LMFDB - Embedded newform 6027.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 861)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +1.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +1.00000 q^{9} +4.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} -4.00000 q^{16} +4.00000 q^{17} +2.00000 q^{18} +7.00000 q^{19} +8.00000 q^{22} +6.00000 q^{23} -5.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -8.00000 q^{29} +1.00000 q^{31} -8.00000 q^{32} +4.00000 q^{33} +8.00000 q^{34} +2.00000 q^{36} +7.00000 q^{37} +14.0000 q^{38} -1.00000 q^{39} -1.00000 q^{41} -7.00000 q^{43} +8.00000 q^{44} +12.0000 q^{46} -4.00000 q^{48} -10.0000 q^{50} +4.00000 q^{51} -2.00000 q^{52} +4.00000 q^{53} +2.00000 q^{54} +7.00000 q^{57} -16.0000 q^{58} +8.00000 q^{59} +2.00000 q^{61} +2.00000 q^{62} -8.00000 q^{64} +8.00000 q^{66} +13.0000 q^{67} +8.00000 q^{68} +6.00000 q^{69} -6.00000 q^{71} +1.00000 q^{73} +14.0000 q^{74} -5.00000 q^{75} +14.0000 q^{76} -2.00000 q^{78} -7.00000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +8.00000 q^{83} -14.0000 q^{86} -8.00000 q^{87} +14.0000 q^{89} +12.0000 q^{92} +1.00000 q^{93} -8.00000 q^{96} -2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	2.00000	0.816497
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	2.00000	0.577350
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	2.00000	0.471405
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	0	0
$22$	8.00000	1.70561
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	−8.00000	−1.41421
$33$	4.00000	0.696311
$34$	8.00000	1.37199
$35$	0	0
$36$	2.00000	0.333333
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	14.0000	2.27110
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	8.00000	1.20605
$45$	0	0
$46$	12.0000	1.76930
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−4.00000	−0.577350
$49$	0	0
$50$	−10.0000	−1.41421
$51$	4.00000	0.560112
$52$	−2.00000	−0.277350
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	2.00000	0.272166
$55$	0	0
$56$	0	0
$57$	7.00000	0.927173
$58$	−16.0000	−2.10090
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	8.00000	0.984732
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	8.00000	0.970143
$69$	6.00000	0.722315
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	14.0000	1.62747
$75$	−5.00000	−0.577350
$76$	14.0000	1.60591
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	0	0
$86$	−14.0000	−1.50966
$87$	−8.00000	−0.857690
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0	0
$92$	12.0000	1.25109
$93$	1.00000	0.103695
$94$	0	0
$95$	0	0
$96$	−8.00000	−0.816497
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	−10.0000	−1.00000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	8.00000	0.792118
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	0	0
$105$	0	0
$106$	8.00000	0.777029
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	2.00000	0.192450
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	14.0000	1.31122
$115$	0	0
$116$	−16.0000	−1.48556
$117$	−1.00000	−0.0924500
$118$	16.0000	1.47292
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	4.00000	0.362143
$123$	−1.00000	−0.0901670
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	0	0
$129$	−7.00000	−0.616316
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	8.00000	0.696311
$133$	0	0
$134$	26.0000	2.24606
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	12.0000	1.02151
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−1.00702
$143$	−4.00000	−0.334497
$144$	−4.00000	−0.333333
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	14.0000	1.15079
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−10.0000	−0.816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	−14.0000	−1.11378
