LMFDB - Embedded newform 882.5.b.a.197.4

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} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.4
Root			$1.16372i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.a.197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +10.8127i q^{5} -22.6274i q^{8} +O(q^{10})$	$q+2.82843i q^{2} -8.00000 q^{4} +10.8127i q^{5} -22.6274i q^{8} -30.5830 q^{10} +230.099i q^{11} +30.9150 q^{13} +64.0000 q^{16} -251.750i q^{17} +583.365 q^{19} -86.5018i q^{20} -650.818 q^{22} +377.166i q^{23} +508.085 q^{25} +87.4409i q^{26} +343.706i q^{29} +1053.02 q^{31} +181.019i q^{32} +712.057 q^{34} -898.729 q^{37} +1650.00i q^{38} +244.664 q^{40} -624.090i q^{41} +645.725 q^{43} -1840.79i q^{44} -1066.79 q^{46} -3867.62i q^{47} +1437.08i q^{50} -247.320 q^{52} +41.7221i q^{53} -2488.00 q^{55} -972.146 q^{58} +3885.52i q^{59} -251.853 q^{61} +2978.38i q^{62} -512.000 q^{64} +334.276i q^{65} +8652.28 q^{67} +2014.00i q^{68} +4603.56i q^{71} -3553.39 q^{73} -2541.99i q^{74} -4666.92 q^{76} +3585.10 q^{79} +692.014i q^{80} +1765.19 q^{82} +1400.50i q^{83} +2722.11 q^{85} +1826.39i q^{86} +5206.55 q^{88} +14202.2i q^{89} -3017.33i q^{92} +10939.3 q^{94} +6307.76i q^{95} -10704.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4}+O(q^{10}) $ 4 * q - 32 * q^4	$ 4 q - 32 q^{4} - 80 q^{10} - 88 q^{13} + 256 q^{16} + 1000 q^{19} - 656 q^{22} + 2244 q^{25} + 1016 q^{31} - 496 q^{34} - 928 q^{37} + 640 q^{40} - 592 q^{43} + 1744 q^{46} + 704 q^{52} - 4216 q^{55} - 1264 q^{58} + 12560 q^{61} - 2048 q^{64} + 16872 q^{67} + 6000 q^{73} - 8000 q^{76} - 15800 q^{79} - 432 q^{82} + 3184 q^{85} + 5248 q^{88} + 17088 q^{94} - 52704 q^{97}+O(q^{100}) $ 4 * q - 32 * q^4 - 80 * q^10 - 88 * q^13 + 256 * q^16 + 1000 * q^19 - 656 * q^22 + 2244 * q^25 + 1016 * q^31 - 496 * q^34 - 928 * q^37 + 640 * q^40 - 592 * q^43 + 1744 * q^46 + 704 * q^52 - 4216 * q^55 - 1264 * q^58 + 12560 * q^61 - 2048 * q^64 + 16872 * q^67 + 6000 * q^73 - 8000 * q^76 - 15800 * q^79 - 432 * q^82 + 3184 * q^85 + 5248 * q^88 + 17088 * q^94 - 52704 * 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
$5$	10.8127i	0.432509i	0.976337	+	0.216255i	$0.0693841\pi$
$5$	10.8127i	0.432509i	−0.976337	+	0.216255i	$0.930616\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	−30.5830	−0.305830
$11$	230.099i	1.90164i	0.309737	+	0.950822i	$0.399759\pi$
$11$	230.099i	1.90164i	−0.309737	+	0.950822i	$0.600241\pi$
$12$	0	0
$13$	30.9150	0.182929	0.0914646	−	0.995808i	$-0.470845\pi$
$13$	30.9150	0.182929	0.0914646	+	0.995808i	$0.470845\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	− 251.750i	− 0.871108i	−0.900163	−	0.435554i	$-0.856552\pi$
$17$	− 251.750i	− 0.871108i	0.900163	−	0.435554i	$-0.143448\pi$
$18$	0	0
$19$	583.365	1.61597	0.807984	−	0.589204i	$-0.200559\pi$
$19$	583.365	1.61597	0.807984	+	0.589204i	$0.200559\pi$
$20$	− 86.5018i	− 0.216255i
$21$	0	0
$22$	−650.818	−1.34467
$23$	377.166i	0.712979i	0.934299	+	0.356490i	$0.116027\pi$
$23$	377.166i	0.712979i	−0.934299	+	0.356490i	$0.883973\pi$
$24$	0	0
$25$	508.085	0.812936
$26$	87.4409i	0.129350i
$27$	0	0
$28$	0	0
$29$	343.706i	0.408687i	0.978899	+	0.204343i	$0.0655059\pi$
$29$	343.706i	0.408687i	−0.978899	+	0.204343i	$0.934494\pi$
$30$	0	0
$31$	1053.02	1.09575	0.547876	−	0.836560i	$-0.315437\pi$
$31$	1053.02	1.09575	0.547876	+	0.836560i	$0.315437\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	712.057	0.615967
$35$	0	0
$36$	0	0
$37$	−898.729	−0.656486	−0.328243	−	0.944593i	$-0.606456\pi$
$37$	−898.729	−0.656486	−0.328243	+	0.944593i	$0.606456\pi$
$38$	1650.00i	1.14266i
$39$	0	0
$40$	244.664	0.152915
$41$	− 624.090i	− 0.371261i	−0.982620	−	0.185630i	$-0.940567\pi$
$41$	− 624.090i	− 0.371261i	0.982620	−	0.185630i	$-0.0594327\pi$
$42$	0	0
$43$	645.725	0.349230	0.174615	−	0.984637i	$-0.444132\pi$
$43$	645.725	0.349230	0.174615	+	0.984637i	$0.444132\pi$
$44$	− 1840.79i	− 0.950822i
$45$	0	0
$46$	−1066.79	−0.504153
$47$	− 3867.62i	− 1.75085i	−0.483355	−	0.875424i	$-0.660582\pi$
$47$	− 3867.62i	− 1.75085i	0.483355	−	0.875424i	$-0.339418\pi$
$48$	0	0
$49$	0	0
$50$	1437.08i	0.574833i
$51$	0	0
$52$	−247.320	−0.0914646
$53$	41.7221i	0.0148530i	0.999972	+	0.00742650i	$0.00236395\pi$
$53$	41.7221i	0.0148530i	−0.999972	+	0.00742650i	$0.997636\pi$
$54$	0	0
$55$	−2488.00	−0.822478
$56$	0	0
$57$	0	0
$58$	−972.146	−0.288985
$59$	3885.52i	1.11621i	0.829771	+	0.558104i	$0.188471\pi$
$59$	3885.52i	1.11621i	−0.829771	+	0.558104i	$0.811529\pi$
$60$	0	0
$61$	−251.853	−0.0676843	−0.0338421	−	0.999427i	$-0.510774\pi$
$61$	−251.853	−0.0676843	−0.0338421	+	0.999427i	$0.510774\pi$
$62$	2978.38i	0.774813i
$63$	0	0
$64$	−512.000	−0.125000
$65$	334.276i	0.0791185i
$66$	0	0
$67$	8652.28	1.92744	0.963720	−	0.266915i	$-0.0860042\pi$
$67$	8652.28	1.92744	0.963720	+	0.266915i	$0.0860042\pi$
$68$	2014.00i	0.435554i
$69$	0	0
$70$	0	0
$71$	4603.56i	0.913223i	0.889666	+	0.456612i	$0.150937\pi$
$71$	4603.56i	0.913223i	−0.889666	+	0.456612i	$0.849063\pi$
