LMFDB - Embedded newform 882.5.b.d.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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.2
Root			$2.57794i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.d.197.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} +44.5764i q^{5} +22.6274i q^{8} +O(q^{10})$	$q-2.82843i q^{2} -8.00000 q^{4} +44.5764i q^{5} +22.6274i q^{8} +126.081 q^{10} -50.9690i q^{11} -43.9190 q^{13} +64.0000 q^{16} -320.520i q^{17} +565.608 q^{19} -356.611i q^{20} -144.162 q^{22} -364.924i q^{23} -1362.05 q^{25} +124.222i q^{26} +263.445i q^{29} -71.0405 q^{31} -181.019i q^{32} -906.567 q^{34} -2159.22 q^{37} -1599.78i q^{38} -1008.65 q^{40} -3083.91i q^{41} +2618.24 q^{43} +407.752i q^{44} -1032.16 q^{46} +3051.66i q^{47} +3852.47i q^{50} +351.352 q^{52} -4166.84i q^{53} +2272.01 q^{55} +745.135 q^{58} +5056.02i q^{59} -280.068 q^{61} +200.933i q^{62} -512.000 q^{64} -1957.75i q^{65} +921.540 q^{67} +2564.16i q^{68} -7866.37i q^{71} +2265.64 q^{73} +6107.18i q^{74} -4524.86 q^{76} +3170.65 q^{79} +2852.89i q^{80} -8722.62 q^{82} +3101.19i q^{83} +14287.6 q^{85} -7405.51i q^{86} +1153.30 q^{88} -1676.73i q^{89} +2919.40i q^{92} +8631.40 q^{94} +25212.7i q^{95} +14578.7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4}+O(q^{10}) $ 4 * q - 32 * q^4	$ 4 q - 32 q^{4} + 208 q^{10} - 472 q^{13} + 256 q^{16} + 40 q^{19} + 16 q^{22} - 1596 q^{25} - 136 q^{31} - 1552 q^{34} - 4192 q^{37} - 1664 q^{40} + 9584 q^{43} - 3536 q^{46} + 3776 q^{52} + 5384 q^{55} - 1168 q^{58} - 4528 q^{61} - 2048 q^{64} - 1944 q^{67} + 21360 q^{73} - 320 q^{76} + 10312 q^{79} - 18000 q^{82} + 29296 q^{85} - 128 q^{88} + 22080 q^{94} + 44832 q^{97}+O(q^{100}) $ 4 * q - 32 * q^4 + 208 * q^10 - 472 * q^13 + 256 * q^16 + 40 * q^19 + 16 * q^22 - 1596 * q^25 - 136 * q^31 - 1552 * q^34 - 4192 * q^37 - 1664 * q^40 + 9584 * q^43 - 3536 * q^46 + 3776 * q^52 + 5384 * q^55 - 1168 * q^58 - 4528 * q^61 - 2048 * q^64 - 1944 * q^67 + 21360 * q^73 - 320 * q^76 + 10312 * q^79 - 18000 * q^82 + 29296 * q^85 - 128 * q^88 + 22080 * q^94 + 44832 * q^97

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
$5$	44.5764i	1.78306i	0.452966	+	0.891528i	$0.350366\pi$
$5$	44.5764i	1.78306i	−0.452966	+	0.891528i	$0.649634\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.6274i	0.353553i
$9$	0	0
$10$	126.081	1.26081
$11$	− 50.9690i	− 0.421231i	−0.977569	−	0.210616i	$-0.932453\pi$
$11$	− 50.9690i	− 0.421231i	0.977569	−	0.210616i	$-0.0675469\pi$
$12$	0	0
$13$	−43.9190	−0.259876	−0.129938	−	0.991522i	$-0.541478\pi$
$13$	−43.9190	−0.259876	−0.129938	+	0.991522i	$0.541478\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	− 320.520i	− 1.10907i	−0.832162	−	0.554533i	$-0.812897\pi$
$17$	− 320.520i	− 1.10907i	0.832162	−	0.554533i	$-0.187103\pi$
$18$	0	0
$19$	565.608	1.56678	0.783390	−	0.621530i	$-0.213489\pi$
$19$	565.608	1.56678	0.783390	+	0.621530i	$0.213489\pi$
$20$	− 356.611i	− 0.891528i
$21$	0	0
$22$	−144.162	−0.297856
$23$	− 364.924i	− 0.689838i	−0.938632	−	0.344919i	$-0.887906\pi$
$23$	− 364.924i	− 0.689838i	0.938632	−	0.344919i	$-0.112094\pi$
$24$	0	0
$25$	−1362.05	−2.17929
$26$	124.222i	0.183760i
$27$	0	0
$28$	0	0
$29$	263.445i	0.313252i	0.987658	+	0.156626i	$0.0500617\pi$
$29$	263.445i	0.313252i	−0.987658	+	0.156626i	$0.949938\pi$
$30$	0	0
$31$	−71.0405	−0.0739235	−0.0369618	−	0.999317i	$-0.511768\pi$
$31$	−71.0405	−0.0739235	−0.0369618	+	0.999317i	$0.511768\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	−906.567	−0.784228
$35$	0	0
$36$	0	0
$37$	−2159.22	−1.57722	−0.788611	−	0.614893i	$-0.789199\pi$
$37$	−2159.22	−1.57722	−0.788611	+	0.614893i	$0.789199\pi$
$38$	− 1599.78i	− 1.10788i
$39$	0	0
$40$	−1008.65	−0.630405
$41$	− 3083.91i	− 1.83457i	−0.398232	−	0.917285i	$-0.630376\pi$
$41$	− 3083.91i	− 1.83457i	0.398232	−	0.917285i	$-0.369624\pi$
$42$	0	0
$43$	2618.24	1.41603	0.708016	−	0.706196i	$-0.249590\pi$
$43$	2618.24	1.41603	0.708016	+	0.706196i	$0.249590\pi$
$44$	407.752i	0.210616i
$45$	0	0
$46$	−1032.16	−0.487789
$47$	3051.66i	1.38147i	0.723109	+	0.690734i	$0.242712\pi$
$47$	3051.66i	1.38147i	−0.723109	+	0.690734i	$0.757288\pi$
$48$	0	0
$49$	0	0
$50$	3852.47i	1.54099i
$51$	0	0
$52$	351.352	0.129938
$53$	− 4166.84i	− 1.48339i	−0.670737	−	0.741695i	$-0.734022\pi$
$53$	− 4166.84i	− 1.48339i	0.670737	−	0.741695i	$-0.265978\pi$
$54$	0	0
$55$	2272.01	0.751079
$56$	0	0
$57$	0	0
$58$	745.135	0.221503
$59$	5056.02i	1.45246i	0.687450	+	0.726231i	$0.258730\pi$
$59$	5056.02i	1.45246i	−0.687450	+	0.726231i	$0.741270\pi$
$60$	0	0
$61$	−280.068	−0.0752669	−0.0376334	−	0.999292i	$-0.511982\pi$
$61$	−280.068	−0.0752669	−0.0376334	+	0.999292i	$0.511982\pi$
$62$	200.933i	0.0522718i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 1957.75i	− 0.463372i
$66$	0	0
$67$	921.540	0.205288	0.102644	−	0.994718i	$-0.467270\pi$
$67$	921.540	0.205288	0.102644	+	0.994718i	$0.467270\pi$
$68$	2564.16i	0.554533i
$69$	0	0
$70$	0	0
$71$	− 7866.37i	− 1.56048i	−0.625482	−	0.780239i	$-0.715098\pi$
