LMFDB - Embedded newform 864.4.f.b.431.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,4,Mod(431,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("864.431");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	864.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$50.9776502450$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.16
Character	$\chi$	$=$	864.431
Dual form			864.4.f.b.431.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.34627 q^{5} +31.3735i q^{7} +O(q^{10})$	$q+5.34627 q^{5} +31.3735i q^{7} +51.3150i q^{11} +4.90945i q^{13} +76.0061i q^{17} -32.6182 q^{19} -96.3546 q^{23} -96.4174 q^{25} +91.4847 q^{29} -229.507i q^{31} +167.731i q^{35} +54.2915i q^{37} -340.220i q^{41} +416.787 q^{43} -593.371 q^{47} -641.297 q^{49} -229.050 q^{53} +274.344i q^{55} -751.749i q^{59} +231.607i q^{61} +26.2473i q^{65} +819.100 q^{67} +20.8051 q^{71} +39.6082 q^{73} -1609.93 q^{77} -739.613i q^{79} +866.479i q^{83} +406.349i q^{85} +777.802i q^{89} -154.027 q^{91} -174.386 q^{95} +490.282 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 48 q^{19} + 600 q^{25} + 432 q^{43} - 816 q^{49} - 1632 q^{67} - 216 q^{73} - 3600 q^{91} + 2280 q^{97}+O(q^{100}) $ 24 * q + 48 * q^19 + 600 * q^25 + 432 * q^43 - 816 * q^49 - 1632 * q^67 - 216 * q^73 - 3600 * q^91 + 2280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.34627	0.478185	0.239092	−	0.970997i	$-0.423150\pi$
$5$	5.34627	0.478185	0.239092	+	0.970997i	$0.423150\pi$
$6$	0	0
$7$	31.3735i	1.69401i	0.531585	+	0.847005i	$0.321597\pi$
$7$	31.3735i	1.69401i	−0.531585	+	0.847005i	$0.678403\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	51.3150i	1.40655i	0.710918	+	0.703275i	$0.248280\pi$
$11$	51.3150i	1.40655i	−0.710918	+	0.703275i	$0.751720\pi$
$12$	0	0
$13$	4.90945i	0.104741i	0.998628	+	0.0523707i	$0.0166777\pi$
$13$	4.90945i	0.104741i	−0.998628	+	0.0523707i	$0.983322\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	76.0061i	1.08436i	0.840261	+	0.542182i	$0.182402\pi$
$17$	76.0061i	1.08436i	−0.840261	+	0.542182i	$0.817598\pi$
$18$	0	0
$19$	−32.6182	−0.393849	−0.196924	−	0.980419i	$-0.563095\pi$
$19$	−32.6182	−0.393849	−0.196924	+	0.980419i	$0.563095\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−96.3546	−0.873536	−0.436768	−	0.899574i	$-0.643877\pi$
$23$	−96.3546	−0.873536	−0.436768	+	0.899574i	$0.643877\pi$
$24$	0	0
$25$	−96.4174	−0.771339
$26$	0	0
$27$	0	0
$28$	0	0
$29$	91.4847	0.585803	0.292901	−	0.956143i	$-0.405379\pi$
$29$	91.4847	0.585803	0.292901	+	0.956143i	$0.405379\pi$
$30$	0	0
$31$	− 229.507i	− 1.32970i	−0.746977	−	0.664850i	$-0.768496\pi$
$31$	− 229.507i	− 1.32970i	0.746977	−	0.664850i	$-0.231504\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	167.731i	0.810050i
$36$	0	0
$37$	54.2915i	0.241229i	0.992699	+	0.120615i	$0.0384865\pi$
$37$	54.2915i	0.241229i	−0.992699	+	0.120615i	$0.961513\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 340.220i	− 1.29594i	−0.761666	−	0.647970i	$-0.775618\pi$
$41$	− 340.220i	− 1.29594i	0.761666	−	0.647970i	$-0.224382\pi$
$42$	0	0
$43$	416.787	1.47813	0.739063	−	0.673636i	$-0.235268\pi$
$43$	416.787	1.47813	0.739063	+	0.673636i	$0.235268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−593.371	−1.84153	−0.920766	−	0.390116i	$-0.872435\pi$
$47$	−593.371	−1.84153	−0.920766	+	0.390116i	$0.872435\pi$
$48$	0	0
$49$	−641.297	−1.86967
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−229.050	−0.593631	−0.296816	−	0.954935i	$-0.595925\pi$
$53$	−229.050	−0.593631	−0.296816	+	0.954935i	$0.595925\pi$
$54$	0	0
$55$	274.344i	0.672591i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 751.749i	− 1.65880i	−0.558653	−	0.829401i	$-0.688682\pi$
$59$	− 751.749i	− 1.65880i	0.558653	−	0.829401i	$-0.311318\pi$
$60$	0	0
$61$	231.607i	0.486135i	0.970009	+	0.243068i	$0.0781537\pi$
$61$	231.607i	0.486135i	−0.970009	+	0.243068i	$0.921846\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	26.2473i	0.0500857i
$66$	0	0
$67$	819.100	1.49357	0.746784	−	0.665067i	$-0.231597\pi$
$67$	819.100	1.49357	0.746784	+	0.665067i	$0.231597\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	20.8051	0.0347763	0.0173881	−	0.999849i	$-0.494465\pi$
$71$	20.8051	0.0347763	0.0173881	+	0.999849i	$0.494465\pi$
$72$	0	0
$73$	39.6082	0.0635040	0.0317520	−	0.999496i	$-0.489891\pi$
$73$	39.6082	0.0635040	0.0317520	+	0.999496i	$0.489891\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1609.93	−2.38271
$78$	0	0
$79$	− 739.613i	− 1.05333i	−0.850073	−	0.526664i	$-0.823442\pi$
$79$	− 739.613i	− 1.05333i	0.850073	−	0.526664i	$-0.176558\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	866.479i	1.14588i	0.819596	+	0.572942i	$0.194198\pi$
