LMFDB - Embedded newform 882.5.c.b.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 14)
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.b.685.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{2} +8.00000 q^{4} +25.3864i q^{5} -22.6274 q^{8} +O(q^{10})$	$q-2.82843 q^{2} +8.00000 q^{4} +25.3864i q^{5} -22.6274 q^{8} -71.8036i q^{10} -43.9706 q^{11} +162.507i q^{13} +64.0000 q^{16} -108.236i q^{17} -449.568i q^{19} +203.091i q^{20} +124.368 q^{22} +435.823 q^{23} -19.4701 q^{25} -459.639i q^{26} -742.118 q^{29} -1039.43i q^{31} -181.019 q^{32} +306.139i q^{34} +986.675 q^{37} +1271.57i q^{38} -574.429i q^{40} +1143.70i q^{41} +2418.82 q^{43} -351.765 q^{44} -1232.69 q^{46} -1345.43i q^{47} +55.0698 q^{50} +1300.05i q^{52} +3566.20 q^{53} -1116.26i q^{55} +2099.03 q^{58} -4285.01i q^{59} +1396.69i q^{61} +2939.96i q^{62} +512.000 q^{64} -4125.47 q^{65} +7265.79 q^{67} -865.891i q^{68} -5987.76 q^{71} -577.493i q^{73} -2790.74 q^{74} -3596.55i q^{76} -3146.23 q^{79} +1624.73i q^{80} -3234.86i q^{82} +4729.96i q^{83} +2747.74 q^{85} -6841.46 q^{86} +994.940 q^{88} -837.909i q^{89} +3486.58 q^{92} +3805.44i q^{94} +11412.9 q^{95} +5622.23i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{4}+O(q^{10}) $ 4 * q + 32 * q^4	$ 4 q + 32 q^{4} - 108 q^{11} + 256 q^{16} + 192 q^{22} - 972 q^{23} + 1144 q^{25} - 3240 q^{29} + 892 q^{37} + 2344 q^{43} - 864 q^{44} - 7680 q^{46} + 3456 q^{50} + 5508 q^{53} - 768 q^{58} + 2048 q^{64} - 6048 q^{65} + 10124 q^{67} - 18792 q^{71} - 8640 q^{74} - 1588 q^{79} + 10380 q^{85} - 20736 q^{86} + 1536 q^{88} - 7776 q^{92} + 17820 q^{95}+O(q^{100}) $ 4 * q + 32 * q^4 - 108 * q^11 + 256 * q^16 + 192 * q^22 - 972 * q^23 + 1144 * q^25 - 3240 * q^29 + 892 * q^37 + 2344 * q^43 - 864 * q^44 - 7680 * q^46 + 3456 * q^50 + 5508 * q^53 - 768 * q^58 + 2048 * q^64 - 6048 * q^65 + 10124 * q^67 - 18792 * q^71 - 8640 * q^74 - 1588 * q^79 + 10380 * q^85 - 20736 * q^86 + 1536 * q^88 - 7776 * q^92 + 17820 * q^95

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 $

$2$	−2.82843	−0.707107
$2$	−2.82843	−0.707107
$3$	0	0
$3$	0	0
$4$	8.00000	0.500000
$5$	25.3864i	1.01546i	0.861517	+	0.507728i	$0.169515\pi$
$5$	25.3864i	1.01546i	−0.861517	+	0.507728i	$0.830485\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−22.6274	−0.353553
$9$	0	0
$10$	− 71.8036i	− 0.718036i
$11$	−43.9706	−0.363393	−0.181697	−	0.983355i	$-0.558159\pi$
$11$	−43.9706	−0.363393	−0.181697	+	0.983355i	$0.558159\pi$
$12$	0	0
$13$	162.507i	0.961579i	0.876836	+	0.480790i	$0.159650\pi$
$13$	162.507i	0.961579i	−0.876836	+	0.480790i	$0.840350\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	− 108.236i	− 0.374521i	−0.982310	−	0.187260i	$-0.940039\pi$
$17$	− 108.236i	− 0.374521i	0.982310	−	0.187260i	$-0.0599608\pi$
$18$	0	0
$19$	− 449.568i	− 1.24534i	−0.782484	−	0.622671i	$-0.786048\pi$
$19$	− 449.568i	− 1.24534i	0.782484	−	0.622671i	$-0.213952\pi$
$20$	203.091i	0.507728i
$21$	0	0
$22$	124.368	0.256958
$23$	435.823	0.823861	0.411931	−	0.911215i	$-0.364855\pi$
$23$	435.823	0.823861	0.411931	+	0.911215i	$0.364855\pi$
$24$	0	0
$25$	−19.4701	−0.0311522
$26$	− 459.639i	− 0.679939i
$27$	0	0
$28$	0	0
$29$	−742.118	−0.882423	−0.441212	−	0.897403i	$-0.645451\pi$
$29$	−742.118	−0.882423	−0.441212	+	0.897403i	$0.645451\pi$
$30$	0	0
$31$	− 1039.43i	− 1.08162i	−0.841146	−	0.540808i	$-0.818119\pi$
$31$	− 1039.43i	− 1.08162i	0.841146	−	0.540808i	$-0.181881\pi$
$32$	−181.019	−0.176777
$33$	0	0
$34$	306.139i	0.264826i
$35$	0	0
$36$	0	0
$37$	986.675	0.720727	0.360364	−	0.932812i	$-0.382653\pi$
$37$	986.675	0.720727	0.360364	+	0.932812i	$0.382653\pi$
$38$	1271.57i	0.880590i
$39$	0	0
$40$	− 574.429i	− 0.359018i
$41$	1143.70i	0.680366i	0.940359	+	0.340183i	$0.110489\pi$
$41$	1143.70i	0.680366i	−0.940359	+	0.340183i	$0.889511\pi$
$42$	0	0
$43$	2418.82	1.30818	0.654089	−	0.756418i	$-0.273052\pi$
$43$	2418.82	1.30818	0.654089	+	0.756418i	$0.273052\pi$
$44$	−351.765	−0.181697
$45$	0	0
$46$	−1232.69	−0.582558
$47$	− 1345.43i	− 0.609066i	−0.952502	−	0.304533i	$-0.901500\pi$
$47$	− 1345.43i	− 0.609066i	0.952502	−	0.304533i	$-0.0985004\pi$
$48$	0	0
$49$	0	0
$50$	55.0698	0.0220279
$51$	0	0
$52$	1300.05i	0.480790i
$53$	3566.20	1.26956	0.634782	−	0.772692i	$-0.281090\pi$
$53$	3566.20	1.26956	0.634782	+	0.772692i	$0.281090\pi$
$54$	0	0
$55$	− 1116.26i	− 0.369010i
$56$	0	0
$57$	0	0
$58$	2099.03	0.623967
$59$	− 4285.01i	− 1.23097i	−0.788148	−	0.615485i	$-0.788960\pi$
$59$	− 4285.01i	− 1.23097i	0.788148	−	0.615485i	$-0.211040\pi$
$60$	0	0
$61$	1396.69i	0.375353i	0.982231	+	0.187677i	$0.0600957\pi$
$61$	1396.69i	0.375353i	−0.982231	+	0.187677i	$0.939904\pi$
$62$	2939.96i	0.764818i
$63$	0	0
$64$	512.000	0.125000
$65$	−4125.47	−0.976442
$66$	0	0
$67$	7265.79	1.61858	0.809288	−	0.587412i	$-0.199853\pi$
$67$	7265.79	1.61858	0.809288	+	0.587412i	$0.199853\pi$
$68$	− 865.891i	− 0.187260i
$69$	0	0
$70$	0	0
$71$	−5987.76	−1.18781	−0.593906	−	0.804534i	$-0.702415\pi$
$71$	−5987.76	−1.18781	−0.593906	+	0.804534i	$0.702415\pi$
$72$	0	0
$73$	− 577.493i	− 0.108368i	−0.998531	−	0.0541840i	$-0.982744\pi$
