LMFDB - Embedded newform 2070.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,4,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2070.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2070.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:	$122.133953712$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -18.0000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -18.0000 q^{7} -8.00000 q^{8} -10.0000 q^{10} +32.0000 q^{11} -47.0000 q^{13} +36.0000 q^{14} +16.0000 q^{16} -20.0000 q^{17} +36.0000 q^{19} +20.0000 q^{20} -64.0000 q^{22} +23.0000 q^{23} +25.0000 q^{25} +94.0000 q^{26} -72.0000 q^{28} +27.0000 q^{29} -33.0000 q^{31} -32.0000 q^{32} +40.0000 q^{34} -90.0000 q^{35} +56.0000 q^{37} -72.0000 q^{38} -40.0000 q^{40} +157.000 q^{41} +18.0000 q^{43} +128.000 q^{44} -46.0000 q^{46} -65.0000 q^{47} -19.0000 q^{49} -50.0000 q^{50} -188.000 q^{52} +14.0000 q^{53} +160.000 q^{55} +144.000 q^{56} -54.0000 q^{58} +744.000 q^{59} +552.000 q^{61} +66.0000 q^{62} +64.0000 q^{64} -235.000 q^{65} -156.000 q^{67} -80.0000 q^{68} +180.000 q^{70} -699.000 q^{71} -609.000 q^{73} -112.000 q^{74} +144.000 q^{76} -576.000 q^{77} -644.000 q^{79} +80.0000 q^{80} -314.000 q^{82} -512.000 q^{83} -100.000 q^{85} -36.0000 q^{86} -256.000 q^{88} +102.000 q^{89} +846.000 q^{91} +92.0000 q^{92} +130.000 q^{94} +180.000 q^{95} +578.000 q^{97} +38.0000 q^{98} +O(q^{100})\)

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	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−18.0000	−0.971909	−0.485954	−	0.873984i	$-0.661528\pi$
$7$	−18.0000	−0.971909	−0.485954	+	0.873984i	$0.661528\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−10.0000	−0.316228
$11$	32.0000	0.877124	0.438562	−	0.898701i	$-0.355488\pi$
$11$	32.0000	0.877124	0.438562	+	0.898701i	$0.355488\pi$
$12$	0	0
$13$	−47.0000	−1.00273	−0.501364	−	0.865237i	$-0.667168\pi$
$13$	−47.0000	−1.00273	−0.501364	+	0.865237i	$0.667168\pi$
$14$	36.0000	0.687243
$15$	0	0
$16$	16.0000	0.250000
$17$	−20.0000	−0.285336	−0.142668	−	0.989771i	$-0.545568\pi$
$17$	−20.0000	−0.285336	−0.142668	+	0.989771i	$0.545568\pi$
$18$	0	0
$19$	36.0000	0.434682	0.217341	−	0.976096i	$-0.430262\pi$
$19$	36.0000	0.434682	0.217341	+	0.976096i	$0.430262\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	−64.0000	−0.620220
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	94.0000	0.709035
$27$	0	0
$28$	−72.0000	−0.485954
$29$	27.0000	0.172889	0.0864444	−	0.996257i	$-0.472450\pi$
$29$	27.0000	0.172889	0.0864444	+	0.996257i	$0.472450\pi$
$30$	0	0
$31$	−33.0000	−0.191193	−0.0955964	−	0.995420i	$-0.530476\pi$
$31$	−33.0000	−0.191193	−0.0955964	+	0.995420i	$0.530476\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	40.0000	0.201763
$35$	−90.0000	−0.434651
$36$	0	0
$37$	56.0000	0.248820	0.124410	−	0.992231i	$-0.460296\pi$
$37$	56.0000	0.248820	0.124410	+	0.992231i	$0.460296\pi$
$38$	−72.0000	−0.307367
$39$	0	0
$40$	−40.0000	−0.158114
$41$	157.000	0.598031	0.299016	−	0.954248i	$-0.403342\pi$
$41$	157.000	0.598031	0.299016	+	0.954248i	$0.403342\pi$
$42$	0	0
$43$	18.0000	0.0638366	0.0319183	−	0.999490i	$-0.489838\pi$
$43$	18.0000	0.0638366	0.0319183	+	0.999490i	$0.489838\pi$
$44$	128.000	0.438562
$45$	0	0
$46$	−46.0000	−0.147442
$47$	−65.0000	−0.201728	−0.100864	−	0.994900i	$-0.532161\pi$
$47$	−65.0000	−0.201728	−0.100864	+	0.994900i	$0.532161\pi$
$48$	0	0
$49$	−19.0000	−0.0553936
$50$	−50.0000	−0.141421
$51$	0	0
$52$	−188.000	−0.501364
$53$	14.0000	0.0362839	0.0181420	−	0.999835i	$-0.494225\pi$
$53$	14.0000	0.0362839	0.0181420	+	0.999835i	$0.494225\pi$
$54$	0	0
$55$	160.000	0.392262
$56$	144.000	0.343622
$57$	0	0
$58$	−54.0000	−0.122251
$59$	744.000	1.64170	0.820852	−	0.571141i	$-0.193499\pi$
$59$	744.000	1.64170	0.820852	+	0.571141i	$0.193499\pi$
$60$	0	0
$61$	552.000	1.15863	0.579314	−	0.815104i	$-0.303320\pi$
$61$	552.000	1.15863	0.579314	+	0.815104i	$0.303320\pi$
$62$	66.0000	0.135194
$63$	0	0
$64$	64.0000	0.125000
$65$	−235.000	−0.448433
$66$	0	0
$67$	−156.000	−0.284454	−0.142227	−	0.989834i	$-0.545426\pi$
$67$	−156.000	−0.284454	−0.142227	+	0.989834i	$0.545426\pi$
$68$	−80.0000	−0.142668
$69$	0	0
$70$	180.000	0.307344
$71$	−699.000	−1.16839	−0.584197	−	0.811612i	$-0.698591\pi$
$71$	−699.000	−1.16839	−0.584197	+	0.811612i	$0.698591\pi$
$72$	0	0
$73$	−609.000	−0.976412	−0.488206	−	0.872728i	$-0.662348\pi$
$73$	−609.000	−0.976412	−0.488206	+	0.872728i	$0.662348\pi$
$74$	−112.000	−0.175942
$75$	0	0
$76$	144.000	0.217341
$77$	−576.000	−0.852484
$78$	0	0
$79$	−644.000	−0.917160	−0.458580	−	0.888653i	$-0.651642\pi$
$79$	−644.000	−0.917160	−0.458580	+	0.888653i	$0.651642\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	−314.000	−0.422872
