LMFDB - Embedded newform 882.5.b.b.197.3

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:	$\Q(\sqrt{-2}, \sqrt{-29})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 28x^{2} + 225 $ x^4 - 28*x^2 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.3
Root			$-3.80789 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.b.197.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} -21.5407i q^{5} -22.6274i q^{8} +O(q^{10})$	$q+2.82843i q^{2} -8.00000 q^{4} -21.5407i q^{5} -22.6274i q^{8} +60.9262 q^{10} +57.9828i q^{11} +304.631 q^{13} +64.0000 q^{16} +323.110i q^{17} -152.315 q^{19} +172.325i q^{20} -164.000 q^{22} -538.815i q^{23} +161.000 q^{25} +861.626i q^{26} +83.4386i q^{29} -761.577 q^{31} +181.019i q^{32} -913.893 q^{34} -600.000 q^{37} -430.813i q^{38} -487.409 q^{40} -1615.55i q^{41} -1160.00 q^{43} -463.862i q^{44} +1524.00 q^{46} +1077.03i q^{47} +455.377i q^{50} -2437.05 q^{52} -1104.50i q^{53} +1248.99 q^{55} -236.000 q^{58} +3661.91i q^{59} -2589.36 q^{61} -2154.07i q^{62} -512.000 q^{64} -6561.95i q^{65} +8278.00 q^{67} -2584.88i q^{68} +1839.89i q^{71} +9291.24 q^{73} -1697.06i q^{74} +1218.52 q^{76} +3082.00 q^{79} -1378.60i q^{80} +4569.46 q^{82} +13570.6i q^{83} +6960.00 q^{85} -3280.98i q^{86} +1312.00 q^{88} -2046.36i q^{89} +4310.52i q^{92} -3046.31 q^{94} +3280.98i q^{95} -761.577 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4}+O(q^{10}) $ 4 * q - 32 * q^4	$ 4 q - 32 q^{4} + 256 q^{16} - 656 q^{22} + 644 q^{25} - 2400 q^{37} - 4640 q^{43} + 6096 q^{46} - 944 q^{58} - 2048 q^{64} + 33112 q^{67} + 12328 q^{79} + 27840 q^{85} + 5248 q^{88}+O(q^{100}) $ 4 * q - 32 * q^4 + 256 * q^16 - 656 * q^22 + 644 * q^25 - 2400 * q^37 - 4640 * q^43 + 6096 * q^46 - 944 * q^58 - 2048 * q^64 + 33112 * q^67 + 12328 * q^79 + 27840 * q^85 + 5248 * q^88

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.82843i	0.707107i
$2$	2.82843i	0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	− 21.5407i	− 0.861626i	−0.902441	−	0.430813i	$-0.858227\pi$
$5$	− 21.5407i	− 0.861626i	0.902441	−	0.430813i	$-0.141773\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	60.9262	0.609262
$11$	57.9828i	0.479196i	0.970872	+	0.239598i	$0.0770157\pi$
$11$	57.9828i	0.479196i	−0.970872	+	0.239598i	$0.922984\pi$
$12$	0	0
$13$	304.631	1.80255	0.901275	−	0.433248i	$-0.142632\pi$
$13$	304.631	1.80255	0.901275	+	0.433248i	$0.142632\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	323.110i	1.11803i	0.829158	+	0.559014i	$0.188820\pi$
$17$	323.110i	1.11803i	−0.829158	+	0.559014i	$0.811180\pi$
$18$	0	0
$19$	−152.315	−0.421926	−0.210963	−	0.977494i	$-0.567660\pi$
$19$	−152.315	−0.421926	−0.210963	+	0.977494i	$0.567660\pi$
$20$	172.325i	0.430813i
$21$	0	0
$22$	−164.000	−0.338843
$23$	− 538.815i	− 1.01855i	−0.860602	−	0.509277i	$-0.829913\pi$
$23$	− 538.815i	− 1.01855i	0.860602	−	0.509277i	$-0.170087\pi$
$24$	0	0
$25$	161.000	0.257600
$26$	861.626i	1.27460i
$27$	0	0
$28$	0	0
$29$	83.4386i	0.0992136i	0.998769	+	0.0496068i	$0.0157968\pi$
$29$	83.4386i	0.0992136i	−0.998769	+	0.0496068i	$0.984203\pi$
$30$	0	0
$31$	−761.577	−0.792484	−0.396242	−	0.918146i	$-0.629686\pi$
$31$	−761.577	−0.792484	−0.396242	+	0.918146i	$0.629686\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	−913.893	−0.790565
$35$	0	0
$36$	0	0
$37$	−600.000	−0.438276	−0.219138	−	0.975694i	$-0.570325\pi$
$37$	−600.000	−0.438276	−0.219138	+	0.975694i	$0.570325\pi$
$38$	− 430.813i	− 0.298347i
$39$	0	0
$40$	−487.409	−0.304631
$41$	− 1615.55i	− 0.961065i	−0.876977	−	0.480532i	$-0.840443\pi$
$41$	− 1615.55i	− 0.961065i	0.876977	−	0.480532i	$-0.159557\pi$
$42$	0	0
$43$	−1160.00	−0.627366	−0.313683	−	0.949528i	$-0.601563\pi$
$43$	−1160.00	−0.627366	−0.313683	+	0.949528i	$0.601563\pi$
$44$	− 463.862i	− 0.239598i
$45$	0	0
$46$	1524.00	0.720227
$47$	1077.03i	0.487566i	0.969830	+	0.243783i	$0.0783884\pi$
$47$	1077.03i	0.487566i	−0.969830	+	0.243783i	$0.921612\pi$
$48$	0	0
$49$	0	0
$50$	455.377i	0.182151i
$51$	0	0
$52$	−2437.05	−0.901275
$53$	− 1104.50i	− 0.393201i	−0.980484	−	0.196600i	$-0.937010\pi$
$53$	− 1104.50i	− 0.393201i	0.980484	−	0.196600i	$-0.0629902\pi$
$54$	0	0
$55$	1248.99	0.412888
$56$	0	0
$57$	0	0
$58$	−236.000	−0.0701546
$59$	3661.91i	1.05197i	0.850493	+	0.525986i	$0.176303\pi$
$59$	3661.91i	1.05197i	−0.850493	+	0.525986i	$0.823697\pi$
$60$	0	0
$61$	−2589.36	−0.695878	−0.347939	−	0.937517i	$-0.613118\pi$
$61$	−2589.36	−0.695878	−0.347939	+	0.937517i	$0.613118\pi$
$62$	− 2154.07i	− 0.560371i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 6561.95i	− 1.55312i
$66$	0	0
$67$	8278.00	1.84406	0.922032	−	0.387115i	$-0.126528\pi$
$67$	8278.00	1.84406	0.922032	+	0.387115i	$0.126528\pi$
$68$	− 2584.88i	− 0.559014i
$69$	0	0
$70$	0	0
$71$	1839.89i	0.364985i	0.983207	+	0.182493i	$0.0584166\pi$
$71$	1839.89i	0.364985i	−0.983207	+	0.182493i	$0.941583\pi$
$72$	0	0
