LMFDB - Embedded newform 882.5.c.a.685.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,Mod(685,882)]

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([0, 1]))

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

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

chi := DirichletCharacter("882.685");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	882.c (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:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.2
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.5.c.a.685.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{2} +8.00000 q^{4} +14.1536i q^{5} -22.6274 q^{8} -40.0324i q^{10} -64.0294 q^{11} +228.919i q^{13} +64.0000 q^{16} -225.455i q^{17} -294.737i q^{19} +113.229i q^{20} +181.103 q^{22} -709.499 q^{23} +424.676 q^{25} -647.481i q^{26} -740.397 q^{29} +666.713i q^{31} -181.019 q^{32} +637.683i q^{34} -833.765 q^{37} +833.642i q^{38} -320.259i q^{40} -2817.60i q^{41} +3066.41 q^{43} -512.235 q^{44} +2006.77 q^{46} +613.726i q^{47} -1201.17 q^{50} +1831.35i q^{52} +1152.60 q^{53} -906.246i q^{55} +2094.16 q^{58} -3492.59i q^{59} +2272.21i q^{61} -1885.75i q^{62} +512.000 q^{64} -3240.03 q^{65} -8674.62 q^{67} -1803.64i q^{68} +353.591 q^{71} -4069.95i q^{73} +2358.24 q^{74} -2357.90i q^{76} +6472.83 q^{79} +905.829i q^{80} +7969.38i q^{82} -8225.83i q^{83} +3191.00 q^{85} -8673.11 q^{86} +1448.82 q^{88} +15538.3i q^{89} -5675.99 q^{92} -1735.88i q^{94} +4171.58 q^{95} -1558.61i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{4} - 324 q^{11} + 256 q^{16} - 192 q^{22} + 624 q^{23} + 952 q^{25} - 2724 q^{29} - 2792 q^{37} - 632 q^{43} - 2592 q^{44} + 9792 q^{46} - 2112 q^{50} - 2076 q^{53} + 672 q^{58} + 2048 q^{64}+ \cdots + 6504 q^{95}+O(q^{100}) $ 4 * q + 32 * q^4 - 324 * q^11 + 256 * q^16 - 192 * q^22 + 624 * q^23 + 952 * q^25 - 2724 * q^29 - 2792 * q^37 - 632 * q^43 - 2592 * q^44 + 9792 * q^46 - 2112 * q^50 - 2076 * q^53 + 672 * q^58 + 2048 * q^64 - 1488 * q^65 - 29200 * q^67 + 9696 * q^71 + 1536 * q^74 - 7948 * q^79 + 1224 * q^85 - 36480 * q^86 - 1536 * q^88 + 4992 * q^92 + 6504 * q^95

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 $

$2$	−2.82843	−0.707107
$2$	−2.82843	−0.707107
$3$	0	0
$3$	0	0
$4$	8.00000	0.500000
$5$	14.1536i	0.566143i	0.959099	+	0.283072i	$0.0913534\pi$
$5$	14.1536i	0.566143i	−0.959099	+	0.283072i	$0.908647\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−22.6274	−0.353553
$9$	0	0
$10$	− 40.0324i	− 0.400324i
$11$	−64.0294	−0.529169	−0.264584	−	0.964363i	$-0.585235\pi$
$11$	−64.0294	−0.529169	−0.264584	+	0.964363i	$0.585235\pi$
$12$	0	0
$13$	228.919i	1.35455i	0.735729	+	0.677276i	$0.236839\pi$
$13$	228.919i	1.35455i	−0.735729	+	0.677276i	$0.763161\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	− 225.455i	− 0.780121i	−0.920789	−	0.390061i	$-0.872454\pi$
$17$	− 225.455i	− 0.780121i	0.920789	−	0.390061i	$-0.127546\pi$
$18$	0	0
$19$	− 294.737i	− 0.816446i	−0.912882	−	0.408223i	$-0.866149\pi$
$19$	− 294.737i	− 0.816446i	0.912882	−	0.408223i	$-0.133851\pi$
$20$	113.229i	0.283072i
$21$	0	0
$22$	181.103	0.374179
$23$	−709.499	−1.34121	−0.670604	−	0.741816i	$-0.733965\pi$
$23$	−709.499	−1.34121	−0.670604	+	0.741816i	$0.733965\pi$
$24$	0	0
$25$	424.676	0.679482
$26$	− 647.481i	− 0.957812i
$27$	0	0
$28$	0	0
$29$	−740.397	−0.880377	−0.440188	−	0.897905i	$-0.645088\pi$
$29$	−740.397	−0.880377	−0.440188	+	0.897905i	$0.645088\pi$
$30$	0	0
$31$	666.713i	0.693770i	0.937908	+	0.346885i	$0.112761\pi$
$31$	666.713i	0.693770i	−0.937908	+	0.346885i	$0.887239\pi$
$32$	−181.019	−0.176777
$33$	0	0
$34$	637.683i	0.551629i
$35$	0	0
$36$	0	0
$37$	−833.765	−0.609032	−0.304516	−	0.952507i	$-0.598495\pi$
$37$	−833.765	−0.609032	−0.304516	+	0.952507i	$0.598495\pi$
$38$	833.642i	0.577315i
$39$	0	0
$40$	− 320.259i	− 0.200162i
$41$	− 2817.60i	− 1.67615i	−0.545558	−	0.838073i	$-0.683682\pi$
$41$	− 2817.60i	− 1.67615i	0.545558	−	0.838073i	$-0.316318\pi$
$42$	0	0
$43$	3066.41	1.65841	0.829207	−	0.558942i	$-0.188793\pi$
$43$	3066.41	1.65841	0.829207	+	0.558942i	$0.188793\pi$
$44$	−512.235	−0.264584
$45$	0	0
$46$	2006.77	0.948377
$47$	613.726i	0.277830i	0.990304	+	0.138915i	$0.0443614\pi$
$47$	613.726i	0.277830i	−0.990304	+	0.138915i	$0.955639\pi$
$48$	0	0
$49$	0	0
$50$	−1201.17	−0.480466
$51$	0	0
$52$	1831.35i	0.677276i
$53$	1152.60	0.410324	0.205162	−	0.978728i	$-0.434228\pi$
$53$	1152.60	0.410324	0.205162	+	0.978728i	$0.434228\pi$
$54$	0	0
$55$	− 906.246i	− 0.299585i
$56$	0	0
$57$	0	0
$58$	2094.16	0.622520
$59$	− 3492.59i	− 1.00333i	−0.865062	−	0.501665i	$-0.832721\pi$
$59$	− 3492.59i	− 1.00333i	0.865062	−	0.501665i	$-0.167279\pi$
$60$	0	0
$61$	2272.21i	0.610645i	0.952249	+	0.305323i	$0.0987643\pi$
$61$	2272.21i	0.610645i	−0.952249	+	0.305323i	$0.901236\pi$
$62$	− 1885.75i	− 0.490569i
$63$	0	0
$64$	512.000	0.125000
$65$	−3240.03	−0.766870
$66$	0	0
$67$	−8674.62	−1.93242	−0.966208	−	0.257764i	$-0.917014\pi$
$67$	−8674.62	−1.93242	−0.966208	+	0.257764i	$0.917014\pi$
