LMFDB - Embedded newform 1089.3.c.i.604.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.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:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.79010463744.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} - 44x^{5} + 108x^{3} + 538x^{2} + 360x + 825 $ x^8 - 6x^6 - 44x^5 + 108x^3 + 538x^2 + 360*x + 825
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.3
Root			$-1.55225 - 2.21772i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.i.604.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50359i q^{2} -2.26795 q^{4} -1.29595 q^{5} +7.07107i q^{7} -4.33634i q^{8} +O(q^{10})$	$q-2.50359i q^{2} -2.26795 q^{4} -1.29595 q^{5} +7.07107i q^{7} -4.33634i q^{8} +3.24453i q^{10} +4.48288i q^{13} +17.7030 q^{14} -19.9282 q^{16} +9.83460i q^{17} -16.0096i q^{19} +2.93915 q^{20} +39.6412 q^{23} -23.3205 q^{25} +11.2233 q^{26} -16.0368i q^{28} -46.7176i q^{29} +34.0000 q^{31} +32.5466i q^{32} +24.6218 q^{34} -9.16377i q^{35} +11.3397 q^{37} -40.0815 q^{38} +5.61969i q^{40} -65.0451i q^{41} +36.7423i q^{43} -99.2451i q^{46} +78.5879 q^{47} -1.00000 q^{49} +58.3849i q^{50} -10.1669i q^{52} -29.3666 q^{53} +30.6626 q^{56} -116.962 q^{58} +66.0686 q^{59} -97.2489i q^{61} -85.1220i q^{62} +1.77053 q^{64} -5.80959i q^{65} +39.0192 q^{67} -22.3044i q^{68} -22.9423 q^{70} -43.6221 q^{71} -75.3794i q^{73} -28.3900i q^{74} +36.3090i q^{76} -61.0614i q^{79} +25.8260 q^{80} -162.846 q^{82} -61.0683i q^{83} -12.7452i q^{85} +91.9877 q^{86} +103.211 q^{89} -31.6987 q^{91} -89.9042 q^{92} -196.752i q^{94} +20.7477i q^{95} +10.6218 q^{97} +2.50359i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4}+O(q^{10}) $ 8 * q - 32 * q^4	$ 8 q - 32 q^{4} - 104 q^{16} - 48 q^{25} + 272 q^{31} - 288 q^{34} + 160 q^{37} - 8 q^{49} - 520 q^{58} + 416 q^{64} + 520 q^{67} + 440 q^{70} + 360 q^{82} - 600 q^{91} - 400 q^{97}+O(q^{100}) $ 8 * q - 32 * q^4 - 104 * q^16 - 48 * q^25 + 272 * q^31 - 288 * q^34 + 160 * q^37 - 8 * q^49 - 520 * q^58 + 416 * q^64 + 520 * q^67 + 440 * q^70 + 360 * q^82 - 600 * q^91 - 400 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.50359i	− 1.25179i	−0.779906	−	0.625897i	$-0.784733\pi$
$2$	− 2.50359i	− 1.25179i	0.779906	−	0.625897i	$-0.215267\pi$
$3$	0	0
$3$	0	0
$4$	−2.26795	−0.566987
$5$	−1.29595	−0.259190	−0.129595	−	0.991567i	$-0.541368\pi$
$5$	−1.29595	−0.259190	−0.129595	+	0.991567i	$0.541368\pi$
$6$	0	0
$7$	7.07107i	1.01015i	0.863075	+	0.505076i	$0.168536\pi$
$7$	7.07107i	1.01015i	−0.863075	+	0.505076i	$0.831464\pi$
$8$	− 4.33634i	− 0.542043i
$9$	0	0
$10$	3.24453i	0.324453i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.48288i	0.344837i	0.985024	+	0.172418i	$0.0551581\pi$
$13$	4.48288i	0.344837i	−0.985024	+	0.172418i	$0.944842\pi$
$14$	17.7030	1.26450
$15$	0	0
$16$	−19.9282	−1.24551
$17$	9.83460i	0.578506i	0.957253	+	0.289253i	$0.0934069\pi$
$17$	9.83460i	0.578506i	−0.957253	+	0.289253i	$0.906593\pi$
$18$	0	0
$19$	− 16.0096i	− 0.842611i	−0.906919	−	0.421306i	$-0.861572\pi$
$19$	− 16.0096i	− 0.842611i	0.906919	−	0.421306i	$-0.138428\pi$
$20$	2.93915	0.146958
$21$	0	0
$22$	0	0
$23$	39.6412	1.72353	0.861765	−	0.507308i	$-0.169359\pi$
$23$	39.6412	1.72353	0.861765	+	0.507308i	$0.169359\pi$
$24$	0	0
$25$	−23.3205	−0.932820
$26$	11.2233	0.431664
$27$	0	0
$28$	− 16.0368i	− 0.572744i
$29$	− 46.7176i	− 1.61095i	−0.592629	−	0.805475i	$-0.701910\pi$
$29$	− 46.7176i	− 1.61095i	0.592629	−	0.805475i	$-0.298090\pi$
$30$	0	0
$31$	34.0000	1.09677	0.548387	−	0.836225i	$-0.315242\pi$
$31$	34.0000	1.09677	0.548387	+	0.836225i	$0.315242\pi$
$32$	32.5466i	1.01708i
$33$	0	0
$34$	24.6218	0.724170
$35$	− 9.16377i	− 0.261822i
$36$	0	0
$37$	11.3397	0.306480	0.153240	−	0.988189i	$-0.451029\pi$
$37$	11.3397	0.306480	0.153240	+	0.988189i	$0.451029\pi$
$38$	−40.0815	−1.05478
$39$	0	0
$40$	5.61969i	0.140492i
$41$	− 65.0451i	− 1.58647i	−0.608918	−	0.793233i	$-0.708396\pi$
$41$	− 65.0451i	− 1.58647i	0.608918	−	0.793233i	$-0.291604\pi$
$42$	0	0
$43$	36.7423i	0.854473i	0.904140	+	0.427237i	$0.140513\pi$
$43$	36.7423i	0.854473i	−0.904140	+	0.427237i	$0.859487\pi$
$44$	0	0
$45$	0	0
$46$	− 99.2451i	− 2.15750i
$47$	78.5879	1.67208	0.836041	−	0.548667i	$-0.184864\pi$
$47$	78.5879	1.67208	0.836041	+	0.548667i	$0.184864\pi$
$48$	0	0
$49$	−1.00000	−0.0204082
$50$	58.3849i	1.16770i
$51$	0	0
$52$	− 10.1669i	− 0.195518i
$53$	−29.3666	−0.554087	−0.277043	−	0.960857i	$-0.589355\pi$
$53$	−29.3666	−0.554087	−0.277043	+	0.960857i	$0.589355\pi$
$54$	0	0
$55$	0	0
$56$	30.6626	0.547546
$57$	0	0
$58$	−116.962	−2.01658
$59$	66.0686	1.11981	0.559904	−	0.828558i	$-0.310838\pi$
$59$	66.0686	1.11981	0.559904	+	0.828558i	$0.310838\pi$
$60$	0	0
$61$	− 97.2489i	− 1.59424i	−0.603818	−	0.797122i	$-0.706355\pi$
$61$	− 97.2489i	− 1.59424i	0.603818	−	0.797122i	$-0.293645\pi$
$62$	− 85.1220i	− 1.37293i
$63$	0	0
$64$	1.77053	0.0276645
$65$	− 5.80959i	− 0.0893784i
$66$	0	0
$67$	39.0192	0.582377	0.291188	−	0.956666i	$-0.405949\pi$