$71$	− 7866.37i	− 1.56048i	0.625482	−	0.780239i	$-0.284902\pi$
$72$	0	0
$73$	2265.64	0.425152	0.212576	−	0.977144i	$-0.431815\pi$
$73$	2265.64	0.425152	0.212576	+	0.977144i	$0.431815\pi$
$74$	6107.18i	1.11526i
$75$	0	0
$76$	−4524.86	−0.783390
$77$	0	0
$78$	0	0
$79$	3170.65	0.508035	0.254018	−	0.967200i	$-0.418248\pi$
$79$	3170.65	0.508035	0.254018	+	0.967200i	$0.418248\pi$
$80$	2852.89i	0.445764i
$81$	0	0
$82$	−8722.62	−1.29724
$83$	3101.19i	0.450166i	0.974340	+	0.225083i	$0.0722653\pi$
$83$	3101.19i	0.450166i	−0.974340	+	0.225083i	$0.927735\pi$
$84$	0	0
$85$	14287.6	1.97752
$86$	− 7405.51i	− 1.00129i
$87$	0	0
$88$	1153.30	0.148928
$89$	− 1676.73i	− 0.211682i	−0.994383	−	0.105841i	$-0.966247\pi$
$89$	− 1676.73i	− 0.211682i	0.994383	−	0.105841i	$-0.0337534\pi$
$90$	0	0
$91$	0	0
$92$	2919.40i	0.344919i
$93$	0	0
$94$	8631.40	0.976845
$95$	25212.7i	2.79366i
$96$	0	0
$97$	14578.7	1.54944	0.774720	−	0.632304i	$-0.217891\pi$
$97$	14578.7	1.54944	0.774720	+	0.632304i	$0.217891\pi$
$98$	0	0
$99$	0	0
$100$	10896.4	1.08964
$101$	11706.4i	1.14757i	0.819005	+	0.573787i	$0.194526\pi$
$101$	11706.4i	1.14757i	−0.819005	+	0.573787i	$0.805474\pi$
$102$	0	0
$103$	8763.07	0.826003	0.413002	−	0.910730i	$-0.364480\pi$
$103$	8763.07	0.826003	0.413002	+	0.910730i	$0.364480\pi$
$104$	− 993.773i	− 0.0918799i
$105$	0	0
$106$	−11785.6	−1.04892
$107$	− 7557.12i	− 0.660068i	−0.943969	−	0.330034i	$-0.892940\pi$
$107$	− 7557.12i	− 0.660068i	0.943969	−	0.330034i	$-0.107060\pi$
$108$	0	0
$109$	20613.5	1.73499	0.867497	−	0.497443i	$-0.165728\pi$
$109$	20613.5	1.73499	0.867497	+	0.497443i	$0.165728\pi$
$110$	− 6426.22i	− 0.531093i
$111$	0	0
$112$	0	0
$113$	633.524i	0.0496142i	0.999692	+	0.0248071i	$0.00789716\pi$
$113$	633.524i	0.0496142i	−0.999692	+	0.0248071i	$0.992103\pi$
$114$	0	0
$115$	16267.0	1.23002
$116$	− 2107.56i	− 0.156626i
$117$	0	0
$118$	14300.6	1.02705
$119$	0	0
$120$	0	0
$121$	12043.2	0.822564
$122$	792.152i	0.0532217i
$123$	0	0
$124$	568.324	0.0369618
$125$	− 32855.2i	− 2.10273i
$126$	0	0
$127$	9531.06	0.590927	0.295463	−	0.955354i	$-0.404526\pi$
$127$	9531.06	0.590927	0.295463	+	0.955354i	$0.404526\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	−5537.35	−0.327654
$131$	− 6504.88i	− 0.379050i	−0.981876	−	0.189525i	$-0.939305\pi$
$131$	− 6504.88i	− 0.379050i	0.981876	−	0.189525i	$-0.0606948\pi$
$132$	0	0
$133$	0	0
$134$	− 2606.51i	− 0.145161i
$135$	0	0
$136$	7252.54	0.392114
$137$	22445.5i	1.19588i	0.801541	+	0.597940i	$0.204014\pi$
$137$	22445.5i	1.19588i	−0.801541	+	0.597940i	$0.795986\pi$
$138$	0	0
$139$	29574.1	1.53067	0.765336	−	0.643631i	$-0.222573\pi$
$139$	29574.1	1.53067	0.765336	+	0.643631i	$0.222573\pi$
$140$	0	0
$141$	0	0
$142$	−22249.4	−1.10342
$143$	2238.51i	0.109468i
$144$	0	0
$145$	−11743.4	−0.558545
$146$	− 6408.19i	− 0.300628i
$147$	0	0
$148$	17273.7	0.788611
$149$	5721.59i	0.257718i	0.991663	+	0.128859i	$0.0411314\pi$
$149$	5721.59i	0.257718i	−0.991663	+	0.128859i	$0.958869\pi$
$150$	0	0
$151$	−44480.4	−1.95081	−0.975405	−	0.220418i	$-0.929258\pi$
$151$	−44480.4	−1.95081	−0.975405	+	0.220418i	$0.929258\pi$
$152$	12798.2i	0.553941i
$153$	0	0
$154$	0	0
$155$	− 3166.73i	− 0.131810i
$156$	0	0
$157$	−4425.33	−0.179534	−0.0897670	−	0.995963i	$-0.528612\pi$
$157$	−4425.33	−0.179534	−0.0897670	+	0.995963i	$0.528612\pi$
$158$	− 8967.95i	− 0.359235i
$159$	0	0
$160$	8069.19	0.315203
$161$	0	0
$162$	0	0
$163$	6414.14	0.241415	0.120707	−	0.992688i	$-0.461484\pi$
$163$	6414.14	0.241415	0.120707	+	0.992688i	$0.461484\pi$
$164$	24671.3i	0.917285i
$165$	0	0
$166$	8771.50	0.318315
$167$	− 27007.7i	− 0.968399i	−0.874958	−	0.484199i	$-0.839111\pi$
$167$	− 27007.7i	− 0.968399i	0.874958	−	0.484199i	$-0.160889\pi$
$168$	0	0
$169$	−26632.1	−0.932465
$170$	− 40411.5i	− 1.39832i
$171$	0	0
$172$	−20945.9	−0.708016
$173$	43268.4i	1.44570i	0.691005	+	0.722850i	$0.257168\pi$
$173$	43268.4i	1.44570i	−0.691005	+	0.722850i	$0.742832\pi$
$174$	0	0
$175$	0	0
$176$	− 3262.02i	− 0.105308i
$177$	0	0
$178$	−4742.51	−0.149682
$179$	− 35603.2i	− 1.11118i	−0.831457	−	0.555589i	$-0.812493\pi$
$179$	− 35603.2i	− 1.11118i	0.831457	−	0.555589i	$-0.187507\pi$
$180$	0	0
$181$	31109.8	0.949598	0.474799	−	0.880094i	$-0.342521\pi$
$181$	31109.8	0.949598	0.474799	+	0.880094i	$0.342521\pi$
$182$	0	0
$183$	0	0
$184$	8257.30	0.243895
$185$	− 96250.0i	− 2.81227i
$186$	0	0
$187$	−16336.6	−0.467173
$188$	− 24413.3i	− 0.690734i
$189$	0	0
$190$	71312.4	1.97541
$191$	− 38815.0i	− 1.06398i	−0.846751	−	0.531990i	$-0.821444\pi$
$191$	− 38815.0i	− 1.06398i	0.846751	−	0.531990i	$-0.178556\pi$
$192$	0	0
$193$	4935.27	0.132494	0.0662471	−	0.997803i	$-0.478897\pi$
$193$	4935.27	0.132494	0.0662471	+	0.997803i	$0.478897\pi$
$194$	− 41234.8i	− 1.09562i
$195$	0	0
$196$	0	0
$197$	18135.9i	0.467311i	0.972319	+	0.233656i	$0.0750689\pi$
$197$	18135.9i	0.467311i	−0.972319	+	0.233656i	$0.924931\pi$
$198$	0	0
$199$	−28645.6	−0.723355	−0.361678	−	0.932303i	$-0.617796\pi$