$68$	− 1803.64i	− 0.390061i
$69$	0	0
$70$	0	0
$71$	353.591	0.0701431	0.0350715	−	0.999385i	$-0.488834\pi$
$71$	353.591	0.0701431	0.0350715	+	0.999385i	$0.488834\pi$
$72$	0	0
$73$	− 4069.95i	− 0.763736i	−0.924217	−	0.381868i	$-0.875281\pi$
$73$	− 4069.95i	− 0.763736i	0.924217	−	0.381868i	$-0.124719\pi$
$74$	2358.24	0.430650
$75$	0	0
$76$	− 2357.90i	− 0.408223i
$77$	0	0
$78$	0	0
$79$	6472.83	1.03715	0.518573	−	0.855033i	$-0.326464\pi$
$79$	6472.83	1.03715	0.518573	+	0.855033i	$0.326464\pi$
$80$	905.829i	0.141536i
$81$	0	0
$82$	7969.38i	1.18521i
$83$	− 8225.83i	− 1.19405i	−0.802222	−	0.597026i	$-0.796349\pi$
$83$	− 8225.83i	− 1.19405i	0.802222	−	0.597026i	$-0.203651\pi$
$84$	0	0
$85$	3191.00	0.441660
$86$	−8673.11	−1.17268
$87$	0	0
$88$	1448.82	0.187089
$89$	15538.3i	1.96166i	0.194878	+	0.980828i	$0.437569\pi$
$89$	15538.3i	1.96166i	−0.194878	+	0.980828i	$0.562431\pi$
$90$	0	0
$91$	0	0
$92$	−5675.99	−0.670604
$93$	0	0
$94$	− 1735.88i	− 0.196455i
$95$	4171.58	0.462225
$96$	0	0
$97$	− 1558.61i	− 0.165651i	−0.996564	−	0.0828254i	$-0.973606\pi$
$97$	− 1558.61i	− 0.165651i	0.996564	−	0.0828254i	$-0.0263944\pi$
$98$	0	0
$99$	0	0
$100$	3397.41	0.339741
$101$	15735.3i	1.54253i	0.636516	+	0.771264i	$0.280375\pi$
$101$	15735.3i	1.54253i	−0.636516	+	0.771264i	$0.719625\pi$
$102$	0	0
$103$	33.7287i	0.00317926i	0.999999	+	0.00158963i	$0.000505995\pi$
$103$	33.7287i	0.00317926i	−0.999999	+	0.00158963i	$0.999494\pi$
$104$	− 5179.85i	− 0.478906i
$105$	0	0
$106$	−3260.05	−0.290143
$107$	5446.46	0.475715	0.237858	−	0.971300i	$-0.423555\pi$
$107$	5446.46	0.475715	0.237858	+	0.971300i	$0.423555\pi$
$108$	0	0
$109$	16697.8	1.40542	0.702709	−	0.711477i	$-0.251974\pi$
$109$	16697.8	1.40542	0.702709	+	0.711477i	$0.251974\pi$
$110$	2563.25i	0.211839i
$111$	0	0
$112$	0	0
$113$	−9455.64	−0.740515	−0.370258	−	0.928929i	$-0.620731\pi$
$113$	−9455.64	−0.740515	−0.370258	+	0.928929i	$0.620731\pi$
$114$	0	0
$115$	− 10041.9i	− 0.759315i
$116$	−5923.18	−0.440188
$117$	0	0
$118$	9878.54i	0.709461i
$119$	0	0
$120$	0	0
$121$	−10541.2	−0.719980
$122$	− 6426.79i	− 0.431792i
$123$	0	0
$124$	5333.70i	0.346885i
$125$	14856.7i	0.950827i
$126$	0	0
$127$	−2380.07	−0.147564	−0.0737822	−	0.997274i	$-0.523507\pi$
$127$	−2380.07	−0.147564	−0.0737822	+	0.997274i	$0.523507\pi$
$128$	−1448.15	−0.0883883
$129$	0	0
$130$	9164.17	0.542259
$131$	4339.79i	0.252887i	0.991974	+	0.126443i	$0.0403562\pi$
$131$	4339.79i	0.252887i	−0.991974	+	0.126443i	$0.959644\pi$
$132$	0	0
$133$	0	0
$134$	24535.5	1.36642
$135$	0	0
$136$	5101.46i	0.275814i
$137$	30759.2	1.63883	0.819414	−	0.573202i	$-0.194299\pi$
$137$	30759.2	1.63883	0.819414	+	0.573202i	$0.194299\pi$
$138$	0	0
$139$	− 27186.6i	− 1.40710i	−0.710644	−	0.703551i	$-0.751596\pi$
$139$	− 27186.6i	− 1.40710i	0.710644	−	0.703551i	$-0.248404\pi$
$140$	0	0
$141$	0	0
$142$	−1000.11	−0.0495987
$143$	− 14657.6i	− 0.716786i
$144$	0	0
$145$	− 10479.3i	− 0.498419i
$146$	11511.6i	0.540043i
$147$	0	0
$148$	−6670.12	−0.304516
$149$	5826.92	0.262462	0.131231	−	0.991352i	$-0.458107\pi$
$149$	5826.92	0.262462	0.131231	+	0.991352i	$0.458107\pi$
$150$	0	0
$151$	25186.8	1.10463	0.552317	−	0.833634i	$-0.313744\pi$
$151$	25186.8	1.10463	0.552317	+	0.833634i	$0.313744\pi$
$152$	6669.14i	0.288657i
$153$	0	0
$154$	0	0
$155$	−9436.37	−0.392773
$156$	0	0
$157$	− 23896.0i	− 0.969453i	−0.874666	−	0.484726i	$-0.838919\pi$
$157$	− 23896.0i	− 0.969453i	0.874666	−	0.484726i	$-0.161081\pi$
$158$	−18307.9	−0.733373
$159$	0	0
$160$	− 2562.07i	− 0.100081i
$161$	0	0
$162$	0	0
$163$	−42585.9	−1.60284	−0.801422	−	0.598100i	$-0.795923\pi$
$163$	−42585.9	−1.60284	−0.801422	+	0.598100i	$0.795923\pi$
$164$	− 22540.8i	− 0.838073i
$165$	0	0
$166$	23266.2i	0.844323i
$167$	− 26356.3i	− 0.945043i	−0.881319	−	0.472521i	$-0.843344\pi$
$167$	− 26356.3i	− 0.945043i	0.881319	−	0.472521i	$-0.156656\pi$
$168$	0	0
$169$	−23843.0	−0.834809
$170$	−9025.50	−0.312301
$171$	0	0
$172$	24531.3	0.829207
$173$	− 28019.2i	− 0.936188i	−0.883679	−	0.468094i	$-0.844941\pi$
$173$	− 28019.2i	− 0.936188i	0.883679	−	0.468094i	$-0.155059\pi$
$174$	0	0
$175$	0	0
$176$	−4097.88	−0.132292
$177$	0	0
$178$	− 43948.9i	− 1.38710i
$179$	−25398.5	−0.792687	−0.396343	−	0.918102i	$-0.629721\pi$
$179$	−25398.5	−0.792687	−0.396343	+	0.918102i	$0.629721\pi$
$180$	0	0
$181$	− 44097.2i	− 1.34603i	−0.739630	−	0.673014i	$-0.764999\pi$
$181$	− 44097.2i	− 1.34603i	0.739630	−	0.673014i	$-0.235001\pi$
$182$	0	0
$183$	0	0
$184$	16054.1	0.474188
$185$	− 11800.8i	− 0.344799i
$186$	0	0
$187$	14435.8i	0.412816i
$188$	4909.81i	0.138915i
$189$	0	0
$190$	−11799.0	−0.326843
$191$	64559.1	1.76966	0.884832	−	0.465911i	$-0.154273\pi$
$191$	64559.1	1.76966	0.884832	+	0.465911i	$0.154273\pi$
$192$	0	0
$193$	36674.6	0.984579	0.492290	−	0.870432i	$-0.336160\pi$
$193$	36674.6	0.984579	0.492290	+	0.870432i	$0.336160\pi$
$194$	4408.41i	0.117133i
$195$	0	0
$196$	0	0
$197$	73147.0	1.88480	0.942398	−	0.334494i	$-0.108566\pi$
$197$	73147.0	1.88480	0.942398	+	0.334494i	$0.108566\pi$
$198$	0	0