$73$	− 577.493i	− 0.108368i	0.998531	−	0.0541840i	$-0.0172557\pi$
$74$	−2790.74	−0.509631
$75$	0	0
$76$	− 3596.55i	− 0.622671i
$77$	0	0
$78$	0	0
$79$	−3146.23	−0.504123	−0.252061	−	0.967711i	$-0.581109\pi$
$79$	−3146.23	−0.504123	−0.252061	+	0.967711i	$0.581109\pi$
$80$	1624.73i	0.253864i
$81$	0	0
$82$	− 3234.86i	− 0.481091i
$83$	4729.96i	0.686596i	0.939227	+	0.343298i	$0.111544\pi$
$83$	4729.96i	0.686596i	−0.939227	+	0.343298i	$0.888456\pi$
$84$	0	0
$85$	2747.74	0.380309
$86$	−6841.46	−0.925021
$87$	0	0
$88$	994.940	0.128479
$89$	− 837.909i	− 0.105783i	−0.998600	−	0.0528916i	$-0.983156\pi$
$89$	− 837.909i	− 0.105783i	0.998600	−	0.0528916i	$-0.0168438\pi$
$90$	0	0
$91$	0	0
$92$	3486.58	0.411931
$93$	0	0
$94$	3805.44i	0.430675i
$95$	11412.9	1.26459
$96$	0	0
$97$	5622.23i	0.597537i	0.954326	+	0.298769i	$0.0965759\pi$
$97$	5622.23i	0.597537i	−0.954326	+	0.298769i	$0.903424\pi$
$98$	0	0
$99$	0	0
$100$	−155.761	−0.0155761
$101$	6585.50i	0.645574i	0.946472	+	0.322787i	$0.104620\pi$
$101$	6585.50i	0.645574i	−0.946472	+	0.322787i	$0.895380\pi$
$102$	0	0
$103$	− 1524.50i	− 0.143698i	−0.997416	−	0.0718492i	$-0.977110\pi$
$103$	− 1524.50i	− 0.143698i	0.997416	−	0.0718492i	$-0.0228900\pi$
$104$	− 3677.11i	− 0.339970i
$105$	0	0
$106$	−10086.7	−0.897717
$107$	−10606.6	−0.926424	−0.463212	−	0.886248i	$-0.653303\pi$
$107$	−10606.6	−0.926424	−0.463212	+	0.886248i	$0.653303\pi$
$108$	0	0
$109$	−13952.3	−1.17433	−0.587167	−	0.809466i	$-0.699757\pi$
$109$	−13952.3	−1.17433	−0.587167	+	0.809466i	$0.699757\pi$
$110$	3157.25i	0.260929i
$111$	0	0
$112$	0	0
$113$	8811.30	0.690054	0.345027	−	0.938593i	$-0.387870\pi$
$113$	8811.30	0.690054	0.345027	+	0.938593i	$0.387870\pi$
$114$	0	0
$115$	11064.0i	0.836595i
$116$	−5936.94	−0.441212
$117$	0	0
$118$	12119.8i	0.870427i
$119$	0	0
$120$	0	0
$121$	−12707.6	−0.867945
$122$	− 3950.44i	− 0.265415i
$123$	0	0
$124$	− 8315.46i	− 0.540808i
$125$	15372.2i	0.983823i
$126$	0	0
$127$	3992.70	0.247548	0.123774	−	0.992310i	$-0.460500\pi$
$127$	3992.70	0.247548	0.123774	+	0.992310i	$0.460500\pi$
$128$	−1448.15	−0.0883883
$129$	0	0
$130$	11668.6	0.690449
$131$	− 20225.0i	− 1.17854i	−0.807935	−	0.589272i	$-0.799415\pi$
$131$	− 20225.0i	− 1.17854i	0.807935	−	0.589272i	$-0.200585\pi$
$132$	0	0
$133$	0	0
$134$	−20550.7	−1.14451
$135$	0	0
$136$	2449.11i	0.132413i
$137$	−25057.7	−1.33506	−0.667530	−	0.744583i	$-0.732648\pi$
$137$	−25057.7	−1.33506	−0.667530	+	0.744583i	$0.732648\pi$
$138$	0	0
$139$	20070.1i	1.03877i	0.854541	+	0.519385i	$0.173839\pi$
$139$	20070.1i	1.03877i	−0.854541	+	0.519385i	$0.826161\pi$
$140$	0	0
$141$	0	0
$142$	16936.0	0.839910
$143$	− 7145.52i	− 0.349431i
$144$	0	0
$145$	− 18839.7i	− 0.896062i
$146$	1633.40i	0.0766277i
$147$	0	0
$148$	7893.40	0.360364
$149$	42707.9	1.92369	0.961846	−	0.273591i	$-0.0882115\pi$
$149$	42707.9	1.92369	0.961846	+	0.273591i	$0.0882115\pi$
$150$	0	0
$151$	29607.4	1.29852	0.649258	−	0.760569i	$-0.275080\pi$
$151$	29607.4	1.29852	0.649258	+	0.760569i	$0.275080\pi$
$152$	10172.6i	0.440295i
$153$	0	0
$154$	0	0
$155$	26387.5	1.09833
$156$	0	0
$157$	12237.1i	0.496452i	0.968702	+	0.248226i	$0.0798476\pi$
$157$	12237.1i	0.496452i	−0.968702	+	0.248226i	$0.920152\pi$
$158$	8898.89	0.356469
$159$	0	0
$160$	− 4595.43i	− 0.179509i
$161$	0	0
$162$	0	0
$163$	21531.1	0.810385	0.405193	−	0.914231i	$-0.367204\pi$
$163$	21531.1	0.810385	0.405193	+	0.914231i	$0.367204\pi$
$164$	9149.56i	0.340183i
$165$	0	0
$166$	− 13378.3i	− 0.485497i
$167$	− 16578.9i	− 0.594461i	−0.954806	−	0.297230i	$-0.903937\pi$
$167$	− 16578.9i	− 0.594461i	0.954806	−	0.297230i	$-0.0960629\pi$
$168$	0	0
$169$	2152.52	0.0753658
$170$	−7771.77	−0.268919
$171$	0	0
$172$	19350.6	0.654089
$173$	− 40383.1i	− 1.34930i	−0.738140	−	0.674648i	$-0.764296\pi$
$173$	− 40383.1i	− 1.34930i	0.738140	−	0.674648i	$-0.235704\pi$
$174$	0	0
$175$	0	0
$176$	−2814.12	−0.0908483
$177$	0	0
$178$	2369.97i	0.0748001i
$179$	53494.4	1.66956	0.834780	−	0.550583i	$-0.185595\pi$
$179$	53494.4	1.66956	0.834780	+	0.550583i	$0.185595\pi$
$180$	0	0
$181$	54202.1i	1.65447i	0.561857	+	0.827235i	$0.310087\pi$
$181$	54202.1i	1.65447i	−0.561857	+	0.827235i	$0.689913\pi$
$182$	0	0
$183$	0	0
$184$	−9861.54	−0.291279
$185$	25048.2i	0.731867i
$186$	0	0
$187$	4759.22i	0.136098i
$188$	− 10763.4i	− 0.304533i
$189$	0	0
$190$	−32280.6	−0.894201
$191$	1203.87	0.0330000	0.0165000	−	0.999864i	$-0.494748\pi$
$191$	1203.87	0.0330000	0.0165000	+	0.999864i	$0.494748\pi$
$192$	0	0
$193$	51312.9	1.37757	0.688783	−	0.724968i	$-0.258145\pi$
$193$	51312.9	1.37757	0.688783	+	0.724968i	$0.258145\pi$
$194$	− 15902.1i	− 0.422523i
$195$	0	0
$196$	0	0
$197$	1456.28	0.0375242	0.0187621	−	0.999824i	$-0.494027\pi$
$197$	1456.28	0.0375242	0.0187621	+	0.999824i	$0.494027\pi$
$198$	0	0
$199$	68758.2i	1.73627i	0.496325	+	0.868137i	$0.334683\pi$
$199$	68758.2i	1.73627i	−0.496325	+	0.868137i	$0.665317\pi$
$200$	440.559	0.0110140
$201$	0	0
$202$	− 18626.6i	− 0.456490i
$203$	0	0
$204$	0	0
$205$	−29034.3	−0.690882
$206$	4311.93i	0.101610i
$207$	0	0
$208$	10400.4i	0.240395i