$83$	866.479i	1.14588i	−0.819596	+	0.572942i	$0.805802\pi$
$84$	0	0
$85$	406.349i	0.518526i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	777.802i	0.926369i	0.886262	+	0.463185i	$0.153293\pi$
$89$	777.802i	0.926369i	−0.886262	+	0.463185i	$0.846707\pi$
$90$	0	0
$91$	−154.027	−0.177433
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−174.386	−0.188332
$96$	0	0
$97$	490.282	0.513202	0.256601	−	0.966517i	$-0.417397\pi$
$97$	490.282	0.513202	0.256601	+	0.966517i	$0.417397\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−952.128	−0.938023	−0.469011	−	0.883192i	$-0.655390\pi$
$101$	−952.128	−0.938023	−0.469011	+	0.883192i	$0.655390\pi$
$102$	0	0
$103$	745.313i	0.712989i	0.934297	+	0.356494i	$0.116028\pi$
$103$	745.313i	0.712989i	−0.934297	+	0.356494i	$0.883972\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 27.3685i	− 0.0247272i	−0.999924	−	0.0123636i	$-0.996064\pi$
$107$	− 27.3685i	− 0.0247272i	0.999924	−	0.0123636i	$-0.00393556\pi$
$108$	0	0
$109$	− 847.307i	− 0.744562i	−0.928120	−	0.372281i	$-0.878576\pi$
$109$	− 847.307i	− 0.744562i	0.928120	−	0.372281i	$-0.121424\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1675.61i	1.39494i	0.716616	+	0.697468i	$0.245690\pi$
$113$	1675.61i	1.39494i	−0.716616	+	0.697468i	$0.754310\pi$
$114$	0	0
$115$	−515.138	−0.417712
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2384.58	−1.83692
$120$	0	0
$121$	−1302.23	−0.978384
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1183.76	−0.847027
$126$	0	0
$127$	− 1339.39i	− 0.935842i	−0.883770	−	0.467921i	$-0.845003\pi$
$127$	− 1339.39i	− 0.935842i	0.883770	−	0.467921i	$-0.154997\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1751.43i	1.16811i	0.811713	+	0.584057i	$0.198535\pi$
$131$	1751.43i	1.16811i	−0.811713	+	0.584057i	$0.801465\pi$
$132$	0	0
$133$	− 1023.35i	− 0.667184i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 189.844i	− 0.118390i	−0.998246	−	0.0591951i	$-0.981147\pi$
$137$	− 189.844i	− 0.118390i	0.998246	−	0.0591951i	$-0.0188534\pi$
$138$	0	0
$139$	734.538	0.448221	0.224111	−	0.974564i	$-0.428052\pi$
$139$	734.538	0.448221	0.224111	+	0.974564i	$0.428052\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−251.929	−0.147324
$144$	0	0
$145$	489.102	0.280122
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1386.42	−0.762282	−0.381141	−	0.924517i	$-0.624469\pi$
$149$	−1386.42	−0.762282	−0.381141	+	0.924517i	$0.624469\pi$
$150$	0	0
$151$	2947.94i	1.58874i	0.607433	+	0.794371i	$0.292199\pi$
$151$	2947.94i	1.58874i	−0.607433	+	0.794371i	$0.707801\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1227.01i	− 0.635842i
$156$	0	0
$157$	1809.82i	0.919997i	0.887920	+	0.459999i	$0.152150\pi$
$157$	1809.82i	0.919997i	−0.887920	+	0.459999i	$0.847850\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3022.98i	− 1.47978i
$162$	0	0
$163$	−2274.91	−1.09316	−0.546579	−	0.837407i	$-0.684070\pi$
$163$	−2274.91	−1.09316	−0.546579	+	0.837407i	$0.684070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2770.18	−1.28361	−0.641806	−	0.766867i	$-0.721815\pi$
$167$	−2770.18	−1.28361	−0.641806	+	0.766867i	$0.721815\pi$
$168$	0	0
$169$	2172.90	0.989029
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2498.16	−1.09787	−0.548936	−	0.835864i	$-0.684967\pi$
$173$	−2498.16	−1.09787	−0.548936	+	0.835864i	$0.684967\pi$
$174$	0	0
$175$	− 3024.95i	− 1.30666i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3199.38i	− 1.33594i	−0.744190	−	0.667968i	$-0.767164\pi$
$179$	− 3199.38i	− 1.33594i	0.744190	−	0.667968i	$-0.232836\pi$
$180$	0	0
$181$	2004.56i	0.823193i	0.911366	+	0.411597i	$0.135029\pi$
$181$	2004.56i	0.823193i	−0.911366	+	0.411597i	$0.864971\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	290.257i	0.115352i
$186$	0	0
$187$	−3900.25	−1.52521
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4254.50	−1.61175	−0.805877	−	0.592084i	$-0.798306\pi$
$191$	−4254.50	−1.61175	−0.805877	+	0.592084i	$0.798306\pi$
$192$	0	0
$193$	−669.854	−0.249830	−0.124915	−	0.992167i	$-0.539866\pi$
$193$	−669.854	−0.249830	−0.124915	+	0.992167i	$0.539866\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4092.97	1.48026	0.740132	−	0.672462i	$-0.234763\pi$
$197$	4092.97	1.48026	0.740132	+	0.672462i	$0.234763\pi$
$198$	0	0
$199$	4026.88i	1.43446i	0.696836	+	0.717231i	$0.254591\pi$
$199$	4026.88i	1.43446i	−0.696836	+	0.717231i	$0.745409\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2870.20i	0.992356i
$204$	0	0
$205$	− 1818.91i	− 0.619698i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1673.80i	− 0.553968i