$73$	9291.24	1.74352	0.871762	−	0.489929i	$-0.162977\pi$
$73$	9291.24	1.74352	0.871762	+	0.489929i	$0.162977\pi$
$74$	− 1697.06i	− 0.309908i
$75$	0	0
$76$	1218.52	0.210963
$77$	0	0
$78$	0	0
$79$	3082.00	0.493831	0.246916	−	0.969037i	$-0.420583\pi$
$79$	3082.00	0.493831	0.246916	+	0.969037i	$0.420583\pi$
$80$	− 1378.60i	− 0.215407i
$81$	0	0
$82$	4569.46	0.679575
$83$	13570.6i	1.96990i	0.172850	+	0.984948i	$0.444702\pi$
$83$	13570.6i	1.96990i	−0.172850	+	0.984948i	$0.555298\pi$
$84$	0	0
$85$	6960.00	0.963322
$86$	− 3280.98i	− 0.443615i
$87$	0	0
$88$	1312.00	0.169421
$89$	− 2046.36i	− 0.258347i	−0.991622	−	0.129173i	$-0.958768\pi$
$89$	− 2046.36i	− 0.258347i	0.991622	−	0.129173i	$-0.0412323\pi$
$90$	0	0
$91$	0	0
$92$	4310.52i	0.509277i
$93$	0	0
$94$	−3046.31	−0.344761
$95$	3280.98i	0.363543i
$96$	0	0
$97$	−761.577	−0.0809414	−0.0404707	−	0.999181i	$-0.512886\pi$
$97$	−761.577	−0.0809414	−0.0404707	+	0.999181i	$0.512886\pi$
$98$	0	0
$99$	0	0
$100$	−1288.00	−0.128800
$101$	− 12816.7i	− 1.25642i	−0.778046	−	0.628208i	$-0.783789\pi$
$101$	− 12816.7i	− 1.25642i	0.778046	−	0.628208i	$-0.216211\pi$
$102$	0	0
$103$	5331.04	0.502502	0.251251	−	0.967922i	$-0.419158\pi$
$103$	5331.04	0.502502	0.251251	+	0.967922i	$0.419158\pi$
$104$	− 6893.01i	− 0.637298i
$105$	0	0
$106$	3124.00	0.278035
$107$	− 15611.5i	− 1.36357i	−0.731553	−	0.681785i	$-0.761204\pi$
$107$	− 15611.5i	− 1.36357i	0.731553	−	0.681785i	$-0.238796\pi$
$108$	0	0
$109$	17080.0	1.43759	0.718795	−	0.695222i	$-0.244694\pi$
$109$	17080.0	1.43759	0.718795	+	0.695222i	$0.244694\pi$
$110$	3532.67i	0.291956i
$111$	0	0
$112$	0	0
$113$	18949.0i	1.48399i	0.670406	+	0.741994i	$0.266120\pi$
$113$	18949.0i	1.48399i	−0.670406	+	0.741994i	$0.733880\pi$
$114$	0	0
$115$	−11606.4	−0.877613
$116$	− 667.509i	− 0.0496068i
$117$	0	0
$118$	−10357.5	−0.743856
$119$	0	0
$120$	0	0
$121$	11279.0	0.770371
$122$	− 7323.82i	− 0.492060i
$123$	0	0
$124$	6092.62	0.396242
$125$	− 16931.0i	− 1.08358i
$126$	0	0
$127$	19958.0	1.23740	0.618699	−	0.785628i	$-0.287660\pi$
$127$	19958.0	1.23740	0.618699	+	0.785628i	$0.287660\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	18560.0	1.09822
$131$	− 21971.5i	− 1.28031i	−0.768244	−	0.640157i	$-0.778869\pi$
$131$	− 21971.5i	− 1.28031i	0.768244	−	0.640157i	$-0.221131\pi$
$132$	0	0
$133$	0	0
$134$	23413.7i	1.30395i
$135$	0	0
$136$	7311.14	0.395282
$137$	11368.9i	0.605726i	0.953034	+	0.302863i	$0.0979424\pi$
$137$	11368.9i	0.605726i	−0.953034	+	0.302863i	$0.902058\pi$
$138$	0	0
$139$	3655.57	0.189202	0.0946010	−	0.995515i	$-0.469842\pi$
$139$	3655.57	0.189202	0.0946010	+	0.995515i	$0.469842\pi$
$140$	0	0
$141$	0	0
$142$	−5204.00	−0.258084
$143$	17663.3i	0.863775i
$144$	0	0
$145$	1797.32	0.0854850
$146$	26279.6i	1.23286i
$147$	0	0
$148$	4800.00	0.219138
$149$	− 13773.0i	− 0.620379i	−0.950675	−	0.310189i	$-0.899608\pi$
$149$	− 13773.0i	− 0.620379i	0.950675	−	0.310189i	$-0.100392\pi$
$150$	0	0
$151$	29200.0	1.28065	0.640323	−	0.768106i	$-0.278801\pi$
$151$	29200.0	1.28065	0.640323	+	0.768106i	$0.278801\pi$
$152$	3446.51i	0.149174i
$153$	0	0
$154$	0	0
$155$	16404.9i	0.682825i
$156$	0	0
$157$	22695.0	0.920727	0.460364	−	0.887730i	$-0.347719\pi$
$157$	22695.0	0.920727	0.460364	+	0.887730i	$0.347719\pi$
$158$	8717.21i	0.349191i
$159$	0	0
$160$	3899.28	0.152315
$161$	0	0
$162$	0	0
$163$	−23562.0	−0.886823	−0.443411	−	0.896318i	$-0.646232\pi$
$163$	−23562.0	−0.886823	−0.443411	+	0.896318i	$0.646232\pi$
$164$	12924.4i	0.480532i
$165$	0	0
$166$	−38383.5	−1.39293
$167$	− 6246.79i	− 0.223988i	−0.993709	−	0.111994i	$-0.964276\pi$
$167$	− 6246.79i	− 0.223988i	0.993709	−	0.111994i	$-0.0357237\pi$
$168$	0	0
$169$	64239.0	2.24919
$170$	19685.9i	0.681171i
$171$	0	0
$172$	9280.00	0.313683
$173$	42542.8i	1.42146i	0.703466	+	0.710729i	$0.251635\pi$
$173$	42542.8i	1.42146i	−0.703466	+	0.710729i	$0.748365\pi$
$174$	0	0
$175$	0	0
$176$	3710.90i	0.119799i
$177$	0	0
$178$	5787.99	0.182679
$179$	46613.9i	1.45482i	0.686203	+	0.727410i	$0.259276\pi$
$179$	46613.9i	1.45482i	−0.686203	+	0.727410i	$0.740724\pi$
$180$	0	0
$181$	6092.62	0.185972	0.0929858	−	0.995667i	$-0.470359\pi$
$181$	6092.62	0.185972	0.0929858	+	0.995667i	$0.470359\pi$
$182$	0	0
$183$	0	0
$184$	−12192.0	−0.360113
$185$	12924.4i	0.377630i
$186$	0	0
$187$	−18734.8	−0.535755
$188$	− 8616.26i	− 0.243783i
$189$	0	0
$190$	−9280.00	−0.257064
$191$	− 45001.7i	− 1.23357i	−0.787134	−	0.616783i	$-0.788436\pi$
$191$	− 45001.7i	− 1.23357i	0.787134	−	0.616783i	$-0.211564\pi$
$192$	0	0
$193$	54080.0	1.45185	0.725926	−	0.687773i	$-0.241412\pi$
$193$	54080.0	1.45185	0.725926	+	0.687773i	$0.241412\pi$
$194$	− 2154.07i	− 0.0572342i
$195$	0	0
$196$	0	0
$197$	− 16940.9i	− 0.436519i	−0.975891	−	0.218259i	$-0.929962\pi$
$197$	− 16940.9i	− 0.436519i	0.975891	−	0.218259i	$-0.0700379\pi$
$198$	0	0
$199$	−58793.8	−1.48465	−0.742327	−	0.670038i	$-0.766278\pi$
$199$	−58793.8	−1.48465	−0.742327	+	0.670038i	$0.766278\pi$