$199$	1401.03i	0.0353786i	0.999844	+	0.0176893i	$0.00563097\pi$
$199$	1401.03i	0.0353786i	−0.999844	+	0.0176893i	$0.994369\pi$
$200$	−9609.33	−0.240233
$201$	0	0
$202$	− 44506.2i	− 1.09073i
$203$	0	0
$204$	0	0
$205$	39879.2	0.948939
$206$	− 95.3993i	− 0.00224807i
$207$	0	0
$208$	14650.8i	0.338638i
$209$	18871.8i	0.432038i
$210$	0	0
$211$	58231.7	1.30796	0.653980	−	0.756512i	$-0.273098\pi$
$211$	58231.7	1.30796	0.653980	+	0.756512i	$0.273098\pi$
$212$	9220.80	0.205162
$213$	0	0
$214$	−15404.9	−0.336381
$215$	43400.6i	0.938900i
$216$	0	0
$217$	0	0
$218$	−47228.4	−0.993781
$219$	0	0
$220$	− 7249.97i	− 0.149793i
$221$	51611.0	1.05671
$222$	0	0
$223$	− 61050.9i	− 1.22767i	−0.789434	−	0.613836i	$-0.789626\pi$
$223$	− 61050.9i	− 1.22767i	0.789434	−	0.613836i	$-0.210374\pi$
$224$	0	0
$225$	0	0
$226$	26744.6	0.523623
$227$	24993.7i	0.485041i	0.970146	+	0.242520i	$0.0779742\pi$
$227$	24993.7i	0.485041i	−0.970146	+	0.242520i	$0.922026\pi$
$228$	0	0
$229$	− 18928.9i	− 0.360956i	−0.983579	−	0.180478i	$-0.942236\pi$
$229$	− 18928.9i	− 0.360956i	0.983579	−	0.180478i	$-0.0577644\pi$
$230$	28402.9i	0.536917i
$231$	0	0
$232$	16753.3	0.311260
$233$	−6684.74	−0.123132	−0.0615662	−	0.998103i	$-0.519610\pi$
$233$	−6684.74	−0.123132	−0.0615662	+	0.998103i	$0.519610\pi$
$234$	0	0
$235$	−8686.42	−0.157291
$236$	− 27940.7i	− 0.501665i
$237$	0	0
$238$	0	0
$239$	96461.0	1.68871	0.844356	−	0.535782i	$-0.179983\pi$
$239$	96461.0	1.68871	0.844356	+	0.535782i	$0.179983\pi$
$240$	0	0
$241$	− 54700.2i	− 0.941792i	−0.882189	−	0.470896i	$-0.843931\pi$
$241$	− 54700.2i	− 0.941792i	0.882189	−	0.470896i	$-0.156069\pi$
$242$	29815.1	0.509103
$243$	0	0
$244$	18177.7i	0.305323i
$245$	0	0
$246$	0	0
$247$	67471.0	1.10592
$248$	− 15086.0i	− 0.245285i
$249$	0	0
$250$	− 42021.0i	− 0.672336i
$251$	− 108137.i	− 1.71643i	−0.513286	−	0.858217i	$-0.671572\pi$
$251$	− 108137.i	− 1.71643i	0.513286	−	0.858217i	$-0.328428\pi$
$252$	0	0
$253$	45428.8	0.709725
$254$	6731.84	0.104344
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 35886.1i	− 0.543325i	−0.962393	−	0.271663i	$-0.912427\pi$
$257$	− 35886.1i	− 0.543325i	0.962393	−	0.271663i	$-0.0875735\pi$
$258$	0	0
$259$	0	0
$260$	−25920.2	−0.383435
$261$	0	0
$262$	− 12274.8i	− 0.178818i
$263$	65431.6	0.945967	0.472984	−	0.881071i	$-0.343177\pi$
$263$	65431.6	0.945967	0.472984	+	0.881071i	$0.343177\pi$
$264$	0	0
$265$	16313.4i	0.232302i
$266$	0	0
$267$	0	0
$268$	−69396.9	−0.966208
$269$	− 69211.3i	− 0.956472i	−0.878231	−	0.478236i	$-0.841276\pi$
$269$	− 69211.3i	− 0.956472i	0.878231	−	0.478236i	$-0.158724\pi$
$270$	0	0
$271$	6550.55i	0.0891947i	0.999005	+	0.0445973i	$0.0142005\pi$
$271$	6550.55i	0.0891947i	−0.999005	+	0.0445973i	$0.985800\pi$
$272$	− 14429.1i	− 0.195030i
$273$	0	0
$274$	−87000.1	−1.15883
$275$	−27191.8	−0.359561
$276$	0	0
$277$	79103.2	1.03094	0.515471	−	0.856907i	$-0.327617\pi$
$277$	79103.2	1.03094	0.515471	+	0.856907i	$0.327617\pi$
$278$	76895.4i	0.994972i
$279$	0	0
$280$	0	0
$281$	−34363.1	−0.435191	−0.217596	−	0.976039i	$-0.569821\pi$
$281$	−34363.1	−0.435191	−0.217596	+	0.976039i	$0.569821\pi$
$282$	0	0
$283$	− 91912.3i	− 1.14763i	−0.818986	−	0.573813i	$-0.805463\pi$
$283$	− 91912.3i	− 1.14763i	0.818986	−	0.573813i	$-0.194537\pi$
$284$	2828.73	0.0350715
$285$	0	0
$286$	41457.8i	0.506844i
$287$	0	0
$288$	0	0
$289$	32691.0	0.391411
$290$	29639.8i	0.352436i
$291$	0	0
$292$	− 32559.6i	− 0.381868i
$293$	23218.2i	0.270453i	0.990815	+	0.135227i	$0.0431763\pi$
$293$	23218.2i	0.270453i	−0.990815	+	0.135227i	$0.956824\pi$
$294$	0	0
$295$	49432.7	0.568028
$296$	18865.9	0.215325
$297$	0	0
$298$	−16481.0	−0.185589
$299$	− 162418.i	− 1.81673i
$300$	0	0
$301$	0	0
$302$	−71239.0	−0.781095
$303$	0	0
$304$	− 18863.2i	− 0.204112i
$305$	−32159.9	−0.345713
$306$	0	0
$307$	66385.9i	0.704367i	0.935931	+	0.352183i	$0.114561\pi$
$307$	66385.9i	0.704367i	−0.935931	+	0.352183i	$0.885439\pi$
$308$	0	0
$309$	0	0
$310$	26690.1	0.277733
$311$	− 95700.5i	− 0.989449i	−0.869050	−	0.494724i	$-0.835269\pi$
$311$	− 95700.5i	− 0.989449i	0.869050	−	0.494724i	$-0.164731\pi$
$312$	0	0
$313$	47904.0i	0.488971i	0.969653	+	0.244486i	$0.0786191\pi$
$313$	47904.0i	0.488971i	−0.969653	+	0.244486i	$0.921381\pi$
$314$	67588.2i	0.685507i
$315$	0	0
$316$	51782.6	0.518573
$317$	34128.2	0.339621	0.169811	−	0.985477i	$-0.445684\pi$
$317$	34128.2	0.339621	0.169811	+	0.985477i	$0.445684\pi$
$318$	0	0
$319$	47407.2	0.465868
$320$	7246.63i	0.0707679i
$321$	0	0
$322$	0	0
$323$	−66450.0	−0.636927
$324$	0	0
$325$	97216.5i	0.920393i
$326$	120451.	1.13338
$327$	0	0
$328$	63755.1i	0.592607i
$329$	0	0
$330$	0	0
$331$	133285.	1.21654	0.608270	−	0.793730i	$-0.291864\pi$
$331$	133285.	1.21654	0.608270	+	0.793730i	$0.291864\pi$
$332$	− 65806.6i	− 0.597026i
$333$	0	0
$334$	74546.9i	0.668246i
$335$	− 122777.i	− 1.09402i
$336$	0	0
$337$	−49734.4	−0.437922	−0.218961	−	0.975734i	$-0.570267\pi$
$337$	−49734.4	−0.437922	−0.218961	+	0.975734i	$0.570267\pi$
$338$	67438.1	0.590299
$339$	0	0
$340$	25528.0	0.220830
$341$	− 42689.2i	− 0.367121i