$209$	19767.8i	0.452549i
$210$	0	0
$211$	−622.821	−0.0139894	−0.00699469	−	0.999976i	$-0.502226\pi$
$211$	−622.821	−0.0139894	−0.00699469	+	0.999976i	$0.502226\pi$
$212$	28529.6	0.634782
$213$	0	0
$214$	30000.1	0.655080
$215$	61405.2i	1.32840i
$216$	0	0
$217$	0	0
$218$	39463.0	0.830380
$219$	0	0
$220$	− 8930.04i	− 0.184505i
$221$	17589.2	0.360131
$222$	0	0
$223$	11509.7i	0.231449i	0.993281	+	0.115725i	$0.0369190\pi$
$223$	11509.7i	0.231449i	−0.993281	+	0.115725i	$0.963081\pi$
$224$	0	0
$225$	0	0
$226$	−24922.1	−0.487942
$227$	56367.2i	1.09389i	0.837168	+	0.546946i	$0.184210\pi$
$227$	56367.2i	1.09389i	−0.837168	+	0.546946i	$0.815790\pi$
$228$	0	0
$229$	42128.5i	0.803351i	0.915782	+	0.401675i	$0.131572\pi$
$229$	42128.5i	0.803351i	−0.915782	+	0.401675i	$0.868428\pi$
$230$	− 31293.6i	− 0.591562i
$231$	0	0
$232$	16792.2	0.311984
$233$	55101.9	1.01497	0.507486	−	0.861660i	$-0.330575\pi$
$233$	55101.9	1.01497	0.507486	+	0.861660i	$0.330575\pi$
$234$	0	0
$235$	34155.6	0.618480
$236$	− 34280.1i	− 0.615485i
$237$	0	0
$238$	0	0
$239$	−23082.5	−0.404098	−0.202049	−	0.979375i	$-0.564760\pi$
$239$	−23082.5	−0.404098	−0.202049	+	0.979375i	$0.564760\pi$
$240$	0	0
$241$	81572.5i	1.40446i	0.711950	+	0.702231i	$0.247812\pi$
$241$	81572.5i	1.40446i	−0.711950	+	0.702231i	$0.752188\pi$
$242$	35942.5	0.613730
$243$	0	0
$244$	11173.5i	0.187677i
$245$	0	0
$246$	0	0
$247$	73057.9	1.19749
$248$	23519.7i	0.382409i
$249$	0	0
$250$	− 43479.2i	− 0.695668i
$251$	27207.6i	0.431859i	0.976409	+	0.215930i	$0.0692782\pi$
$251$	27207.6i	0.431859i	−0.976409	+	0.215930i	$0.930722\pi$
$252$	0	0
$253$	−19163.4	−0.299385
$254$	−11293.1	−0.175043
$255$	0	0
$256$	4096.00	0.0625000
$257$	107929.i	1.63408i	0.576584	+	0.817038i	$0.304385\pi$
$257$	107929.i	1.63408i	−0.576584	+	0.817038i	$0.695615\pi$
$258$	0	0
$259$	0	0
$260$	−33003.7	−0.488221
$261$	0	0
$262$	57204.9i	0.833356i
$263$	40144.1	0.580377	0.290188	−	0.956970i	$-0.406282\pi$
$263$	40144.1	0.580377	0.290188	+	0.956970i	$0.406282\pi$
$264$	0	0
$265$	90533.1i	1.28919i
$266$	0	0
$267$	0	0
$268$	58126.3	0.809288
$269$	− 105354.i	− 1.45595i	−0.685606	−	0.727973i	$-0.740462\pi$
$269$	− 105354.i	− 1.45595i	0.685606	−	0.727973i	$-0.259538\pi$
$270$	0	0
$271$	15229.7i	0.207373i	0.994610	+	0.103687i	$0.0330639\pi$
$271$	15229.7i	0.207373i	−0.994610	+	0.103687i	$0.966936\pi$
$272$	− 6927.13i	− 0.0936301i
$273$	0	0
$274$	70874.0	0.944030
$275$	856.113	0.0113205
$276$	0	0
$277$	−9871.62	−0.128656	−0.0643278	−	0.997929i	$-0.520490\pi$
$277$	−9871.62	−0.128656	−0.0643278	+	0.997929i	$0.520490\pi$
$278$	− 56766.7i	− 0.734521i
$279$	0	0
$280$	0	0
$281$	155117.	1.96448	0.982240	−	0.187631i	$-0.0600810\pi$
$281$	155117.	1.96448	0.982240	+	0.187631i	$0.0600810\pi$
$282$	0	0
$283$	− 109662.i	− 1.36925i	−0.728895	−	0.684626i	$-0.759966\pi$
$283$	− 109662.i	− 1.36925i	0.728895	−	0.684626i	$-0.240034\pi$
$284$	−47902.1	−0.593906
$285$	0	0
$286$	20210.6i	0.247085i
$287$	0	0
$288$	0	0
$289$	71805.9	0.859734
$290$	53286.7i	0.633612i
$291$	0	0
$292$	− 4619.94i	− 0.0541840i
$293$	89912.4i	1.04733i	0.851924	+	0.523666i	$0.175436\pi$
$293$	89912.4i	1.04733i	−0.851924	+	0.523666i	$0.824564\pi$
$294$	0	0
$295$	108781.	1.25000
$296$	−22325.9	−0.254815
$297$	0	0
$298$	−120796.	−1.36026
$299$	70824.1i	0.792208i
$300$	0	0
$301$	0	0
$302$	−83742.5	−0.918189
$303$	0	0
$304$	− 28772.4i	− 0.311335i
$305$	−35456.9	−0.381155
$306$	0	0
$307$	− 141681.i	− 1.50327i	−0.659582	−	0.751633i	$-0.729267\pi$
$307$	− 141681.i	− 1.50327i	0.659582	−	0.751633i	$-0.270733\pi$
$308$	0	0
$309$	0	0
$310$	−74635.1	−0.776639
$311$	85645.6i	0.885491i	0.896647	+	0.442746i	$0.145995\pi$
$311$	85645.6i	0.885491i	−0.896647	+	0.442746i	$0.854005\pi$
$312$	0	0
$313$	− 17831.6i	− 0.182012i	−0.995850	−	0.0910061i	$-0.970992\pi$
$313$	− 17831.6i	− 0.182012i	0.995850	−	0.0910061i	$-0.0290083\pi$
$314$	− 34611.6i	− 0.351045i
$315$	0	0
$316$	−25169.8	−0.252061
$317$	−80805.2	−0.804120	−0.402060	−	0.915613i	$-0.631706\pi$
$317$	−80805.2	−0.804120	−0.402060	+	0.915613i	$0.631706\pi$
$318$	0	0
$319$	32631.3	0.320666
$320$	12997.8i	0.126932i
$321$	0	0
$322$	0	0
$323$	−48659.7	−0.466406
$324$	0	0
$325$	− 3164.03i	− 0.0299553i
$326$	−60899.2	−0.573029
$327$	0	0
$328$	− 25878.9i	− 0.240546i
$329$	0	0
$330$	0	0
$331$	72804.3	0.664509	0.332254	−	0.943190i	$-0.392191\pi$
$331$	72804.3	0.664509	0.332254	+	0.943190i	$0.392191\pi$
$332$	37839.7i	0.343298i
$333$	0	0
$334$	46892.3i	0.420347i
$335$	184452.i	1.64359i
$336$	0	0
$337$	126538.	1.11419	0.557095	−	0.830449i	$-0.311916\pi$
$337$	126538.	1.11419	0.557095	+	0.830449i	$0.311916\pi$
$338$	−6088.25	−0.0532917
$339$	0	0
$340$	21981.9	0.190155
$341$	45704.5i	0.393052i
$342$	0	0
$343$	0	0
$344$	−54731.7	−0.462511
$345$	0	0
$346$	114221.i	0.954096i
$347$	66949.3	0.556016	0.278008	−	0.960579i	$-0.410326\pi$
$347$	66949.3	0.556016	0.278008	+	0.960579i	$0.410326\pi$
$348$	0	0
$349$	25527.5i	0.209583i	0.994494	+	0.104792i	$0.0334176\pi$
$349$	25527.5i	0.209583i	−0.994494	+	0.104792i	$0.966582\pi$
$350$	0	0
$351$	0	0
$352$	7959.52	0.0642394