$72$	0	0
$73$	−3553.39	−0.666801	−0.333401	−	0.942785i	$-0.608196\pi$
$73$	−3553.39	−0.666801	−0.333401	+	0.942785i	$0.608196\pi$
$74$	− 2541.99i	− 0.464206i
$75$	0	0
$76$	−4666.92	−0.807984
$77$	0	0
$78$	0	0
$79$	3585.10	0.574443	0.287222	−	0.957864i	$-0.407268\pi$
$79$	3585.10	0.574443	0.287222	+	0.957864i	$0.407268\pi$
$80$	692.014i	0.108127i
$81$	0	0
$82$	1765.19	0.262521
$83$	1400.50i	0.203295i	0.994820	+	0.101648i	$0.0324114\pi$
$83$	1400.50i	0.203295i	−0.994820	+	0.101648i	$0.967589\pi$
$84$	0	0
$85$	2722.11	0.376762
$86$	1826.39i	0.246943i
$87$	0	0
$88$	5206.55	0.672333
$89$	14202.2i	1.79298i	0.443062	+	0.896491i	$0.353892\pi$
$89$	14202.2i	1.79298i	−0.443062	+	0.896491i	$0.646108\pi$
$90$	0	0
$91$	0	0
$92$	− 3017.33i	− 0.356490i
$93$	0	0
$94$	10939.3	1.23804
$95$	6307.76i	0.698921i
$96$	0	0
$97$	−10704.9	−1.13773	−0.568863	−	0.822432i	$-0.692617\pi$
$97$	−10704.9	−1.13773	−0.568863	+	0.822432i	$0.692617\pi$
$98$	0	0
$99$	0	0
$100$	−4064.68	−0.406468
$101$	11335.5i	1.11122i	0.831444	+	0.555608i	$0.187515\pi$
$101$	11335.5i	1.11122i	−0.831444	+	0.555608i	$0.812485\pi$
$102$	0	0
$103$	14992.7	1.41320	0.706601	−	0.707613i	$-0.250228\pi$
$103$	14992.7	1.41320	0.706601	+	0.707613i	$0.250228\pi$
$104$	− 699.527i	− 0.0646752i
$105$	0	0
$106$	−118.008	−0.0105027
$107$	− 2625.18i	− 0.229294i	−0.993406	−	0.114647i	$-0.963426\pi$
$107$	− 2625.18i	− 0.229294i	0.993406	−	0.114647i	$-0.0365737\pi$
$108$	0	0
$109$	−8817.40	−0.742143	−0.371072	−	0.928604i	$-0.621010\pi$
$109$	−8817.40	−0.742143	−0.371072	+	0.928604i	$0.621010\pi$
$110$	− 7037.12i	− 0.581580i
$111$	0	0
$112$	0	0
$113$	8421.88i	0.659557i	0.944058	+	0.329778i	$0.106974\pi$
$113$	8421.88i	0.659557i	−0.944058	+	0.329778i	$0.893026\pi$
$114$	0	0
$115$	−4078.19	−0.308370
$116$	− 2749.65i	− 0.204343i
$117$	0	0
$118$	−10989.9	−0.789278
$119$	0	0
$120$	0	0
$121$	−38304.5	−2.61625
$122$	− 712.348i	− 0.0478600i
$123$	0	0
$124$	−8424.14	−0.547876
$125$	12251.7i	0.784111i
$126$	0	0
$127$	−16458.6	−1.02044	−0.510219	−	0.860045i	$-0.670436\pi$
$127$	−16458.6	−1.02044	−0.510219	+	0.860045i	$0.670436\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	−945.474	−0.0559452
$131$	− 19021.0i	− 1.10839i	−0.832388	−	0.554194i	$-0.813027\pi$
$131$	− 19021.0i	− 1.10839i	0.832388	−	0.554194i	$-0.186973\pi$
$132$	0	0
$133$	0	0
$134$	24472.3i	1.36291i
$135$	0	0
$136$	−5696.46	−0.307983
$137$	− 10371.8i	− 0.552605i	−0.961071	−	0.276302i	$-0.910891\pi$
$137$	− 10371.8i	− 0.552605i	0.961071	−	0.276302i	$-0.0891091\pi$
$138$	0	0
$139$	−26702.3	−1.38203	−0.691017	−	0.722839i	$-0.742837\pi$
$139$	−26702.3	−1.38203	−0.691017	+	0.722839i	$0.742837\pi$
$140$	0	0
$141$	0	0
$142$	−13020.8	−0.645746
$143$	7113.52i	0.347866i
$144$	0	0
$145$	−3716.39	−0.176761
$146$	− 10050.5i	− 0.471500i
$147$	0	0
$148$	7189.83	0.328243
$149$	23722.9i	1.06855i	0.845310	+	0.534276i	$0.179416\pi$
$149$	23722.9i	1.06855i	−0.845310	+	0.534276i	$0.820584\pi$
$150$	0	0
$151$	24177.3	1.06036	0.530180	−	0.847885i	$-0.322124\pi$
$151$	24177.3	1.06036	0.530180	+	0.847885i	$0.322124\pi$
$152$	− 13200.0i	− 0.571331i
$153$	0	0
$154$	0	0
$155$	11386.0i	0.473922i
$156$	0	0
$157$	45646.3	1.85185	0.925926	−	0.377704i	$-0.123286\pi$
$157$	45646.3	1.85185	0.925926	+	0.377704i	$0.123286\pi$
$158$	10140.2i	0.406193i
$159$	0	0
$160$	−1957.31	−0.0764575
$161$	0	0
$162$	0	0
$163$	−9360.10	−0.352294	−0.176147	−	0.984364i	$-0.556363\pi$
$163$	−9360.10	−0.352294	−0.176147	+	0.984364i	$0.556363\pi$
$164$	4992.72i	0.185630i
$165$	0	0
$166$	−3961.21	−0.143751
$167$	25422.6i	0.911565i	0.890091	+	0.455782i	$0.150640\pi$
$167$	25422.6i	0.911565i	−0.890091	+	0.455782i	$0.849360\pi$
$168$	0	0
$169$	−27605.3	−0.966537
$170$	7699.28i	0.266411i
$171$	0	0
$172$	−5165.80	−0.174615
$173$	36179.5i	1.20885i	0.796664	+	0.604423i	$0.206596\pi$
$173$	36179.5i	1.20885i	−0.796664	+	0.604423i	$0.793404\pi$
$174$	0	0
$175$	0	0
$176$	14726.3i	0.475411i
$177$	0	0
$178$	−40169.9	−1.26783
$179$	− 7814.61i	− 0.243894i	−0.992537	−	0.121947i	$-0.961086\pi$
$179$	− 7814.61i	− 0.243894i	0.992537	−	0.121947i	$-0.0389138\pi$
$180$	0	0
$181$	−30286.5	−0.924468	−0.462234	−	0.886758i	$-0.652952\pi$
$181$	−30286.5	−0.924468	−0.462234	+	0.886758i	$0.652952\pi$
$182$	0	0
$183$	0	0
$184$	8534.29	0.252076
$185$	− 9717.71i	− 0.283936i
$186$	0	0
$187$	57927.5	1.65654
$188$	30941.0i	0.875424i
$189$	0	0
$190$	−17841.0	−0.494212
$191$	48053.6i	1.31722i	0.752484	+	0.658611i	$0.228856\pi$
$191$	48053.6i	1.31722i	−0.752484	+	0.658611i	$0.771144\pi$
$192$	0	0
$193$	53655.0	1.44044	0.720220	−	0.693746i	$-0.244041\pi$
$193$	53655.0	1.44044	0.720220	+	0.693746i	$0.244041\pi$
$194$	− 30277.9i	− 0.804494i
$195$	0	0
$196$	0	0
$197$	− 52274.9i	− 1.34698i	−0.739197	−	0.673489i	$-0.764795\pi$
$197$	− 52274.9i	− 1.34698i	0.739197	−	0.673489i	$-0.235205\pi$
$198$	0	0
$199$	−72538.5	−1.83173	−0.915867	−	0.401481i	$-0.868496\pi$
$199$	−72538.5	−1.83173	−0.915867	+	0.401481i	$0.868496\pi$
$200$	− 11496.7i	− 0.287416i
