+	0.872974i	$0.337811\pi$
$564$	0	0
$565$	3467.05	0.258159
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7591.83	0.559342	0.279671	−	0.960096i	$-0.409775\pi$
$569$	7591.83	0.559342	0.279671	+	0.960096i	$0.409775\pi$
$570$	0	0
$571$	−9344.83	−0.684884	−0.342442	−	0.939539i	$-0.611254\pi$
$571$	−9344.83	−0.684884	−0.342442	+	0.939539i	$0.611254\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1903.24	0.138036
$576$	0	0
$577$	−20553.5	−1.48294	−0.741469	−	0.670987i	$-0.765870\pi$
$577$	−20553.5	−1.48294	−0.741469	+	0.670987i	$0.765870\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6095.61	−0.435264
$582$	0	0
$583$	4353.69	0.309282
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4842.95	−0.340528	−0.170264	−	0.985398i	$-0.554462\pi$
$587$	−4842.95	−0.340528	−0.170264	+	0.985398i	$0.554462\pi$
$588$	0	0
$589$	−4220.14	−0.295226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15403.8	1.06671	0.533355	−	0.845891i	$-0.320931\pi$
$593$	15403.8	1.06671	0.533355	+	0.845891i	$0.320931\pi$
$594$	0	0
$595$	−1294.31	−0.0891793
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3971.66	0.270914	0.135457	−	0.990783i	$-0.456750\pi$
$599$	3971.66	0.270914	0.135457	+	0.990783i	$0.456750\pi$
$600$	0	0
$601$	−20636.8	−1.40065	−0.700326	−	0.713824i	$-0.746962\pi$
$601$	−20636.8	−1.40065	−0.700326	+	0.713824i	$0.746962\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3190.87	−0.214425
$606$	0	0
$607$	−18039.1	−1.20624	−0.603118	−	0.797652i	$-0.706075\pi$
$607$	−18039.1	−1.20624	−0.603118	+	0.797652i	$0.706075\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	23029.3	1.52482
$612$	0	0
$613$	−14654.4	−0.965559	−0.482779	−	0.875742i	$-0.660373\pi$
$613$	−14654.4	−0.965559	−0.482779	+	0.875742i	$0.660373\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19933.8	−1.30066	−0.650329	−	0.759653i	$-0.725369\pi$
$617$	−19933.8	−1.30066	−0.650329	+	0.759653i	$0.725369\pi$
$618$	0	0
$619$	10548.9	0.684969	0.342485	−	0.939523i	$-0.388732\pi$
$619$	10548.9	0.684969	0.342485	+	0.939523i	$0.388732\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6071.18	0.390428
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2051.48	−0.130044
$630$	0	0
$631$	−23088.8	−1.45666	−0.728330	−	0.685227i	$-0.759703\pi$
$631$	−23088.8	−1.45666	−0.728330	+	0.685227i	$0.759703\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4167.89	−0.260469
$636$	0	0
$637$	12104.4	0.752892
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−711.767	−0.0438582	−0.0219291	−	0.999760i	$-0.506981\pi$
$641$	−711.767	−0.0438582	−0.0219291	+	0.999760i	$0.506981\pi$
$642$	0	0
$643$	6867.64	0.421203	0.210601	−	0.977572i	$-0.432458\pi$
$643$	6867.64	0.421203	0.210601	+	0.977572i	$0.432458\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1122.52	0.0682086	0.0341043	−	0.999418i	$-0.489142\pi$
$647$	1122.52	0.0682086	0.0341043	+	0.999418i	$0.489142\pi$
$648$	0	0
$649$	−23967.9	−1.44965
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5153.57	0.308843	0.154422	−	0.988005i	$-0.450649\pi$
$653$	5153.57	0.308843	0.154422	+	0.988005i	$0.450649\pi$
$654$	0	0
$655$	−1768.60	−0.105504
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28258.6	−1.67041	−0.835205	−	0.549939i	$-0.814651\pi$
$659$	−28258.6	−1.67041	−0.835205	+	0.549939i	$0.814651\pi$
$660$	0	0
$661$	2401.89	0.141335	0.0706676	−	0.997500i	$-0.477487\pi$
$661$	2401.89	0.141335	0.0706676	+	0.997500i	$0.477487\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1608.21	0.0937800
$666$	0	0
$667$	−11876.8	−0.689463
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−23176.9	−1.33343
$672$	0	0
$673$	−21090.9	−1.20801	−0.604007	−	0.796979i	$-0.706430\pi$
$673$	−21090.9	−1.20801	−0.604007	+	0.796979i	$0.706430\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9098.10	−0.516497	−0.258249	−	0.966079i	$-0.583145\pi$
