LMFDB - Embedded newform 70.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [70,2,Mod(1,70)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(70, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("70.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 70 = 2 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	70.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:	$0.558952814149$
Analytic rank:	$0$
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	$\chi$	$=$	70.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} -6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} -3.00000 q^{36} -10.0000 q^{37} -1.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +3.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -6.00000 q^{52} -2.00000 q^{53} -4.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} -8.00000 q^{59} -14.0000 q^{61} +8.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -12.0000 q^{67} +2.00000 q^{68} +1.00000 q^{70} -16.0000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} -4.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} +8.00000 q^{83} -2.00000 q^{85} +4.00000 q^{86} +4.00000 q^{88} +10.0000 q^{89} +3.00000 q^{90} +6.00000 q^{91} +8.00000 q^{94} +2.00000 q^{97} +1.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−3.00000	−1.00000
$10$	−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	$0.293963\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−3.00000	−0.707107
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.00000	−1.17670
$27$	0	0
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	1.00000	0.169031
$36$	−3.00000	−0.500000
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	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
$45$	3.00000	0.447214
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−6.00000	−0.832050
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	−1.00000	−0.133631
$57$	0	0
$58$	6.00000	0.787839
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	8.00000	1.01600
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	1.00000	0.119523
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	−3.00000	−0.353553
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−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
$81$	9.00000	1.00000
$82$	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
$85$	−2.00000	−0.216930
$86$	4.00000	0.431331
$87$	0	0
$88$	4.00000	0.426401
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	3.00000	0.316228
$91$	6.00000	0.628971
$92$	0	0
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.00000	0.101015
$99$	−12.0000	−1.20605
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	18.0000	1.66410
$118$	−8.00000	−0.736460
$119$	−2.00000	−0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	−14.0000	−1.26750
$123$	0	0
$124$	8.00000	0.718421
$125$	−1.00000	−0.0894427
$126$	3.00000	0.267261
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	6.00000	0.526235
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	0	0
$134$	−12.0000	−1.03664
$135$	0	0
$136$	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
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−16.0000	−1.34269
$143$	−24.0000	−2.00698
$144$	−3.00000	−0.250000
$145$	−6.00000	−0.498273
$146$	2.00000	0.165521
$147$	0	0
$148$	−10.0000	−0.821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	−4.00000	−0.322329
$155$	−8.00000	−0.642575
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	9.00000	0.707107
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	8.00000	0.620920
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−2.00000	−0.153393
$171$	0	0
$172$	4.00000	0.304997
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	4.00000	0.301511
$177$	0	0
$178$	10.0000	0.749532
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	3.00000	0.223607
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	0	0
$185$	10.0000	0.735215
$186$	0	0
$187$	8.00000	0.585018
$188$	8.00000	0.583460
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	−12.0000	−0.852803
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−6.00000	−0.422159
$203$	−6.00000	−0.421117
$204$	0	0