$200$	− 3643.01i	− 0.0910754i
$201$	0	0
$202$	36251.1	0.888420
$203$	0	0
$204$	0	0
$205$	−34800.0	−0.828079
$206$	15078.5i	0.355322i
$207$	0	0
$208$	19496.4	0.450637
$209$	− 8831.67i	− 0.202186i
$210$	0	0
$211$	65480.0	1.47077	0.735383	−	0.677651i	$-0.237002\pi$
$211$	65480.0	1.47077	0.735383	+	0.677651i	$0.237002\pi$
$212$	8836.01i	0.196600i
$213$	0	0
$214$	44156.0	0.964189
$215$	24987.2i	0.540555i
$216$	0	0
$217$	0	0
$218$	48309.5i	1.01653i
$219$	0	0
$220$	−9991.89	−0.206444
$221$	98429.3i	2.01530i
$222$	0	0
$223$	−49654.8	−0.998509	−0.499254	−	0.866455i	$-0.666393\pi$
$223$	−49654.8	−0.998509	−0.499254	+	0.866455i	$0.666393\pi$
$224$	0	0
$225$	0	0
$226$	−53596.0	−1.04934
$227$	− 68283.9i	− 1.32515i	−0.748994	−	0.662577i	$-0.769463\pi$
$227$	− 68283.9i	− 1.32515i	0.748994	−	0.662577i	$-0.230537\pi$
$228$	0	0
$229$	49654.8	0.946871	0.473435	−	0.880829i	$-0.343014\pi$
$229$	49654.8	0.946871	0.473435	+	0.880829i	$0.343014\pi$
$230$	− 32828.0i	− 0.620566i
$231$	0	0
$232$	1888.00	0.0350773
$233$	5488.56i	0.101099i	0.998722	+	0.0505495i	$0.0160973\pi$
$233$	5488.56i	0.101099i	−0.998722	+	0.0505495i	$0.983903\pi$
$234$	0	0
$235$	23200.0	0.420100
$236$	− 29295.3i	− 0.525986i
$237$	0	0
$238$	0	0
$239$	− 38323.8i	− 0.670923i	−0.942054	−	0.335461i	$-0.891108\pi$
$239$	− 38323.8i	− 0.670923i	0.942054	−	0.335461i	$-0.108892\pi$
$240$	0	0
$241$	−53767.4	−0.925731	−0.462865	−	0.886429i	$-0.653179\pi$
$241$	−53767.4	−0.925731	−0.462865	+	0.886429i	$0.653179\pi$
$242$	31901.8i	0.544734i
$243$	0	0
$244$	20714.9	0.347939
$245$	0	0
$246$	0	0
$247$	−46400.0	−0.760544
$248$	17232.5i	0.280185i
$249$	0	0
$250$	47888.0	0.766208
$251$	18525.0i	0.294042i	0.989133	+	0.147021i	$0.0469686\pi$
$251$	18525.0i	0.294042i	−0.989133	+	0.147021i	$0.953031\pi$
$252$	0	0
$253$	31242.0	0.488088
$254$	56449.7i	0.874973i
$255$	0	0
$256$	4096.00	0.0625000
$257$	10231.8i	0.154912i	0.996996	+	0.0774562i	$0.0246798\pi$
$257$	10231.8i	0.154912i	−0.996996	+	0.0774562i	$0.975320\pi$
$258$	0	0
$259$	0	0
$260$	52495.6i	0.776562i
$261$	0	0
$262$	62144.7	0.905319
$263$	115823.i	1.67449i	0.546829	+	0.837244i	$0.315835\pi$
$263$	115823.i	1.67449i	−0.546829	+	0.837244i	$0.684165\pi$
$264$	0	0
$265$	−23791.7	−0.338792
$266$	0	0
$267$	0	0
$268$	−66224.0	−0.922032
$269$	38880.9i	0.537318i	0.963235	+	0.268659i	$0.0865805\pi$
$269$	38880.9i	0.537318i	−0.963235	+	0.268659i	$0.913419\pi$
$270$	0	0
$271$	−117131.	−1.59489	−0.797447	−	0.603389i	$-0.793817\pi$
$271$	−117131.	−1.59489	−0.797447	+	0.603389i	$0.793817\pi$
$272$	20679.0i	0.279507i
$273$	0	0
$274$	−32156.0	−0.428313
$275$	9335.22i	0.123441i
$276$	0	0
$277$	−2558.00	−0.0333381	−0.0166691	−	0.999861i	$-0.505306\pi$
$277$	−2558.00	−0.0333381	−0.0166691	+	0.999861i	$0.505306\pi$
$278$	10339.5i	0.133786i
$279$	0	0
$280$	0	0
$281$	− 44857.4i	− 0.568096i	−0.958810	−	0.284048i	$-0.908322\pi$
$281$	− 44857.4i	− 0.568096i	0.958810	−	0.284048i	$-0.0916775\pi$
$282$	0	0
$283$	152772.	1.90753	0.953766	−	0.300549i	$-0.0971698\pi$
$283$	152772.	1.90753	0.953766	+	0.300549i	$0.0971698\pi$
$284$	− 14719.1i	− 0.182493i
$285$	0	0
$286$	−49959.5	−0.610781
$287$	0	0
$288$	0	0
$289$	−20879.0	−0.249985
$290$	5083.60i	0.0604470i
$291$	0	0
$292$	−74329.9	−0.871762
$293$	77223.3i	0.899524i	0.893148	+	0.449762i	$0.148491\pi$
$293$	77223.3i	0.899524i	−0.893148	+	0.449762i	$0.851509\pi$
$294$	0	0
$295$	78880.0	0.906406
$296$	13576.5i	0.154954i
$297$	0	0
$298$	38956.0	0.438674
$299$	− 164140.i	− 1.83600i
$300$	0	0
$301$	0	0
$302$	82590.1i	0.905553i
$303$	0	0
$304$	−9748.19	−0.105482
$305$	55776.6i	0.599587i
$306$	0	0
$307$	119872.	1.27187	0.635934	−	0.771744i	$-0.280615\pi$
$307$	119872.	1.27187	0.635934	+	0.771744i	$0.280615\pi$
$308$	0	0
$309$	0	0
$310$	−46400.0	−0.482830
$311$	− 82500.7i	− 0.852976i	−0.904493	−	0.426488i	$-0.859751\pi$
$311$	− 82500.7i	− 0.852976i	0.904493	−	0.426488i	$-0.140249\pi$
$312$	0	0
$313$	170898.	1.74441	0.872204	−	0.489142i	$-0.162690\pi$
$313$	170898.	1.74441	0.872204	+	0.489142i	$0.162690\pi$
$314$	64191.2i	0.651052i
$315$	0	0
$316$	−24656.0	−0.246916
$317$	− 108472.i	− 1.07944i	−0.841845	−	0.539719i	$-0.818531\pi$
$317$	− 108472.i	− 1.07944i	0.841845	−	0.539719i	$-0.181469\pi$
$318$	0	0
$319$	−4838.00	−0.0475428
$320$	11028.8i	0.107703i
$321$	0	0
$322$	0	0
$323$	− 49214.6i	− 0.471725i
$324$	0	0
$325$	49045.6	0.464337
$326$	− 66643.4i	− 0.627079i
$327$	0	0
$328$	−36555.7	−0.339788
$329$	0	0
$330$	0	0
$331$	123320.	1.12558	0.562792	−	0.826599i	$-0.309727\pi$
$331$	123320.	1.12558	0.562792	+	0.826599i	$0.309727\pi$
$332$	− 108565.i	− 0.984948i
$333$	0	0
$334$	17668.6	0.158383
$335$	− 178314.i	− 1.58889i
$336$	0	0
$337$	183920.	1.61946	0.809728	−	0.586805i	$-0.199615\pi$
$337$	183920.	1.61946	0.809728	+	0.586805i	$0.199615\pi$
$338$	181695.i	1.59041i
$339$	0	0
$340$	−55680.0	−0.481661
$341$	− 44158.4i	− 0.379756i