$67$	39.0192	0.582377	0.291188	+	0.956666i	$0.405949\pi$
$68$	− 22.3044i	− 0.328005i
$69$	0	0
$70$	−22.9423	−0.327747
$71$	−43.6221	−0.614395	−0.307198	−	0.951646i	$-0.599391\pi$
$71$	−43.6221	−0.614395	−0.307198	+	0.951646i	$0.599391\pi$
$72$	0	0
$73$	− 75.3794i	− 1.03259i	−0.856410	−	0.516297i	$-0.827310\pi$
$73$	− 75.3794i	− 1.03259i	0.856410	−	0.516297i	$-0.172690\pi$
$74$	− 28.3900i	− 0.383649i
$75$	0	0
$76$	36.3090i	0.477750i
$77$	0	0
$78$	0	0
$79$	− 61.0614i	− 0.772929i	−0.922304	−	0.386464i	$-0.873696\pi$
$79$	− 61.0614i	− 0.772929i	0.922304	−	0.386464i	$-0.126304\pi$
$80$	25.8260	0.322825
$81$	0	0
$82$	−162.846	−1.98593
$83$	− 61.0683i	− 0.735762i	−0.929873	−	0.367881i	$-0.880083\pi$
$83$	− 61.0683i	− 0.735762i	0.929873	−	0.367881i	$-0.119917\pi$
$84$	0	0
$85$	− 12.7452i	− 0.149943i
$86$	91.9877	1.06962
$87$	0	0
$88$	0	0
$89$	103.211	1.15967	0.579837	−	0.814733i	$-0.303116\pi$
$89$	103.211	1.15967	0.579837	+	0.814733i	$0.303116\pi$
$90$	0	0
$91$	−31.6987	−0.348338
$92$	−89.9042	−0.977219
$93$	0	0
$94$	− 196.752i	− 2.09310i
$95$	20.7477i	0.218397i
$96$	0	0
$97$	10.6218	0.109503	0.0547514	−	0.998500i	$-0.482563\pi$
$97$	10.6218	0.109503	0.0547514	+	0.998500i	$0.482563\pi$
$98$	2.50359i	0.0255468i
$99$	0	0
$100$	52.8897	0.528897
$101$	52.5272i	0.520071i	0.965599	+	0.260035i	$0.0837343\pi$
$101$	52.5272i	0.520071i	−0.965599	+	0.260035i	$0.916266\pi$
$102$	0	0
$103$	−167.583	−1.62702	−0.813511	−	0.581549i	$-0.802447\pi$
$103$	−167.583	−1.62702	−0.813511	+	0.581549i	$0.802447\pi$
$104$	19.4393	0.186916
$105$	0	0
$106$	73.5219i	0.693602i
$107$	88.2964i	0.825200i	0.910912	+	0.412600i	$0.135379\pi$
$107$	88.2964i	0.825200i	−0.910912	+	0.412600i	$0.864621\pi$
$108$	0	0
$109$	108.627i	0.996578i	0.867011	+	0.498289i	$0.166038\pi$
$109$	108.627i	0.996578i	−0.867011	+	0.498289i	$0.833962\pi$
$110$	0	0
$111$	0	0
$112$	− 140.914i	− 1.25816i
$113$	−37.6507	−0.333192	−0.166596	−	0.986025i	$-0.553278\pi$
$113$	−37.6507	−0.333192	−0.166596	+	0.986025i	$0.553278\pi$
$114$	0	0
$115$	−51.3731	−0.446722
$116$	105.953i	0.913389i
$117$	0	0
$118$	− 165.409i	− 1.40177i
$119$	−69.5411	−0.584379
$120$	0	0
$121$	0	0
$122$	−243.471	−1.99567
$123$	0	0
$124$	−77.1103	−0.621857
$125$	62.6211	0.500969
$126$	0	0
$127$	− 83.4658i	− 0.657211i	−0.944467	−	0.328606i	$-0.893421\pi$
$127$	− 83.4658i	− 0.657211i	0.944467	−	0.328606i	$-0.106579\pi$
$128$	125.754i	0.982452i
$129$	0	0
$130$	−14.5448	−0.111883
$131$	− 90.9797i	− 0.694502i	−0.937772	−	0.347251i	$-0.887115\pi$
$131$	− 90.9797i	− 0.694502i	0.937772	−	0.347251i	$-0.112885\pi$
$132$	0	0
$133$	113.205	0.851166
$134$	− 97.6881i	− 0.729015i
$135$	0	0
$136$	42.6462	0.313575
$137$	163.240	1.19153	0.595767	−	0.803157i	$-0.296848\pi$
$137$	163.240	1.19153	0.595767	+	0.803157i	$0.296848\pi$
$138$	0	0
$139$	12.9310i	0.0930287i	0.998918	+	0.0465144i	$0.0148113\pi$
$139$	12.9310i	0.0930287i	−0.998918	+	0.0465144i	$0.985189\pi$
$140$	20.7830i	0.148450i
$141$	0	0
$142$	109.212i	0.769096i
$143$	0	0
$144$	0	0
$145$	60.5437i	0.417543i
$146$	−188.719	−1.29259
$147$	0	0
$148$	−25.7180	−0.173770
$149$	− 286.115i	− 1.92024i	−0.279597	−	0.960118i	$-0.590201\pi$
$149$	− 286.115i	− 1.92024i	0.279597	−	0.960118i	$-0.409799\pi$
$150$	0	0
$151$	129.905i	0.860295i	0.902759	+	0.430148i	$0.141538\pi$
$151$	129.905i	0.860295i	−0.902759	+	0.430148i	$0.858462\pi$
$152$	−69.4231	−0.456731
$153$	0	0
$154$	0	0
$155$	−44.0624	−0.284273
$156$	0	0
$157$	−240.526	−1.53201	−0.766005	−	0.642835i	$-0.777758\pi$
$157$	−240.526	−1.53201	−0.766005	+	0.642835i	$0.777758\pi$
$158$	−152.872	−0.967547
$159$	0	0
$160$	− 42.1789i	− 0.263618i
$161$	280.305i	1.74103i
$162$	0	0
$163$	224.904	1.37978	0.689889	−	0.723915i	$-0.257659\pi$
$163$	224.904	1.37978	0.689889	+	0.723915i	$0.257659\pi$
$164$	147.519i	0.899506i
$165$	0	0
$166$	−152.890	−0.921022
$167$	161.607i	0.967704i	0.875150	+	0.483852i	$0.160763\pi$
$167$	161.607i	0.967704i	−0.875150	+	0.483852i	$0.839237\pi$
$168$	0	0
$169$	148.904	0.881088
$170$	−31.9086	−0.187698
$171$	0	0
$172$	− 83.3298i	− 0.484475i
$173$	101.976i	0.589458i	0.955581	+	0.294729i	$0.0952294\pi$
$173$	101.976i	0.589458i	−0.955581	+	0.294729i	$0.904771\pi$
$174$	0	0
$175$	− 164.901i	− 0.942291i
$176$	0	0
$177$	0	0
$178$	− 258.398i	− 1.45167i
$179$	−331.614	−1.85259	−0.926297	−	0.376795i	$-0.877026\pi$
$179$	−331.614	−1.85259	−0.926297	+	0.376795i	$0.877026\pi$
$180$	0	0
$181$	−91.3397	−0.504639	−0.252320	−	0.967644i	$-0.581193\pi$
$181$	−91.3397	−0.504639	−0.252320	+	0.967644i	$0.581193\pi$
$182$	79.3605i	0.436047i
$183$	0	0
$184$	− 171.898i	− 0.934226i
$185$	−14.6958	−0.0794366
$186$	0	0
$187$	0	0
$188$	−178.233	−0.948049
$189$	0	0
$190$	51.9437	0.273388
$191$	25.9190	0.135702	0.0678509	−	0.997695i	$-0.478386\pi$
$191$	25.9190	0.135702	0.0678509	+	0.997695i	$0.478386\pi$
$192$	0	0
$193$	237.184i	1.22893i	0.788942	+	0.614467i	$0.210629\pi$
$193$	237.184i	1.22893i	−0.788942	+	0.614467i	$0.789371\pi$
$194$	− 26.5925i	− 0.137075i
$195$	0	0
$196$	2.26795	0.0115712
