LMFDB - Embedded newform 882.5.b.j.197.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,Mod(197,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	882.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$91.1723074400$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.224054542336.12
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 199x^{4} + 14641 $ x^8 + 199*x^4 + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.2
Root			$1.96485 - 2.67196i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.j.197.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -12.6723i q^{5} +22.6274i q^{8} +O(q^{10})$	$q-2.82843i q^{2} -8.00000 q^{4} -12.6723i q^{5} +22.6274i q^{8} -35.8427 q^{10} +166.462i q^{11} +140.982 q^{13} +64.0000 q^{16} -38.7401i q^{17} -359.282 q^{19} +101.379i q^{20} +470.825 q^{22} -22.2119i q^{23} +464.412 q^{25} -398.757i q^{26} -83.2708i q^{29} -20.6539 q^{31} -181.019i q^{32} -109.574 q^{34} -1361.06 q^{37} +1016.20i q^{38} +286.742 q^{40} -2210.66i q^{41} -1459.06 q^{43} -1331.69i q^{44} -62.8248 q^{46} -1415.45i q^{47} -1313.56i q^{50} -1127.86 q^{52} -2591.76i q^{53} +2109.46 q^{55} -235.525 q^{58} +2773.93i q^{59} -1734.29 q^{61} +58.4181i q^{62} -512.000 q^{64} -1786.57i q^{65} -952.113 q^{67} +309.921i q^{68} -3845.91i q^{71} -3605.64 q^{73} +3849.66i q^{74} +2874.26 q^{76} -6839.18 q^{79} -811.028i q^{80} -6252.69 q^{82} +9448.26i q^{83} -490.927 q^{85} +4126.85i q^{86} -3766.60 q^{88} -13655.2i q^{89} +177.695i q^{92} -4003.49 q^{94} +4552.94i q^{95} +10360.4 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4}+O(q^{10}) $ 8 * q - 64 * q^4	$ 8 q - 64 q^{4} + 512 q^{16} - 640 q^{22} + 1512 q^{25} + 128 q^{37} - 656 q^{43} + 3904 q^{46} - 15104 q^{58} - 4096 q^{64} - 23040 q^{67} - 59120 q^{79} - 52400 q^{85} + 5120 q^{88}+O(q^{100}) $ 8 * q - 64 * q^4 + 512 * q^16 - 640 * q^22 + 1512 * q^25 + 128 * q^37 - 656 * q^43 + 3904 * q^46 - 15104 * q^58 - 4096 * q^64 - 23040 * q^67 - 59120 * q^79 - 52400 * q^85 + 5120 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	− 12.6723i	− 0.506893i	−0.967349	−	0.253446i	$-0.918436\pi$
$5$	− 12.6723i	− 0.506893i	0.967349	−	0.253446i	$-0.0815641\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.6274i	0.353553i
$9$	0	0
$10$	−35.8427	−0.358427
$11$	166.462i	1.37572i	0.725845	+	0.687858i	$0.241449\pi$
$11$	166.462i	1.37572i	−0.725845	+	0.687858i	$0.758551\pi$
$12$	0	0
$13$	140.982	0.834212	0.417106	−	0.908858i	$-0.363044\pi$
$13$	140.982	0.834212	0.417106	+	0.908858i	$0.363044\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	− 38.7401i	− 0.134049i	−0.997751	−	0.0670244i	$-0.978649\pi$
$17$	− 38.7401i	− 0.134049i	0.997751	−	0.0670244i	$-0.0213505\pi$
$18$	0	0
$19$	−359.282	−0.995241	−0.497621	−	0.867395i	$-0.665793\pi$
$19$	−359.282	−0.995241	−0.497621	+	0.867395i	$0.665793\pi$
$20$	101.379i	0.253446i
$21$	0	0
$22$	470.825	0.972779
$23$	− 22.2119i	− 0.0419885i	−0.999780	−	0.0209943i	$-0.993317\pi$
$23$	− 22.2119i	− 0.0419885i	0.999780	−	0.0209943i	$-0.00668317\pi$
$24$	0	0
$25$	464.412	0.743060
$26$	− 398.757i	− 0.589877i
$27$	0	0
$28$	0	0
$29$	− 83.2708i	− 0.0990141i	−0.998774	−	0.0495070i	$-0.984235\pi$
$29$	− 83.2708i	− 0.0990141i	0.998774	−	0.0495070i	$-0.0157650\pi$
$30$	0	0
$31$	−20.6539	−0.0214921	−0.0107461	−	0.999942i	$-0.503421\pi$
$31$	−20.6539	−0.0214921	−0.0107461	+	0.999942i	$0.503421\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	−109.574	−0.0947868
$35$	0	0
$36$	0	0
$37$	−1361.06	−0.994202	−0.497101	−	0.867693i	$-0.665602\pi$
$37$	−1361.06	−0.994202	−0.497101	+	0.867693i	$0.665602\pi$
$38$	1016.20i	0.703742i
$39$	0	0
$40$	286.742	0.179214
$41$	− 2210.66i	− 1.31509i	−0.753417	−	0.657543i	$-0.771596\pi$
$41$	− 2210.66i	− 1.31509i	0.753417	−	0.657543i	$-0.228404\pi$
$42$	0	0
$43$	−1459.06	−0.789109	−0.394554	−	0.918873i	$-0.629101\pi$
$43$	−1459.06	−0.789109	−0.394554	+	0.918873i	$0.629101\pi$
$44$	− 1331.69i	− 0.687858i
$45$	0	0
$46$	−62.8248	−0.0296904
$47$	− 1415.45i	− 0.640763i	−0.947289	−	0.320382i	$-0.896189\pi$
$47$	− 1415.45i	− 0.640763i	0.947289	−	0.320382i	$-0.103811\pi$
$48$	0	0
$49$	0	0
$50$	− 1313.56i	− 0.525423i
$51$	0	0
$52$	−1127.86	−0.417106
$53$	− 2591.76i	− 0.922662i	−0.887228	−	0.461331i	$-0.847372\pi$
$53$	− 2591.76i	− 0.922662i	0.887228	−	0.461331i	$-0.152628\pi$
$54$	0	0
$55$	2109.46	0.697341
$56$	0	0
$57$	0	0
$58$	−235.525	−0.0700135
$59$	2773.93i	0.796878i	0.917195	+	0.398439i	$0.130448\pi$
$59$	2773.93i	0.796878i	−0.917195	+	0.398439i	$0.869552\pi$
$60$	0	0
$61$	−1734.29	−0.466081	−0.233041	−	0.972467i	$-0.574868\pi$
$61$	−1734.29	−0.466081	−0.233041	+	0.972467i	$0.574868\pi$
$62$	58.4181i	0.0151972i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 1786.57i	− 0.422856i
$66$	0	0
$67$	−952.113	−0.212099	−0.106050	−	0.994361i	$-0.533820\pi$
$67$	−952.113	−0.212099	−0.106050	+	0.994361i	$0.533820\pi$
$68$	309.921i	0.0670244i
$69$	0	0
$70$	0	0