$83$	−512.000	−0.677100	−0.338550	−	0.940948i	$-0.609936\pi$
$83$	−512.000	−0.677100	−0.338550	+	0.940948i	$0.609936\pi$
$84$	0	0
$85$	−100.000	−0.127606
$86$	−36.0000	−0.0451393
$87$	0	0
$88$	−256.000	−0.310110
$89$	102.000	0.121483	0.0607415	−	0.998154i	$-0.480653\pi$
$89$	102.000	0.121483	0.0607415	+	0.998154i	$0.480653\pi$
$90$	0	0
$91$	846.000	0.974559
$92$	92.0000	0.104257
$93$	0	0
$94$	130.000	0.142643
$95$	180.000	0.194396
$96$	0	0
$97$	578.000	0.605021	0.302510	−	0.953146i	$-0.402175\pi$
$97$	578.000	0.605021	0.302510	+	0.953146i	$0.402175\pi$
$98$	38.0000	0.0391692
$99$	0	0
$100$	100.000	0.100000
$101$	6.00000	0.00591111	0.00295556	−	0.999996i	$-0.499059\pi$
$101$	6.00000	0.00591111	0.00295556	+	0.999996i	$0.499059\pi$
$102$	0	0
$103$	−160.000	−0.153061	−0.0765304	−	0.997067i	$-0.524384\pi$
$103$	−160.000	−0.153061	−0.0765304	+	0.997067i	$0.524384\pi$
$104$	376.000	0.354518
$105$	0	0
$106$	−28.0000	−0.0256566
$107$	−380.000	−0.343327	−0.171663	−	0.985156i	$-0.554914\pi$
$107$	−380.000	−0.343327	−0.171663	+	0.985156i	$0.554914\pi$
$108$	0	0
$109$	250.000	0.219685	0.109842	−	0.993949i	$-0.464965\pi$
$109$	250.000	0.219685	0.109842	+	0.993949i	$0.464965\pi$
$110$	−320.000	−0.277371
$111$	0	0
$112$	−288.000	−0.242977
$113$	390.000	0.324674	0.162337	−	0.986735i	$-0.448097\pi$
$113$	390.000	0.324674	0.162337	+	0.986735i	$0.448097\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	108.000	0.0864444
$117$	0	0
$118$	−1488.00	−1.16086
$119$	360.000	0.277321
$120$	0	0
$121$	−307.000	−0.230654
$122$	−1104.00	−0.819274
$123$	0	0
$124$	−132.000	−0.0955964
$125$	125.000	0.0894427
$126$	0	0
$127$	−769.000	−0.537305	−0.268652	−	0.963237i	$-0.586578\pi$
$127$	−769.000	−0.537305	−0.268652	+	0.963237i	$0.586578\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	470.000	0.317090
$131$	213.000	0.142060	0.0710301	−	0.997474i	$-0.477371\pi$
$131$	213.000	0.142060	0.0710301	+	0.997474i	$0.477371\pi$
$132$	0	0
$133$	−648.000	−0.422472
$134$	312.000	0.201140
$135$	0	0
$136$	160.000	0.100882
$137$	−2836.00	−1.76858	−0.884291	−	0.466936i	$-0.845358\pi$
$137$	−2836.00	−1.76858	−0.884291	+	0.466936i	$0.845358\pi$
$138$	0	0
$139$	−1631.00	−0.995249	−0.497625	−	0.867393i	$-0.665794\pi$
$139$	−1631.00	−0.995249	−0.497625	+	0.867393i	$0.665794\pi$
$140$	−360.000	−0.217325
$141$	0	0
$142$	1398.00	0.826180
$143$	−1504.00	−0.879516
$144$	0	0
$145$	135.000	0.0773182
$146$	1218.00	0.690427
$147$	0	0
$148$	224.000	0.124410
$149$	1966.00	1.08095	0.540473	−	0.841361i	$-0.318245\pi$
$149$	1966.00	1.08095	0.540473	+	0.841361i	$0.318245\pi$
$150$	0	0
$151$	35.0000	0.0188626	0.00943132	−	0.999956i	$-0.496998\pi$
$151$	35.0000	0.0188626	0.00943132	+	0.999956i	$0.496998\pi$
$152$	−288.000	−0.153683
$153$	0	0
$154$	1152.00	0.602797
$155$	−165.000	−0.0855040
$156$	0	0
$157$	1702.00	0.865187	0.432594	−	0.901589i	$-0.357599\pi$
$157$	1702.00	0.865187	0.432594	+	0.901589i	$0.357599\pi$
$158$	1288.00	0.648530
$159$	0	0
$160$	−160.000	−0.0790569
$161$	−414.000	−0.202657
$162$	0	0
$163$	−2045.00	−0.982680	−0.491340	−	0.870968i	$-0.663493\pi$
$163$	−2045.00	−0.982680	−0.491340	+	0.870968i	$0.663493\pi$
$164$	628.000	0.299016
$165$	0	0
$166$	1024.00	0.478782
$167$	−1016.00	−0.470781	−0.235391	−	0.971901i	$-0.575637\pi$
$167$	−1016.00	−0.470781	−0.235391	+	0.971901i	$0.575637\pi$
$168$	0	0
$169$	12.0000	0.00546199
$170$	200.000	0.0902312
$171$	0	0
$172$	72.0000	0.0319183
$173$	−598.000	−0.262804	−0.131402	−	0.991329i	$-0.541948\pi$
$173$	−598.000	−0.262804	−0.131402	+	0.991329i	$0.541948\pi$
$174$	0	0
$175$	−450.000	−0.194382
$176$	512.000	0.219281
$177$	0	0
$178$	−204.000	−0.0859014
$179$	4607.00	1.92371	0.961853	−	0.273567i	$-0.0882036\pi$
$179$	4607.00	1.92371	0.961853	+	0.273567i	$0.0882036\pi$
$180$	0	0
$181$	−1212.00	−0.497720	−0.248860	−	0.968540i	$-0.580056\pi$
$181$	−1212.00	−0.497720	−0.248860	+	0.968540i	$0.580056\pi$
$182$	−1692.00	−0.689117
$183$	0	0
$184$	−184.000	−0.0737210
$185$	280.000	0.111276
$186$	0	0
$187$	−640.000	−0.250275
$188$	−260.000	−0.100864
$189$	0	0
$190$	−360.000	−0.137459
$191$	1058.00	0.400807	0.200404	−	0.979713i	$-0.435775\pi$
$191$	1058.00	0.400807	0.200404	+	0.979713i	$0.435775\pi$
$192$	0	0
$193$	1047.00	0.390491	0.195245	−	0.980754i	$-0.437450\pi$
$193$	1047.00	0.390491	0.195245	+	0.980754i	$0.437450\pi$
$194$	−1156.00	−0.427814
$195$	0	0
$196$	−76.0000	−0.0276968
$197$	−251.000	−0.0907767	−0.0453883	−	0.998969i	$-0.514453\pi$
$197$	−251.000	−0.0907767	−0.0453883	+	0.998969i	$0.514453\pi$
$198$	0	0
$199$	−3508.00	−1.24963	−0.624813	−	0.780775i	$-0.714825\pi$
$199$	−3508.00	−1.24963	−0.624813	+	0.780775i	$0.714825\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	−12.0000	−0.00417979
