LMFDB - Embedded newform 2160.4.h.c.431.5

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)

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")

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);

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 = [12,0,0,0,0,0,0,0,0,0,0,0,-192] 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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 8 x^{10} - 100 x^{9} + 160 x^{8} - 522 x^{7} + 3906 x^{6} - 4698 x^{5} + \cdots + 531441 $ x^12 - 2x^11 + 8x^10 - 100x^9 + 160x^8 - 522x^7 + 3906x^6 - 4698x^5 + 12960x^4 - 72900x^3 + 52488x^2 - 118098*x + 531441
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.5
Root			$2.98614 - 0.288063i$ of defining polynomial
Character	$\chi$	$=$	2160.431
Dual form			2160.4.h.c.431.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000i q^{5} +7.56760i q^{7} -32.1145 q^{11} -70.8060 q^{13} +78.3224i q^{17} +58.7777i q^{19} -126.204 q^{23} -25.0000 q^{25} -131.149i q^{29} -294.845i q^{31} +37.8380 q^{35} -242.687 q^{37} -32.2895i q^{41} +178.139i q^{43} +202.357 q^{47} +285.731 q^{49} +754.105i q^{53} +160.572i q^{55} +674.076 q^{59} +228.588 q^{61} +354.030i q^{65} -150.745i q^{67} -902.379 q^{71} +1053.15 q^{73} -243.030i q^{77} -679.193i q^{79} +898.485 q^{83} +391.612 q^{85} -546.141i q^{89} -535.831i q^{91} +293.889 q^{95} -1021.12 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 192 q^{13} - 300 q^{25} + 936 q^{37} + 2820 q^{49} - 492 q^{61} + 4488 q^{73} + 900 q^{85} + 2544 q^{97}+O(q^{100}) $ 12 * q - 192 * q^13 - 300 * q^25 + 936 * q^37 + 2820 * q^49 - 492 * q^61 + 4488 * q^73 + 900 * q^85 + 2544 * 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
$7$	7.56760i	0.408612i	0.978907	+	0.204306i	$0.0654938\pi$
$7$	7.56760i	0.408612i	−0.978907	+	0.204306i	$0.934506\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−32.1145	−0.880262	−0.440131	−	0.897934i	$-0.645068\pi$
$11$	−32.1145	−0.880262	−0.440131	+	0.897934i	$0.645068\pi$
$12$	0	0
$13$	−70.8060	−1.51062	−0.755309	−	0.655369i	$-0.772513\pi$
$13$	−70.8060	−1.51062	−0.755309	+	0.655369i	$0.772513\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	78.3224i	1.11741i	0.829366	+	0.558705i	$0.188702\pi$
$17$	78.3224i	1.11741i	−0.829366	+	0.558705i	$0.811298\pi$
$18$	0	0
$19$	58.7777i	0.709712i	0.934921	+	0.354856i	$0.115470\pi$
$19$	58.7777i	0.709712i	−0.934921	+	0.354856i	$0.884530\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−126.204	−1.14414	−0.572071	−	0.820204i	$-0.693860\pi$
$23$	−126.204	−1.14414	−0.572071	+	0.820204i	$0.693860\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 131.149i	− 0.839787i	−0.907573	−	0.419893i	$-0.862067\pi$
$29$	− 131.149i	− 0.839787i	0.907573	−	0.419893i	$-0.137933\pi$
$30$	0	0
$31$	− 294.845i	− 1.70825i	−0.520068	−	0.854125i	$-0.674093\pi$
$31$	− 294.845i	− 1.70825i	0.520068	−	0.854125i	$-0.325907\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	37.8380	0.182737
$36$	0	0
$37$	−242.687	−1.07831	−0.539155	−	0.842207i	$-0.681256\pi$
$37$	−242.687	−1.07831	−0.539155	+	0.842207i	$0.681256\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 32.2895i	− 0.122994i	−0.998107	−	0.0614972i	$-0.980412\pi$
$41$	− 32.2895i	− 0.122994i	0.998107	−	0.0614972i	$-0.0195875\pi$
$42$	0	0
$43$	178.139i	0.631767i	0.948798	+	0.315884i	$0.102301\pi$
$43$	178.139i	0.631767i	−0.948798	+	0.315884i	$0.897699\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	202.357	0.628017	0.314008	−	0.949420i	$-0.398328\pi$
$47$	202.357	0.628017	0.314008	+	0.949420i	$0.398328\pi$
$48$	0	0
$49$	285.731	0.833036
$50$	0	0
$51$	0	0
$52$	0	0
$53$	754.105i	1.95442i	0.212276	+	0.977210i	$0.431912\pi$
$53$	754.105i	1.95442i	−0.212276	+	0.977210i	$0.568088\pi$
$54$	0	0
$55$	160.572i	0.393665i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	674.076	1.48741	0.743706	−	0.668507i	$-0.233066\pi$
$59$	674.076	1.48741	0.743706	+	0.668507i	$0.233066\pi$
$60$	0	0
$61$	228.588	0.479798	0.239899	−	0.970798i	$-0.422886\pi$
$61$	228.588	0.479798	0.239899	+	0.970798i	$0.422886\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	354.030i	0.675569i
$66$	0	0
$67$	− 150.745i	− 0.274873i	−0.990511	−	0.137436i	$-0.956114\pi$
$67$	− 150.745i	− 0.274873i	0.990511	−	0.137436i	$-0.0438863\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−902.379	−1.50835	−0.754174	−	0.656674i	$-0.771963\pi$
$71$	−902.379	−1.50835	−0.754174	+	0.656674i	$0.771963\pi$
$72$	0	0
$73$	1053.15	1.68851	0.844257	−	0.535939i	$-0.180042\pi$
$73$	1053.15	1.68851	0.844257	+	0.535939i	$0.180042\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 243.030i	− 0.359686i