$199$	−28645.6	−0.723355	−0.361678	+	0.932303i	$0.617796\pi$
$200$	− 30819.8i	− 0.770494i
$201$	0	0
$202$	33110.7	0.811457
$203$	0	0
$204$	0	0
$205$	137470.	3.27114
$206$	− 24785.7i	− 0.584072i
$207$	0	0
$208$	−2810.81	−0.0649689
$209$	− 28828.5i	− 0.659977i
$210$	0	0
$211$	62987.2	1.41478	0.707388	−	0.706826i	$-0.249873\pi$
$211$	62987.2	1.41478	0.707388	+	0.706826i	$0.249873\pi$
$212$	33334.8i	0.741695i
$213$	0	0
$214$	−21374.8	−0.466739
$215$	116712.i	2.52486i
$216$	0	0
$217$	0	0
$218$	− 58303.6i	− 1.22683i
$219$	0	0
$220$	−18176.1	−0.375539
$221$	14076.9i	0.288219i
$222$	0	0
$223$	−42250.3	−0.849611	−0.424806	−	0.905285i	$-0.639658\pi$
$223$	−42250.3	−0.849611	−0.424806	+	0.905285i	$0.639658\pi$
$224$	0	0
$225$	0	0
$226$	1791.88	0.0350826
$227$	− 2129.31i	− 0.0413226i	−0.999787	−	0.0206613i	$-0.993423\pi$
$227$	− 2129.31i	− 0.0413226i	0.999787	−	0.0206613i	$-0.00657716\pi$
$228$	0	0
$229$	−49050.1	−0.935339	−0.467670	−	0.883903i	$-0.654906\pi$
$229$	−49050.1	−0.935339	−0.467670	+	0.883903i	$0.654906\pi$
$230$	− 46010.0i	− 0.869755i
$231$	0	0
$232$	−5961.08	−0.110751
$233$	− 20829.6i	− 0.383679i	−0.981426	−	0.191840i	$-0.938555\pi$
$233$	− 20829.6i	− 0.383679i	0.981426	−	0.191840i	$-0.0614454\pi$
$234$	0	0
$235$	−136032.	−2.46323
$236$	− 40448.2i	− 0.726231i
$237$	0	0
$238$	0	0
$239$	− 102950.i	− 1.80232i	−0.433488	−	0.901159i	$-0.642717\pi$
$239$	− 102950.i	− 1.80232i	0.433488	−	0.901159i	$-0.357283\pi$
$240$	0	0
$241$	67171.7	1.15652	0.578259	−	0.815853i	$-0.303732\pi$
$241$	67171.7	1.15652	0.578259	+	0.815853i	$0.303732\pi$
$242$	− 34063.2i	− 0.581641i
$243$	0	0
$244$	2240.54	0.0376334
$245$	0	0
$246$	0	0
$247$	−24840.9	−0.407168
$248$	− 1607.46i	− 0.0261359i
$249$	0	0
$250$	−92928.5	−1.48686
$251$	78073.0i	1.23923i	0.784904	+	0.619617i	$0.212712\pi$
$251$	78073.0i	1.23923i	−0.784904	+	0.619617i	$0.787288\pi$
$252$	0	0
$253$	−18599.8	−0.290581
$254$	− 26957.9i	− 0.417848i
$255$	0	0
$256$	4096.00	0.0625000
$257$	1627.00i	0.0246333i	0.999924	+	0.0123166i	$0.00392060\pi$
$257$	1627.00i	0.0246333i	−0.999924	+	0.0123166i	$0.996079\pi$
$258$	0	0
$259$	0	0
$260$	15662.0i	0.231686i
$261$	0	0
$262$	−18398.6	−0.268029
$263$	121788.i	1.76073i	0.474292	+	0.880367i	$0.342704\pi$
$263$	121788.i	1.76073i	−0.474292	+	0.880367i	$0.657296\pi$
$264$	0	0
$265$	185743.	2.64497
$266$	0	0
$267$	0	0
$268$	−7372.32	−0.102644
$269$	− 51811.0i	− 0.716007i	−0.933720	−	0.358004i	$-0.883458\pi$
$269$	− 51811.0i	− 0.716007i	0.933720	−	0.358004i	$-0.116542\pi$
$270$	0	0
$271$	65656.0	0.893996	0.446998	−	0.894535i	$-0.352493\pi$
$271$	65656.0	0.893996	0.446998	+	0.894535i	$0.352493\pi$
$272$	− 20513.3i	− 0.277266i
$273$	0	0
$274$	63485.3	0.845614
$275$	69422.5i	0.917983i
$276$	0	0
$277$	112957.	1.47216	0.736081	−	0.676894i	$-0.236674\pi$
$277$	112957.	1.47216	0.736081	+	0.676894i	$0.236674\pi$
$278$	− 83648.2i	− 1.08235i
$279$	0	0
$280$	0	0
$281$	− 72488.0i	− 0.918022i	−0.888431	−	0.459011i	$-0.848204\pi$
$281$	− 72488.0i	− 0.918022i	0.888431	−	0.459011i	$-0.151796\pi$
$282$	0	0
$283$	−66170.0	−0.826205	−0.413103	−	0.910684i	$-0.635555\pi$
$283$	−66170.0	−0.826205	−0.413103	+	0.910684i	$0.635555\pi$
$284$	62930.9i	0.780239i
$285$	0	0
$286$	6331.45	0.0774054
$287$	0	0
$288$	0	0
$289$	−19212.0	−0.230026
$290$	33215.4i	0.394951i
$291$	0	0
$292$	−18125.1	−0.212576
$293$	− 17410.4i	− 0.202802i	−0.994846	−	0.101401i	$-0.967667\pi$
$293$	− 17410.4i	− 0.202802i	0.994846	−	0.101401i	$-0.0323325\pi$
$294$	0	0
$295$	−225379.	−2.58982
$296$	− 48857.5i	− 0.557632i
$297$	0	0
$298$	16183.1	0.182234
$299$	16027.1i	0.179272i
$300$	0	0
$301$	0	0
$302$	125810.i	1.37943i
$303$	0	0
$304$	36198.9	0.391695
$305$	− 12484.4i	− 0.134205i
$306$	0	0
$307$	139697.	1.48221	0.741107	−	0.671386i	$-0.234301\pi$
$307$	139697.	1.48221	0.741107	+	0.671386i	$0.234301\pi$
$308$	0	0
$309$	0	0
$310$	−8956.86	−0.0932036
$311$	− 14782.3i	− 0.152835i	−0.997076	−	0.0764175i	$-0.975652\pi$
$311$	− 14782.3i	− 0.152835i	0.997076	−	0.0764175i	$-0.0243482\pi$
$312$	0	0
$313$	−109301.	−1.11567	−0.557836	−	0.829951i	$-0.688368\pi$
$313$	−109301.	−1.11567	−0.557836	+	0.829951i	$0.688368\pi$
$314$	12516.7i	0.126950i
$315$	0	0
$316$	−25365.2	−0.254018
$317$	113824.i	1.13270i	0.824164	+	0.566351i	$0.191645\pi$
$317$	113824.i	1.13270i	−0.824164	+	0.566351i	$0.808355\pi$
$318$	0	0
$319$	13427.5	0.131952
$320$	− 22823.1i	− 0.222882i
$321$	0	0
$322$	0	0
$323$	− 181289.i	− 1.73766i
$324$	0	0
$325$	59820.0	0.566343
$326$	− 18141.9i	− 0.170706i
$327$	0	0
$328$	69781.0	0.648618
$329$	0	0
$330$	0	0
$331$	83037.1	0.757908	0.378954	−	0.925416i	$-0.376284\pi$
$331$	83037.1	0.757908	0.378954	+	0.925416i	$0.376284\pi$
$332$	− 24809.6i	− 0.225083i
$333$	0	0
$334$	−76389.2	−0.684761
$335$	41078.9i	0.366041i
$336$	0	0
$337$	117911.	1.03823	0.519117	−	0.854703i	$-0.326261\pi$
$337$	117911.	1.03823	0.519117	+	0.854703i	$0.326261\pi$
$338$	75327.0i	0.659352i
$339$	0	0
$340$	−114301.	−0.988762
$341$	3620.86i	0.0311389i
$342$	0	0
$343$	0	0