$203$	−486.000	−0.168032
$204$	0	0
$205$	785.000	0.267448
$206$	320.000	0.108230
$207$	0	0
$208$	−752.000	−0.250682
$209$	1152.00	0.381270
$210$	0	0
$211$	−3296.00	−1.07538	−0.537692	−	0.843141i	$-0.680704\pi$
$211$	−3296.00	−1.07538	−0.537692	+	0.843141i	$0.680704\pi$
$212$	56.0000	0.0181420
$213$	0	0
$214$	760.000	0.242769
$215$	90.0000	0.0285486
$216$	0	0
$217$	594.000	0.185822
$218$	−500.000	−0.155341
$219$	0	0
$220$	640.000	0.196131
$221$	940.000	0.286114
$222$	0	0
$223$	−2720.00	−0.816792	−0.408396	−	0.912805i	$-0.633912\pi$
$223$	−2720.00	−0.816792	−0.408396	+	0.912805i	$0.633912\pi$
$224$	576.000	0.171811
$225$	0	0
$226$	−780.000	−0.229579
$227$	4134.00	1.20874	0.604368	−	0.796705i	$-0.293426\pi$
$227$	4134.00	1.20874	0.604368	+	0.796705i	$0.293426\pi$
$228$	0	0
$229$	−4510.00	−1.30144	−0.650719	−	0.759319i	$-0.725532\pi$
$229$	−4510.00	−1.30144	−0.650719	+	0.759319i	$0.725532\pi$
$230$	−230.000	−0.0659380
$231$	0	0
$232$	−216.000	−0.0611254
$233$	5003.00	1.40668	0.703342	−	0.710852i	$-0.251690\pi$
$233$	5003.00	1.40668	0.703342	+	0.710852i	$0.251690\pi$
$234$	0	0
$235$	−325.000	−0.0902156
$236$	2976.00	0.820852
$237$	0	0
$238$	−720.000	−0.196095
$239$	6309.00	1.70751	0.853756	−	0.520674i	$-0.174319\pi$
$239$	6309.00	1.70751	0.853756	+	0.520674i	$0.174319\pi$
$240$	0	0
$241$	3038.00	0.812012	0.406006	−	0.913871i	$-0.366921\pi$
$241$	3038.00	0.812012	0.406006	+	0.913871i	$0.366921\pi$
$242$	614.000	0.163097
$243$	0	0
$244$	2208.00	0.579314
$245$	−95.0000	−0.0247728
$246$	0	0
$247$	−1692.00	−0.435868
$248$	264.000	0.0675968
$249$	0	0
$250$	−250.000	−0.0632456
$251$	1332.00	0.334961	0.167480	−	0.985875i	$-0.446437\pi$
$251$	1332.00	0.334961	0.167480	+	0.985875i	$0.446437\pi$
$252$	0	0
$253$	736.000	0.182893
$254$	1538.00	0.379932
$255$	0	0
$256$	256.000	0.0625000
$257$	−3301.00	−0.801209	−0.400605	−	0.916251i	$-0.631200\pi$
$257$	−3301.00	−0.801209	−0.400605	+	0.916251i	$0.631200\pi$
$258$	0	0
$259$	−1008.00	−0.241830
$260$	−940.000	−0.224217
$261$	0	0
$262$	−426.000	−0.100452
$263$	−2072.00	−0.485798	−0.242899	−	0.970052i	$-0.578098\pi$
$263$	−2072.00	−0.485798	−0.242899	+	0.970052i	$0.578098\pi$
$264$	0	0
$265$	70.0000	0.0162267
$266$	1296.00	0.298733
$267$	0	0
$268$	−624.000	−0.142227
$269$	−5721.00	−1.29671	−0.648356	−	0.761337i	$-0.724543\pi$
$269$	−5721.00	−1.29671	−0.648356	+	0.761337i	$0.724543\pi$
$270$	0	0
$271$	−5900.00	−1.32251	−0.661254	−	0.750162i	$-0.729975\pi$
$271$	−5900.00	−1.32251	−0.661254	+	0.750162i	$0.729975\pi$
$272$	−320.000	−0.0713340
$273$	0	0
$274$	5672.00	1.25058
$275$	800.000	0.175425
$276$	0	0
$277$	6371.00	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	6371.00	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	3262.00	0.703747
$279$	0	0
$280$	720.000	0.153672
$281$	−3190.00	−0.677222	−0.338611	−	0.940926i	$-0.609957\pi$
$281$	−3190.00	−0.677222	−0.338611	+	0.940926i	$0.609957\pi$
$282$	0	0
$283$	−4226.00	−0.887667	−0.443833	−	0.896109i	$-0.646382\pi$
$283$	−4226.00	−0.887667	−0.443833	+	0.896109i	$0.646382\pi$
$284$	−2796.00	−0.584197
$285$	0	0
$286$	3008.00	0.621912
$287$	−2826.00	−0.581232
$288$	0	0
$289$	−4513.00	−0.918583
$290$	−270.000	−0.0546722
$291$	0	0
$292$	−2436.00	−0.488206
$293$	−6048.00	−1.20590	−0.602949	−	0.797780i	$-0.706008\pi$
$293$	−6048.00	−1.20590	−0.602949	+	0.797780i	$0.706008\pi$
$294$	0	0
$295$	3720.00	0.734192
$296$	−448.000	−0.0879712
$297$	0	0
$298$	−3932.00	−0.764344
$299$	−1081.00	−0.209083
$300$	0	0
$301$	−324.000	−0.0620434
$302$	−70.0000	−0.0133379
$303$	0	0
$304$	576.000	0.108671
$305$	2760.00	0.518155
$306$	0	0
$307$	8628.00	1.60399	0.801997	−	0.597328i	$-0.203771\pi$
$307$	8628.00	1.60399	0.801997	+	0.597328i	$0.203771\pi$
$308$	−2304.00	−0.426242
$309$	0	0
$310$	330.000	0.0604605
$311$	−8247.00	−1.50368	−0.751840	−	0.659346i	$-0.770833\pi$
$311$	−8247.00	−1.50368	−0.751840	+	0.659346i	$0.770833\pi$
$312$	0	0
$313$	2620.00	0.473135	0.236567	−	0.971615i	$-0.423978\pi$
$313$	2620.00	0.473135	0.236567	+	0.971615i	$0.423978\pi$
$314$	−3404.00	−0.611780
$315$	0	0
$316$	−2576.00	−0.458580
$317$	−9906.00	−1.75513	−0.877565	−	0.479457i	$-0.840834\pi$
$317$	−9906.00	−1.75513	−0.877565	+	0.479457i	$0.840834\pi$
$318$	0	0
$319$	864.000	0.151645
$320$	320.000	0.0559017
$321$	0	0
$322$	828.000	0.143300
$323$	−720.000	−0.124031
$324$	0	0
$325$	−1175.00	−0.200545
$326$	4090.00	0.694859
$327$	0	0
$328$	−1256.00	−0.211436
$329$	1170.00	0.196061
$330$	0	0
$331$	−8115.00	−1.34756	−0.673778	−	0.738934i	$-0.735329\pi$
$331$	−8115.00	−1.34756	−0.673778	+	0.738934i	$0.735329\pi$
$332$	−2048.00	−0.338550
$333$	0	0
$334$	2032.00	0.332892
$335$	−780.000	−0.127212