$159$	4.00000	0.317221
$160$	0	0
$161$	0	0
$162$	2.00000	0.157135
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	16.0000	1.24184
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	7.00000	0.535303
$172$	−14.0000	−1.06749
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	−16.0000	−1.21296
$175$	0	0
$176$	−16.0000	−1.20605
$177$	8.00000	0.601317
$178$	28.0000	2.09869
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	0	0
$181$	−21.0000	−1.56092	−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000	−1.56092	−0.780459	+	0.625207i	$0.785014\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	0	0
$186$	2.00000	0.146647
$187$	16.0000	1.17004
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−8.00000	−0.577350
$193$	−15.0000	−1.07972	−0.539862	−	0.841754i	$-0.681524\pi$
$193$	−15.0000	−1.07972	−0.539862	+	0.841754i	$0.681524\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	0	0
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	8.00000	0.568535
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	−28.0000	−1.97007
$203$	0	0
$204$	8.00000	0.560112
$205$	0	0
$206$	18.0000	1.25412
$207$	6.00000	0.417029
$208$	4.00000	0.277350
$209$	28.0000	1.93680
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	8.00000	0.549442
$213$	−6.00000	−0.411113
$214$	−28.0000	−1.91404
$215$	0	0
$216$	0	0
$217$	0	0
$218$	22.0000	1.49003
$219$	1.00000	0.0675737
$220$	0	0
$221$	−4.00000	−0.269069
$222$	14.0000	0.939618
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−16.0000	−1.06430
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	14.0000	0.927173
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	16.0000	1.04151
$237$	−7.00000	−0.454699
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	10.0000	0.642824
$243$	1.00000	0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	−2.00000	−0.127515
$247$	−7.00000	−0.445399
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	34.0000	2.13335
$255$	0	0
$256$	16.0000	1.00000
$257$	−28.0000	−1.74659	−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000	−1.74659	−0.873296	+	0.487190i	$0.838022\pi$
$258$	−14.0000	−0.871602
$259$	0	0
$260$	0	0
$261$	−8.00000	−0.495188
$262$	16.0000	0.988483
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.0000	0.856786
$268$	26.0000	1.58820
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−16.0000	−0.970143
$273$	0	0
$274$	−24.0000	−1.44989
$275$	−20.0000	−1.20605
$276$	12.0000	0.722315
$277$	−23.0000	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$277$	−23.0000	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$278$	2.00000	0.119952
$279$	1.00000	0.0598684
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	−31.0000	−1.84276	−0.921379	−	0.388664i	$-0.872937\pi$
$283$	−31.0000	−1.84276	−0.921379	+	0.388664i	$0.872937\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−8.00000	−0.473050
$287$	0	0
$288$	−8.00000	−0.471405
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−2.00000	−0.117242
$292$	2.00000	0.117041
$293$	−34.0000	−1.98630	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	−34.0000	−1.98630	−0.993151	+	0.116841i	$0.962723\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.00000	0.232104
$298$	−12.0000	−0.695141
$299$	−6.00000	−0.346989
$300$	−10.0000	−0.577350
$301$	0	0
$302$	32.0000	1.84139
$303$	−14.0000	−0.804279
$304$	−28.0000	−1.60591
$305$	0	0
$306$	8.00000	0.457330
$307$	11.0000	0.627803	0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000	0.627803	0.313902	+	0.949456i	$0.398364\pi$
$308$	0	0
$309$	9.00000	0.511992
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−15.0000	−0.847850	−0.423925	−	0.905697i	$-0.639348\pi$