$201$	0	0
$202$	−32061.7	−0.785749
$203$	0	0
$204$	0	0
$205$	6748.11	0.160574
$206$	42405.6i	0.999284i
$207$	0	0
$208$	1978.56	0.0457323
$209$	134232.i	3.07300i
$210$	0	0
$211$	−33054.7	−0.742453	−0.371226	−	0.928542i	$-0.621063\pi$
$211$	−33054.7	−0.742453	−0.371226	+	0.928542i	$0.621063\pi$
$212$	− 333.777i	− 0.00742650i
$213$	0	0
$214$	7425.14	0.162135
$215$	6982.05i	0.151045i
$216$	0	0
$217$	0	0
$218$	− 24939.4i	− 0.524775i
$219$	0	0
$220$	19904.0	0.411239
$221$	− 7782.87i	− 0.159351i
$222$	0	0
$223$	−71644.9	−1.44071	−0.720353	−	0.693607i	$-0.756020\pi$
$223$	−71644.9	−1.44071	−0.720353	+	0.693607i	$0.756020\pi$
$224$	0	0
$225$	0	0
$226$	−23820.7	−0.466377
$227$	− 46369.4i	− 0.899869i	−0.893061	−	0.449935i	$-0.851447\pi$
$227$	− 46369.4i	− 0.899869i	0.893061	−	0.449935i	$-0.148553\pi$
$228$	0	0
$229$	19028.6	0.362858	0.181429	−	0.983404i	$-0.441928\pi$
$229$	19028.6	0.362858	0.181429	+	0.983404i	$0.441928\pi$
$230$	− 11534.9i	− 0.218051i
$231$	0	0
$232$	7777.17	0.144493
$233$	− 12296.0i	− 0.226492i	−0.993567	−	0.113246i	$-0.963875\pi$
$233$	− 12296.0i	− 0.226492i	0.993567	−	0.113246i	$-0.0361248\pi$
$234$	0	0
$235$	41819.6	0.757258
$236$	− 31084.2i	− 0.558104i
$237$	0	0
$238$	0	0
$239$	76010.4i	1.33069i	0.746535	+	0.665346i	$0.231716\pi$
$239$	76010.4i	1.33069i	−0.746535	+	0.665346i	$0.768284\pi$
$240$	0	0
$241$	−76112.7	−1.31046	−0.655229	−	0.755430i	$-0.727428\pi$
$241$	−76112.7	−1.31046	−0.655229	+	0.755430i	$0.727428\pi$
$242$	− 108342.i	− 1.84997i
$243$	0	0
$244$	2014.83	0.0338421
$245$	0	0
$246$	0	0
$247$	18034.7	0.295608
$248$	− 23827.1i	− 0.387407i
$249$	0	0
$250$	−34653.1	−0.554450
$251$	− 37358.8i	− 0.592988i	−0.955035	−	0.296494i	$-0.904183\pi$
$251$	− 37358.8i	− 0.592988i	0.955035	−	0.296494i	$-0.0958174\pi$
$252$	0	0
$253$	−86785.5	−1.35583
$254$	− 46552.0i	− 0.721558i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 60153.1i	− 0.910735i	−0.890304	−	0.455367i	$-0.849508\pi$
$257$	− 60153.1i	− 0.910735i	0.890304	−	0.455367i	$-0.150492\pi$
$258$	0	0
$259$	0	0
$260$	− 2674.21i	− 0.0395593i
$261$	0	0
$262$	53799.6	0.783748
$263$	− 9342.68i	− 0.135070i	−0.997717	−	0.0675352i	$-0.978487\pi$
$263$	− 9342.68i	− 0.135070i	0.997717	−	0.0675352i	$-0.0215135\pi$
$264$	0	0
$265$	−451.129	−0.00642406
$266$	0	0
$267$	0	0
$268$	−69218.2	−0.963720
$269$	341.689i	0.00472201i	0.999997	+	0.00236100i	$0.000751532\pi$
$269$	341.689i	0.00472201i	−0.999997	+	0.00236100i	$0.999248\pi$
$270$	0	0
$271$	−20686.5	−0.281675	−0.140837	−	0.990033i	$-0.544979\pi$
$271$	−20686.5	−0.281675	−0.140837	+	0.990033i	$0.544979\pi$
$272$	− 16112.0i	− 0.217777i
$273$	0	0
$274$	29336.0	0.390751
$275$	116910.i	1.54592i
$276$	0	0
$277$	−28965.1	−0.377499	−0.188750	−	0.982025i	$-0.560443\pi$
$277$	−28965.1	−0.377499	−0.188750	+	0.982025i	$0.560443\pi$
$278$	− 75525.4i	− 0.977245i
$279$	0	0
$280$	0	0
$281$	18123.3i	0.229522i	0.993393	+	0.114761i	$0.0366102\pi$
$281$	18123.3i	0.229522i	−0.993393	+	0.114761i	$0.963390\pi$
$282$	0	0
$283$	−99422.3	−1.24140	−0.620699	−	0.784049i	$-0.713151\pi$
$283$	−99422.3	−1.24140	−0.620699	+	0.784049i	$0.713151\pi$
$284$	− 36828.5i	− 0.456612i
$285$	0	0
$286$	−20120.1	−0.245979
$287$	0	0
$288$	0	0
$289$	20142.8	0.241170
$290$	− 10511.6i	− 0.124989i
$291$	0	0
$292$	28427.1	0.333401
$293$	− 51861.3i	− 0.604099i	−0.953292	−	0.302050i	$-0.902329\pi$
$293$	− 51861.3i	− 0.604099i	0.953292	−	0.302050i	$-0.0976708\pi$
$294$	0	0
$295$	−42013.1	−0.482770
$296$	20335.9i	0.232103i
$297$	0	0
$298$	−67098.6	−0.755581
$299$	11660.1i	0.130425i
$300$	0	0
$301$	0	0
$302$	68383.7i	0.749788i
$303$	0	0
$304$	37335.3	0.403992
$305$	− 2723.22i	− 0.0292741i
$306$	0	0
$307$	−72049.6	−0.764460	−0.382230	−	0.924067i	$-0.624844\pi$
$307$	−72049.6	−0.764460	−0.382230	+	0.924067i	$0.624844\pi$
$308$	0	0
$309$	0	0
$310$	−32204.4	−0.335114
$311$	− 111740.i	− 1.15528i	−0.816292	−	0.577640i	$-0.803974\pi$
$311$	− 111740.i	− 1.15528i	0.816292	−	0.577640i	$-0.196026\pi$
$312$	0	0
$313$	183743.	1.87552	0.937761	−	0.347282i	$-0.112895\pi$
$313$	183743.	1.87552	0.937761	+	0.347282i	$0.112895\pi$
$314$	129107.i	1.30946i
$315$	0	0
$316$	−28680.8	−0.287222
$317$	87137.0i	0.867130i	0.901122	+	0.433565i	$0.142744\pi$
$317$	87137.0i	0.867130i	−0.901122	+	0.433565i	$0.857256\pi$
$318$	0	0
$319$	−79086.3	−0.777177
$320$	− 5536.12i	− 0.0540636i
$321$	0	0
$322$	0	0
$323$	− 146862.i	− 1.40768i
$324$	0	0
$325$	15707.5	0.148710
$326$	− 26474.4i	− 0.249110i
$327$	0	0
$328$	−14121.5	−0.131261
$329$	0	0
$330$	0	0
$331$	−78898.4	−0.720132	−0.360066	−	0.932927i	$-0.617246\pi$
$331$	−78898.4	−0.720132	−0.360066	+	0.932927i	$0.617246\pi$
$332$	− 11204.0i	− 0.101648i
$333$	0	0
$334$	−71906.1	−0.644574
$335$	93554.7i	0.833635i
$336$	0	0
$337$	−107785.	−0.949074	−0.474537	−	0.880236i	$-0.657384\pi$
$337$	−107785.	−0.949074	−0.474537	+	0.880236i	$0.657384\pi$
$338$	− 78079.5i	− 0.683445i
$339$	0	0
$340$	−21776.9	−0.188381
$341$	242298.i	2.08373i
$342$	0	0
$343$	0	0
$344$	− 14611.1i	− 0.123471i
