LMFDB - Embedded newform 1110.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1110.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.a (trivial)

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:	yes
Analytic conductor:	$8.86339462436$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{20} +4.00000 q^{21} -4.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} -8.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} -1.00000 q^{45} -12.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +4.00000 q^{56} -8.00000 q^{57} +6.00000 q^{58} +1.00000 q^{60} -6.00000 q^{61} +8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} -4.00000 q^{70} -1.00000 q^{72} -6.00000 q^{73} +1.00000 q^{74} -1.00000 q^{75} +8.00000 q^{76} -16.0000 q^{77} +2.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} +4.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} +6.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} +1.00000 q^{90} -8.00000 q^{91} +8.00000 q^{93} +12.0000 q^{94} -8.00000 q^{95} +1.00000 q^{96} +6.00000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	4.00000	1.06904
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−1.00000	−0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	−1.00000	−0.223607
$21$	4.00000	0.872872
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	−4.00000	−0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−1.00000	−0.182574
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	2.00000	0.342997
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−8.00000	−1.29777
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−4.00000	−0.617213
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	4.00000	0.603023
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	2.00000	0.280056
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	−4.00000	−0.539360
$56$	4.00000	0.534522
$57$	−8.00000	−1.05963
$58$	6.00000	0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	8.00000	1.01600
$63$	−4.00000	−0.503953
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	4.00000	0.492366
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−4.00000	−0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	1.00000	0.116248
$75$	−1.00000	−0.115470
$76$	8.00000	0.917663
$77$	−16.0000	−1.82337
$78$	2.00000	0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	4.00000	0.436436
$85$	2.00000	0.216930
$86$	−4.00000	−0.431331
$87$	6.00000	0.643268
$88$	−4.00000	−0.426401
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	1.00000	0.105409
$91$	−8.00000	−0.838628
$92$	0	0
$93$	8.00000	0.829561
$94$	12.0000	1.23771
$95$	−8.00000	−0.820783
$96$	1.00000	0.102062
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−9.00000	−0.909137
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−2.00000	−0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−2.00000	−0.196116
$105$	−4.00000	−0.390360
$106$	6.00000	0.582772
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−1.00000	−0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	4.00000	0.381385
$111$	1.00000	0.0949158
$112$	−4.00000	−0.377964
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	0	0
$119$	8.00000	0.733359
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	6.00000	0.543214
$123$	6.00000	0.541002
$124$	−8.00000	−0.718421
$125$	−1.00000	−0.0894427
$126$	4.00000	0.356348
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	2.00000	0.175412
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	−4.00000	−0.348155
$133$	−32.0000	−2.77475
$134$	−12.0000	−1.03664
$135$	1.00000	0.0860663
$136$	2.00000	0.171499
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	4.00000	0.338062
$141$	12.0000	1.01058
$142$	0	0
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	6.00000	0.496564
$147$	−9.00000	−0.742307