$353$	56575.6i	0.454025i	0.973892	+	0.227013i	$0.0728959\pi$
$353$	56575.6i	0.454025i	−0.973892	+	0.227013i	$0.927104\pi$
$354$	0	0
$355$	− 152008.i	− 1.20617i
$356$	− 6703.27i	− 0.0528916i
$357$	0	0
$358$	−151305.	−1.18056
$359$	−37804.8	−0.293331	−0.146665	−	0.989186i	$-0.546854\pi$
$359$	−37804.8	−0.293331	−0.146665	+	0.989186i	$0.546854\pi$
$360$	0	0
$361$	−71790.8	−0.550876
$362$	− 153307.i	− 1.16989i
$363$	0	0
$364$	0	0
$365$	14660.5	0.110043
$366$	0	0
$367$	10816.4i	0.0803065i	0.999194	+	0.0401532i	$0.0127846\pi$
$367$	10816.4i	0.0803065i	−0.999194	+	0.0401532i	$0.987215\pi$
$368$	27892.6	0.205965
$369$	0	0
$370$	− 70846.9i	− 0.517508i
$371$	0	0
$372$	0	0
$373$	−31023.4	−0.222983	−0.111491	−	0.993765i	$-0.535563\pi$
$373$	−31023.4	−0.222983	−0.111491	+	0.993765i	$0.535563\pi$
$374$	− 13461.1i	− 0.0962359i
$375$	0	0
$376$	30443.6i	0.215337i
$377$	− 120599.i	− 0.848519i
$378$	0	0
$379$	−98527.5	−0.685929	−0.342964	−	0.939348i	$-0.611431\pi$
$379$	−98527.5	−0.685929	−0.342964	+	0.939348i	$0.611431\pi$
$380$	91303.4	0.632295
$381$	0	0
$382$	−3405.07	−0.0233345
$383$	− 5143.33i	− 0.0350628i	−0.999846	−	0.0175314i	$-0.994419\pi$
$383$	− 5143.33i	− 0.0350628i	0.999846	−	0.0175314i	$-0.00558071\pi$
$384$	0	0
$385$	0	0
$386$	−145135.	−0.974086
$387$	0	0
$388$	44977.8i	0.298769i
$389$	−86282.0	−0.570192	−0.285096	−	0.958499i	$-0.592025\pi$
$389$	−86282.0	−0.570192	−0.285096	+	0.958499i	$0.592025\pi$
$390$	0	0
$391$	− 47171.9i	− 0.308553i
$392$	0	0
$393$	0	0
$394$	−4118.97	−0.0265336
$395$	− 79871.5i	− 0.511915i
$396$	0	0
$397$	155926.i	0.989323i	0.869086	+	0.494661i	$0.164708\pi$
$397$	155926.i	0.989323i	−0.869086	+	0.494661i	$0.835292\pi$
$398$	− 194478.i	− 1.22773i
$399$	0	0
$400$	−1246.09	−0.00778805
$401$	−34614.5	−0.215263	−0.107631	−	0.994191i	$-0.534327\pi$
$401$	−34614.5	−0.215263	−0.107631	+	0.994191i	$0.534327\pi$
$402$	0	0
$403$	168915.	1.04006
$404$	52684.0i	0.322787i
$405$	0	0
$406$	0	0
$407$	−43384.7	−0.261907
$408$	0	0
$409$	− 313788.i	− 1.87581i	−0.346886	−	0.937907i	$-0.612761\pi$
$409$	− 313788.i	− 1.87581i	0.346886	−	0.937907i	$-0.387239\pi$
$410$	82121.5	0.488528
$411$	0	0
$412$	− 12196.0i	− 0.0718492i
$413$	0	0
$414$	0	0
$415$	−120077.	−0.697208
$416$	− 29416.9i	− 0.169985i
$417$	0	0
$418$	− 55911.7i	− 0.320000i
$419$	7324.74i	0.0417219i	0.999782	+	0.0208609i	$0.00664073\pi$
$419$	7324.74i	0.0417219i	−0.999782	+	0.0208609i	$0.993359\pi$
$420$	0	0
$421$	−25178.3	−0.142057	−0.0710285	−	0.997474i	$-0.522628\pi$
$421$	−25178.3	−0.142057	−0.0710285	+	0.997474i	$0.522628\pi$
$422$	1761.60	0.00989198
$423$	0	0
$424$	−80694.0	−0.448858
$425$	2107.38i	0.0116671i
$426$	0	0
$427$	0	0
$428$	−84853.0	−0.463212
$429$	0	0
$430$	− 173680.i	− 0.939319i
$431$	−13827.2	−0.0744353	−0.0372176	−	0.999307i	$-0.511849\pi$
$431$	−13827.2	−0.0744353	−0.0372176	+	0.999307i	$0.511849\pi$
$432$	0	0
$433$	− 47438.8i	− 0.253022i	−0.991965	−	0.126511i	$-0.959622\pi$
$433$	− 47438.8i	− 0.253022i	0.991965	−	0.126511i	$-0.0403779\pi$
$434$	0	0
$435$	0	0
$436$	−111618.	−0.587167
$437$	− 195932.i	− 1.02599i
$438$	0	0
$439$	− 109814.i	− 0.569808i	−0.958556	−	0.284904i	$-0.908038\pi$
$439$	− 109814.i	− 0.569808i	0.958556	−	0.284904i	$-0.0919618\pi$
$440$	25258.0i	0.130465i
$441$	0	0
$442$	−49749.7	−0.254651
$443$	82560.7	0.420694	0.210347	−	0.977627i	$-0.432541\pi$
$443$	82560.7	0.420694	0.210347	+	0.977627i	$0.432541\pi$
$444$	0	0
$445$	21271.5	0.107418
$446$	− 32554.5i	− 0.163659i
$447$	0	0
$448$	0	0
$449$	330438.	1.63907	0.819534	−	0.573030i	$-0.194232\pi$
$449$	330438.	1.63907	0.819534	+	0.573030i	$0.194232\pi$
$450$	0	0
$451$	− 50288.9i	− 0.247240i
$452$	70490.4	0.345027
$453$	0	0
$454$	− 159430.i	− 0.773499i
$455$	0	0
$456$	0	0
$457$	−373287.	−1.78735	−0.893677	−	0.448711i	$-0.851883\pi$
$457$	−373287.	−1.78735	−0.893677	+	0.448711i	$0.851883\pi$
$458$	− 119157.i	− 0.568055i
$459$	0	0
$460$	88511.8i	0.418298i
$461$	20355.2i	0.0957798i	0.998853	+	0.0478899i	$0.0152497\pi$
$461$	20355.2i	0.0957798i	−0.998853	+	0.0478899i	$0.984750\pi$
$462$	0	0
$463$	−31552.7	−0.147189	−0.0735944	−	0.997288i	$-0.523447\pi$
$463$	−31552.7	−0.147189	−0.0735944	+	0.997288i	$0.523447\pi$
$464$	−47495.5	−0.220606
$465$	0	0
$466$	−155852.	−0.717694
$467$	− 241854.i	− 1.10897i	−0.832193	−	0.554486i	$-0.812915\pi$
$467$	− 241854.i	− 1.10897i	0.832193	−	0.554486i	$-0.187085\pi$
$468$	0	0
$469$	0	0
$470$	−96606.6	−0.437332
$471$	0	0
$472$	96958.7i	0.435214i
$473$	−106357.	−0.475383
$474$	0	0
$475$	8753.16i	0.0387951i
$476$	0	0
$477$	0	0
$478$	65287.1	0.285740
$479$	− 272083.i	− 1.18585i	−0.805257	−	0.592926i	$-0.797973\pi$
$479$	− 272083.i	− 1.18585i	0.805257	−	0.592926i	$-0.202027\pi$
$480$	0	0
$481$	160342.i	0.693036i
$482$	− 230722.i	− 0.993104i
$483$	0	0
$484$	−101661.	−0.433973
$485$	−142728.	−0.606773
$486$	0	0
$487$	−407746.	−1.71922	−0.859611	−	0.510949i	$-0.829294\pi$
$487$	−407746.	−1.71922	−0.859611	+	0.510949i	$0.829294\pi$
$488$	− 31603.5i	− 0.132707i
$489$	0	0
$490$	0	0
$491$	286772.	1.18953	0.594763	−	0.803901i	$-0.297246\pi$
$491$	286772.	1.18953	0.594763	+	0.803901i	$0.297246\pi$