$342$	0	0
$343$	0	0
$344$	−69384.9	−0.586338
$345$	0	0
$346$	79250.2i	0.661985i
$347$	−18769.5	−0.155882	−0.0779408	−	0.996958i	$-0.524835\pi$
$347$	−18769.5	−0.155882	−0.0779408	+	0.996958i	$0.524835\pi$
$348$	0	0
$349$	4574.17i	0.0375545i	0.999824	+	0.0187772i	$0.00597733\pi$
$349$	4574.17i	0.0375545i	−0.999824	+	0.0187772i	$0.994023\pi$
$350$	0	0
$351$	0	0
$352$	11590.6	0.0935447
$353$	60893.0i	0.488673i	0.969691	+	0.244336i	$0.0785701\pi$
$353$	60893.0i	0.488673i	−0.969691	+	0.244336i	$0.921430\pi$
$354$	0	0
$355$	5004.58i	0.0397110i
$356$	124306.i	0.980828i
$357$	0	0
$358$	71837.7	0.560514
$359$	−232118.	−1.80103	−0.900513	−	0.434830i	$-0.856808\pi$
$359$	−232118.	−1.80103	−0.900513	+	0.434830i	$0.856808\pi$
$360$	0	0
$361$	43451.1	0.333416
$362$	124726.i	0.951786i
$363$	0	0
$364$	0	0
$365$	57604.4	0.432384
$366$	0	0
$367$	144718.i	1.07446i	0.843436	+	0.537229i	$0.180529\pi$
$367$	144718.i	1.07446i	−0.843436	+	0.537229i	$0.819471\pi$
$368$	−45407.9	−0.335302
$369$	0	0
$370$	33377.6i	0.243810i
$371$	0	0
$372$	0	0
$373$	−148883.	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$373$	−148883.	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$374$	− 40830.5i	− 0.291905i
$375$	0	0
$376$	− 13887.0i	− 0.0982276i
$377$	− 169491.i	− 1.19252i
$378$	0	0
$379$	140667.	0.979299	0.489649	−	0.871919i	$-0.337125\pi$
$379$	140667.	0.979299	0.489649	+	0.871919i	$0.337125\pi$
$380$	33372.7	0.231113
$381$	0	0
$382$	−182601.	−1.25134
$383$	− 29322.3i	− 0.199895i	−0.994993	−	0.0999473i	$-0.968133\pi$
$383$	− 29322.3i	− 0.199895i	0.994993	−	0.0999473i	$-0.0318674\pi$
$384$	0	0
$385$	0	0
$386$	−103731.	−0.696202
$387$	0	0
$388$	− 12468.9i	− 0.0828254i
$389$	−132420.	−0.875092	−0.437546	−	0.899196i	$-0.644152\pi$
$389$	−132420.	−0.875092	−0.437546	+	0.899196i	$0.644152\pi$
$390$	0	0
$391$	159960.i	1.04630i
$392$	0	0
$393$	0	0
$394$	−206891.	−1.33275
$395$	91613.7i	0.587173i
$396$	0	0
$397$	34227.6i	0.217168i	0.994087	+	0.108584i	$0.0346316\pi$
$397$	34227.6i	0.217168i	−0.994087	+	0.108584i	$0.965368\pi$
$398$	− 3962.70i	− 0.0250164i
$399$	0	0
$400$	27179.3	0.169870
$401$	179081.	1.11368	0.556841	−	0.830619i	$-0.312013\pi$
$401$	179081.	1.11368	0.556841	+	0.830619i	$0.312013\pi$
$402$	0	0
$403$	−152623.	−0.939747
$404$	125883.i	0.771264i
$405$	0	0
$406$	0	0
$407$	53385.5	0.322281
$408$	0	0
$409$	− 49766.8i	− 0.297504i	−0.988875	−	0.148752i	$-0.952474\pi$
$409$	− 49766.8i	− 0.297504i	0.988875	−	0.148752i	$-0.0475257\pi$
$410$	−112795.	−0.671001
$411$	0	0
$412$	269.830i	0.00158963i
$413$	0	0
$414$	0	0
$415$	116425.	0.676005
$416$	− 41438.8i	− 0.239453i
$417$	0	0
$418$	− 53377.6i	− 0.305497i
$419$	− 43951.2i	− 0.250347i	−0.992135	−	0.125174i	$-0.960051\pi$
$419$	− 43951.2i	− 0.250347i	0.992135	−	0.125174i	$-0.0399488\pi$
$420$	0	0
$421$	−218257.	−1.23141	−0.615707	−	0.787975i	$-0.711129\pi$
$421$	−218257.	−1.23141	−0.615707	+	0.787975i	$0.711129\pi$
$422$	−164704.	−0.924867
$423$	0	0
$424$	−26080.4	−0.145071
$425$	− 95745.4i	− 0.530078i
$426$	0	0
$427$	0	0
$428$	43571.7	0.237858
$429$	0	0
$430$	− 122756.i	− 0.663902i
$431$	76739.6	0.413109	0.206555	−	0.978435i	$-0.433775\pi$
$431$	76739.6	0.413109	0.206555	+	0.978435i	$0.433775\pi$
$432$	0	0
$433$	− 216713.i	− 1.15587i	−0.816083	−	0.577935i	$-0.803859\pi$
$433$	− 216713.i	− 1.15587i	0.816083	−	0.577935i	$-0.196141\pi$
$434$	0	0
$435$	0	0
$436$	133582.	0.702709
$437$	209116.i	1.09502i
$438$	0	0
$439$	23762.1i	0.123298i	0.998098	+	0.0616489i	$0.0196359\pi$
$439$	23762.1i	0.123298i	−0.998098	+	0.0616489i	$0.980364\pi$
$440$	20506.0i	0.105919i
$441$	0	0
$442$	−145978.	−0.747210
$443$	−174865.	−0.891035	−0.445517	−	0.895273i	$-0.646980\pi$
$443$	−174865.	−0.891035	−0.445517	+	0.895273i	$0.646980\pi$
$444$	0	0
$445$	−219922.	−1.11058
$446$	172678.i	0.868095i
$447$	0	0
$448$	0	0
$449$	195154.	0.968023	0.484011	−	0.875062i	$-0.339179\pi$
$449$	195154.	0.968023	0.484011	+	0.875062i	$0.339179\pi$
$450$	0	0
$451$	180409.i	0.886965i
$452$	−75645.1	−0.370258
$453$	0	0
$454$	− 70692.8i	− 0.342976i
$455$	0	0
$456$	0	0
$457$	58472.2	0.279974	0.139987	−	0.990153i	$-0.455294\pi$
$457$	58472.2	0.279974	0.139987	+	0.990153i	$0.455294\pi$
$458$	53539.0i	0.255234i
$459$	0	0
$460$	− 80335.6i	− 0.379658i
$461$	307243.i	1.44571i	0.691002	+	0.722853i	$0.257169\pi$
$461$	307243.i	1.44571i	−0.691002	+	0.722853i	$0.742831\pi$
$462$	0	0
$463$	−17772.2	−0.0829049	−0.0414524	−	0.999140i	$-0.513199\pi$
$463$	−17772.2	−0.0829049	−0.0414524	+	0.999140i	$0.513199\pi$
$464$	−47385.4	−0.220094
$465$	0	0
$466$	18907.3	0.0870678
$467$	− 64965.1i	− 0.297883i	−0.988846	−	0.148942i	$-0.952413\pi$
$467$	− 64965.1i	− 0.297883i	0.988846	−	0.148942i	$-0.0475867\pi$
$468$	0	0
$469$	0	0
$470$	24568.9	0.111222
$471$	0	0
$472$	79028.3i	0.354731i
$473$	−196340.	−0.877581
$474$	0	0
$475$	− 125168.i	− 0.554760i
$476$	0	0
$477$	0	0
$478$	−272833.	−1.19410
$479$	210937.i	0.919351i	0.888087	+	0.459675i	$0.152034\pi$
$479$	210937.i	0.919351i	−0.888087	+	0.459675i	$0.847966\pi$
$480$	0	0