$148$	−1.00000	−0.0821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	1.00000	0.0816497
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−8.00000	−0.648886
$153$	−2.00000	−0.161690
$154$	16.0000	1.28932
$155$	8.00000	0.642575
$156$	−2.00000	−0.160128
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	8.00000	0.636446
$159$	6.00000	0.475831
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−6.00000	−0.468521
$165$	4.00000	0.311400
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−4.00000	−0.308607
$169$	−9.00000	−0.692308
$170$	−2.00000	−0.153393
$171$	8.00000	0.611775
$172$	4.00000	0.304997
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	−6.00000	−0.454859
$175$	−4.00000	−0.302372
$176$	4.00000	0.301511
$177$	0	0
$178$	14.0000	1.04934
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	−1.00000	−0.0745356
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	8.00000	0.592999
$183$	6.00000	0.443533
$184$	0	0
$185$	1.00000	0.0735215
$186$	−8.00000	−0.586588
$187$	−8.00000	−0.585018
$188$	−12.0000	−0.875190
$189$	4.00000	0.290957
$190$	8.00000	0.580381
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	−1.00000	−0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−6.00000	−0.430775
$195$	2.00000	0.143223
$196$	9.00000	0.642857
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−4.00000	−0.284268
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−1.00000	−0.0707107
$201$	−12.0000	−0.846415
$202$	10.0000	0.703598
$203$	24.0000	1.68447
$204$	2.00000	0.140028
$205$	6.00000	0.419058
$206$	−8.00000	−0.557386
$207$	0	0
$208$	2.00000	0.138675
$209$	32.0000	2.21349
$210$	4.00000	0.276026
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−4.00000	−0.273434
$215$	−4.00000	−0.272798
$216$	1.00000	0.0680414
$217$	32.0000	2.17230
$218$	−2.00000	−0.135457
$219$	6.00000	0.405442
$220$	−4.00000	−0.269680
$221$	−4.00000	−0.269069
$222$	−1.00000	−0.0671156
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	4.00000	0.267261
$225$	1.00000	0.0666667
$226$	18.0000	1.19734
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−8.00000	−0.529813
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	6.00000	0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−2.00000	−0.130744
$235$	12.0000	0.782794
$236$	0	0
$237$	8.00000	0.519656
$238$	−8.00000	−0.518563
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	1.00000	0.0645497
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	−9.00000	−0.574989
$246$	−6.00000	−0.382546
$247$	16.0000	1.01806
$248$	8.00000	0.508001
$249$	−4.00000	−0.253490
$250$	1.00000	0.0632456
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	−4.00000	−0.251976
$253$	0	0
$254$	−4.00000	−0.250982
$255$	−2.00000	−0.125245
$256$	1.00000	0.0625000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	4.00000	0.249029
$259$	4.00000	0.248548
$260$	−2.00000	−0.124035
$261$	−6.00000	−0.371391
$262$	16.0000	0.988483
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	4.00000	0.246183
$265$	6.00000	0.368577
$266$	32.0000	1.96205
$267$	14.0000	0.856786
$268$	12.0000	0.733017
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	−1.00000	−0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−2.00000	−0.121268
$273$	8.00000	0.484182
$274$	6.00000	0.362473
$275$	4.00000	0.241209
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	4.00000	0.239904
$279$	−8.00000	−0.478947
$280$	−4.00000	−0.239046
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	−12.0000	−0.714590
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	−8.00000	−0.473050
$287$	24.0000	1.41668
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−6.00000	−0.352332
$291$	−6.00000	−0.351726
$292$	−6.00000	−0.351123
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	1.00000	0.0581238
$297$	−4.00000	−0.232104