$71$	− 3845.91i	− 0.762926i	−0.924384	−	0.381463i	$-0.875420\pi$
$71$	− 3845.91i	− 0.762926i	0.924384	−	0.381463i	$-0.124580\pi$
$72$	0	0
$73$	−3605.64	−0.676607	−0.338303	−	0.941037i	$-0.609853\pi$
$73$	−3605.64	−0.676607	−0.338303	+	0.941037i	$0.609853\pi$
$74$	3849.66i	0.703007i
$75$	0	0
$76$	2874.26	0.497621
$77$	0	0
$78$	0	0
$79$	−6839.18	−1.09585	−0.547923	−	0.836529i	$-0.684581\pi$
$79$	−6839.18	−1.09585	−0.547923	+	0.836529i	$0.684581\pi$
$80$	− 811.028i	− 0.126723i
$81$	0	0
$82$	−6252.69	−0.929907
$83$	9448.26i	1.37150i	0.727837	+	0.685750i	$0.240526\pi$
$83$	9448.26i	1.37150i	−0.727837	+	0.685750i	$0.759474\pi$
$84$	0	0
$85$	−490.927	−0.0679483
$86$	4126.85i	0.557984i
$87$	0	0
$88$	−3766.60	−0.486389
$89$	− 13655.2i	− 1.72392i	−0.506976	−	0.861960i	$-0.669237\pi$
$89$	− 13655.2i	− 1.72392i	0.506976	−	0.861960i	$-0.330763\pi$
$90$	0	0
$91$	0	0
$92$	177.695i	0.0209943i
$93$	0	0
$94$	−4003.49	−0.453088
$95$	4552.94i	0.504481i
$96$	0	0
$97$	10360.4	1.10112	0.550561	−	0.834795i	$-0.314414\pi$
$97$	10360.4	1.10112	0.550561	+	0.834795i	$0.314414\pi$
$98$	0	0
$99$	0	0
$100$	−3715.30	−0.371530
$101$	13850.7i	1.35778i	0.734240	+	0.678890i	$0.237539\pi$
$101$	13850.7i	1.35778i	−0.734240	+	0.678890i	$0.762461\pi$
$102$	0	0
$103$	15678.8	1.47788	0.738941	−	0.673771i	$-0.235326\pi$
$103$	15678.8	1.47788	0.738941	+	0.673771i	$0.235326\pi$
$104$	3190.06i	0.294939i
$105$	0	0
$106$	−7330.60	−0.652421
$107$	9059.88i	0.791325i	0.918396	+	0.395663i	$0.129485\pi$
$107$	9059.88i	0.791325i	−0.918396	+	0.395663i	$0.870515\pi$
$108$	0	0
$109$	−8842.23	−0.744232	−0.372116	−	0.928186i	$-0.621368\pi$
$109$	−8842.23	−0.744232	−0.372116	+	0.928186i	$0.621368\pi$
$110$	− 5966.44i	− 0.493094i
$111$	0	0
$112$	0	0
$113$	− 11117.8i	− 0.870688i	−0.900264	−	0.435344i	$-0.856627\pi$
$113$	− 11117.8i	− 0.870688i	0.900264	−	0.435344i	$-0.143373\pi$
$114$	0	0
$115$	−281.477	−0.0212837
$116$	666.167i	0.0495070i
$117$	0	0
$118$	7845.87	0.563478
$119$	0	0
$120$	0	0
$121$	−13068.5	−0.892596
$122$	4905.31i	0.329569i
$123$	0	0
$124$	165.231	0.0107461
$125$	− 13805.4i	− 0.883544i
$126$	0	0
$127$	−26937.3	−1.67012	−0.835059	−	0.550160i	$-0.814567\pi$
$127$	−26937.3	−1.67012	−0.835059	+	0.550160i	$0.814567\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	−5053.18	−0.299004
$131$	− 10883.3i	− 0.634191i	−0.948394	−	0.317095i	$-0.897292\pi$
$131$	− 10883.3i	− 0.634191i	0.948394	−	0.317095i	$-0.102708\pi$
$132$	0	0
$133$	0	0
$134$	2692.98i	0.149977i
$135$	0	0
$136$	876.588	0.0473934
$137$	− 6806.31i	− 0.362636i	−0.983425	−	0.181318i	$-0.941964\pi$
$137$	− 6806.31i	− 0.362636i	0.983425	−	0.181318i	$-0.0580363\pi$
$138$	0	0
$139$	−18840.7	−0.975139	−0.487570	−	0.873084i	$-0.662116\pi$
$139$	−18840.7	−0.975139	−0.487570	+	0.873084i	$0.662116\pi$
$140$	0	0
$141$	0	0
$142$	−10877.9	−0.539470
$143$	23468.1i	1.14764i
$144$	0	0
$145$	−1055.23	−0.0501895
$146$	10198.3i	0.478433i
$147$	0	0
$148$	10888.5	0.497101
$149$	15302.7i	0.689279i	0.938735	+	0.344640i	$0.111999\pi$
$149$	15302.7i	0.689279i	−0.938735	+	0.344640i	$0.888001\pi$
$150$	0	0
$151$	−44025.0	−1.93084	−0.965418	−	0.260708i	$-0.916044\pi$
$151$	−44025.0	−1.93084	−0.965418	+	0.260708i	$0.916044\pi$
$152$	− 8129.63i	− 0.351871i
$153$	0	0
$154$	0	0
$155$	261.733i	0.0108942i
$156$	0	0
$157$	−45077.6	−1.82878	−0.914390	−	0.404834i	$-0.867329\pi$
$157$	−45077.6	−1.82878	−0.914390	+	0.404834i	$0.867329\pi$
$158$	19344.1i	0.774880i
$159$	0	0
$160$	−2293.93	−0.0896068
$161$	0	0
$162$	0	0
$163$	−49796.7	−1.87424	−0.937119	−	0.349009i	$-0.886518\pi$
$163$	−49796.7	−1.87424	−0.937119	+	0.349009i	$0.886518\pi$
$164$	17685.3i	0.657543i
$165$	0	0
$166$	26723.7	0.969797
$167$	− 19878.6i	− 0.712775i	−0.934338	−	0.356388i	$-0.884008\pi$
$167$	− 19878.6i	− 0.712775i	0.934338	−	0.356388i	$-0.115992\pi$
$168$	0	0
$169$	−8685.10	−0.304090
$170$	1388.55i	0.0480467i
$171$	0	0
$172$	11672.5	0.394554
$173$	− 2800.14i	− 0.0935595i	−0.998905	−	0.0467797i	$-0.985104\pi$
$173$	− 2800.14i	− 0.0935595i	0.998905	−	0.0467797i	$-0.0148959\pi$
$174$	0	0
$175$	0	0
$176$	10653.5i	0.343929i
$177$	0	0
$178$	−38622.7	−1.21900
$179$	38943.0i	1.21541i	0.794163	+	0.607705i	$0.207910\pi$
$179$	38943.0i	1.21541i	−0.794163	+	0.607705i	$0.792090\pi$
$180$	0	0
$181$	45313.6	1.38316	0.691578	−	0.722302i	$-0.256916\pi$
$181$	45313.6	1.38316	0.691578	+	0.722302i	$0.256916\pi$
$182$	0	0
$183$	0	0
$184$	502.599	0.0148452
$185$	17247.8i	0.503953i
$186$	0	0
$187$	6448.74	0.184413
$188$	11323.6i	0.320382i
$189$	0	0
$190$	12877.6	0.356722
$191$	14139.4i	0.387582i	0.981043	+	0.193791i	$0.0620783\pi$
$191$	14139.4i	0.387582i	−0.981043	+	0.193791i	$0.937922\pi$
$192$	0	0
$193$	−18480.7	−0.496138	−0.248069	−	0.968742i	$-0.579796\pi$
$193$	−18480.7	−0.496138	−0.248069	+	0.968742i	$0.579796\pi$
$194$	− 29303.8i	− 0.778610i
$195$	0	0
$196$	0	0
$197$	29573.5i	0.762027i	0.924569	+	0.381014i	$0.124425\pi$