$336$	0	0
$337$	−7586.00	−1.22622	−0.613109	−	0.789998i	$-0.710082\pi$
$337$	−7586.00	−1.22622	−0.613109	+	0.789998i	$0.710082\pi$
$338$	−24.0000	−0.00386221
$339$	0	0
$340$	−400.000	−0.0638031
$341$	−1056.00	−0.167700
$342$	0	0
$343$	6516.00	1.02575
$344$	−144.000	−0.0225697
$345$	0	0
$346$	1196.00	0.185831
$347$	−1356.00	−0.209781	−0.104890	−	0.994484i	$-0.533449\pi$
$347$	−1356.00	−0.209781	−0.104890	+	0.994484i	$0.533449\pi$
$348$	0	0
$349$	6649.00	1.01981	0.509904	−	0.860231i	$-0.329681\pi$
$349$	6649.00	1.01981	0.509904	+	0.860231i	$0.329681\pi$
$350$	900.000	0.137449
$351$	0	0
$352$	−1024.00	−0.155055
$353$	−10691.0	−1.61197	−0.805984	−	0.591938i	$-0.798363\pi$
$353$	−10691.0	−1.61197	−0.805984	+	0.591938i	$0.798363\pi$
$354$	0	0
$355$	−3495.00	−0.522522
$356$	408.000	0.0607415
$357$	0	0
$358$	−9214.00	−1.36027
$359$	6420.00	0.943829	0.471915	−	0.881644i	$-0.343563\pi$
$359$	6420.00	0.943829	0.471915	+	0.881644i	$0.343563\pi$
$360$	0	0
$361$	−5563.00	−0.811051
$362$	2424.00	0.351941
$363$	0	0
$364$	3384.00	0.487280
$365$	−3045.00	−0.436665
$366$	0	0
$367$	−524.000	−0.0745302	−0.0372651	−	0.999305i	$-0.511865\pi$
$367$	−524.000	−0.0745302	−0.0372651	+	0.999305i	$0.511865\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	−560.000	−0.0786838
$371$	−252.000	−0.0352647
$372$	0	0
$373$	5566.00	0.772645	0.386322	−	0.922364i	$-0.373745\pi$
$373$	5566.00	0.772645	0.386322	+	0.922364i	$0.373745\pi$
$374$	1280.00	0.176971
$375$	0	0
$376$	520.000	0.0713217
$377$	−1269.00	−0.173360
$378$	0	0
$379$	2240.00	0.303591	0.151796	−	0.988412i	$-0.451494\pi$
$379$	2240.00	0.303591	0.151796	+	0.988412i	$0.451494\pi$
$380$	720.000	0.0971979
$381$	0	0
$382$	−2116.00	−0.283414
$383$	−8778.00	−1.17111	−0.585555	−	0.810633i	$-0.699123\pi$
$383$	−8778.00	−1.17111	−0.585555	+	0.810633i	$0.699123\pi$
$384$	0	0
$385$	−2880.00	−0.381243
$386$	−2094.00	−0.276119
$387$	0	0
$388$	2312.00	0.302510
$389$	−4056.00	−0.528656	−0.264328	−	0.964433i	$-0.585150\pi$
$389$	−4056.00	−0.528656	−0.264328	+	0.964433i	$0.585150\pi$
$390$	0	0
$391$	−460.000	−0.0594967
$392$	152.000	0.0195846
$393$	0	0
$394$	502.000	0.0641888
$395$	−3220.00	−0.410167
$396$	0	0
$397$	−9151.00	−1.15687	−0.578433	−	0.815730i	$-0.696335\pi$
$397$	−9151.00	−1.15687	−0.578433	+	0.815730i	$0.696335\pi$
$398$	7016.00	0.883619
$399$	0	0
$400$	400.000	0.0500000
$401$	−15930.0	−1.98381	−0.991903	−	0.126997i	$-0.959466\pi$
$401$	−15930.0	−1.98381	−0.991903	+	0.126997i	$0.959466\pi$
$402$	0	0
$403$	1551.00	0.191714
$404$	24.0000	0.00295556
$405$	0	0
$406$	972.000	0.118817
$407$	1792.00	0.218246
$408$	0	0
$409$	−5891.00	−0.712203	−0.356102	−	0.934447i	$-0.615894\pi$
$409$	−5891.00	−0.712203	−0.356102	+	0.934447i	$0.615894\pi$
$410$	−1570.00	−0.189114
$411$	0	0
$412$	−640.000	−0.0765304
$413$	−13392.0	−1.59559
$414$	0	0
$415$	−2560.00	−0.302808
$416$	1504.00	0.177259
$417$	0	0
$418$	−2304.00	−0.269599
$419$	−15282.0	−1.78180	−0.890900	−	0.454199i	$-0.849926\pi$
$419$	−15282.0	−1.78180	−0.890900	+	0.454199i	$0.849926\pi$
$420$	0	0
$421$	−10934.0	−1.26577	−0.632887	−	0.774244i	$-0.718130\pi$
$421$	−10934.0	−1.26577	−0.632887	+	0.774244i	$0.718130\pi$
$422$	6592.00	0.760411
$423$	0	0
$424$	−112.000	−0.0128283
$425$	−500.000	−0.0570672
$426$	0	0
$427$	−9936.00	−1.12608
$428$	−1520.00	−0.171663
$429$	0	0
$430$	−180.000	−0.0201869
$431$	2794.00	0.312256	0.156128	−	0.987737i	$-0.450099\pi$
$431$	2794.00	0.312256	0.156128	+	0.987737i	$0.450099\pi$
$432$	0	0
$433$	−15062.0	−1.67167	−0.835835	−	0.548980i	$-0.815016\pi$
$433$	−15062.0	−1.67167	−0.835835	+	0.548980i	$0.815016\pi$
$434$	−1188.00	−0.131396
$435$	0	0
$436$	1000.00	0.109842
$437$	828.000	0.0906376
$438$	0	0
$439$	261.000	0.0283755	0.0141878	−	0.999899i	$-0.495484\pi$
$439$	261.000	0.0283755	0.0141878	+	0.999899i	$0.495484\pi$
$440$	−1280.00	−0.138685
$441$	0	0
$442$	−1880.00	−0.202313
$443$	7083.00	0.759647	0.379823	−	0.925059i	$-0.375985\pi$
$443$	7083.00	0.759647	0.379823	+	0.925059i	$0.375985\pi$
$444$	0	0
$445$	510.000	0.0543288
$446$	5440.00	0.577559
$447$	0	0
$448$	−1152.00	−0.121489
$449$	10370.0	1.08996	0.544978	−	0.838450i	$-0.316538\pi$
$449$	10370.0	1.08996	0.544978	+	0.838450i	$0.316538\pi$
$450$	0	0
$451$	5024.00	0.524547
$452$	1560.00	0.162337
$453$	0	0
$454$	−8268.00	−0.854706
$455$	4230.00	0.435836
$456$	0	0
$457$	−10496.0	−1.07436	−0.537180	−	0.843468i	$-0.680510\pi$
$457$	−10496.0	−1.07436	−0.537180	+	0.843468i	$0.680510\pi$
$458$	9020.00	0.920255
$459$	0	0
$460$	460.000	0.0466252
$461$	−18021.0	−1.82065	−0.910327	−	0.413889i	$-0.864170\pi$
$461$	−18021.0	−1.82065	−0.910327	+	0.413889i	$0.864170\pi$
$462$	0	0
$463$	−17188.0	−1.72526	−0.862629	−	0.505838i	$-0.831183\pi$