$344$	59244.1i	0.500643i
$345$	0	0
$346$	122381.	1.02226
$347$	183327.i	1.52254i	0.648437	+	0.761268i	$0.275423\pi$
$347$	183327.i	1.52254i	−0.648437	+	0.761268i	$0.724577\pi$
$348$	0	0
$349$	−66776.9	−0.548246	−0.274123	−	0.961695i	$-0.588387\pi$
$349$	−66776.9	−0.548246	−0.274123	+	0.961695i	$0.588387\pi$
$350$	0	0
$351$	0	0
$352$	−9226.37	−0.0744639
$353$	− 6419.70i	− 0.0515187i	−0.999668	−	0.0257594i	$-0.991800\pi$
$353$	− 6419.70i	− 0.0515187i	0.999668	−	0.0257594i	$-0.00820037\pi$
$354$	0	0
$355$	350654.	2.78242
$356$	13413.8i	0.105841i
$357$	0	0
$358$	−100701.	−0.785721
$359$	79303.7i	0.615325i	0.951496	+	0.307663i	$0.0995468\pi$
$359$	79303.7i	0.615325i	−0.951496	+	0.307663i	$0.900453\pi$
$360$	0	0
$361$	189591.	1.45480
$362$	− 87991.8i	− 0.671467i
$363$	0	0
$364$	0	0
$365$	100994.i	0.758070i
$366$	0	0
$367$	−112008.	−0.831602	−0.415801	−	0.909456i	$-0.636499\pi$
$367$	−112008.	−0.831602	−0.415801	+	0.909456i	$0.636499\pi$
$368$	− 23355.2i	− 0.172460i
$369$	0	0
$370$	−272236.	−1.98858
$371$	0	0
$372$	0	0
$373$	92552.5	0.665228	0.332614	−	0.943063i	$-0.392069\pi$
$373$	92552.5	0.665228	0.332614	+	0.943063i	$0.392069\pi$
$374$	46206.8i	0.330341i
$375$	0	0
$376$	−69051.2	−0.488423
$377$	− 11570.2i	− 0.0814065i
$378$	0	0
$379$	216941.	1.51030	0.755151	−	0.655551i	$-0.227564\pi$
$379$	216941.	1.51030	0.755151	+	0.655551i	$0.227564\pi$
$380$	− 201702.i	− 1.39683i
$381$	0	0
$382$	−109786.	−0.752347
$383$	− 71061.2i	− 0.484434i	−0.970222	−	0.242217i	$-0.922125\pi$
$383$	− 71061.2i	− 0.484434i	0.970222	−	0.242217i	$-0.0778747\pi$
$384$	0	0
$385$	0	0
$386$	− 13959.1i	− 0.0936875i
$387$	0	0
$388$	−116629.	−0.774720
$389$	− 222556.i	− 1.47075i	−0.677660	−	0.735376i	$-0.737006\pi$
$389$	− 222556.i	− 1.47075i	0.677660	−	0.735376i	$-0.262994\pi$
$390$	0	0
$391$	−116966.	−0.765076
$392$	0	0
$393$	0	0
$394$	51296.0	0.330439
$395$	141336.i	0.905855i
$396$	0	0
$397$	265763.	1.68622	0.843110	−	0.537741i	$-0.180722\pi$
$397$	265763.	1.68622	0.843110	+	0.537741i	$0.180722\pi$
$398$	81021.9i	0.511489i
$399$	0	0
$400$	−87171.4	−0.544821
$401$	− 157662.i	− 0.980477i	−0.871588	−	0.490238i	$-0.836910\pi$
$401$	− 157662.i	− 0.980477i	0.871588	−	0.490238i	$-0.163090\pi$
$402$	0	0
$403$	3120.03	0.0192109
$404$	− 93651.1i	− 0.573787i
$405$	0	0
$406$	0	0
$407$	110053.i	0.664375i
$408$	0	0
$409$	−239317.	−1.43063	−0.715316	−	0.698801i	$-0.753717\pi$
$409$	−239317.	−1.43063	−0.715316	+	0.698801i	$0.753717\pi$
$410$	− 388823.i	− 2.31304i
$411$	0	0
$412$	−70104.5	−0.413002
$413$	0	0
$414$	0	0
$415$	−138240.	−0.802671
$416$	7950.18i	0.0459399i
$417$	0	0
$418$	−81539.2	−0.466674
$419$	30988.1i	0.176509i	0.996098	+	0.0882546i	$0.0281289\pi$
$419$	30988.1i	0.176509i	−0.996098	+	0.0882546i	$0.971871\pi$
$420$	0	0
$421$	−101278.	−0.571414	−0.285707	−	0.958317i	$-0.592228\pi$
$421$	−101278.	−0.571414	−0.285707	+	0.958317i	$0.592228\pi$
$422$	− 178155.i	− 1.00040i
$423$	0	0
$424$	94284.9	0.524458
$425$	436565.i	2.41697i
$426$	0	0
$427$	0	0
$428$	60457.0i	0.330034i
$429$	0	0
$430$	330111.	1.78535
$431$	− 32616.3i	− 0.175582i	−0.996139	−	0.0877910i	$-0.972019\pi$
$431$	− 32616.3i	− 0.175582i	0.996139	−	0.0877910i	$-0.0279808\pi$
$432$	0	0
$433$	−87431.7	−0.466330	−0.233165	−	0.972437i	$-0.574908\pi$
$433$	−87431.7	−0.466330	−0.233165	+	0.972437i	$0.574908\pi$
$434$	0	0
$435$	0	0
$436$	−164908.	−0.867497
$437$	− 206404.i	− 1.08083i
$438$	0	0
$439$	−82469.8	−0.427923	−0.213962	−	0.976842i	$-0.568637\pi$
$439$	−82469.8	−0.427923	−0.213962	+	0.976842i	$0.568637\pi$
$440$	51409.8i	0.265546i
$441$	0	0
$442$	39815.5	0.203802
$443$	9157.86i	0.0466645i	0.999728	+	0.0233322i	$0.00742756\pi$
$443$	9157.86i	0.0466645i	−0.999728	+	0.0233322i	$0.992572\pi$
$444$	0	0
$445$	74742.6	0.377440
$446$	119502.i	0.600766i
$447$	0	0
$448$	0	0
$449$	71190.4i	0.353125i	0.984289	+	0.176563i	$0.0564978\pi$
$449$	71190.4i	0.353125i	−0.984289	+	0.176563i	$0.943502\pi$
$450$	0	0
$451$	−157184.	−0.772778
$452$	− 5068.19i	− 0.0248071i
$453$	0	0
$454$	−6022.60	−0.0292195
$455$	0	0
$456$	0	0
$457$	−247187.	−1.18357	−0.591785	−	0.806096i	$-0.701577\pi$
$457$	−247187.	−1.18357	−0.591785	+	0.806096i	$0.701577\pi$
$458$	138735.i	0.661385i
$459$	0	0
$460$	−130136.	−0.615010
$461$	− 121488.i	− 0.571653i	−0.958281	−	0.285827i	$-0.907732\pi$
$461$	− 121488.i	− 0.571653i	0.958281	−	0.285827i	$-0.0922681\pi$
$462$	0	0
$463$	188335.	0.878558	0.439279	−	0.898351i	$-0.355234\pi$
$463$	188335.	0.878558	0.439279	+	0.898351i	$0.355234\pi$
$464$	16860.5i	0.0783130i
$465$	0	0
$466$	−58914.9	−0.271302
$467$	− 123077.i	− 0.564344i	−0.959364	−	0.282172i	$-0.908945\pi$
$467$	− 123077.i	− 0.564344i	0.959364	−	0.282172i	$-0.0910548\pi$
$468$	0	0
$469$	0	0
$470$	384757.i	1.74177i
$471$	0	0
$472$	−114405.	−0.513523
$473$	− 133449.i	− 0.596477i
$474$	0	0
$475$	−770388.	−3.41446
$476$	0	0
$477$	0	0
$478$	−291187.	−1.27443
$479$	170838.i	0.744583i	0.928116	+	0.372291i	$0.121428\pi$
$479$	170838.i	0.744583i	−0.928116	+	0.372291i	$0.878572\pi$