$210$	0	0
$211$	2003.77	0.653767	0.326884	−	0.945065i	$-0.394001\pi$
$211$	2003.77	0.653767	0.326884	+	0.945065i	$0.394001\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2228.26	0.706818
$216$	0	0
$217$	7200.44	2.25252
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−373.148	−0.113578
$222$	0	0
$223$	− 1256.58i	− 0.377339i	−0.982041	−	0.188670i	$-0.939582\pi$
$223$	− 1256.58i	− 0.377339i	0.982041	−	0.188670i	$-0.0604175\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1718.54i	0.502482i	0.967925	+	0.251241i	$0.0808386\pi$
$227$	1718.54i	0.502482i	−0.967925	+	0.251241i	$0.919161\pi$
$228$	0	0
$229$	− 6743.82i	− 1.94605i	−0.230710	−	0.973023i	$-0.574105\pi$
$229$	− 6743.82i	− 1.94605i	0.230710	−	0.973023i	$-0.425895\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	985.152i	0.276993i	0.990363	+	0.138497i	$0.0442270\pi$
$233$	985.152i	0.276993i	−0.990363	+	0.138497i	$0.955773\pi$
$234$	0	0
$235$	−3172.32	−0.880592
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2339.18	−0.633092	−0.316546	−	0.948577i	$-0.602523\pi$
$239$	−2339.18	−0.633092	−0.316546	+	0.948577i	$0.602523\pi$
$240$	0	0
$241$	−107.024	−0.0286059	−0.0143029	−	0.999898i	$-0.504553\pi$
$241$	−107.024	−0.0286059	−0.0143029	+	0.999898i	$0.504553\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3428.54	−0.894047
$246$	0	0
$247$	− 160.138i	− 0.0412523i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4821.60i	1.21250i	0.795275	+	0.606248i	$0.207326\pi$
$251$	4821.60i	1.21250i	−0.795275	+	0.606248i	$0.792674\pi$
$252$	0	0
$253$	− 4944.44i	− 1.22867i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2959.11i	0.718226i	0.933294	+	0.359113i	$0.116921\pi$
$257$	2959.11i	0.718226i	−0.933294	+	0.359113i	$0.883079\pi$
$258$	0	0
$259$	−1703.32	−0.408644
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3703.31	0.868274	0.434137	−	0.900847i	$-0.357053\pi$
$263$	3703.31	0.868274	0.434137	+	0.900847i	$0.357053\pi$
$264$	0	0
$265$	−1224.56	−0.283865
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6140.24	1.39174	0.695868	−	0.718170i	$-0.255020\pi$
$269$	6140.24	1.39174	0.695868	+	0.718170i	$0.255020\pi$
$270$	0	0
$271$	1756.25i	0.393669i	0.980437	+	0.196835i	$0.0630662\pi$
$271$	1756.25i	0.393669i	−0.980437	+	0.196835i	$0.936934\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4947.66i	− 1.08493i
$276$	0	0
$277$	7914.36i	1.71671i	0.513058	+	0.858354i	$0.328512\pi$
$277$	7914.36i	1.71671i	−0.513058	+	0.858354i	$0.671488\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8214.74i	1.74395i	0.489549	+	0.871976i	$0.337162\pi$
$281$	8214.74i	1.74395i	−0.489549	+	0.871976i	$0.662838\pi$
$282$	0	0
$283$	6234.35	1.30952	0.654759	−	0.755838i	$-0.272770\pi$
$283$	6234.35	1.30952	0.654759	+	0.755838i	$0.272770\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10673.9	2.19533
$288$	0	0
$289$	−863.927	−0.175845
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2144.16	0.427519	0.213759	−	0.976886i	$-0.431429\pi$
$293$	2144.16	0.427519	0.213759	+	0.976886i	$0.431429\pi$
$294$	0	0
$295$	− 4019.05i	− 0.793214i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 473.049i	− 0.0914953i
$300$	0	0
$301$	13076.1i	2.50396i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1238.23i	0.232463i
$306$	0	0
$307$	458.340	0.0852080	0.0426040	−	0.999092i	$-0.486435\pi$
$307$	458.340	0.0852080	0.0426040	+	0.999092i	$0.486435\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	441.865	0.0805656	0.0402828	−	0.999188i	$-0.487174\pi$
$311$	441.865	0.0805656	0.0402828	+	0.999188i	$0.487174\pi$
$312$	0	0
$313$	−1758.31	−0.317526	−0.158763	−	0.987317i	$-0.550751\pi$
$313$	−1758.31	−0.317526	−0.158763	+	0.987317i	$0.550751\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8501.65	−1.50631	−0.753155	−	0.657843i	$-0.771469\pi$
$317$	−8501.65	−1.50631	−0.753155	+	0.657843i	$0.771469\pi$
$318$	0	0
$319$	4694.54i	0.823961i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2479.18i	− 0.427075i
$324$	0	0
$325$	− 473.357i	− 0.0807911i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 18616.1i	− 3.11957i
$330$	0	0
$331$	7281.52	1.20915	0.604575	−	0.796549i	$-0.293343\pi$
$331$	7281.52	1.20915	0.604575	+	0.796549i	$0.293343\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4379.13	0.714201
$336$	0	0
$337$	−8529.49	−1.37873	−0.689363	−	0.724416i	$-0.742110\pi$
$337$	−8529.49	−1.37873	−0.689363	+	0.724416i	$0.742110\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11777.2	1.87029
$342$	0	0
$343$	− 9358.61i	− 1.47323i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4040.21i	− 0.625042i	−0.949911	−	0.312521i	$-0.898826\pi$