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−16.0000	−0.922225
$302$	16.0000	0.920697
$303$	10.0000	0.574485
$304$	8.00000	0.458831
$305$	6.00000	0.343559
$306$	2.00000	0.114332
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	−16.0000	−0.911685
$309$	−8.00000	−0.455104
$310$	−8.00000	−0.454369
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	2.00000	0.113228
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	22.0000	1.24153
$315$	4.00000	0.225374
$316$	−8.00000	−0.450035
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	−6.00000	−0.336463
$319$	−24.0000	−1.34374
$320$	−1.00000	−0.0559017
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−16.0000	−0.890264
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	12.0000	0.664619
$327$	−2.00000	−0.110600
$328$	6.00000	0.331295
$329$	48.0000	2.64633
$330$	−4.00000	−0.220193
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	4.00000	0.219529
$333$	−1.00000	−0.0547997
$334$	8.00000	0.437741
$335$	−12.0000	−0.655630
$336$	4.00000	0.218218
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	9.00000	0.489535
$339$	18.0000	0.977626
$340$	2.00000	0.108465
$341$	−32.0000	−1.73290
$342$	−8.00000	−0.432590
$343$	−8.00000	−0.431959
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−2.00000	−0.107521
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	6.00000	0.321634
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	4.00000	0.213809
$351$	−2.00000	−0.106752
$352$	−4.00000	−0.213201
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	−8.00000	−0.423405
$358$	−24.0000	−1.26844
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	1.00000	0.0527046
$361$	45.0000	2.36842
$362$	−14.0000	−0.735824
$363$	−5.00000	−0.262432
$364$	−8.00000	−0.419314
$365$	6.00000	0.314054
$366$	−6.00000	−0.313625
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	−1.00000	−0.0519875
$371$	24.0000	1.24602
$372$	8.00000	0.414781
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	8.00000	0.413670
$375$	1.00000	0.0516398
$376$	12.0000	0.618853
$377$	−12.0000	−0.618031
$378$	−4.00000	−0.205738
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	−8.00000	−0.410391
$381$	−4.00000	−0.204926
$382$	−24.0000	−1.22795
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	1.00000	0.0510310
$385$	16.0000	0.815436
$386$	−14.0000	−0.712581
$387$	4.00000	0.203331
$388$	6.00000	0.304604
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	−9.00000	−0.454569
$393$	16.0000	0.807093
$394$	−2.00000	−0.100759
$395$	8.00000	0.402524
$396$	4.00000	0.201008
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−16.0000	−0.802008
$399$	32.0000	1.60200
$400$	1.00000	0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	12.0000	0.598506
$403$	−16.0000	−0.797017
$404$	−10.0000	−0.497519
$405$	−1.00000	−0.0496904
$406$	−24.0000	−1.19110
$407$	−4.00000	−0.198273
$408$	−2.00000	−0.0990148
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	−6.00000	−0.296319
$411$	6.00000	0.295958
$412$	8.00000	0.394132
$413$	0	0
$414$	0	0
$415$	−4.00000	−0.196352
$416$	−2.00000	−0.0980581
$417$	4.00000	0.195881
$418$	−32.0000	−1.56517
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−4.00000	−0.195180
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	20.0000	0.973585
$423$	−12.0000	−0.583460
$424$	6.00000	0.291386
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	24.0000	1.16144
$428$	4.00000	0.193347
$429$	−8.00000	−0.386244
$430$	4.00000	0.192897
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−1.00000	−0.0481125
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	−32.0000	−1.53605
$435$	−6.00000	−0.287678
$436$	2.00000	0.0957826
$437$	0	0
$438$	−6.00000	−0.286691
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	4.00000	0.190693
$441$	9.00000	0.428571
$442$	4.00000	0.190261
