LMFDB - Embedded newform 108.5.c.c.53.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(53,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, 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("108.53");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.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:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	108.53
Dual form			108.5.c.c.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{5} +5.00000 q^{7} +O(q^{10})$	$q+9.00000i q^{5} +5.00000 q^{7} +117.000i q^{11} -34.0000 q^{13} +450.000i q^{17} -64.0000 q^{19} +612.000i q^{23} +544.000 q^{25} +1062.00i q^{29} -697.000 q^{31} +45.0000i q^{35} -748.000 q^{37} +684.000i q^{41} +2618.00 q^{43} -2646.00i q^{47} -2376.00 q^{49} -1071.00i q^{53} -1053.00 q^{55} -5814.00i q^{59} +6404.00 q^{61} -306.000i q^{65} -5218.00 q^{67} -6570.00i q^{71} -4519.00 q^{73} +585.000i q^{77} +7502.00 q^{79} -5481.00i q^{83} -4050.00 q^{85} +8874.00i q^{89} -170.000 q^{91} -576.000i q^{95} +10571.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{7}+O(q^{10}) $ 2 * q + 10 * q^7	$ 2 q + 10 q^{7} - 68 q^{13} - 128 q^{19} + 1088 q^{25} - 1394 q^{31} - 1496 q^{37} + 5236 q^{43} - 4752 q^{49} - 2106 q^{55} + 12808 q^{61} - 10436 q^{67} - 9038 q^{73} + 15004 q^{79} - 8100 q^{85} - 340 q^{91} + 21142 q^{97}+O(q^{100}) $ 2 * q + 10 * q^7 - 68 * q^13 - 128 * q^19 + 1088 * q^25 - 1394 * q^31 - 1496 * q^37 + 5236 * q^43 - 4752 * q^49 - 2106 * q^55 + 12808 * q^61 - 10436 * q^67 - 9038 * q^73 + 15004 * q^79 - 8100 * q^85 - 340 * q^91 + 21142 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	9.00000i	0.360000i	0.983667	+	0.180000i	$0.0576098\pi$
$5$	9.00000i	0.360000i	−0.983667	+	0.180000i	$0.942390\pi$
$6$	0	0
$7$	5.00000	0.102041	0.0510204	−	0.998698i	$-0.483753\pi$
$7$	5.00000	0.102041	0.0510204	+	0.998698i	$0.483753\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	117.000i	0.966942i	0.875360	+	0.483471i	$0.160624\pi$
$11$	117.000i	0.966942i	−0.875360	+	0.483471i	$0.839376\pi$
$12$	0	0
$13$	−34.0000	−0.201183	−0.100592	−	0.994928i	$-0.532074\pi$
$13$	−34.0000	−0.201183	−0.100592	+	0.994928i	$0.532074\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	450.000i	1.55709i	0.627587	+	0.778547i	$0.284043\pi$
$17$	450.000i	1.55709i	−0.627587	+	0.778547i	$0.715957\pi$
$18$	0	0
$19$	−64.0000	−0.177285	−0.0886427	−	0.996063i	$-0.528253\pi$
$19$	−64.0000	−0.177285	−0.0886427	+	0.996063i	$0.528253\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	612.000i	1.15690i	0.815718	+	0.578450i	$0.196342\pi$
$23$	612.000i	1.15690i	−0.815718	+	0.578450i	$0.803658\pi$
$24$	0	0
$25$	544.000	0.870400
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1062.00i	1.26278i	0.775464	+	0.631391i	$0.217516\pi$
$29$	1062.00i	1.26278i	−0.775464	+	0.631391i	$0.782484\pi$
$30$	0	0
$31$	−697.000	−0.725286	−0.362643	−	0.931928i	$-0.618126\pi$
$31$	−697.000	−0.725286	−0.362643	+	0.931928i	$0.618126\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	45.0000i	0.0367347i
$36$	0	0
$37$	−748.000	−0.546384	−0.273192	−	0.961959i	$-0.588079\pi$
$37$	−748.000	−0.546384	−0.273192	+	0.961959i	$0.588079\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	684.000i	0.406901i	0.979085	+	0.203450i	$0.0652155\pi$
$41$	684.000i	0.406901i	−0.979085	+	0.203450i	$0.934784\pi$
$42$	0	0
$43$	2618.00	1.41590	0.707950	−	0.706262i	$-0.249620\pi$
$43$	2618.00	1.41590	0.707950	+	0.706262i	$0.249620\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2646.00i	− 1.19783i	−0.800814	−	0.598914i	$-0.795599\pi$
$47$	− 2646.00i	− 1.19783i	0.800814	−	0.598914i	$-0.204401\pi$
$48$	0	0
$49$	−2376.00	−0.989588
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1071.00i	− 0.381274i	−0.981661	−	0.190637i	$-0.938945\pi$
$53$	− 1071.00i	− 0.381274i	0.981661	−	0.190637i	$-0.0610554\pi$
$54$	0	0
$55$	−1053.00	−0.348099
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5814.00i	− 1.67021i	−0.550091	−	0.835105i	$-0.685407\pi$
$59$	− 5814.00i	− 1.67021i	0.550091	−	0.835105i	$-0.314593\pi$
$60$	0	0
$61$	6404.00	1.72104	0.860521	−	0.509414i	$-0.170138\pi$
$61$	6404.00	1.72104	0.860521	+	0.509414i	$0.170138\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 306.000i	− 0.0724260i
$66$	0	0
$67$	−5218.00	−1.16240	−0.581198	−	0.813762i	$-0.697416\pi$
$67$	−5218.00	−1.16240	−0.581198	+	0.813762i	$0.697416\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 6570.00i	− 1.30331i	−0.758514	−	0.651656i	$-0.774074\pi$
$71$	− 6570.00i	− 1.30331i	0.758514	−	0.651656i	$-0.225926\pi$
$72$	0	0
$73$	−4519.00	−0.848002	−0.424001	−	0.905662i	$-0.639375\pi$