$347$	− 4040.21i	− 0.625042i	0.949911	−	0.312521i	$-0.101174\pi$
$348$	0	0
$349$	− 8536.83i	− 1.30936i	−0.755907	−	0.654679i	$-0.772804\pi$
$349$	− 8536.83i	− 1.30936i	0.755907	−	0.654679i	$-0.227196\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	995.881i	0.150157i	0.997178	+	0.0750785i	$0.0239207\pi$
$353$	995.881i	0.150157i	−0.997178	+	0.0750785i	$0.976079\pi$
$354$	0	0
$355$	111.230	0.0166295
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10756.7	1.58139	0.790695	−	0.612210i	$-0.209719\pi$
$359$	10756.7	1.58139	0.790695	+	0.612210i	$0.209719\pi$
$360$	0	0
$361$	−5795.05	−0.844883
$362$	0	0
$363$	0	0
$364$	0	0
$365$	211.756	0.0303666
$366$	0	0
$367$	5839.99i	0.830640i	0.909675	+	0.415320i	$0.136330\pi$
$367$	5839.99i	0.830640i	−0.909675	+	0.415320i	$0.863670\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 7186.11i	− 1.00562i
$372$	0	0
$373$	− 4710.48i	− 0.653886i	−0.945044	−	0.326943i	$-0.893981\pi$
$373$	− 4710.48i	− 0.653886i	0.945044	−	0.326943i	$-0.106019\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	449.140i	0.0613578i
$378$	0	0
$379$	−115.231	−0.0156174	−0.00780872	−	0.999970i	$-0.502486\pi$
$379$	−115.231	−0.0156174	−0.00780872	+	0.999970i	$0.502486\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−504.187	−0.0672656	−0.0336328	−	0.999434i	$-0.510708\pi$
$383$	−504.187	−0.0672656	−0.0336328	+	0.999434i	$0.510708\pi$
$384$	0	0
$385$	−8607.12	−1.13938
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1343.26	0.175080	0.0875401	−	0.996161i	$-0.472099\pi$
$389$	1343.26	0.175080	0.0875401	+	0.996161i	$0.472099\pi$
$390$	0	0
$391$	− 7323.54i	− 0.947231i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 3954.17i	− 0.503686i
$396$	0	0
$397$	− 547.594i	− 0.0692266i	−0.999401	−	0.0346133i	$-0.988980\pi$
$397$	− 547.594i	− 0.0692266i	0.999401	−	0.0346133i	$-0.0110200\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6587.39i	0.820346i	0.912008	+	0.410173i	$0.134532\pi$
$401$	6587.39i	0.820346i	−0.912008	+	0.410173i	$0.865468\pi$
$402$	0	0
$403$	1126.75	0.139275
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2785.97	−0.339301
$408$	0	0
$409$	8769.93	1.06026	0.530129	−	0.847917i	$-0.322144\pi$
$409$	8769.93	1.06026	0.530129	+	0.847917i	$0.322144\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	23585.0	2.81003
$414$	0	0
$415$	4632.43i	0.547945i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3508.29i	0.409048i	0.978862	+	0.204524i	$0.0655646\pi$
$419$	3508.29i	0.409048i	−0.978862	+	0.204524i	$0.934435\pi$
$420$	0	0
$421$	− 1973.08i	− 0.228413i	−0.993457	−	0.114206i	$-0.963567\pi$
$421$	− 1973.08i	− 0.228413i	0.993457	−	0.114206i	$-0.0364325\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 7328.31i	− 0.836413i
$426$	0	0
$427$	−7266.33	−0.823518
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12651.4	1.41391	0.706956	−	0.707258i	$-0.250068\pi$
$431$	12651.4	1.41391	0.706956	+	0.707258i	$0.250068\pi$
$432$	0	0
$433$	−7620.49	−0.845767	−0.422884	−	0.906184i	$-0.638982\pi$
$433$	−7620.49	−0.845767	−0.422884	+	0.906184i	$0.638982\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3142.91	0.344041
$438$	0	0
$439$	− 2823.61i	− 0.306978i	−0.988150	−	0.153489i	$-0.950949\pi$
$439$	− 2823.61i	− 0.306978i	0.988150	−	0.153489i	$-0.0490510\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13007.9i	1.39509i	0.716543	+	0.697543i	$0.245723\pi$
$443$	13007.9i	1.39509i	−0.716543	+	0.697543i	$0.754277\pi$
$444$	0	0
$445$	4158.34i	0.442975i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 12452.1i	− 1.30879i	−0.756151	−	0.654397i	$-0.772922\pi$
$449$	− 12452.1i	− 1.30879i	0.756151	−	0.654397i	$-0.227078\pi$
$450$	0	0
$451$	17458.4	1.82280
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−823.468	−0.0848457
$456$	0	0
$457$	12468.0	1.27621	0.638105	−	0.769949i	$-0.279719\pi$
$457$	12468.0	1.27621	0.638105	+	0.769949i	$0.279719\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7335.04	0.741056	0.370528	−	0.928821i	$-0.379177\pi$
$461$	7335.04	0.741056	0.370528	+	0.928821i	$0.379177\pi$
$462$	0	0
$463$	4545.68i	0.456275i	0.973629	+	0.228138i	$0.0732637\pi$
$463$	4545.68i	0.456275i	−0.973629	+	0.228138i	$0.926736\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11514.8i	1.14099i	0.821303	+	0.570493i	$0.193248\pi$
$467$	11514.8i	1.14099i	−0.821303	+	0.570493i	$0.806752\pi$
$468$	0	0
$469$	25698.0i	2.53012i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	21387.4i	2.07906i
$474$	0	0
$475$	3144.96	0.303791
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17579.7	−1.67691	−0.838454	−	0.544973i	$-0.816540\pi$