$492$	0	0
$493$	80324.2i	0.330486i
$494$	−206639.	−0.846757
$495$	0	0
$496$	− 66523.7i	− 0.270404i
$497$	0	0
$498$	0	0
$499$	351435.	1.41138	0.705690	−	0.708520i	$-0.250637\pi$
$499$	351435.	1.41138	0.705690	+	0.708520i	$0.250637\pi$
$500$	122978.i	0.491911i
$501$	0	0
$502$	− 76954.6i	− 0.305371i
$503$	116045.i	0.458660i	0.973349	+	0.229330i	$0.0736535\pi$
$503$	116045.i	0.458660i	−0.973349	+	0.229330i	$0.926347\pi$
$504$	0	0
$505$	−167182.	−0.655552
$506$	54202.2	0.211697
$507$	0	0
$508$	31941.6	0.123774
$509$	83173.8i	0.321034i	0.987033	+	0.160517i	$0.0513162\pi$
$509$	83173.8i	0.321034i	−0.987033	+	0.160517i	$0.948684\pi$
$510$	0	0
$511$	0	0
$512$	−11585.2	−0.0441942
$513$	0	0
$514$	− 305269.i	− 1.15547i
$515$	38701.5	0.145920
$516$	0	0
$517$	59159.2i	0.221330i
$518$	0	0
$519$	0	0
$520$	93348.7	0.345224
$521$	− 202013.i	− 0.744224i	−0.928188	−	0.372112i	$-0.878634\pi$
$521$	− 202013.i	− 0.744224i	0.928188	−	0.372112i	$-0.121366\pi$
$522$	0	0
$523$	27867.6i	0.101882i	0.998702	+	0.0509408i	$0.0162220\pi$
$523$	27867.6i	0.101882i	−0.998702	+	0.0509408i	$0.983778\pi$
$524$	− 161800.i	− 0.589272i
$525$	0	0
$526$	−113545.	−0.410389
$527$	−112505.	−0.405087
$528$	0	0
$529$	−89899.7	−0.321253
$530$	− 256066.i	− 0.911592i
$531$	0	0
$532$	0	0
$533$	−185858.	−0.654226
$534$	0	0
$535$	− 269264.i	− 0.940743i
$536$	−164406.	−0.572253
$537$	0	0
$538$	297985.i	1.02951i
$539$	0	0
$540$	0	0
$541$	−37993.6	−0.129812	−0.0649062	−	0.997891i	$-0.520675\pi$
$541$	−37993.6	−0.129812	−0.0649062	+	0.997891i	$0.520675\pi$
$542$	− 43076.1i	− 0.146635i
$543$	0	0
$544$	19592.9i	0.0662065i
$545$	− 354198.i	− 1.19249i
$546$	0	0
$547$	360160.	1.20371	0.601854	−	0.798606i	$-0.294429\pi$
$547$	360160.	1.20371	0.601854	+	0.798606i	$0.294429\pi$
$548$	−200462.	−0.667530
$549$	0	0
$550$	−2421.45	−0.00800480
$551$	333633.i	1.09892i
$552$	0	0
$553$	0	0
$554$	27921.1	0.0909732
$555$	0	0
$556$	160560.i	0.519385i
$557$	−310543.	−1.00095	−0.500474	−	0.865751i	$-0.666841\pi$
$557$	−310543.	−1.00095	−0.500474	+	0.865751i	$0.666841\pi$
$558$	0	0
$559$	393075.i	1.25792i
$560$	0	0
$561$	0	0
$562$	−438738.	−1.38910
$563$	− 16689.4i	− 0.0526530i	−0.999653	−	0.0263265i	$-0.991619\pi$
$563$	− 16689.4i	− 0.0526530i	0.999653	−	0.0263265i	$-0.00838095\pi$
$564$	0	0
$565$	223687.i	0.700720i
$566$	310171.i	0.968207i
$567$	0	0
$568$	135488.	0.419955
$569$	130262.	0.402341	0.201170	−	0.979556i	$-0.435526\pi$
$569$	130262.	0.402341	0.201170	+	0.979556i	$0.435526\pi$
$570$	0	0
$571$	−498239.	−1.52815	−0.764073	−	0.645129i	$-0.776804\pi$
$571$	−498239.	−1.52815	−0.764073	+	0.645129i	$0.776804\pi$
$572$	− 57164.1i	− 0.174716i
$573$	0	0
$574$	0	0
$575$	−8485.52	−0.0256651
$576$	0	0
$577$	− 33716.2i	− 0.101271i	−0.998717	−	0.0506357i	$-0.983875\pi$
$577$	− 33716.2i	− 0.101271i	0.998717	−	0.0506357i	$-0.0161247\pi$
$578$	−203098.	−0.607924
$579$	0	0
$580$	− 150718.i	− 0.448031i
$581$	0	0
$582$	0	0
$583$	−156808.	−0.461350
$584$	13067.2i	0.0383139i
$585$	0	0
$586$	− 254311.i	− 0.740576i
$587$	356809.i	1.03552i	0.855525	+	0.517762i	$0.173235\pi$
$587$	356809.i	1.03552i	−0.855525	+	0.517762i	$0.826765\pi$
$588$	0	0
$589$	−467296.	−1.34698
$590$	−307679.	−0.883881
$591$	0	0
$592$	63147.2	0.180182
$593$	− 33339.2i	− 0.0948082i	−0.998876	−	0.0474041i	$-0.984905\pi$
$593$	− 33339.2i	− 0.0948082i	0.998876	−	0.0474041i	$-0.0150949\pi$
$594$	0	0
$595$	0	0
$596$	341663.	0.961846
$597$	0	0
$598$	− 200321.i	− 0.560175i
$599$	−324330.	−0.903927	−0.451963	−	0.892036i	$-0.649276\pi$
$599$	−324330.	−0.903927	−0.451963	+	0.892036i	$0.649276\pi$
$600$	0	0
$601$	− 66322.0i	− 0.183615i	−0.995777	−	0.0918076i	$-0.970736\pi$
$601$	− 66322.0i	− 0.183615i	0.995777	−	0.0918076i	$-0.0292645\pi$
$602$	0	0
$603$	0	0
$604$	236860.	0.649258
$605$	− 322600.i	− 0.881361i
$606$	0	0
$607$	487655.i	1.32354i	0.749709	+	0.661768i	$0.230194\pi$
$607$	487655.i	1.32354i	−0.749709	+	0.661768i	$0.769806\pi$
$608$	81380.6i	0.220147i
$609$	0	0
$610$	100287.	0.269517
$611$	218641.	0.585665
$612$	0	0
$613$	−341510.	−0.908831	−0.454415	−	0.890790i	$-0.650152\pi$
$613$	−341510.	−0.908831	−0.454415	+	0.890790i	$0.650152\pi$
$614$	400735.i	1.06297i
$615$	0	0
$616$	0	0
$617$	−240873.	−0.632728	−0.316364	−	0.948638i	$-0.602462\pi$
$617$	−240873.	−0.632728	−0.316364	+	0.948638i	$0.602462\pi$
$618$	0	0
$619$	− 154335.i	− 0.402794i	−0.979510	−	0.201397i	$-0.935452\pi$
$619$	− 154335.i	− 0.402794i	0.979510	−	0.201397i	$-0.0645481\pi$
$620$	211100.	0.549167
$621$	0	0
$622$	− 242242.i	− 0.626137i
$623$	0	0
$624$	0	0
$625$	−402415.	−1.03018
$626$	50435.2i	0.128702i
$627$	0	0
$628$	97896.4i	0.248226i
$629$	− 106794.i	− 0.269927i
$630$	0	0
$631$	220248.	0.553164	0.276582	−	0.960990i	$-0.410798\pi$
$631$	220248.	0.553164	0.276582	+	0.960990i	$0.410798\pi$
$632$	71191.1	0.178234
$633$	0	0
$634$	228552.	0.568599
$635$	101360.i	0.251374i
$636$	0	0
$637$	0	0
$638$	−92295.4	−0.226745
$639$	0	0
$640$	− 36763.5i	− 0.0897545i
$641$	681208.	1.65792	0.828960	−	0.559308i	$-0.188933\pi$
$641$	681208.	1.65792	0.828960	+	0.559308i	$0.188933\pi$