$313$	−15.0000	−0.847850	−0.423925	+	0.905697i	$0.639348\pi$
$314$	44.0000	2.48306
$315$	0	0
$316$	−14.0000	−0.787562
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	8.00000	0.448618
$319$	−32.0000	−1.79166
$320$	0	0
$321$	−14.0000	−0.781404
$322$	0	0
$323$	28.0000	1.55796
$324$	2.00000	0.111111
$325$	5.00000	0.277350
$326$	−24.0000	−1.32924
$327$	11.0000	0.608301
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.0000	−0.824475	−0.412237	−	0.911077i	$-0.635253\pi$
$331$	−15.0000	−0.824475	−0.412237	+	0.911077i	$0.635253\pi$
$332$	16.0000	0.878114
$333$	7.00000	0.383598
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	−24.0000	−1.30543
$339$	−8.00000	−0.434500
$340$	0	0
$341$	4.00000	0.216612
$342$	14.0000	0.757033
$343$	0	0
$344$	0	0
$345$	0	0
$346$	24.0000	1.29025
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−16.0000	−0.857690
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−32.0000	−1.70561
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	16.0000	0.850390
$355$	0	0
$356$	28.0000	1.48400
$357$	0	0
$358$	−4.00000	−0.211407
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−42.0000	−2.20747
$363$	5.00000	0.262432
$364$	0	0
$365$	0	0
$366$	4.00000	0.209083
$367$	−25.0000	−1.30499	−0.652495	−	0.757793i	$-0.726278\pi$
$367$	−25.0000	−1.30499	−0.652495	+	0.757793i	$0.726278\pi$
$368$	−24.0000	−1.25109
$369$	−1.00000	−0.0520579
$370$	0	0
$371$	0	0
$372$	2.00000	0.103695
$373$	−29.0000	−1.50156	−0.750782	−	0.660551i	$-0.770323\pi$
$373$	−29.0000	−1.50156	−0.750782	+	0.660551i	$0.770323\pi$
$374$	32.0000	1.65468
$375$	0	0
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	−21.0000	−1.07870	−0.539349	−	0.842082i	$-0.681330\pi$
$379$	−21.0000	−1.07870	−0.539349	+	0.842082i	$0.681330\pi$
$380$	0	0
$381$	17.0000	0.870936
$382$	24.0000	1.22795
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	0	0
$386$	−30.0000	−1.52696
$387$	−7.00000	−0.355830
$388$	−4.00000	−0.203069
$389$	28.0000	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$389$	28.0000	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	8.00000	0.403547
$394$	16.0000	0.806068
$395$	0	0
$396$	8.00000	0.402015
$397$	1.00000	0.0501886	0.0250943	−	0.999685i	$-0.492011\pi$
$397$	1.00000	0.0501886	0.0250943	+	0.999685i	$0.492011\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	20.0000	1.00000
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	26.0000	1.29676
$403$	−1.00000	−0.0498135
$404$	−28.0000	−1.39305
$405$	0	0
$406$	0	0
$407$	28.0000	1.38791
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	18.0000	0.886796
$413$	0	0
$414$	12.0000	0.589768
$415$	0	0
$416$	8.00000	0.392232
$417$	1.00000	0.0489702
$418$	56.0000	2.73905
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	−5.00000	−0.243685	−0.121843	−	0.992549i	$-0.538880\pi$
$421$	−5.00000	−0.243685	−0.121843	+	0.992549i	$0.538880\pi$
$422$	−32.0000	−1.55774
$423$	0	0
$424$	0	0
$425$	−20.0000	−0.970143
$426$	−12.0000	−0.581402
$427$	0	0
$428$	−28.0000	−1.35343
$429$	−4.00000	−0.193122
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	−4.00000	−0.192450
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	0	0
$436$	22.0000	1.05361
$437$	42.0000	2.00913
$438$	2.00000	0.0955637
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	−8.00000	−0.380521
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	14.0000	0.664411
$445$	0	0
$446$	40.0000	1.89405
$447$	−6.00000	−0.283790
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	−10.0000	−0.471405
$451$	−4.00000	−0.188353
$452$	−16.0000	−0.752577
$453$	16.0000	0.751746
$454$	−36.0000	−1.68956
$455$	0	0
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	30.0000	1.40181