$73$	−4519.00	−0.848002	−0.424001	+	0.905662i	$0.639375\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	585.000i	0.0986676i
$78$	0	0
$79$	7502.00	1.20205	0.601025	−	0.799230i	$-0.294759\pi$
$79$	7502.00	1.20205	0.601025	+	0.799230i	$0.294759\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5481.00i	− 0.795616i	−0.917469	−	0.397808i	$-0.869771\pi$
$83$	− 5481.00i	− 0.795616i	0.917469	−	0.397808i	$-0.130229\pi$
$84$	0	0
$85$	−4050.00	−0.560554
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8874.00i	1.12031i	0.828387	+	0.560157i	$0.189259\pi$
$89$	8874.00i	1.12031i	−0.828387	+	0.560157i	$0.810741\pi$
$90$	0	0
$91$	−170.000	−0.0205289
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 576.000i	− 0.0638227i
$96$	0	0
$97$	10571.0	1.12350	0.561749	−	0.827307i	$-0.310129\pi$
$97$	10571.0	1.12350	0.561749	+	0.827307i	$0.310129\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13113.0i	1.28546i	0.766092	+	0.642731i	$0.222199\pi$
$101$	13113.0i	1.28546i	−0.766092	+	0.642731i	$0.777801\pi$
$102$	0	0
$103$	−5830.00	−0.549533	−0.274767	−	0.961511i	$-0.588601\pi$
$103$	−5830.00	−0.549533	−0.274767	+	0.961511i	$0.588601\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1089.00i	0.0951175i	0.998868	+	0.0475587i	$0.0151441\pi$
$107$	1089.00i	0.0951175i	−0.998868	+	0.0475587i	$0.984856\pi$
$108$	0	0
$109$	−5020.00	−0.422523	−0.211262	−	0.977430i	$-0.567757\pi$
$109$	−5020.00	−0.422523	−0.211262	+	0.977430i	$0.567757\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	21384.0i	1.67468i	0.546682	+	0.837340i	$0.315891\pi$
$113$	21384.0i	1.67468i	−0.546682	+	0.837340i	$0.684109\pi$
$114$	0	0
$115$	−5508.00	−0.416484
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2250.00i	0.158887i
$120$	0	0
$121$	952.000	0.0650229
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10521.0i	0.673344i
$126$	0	0
$127$	9227.00	0.572075	0.286038	−	0.958218i	$-0.407662\pi$
$127$	9227.00	0.572075	0.286038	+	0.958218i	$0.407662\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4275.00i	0.249111i	0.992213	+	0.124556i	$0.0397505\pi$
$131$	4275.00i	0.249111i	−0.992213	+	0.124556i	$0.960249\pi$
$132$	0	0
$133$	−320.000	−0.0180903
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11322.0i	0.603229i	0.953430	+	0.301614i	$0.0975255\pi$
$137$	11322.0i	0.603229i	−0.953430	+	0.301614i	$0.902474\pi$
$138$	0	0
$139$	9812.00	0.507841	0.253921	−	0.967225i	$-0.418280\pi$
$139$	9812.00	0.507841	0.253921	+	0.967225i	$0.418280\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 3978.00i	− 0.194533i
$144$	0	0
$145$	−9558.00	−0.454602
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 16479.0i	− 0.742264i	−0.928580	−	0.371132i	$-0.878970\pi$
$149$	− 16479.0i	− 0.742264i	0.928580	−	0.371132i	$-0.121030\pi$
$150$	0	0
$151$	−25555.0	−1.12078	−0.560392	−	0.828227i	$-0.689350\pi$
$151$	−25555.0	−1.12078	−0.560392	+	0.828227i	$0.689350\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 6273.00i	− 0.261103i
$156$	0	0
$157$	21164.0	0.858615	0.429307	−	0.903158i	$-0.358758\pi$
$157$	21164.0	0.858615	0.429307	+	0.903158i	$0.358758\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3060.00i	0.118051i
$162$	0	0
$163$	33830.0	1.27329	0.636644	−	0.771158i	$-0.280322\pi$
$163$	33830.0	1.27329	0.636644	+	0.771158i	$0.280322\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 27378.0i	− 0.981677i	−0.871250	−	0.490839i	$-0.836690\pi$
$167$	− 27378.0i	− 0.981677i	0.871250	−	0.490839i	$-0.163310\pi$
$168$	0	0
$169$	−27405.0	−0.959525
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 46197.0i	− 1.54355i	−0.635894	−	0.771777i	$-0.719368\pi$
$173$	− 46197.0i	− 1.54355i	0.635894	−	0.771777i	$-0.280632\pi$
$174$	0	0
$175$	2720.00	0.0888163
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 20781.0i	− 0.648575i	−0.945959	−	0.324288i	$-0.894875\pi$
$179$	− 20781.0i	− 0.648575i	0.945959	−	0.324288i	$-0.105125\pi$
$180$	0	0
$181$	−19504.0	−0.595342	−0.297671	−	0.954669i	$-0.596210\pi$
$181$	−19504.0	−0.595342	−0.297671	+	0.954669i	$0.596210\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 6732.00i	− 0.196698i
$186$	0	0
$187$	−52650.0	−1.50562
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7038.00i	0.192922i	0.995337	+	0.0964612i	$0.0307524\pi$
$191$	7038.00i	0.192922i	−0.995337	+	0.0964612i	$0.969248\pi$
$192$	0	0
$193$	51527.0	1.38331	0.691656	−	0.722227i	$-0.256881\pi$
$193$	51527.0	1.38331	0.691656	+	0.722227i	$0.256881\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 60435.0i	− 1.55724i	−0.627495	−	0.778621i	$-0.715920\pi$
$197$	− 60435.0i	− 1.55724i	0.627495	−	0.778621i	$-0.284080\pi$
$198$	0	0