$480$	0	0
$481$	94830.5	0.409881
$482$	− 189990.i	− 0.817782i
$483$	0	0
$484$	−96345.3	−0.411282
$485$	649865.i	2.76274i
$486$	0	0
$487$	206151.	0.869217	0.434609	−	0.900619i	$-0.356887\pi$
$487$	206151.	0.869217	0.434609	+	0.900619i	$0.356887\pi$
$488$	− 6337.22i	− 0.0266109i
$489$	0	0
$490$	0	0
$491$	− 19775.3i	− 0.0820278i	−0.999159	−	0.0410139i	$-0.986941\pi$
$491$	− 19775.3i	− 0.0820278i	0.999159	−	0.0410139i	$-0.0130588\pi$
$492$	0	0
$493$	84439.3	0.347417
$494$	70260.7i	0.287911i
$495$	0	0
$496$	−4546.59	−0.0184809
$497$	0	0
$498$	0	0
$499$	−24954.7	−0.100219	−0.0501097	−	0.998744i	$-0.515957\pi$
$499$	−24954.7	−0.100219	−0.0501097	+	0.998744i	$0.515957\pi$
$500$	262841.i	1.05137i
$501$	0	0
$502$	220824.	0.876271
$503$	7388.99i	0.0292045i	0.999893	+	0.0146022i	$0.00464820\pi$
$503$	7388.99i	0.0292045i	−0.999893	+	0.0146022i	$0.995352\pi$
$504$	0	0
$505$	−521829.	−2.04619
$506$	52608.3i	0.205472i
$507$	0	0
$508$	−76248.5	−0.295463
$509$	66465.2i	0.256542i	0.991739	+	0.128271i	$0.0409428\pi$
$509$	66465.2i	0.256542i	−0.991739	+	0.128271i	$0.959057\pi$
$510$	0	0
$511$	0	0
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	4601.86	0.0174183
$515$	390626.i	1.47281i
$516$	0	0
$517$	155540.	0.581917
$518$	0	0
$519$	0	0
$520$	44298.8	0.163827
$521$	384689.i	1.41721i	0.705605	+	0.708605i	$0.250675\pi$
$521$	384689.i	1.41721i	−0.705605	+	0.708605i	$0.749325\pi$
$522$	0	0
$523$	296778.	1.08500	0.542498	−	0.840057i	$-0.317478\pi$
$523$	296778.	1.08500	0.542498	+	0.840057i	$0.317478\pi$
$524$	52039.0i	0.189525i
$525$	0	0
$526$	344469.	1.24503
$527$	22769.9i	0.0819860i
$528$	0	0
$529$	146671.	0.524123
$530$	− 525360.i	− 1.87027i
$531$	0	0
$532$	0	0
$533$	135442.i	0.476760i
$534$	0	0
$535$	336869.	1.17694
$536$	20852.1i	0.0725804i
$537$	0	0
$538$	−146544.	−0.506294
$539$	0	0
$540$	0	0
$541$	−312849.	−1.06891	−0.534453	−	0.845198i	$-0.679482\pi$
$541$	−312849.	−1.06891	−0.534453	+	0.845198i	$0.679482\pi$
$542$	− 185703.i	− 0.632151i
$543$	0	0
$544$	−58020.3	−0.196057
$545$	918873.i	3.09359i
$546$	0	0
$547$	−466647.	−1.55960	−0.779801	−	0.626027i	$-0.784680\pi$
$547$	−466647.	−1.55960	−0.779801	+	0.626027i	$0.784680\pi$
$548$	− 179564.i	− 0.597940i
$549$	0	0
$550$	196356.	0.649112
$551$	149006.i	0.490797i
$552$	0	0
$553$	0	0
$554$	− 319492.i	− 1.04098i
$555$	0	0
$556$	−236593.	−0.765336
$557$	481180.i	1.55095i	0.631380	+	0.775474i	$0.282489\pi$
$557$	481180.i	1.55095i	−0.631380	+	0.775474i	$0.717511\pi$
$558$	0	0
$559$	−114991.	−0.367992
$560$	0	0
$561$	0	0
$562$	−205027.	−0.649140
$563$	− 516819.i	− 1.63050i	−0.579106	−	0.815252i	$-0.696598\pi$
$563$	− 516819.i	− 1.63050i	0.579106	−	0.815252i	$-0.303402\pi$
$564$	0	0
$565$	−28240.2	−0.0884649
$566$	187157.i	0.584215i
$567$	0	0
$568$	177996.	0.551712
$569$	− 92366.1i	− 0.285291i	−0.989774	−	0.142645i	$-0.954439\pi$
$569$	− 92366.1i	− 0.285291i	0.989774	−	0.142645i	$-0.0455609\pi$
$570$	0	0
$571$	285508.	0.875681	0.437841	−	0.899053i	$-0.355743\pi$
$571$	285508.	0.875681	0.437841	+	0.899053i	$0.355743\pi$
$572$	− 17908.0i	− 0.0547339i
$573$	0	0
$574$	0	0
$575$	497047.i	1.50335i
$576$	0	0
$577$	612927.	1.84101	0.920507	−	0.390726i	$-0.127776\pi$
$577$	612927.	1.84101	0.920507	+	0.390726i	$0.127776\pi$
$578$	54339.8i	0.162653i
$579$	0	0
$580$	93947.3	0.279273
$581$	0	0
$582$	0	0
$583$	−212380.	−0.624851
$584$	51265.5i	0.150314i
$585$	0	0
$586$	−49243.9	−0.143403
$587$	117754.i	0.341743i	0.985293	+	0.170871i	$0.0546583\pi$
$587$	117754.i	0.341743i	−0.985293	+	0.170871i	$0.945342\pi$
$588$	0	0
$589$	−40181.1	−0.115822
$590$	637469.i	1.83128i
$591$	0	0
$592$	−138190.	−0.394305
$593$	− 200252.i	− 0.569466i	−0.958607	−	0.284733i	$-0.908095\pi$
$593$	− 200252.i	− 0.569466i	0.958607	−	0.284733i	$-0.0919050\pi$
$594$	0	0
$595$	0	0
$596$	− 45772.7i	− 0.128859i
$597$	0	0
$598$	45331.5	0.126764
$599$	35360.4i	0.0985516i	0.998785	+	0.0492758i	$0.0156913\pi$
$599$	35360.4i	0.0985516i	−0.998785	+	0.0492758i	$0.984309\pi$
$600$	0	0
$601$	21640.2	0.0599119	0.0299559	−	0.999551i	$-0.490463\pi$
$601$	21640.2	0.0599119	0.0299559	+	0.999551i	$0.490463\pi$
$602$	0	0
$603$	0	0
$604$	355844.	0.975405
$605$	536841.i	1.46668i
$606$	0	0
$607$	−436076.	−1.18355	−0.591773	−	0.806105i	$-0.701572\pi$
$607$	−436076.	−1.18355	−0.591773	+	0.806105i	$0.701572\pi$
$608$	− 102386.i	− 0.276970i
$609$	0	0
$610$	−35311.3	−0.0948973
$611$	− 134026.i	− 0.359010i
$612$	0	0
$613$	17161.1	0.0456693	0.0228346	−	0.999739i	$-0.492731\pi$
$613$	17161.1	0.0456693	0.0228346	+	0.999739i	$0.492731\pi$
$614$	− 395124.i	− 1.04808i
$615$	0	0
$616$	0	0
$617$	− 473055.i	− 1.24263i	−0.783562	−	0.621314i	$-0.786599\pi$
$617$	− 473055.i	− 1.24263i	0.783562	−	0.621314i	$-0.213401\pi$
$618$	0	0
$619$	−49709.6	−0.129736	−0.0648678	−	0.997894i	$-0.520663\pi$
$619$	−49709.6	−0.129736	−0.0648678	+	0.997894i	$0.520663\pi$
$620$	25333.8i	0.0659049i
$621$	0	0
$622$	−41810.8	−0.108071
$623$	0	0
$624$	0	0
$625$	613281.	1.57000
$626$	309151.i	0.788899i