$205$	−2.00000	−0.139686
$206$	16.0000	1.11477
$207$	0	0
$208$	−6.00000	−0.416025
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	12.0000	0.820303
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−8.00000	−0.543075
$218$	6.00000	0.406371
$219$	0	0
$220$	−4.00000	−0.269680
$221$	−12.0000	−0.807207
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−1.00000	−0.0668153
$225$	−3.00000	−0.200000
$226$	2.00000	0.133038
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	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	$0.562969\pi$
$234$	18.0000	1.17670
$235$	−8.00000	−0.521862
$236$	−8.00000	−0.520756
$237$	0	0
$238$	−2.00000	−0.129641
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	−14.0000	−0.896258
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	8.00000	0.508001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	3.00000	0.188982
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	6.00000	0.372104
$261$	−18.0000	−1.11417
$262$	−16.0000	−0.988483
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	0	0
$268$	−12.0000	−0.733017
$269$	−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
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−6.00000	−0.362473
$275$	4.00000	0.241209
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	16.0000	0.959616
$279$	−24.0000	−1.43684
$280$	1.00000	0.0597614
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	32.0000	1.90220	0.951101	−	0.308879i	$-0.0999539\pi$
$283$	32.0000	1.90220	0.951101	+	0.308879i	$0.0999539\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	−24.0000	−1.41915
$287$	−2.00000	−0.118056
$288$	−3.00000	−0.176777
$289$	−13.0000	−0.764706
$290$	−6.00000	−0.352332
$291$	0	0
$292$	2.00000	0.117041
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	−10.0000	−0.581238
$297$	0	0
$298$	6.00000	0.347571
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	8.00000	0.460348
$303$	0	0
$304$	0	0
$305$	14.0000	0.801638
$306$	−6.00000	−0.342997
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	−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	$0.738219\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	10.0000	0.564333
$315$	−3.00000	−0.169031
$316$	−8.00000	−0.450035
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	0	0
$324$	9.00000	0.500000
$325$	−6.00000	−0.332820
$326$	−4.00000	−0.221540
$327$	0	0
$328$	2.00000	0.110432
$329$	−8.00000	−0.441054
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	8.00000	0.439057
$333$	30.0000	1.64399
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	23.0000	1.25104
$339$	0	0
$340$	−2.00000	−0.108465
$341$	32.0000	1.73290
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	0	0
$346$	−22.0000	−1.18273
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	4.00000	0.213201
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	10.0000	0.529999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	3.00000	0.158114
$361$	−19.0000	−1.00000
$362$	−14.0000	−0.735824
$363$	0	0
$364$	6.00000	0.314485
$365$	−2.00000	−0.104685
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	10.0000	0.519875
$371$	2.00000	0.103835
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	8.00000	0.412568
$377$	−36.0000	−1.85409
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	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	$0.289894\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	2.00000	0.101797
$387$	−12.0000	−0.609994
$388$	2.00000	0.101535
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	0	0
$392$	1.00000	0.0505076
$393$	0	0
$394$	14.0000	0.705310
$395$	8.00000	0.402524
$396$	−12.0000	−0.603023
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	−48.0000	−2.39105
$404$	−6.00000	−0.298511
$405$	−9.00000	−0.447214
$406$	−6.00000	−0.297775
$407$	−40.0000	−1.98273
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	−2.00000	−0.0987730
$411$	0	0
$412$	16.0000	0.788263
$413$	8.00000	0.393654
$414$	0	0
$415$	−8.00000	−0.392705
$416$	−6.00000	−0.294174
$417$	0	0
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000	0.194717
$423$	−24.0000	−1.16692
$424$	−2.00000	−0.0971286
$425$	2.00000	0.0970143
$426$	0	0
$427$	14.0000	0.677507