$199$	45665.0	1.15313	0.576564	−	0.817052i	$-0.304393\pi$
$199$	45665.0	1.15313	0.576564	+	0.817052i	$0.304393\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5310.00i	0.128855i
$204$	0	0
$205$	−6156.00	−0.146484
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 7488.00i	− 0.171425i
$210$	0	0
$211$	13124.0	0.294782	0.147391	−	0.989078i	$-0.452912\pi$
$211$	13124.0	0.294782	0.147391	+	0.989078i	$0.452912\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	23562.0i	0.509724i
$216$	0	0
$217$	−3485.00	−0.0740088
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 15300.0i	− 0.313261i
$222$	0	0
$223$	−78850.0	−1.58559	−0.792797	−	0.609486i	$-0.791376\pi$
$223$	−78850.0	−1.58559	−0.792797	+	0.609486i	$0.791376\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22302.0i	0.432805i	0.976304	+	0.216402i	$0.0694323\pi$
$227$	22302.0i	0.432805i	−0.976304	+	0.216402i	$0.930568\pi$
$228$	0	0
$229$	8870.00	0.169142	0.0845712	−	0.996417i	$-0.473048\pi$
$229$	8870.00	0.169142	0.0845712	+	0.996417i	$0.473048\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26010.0i	0.479103i	0.970884	+	0.239551i	$0.0770003\pi$
$233$	26010.0i	0.479103i	−0.970884	+	0.239551i	$0.923000\pi$
$234$	0	0
$235$	23814.0	0.431218
$236$	0	0
$237$	0	0
$238$	0	0
$239$	64152.0i	1.12309i	0.827446	+	0.561545i	$0.189793\pi$
$239$	64152.0i	1.12309i	−0.827446	+	0.561545i	$0.810207\pi$
$240$	0	0
$241$	33422.0	0.575438	0.287719	−	0.957715i	$-0.407103\pi$
$241$	33422.0	0.575438	0.287719	+	0.957715i	$0.407103\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 21384.0i	− 0.356252i
$246$	0	0
$247$	2176.00	0.0356669
$248$	0	0
$249$	0	0
$250$	0	0
$251$	110466.i	1.75340i	0.481037	+	0.876700i	$0.340260\pi$
$251$	110466.i	1.75340i	−0.481037	+	0.876700i	$0.659740\pi$
$252$	0	0
$253$	−71604.0	−1.11866
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15174.0i	0.229739i	0.993381	+	0.114869i	$0.0366449\pi$
$257$	15174.0i	0.229739i	−0.993381	+	0.114869i	$0.963355\pi$
$258$	0	0
$259$	−3740.00	−0.0557535
$260$	0	0
$261$	0	0
$262$	0	0
$263$	69948.0i	1.01126i	0.862750	+	0.505631i	$0.168740\pi$
$263$	69948.0i	1.01126i	−0.862750	+	0.505631i	$0.831260\pi$
$264$	0	0
$265$	9639.00	0.137259
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 50346.0i	− 0.695762i	−0.937539	−	0.347881i	$-0.886901\pi$
$269$	− 50346.0i	− 0.695762i	0.937539	−	0.347881i	$-0.113099\pi$
$270$	0	0
$271$	108323.	1.47497	0.737483	−	0.675366i	$-0.236014\pi$
$271$	108323.	1.47497	0.737483	+	0.675366i	$0.236014\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	63648.0i	0.841626i
$276$	0	0
$277$	30716.0	0.400318	0.200159	−	0.979763i	$-0.435854\pi$
$277$	30716.0	0.400318	0.200159	+	0.979763i	$0.435854\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	129258.i	1.63699i	0.574517	+	0.818493i	$0.305190\pi$
$281$	129258.i	1.63699i	−0.574517	+	0.818493i	$0.694810\pi$
$282$	0	0
$283$	−63976.0	−0.798811	−0.399406	−	0.916774i	$-0.630783\pi$
$283$	−63976.0	−0.798811	−0.399406	+	0.916774i	$0.630783\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3420.00i	0.0415205i
$288$	0	0
$289$	−118979.	−1.42454
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 108630.i	− 1.26536i	−0.774413	−	0.632681i	$-0.781955\pi$
$293$	− 108630.i	− 1.26536i	0.774413	−	0.632681i	$-0.218045\pi$
$294$	0	0
$295$	52326.0	0.601275
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 20808.0i	− 0.232749i
$300$	0	0
$301$	13090.0	0.144480
$302$	0	0
$303$	0	0
$304$	0	0
$305$	57636.0i	0.619575i
$306$	0	0
$307$	6410.00	0.0680113	0.0340057	−	0.999422i	$-0.489174\pi$
$307$	6410.00	0.0680113	0.0340057	+	0.999422i	$0.489174\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	70992.0i	0.733987i	0.930223	+	0.366994i	$0.119613\pi$
$311$	70992.0i	0.733987i	−0.930223	+	0.366994i	$0.880387\pi$
$312$	0	0
$313$	160961.	1.64298	0.821489	−	0.570224i	$-0.193143\pi$
$313$	160961.	1.64298	0.821489	+	0.570224i	$0.193143\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 194733.i	− 1.93785i	−0.247347	−	0.968927i	$-0.579559\pi$
$317$	− 194733.i	− 1.93785i	0.247347	−	0.968927i	$-0.420441\pi$
$318$	0	0
$319$	−124254.	−1.22104
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 28800.0i	− 0.276050i
$324$	0	0
$325$	−18496.0	−0.175110
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 13230.0i	− 0.122227i
$330$	0	0
$331$	−33286.0	−0.303812	−0.151906	−	0.988395i	$-0.548541\pi$
$331$	−33286.0	−0.303812	−0.151906	+	0.988395i	$0.548541\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 46962.0i	− 0.418463i
$336$	0	0
$337$	−127690.	−1.12434	−0.562169	−	0.827022i	$-0.690033\pi$