$627$	0	0
$628$	35402.7	0.0897670
$629$	692072.i	1.74924i
$630$	0	0
$631$	−640140.	−1.60774	−0.803871	−	0.594804i	$-0.797230\pi$
$631$	−640140.	−1.60774	−0.803871	+	0.594804i	$0.797230\pi$
$632$	71743.6i	0.179618i
$633$	0	0
$634$	321943.	0.800942
$635$	424860.i	1.05365i
$636$	0	0
$637$	0	0
$638$	− 37978.8i	− 0.0933038i
$639$	0	0
$640$	−64553.5	−0.157601
$641$	− 203028.i	− 0.494129i	−0.968999	−	0.247065i	$-0.920534\pi$
$641$	− 203028.i	− 0.494129i	0.968999	−	0.247065i	$-0.0794660\pi$
$642$	0	0
$643$	−125822.	−0.304323	−0.152162	−	0.988356i	$-0.548623\pi$
$643$	−125822.	−0.304323	−0.152162	+	0.988356i	$0.548623\pi$
$644$	0	0
$645$	0	0
$646$	−512761.	−1.22871
$647$	− 196756.i	− 0.470022i	−0.971993	−	0.235011i	$-0.924487\pi$
$647$	− 196756.i	− 0.470022i	0.971993	−	0.235011i	$-0.0755127\pi$
$648$	0	0
$649$	257700.	0.611823
$650$	− 169196.i	− 0.400465i
$651$	0	0
$652$	−51313.1	−0.120707
$653$	− 502941.i	− 1.17948i	−0.807593	−	0.589741i	$-0.799230\pi$
$653$	− 502941.i	− 1.17948i	0.807593	−	0.589741i	$-0.200770\pi$
$654$	0	0
$655$	289964.	0.675867
$656$	− 197370.i	− 0.458642i
$657$	0	0
$658$	0	0
$659$	− 94760.4i	− 0.218201i	−0.994031	−	0.109100i	$-0.965203\pi$
$659$	− 94760.4i	− 0.218201i	0.994031	−	0.109100i	$-0.0347970\pi$
$660$	0	0
$661$	−546928.	−1.25178	−0.625889	−	0.779912i	$-0.715264\pi$
$661$	−546928.	−1.25178	−0.625889	+	0.779912i	$0.715264\pi$
$662$	− 234864.i	− 0.535922i
$663$	0	0
$664$	−70172.0	−0.159158
$665$	0	0
$666$	0	0
$667$	96137.4	0.216093
$668$	216061.i	0.484199i
$669$	0	0
$670$	116189.	0.258830
$671$	14274.8i	0.0317048i
$672$	0	0
$673$	249645.	0.551179	0.275590	−	0.961275i	$-0.411127\pi$
$673$	249645.	0.551179	0.275590	+	0.961275i	$0.411127\pi$
$674$	− 333503.i	− 0.734143i
$675$	0	0
$676$	213057.	0.466232
$677$	− 380869.i	− 0.830996i	−0.909594	−	0.415498i	$-0.863607\pi$
$677$	− 380869.i	− 0.830996i	0.909594	−	0.415498i	$-0.136393\pi$
$678$	0	0
$679$	0	0
$680$	323292.i	0.699161i
$681$	0	0
$682$	10241.3	0.0220185
$683$	− 792411.i	− 1.69867i	−0.527854	−	0.849335i	$-0.677003\pi$
$683$	− 792411.i	− 1.69867i	0.527854	−	0.849335i	$-0.322997\pi$
$684$	0	0
$685$	−1.00054e6	−2.13232
$686$	0	0
$687$	0	0
$688$	167568.	0.354008
$689$	183003.i	0.385497i
$690$	0	0
$691$	81763.5	0.171239	0.0856196	−	0.996328i	$-0.472713\pi$
$691$	81763.5	0.171239	0.0856196	+	0.996328i	$0.472713\pi$
$692$	− 346147.i	− 0.722850i
$693$	0	0
$694$	518527.	1.07660
$695$	1.31831e6i	2.72927i
$696$	0	0
$697$	−988455.	−2.03466
$698$	188874.i	0.387668i
$699$	0	0
$700$	0	0
$701$	420739.i	0.856202i	0.903731	+	0.428101i	$0.140817\pi$
$701$	420739.i	0.856202i	−0.903731	+	0.428101i	$0.859183\pi$
$702$	0	0
$703$	−1.22127e6	−2.47116
$704$	26096.1i	0.0526539i
$705$	0	0
$706$	−18157.7	−0.0364293
$707$	0	0
$708$	0	0
$709$	−173701.	−0.345549	−0.172774	−	0.984961i	$-0.555273\pi$
$709$	−173701.	−0.345549	−0.172774	+	0.984961i	$0.555273\pi$
$710$	− 991800.i	− 1.96747i
$711$	0	0
$712$	37940.1	0.0748408
$713$	25924.4i	0.0509953i
$714$	0	0
$715$	−99784.5	−0.195187
$716$	284826.i	0.555589i
$717$	0	0
$718$	224305.	0.435101
$719$	298584.i	0.577576i	0.957393	+	0.288788i	$0.0932522\pi$
$719$	298584.i	0.577576i	−0.957393	+	0.288788i	$0.906748\pi$
$720$	0	0
$721$	0	0
$722$	− 536245.i	− 1.02870i
$723$	0	0
$724$	−248878.	−0.474799
$725$	− 358826.i	− 0.682665i
$726$	0	0
$727$	632914.	1.19750	0.598750	−	0.800936i	$-0.295664\pi$
$727$	632914.	1.19750	0.598750	+	0.800936i	$0.295664\pi$
$728$	0	0
$729$	0	0
$730$	285654.	0.536037
$731$	− 839199.i	− 1.57047i
$732$	0	0
$733$	−449820.	−0.837203	−0.418601	−	0.908170i	$-0.637480\pi$
$733$	−449820.	−0.837203	−0.418601	+	0.908170i	$0.637480\pi$
$734$	316805.i	0.588031i
$735$	0	0
$736$	−66058.4	−0.121947
$737$	− 46969.9i	− 0.0864739i
$738$	0	0
$739$	156096.	0.285827	0.142914	−	0.989735i	$-0.454353\pi$
$739$	156096.	0.285827	0.142914	+	0.989735i	$0.454353\pi$
$740$	770000.i	1.40614i
$741$	0	0
$742$	0	0
$743$	− 426073.i	− 0.771803i	−0.922540	−	0.385901i	$-0.873890\pi$
$743$	− 426073.i	− 0.771803i	0.922540	−	0.385901i	$-0.126110\pi$
$744$	0	0
$745$	−255048.	−0.459525
$746$	− 261778.i	− 0.470387i
$747$	0	0
$748$	130693.	0.233587
$749$	0	0
$750$	0	0
$751$	−328471.	−0.582395	−0.291197	−	0.956663i	$-0.594054\pi$
$751$	−328471.	−0.582395	−0.291197	+	0.956663i	$0.594054\pi$
$752$	195306.i	0.345367i
$753$	0	0
$754$	−32725.5	−0.0575631
$755$	− 1.98278e6i	− 3.47840i
$756$	0	0
$757$	14089.4	0.0245867	0.0122933	−	0.999924i	$-0.496087\pi$
$757$	14089.4	0.0245867	0.0122933	+	0.999924i	$0.496087\pi$
$758$	− 613603.i	− 1.06794i
$759$	0	0
$760$	−570499.	−0.987707
$761$	− 368812.i	− 0.636849i	−0.947948	−	0.318424i	$-0.896846\pi$
$761$	− 368812.i	− 0.636849i	0.947948	−	0.318424i	$-0.103154\pi$
$762$	0	0
$763$	0	0
$764$	310520.i	0.531990i
$765$	0	0
$766$	−200991.	−0.342547
$767$	− 222055.i	− 0.377459i
$768$	0	0
$769$	762394.	1.28922	0.644610	−	0.764512i	$-0.277020\pi$
$769$	762394.	1.28922	0.644610	+	0.764512i	$0.277020\pi$
$770$	0	0
$771$	0	0
$772$	−39482.2	−0.0662471