$197$	− 290.403i	− 1.47413i	−0.675823	−	0.737064i	$-0.736212\pi$
$197$	− 290.403i	− 1.47413i	0.675823	−	0.737064i	$-0.263788\pi$
$198$	0	0
$199$	247.976	1.24611	0.623054	−	0.782179i	$-0.285892\pi$
$199$	247.976	1.24611	0.623054	+	0.782179i	$0.285892\pi$
$200$	101.126i	0.505628i
$201$	0	0
$202$	131.506	0.651022
$203$	330.343	1.62731
$204$	0	0
$205$	84.2953i	0.411197i
$206$	419.559i	2.03670i
$207$	0	0
$208$	− 89.3357i	− 0.429499i
$209$	0	0
$210$	0	0
$211$	139.858i	0.662836i	0.943484	+	0.331418i	$0.107527\pi$
$211$	139.858i	0.662836i	−0.943484	+	0.331418i	$0.892473\pi$
$212$	66.6020	0.314160
$213$	0	0
$214$	221.058	1.03298
$215$	− 47.6163i	− 0.221471i
$216$	0	0
$217$	240.416i	1.10791i
$218$	271.957	1.24751
$219$	0	0
$220$	0	0
$221$	−44.0873	−0.199490
$222$	0	0
$223$	83.2051	0.373117	0.186558	−	0.982444i	$-0.440267\pi$
$223$	83.2051	0.373117	0.186558	+	0.982444i	$0.440267\pi$
$224$	−230.139	−1.02741
$225$	0	0
$226$	94.2619i	0.417088i
$227$	162.685i	0.716674i	0.933592	+	0.358337i	$0.116656\pi$
$227$	162.685i	0.716674i	−0.933592	+	0.358337i	$0.883344\pi$
$228$	0	0
$229$	−71.0141	−0.310105	−0.155053	−	0.987906i	$-0.549555\pi$
$229$	−71.0141	−0.310105	−0.155053	+	0.987906i	$0.549555\pi$
$230$	128.617i	0.559204i
$231$	0	0
$232$	−202.583	−0.873204
$233$	− 60.9848i	− 0.261738i	−0.991400	−	0.130869i	$-0.958223\pi$
$233$	− 60.9848i	− 0.261738i	0.991400	−	0.130869i	$-0.0417767\pi$
$234$	0	0
$235$	−101.846	−0.433388
$236$	−149.840	−0.634916
$237$	0	0
$238$	174.102i	0.731522i
$239$	− 275.395i	− 1.15228i	−0.817351	−	0.576139i	$-0.804559\pi$
$239$	− 275.395i	− 1.15228i	0.817351	−	0.576139i	$-0.195441\pi$
$240$	0	0
$241$	339.587i	1.40908i	0.709667	+	0.704538i	$0.248846\pi$
$241$	339.587i	1.40908i	−0.709667	+	0.704538i	$0.751154\pi$
$242$	0	0
$243$	0	0
$244$	220.556i	0.903916i
$245$	1.29595	0.00528960
$246$	0	0
$247$	71.7691	0.290563
$248$	− 147.436i	− 0.594498i
$249$	0	0
$250$	− 156.777i	− 0.627109i
$251$	−253.516	−1.01003	−0.505013	−	0.863112i	$-0.668512\pi$
$251$	−253.516	−1.01003	−0.505013	+	0.863112i	$0.668512\pi$
$252$	0	0
$253$	0	0
$254$	−208.964	−0.822693
$255$	0	0
$256$	321.918	1.25749
$257$	168.517	0.655708	0.327854	−	0.944728i	$-0.393675\pi$
$257$	168.517	0.655708	0.327854	+	0.944728i	$0.393675\pi$
$258$	0	0
$259$	80.1841i	0.309591i
$260$	13.1759i	0.0506764i
$261$	0	0
$262$	−227.776	−0.869373
$263$	− 220.219i	− 0.837336i	−0.908139	−	0.418668i	$-0.862497\pi$
$263$	− 220.219i	− 0.837336i	0.908139	−	0.418668i	$-0.137503\pi$
$264$	0	0
$265$	38.0577	0.143614
$266$	− 283.419i	− 1.06548i
$267$	0	0
$268$	−88.4936	−0.330200
$269$	−388.661	−1.44484	−0.722418	−	0.691456i	$-0.756969\pi$
$269$	−388.661	−1.44484	−0.722418	+	0.691456i	$0.756969\pi$
$270$	0	0
$271$	− 273.146i	− 1.00792i	−0.863727	−	0.503960i	$-0.831876\pi$
$271$	− 273.146i	− 1.00792i	0.863727	−	0.503960i	$-0.168124\pi$
$272$	− 195.986i	− 0.720536i
$273$	0	0
$274$	− 408.686i	− 1.49155i
$275$	0	0
$276$	0	0
$277$	469.007i	1.69316i	0.532258	+	0.846582i	$0.321344\pi$
$277$	469.007i	1.69316i	−0.532258	+	0.846582i	$0.678656\pi$
$278$	32.3739	0.116453
$279$	0	0
$280$	−39.7372	−0.141919
$281$	− 70.8547i	− 0.252152i	−0.992021	−	0.126076i	$-0.959762\pi$
$281$	− 70.8547i	− 0.252152i	0.992021	−	0.126076i	$-0.0402383\pi$
$282$	0	0
$283$	113.001i	0.399297i	0.979868	+	0.199649i	$0.0639800\pi$
$283$	113.001i	0.399297i	−0.979868	+	0.199649i	$0.936020\pi$
$284$	98.9327	0.348354
$285$	0	0
$286$	0	0
$287$	459.938	1.60257
$288$	0	0
$289$	192.281	0.665331
$290$	151.577	0.522678
$291$	0	0
$292$	170.957i	0.585468i
$293$	− 205.150i	− 0.700170i	−0.936718	−	0.350085i	$-0.886153\pi$
$293$	− 205.150i	− 0.700170i	0.936718	−	0.350085i	$-0.113847\pi$
$294$	0	0
$295$	−85.6218	−0.290243
$296$	− 49.1730i	− 0.166125i
$297$	0	0
$298$	−716.314	−2.40374
$299$	177.707i	0.594336i
$300$	0	0
$301$	−259.808	−0.863148
$302$	325.227	1.07691
$303$	0	0
$304$	319.043i	1.04948i
$305$	126.030i	0.413213i
$306$	0	0
$307$	− 420.339i	− 1.36918i	−0.728928	−	0.684591i	$-0.759981\pi$
$307$	− 420.339i	− 1.36918i	0.728928	−	0.684591i	$-0.240019\pi$
$308$	0	0
$309$	0	0
$310$	110.314i	0.355852i
$311$	431.818	1.38848	0.694241	−	0.719743i	$-0.255740\pi$
$311$	431.818	1.38848	0.694241	+	0.719743i	$0.255740\pi$
$312$	0	0
$313$	315.000	1.00639	0.503195	−	0.864173i	$-0.332158\pi$
$313$	315.000	1.00639	0.503195	+	0.864173i	$0.332158\pi$
$314$	602.177i	1.91776i
$315$	0	0
$316$	138.484i	0.438241i
$317$	113.672	0.358586	0.179293	−	0.983796i	$-0.442619\pi$
$317$	113.672	0.358586	0.179293	+	0.983796i	$0.442619\pi$
$318$	0	0
$319$	0	0
$320$	−2.29452	−0.00717037
$321$	0	0
$322$	701.769	2.17941
$323$	157.448	0.487456
$324$	0	0
$325$	− 104.543i	− 0.321671i
$326$	− 563.066i	− 1.72720i
$327$	0	0
$328$	−282.058	−0.859932
$329$	555.700i	1.68906i
$330$	0	0
$331$	285.685	0.863096	0.431548	−	0.902090i	$-0.357968\pi$
$331$	285.685	0.863096	0.431548	+	0.902090i	$0.357968\pi$
$332$	138.500i	0.417168i
$333$	0	0
$334$	404.596	1.21137
$335$	−50.5671	−0.150946
$336$	0	0