$337$	−127690.	−1.12434	−0.562169	+	0.827022i	$0.690033\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 81549.0i	− 0.701310i
$342$	0	0
$343$	−23885.0	−0.203019
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 138807.i	− 1.15280i	−0.817169	−	0.576398i	$-0.804458\pi$
$347$	− 138807.i	− 1.15280i	0.817169	−	0.576398i	$-0.195542\pi$
$348$	0	0
$349$	203792.	1.67316	0.836578	−	0.547848i	$-0.184553\pi$
$349$	203792.	1.67316	0.836578	+	0.547848i	$0.184553\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32328.0i	0.259436i	0.991551	+	0.129718i	$0.0414071\pi$
$353$	32328.0i	0.259436i	−0.991551	+	0.129718i	$0.958593\pi$
$354$	0	0
$355$	59130.0	0.469193
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 185922.i	− 1.44259i	−0.692630	−	0.721293i	$-0.743548\pi$
$359$	− 185922.i	− 1.44259i	0.692630	−	0.721293i	$-0.256452\pi$
$360$	0	0
$361$	−126225.	−0.968570
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 40671.0i	− 0.305281i
$366$	0	0
$367$	151079.	1.12169	0.560844	−	0.827922i	$-0.310477\pi$
$367$	151079.	1.12169	0.560844	+	0.827922i	$0.310477\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 5355.00i	− 0.0389056i
$372$	0	0
$373$	−6478.00	−0.0465611	−0.0232806	−	0.999729i	$-0.507411\pi$
$373$	−6478.00	−0.0465611	−0.0232806	+	0.999729i	$0.507411\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 36108.0i	− 0.254051i
$378$	0	0
$379$	99008.0	0.689274	0.344637	−	0.938736i	$-0.388002\pi$
$379$	99008.0	0.689274	0.344637	+	0.938736i	$0.388002\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 91062.0i	− 0.620783i	−0.950609	−	0.310391i	$-0.899540\pi$
$383$	− 91062.0i	− 0.620783i	0.950609	−	0.310391i	$-0.100460\pi$
$384$	0	0
$385$	−5265.00	−0.0355203
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 95319.0i	− 0.629913i	−0.949106	−	0.314956i	$-0.898010\pi$
$389$	− 95319.0i	− 0.629913i	0.949106	−	0.314956i	$-0.101990\pi$
$390$	0	0
$391$	−275400.	−1.80140
$392$	0	0
$393$	0	0
$394$	0	0
$395$	67518.0i	0.432738i
$396$	0	0
$397$	−163438.	−1.03698	−0.518492	−	0.855083i	$-0.673506\pi$
$397$	−163438.	−1.03698	−0.518492	+	0.855083i	$0.673506\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	284616.i	1.76999i	0.465601	+	0.884994i	$0.345838\pi$
$401$	284616.i	1.76999i	−0.465601	+	0.884994i	$0.654162\pi$
$402$	0	0
$403$	23698.0	0.145916
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 87516.0i	− 0.528322i
$408$	0	0
$409$	107525.	0.642781	0.321390	−	0.946947i	$-0.395850\pi$
$409$	107525.	0.642781	0.321390	+	0.946947i	$0.395850\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 29070.0i	− 0.170430i
$414$	0	0
$415$	49329.0	0.286422
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 150282.i	− 0.856010i	−0.903776	−	0.428005i	$-0.859217\pi$
$419$	− 150282.i	− 0.856010i	0.903776	−	0.428005i	$-0.140783\pi$
$420$	0	0
$421$	134420.	0.758402	0.379201	−	0.925314i	$-0.376199\pi$
$421$	134420.	0.758402	0.379201	+	0.925314i	$0.376199\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	244800.i	1.35529i
$426$	0	0
$427$	32020.0	0.175617
$428$	0	0
$429$	0	0
$430$	0	0
$431$	237726.i	1.27974i	0.768483	+	0.639871i	$0.221012\pi$
$431$	237726.i	1.27974i	−0.768483	+	0.639871i	$0.778988\pi$
$432$	0	0
$433$	112187.	0.598366	0.299183	−	0.954196i	$-0.403286\pi$
$433$	112187.	0.598366	0.299183	+	0.954196i	$0.403286\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 39168.0i	− 0.205101i
$438$	0	0
$439$	−204643.	−1.06186	−0.530931	−	0.847415i	$-0.678158\pi$
$439$	−204643.	−1.06186	−0.530931	+	0.847415i	$0.678158\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	50490.0i	0.257275i	0.991692	+	0.128638i	$0.0410604\pi$
$443$	50490.0i	0.257275i	−0.991692	+	0.128638i	$0.958940\pi$
$444$	0	0
$445$	−79866.0	−0.403313
$446$	0	0
$447$	0	0
$448$	0	0
$449$	363528.i	1.80321i	0.432565	+	0.901603i	$0.357609\pi$
$449$	363528.i	1.80321i	−0.432565	+	0.901603i	$0.642391\pi$
$450$	0	0
$451$	−80028.0	−0.393449
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1530.00i	− 0.00739041i
$456$	0	0
$457$	6677.00	0.0319705	0.0159852	−	0.999872i	$-0.494912\pi$
$457$	6677.00	0.0319705	0.0159852	+	0.999872i	$0.494912\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	229347.i	1.07917i	0.841930	+	0.539587i	$0.181419\pi$
$461$	229347.i	1.07917i	−0.841930	+	0.539587i	$0.818581\pi$
$462$	0	0
$463$	238799.	1.11396	0.556981	−	0.830525i	$-0.311960\pi$
$463$	238799.	1.11396	0.556981	+	0.830525i	$0.311960\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 263133.i	− 1.20654i	−0.797537	−	0.603270i	$-0.793864\pi$
$467$	− 263133.i	− 1.20654i	0.797537	−	0.603270i	$-0.206136\pi$
$468$	0	0
$469$	−26090.0	−0.118612
$470$	0	0
$471$	0	0
$472$	0	0