$642$	0	0
$643$	− 572102.i	− 1.38373i	−0.722027	−	0.691865i	$-0.756789\pi$
$643$	− 572102.i	− 1.38373i	0.722027	−	0.691865i	$-0.243211\pi$
$644$	0	0
$645$	0	0
$646$	137630.	0.329799
$647$	607275.i	1.45070i	0.688382	+	0.725348i	$0.258321\pi$
$647$	607275.i	1.45070i	−0.688382	+	0.725348i	$0.741679\pi$
$648$	0	0
$649$	188414.i	0.447326i
$650$	8949.23i	0.0211816i
$651$	0	0
$652$	172249.	0.405193
$653$	484670.	1.13663	0.568316	−	0.822810i	$-0.307595\pi$
$653$	484670.	1.13663	0.568316	+	0.822810i	$0.307595\pi$
$654$	0	0
$655$	513440.	1.19676
$656$	73196.5i	0.170092i
$657$	0	0
$658$	0	0
$659$	−329627.	−0.759017	−0.379509	−	0.925188i	$-0.623907\pi$
$659$	−329627.	−0.759017	−0.379509	+	0.925188i	$0.623907\pi$
$660$	0	0
$661$	− 210608.i	− 0.482028i	−0.970522	−	0.241014i	$-0.922520\pi$
$661$	− 210608.i	− 0.482028i	0.970522	−	0.241014i	$-0.0774800\pi$
$662$	−205922.	−0.469879
$663$	0	0
$664$	− 107027.i	− 0.242748i
$665$	0	0
$666$	0	0
$667$	−323432.	−0.726994
$668$	− 132631.i	− 0.297230i
$669$	0	0
$670$	− 521710.i	− 1.16220i
$671$	− 61413.2i	− 0.136401i
$672$	0	0
$673$	94709.8	0.209105	0.104553	−	0.994519i	$-0.466659\pi$
$673$	94709.8	0.209105	0.104553	+	0.994519i	$0.466659\pi$
$674$	−357902.	−0.787852
$675$	0	0
$676$	17220.2	0.0376829
$677$	538552.i	1.17503i	0.809212	+	0.587517i	$0.199894\pi$
$677$	538552.i	1.17503i	−0.809212	+	0.587517i	$0.800106\pi$
$678$	0	0
$679$	0	0
$680$	−62174.1	−0.134460
$681$	0	0
$682$	− 129272.i	− 0.277930i
$683$	129072.	0.276688	0.138344	−	0.990384i	$-0.455822\pi$
$683$	129072.	0.276688	0.138344	+	0.990384i	$0.455822\pi$
$684$	0	0
$685$	− 636126.i	− 1.35570i
$686$	0	0
$687$	0	0
$688$	154805.	0.327044
$689$	579532.i	1.22079i
$690$	0	0
$691$	− 130713.i	− 0.273756i	−0.990588	−	0.136878i	$-0.956293\pi$
$691$	− 130713.i	− 0.273756i	0.990588	−	0.136878i	$-0.0437068\pi$
$692$	− 323064.i	− 0.674648i
$693$	0	0
$694$	−189361.	−0.393162
$695$	−509507.	−1.05483
$696$	0	0
$697$	123790.	0.254811
$698$	− 72202.6i	− 0.148198i
$699$	0	0
$700$	0	0
$701$	182501.	0.371389	0.185694	−	0.982608i	$-0.440547\pi$
$701$	182501.	0.371389	0.185694	+	0.982608i	$0.440547\pi$
$702$	0	0
$703$	− 443578.i	− 0.897552i
$704$	−22512.9	−0.0454241
$705$	0	0
$706$	− 160020.i	− 0.321044i
$707$	0	0
$708$	0	0
$709$	813798.	1.61892	0.809458	−	0.587178i	$-0.199761\pi$
$709$	813798.	1.61892	0.809458	+	0.587178i	$0.199761\pi$
$710$	429943.i	0.852892i
$711$	0	0
$712$	18959.7i	0.0374000i
$713$	− 453008.i	− 0.891101i
$714$	0	0
$715$	181399.	0.354832
$716$	427955.	0.834780
$717$	0	0
$718$	106928.	0.207416
$719$	− 420567.i	− 0.813537i	−0.913531	−	0.406769i	$-0.866656\pi$
$719$	− 420567.i	− 0.813537i	0.913531	−	0.406769i	$-0.133344\pi$
$720$	0	0
$721$	0	0
$722$	203055.	0.389528
$723$	0	0
$724$	433616.i	0.827235i
$725$	14449.1	0.0274894
$726$	0	0
$727$	− 172948.i	− 0.327225i	−0.986525	−	0.163613i	$-0.947685\pi$
$727$	− 172948.i	− 0.327225i	0.986525	−	0.163613i	$-0.0523147\pi$
$728$	0	0
$729$	0	0
$730$	−41466.1	−0.0778121
$731$	− 261805.i	− 0.489939i
$732$	0	0
$733$	− 131562.i	− 0.244862i	−0.992477	−	0.122431i	$-0.960931\pi$
$733$	− 131562.i	− 0.244862i	0.992477	−	0.122431i	$-0.0390690\pi$
$734$	− 30593.4i	− 0.0567852i
$735$	0	0
$736$	−78892.3	−0.145639
$737$	−319481.	−0.588179
$738$	0	0
$739$	−489768.	−0.896812	−0.448406	−	0.893830i	$-0.648008\pi$
$739$	−489768.	−0.896812	−0.448406	+	0.893830i	$0.648008\pi$
$740$	200385.i	0.365934i
$741$	0	0
$742$	0	0
$743$	−580258.	−1.05110	−0.525549	−	0.850763i	$-0.676140\pi$
$743$	−580258.	−1.05110	−0.525549	+	0.850763i	$0.676140\pi$
$744$	0	0
$745$	1.08420e6i	1.95343i
$746$	87747.4	0.157673
$747$	0	0
$748$	38073.7i	0.0680491i
$749$	0	0
$750$	0	0
$751$	437397.	0.775525	0.387763	−	0.921759i	$-0.373248\pi$
$751$	437397.	0.775525	0.387763	+	0.921759i	$0.373248\pi$
$752$	− 86107.4i	− 0.152267i
$753$	0	0
$754$	341106.i	0.599994i
$755$	751627.i	1.31859i
$756$	0	0
$757$	−24171.7	−0.0421809	−0.0210905	−	0.999778i	$-0.506714\pi$
$757$	−24171.7	−0.0421809	−0.0210905	+	0.999778i	$0.506714\pi$
$758$	278678.	0.485025
$759$	0	0
$760$	−258245.	−0.447100
$761$	− 648221.i	− 1.11932i	−0.828723	−	0.559659i	$-0.810932\pi$
$761$	− 648221.i	− 1.11932i	0.828723	−	0.559659i	$-0.189068\pi$
$762$	0	0
$763$	0	0
$764$	9630.98	0.0165000
$765$	0	0
$766$	14547.5i	0.0247932i
$767$	696343.	1.18368
$768$	0	0
$769$	179564.i	0.303646i	0.988408	+	0.151823i	$0.0485143\pi$
$769$	179564.i	0.303646i	−0.988408	+	0.151823i	$0.951486\pi$
$770$	0	0
$771$	0	0
$772$	410503.	0.688783
$773$	− 610915.i	− 1.02240i	−0.859461	−	0.511201i	$-0.829201\pi$
$773$	− 610915.i	− 1.02240i	0.859461	−	0.511201i	$-0.170799\pi$
$774$	0	0
$775$	20237.9i	0.0336947i
$776$	− 127216.i	− 0.211261i
$777$	0	0
$778$	244042.	0.403186
$779$	514169.	0.847288
$780$	0	0
$781$	263285.	0.431643
$782$	133422.i	0.218180i
$783$	0	0
$784$	0	0
$785$	−310655.	−0.504126
$786$	0	0
$787$	1.19155e6i	1.92381i	0.273384	+	0.961905i	$0.411857\pi$
$787$	1.19155e6i	1.92381i	−0.273384	+	0.961905i	$0.588143\pi$
$788$	11650.2	0.0187621
$789$	0	0
$790$	225911.i	0.361979i
$791$	0	0
$792$	0	0
$793$	−226972.	−0.360932