$337$	− 533.154i	− 1.58206i	−0.611778	−	0.791030i	$-0.709545\pi$
$337$	− 533.154i	− 1.58206i	0.611778	−	0.791030i	$-0.290455\pi$
$338$	− 372.794i	− 1.10294i
$339$	0	0
$340$	28.9054i	0.0850159i
$341$	0	0
$342$	0	0
$343$	339.411i	0.989537i
$344$	159.327	0.463161
$345$	0	0
$346$	255.306	0.737880
$347$	− 21.7299i	− 0.0626221i	−0.999510	−	0.0313110i	$-0.990032\pi$
$347$	− 21.7299i	− 0.0626221i	0.999510	−	0.0313110i	$-0.00996824\pi$
$348$	0	0
$349$	514.011i	1.47281i	0.676540	+	0.736406i	$0.263479\pi$
$349$	514.011i	1.47281i	−0.676540	+	0.736406i	$0.736521\pi$
$350$	−412.844	−1.17955
$351$	0	0
$352$	0	0
$353$	458.983	1.30024	0.650118	−	0.759834i	$-0.274720\pi$
$353$	458.983	1.30024	0.650118	+	0.759834i	$0.274720\pi$
$354$	0	0
$355$	56.5321	0.159245
$356$	−234.077	−0.657520
$357$	0	0
$358$	830.225i	2.31906i
$359$	− 302.886i	− 0.843693i	−0.906667	−	0.421847i	$-0.861382\pi$
$359$	− 302.886i	− 0.843693i	0.906667	−	0.421847i	$-0.138618\pi$
$360$	0	0
$361$	104.692	0.290006
$362$	228.677i	0.631704i
$363$	0	0
$364$	71.8911	0.197503
$365$	97.6881i	0.267639i
$366$	0	0
$367$	−173.468	−0.472665	−0.236332	−	0.971672i	$-0.575945\pi$
$367$	−173.468	−0.472665	−0.236332	+	0.971672i	$0.575945\pi$
$368$	−789.977	−2.14668
$369$	0	0
$370$	36.7921i	0.0994382i
$371$	− 207.653i	− 0.559712i
$372$	0	0
$373$	306.830i	0.822600i	0.911500	+	0.411300i	$0.134925\pi$
$373$	306.830i	0.822600i	−0.911500	+	0.411300i	$0.865075\pi$
$374$	0	0
$375$	0	0
$376$	− 340.784i	− 0.906340i
$377$	209.429	0.555515
$378$	0	0
$379$	518.711	1.36863	0.684316	−	0.729186i	$-0.260101\pi$
$379$	518.711	1.36863	0.684316	+	0.729186i	$0.260101\pi$
$380$	− 47.0547i	− 0.123828i
$381$	0	0
$382$	− 64.8906i	− 0.169871i
$383$	440.914	1.15121	0.575606	−	0.817727i	$-0.304766\pi$
$383$	440.914	1.15121	0.575606	+	0.817727i	$0.304766\pi$
$384$	0	0
$385$	0	0
$386$	593.812	1.53837
$387$	0	0
$388$	−24.0897	−0.0620867
$389$	296.673	0.762656	0.381328	−	0.924440i	$-0.375467\pi$
$389$	296.673	0.762656	0.381328	+	0.924440i	$0.375467\pi$
$390$	0	0
$391$	389.855i	0.997072i
$392$	4.33634i	0.0110621i
$393$	0	0
$394$	−727.050	−1.84530
$395$	79.1326i	0.200336i
$396$	0	0
$397$	−448.731	−1.13030	−0.565152	−	0.824987i	$-0.691182\pi$
$397$	−448.731	−1.13030	−0.565152	+	0.824987i	$0.691182\pi$
$398$	− 620.829i	− 1.55987i
$399$	0	0
$400$	464.736	1.16184
$401$	88.0500	0.219576	0.109788	−	0.993955i	$-0.464983\pi$
$401$	88.0500	0.219576	0.109788	+	0.993955i	$0.464983\pi$
$402$	0	0
$403$	152.418i	0.378208i
$404$	− 119.129i	− 0.294874i
$405$	0	0
$406$	− 827.043i	− 2.03705i
$407$	0	0
$408$	0	0
$409$	416.277i	1.01779i	0.860828	+	0.508895i	$0.169946\pi$
$409$	416.277i	1.01779i	−0.860828	+	0.508895i	$0.830054\pi$
$410$	211.041	0.514734
$411$	0	0
$412$	380.070	0.922501
$413$	467.176i	1.13118i
$414$	0	0
$415$	79.1415i	0.190703i
$416$	−145.903	−0.350727
$417$	0	0
$418$	0	0
$419$	−256.059	−0.611118	−0.305559	−	0.952173i	$-0.598843\pi$
$419$	−256.059	−0.611118	−0.305559	+	0.952173i	$0.598843\pi$
$420$	0	0
$421$	200.808	0.476978	0.238489	−	0.971145i	$-0.423348\pi$
$421$	200.808	0.476978	0.238489	+	0.971145i	$0.423348\pi$
$422$	350.148	0.829734
$423$	0	0
$424$	127.344i	0.300339i
$425$	− 229.348i	− 0.539642i
$426$	0	0
$427$	687.654	1.61043
$428$	− 200.252i	− 0.467878i
$429$	0	0
$430$	−119.212	−0.277236
$431$	− 471.429i	− 1.09380i	−0.837197	−	0.546901i	$-0.815807\pi$
$431$	− 471.429i	− 1.09380i	0.837197	−	0.546901i	$-0.184193\pi$
$432$	0	0
$433$	380.429	0.878590	0.439295	−	0.898343i	$-0.355228\pi$
$433$	380.429	0.878590	0.439295	+	0.898343i	$0.355228\pi$
$434$	601.903	1.38687
$435$	0	0
$436$	− 246.361i	− 0.565047i
$437$	− 634.640i	− 1.45227i
$438$	0	0
$439$	178.151i	0.405811i	0.979198	+	0.202906i	$0.0650384\pi$
$439$	178.151i	0.405811i	−0.979198	+	0.202906i	$0.934962\pi$
$440$	0	0
$441$	0	0
$442$	110.376i	0.249720i
$443$	630.904	1.42416	0.712082	−	0.702097i	$-0.247753\pi$
$443$	630.904	1.42416	0.712082	+	0.702097i	$0.247753\pi$
$444$	0	0
$445$	−133.756	−0.300576
$446$	− 208.311i	− 0.467065i
$447$	0	0
$448$	12.5195i	0.0279453i
$449$	−621.342	−1.38384	−0.691918	−	0.721976i	$-0.743234\pi$
$449$	−621.342	−1.38384	−0.691918	+	0.721976i	$0.743234\pi$
$450$	0	0
$451$	0	0
$452$	85.3899	0.188916
$453$	0	0
$454$	407.296	0.897128
$455$	41.0800	0.0902858
$456$	0	0
$457$	1.06515i	0.00233075i	0.999999	+	0.00116537i	$0.000370950\pi$
$457$	1.06515i	0.00233075i	−0.999999	+	0.00116537i	$0.999629\pi$
$458$	177.790i	0.388188i
$459$	0	0
$460$	116.512	0.253286
$461$	− 165.189i	− 0.358327i	−0.983819	−	0.179163i	$-0.942661\pi$
$461$	− 165.189i	− 0.358327i	0.983819	−	0.179163i	$-0.0573391\pi$
$462$	0	0
$463$	212.032	0.457952	0.228976	−	0.973432i	$-0.426462\pi$
$463$	212.032	0.457952	0.228976	+	0.973432i	$0.426462\pi$
$464$	930.997i	2.00646i
$465$	0	0
$466$	−152.681	−0.327641
$467$	−578.694	−1.23917	−0.619587	−	0.784928i	$-0.712700\pi$
$467$	−578.694	−1.23917	−0.619587	+	0.784928i	$0.712700\pi$
$468$	0	0
$469$	275.908i	0.588289i
$470$	254.981i	0.542512i
$471$	0	0
$472$	− 286.496i	− 0.606983i
$473$	0	0
$474$	0	0