$677$	−9098.10	−0.516497	−0.258249	+	0.966079i	$0.583145\pi$
$678$	0	0
$679$	6835.50	0.386336
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9488.98	0.531604	0.265802	−	0.964028i	$-0.414363\pi$
$683$	9488.98	0.531604	0.265802	+	0.964028i	$0.414363\pi$
$684$	0	0
$685$	7549.49	0.421097
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4991.69	0.276006
$690$	0	0
$691$	21798.7	1.20009	0.600043	−	0.799967i	$-0.295150\pi$
$691$	21798.7	1.20009	0.600043	+	0.799967i	$0.295150\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3092.42	−0.168780
$696$	0	0
$697$	8238.42	0.447708
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13799.3	0.743499	0.371750	−	0.928333i	$-0.378758\pi$
$701$	13799.3	0.743499	0.371750	+	0.928333i	$0.378758\pi$
$702$	0	0
$703$	2549.01	0.136753
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12492.0	0.664510
$708$	0	0
$709$	1561.32	0.0827030	0.0413515	−	0.999145i	$-0.486834\pi$
$709$	1561.32	0.0827030	0.0413515	+	0.999145i	$0.486834\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10239.8	−0.537846
$714$	0	0
$715$	−11288.7	−0.590452
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−14008.2	−0.726590	−0.363295	−	0.931674i	$-0.618348\pi$
$719$	−14008.2	−0.726590	−0.363295	+	0.931674i	$0.618348\pi$
$720$	0	0
$721$	−2376.16	−0.122736
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3900.19	−0.199793
$726$	0	0
$727$	−20977.4	−1.07016	−0.535081	−	0.844801i	$-0.679719\pi$
$727$	−20977.4	−1.07016	−0.535081	+	0.844801i	$0.679719\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	10669.4	0.539838
$732$	0	0
$733$	20802.8	1.04825	0.524126	−	0.851641i	$-0.324392\pi$
$733$	20802.8	1.04825	0.524126	+	0.851641i	$0.324392\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5736.49	−0.286711
$738$	0	0
$739$	−3111.47	−0.154881	−0.0774407	−	0.996997i	$-0.524675\pi$
$739$	−3111.47	−0.154881	−0.0774407	+	0.996997i	$0.524675\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5449.19	0.269060	0.134530	−	0.990910i	$-0.457048\pi$
$743$	5449.19	0.269060	0.134530	+	0.990910i	$0.457048\pi$
$744$	0	0
$745$	12027.0	0.591459
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20297.4	−0.990188
$750$	0	0
$751$	−28975.5	−1.40790	−0.703949	−	0.710251i	$-0.748581\pi$
$751$	−28975.5	−1.40790	−0.703949	+	0.710251i	$0.748581\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8005.64	0.385901
$756$	0	0
$757$	20654.2	0.991666	0.495833	−	0.868418i	$-0.334863\pi$
$757$	20654.2	0.991666	0.495833	+	0.868418i	$0.334863\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13619.4	0.648757	0.324378	−	0.945927i	$-0.394845\pi$
$761$	13619.4	0.648757	0.324378	+	0.945927i	$0.394845\pi$
$762$	0	0
$763$	−11024.5	−0.523083
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−27480.2	−1.29368
$768$	0	0
$769$	−15267.8	−0.715959	−0.357980	−	0.933729i	$-0.616534\pi$
$769$	−15267.8	−0.715959	−0.357980	+	0.933729i	$0.616534\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16108.3	0.749514	0.374757	−	0.927123i	$-0.377726\pi$
$773$	16108.3	0.749514	0.374757	+	0.927123i	$0.377726\pi$
$774$	0	0
$775$	−3362.62	−0.155857
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10236.4	−0.470805
$780$	0	0
$781$	−1180.54	−0.0540882
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16042.0	−0.729379
$786$	0	0
$787$	−25511.0	−1.15549	−0.577743	−	0.816219i	$-0.696067\pi$
$787$	−25511.0	−1.15549	−0.577743	+	0.816219i	$0.696067\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7108.43	0.319528
$792$	0	0
$793$	−26573.3	−1.18997
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26626.1	−1.18337	−0.591685	−	0.806169i	$-0.701537\pi$
$797$	−26626.1	−1.18337	−0.591685	+	0.806169i	$0.701537\pi$
$798$	0	0
$799$	11429.7	0.506074
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6561.90	0.288374
$804$	0	0
$805$	3902.18	0.170849