$481$	− 190865.i	− 0.824965i
$482$	154716.i	0.665948i
$483$	0	0
$484$	−84329.8	−0.359990
$485$	22059.9	0.0937821
$486$	0	0
$487$	−228888.	−0.965084	−0.482542	−	0.875873i	$-0.660286\pi$
$487$	−228888.	−0.965084	−0.482542	+	0.875873i	$0.660286\pi$
$488$	− 51414.3i	− 0.215896i
$489$	0	0
$490$	0	0
$491$	−140350.	−0.582169	−0.291085	−	0.956697i	$-0.594016\pi$
$491$	−140350.	−0.582169	−0.291085	+	0.956697i	$0.594016\pi$
$492$	0	0
$493$	166926.i	0.686801i
$494$	−190837.	−0.782002
$495$	0	0
$496$	42669.6i	0.173442i
$497$	0	0
$498$	0	0
$499$	−345327.	−1.38685	−0.693424	−	0.720529i	$-0.743899\pi$
$499$	−345327.	−1.38685	−0.693424	+	0.720529i	$0.743899\pi$
$500$	118853.i	0.475414i
$501$	0	0
$502$	305858.i	1.21370i
$503$	− 58979.0i	− 0.233110i	−0.993184	−	0.116555i	$-0.962815\pi$
$503$	− 58979.0i	− 0.233110i	0.993184	−	0.116555i	$-0.0371851\pi$
$504$	0	0
$505$	−222711.	−0.873291
$506$	−128492.	−0.501852
$507$	0	0
$508$	−19040.5	−0.0737822
$509$	− 23848.7i	− 0.0920511i	−0.998940	−	0.0460255i	$-0.985344\pi$
$509$	− 23848.7i	− 0.0920511i	0.998940	−	0.0460255i	$-0.0146556\pi$
$510$	0	0
$511$	0	0
$512$	−11585.2	−0.0441942
$513$	0	0
$514$	101501.i	0.384189i
$515$	−477.382	−0.00179991
$516$	0	0
$517$	− 39296.5i	− 0.147019i
$518$	0	0
$519$	0	0
$520$	73313.4	0.271129
$521$	481708.i	1.77463i	0.461163	+	0.887315i	$0.347432\pi$
$521$	481708.i	1.77463i	−0.461163	+	0.887315i	$0.652568\pi$
$522$	0	0
$523$	− 461410.i	− 1.68688i	−0.537226	−	0.843438i	$-0.680528\pi$
$523$	− 461410.i	− 1.68688i	0.537226	−	0.843438i	$-0.319472\pi$
$524$	34718.3i	0.126443i
$525$	0	0
$526$	−185068.	−0.668900
$527$	150314.	0.541225
$528$	0	0
$529$	223547.	0.798837
$530$	− 46141.3i	− 0.164262i
$531$	0	0
$532$	0	0
$533$	645003.	2.27043
$534$	0	0
$535$	77086.9i	0.269323i
$536$	196284.	0.683212
$537$	0	0
$538$	195759.i	0.676328i
$539$	0	0
$540$	0	0
$541$	−508196.	−1.73635	−0.868174	−	0.496260i	$-0.834706\pi$
$541$	−508196.	−1.73635	−0.868174	+	0.496260i	$0.834706\pi$
$542$	− 18527.7i	− 0.0630702i
$543$	0	0
$544$	40811.7i	0.137907i
$545$	236333.i	0.795668i
$546$	0	0
$547$	40170.8	0.134257	0.0671283	−	0.997744i	$-0.478616\pi$
$547$	40170.8	0.134257	0.0671283	+	0.997744i	$0.478616\pi$
$548$	246073.	0.819414
$549$	0	0
$550$	76910.0	0.254248
$551$	218222.i	0.718780i
$552$	0	0
$553$	0	0
$554$	−223738.	−0.728987
$555$	0	0
$556$	− 217493.i	− 0.703551i
$557$	78460.4	0.252895	0.126447	−	0.991973i	$-0.459642\pi$
$557$	78460.4	0.252895	0.126447	+	0.991973i	$0.459642\pi$
$558$	0	0
$559$	701959.i	2.24641i
$560$	0	0
$561$	0	0
$562$	97193.7	0.307727
$563$	− 410625.i	− 1.29547i	−0.761864	−	0.647737i	$-0.775716\pi$
$563$	− 410625.i	− 1.29547i	0.761864	−	0.647737i	$-0.224284\pi$
$564$	0	0
$565$	− 133831.i	− 0.419238i
$566$	259967.i	0.811495i
$567$	0	0
$568$	−8000.86	−0.0247993
$569$	148809.	0.459625	0.229812	−	0.973235i	$-0.426189\pi$
$569$	148809.	0.459625	0.229812	+	0.973235i	$0.426189\pi$
$570$	0	0
$571$	72292.6	0.221729	0.110864	−	0.993836i	$-0.464638\pi$
$571$	72292.6	0.221729	0.110864	+	0.993836i	$0.464638\pi$
$572$	− 117261.i	− 0.358393i
$573$	0	0
$574$	0	0
$575$	−301307.	−0.911326
$576$	0	0
$577$	557335.i	1.67404i	0.547176	+	0.837018i	$0.315703\pi$
$577$	557335.i	1.67404i	−0.547176	+	0.837018i	$0.684297\pi$
$578$	−92464.2	−0.276769
$579$	0	0
$580$	− 83834.1i	− 0.249210i
$581$	0	0
$582$	0	0
$583$	−73800.4	−0.217131
$584$	92092.4i	0.270021i
$585$	0	0
$586$	− 65670.9i	− 0.191239i
$587$	− 308119.i	− 0.894217i	−0.894480	−	0.447109i	$-0.852454\pi$
$587$	− 308119.i	− 0.894217i	0.894480	−	0.447109i	$-0.147546\pi$
$588$	0	0
$589$	196505.	0.566426
$590$	−139817.	−0.401657
$591$	0	0
$592$	−53360.9	−0.152258
$593$	231870.i	0.659378i	0.944090	+	0.329689i	$0.106944\pi$
$593$	231870.i	0.659378i	−0.944090	+	0.329689i	$0.893056\pi$
$594$	0	0
$595$	0	0
$596$	46615.3	0.131231
$597$	0	0
$598$	459387.i	1.28462i
$599$	334083.	0.931110	0.465555	−	0.885019i	$-0.345855\pi$
$599$	334083.	0.931110	0.465555	+	0.885019i	$0.345855\pi$
$600$	0	0
$601$	− 645072.i	− 1.78591i	−0.450147	−	0.892955i	$-0.648628\pi$
$601$	− 645072.i	− 1.78591i	0.450147	−	0.892955i	$-0.351372\pi$
$602$	0	0
$603$	0	0
$604$	201494.	0.552317
$605$	− 149196.i	− 0.407612i
$606$	0	0
$607$	139306.i	0.378088i	0.981969	+	0.189044i	$0.0605388\pi$
$607$	139306.i	0.378088i	−0.981969	+	0.189044i	$0.939461\pi$
$608$	53353.1i	0.144329i
$609$	0	0
$610$	90962.0	0.244456
$611$	−140494.	−0.376335
$612$	0	0
$613$	−407331.	−1.08399	−0.541997	−	0.840381i	$-0.682331\pi$
$613$	−407331.	−1.08399	−0.541997	+	0.840381i	$0.682331\pi$
$614$	− 187768.i	− 0.498063i
$615$	0	0
$616$	0	0
$617$	−276504.	−0.726325	−0.363163	−	0.931726i	$-0.618303\pi$
$617$	−276504.	−0.726325	−0.363163	+	0.931726i	$0.618303\pi$
$618$	0	0
$619$	− 223901.i	− 0.584352i	−0.956365	−	0.292176i	$-0.905621\pi$
$619$	− 223901.i	− 0.584352i	0.956365	−	0.292176i	$-0.0943793\pi$
$620$	−75491.0	−0.196387
$621$	0	0
$622$	270682.i	0.699646i
$623$	0	0
$624$	0	0
$625$	55147.5	0.141178
$626$	− 135493.i	− 0.345755i
$627$	0	0
$628$	− 191168.i	− 0.484726i
$629$	187976.i	0.475119i