$428$	12.0000	0.580042
$429$	0	0
$430$	−4.00000	−0.192897
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	6.00000	0.287348
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	−4.00000	−0.190693
$441$	−3.00000	−0.142857
$442$	−12.0000	−0.570782
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	16.0000	0.757622
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−3.00000	−0.141421
$451$	8.00000	0.376705
$452$	2.00000	0.0940721
$453$	0	0
$454$	8.00000	0.375459
$455$	−6.00000	−0.281284
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	−14.0000	−0.654177
$459$	0	0
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−40.0000	−1.85098	−0.925490	−	0.378773i	$-0.876346\pi$
$467$	−40.0000	−1.85098	−0.925490	+	0.378773i	$0.876346\pi$
$468$	18.0000	0.832050
$469$	12.0000	0.554109
$470$	−8.00000	−0.369012
$471$	0	0
$472$	−8.00000	−0.368230
$473$	16.0000	0.735681
$474$	0	0
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	6.00000	0.274721
$478$	16.0000	0.731823
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	60.0000	2.73576
$482$	10.0000	0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−14.0000	−0.633750
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	12.0000	0.539360
$496$	8.00000	0.359211
$497$	16.0000	0.717698
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	3.00000	0.133631
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	−22.0000	−0.970378
$515$	−16.0000	−0.705044
$516$	0	0
$517$	32.0000	1.40736
$518$	10.0000	0.439375
$519$	0	0
$520$	6.00000	0.263117
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	−18.0000	−0.787839
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	8.00000	0.348817
$527$	16.0000	0.696971
$528$	0	0
$529$	−23.0000	−1.00000
$530$	2.00000	0.0868744
$531$	24.0000	1.04151
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−12.0000	−0.518805
$536$	−12.0000	−0.518321
$537$	0	0
$538$	−6.00000	−0.258678
$539$	4.00000	0.172292
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	0	0
$544$	2.00000	0.0857493
$545$	−6.00000	−0.257012
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−6.00000	−0.256307
$549$	42.0000	1.79252
$550$	4.00000	0.170561
$551$	0	0
$552$	0	0
$553$	8.00000	0.340195
$554$	−18.0000	−0.764747
$555$	0	0
$556$	16.0000	0.678551
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	−24.0000	−1.01600
$559$	−24.0000	−1.01509
$560$	1.00000	0.0422577
$561$	0	0
$562$	26.0000	1.09674
$563$	−16.0000	−0.674320	−0.337160	−	0.941447i	$-0.609466\pi$
$563$	−16.0000	−0.674320	−0.337160	+	0.941447i	$0.609466\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	32.0000	1.34506
$567$	−9.00000	−0.377964
$568$	−16.0000	−0.671345
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	−24.0000	−1.00349
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	0	0
$576$	−3.00000	−0.125000
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	−6.00000	−0.249136
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−8.00000	−0.331326
$584$	2.00000	0.0827606
$585$	−18.0000	−0.744208
$586$	10.0000	0.413096
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	0	0
$590$	8.00000	0.329355
$591$	0	0
$592$	−10.0000	−0.410997
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	−4.00000	−0.163028
$603$	36.0000	1.46603
$604$	8.00000	0.325515
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	0	0
$610$	14.0000	0.566843
$611$	−48.0000	−1.94187
$612$	−6.00000	−0.242536
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	−24.0000	−0.962312
$623$	−10.0000	−0.400642
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	10.0000	0.399043
$629$	−20.0000	−0.797452
$630$	−3.00000	−0.119523
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	22.0000	0.873732
$635$	8.00000	0.317470
$636$	0	0
$637$	−6.00000	−0.237729
$638$	24.0000	0.950169
$639$	48.0000	1.89885
$640$	−1.00000	−0.0395285
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	9.00000	0.353553
$649$	−32.0000	−1.25611
$650$	−6.00000	−0.235339
$651$	0	0
$652$	−4.00000	−0.156652
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	2.00000	0.0780869
$657$	−6.00000	−0.234082
$658$	−8.00000	−0.311872
$659$	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