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18935.5	0.822911	0.411456	−	0.911430i	$-0.365020\pi$
$809$	18935.5	0.822911	0.411456	+	0.911430i	$0.365020\pi$
$810$	0	0
$811$	18366.2	0.795220	0.397610	−	0.917554i	$-0.369840\pi$
$811$	18366.2	0.795220	0.397610	+	0.917554i	$0.369840\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1962.41	−0.0843437
$816$	0	0
$817$	−13256.9	−0.567688
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−32495.2	−1.38135	−0.690675	−	0.723165i	$-0.742687\pi$
$821$	−32495.2	−1.38135	−0.690675	+	0.723165i	$0.742687\pi$
$822$	0	0
$823$	11442.4	0.484637	0.242318	−	0.970197i	$-0.422092\pi$
$823$	11442.4	0.484637	0.242318	+	0.970197i	$0.422092\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32378.4	1.36143	0.680717	−	0.732546i	$-0.261668\pi$
$827$	32378.4	1.36143	0.680717	+	0.732546i	$0.261668\pi$
$828$	0	0
$829$	−35365.1	−1.48164	−0.740821	−	0.671702i	$-0.765563\pi$
$829$	−35365.1	−1.48164	−0.740821	+	0.671702i	$0.765563\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6007.53	0.249878
$834$	0	0
$835$	1304.09	0.0540477
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6335.06	0.260680	0.130340	−	0.991469i	$-0.458393\pi$
$839$	6335.06	0.260680	0.130340	+	0.991469i	$0.458393\pi$
$840$	0	0
$841$	−50.5692	−0.00207344
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1957.95	−0.0797107
$846$	0	0
$847$	−6542.18	−0.265398
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6184.94	0.249139
$852$	0	0
$853$	4491.51	0.180289	0.0901445	−	0.995929i	$-0.471267\pi$
$853$	4491.51	0.180289	0.0901445	+	0.995929i	$0.471267\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10823.8	−0.431430	−0.215715	−	0.976456i	$-0.569208\pi$
$857$	−10823.8	−0.431430	−0.215715	+	0.976456i	$0.569208\pi$
$858$	0	0
$859$	45625.9	1.81226	0.906132	−	0.422995i	$-0.139021\pi$
$859$	45625.9	1.81226	0.906132	+	0.422995i	$0.139021\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29567.9	1.16629	0.583143	−	0.812370i	$-0.301823\pi$
$863$	29567.9	1.16629	0.583143	+	0.812370i	$0.301823\pi$
$864$	0	0
$865$	−1024.35	−0.0402647
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−48320.9	−1.88628
$870$	0	0
$871$	−6577.12	−0.255864
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1281.43	0.0495087
$876$	0	0
$877$	−1813.37	−0.0698212	−0.0349106	−	0.999390i	$-0.511115\pi$
$877$	−1813.37	−0.0698212	−0.0349106	+	0.999390i	$0.511115\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−29307.8	−1.12078	−0.560390	−	0.828229i	$-0.689349\pi$
$881$	−29307.8	−1.12078	−0.560390	+	0.828229i	$0.689349\pi$
$882$	0	0
$883$	−48560.6	−1.85073	−0.925365	−	0.379077i	$-0.876241\pi$
$883$	−48560.6	−1.85073	−0.925365	+	0.379077i	$0.876241\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10983.5	0.415770	0.207885	−	0.978153i	$-0.433342\pi$
$887$	10983.5	0.415770	0.207885	+	0.978153i	$0.433342\pi$
$888$	0	0
$889$	−8545.36	−0.322387
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−14201.6	−0.532182
$894$	0	0
$895$	9184.62	0.343026
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20983.8	0.778476
$900$	0	0
$901$	2477.43	0.0916039
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14403.4	0.529043
$906$	0	0
$907$	−41652.9	−1.52488	−0.762438	−	0.647061i	$-0.775998\pi$
$907$	−41652.9	−1.52488	−0.762438	+	0.647061i	$0.775998\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	7771.00	0.282618	0.141309	−	0.989966i	$-0.454869\pi$
$911$	7771.00	0.282618	0.141309	+	0.989966i	$0.454869\pi$
$912$	0	0
$913$	−26386.1	−0.956465
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3626.14	−0.130584
$918$	0	0
$919$	11284.4	0.405045	0.202523	−	0.979278i	$-0.435086\pi$
$919$	11284.4	0.405045	0.202523	+	0.979278i	$0.435086\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1353.53	−0.0482688
$924$	0	0
$925$	2031.06	0.0721954
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50206.4	−1.77311	−0.886555	−	0.462623i	$-0.846908\pi$