$630$	0	0
$631$	299528.	0.752278	0.376139	−	0.926563i	$-0.377251\pi$
$631$	299528.	0.752278	0.376139	+	0.926563i	$0.377251\pi$
$632$	−146463.	−0.366686
$633$	0	0
$634$	−96529.1	−0.240148
$635$	− 33686.4i	− 0.0835425i
$636$	0	0
$637$	0	0
$638$	−134088.	−0.329418
$639$	0	0
$640$	− 20496.6i	− 0.0500405i
$641$	−577884.	−1.40645	−0.703226	−	0.710967i	$-0.748258\pi$
$641$	−577884.	−1.40645	−0.703226	+	0.710967i	$0.748258\pi$
$642$	0	0
$643$	135320.i	0.327295i	0.986519	+	0.163647i	$0.0523259\pi$
$643$	135320.i	0.327295i	−0.986519	+	0.163647i	$0.947674\pi$
$644$	0	0
$645$	0	0
$646$	187949.	0.450375
$647$	218100.i	0.521011i	0.965472	+	0.260506i	$0.0838892\pi$
$647$	218100.i	0.521011i	−0.965472	+	0.260506i	$0.916111\pi$
$648$	0	0
$649$	223629.i	0.530931i
$650$	− 274970.i	− 0.650816i
$651$	0	0
$652$	−340688.	−0.801422
$653$	−144254.	−0.338301	−0.169150	−	0.985590i	$-0.554102\pi$
$653$	−144254.	−0.338301	−0.169150	+	0.985590i	$0.554102\pi$
$654$	0	0
$655$	−61423.6	−0.143170
$656$	− 180327.i	− 0.419037i
$657$	0	0
$658$	0	0
$659$	159392.	0.367024	0.183512	−	0.983017i	$-0.441253\pi$
$659$	159392.	0.367024	0.183512	+	0.983017i	$0.441253\pi$
$660$	0	0
$661$	− 352962.i	− 0.807839i	−0.914795	−	0.403919i	$-0.867648\pi$
$661$	− 352962.i	− 0.807839i	0.914795	−	0.403919i	$-0.132352\pi$
$662$	−376988.	−0.860223
$663$	0	0
$664$	186129.i	0.422161i
$665$	0	0
$666$	0	0
$667$	525311.	1.18077
$668$	− 210850.i	− 0.472521i
$669$	0	0
$670$	347265.i	0.773592i
$671$	− 145488.i	− 0.323135i
$672$	0	0
$673$	504858.	1.11465	0.557326	−	0.830294i	$-0.311827\pi$
$673$	504858.	1.11465	0.557326	+	0.830294i	$0.311827\pi$
$674$	140670.	0.309658
$675$	0	0
$676$	−190744.	−0.417404
$677$	195706.i	0.426999i	0.976943	+	0.213500i	$0.0684862\pi$
$677$	195706.i	0.426999i	−0.976943	+	0.213500i	$0.931514\pi$
$678$	0	0
$679$	0	0
$680$	−72204.0	−0.156150
$681$	0	0
$682$	120743.i	0.259594i
$683$	234441.	0.502566	0.251283	−	0.967914i	$-0.419148\pi$
$683$	234441.	0.502566	0.251283	+	0.967914i	$0.419148\pi$
$684$	0	0
$685$	435352.i	0.927812i
$686$	0	0
$687$	0	0
$688$	196250.	0.414603
$689$	263852.i	0.555805i
$690$	0	0
$691$	− 89309.2i	− 0.187042i	−0.995617	−	0.0935212i	$-0.970188\pi$
$691$	− 89309.2i	− 0.187042i	0.995617	−	0.0935212i	$-0.0298123\pi$
$692$	− 224153.i	− 0.468094i
$693$	0	0
$694$	53088.3	0.110225
$695$	384788.	0.796622
$696$	0	0
$697$	−635243.	−1.30760
$698$	− 12937.7i	− 0.0265550i
$699$	0	0
$700$	0	0
$701$	−122213.	−0.248704	−0.124352	−	0.992238i	$-0.539685\pi$
$701$	−122213.	−0.248704	−0.124352	+	0.992238i	$0.539685\pi$
$702$	0	0
$703$	245741.i	0.497242i
$704$	−32783.1	−0.0661461
$705$	0	0
$706$	− 172231.i	− 0.345544i
$707$	0	0
$708$	0	0
$709$	84019.8	0.167143	0.0835717	−	0.996502i	$-0.473367\pi$
$709$	84019.8	0.167143	0.0835717	+	0.996502i	$0.473367\pi$
$710$	− 14155.1i	− 0.0280799i
$711$	0	0
$712$	− 351591.i	− 0.693550i
$713$	− 473032.i	− 0.930489i
$714$	0	0
$715$	207457.	0.405804
$716$	−203188.	−0.396343
$717$	0	0
$718$	656529.	1.27352
$719$	− 759119.i	− 1.46843i	−0.678919	−	0.734213i	$-0.737551\pi$
$719$	− 759119.i	− 1.46843i	0.678919	−	0.734213i	$-0.262449\pi$
$720$	0	0
$721$	0	0
$722$	−122898.	−0.235760
$723$	0	0
$724$	− 352778.i	− 0.673014i
$725$	−314429.	−0.598200
$726$	0	0
$727$	− 92384.1i	− 0.174795i	−0.996174	−	0.0873974i	$-0.972145\pi$
$727$	− 92384.1i	− 0.174795i	0.996174	−	0.0873974i	$-0.0278550\pi$
$728$	0	0
$729$	0	0
$730$	−162930.	−0.305742
$731$	− 691337.i	− 1.29376i
$732$	0	0
$733$	3352.68i	0.00623999i	0.999995	+	0.00312000i	$0.000993127\pi$
$733$	3352.68i	0.00623999i	−0.999995	+	0.00312000i	$0.999007\pi$
$734$	− 409324.i	− 0.759757i
$735$	0	0
$736$	128433.	0.237094
$737$	555431.	1.02257
$738$	0	0
$739$	−27185.6	−0.0497795	−0.0248897	−	0.999690i	$-0.507923\pi$
$739$	−27185.6	−0.0497795	−0.0248897	+	0.999690i	$0.507923\pi$
$740$	− 94406.0i	− 0.172400i
$741$	0	0
$742$	0	0
$743$	−773801.	−1.40169	−0.700845	−	0.713314i	$-0.747193\pi$
$743$	−773801.	−1.40169	−0.700845	+	0.713314i	$0.747193\pi$
$744$	0	0
$745$	82471.7i	0.148591i
$746$	421105.	0.756681
$747$	0	0
$748$	115486.i	0.206408i
$749$	0	0
$750$	0	0
$751$	197844.	0.350787	0.175394	−	0.984498i	$-0.443880\pi$
$751$	197844.	0.350787	0.175394	+	0.984498i	$0.443880\pi$
$752$	39278.5i	0.0694574i
$753$	0	0
$754$	479393.i	0.843236i
$755$	356483.i	0.625381i
$756$	0	0
$757$	−770706.	−1.34492	−0.672461	−	0.740132i	$-0.734763\pi$
$757$	−770706.	−1.34492	−0.672461	+	0.740132i	$0.734763\pi$
$758$	−397868.	−0.692469
$759$	0	0
$760$	−94392.2	−0.163421
$761$	152140.i	0.262708i	0.991336	+	0.131354i	$0.0419325\pi$
$761$	152140.i	0.262708i	−0.991336	+	0.131354i	$0.958068\pi$
$762$	0	0
$763$	0	0
$764$	516473.	0.884832
$765$	0	0
$766$	82936.1i	0.141347i
$767$	799521.	1.35906
$768$	0	0
$769$	961897.i	1.62658i	0.581857	+	0.813291i	$0.302326\pi$
$769$	961897.i	1.62658i	−0.581857	+	0.813291i	$0.697674\pi$
$770$	0	0
$771$	0	0
$772$	293397.	0.492290
$773$	− 151069.i	− 0.252823i	−0.991978	−	0.126411i	$-0.959654\pi$
$773$	− 151069.i	− 0.252823i	0.991978	−	0.126411i	$-0.0403459\pi$