$197$	29573.5i	0.762027i	−0.924569	+	0.381014i	$0.875575\pi$
$198$	0	0
$199$	−41458.1	−1.04690	−0.523448	−	0.852058i	$-0.675355\pi$
$199$	−41458.1	−1.04690	−0.523448	+	0.852058i	$0.675355\pi$
$200$	10508.5i	0.262711i
$201$	0	0
$202$	39175.7	0.960095
$203$	0	0
$204$	0	0
$205$	−28014.2	−0.666608
$206$	− 44346.5i	− 1.04502i
$207$	0	0
$208$	9022.84	0.208553
$209$	− 59806.7i	− 1.36917i
$210$	0	0
$211$	29165.3	0.655090	0.327545	−	0.944836i	$-0.393779\pi$
$211$	29165.3	0.655090	0.327545	+	0.944836i	$0.393779\pi$
$212$	20734.1i	0.461331i
$213$	0	0
$214$	25625.2	0.559551
$215$	18489.7i	0.399993i
$216$	0	0
$217$	0	0
$218$	25009.6i	0.526252i
$219$	0	0
$220$	−16875.6	−0.348670
$221$	− 5461.65i	− 0.111825i
$222$	0	0
$223$	−18360.3	−0.369208	−0.184604	−	0.982813i	$-0.559100\pi$
$223$	−18360.3	−0.369208	−0.184604	+	0.982813i	$0.559100\pi$
$224$	0	0
$225$	0	0
$226$	−31445.9	−0.615670
$227$	59117.7i	1.14727i	0.819111	+	0.573635i	$0.194467\pi$
$227$	59117.7i	1.14727i	−0.819111	+	0.573635i	$0.805533\pi$
$228$	0	0
$229$	95795.1	1.82672	0.913361	−	0.407150i	$-0.133478\pi$
$229$	95795.1	1.82672	0.913361	+	0.407150i	$0.133478\pi$
$230$	796.136i	0.0150498i
$231$	0	0
$232$	1884.20	0.0350068
$233$	33711.0i	0.620955i	0.950581	+	0.310478i	$0.100489\pi$
$233$	33711.0i	0.620955i	−0.950581	+	0.310478i	$0.899511\pi$
$234$	0	0
$235$	−17937.0	−0.324798
$236$	− 22191.5i	− 0.398439i
$237$	0	0
$238$	0	0
$239$	5136.58i	0.0899246i	0.998989	+	0.0449623i	$0.0143168\pi$
$239$	5136.58i	0.0899246i	−0.998989	+	0.0449623i	$0.985683\pi$
$240$	0	0
$241$	−46159.8	−0.794748	−0.397374	−	0.917657i	$-0.630078\pi$
$241$	−46159.8	−0.794748	−0.397374	+	0.917657i	$0.630078\pi$
$242$	36963.3i	0.631161i
$243$	0	0
$244$	13874.3	0.233041
$245$	0	0
$246$	0	0
$247$	−50652.3	−0.830243
$248$	− 467.345i	− 0.00759861i
$249$	0	0
$250$	−39047.5	−0.624760
$251$	− 97957.3i	− 1.55485i	−0.628974	−	0.777426i	$-0.716525\pi$
$251$	− 97957.3i	− 1.55485i	0.628974	−	0.777426i	$-0.283475\pi$
$252$	0	0
$253$	3697.44	0.0577643
$254$	76190.3i	1.18095i
$255$	0	0
$256$	4096.00	0.0625000
$257$	77010.2i	1.16596i	0.812488	+	0.582978i	$0.198113\pi$
$257$	77010.2i	1.16596i	−0.812488	+	0.582978i	$0.801887\pi$
$258$	0	0
$259$	0	0
$260$	14292.5i	0.211428i
$261$	0	0
$262$	−30782.7	−0.448440
$263$	− 82173.7i	− 1.18801i	−0.804460	−	0.594007i	$-0.797545\pi$
$263$	− 82173.7i	− 1.18801i	0.804460	−	0.594007i	$-0.202455\pi$
$264$	0	0
$265$	−32843.6	−0.467691
$266$	0	0
$267$	0	0
$268$	7616.90	0.106050
$269$	115507.i	1.59626i	0.602488	+	0.798128i	$0.294176\pi$
$269$	115507.i	1.59626i	−0.602488	+	0.798128i	$0.705824\pi$
$270$	0	0
$271$	60189.3	0.819559	0.409780	−	0.912185i	$-0.365606\pi$
$271$	60189.3	0.819559	0.409780	+	0.912185i	$0.365606\pi$
$272$	− 2479.37i	− 0.0335122i
$273$	0	0
$274$	−19251.2	−0.256422
$275$	77306.9i	1.02224i
$276$	0	0
$277$	−113652.	−1.48122	−0.740609	−	0.671936i	$-0.765463\pi$
$277$	−113652.	−1.48122	−0.740609	+	0.671936i	$0.765463\pi$
$278$	53289.4i	0.689528i
$279$	0	0
$280$	0	0
$281$	− 64605.4i	− 0.818194i	−0.912491	−	0.409097i	$-0.865844\pi$
$281$	− 64605.4i	− 0.818194i	0.912491	−	0.409097i	$-0.134156\pi$
$282$	0	0
$283$	−75243.4	−0.939497	−0.469749	−	0.882800i	$-0.655655\pi$
$283$	−75243.4	−0.939497	−0.469749	+	0.882800i	$0.655655\pi$
$284$	30767.3i	0.381463i
$285$	0	0
$286$	66377.8	0.811504
$287$	0	0
$288$	0	0
$289$	82020.2	0.982031
$290$	2984.65i	0.0354893i
$291$	0	0
$292$	28845.1	0.338303
$293$	− 62577.9i	− 0.728930i	−0.931217	−	0.364465i	$-0.881252\pi$
$293$	− 62577.9i	− 0.728930i	0.931217	−	0.364465i	$-0.118748\pi$
$294$	0	0
$295$	35152.1	0.403932
$296$	− 30797.3i	− 0.351503i
$297$	0	0
$298$	43282.6	0.487394
$299$	− 3131.48i	− 0.0350274i
$300$	0	0
$301$	0	0
$302$	124521.i	1.36531i
$303$	0	0
$304$	−22994.1	−0.248810
$305$	21977.5i	0.236253i
$306$	0	0
$307$	−41292.1	−0.438117	−0.219059	−	0.975712i	$-0.570299\pi$
$307$	−41292.1	−0.438117	−0.219059	+	0.975712i	$0.570299\pi$
$308$	0	0
$309$	0	0
$310$	740.293	0.00770336
$311$	− 18468.2i	− 0.190943i	−0.995432	−	0.0954715i	$-0.969564\pi$
$311$	− 18468.2i	− 0.190943i	0.995432	−	0.0954715i	$-0.0304359\pi$
$312$	0	0
$313$	70159.4	0.716139	0.358069	−	0.933695i	$-0.383435\pi$
$313$	70159.4	0.716139	0.358069	+	0.933695i	$0.383435\pi$
$314$	127499.i	1.29314i
$315$	0	0
$316$	54713.4	0.547923
$317$	− 36037.5i	− 0.358622i	−0.983792	−	0.179311i	$-0.942613\pi$
$317$	− 36037.5i	− 0.358622i	0.983792	−	0.179311i	$-0.0573868\pi$
$318$	0	0
$319$	13861.4	0.136215
$320$	6488.23i	0.0633616i
$321$	0	0
$322$	0	0
$323$	13918.6i	0.133411i
$324$	0	0
$325$	65473.7	0.619870
$326$	140846.i	1.32529i
$327$	0	0
$328$	50021.5	0.464953
$329$	0	0
$330$	0	0
$331$	32479.8	0.296454	0.148227	−	0.988953i	$-0.452643\pi$
$331$	32479.8	0.296454	0.148227	+	0.988953i	$0.452643\pi$
$332$	− 75586.1i	− 0.685750i
$333$	0	0
$334$	−56225.1	−0.504008
$335$	12065.5i	0.107511i
$336$	0	0
$337$	−156861.	−1.38119	−0.690597	−	0.723240i	$-0.742652\pi$