$929$	−50206.4	−1.77311	−0.886555	+	0.462623i	$0.846908\pi$
$930$	0	0
$931$	−7464.47	−0.262769
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5602.70	−0.195966
$936$	0	0
$937$	16233.7	0.565988	0.282994	−	0.959122i	$-0.408672\pi$
$937$	16233.7	0.565988	0.282994	+	0.959122i	$0.408672\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	33113.6	1.14715	0.573577	−	0.819152i	$-0.305556\pi$
$941$	33113.6	1.14715	0.573577	+	0.819152i	$0.305556\pi$
$942$	0	0
$943$	−24837.7	−0.857717
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19025.7	−0.652855	−0.326427	−	0.945222i	$-0.605845\pi$
$947$	−19025.7	−0.652855	−0.326427	+	0.945222i	$0.605845\pi$
$948$	0	0
$949$	7523.49	0.257347
$950$	0	0
$951$	0	0
$952$	0	0
$953$	46547.9	1.58220	0.791098	−	0.611689i	$-0.209510\pi$
$953$	46547.9	1.58220	0.791098	+	0.611689i	$0.209510\pi$
$954$	0	0
$955$	−3741.21	−0.126767
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15478.6	0.521199
$960$	0	0
$961$	−11699.4	−0.392716
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3108.10	−0.103682
$966$	0	0
$967$	1793.49	0.0596430	0.0298215	−	0.999555i	$-0.490506\pi$
$967$	1793.49	0.0596430	0.0298215	+	0.999555i	$0.490506\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	34899.8	1.15344	0.576719	−	0.816942i	$-0.304333\pi$
$971$	34899.8	1.15344	0.576719	+	0.816942i	$0.304333\pi$
$972$	0	0
$973$	−6340.34	−0.208902
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31360.4	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$977$	31360.4	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$978$	0	0
$979$	26280.4	0.857941
$980$	0	0
$981$	0	0
$982$	0	0
$983$	56392.0	1.82973	0.914866	−	0.403758i	$-0.132296\pi$
$983$	56392.0	1.82973	0.914866	+	0.403758i	$0.132296\pi$
$984$	0	0
$985$	2941.22	0.0951421
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32166.8	−1.03422
$990$	0	0
$991$	−10086.0	−0.323303	−0.161652	−	0.986848i	$-0.551682\pi$
$991$	−10086.0	−0.323303	−0.161652	+	0.986848i	$0.551682\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−20076.6	−0.639668
$996$	0	0
$997$	33875.2	1.07607	0.538033	−	0.842924i	$-0.319168\pi$
$997$	33875.2	1.07607	0.538033	+	0.842924i	$0.319168\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bj.1.2		3
3.2	odd	2		2160.4.a.br.1.2		3
4.3	odd	2		1080.4.a.e.1.2	✓	3
12.11	even	2		1080.4.a.k.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.e.1.2	✓	3	4.3	odd	2
1080.4.a.k.1.2	yes	3	12.11	even	2
2160.4.a.bj.1.2		3	1.1	even	1	trivial
2160.4.a.br.1.2		3	3.2	odd	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -10.2514 q^{7} -44.3754 q^{11} -50.8782 q^{13} -25.2514 q^{17} +31.3754 q^{19} +76.1296 q^{23} +25.0000 q^{25} -156.008 q^{29} -134.505 q^{31} +51.2571 q^{35} +81.2423 q^{37} -326.256 q^{41} -422.526 q^{43} -452.635 q^{47} -237.909 q^{49} -98.1105 q^{53} +221.877 q^{55} +540.118 q^{59} +522.292 q^{61} +254.391 q^{65} +129.272 q^{67} +26.6034 q^{71} -147.873 q^{73} +454.910 q^{77} +1088.91 q^{79} +594.611 q^{83} +126.257 q^{85} -592.229 q^{89} +521.573 q^{91} -156.877 q^{95} -666.786 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} + 8 q^{7} - 10 q^{11} + 48 q^{13} - 37 q^{17} - 29 q^{19} - 11 q^{23} + 75 q^{25} - 28 q^{29} - 41 q^{31} - 40 q^{35} + 230 q^{37} - 370 q^{41} + 130 q^{43} - 56 q^{47} + 547 q^{49} - 805 q^{53}+ \cdots + 576 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 8 * q^7 - 10 * q^11 + 48 * q^13 - 37 * q^17 - 29 * q^19 - 11 * q^23 + 75 * q^25 - 28 * q^29 - 41 * q^31 - 40 * q^35 + 230 * q^37 - 370 * q^41 + 130 * q^43 - 56 * q^47 + 547 * q^49 - 805 * q^53 + 50 * q^55 + 576 * q^59 - 257 * q^61 - 240 * q^65 + 14 * q^67 + 1238 * q^71 - 398 * q^73 - 1296 * q^77 + 321 * q^79 + 687 * q^83 + 185 * q^85 - 2358 * q^89 + 1968 * q^91 + 145 * q^95 + 576 * q^97

Introduction

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