$774$	0	0
$775$	283137.i	0.471404i
$776$	35267.3i	0.0585664i
$777$	0	0
$778$	374540.	0.618783
$779$	−830452.	−1.36848
$780$	0	0
$781$	−22640.3	−0.0371175
$782$	− 452435.i	− 0.739849i
$783$	0	0
$784$	0	0
$785$	338215.	0.548849
$786$	0	0
$787$	198891.i	0.321118i	0.987026	+	0.160559i	$0.0513297\pi$
$787$	198891.i	0.321118i	−0.987026	+	0.160559i	$0.948670\pi$
$788$	585176.	0.942398
$789$	0	0
$790$	− 259123.i	− 0.415194i
$791$	0	0
$792$	0	0
$793$	−520153.	−0.827150
$794$	− 96810.3i	− 0.153561i
$795$	0	0
$796$	11208.2i	0.0176893i
$797$	990756.i	1.55973i	0.625947	+	0.779866i	$0.284713\pi$
$797$	990756.i	1.55973i	−0.625947	+	0.779866i	$0.715287\pi$
$798$	0	0
$799$	138368.	0.216741
$800$	−76874.6	−0.120117
$801$	0	0
$802$	−506518.	−0.787492
$803$	260597.i	0.404145i
$804$	0	0
$805$	0	0
$806$	431684.	0.664501
$807$	0	0
$808$	− 356050.i	− 0.545366i
$809$	92288.6	0.141010	0.0705052	−	0.997511i	$-0.477539\pi$
$809$	92288.6	0.141010	0.0705052	+	0.997511i	$0.477539\pi$
$810$	0	0
$811$	− 617125.i	− 0.938277i	−0.883125	−	0.469139i	$-0.844564\pi$
$811$	− 617125.i	− 0.938277i	0.883125	−	0.469139i	$-0.155436\pi$
$812$	0	0
$813$	0	0
$814$	−150997.	−0.227887
$815$	− 602743.i	− 0.907439i
$816$	0	0
$817$	− 903784.i	− 1.35401i
$818$	140762.i	0.210367i
$819$	0	0
$820$	319033.	0.474469
$821$	−307066.	−0.455559	−0.227780	−	0.973713i	$-0.573147\pi$
$821$	−307066.	−0.455559	−0.227780	+	0.973713i	$0.573147\pi$
$822$	0	0
$823$	846858.	1.25029	0.625146	−	0.780508i	$-0.285040\pi$
$823$	846858.	1.25029	0.625146	+	0.780508i	$0.285040\pi$
$824$	− 763.194i	− 0.00112404i
$825$	0	0
$826$	0	0
$827$	−294059.	−0.429956	−0.214978	−	0.976619i	$-0.568968\pi$
$827$	−294059.	−0.429956	−0.214978	+	0.976619i	$0.568968\pi$
$828$	0	0
$829$	− 1.03438e6i	− 1.50513i	−0.658520	−	0.752563i	$-0.728817\pi$
$829$	− 1.03438e6i	− 1.50513i	0.658520	−	0.752563i	$-0.271183\pi$
$830$	−329299.	−0.478007
$831$	0	0
$832$	117207.i	0.169319i
$833$	0	0
$834$	0	0
$835$	373036.	0.535030
$836$	150975.i	0.216019i
$837$	0	0
$838$	124313.i	0.177022i
$839$	− 307258.i	− 0.436495i	−0.975893	−	0.218248i	$-0.929966\pi$
$839$	− 307258.i	− 0.436495i	0.975893	−	0.218248i	$-0.0700340\pi$
$840$	0	0
$841$	−159093.	−0.224937
$842$	617324.	0.870741
$843$	0	0
$844$	465853.	0.653980
$845$	− 337463.i	− 0.472621i
$846$	0	0
$847$	0	0
$848$	73766.4	0.102581
$849$	0	0
$850$	270809.i	0.374822i
$851$	591555.	0.816838
$852$	0	0
$853$	70737.4i	0.0972190i	0.998818	+	0.0486095i	$0.0154790\pi$
$853$	70737.4i	0.0972190i	−0.998818	+	0.0486095i	$0.984521\pi$
$854$	0	0
$855$	0	0
$856$	−123239.	−0.168191
$857$	329863.i	0.449130i	0.974459	+	0.224565i	$0.0720961\pi$
$857$	329863.i	0.449130i	−0.974459	+	0.224565i	$0.927904\pi$
$858$	0	0
$859$	− 69962.5i	− 0.0948154i	−0.998876	−	0.0474077i	$-0.984904\pi$
$859$	− 69962.5i	− 0.0948154i	0.998876	−	0.0474077i	$-0.0150960\pi$
$860$	347205.i	0.469450i
$861$	0	0
$862$	−217052.	−0.292113
$863$	772223.	1.03686	0.518431	−	0.855119i	$-0.326516\pi$
$863$	772223.	1.03686	0.518431	+	0.855119i	$0.326516\pi$
$864$	0	0
$865$	396572.	0.530017
$866$	612957.i	0.817324i
$867$	0	0
$868$	0	0
$869$	−414451.	−0.548825
$870$	0	0
$871$	− 1.98579e6i	− 2.61756i
$872$	−377827.	−0.496890
$873$	0	0
$874$	− 591468.i	− 0.774299i
$875$	0	0
$876$	0	0
$877$	242520.	0.315317	0.157659	−	0.987494i	$-0.449605\pi$
$877$	242520.	0.315317	0.157659	+	0.987494i	$0.449605\pi$
$878$	− 67209.3i	− 0.0871847i
$879$	0	0
$880$	− 57999.7i	− 0.0748963i
$881$	− 646528.i	− 0.832982i	−0.909140	−	0.416491i	$-0.863260\pi$
$881$	− 646528.i	− 0.832982i	0.909140	−	0.416491i	$-0.136740\pi$
$882$	0	0
$883$	−461877.	−0.592387	−0.296193	−	0.955128i	$-0.595717\pi$
$883$	−461877.	−0.592387	−0.296193	+	0.955128i	$0.595717\pi$
$884$	412888.	0.528357
$885$	0	0
$886$	494592.	0.630057
$887$	215226.i	0.273557i	0.990602	+	0.136778i	$0.0436748\pi$
$887$	215226.i	0.273557i	−0.990602	+	0.136778i	$0.956325\pi$
$888$	0	0
$889$	0	0
$890$	622034.	0.785297
$891$	0	0
$892$	− 488407.i	− 0.613836i
$893$	180888.	0.226833
$894$	0	0
$895$	− 359479.i	− 0.448774i
$896$	0	0
$897$	0	0
$898$	−551980.	−0.684496
$899$	− 493632.i	− 0.610779i
$900$	0	0
$901$	− 259860.i	− 0.320103i
$902$	− 510275.i	− 0.627179i
$903$	0	0
$904$	213957.	0.261812
$905$	624134.	0.762045
$906$	0	0
$907$	1.09827e6	1.33504	0.667519	−	0.744592i	$-0.267356\pi$
$907$	1.09827e6	1.33504	0.667519	+	0.744592i	$0.267356\pi$
$908$	199949.i	0.242520i
$909$	0	0
$910$	0	0
$911$	1.32312e6	1.59427	0.797133	−	0.603803i	$-0.206349\pi$
$911$	1.32312e6	1.59427	0.797133	+	0.603803i	$0.206349\pi$
$912$	0	0
$913$	526695.i	0.631855i
$914$	−165384.	−0.197971
$915$	0	0
$916$	− 151431.i	− 0.180478i
$917$	0	0
$918$	0	0
$919$	516528.	0.611594	0.305797	−	0.952097i	$-0.401077\pi$
$919$	516528.	0.611594	0.305797	+	0.952097i	$0.401077\pi$
$920$	227223.i	0.268459i
$921$	0	0
$922$	− 869014.i	− 1.02227i
$923$	80943.8i	0.0950124i
$924$	0	0
$925$	−354080.	−0.413826
$926$	50267.5	0.0586226
$927$	0	0
$928$	134026.	0.155630
$929$	− 1.23939e6i	− 1.43608i	−0.696003	−	0.718039i	$-0.745040\pi$