$78$	0	0
$79$	− 679.193i	− 0.967281i	−0.875267	−	0.483640i	$-0.839314\pi$
$79$	− 679.193i	− 0.967281i	0.875267	−	0.483640i	$-0.160686\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	898.485	1.18821	0.594105	−	0.804387i	$-0.297506\pi$
$83$	898.485	1.18821	0.594105	+	0.804387i	$0.297506\pi$
$84$	0	0
$85$	391.612	0.499721
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 546.141i	− 0.650458i	−0.945635	−	0.325229i	$-0.894559\pi$
$89$	− 546.141i	− 0.650458i	0.945635	−	0.325229i	$-0.105441\pi$
$90$	0	0
$91$	− 535.831i	− 0.617257i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	293.889	0.317393
$96$	0	0
$97$	−1021.12	−1.06885	−0.534426	−	0.845215i	$-0.679472\pi$
$97$	−1021.12	−1.06885	−0.534426	+	0.845215i	$0.679472\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 637.176i	− 0.627737i	−0.949467	−	0.313868i	$-0.898375\pi$
$101$	− 637.176i	− 0.627737i	0.949467	−	0.313868i	$-0.101625\pi$
$102$	0	0
$103$	− 1653.02i	− 1.58133i	−0.612249	−	0.790665i	$-0.709735\pi$
$103$	− 1653.02i	− 1.58133i	0.612249	−	0.790665i	$-0.290265\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−796.897	−0.719990	−0.359995	−	0.932954i	$-0.617222\pi$
$107$	−796.897	−0.719990	−0.359995	+	0.932954i	$0.617222\pi$
$108$	0	0
$109$	1634.83	1.43659	0.718295	−	0.695738i	$-0.244923\pi$
$109$	1634.83	1.43659	0.718295	+	0.695738i	$0.244923\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	341.304i	0.284134i	0.989857	+	0.142067i	$0.0453749\pi$
$113$	341.304i	0.284134i	−0.989857	+	0.142067i	$0.954625\pi$
$114$	0	0
$115$	631.018i	0.511676i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−592.713	−0.456587
$120$	0	0
$121$	−299.660	−0.225139
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000i	0.0894427i
$126$	0	0
$127$	879.158i	0.614273i	0.951665	+	0.307137i	$0.0993708\pi$
$127$	879.158i	0.614273i	−0.951665	+	0.307137i	$0.900629\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	564.363	0.376402	0.188201	−	0.982131i	$-0.439734\pi$
$131$	564.363	0.376402	0.188201	+	0.982131i	$0.439734\pi$
$132$	0	0
$133$	−444.806	−0.289997
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2214.77i	− 1.38117i	−0.723251	−	0.690585i	$-0.757353\pi$
$137$	− 2214.77i	− 1.38117i	0.723251	−	0.690585i	$-0.242647\pi$
$138$	0	0
$139$	2824.39i	1.72346i	0.507364	+	0.861732i	$0.330620\pi$
$139$	2824.39i	1.72346i	−0.507364	+	0.861732i	$0.669380\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2273.90	1.32974
$144$	0	0
$145$	−655.746	−0.375564
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1392.40i	0.765569i	0.923838	+	0.382784i	$0.125035\pi$
$149$	1392.40i	0.765569i	−0.923838	+	0.382784i	$0.874965\pi$
$150$	0	0
$151$	− 320.060i	− 0.172491i	−0.996274	−	0.0862455i	$-0.972513\pi$
$151$	− 320.060i	− 0.172491i	0.996274	−	0.0862455i	$-0.0274870\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1474.23	−0.763953
$156$	0	0
$157$	3486.84	1.77249	0.886243	−	0.463220i	$-0.153306\pi$
$157$	3486.84	1.77249	0.886243	+	0.463220i	$0.153306\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 955.059i	− 0.467511i
$162$	0	0
$163$	− 529.223i	− 0.254306i	−0.991883	−	0.127153i	$-0.959416\pi$
$163$	− 529.223i	− 0.254306i	0.991883	−	0.127153i	$-0.0405840\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1338.03	−0.620000	−0.310000	−	0.950737i	$-0.600329\pi$
$167$	−1338.03	−0.620000	−0.310000	+	0.950737i	$0.600329\pi$
$168$	0	0
$169$	2816.48	1.28197
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2724.46i	1.19732i	0.801002	+	0.598661i	$0.204300\pi$
$173$	2724.46i	1.19732i	−0.801002	+	0.598661i	$0.795700\pi$
$174$	0	0
$175$	− 189.190i	− 0.0817224i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1754.95	−0.732801	−0.366400	−	0.930457i	$-0.619410\pi$
$179$	−1754.95	−0.732801	−0.366400	+	0.930457i	$0.619410\pi$
$180$	0	0
$181$	1400.27	0.575034	0.287517	−	0.957776i	$-0.407170\pi$
$181$	1400.27	0.575034	0.287517	+	0.957776i	$0.407170\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1213.43i	0.482235i
$186$	0	0
$187$	− 2515.28i	− 0.983614i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2354.25	−0.891870	−0.445935	−	0.895065i	$-0.647129\pi$
$191$	−2354.25	−0.891870	−0.445935	+	0.895065i	$0.647129\pi$
$192$	0	0
$193$	−538.288	−0.200761	−0.100380	−	0.994949i	$-0.532006\pi$
$193$	−538.288	−0.200761	−0.100380	+	0.994949i	$0.532006\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3671.59i	1.32787i	0.747791	+	0.663935i	$0.231115\pi$
$197$	3671.59i	1.32787i	−0.747791	+	0.663935i	$0.768885\pi$
$198$	0	0
$199$	− 2586.96i	− 0.921531i	−0.887522	−	0.460765i	$-0.847575\pi$