$475$	373.352i	0.786005i
$476$	157.716	0.331336
$477$	0	0
$478$	−689.474	−1.44242
$479$	144.165i	0.300970i	0.988612	+	0.150485i	$0.0480836\pi$
$479$	144.165i	0.300970i	−0.988612	+	0.150485i	$0.951916\pi$
$480$	0	0
$481$	50.8347i	0.105685i
$482$	850.186	1.76387
$483$	0	0
$484$	0	0
$485$	−13.7653	−0.0283821
$486$	0	0
$487$	−452.750	−0.929671	−0.464836	−	0.885397i	$-0.653887\pi$
$487$	−452.750	−0.929671	−0.464836	+	0.885397i	$0.653887\pi$
$488$	−421.704	−0.864148
$489$	0	0
$490$	− 3.24453i	− 0.00662149i
$491$	− 116.674i	− 0.237624i	−0.992917	−	0.118812i	$-0.962091\pi$
$491$	− 116.674i	− 0.237624i	0.992917	−	0.118812i	$-0.0379086\pi$
$492$	0	0
$493$	459.449	0.931944
$494$	− 179.680i	− 0.363725i
$495$	0	0
$496$	−677.559	−1.36605
$497$	− 308.455i	− 0.620633i
$498$	0	0
$499$	−347.058	−0.695506	−0.347753	−	0.937586i	$-0.613055\pi$
$499$	−347.058	−0.695506	−0.347753	+	0.937586i	$0.613055\pi$
$500$	−142.021	−0.284043
$501$	0	0
$502$	634.701i	1.26434i
$503$	865.102i	1.71988i	0.510392	+	0.859942i	$0.329500\pi$
$503$	865.102i	1.71988i	−0.510392	+	0.859942i	$0.670500\pi$
$504$	0	0
$505$	− 68.0727i	− 0.134797i
$506$	0	0
$507$	0	0
$508$	189.296i	0.372630i
$509$	−680.591	−1.33711	−0.668557	−	0.743661i	$-0.733088\pi$
$509$	−680.591	−1.33711	−0.668557	+	0.743661i	$0.733088\pi$
$510$	0	0
$511$	533.013	1.04308
$512$	− 302.934i	− 0.591668i
$513$	0	0
$514$	− 421.897i	− 0.820811i
$515$	217.180	0.421709
$516$	0	0
$517$	0	0
$518$	200.748	0.387544
$519$	0	0
$520$	−25.1924	−0.0484469
$521$	−805.783	−1.54661	−0.773304	−	0.634035i	$-0.781397\pi$
$521$	−805.783	−1.54661	−0.773304	+	0.634035i	$0.781397\pi$
$522$	0	0
$523$	150.487i	0.287737i	0.989597	+	0.143869i	$0.0459543\pi$
$523$	150.487i	0.287737i	−0.989597	+	0.143869i	$0.954046\pi$
$524$	206.337i	0.393774i
$525$	0	0
$526$	−551.338	−1.04817
$527$	334.376i	0.634490i
$528$	0	0
$529$	1042.42	1.97055
$530$	− 95.2808i	− 0.179775i
$531$	0	0
$532$	−256.743	−0.482600
$533$	291.589	0.547072
$534$	0	0
$535$	− 114.428i	− 0.213884i
$536$	− 169.201i	− 0.315673i
$537$	0	0
$538$	973.047i	1.80864i
$539$	0	0
$540$	0	0
$541$	402.298i	0.743620i	0.928309	+	0.371810i	$0.121263\pi$
$541$	402.298i	0.743620i	−0.928309	+	0.371810i	$0.878737\pi$
$542$	−683.846	−1.26171
$543$	0	0
$544$	−320.083	−0.588388
$545$	− 140.775i	− 0.258303i
$546$	0	0
$547$	− 502.925i	− 0.919424i	−0.888068	−	0.459712i	$-0.847953\pi$
$547$	− 502.925i	− 0.919424i	0.888068	−	0.459712i	$-0.152047\pi$
$548$	−370.220	−0.675584
$549$	0	0
$550$	0	0
$551$	−747.930	−1.35741
$552$	0	0
$553$	431.769	0.780776
$554$	1174.20	2.11949
$555$	0	0
$556$	− 29.3268i	− 0.0527461i
$557$	215.247i	0.386441i	0.981155	+	0.193220i	$0.0618932\pi$
$557$	215.247i	0.386441i	−0.981155	+	0.193220i	$0.938107\pi$
$558$	0	0
$559$	−164.711	−0.294654
$560$	182.617i	0.326102i
$561$	0	0
$562$	−177.391	−0.315642
$563$	282.068i	0.501008i	0.968116	+	0.250504i	$0.0805964\pi$
$563$	282.068i	0.501008i	−0.968116	+	0.250504i	$0.919404\pi$
$564$	0	0
$565$	48.7935	0.0863602
$566$	282.908	0.499837
$567$	0	0
$568$	189.160i	0.333028i
$569$	69.0572i	0.121366i	0.998157	+	0.0606830i	$0.0193279\pi$
$569$	69.0572i	0.121366i	−0.998157	+	0.0606830i	$0.980672\pi$
$570$	0	0
$571$	184.504i	0.323125i	0.986863	+	0.161562i	$0.0516533\pi$
$571$	184.504i	0.323125i	−0.986863	+	0.161562i	$0.948347\pi$
$572$	0	0
$573$	0	0
$574$	− 1151.50i	− 2.00609i
$575$	−924.452	−1.60774
$576$	0	0
$577$	−879.519	−1.52430	−0.762148	−	0.647403i	$-0.775855\pi$
$577$	−879.519	−1.52430	−0.762148	+	0.647403i	$0.775855\pi$
$578$	− 481.391i	− 0.832857i
$579$	0	0
$580$	− 137.310i	− 0.236742i
$581$	431.818	0.743232
$582$	0	0
$583$	0	0
$584$	−326.871	−0.559710
$585$	0	0
$586$	−513.610	−0.876468
$587$	−574.881	−0.979354	−0.489677	−	0.871904i	$-0.662885\pi$
$587$	−574.881	−0.979354	−0.489677	+	0.871904i	$0.662885\pi$
$588$	0	0
$589$	− 544.327i	− 0.924154i
$590$	214.362i	0.363325i
$591$	0	0
$592$	−225.981	−0.381724
$593$	− 706.849i	− 1.19199i	−0.802989	−	0.595994i	$-0.796758\pi$
$593$	− 706.849i	− 1.19199i	0.802989	−	0.595994i	$-0.203242\pi$
$594$	0	0
$595$	90.1220	0.151465
$596$	648.894i	1.08875i
$597$	0	0
$598$	444.904	0.743986
$599$	686.605	1.14625	0.573126	−	0.819467i	$-0.305730\pi$
$599$	686.605	1.14625	0.573126	+	0.819467i	$0.305730\pi$
$600$	0	0
$601$	60.5165i	0.100693i	0.998732	+	0.0503465i	$0.0160326\pi$
$601$	60.5165i	0.100693i	−0.998732	+	0.0503465i	$0.983967\pi$
$602$	650.451i	1.08048i
$603$	0	0
$604$	− 294.617i	− 0.487776i
$605$	0	0
$606$	0	0
$607$	− 284.502i	− 0.468701i	−0.972152	−	0.234351i	$-0.924704\pi$
$607$	− 284.502i	− 0.468701i	0.972152	−	0.234351i	$-0.0752964\pi$
$608$	521.059	0.857005
$609$	0	0
$610$	315.527	0.517257
$611$	352.300i	0.576595i
$612$	0	0
$613$	925.208i	1.50931i	0.656121	+	0.754656i	$0.272196\pi$
$613$	925.208i	1.50931i	−0.656121	+	0.754656i	$0.727804\pi$
$614$	−1052.35	−1.71393
$615$	0	0
$616$	0	0
$617$	−1158.11	−1.87700	−0.938499	−	0.345283i	$-0.887783\pi$
$617$	−1158.11	−1.87700	−0.938499	+	0.345283i	$0.887783\pi$
$618$	0	0