$342$	0	0
$343$	0	0
$344$	26247.8i	0.221807i
$345$	0	0
$346$	−120329.	−1.00512
$347$	52383.9i	0.435050i	0.976055	+	0.217525i	$0.0697983\pi$
$347$	52383.9i	0.435050i	−0.976055	+	0.217525i	$0.930202\pi$
$348$	0	0
$349$	−105555.	−0.866615	−0.433308	−	0.901246i	$-0.642654\pi$
$349$	−105555.	−0.866615	−0.433308	+	0.901246i	$0.642654\pi$
$350$	0	0
$351$	0	0
$352$	−10496.0	−0.0847107
$353$	176095.i	1.41318i	0.707623	+	0.706590i	$0.249767\pi$
$353$	176095.i	1.41318i	−0.707623	+	0.706590i	$0.750233\pi$
$354$	0	0
$355$	39632.5	0.314481
$356$	16370.9i	0.129173i
$357$	0	0
$358$	−131844.	−1.02871
$359$	22767.4i	0.176655i	0.996092	+	0.0883273i	$0.0281521\pi$
$359$	22767.4i	0.176655i	−0.996092	+	0.0883273i	$0.971848\pi$
$360$	0	0
$361$	−107121.	−0.821978
$362$	17232.5i	0.131502i
$363$	0	0
$364$	0	0
$365$	− 200140.i	− 1.50227i
$366$	0	0
$367$	−210500.	−1.56286	−0.781430	−	0.623993i	$-0.785509\pi$
$367$	−210500.	−1.56286	−0.781430	+	0.623993i	$0.785509\pi$
$368$	− 34484.2i	− 0.254639i
$369$	0	0
$370$	−36555.7	−0.267025
$371$	0	0
$372$	0	0
$373$	45358.0	0.326014	0.163007	−	0.986625i	$-0.447881\pi$
$373$	45358.0	0.326014	0.163007	+	0.986625i	$0.447881\pi$
$374$	− 52990.0i	− 0.378836i
$375$	0	0
$376$	24370.5	0.172381
$377$	25418.0i	0.178837i
$378$	0	0
$379$	8520.00	0.0593145	0.0296573	−	0.999560i	$-0.490558\pi$
$379$	8520.00	0.0593145	0.0296573	+	0.999560i	$0.490558\pi$
$380$	− 26247.8i	− 0.181771i
$381$	0	0
$382$	127284.	0.872262
$383$	232639.i	1.58593i	0.609264	+	0.792967i	$0.291465\pi$
$383$	232639.i	1.58593i	−0.609264	+	0.792967i	$0.708535\pi$
$384$	0	0
$385$	0	0
$386$	152961.i	1.02661i
$387$	0	0
$388$	6092.62	0.0404707
$389$	− 220757.i	− 1.45887i	−0.684051	−	0.729434i	$-0.739784\pi$
$389$	− 220757.i	− 1.45887i	0.684051	−	0.729434i	$-0.260216\pi$
$390$	0	0
$391$	174097.	1.13877
$392$	0	0
$393$	0	0
$394$	47916.0	0.308666
$395$	− 66388.3i	− 0.425498i
$396$	0	0
$397$	−106164.	−0.673590	−0.336795	−	0.941578i	$-0.609343\pi$
$397$	−106164.	−0.673590	−0.336795	+	0.941578i	$0.609343\pi$
$398$	− 166294.i	− 1.04981i
$399$	0	0
$400$	10304.0	0.0644000
$401$	− 79027.7i	− 0.491463i	−0.969338	−	0.245731i	$-0.920972\pi$
$401$	− 79027.7i	− 0.491463i	0.969338	−	0.245731i	$-0.0790281\pi$
$402$	0	0
$403$	−232000.	−1.42849
$404$	102534.i	0.628208i
$405$	0	0
$406$	0	0
$407$	− 34789.7i	− 0.210020i
$408$	0	0
$409$	165872.	0.991574	0.495787	−	0.868444i	$-0.334880\pi$
$409$	165872.	0.991574	0.495787	+	0.868444i	$0.334880\pi$
$410$	− 98429.3i	− 0.585540i
$411$	0	0
$412$	−42648.3	−0.251251
$413$	0	0
$414$	0	0
$415$	292320.	1.69731
$416$	55144.1i	0.318649i
$417$	0	0
$418$	24979.7	0.142967
$419$	293815.i	1.67358i	0.547527	+	0.836788i	$0.315569\pi$
$419$	293815.i	1.67358i	−0.547527	+	0.836788i	$0.684431\pi$
$420$	0	0
$421$	−182082.	−1.02731	−0.513657	−	0.857996i	$-0.671709\pi$
$421$	−182082.	−1.02731	−0.513657	+	0.857996i	$0.671709\pi$
$422$	185205.i	1.03999i
$423$	0	0
$424$	−24992.0	−0.139017
$425$	52020.7i	0.288004i
$426$	0	0
$427$	0	0
$428$	124892.i	0.681785i
$429$	0	0
$430$	−70674.4	−0.382230
$431$	− 192928.i	− 1.03858i	−0.854597	−	0.519292i	$-0.826196\pi$
$431$	− 192928.i	− 1.03858i	0.854597	−	0.519292i	$-0.173804\pi$
$432$	0	0
$433$	−16145.4	−0.0861141	−0.0430570	−	0.999073i	$-0.513710\pi$
$433$	−16145.4	−0.0861141	−0.0430570	+	0.999073i	$0.513710\pi$
$434$	0	0
$435$	0	0
$436$	−136640.	−0.718795
$437$	82069.9i	0.429755i
$438$	0	0
$439$	−305240.	−1.58384	−0.791922	−	0.610622i	$-0.790920\pi$
$439$	−305240.	−1.58384	−0.791922	+	0.610622i	$0.790920\pi$
$440$	− 28261.3i	− 0.145978i
$441$	0	0
$442$	−278400.	−1.42503
$443$	− 328266.i	− 1.67270i	−0.548195	−	0.836350i	$-0.684685\pi$
$443$	− 328266.i	− 1.67270i	0.548195	−	0.836350i	$-0.315315\pi$
$444$	0	0
$445$	−44080.0	−0.222598
$446$	− 140445.i	− 0.706052i
$447$	0	0
$448$	0	0
$449$	− 129091.i	− 0.640328i	−0.947362	−	0.320164i	$-0.896262\pi$
$449$	− 129091.i	− 0.640328i	0.947362	−	0.320164i	$-0.103738\pi$
$450$	0	0
$451$	93674.0	0.460539
$452$	− 151592.i	− 0.741994i
$453$	0	0
$454$	193136.	0.937026
$455$	0	0
$456$	0	0
$457$	242240.	1.15988	0.579941	−	0.814659i	$-0.303076\pi$
$457$	242240.	1.15988	0.579941	+	0.814659i	$0.303076\pi$
$458$	140445.i	0.669539i
$459$	0	0
$460$	92851.5	0.438807
$461$	− 57836.7i	− 0.272146i	−0.990699	−	0.136073i	$-0.956552\pi$
$461$	− 57836.7i	− 0.272146i	0.990699	−	0.136073i	$-0.0434481\pi$
$462$	0	0
$463$	−100438.	−0.468529	−0.234264	−	0.972173i	$-0.575268\pi$
$463$	−100438.	−0.468529	−0.234264	+	0.972173i	$0.575268\pi$
$464$	5340.07i	0.0248034i
$465$	0	0
$466$	−15524.0	−0.0714878
$467$	− 138722.i	− 0.636079i	−0.948077	−	0.318040i	$-0.896976\pi$
$467$	− 138722.i	− 0.636079i	0.948077	−	0.318040i	$-0.103024\pi$
$468$	0	0
$469$	0	0
$470$	65619.5i	0.297055i
$471$	0	0
$472$	82859.6	0.371928
$473$	− 67260.0i	− 0.300632i
$474$	0	0
$475$	−24522.8	−0.108688
$476$	0	0
$477$	0	0
$478$	108396.	0.474414