$459$	4.00000	0.186704
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	29.0000	1.34774	0.673872	−	0.738848i	$-0.264630\pi$
$463$	29.0000	1.34774	0.673872	+	0.738848i	$0.264630\pi$
$464$	32.0000	1.48556
$465$	0	0
$466$	−8.00000	−0.370593
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	0	0
$471$	22.0000	1.01371
$472$	0	0
$473$	−28.0000	−1.28744
$474$	−14.0000	−0.643041
$475$	−35.0000	−1.60591
$476$	0	0
$477$	4.00000	0.183147
$478$	−12.0000	−0.548867
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	−44.0000	−2.00415
$483$	0	0
$484$	10.0000	0.454545
$485$	0	0
$486$	2.00000	0.0907218
$487$	31.0000	1.40474	0.702372	−	0.711810i	$-0.252124\pi$
$487$	31.0000	1.40474	0.702372	+	0.711810i	$0.252124\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	−2.00000	−0.0901670
$493$	−32.0000	−1.44121
$494$	−14.0000	−0.629890
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	16.0000	0.716977
$499$	−37.0000	−1.65635	−0.828174	−	0.560471i	$-0.810620\pi$
$499$	−37.0000	−1.65635	−0.828174	+	0.560471i	$0.810620\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	52.0000	2.32087
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	48.0000	2.13386
$507$	−12.0000	−0.532939
$508$	34.0000	1.50851
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	0	0
$512$	32.0000	1.41421
$513$	7.00000	0.309058
$514$	−56.0000	−2.47005
$515$	0	0
$516$	−14.0000	−0.616316
$517$	0	0
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	−16.0000	−0.700301
$523$	−5.00000	−0.218635	−0.109317	−	0.994007i	$-0.534866\pi$
$523$	−5.00000	−0.218635	−0.109317	+	0.994007i	$0.534866\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	36.0000	1.56967
$527$	4.00000	0.174243
$528$	−16.0000	−0.696311
$529$	13.0000	0.565217
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	1.00000	0.0433148
$534$	28.0000	1.21168
$535$	0	0
$536$	0	0
$537$	−2.00000	−0.0863064
$538$	−12.0000	−0.517357
$539$	0	0
$540$	0	0
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	16.0000	0.687259
$543$	−21.0000	−0.901196
$544$	−32.0000	−1.37199
$545$	0	0
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−24.0000	−1.02523
$549$	2.00000	0.0853579
$550$	−40.0000	−1.70561
$551$	−56.0000	−2.38568
$552$	0	0
$553$	0	0
$554$	−46.0000	−1.95435
$555$	0	0
$556$	2.00000	0.0848189
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	2.00000	0.0846668
$559$	7.00000	0.296068
$560$	0	0
$561$	16.0000	0.675521
$562$	−36.0000	−1.51857
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	0	0
$565$	0	0
$566$	−62.0000	−2.60605
$567$	0	0
$568$	0	0
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	−8.00000	−0.334497
$573$	12.0000	0.501307
$574$	0	0
$575$	−30.0000	−1.25109
$576$	−8.00000	−0.333333
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	−2.00000	−0.0831890
$579$	−15.0000	−0.623379
$580$	0	0
$581$	0	0
$582$	−4.00000	−0.165805
$583$	16.0000	0.662652
$584$	0	0
$585$	0	0
$586$	−68.0000	−2.80905
$587$	−10.0000	−0.412744	−0.206372	−	0.978474i	$-0.566166\pi$
$587$	−10.0000	−0.412744	−0.206372	+	0.978474i	$0.566166\pi$
$588$	0	0
$589$	7.00000	0.288430
$590$	0	0
$591$	8.00000	0.329076
$592$	−28.0000	−1.15079
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	8.00000	0.328244
$595$	0	0
$596$	−12.0000	−0.491539
$597$	8.00000	0.327418
$598$	−12.0000	−0.490716
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	1.00000	0.0407909	0.0203954	−	0.999792i	$-0.493507\pi$
$601$	1.00000	0.0407909	0.0203954	+	0.999792i	$0.493507\pi$
$602$	0	0
$603$	13.0000	0.529401
$604$	32.0000	1.30206
$605$	0	0
$606$	−28.0000	−1.13742
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	−56.0000	−2.27110
$609$	0	0
$610$	0	0
$611$	0	0
$612$	8.00000	0.323381