$345$	0	0
$346$	−102331.	−0.854783
$347$	203518.i	1.69022i	0.534589	+	0.845112i	$0.320467\pi$
$347$	203518.i	1.69022i	−0.534589	+	0.845112i	$0.679533\pi$
$348$	0	0
$349$	213886.	1.75603	0.878013	−	0.478636i	$-0.158869\pi$
$349$	213886.	1.75603	0.878013	+	0.478636i	$0.158869\pi$
$350$	0	0
$351$	0	0
$352$	−41652.4	−0.336166
$353$	− 92369.6i	− 0.741276i	−0.928777	−	0.370638i	$-0.879139\pi$
$353$	− 92369.6i	− 0.741276i	0.928777	−	0.370638i	$-0.120861\pi$
$354$	0	0
$355$	−49777.0	−0.394977
$356$	− 113618.i	− 0.896491i
$357$	0	0
$358$	22103.0	0.172459
$359$	37359.7i	0.289877i	0.989441	+	0.144939i	$0.0462985\pi$
$359$	37359.7i	0.289877i	−0.989441	+	0.144939i	$0.953702\pi$
$360$	0	0
$361$	209993.	1.61135
$362$	− 85663.2i	− 0.653698i
$363$	0	0
$364$	0	0
$365$	− 38421.8i	− 0.288398i
$366$	0	0
$367$	152338.	1.13104	0.565519	−	0.824735i	$-0.308676\pi$
$367$	152338.	1.13104	0.565519	+	0.824735i	$0.308676\pi$
$368$	24138.6i	0.178245i
$369$	0	0
$370$	27485.8	0.200773
$371$	0	0
$372$	0	0
$373$	204868.	1.47250	0.736251	−	0.676708i	$-0.236594\pi$
$373$	204868.	1.47250	0.736251	+	0.676708i	$0.236594\pi$
$374$	163844.i	1.17135i
$375$	0	0
$376$	−87514.3	−0.619018
$377$	10625.7i	0.0747607i
$378$	0	0
$379$	260680.	1.81480	0.907402	−	0.420264i	$-0.138062\pi$
$379$	260680.	1.81480	0.907402	+	0.420264i	$0.138062\pi$
$380$	− 50462.1i	− 0.349460i
$381$	0	0
$382$	−135916.	−0.931416
$383$	192079.i	1.30943i	0.755876	+	0.654715i	$0.227211\pi$
$383$	192079.i	1.30943i	−0.755876	+	0.654715i	$0.772789\pi$
$384$	0	0
$385$	0	0
$386$	151759.i	1.01855i
$387$	0	0
$388$	85638.9	0.568863
$389$	− 183420.i	− 1.21213i	−0.795416	−	0.606063i	$-0.792748\pi$
$389$	− 183420.i	− 1.21213i	0.795416	−	0.606063i	$-0.207252\pi$
$390$	0	0
$391$	94951.7	0.621082
$392$	0	0
$393$	0	0
$394$	147856.	0.952458
$395$	38764.7i	0.248452i
$396$	0	0
$397$	48782.9	0.309518	0.154759	−	0.987952i	$-0.450540\pi$
$397$	48782.9	0.309518	0.154759	+	0.987952i	$0.450540\pi$
$398$	− 205170.i	− 1.29523i
$399$	0	0
$400$	32517.4	0.203234
$401$	− 65557.1i	− 0.407691i	−0.979003	−	0.203846i	$-0.934656\pi$
$401$	− 65557.1i	− 0.407691i	0.979003	−	0.203846i	$-0.0653440\pi$
$402$	0	0
$403$	32554.0	0.200445
$404$	− 90684.2i	− 0.555608i
$405$	0	0
$406$	0	0
$407$	− 206797.i	− 1.24840i
$408$	0	0
$409$	−26310.3	−0.157282	−0.0786409	−	0.996903i	$-0.525058\pi$
$409$	−26310.3	−0.157282	−0.0786409	+	0.996903i	$0.525058\pi$
$410$	19086.5i	0.113543i
$411$	0	0
$412$	−119941.	−0.706601
$413$	0	0
$414$	0	0
$415$	−15143.2	−0.0879270
$416$	5596.22i	0.0323376i
$417$	0	0
$418$	−379664.	−2.17294
$419$	79001.2i	0.449993i	0.974360	+	0.224996i	$0.0722370\pi$
$419$	79001.2i	0.449993i	−0.974360	+	0.224996i	$0.927763\pi$
$420$	0	0
$421$	248182.	1.40025	0.700126	−	0.714019i	$-0.253127\pi$
$421$	248182.	1.40025	0.700126	+	0.714019i	$0.253127\pi$
$422$	− 93492.9i	− 0.524994i
$423$	0	0
$424$	944.063	0.00525133
$425$	− 127911.i	− 0.708155i
$426$	0	0
$427$	0	0
$428$	21001.5i	0.114647i
$429$	0	0
$430$	−19748.2	−0.106805
$431$	330005.i	1.77650i	0.459359	+	0.888251i	$0.348079\pi$
$431$	330005.i	1.77650i	−0.459359	+	0.888251i	$0.651921\pi$
$432$	0	0
$433$	−177919.	−0.948958	−0.474479	−	0.880267i	$-0.657364\pi$
$433$	−177919.	−0.948958	−0.474479	+	0.880267i	$0.657364\pi$
$434$	0	0
$435$	0	0
$436$	70539.2	0.371072
$437$	220025.i	1.15215i
$438$	0	0
$439$	−51931.2	−0.269463	−0.134732	−	0.990882i	$-0.543017\pi$
$439$	−51931.2	−0.269463	−0.134732	+	0.990882i	$0.543017\pi$
$440$	56297.0i	0.290790i
$441$	0	0
$442$	22013.3	0.112678
$443$	− 175125.i	− 0.892362i	−0.894943	−	0.446181i	$-0.852784\pi$
$443$	− 175125.i	− 0.892362i	0.894943	−	0.446181i	$-0.147216\pi$
$444$	0	0
$445$	−153565.	−0.775481
$446$	− 202642.i	− 1.01873i
$447$	0	0
$448$	0	0
$449$	71607.4i	0.355194i	0.984103	+	0.177597i	$0.0568323\pi$
$449$	71607.4i	0.355194i	−0.984103	+	0.177597i	$0.943168\pi$
$450$	0	0
$451$	143602.	0.706006
$452$	− 67375.1i	− 0.329778i
$453$	0	0
$454$	131152.	0.636304
$455$	0	0
$456$	0	0
$457$	−119636.	−0.572836	−0.286418	−	0.958105i	$-0.592465\pi$
$457$	−119636.	−0.572836	−0.286418	+	0.958105i	$0.592465\pi$
$458$	53821.1i	0.256580i
$459$	0	0
$460$	32625.5	0.154185
$461$	− 246383.i	− 1.15933i	−0.814853	−	0.579667i	$-0.803183\pi$
$461$	− 246383.i	− 1.15933i	0.814853	−	0.579667i	$-0.196817\pi$
$462$	0	0
$463$	−65371.6	−0.304949	−0.152474	−	0.988307i	$-0.548724\pi$
$463$	−65371.6	−0.304949	−0.152474	+	0.988307i	$0.548724\pi$
$464$	21997.2i	0.102172i
$465$	0	0
$466$	34778.4	0.160154
$467$	358794.i	1.64517i	0.568640	+	0.822586i	$0.307470\pi$
$467$	358794.i	1.64517i	−0.568640	+	0.822586i	$0.692530\pi$
$468$	0	0
$469$	0	0
$470$	118284.i	0.535462i
$471$	0	0
$472$	87919.3	0.394639
$473$	148581.i	0.664110i
$474$	0	0
$475$	296399.	1.31368
$476$	0	0
$477$	0	0
$478$	−214990.	−0.940941
$479$	153309.i	0.668185i	0.942540	+	0.334093i	$0.108430\pi$
$479$	153309.i	0.668185i	−0.942540	+	0.334093i	$0.891570\pi$
$480$	0	0
$481$	−27784.2	−0.120090
$482$	− 215279.i	− 0.926634i
$483$	0	0