$337$	−156861.	−1.38119	−0.690597	+	0.723240i	$0.742652\pi$
$338$	24565.2i	0.215024i
$339$	0	0
$340$	3927.41	0.0339742
$341$	− 3438.09i	− 0.0295671i
$342$	0	0
$343$	0	0
$344$	− 33014.8i	− 0.278992i
$345$	0	0
$346$	−7920.00	−0.0661565
$347$	− 97556.6i	− 0.810210i	−0.914270	−	0.405105i	$-0.867235\pi$
$347$	− 97556.6i	− 0.810210i	0.914270	−	0.405105i	$-0.132765\pi$
$348$	0	0
$349$	−84314.3	−0.692230	−0.346115	−	0.938192i	$-0.612499\pi$
$349$	−84314.3	−0.692230	−0.346115	+	0.938192i	$0.612499\pi$
$350$	0	0
$351$	0	0
$352$	30132.8	0.243195
$353$	− 161034.i	− 1.29232i	−0.763204	−	0.646158i	$-0.776375\pi$
$353$	− 161034.i	− 1.29232i	0.763204	−	0.646158i	$-0.223625\pi$
$354$	0	0
$355$	−48736.6	−0.386722
$356$	109241.i	0.861960i
$357$	0	0
$358$	110147.	0.859425
$359$	124144.i	0.963246i	0.876378	+	0.481623i	$0.159953\pi$
$359$	124144.i	0.963246i	−0.876378	+	0.481623i	$0.840047\pi$
$360$	0	0
$361$	−1237.34	−0.00949457
$362$	− 128166.i	− 0.978039i
$363$	0	0
$364$	0	0
$365$	45691.8i	0.342967i
$366$	0	0
$367$	94483.2	0.701492	0.350746	−	0.936471i	$-0.385928\pi$
$367$	94483.2	0.701492	0.350746	+	0.936471i	$0.385928\pi$
$368$	− 1421.56i	− 0.0104971i
$369$	0	0
$370$	48784.2	0.356349
$371$	0	0
$372$	0	0
$373$	−248173.	−1.78376	−0.891881	−	0.452270i	$-0.850615\pi$
$373$	−248173.	−1.78376	−0.891881	+	0.452270i	$0.850615\pi$
$374$	− 18239.8i	− 0.130400i
$375$	0	0
$376$	32027.9	0.226544
$377$	− 11739.7i	− 0.0825988i
$378$	0	0
$379$	−16745.0	−0.116575	−0.0582877	−	0.998300i	$-0.518564\pi$
$379$	−16745.0	−0.116575	−0.0582877	+	0.998300i	$0.518564\pi$
$380$	− 36423.5i	− 0.252240i
$381$	0	0
$382$	39992.2	0.274062
$383$	130421.i	0.889101i	0.895754	+	0.444550i	$0.146637\pi$
$383$	130421.i	0.889101i	−0.895754	+	0.444550i	$0.853363\pi$
$384$	0	0
$385$	0	0
$386$	52271.2i	0.350823i
$387$	0	0
$388$	−82883.6	−0.550561
$389$	− 45385.9i	− 0.299931i	−0.988691	−	0.149966i	$-0.952084\pi$
$389$	− 45385.9i	− 0.299931i	0.988691	−	0.149966i	$-0.0479163\pi$
$390$	0	0
$391$	−860.493	−0.00562851
$392$	0	0
$393$	0	0
$394$	83646.5	0.538835
$395$	86668.2i	0.555476i
$396$	0	0
$397$	58752.9	0.372776	0.186388	−	0.982476i	$-0.440322\pi$
$397$	58752.9	0.372776	0.186388	+	0.982476i	$0.440322\pi$
$398$	117261.i	0.740267i
$399$	0	0
$400$	29722.4	0.185765
$401$	− 269379.i	− 1.67523i	−0.546258	−	0.837617i	$-0.683948\pi$
$401$	− 269379.i	− 1.67523i	0.546258	−	0.837617i	$-0.316052\pi$
$402$	0	0
$403$	−2911.83	−0.0179290
$404$	− 110806.i	− 0.678890i
$405$	0	0
$406$	0	0
$407$	− 226565.i	− 1.36774i
$408$	0	0
$409$	308049.	1.84150	0.920752	−	0.390148i	$-0.127576\pi$
$409$	308049.	1.84150	0.920752	+	0.390148i	$0.127576\pi$
$410$	79236.1i	0.471363i
$411$	0	0
$412$	−125431.	−0.738941
$413$	0	0
$414$	0	0
$415$	119731.	0.695203
$416$	− 25520.5i	− 0.147469i
$417$	0	0
$418$	−169159.	−0.968150
$419$	− 234381.i	− 1.33504i	−0.744592	−	0.667520i	$-0.767356\pi$
$419$	− 234381.i	− 1.33504i	0.744592	−	0.667520i	$-0.232644\pi$
$420$	0	0
$421$	−120240.	−0.678401	−0.339200	−	0.940714i	$-0.610156\pi$
$421$	−120240.	−0.678401	−0.339200	+	0.940714i	$0.610156\pi$
$422$	− 82491.8i	− 0.463219i
$423$	0	0
$424$	58644.8	0.326210
$425$	− 17991.4i	− 0.0996063i
$426$	0	0
$427$	0	0
$428$	− 72479.1i	− 0.395663i
$429$	0	0
$430$	52296.8	0.282838
$431$	− 229940.i	− 1.23783i	−0.785460	−	0.618913i	$-0.787573\pi$
$431$	− 229940.i	− 1.23783i	0.785460	−	0.618913i	$-0.212427\pi$
$432$	0	0
$433$	−297018.	−1.58419	−0.792095	−	0.610398i	$-0.791010\pi$
$433$	−297018.	−1.58419	−0.792095	+	0.610398i	$0.791010\pi$
$434$	0	0
$435$	0	0
$436$	70737.8	0.372116
$437$	7980.35i	0.0417887i
$438$	0	0
$439$	−281472.	−1.46051	−0.730257	−	0.683172i	$-0.760600\pi$
$439$	−281472.	−1.46051	−0.730257	+	0.683172i	$0.760600\pi$
$440$	47731.5i	0.246547i
$441$	0	0
$442$	−15447.9	−0.0790723
$443$	− 332821.i	− 1.69591i	−0.530068	−	0.847955i	$-0.677833\pi$
$443$	− 332821.i	− 1.69591i	0.530068	−	0.847955i	$-0.322167\pi$
$444$	0	0
$445$	−173043.	−0.873842
$446$	51930.9i	0.261069i
$447$	0	0
$448$	0	0
$449$	− 13253.5i	− 0.0657413i	−0.999460	−	0.0328706i	$-0.989535\pi$
$449$	− 13253.5i	− 0.0657413i	0.999460	−	0.0328706i	$-0.0104649\pi$
$450$	0	0
$451$	367990.	1.80919
$452$	88942.6i	0.435344i
$453$	0	0
$454$	167210.	0.811243
$455$	0	0
$456$	0	0
$457$	267097.	1.27890	0.639449	−	0.768833i	$-0.279162\pi$
$457$	267097.	1.27890	0.639449	+	0.768833i	$0.279162\pi$
$458$	− 270950.i	− 1.29169i
$459$	0	0
$460$	2251.81	0.0106418
$461$	3390.66i	0.0159545i	0.999968	+	0.00797724i	$0.00253926\pi$
$461$	3390.66i	0.0159545i	−0.999968	+	0.00797724i	$0.997461\pi$
$462$	0	0
$463$	−102480.	−0.478054	−0.239027	−	0.971013i	$-0.576828\pi$
$463$	−102480.	−0.478054	−0.239027	+	0.971013i	$0.576828\pi$
$464$	− 5329.33i	− 0.0247535i
$465$	0	0
$466$	95349.2	0.439082
$467$	− 62515.0i	− 0.286649i	−0.989676	−	0.143324i	$-0.954221\pi$
$467$	− 62515.0i	− 0.286649i	0.989676	−	0.143324i	$-0.0457793\pi$
$468$	0	0
$469$	0	0
$470$	50733.4i	0.229667i