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	22.0000	0.887848
$615$	0	0
$616$	0	0
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	18.0000	0.724066
$619$	15.0000	0.602901	0.301450	−	0.953482i	$-0.402529\pi$
$619$	15.0000	0.602901	0.301450	+	0.953482i	$0.402529\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	24.0000	0.962312
$623$	0	0
$624$	4.00000	0.160128
$625$	25.0000	1.00000
$626$	−30.0000	−1.19904
$627$	28.0000	1.11821
$628$	44.0000	1.75579
$629$	28.0000	1.11643
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	−16.0000	−0.635943
$634$	−44.0000	−1.74746
$635$	0	0
$636$	8.00000	0.317221
$637$	0	0
$638$	−64.0000	−2.53378
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	−28.0000	−1.10507
$643$	−37.0000	−1.45914	−0.729569	−	0.683907i	$-0.760279\pi$
$643$	−37.0000	−1.45914	−0.729569	+	0.683907i	$0.760279\pi$
$644$	0	0
$645$	0	0
$646$	56.0000	2.20329
$647$	−46.0000	−1.80845	−0.904223	−	0.427060i	$-0.859549\pi$
$647$	−46.0000	−1.80845	−0.904223	+	0.427060i	$0.859549\pi$
$648$	0	0
$649$	32.0000	1.25611
$650$	10.0000	0.392232
$651$	0	0
$652$	−24.0000	−0.939913
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	22.0000	0.860268
$655$	0	0
$656$	4.00000	0.156174
$657$	1.00000	0.0390137
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−1.00000	−0.0388955	−0.0194477	−	0.999811i	$-0.506191\pi$
$661$	−1.00000	−0.0388955	−0.0194477	+	0.999811i	$0.506191\pi$
$662$	−30.0000	−1.16598
$663$	−4.00000	−0.155347
$664$	0	0
$665$	0	0
$666$	14.0000	0.542489
$667$	−48.0000	−1.85857
$668$	12.0000	0.464294
$669$	20.0000	0.773245
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	−23.0000	−0.886585	−0.443292	−	0.896377i	$-0.646190\pi$
$673$	−23.0000	−0.886585	−0.443292	+	0.896377i	$0.646190\pi$
$674$	−46.0000	−1.77185
$675$	−5.00000	−0.192450
$676$	−24.0000	−0.923077
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	−16.0000	−0.614476
$679$	0	0
$680$	0	0
$681$	−18.0000	−0.689761
$682$	8.00000	0.306336
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	14.0000	0.535303
$685$	0	0
$686$	0	0
$687$	15.0000	0.572286
$688$	28.0000	1.06749
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−3.00000	−0.114125	−0.0570627	−	0.998371i	$-0.518173\pi$
$691$	−3.00000	−0.114125	−0.0570627	+	0.998371i	$0.518173\pi$
$692$	24.0000	0.912343
$693$	0	0
$694$	−24.0000	−0.911028
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	−68.0000	−2.57384
$699$	−4.00000	−0.151294
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	−2.00000	−0.0754851
$703$	49.0000	1.84807
$704$	−32.0000	−1.20605
$705$	0	0
$706$	−20.0000	−0.752710
$707$	0	0
$708$	16.0000	0.601317
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	−7.00000	−0.262521
$712$	0	0
$713$	6.00000	0.224702
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	−6.00000	−0.224074
$718$	20.0000	0.746393
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	0	0
$722$	60.0000	2.23297
$723$	−22.0000	−0.818189
$724$	−42.0000	−1.56092
$725$	40.0000	1.48556
$726$	10.0000	0.371135
$727$	1.00000	0.0370879	0.0185440	−	0.999828i	$-0.494097\pi$
$727$	1.00000	0.0370879	0.0185440	+	0.999828i	$0.494097\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−28.0000	−1.03562
$732$	4.00000	0.147844
$733$	−19.0000	−0.701781	−0.350891	−	0.936416i	$-0.614121\pi$
$733$	−19.0000	−0.701781	−0.350891	+	0.936416i	$0.614121\pi$
$734$	−50.0000	−1.84553
$735$	0	0
$736$	−48.0000	−1.76930
$737$	52.0000	1.91544
$738$	−2.00000	−0.0736210
$739$	−19.0000	−0.698926	−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000	−0.698926	−0.349463	+	0.936950i	$0.613636\pi$
$740$	0	0
$741$	−7.00000	−0.257151
$742$	0	0
$743$	22.0000	0.807102	0.403551	−	0.914957i	$-0.367776\pi$
$743$	22.0000	0.807102	0.403551	+	0.914957i	$0.367776\pi$