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	1.00000	0.0474579
$445$	14.0000	0.663664
$446$	12.0000	0.568216
$447$	−6.00000	−0.283790
$448$	−4.00000	−0.188982
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	−1.00000	−0.0471405
$451$	−24.0000	−1.13012
$452$	−18.0000	−0.846649
$453$	16.0000	0.751746
$454$	−12.0000	−0.563188
$455$	8.00000	0.375046
$456$	8.00000	0.374634
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	−14.0000	−0.654177
$459$	2.00000	0.0933520
$460$	0	0
$461$	−38.0000	−1.76984	−0.884918	−	0.465746i	$-0.845786\pi$
$461$	−38.0000	−1.76984	−0.884918	+	0.465746i	$0.845786\pi$
$462$	−16.0000	−0.744387
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−6.00000	−0.278543
$465$	−8.00000	−0.370991
$466$	6.00000	0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	2.00000	0.0924500
$469$	−48.0000	−2.21643
$470$	−12.0000	−0.553519
$471$	22.0000	1.01371
$472$	0	0
$473$	16.0000	0.735681
$474$	−8.00000	−0.367452
$475$	8.00000	0.367065
$476$	8.00000	0.366679
$477$	−6.00000	−0.274721
$478$	8.00000	0.365911
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	−1.00000	−0.0456435
$481$	−2.00000	−0.0911922
$482$	14.0000	0.637683
$483$	0	0
$484$	5.00000	0.227273
$485$	−6.00000	−0.272446
$486$	1.00000	0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	6.00000	0.271607
$489$	12.0000	0.542659
$490$	9.00000	0.406579
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	6.00000	0.270501
$493$	12.0000	0.540453
$494$	−16.0000	−0.719874
$495$	−4.00000	−0.179787
$496$	−8.00000	−0.359211
$497$	0	0
$498$	4.00000	0.179244
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	−1.00000	−0.0447214
$501$	8.00000	0.357414
$502$	16.0000	0.714115
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	4.00000	0.178174
$505$	10.0000	0.444994
$506$	0	0
$507$	9.00000	0.399704
$508$	4.00000	0.177471
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	2.00000	0.0885615
$511$	24.0000	1.06170
$512$	−1.00000	−0.0441942
$513$	−8.00000	−0.353209
$514$	−14.0000	−0.617514
$515$	−8.00000	−0.352522
$516$	−4.00000	−0.176090
$517$	−48.0000	−2.11104
$518$	−4.00000	−0.175750
$519$	−2.00000	−0.0877903
$520$	2.00000	0.0877058
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	6.00000	0.262613
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−16.0000	−0.698963
$525$	4.00000	0.174574
$526$	4.00000	0.174408
$527$	16.0000	0.696971
$528$	−4.00000	−0.174078
$529$	−23.0000	−1.00000
$530$	−6.00000	−0.260623
$531$	0	0
$532$	−32.0000	−1.38738
$533$	−12.0000	−0.519778
$534$	−14.0000	−0.605839
$535$	−4.00000	−0.172935
$536$	−12.0000	−0.518321
$537$	−24.0000	−1.03568
$538$	−30.0000	−1.29339
$539$	36.0000	1.55063
$540$	1.00000	0.0430331
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	−8.00000	−0.343629
$543$	−14.0000	−0.600798
$544$	2.00000	0.0857493
$545$	−2.00000	−0.0856706
$546$	−8.00000	−0.342368
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	−6.00000	−0.256307
$549$	−6.00000	−0.256074
$550$	−4.00000	−0.170561
$551$	−48.0000	−2.04487
$552$	0	0
$553$	32.0000	1.36078
$554$	22.0000	0.934690
$555$	−1.00000	−0.0424476
$556$	−4.00000	−0.169638
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	8.00000	0.338667
$559$	8.00000	0.338364
$560$	4.00000	0.169031
$561$	8.00000	0.337760
$562$	14.0000	0.590554
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	12.0000	0.505291
$565$	18.0000	0.757266
$566$	−12.0000	−0.504398
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	−8.00000	−0.335083
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	8.00000	0.334497
$573$	−24.0000	−1.00261
$574$	−24.0000	−1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	13.0000	0.540729
$579$	−14.0000	−0.581820
$580$	6.00000	0.249136
$581$	−16.0000	−0.663792
$582$	6.00000	0.248708
$583$	−24.0000	−0.993978
$584$	6.00000	0.248282