$471$	0	0
$472$	−62766.9	−0.281739
$473$	− 242878.i	− 1.08559i
$474$	0	0
$475$	−166855.	−0.739524
$476$	0	0
$477$	0	0
$478$	14528.5	0.0635863
$479$	247519.i	1.07879i	0.842052	+	0.539396i	$0.181347\pi$
$479$	247519.i	1.07879i	−0.842052	+	0.539396i	$0.818653\pi$
$480$	0	0
$481$	−191885.	−0.829375
$482$	130560.i	0.561972i
$483$	0	0
$484$	104548.	0.446298
$485$	− 131291.i	− 0.558150i
$486$	0	0
$487$	258363.	1.08936	0.544680	−	0.838644i	$-0.316651\pi$
$487$	258363.	1.08936	0.544680	+	0.838644i	$0.316651\pi$
$488$	− 39242.5i	− 0.164785i
$489$	0	0
$490$	0	0
$491$	− 239798.i	− 0.994676i	−0.867557	−	0.497338i	$-0.834311\pi$
$491$	− 239798.i	− 0.994676i	0.867557	−	0.497338i	$-0.165689\pi$
$492$	0	0
$493$	−3225.92	−0.0132727
$494$	143266.i	0.587070i
$495$	0	0
$496$	−1321.85	−0.00537303
$497$	0	0
$498$	0	0
$499$	33013.7	0.132584	0.0662922	−	0.997800i	$-0.478883\pi$
$499$	33013.7	0.132584	0.0662922	+	0.997800i	$0.478883\pi$
$500$	110443.i	0.441772i
$501$	0	0
$502$	−277065.	−1.09945
$503$	260862.i	1.03104i	0.856878	+	0.515519i	$0.172401\pi$
$503$	260862.i	1.03104i	−0.856878	+	0.515519i	$0.827599\pi$
$504$	0	0
$505$	175521.	0.688248
$506$	− 10457.9i	− 0.0408455i
$507$	0	0
$508$	215499.	0.835059
$509$	− 60997.8i	− 0.235439i	−0.993047	−	0.117720i	$-0.962442\pi$
$509$	− 60997.8i	− 0.235439i	0.993047	−	0.117720i	$-0.0375584\pi$
$510$	0	0
$511$	0	0
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	217818.	0.824455
$515$	− 198687.i	− 0.749127i
$516$	0	0
$517$	235618.	0.881509
$518$	0	0
$519$	0	0
$520$	40425.4	0.149502
$521$	− 365690.i	− 1.34722i	−0.739087	−	0.673609i	$-0.764743\pi$
$521$	− 365690.i	− 1.34722i	0.739087	−	0.673609i	$-0.235257\pi$
$522$	0	0
$523$	−101715.	−0.371863	−0.185931	−	0.982563i	$-0.559530\pi$
$523$	−101715.	−0.371863	−0.185931	+	0.982563i	$0.559530\pi$
$524$	87066.8i	0.317095i
$525$	0	0
$526$	−232422.	−0.840053
$527$	800.135i	0.00288099i
$528$	0	0
$529$	279348.	0.998237
$530$	92895.7i	0.330707i
$531$	0	0
$532$	0	0
$533$	− 311663.i	− 1.09706i
$534$	0	0
$535$	114810.	0.401117
$536$	− 21543.9i	− 0.0749884i
$537$	0	0
$538$	326702.	1.12872
$539$	0	0
$540$	0	0
$541$	288439.	0.985508	0.492754	−	0.870169i	$-0.335990\pi$
$541$	288439.	0.985508	0.492754	+	0.870169i	$0.335990\pi$
$542$	− 170241.i	− 0.579516i
$543$	0	0
$544$	−7012.71	−0.0236967
$545$	112051.i	0.377246i
$546$	0	0
$547$	122429.	0.409175	0.204587	−	0.978848i	$-0.434415\pi$
$547$	122429.	0.409175	0.204587	+	0.978848i	$0.434415\pi$
$548$	54450.5i	0.181318i
$549$	0	0
$550$	218657.	0.722833
$551$	29917.7i	0.0985429i
$552$	0	0
$553$	0	0
$554$	321457.i	1.04738i
$555$	0	0
$556$	150725.	0.487570
$557$	− 188468.i	− 0.607473i	−0.952756	−	0.303736i	$-0.901766\pi$
$557$	− 188468.i	− 0.607473i	0.952756	−	0.303736i	$-0.0982342\pi$
$558$	0	0
$559$	−205701.	−0.658284
$560$	0	0
$561$	0	0
$562$	−182732.	−0.578550
$563$	− 274483.i	− 0.865960i	−0.901403	−	0.432980i	$-0.857462\pi$
$563$	− 274483.i	− 0.865960i	0.901403	−	0.432980i	$-0.142538\pi$
$564$	0	0
$565$	−140889.	−0.441346
$566$	212820.i	0.664325i
$567$	0	0
$568$	87023.0	0.269735
$569$	− 296571.i	− 0.916018i	−0.888948	−	0.458009i	$-0.848563\pi$
$569$	− 296571.i	− 0.916018i	0.888948	−	0.458009i	$-0.151437\pi$
$570$	0	0
$571$	34109.6	0.104618	0.0523088	−	0.998631i	$-0.483342\pi$
$571$	34109.6	0.104618	0.0523088	+	0.998631i	$0.483342\pi$
$572$	− 187745.i	− 0.573820i
$573$	0	0
$574$	0	0
$575$	− 10315.5i	− 0.0312000i
$576$	0	0
$577$	304240.	0.913828	0.456914	−	0.889511i	$-0.348955\pi$
$577$	304240.	0.913828	0.456914	+	0.889511i	$0.348955\pi$
$578$	− 231988.i	− 0.694401i
$579$	0	0
$580$	8441.87	0.0250948
$581$	0	0
$582$	0	0
$583$	431428.	1.26932
$584$	− 81586.3i	− 0.239217i
$585$	0	0
$586$	−176997.	−0.515431
$587$	− 200013.i	− 0.580474i	−0.956955	−	0.290237i	$-0.906266\pi$
$587$	− 200013.i	− 0.580474i	0.956955	−	0.290237i	$-0.0937341\pi$
$588$	0	0
$589$	7420.59	0.0213898
$590$	− 99425.3i	− 0.285623i
$591$	0	0
$592$	−87108.0	−0.248550
$593$	472577.i	1.34389i	0.740602	+	0.671944i	$0.234540\pi$
$593$	472577.i	1.34389i	−0.740602	+	0.671944i	$0.765460\pi$
$594$	0	0
$595$	0	0
$596$	− 122422.i	− 0.344640i
$597$	0	0
$598$	−8857.17	−0.0247681
$599$	254544.i	0.709429i	0.934975	+	0.354714i	$0.115422\pi$
$599$	254544.i	0.709429i	−0.934975	+	0.354714i	$0.884578\pi$
$600$	0	0
$601$	−46871.2	−0.129765	−0.0648824	−	0.997893i	$-0.520667\pi$
$601$	−46871.2	−0.129765	−0.0648824	+	0.997893i	$0.520667\pi$
$602$	0	0
$603$	0	0
$604$	352200.	0.965418
$605$	165608.i	0.452451i
$606$	0	0
$607$	299526.	0.812937	0.406468	−	0.913665i	$-0.366760\pi$
$607$	299526.	0.812937	0.406468	+	0.913665i	$0.366760\pi$
$608$	65037.0i	0.175935i
$609$	0	0
$610$	62161.6	0.167056
$611$	− 199552.i	− 0.534533i
$612$	0	0
$613$	−192711.	−0.512845	−0.256422	−	0.966565i	$-0.582544\pi$
$613$	−192711.	−0.512845	−0.256422	+	0.966565i	$0.582544\pi$
$614$	116792.i	0.309796i
$615$	0	0
$616$	0	0
$617$	− 470863.i	− 1.23687i	−0.785835	−	0.618436i	$-0.787767\pi$