$744$	0	0
$745$	0	0
$746$	−58.0000	−2.12353
$747$	8.00000	0.292705
$748$	32.0000	1.17004
$749$	0	0
$750$	0	0
$751$	7.00000	0.255434	0.127717	−	0.991811i	$-0.459235\pi$
$751$	7.00000	0.255434	0.127717	+	0.991811i	$0.459235\pi$
$752$	0	0
$753$	26.0000	0.947493
$754$	16.0000	0.582686
$755$	0	0
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−42.0000	−1.52551
$759$	24.0000	0.871145
$760$	0	0
$761$	24.0000	0.869999	0.435000	−	0.900431i	$-0.356748\pi$
$761$	24.0000	0.869999	0.435000	+	0.900431i	$0.356748\pi$
$762$	34.0000	1.23169
$763$	0	0
$764$	24.0000	0.868290
$765$	0	0
$766$	−40.0000	−1.44526
$767$	−8.00000	−0.288863
$768$	16.0000	0.577350
$769$	7.00000	0.252426	0.126213	−	0.992003i	$-0.459718\pi$
$769$	7.00000	0.252426	0.126213	+	0.992003i	$0.459718\pi$
$770$	0	0
$771$	−28.0000	−1.00840
$772$	−30.0000	−1.07972
$773$	20.0000	0.719350	0.359675	−	0.933078i	$-0.382888\pi$
$773$	20.0000	0.719350	0.359675	+	0.933078i	$0.382888\pi$
$774$	−14.0000	−0.503220
$775$	−5.00000	−0.179605
$776$	0	0
$777$	0	0
$778$	56.0000	2.00770
$779$	−7.00000	−0.250801
$780$	0	0
$781$	−24.0000	−0.858788
$782$	48.0000	1.71648
$783$	−8.00000	−0.285897
$784$	0	0
$785$	0	0
$786$	16.0000	0.570701
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	16.0000	0.569976
$789$	18.0000	0.640817
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	2.00000	0.0709773
$795$	0	0
$796$	16.0000	0.567105
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	0	0
$800$	40.0000	1.41421
$801$	14.0000	0.494666
$802$	−56.0000	−1.97743
$803$	4.00000	0.141157
$804$	26.0000	0.916949
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	−6.00000	−0.211210
$808$	0	0
$809$	28.0000	0.984428	0.492214	−	0.870474i	$-0.336188\pi$
$809$	28.0000	0.984428	0.492214	+	0.870474i	$0.336188\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	56.0000	1.96280
$815$	0	0
$816$	−16.0000	−0.560112
$817$	−49.0000	−1.71429
$818$	−14.0000	−0.489499
$819$	0	0
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	−24.0000	−0.837096
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	−20.0000	−0.696311
$826$	0	0
$827$	14.0000	0.486828	0.243414	−	0.969923i	$-0.421733\pi$
$827$	14.0000	0.486828	0.243414	+	0.969923i	$0.421733\pi$
$828$	12.0000	0.417029
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	−23.0000	−0.797861
$832$	8.00000	0.277350
$833$	0	0
$834$	2.00000	0.0692543
$835$	0	0
$836$	56.0000	1.93680
$837$	1.00000	0.0345651
$838$	36.0000	1.24360
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−10.0000	−0.344623
$843$	−18.0000	−0.619953
$844$	−32.0000	−1.10149
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−16.0000	−0.549442
$849$	−31.0000	−1.06392
$850$	−40.0000	−1.37199
$851$	42.0000	1.43974
$852$	−12.0000	−0.411113
$853$	51.0000	1.74621	0.873103	−	0.487535i	$-0.162104\pi$
$853$	51.0000	1.74621	0.873103	+	0.487535i	$0.162104\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.0000	1.63965	0.819824	−	0.572615i	$-0.194071\pi$
$857$	48.0000	1.63965	0.819824	+	0.572615i	$0.194071\pi$
$858$	−8.00000	−0.273115
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	0	0
$862$	32.0000	1.08992
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	−8.00000	−0.272166
$865$	0	0
$866$	22.0000	0.747590
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−28.0000	−0.949835
$870$	0	0
$871$	−13.0000	−0.440488
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	84.0000	2.84134
$875$	0	0
$876$	2.00000	0.0675737
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	32.0000	1.07995
$879$	−34.0000	−1.14679
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	0	0
$883$	−35.0000	−1.17784	−0.588922	−	0.808190i	$-0.700447\pi$