$479$	215837.i	0.940710i	0.882477	+	0.470355i	$0.155874\pi$
$479$	215837.i	0.940710i	−0.882477	+	0.470355i	$0.844126\pi$
$480$	0	0
$481$	−182779.	−0.790015
$482$	− 152077.i	− 0.654590i
$483$	0	0
$484$	−90232.0	−0.385185
$485$	16404.9i	0.0697412i
$486$	0	0
$487$	−356240.	−1.50205	−0.751026	−	0.660273i	$-0.770441\pi$
$487$	−356240.	−1.50205	−0.751026	+	0.660273i	$0.770441\pi$
$488$	58590.6i	0.246030i
$489$	0	0
$490$	0	0
$491$	− 256540.i	− 1.06412i	−0.846706	−	0.532061i	$-0.821418\pi$
$491$	− 256540.i	− 1.06412i	0.846706	−	0.532061i	$-0.178582\pi$
$492$	0	0
$493$	−26959.8	−0.110923
$494$	− 131239.i	− 0.537785i
$495$	0	0
$496$	−48740.9	−0.198121
$497$	0	0
$498$	0	0
$499$	118440.	0.475661	0.237830	−	0.971307i	$-0.423564\pi$
$499$	118440.	0.475661	0.237830	+	0.971307i	$0.423564\pi$
$500$	135448.i	0.541791i
$501$	0	0
$502$	−52396.5	−0.207919
$503$	− 141953.i	− 0.561059i	−0.959845	−	0.280529i	$-0.909490\pi$
$503$	− 141953.i	− 0.561059i	0.959845	−	0.280529i	$-0.0905100\pi$
$504$	0	0
$505$	−276080.	−1.08256
$506$	88365.7i	0.345130i
$507$	0	0
$508$	−159664.	−0.618699
$509$	119443.i	0.461026i	0.973069	+	0.230513i	$0.0740403\pi$
$509$	119443.i	0.461026i	−0.973069	+	0.230513i	$0.925960\pi$
$510$	0	0
$511$	0	0
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	−28939.9	−0.109540
$515$	− 114834.i	− 0.432969i
$516$	0	0
$517$	−62449.3	−0.233640
$518$	0	0
$519$	0	0
$520$	−148480.	−0.549112
$521$	− 360698.i	− 1.32883i	−0.747365	−	0.664414i	$-0.768681\pi$
$521$	− 360698.i	− 1.32883i	0.747365	−	0.664414i	$-0.231319\pi$
$522$	0	0
$523$	523965.	1.91557	0.957787	−	0.287478i	$-0.0928168\pi$
$523$	523965.	1.91557	0.957787	+	0.287478i	$0.0928168\pi$
$524$	175772.i	0.640157i
$525$	0	0
$526$	−327596.	−1.18404
$527$	− 246073.i	− 0.886019i
$528$	0	0
$529$	−10481.0	−0.0374534
$530$	− 67293.0i	− 0.239562i
$531$	0	0
$532$	0	0
$533$	− 492146.i	− 1.73237i
$534$	0	0
$535$	−336282.	−1.17489
$536$	− 187310.i	− 0.651975i
$537$	0	0
$538$	−109972.	−0.379941
$539$	0	0
$540$	0	0
$541$	397762.	1.35903	0.679515	−	0.733662i	$-0.262190\pi$
$541$	397762.	1.35903	0.679515	+	0.733662i	$0.262190\pi$
$542$	− 331295.i	− 1.12776i
$543$	0	0
$544$	−58489.1	−0.197641
$545$	− 367914.i	− 1.23866i
$546$	0	0
$547$	−371318.	−1.24100	−0.620499	−	0.784207i	$-0.713070\pi$
$547$	−371318.	−1.24100	−0.620499	+	0.784207i	$0.713070\pi$
$548$	− 90950.9i	− 0.302863i
$549$	0	0
$550$	−26404.0	−0.0872860
$551$	− 12709.0i	− 0.0418608i
$552$	0	0
$553$	0	0
$554$	− 7235.12i	− 0.0235736i
$555$	0	0
$556$	−29244.6	−0.0946010
$557$	− 374002.i	− 1.20549i	−0.797935	−	0.602744i	$-0.794074\pi$
$557$	− 374002.i	− 1.20549i	0.797935	−	0.602744i	$-0.205926\pi$
$558$	0	0
$559$	−353372.	−1.13086
$560$	0	0
$561$	0	0
$562$	126876.	0.401705
$563$	− 210452.i	− 0.663952i	−0.943288	−	0.331976i	$-0.892285\pi$
$563$	− 210452.i	− 0.663952i	0.943288	−	0.331976i	$-0.107715\pi$
$564$	0	0
$565$	408175.	1.27864
$566$	432106.i	1.34883i
$567$	0	0
$568$	41632.0	0.129042
$569$	− 126825.i	− 0.391725i	−0.980631	−	0.195862i	$-0.937249\pi$
$569$	− 126825.i	− 0.391725i	0.980631	−	0.195862i	$-0.0627506\pi$
$570$	0	0
$571$	−119082.	−0.365236	−0.182618	−	0.983184i	$-0.558457\pi$
$571$	−119082.	−0.365236	−0.182618	+	0.983184i	$0.558457\pi$
$572$	− 141307.i	− 0.431888i
$573$	0	0
$574$	0	0
$575$	− 86749.3i	− 0.262380i
$576$	0	0
$577$	89256.9	0.268096	0.134048	−	0.990975i	$-0.457202\pi$
$577$	89256.9	0.268096	0.134048	+	0.990975i	$0.457202\pi$
$578$	− 59054.7i	− 0.176766i
$579$	0	0
$580$	−14378.6	−0.0427425
$581$	0	0
$582$	0	0
$583$	64042.0	0.188420
$584$	− 210237.i	− 0.616429i
$585$	0	0
$586$	−218420.	−0.636060
$587$	465063.i	1.34969i	0.737958	+	0.674847i	$0.235790\pi$
$587$	465063.i	1.34969i	−0.737958	+	0.674847i	$0.764210\pi$
$588$	0	0
$589$	116000.	0.334370
$590$	223106.i	0.640926i
$591$	0	0
$592$	−38400.0	−0.109569
$593$	− 232316.i	− 0.660647i	−0.943868	−	0.330324i	$-0.892842\pi$
$593$	− 232316.i	− 0.660647i	0.943868	−	0.330324i	$-0.107158\pi$
$594$	0	0
$595$	0	0
$596$	110184.i	0.310189i
$597$	0	0
$598$	464258.	1.29824
$599$	38323.8i	0.106811i	0.998573	+	0.0534053i	$0.0170075\pi$
$599$	38323.8i	0.106811i	−0.998573	+	0.0534053i	$0.982992\pi$
$600$	0	0
$601$	340273.	0.942059	0.471030	−	0.882117i	$-0.343882\pi$
$601$	340273.	0.942059	0.471030	+	0.882117i	$0.343882\pi$
$602$	0	0
$603$	0	0
$604$	−233600.	−0.640323
$605$	− 242957.i	− 0.663772i
$606$	0	0
$607$	−4569.46	−0.0124019	−0.00620094	−	0.999981i	$-0.501974\pi$
$607$	−4569.46	−0.0124019	−0.00620094	+	0.999981i	$0.501974\pi$
$608$	− 27572.0i	− 0.0745868i
$609$	0	0
$610$	−157760.	−0.423972
$611$	328098.i	0.878862i
$612$	0	0
$613$	−564562.	−1.50242	−0.751209	−	0.660065i	$-0.770529\pi$
$613$	−564562.	−1.50242	−0.751209	+	0.660065i	$0.770529\pi$
$614$	339050.i	0.899346i
$615$	0	0
$616$	0	0
$617$	− 346537.i	− 0.910290i	−0.890417	−	0.455145i	$-0.849587\pi$
$617$	− 346537.i	− 0.910290i	0.890417	−	0.455145i	$-0.150413\pi$
$618$	0	0