$484$	306436.	1.30813
$485$	− 115749.i	− 0.492077i
$486$	0	0
$487$	−8247.68	−0.0347755	−0.0173878	−	0.999849i	$-0.505535\pi$
$487$	−8247.68	−0.0347755	−0.0173878	+	0.999849i	$0.505535\pi$
$488$	5698.79i	0.0239300i
$489$	0	0
$490$	0	0
$491$	− 204324.i	− 0.847532i	−0.905772	−	0.423766i	$-0.860708\pi$
$491$	− 204324.i	− 0.847532i	0.905772	−	0.423766i	$-0.139292\pi$
$492$	0	0
$493$	86528.0	0.356011
$494$	51009.9i	0.209026i
$495$	0	0
$496$	67393.1	0.273938
$497$	0	0
$498$	0	0
$499$	375253.	1.50703	0.753517	−	0.657429i	$-0.228356\pi$
$499$	375253.	1.50703	0.753517	+	0.657429i	$0.228356\pi$
$500$	− 98013.9i	− 0.392056i
$501$	0	0
$502$	105667.	0.419306
$503$	− 422774.i	− 1.67098i	−0.549502	−	0.835492i	$-0.685183\pi$
$503$	− 422774.i	− 1.67098i	0.549502	−	0.835492i	$-0.314817\pi$
$504$	0	0
$505$	−122568.	−0.480611
$506$	− 245467.i	− 0.958719i
$507$	0	0
$508$	131669.	0.510219
$509$	225791.i	0.871508i	0.900066	+	0.435754i	$0.143518\pi$
$509$	225791.i	0.871508i	−0.900066	+	0.435754i	$0.856482\pi$
$510$	0	0
$511$	0	0
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	170139.	0.643987
$515$	162111.i	0.611222i
$516$	0	0
$517$	889936.	3.32949
$518$	0	0
$519$	0	0
$520$	7563.80	0.0279726
$521$	− 160490.i	− 0.591254i	−0.955304	−	0.295627i	$-0.904472\pi$
$521$	− 160490.i	− 0.591254i	0.955304	−	0.295627i	$-0.0955285\pi$
$522$	0	0
$523$	396168.	1.44836	0.724179	−	0.689612i	$-0.242219\pi$
$523$	396168.	1.44836	0.724179	+	0.689612i	$0.242219\pi$
$524$	152168.i	0.554194i
$525$	0	0
$526$	26425.1	0.0955092
$527$	− 265097.i	− 0.954518i
$528$	0	0
$529$	137587.	0.491660
$530$	− 1275.99i	− 0.00454249i
$531$	0	0
$532$	0	0
$533$	− 19293.7i	− 0.0679144i
$534$	0	0
$535$	28385.4	0.0991716
$536$	− 195779.i	− 0.681453i
$537$	0	0
$538$	−966.443	−0.00333896
$539$	0	0
$540$	0	0
$541$	480803.	1.64275	0.821377	−	0.570386i	$-0.193207\pi$
$541$	480803.	1.64275	0.821377	+	0.570386i	$0.193207\pi$
$542$	− 58510.2i	− 0.199174i
$543$	0	0
$544$	45571.7	0.153992
$545$	− 95340.2i	− 0.320984i
$546$	0	0
$547$	−315421.	−1.05418	−0.527092	−	0.849808i	$-0.676718\pi$
$547$	−315421.	−1.05418	−0.527092	+	0.849808i	$0.676718\pi$
$548$	82974.7i	0.276302i
$549$	0	0
$550$	−330671.	−1.09313
$551$	200506.i	0.660425i
$552$	0	0
$553$	0	0
$554$	− 81925.8i	− 0.266932i
$555$	0	0
$556$	213618.	0.691017
$557$	571699.i	1.84271i	0.388722	+	0.921355i	$0.372917\pi$
$557$	571699.i	1.84271i	−0.388722	+	0.921355i	$0.627083\pi$
$558$	0	0
$559$	19962.6	0.0638843
$560$	0	0
$561$	0	0
$562$	−51260.4	−0.162297
$563$	− 5270.49i	− 0.0166278i	−0.999965	−	0.00831388i	$-0.997354\pi$
$563$	− 5270.49i	− 0.0166278i	0.999965	−	0.00831388i	$-0.00264642\pi$
$564$	0	0
$565$	−91063.5	−0.285264
$566$	− 281209.i	− 0.877801i
$567$	0	0
$568$	104167.	0.322873
$569$	532406.i	1.64444i	0.569170	+	0.822220i	$0.307265\pi$
$569$	532406.i	1.64444i	−0.569170	+	0.822220i	$0.692735\pi$
$570$	0	0
$571$	−374717.	−1.14929	−0.574646	−	0.818402i	$-0.694860\pi$
$571$	−374717.	−1.14929	−0.574646	+	0.818402i	$0.694860\pi$
$572$	− 56908.1i	− 0.173933i
$573$	0	0
$574$	0	0
$575$	191632.i	0.579607i
$576$	0	0
$577$	−174028.	−0.522719	−0.261359	−	0.965242i	$-0.584171\pi$
$577$	−174028.	−0.522719	−0.261359	+	0.965242i	$0.584171\pi$
$578$	56972.4i	0.170533i
$579$	0	0
$580$	29731.2	0.0883804
$581$	0	0
$582$	0	0
$583$	−9600.21	−0.0282451
$584$	80403.9i	0.235750i
$585$	0	0
$586$	146686.	0.427163
$587$	− 378974.i	− 1.09985i	−0.835215	−	0.549924i	$-0.814657\pi$
$587$	− 378974.i	− 1.09985i	0.835215	−	0.549924i	$-0.185343\pi$
$588$	0	0
$589$	614293.	1.77070
$590$	− 118831.i	− 0.341370i
$591$	0	0
$592$	−57518.7	−0.164121
$593$	− 173221.i	− 0.492596i	−0.969194	−	0.246298i	$-0.920786\pi$
$593$	− 173221.i	− 0.492596i	0.969194	−	0.246298i	$-0.0792142\pi$
$594$	0	0
$595$	0	0
$596$	− 189784.i	− 0.534276i
$597$	0	0
$598$	−32979.7	−0.0922242
$599$	− 502186.i	− 1.39962i	−0.714327	−	0.699812i	$-0.753267\pi$
$599$	− 502186.i	− 1.39962i	0.714327	−	0.699812i	$-0.246733\pi$
$600$	0	0
$601$	−336698.	−0.932162	−0.466081	−	0.884742i	$-0.654335\pi$
$601$	−336698.	−0.932162	−0.466081	+	0.884742i	$0.654335\pi$
$602$	0	0
$603$	0	0
$604$	−193418.	−0.530180
$605$	− 414177.i	− 1.13155i
$606$	0	0
$607$	93133.7	0.252772	0.126386	−	0.991981i	$-0.459662\pi$
$607$	93133.7	0.252772	0.126386	+	0.991981i	$0.459662\pi$
$608$	105600.i	0.285666i
$609$	0	0
$610$	7702.43	0.0206999
$611$	− 119568.i	− 0.320281i
$612$	0	0
$613$	−106256.	−0.282768	−0.141384	−	0.989955i	$-0.545155\pi$
$613$	−106256.	−0.282768	−0.141384	+	0.989955i	$0.545155\pi$
$614$	− 203787.i	− 0.540555i
$615$	0	0
$616$	0	0
$617$	410880.i	1.07931i	0.841888	+	0.539653i	$0.181444\pi$
$617$	410880.i	1.07931i	−0.841888	+	0.539653i	$0.818556\pi$
$618$	0	0
$619$	−122555.	−0.319853	−0.159927	−	0.987129i	$-0.551126\pi$
$619$	−122555.	−0.319853	−0.159927	+	0.987129i	$0.551126\pi$
$620$	− 91087.9i	− 0.236961i
$621$	0	0
$622$	316048.	0.816906
$623$	0	0
$624$	0	0
$625$	185078.	0.473801
$626$	519704.i	1.32619i
$627$	0	0
$628$	−365171.	−0.925926
$629$	226255.i	0.571870i