$199$	− 2586.96i	− 0.921531i	0.887522	−	0.460765i	$-0.152425\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	992.486	0.343147
$204$	0	0
$205$	−161.448	−0.0550048
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1887.62i	− 0.624733i
$210$	0	0
$211$	− 1978.53i	− 0.645533i	−0.946479	−	0.322767i	$-0.895387\pi$
$211$	− 1978.53i	− 0.645533i	0.946479	−	0.322767i	$-0.104613\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	890.696	0.282535
$216$	0	0
$217$	2231.27	0.698012
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 5545.69i	− 1.68798i
$222$	0	0
$223$	− 3714.15i	− 1.11533i	−0.830067	−	0.557664i	$-0.811698\pi$
$223$	− 3714.15i	− 1.11533i	0.830067	−	0.557664i	$-0.188302\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1493.47	−0.436676	−0.218338	−	0.975873i	$-0.570063\pi$
$227$	−1493.47	−0.436676	−0.218338	+	0.975873i	$0.570063\pi$
$228$	0	0
$229$	2051.91	0.592115	0.296058	−	0.955170i	$-0.404328\pi$
$229$	2051.91	0.592115	0.296058	+	0.955170i	$0.404328\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1935.06i	− 0.544078i	−0.962286	−	0.272039i	$-0.912302\pi$
$233$	− 1935.06i	− 0.544078i	0.962286	−	0.272039i	$-0.0876980\pi$
$234$	0	0
$235$	− 1011.78i	− 0.280858i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	478.344	0.129462	0.0647312	−	0.997903i	$-0.479381\pi$
$239$	478.344	0.129462	0.0647312	+	0.997903i	$0.479381\pi$
$240$	0	0
$241$	6911.10	1.84723	0.923616	−	0.383318i	$-0.125219\pi$
$241$	6911.10	1.84723	0.923616	+	0.383318i	$0.125219\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1428.66i	− 0.372545i
$246$	0	0
$247$	− 4161.81i	− 1.07210i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4173.23	1.04945	0.524725	−	0.851272i	$-0.324168\pi$
$251$	4173.23	1.04945	0.524725	+	0.851272i	$0.324168\pi$
$252$	0	0
$253$	4052.97	1.00715
$254$	0	0
$255$	0	0
$256$	0	0
$257$	925.220i	0.224567i	0.993676	+	0.112283i	$0.0358164\pi$
$257$	925.220i	0.224567i	−0.993676	+	0.112283i	$0.964184\pi$
$258$	0	0
$259$	− 1836.56i	− 0.440610i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4377.50	−1.02634	−0.513171	−	0.858286i	$-0.671529\pi$
$263$	−4377.50	−1.02634	−0.513171	+	0.858286i	$0.671529\pi$
$264$	0	0
$265$	3770.52	0.874043
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 7418.00i	− 1.68135i	−0.541538	−	0.840676i	$-0.682158\pi$
$269$	− 7418.00i	− 1.68135i	0.541538	−	0.840676i	$-0.317842\pi$
$270$	0	0
$271$	4543.85i	1.01852i	0.860612	+	0.509261i	$0.170081\pi$
$271$	4543.85i	1.01852i	−0.860612	+	0.509261i	$0.829919\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	802.862	0.176052
$276$	0	0
$277$	7997.72	1.73479	0.867394	−	0.497622i	$-0.165793\pi$
$277$	7997.72	1.73479	0.867394	+	0.497622i	$0.165793\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7823.65i	1.66092i	0.557075	+	0.830462i	$0.311923\pi$
$281$	7823.65i	1.66092i	−0.557075	+	0.830462i	$0.688077\pi$
$282$	0	0
$283$	− 6934.81i	− 1.45665i	−0.685232	−	0.728325i	$-0.740299\pi$
$283$	− 6934.81i	− 1.45665i	0.685232	−	0.728325i	$-0.259701\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	244.354	0.0502570
$288$	0	0
$289$	−1221.40	−0.248606
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 3629.61i	− 0.723700i	−0.932236	−	0.361850i	$-0.882145\pi$
$293$	− 3629.61i	− 0.723700i	0.932236	−	0.361850i	$-0.117855\pi$
$294$	0	0
$295$	− 3370.38i	− 0.665191i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8935.97	1.72836
$300$	0	0
$301$	−1348.09	−0.258148
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1142.94i	− 0.214572i
$306$	0	0
$307$	− 1302.76i	− 0.242190i	−0.992641	−	0.121095i	$-0.961359\pi$
$307$	− 1302.76i	− 0.242190i	0.992641	−	0.121095i	$-0.0386406\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8231.35	1.50083	0.750413	−	0.660969i	$-0.229855\pi$
$311$	8231.35	1.50083	0.750413	+	0.660969i	$0.229855\pi$
$312$	0	0
$313$	3316.19	0.598856	0.299428	−	0.954119i	$-0.403204\pi$
$313$	3316.19	0.598856	0.299428	+	0.954119i	$0.403204\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1687.78i	0.299039i	0.988759	+	0.149519i	$0.0477727\pi$
$317$	1687.78i	0.299039i	−0.988759	+	0.149519i	$0.952227\pi$
$318$	0	0
$319$	4211.79i	0.739232i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4603.61	−0.793040
$324$	0	0
$325$	1770.15	0.302124
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1531.36i	0.256615i
$330$	0	0
$331$	− 8155.71i	− 1.35432i	−0.735838	−	0.677158i	$-0.763211\pi$
$331$	− 8155.71i	− 1.35432i	0.735838	−	0.677158i	$-0.236789\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−753.727	−0.122927
$336$	0	0