$619$	164196.	0.428530	0.214265	−	0.976776i	$-0.431264\pi$
$619$	164196.	0.428530	0.214265	+	0.976776i	$0.431264\pi$
$620$	− 131239.i	− 0.341413i
$621$	0	0
$622$	233347.	0.603145
$623$	0	0
$624$	0	0
$625$	−264079.	−0.676042
$626$	483372.i	1.23348i
$627$	0	0
$628$	−181560.	−0.460364
$629$	− 193866.i	− 0.490005i
$630$	0	0
$631$	−394122.	−0.989856	−0.494928	−	0.868934i	$-0.664806\pi$
$631$	−394122.	−0.989856	−0.494928	+	0.868934i	$0.664806\pi$
$632$	− 69737.7i	− 0.174596i
$633$	0	0
$634$	306804.	0.763278
$635$	− 429908.i	− 1.06618i
$636$	0	0
$637$	0	0
$638$	− 13683.9i	− 0.0336178i
$639$	0	0
$640$	−31194.2	−0.0761577
$641$	− 346368.i	− 0.842988i	−0.906831	−	0.421494i	$-0.861506\pi$
$641$	− 346368.i	− 0.842988i	0.906831	−	0.421494i	$-0.138494\pi$
$642$	0	0
$643$	−54376.6	−0.131520	−0.0657598	−	0.997835i	$-0.520947\pi$
$643$	−54376.6	−0.131520	−0.0657598	+	0.997835i	$0.520947\pi$
$644$	0	0
$645$	0	0
$646$	139200.	0.333560
$647$	730875.i	1.74596i	0.487756	+	0.872980i	$0.337816\pi$
$647$	730875.i	1.74596i	−0.487756	+	0.872980i	$0.662184\pi$
$648$	0	0
$649$	−212328.	−0.504101
$650$	138722.i	0.328336i
$651$	0	0
$652$	188496.	0.443411
$653$	488836.i	1.14640i	0.819415	+	0.573200i	$0.194298\pi$
$653$	488836.i	1.14640i	−0.819415	+	0.573200i	$0.805702\pi$
$654$	0	0
$655$	−473280.	−1.10315
$656$	− 103395.i	− 0.240266i
$657$	0	0
$658$	0	0
$659$	6560.54i	0.0151067i	0.999971	+	0.00755333i	$0.00240432\pi$
$659$	6560.54i	0.0151067i	−0.999971	+	0.00755333i	$0.997596\pi$
$660$	0	0
$661$	−198467.	−0.454240	−0.227120	−	0.973867i	$-0.572931\pi$
$661$	−198467.	−0.454240	−0.227120	+	0.973867i	$0.572931\pi$
$662$	348802.i	0.795907i
$663$	0	0
$664$	307068.	0.696464
$665$	0	0
$666$	0	0
$667$	44958.0	0.101054
$668$	49974.3i	0.111994i
$669$	0	0
$670$	504347.	1.12352
$671$	− 150138.i	− 0.333462i
$672$	0	0
$673$	224240.	0.495089	0.247544	−	0.968877i	$-0.420376\pi$
$673$	224240.	0.495089	0.247544	+	0.968877i	$0.420376\pi$
$674$	520204.i	1.14513i
$675$	0	0
$676$	−513912.	−1.12459
$677$	213576.i	0.465988i	0.972478	+	0.232994i	$0.0748522\pi$
$677$	213576.i	0.465988i	−0.972478	+	0.232994i	$0.925148\pi$
$678$	0	0
$679$	0	0
$680$	− 157487.i	− 0.340586i
$681$	0	0
$682$	124899.	0.268528
$683$	433031.i	0.928277i	0.885763	+	0.464138i	$0.153636\pi$
$683$	433031.i	0.928277i	−0.885763	+	0.464138i	$0.846364\pi$
$684$	0	0
$685$	244893.	0.521909
$686$	0	0
$687$	0	0
$688$	−74240.0	−0.156842
$689$	− 336465.i	− 0.708764i
$690$	0	0
$691$	−9443.56	−0.0197779	−0.00988894	−	0.999951i	$-0.503148\pi$
$691$	−9443.56	−0.0197779	−0.00988894	+	0.999951i	$0.503148\pi$
$692$	− 340342.i	− 0.710729i
$693$	0	0
$694$	−148164.	−0.307627
$695$	− 78743.4i	− 0.163021i
$696$	0	0
$697$	522000.	1.07450
$698$	− 298554.i	− 0.612790i
$699$	0	0
$700$	0	0
$701$	458232.i	0.932501i	0.884653	+	0.466251i	$0.154396\pi$
$701$	458232.i	0.932501i	−0.884653	+	0.466251i	$0.845604\pi$
$702$	0	0
$703$	91389.3	0.184920
$704$	− 29687.2i	− 0.0598995i
$705$	0	0
$706$	−498072.	−0.999269
$707$	0	0
$708$	0	0
$709$	−84920.0	−0.168934	−0.0844671	−	0.996426i	$-0.526919\pi$
$709$	−84920.0	−0.168934	−0.0844671	+	0.996426i	$0.526919\pi$
$710$	112098.i	0.222372i
$711$	0	0
$712$	−46303.9	−0.0913393
$713$	410350.i	0.807188i
$714$	0	0
$715$	380480.	0.744252
$716$	− 372911.i	− 0.727410i
$717$	0	0
$718$	−64396.0	−0.124914
$719$	469156.i	0.907526i	0.891122	+	0.453763i	$0.149919\pi$
$719$	469156.i	0.907526i	−0.891122	+	0.453763i	$0.850081\pi$
$720$	0	0
$721$	0	0
$722$	− 302984.i	− 0.581226i
$723$	0	0
$724$	−48740.9	−0.0929858
$725$	13433.6i	0.0255574i
$726$	0	0
$727$	−755028.	−1.42855	−0.714273	−	0.699867i	$-0.753242\pi$
$727$	−755028.	−1.42855	−0.714273	+	0.699867i	$0.753242\pi$
$728$	0	0
$729$	0	0
$730$	566080.	1.06226
$731$	− 374807.i	− 0.701412i
$732$	0	0
$733$	−829205.	−1.54331	−0.771657	−	0.636039i	$-0.780572\pi$
$733$	−829205.	−1.54331	−0.771657	+	0.636039i	$0.780572\pi$
$734$	− 595384.i	− 1.10511i
$735$	0	0
$736$	97536.0	0.180057
$737$	479981.i	0.883668i
$738$	0	0
$739$	−396758.	−0.726502	−0.363251	−	0.931691i	$-0.618333\pi$
$739$	−396758.	−0.726502	−0.363251	+	0.931691i	$0.618333\pi$
$740$	− 103395.i	− 0.188815i
$741$	0	0
$742$	0	0
$743$	335818.i	0.608311i	0.952622	+	0.304156i	$0.0983743\pi$
$743$	335818.i	0.608311i	−0.952622	+	0.304156i	$0.901626\pi$
$744$	0	0
$745$	−296680.	−0.534535
$746$	128292.i	0.230527i
$747$	0	0
$748$	149878.	0.267877
$749$	0	0
$750$	0	0
$751$	−223680.	−0.396595	−0.198298	−	0.980142i	$-0.563541\pi$
$751$	−223680.	−0.396595	−0.198298	+	0.980142i	$0.563541\pi$
$752$	68930.1i	0.121891i
$753$	0	0
$754$	−71892.9	−0.126457
$755$	− 628987.i	− 1.10344i
$756$	0	0
$757$	721720.	1.25944	0.629719	−	0.776823i	$-0.283170\pi$
$757$	721720.	1.25944	0.629719	+	0.776823i	$0.283170\pi$
$758$	24098.2i	0.0419417i
$759$	0	0
$760$	74240.0	0.128532
$761$	773633.i	1.33587i	0.744218	+	0.667937i	$0.232822\pi$
$761$	773633.i	1.33587i	−0.744218	+	0.667937i	$0.767178\pi$