$630$	0	0
$631$	138825.	0.348666	0.174333	−	0.984687i	$-0.444223\pi$
$631$	138825.	0.348666	0.174333	+	0.984687i	$0.444223\pi$
$632$	− 81121.5i	− 0.203096i
$633$	0	0
$634$	−246461.	−0.613153
$635$	− 177963.i	− 0.441348i
$636$	0	0
$637$	0	0
$638$	− 223690.i	− 0.549547i
$639$	0	0
$640$	15658.5	0.0382288
$641$	163474.i	0.397862i	0.980014	+	0.198931i	$0.0637469\pi$
$641$	163474.i	0.397862i	−0.980014	+	0.198931i	$0.936253\pi$
$642$	0	0
$643$	141425.	0.342062	0.171031	−	0.985266i	$-0.445290\pi$
$643$	141425.	0.342062	0.171031	+	0.985266i	$0.445290\pi$
$644$	0	0
$645$	0	0
$646$	415389.	0.995383
$647$	− 230661.i	− 0.551017i	−0.961299	−	0.275508i	$-0.911154\pi$
$647$	− 230661.i	− 0.551017i	0.961299	−	0.275508i	$-0.0888462\pi$
$648$	0	0
$649$	−894054.	−2.12263
$650$	44427.4i	0.105154i
$651$	0	0
$652$	74880.8	0.176147
$653$	− 586067.i	− 1.37442i	−0.726456	−	0.687212i	$-0.758834\pi$
$653$	− 586067.i	− 1.37442i	0.726456	−	0.687212i	$-0.241166\pi$
$654$	0	0
$655$	205669.	0.479387
$656$	− 39941.7i	− 0.0928152i
$657$	0	0
$658$	0	0
$659$	− 174639.i	− 0.402134i	−0.979577	−	0.201067i	$-0.935559\pi$
$659$	− 174639.i	− 0.402134i	0.979577	−	0.201067i	$-0.0644409\pi$
$660$	0	0
$661$	604693.	1.38399	0.691993	−	0.721904i	$-0.256733\pi$
$661$	604693.	1.38399	0.691993	+	0.721904i	$0.256733\pi$
$662$	− 223158.i	− 0.509210i
$663$	0	0
$664$	31689.7	0.0718757
$665$	0	0
$666$	0	0
$667$	−129634.	−0.291385
$668$	− 203381.i	− 0.455782i
$669$	0	0
$670$	−264613.	−0.589469
$671$	− 57951.2i	− 0.128711i
$672$	0	0
$673$	341511.	0.754006	0.377003	−	0.926212i	$-0.376955\pi$
$673$	341511.	0.754006	0.377003	+	0.926212i	$0.376955\pi$
$674$	− 304863.i	− 0.671096i
$675$	0	0
$676$	220842.	0.483268
$677$	− 4280.72i	− 0.00933983i	−0.999989	−	0.00466992i	$-0.998514\pi$
$677$	− 4280.72i	− 0.00933983i	0.999989	−	0.00466992i	$-0.00148649\pi$
$678$	0	0
$679$	0	0
$680$	− 61594.2i	− 0.133206i
$681$	0	0
$682$	−685323.	−1.47342
$683$	− 492177.i	− 1.05507i	−0.849534	−	0.527533i	$-0.823117\pi$
$683$	− 492177.i	− 1.05507i	0.849534	−	0.527533i	$-0.176883\pi$
$684$	0	0
$685$	112148.	0.239007
$686$	0	0
$687$	0	0
$688$	41326.4	0.0873074
$689$	1289.84i	0.00271705i
$690$	0	0
$691$	−72024.2	−0.150842	−0.0754210	−	0.997152i	$-0.524030\pi$
$691$	−72024.2	−0.150842	−0.0754210	+	0.997152i	$0.524030\pi$
$692$	− 289436.i	− 0.604423i
$693$	0	0
$694$	−575637.	−1.19517
$695$	− 288724.i	− 0.597742i
$696$	0	0
$697$	−157115.	−0.323408
$698$	604960.i	1.24170i
$699$	0	0
$700$	0	0
$701$	− 903262.i	− 1.83814i	−0.394098	−	0.919068i	$-0.628943\pi$
$701$	− 903262.i	− 1.83814i	0.394098	−	0.919068i	$-0.371057\pi$
$702$	0	0
$703$	−524287.	−1.06086
$704$	− 117811.i	− 0.237706i
$705$	0	0
$706$	261261.	0.524161
$707$	0	0
$708$	0	0
$709$	73876.9	0.146966	0.0734829	−	0.997296i	$-0.476589\pi$
$709$	73876.9	0.146966	0.0734829	+	0.997296i	$0.476589\pi$
$710$	− 140791.i	− 0.279291i
$711$	0	0
$712$	321359.	0.633915
$713$	397162.i	0.781248i
$714$	0	0
$715$	−76916.5	−0.150455
$716$	62516.8i	0.121947i
$717$	0	0
$718$	−105669.	−0.204974
$719$	296844.i	0.574210i	0.957899	+	0.287105i	$0.0926929\pi$
$719$	296844.i	0.574210i	−0.957899	+	0.287105i	$0.907307\pi$
$720$	0	0
$721$	0	0
$722$	593951.i	1.13940i
$723$	0	0
$724$	242292.	0.462234
$725$	174632.i	0.332236i
$726$	0	0
$727$	−256007.	−0.484377	−0.242188	−	0.970229i	$-0.577865\pi$
$727$	−256007.	−0.484377	−0.242188	+	0.970229i	$0.577865\pi$
$728$	0	0
$729$	0	0
$730$	108673.	0.203928
$731$	− 162562.i	− 0.304217i
$732$	0	0
$733$	−164445.	−0.306064	−0.153032	−	0.988221i	$-0.548904\pi$
$733$	−164445.	−0.306064	−0.153032	+	0.988221i	$0.548904\pi$
$734$	430878.i	0.799765i
$735$	0	0
$736$	−68274.4	−0.126038
$737$	1.99088e6i	3.66531i
$738$	0	0
$739$	606890.	1.11127	0.555637	−	0.831425i	$-0.312475\pi$
$739$	606890.	1.11127	0.555637	+	0.831425i	$0.312475\pi$
$740$	77741.7i	0.141968i
$741$	0	0
$742$	0	0
$743$	− 945065.i	− 1.71192i	−0.517040	−	0.855961i	$-0.672966\pi$
$743$	− 945065.i	− 1.71192i	0.517040	−	0.855961i	$-0.327034\pi$
$744$	0	0
$745$	−256510.	−0.462159
$746$	579454.i	1.04122i
$747$	0	0
$748$	−463420.	−0.828269
$749$	0	0
$750$	0	0
$751$	−329864.	−0.584864	−0.292432	−	0.956286i	$-0.594465\pi$
$751$	−329864.	−0.584864	−0.292432	+	0.956286i	$0.594465\pi$
$752$	− 247528.i	− 0.437712i
$753$	0	0
$754$	−30053.9	−0.0528638
$755$	261422.i	0.458616i
$756$	0	0
$757$	−583643.	−1.01849	−0.509244	−	0.860622i	$-0.670075\pi$
$757$	−583643.	−1.01849	−0.509244	+	0.860622i	$0.670075\pi$
$758$	737315.i	1.28326i
$759$	0	0
$760$	142728.	0.247106
$761$	31562.5i	0.0545008i	0.999629	+	0.0272504i	$0.00867514\pi$
$761$	31562.5i	0.0545008i	−0.999629	+	0.0272504i	$0.991325\pi$
$762$	0	0
$763$	0	0
$764$	− 384428.i	− 0.658611i
$765$	0	0
$766$	−543281.	−0.925906
$767$	120121.i	0.204187i
$768$	0	0
$769$	−335400.	−0.567166	−0.283583	−	0.958948i	$-0.591523\pi$
$769$	−335400.	−0.567166	−0.283583	+	0.958948i	$0.591523\pi$
$770$	0	0
$771$	0	0
$772$	−429240.	−0.720220
$773$	− 153794.i	− 0.257383i	−0.991685	−	0.128692i	$-0.958922\pi$