$773$	− 646391.i	− 1.08177i	−0.841096	−	0.540887i	$-0.818089\pi$
$773$	− 646391.i	− 1.08177i	0.841096	−	0.540887i	$-0.181911\pi$
$774$	0	0
$775$	96761.0	0.161100
$776$	329878.i	0.547810i
$777$	0	0
$778$	−629482.	−1.03998
$779$	− 1.74428e6i	− 2.87437i
$780$	0	0
$781$	−400941.	−0.657322
$782$	330829.i	0.540990i
$783$	0	0
$784$	0	0
$785$	− 197265.i	− 0.320119i
$786$	0	0
$787$	1.06912e6	1.72614	0.863069	−	0.505086i	$-0.168539\pi$
$787$	1.06912e6	1.72614	0.863069	+	0.505086i	$0.168539\pi$
$788$	− 145087.i	− 0.233656i
$789$	0	0
$790$	399759.	0.640536
$791$	0	0
$792$	0	0
$793$	12300.3	0.0195600
$794$	− 751693.i	− 1.19234i
$795$	0	0
$796$	229165.	0.361678
$797$	124813.i	0.196491i	0.995162	+	0.0982454i	$0.0313230\pi$
$797$	124813.i	0.196491i	−0.995162	+	0.0982454i	$0.968677\pi$
$798$	0	0
$799$	978118.	1.53214
$800$	246558.i	0.385247i
$801$	0	0
$802$	−445934.	−0.693302
$803$	− 115477.i	− 0.179087i
$804$	0	0
$805$	0	0
$806$	− 8824.77i	− 0.0135842i
$807$	0	0
$808$	−264885.	−0.405728
$809$	− 1.05471e6i	− 1.61152i	−0.592240	−	0.805762i	$-0.701756\pi$
$809$	− 1.05471e6i	− 1.61152i	0.592240	−	0.805762i	$-0.298244\pi$
$810$	0	0
$811$	−238930.	−0.363270	−0.181635	−	0.983366i	$-0.558139\pi$
$811$	−238930.	−0.363270	−0.181635	+	0.983366i	$0.558139\pi$
$812$	0	0
$813$	0	0
$814$	311277.	0.469784
$815$	285919.i	0.430455i
$816$	0	0
$817$	1.48090e6	2.21861
$818$	676892.i	1.01161i
$819$	0	0
$820$	−1.09976e6	−1.63557
$821$	684946.i	1.01618i	0.861305	+	0.508089i	$0.169648\pi$
$821$	684946.i	1.01618i	−0.861305	+	0.508089i	$0.830352\pi$
$822$	0	0
$823$	280873.	0.414677	0.207338	−	0.978269i	$-0.433520\pi$
$823$	280873.	0.414677	0.207338	+	0.978269i	$0.433520\pi$
$824$	198286.i	0.292036i
$825$	0	0
$826$	0	0
$827$	66707.0i	0.0975350i	0.998810	+	0.0487675i	$0.0155293\pi$
$827$	66707.0i	0.0975350i	−0.998810	+	0.0487675i	$0.984471\pi$
$828$	0	0
$829$	−204815.	−0.298025	−0.149013	−	0.988835i	$-0.547609\pi$
$829$	−204815.	−0.298025	−0.149013	+	0.988835i	$0.547609\pi$
$830$	391002.i	0.567574i
$831$	0	0
$832$	22486.5	0.0324844
$833$	0	0
$834$	0	0
$835$	1.20390e6	1.72671
$836$	230628.i	0.329989i
$837$	0	0
$838$	87647.6	0.124811
$839$	− 686968.i	− 0.975916i	−0.872867	−	0.487958i	$-0.837742\pi$
$839$	− 686968.i	− 0.975916i	0.872867	−	0.487958i	$-0.162258\pi$
$840$	0	0
$841$	637878.	0.901873
$842$	286458.i	0.404051i
$843$	0	0
$844$	−503898.	−0.707388
$845$	− 1.18716e6i	− 1.66264i
$846$	0	0
$847$	0	0
$848$	− 266678.i	− 0.370848i
$849$	0	0
$850$	1.23479e6	1.70906
$851$	787950.i	1.08803i
$852$	0	0
$853$	873982.	1.20117	0.600585	−	0.799561i	$-0.294935\pi$
$853$	873982.	1.20117	0.600585	+	0.799561i	$0.294935\pi$
$854$	0	0
$855$	0	0
$856$	170998.	0.233369
$857$	− 380758.i	− 0.518426i	−0.965820	−	0.259213i	$-0.916537\pi$
$857$	− 380758.i	− 0.518426i	0.965820	−	0.259213i	$-0.0834632\pi$
$858$	0	0
$859$	−1.19101e6	−1.61410	−0.807050	−	0.590483i	$-0.798937\pi$
$859$	−1.19101e6	−1.61410	−0.807050	+	0.590483i	$0.798937\pi$
$860$	− 933694.i	− 1.26243i
$861$	0	0
$862$	−92252.8	−0.124155
$863$	− 265122.i	− 0.355979i	−0.984032	−	0.177989i	$-0.943041\pi$
$863$	− 265122.i	− 0.355979i	0.984032	−	0.177989i	$-0.0569592\pi$
$864$	0	0
$865$	−1.92875e6	−2.57776
$866$	247294.i	0.329745i
$867$	0	0
$868$	0	0
$869$	− 161605.i	− 0.214000i
$870$	0	0
$871$	−40473.1	−0.0533494
$872$	466429.i	0.613413i
$873$	0	0
$874$	−583799.	−0.764259
$875$	0	0
$876$	0	0
$877$	687972.	0.894483	0.447241	−	0.894413i	$-0.352407\pi$
$877$	687972.	0.894483	0.447241	+	0.894413i	$0.352407\pi$
$878$	233260.i	0.302587i
$879$	0	0
$880$	145409.	0.187770
$881$	− 230909.i	− 0.297502i	−0.988875	−	0.148751i	$-0.952475\pi$
$881$	− 230909.i	− 0.297502i	0.988875	−	0.148751i	$-0.0475253\pi$
$882$	0	0
$883$	95623.9	0.122644	0.0613218	−	0.998118i	$-0.480468\pi$
$883$	95623.9	0.122644	0.0613218	+	0.998118i	$0.480468\pi$
$884$	− 112615.i	− 0.144109i
$885$	0	0
$886$	25902.3	0.0329968
$887$	− 399587.i	− 0.507883i	−0.967220	−	0.253941i	$-0.918273\pi$
$887$	− 399587.i	− 0.507883i	0.967220	−	0.253941i	$-0.0817271\pi$
$888$	0	0
$889$	0	0
$890$	− 211404.i	− 0.266890i
$891$	0	0
$892$	338003.	0.424806
$893$	1.72604e6i	2.16446i
$894$	0	0
$895$	1.58706e6	1.98129
$896$	0	0
$897$	0	0
$898$	201357.	0.249697
$899$	− 18715.3i	− 0.0231567i
$900$	0	0
$901$	−1.33556e6	−1.64518
$902$	444583.i	0.546437i
$903$	0	0
$904$	−14335.0	−0.0175413
$905$	1.38676e6i	1.69319i
$906$	0	0
$907$	724819.	0.881079	0.440539	−	0.897733i	$-0.354787\pi$
$907$	724819.	0.881079	0.440539	+	0.897733i	$0.354787\pi$
$908$	17034.5i	0.0206613i
$909$	0	0
$910$	0	0
$911$	− 571014.i	− 0.688035i	−0.938963	−	0.344017i	$-0.888212\pi$
$911$	− 571014.i	− 0.688035i	0.938963	−	0.344017i	$-0.111788\pi$
$912$	0	0
$913$	158065.	0.189624
$914$	699152.i	0.836910i
$915$	0	0
$916$	392401.	0.467670
$917$	0	0
$918$	0	0
$919$	−1.45949e6	−1.72811	−0.864055	−	0.503397i	$-0.832083\pi$
$919$	−1.45949e6	−1.72811	−0.864055	+	0.503397i	$0.832083\pi$
$920$	368080.i	0.434878i
$921$	0	0
$922$	−343621.	−0.404220