$762$	0	0
$763$	0	0
$764$	360014.i	0.616783i
$765$	0	0
$766$	−658003.	−1.12142
$767$	1.11553e6i	1.89623i
$768$	0	0
$769$	−170593.	−0.288476	−0.144238	−	0.989543i	$-0.546073\pi$
$769$	−170593.	−0.288476	−0.144238	+	0.989543i	$0.546073\pi$
$770$	0	0
$771$	0	0
$772$	−432640.	−0.725926
$773$	189019.i	0.316335i	0.987412	+	0.158167i	$0.0505586\pi$
$773$	189019.i	0.316335i	−0.987412	+	0.158167i	$0.949441\pi$
$774$	0	0
$775$	−122614.	−0.204144
$776$	17232.5i	0.0286171i
$777$	0	0
$778$	624396.	1.03158
$779$	246073.i	0.405499i
$780$	0	0
$781$	−106682.	−0.174900
$782$	492419.i	0.805233i
$783$	0	0
$784$	0	0
$785$	− 488865.i	− 0.793323i
$786$	0	0
$787$	−262592.	−0.423967	−0.211983	−	0.977273i	$-0.567992\pi$
$787$	−262592.	−0.423967	−0.211983	+	0.977273i	$0.567992\pi$
$788$	135527.i	0.218259i
$789$	0	0
$790$	187775.	0.300872
$791$	0	0
$792$	0	0
$793$	−788800.	−1.25436
$794$	− 300277.i	− 0.476300i
$795$	0	0
$796$	470350.	0.742327
$797$	870135.i	1.36984i	0.728618	+	0.684920i	$0.240163\pi$
$797$	870135.i	1.36984i	−0.728618	+	0.684920i	$0.759837\pi$
$798$	0	0
$799$	−348000.	−0.545112
$800$	29144.1i	0.0455377i
$801$	0	0
$802$	223524.	0.347516
$803$	538732.i	0.835491i
$804$	0	0
$805$	0	0
$806$	− 656195.i	− 1.01010i
$807$	0	0
$808$	−290009.	−0.444210
$809$	− 99672.4i	− 0.152292i	−0.997097	−	0.0761461i	$-0.975738\pi$
$809$	− 99672.4i	− 0.152292i	0.997097	−	0.0761461i	$-0.0242615\pi$
$810$	0	0
$811$	524879.	0.798027	0.399013	−	0.916945i	$-0.369353\pi$
$811$	524879.	0.798027	0.399013	+	0.916945i	$0.369353\pi$
$812$	0	0
$813$	0	0
$814$	98400.0	0.148507
$815$	507541.i	0.764110i
$816$	0	0
$817$	176686.	0.264702
$818$	469156.i	0.701149i
$819$	0	0
$820$	278400.	0.414039
$821$	382490.i	0.567457i	0.958905	+	0.283729i	$0.0915715\pi$
$821$	382490.i	0.567457i	−0.958905	+	0.283729i	$0.908428\pi$
$822$	0	0
$823$	1.00724e6	1.48708	0.743540	−	0.668692i	$-0.233145\pi$
$823$	1.00724e6	1.48708	0.743540	+	0.668692i	$0.233145\pi$
$824$	− 120628.i	− 0.177661i
$825$	0	0
$826$	0	0
$827$	1.29576e6i	1.89458i	0.320379	+	0.947290i	$0.396190\pi$
$827$	1.29576e6i	1.89458i	−0.320379	+	0.947290i	$0.603810\pi$
$828$	0	0
$829$	−61535.4	−0.0895398	−0.0447699	−	0.998997i	$-0.514255\pi$
$829$	−61535.4	−0.0895398	−0.0447699	+	0.998997i	$0.514255\pi$
$830$	826806.i	1.20018i
$831$	0	0
$832$	−155971.	−0.225319
$833$	0	0
$834$	0	0
$835$	−134560.	−0.192994
$836$	70653.4i	0.101093i
$837$	0	0
$838$	−831033.	−1.18340
$839$	− 600984.i	− 0.853767i	−0.904307	−	0.426883i	$-0.859611\pi$
$839$	− 600984.i	− 0.853767i	0.904307	−	0.426883i	$-0.140389\pi$
$840$	0	0
$841$	700319.	0.990157
$842$	− 515006.i	− 0.726420i
$843$	0	0
$844$	−523840.	−0.735383
$845$	− 1.38375e6i	− 1.93796i
$846$	0	0
$847$	0	0
$848$	− 70688.1i	− 0.0983002i
$849$	0	0
$850$	−147137.	−0.203649
$851$	323289.i	0.446408i
$852$	0	0
$853$	974971.	1.33997	0.669983	−	0.742376i	$-0.266301\pi$
$853$	974971.	1.33997	0.669983	+	0.742376i	$0.266301\pi$
$854$	0	0
$855$	0	0
$856$	−353248.	−0.482095
$857$	914078.i	1.24458i	0.782788	+	0.622288i	$0.213797\pi$
$857$	914078.i	1.24458i	−0.782788	+	0.622288i	$0.786203\pi$
$858$	0	0
$859$	918005.	1.24411	0.622055	−	0.782973i	$-0.286298\pi$
$859$	918005.	1.24411	0.622055	+	0.782973i	$0.286298\pi$
$860$	− 199897.i	− 0.270278i
$861$	0	0
$862$	545684.	0.734390
$863$	420331.i	0.564378i	0.959359	+	0.282189i	$0.0910605\pi$
$863$	420331.i	0.564378i	−0.959359	+	0.282189i	$0.908939\pi$
$864$	0	0
$865$	916400.	1.22477
$866$	− 45666.2i	− 0.0608918i
$867$	0	0
$868$	0	0
$869$	178703.i	0.236642i
$870$	0	0
$871$	2.52173e6	3.32402
$872$	− 386476.i	− 0.508265i
$873$	0	0
$874$	−232129.	−0.303883
$875$	0	0
$876$	0	0
$877$	−886158.	−1.15216	−0.576079	−	0.817394i	$-0.695418\pi$
$877$	−886158.	−1.15216	−0.576079	+	0.817394i	$0.695418\pi$
$878$	− 863350.i	− 1.11995i
$879$	0	0
$880$	79935.2	0.103222
$881$	341958.i	0.440576i	0.975435	+	0.220288i	$0.0706997\pi$
$881$	341958.i	0.440576i	−0.975435	+	0.220288i	$0.929300\pi$
$882$	0	0
$883$	−671720.	−0.861523	−0.430761	−	0.902466i	$-0.641755\pi$
$883$	−671720.	−0.861523	−0.430761	+	0.902466i	$0.641755\pi$
$884$	− 787434.i	− 1.00765i
$885$	0	0
$886$	928476.	1.18278
$887$	− 327633.i	− 0.416429i	−0.978083	−	0.208214i	$-0.933235\pi$
$887$	− 327633.i	− 0.416429i	0.978083	−	0.208214i	$-0.0667652\pi$
$888$	0	0
$889$	0	0
$890$	− 124677.i	− 0.157401i
$891$	0	0
$892$	397239.	0.499254
$893$	− 164049.i	− 0.205717i
$894$	0	0
$895$	1.00409e6	1.25351
$896$	0	0
$897$	0	0
$898$	365124.	0.452780
$899$	− 63544.9i	− 0.0786252i
$900$	0	0
$901$	356875.	0.439609
$902$	264950.i	0.325650i
$903$	0	0
$904$	428768.	0.524669
$905$	− 131239.i	− 0.160238i
$906$	0	0
$907$	762760.	0.927200	0.463600	−	0.886045i	$-0.346558\pi$
$907$	762760.	0.927200	0.463600	+	0.886045i	$0.346558\pi$
$908$	546271.i	0.662577i
$909$	0	0
$910$	0	0
$911$	529059.i	0.637481i	0.947842	+	0.318740i	$0.103260\pi$