$617$	− 470863.i	− 1.23687i	0.785835	−	0.618436i	$-0.212233\pi$
$618$	0	0
$619$	518245.	1.35255	0.676276	−	0.736648i	$-0.263593\pi$
$619$	518245.	1.35255	0.676276	+	0.736648i	$0.263593\pi$
$620$	− 2093.86i	− 0.00544710i
$621$	0	0
$622$	−52235.9	−0.135017
$623$	0	0
$624$	0	0
$625$	115312.	0.295198
$626$	− 198441.i	− 0.506386i
$627$	0	0
$628$	360621.	0.914390
$629$	52727.7i	0.133272i
$630$	0	0
$631$	−264043.	−0.663157	−0.331578	−	0.943428i	$-0.607581\pi$
$631$	−264043.	−0.663157	−0.331578	+	0.943428i	$0.607581\pi$
$632$	− 154753.i	− 0.387440i
$633$	0	0
$634$	−101930.	−0.253584
$635$	341358.i	0.846570i
$636$	0	0
$637$	0	0
$638$	− 39206.0i	− 0.0963188i
$639$	0	0
$640$	18351.5	0.0448034
$641$	− 757333.i	− 1.84319i	−0.388149	−	0.921597i	$-0.626885\pi$
$641$	− 757333.i	− 1.84319i	0.388149	−	0.921597i	$-0.373115\pi$
$642$	0	0
$643$	−317514.	−0.767965	−0.383982	−	0.923340i	$-0.625448\pi$
$643$	−317514.	−0.767965	−0.383982	+	0.923340i	$0.625448\pi$
$644$	0	0
$645$	0	0
$646$	39367.8	0.0943358
$647$	41488.0i	0.0991092i	0.998771	+	0.0495546i	$0.0157802\pi$
$647$	41488.0i	0.0991092i	−0.998771	+	0.0495546i	$0.984220\pi$
$648$	0	0
$649$	−461754.	−1.09628
$650$	− 185188.i	− 0.438314i
$651$	0	0
$652$	398373.	0.937119
$653$	226789.i	0.531857i	0.963993	+	0.265928i	$0.0856785\pi$
$653$	226789.i	0.531857i	−0.963993	+	0.265928i	$0.914321\pi$
$654$	0	0
$655$	−137917.	−0.321467
$656$	− 141482.i	− 0.328772i
$657$	0	0
$658$	0	0
$659$	348934.i	0.803474i	0.915755	+	0.401737i	$0.131593\pi$
$659$	348934.i	0.803474i	−0.915755	+	0.401737i	$0.868407\pi$
$660$	0	0
$661$	−739225.	−1.69190	−0.845948	−	0.533266i	$-0.820964\pi$
$661$	−739225.	−1.69190	−0.845948	+	0.533266i	$0.820964\pi$
$662$	− 91866.7i	− 0.209624i
$663$	0	0
$664$	−213790.	−0.484898
$665$	0	0
$666$	0	0
$667$	−1849.61	−0.00415746
$668$	159029.i	0.356388i
$669$	0	0
$670$	34126.3	0.0760221
$671$	− 288693.i	− 0.641196i
$672$	0	0
$673$	−4304.77	−0.00950429	−0.00475214	−	0.999989i	$-0.501513\pi$
$673$	−4304.77	−0.00950429	−0.00475214	+	0.999989i	$0.501513\pi$
$674$	443670.i	0.976652i
$675$	0	0
$676$	69480.8	0.152045
$677$	392165.i	0.855641i	0.903864	+	0.427820i	$0.140718\pi$
$677$	392165.i	0.855641i	−0.903864	+	0.427820i	$0.859282\pi$
$678$	0	0
$679$	0	0
$680$	− 11108.4i	− 0.0240234i
$681$	0	0
$682$	−9724.38	−0.0209071
$683$	881336.i	1.88930i	0.328085	+	0.944648i	$0.393597\pi$
$683$	881336.i	1.88930i	−0.328085	+	0.944648i	$0.606403\pi$
$684$	0	0
$685$	−86251.8	−0.183817
$686$	0	0
$687$	0	0
$688$	−93380.0	−0.197277
$689$	− 365391.i	− 0.769696i
$690$	0	0
$691$	−244166.	−0.511364	−0.255682	−	0.966761i	$-0.582300\pi$
$691$	−244166.	−0.511364	−0.255682	+	0.966761i	$0.582300\pi$
$692$	22401.1i	0.0467797i
$693$	0	0
$694$	−275932.	−0.572905
$695$	238755.i	0.494291i
$696$	0	0
$697$	−85641.2	−0.176286
$698$	238477.i	0.489480i
$699$	0	0
$700$	0	0
$701$	651477.i	1.32575i	0.748728	+	0.662877i	$0.230665\pi$
$701$	651477.i	1.32575i	−0.748728	+	0.662877i	$0.769335\pi$
$702$	0	0
$703$	489005.	0.989471
$704$	− 85228.4i	− 0.171965i
$705$	0	0
$706$	−455473.	−0.913805
$707$	0	0
$708$	0	0
$709$	−509544.	−1.01365	−0.506827	−	0.862048i	$-0.669182\pi$
$709$	−509544.	−1.01365	−0.506827	+	0.862048i	$0.669182\pi$
$710$	137848.i	0.273453i
$711$	0	0
$712$	308981.	0.609498
$713$	458.764i	0 0.000902422i
$714$	0	0
$715$	297395.	0.581730
$716$	− 311544.i	− 0.607705i
$717$	0	0
$718$	351133.	0.681118
$719$	− 449003.i	− 0.868544i	−0.900782	−	0.434272i	$-0.857006\pi$
$719$	− 449003.i	− 0.868544i	0.900782	−	0.434272i	$-0.142994\pi$
$720$	0	0
$721$	0	0
$722$	3499.73i	0.00671367i
$723$	0	0
$724$	−362509.	−0.691578
$725$	− 38672.0i	− 0.0735734i
$726$	0	0
$727$	−154130.	−0.291620	−0.145810	−	0.989313i	$-0.546579\pi$
$727$	−154130.	−0.291620	−0.145810	+	0.989313i	$0.546579\pi$
$728$	0	0
$729$	0	0
$730$	129236.	0.242514
$731$	56524.2i	0.105779i
$732$	0	0
$733$	807161.	1.50228	0.751142	−	0.660141i	$-0.229503\pi$
$733$	807161.	1.50228	0.751142	+	0.660141i	$0.229503\pi$
$734$	− 267239.i	− 0.496030i
$735$	0	0
$736$	−4020.79	−0.00742259
$737$	− 158490.i	− 0.291788i
$738$	0	0
$739$	127146.	0.232816	0.116408	−	0.993201i	$-0.462862\pi$
$739$	127146.	0.232816	0.116408	+	0.993201i	$0.462862\pi$
$740$	− 137982.i	− 0.251977i
$741$	0	0
$742$	0	0
$743$	930586.i	1.68569i	0.538153	+	0.842847i	$0.319122\pi$
$743$	930586.i	1.68569i	−0.538153	+	0.842847i	$0.680878\pi$
$744$	0	0
$745$	193921.	0.349391
$746$	701940.i	1.26131i
$747$	0	0
$748$	−51590.0	−0.0922066
$749$	0	0
$750$	0	0
$751$	353228.	0.626289	0.313145	−	0.949705i	$-0.398618\pi$
$751$	353228.	0.626289	0.313145	+	0.949705i	$0.398618\pi$
$752$	− 90588.6i	− 0.160191i
$753$	0	0
$754$	−33204.8	−0.0584062
$755$	557898.i	0.978726i
$756$	0	0
$757$	258673.	0.451398	0.225699	−	0.974197i	$-0.427533\pi$
$757$	258673.	0.451398	0.225699	+	0.974197i	$0.427533\pi$
$758$	47362.1i	0.0824313i
$759$	0	0
$760$	−103021.	−0.178361