$479$	−17579.7	−1.67691	−0.838454	+	0.544973i	$0.816540\pi$
$480$	0	0
$481$	−266.542	−0.0252667
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2621.18	0.245405
$486$	0	0
$487$	− 1599.21i	− 0.148803i	−0.997228	−	0.0744014i	$-0.976295\pi$
$487$	− 1599.21i	− 0.148803i	0.997228	−	0.0744014i	$-0.0237046\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5007.83i	0.460285i	0.973157	+	0.230143i	$0.0739193\pi$
$491$	5007.83i	0.460285i	−0.973157	+	0.230143i	$0.926081\pi$
$492$	0	0
$493$	6953.40i	0.635224i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	652.730i	0.0589113i
$498$	0	0
$499$	−13048.9	−1.17064	−0.585320	−	0.810802i	$-0.699031\pi$
$499$	−13048.9	−1.17064	−0.585320	+	0.810802i	$0.699031\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6424.25	0.569470	0.284735	−	0.958606i	$-0.408094\pi$
$503$	6424.25	0.569470	0.284735	+	0.958606i	$0.408094\pi$
$504$	0	0
$505$	−5090.33	−0.448548
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2904.65	0.252940	0.126470	−	0.991970i	$-0.459635\pi$
$509$	2904.65	0.252940	0.126470	+	0.991970i	$0.459635\pi$
$510$	0	0
$511$	1242.65i	0.107576i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3984.64i	0.340940i
$516$	0	0
$517$	− 30448.8i	− 2.59021i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2118.60i	0.178153i	0.996025	+	0.0890765i	$0.0283916\pi$
$521$	2118.60i	0.178153i	−0.996025	+	0.0890765i	$0.971608\pi$
$522$	0	0
$523$	4864.27	0.406692	0.203346	−	0.979107i	$-0.434818\pi$
$523$	4864.27	0.406692	0.203346	+	0.979107i	$0.434818\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17443.9	1.44188
$528$	0	0
$529$	−2882.78	−0.236935
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1670.30	0.135738
$534$	0	0
$535$	− 146.319i	− 0.0118242i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 32908.1i	− 2.62978i
$540$	0	0
$541$	− 19128.9i	− 1.52018i	−0.649819	−	0.760089i	$-0.725155\pi$
$541$	− 19128.9i	− 1.52018i	0.649819	−	0.760089i	$-0.274845\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 4529.93i	− 0.356038i
$546$	0	0
$547$	6067.06	0.474239	0.237119	−	0.971480i	$-0.423797\pi$
$547$	6067.06	0.474239	0.237119	+	0.971480i	$0.423797\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2984.07	−0.230718
$552$	0	0
$553$	23204.3	1.78435
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8164.41	−0.621072	−0.310536	−	0.950562i	$-0.600509\pi$
$557$	−8164.41	−0.621072	−0.310536	+	0.950562i	$0.600509\pi$
$558$	0	0
$559$	2046.20i	0.154821i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14439.3i	1.08089i	0.841379	+	0.540446i	$0.181744\pi$
$563$	14439.3i	1.08089i	−0.841379	+	0.540446i	$0.818256\pi$
$564$	0	0
$565$	8958.24i	0.667037i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 8787.31i	− 0.647422i	−0.946156	−	0.323711i	$-0.895069\pi$
$569$	− 8787.31i	− 0.647422i	0.946156	−	0.323711i	$-0.104931\pi$
$570$	0	0
$571$	−9655.32	−0.707640	−0.353820	−	0.935313i	$-0.615117\pi$
$571$	−9655.32	−0.707640	−0.353820	+	0.935313i	$0.615117\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9290.27	0.673793
$576$	0	0
$577$	−12495.1	−0.901523	−0.450761	−	0.892645i	$-0.648847\pi$
$577$	−12495.1	−0.901523	−0.450761	+	0.892645i	$0.648847\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−27184.5	−1.94114
$582$	0	0
$583$	− 11753.7i	− 0.834972i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 19689.8i	− 1.38447i	−0.721670	−	0.692237i	$-0.756625\pi$
$587$	− 19689.8i	− 1.38447i	0.721670	−	0.692237i	$-0.243375\pi$
$588$	0	0
$589$	7486.11i	0.523701i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 3166.24i	− 0.219261i	−0.993972	−	0.109631i	$-0.965033\pi$
$593$	− 3166.24i	− 0.219261i	0.993972	−	0.109631i	$-0.0349668\pi$
$594$	0	0
$595$	−12748.6	−0.878389
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14634.3	−0.998232	−0.499116	−	0.866535i	$-0.666342\pi$
$599$	−14634.3	−0.998232	−0.499116	+	0.866535i	$0.666342\pi$
$600$	0	0
$601$	11085.4	0.752384	0.376192	−	0.926542i	$-0.377233\pi$
$601$	11085.4	0.752384	0.376192	+	0.926542i	$0.377233\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6962.06	−0.467848
$606$	0	0
$607$	13823.4i	0.924338i	0.886792	+	0.462169i	$0.152929\pi$
$607$	13823.4i	0.924338i	−0.886792	+	0.462169i	$0.847071\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2913.12i	− 0.192884i
$612$	0	0
$613$	− 7316.01i	− 0.482041i	−0.970520	−	0.241020i	$-0.922518\pi$
$613$	− 7316.01i	− 0.482041i	0.970520	−	0.241020i	$-0.0774820\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4739.21i	0.309228i	0.987975	+	0.154614i	$0.0494133\pi$
$617$	4739.21i	0.309228i	−0.987975	+	0.154614i	$0.950587\pi$