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	8.00000	0.310460
$665$	0	0
$666$	30.0000	1.16248
$667$	0	0
$668$	0	0
$669$	0	0
$670$	12.0000	0.463600
$671$	−56.0000	−2.16186
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	23.0000	0.884615
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	−2.00000	−0.0766965
$681$	0	0
$682$	32.0000	1.22534
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	4.00000	0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	−22.0000	−0.836315
$693$	12.0000	0.455842
$694$	4.00000	0.151838
$695$	−16.0000	−0.606915
$696$	0	0
$697$	4.00000	0.151511
$698$	10.0000	0.378506
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	0	0
$704$	4.00000	0.150756
$705$	0	0
$706$	−6.00000	−0.225813
$707$	6.00000	0.225653
$708$	0	0
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	16.0000	0.600469
$711$	24.0000	0.900070
$712$	10.0000	0.374766
$713$	0	0
$714$	0	0
$715$	24.0000	0.897549
$716$	−12.0000	−0.448461
$717$	0	0
$718$	8.00000	0.298557
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	3.00000	0.111803
$721$	−16.0000	−0.595871
$722$	−19.0000	−0.707107
$723$	0	0
$724$	−14.0000	−0.520306
$725$	6.00000	0.222834
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	6.00000	0.222375
$729$	−27.0000	−1.00000
$730$	−2.00000	−0.0740233
$731$	8.00000	0.295891
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	0	0
$737$	−48.0000	−1.76810
$738$	−6.00000	−0.220863
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	10.0000	0.367607
$741$	0	0
$742$	2.00000	0.0734223
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	14.0000	0.512576
$747$	−24.0000	−0.878114
$748$	8.00000	0.292509
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	−36.0000	−1.31104
$755$	−8.00000	−0.291150
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−12.0000	−0.435860
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	−6.00000	−0.217215
$764$	24.0000	0.868290
$765$	6.00000	0.216930
$766$	24.0000	0.867155
$767$	48.0000	1.73318
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	4.00000	0.144150
$771$	0	0
$772$	2.00000	0.0719816
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	−12.0000	−0.431331
$775$	8.00000	0.287368
$776$	2.00000	0.0717958
$777$	0	0
$778$	14.0000	0.501924
$779$	0	0
$780$	0	0
$781$	−64.0000	−2.29010
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	14.0000	0.498729
$789$	0	0
$790$	8.00000	0.284627
$791$	−2.00000	−0.0711118
$792$	−12.0000	−0.426401
$793$	84.0000	2.98293
$794$	10.0000	0.354887
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	1.00000	0.0353553
$801$	−30.0000	−1.06000
$802$	−14.0000	−0.494357
$803$	8.00000	0.282314
$804$	0	0
$805$	0	0
$806$	−48.0000	−1.69073
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−9.00000	−0.316228
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	−40.0000	−1.40200
$815$	4.00000	0.140114
$816$	0	0
$817$	0	0
$818$	−30.0000	−1.04893
$819$	−18.0000	−0.628971
$820$	−2.00000	−0.0698430
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	8.00000	0.278356
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	−6.00000	−0.208013
$833$	2.00000	0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	24.0000	0.829066
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	0	0
$844$	4.00000	0.137686
$845$	−23.0000	−0.791224
$846$	−24.0000	−0.825137
$847$	−5.00000	−0.171802
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	2.00000	0.0685994
$851$	0	0
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	14.0000	0.479070
$855$	0	0
$856$	12.0000	0.410152
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	−48.0000	−1.63774	−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000	−1.63774	−0.818869	+	0.573980i	$0.805399\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	2.00000	0.0679628
$867$	0	0
$868$	−8.00000	−0.271538
$869$	−32.0000	−1.08553
$870$	0	0
$871$	72.0000	2.43963
$872$	6.00000	0.203186
$873$	−6.00000	−0.203069
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	−4.00000	−0.134840
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	−3.00000	−0.101015
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−20.0000	−0.671913
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	−10.0000	−0.335201