$794$	− 441026.i	− 0.699557i
$795$	0	0
$796$	550066.i	0.868137i
$797$	− 900495.i	− 1.41764i	−0.705392	−	0.708818i	$-0.749229\pi$
$797$	− 900495.i	− 1.41764i	0.705392	−	0.708818i	$-0.250771\pi$
$798$	0	0
$799$	−145624.	−0.228108
$800$	3524.47	0.00550698
$801$	0	0
$802$	97904.5	0.152214
$803$	25392.7i	0.0393802i
$804$	0	0
$805$	0	0
$806$	−477764.	−0.735433
$807$	0	0
$808$	− 149013.i	− 0.228245i
$809$	−555764.	−0.849168	−0.424584	−	0.905389i	$-0.639580\pi$
$809$	−555764.	−0.849168	−0.424584	+	0.905389i	$0.639580\pi$
$810$	0	0
$811$	− 70954.2i	− 0.107879i	−0.998544	−	0.0539395i	$-0.982822\pi$
$811$	− 70954.2i	− 0.107879i	0.998544	−	0.0539395i	$-0.0171778\pi$
$812$	0	0
$813$	0	0
$814$	122710.	0.185196
$815$	546598.i	0.822911i
$816$	0	0
$817$	− 1.08743e6i	− 1.62913i
$818$	887527.i	1.32640i
$819$	0	0
$820$	−232275.	−0.345441
$821$	624712.	0.926816	0.463408	−	0.886145i	$-0.346626\pi$
$821$	624712.	0.926816	0.463408	+	0.886145i	$0.346626\pi$
$822$	0	0
$823$	−210059.	−0.310128	−0.155064	−	0.987904i	$-0.549558\pi$
$823$	−210059.	−0.310128	−0.155064	+	0.987904i	$0.549558\pi$
$824$	34495.4i	0.0508051i
$825$	0	0
$826$	0	0
$827$	880910.	1.28801	0.644007	−	0.765020i	$-0.277271\pi$
$827$	880910.	1.28801	0.644007	+	0.765020i	$0.277271\pi$
$828$	0	0
$829$	− 1.29270e6i	− 1.88100i	−0.339799	−	0.940498i	$-0.610359\pi$
$829$	− 1.29270e6i	− 1.88100i	0.339799	−	0.940498i	$-0.389641\pi$
$830$	339628.	0.493001
$831$	0	0
$832$	83203.5i	0.120197i
$833$	0	0
$834$	0	0
$835$	420879.	0.603649
$836$	158142.i	0.226274i
$837$	0	0
$838$	− 20717.5i	− 0.0295018i
$839$	1.11586e6i	1.58520i	0.609741	+	0.792601i	$0.291274\pi$
$839$	1.11586e6i	1.58520i	−0.609741	+	0.792601i	$0.708726\pi$
$840$	0	0
$841$	−156542.	−0.221330
$842$	71215.1	0.100449
$843$	0	0
$844$	−4982.57	−0.00699469
$845$	54644.8i	0.0765307i
$846$	0	0
$847$	0	0
$848$	228237.	0.317391
$849$	0	0
$850$	− 5960.56i	− 0.00824991i
$851$	430015.	0.593779
$852$	0	0
$853$	1.28959e6i	1.77236i	0.463341	+	0.886180i	$0.346650\pi$
$853$	1.28959e6i	1.77236i	−0.463341	+	0.886180i	$0.653350\pi$
$854$	0	0
$855$	0	0
$856$	240001.	0.327540
$857$	654547.i	0.891208i	0.895230	+	0.445604i	$0.147011\pi$
$857$	654547.i	0.891208i	−0.895230	+	0.445604i	$0.852989\pi$
$858$	0	0
$859$	− 244561.i	− 0.331437i	−0.986173	−	0.165718i	$-0.947006\pi$
$859$	− 244561.i	− 0.331437i	0.986173	−	0.165718i	$-0.0529943\pi$
$860$	491242.i	0.664199i
$861$	0	0
$862$	39109.2	0.0526337
$863$	465277.	0.624726	0.312363	−	0.949963i	$-0.398879\pi$
$863$	465277.	0.624726	0.312363	+	0.949963i	$0.398879\pi$
$864$	0	0
$865$	1.02518e6	1.37015
$866$	134177.i	0.178913i
$867$	0	0
$868$	0	0
$869$	138342.	0.183195
$870$	0	0
$871$	1.18074e6i	1.55639i
$872$	315704.	0.415190
$873$	0	0
$874$	554179.i	0.725484i
$875$	0	0
$876$	0	0
$877$	304172.	0.395475	0.197738	−	0.980255i	$-0.436641\pi$
$877$	304172.	0.395475	0.197738	+	0.980255i	$0.436641\pi$
$878$	310601.i	0.402915i
$879$	0	0
$880$	− 71440.3i	− 0.0922525i
$881$	− 697876.i	− 0.899138i	−0.893246	−	0.449569i	$-0.851578\pi$
$881$	− 697876.i	− 0.899138i	0.893246	−	0.449569i	$-0.148422\pi$
$882$	0	0
$883$	891773.	1.14375	0.571877	−	0.820339i	$-0.306215\pi$
$883$	891773.	1.14375	0.571877	+	0.820339i	$0.306215\pi$
$884$	140713.	0.180066
$885$	0	0
$886$	−233517.	−0.297475
$887$	− 1.26256e6i	− 1.60473i	−0.596831	−	0.802367i	$-0.703574\pi$
$887$	− 1.26256e6i	− 1.60473i	0.596831	−	0.802367i	$-0.296426\pi$
$888$	0	0
$889$	0	0
$890$	−60164.9	−0.0759562
$891$	0	0
$892$	92077.9i	0.115725i
$893$	−604862.	−0.758496
$894$	0	0
$895$	1.35803e6i	1.69537i
$896$	0	0
$897$	0	0
$898$	−934619.	−1.15900
$899$	771382.i	0.954443i
$900$	0	0
$901$	− 385993.i	− 0.475477i
$902$	142239.i	0.174825i
$903$	0	0
$904$	−199377.	−0.243971
$905$	−1.37600e6	−1.68004
$906$	0	0
$907$	951981.	1.15721	0.578607	−	0.815607i	$-0.303597\pi$
$907$	951981.	1.15721	0.578607	+	0.815607i	$0.303597\pi$
$908$	450938.i	0.546946i
$909$	0	0
$910$	0	0
$911$	743243.	0.895559	0.447779	−	0.894144i	$-0.352215\pi$
$911$	743243.	0.895559	0.447779	+	0.894144i	$0.352215\pi$
$912$	0	0
$913$	− 207979.i	− 0.249504i
$914$	1.05582e6	1.26385
$915$	0	0
$916$	337028.i	0.401675i
$917$	0	0
$918$	0	0
$919$	−524300.	−0.620796	−0.310398	−	0.950607i	$-0.600462\pi$
$919$	−524300.	−0.620796	−0.310398	+	0.950607i	$0.600462\pi$
$920$	− 250349.i	− 0.295781i
$921$	0	0
$922$	− 57573.3i	− 0.0677266i
$923$	− 973052.i	− 1.14218i
$924$	0	0
$925$	−19210.7	−0.0224522
$926$	89244.6	0.104078
$927$	0	0
$928$	134338.	0.155992
$929$	979392.i	1.13482i	0.823437	+	0.567408i	$0.192054\pi$
$929$	979392.i	1.13482i	−0.823437	+	0.567408i	$0.807946\pi$
$930$	0	0
$931$	0	0
$932$	440815.	0.507486
$933$	0	0
$934$	684068.i	0.784161i
$935$	−120819.	−0.138202
$936$	0	0
$937$	− 1.13988e6i	− 1.29831i	−0.760656	−	0.649155i	$-0.775123\pi$
$937$	− 1.13988e6i	− 1.29831i	0.760656	−	0.649155i	$-0.224877\pi$
$938$	0	0
$939$	0	0
$940$	273245.	0.309240
$941$	− 801121.i	− 0.904730i	−0.891833	−	0.452365i	$-0.850580\pi$
$941$	− 801121.i	− 0.904730i	0.891833	−	0.452365i	$-0.149420\pi$
$942$	0	0
$943$	498448.i	0.560527i
$944$	− 274240.i	− 0.307743i