$585$	−2.00000	−0.0826898
$586$	14.0000	0.578335
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−9.00000	−0.371154
$589$	−64.0000	−2.63707
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	−1.00000	−0.0410997
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	4.00000	0.164122
$595$	−8.00000	−0.327968
$596$	6.00000	0.245770
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	1.00000	0.0408248
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	16.0000	0.652111
$603$	12.0000	0.488678
$604$	−16.0000	−0.651031
$605$	−5.00000	−0.203279
$606$	−10.0000	−0.406222
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	−8.00000	−0.324443
$609$	−24.0000	−0.972529
$610$	−6.00000	−0.242933
$611$	−24.0000	−0.970936
$612$	−2.00000	−0.0808452
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	−20.0000	−0.807134
$615$	−6.00000	−0.241943
$616$	16.0000	0.644658
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	8.00000	0.321807
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	24.0000	0.962312
$623$	56.0000	2.24359
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	−32.0000	−1.27796
$628$	−22.0000	−0.877896
$629$	2.00000	0.0797452
$630$	−4.00000	−0.159364
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	8.00000	0.318223
$633$	20.0000	0.794929
$634$	30.0000	1.19145
$635$	−4.00000	−0.158735
$636$	6.00000	0.237915
$637$	18.0000	0.713186
$638$	24.0000	0.950169
$639$	0	0
$640$	1.00000	0.0395285
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	4.00000	0.157867
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	16.0000	0.629512
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	−32.0000	−1.25418
$652$	−12.0000	−0.469956
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	2.00000	0.0782062
$655$	16.0000	0.625172
$656$	−6.00000	−0.234261
$657$	−6.00000	−0.234082
$658$	−48.0000	−1.87123
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	4.00000	0.155700
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	8.00000	0.310929
$663$	4.00000	0.155347
$664$	−4.00000	−0.155230
$665$	32.0000	1.24091
$666$	1.00000	0.0387492
$667$	0	0
$668$	−8.00000	−0.309529
$669$	12.0000	0.463947
$670$	12.0000	0.463600
$671$	−24.0000	−0.926510
$672$	−4.00000	−0.154303
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	−2.00000	−0.0770371
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	34.0000	1.30673	0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000	1.30673	0.653363	+	0.757045i	$0.273358\pi$
$678$	−18.0000	−0.691286
$679$	−24.0000	−0.921035
$680$	−2.00000	−0.0766965
$681$	−12.0000	−0.459841
$682$	32.0000	1.22534
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	8.00000	0.305888
$685$	6.00000	0.229248
$686$	8.00000	0.305441
$687$	−14.0000	−0.534133
$688$	4.00000	0.152499
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	2.00000	0.0760286
$693$	−16.0000	−0.607790
$694$	12.0000	0.455514
$695$	4.00000	0.151729
$696$	−6.00000	−0.227429
$697$	12.0000	0.454532
$698$	2.00000	0.0757011
$699$	6.00000	0.226941
$700$	−4.00000	−0.151186
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	2.00000	0.0754851
$703$	−8.00000	−0.301726
$704$	4.00000	0.150756
$705$	−12.0000	−0.451946
$706$	26.0000	0.978523
$707$	40.0000	1.50435
$708$	0	0
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	14.0000	0.524672
$713$	0	0
$714$	8.00000	0.299392
$715$	−8.00000	−0.299183
$716$	24.0000	0.896922
$717$	8.00000	0.298765
$718$	−24.0000	−0.895672
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	−1.00000	−0.0372678
$721$	−32.0000	−1.19174
$722$	−45.0000	−1.67473
$723$	14.0000	0.520666
$724$	14.0000	0.520306
$725$	−6.00000	−0.222834
$726$	5.00000	0.185567
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	8.00000	0.296500
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	−8.00000	−0.295891