$463$	−17188.0	−1.72526	−0.862629	+	0.505838i	$0.831183\pi$
$464$	432.000	0.0432222
$465$	0	0
$466$	−10006.0	−0.994676
$467$	15246.0	1.51071	0.755354	−	0.655317i	$-0.227465\pi$
$467$	15246.0	1.51071	0.755354	+	0.655317i	$0.227465\pi$
$468$	0	0
$469$	2808.00	0.276464
$470$	650.000	0.0637921
$471$	0	0
$472$	−5952.00	−0.580430
$473$	576.000	0.0559926
$474$	0	0
$475$	900.000	0.0869365
$476$	1440.00	0.138660
$477$	0	0
$478$	−12618.0	−1.20739
$479$	8556.00	0.816145	0.408073	−	0.912949i	$-0.366201\pi$
$479$	8556.00	0.816145	0.408073	+	0.912949i	$0.366201\pi$
$480$	0	0
$481$	−2632.00	−0.249499
$482$	−6076.00	−0.574179
$483$	0	0
$484$	−1228.00	−0.115327
$485$	2890.00	0.270573
$486$	0	0
$487$	−1805.00	−0.167951	−0.0839757	−	0.996468i	$-0.526762\pi$
$487$	−1805.00	−0.167951	−0.0839757	+	0.996468i	$0.526762\pi$
$488$	−4416.00	−0.409637
$489$	0	0
$490$	190.000	0.0175170
$491$	−5245.00	−0.482085	−0.241042	−	0.970515i	$-0.577489\pi$
$491$	−5245.00	−0.482085	−0.241042	+	0.970515i	$0.577489\pi$
$492$	0	0
$493$	−540.000	−0.0493314
$494$	3384.00	0.308205
$495$	0	0
$496$	−528.000	−0.0477982
$497$	12582.0	1.13557
$498$	0	0
$499$	9027.00	0.809828	0.404914	−	0.914355i	$-0.367302\pi$
$499$	9027.00	0.809828	0.404914	+	0.914355i	$0.367302\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	−2664.00	−0.236853
$503$	−3522.00	−0.312203	−0.156102	−	0.987741i	$-0.549893\pi$
$503$	−3522.00	−0.312203	−0.156102	+	0.987741i	$0.549893\pi$
$504$	0	0
$505$	30.0000	0.00264353
$506$	−1472.00	−0.129325
$507$	0	0
$508$	−3076.00	−0.268652
$509$	−3949.00	−0.343883	−0.171941	−	0.985107i	$-0.555004\pi$
$509$	−3949.00	−0.343883	−0.171941	+	0.985107i	$0.555004\pi$
$510$	0	0
$511$	10962.0	0.948983
$512$	−512.000	−0.0441942
$513$	0	0
$514$	6602.00	0.566540
$515$	−800.000	−0.0684509
$516$	0	0
$517$	−2080.00	−0.176941
$518$	2016.00	0.171000
$519$	0	0
$520$	1880.00	0.158545
$521$	−3236.00	−0.272115	−0.136057	−	0.990701i	$-0.543443\pi$
$521$	−3236.00	−0.272115	−0.136057	+	0.990701i	$0.543443\pi$
$522$	0	0
$523$	12394.0	1.03624	0.518118	−	0.855309i	$-0.326633\pi$
$523$	12394.0	1.03624	0.518118	+	0.855309i	$0.326633\pi$
$524$	852.000	0.0710301
$525$	0	0
$526$	4144.00	0.343511
$527$	660.000	0.0545542
$528$	0	0
$529$	529.000	0.0434783
$530$	−140.000	−0.0114740
$531$	0	0
$532$	−2592.00	−0.211236
$533$	−7379.00	−0.599662
$534$	0	0
$535$	−1900.00	−0.153540
$536$	1248.00	0.100570
$537$	0	0
$538$	11442.0	0.916914
$539$	−608.000	−0.0485870
$540$	0	0
$541$	−7159.00	−0.568927	−0.284463	−	0.958687i	$-0.591815\pi$
$541$	−7159.00	−0.568927	−0.284463	+	0.958687i	$0.591815\pi$
$542$	11800.0	0.935154
$543$	0	0
$544$	640.000	0.0504408
$545$	1250.00	0.0982461
$546$	0	0
$547$	−19761.0	−1.54464	−0.772321	−	0.635232i	$-0.780904\pi$
$547$	−19761.0	−1.54464	−0.772321	+	0.635232i	$0.780904\pi$
$548$	−11344.0	−0.884291
$549$	0	0
$550$	−1600.00	−0.124044
$551$	972.000	0.0751517
$552$	0	0
$553$	11592.0	0.891396
$554$	−12742.0	−0.977176
$555$	0	0
$556$	−6524.00	−0.497625
$557$	−18010.0	−1.37003	−0.685016	−	0.728528i	$-0.740205\pi$
$557$	−18010.0	−1.37003	−0.685016	+	0.728528i	$0.740205\pi$
$558$	0	0
$559$	−846.000	−0.0640107
$560$	−1440.00	−0.108663
$561$	0	0
$562$	6380.00	0.478868
$563$	2648.00	0.198224	0.0991118	−	0.995076i	$-0.468400\pi$
$563$	2648.00	0.198224	0.0991118	+	0.995076i	$0.468400\pi$
$564$	0	0
$565$	1950.00	0.145198
$566$	8452.00	0.627675
$567$	0	0
$568$	5592.00	0.413090
$569$	1566.00	0.115378	0.0576890	−	0.998335i	$-0.481627\pi$
$569$	1566.00	0.115378	0.0576890	+	0.998335i	$0.481627\pi$
$570$	0	0
$571$	2864.00	0.209903	0.104952	−	0.994477i	$-0.466531\pi$
$571$	2864.00	0.209903	0.104952	+	0.994477i	$0.466531\pi$
$572$	−6016.00	−0.439758
$573$	0	0
$574$	5652.00	0.410993
$575$	575.000	0.0417029
$576$	0	0
$577$	−929.000	−0.0670273	−0.0335137	−	0.999438i	$-0.510670\pi$
$577$	−929.000	−0.0670273	−0.0335137	+	0.999438i	$0.510670\pi$
$578$	9026.00	0.649537
$579$	0	0
$580$	540.000	0.0386591
$581$	9216.00	0.658079
$582$	0	0
$583$	448.000	0.0318255
$584$	4872.00	0.345214
$585$	0	0
$586$	12096.0	0.852698
$587$	19499.0	1.37106	0.685528	−	0.728046i	$-0.259571\pi$
$587$	19499.0	1.37106	0.685528	+	0.728046i	$0.259571\pi$
$588$	0	0
$589$	−1188.00	−0.0831081
$590$	−7440.00	−0.519152
$591$	0	0
$592$	896.000	0.0622050
$593$	−6570.00	−0.454971	−0.227485	−	0.973782i	$-0.573050\pi$
$593$	−6570.00	−0.454971	−0.227485	+	0.973782i	$0.573050\pi$
$594$	0	0
$595$	1800.00	0.124022
$596$	7864.00	0.540473
$597$	0	0
$598$	2162.00	0.147844
$599$	−1880.00	−0.128238	−0.0641191	−	0.997942i	$-0.520424\pi$
$599$	−1880.00	−0.128238	−0.0641191	+	0.997942i	$0.520424\pi$
$600$	0	0
$601$	3701.00	0.251193	0.125596	−	0.992081i	$-0.459916\pi$