$883$	−35.0000	−1.17784	−0.588922	+	0.808190i	$0.700447\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	48.0000	1.61259
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.00000	0.134005
$892$	40.0000	1.33930
$893$	0	0
$894$	−12.0000	−0.401340
$895$	0	0
$896$	0	0
$897$	−6.00000	−0.200334
$898$	24.0000	0.800890
$899$	−8.00000	−0.266815
$900$	−10.0000	−0.333333
$901$	16.0000	0.533037
$902$	−8.00000	−0.266371
$903$	0	0
$904$	0	0
$905$	0	0
$906$	32.0000	1.06313
$907$	23.0000	0.763702	0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000	0.763702	0.381851	+	0.924224i	$0.375287\pi$
$908$	−36.0000	−1.19470
$909$	−14.0000	−0.464351
$910$	0	0
$911$	60.0000	1.98789	0.993944	−	0.109885i	$-0.0350482\pi$
$911$	60.0000	1.98789	0.993944	+	0.109885i	$0.0350482\pi$
$912$	−28.0000	−0.927173
$913$	32.0000	1.05905
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	30.0000	0.991228
$917$	0	0
$918$	8.00000	0.264039
$919$	−7.00000	−0.230909	−0.115454	−	0.993313i	$-0.536832\pi$
$919$	−7.00000	−0.230909	−0.115454	+	0.993313i	$0.536832\pi$
$920$	0	0
$921$	11.0000	0.362462
$922$	−60.0000	−1.97599
$923$	6.00000	0.197492
$924$	0	0
$925$	−35.0000	−1.15079
$926$	58.0000	1.90600
$927$	9.00000	0.295599
$928$	64.0000	2.10090
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	0	0
$932$	−8.00000	−0.262049
$933$	12.0000	0.392862
$934$	−48.0000	−1.57061
$935$	0	0
$936$	0	0
$937$	−39.0000	−1.27407	−0.637037	−	0.770833i	$-0.719840\pi$
$937$	−39.0000	−1.27407	−0.637037	+	0.770833i	$0.719840\pi$
$938$	0	0
$939$	−15.0000	−0.489506
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	44.0000	1.43360
$943$	−6.00000	−0.195387
$944$	−32.0000	−1.04151
$945$	0	0
$946$	−56.0000	−1.82072
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	−14.0000	−0.454699
$949$	−1.00000	−0.0324614
$950$	−70.0000	−2.27110
$951$	−22.0000	−0.713399
$952$	0	0
$953$	−32.0000	−1.03658	−0.518291	−	0.855204i	$-0.673432\pi$
$953$	−32.0000	−1.03658	−0.518291	+	0.855204i	$0.673432\pi$
$954$	8.00000	0.259010
$955$	0	0
$956$	−12.0000	−0.388108
$957$	−32.0000	−1.03441
$958$	−12.0000	−0.387702
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	−14.0000	−0.451378
$963$	−14.0000	−0.451144
$964$	−44.0000	−1.41714
$965$	0	0
$966$	0	0
$967$	37.0000	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$967$	37.0000	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$968$	0	0
$969$	28.0000	0.899490
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	2.00000	0.0641500
$973$	0	0
$974$	62.0000	1.98661
$975$	5.00000	0.160128
$976$	−8.00000	−0.256074
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	−24.0000	−0.767435
$979$	56.0000	1.78977
$980$	0	0
$981$	11.0000	0.351203
$982$	56.0000	1.78703
$983$	−14.0000	−0.446531	−0.223265	−	0.974758i	$-0.571672\pi$
$983$	−14.0000	−0.446531	−0.223265	+	0.974758i	$0.571672\pi$
$984$	0	0
$985$	0	0
$986$	−64.0000	−2.03818
$987$	0	0
$988$	−14.0000	−0.445399
$989$	−42.0000	−1.33552
$990$	0	0
$991$	−5.00000	−0.158830	−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000	−0.158830	−0.0794151	+	0.996842i	$0.525305\pi$
$992$	−8.00000	−0.254000
$993$	−15.0000	−0.476011
$994$	0	0
$995$	0	0
$996$	16.0000	0.506979
$997$	−55.0000	−1.74187	−0.870934	−	0.491400i	$-0.836485\pi$
$997$	−55.0000	−1.74187	−0.870934	+	0.491400i	$0.836485\pi$
$998$	−74.0000	−2.34243
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.h.1.1		1
7.2	even	3		861.2.i.a.739.1	yes	2
7.4	even	3		861.2.i.a.247.1	✓	2
7.6	odd	2		6027.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.i.a.247.1	✓	2	7.4	even	3
861.2.i.a.739.1	yes	2	7.2	even	3
6027.2.a.g.1.1		1	7.6	odd	2
6027.2.a.h.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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