$761$	44043.2i	0.0760518i	0.999277	+	0.0380259i	$0.0121069\pi$
$761$	44043.2i	0.0760518i	−0.999277	+	0.0380259i	$0.987893\pi$
$762$	0	0
$763$	0	0
$764$	− 113115.i	− 0.193791i
$765$	0	0
$766$	368887.	0.628689
$767$	391074.i	0.664766i
$768$	0	0
$769$	438155.	0.740927	0.370463	−	0.928847i	$-0.379199\pi$
$769$	438155.	0.740927	0.370463	+	0.928847i	$0.379199\pi$
$770$	0	0
$771$	0	0
$772$	147845.	0.248069
$773$	− 759856.i	− 1.27166i	−0.771827	−	0.635832i	$-0.780657\pi$
$773$	− 759856.i	− 1.27166i	0.771827	−	0.635832i	$-0.219343\pi$
$774$	0	0
$775$	−9591.94	−0.0159699
$776$	234430.i	0.389305i
$777$	0	0
$778$	−128371.	−0.212083
$779$	794251.i	1.30883i
$780$	0	0
$781$	640197.	1.04957
$782$	2433.84i	0.00397996i
$783$	0	0
$784$	0	0
$785$	571238.i	0.926995i
$786$	0	0
$787$	−865390.	−1.39721	−0.698606	−	0.715507i	$-0.746196\pi$
$787$	−865390.	−1.39721	−0.698606	+	0.715507i	$0.746196\pi$
$788$	− 236588.i	− 0.381014i
$789$	0	0
$790$	245135.	0.392781
$791$	0	0
$792$	0	0
$793$	−244503.	−0.388811
$794$	− 166178.i	− 0.263593i
$795$	0	0
$796$	331665.	0.523448
$797$	− 435436.i	− 0.685501i	−0.939427	−	0.342750i	$-0.888642\pi$
$797$	− 435436.i	− 0.685501i	0.939427	−	0.342750i	$-0.111358\pi$
$798$	0	0
$799$	−54834.5	−0.0858936
$800$	− 84067.6i	− 0.131356i
$801$	0	0
$802$	−761920.	−1.18457
$803$	− 600201.i	− 0.930819i
$804$	0	0
$805$	0	0
$806$	8235.90i	0.0126777i
$807$	0	0
$808$	−313406.	−0.480047
$809$	593479.i	0.906794i	0.891309	+	0.453397i	$0.149788\pi$
$809$	593479.i	0.906794i	−0.891309	+	0.453397i	$0.850212\pi$
$810$	0	0
$811$	95960.2	0.145898	0.0729490	−	0.997336i	$-0.476759\pi$
$811$	95960.2	0.145898	0.0729490	+	0.997336i	$0.476759\pi$
$812$	0	0
$813$	0	0
$814$	−640822.	−0.967138
$815$	631039.i	0.950038i
$816$	0	0
$817$	524215.	0.785354
$818$	− 871293.i	− 1.30214i
$819$	0	0
$820$	224114.	0.333304
$821$	280951.i	0.416816i	0.978042	+	0.208408i	$0.0668282\pi$
$821$	280951.i	0.416816i	−0.978042	+	0.208408i	$0.933172\pi$
$822$	0	0
$823$	−579343.	−0.855335	−0.427668	−	0.903936i	$-0.640665\pi$
$823$	−579343.	−0.855335	−0.427668	+	0.903936i	$0.640665\pi$
$824$	354772.i	0.522510i
$825$	0	0
$826$	0	0
$827$	73153.5i	0.106961i	0.998569	+	0.0534803i	$0.0170314\pi$
$827$	73153.5i	0.106961i	−0.998569	+	0.0534803i	$0.982969\pi$
$828$	0	0
$829$	−1.13687e6	−1.65425	−0.827125	−	0.562018i	$-0.810025\pi$
$829$	−1.13687e6	−1.65425	−0.827125	+	0.562018i	$0.810025\pi$
$830$	− 338651.i	− 0.491583i
$831$	0	0
$832$	−72182.7	−0.104277
$833$	0	0
$834$	0	0
$835$	−251908.	−0.361300
$836$	478454.i	0.684585i
$837$	0	0
$838$	−662929.	−0.944015
$839$	− 530489.i	− 0.753620i	−0.926291	−	0.376810i	$-0.877021\pi$
$839$	− 530489.i	− 0.753620i	0.926291	−	0.376810i	$-0.122979\pi$
$840$	0	0
$841$	700347.	0.990196
$842$	340091.i	0.479702i
$843$	0	0
$844$	−233322.	−0.327545
$845$	110060.i	0.154141i
$846$	0	0
$847$	0	0
$848$	− 165873.i	− 0.230666i
$849$	0	0
$850$	−50887.3	−0.0704323
$851$	30231.8i	0.0417451i
$852$	0	0
$853$	160401.	0.220450	0.110225	−	0.993907i	$-0.464843\pi$
$853$	160401.	0.220450	0.110225	+	0.993907i	$0.464843\pi$
$854$	0	0
$855$	0	0
$856$	−205002.	−0.279776
$857$	566167.i	0.770873i	0.922734	+	0.385436i	$0.125949\pi$
$857$	566167.i	0.770873i	−0.922734	+	0.385436i	$0.874051\pi$
$858$	0	0
$859$	993543.	1.34648	0.673241	−	0.739424i	$-0.264902\pi$
$859$	993543.	1.34648	0.673241	+	0.739424i	$0.264902\pi$
$860$	− 147918.i	− 0.199997i
$861$	0	0
$862$	−650368.	−0.875275
$863$	341066.i	0.457949i	0.973432	+	0.228974i	$0.0735372\pi$
$863$	341066.i	0.457949i	−0.973432	+	0.228974i	$0.926463\pi$
$864$	0	0
$865$	−35484.3	−0.0474246
$866$	840094.i	1.12019i
$867$	0	0
$868$	0	0
$869$	− 1.13846e6i	− 1.50757i
$870$	0	0
$871$	−134231.	−0.176936
$872$	− 200077.i	− 0.263126i
$873$	0	0
$874$	22571.8	0.0295491
$875$	0	0
$876$	0	0
$877$	−1.40566e6	−1.82760	−0.913798	−	0.406169i	$-0.866864\pi$
$877$	−1.40566e6	−1.82760	−0.913798	+	0.406169i	$0.866864\pi$
$878$	796123.i	1.03274i
$879$	0	0
$880$	135005.	0.174335
$881$	1.09152e6i	1.40631i	0.711036	+	0.703155i	$0.248226\pi$
$881$	1.09152e6i	1.40631i	−0.711036	+	0.703155i	$0.751774\pi$
$882$	0	0
$883$	60422.5	0.0774957	0.0387478	−	0.999249i	$-0.487663\pi$
$883$	60422.5	0.0774957	0.0387478	+	0.999249i	$0.487663\pi$
$884$	43693.2i	0.0559126i
$885$	0	0
$886$	−941359.	−1.19919
$887$	− 362657.i	− 0.460945i	−0.973079	−	0.230472i	$-0.925973\pi$
$887$	− 362657.i	− 0.460945i	0.973079	−	0.230472i	$-0.0740271\pi$
$888$	0	0
$889$	0	0
$890$	489438.i	0.617900i
$891$	0	0
$892$	146883.	0.184604
$893$	508545.i	0.637714i
$894$	0	0
$895$	493498.	0.616083
$896$	0	0
$897$	0	0
$898$	−37486.6	−0.0464861
$899$	1719.87i	0.00212802i
$900$	0	0
$901$	−100405.	−0.123682
$902$	− 1.04083e6i	− 1.27929i
$903$	0	0
$904$	251568.	0.307835
$905$	− 574228.i	− 0.701112i
$906$	0	0
$907$	−24950.6	−0.0303296	−0.0151648	−	0.999885i	$-0.504827\pi$
$907$	−24950.6	−0.0303296	−0.0151648	+	0.999885i	$0.504827\pi$
$908$	− 472942.i	− 0.573635i
$909$	0	0
$910$	0	0