$891$	36.0000	1.20605
$892$	16.0000	0.535720
$893$	0	0
$894$	0	0
$895$	12.0000	0.401116
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−30.0000	−1.00111
$899$	48.0000	1.60089
$900$	−3.00000	−0.100000
$901$	−4.00000	−0.133259
$902$	8.00000	0.266371
$903$	0	0
$904$	2.00000	0.0665190
$905$	14.0000	0.465376
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	8.00000	0.265489
$909$	18.0000	0.597022
$910$	−6.00000	−0.198898
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	32.0000	1.05905
$914$	10.0000	0.330771
$915$	0	0
$916$	−14.0000	−0.462573
$917$	16.0000	0.528367
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	0	0
$922$	10.0000	0.329332
$923$	96.0000	3.15988
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−16.0000	−0.525793
$927$	−48.0000	−1.57653
$928$	6.00000	0.196960
$929$	58.0000	1.90292	0.951459	−	0.307775i	$-0.0995844\pi$
$929$	58.0000	1.90292	0.951459	+	0.307775i	$0.0995844\pi$
$930$	0	0
$931$	0	0
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−40.0000	−1.30884
$935$	−8.00000	−0.261628
$936$	18.0000	0.588348
$937$	50.0000	1.63343	0.816714	−	0.577042i	$-0.195793\pi$
$937$	50.0000	1.63343	0.816714	+	0.577042i	$0.195793\pi$
$938$	12.0000	0.391814
$939$	0	0
$940$	−8.00000	−0.260931
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	0	0
$943$	0	0
$944$	−8.00000	−0.260378
$945$	0	0
$946$	16.0000	0.520205
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	−2.00000	−0.0648204
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	6.00000	0.194257
$955$	−24.0000	−0.776622
$956$	16.0000	0.517477
$957$	0	0
$958$	24.0000	0.775405
$959$	6.00000	0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	60.0000	1.93448
$963$	−36.0000	−1.16008
$964$	10.0000	0.322078
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	32.0000	1.02535
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	−1.00000	−0.0319438
$981$	−18.0000	−0.574696
$982$	12.0000	0.382935
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	0	0
$985$	−14.0000	−0.446077
$986$	12.0000	0.382158
$987$	0	0
$988$	0	0
$989$	0	0
$990$	12.0000	0.381385
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	16.0000	0.507489
$995$	16.0000	0.507234
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−12.0000	−0.379853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	70.2.a.a.1.1	✓	1
3.2	odd	2		630.2.a.d.1.1		1
4.3	odd	2		560.2.a.d.1.1		1
5.2	odd	4		350.2.c.b.99.2		2
5.3	odd	4		350.2.c.b.99.1		2
5.4	even	2		350.2.a.b.1.1		1
7.2	even	3		490.2.e.d.361.1		2
7.3	odd	6		490.2.e.c.471.1		2
7.4	even	3		490.2.e.d.471.1		2
7.5	odd	6		490.2.e.c.361.1		2
7.6	odd	2		490.2.a.h.1.1		1
8.3	odd	2		2240.2.a.q.1.1		1
8.5	even	2		2240.2.a.n.1.1		1
11.10	odd	2		8470.2.a.j.1.1		1
12.11	even	2		5040.2.a.bm.1.1		1
15.2	even	4		3150.2.g.c.2899.1		2
15.8	even	4		3150.2.g.c.2899.2		2
15.14	odd	2		3150.2.a.bj.1.1		1
20.3	even	4		2800.2.g.n.449.1		2
20.7	even	4		2800.2.g.n.449.2		2
20.19	odd	2		2800.2.a.m.1.1		1
21.20	even	2		4410.2.a.b.1.1		1
28.27	even	2		3920.2.a.t.1.1		1
35.13	even	4		2450.2.c.k.99.1		2
35.27	even	4		2450.2.c.k.99.2		2
35.34	odd	2		2450.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.a.a.1.1	✓	1	1.1	even	1	trivial
350.2.a.b.1.1		1	5.4	even	2
350.2.c.b.99.1		2	5.3	odd	4
350.2.c.b.99.2		2	5.2	odd	4
490.2.a.h.1.1		1	7.6	odd	2
490.2.e.c.361.1		2	7.5	odd	6
490.2.e.c.471.1		2	7.3	odd	6
490.2.e.d.361.1		2	7.2	even	3
490.2.e.d.471.1		2	7.4	even	3
560.2.a.d.1.1		1	4.3	odd	2
630.2.a.d.1.1		1	3.2	odd	2
2240.2.a.n.1.1		1	8.5	even	2
2240.2.a.q.1.1		1	8.3	odd	2
2450.2.a.l.1.1		1	35.34	odd	2
2450.2.c.k.99.1		2	35.13	even	4
2450.2.c.k.99.2		2	35.27	even	4
2800.2.a.m.1.1		1	20.19	odd	2
2800.2.g.n.449.1		2	20.3	even	4
2800.2.g.n.449.2		2	20.7	even	4
3150.2.a.bj.1.1		1	15.14	odd	2
3150.2.g.c.2899.1		2	15.2	even	4
3150.2.g.c.2899.2		2	15.8	even	4
3920.2.a.t.1.1		1	28.27	even	2
4410.2.a.b.1.1		1	21.20	even	2
5040.2.a.bm.1.1		1	12.11	even	2
8470.2.a.j.1.1		1	11.10	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:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	70.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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