$618$	0	0
$619$	−13238.6	−0.859620	−0.429810	−	0.902919i	$-0.641420\pi$
$619$	−13238.6	−0.859620	−0.429810	+	0.902919i	$0.641420\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−24402.4	−1.56928
$624$	0	0
$625$	5723.50	0.366304
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4126.49	−0.261580
$630$	0	0
$631$	− 7023.61i	− 0.443115i	−0.975147	−	0.221558i	$-0.928886\pi$
$631$	− 7023.61i	− 0.443115i	0.975147	−	0.221558i	$-0.0711141\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 7160.75i	− 0.447505i
$636$	0	0
$637$	− 3148.42i	− 0.195832i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 4093.28i	− 0.252223i	−0.992016	−	0.126112i	$-0.959750\pi$
$641$	− 4093.28i	− 0.252223i	0.992016	−	0.126112i	$-0.0402497\pi$
$642$	0	0
$643$	−21554.7	−1.32198	−0.660990	−	0.750395i	$-0.729863\pi$
$643$	−21554.7	−1.32198	−0.660990	+	0.750395i	$0.729863\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11733.0	0.712940	0.356470	−	0.934307i	$-0.383980\pi$
$647$	11733.0	0.712940	0.356470	+	0.934307i	$0.383980\pi$
$648$	0	0
$649$	38576.0	2.33319
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3463.08	−0.207536	−0.103768	−	0.994602i	$-0.533090\pi$
$653$	−3463.08	−0.207536	−0.103768	+	0.994602i	$0.533090\pi$
$654$	0	0
$655$	9363.59i	0.558574i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8365.15i	0.494476i	0.968955	+	0.247238i	$0.0795230\pi$
$659$	8365.15i	0.494476i	−0.968955	+	0.247238i	$0.920477\pi$
$660$	0	0
$661$	18883.1i	1.11115i	0.831467	+	0.555574i	$0.187502\pi$
$661$	18883.1i	1.11115i	−0.831467	+	0.555574i	$0.812498\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 5471.09i	− 0.319037i
$666$	0	0
$667$	−8814.98	−0.511720
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11884.9	−0.683774
$672$	0	0
$673$	−28007.6	−1.60418	−0.802092	−	0.597201i	$-0.796280\pi$
$673$	−28007.6	−1.60418	−0.802092	+	0.597201i	$0.796280\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18077.1	−1.02623	−0.513117	−	0.858318i	$-0.671509\pi$
$677$	−18077.1	−1.02623	−0.513117	+	0.858318i	$0.671509\pi$
$678$	0	0
$679$	15381.9i	0.869370i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21689.5i	1.21512i	0.794275	+	0.607559i	$0.207851\pi$
$683$	21689.5i	1.21512i	−0.794275	+	0.607559i	$0.792149\pi$
$684$	0	0
$685$	− 1014.96i	− 0.0566123i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1124.51i	− 0.0621778i
$690$	0	0
$691$	−8979.75	−0.494364	−0.247182	−	0.968969i	$-0.579505\pi$
$691$	−8979.75	−0.494364	−0.247182	+	0.968969i	$0.579505\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3927.04	0.214332
$696$	0	0
$697$	25858.8	1.40527
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17863.9	−0.962495	−0.481247	−	0.876585i	$-0.659816\pi$
$701$	−17863.9	−0.962495	−0.481247	+	0.876585i	$0.659816\pi$
$702$	0	0
$703$	− 1770.89i	− 0.0950078i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 29871.6i	− 1.58902i
$708$	0	0
$709$	− 24502.0i	− 1.29787i	−0.760842	−	0.648937i	$-0.775214\pi$
$709$	− 24502.0i	− 1.29787i	0.760842	−	0.648937i	$-0.224786\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22114.1i	1.16154i
$714$	0	0
$715$	−1346.88	−0.0704481
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27106.8	−1.40600	−0.702999	−	0.711191i	$-0.748156\pi$
$719$	−27106.8	−1.40600	−0.702999	+	0.711191i	$0.748156\pi$
$720$	0	0
$721$	−23383.1	−1.20781
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8820.72	−0.451853
$726$	0	0
$727$	22287.0i	1.13697i	0.822693	+	0.568486i	$0.192471\pi$
$727$	22287.0i	1.13697i	−0.822693	+	0.568486i	$0.807529\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	31678.4i	1.60283i
$732$	0	0
$733$	3562.98i	0.179538i	0.995963	+	0.0897691i	$0.0286129\pi$
$733$	3562.98i	0.179538i	−0.995963	+	0.0897691i	$0.971387\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42032.1i	2.10078i
$738$	0	0
$739$	11174.7	0.556251	0.278126	−	0.960545i	$-0.410287\pi$
$739$	11174.7	0.556251	0.278126	+	0.960545i	$0.410287\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	599.814	0.0296165	0.0148082	−	0.999890i	$-0.495286\pi$
$743$	599.814	0.0296165	0.0148082	+	0.999890i	$0.495286\pi$
$744$	0	0
$745$	−7412.17	−0.364511
$746$	0	0
$747$	0	0
$748$	0	0
$749$	858.646	0.0418882
$750$	0	0
$751$	− 27113.3i	− 1.31742i	−0.752399	−	0.658708i	$-0.771103\pi$
$751$	− 27113.3i	− 1.31742i	0.752399	−	0.658708i	$-0.228897\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15760.5i	0.759712i
$756$	0	0
$757$	6977.68i	0.335017i	0.985871	+	0.167509i	$0.0535722\pi$
$757$	6977.68i	0.335017i	−0.985871	+	0.167509i	$0.946428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 2313.56i	− 0.110206i	−0.998481	−	0.0551028i	$-0.982451\pi$