$619$	−774.186	−1.25070	−0.625352	−	0.780343i	$-0.715045\pi$
$619$	−774.186	−1.25070	−0.625352	+	0.780343i	$0.715045\pi$
$620$	99.9312	0.161179
$621$	0	0
$622$	− 1081.09i	− 1.73809i
$623$	729.812i	1.17145i
$624$	0	0
$625$	501.859	0.802974
$626$	− 788.630i	− 1.25979i
$627$	0	0
$628$	545.500	0.868630
$629$	111.522i	0.177300i
$630$	0	0
$631$	−740.922	−1.17420	−0.587101	−	0.809514i	$-0.699731\pi$
$631$	−740.922	−1.17420	−0.587101	+	0.809514i	$0.699731\pi$
$632$	−264.783	−0.418960
$633$	0	0
$634$	− 284.587i	− 0.448875i
$635$	108.168i	0.170343i
$636$	0	0
$637$	− 4.48288i	− 0.00703748i
$638$	0	0
$639$	0	0
$640$	− 162.971i	− 0.254642i
$641$	−251.780	−0.392793	−0.196396	−	0.980525i	$-0.562924\pi$
$641$	−251.780	−0.392793	−0.196396	+	0.980525i	$0.562924\pi$
$642$	0	0
$643$	−409.808	−0.637337	−0.318668	−	0.947866i	$-0.603236\pi$
$643$	−409.808	−0.637337	−0.318668	+	0.947866i	$0.603236\pi$
$644$	− 635.719i	− 0.987141i
$645$	0	0
$646$	− 394.185i	− 0.610194i
$647$	−229.699	−0.355022	−0.177511	−	0.984119i	$-0.556805\pi$
$647$	−229.699	−0.355022	−0.177511	+	0.984119i	$0.556805\pi$
$648$	0	0
$649$	0	0
$650$	−261.732	−0.402665
$651$	0	0
$652$	−510.070	−0.782317
$653$	−947.407	−1.45085	−0.725427	−	0.688299i	$-0.758358\pi$
$653$	−947.407	−1.45085	−0.725427	+	0.688299i	$0.758358\pi$
$654$	0	0
$655$	117.905i	0.180008i
$656$	1296.23i	1.97596i
$657$	0	0
$658$	1391.24	2.11435
$659$	583.849i	0.885962i	0.896531	+	0.442981i	$0.146079\pi$
$659$	583.849i	0.885962i	−0.896531	+	0.442981i	$0.853921\pi$
$660$	0	0
$661$	−269.115	−0.407134	−0.203567	−	0.979061i	$-0.565253\pi$
$661$	−269.115	−0.407134	−0.203567	+	0.979061i	$0.565253\pi$
$662$	− 715.236i	− 1.08042i
$663$	0	0
$664$	−264.813	−0.398814
$665$	−146.708	−0.220614
$666$	0	0
$667$	− 1851.94i	− 2.77652i
$668$	− 366.515i	− 0.548676i
$669$	0	0
$670$	126.599i	0.188954i
$671$	0	0
$672$	0	0
$673$	− 1050.64i	− 1.56113i	−0.625073	−	0.780567i	$-0.714931\pi$
$673$	− 1050.64i	− 1.56113i	0.625073	−	0.780567i	$-0.285069\pi$
$674$	−1334.80	−1.98041
$675$	0	0
$676$	−337.706	−0.499565
$677$	− 209.175i	− 0.308973i	−0.987995	−	0.154486i	$-0.950628\pi$
$677$	− 209.175i	− 0.308973i	0.987995	−	0.154486i	$-0.0493723\pi$
$678$	0	0
$679$	75.1073i	0.110615i
$680$	−55.2674	−0.0812756
$681$	0	0
$682$	0	0
$683$	−284.569	−0.416646	−0.208323	−	0.978060i	$-0.566801\pi$
$683$	−284.569	−0.416646	−0.208323	+	0.978060i	$0.566801\pi$
$684$	0	0
$685$	−211.551	−0.308834
$686$	849.746	1.23870
$687$	0	0
$688$	− 732.209i	− 1.06426i
$689$	− 131.647i	− 0.191070i
$690$	0	0
$691$	−1338.13	−1.93652	−0.968259	−	0.249948i	$-0.919586\pi$
$691$	−1338.13	−1.93652	−0.968259	+	0.249948i	$0.919586\pi$
$692$	− 231.277i	− 0.334215i
$693$	0	0
$694$	−54.4026	−0.0783899
$695$	− 16.7579i	− 0.0241122i
$696$	0	0
$697$	639.693	0.917780
$698$	1286.87	1.84366
$699$	0	0
$700$	373.987i	0.534267i
$701$	996.941i	1.42217i	0.703106	+	0.711085i	$0.251796\pi$
$701$	996.941i	1.42217i	−0.703106	+	0.711085i	$0.748204\pi$
$702$	0	0
$703$	− 181.545i	− 0.258243i
$704$	0	0
$705$	0	0
$706$	− 1149.10i	− 1.62763i
$707$	−371.423	−0.525351
$708$	0	0
$709$	738.820	1.04206	0.521030	−	0.853538i	$-0.325548\pi$
$709$	738.820	1.04206	0.521030	+	0.853538i	$0.325548\pi$
$710$	− 141.533i	− 0.199342i
$711$	0	0
$712$	− 447.558i	− 0.628592i
$713$	1347.80	1.89032
$714$	0	0
$715$	0	0
$716$	752.084	1.05040
$717$	0	0
$718$	−758.301	−1.05613
$719$	−251.656	−0.350008	−0.175004	−	0.984568i	$-0.555994\pi$
$719$	−251.656	−0.350008	−0.175004	+	0.984568i	$0.555994\pi$
$720$	0	0
$721$	− 1184.99i	− 1.64354i
$722$	− 262.106i	− 0.363028i
$723$	0	0
$724$	207.154	0.286124
$725$	1089.48i	1.50273i
$726$	0	0
$727$	611.506	0.841137	0.420568	−	0.907261i	$-0.361831\pi$
$727$	611.506	0.841137	0.420568	+	0.907261i	$0.361831\pi$
$728$	137.456i	0.188814i
$729$	0	0
$730$	244.571	0.335028
$731$	−361.346	−0.494318
$732$	0	0
$733$	1156.89i	1.57830i	0.614200	+	0.789150i	$0.289479\pi$
$733$	1156.89i	1.57830i	−0.614200	+	0.789150i	$0.710521\pi$
$734$	434.292i	0.591678i
$735$	0	0
$736$	1290.19i	1.75297i
$737$	0	0
$738$	0	0
$739$	− 70.8593i	− 0.0958854i	−0.998850	−	0.0479427i	$-0.984734\pi$
$739$	− 70.8593i	− 0.0958854i	0.998850	−	0.0479427i	$-0.0152665\pi$
$740$	33.3293	0.0450395
$741$	0	0
$742$	−519.878	−0.700644
$743$	65.5138i	0.0881747i	0.999028	+	0.0440874i	$0.0140380\pi$
$743$	65.5138i	0.0881747i	−0.999028	+	0.0440874i	$0.985962\pi$
$744$	0	0
$745$	370.791i	0.497707i
$746$	768.176	1.02973
$747$	0	0
$748$	0	0
$749$	−624.350	−0.833578
$750$	0	0
$751$	−905.947	−1.20632	−0.603161	−	0.797620i	$-0.706092\pi$
$751$	−905.947	−1.20632	−0.603161	+	0.797620i	$0.706092\pi$
$752$	−1566.11	−2.08260
$753$	0	0
$754$	− 524.324i	− 0.695390i
$755$	− 168.350i	− 0.222980i
$756$	0	0
$757$	118.205	0.156149	0.0780747	−	0.996948i	$-0.475123\pi$
$757$	118.205	0.156149	0.0780747	+	0.996948i	$0.475123\pi$
$758$	− 1298.64i	− 1.71324i
$759$	0	0
$760$	89.9691	0.118380
$761$	− 48.5151i	− 0.0637517i	−0.999492	−	0.0318759i	$-0.989852\pi$
$761$	− 48.5151i	− 0.0637517i	0.999492	−	0.0318759i	$-0.0101481\pi$