$473$	306306.i	1.36909i
$474$	0	0
$475$	−34816.0	−0.154309
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9342.00i	0.0407163i	0.999793	+	0.0203582i	$0.00648066\pi$
$479$	9342.00i	0.0407163i	−0.999793	+	0.0203582i	$0.993519\pi$
$480$	0	0
$481$	25432.0	0.109923
$482$	0	0
$483$	0	0
$484$	0	0
$485$	95139.0i	0.404460i
$486$	0	0
$487$	331262.	1.39673	0.698367	−	0.715740i	$-0.253910\pi$
$487$	331262.	1.39673	0.698367	+	0.715740i	$0.253910\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	396297.i	1.64383i	0.569608	+	0.821917i	$0.307095\pi$
$491$	396297.i	1.64383i	−0.569608	+	0.821917i	$0.692905\pi$
$492$	0	0
$493$	−477900.	−1.96627
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 32850.0i	− 0.132991i
$498$	0	0
$499$	45050.0	0.180923	0.0904615	−	0.995900i	$-0.471166\pi$
$499$	45050.0	0.180923	0.0904615	+	0.995900i	$0.471166\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 233172.i	− 0.921596i	−0.887505	−	0.460798i	$-0.847563\pi$
$503$	− 233172.i	− 0.921596i	0.887505	−	0.460798i	$-0.152437\pi$
$504$	0	0
$505$	−118017.	−0.462766
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 163449.i	− 0.630880i	−0.948946	−	0.315440i	$-0.897848\pi$
$509$	− 163449.i	− 0.630880i	0.948946	−	0.315440i	$-0.102152\pi$
$510$	0	0
$511$	−22595.0	−0.0865308
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 52470.0i	− 0.197832i
$516$	0	0
$517$	309582.	1.15823
$518$	0	0
$519$	0	0
$520$	0	0
$521$	384606.i	1.41690i	0.705759	+	0.708452i	$0.250606\pi$
$521$	384606.i	1.41690i	−0.705759	+	0.708452i	$0.749394\pi$
$522$	0	0
$523$	−214642.	−0.784714	−0.392357	−	0.919813i	$-0.628340\pi$
$523$	−214642.	−0.784714	−0.392357	+	0.919813i	$0.628340\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 313650.i	− 1.12934i
$528$	0	0
$529$	−94703.0	−0.338417
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 23256.0i	− 0.0818617i
$534$	0	0
$535$	−9801.00	−0.0342423
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 277992.i	− 0.956874i
$540$	0	0
$541$	545156.	1.86263	0.931314	−	0.364217i	$-0.118663\pi$
$541$	545156.	1.86263	0.931314	+	0.364217i	$0.118663\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 45180.0i	− 0.152108i
$546$	0	0
$547$	−491422.	−1.64240	−0.821202	−	0.570638i	$-0.806696\pi$
$547$	−491422.	−1.64240	−0.821202	+	0.570638i	$0.806696\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 67968.0i	− 0.223873i
$552$	0	0
$553$	37510.0	0.122658
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 445077.i	− 1.43458i	−0.696775	−	0.717290i	$-0.745382\pi$
$557$	− 445077.i	− 1.43458i	0.696775	−	0.717290i	$-0.254618\pi$
$558$	0	0
$559$	−89012.0	−0.284856
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 153765.i	− 0.485111i	−0.970138	−	0.242555i	$-0.922014\pi$
$563$	− 153765.i	− 0.485111i	0.970138	−	0.242555i	$-0.0779856\pi$
$564$	0	0
$565$	−192456.	−0.602885
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 269712.i	− 0.833059i	−0.909122	−	0.416529i	$-0.863246\pi$
$569$	− 269712.i	− 0.833059i	0.909122	−	0.416529i	$-0.136754\pi$
$570$	0	0
$571$	589718.	1.80872	0.904362	−	0.426767i	$-0.140347\pi$
$571$	589718.	1.80872	0.904362	+	0.426767i	$0.140347\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	332928.i	1.00697i
$576$	0	0
$577$	184094.	0.552953	0.276476	−	0.961021i	$-0.410833\pi$
$577$	184094.	0.552953	0.276476	+	0.961021i	$0.410833\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 27405.0i	− 0.0811853i
$582$	0	0
$583$	125307.	0.368670
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 411543.i	− 1.19437i	−0.802103	−	0.597185i	$-0.796286\pi$
$587$	− 411543.i	− 1.19437i	0.802103	−	0.597185i	$-0.203714\pi$
$588$	0	0
$589$	44608.0	0.128583
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 339966.i	− 0.966777i	−0.875406	−	0.483388i	$-0.839406\pi$
$593$	− 339966.i	− 0.966777i	0.875406	−	0.483388i	$-0.160594\pi$
$594$	0	0
$595$	−20250.0	−0.0571994
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 299574.i	− 0.834931i	−0.908693	−	0.417465i	$-0.862918\pi$
$599$	− 299574.i	− 0.834931i	0.908693	−	0.417465i	$-0.137082\pi$
$600$	0	0
$601$	−516115.	−1.42889	−0.714443	−	0.699694i	$-0.753320\pi$
$601$	−516115.	−1.42889	−0.714443	+	0.699694i	$0.753320\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8568.00i	0.0234082i
$606$	0	0
$607$	−486574.	−1.32060	−0.660300	−	0.751002i	$-0.729571\pi$
$607$	−486574.	−1.32060	−0.660300	+	0.751002i	$0.729571\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	89964.0i	0.240983i
$612$	0	0
$613$	−189550.	−0.504432	−0.252216	−	0.967671i	$-0.581159\pi$
$613$	−189550.	−0.504432	−0.252216	+	0.967671i	$0.581159\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 148248.i	− 0.389420i	−0.980861	−	0.194710i	$-0.937623\pi$