$337$	−3784.89	−0.611798	−0.305899	−	0.952064i	$-0.598957\pi$
$337$	−3784.89	−0.611798	−0.305899	+	0.952064i	$0.598957\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9468.80i	1.50371i
$342$	0	0
$343$	4757.99i	0.749001i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3510.97	−0.543166	−0.271583	−	0.962415i	$-0.587547\pi$
$347$	−3510.97	−0.543166	−0.271583	+	0.962415i	$0.587547\pi$
$348$	0	0
$349$	−5368.79	−0.823452	−0.411726	−	0.911308i	$-0.635074\pi$
$349$	−5368.79	−0.823452	−0.411726	+	0.911308i	$0.635074\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 794.397i	− 0.119778i	−0.998205	−	0.0598888i	$-0.980925\pi$
$353$	− 794.397i	− 0.119778i	0.998205	−	0.0598888i	$-0.0190746\pi$
$354$	0	0
$355$	4511.90i	0.674554i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9183.57	−1.35011	−0.675056	−	0.737767i	$-0.735880\pi$
$359$	−9183.57	−1.35011	−0.675056	+	0.737767i	$0.735880\pi$
$360$	0	0
$361$	3404.18	0.496309
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 5265.73i	− 0.755126i
$366$	0	0
$367$	525.371i	0.0747251i	0.999302	+	0.0373626i	$0.0118956\pi$
$367$	525.371i	0.0747251i	−0.999302	+	0.0373626i	$0.988104\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5706.76	−0.798599
$372$	0	0
$373$	4230.34	0.587235	0.293617	−	0.955923i	$-0.405141\pi$
$373$	4230.34	0.587235	0.293617	+	0.955923i	$0.405141\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9286.15i	1.26860i
$378$	0	0
$379$	− 11254.9i	− 1.52539i	−0.646758	−	0.762696i	$-0.723875\pi$
$379$	− 11254.9i	− 1.52539i	0.646758	−	0.762696i	$-0.276125\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3249.64	0.433548	0.216774	−	0.976222i	$-0.430447\pi$
$383$	3249.64	0.433548	0.216774	+	0.976222i	$0.430447\pi$
$384$	0	0
$385$	−1215.15	−0.160856
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1747.00i	0.227703i	0.993498	+	0.113852i	$0.0363189\pi$
$389$	1747.00i	0.227703i	−0.993498	+	0.113852i	$0.963681\pi$
$390$	0	0
$391$	− 9884.58i	− 1.27848i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3395.96	−0.432581
$396$	0	0
$397$	12783.9	1.61613	0.808067	−	0.589091i	$-0.200514\pi$
$397$	12783.9	1.61613	0.808067	+	0.589091i	$0.200514\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5751.27i	0.716221i	0.933679	+	0.358111i	$0.116579\pi$
$401$	5751.27i	0.716221i	−0.933679	+	0.358111i	$0.883421\pi$
$402$	0	0
$403$	20876.8i	2.58051i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7793.76	0.949195
$408$	0	0
$409$	12328.2	1.49044	0.745221	−	0.666818i	$-0.232344\pi$
$409$	12328.2	1.49044	0.745221	+	0.666818i	$0.232344\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5101.14i	0.607774i
$414$	0	0
$415$	− 4492.42i	− 0.531384i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8716.10	1.01625	0.508126	−	0.861283i	$-0.330339\pi$
$419$	8716.10	1.01625	0.508126	+	0.861283i	$0.330339\pi$
$420$	0	0
$421$	4896.49	0.566842	0.283421	−	0.958996i	$-0.408531\pi$
$421$	4896.49	0.566842	0.283421	+	0.958996i	$0.408531\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1958.06i	− 0.223482i
$426$	0	0
$427$	1729.86i	0.196051i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4670.18	−0.521937	−0.260968	−	0.965347i	$-0.584042\pi$
$431$	−4670.18	−0.521937	−0.260968	+	0.965347i	$0.584042\pi$
$432$	0	0
$433$	8626.74	0.957448	0.478724	−	0.877966i	$-0.341100\pi$
$433$	8626.74	0.957448	0.478724	+	0.877966i	$0.341100\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7417.96i	− 0.812012i
$438$	0	0
$439$	− 10754.0i	− 1.16916i	−0.811336	−	0.584580i	$-0.801259\pi$
$439$	− 10754.0i	− 1.16916i	0.811336	−	0.584580i	$-0.198741\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12700.2	1.36209	0.681045	−	0.732242i	$-0.261526\pi$
$443$	12700.2	1.36209	0.681045	+	0.732242i	$0.261526\pi$
$444$	0	0
$445$	−2730.70	−0.290894
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3123.57i	0.328308i	0.986435	+	0.164154i	$0.0524895\pi$
$449$	3123.57i	0.328308i	−0.986435	+	0.164154i	$0.947511\pi$
$450$	0	0
$451$	1036.96i	0.108267i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2679.16	−0.276046
$456$	0	0
$457$	232.538	0.0238024	0.0119012	−	0.999929i	$-0.496212\pi$
$457$	232.538	0.0238024	0.0119012	+	0.999929i	$0.496212\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3804.93i	0.384411i	0.981355	+	0.192205i	$0.0615640\pi$
$461$	3804.93i	0.384411i	−0.981355	+	0.192205i	$0.938436\pi$
$462$	0	0
$463$	− 11122.2i	− 1.11640i	−0.829705	−	0.558202i	$-0.811492\pi$
$463$	− 11122.2i	− 1.11640i	0.829705	−	0.558202i	$-0.188508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5105.56	−0.505904	−0.252952	−	0.967479i	$-0.581401\pi$