$762$	0	0
$763$	−768.109	−1.00670
$764$	−58.7831	−0.0769412
$765$	0	0
$766$	− 1103.87i	− 1.44108i
$767$	296.178i	0.386151i
$768$	0	0
$769$	1144.24i	1.48795i	0.668206	+	0.743976i	$0.267062\pi$
$769$	1144.24i	1.48795i	−0.668206	+	0.743976i	$0.732938\pi$
$770$	0	0
$771$	0	0
$772$	− 537.922i	− 0.696790i
$773$	1180.17	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$773$	1180.17	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$774$	0	0
$775$	−792.897	−1.02309
$776$	− 46.0596i	− 0.0593552i
$777$	0	0
$778$	− 742.748i	− 0.954688i
$779$	−1041.35	−1.33677
$780$	0	0
$781$	0	0
$782$	976.036	1.24813
$783$	0	0
$784$	19.9282	0.0254186
$785$	311.710	0.397082
$786$	0	0
$787$	− 114.796i	− 0.145866i	−0.997337	−	0.0729328i	$-0.976764\pi$
$787$	− 114.796i	− 0.145866i	0.997337	−	0.0729328i	$-0.0232358\pi$
$788$	658.620i	0.835812i
$789$	0	0
$790$	198.115	0.250779
$791$	− 266.231i	− 0.336575i
$792$	0	0
$793$	435.955	0.549754
$794$	1123.44i	1.41491i
$795$	0	0
$796$	−562.396	−0.706528
$797$	5.08408	0.00637903	0.00318951	−	0.999995i	$-0.498985\pi$
$797$	5.08408	0.00637903	0.00318951	+	0.999995i	$0.498985\pi$
$798$	0	0
$799$	772.880i	0.967309i
$800$	− 759.004i	− 0.948755i
$801$	0	0
$802$	− 220.441i	− 0.274864i
$803$	0	0
$804$	0	0
$805$	− 363.262i	− 0.451258i
$806$	381.591	0.473438
$807$	0	0
$808$	227.776	0.281901
$809$	905.063i	1.11874i	0.828917	+	0.559371i	$0.188957\pi$
$809$	905.063i	1.11874i	−0.828917	+	0.559371i	$0.811043\pi$
$810$	0	0
$811$	− 800.505i	− 0.987059i	−0.869729	−	0.493529i	$-0.835707\pi$
$811$	− 800.505i	− 0.987059i	0.869729	−	0.493529i	$-0.164293\pi$
$812$	−749.201	−0.922662
$813$	0	0
$814$	0	0
$815$	−291.465	−0.357625
$816$	0	0
$817$	588.231	0.719989
$818$	1042.18	1.27406
$819$	0	0
$820$	− 191.178i	− 0.233143i
$821$	409.256i	0.498485i	0.968441	+	0.249242i	$0.0801816\pi$
$821$	409.256i	0.498485i	−0.968441	+	0.249242i	$0.919818\pi$
$822$	0	0
$823$	283.468	0.344432	0.172216	−	0.985059i	$-0.444907\pi$
$823$	283.468	0.344432	0.172216	+	0.985059i	$0.444907\pi$
$824$	726.698i	0.881915i
$825$	0	0
$826$	1169.62	1.41600
$827$	− 1000.99i	− 1.21039i	−0.796079	−	0.605193i	$-0.793096\pi$
$827$	− 1000.99i	− 1.21039i	0.796079	−	0.605193i	$-0.206904\pi$
$828$	0	0
$829$	127.546	0.153855	0.0769277	−	0.997037i	$-0.475489\pi$
$829$	127.546	0.153855	0.0769277	+	0.997037i	$0.475489\pi$
$830$	198.138	0.238720
$831$	0	0
$832$	7.93705i	0.00953973i
$833$	− 9.83460i	− 0.0118062i
$834$	0	0
$835$	− 209.434i	− 0.250820i
$836$	0	0
$837$	0	0
$838$	641.065i	0.764994i
$839$	−171.946	−0.204942	−0.102471	−	0.994736i	$-0.532675\pi$
$839$	−171.946	−0.204942	−0.102471	+	0.994736i	$0.532675\pi$
$840$	0	0
$841$	−1341.53	−1.59516
$842$	− 502.739i	− 0.597078i
$843$	0	0
$844$	− 317.192i	− 0.375820i
$845$	−192.972	−0.228369
$846$	0	0
$847$	0	0
$848$	585.224	0.690122
$849$	0	0
$850$	−574.192	−0.675520
$851$	449.521	0.528227
$852$	0	0
$853$	240.131i	0.281513i	0.990044	+	0.140757i	$0.0449535\pi$
$853$	240.131i	0.281513i	−0.990044	+	0.140757i	$0.955046\pi$
$854$	− 1721.60i	− 2.01593i
$855$	0	0
$856$	382.883	0.447293
$857$	200.752i	0.234250i	0.993117	+	0.117125i	$0.0373678\pi$
$857$	200.752i	0.234250i	−0.993117	+	0.117125i	$0.962632\pi$
$858$	0	0
$859$	−681.699	−0.793596	−0.396798	−	0.917906i	$-0.629879\pi$
$859$	−681.699	−0.793596	−0.396798	+	0.917906i	$0.629879\pi$
$860$	107.991i	0.125571i
$861$	0	0
$862$	−1180.26	−1.36921
$863$	−544.591	−0.631044	−0.315522	−	0.948918i	$-0.602180\pi$
$863$	−544.591	−0.631044	−0.315522	+	0.948918i	$0.602180\pi$
$864$	0	0
$865$	− 132.156i	− 0.152782i
$866$	− 952.438i	− 1.09981i
$867$	0	0
$868$	− 545.252i	− 0.628170i
$869$	0	0
$870$	0	0
$871$	174.918i	0.200825i
$872$	471.044	0.540188
$873$	0	0
$874$	−1588.88	−1.81794
$875$	442.798i	0.506055i
$876$	0	0
$877$	1054.93i	1.20288i	0.798918	+	0.601440i	$0.205406\pi$
$877$	1054.93i	1.20288i	−0.798918	+	0.601440i	$0.794594\pi$
$878$	446.017	0.507992
$879$	0	0
$880$	0	0
$881$	−7.75078	−0.00879771	−0.00439885	−	0.999990i	$-0.501400\pi$
$881$	−7.75078	−0.00879771	−0.00439885	+	0.999990i	$0.501400\pi$
$882$	0	0
$883$	−1127.60	−1.27701	−0.638506	−	0.769617i	$-0.720447\pi$
$883$	−1127.60	−1.27701	−0.638506	+	0.769617i	$0.720447\pi$
$884$	99.9878	0.113108
$885$	0	0
$886$	− 1579.52i	− 1.78276i
$887$	− 1098.50i	− 1.23844i	−0.785216	−	0.619222i	$-0.787448\pi$
$887$	− 1098.50i	− 1.23844i	0.785216	−	0.619222i	$-0.212552\pi$
$888$	0	0
$889$	590.192	0.663883
$890$	334.871i	0.376259i
$891$	0	0
$892$	−188.705	−0.211553
$893$	− 1258.16i	− 1.40892i
$894$	0	0
$895$	429.756	0.480174
$896$	−889.214	−0.992426
$897$	0	0
$898$	1555.59i	1.73228i
$899$	− 1588.40i	− 1.76685i
$900$	0	0
$901$	− 288.809i	− 0.320543i
$902$	0	0
$903$	0	0
$904$	163.266i	0.180604i
$905$	118.372	0.130798
$906$	0	0
$907$	193.801	0.213673	0.106836	−	0.994277i	$-0.465928\pi$
$907$	193.801	0.213673	0.106836	+	0.994277i	$0.465928\pi$
$908$	− 368.961i	− 0.406345i
$909$	0	0
$910$	− 102.847i	− 0.113019i
$911$	1118.76	1.22806	0.614031	−	0.789282i	$-0.289547\pi$