$773$	− 153794.i	− 0.257383i	0.991685	−	0.128692i	$-0.0410777\pi$
$774$	0	0
$775$	535022.	0.890776
$776$	242224.i	0.402247i
$777$	0	0
$778$	518791.	0.857103
$779$	− 364072.i	− 0.599946i
$780$	0	0
$781$	−1.05927e6	−1.73663
$782$	268564.i	0.439171i
$783$	0	0
$784$	0	0
$785$	493561.i	0.800943i
$786$	0	0
$787$	365655.	0.590367	0.295184	−	0.955441i	$-0.404619\pi$
$787$	365655.	0.590367	0.295184	+	0.955441i	$0.404619\pi$
$788$	418199.i	0.673489i
$789$	0	0
$790$	−109643.	−0.175682
$791$	0	0
$792$	0	0
$793$	−7786.05	−0.0123814
$794$	137979.i	0.218863i
$795$	0	0
$796$	580308.	0.915867
$797$	670503.i	1.05556i	0.849380	+	0.527782i	$0.176976\pi$
$797$	670503.i	1.05556i	−0.849380	+	0.527782i	$0.823024\pi$
$798$	0	0
$799$	−973676.	−1.52518
$800$	91973.2i	0.143708i
$801$	0	0
$802$	185424.	0.288281
$803$	− 817630.i	− 1.26802i
$804$	0	0
$805$	0	0
$806$	92076.7i	0.141736i
$807$	0	0
$808$	256494.	0.392875
$809$	− 293467.i	− 0.448396i	−0.974544	−	0.224198i	$-0.928024\pi$
$809$	− 293467.i	− 0.448396i	0.974544	−	0.224198i	$-0.0719763\pi$
$810$	0	0
$811$	−103050.	−0.156678	−0.0783390	−	0.996927i	$-0.524962\pi$
$811$	−103050.	−0.156678	−0.0783390	+	0.996927i	$0.524962\pi$
$812$	0	0
$813$	0	0
$814$	584909.	0.882754
$815$	− 101208.i	− 0.152370i
$816$	0	0
$817$	376693.	0.564344
$818$	− 74416.6i	− 0.111215i
$819$	0	0
$820$	−53984.9	−0.0802868
$821$	− 730796.i	− 1.08420i	−0.840314	−	0.542100i	$-0.817629\pi$
$821$	− 730796.i	− 1.08420i	0.840314	−	0.542100i	$-0.182371\pi$
$822$	0	0
$823$	444974.	0.656954	0.328477	−	0.944512i	$-0.393465\pi$
$823$	444974.	0.656954	0.328477	+	0.944512i	$0.393465\pi$
$824$	− 339245.i	− 0.499642i
$825$	0	0
$826$	0	0
$827$	757726.i	1.10790i	0.832549	+	0.553951i	$0.186880\pi$
$827$	757726.i	1.10790i	−0.832549	+	0.553951i	$0.813120\pi$
$828$	0	0
$829$	−640544.	−0.932051	−0.466026	−	0.884771i	$-0.654315\pi$
$829$	−640544.	−0.932051	−0.466026	+	0.884771i	$0.654315\pi$
$830$	− 42831.5i	− 0.0621738i
$831$	0	0
$832$	−15828.5	−0.0228661
$833$	0	0
$834$	0	0
$835$	−274888.	−0.394260
$836$	− 1.07385e6i	− 1.53650i
$837$	0	0
$838$	−223449.	−0.318193
$839$	− 822723.i	− 1.16877i	−0.811476	−	0.584386i	$-0.801335\pi$
$839$	− 822723.i	− 1.16877i	0.811476	−	0.584386i	$-0.198665\pi$
$840$	0	0
$841$	589147.	0.832975
$842$	701965.i	0.990128i
$843$	0	0
$844$	264438.	0.371226
$845$	− 298488.i	− 0.418036i
$846$	0	0
$847$	0	0
$848$	2670.21i	0.00371325i
$849$	0	0
$850$	361786.	0.500741
$851$	− 338970.i	− 0.468061i
$852$	0	0
$853$	343192.	0.471671	0.235835	−	0.971793i	$-0.424217\pi$
$853$	343192.	0.471671	0.235835	+	0.971793i	$0.424217\pi$
$854$	0	0
$855$	0	0
$856$	−59401.1	−0.0810676
$857$	472404.i	0.643209i	0.946874	+	0.321605i	$0.104222\pi$
$857$	472404.i	0.643209i	−0.946874	+	0.321605i	$0.895778\pi$
$858$	0	0
$859$	−436818.	−0.591989	−0.295995	−	0.955190i	$-0.595651\pi$
$859$	−436818.	−0.591989	−0.295995	+	0.955190i	$0.595651\pi$
$860$	− 55856.4i	− 0.0755225i
$861$	0	0
$862$	−933394.	−1.25618
$863$	210991.i	0.283298i	0.989917	+	0.141649i	$0.0452404\pi$
$863$	210991.i	0.283298i	−0.989917	+	0.141649i	$0.954760\pi$
$864$	0	0
$865$	−391200.	−0.522837
$866$	− 503232.i	− 0.671015i
$867$	0	0
$868$	0	0
$869$	824928.i	1.09239i
$870$	0	0
$871$	267485.	0.352585
$872$	199515.i	0.262387i
$873$	0	0
$874$	−622326.	−0.814695
$875$	0	0
$876$	0	0
$877$	−1.12436e6	−1.46186	−0.730929	−	0.682454i	$-0.760913\pi$
$877$	−1.12436e6	−1.46186	−0.730929	+	0.682454i	$0.760913\pi$
$878$	− 146884.i	− 0.190539i
$879$	0	0
$880$	−159232.	−0.205620
$881$	1.05411e6i	1.35811i	0.734087	+	0.679055i	$0.237610\pi$
$881$	1.05411e6i	1.35811i	−0.734087	+	0.679055i	$0.762390\pi$
$882$	0	0
$883$	−720990.	−0.924715	−0.462357	−	0.886694i	$-0.652996\pi$
$883$	−720990.	−0.924715	−0.462357	+	0.886694i	$0.652996\pi$
$884$	62262.9i	0.0796756i
$885$	0	0
$886$	495329.	0.630995
$887$	− 835959.i	− 1.06252i	−0.847208	−	0.531261i	$-0.821718\pi$
$887$	− 835959.i	− 1.06252i	0.847208	−	0.531261i	$-0.178282\pi$
$888$	0	0
$889$	0	0
$890$	− 434346.i	− 0.548348i
$891$	0	0
$892$	573159.	0.720353
$893$	− 2.25624e6i	− 2.82932i
$894$	0	0
$895$	84497.2	0.105486
$896$	0	0
$897$	0	0
$898$	−202536.	−0.251160
$899$	361928.i	0.447819i
$900$	0	0
$901$	10503.5	0.0129386
$902$	406169.i	0.499222i
$903$	0	0
$904$	190565.	0.233189
$905$	− 327480.i	− 0.399841i
$906$	0	0
$907$	64528.5	0.0784400	0.0392200	−	0.999231i	$-0.487513\pi$
$907$	64528.5	0.0784400	0.0392200	+	0.999231i	$0.487513\pi$
$908$	370955.i	0.449935i
$909$	0	0
$910$	0	0
$911$	812433.i	0.978928i	0.872024	+	0.489464i	$0.162807\pi$
$911$	812433.i	0.978928i	−0.872024	+	0.489464i	$0.837193\pi$
$912$	0	0
$913$	−322254.	−0.386595
$914$	− 338382.i	− 0.405056i
$915$	0	0
$916$	−152229.	−0.181429
$917$	0	0
$918$	0	0
$919$	−1.10930e6	−1.31346	−0.656732	−	0.754124i	$-0.728062\pi$
$919$	−1.10930e6	−1.31346	−0.656732	+	0.754124i	$0.728062\pi$
$920$	92279.0i	0.109025i
$921$	0	0
$922$	696876.	0.819773
$923$	142319.i	0.167055i
$924$	0	0
$925$	−456631.	−0.533681
$926$	− 184899.i	− 0.215631i