$601$	3701.00	0.251193	0.125596	+	0.992081i	$0.459916\pi$
$602$	648.000	0.0438713
$603$	0	0
$604$	140.000	0.00943132
$605$	−1535.00	−0.103151
$606$	0	0
$607$	−3080.00	−0.205953	−0.102976	−	0.994684i	$-0.532837\pi$
$607$	−3080.00	−0.205953	−0.102976	+	0.994684i	$0.532837\pi$
$608$	−1152.00	−0.0768417
$609$	0	0
$610$	−5520.00	−0.366391
$611$	3055.00	0.202278
$612$	0	0
$613$	24004.0	1.58159	0.790793	−	0.612083i	$-0.209668\pi$
$613$	24004.0	1.58159	0.790793	+	0.612083i	$0.209668\pi$
$614$	−17256.0	−1.13419
$615$	0	0
$616$	4608.00	0.301399
$617$	−780.000	−0.0508940	−0.0254470	−	0.999676i	$-0.508101\pi$
$617$	−780.000	−0.0508940	−0.0254470	+	0.999676i	$0.508101\pi$
$618$	0	0
$619$	21892.0	1.42151	0.710754	−	0.703440i	$-0.248354\pi$
$619$	21892.0	1.42151	0.710754	+	0.703440i	$0.248354\pi$
$620$	−660.000	−0.0427520
$621$	0	0
$622$	16494.0	1.06326
$623$	−1836.00	−0.118070
$624$	0	0
$625$	625.000	0.0400000
$626$	−5240.00	−0.334557
$627$	0	0
$628$	6808.00	0.432594
$629$	−1120.00	−0.0709973
$630$	0	0
$631$	8050.00	0.507869	0.253935	−	0.967221i	$-0.418275\pi$
$631$	8050.00	0.507869	0.253935	+	0.967221i	$0.418275\pi$
$632$	5152.00	0.324265
$633$	0	0
$634$	19812.0	1.24106
$635$	−3845.00	−0.240290
$636$	0	0
$637$	893.000	0.0555447
$638$	−1728.00	−0.107229
$639$	0	0
$640$	−640.000	−0.0395285
$641$	25890.0	1.59531	0.797655	−	0.603114i	$-0.206074\pi$
$641$	25890.0	1.59531	0.797655	+	0.603114i	$0.206074\pi$
$642$	0	0
$643$	4774.00	0.292797	0.146398	−	0.989226i	$-0.453232\pi$
$643$	4774.00	0.292797	0.146398	+	0.989226i	$0.453232\pi$
$644$	−1656.00	−0.101328
$645$	0	0
$646$	1440.00	0.0877029
$647$	−3349.00	−0.203497	−0.101749	−	0.994810i	$-0.532444\pi$
$647$	−3349.00	−0.203497	−0.101749	+	0.994810i	$0.532444\pi$
$648$	0	0
$649$	23808.0	1.43998
$650$	2350.00	0.141807
$651$	0	0
$652$	−8180.00	−0.491340
$653$	−24813.0	−1.48699	−0.743497	−	0.668739i	$-0.766834\pi$
$653$	−24813.0	−1.48699	−0.743497	+	0.668739i	$0.766834\pi$
$654$	0	0
$655$	1065.00	0.0635313
$656$	2512.00	0.149508
$657$	0	0
$658$	−2340.00	−0.138636
$659$	−18180.0	−1.07465	−0.537323	−	0.843376i	$-0.680565\pi$
$659$	−18180.0	−1.07465	−0.537323	+	0.843376i	$0.680565\pi$
$660$	0	0
$661$	−29250.0	−1.72117	−0.860585	−	0.509307i	$-0.829902\pi$
$661$	−29250.0	−1.72117	−0.860585	+	0.509307i	$0.829902\pi$
$662$	16230.0	0.952865
$663$	0	0
$664$	4096.00	0.239391
$665$	−3240.00	−0.188935
$666$	0	0
$667$	621.000	0.0360498
$668$	−4064.00	−0.235391
$669$	0	0
$670$	1560.00	0.0899523
$671$	17664.0	1.01626
$672$	0	0
$673$	−23027.0	−1.31891	−0.659454	−	0.751745i	$-0.729213\pi$
$673$	−23027.0	−1.31891	−0.659454	+	0.751745i	$0.729213\pi$
$674$	15172.0	0.867068
$675$	0	0
$676$	48.0000	0.00273100
$677$	−20106.0	−1.14141	−0.570706	−	0.821154i	$-0.693331\pi$
$677$	−20106.0	−1.14141	−0.570706	+	0.821154i	$0.693331\pi$
$678$	0	0
$679$	−10404.0	−0.588025
$680$	800.000	0.0451156
$681$	0	0
$682$	2112.00	0.118582
$683$	18745.0	1.05016	0.525079	−	0.851054i	$-0.324036\pi$
$683$	18745.0	1.05016	0.525079	+	0.851054i	$0.324036\pi$
$684$	0	0
$685$	−14180.0	−0.790934
$686$	−13032.0	−0.725312
$687$	0	0
$688$	288.000	0.0159592
$689$	−658.000	−0.0363829
$690$	0	0
$691$	24424.0	1.34462	0.672310	−	0.740270i	$-0.265302\pi$
$691$	24424.0	1.34462	0.672310	+	0.740270i	$0.265302\pi$
$692$	−2392.00	−0.131402
$693$	0	0
$694$	2712.00	0.148337
$695$	−8155.00	−0.445089
$696$	0	0
$697$	−3140.00	−0.170640
$698$	−13298.0	−0.721113
$699$	0	0
$700$	−1800.00	−0.0971909
$701$	27278.0	1.46972	0.734862	−	0.678217i	$-0.237247\pi$
$701$	27278.0	1.46972	0.734862	+	0.678217i	$0.237247\pi$
$702$	0	0
$703$	2016.00	0.108158
$704$	2048.00	0.109640
$705$	0	0
$706$	21382.0	1.13983
$707$	−108.000	−0.00574506
$708$	0	0
$709$	12214.0	0.646977	0.323488	−	0.946232i	$-0.395144\pi$
$709$	12214.0	0.646977	0.323488	+	0.946232i	$0.395144\pi$
$710$	6990.00	0.369479
$711$	0	0
$712$	−816.000	−0.0429507
$713$	−759.000	−0.0398664
$714$	0	0
$715$	−7520.00	−0.393332
$716$	18428.0	0.961853
$717$	0	0
$718$	−12840.0	−0.667388
$719$	−12932.0	−0.670768	−0.335384	−	0.942082i	$-0.608866\pi$
$719$	−12932.0	−0.670768	−0.335384	+	0.942082i	$0.608866\pi$
$720$	0	0
$721$	2880.00	0.148761
$722$	11126.0	0.573500
$723$	0	0
$724$	−4848.00	−0.248860
$725$	675.000	0.0345778
$726$	0	0
$727$	10046.0	0.512497	0.256249	−	0.966611i	$-0.417513\pi$
$727$	10046.0	0.512497	0.256249	+	0.966611i	$0.417513\pi$
$728$	−6768.00	−0.344559
$729$	0	0
$730$	6090.00	0.308769
$731$	−360.000	−0.0182149
$732$	0	0
$733$	−5924.00	−0.298510	−0.149255	−	0.988799i	$-0.547688\pi$
$733$	−5924.00	−0.298510	−0.149255	+	0.988799i	$0.547688\pi$
$734$	1048.00	0.0527008
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−4992.00	−0.249502
$738$	0	0