$617$	− 148248.i	− 0.389420i	0.980861	−	0.194710i	$-0.0623766\pi$
$618$	0	0
$619$	11390.0	0.0297264	0.0148632	−	0.999890i	$-0.495269\pi$
$619$	11390.0	0.0297264	0.0148632	+	0.999890i	$0.495269\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	44370.0i	0.114318i
$624$	0	0
$625$	245311.	0.627996
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 336600.i	− 0.850771i
$630$	0	0
$631$	−6715.00	−0.0168650	−0.00843252	−	0.999964i	$-0.502684\pi$
$631$	−6715.00	−0.0168650	−0.00843252	+	0.999964i	$0.502684\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	83043.0i	0.205947i
$636$	0	0
$637$	80784.0	0.199089
$638$	0	0
$639$	0	0
$640$	0	0
$641$	441108.i	1.07357i	0.843720	+	0.536783i	$0.180361\pi$
$641$	441108.i	1.07357i	−0.843720	+	0.536783i	$0.819639\pi$
$642$	0	0
$643$	547448.	1.32410	0.662050	−	0.749459i	$-0.269687\pi$
$643$	547448.	1.32410	0.662050	+	0.749459i	$0.269687\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42228.0i	0.100877i	0.998727	+	0.0504385i	$0.0160619\pi$
$647$	42228.0i	0.100877i	−0.998727	+	0.0504385i	$0.983938\pi$
$648$	0	0
$649$	680238.	1.61500
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 258993.i	− 0.607382i	−0.952771	−	0.303691i	$-0.901781\pi$
$653$	− 258993.i	− 0.607382i	0.952771	−	0.303691i	$-0.0982190\pi$
$654$	0	0
$655$	−38475.0	−0.0896801
$656$	0	0
$657$	0	0
$658$	0	0
$659$	313083.i	0.720923i	0.932774	+	0.360461i	$0.117381\pi$
$659$	313083.i	0.720923i	−0.932774	+	0.360461i	$0.882619\pi$
$660$	0	0
$661$	−686320.	−1.57081	−0.785405	−	0.618982i	$-0.787545\pi$
$661$	−686320.	−1.57081	−0.785405	+	0.618982i	$0.787545\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 2880.00i	− 0.00651252i
$666$	0	0
$667$	−649944.	−1.46091
$668$	0	0
$669$	0	0
$670$	0	0
$671$	749268.i	1.66415i
$672$	0	0
$673$	214727.	0.474085	0.237043	−	0.971499i	$-0.423822\pi$
$673$	214727.	0.474085	0.237043	+	0.971499i	$0.423822\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 343638.i	− 0.749763i	−0.927073	−	0.374881i	$-0.877683\pi$
$677$	− 343638.i	− 0.749763i	0.927073	−	0.374881i	$-0.122317\pi$
$678$	0	0
$679$	52855.0	0.114643
$680$	0	0
$681$	0	0
$682$	0	0
$683$	626238.i	1.34245i	0.741254	+	0.671225i	$0.234231\pi$
$683$	626238.i	1.34245i	−0.741254	+	0.671225i	$0.765769\pi$
$684$	0	0
$685$	−101898.	−0.217162
$686$	0	0
$687$	0	0
$688$	0	0
$689$	36414.0i	0.0767061i
$690$	0	0
$691$	684476.	1.43351	0.716757	−	0.697323i	$-0.245626\pi$
$691$	684476.	1.43351	0.716757	+	0.697323i	$0.245626\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	88308.0i	0.182823i
$696$	0	0
$697$	−307800.	−0.633582
$698$	0	0
$699$	0	0
$700$	0	0
$701$	809523.i	1.64738i	0.567042	+	0.823689i	$0.308088\pi$
$701$	809523.i	1.64738i	−0.567042	+	0.823689i	$0.691912\pi$
$702$	0	0
$703$	47872.0	0.0968659
$704$	0	0
$705$	0	0
$706$	0	0
$707$	65565.0i	0.131170i
$708$	0	0
$709$	−319648.	−0.635886	−0.317943	−	0.948110i	$-0.602992\pi$
$709$	−319648.	−0.635886	−0.317943	+	0.948110i	$0.602992\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 426564.i	− 0.839083i
$714$	0	0
$715$	35802.0	0.0700318
$716$	0	0
$717$	0	0
$718$	0	0
$719$	55836.0i	0.108008i	0.998541	+	0.0540041i	$0.0171984\pi$
$719$	55836.0i	0.108008i	−0.998541	+	0.0540041i	$0.982802\pi$
$720$	0	0
$721$	−29150.0	−0.0560748
$722$	0	0
$723$	0	0
$724$	0	0
$725$	577728.i	1.09913i
$726$	0	0
$727$	863873.	1.63449	0.817243	−	0.576294i	$-0.195502\pi$
$727$	863873.	1.63449	0.817243	+	0.576294i	$0.195502\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.17810e6i	2.20469i
$732$	0	0
$733$	207608.	0.386399	0.193200	−	0.981159i	$-0.438114\pi$
$733$	207608.	0.386399	0.193200	+	0.981159i	$0.438114\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 610506.i	− 1.12397i
$738$	0	0
$739$	−803590.	−1.47145	−0.735725	−	0.677280i	$-0.763159\pi$
$739$	−803590.	−1.47145	−0.735725	+	0.677280i	$0.763159\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	619650.i	1.12245i	0.827662	+	0.561227i	$0.189671\pi$
$743$	619650.i	1.12245i	−0.827662	+	0.561227i	$0.810329\pi$
$744$	0	0
$745$	148311.	0.267215
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5445.00i	0.00970587i
$750$	0	0
$751$	−1.06598e6	−1.89004	−0.945020	−	0.327011i	$-0.893959\pi$
$751$	−1.06598e6	−1.89004	−0.945020	+	0.327011i	$0.893959\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 229995.i	− 0.403482i
$756$	0	0
$757$	750410.	1.30950	0.654752	−	0.755844i	$-0.272773\pi$
$757$	750410.	1.30950	0.654752	+	0.755844i	$0.272773\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 413208.i	− 0.713509i	−0.934198	−	0.356754i	$-0.883883\pi$