$467$	−5105.56	−0.505904	−0.252952	+	0.967479i	$0.581401\pi$
$468$	0	0
$469$	1140.78	0.112316
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 5720.85i	− 0.556121i
$474$	0	0
$475$	− 1469.44i	− 0.141942i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17479.7	−1.66736	−0.833682	−	0.552245i	$-0.813771\pi$
$479$	−17479.7	−1.66736	−0.833682	+	0.552245i	$0.813771\pi$
$480$	0	0
$481$	17183.7	1.62891
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5105.59i	0.478006i
$486$	0	0
$487$	− 16203.2i	− 1.50768i	−0.657060	−	0.753838i	$-0.728200\pi$
$487$	− 16203.2i	− 1.50768i	0.657060	−	0.753838i	$-0.271800\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11651.6	1.07094	0.535468	−	0.844556i	$-0.320135\pi$
$491$	11651.6	1.07094	0.535468	+	0.844556i	$0.320135\pi$
$492$	0	0
$493$	10271.9	0.938386
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6828.85i	− 0.616329i
$498$	0	0
$499$	− 1814.85i	− 0.162813i	−0.996681	−	0.0814067i	$-0.974059\pi$
$499$	− 1814.85i	− 0.162813i	0.996681	−	0.0814067i	$-0.0259413\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16771.0	−1.48665	−0.743324	−	0.668932i	$-0.766752\pi$
$503$	−16771.0	−1.48665	−0.743324	+	0.668932i	$0.766752\pi$
$504$	0	0
$505$	−3185.88	−0.280732
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 16526.3i	− 1.43913i	−0.694425	−	0.719565i	$-0.744341\pi$
$509$	− 16526.3i	− 1.43913i	0.694425	−	0.719565i	$-0.255659\pi$
$510$	0	0
$511$	7969.79i	0.689947i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8265.11	−0.707192
$516$	0	0
$517$	−6498.59	−0.552819
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 12039.9i	− 1.01243i	−0.862406	−	0.506217i	$-0.831043\pi$
$521$	− 12039.9i	− 1.01243i	0.862406	−	0.506217i	$-0.168957\pi$
$522$	0	0
$523$	7346.21i	0.614202i	0.951677	+	0.307101i	$0.0993589\pi$
$523$	7346.21i	0.614202i	−0.951677	+	0.307101i	$0.900641\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23093.0	1.90882
$528$	0	0
$529$	3760.37	0.309063
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2286.29i	0.185798i
$534$	0	0
$535$	3984.48i	0.321989i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9176.12	−0.733290
$540$	0	0
$541$	−11949.5	−0.949626	−0.474813	−	0.880087i	$-0.657484\pi$
$541$	−11949.5	−0.949626	−0.474813	+	0.880087i	$0.657484\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8174.15i	− 0.642463i
$546$	0	0
$547$	4682.52i	0.366015i	0.983112	+	0.183007i	$0.0585832\pi$
$547$	4682.52i	0.366015i	−0.983112	+	0.183007i	$0.941417\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7708.65	0.596007
$552$	0	0
$553$	5139.86	0.395243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6922.90i	0.526629i	0.964710	+	0.263315i	$0.0848157\pi$
$557$	6922.90i	0.526629i	−0.964710	+	0.263315i	$0.915184\pi$
$558$	0	0
$559$	− 12613.3i	− 0.954359i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	21091.2	1.57884	0.789422	−	0.613851i	$-0.210380\pi$
$563$	21091.2	1.57884	0.789422	+	0.613851i	$0.210380\pi$
$564$	0	0
$565$	1706.52	0.127069
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8030.35i	0.591652i	0.955242	+	0.295826i	$0.0955948\pi$
$569$	8030.35i	0.591652i	−0.955242	+	0.295826i	$0.904405\pi$
$570$	0	0
$571$	16492.1i	1.20871i	0.796716	+	0.604354i	$0.206569\pi$
$571$	16492.1i	1.20871i	−0.796716	+	0.604354i	$0.793431\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3155.09	0.228829
$576$	0	0
$577$	−1853.85	−0.133755	−0.0668776	−	0.997761i	$-0.521304\pi$
$577$	−1853.85	−0.133755	−0.0668776	+	0.997761i	$0.521304\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6799.37i	0.485517i
$582$	0	0
$583$	− 24217.7i	− 1.72040i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14358.6	−1.00962	−0.504808	−	0.863232i	$-0.668437\pi$
$587$	−14358.6	−1.00962	−0.504808	+	0.863232i	$0.668437\pi$
$588$	0	0
$589$	17330.3	1.21237
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3228.40i	0.223566i	0.993733	+	0.111783i	$0.0356561\pi$
$593$	3228.40i	0.223566i	−0.993733	+	0.111783i	$0.964344\pi$
$594$	0	0
$595$	2963.56i	0.204192i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4904.57	−0.334550	−0.167275	−	0.985910i	$-0.553497\pi$
$599$	−4904.57	−0.334550	−0.167275	+	0.985910i	$0.553497\pi$
$600$	0	0
$601$	−12459.3	−0.845634	−0.422817	−	0.906215i	$-0.638959\pi$
$601$	−12459.3	−0.845634	−0.422817	+	0.906215i	$0.638959\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1498.30i	0.100685i
$606$	0	0
$607$	17076.7i	1.14188i	0.820991	+	0.570942i	$0.193422\pi$
$607$	17076.7i	1.14188i	−0.820991	+	0.570942i	$0.806578\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−14328.1	−0.948694