$739$	829.000	0.0412656	0.0206328	−	0.999787i	$-0.493432\pi$
$739$	829.000	0.0412656	0.0206328	+	0.999787i	$0.493432\pi$
$740$	1120.00	0.0556379
$741$	0	0
$742$	504.000	0.0249359
$743$	−7072.00	−0.349188	−0.174594	−	0.984641i	$-0.555861\pi$
$743$	−7072.00	−0.349188	−0.174594	+	0.984641i	$0.555861\pi$
$744$	0	0
$745$	9830.00	0.483414
$746$	−11132.0	−0.546342
$747$	0	0
$748$	−2560.00	−0.125138
$749$	6840.00	0.333682
$750$	0	0
$751$	16234.0	0.788798	0.394399	−	0.918939i	$-0.370953\pi$
$751$	16234.0	0.788798	0.394399	+	0.918939i	$0.370953\pi$
$752$	−1040.00	−0.0504320
$753$	0	0
$754$	2538.00	0.122584
$755$	175.000	0.00843563
$756$	0	0
$757$	9128.00	0.438260	0.219130	−	0.975696i	$-0.429678\pi$
$757$	9128.00	0.438260	0.219130	+	0.975696i	$0.429678\pi$
$758$	−4480.00	−0.214671
$759$	0	0
$760$	−1440.00	−0.0687293
$761$	−165.000	−0.00785972	−0.00392986	−	0.999992i	$-0.501251\pi$
$761$	−165.000	−0.00785972	−0.00392986	+	0.999992i	$0.501251\pi$
$762$	0	0
$763$	−4500.00	−0.213514
$764$	4232.00	0.200404
$765$	0	0
$766$	17556.0	0.828099
$767$	−34968.0	−1.64618
$768$	0	0
$769$	−20834.0	−0.976974	−0.488487	−	0.872571i	$-0.662451\pi$
$769$	−20834.0	−0.976974	−0.488487	+	0.872571i	$0.662451\pi$
$770$	5760.00	0.269579
$771$	0	0
$772$	4188.00	0.195245
$773$	31782.0	1.47881	0.739404	−	0.673262i	$-0.235107\pi$
$773$	31782.0	1.47881	0.739404	+	0.673262i	$0.235107\pi$
$774$	0	0
$775$	−825.000	−0.0382385
$776$	−4624.00	−0.213907
$777$	0	0
$778$	8112.00	0.373817
$779$	5652.00	0.259954
$780$	0	0
$781$	−22368.0	−1.02483
$782$	920.000	0.0420705
$783$	0	0
$784$	−304.000	−0.0138484
$785$	8510.00	0.386923
$786$	0	0
$787$	33104.0	1.49940	0.749701	−	0.661776i	$-0.230197\pi$
$787$	33104.0	1.49940	0.749701	+	0.661776i	$0.230197\pi$
$788$	−1004.00	−0.0453883
$789$	0	0
$790$	6440.00	0.290032
$791$	−7020.00	−0.315553
$792$	0	0
$793$	−25944.0	−1.16179
$794$	18302.0	0.818027
$795$	0	0
$796$	−14032.0	−0.624813
$797$	−4736.00	−0.210486	−0.105243	−	0.994447i	$-0.533562\pi$
$797$	−4736.00	−0.210486	−0.105243	+	0.994447i	$0.533562\pi$
$798$	0	0
$799$	1300.00	0.0575603
$800$	−800.000	−0.0353553
$801$	0	0
$802$	31860.0	1.40276
$803$	−19488.0	−0.856434
$804$	0	0
$805$	−2070.00	−0.0906309
$806$	−3102.00	−0.135562
$807$	0	0
$808$	−48.0000	−0.00208989
$809$	7470.00	0.324637	0.162318	−	0.986738i	$-0.448103\pi$
$809$	7470.00	0.324637	0.162318	+	0.986738i	$0.448103\pi$
$810$	0	0
$811$	19919.0	0.862455	0.431227	−	0.902243i	$-0.358081\pi$
$811$	19919.0	0.862455	0.431227	+	0.902243i	$0.358081\pi$
$812$	−1944.00	−0.0840160
$813$	0	0
$814$	−3584.00	−0.154323
$815$	−10225.0	−0.439468
$816$	0	0
$817$	648.000	0.0277487
$818$	11782.0	0.503604
$819$	0	0
$820$	3140.00	0.133724
$821$	22694.0	0.964709	0.482354	−	0.875976i	$-0.339782\pi$
$821$	22694.0	0.964709	0.482354	+	0.875976i	$0.339782\pi$
$822$	0	0
$823$	−31907.0	−1.35141	−0.675704	−	0.737173i	$-0.736160\pi$
$823$	−31907.0	−1.35141	−0.675704	+	0.737173i	$0.736160\pi$
$824$	1280.00	0.0541152
$825$	0	0
$826$	26784.0	1.12825
$827$	15236.0	0.640638	0.320319	−	0.947310i	$-0.396210\pi$
$827$	15236.0	0.640638	0.320319	+	0.947310i	$0.396210\pi$
$828$	0	0
$829$	27286.0	1.14316	0.571581	−	0.820545i	$-0.306330\pi$
$829$	27286.0	1.14316	0.571581	+	0.820545i	$0.306330\pi$
$830$	5120.00	0.214118
$831$	0	0
$832$	−3008.00	−0.125341
$833$	380.000	0.0158058
$834$	0	0
$835$	−5080.00	−0.210540
$836$	4608.00	0.190635
$837$	0	0
$838$	30564.0	1.25992
$839$	−23054.0	−0.948644	−0.474322	−	0.880351i	$-0.657307\pi$
$839$	−23054.0	−0.948644	−0.474322	+	0.880351i	$0.657307\pi$
$840$	0	0
$841$	−23660.0	−0.970109
$842$	21868.0	0.895037
$843$	0	0
$844$	−13184.0	−0.537692
$845$	60.0000	0.00244268
$846$	0	0
$847$	5526.00	0.224174
$848$	224.000	0.00907098
$849$	0	0
$850$	1000.00	0.0403526
$851$	1288.00	0.0518826
$852$	0	0
$853$	−34506.0	−1.38507	−0.692534	−	0.721385i	$-0.743506\pi$
$853$	−34506.0	−1.38507	−0.692534	+	0.721385i	$0.743506\pi$
$854$	19872.0	0.796260
$855$	0	0
$856$	3040.00	0.121384
$857$	22263.0	0.887386	0.443693	−	0.896179i	$-0.353668\pi$
$857$	22263.0	0.887386	0.443693	+	0.896179i	$0.353668\pi$
$858$	0	0
$859$	12851.0	0.510443	0.255221	−	0.966883i	$-0.417852\pi$
$859$	12851.0	0.510443	0.255221	+	0.966883i	$0.417852\pi$
$860$	360.000	0.0142743
$861$	0	0
$862$	−5588.00	−0.220798
$863$	−15723.0	−0.620182	−0.310091	−	0.950707i	$-0.600360\pi$
$863$	−15723.0	−0.620182	−0.310091	+	0.950707i	$0.600360\pi$
$864$	0	0
$865$	−2990.00	−0.117530
$866$	30124.0	1.18205
$867$	0	0
$868$	2376.00	0.0929109
$869$	−20608.0	−0.804463
$870$	0	0
$871$	7332.00	0.285230
$872$	−2000.00	−0.0776704
$873$	0	0
$874$	−1656.00	−0.0640904
$875$	−2250.00	−0.0869302
$876$	0	0
$877$	−886.000	−0.0341141	−0.0170571	−	0.999855i	$-0.505430\pi$