$761$	− 413208.i	− 0.713509i	0.934198	−	0.356754i	$-0.116117\pi$
$762$	0	0
$763$	−25100.0	−0.0431146
$764$	0	0
$765$	0	0
$766$	0	0
$767$	197676.i	0.336019i
$768$	0	0
$769$	−733381.	−1.24016	−0.620079	−	0.784539i	$-0.712899\pi$
$769$	−733381.	−1.24016	−0.620079	+	0.784539i	$0.712899\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	231822.i	0.387968i	0.981005	+	0.193984i	$0.0621410\pi$
$773$	231822.i	0.387968i	−0.981005	+	0.193984i	$0.937859\pi$
$774$	0	0
$775$	−379168.	−0.631289
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 43776.0i	− 0.0721375i
$780$	0	0
$781$	768690.	1.26023
$782$	0	0
$783$	0	0
$784$	0	0
$785$	190476.i	0.309101i
$786$	0	0
$787$	502724.	0.811671	0.405836	−	0.913946i	$-0.366981\pi$
$787$	502724.	0.811671	0.405836	+	0.913946i	$0.366981\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	106920.i	0.170886i
$792$	0	0
$793$	−217736.	−0.346245
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 596241.i	− 0.938653i	−0.883025	−	0.469327i	$-0.844497\pi$
$797$	− 596241.i	− 0.938653i	0.883025	−	0.469327i	$-0.155503\pi$
$798$	0	0
$799$	1.19070e6	1.86513
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 528723.i	− 0.819968i
$804$	0	0
$805$	−27540.0	−0.0424984
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 376038.i	− 0.574559i	−0.957847	−	0.287280i	$-0.907249\pi$
$809$	− 376038.i	− 0.574559i	0.957847	−	0.287280i	$-0.0927509\pi$
$810$	0	0
$811$	−331072.	−0.503362	−0.251681	−	0.967810i	$-0.580983\pi$
$811$	−331072.	−0.503362	−0.251681	+	0.967810i	$0.580983\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	304470.i	0.458384i
$816$	0	0
$817$	−167552.	−0.251018
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 274626.i	− 0.407432i	−0.979030	−	0.203716i	$-0.934698\pi$
$821$	− 274626.i	− 0.407432i	0.979030	−	0.203716i	$-0.0653019\pi$
$822$	0	0
$823$	−541195.	−0.799013	−0.399507	−	0.916730i	$-0.630819\pi$
$823$	−541195.	−0.799013	−0.399507	+	0.916730i	$0.630819\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	358362.i	0.523975i	0.965071	+	0.261988i	$0.0843780\pi$
$827$	358362.i	0.523975i	−0.965071	+	0.261988i	$0.915622\pi$
$828$	0	0
$829$	39626.0	0.0576595	0.0288298	−	0.999584i	$-0.490822\pi$
$829$	39626.0	0.0576595	0.0288298	+	0.999584i	$0.490822\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1.06920e6i	− 1.54088i
$834$	0	0
$835$	246402.	0.353404
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1.05543e6i	− 1.49936i	−0.661801	−	0.749679i	$-0.730208\pi$
$839$	− 1.05543e6i	− 1.49936i	0.661801	−	0.749679i	$-0.269792\pi$
$840$	0	0
$841$	−420563.	−0.594619
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 246645.i	− 0.345429i
$846$	0	0
$847$	4760.00	0.00663499
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 457776.i	− 0.632112i
$852$	0	0
$853$	−33034.0	−0.0454008	−0.0227004	−	0.999742i	$-0.507226\pi$
$853$	−33034.0	−0.0454008	−0.0227004	+	0.999742i	$0.507226\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 456246.i	− 0.621209i	−0.950539	−	0.310604i	$-0.899469\pi$
$857$	− 456246.i	− 0.621209i	0.950539	−	0.310604i	$-0.100531\pi$
$858$	0	0
$859$	343604.	0.465663	0.232832	−	0.972517i	$-0.425201\pi$
$859$	343604.	0.465663	0.232832	+	0.972517i	$0.425201\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 74556.0i	− 0.100106i	−0.998747	−	0.0500531i	$-0.984061\pi$
$863$	− 74556.0i	− 0.100106i	0.998747	−	0.0500531i	$-0.0159391\pi$
$864$	0	0
$865$	415773.	0.555679
$866$	0	0
$867$	0	0
$868$	0	0
$869$	877734.i	1.16231i
$870$	0	0
$871$	177412.	0.233855
$872$	0	0
$873$	0	0
$874$	0	0
$875$	52605.0i	0.0687086i
$876$	0	0
$877$	−696094.	−0.905042	−0.452521	−	0.891754i	$-0.649475\pi$
$877$	−696094.	−0.905042	−0.452521	+	0.891754i	$0.649475\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 672426.i	− 0.866349i	−0.901310	−	0.433174i	$-0.857393\pi$
$881$	− 672426.i	− 0.866349i	0.901310	−	0.433174i	$-0.142607\pi$
$882$	0	0
$883$	−1.44813e6	−1.85731	−0.928657	−	0.370938i	$-0.879036\pi$
$883$	−1.44813e6	−1.85731	−0.928657	+	0.370938i	$0.879036\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.50964e6i	− 1.91879i	−0.282071	−	0.959393i	$-0.591022\pi$
$887$	− 1.50964e6i	− 1.91879i	0.282071	−	0.959393i	$-0.408978\pi$
$888$	0	0
$889$	46135.0	0.0583750
$890$	0	0
$891$	0	0
$892$	0	0
$893$	169344.i	0.212357i
$894$	0	0
$895$	187029.	0.233487
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 740214.i	− 0.915879i
$900$	0	0
$901$	481950.	0.593680
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 175536.i	− 0.214323i
$906$	0	0
$907$	374828.	0.455635	0.227818	−	0.973704i	$-0.426841\pi$