$923$	345483.i	0.405530i
$924$	0	0
$925$	2.94097e6	3.43721
$926$	− 532693.i	− 0.621234i
$927$	0	0
$928$	47688.6	0.0553756
$929$	571275.i	0.661932i	0.943643	+	0.330966i	$0.107375\pi$
$929$	571275.i	0.661932i	−0.943643	+	0.330966i	$0.892625\pi$
$930$	0	0
$931$	0	0
$932$	166637.i	0.191840i
$933$	0	0
$934$	−348115.	−0.399051
$935$	− 728225.i	− 0.832995i
$936$	0	0
$937$	1.19621e6	1.36247	0.681235	−	0.732065i	$-0.261443\pi$
$937$	1.19621e6	1.36247	0.681235	+	0.732065i	$0.261443\pi$
$938$	0	0
$939$	0	0
$940$	1.08826e6	1.23162
$941$	− 17545.6i	− 0.0198148i	−0.999951	−	0.00990741i	$-0.996846\pi$
$941$	− 17545.6i	− 0.0198148i	0.999951	−	0.00990741i	$-0.00315368\pi$
$942$	0	0
$943$	−1.12539e6	−1.26556
$944$	323585.i	0.363116i
$945$	0	0
$946$	−377451.	−0.421773
$947$	− 1.46399e6i	− 1.63244i	−0.577739	−	0.816222i	$-0.696065\pi$
$947$	− 1.46399e6i	− 1.63244i	0.577739	−	0.816222i	$-0.303935\pi$
$948$	0	0
$949$	−99504.4	−0.110487
$950$	2.17899e6i	2.41439i
$951$	0	0
$952$	0	0
$953$	279568.i	0.307824i	0.988085	+	0.153912i	$0.0491872\pi$
$953$	279568.i	0.307824i	−0.988085	+	0.153912i	$0.950813\pi$
$954$	0	0
$955$	1.73023e6	1.89713
$956$	823602.i	0.901159i
$957$	0	0
$958$	483202.	0.526500
$959$	0	0
$960$	0	0
$961$	−918474.	−0.994535
$962$	− 268221.i	− 0.289830i
$963$	0	0
$964$	−537374.	−0.578259
$965$	219997.i	0.236244i
$966$	0	0
$967$	−1.07601e6	−1.15070	−0.575351	−	0.817907i	$-0.695134\pi$
$967$	−1.07601e6	−1.15070	−0.575351	+	0.817907i	$0.695134\pi$
$968$	272506.i	0.290820i
$969$	0	0
$970$	1.83810e6	1.95355
$971$	− 1.43007e6i	− 1.51677i	−0.651808	−	0.758384i	$-0.725989\pi$
$971$	− 1.43007e6i	− 1.51677i	0.651808	−	0.758384i	$-0.274011\pi$
$972$	0	0
$973$	0	0
$974$	− 583084.i	− 0.614630i
$975$	0	0
$976$	−17924.4	−0.0188167
$977$	1.43435e6i	1.50268i	0.659914	+	0.751341i	$0.270593\pi$
$977$	1.43435e6i	1.50268i	−0.659914	+	0.751341i	$0.729407\pi$
$978$	0	0
$979$	−85461.2	−0.0891669
$980$	0	0
$981$	0	0
$982$	−55933.1	−0.0580024
$983$	1.28283e6i	1.32758i	0.747918	+	0.663791i	$0.231054\pi$
$983$	1.28283e6i	1.32758i	−0.747918	+	0.663791i	$0.768946\pi$
$984$	0	0
$985$	−808432.	−0.833242
$986$	− 238830.i	− 0.245661i
$987$	0	0
$988$	198727.	0.203584
$989$	− 955461.i	− 0.976833i
$990$	0	0
$991$	1.75222e6	1.78419	0.892097	−	0.451843i	$-0.149233\pi$
$991$	1.75222e6	1.78419	0.892097	+	0.451843i	$0.149233\pi$
$992$	12859.7i	0.0130680i
$993$	0	0
$994$	0	0
$995$	− 1.27692e6i	− 1.28978i
$996$	0	0
$997$	−1.04474e6	−1.05103	−0.525517	−	0.850783i	$-0.676128\pi$
$997$	−1.04474e6	−1.05103	−0.525517	+	0.850783i	$0.676128\pi$
$998$	70582.7i	0.0708659i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.d.197.2		4
3.2	odd	2	inner	882.5.b.d.197.3		4
7.6	odd	2		126.5.b.a.71.1	✓	4
21.20	even	2		126.5.b.a.71.4	yes	4
28.27	even	2		1008.5.d.b.449.1		4
84.83	odd	2		1008.5.d.b.449.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.5.b.a.71.1	✓	4	7.6	odd	2
126.5.b.a.71.4	yes	4	21.20	even	2
882.5.b.d.197.2		4	1.1	even	1	trivial
882.5.b.d.197.3		4	3.2	odd	2	inner
1008.5.d.b.449.1		4	28.27	even	2
1008.5.d.b.449.4		4	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} +44.5764i q^{5} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} +44.5764i q^{5} +22.6274i q^{8} +126.081 q^{10} -50.9690i q^{11} -43.9190 q^{13} +64.0000 q^{16} -320.520i q^{17} +565.608 q^{19} -356.611i q^{20} -144.162 q^{22} -364.924i q^{23} -1362.05 q^{25} +124.222i q^{26} +263.445i q^{29} -71.0405 q^{31} -181.019i q^{32} -906.567 q^{34} -2159.22 q^{37} -1599.78i q^{38} -1008.65 q^{40} -3083.91i q^{41} +2618.24 q^{43} +407.752i q^{44} -1032.16 q^{46} +3051.66i q^{47} +3852.47i q^{50} +351.352 q^{52} -4166.84i q^{53} +2272.01 q^{55} +745.135 q^{58} +5056.02i q^{59} -280.068 q^{61} +200.933i q^{62} -512.000 q^{64} -1957.75i q^{65} +921.540 q^{67} +2564.16i q^{68} -7866.37i q^{71} +2265.64 q^{73} +6107.18i q^{74} -4524.86 q^{76} +3170.65 q^{79} +2852.89i q^{80} -8722.62 q^{82} +3101.19i q^{83} +14287.6 q^{85} -7405.51i q^{86} +1153.30 q^{88} -1676.73i q^{89} +2919.40i q^{92} +8631.40 q^{94} +25212.7i q^{95} +14578.7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4}+O(q^{10}) \) 4 * q - 32 * q^4	\( 4 q - 32 q^{4} + 208 q^{10} - 472 q^{13} + 256 q^{16} + 40 q^{19} + 16 q^{22} - 1596 q^{25} - 136 q^{31} - 1552 q^{34} - 4192 q^{37} - 1664 q^{40} + 9584 q^{43} - 3536 q^{46} + 3776 q^{52} + 5384 q^{55} - 1168 q^{58} - 4528 q^{61} - 2048 q^{64} - 1944 q^{67} + 21360 q^{73} - 320 q^{76} + 10312 q^{79} - 18000 q^{82} + 29296 q^{85} - 128 q^{88} + 22080 q^{94} + 44832 q^{97}+O(q^{100}) \) 4 * q - 32 * q^4 + 208 * q^10 - 472 * q^13 + 256 * q^16 + 40 * q^19 + 16 * q^22 - 1596 * q^25 - 136 * q^31 - 1552 * q^34 - 4192 * q^37 - 1664 * q^40 + 9584 * q^43 - 3536 * q^46 + 3776 * q^52 + 5384 * q^55 - 1168 * q^58 - 4528 * q^61 - 2048 * q^64 - 1944 * q^67 + 21360 * q^73 - 320 * q^76 + 10312 * q^79 - 18000 * q^82 + 29296 * q^85 - 128 * q^88 + 22080 * q^94 + 44832 * q^97

Introduction

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