$877$	−886.000	−0.0341141	−0.0170571	+	0.999855i	$0.505430\pi$
$878$	−522.000	−0.0200645
$879$	0	0
$880$	2560.00	0.0980654
$881$	37120.0	1.41953	0.709764	−	0.704439i	$-0.248801\pi$
$881$	37120.0	1.41953	0.709764	+	0.704439i	$0.248801\pi$
$882$	0	0
$883$	−7524.00	−0.286753	−0.143376	−	0.989668i	$-0.545796\pi$
$883$	−7524.00	−0.286753	−0.143376	+	0.989668i	$0.545796\pi$
$884$	3760.00	0.143057
$885$	0	0
$886$	−14166.0	−0.537151
$887$	−9221.00	−0.349054	−0.174527	−	0.984652i	$-0.555840\pi$
$887$	−9221.00	−0.349054	−0.174527	+	0.984652i	$0.555840\pi$
$888$	0	0
$889$	13842.0	0.522211
$890$	−1020.00	−0.0384163
$891$	0	0
$892$	−10880.0	−0.408396
$893$	−2340.00	−0.0876877
$894$	0	0
$895$	23035.0	0.860307
$896$	2304.00	0.0859054
$897$	0	0
$898$	−20740.0	−0.770716
$899$	−891.000	−0.0330551
$900$	0	0
$901$	−280.000	−0.0103531
$902$	−10048.0	−0.370911
$903$	0	0
$904$	−3120.00	−0.114789
$905$	−6060.00	−0.222587
$906$	0	0
$907$	29116.0	1.06591	0.532955	−	0.846143i	$-0.321081\pi$
$907$	29116.0	1.06591	0.532955	+	0.846143i	$0.321081\pi$
$908$	16536.0	0.604368
$909$	0	0
$910$	−8460.00	−0.308183
$911$	−11440.0	−0.416053	−0.208026	−	0.978123i	$-0.566704\pi$
$911$	−11440.0	−0.416053	−0.208026	+	0.978123i	$0.566704\pi$
$912$	0	0
$913$	−16384.0	−0.593901
$914$	20992.0	0.759687
$915$	0	0
$916$	−18040.0	−0.650719
$917$	−3834.00	−0.138070
$918$	0	0
$919$	2958.00	0.106176	0.0530878	−	0.998590i	$-0.483094\pi$
$919$	2958.00	0.106176	0.0530878	+	0.998590i	$0.483094\pi$
$920$	−920.000	−0.0329690
$921$	0	0
$922$	36042.0	1.28740
$923$	32853.0	1.17158
$924$	0	0
$925$	1400.00	0.0497640
$926$	34376.0	1.21994
$927$	0	0
$928$	−864.000	−0.0305627
$929$	−20907.0	−0.738360	−0.369180	−	0.929358i	$-0.620361\pi$
$929$	−20907.0	−0.738360	−0.369180	+	0.929358i	$0.620361\pi$
$930$	0	0
$931$	−684.000	−0.0240786
$932$	20012.0	0.703342
$933$	0	0
$934$	−30492.0	−1.06823
$935$	−3200.00	−0.111926
$936$	0	0
$937$	9748.00	0.339865	0.169932	−	0.985456i	$-0.445645\pi$
$937$	9748.00	0.339865	0.169932	+	0.985456i	$0.445645\pi$
$938$	−5616.00	−0.195489
$939$	0	0
$940$	−1300.00	−0.0451078
$941$	−19624.0	−0.679834	−0.339917	−	0.940455i	$-0.610399\pi$
$941$	−19624.0	−0.679834	−0.339917	+	0.940455i	$0.610399\pi$
$942$	0	0
$943$	3611.00	0.124698
$944$	11904.0	0.410426
$945$	0	0
$946$	−1152.00	−0.0395928
$947$	41859.0	1.43636	0.718181	−	0.695856i	$-0.244975\pi$
$947$	41859.0	1.43636	0.718181	+	0.695856i	$0.244975\pi$
$948$	0	0
$949$	28623.0	0.979075
$950$	−1800.00	−0.0614734
$951$	0	0
$952$	−2880.00	−0.0980476
$953$	−29226.0	−0.993413	−0.496707	−	0.867918i	$-0.665458\pi$
$953$	−29226.0	−0.993413	−0.496707	+	0.867918i	$0.665458\pi$
$954$	0	0
$955$	5290.00	0.179246
$956$	25236.0	0.853756
$957$	0	0
$958$	−17112.0	−0.577102
$959$	51048.0	1.71890
$960$	0	0
$961$	−28702.0	−0.963445
$962$	5264.00	0.176422
$963$	0	0
$964$	12152.0	0.406006
$965$	5235.00	0.174633
$966$	0	0
$967$	−29849.0	−0.992636	−0.496318	−	0.868141i	$-0.665315\pi$
$967$	−29849.0	−0.992636	−0.496318	+	0.868141i	$0.665315\pi$
$968$	2456.00	0.0815484
$969$	0	0
$970$	−5780.00	−0.191324
$971$	9390.00	0.310339	0.155170	−	0.987888i	$-0.450408\pi$
$971$	9390.00	0.310339	0.155170	+	0.987888i	$0.450408\pi$
$972$	0	0
$973$	29358.0	0.967291
$974$	3610.00	0.118760
$975$	0	0
$976$	8832.00	0.289657
$977$	−33536.0	−1.09817	−0.549085	−	0.835767i	$-0.685024\pi$
$977$	−33536.0	−1.09817	−0.549085	+	0.835767i	$0.685024\pi$
$978$	0	0
$979$	3264.00	0.106556
$980$	−380.000	−0.0123864
$981$	0	0
$982$	10490.0	0.340885
$983$	−28994.0	−0.940758	−0.470379	−	0.882465i	$-0.655883\pi$
$983$	−28994.0	−0.940758	−0.470379	+	0.882465i	$0.655883\pi$
$984$	0	0
$985$	−1255.00	−0.0405966
$986$	1080.00	0.0348826
$987$	0	0
$988$	−6768.00	−0.217934
$989$	414.000	0.0133109
$990$	0	0
$991$	11272.0	0.361319	0.180659	−	0.983546i	$-0.442177\pi$
$991$	11272.0	0.361319	0.180659	+	0.983546i	$0.442177\pi$
$992$	1056.00	0.0337984
$993$	0	0
$994$	−25164.0	−0.802971
$995$	−17540.0	−0.558850
$996$	0	0
$997$	61186.0	1.94361	0.971805	−	0.235784i	$-0.0757658\pi$
$997$	61186.0	1.94361	0.971805	+	0.235784i	$0.0757658\pi$
$998$	−18054.0	−0.572635
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.e.1.1		1
3.2	odd	2		230.4.a.e.1.1	✓	1
12.11	even	2		1840.4.a.d.1.1		1
15.2	even	4		1150.4.b.f.599.2		2
15.8	even	4		1150.4.b.f.599.1		2
15.14	odd	2		1150.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.e.1.1	✓	1	3.2	odd	2
1150.4.a.b.1.1		1	15.14	odd	2
1150.4.b.f.599.1		2	15.8	even	4
1150.4.b.f.599.2		2	15.2	even	4
1840.4.a.d.1.1		1	12.11	even	2
2070.4.a.e.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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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