$732$	6.00000	0.221766
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	28.0000	1.03350
$735$	9.00000	0.331970
$736$	0	0
$737$	48.0000	1.76810
$738$	6.00000	0.220863
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	1.00000	0.0367607
$741$	−16.0000	−0.587775
$742$	−24.0000	−0.881068
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	−8.00000	−0.293294
$745$	−6.00000	−0.219823
$746$	−34.0000	−1.24483
$747$	4.00000	0.146352
$748$	−8.00000	−0.292509
$749$	−16.0000	−0.584627
$750$	−1.00000	−0.0365148
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	−12.0000	−0.437595
$753$	16.0000	0.583072
$754$	12.0000	0.437014
$755$	16.0000	0.582300
$756$	4.00000	0.145479
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	36.0000	1.30758
$759$	0	0
$760$	8.00000	0.290191
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	4.00000	0.144905
$763$	−8.00000	−0.289619
$764$	24.0000	0.868290
$765$	2.00000	0.0723102
$766$	−24.0000	−0.867155
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	−16.0000	−0.576600
$771$	−14.0000	−0.504198
$772$	14.0000	0.503871
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	−4.00000	−0.143777
$775$	−8.00000	−0.287368
$776$	−6.00000	−0.215387
$777$	−4.00000	−0.143499
$778$	−34.0000	−1.21896
$779$	−48.0000	−1.71978
$780$	2.00000	0.0716115
$781$	0	0
$782$	0	0
$783$	6.00000	0.214423
$784$	9.00000	0.321429
$785$	22.0000	0.785214
$786$	−16.0000	−0.570701
$787$	−52.0000	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$788$	2.00000	0.0712470
$789$	4.00000	0.142404
$790$	−8.00000	−0.284627
$791$	72.0000	2.56003
$792$	−4.00000	−0.142134
$793$	−12.0000	−0.426132
$794$	14.0000	0.496841
$795$	−6.00000	−0.212798
$796$	16.0000	0.567105
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	−32.0000	−1.13279
$799$	24.0000	0.849059
$800$	−1.00000	−0.0353553
$801$	−14.0000	−0.494666
$802$	6.00000	0.211867
$803$	−24.0000	−0.846942
$804$	−12.0000	−0.423207
$805$	0	0
$806$	16.0000	0.563576
$807$	−30.0000	−1.05605
$808$	10.0000	0.351799
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	1.00000	0.0351364
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	24.0000	0.842235
$813$	−8.00000	−0.280572
$814$	4.00000	0.140200
$815$	12.0000	0.420342
$816$	2.00000	0.0700140
$817$	32.0000	1.11954
$818$	−2.00000	−0.0699284
$819$	−8.00000	−0.279543
$820$	6.00000	0.209529
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−6.00000	−0.209274
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	−8.00000	−0.278693
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	4.00000	0.138842
$831$	22.0000	0.763172
$832$	2.00000	0.0693375
$833$	−18.0000	−0.623663
$834$	−4.00000	−0.138509
$835$	8.00000	0.276851
$836$	32.0000	1.10674
$837$	8.00000	0.276520
$838$	12.0000	0.414533
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	4.00000	0.138013
$841$	7.00000	0.241379
$842$	22.0000	0.758170
$843$	14.0000	0.482186
$844$	−20.0000	−0.688428
$845$	9.00000	0.309609
$846$	12.0000	0.412568
$847$	−20.0000	−0.687208
$848$	−6.00000	−0.206041
$849$	−12.0000	−0.411839
$850$	2.00000	0.0685994
$851$	0	0
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	−24.0000	−0.821263
$855$	−8.00000	−0.273594
$856$	−4.00000	−0.136717
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	8.00000	0.273115
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	−4.00000	−0.136399
$861$	−24.0000	−0.817918
$862$	−24.0000	−0.817443
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	1.00000	0.0340207
$865$	−2.00000	−0.0680020
$866$	30.0000	1.01944
$867$	13.0000	0.441503
$868$	32.0000	1.08615
$869$	−32.0000	−1.08553
$870$	6.00000	0.203419
$871$	24.0000	0.813209
$872$	−2.00000	−0.0677285
$873$	6.00000	0.203069
$874$	0	0
$875$	4.00000	0.135225
$876$	6.00000	0.202721
$877$	58.0000	1.95852	0.979260	−	0.202606i	$-0.0649409\pi$