$761$	− 2313.56i	− 0.110206i	0.998481	−	0.0551028i	$-0.0175487\pi$
$762$	0	0
$763$	26583.0	1.26130
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3690.68	0.173745
$768$	0	0
$769$	−4046.15	−0.189737	−0.0948687	−	0.995490i	$-0.530243\pi$
$769$	−4046.15	−0.189737	−0.0948687	+	0.995490i	$0.530243\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1325.18	−0.0616604	−0.0308302	−	0.999525i	$-0.509815\pi$
$773$	−1325.18	−0.0616604	−0.0308302	+	0.999525i	$0.509815\pi$
$774$	0	0
$775$	22128.5i	1.02565i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11097.4i	0.510404i
$780$	0	0
$781$	1067.61i	0.0489145i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9675.80i	0.439929i
$786$	0	0
$787$	29285.3	1.32644	0.663221	−	0.748424i	$-0.269189\pi$
$787$	29285.3	1.32644	0.663221	+	0.748424i	$0.269189\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−52569.6	−2.36303
$792$	0	0
$793$	−1137.06	−0.0509185
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19355.3	−0.860228	−0.430114	−	0.902775i	$-0.641527\pi$
$797$	−19355.3	−0.860228	−0.430114	+	0.902775i	$0.641527\pi$
$798$	0	0
$799$	− 45099.8i	− 1.99689i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2032.50i	0.0893216i
$804$	0	0
$805$	− 16161.7i	− 0.707608i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 9344.89i	− 0.406117i	−0.979167	−	0.203059i	$-0.934912\pi$
$809$	− 9344.89i	− 0.406117i	0.979167	−	0.203059i	$-0.0650882\pi$
$810$	0	0
$811$	7747.11	0.335435	0.167717	−	0.985835i	$-0.446360\pi$
$811$	7747.11	0.335435	0.167717	+	0.985835i	$0.446360\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12162.3	−0.522732
$816$	0	0
$817$	−13594.8	−0.582158
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18279.5	−0.777049	−0.388525	−	0.921438i	$-0.627015\pi$
$821$	−18279.5	−0.777049	−0.388525	+	0.921438i	$0.627015\pi$
$822$	0	0
$823$	6867.66i	0.290877i	0.989367	+	0.145438i	$0.0464592\pi$
$823$	6867.66i	0.290877i	−0.989367	+	0.145438i	$0.953541\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28105.6i	1.18177i	0.806754	+	0.590887i	$0.201222\pi$
$827$	28105.6i	1.18177i	−0.806754	+	0.590887i	$0.798778\pi$
$828$	0	0
$829$	15815.7i	0.662607i	0.943524	+	0.331304i	$0.107488\pi$
$829$	15815.7i	0.662607i	−0.943524	+	0.331304i	$0.892512\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 48742.5i	− 2.02740i
$834$	0	0
$835$	−14810.1	−0.613804
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10718.6	−0.441057	−0.220529	−	0.975380i	$-0.570778\pi$
$839$	−10718.6	−0.441057	−0.220529	+	0.975380i	$0.570778\pi$
$840$	0	0
$841$	−16019.5	−0.656835
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11616.9	0.472939
$846$	0	0
$847$	− 40855.5i	− 1.65739i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 5231.24i	− 0.210722i
$852$	0	0
$853$	44682.8i	1.79356i	0.442474	+	0.896781i	$0.354101\pi$
$853$	44682.8i	1.79356i	−0.442474	+	0.896781i	$0.645899\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18607.3i	− 0.741671i	−0.928698	−	0.370836i	$-0.879071\pi$
$857$	− 18607.3i	− 0.741671i	0.928698	−	0.370836i	$-0.120929\pi$
$858$	0	0
$859$	−7890.17	−0.313398	−0.156699	−	0.987646i	$-0.550085\pi$
$859$	−7890.17	−0.313398	−0.156699	+	0.987646i	$0.550085\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13106.2	−0.516965	−0.258482	−	0.966016i	$-0.583222\pi$
$863$	−13106.2	−0.516965	−0.258482	+	0.966016i	$0.583222\pi$
$864$	0	0
$865$	−13355.9	−0.524986
$866$	0	0
$867$	0	0
$868$	0	0
$869$	37953.2	1.48156
$870$	0	0
$871$	4021.33i	0.156438i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 37138.6i	− 1.43487i
$876$	0	0
$877$	− 13098.1i	− 0.504322i	−0.967685	−	0.252161i	$-0.918859\pi$
$877$	− 13098.1i	− 0.504322i	0.967685	−	0.252161i	$-0.0811412\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3651.26i	0.139630i	0.997560	+	0.0698151i	$0.0222409\pi$
$881$	3651.26i	0.139630i	−0.997560	+	0.0698151i	$0.977759\pi$
$882$	0	0
$883$	35395.5	1.34898	0.674492	−	0.738282i	$-0.264363\pi$
$883$	35395.5	1.34898	0.674492	+	0.738282i	$0.264363\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39080.7	−1.47937	−0.739685	−	0.672953i	$-0.765026\pi$
$887$	−39080.7	−1.47937	−0.739685	+	0.672953i	$0.765026\pi$
$888$	0	0
$889$	42021.4	1.58532
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19354.7	0.725285
$894$	0	0
$895$	− 17104.7i	− 0.638824i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 20996.4i	− 0.778942i
$900$	0	0
$901$	− 17409.2i	− 0.643713i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10716.9i	0.393638i
$906$	0	0
$907$	26549.1	0.971939	0.485969	−	0.873976i	$-0.338467\pi$