$612$	0	0
$613$	−5178.25	−0.341187	−0.170593	−	0.985341i	$-0.554568\pi$
$613$	−5178.25	−0.341187	−0.170593	+	0.985341i	$0.554568\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4037.33i	− 0.263431i	−0.991288	−	0.131715i	$-0.957951\pi$
$617$	− 4037.33i	− 0.263431i	0.991288	−	0.131715i	$-0.0420485\pi$
$618$	0	0
$619$	17079.7i	1.10903i	0.832174	+	0.554515i	$0.187096\pi$
$619$	17079.7i	1.10903i	−0.832174	+	0.554515i	$0.812904\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4132.98	0.265785
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 19007.8i	− 1.20491i
$630$	0	0
$631$	− 7197.29i	− 0.454072i	−0.973886	−	0.227036i	$-0.927096\pi$
$631$	− 7197.29i	− 0.454072i	0.973886	−	0.227036i	$-0.0729035\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4395.79	0.274711
$636$	0	0
$637$	−20231.5	−1.25840
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3352.62i	0.206584i	0.994651	+	0.103292i	$0.0329376\pi$
$641$	3352.62i	0.206584i	−0.994651	+	0.103292i	$0.967062\pi$
$642$	0	0
$643$	2845.94i	0.174546i	0.996184	+	0.0872728i	$0.0278152\pi$
$643$	2845.94i	0.174546i	−0.996184	+	0.0872728i	$0.972185\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20674.6	1.25626	0.628130	−	0.778108i	$-0.283820\pi$
$647$	20674.6	1.25626	0.628130	+	0.778108i	$0.283820\pi$
$648$	0	0
$649$	−21647.6	−1.30931
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32959.7i	1.97521i	0.156963	+	0.987605i	$0.449830\pi$
$653$	32959.7i	1.97521i	−0.156963	+	0.987605i	$0.550170\pi$
$654$	0	0
$655$	− 2821.81i	− 0.168332i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10854.1	−0.641604	−0.320802	−	0.947146i	$-0.603952\pi$
$659$	−10854.1	−0.641604	−0.320802	+	0.947146i	$0.603952\pi$
$660$	0	0
$661$	−16856.0	−0.991864	−0.495932	−	0.868361i	$-0.665173\pi$
$661$	−16856.0	−0.991864	−0.495932	+	0.868361i	$0.665173\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2224.03i	0.129691i
$666$	0	0
$667$	16551.5i	0.960836i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7340.98	−0.422348
$672$	0	0
$673$	−11931.8	−0.683415	−0.341707	−	0.939806i	$-0.611005\pi$
$673$	−11931.8	−0.683415	−0.341707	+	0.939806i	$0.611005\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 10320.9i	− 0.585915i	−0.956125	−	0.292957i	$-0.905361\pi$
$677$	− 10320.9i	− 0.585915i	0.956125	−	0.292957i	$-0.0946394\pi$
$678$	0	0
$679$	− 7727.41i	− 0.436746i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	601.636	0.0337056	0.0168528	−	0.999858i	$-0.494635\pi$
$683$	601.636	0.0337056	0.0168528	+	0.999858i	$0.494635\pi$
$684$	0	0
$685$	−11073.8	−0.617678
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 53395.1i	− 2.95238i
$690$	0	0
$691$	5702.87i	0.313961i	0.987602	+	0.156981i	$0.0501760\pi$
$691$	5702.87i	0.313961i	−0.987602	+	0.156981i	$0.949824\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14121.9	0.770756
$696$	0	0
$697$	2528.99	0.137435
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3552.26i	0.191394i	0.995411	+	0.0956968i	$0.0305079\pi$
$701$	3552.26i	0.191394i	−0.995411	+	0.0956968i	$0.969492\pi$
$702$	0	0
$703$	− 14264.6i	− 0.765289i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4821.90	0.256501
$708$	0	0
$709$	1674.89	0.0887188	0.0443594	−	0.999016i	$-0.485875\pi$
$709$	1674.89	0.0887188	0.0443594	+	0.999016i	$0.485875\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	37210.5i	1.95448i
$714$	0	0
$715$	− 11369.5i	− 0.594678i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6862.29	−0.355939	−0.177970	−	0.984036i	$-0.556953\pi$
$719$	−6862.29	−0.355939	−0.177970	+	0.984036i	$0.556953\pi$
$720$	0	0
$721$	12509.4	0.646151
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3278.73i	0.167957i
$726$	0	0
$727$	− 4516.84i	− 0.230427i	−0.993341	−	0.115213i	$-0.963245\pi$
$727$	− 4516.84i	− 0.230427i	0.993341	−	0.115213i	$-0.0367552\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13952.3	−0.705943
$732$	0	0
$733$	−32489.6	−1.63715	−0.818574	−	0.574402i	$-0.805235\pi$
$733$	−32489.6	−1.63715	−0.818574	+	0.574402i	$0.805235\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4841.11i	0.241960i
$738$	0	0
$739$	14068.1i	0.700273i	0.936699	+	0.350137i	$0.113865\pi$
$739$	14068.1i	0.700273i	−0.936699	+	0.350137i	$0.886135\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24655.3	1.21738	0.608691	−	0.793408i	$-0.291695\pi$
$743$	24655.3	1.21738	0.608691	+	0.793408i	$0.291695\pi$
$744$	0	0
$745$	6961.99	0.342373
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 6030.60i	− 0.294197i
$750$	0	0
$751$	− 28039.5i	− 1.36242i	−0.732088	−	0.681210i	$-0.761454\pi$