$877$	58.0000	1.95852	0.979260	+	0.202606i	$0.0649409\pi$
$878$	−16.0000	−0.539974
$879$	14.0000	0.472208
$880$	−4.00000	−0.134840
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	−9.00000	−0.303046
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−20.0000	−0.671913
$887$	−52.0000	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$887$	−52.0000	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$888$	−1.00000	−0.0335578
$889$	−16.0000	−0.536623
$890$	−14.0000	−0.469281
$891$	4.00000	0.134005
$892$	−12.0000	−0.401790
$893$	−96.0000	−3.21252
$894$	6.00000	0.200670
$895$	−24.0000	−0.802232
$896$	4.00000	0.133631
$897$	0	0
$898$	22.0000	0.734150
$899$	48.0000	1.60089
$900$	1.00000	0.0333333
$901$	12.0000	0.399778
$902$	24.0000	0.799113
$903$	16.0000	0.532447
$904$	18.0000	0.598671
$905$	−14.0000	−0.465376
$906$	−16.0000	−0.531564
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	12.0000	0.398234
$909$	−10.0000	−0.331679
$910$	−8.00000	−0.265197
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−8.00000	−0.264906
$913$	16.0000	0.529523
$914$	−14.0000	−0.463079
$915$	−6.00000	−0.198354
$916$	14.0000	0.462573
$917$	64.0000	2.11347
$918$	−2.00000	−0.0660098
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	38.0000	1.25146
$923$	0	0
$924$	16.0000	0.526361
$925$	−1.00000	−0.0328798
$926$	8.00000	0.262896
$927$	8.00000	0.262754
$928$	6.00000	0.196960
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	8.00000	0.262330
$931$	72.0000	2.35970
$932$	−6.00000	−0.196537
$933$	24.0000	0.785725
$934$	12.0000	0.392652
$935$	8.00000	0.261628
$936$	−2.00000	−0.0653720
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	48.0000	1.56726
$939$	−22.0000	−0.717943
$940$	12.0000	0.391397
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	−22.0000	−0.716799
$943$	0	0
$944$	0	0
$945$	−4.00000	−0.130120
$946$	−16.0000	−0.520205
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	8.00000	0.259828
$949$	−12.0000	−0.389536
$950$	−8.00000	−0.259554
$951$	30.0000	0.972817
$952$	−8.00000	−0.259281
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	6.00000	0.194257
$955$	−24.0000	−0.776622
$956$	−8.00000	−0.258738
$957$	24.0000	0.775810
$958$	−24.0000	−0.775405
$959$	24.0000	0.775000
$960$	1.00000	0.0322749
$961$	33.0000	1.06452
$962$	2.00000	0.0644826
$963$	4.00000	0.128898
$964$	−14.0000	−0.450910
$965$	−14.0000	−0.450676
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	−5.00000	−0.160706
$969$	16.0000	0.513994
$970$	6.00000	0.192648
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	−1.00000	−0.0320750
$973$	16.0000	0.512936
$974$	−8.00000	−0.256337
$975$	−2.00000	−0.0640513
$976$	−6.00000	−0.192055
$977$	−50.0000	−1.59964	−0.799821	−	0.600239i	$-0.795072\pi$
$977$	−50.0000	−1.59964	−0.799821	+	0.600239i	$0.795072\pi$
$978$	−12.0000	−0.383718
$979$	−56.0000	−1.78977
$980$	−9.00000	−0.287494
$981$	2.00000	0.0638551
$982$	12.0000	0.382935
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	−6.00000	−0.191273
$985$	−2.00000	−0.0637253
$986$	−12.0000	−0.382158
$987$	−48.0000	−1.52786
$988$	16.0000	0.509028
$989$	0	0
$990$	4.00000	0.127128
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	8.00000	0.254000
$993$	8.00000	0.253872
$994$	0	0
$995$	−16.0000	−0.507234
$996$	−4.00000	−0.126745
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	−32.0000	−1.01294
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.a.a.1.1	✓	1
3.2	odd	2		3330.2.a.s.1.1		1
4.3	odd	2		8880.2.a.w.1.1		1
5.4	even	2		5550.2.a.br.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.a.1.1	✓	1	1.1	even	1	trivial
3330.2.a.s.1.1		1	3.2	odd	2
5550.2.a.br.1.1		1	5.4	even	2
8880.2.a.w.1.1		1	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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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