$945$	0	0
$946$	300823.	0.336146
$947$	−680286.	−0.758563	−0.379282	−	0.925281i	$-0.623829\pi$
$947$	−680286.	−0.758563	−0.379282	+	0.925281i	$0.623829\pi$
$948$	0	0
$949$	93846.5	0.104204
$950$	− 24757.7i	− 0.0274323i
$951$	0	0
$952$	0	0
$953$	394774.	0.434673	0.217337	−	0.976097i	$-0.430263\pi$
$953$	394774.	0.434673	0.217337	+	0.976097i	$0.430263\pi$
$954$	0	0
$955$	30562.0i	0.0335101i
$956$	−184660.	−0.202049
$957$	0	0
$958$	769567.i	0.838524i
$959$	0	0
$960$	0	0
$961$	−156900.	−0.169893
$962$	− 453514.i	− 0.490050i
$963$	0	0
$964$	652580.i	0.702231i
$965$	1.30265e6i	1.39886i
$966$	0	0
$967$	−604328.	−0.646279	−0.323140	−	0.946351i	$-0.604738\pi$
$967$	−604328.	−0.646279	−0.323140	+	0.946351i	$0.604738\pi$
$968$	287540.	0.306865
$969$	0	0
$970$	403696.	0.429053
$971$	758986.i	0.804999i	0.915420	+	0.402499i	$0.131858\pi$
$971$	758986.i	0.804999i	−0.915420	+	0.402499i	$0.868142\pi$
$972$	0	0
$973$	0	0
$974$	1.15328e6	1.21567
$975$	0	0
$976$	89388.1i	0.0938383i
$977$	−1.22518e6	−1.28355	−0.641773	−	0.766894i	$-0.721801\pi$
$977$	−1.22518e6	−1.28355	−0.641773	+	0.766894i	$0.721801\pi$
$978$	0	0
$979$	36843.3i	0.0384409i
$980$	0	0
$981$	0	0
$982$	−811115.	−0.841123
$983$	1.20047e6i	1.24235i	0.783670	+	0.621177i	$0.213345\pi$
$983$	1.20047e6i	1.24235i	−0.783670	+	0.621177i	$0.786655\pi$
$984$	0	0
$985$	36969.6i	0.0381042i
$986$	− 227191.i	− 0.233689i
$987$	0	0
$988$	584464.	0.598747
$989$	1.05418e6	1.07776
$990$	0	0
$991$	1.80993e6	1.84295	0.921475	−	0.388438i	$-0.126985\pi$
$991$	1.80993e6	1.84295	0.921475	+	0.388438i	$0.126985\pi$
$992$	188157.i	0.191204i
$993$	0	0
$994$	0	0
$995$	−1.74552e6	−1.76311
$996$	0	0
$997$	749289.i	0.753805i	0.926253	+	0.376902i	$0.123011\pi$
$997$	749289.i	0.753805i	−0.926253	+	0.376902i	$0.876989\pi$
$998$	−994009.	−0.997997
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.c.b.685.2		4
3.2	odd	2		98.5.b.b.97.3		4
7.4	even	3		126.5.n.a.19.2		4
7.5	odd	6		126.5.n.a.73.2		4
7.6	odd	2	inner	882.5.c.b.685.1		4
12.11	even	2		784.5.c.b.97.4		4
21.2	odd	6		98.5.d.a.31.1		4
21.5	even	6		14.5.d.a.3.1	✓	4
21.11	odd	6		14.5.d.a.5.1	yes	4
21.17	even	6		98.5.d.a.19.1		4
21.20	even	2		98.5.b.b.97.4		4
84.11	even	6		112.5.s.b.33.2		4
84.47	odd	6		112.5.s.b.17.2		4
84.83	odd	2		784.5.c.b.97.1		4
105.32	even	12		350.5.i.a.299.2		8
105.47	odd	12		350.5.i.a.199.3		8
105.53	even	12		350.5.i.a.299.3		8
105.68	odd	12		350.5.i.a.199.2		8
105.74	odd	6		350.5.k.a.201.2		4
105.89	even	6		350.5.k.a.101.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.5.d.a.3.1	✓	4	21.5	even	6
14.5.d.a.5.1	yes	4	21.11	odd	6
98.5.b.b.97.3		4	3.2	odd	2
98.5.b.b.97.4		4	21.20	even	2
98.5.d.a.19.1		4	21.17	even	6
98.5.d.a.31.1		4	21.2	odd	6
112.5.s.b.17.2		4	84.47	odd	6
112.5.s.b.33.2		4	84.11	even	6
126.5.n.a.19.2		4	7.4	even	3
126.5.n.a.73.2		4	7.5	odd	6
350.5.i.a.199.2		8	105.68	odd	12
350.5.i.a.199.3		8	105.47	odd	12
350.5.i.a.299.2		8	105.32	even	12
350.5.i.a.299.3		8	105.53	even	12
350.5.k.a.101.2		4	105.89	even	6
350.5.k.a.201.2		4	105.74	odd	6
784.5.c.b.97.1		4	84.83	odd	2
784.5.c.b.97.4		4	12.11	even	2
882.5.c.b.685.1		4	7.6	odd	2	inner
882.5.c.b.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)

\(f(q)\)	\(=\)	\(q-2.82843 q^{2} +8.00000 q^{4} +25.3864i q^{5} -22.6274 q^{8} +O(q^{10})\)	\(q-2.82843 q^{2} +8.00000 q^{4} +25.3864i q^{5} -22.6274 q^{8} -71.8036i q^{10} -43.9706 q^{11} +162.507i q^{13} +64.0000 q^{16} -108.236i q^{17} -449.568i q^{19} +203.091i q^{20} +124.368 q^{22} +435.823 q^{23} -19.4701 q^{25} -459.639i q^{26} -742.118 q^{29} -1039.43i q^{31} -181.019 q^{32} +306.139i q^{34} +986.675 q^{37} +1271.57i q^{38} -574.429i q^{40} +1143.70i q^{41} +2418.82 q^{43} -351.765 q^{44} -1232.69 q^{46} -1345.43i q^{47} +55.0698 q^{50} +1300.05i q^{52} +3566.20 q^{53} -1116.26i q^{55} +2099.03 q^{58} -4285.01i q^{59} +1396.69i q^{61} +2939.96i q^{62} +512.000 q^{64} -4125.47 q^{65} +7265.79 q^{67} -865.891i q^{68} -5987.76 q^{71} -577.493i q^{73} -2790.74 q^{74} -3596.55i q^{76} -3146.23 q^{79} +1624.73i q^{80} -3234.86i q^{82} +4729.96i q^{83} +2747.74 q^{85} -6841.46 q^{86} +994.940 q^{88} -837.909i q^{89} +3486.58 q^{92} +3805.44i q^{94} +11412.9 q^{95} +5622.23i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{4}+O(q^{10}) \) 4 * q + 32 * q^4	\( 4 q + 32 q^{4} - 108 q^{11} + 256 q^{16} + 192 q^{22} - 972 q^{23} + 1144 q^{25} - 3240 q^{29} + 892 q^{37} + 2344 q^{43} - 864 q^{44} - 7680 q^{46} + 3456 q^{50} + 5508 q^{53} - 768 q^{58} + 2048 q^{64} - 6048 q^{65} + 10124 q^{67} - 18792 q^{71} - 8640 q^{74} - 1588 q^{79} + 10380 q^{85} - 20736 q^{86} + 1536 q^{88} - 7776 q^{92} + 17820 q^{95}+O(q^{100}) \) 4 * q + 32 * q^4 - 108 * q^11 + 256 * q^16 + 192 * q^22 - 972 * q^23 + 1144 * q^25 - 3240 * q^29 + 892 * q^37 + 2344 * q^43 - 864 * q^44 - 7680 * q^46 + 3456 * q^50 + 5508 * q^53 - 768 * q^58 + 2048 * q^64 - 6048 * q^65 + 10124 * q^67 - 18792 * q^71 - 8640 * q^74 - 1588 * q^79 + 10380 * q^85 - 20736 * q^86 + 1536 * q^88 - 7776 * q^92 + 17820 * q^95

Introduction

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