$907$	26549.1	0.971939	0.485969	+	0.873976i	$0.338467\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25921.3	0.942713	0.471356	−	0.881943i	$-0.343765\pi$
$911$	25921.3	0.942713	0.471356	+	0.881943i	$0.343765\pi$
$912$	0	0
$913$	−44463.4	−1.61174
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−54948.4	−1.97880
$918$	0	0
$919$	28260.7i	1.01440i	0.861828	+	0.507200i	$0.169320\pi$
$919$	28260.7i	1.01440i	−0.861828	+	0.507200i	$0.830680\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	102.142i	0.00364251i
$924$	0	0
$925$	− 5234.65i	− 0.186069i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21212.5i	0.749151i	0.927197	+	0.374575i	$0.122211\pi$
$929$	21212.5i	0.749151i	−0.927197	+	0.374575i	$0.877789\pi$
$930$	0	0
$931$	20917.9	0.736367
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−20851.8	−0.729333
$936$	0	0
$937$	9169.33	0.319689	0.159845	−	0.987142i	$-0.448901\pi$
$937$	9169.33	0.319689	0.159845	+	0.987142i	$0.448901\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2400.62	0.0831646	0.0415823	−	0.999135i	$-0.486760\pi$
$941$	2400.62	0.0831646	0.0415823	+	0.999135i	$0.486760\pi$
$942$	0	0
$943$	32781.8i	1.13205i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 31318.5i	− 1.07467i	−0.843368	−	0.537336i	$-0.819431\pi$
$947$	− 31318.5i	− 1.07467i	0.843368	−	0.537336i	$-0.180569\pi$
$948$	0	0
$949$	194.455i	0.00665150i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 48391.0i	− 1.64485i	−0.568877	−	0.822423i	$-0.692622\pi$
$953$	− 48391.0i	− 1.64485i	0.568877	−	0.822423i	$-0.307378\pi$
$954$	0	0
$955$	−22745.7	−0.770716
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5956.06	0.200554
$960$	0	0
$961$	−22882.5	−0.768101
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3581.22	−0.119465
$966$	0	0
$967$	17813.6i	0.592396i	0.955127	+	0.296198i	$0.0957189\pi$
$967$	17813.6i	0.592396i	−0.955127	+	0.296198i	$0.904281\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5916.67i	0.195546i	0.995209	+	0.0977729i	$0.0311719\pi$
$971$	5916.67i	0.195546i	−0.995209	+	0.0977729i	$0.968828\pi$
$972$	0	0
$973$	23045.0i	0.759291i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 5580.76i	− 0.182748i	−0.995817	−	0.0913738i	$-0.970874\pi$
$977$	− 5580.76i	− 0.182748i	0.995817	−	0.0913738i	$-0.0291258\pi$
$978$	0	0
$979$	−39912.9	−1.30298
$980$	0	0
$981$	0	0
$982$	0	0
$983$	60174.9	1.95247	0.976237	−	0.216707i	$-0.0695316\pi$
$983$	60174.9	1.95247	0.976237	+	0.216707i	$0.0695316\pi$
$984$	0	0
$985$	21882.1	0.707839
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−40159.4	−1.29120
$990$	0	0
$991$	9260.95i	0.296855i	0.988923	+	0.148428i	$0.0474212\pi$
$991$	9260.95i	0.296855i	−0.988923	+	0.148428i	$0.952579\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21528.8i	0.685938i
$996$	0	0
$997$	− 3944.11i	− 0.125287i	−0.998036	−	0.0626435i	$-0.980047\pi$
$997$	− 3944.11i	− 0.125287i	0.998036	−	0.0626435i	$-0.0199531\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.4.f.b.431.16		24
3.2	odd	2	inner	864.4.f.b.431.10		24
4.3	odd	2		216.4.f.b.107.24	yes	24
8.3	odd	2	inner	864.4.f.b.431.9		24
8.5	even	2		216.4.f.b.107.2	yes	24
12.11	even	2		216.4.f.b.107.1	✓	24
24.5	odd	2		216.4.f.b.107.23	yes	24
24.11	even	2	inner	864.4.f.b.431.15		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.4.f.b.107.1	✓	24	12.11	even	2
216.4.f.b.107.2	yes	24	8.5	even	2
216.4.f.b.107.23	yes	24	24.5	odd	2
216.4.f.b.107.24	yes	24	4.3	odd	2
864.4.f.b.431.9		24	8.3	odd	2	inner
864.4.f.b.431.10		24	3.2	odd	2	inner
864.4.f.b.431.15		24	24.11	even	2	inner
864.4.f.b.431.16		24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	864.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.34627 q^{5} +31.3735i q^{7} +O(q^{10})\)	\(q+5.34627 q^{5} +31.3735i q^{7} +51.3150i q^{11} +4.90945i q^{13} +76.0061i q^{17} -32.6182 q^{19} -96.3546 q^{23} -96.4174 q^{25} +91.4847 q^{29} -229.507i q^{31} +167.731i q^{35} +54.2915i q^{37} -340.220i q^{41} +416.787 q^{43} -593.371 q^{47} -641.297 q^{49} -229.050 q^{53} +274.344i q^{55} -751.749i q^{59} +231.607i q^{61} +26.2473i q^{65} +819.100 q^{67} +20.8051 q^{71} +39.6082 q^{73} -1609.93 q^{77} -739.613i q^{79} +866.479i q^{83} +406.349i q^{85} +777.802i q^{89} -154.027 q^{91} -174.386 q^{95} +490.282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 48 q^{19} + 600 q^{25} + 432 q^{43} - 816 q^{49} - 1632 q^{67} - 216 q^{73} - 3600 q^{91} + 2280 q^{97}+O(q^{100}) \) 24 * q + 48 * q^19 + 600 * q^25 + 432 * q^43 - 816 * q^49 - 1632 * q^67 - 216 * q^73 - 3600 * q^91 + 2280 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more