$907$	374828.	0.455635	0.227818	+	0.973704i	$0.426841\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.21149e6i	1.45977i	0.683572	+	0.729883i	$0.260425\pi$
$911$	1.21149e6i	1.45977i	−0.683572	+	0.729883i	$0.739575\pi$
$912$	0	0
$913$	641277.	0.769315
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21375.0i	0.0254195i
$918$	0	0
$919$	656777.	0.777655	0.388827	−	0.921311i	$-0.372880\pi$
$919$	656777.	0.777655	0.388827	+	0.921311i	$0.372880\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	223380.i	0.262205i
$924$	0	0
$925$	−406912.	−0.475573
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 578988.i	− 0.670870i	−0.942063	−	0.335435i	$-0.891117\pi$
$929$	− 578988.i	− 0.670870i	0.942063	−	0.335435i	$-0.108883\pi$
$930$	0	0
$931$	152064.	0.175439
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 473850.i	− 0.542023i
$936$	0	0
$937$	195173.	0.222301	0.111150	−	0.993804i	$-0.464547\pi$
$937$	195173.	0.222301	0.111150	+	0.993804i	$0.464547\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.31932e6i	1.48995i	0.667095	+	0.744973i	$0.267538\pi$
$941$	1.31932e6i	1.48995i	−0.667095	+	0.744973i	$0.732462\pi$
$942$	0	0
$943$	−418608.	−0.470743
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 972621.i	− 1.08454i	−0.840206	−	0.542268i	$-0.817566\pi$
$947$	− 972621.i	− 1.08454i	0.840206	−	0.542268i	$-0.182434\pi$
$948$	0	0
$949$	153646.	0.170604
$950$	0	0
$951$	0	0
$952$	0	0
$953$	534384.i	0.588393i	0.955745	+	0.294197i	$0.0950520\pi$
$953$	534384.i	0.588393i	−0.955745	+	0.294197i	$0.904948\pi$
$954$	0	0
$955$	−63342.0	−0.0694520
$956$	0	0
$957$	0	0
$958$	0	0
$959$	56610.0i	0.0615540i
$960$	0	0
$961$	−437712.	−0.473960
$962$	0	0
$963$	0	0
$964$	0	0
$965$	463743.i	0.497992i
$966$	0	0
$967$	−783619.	−0.838015	−0.419008	−	0.907983i	$-0.637622\pi$
$967$	−783619.	−0.838015	−0.419008	+	0.907983i	$0.637622\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	276777.i	0.293556i	0.989169	+	0.146778i	$0.0468904\pi$
$971$	276777.i	0.293556i	−0.989169	+	0.146778i	$0.953110\pi$
$972$	0	0
$973$	49060.0	0.0518205
$974$	0	0
$975$	0	0
$976$	0	0
$977$	648234.i	0.679114i	0.940585	+	0.339557i	$0.110277\pi$
$977$	648234.i	0.679114i	−0.940585	+	0.339557i	$0.889723\pi$
$978$	0	0
$979$	−1.03826e6	−1.08328
$980$	0	0
$981$	0	0
$982$	0	0
$983$	290862.i	0.301009i	0.988609	+	0.150505i	$0.0480899\pi$
$983$	290862.i	0.301009i	−0.988609	+	0.150505i	$0.951910\pi$
$984$	0	0
$985$	543915.	0.560607
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.60222e6i	1.63806i
$990$	0	0
$991$	881243.	0.897322	0.448661	−	0.893702i	$-0.351901\pi$
$991$	881243.	0.897322	0.448661	+	0.893702i	$0.351901\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	410985.i	0.415126i
$996$	0	0
$997$	−690166.	−0.694326	−0.347163	−	0.937805i	$-0.612855\pi$
$997$	−690166.	−0.694326	−0.347163	+	0.937805i	$0.612855\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.c.c.53.2	yes	2
3.2	odd	2	inner	108.5.c.c.53.1	✓	2
4.3	odd	2		432.5.e.d.161.2		2
9.2	odd	6		324.5.g.d.53.2		4
9.4	even	3		324.5.g.d.269.2		4
9.5	odd	6		324.5.g.d.269.1		4
9.7	even	3		324.5.g.d.53.1		4
12.11	even	2		432.5.e.d.161.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.c.c.53.1	✓	2	3.2	odd	2	inner
108.5.c.c.53.2	yes	2	1.1	even	1	trivial
324.5.g.d.53.1		4	9.7	even	3
324.5.g.d.53.2		4	9.2	odd	6
324.5.g.d.269.1		4	9.5	odd	6
324.5.g.d.269.2		4	9.4	even	3
432.5.e.d.161.1		2	12.11	even	2
432.5.e.d.161.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{5} +5.00000 q^{7} +O(q^{10})\)	\(q+9.00000i q^{5} +5.00000 q^{7} +117.000i q^{11} -34.0000 q^{13} +450.000i q^{17} -64.0000 q^{19} +612.000i q^{23} +544.000 q^{25} +1062.00i q^{29} -697.000 q^{31} +45.0000i q^{35} -748.000 q^{37} +684.000i q^{41} +2618.00 q^{43} -2646.00i q^{47} -2376.00 q^{49} -1071.00i q^{53} -1053.00 q^{55} -5814.00i q^{59} +6404.00 q^{61} -306.000i q^{65} -5218.00 q^{67} -6570.00i q^{71} -4519.00 q^{73} +585.000i q^{77} +7502.00 q^{79} -5481.00i q^{83} -4050.00 q^{85} +8874.00i q^{89} -170.000 q^{91} -576.000i q^{95} +10571.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{7}+O(q^{10}) \) 2 * q + 10 * q^7	\( 2 q + 10 q^{7} - 68 q^{13} - 128 q^{19} + 1088 q^{25} - 1394 q^{31} - 1496 q^{37} + 5236 q^{43} - 4752 q^{49} - 2106 q^{55} + 12808 q^{61} - 10436 q^{67} - 9038 q^{73} + 15004 q^{79} - 8100 q^{85} - 340 q^{91} + 21142 q^{97}+O(q^{100}) \) 2 * q + 10 * q^7 - 68 * q^13 - 128 * q^19 + 1088 * q^25 - 1394 * q^31 - 1496 * q^37 + 5236 * q^43 - 4752 * q^49 - 2106 * q^55 + 12808 * q^61 - 10436 * q^67 - 9038 * q^73 + 15004 * q^79 - 8100 * q^85 - 340 * q^91 + 21142 * q^97

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