$751$	− 28039.5i	− 1.36242i	0.732088	−	0.681210i	$-0.238546\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1600.30	−0.0771403
$756$	0	0
$757$	3462.15	0.166227	0.0831137	−	0.996540i	$-0.473514\pi$
$757$	3462.15	0.166227	0.0831137	+	0.996540i	$0.473514\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 27915.3i	− 1.32974i	−0.746961	−	0.664868i	$-0.768488\pi$
$761$	− 27915.3i	− 1.32974i	0.746961	−	0.664868i	$-0.231512\pi$
$762$	0	0
$763$	12371.7i	0.587008i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−47728.6	−2.24691
$768$	0	0
$769$	−36403.1	−1.70706	−0.853529	−	0.521045i	$-0.825543\pi$
$769$	−36403.1	−1.70706	−0.853529	+	0.521045i	$0.825543\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34789.2i	1.61873i	0.587305	+	0.809366i	$0.300189\pi$
$773$	34789.2i	1.61873i	−0.587305	+	0.809366i	$0.699811\pi$
$774$	0	0
$775$	7371.13i	0.341650i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1897.90	0.0872907
$780$	0	0
$781$	28979.5	1.32774
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 17434.2i	− 0.792680i
$786$	0	0
$787$	1455.87i	0.0659416i	0.999456	+	0.0329708i	$0.0104968\pi$
$787$	1455.87i	0.0659416i	−0.999456	+	0.0329708i	$0.989503\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2582.85	−0.116101
$792$	0	0
$793$	−16185.4	−0.724792
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30305.4i	− 1.34689i	−0.739237	−	0.673446i	$-0.764814\pi$
$797$	− 30305.4i	− 1.34689i	0.739237	−	0.673446i	$-0.235186\pi$
$798$	0	0
$799$	15849.1i	0.701752i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−33821.3	−1.48633
$804$	0	0
$805$	−4775.30	−0.209077
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12798.0i	0.556186i	0.960554	+	0.278093i	$0.0897024\pi$
$809$	12798.0i	0.556186i	−0.960554	+	0.278093i	$0.910298\pi$
$810$	0	0
$811$	20816.1i	0.901296i	0.892702	+	0.450648i	$0.148807\pi$
$811$	20816.1i	0.901296i	−0.892702	+	0.450648i	$0.851193\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2646.11	−0.113729
$816$	0	0
$817$	−10470.6	−0.448373
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 21674.7i	− 0.921377i	−0.887562	−	0.460689i	$-0.847602\pi$
$821$	− 21674.7i	− 0.921377i	0.887562	−	0.460689i	$-0.152398\pi$
$822$	0	0
$823$	18068.3i	0.765274i	0.923899	+	0.382637i	$0.124984\pi$
$823$	18068.3i	0.765274i	−0.923899	+	0.382637i	$0.875016\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26165.2	−1.10019	−0.550093	−	0.835104i	$-0.685408\pi$
$827$	−26165.2	−1.10019	−0.550093	+	0.835104i	$0.685408\pi$
$828$	0	0
$829$	32877.9	1.37744	0.688719	−	0.725028i	$-0.258173\pi$
$829$	32877.9	1.37744	0.688719	+	0.725028i	$0.258173\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	22379.2i	0.930843i
$834$	0	0
$835$	6690.16i	0.277272i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10340.9	−0.425515	−0.212758	−	0.977105i	$-0.568244\pi$
$839$	−10340.9	−0.425515	−0.212758	+	0.977105i	$0.568244\pi$
$840$	0	0
$841$	7188.87	0.294758
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 14082.4i	− 0.573314i
$846$	0	0
$847$	− 2267.71i	− 0.0919944i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	30627.9	1.23374
$852$	0	0
$853$	5312.72	0.213252	0.106626	−	0.994299i	$-0.465995\pi$
$853$	5312.72	0.213252	0.106626	+	0.994299i	$0.465995\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38844.7i	1.54832i	0.632992	+	0.774159i	$0.281827\pi$
$857$	38844.7i	1.54832i	−0.632992	+	0.774159i	$0.718173\pi$
$858$	0	0
$859$	38513.1i	1.52974i	0.644183	+	0.764872i	$0.277198\pi$
$859$	38513.1i	1.52974i	−0.644183	+	0.764872i	$0.722802\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24453.7	0.964557	0.482279	−	0.876018i	$-0.339809\pi$
$863$	24453.7	0.964557	0.482279	+	0.876018i	$0.339809\pi$
$864$	0	0
$865$	13622.3	0.535459
$866$	0	0
$867$	0	0
$868$	0	0
$869$	21811.9i	0.851460i
$870$	0	0
$871$	10673.7i	0.415228i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−945.950	−0.0365474
$876$	0	0
$877$	20329.8	0.782767	0.391384	−	0.920228i	$-0.371997\pi$
$877$	20329.8	0.782767	0.391384	+	0.920228i	$0.371997\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47887.5i	1.83129i	0.401984	+	0.915647i	$0.368321\pi$
$881$	47887.5i	1.83129i	−0.401984	+	0.915647i	$0.631679\pi$
$882$	0	0
$883$	− 14012.7i	− 0.534048i	−0.963690	−	0.267024i	$-0.913960\pi$
$883$	− 14012.7i	− 0.534048i	0.963690	−	0.267024i	$-0.0860403\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	50634.6	1.91674	0.958368	−	0.285537i	$-0.0921720\pi$
$887$	50634.6	1.91674	0.958368	+	0.285537i	$0.0921720\pi$
$888$	0	0
$889$	−6653.12	−0.250999
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11894.1i	0.445711i