$929$	− 1.23939e6i	− 1.43608i	0.696003	−	0.718039i	$-0.254960\pi$
$930$	0	0
$931$	0	0
$932$	−53477.9	−0.0615662
$933$	0	0
$934$	183749.i	0.210635i
$935$	−204318.	−0.233713
$936$	0	0
$937$	− 1.15260e6i	− 1.31281i	−0.754411	−	0.656403i	$-0.772077\pi$
$937$	− 1.15260e6i	− 1.31281i	0.754411	−	0.656403i	$-0.227923\pi$
$938$	0	0
$939$	0	0
$940$	−69491.3	−0.0786457
$941$	107507.i	0.121411i	0.998156	+	0.0607057i	$0.0193351\pi$
$941$	107507.i	0.121411i	−0.998156	+	0.0607057i	$0.980665\pi$
$942$	0	0
$943$	1.99909e6i	2.24806i
$944$	− 223526.i	− 0.250832i
$945$	0	0
$946$	555334.	0.620543
$947$	412614.	0.460092	0.230046	−	0.973180i	$-0.426112\pi$
$947$	412614.	0.460092	0.230046	+	0.973180i	$0.426112\pi$
$948$	0	0
$949$	931689.	1.03452
$950$	354028.i	0.392275i
$951$	0	0
$952$	0	0
$953$	−1.43052e6	−1.57510	−0.787550	−	0.616251i	$-0.788651\pi$
$953$	−1.43052e6	−1.57510	−0.787550	+	0.616251i	$0.788651\pi$
$954$	0	0
$955$	913742.i	1.00188i
$956$	771688.	0.844356
$957$	0	0
$958$	− 596619.i	− 0.650079i
$959$	0	0
$960$	0	0
$961$	479015.	0.518683
$962$	539847.i	0.583338i
$963$	0	0
$964$	− 437602.i	− 0.470896i
$965$	519077.i	0.557413i
$966$	0	0
$967$	−344533.	−0.368449	−0.184225	−	0.982884i	$-0.558977\pi$
$967$	−344533.	−0.368449	−0.184225	+	0.982884i	$0.558977\pi$
$968$	238521.	0.254551
$969$	0	0
$970$	−62394.8	−0.0663139
$971$	1.45316e6i	1.54126i	0.637283	+	0.770630i	$0.280058\pi$
$971$	1.45316e6i	1.54126i	−0.637283	+	0.770630i	$0.719942\pi$
$972$	0	0
$973$	0	0
$974$	647393.	0.682417
$975$	0	0
$976$	145422.i	0.152661i
$977$	−1.54571e6	−1.61934	−0.809669	−	0.586887i	$-0.800353\pi$
$977$	−1.54571e6	−1.61934	−0.809669	+	0.586887i	$0.800353\pi$
$978$	0	0
$979$	− 994907.i	− 1.03805i
$980$	0	0
$981$	0	0
$982$	396970.	0.411656
$983$	134268.i	0.138952i	0.997584	+	0.0694761i	$0.0221328\pi$
$983$	134268.i	0.138952i	−0.997584	+	0.0694761i	$0.977867\pi$
$984$	0	0
$985$	1.03529e6i	1.06706i
$986$	− 472139.i	− 0.485641i
$987$	0	0
$988$	539768.	0.552959
$989$	−2.17561e6	−2.22428
$990$	0	0
$991$	−791191.	−0.805627	−0.402814	−	0.915282i	$-0.631968\pi$
$991$	−791191.	−0.805627	−0.402814	+	0.915282i	$0.631968\pi$
$992$	− 120688.i	− 0.122642i
$993$	0	0
$994$	0	0
$995$	−19829.5	−0.0200293
$996$	0	0
$997$	− 672757.i	− 0.676811i	−0.941000	−	0.338406i	$-0.890112\pi$
$997$	− 672757.i	− 0.676811i	0.941000	−	0.338406i	$-0.109888\pi$
$998$	976732.	0.980650
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.c.a.685.2		4
3.2	odd	2		294.5.c.a.97.4		4
7.2	even	3		126.5.n.b.73.2		4
7.3	odd	6		126.5.n.b.19.2		4
7.6	odd	2	inner	882.5.c.a.685.1		4
21.2	odd	6		42.5.g.a.31.1	yes	4
21.5	even	6		294.5.g.c.31.1		4
21.11	odd	6		294.5.g.c.19.1		4
21.17	even	6		42.5.g.a.19.1	✓	4
21.20	even	2		294.5.c.a.97.3		4
84.23	even	6		336.5.bh.d.241.2		4
84.59	odd	6		336.5.bh.d.145.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.5.g.a.19.1	✓	4	21.17	even	6
42.5.g.a.31.1	yes	4	21.2	odd	6
126.5.n.b.19.2		4	7.3	odd	6
126.5.n.b.73.2		4	7.2	even	3
294.5.c.a.97.3		4	21.20	even	2
294.5.c.a.97.4		4	3.2	odd	2
294.5.g.c.19.1		4	21.11	odd	6
294.5.g.c.31.1		4	21.5	even	6
336.5.bh.d.145.2		4	84.59	odd	6
336.5.bh.d.241.2		4	84.23	even	6
882.5.c.a.685.1		4	7.6	odd	2	inner
882.5.c.a.685.2		4	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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	882.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843 q^{2} +8.00000 q^{4} +14.1536i q^{5} -22.6274 q^{8} -40.0324i q^{10} -64.0294 q^{11} +228.919i q^{13} +64.0000 q^{16} -225.455i q^{17} -294.737i q^{19} +113.229i q^{20} +181.103 q^{22} -709.499 q^{23} +424.676 q^{25} -647.481i q^{26} -740.397 q^{29} +666.713i q^{31} -181.019 q^{32} +637.683i q^{34} -833.765 q^{37} +833.642i q^{38} -320.259i q^{40} -2817.60i q^{41} +3066.41 q^{43} -512.235 q^{44} +2006.77 q^{46} +613.726i q^{47} -1201.17 q^{50} +1831.35i q^{52} +1152.60 q^{53} -906.246i q^{55} +2094.16 q^{58} -3492.59i q^{59} +2272.21i q^{61} -1885.75i q^{62} +512.000 q^{64} -3240.03 q^{65} -8674.62 q^{67} -1803.64i q^{68} +353.591 q^{71} -4069.95i q^{73} +2358.24 q^{74} -2357.90i q^{76} +6472.83 q^{79} +905.829i q^{80} +7969.38i q^{82} -8225.83i q^{83} +3191.00 q^{85} -8673.11 q^{86} +1448.82 q^{88} +15538.3i q^{89} -5675.99 q^{92} -1735.88i q^{94} +4171.58 q^{95} -1558.61i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{4} - 324 q^{11} + 256 q^{16} - 192 q^{22} + 624 q^{23} + 952 q^{25} - 2724 q^{29} - 2792 q^{37} - 632 q^{43} - 2592 q^{44} + 9792 q^{46} - 2112 q^{50} - 2076 q^{53} + 672 q^{58} + 2048 q^{64}+ \cdots + 6504 q^{95}+O(q^{100}) \) 4 * q + 32 * q^4 - 324 * q^11 + 256 * q^16 - 192 * q^22 + 624 * q^23 + 952 * q^25 - 2724 * q^29 - 2792 * q^37 - 632 * q^43 - 2592 * q^44 + 9792 * q^46 - 2112 * q^50 - 2076 * q^53 + 672 * q^58 + 2048 * q^64 - 1488 * q^65 - 29200 * q^67 + 9696 * q^71 + 1536 * q^74 - 7948 * q^79 + 1224 * q^85 - 36480 * q^86 - 1536 * q^88 + 4992 * q^92 + 6504 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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