$927$	0	0
$928$	−62217.4	−0.0722463
$929$	− 1.51550e6i	− 1.75600i	−0.478657	−	0.878002i	$-0.658876\pi$
$929$	− 1.51550e6i	− 1.75600i	0.478657	−	0.878002i	$-0.341124\pi$
$930$	0	0
$931$	0	0
$932$	98368.1i	0.113246i
$933$	0	0
$934$	−1.01482e6	−1.16331
$935$	626354.i	0.716468i
$936$	0	0
$937$	−1.48591e6	−1.69244	−0.846222	−	0.532830i	$-0.821128\pi$
$937$	−1.48591e6	−1.69244	−0.846222	+	0.532830i	$0.821128\pi$
$938$	0	0
$939$	0	0
$940$	−334556.	−0.378629
$941$	1.54522e6i	1.74506i	0.488562	+	0.872529i	$0.337522\pi$
$941$	1.54522e6i	1.74506i	−0.488562	+	0.872529i	$0.662478\pi$
$942$	0	0
$943$	235385.	0.264701
$944$	248673.i	0.279052i
$945$	0	0
$946$	−420250.	−0.469597
$947$	− 456912.i	− 0.509486i	−0.967009	−	0.254743i	$-0.918009\pi$
$947$	− 456912.i	− 0.509486i	0.967009	−	0.254743i	$-0.0819909\pi$
$948$	0	0
$949$	−109853.	−0.121977
$950$	838342.i	0.928911i
$951$	0	0
$952$	0	0
$953$	935656.i	1.03022i	0.857124	+	0.515111i	$0.172249\pi$
$953$	935656.i	1.03022i	−0.857124	+	0.515111i	$0.827751\pi$
$954$	0	0
$955$	−519590.	−0.569710
$956$	− 608083.i	− 0.665346i
$957$	0	0
$958$	−433623.	−0.472478
$959$	0	0
$960$	0	0
$961$	185324.	0.200671
$962$	− 78585.7i	− 0.0849168i
$963$	0	0
$964$	608902.	0.655229
$965$	580156.i	0.623003i
$966$	0	0
$967$	796333.	0.851612	0.425806	−	0.904815i	$-0.359991\pi$
$967$	796333.	0.851612	0.425806	+	0.904815i	$0.359991\pi$
$968$	866733.i	0.924985i
$969$	0	0
$970$	327387.	0.347951
$971$	1.38318e6i	1.46704i	0.679670	+	0.733518i	$0.262123\pi$
$971$	1.38318e6i	1.46704i	−0.679670	+	0.733518i	$0.737877\pi$
$972$	0	0
$973$	0	0
$974$	− 23328.0i	− 0.0245900i
$975$	0	0
$976$	−16118.6	−0.0169211
$977$	− 894177.i	− 0.936773i	−0.883524	−	0.468387i	$-0.844835\pi$
$977$	− 894177.i	− 0.936773i	0.883524	−	0.468387i	$-0.155165\pi$
$978$	0	0
$979$	−3.26792e6	−3.40962
$980$	0	0
$981$	0	0
$982$	577915.	0.599296
$983$	1.19701e6i	1.23877i	0.785086	+	0.619387i	$0.212619\pi$
$983$	1.19701e6i	1.23877i	−0.785086	+	0.619387i	$0.787381\pi$
$984$	0	0
$985$	565234.	0.582580
$986$	244738.i	0.251737i
$987$	0	0
$988$	−144278.	−0.147804
$989$	243546.i	0.248993i
$990$	0	0
$991$	771624.	0.785703	0.392851	−	0.919602i	$-0.371489\pi$
$991$	771624.	0.785703	0.392851	+	0.919602i	$0.371489\pi$
$992$	190616.i	0.193703i
$993$	0	0
$994$	0	0
$995$	− 784339.i	− 0.792242i
$996$	0	0
$997$	587762.	0.591304	0.295652	−	0.955296i	$-0.404463\pi$
$997$	587762.	0.591304	0.295652	+	0.955296i	$0.404463\pi$
$998$	1.06138e6i	1.06563i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.a.197.4		4
3.2	odd	2	inner	882.5.b.a.197.1		4
7.6	odd	2		126.5.b.b.71.3	yes	4
21.20	even	2		126.5.b.b.71.2	✓	4
28.27	even	2		1008.5.d.a.449.1		4
84.83	odd	2		1008.5.d.a.449.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.5.b.b.71.2	✓	4	21.20	even	2
126.5.b.b.71.3	yes	4	7.6	odd	2
882.5.b.a.197.1		4	3.2	odd	2	inner
882.5.b.a.197.4		4	1.1	even	1	trivial
1008.5.d.a.449.1		4	28.27	even	2
1008.5.d.a.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)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +10.8127i q^{5} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} +10.8127i q^{5} -22.6274i q^{8} -30.5830 q^{10} +230.099i q^{11} +30.9150 q^{13} +64.0000 q^{16} -251.750i q^{17} +583.365 q^{19} -86.5018i q^{20} -650.818 q^{22} +377.166i q^{23} +508.085 q^{25} +87.4409i q^{26} +343.706i q^{29} +1053.02 q^{31} +181.019i q^{32} +712.057 q^{34} -898.729 q^{37} +1650.00i q^{38} +244.664 q^{40} -624.090i q^{41} +645.725 q^{43} -1840.79i q^{44} -1066.79 q^{46} -3867.62i q^{47} +1437.08i q^{50} -247.320 q^{52} +41.7221i q^{53} -2488.00 q^{55} -972.146 q^{58} +3885.52i q^{59} -251.853 q^{61} +2978.38i q^{62} -512.000 q^{64} +334.276i q^{65} +8652.28 q^{67} +2014.00i q^{68} +4603.56i q^{71} -3553.39 q^{73} -2541.99i q^{74} -4666.92 q^{76} +3585.10 q^{79} +692.014i q^{80} +1765.19 q^{82} +1400.50i q^{83} +2722.11 q^{85} +1826.39i q^{86} +5206.55 q^{88} +14202.2i q^{89} -3017.33i q^{92} +10939.3 q^{94} +6307.76i q^{95} -10704.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4}+O(q^{10}) \) 4 * q - 32 * q^4	\( 4 q - 32 q^{4} - 80 q^{10} - 88 q^{13} + 256 q^{16} + 1000 q^{19} - 656 q^{22} + 2244 q^{25} + 1016 q^{31} - 496 q^{34} - 928 q^{37} + 640 q^{40} - 592 q^{43} + 1744 q^{46} + 704 q^{52} - 4216 q^{55} - 1264 q^{58} + 12560 q^{61} - 2048 q^{64} + 16872 q^{67} + 6000 q^{73} - 8000 q^{76} - 15800 q^{79} - 432 q^{82} + 3184 q^{85} + 5248 q^{88} + 17088 q^{94} - 52704 q^{97}+O(q^{100}) \) 4 * q - 32 * q^4 - 80 * q^10 - 88 * q^13 + 256 * q^16 + 1000 * q^19 - 656 * q^22 + 2244 * q^25 + 1016 * q^31 - 496 * q^34 - 928 * q^37 + 640 * q^40 - 592 * q^43 + 1744 * q^46 + 704 * q^52 - 4216 * q^55 - 1264 * q^58 + 12560 * q^61 - 2048 * q^64 + 16872 * q^67 + 6000 * q^73 - 8000 * q^76 - 15800 * q^79 - 432 * q^82 + 3184 * q^85 + 5248 * q^88 + 17088 * q^94 - 52704 * q^97

Introduction

L-functions

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Database

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