$894$	0	0
$895$	8774.76i	0.327718i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−38668.7	−1.43457
$900$	0	0
$901$	−59063.3	−2.18389
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 7001.34i	− 0.257163i
$906$	0	0
$907$	− 16896.5i	− 0.618567i	−0.950970	−	0.309283i	$-0.899911\pi$
$907$	− 16896.5i	− 0.618567i	0.950970	−	0.309283i	$-0.100089\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28351.8	1.03111	0.515553	−	0.856858i	$-0.327587\pi$
$911$	28351.8	1.03111	0.515553	+	0.856858i	$0.327587\pi$
$912$	0	0
$913$	−28854.4	−1.04594
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4270.87i	0.153802i
$918$	0	0
$919$	38075.6i	1.36670i	0.730091	+	0.683350i	$0.239478\pi$
$919$	38075.6i	1.36670i	−0.730091	+	0.683350i	$0.760522\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	63893.8	2.27854
$924$	0	0
$925$	6067.17	0.215662
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 44190.6i	− 1.56065i	−0.625373	−	0.780326i	$-0.715053\pi$
$929$	− 44190.6i	− 1.56065i	0.625373	−	0.780326i	$-0.284947\pi$
$930$	0	0
$931$	16794.6i	0.591216i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12576.4	−0.439886
$936$	0	0
$937$	−26332.2	−0.918074	−0.459037	−	0.888417i	$-0.651805\pi$
$937$	−26332.2	−0.918074	−0.459037	+	0.888417i	$0.651805\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7342.98i	0.254383i	0.991878	+	0.127191i	$0.0405963\pi$
$941$	7342.98i	0.254383i	−0.991878	+	0.127191i	$0.959404\pi$
$942$	0	0
$943$	4075.05i	0.140723i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	54421.3	1.86743	0.933714	−	0.358019i	$-0.116548\pi$
$947$	54421.3	1.86743	0.933714	+	0.358019i	$0.116548\pi$
$948$	0	0
$949$	−74569.0	−2.55070
$950$	0	0
$951$	0	0
$952$	0	0
$953$	7391.32i	0.251237i	0.992079	+	0.125618i	$0.0400915\pi$
$953$	7391.32i	0.251237i	−0.992079	+	0.125618i	$0.959909\pi$
$954$	0	0
$955$	11771.2i	0.398856i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16760.5	0.564363
$960$	0	0
$961$	−57142.7	−1.91812
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2691.44i	0.0897829i
$966$	0	0
$967$	− 31710.4i	− 1.05454i	−0.849699	−	0.527268i	$-0.823216\pi$
$967$	− 31710.4i	− 1.05454i	0.849699	−	0.527268i	$-0.176784\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−16757.8	−0.553846	−0.276923	−	0.960892i	$-0.589315\pi$
$971$	−16757.8	−0.553846	−0.276923	+	0.960892i	$0.589315\pi$
$972$	0	0
$973$	−21373.8	−0.704228
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 8207.77i	− 0.268772i	−0.990929	−	0.134386i	$-0.957094\pi$
$977$	− 8207.77i	− 0.268772i	0.990929	−	0.134386i	$-0.0429062\pi$
$978$	0	0
$979$	17539.0i	0.572574i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8219.39	0.266692	0.133346	−	0.991070i	$-0.457428\pi$
$983$	8219.39	0.266692	0.133346	+	0.991070i	$0.457428\pi$
$984$	0	0
$985$	18358.0	0.593841
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 22481.8i	− 0.722832i
$990$	0	0
$991$	− 25875.6i	− 0.829430i	−0.909951	−	0.414715i	$-0.863881\pi$
$991$	− 25875.6i	− 0.829430i	0.909951	−	0.414715i	$-0.136119\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−12934.8	−0.412121
$996$	0	0
$997$	−47593.2	−1.51183	−0.755914	−	0.654671i	$-0.772807\pi$
$997$	−47593.2	−1.51183	−0.755914	+	0.654671i	$0.772807\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.h.c.431.5	yes	12
3.2	odd	2	inner	2160.4.h.c.431.11	yes	12
4.3	odd	2	inner	2160.4.h.c.431.2	✓	12
12.11	even	2	inner	2160.4.h.c.431.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.4.h.c.431.2	✓	12	4.3	odd	2	inner
2160.4.h.c.431.5	yes	12	1.1	even	1	trivial
2160.4.h.c.431.8	yes	12	12.11	even	2	inner
2160.4.h.c.431.11	yes	12	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)

\(f(q)\)	\(=\)	\(q-5.00000i q^{5} +7.56760i q^{7} -32.1145 q^{11} -70.8060 q^{13} +78.3224i q^{17} +58.7777i q^{19} -126.204 q^{23} -25.0000 q^{25} -131.149i q^{29} -294.845i q^{31} +37.8380 q^{35} -242.687 q^{37} -32.2895i q^{41} +178.139i q^{43} +202.357 q^{47} +285.731 q^{49} +754.105i q^{53} +160.572i q^{55} +674.076 q^{59} +228.588 q^{61} +354.030i q^{65} -150.745i q^{67} -902.379 q^{71} +1053.15 q^{73} -243.030i q^{77} -679.193i q^{79} +898.485 q^{83} +391.612 q^{85} -546.141i q^{89} -535.831i q^{91} +293.889 q^{95} -1021.12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 192 q^{13} - 300 q^{25} + 936 q^{37} + 2820 q^{49} - 492 q^{61} + 4488 q^{73} + 900 q^{85} + 2544 q^{97}+O(q^{100}) \) 12 * q - 192 * q^13 - 300 * q^25 + 936 * q^37 + 2820 * q^49 - 492 * q^61 + 4488 * q^73 + 900 * q^85 + 2544 * q^97

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