$911$	− 657747.i	− 0.792542i	−0.918134	−	0.396271i	$-0.870304\pi$
$911$	− 657747.i	− 0.792542i	0.918134	−	0.396271i	$-0.129696\pi$
$912$	0	0
$913$	−1.57277e6	−1.88679
$914$	− 755464.i	− 0.904318i
$915$	0	0
$916$	−766361.	−0.913361
$917$	0	0
$918$	0	0
$919$	565588.	0.669683	0.334842	−	0.942274i	$-0.391317\pi$
$919$	565588.	0.669683	0.334842	+	0.942274i	$0.391317\pi$
$920$	− 6369.09i	− 0.00752492i
$921$	0	0
$922$	9590.25	0.0112815
$923$	− 542204.i	− 0.636442i
$924$	0	0
$925$	−632094.	−0.738751
$926$	289857.i	0.338035i
$927$	0	0
$928$	−15073.6	−0.0175034
$929$	338139.i	0.391800i	0.980624	+	0.195900i	$0.0627628\pi$
$929$	338139.i	0.391800i	−0.980624	+	0.195900i	$0.937237\pi$
$930$	0	0
$931$	0	0
$932$	− 269688.i	− 0.310478i
$933$	0	0
$934$	−176819.	−0.202691
$935$	− 81720.5i	− 0.0934777i
$936$	0	0
$937$	600379.	0.683827	0.341914	−	0.939731i	$-0.388925\pi$
$937$	600379.	0.683827	0.341914	+	0.939731i	$0.388925\pi$
$938$	0	0
$939$	0	0
$940$	143496.	0.162399
$941$	84374.7i	0.0952869i	0.998864	+	0.0476434i	$0.0151711\pi$
$941$	84374.7i	0.0952869i	−0.998864	+	0.0476434i	$0.984829\pi$
$942$	0	0
$943$	−49103.1	−0.0552186
$944$	177532.i	0.199219i
$945$	0	0
$946$	−686963.	−0.767628
$947$	1.71415e6i	1.91138i	0.294369	+	0.955692i	$0.404891\pi$
$947$	1.71415e6i	1.91138i	−0.294369	+	0.955692i	$0.595109\pi$
$948$	0	0
$949$	−508330.	−0.564434
$950$	471937.i	0.522922i
$951$	0	0
$952$	0	0
$953$	928678.i	1.02254i	0.859421	+	0.511269i	$0.170824\pi$
$953$	928678.i	1.02254i	−0.859421	+	0.511269i	$0.829176\pi$
$954$	0	0
$955$	179179.	0.196462
$956$	− 41092.7i	− 0.0449623i
$957$	0	0
$958$	700090.	0.762821
$959$	0	0
$960$	0	0
$961$	−923094.	−0.999538
$962$	542733.i	0.586457i
$963$	0	0
$964$	369278.	0.397374
$965$	234193.i	0.251489i
$966$	0	0
$967$	−729610.	−0.780257	−0.390129	−	0.920760i	$-0.627569\pi$
$967$	−729610.	−0.780257	−0.390129	+	0.920760i	$0.627569\pi$
$968$	− 295706.i	− 0.315580i
$969$	0	0
$970$	−371347.	−0.394672
$971$	− 1.56670e6i	− 1.66168i	−0.556511	−	0.830841i	$-0.687860\pi$
$971$	− 1.56670e6i	− 1.66168i	0.556511	−	0.830841i	$-0.312140\pi$
$972$	0	0
$973$	0	0
$974$	− 730760.i	− 0.770294i
$975$	0	0
$976$	−110995.	−0.116520
$977$	239837.i	0.251263i	0.992077	+	0.125631i	$0.0400956\pi$
$977$	239837.i	0.251263i	−0.992077	+	0.125631i	$0.959904\pi$
$978$	0	0
$979$	2.27306e6	2.37163
$980$	0	0
$981$	0	0
$982$	−678250.	−0.703342
$983$	1.42682e6i	1.47660i	0.674472	+	0.738300i	$0.264371\pi$
$983$	1.42682e6i	1.47660i	−0.674472	+	0.738300i	$0.735629\pi$
$984$	0	0
$985$	374765.	0.386266
$986$	9124.28i	0.00938523i
$987$	0	0
$988$	405218.	0.415121
$989$	32408.6i	0.0331335i
$990$	0	0
$991$	1.57484e6	1.60357	0.801786	−	0.597612i	$-0.203884\pi$
$991$	1.57484e6	1.60357	0.801786	+	0.597612i	$0.203884\pi$
$992$	3738.76i	0.00379931i
$993$	0	0
$994$	0	0
$995$	525370.i	0.530664i
$996$	0	0
$997$	−612478.	−0.616169	−0.308084	−	0.951359i	$-0.599688\pi$
$997$	−612478.	−0.616169	−0.308084	+	0.951359i	$0.599688\pi$
$998$	− 93376.7i	− 0.0937514i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.j.197.2	✓	8
3.2	odd	2	inner	882.5.b.j.197.7	yes	8
7.6	odd	2	inner	882.5.b.j.197.3	yes	8
21.20	even	2	inner	882.5.b.j.197.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.5.b.j.197.2	✓	8	1.1	even	1	trivial
882.5.b.j.197.3	yes	8	7.6	odd	2	inner
882.5.b.j.197.6	yes	8	21.20	even	2	inner
882.5.b.j.197.7	yes	8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -12.6723i q^{5} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -12.6723i q^{5} +22.6274i q^{8} -35.8427 q^{10} +166.462i q^{11} +140.982 q^{13} +64.0000 q^{16} -38.7401i q^{17} -359.282 q^{19} +101.379i q^{20} +470.825 q^{22} -22.2119i q^{23} +464.412 q^{25} -398.757i q^{26} -83.2708i q^{29} -20.6539 q^{31} -181.019i q^{32} -109.574 q^{34} -1361.06 q^{37} +1016.20i q^{38} +286.742 q^{40} -2210.66i q^{41} -1459.06 q^{43} -1331.69i q^{44} -62.8248 q^{46} -1415.45i q^{47} -1313.56i q^{50} -1127.86 q^{52} -2591.76i q^{53} +2109.46 q^{55} -235.525 q^{58} +2773.93i q^{59} -1734.29 q^{61} +58.4181i q^{62} -512.000 q^{64} -1786.57i q^{65} -952.113 q^{67} +309.921i q^{68} -3845.91i q^{71} -3605.64 q^{73} +3849.66i q^{74} +2874.26 q^{76} -6839.18 q^{79} -811.028i q^{80} -6252.69 q^{82} +9448.26i q^{83} -490.927 q^{85} +4126.85i q^{86} -3766.60 q^{88} -13655.2i q^{89} +177.695i q^{92} -4003.49 q^{94} +4552.94i q^{95} +10360.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 8 * q - 64 * q^4	\( 8 q - 64 q^{4} + 512 q^{16} - 640 q^{22} + 1512 q^{25} + 128 q^{37} - 656 q^{43} + 3904 q^{46} - 15104 q^{58} - 4096 q^{64} - 23040 q^{67} - 59120 q^{79} - 52400 q^{85} + 5120 q^{88}+O(q^{100}) \) 8 * q - 64 * q^4 + 512 * q^16 - 640 * q^22 + 1512 * q^25 + 128 * q^37 - 656 * q^43 + 3904 * q^46 - 15104 * q^58 - 4096 * q^64 - 23040 * q^67 - 59120 * q^79 - 52400 * q^85 + 5120 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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