$911$	529059.i	0.637481i	−0.947842	+	0.318740i	$0.896740\pi$
$912$	0	0
$913$	−786862.	−0.943967
$914$	685158.i	0.820160i
$915$	0	0
$916$	−397239.	−0.473435
$917$	0	0
$918$	0	0
$919$	−422880.	−0.500710	−0.250355	−	0.968154i	$-0.580547\pi$
$919$	−422880.	−0.500710	−0.250355	+	0.968154i	$0.580547\pi$
$920$	262624.i	0.310283i
$921$	0	0
$922$	163587.	0.192436
$923$	560488.i	0.657905i
$924$	0	0
$925$	−96600.0	−0.112900
$926$	− 284082.i	− 0.331300i
$927$	0	0
$928$	−15104.0	−0.0175386
$929$	− 1.68631e6i	− 1.95392i	−0.213429	−	0.976959i	$-0.568463\pi$
$929$	− 1.68631e6i	− 1.95392i	0.213429	−	0.976959i	$-0.431537\pi$
$930$	0	0
$931$	0	0
$932$	− 43908.5i	− 0.0505495i
$933$	0	0
$934$	392365.	0.449776
$935$	403560.i	0.461620i
$936$	0	0
$937$	−1.31235e6	−1.49476	−0.747378	−	0.664399i	$-0.768688\pi$
$937$	−1.31235e6	−1.49476	−0.747378	+	0.664399i	$0.768688\pi$
$938$	0	0
$939$	0	0
$940$	−185600.	−0.210050
$941$	− 362852.i	− 0.409780i	−0.978785	−	0.204890i	$-0.934316\pi$
$941$	− 362852.i	− 0.409780i	0.978785	−	0.204890i	$-0.0656837\pi$
$942$	0	0
$943$	−870483.	−0.978897
$944$	234362.i	0.262993i
$945$	0	0
$946$	190240.	0.212579
$947$	− 1.05772e6i	− 1.17942i	−0.807614	−	0.589712i	$-0.799241\pi$
$947$	− 1.05772e6i	− 1.17942i	0.807614	−	0.589712i	$-0.200759\pi$
$948$	0	0
$949$	2.83040e6	3.14279
$950$	− 69360.9i	− 0.0768542i
$951$	0	0
$952$	0	0
$953$	1.55377e6i	1.71081i	0.517963	+	0.855403i	$0.326691\pi$
$953$	1.55377e6i	1.71081i	−0.517963	+	0.855403i	$0.673309\pi$
$954$	0	0
$955$	−969366.	−1.06287
$956$	306590.i	0.335461i
$957$	0	0
$958$	−610480.	−0.665182
$959$	0	0
$960$	0	0
$961$	−343521.	−0.371969
$962$	− 516976.i	− 0.558625i
$963$	0	0
$964$	430139.	0.462865
$965$	− 1.16492e6i	− 1.25095i
$966$	0	0
$967$	1.09892e6	1.17521	0.587603	−	0.809149i	$-0.300072\pi$
$967$	1.09892e6	1.17521	0.587603	+	0.809149i	$0.300072\pi$
$968$	− 255215.i	− 0.272367i
$969$	0	0
$970$	−46400.0	−0.0493145
$971$	1.63472e6i	1.73382i	0.498461	+	0.866912i	$0.333899\pi$
$971$	1.63472e6i	1.73382i	−0.498461	+	0.866912i	$0.666101\pi$
$972$	0	0
$973$	0	0
$974$	− 1.00760e6i	− 1.06211i
$975$	0	0
$976$	−165719.	−0.173970
$977$	82025.8i	0.0859333i	0.999077	+	0.0429666i	$0.0136809\pi$
$977$	82025.8i	0.0859333i	−0.999077	+	0.0429666i	$0.986319\pi$
$978$	0	0
$979$	118654.	0.123799
$980$	0	0
$981$	0	0
$982$	725604.	0.752448
$983$	− 1.51883e6i	− 1.57182i	−0.618341	−	0.785910i	$-0.712195\pi$
$983$	− 1.51883e6i	− 1.57182i	0.618341	−	0.785910i	$-0.287805\pi$
$984$	0	0
$985$	−364917.	−0.376116
$986$	− 76253.9i	− 0.0784347i
$987$	0	0
$988$	371200.	0.380272
$989$	625026.i	0.639007i
$990$	0	0
$991$	−497680.	−0.506761	−0.253380	−	0.967367i	$-0.581542\pi$
$991$	−497680.	−0.506761	−0.253380	+	0.967367i	$0.581542\pi$
$992$	− 137860.i	− 0.140093i
$993$	0	0
$994$	0	0
$995$	1.26646e6i	1.27922i
$996$	0	0
$997$	−203950.	−0.205180	−0.102590	−	0.994724i	$-0.532713\pi$
$997$	−203950.	−0.205180	−0.102590	+	0.994724i	$0.532713\pi$
$998$	334999.i	0.336343i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.b.197.3	yes	4
3.2	odd	2	inner	882.5.b.b.197.2	yes	4
7.6	odd	2	inner	882.5.b.b.197.4	yes	4
21.20	even	2	inner	882.5.b.b.197.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.5.b.b.197.1	✓	4	21.20	even	2	inner
882.5.b.b.197.2	yes	4	3.2	odd	2	inner
882.5.b.b.197.3	yes	4	1.1	even	1	trivial
882.5.b.b.197.4	yes	4	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} -21.5407i q^{5} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} -21.5407i q^{5} -22.6274i q^{8} +60.9262 q^{10} +57.9828i q^{11} +304.631 q^{13} +64.0000 q^{16} +323.110i q^{17} -152.315 q^{19} +172.325i q^{20} -164.000 q^{22} -538.815i q^{23} +161.000 q^{25} +861.626i q^{26} +83.4386i q^{29} -761.577 q^{31} +181.019i q^{32} -913.893 q^{34} -600.000 q^{37} -430.813i q^{38} -487.409 q^{40} -1615.55i q^{41} -1160.00 q^{43} -463.862i q^{44} +1524.00 q^{46} +1077.03i q^{47} +455.377i q^{50} -2437.05 q^{52} -1104.50i q^{53} +1248.99 q^{55} -236.000 q^{58} +3661.91i q^{59} -2589.36 q^{61} -2154.07i q^{62} -512.000 q^{64} -6561.95i q^{65} +8278.00 q^{67} -2584.88i q^{68} +1839.89i q^{71} +9291.24 q^{73} -1697.06i q^{74} +1218.52 q^{76} +3082.00 q^{79} -1378.60i q^{80} +4569.46 q^{82} +13570.6i q^{83} +6960.00 q^{85} -3280.98i q^{86} +1312.00 q^{88} -2046.36i q^{89} +4310.52i q^{92} -3046.31 q^{94} +3280.98i q^{95} -761.577 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4}+O(q^{10}) \) 4 * q - 32 * q^4	\( 4 q - 32 q^{4} + 256 q^{16} - 656 q^{22} + 644 q^{25} - 2400 q^{37} - 4640 q^{43} + 6096 q^{46} - 944 q^{58} - 2048 q^{64} + 33112 q^{67} + 12328 q^{79} + 27840 q^{85} + 5248 q^{88}+O(q^{100}) \) 4 * q - 32 * q^4 + 256 * q^16 - 656 * q^22 + 644 * q^25 - 2400 * q^37 - 4640 * q^43 + 6096 * q^46 - 944 * q^58 - 2048 * q^64 + 33112 * q^67 + 12328 * q^79 + 27840 * q^85 + 5248 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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