$911$	1118.76	1.22806	0.614031	+	0.789282i	$0.289547\pi$
$912$	0	0
$913$	0	0
$914$	2.66670	0.00291761
$915$	0	0
$916$	161.056	0.175826
$917$	643.324	0.701553
$918$	0	0
$919$	836.589i	0.910325i	0.890408	+	0.455163i	$0.150419\pi$
$919$	836.589i	0.910325i	−0.890408	+	0.455163i	$0.849581\pi$
$920$	222.771i	0.242143i
$921$	0	0
$922$	−413.564	−0.448551
$923$	− 195.552i	− 0.211866i
$924$	0	0
$925$	−264.449	−0.285890
$926$	− 530.840i	− 0.573262i
$927$	0	0
$928$	1520.50	1.63847
$929$	−1044.26	−1.12407	−0.562036	−	0.827113i	$-0.689982\pi$
$929$	−1044.26	−1.12407	−0.562036	+	0.827113i	$0.689982\pi$
$930$	0	0
$931$	16.0096i	0.0171962i
$932$	138.311i	0.148402i
$933$	0	0
$934$	1448.81i	1.55119i
$935$	0	0
$936$	0	0
$937$	− 508.188i	− 0.542356i	−0.962529	−	0.271178i	$-0.912587\pi$
$937$	− 508.188i	− 0.542356i	0.962529	−	0.271178i	$-0.0874132\pi$
$938$	690.759	0.736417
$939$	0	0
$940$	230.982	0.245725
$941$	654.463i	0.695498i	0.937588	+	0.347749i	$0.113054\pi$
$941$	654.463i	0.695498i	−0.937588	+	0.347749i	$0.886946\pi$
$942$	0	0
$943$	− 2578.46i	− 2.73432i
$944$	−1316.63	−1.39473
$945$	0	0
$946$	0	0
$947$	1281.12	1.35282	0.676411	−	0.736524i	$-0.263534\pi$
$947$	1281.12	1.35282	0.676411	+	0.736524i	$0.263534\pi$
$948$	0	0
$949$	337.917	0.356076
$950$	934.720	0.983916
$951$	0	0
$952$	301.554i	0.316758i
$953$	− 940.126i	− 0.986491i	−0.869890	−	0.493245i	$-0.835810\pi$
$953$	− 940.126i	− 0.986491i	0.869890	−	0.493245i	$-0.164190\pi$
$954$	0	0
$955$	−33.5898	−0.0351726
$956$	624.581i	0.653327i
$957$	0	0
$958$	360.929	0.376753
$959$	1154.28i	1.20363i
$960$	0	0
$961$	195.000	0.202914
$962$	127.269	0.132296
$963$	0	0
$964$	− 770.166i	− 0.798928i
$965$	− 307.380i	− 0.318528i
$966$	0	0
$967$	− 557.871i	− 0.576909i	−0.957494	−	0.288455i	$-0.906859\pi$
$967$	− 557.871i	− 0.576909i	0.957494	−	0.288455i	$-0.0931414\pi$
$968$	0	0
$969$	0	0
$970$	34.4627i	0.0355285i
$971$	1079.20	1.11144	0.555718	−	0.831371i	$-0.312443\pi$
$971$	1079.20	1.11144	0.555718	+	0.831371i	$0.312443\pi$
$972$	0	0
$973$	−91.4359	−0.0939732
$974$	1133.50i	1.16376i
$975$	0	0
$976$	1938.00i	1.98565i
$977$	−422.306	−0.432248	−0.216124	−	0.976366i	$-0.569341\pi$
$977$	−422.306	−0.432248	−0.216124	+	0.976366i	$0.569341\pi$
$978$	0	0
$979$	0	0
$980$	−2.93915	−0.00299914
$981$	0	0
$982$	−292.102	−0.297457
$983$	−1572.84	−1.60004	−0.800019	−	0.599975i	$-0.795177\pi$
$983$	−1572.84	−1.60004	−0.800019	+	0.599975i	$0.795177\pi$
$984$	0	0
$985$	376.349i	0.382080i
$986$	− 1150.27i	− 1.16660i
$987$	0	0
$988$	−162.769	−0.164746
$989$	1456.51i	1.47271i
$990$	0	0
$991$	50.6025	0.0510621	0.0255310	−	0.999674i	$-0.491872\pi$
$991$	50.6025	0.0510621	0.0255310	+	0.999674i	$0.491872\pi$
$992$	1106.59i	1.11551i
$993$	0	0
$994$	−772.243	−0.776905
$995$	−321.365	−0.322979
$996$	0	0
$997$	623.591i	0.625468i	0.949841	+	0.312734i	$0.101245\pi$
$997$	623.591i	0.625468i	−0.949841	+	0.312734i	$0.898755\pi$
$998$	868.889i	0.870631i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.i.604.3	✓	8
3.2	odd	2	inner	1089.3.c.i.604.6	yes	8
11.10	odd	2	inner	1089.3.c.i.604.5	yes	8
33.32	even	2	inner	1089.3.c.i.604.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.3.c.i.604.3	✓	8	1.1	even	1	trivial
1089.3.c.i.604.4	yes	8	33.32	even	2	inner
1089.3.c.i.604.5	yes	8	11.10	odd	2	inner
1089.3.c.i.604.6	yes	8	3.2	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.50359i q^{2} -2.26795 q^{4} -1.29595 q^{5} +7.07107i q^{7} -4.33634i q^{8} +O(q^{10})\)	\(q-2.50359i q^{2} -2.26795 q^{4} -1.29595 q^{5} +7.07107i q^{7} -4.33634i q^{8} +3.24453i q^{10} +4.48288i q^{13} +17.7030 q^{14} -19.9282 q^{16} +9.83460i q^{17} -16.0096i q^{19} +2.93915 q^{20} +39.6412 q^{23} -23.3205 q^{25} +11.2233 q^{26} -16.0368i q^{28} -46.7176i q^{29} +34.0000 q^{31} +32.5466i q^{32} +24.6218 q^{34} -9.16377i q^{35} +11.3397 q^{37} -40.0815 q^{38} +5.61969i q^{40} -65.0451i q^{41} +36.7423i q^{43} -99.2451i q^{46} +78.5879 q^{47} -1.00000 q^{49} +58.3849i q^{50} -10.1669i q^{52} -29.3666 q^{53} +30.6626 q^{56} -116.962 q^{58} +66.0686 q^{59} -97.2489i q^{61} -85.1220i q^{62} +1.77053 q^{64} -5.80959i q^{65} +39.0192 q^{67} -22.3044i q^{68} -22.9423 q^{70} -43.6221 q^{71} -75.3794i q^{73} -28.3900i q^{74} +36.3090i q^{76} -61.0614i q^{79} +25.8260 q^{80} -162.846 q^{82} -61.0683i q^{83} -12.7452i q^{85} +91.9877 q^{86} +103.211 q^{89} -31.6987 q^{91} -89.9042 q^{92} -196.752i q^{94} +20.7477i q^{95} +10.6218 q^{97} +2.50359i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4}+O(q^{10}) \) 8 * q - 32 * q^4	\( 8 q - 32 q^{4} - 104 q^{16} - 48 q^{25} + 272 q^{31} - 288 q^{34} + 160 q^{37} - 8 q^{49} - 520 q^{58} + 416 q^{64} + 520 q^{67} + 440 q^{70} + 360 q^{82} - 600 q^{91} - 400 q^{97}+O(q^{100}) \) 8 * q - 32 * q^4 - 104 * q^16 - 48 * q^25 + 272 * q^31 - 288 * q^34 + 160 * q^37 - 8 * q^49 - 520 * q^58 + 416 * q^64 + 520 * q^67 + 440 * q^70 + 360 * q^82 - 600 * q^91 - 400 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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