LMFDB - Embedded newform 2070.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,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, 2, names="a")

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

chi := DirichletCharacter("2070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
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:	$16.5290332184$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.82843 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.82843 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} -2.82843 q^{13} -4.82843 q^{14} +1.00000 q^{16} -4.82843 q^{17} -1.00000 q^{20} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +2.82843 q^{26} +4.82843 q^{28} +0.828427 q^{29} -9.65685 q^{31} -1.00000 q^{32} +4.82843 q^{34} -4.82843 q^{35} -6.82843 q^{37} +1.00000 q^{40} -5.65685 q^{41} -8.48528 q^{43} -2.00000 q^{44} +1.00000 q^{46} +16.3137 q^{49} -1.00000 q^{50} -2.82843 q^{52} +9.31371 q^{53} +2.00000 q^{55} -4.82843 q^{56} -0.828427 q^{58} -10.0000 q^{59} -3.17157 q^{61} +9.65685 q^{62} +1.00000 q^{64} +2.82843 q^{65} +1.17157 q^{67} -4.82843 q^{68} +4.82843 q^{70} +2.82843 q^{71} +9.31371 q^{73} +6.82843 q^{74} -9.65685 q^{77} +2.82843 q^{79} -1.00000 q^{80} +5.65685 q^{82} +6.82843 q^{83} +4.82843 q^{85} +8.48528 q^{86} +2.00000 q^{88} +5.65685 q^{89} -13.6569 q^{91} -1.00000 q^{92} +8.82843 q^{97} -16.3137 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 2 q^{20} + 4 q^{22} - 2 q^{23} + 2 q^{25} + 4 q^{28} - 4 q^{29} - 8 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{35} - 8 q^{37} + 2 q^{40} - 4 q^{44} + 2 q^{46} + 10 q^{49} - 2 q^{50} - 4 q^{53} + 4 q^{55} - 4 q^{56} + 4 q^{58} - 20 q^{59} - 12 q^{61} + 8 q^{62} + 2 q^{64} + 8 q^{67} - 4 q^{68} + 4 q^{70} - 4 q^{73} + 8 q^{74} - 8 q^{77} - 2 q^{80} + 8 q^{83} + 4 q^{85} + 4 q^{88} - 16 q^{91} - 2 q^{92} + 12 q^{97} - 10 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 - 4 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^17 - 2 * q^20 + 4 * q^22 - 2 * q^23 + 2 * q^25 + 4 * q^28 - 4 * q^29 - 8 * q^31 - 2 * q^32 + 4 * q^34 - 4 * q^35 - 8 * q^37 + 2 * q^40 - 4 * q^44 + 2 * q^46 + 10 * q^49 - 2 * q^50 - 4 * q^53 + 4 * q^55 - 4 * q^56 + 4 * q^58 - 20 * q^59 - 12 * q^61 + 8 * q^62 + 2 * q^64 + 8 * q^67 - 4 * q^68 + 4 * q^70 - 4 * q^73 + 8 * q^74 - 8 * q^77 - 2 * q^80 + 8 * q^83 + 4 * q^85 + 4 * q^88 - 16 * q^91 - 2 * q^92 + 12 * q^97 - 10 * q^98

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
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	−4.82843	−1.29045
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.00000	0.426401
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	2.82843	0.554700
$27$	0	0
$28$	4.82843	0.912487
$29$	0.828427	0.153835	0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427	0.153835	0.0769175	+	0.997037i	$0.475492\pi$
$30$	0	0
$31$	−9.65685	−1.73442	−0.867211	−	0.497941i	$-0.834090\pi$
$31$	−9.65685	−1.73442	−0.867211	+	0.497941i	$0.834090\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.82843	0.828068
$35$	−4.82843	−0.816153
$36$	0	0
$37$	−6.82843	−1.12259	−0.561293	−	0.827617i	$-0.689696\pi$
$37$	−6.82843	−1.12259	−0.561293	+	0.827617i	$0.689696\pi$
$38$	0	0
$39$	0	0
$40$	1.00000	0.158114
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	−8.48528	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	1.00000	0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.82843	−0.392232
$53$	9.31371	1.27934	0.639668	−	0.768651i	$-0.279072\pi$
$53$	9.31371	1.27934	0.639668	+	0.768651i	$0.279072\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	−4.82843	−0.645226
$57$	0	0
$58$	−0.828427	−0.108778
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−3.17157	−0.406078	−0.203039	−	0.979171i	$-0.565082\pi$
$61$	−3.17157	−0.406078	−0.203039	+	0.979171i	$0.565082\pi$
$62$	9.65685	1.22642
$63$	0	0
$64$	1.00000	0.125000
$65$	2.82843	0.350823
$66$	0	0
$67$	1.17157	0.143130	0.0715652	−	0.997436i	$-0.477201\pi$
$67$	1.17157	0.143130	0.0715652	+	0.997436i	$0.477201\pi$
$68$	−4.82843	−0.585533
$69$	0	0
$70$	4.82843	0.577107
$71$	2.82843	0.335673	0.167836	−	0.985815i	$-0.446322\pi$
$71$	2.82843	0.335673	0.167836	+	0.985815i	$0.446322\pi$
$72$	0	0
$73$	9.31371	1.09009	0.545044	−	0.838408i	$-0.316513\pi$
$73$	9.31371	1.09009	0.545044	+	0.838408i	$0.316513\pi$
$74$	6.82843	0.793789
$75$	0	0
$76$	0	0
$77$	−9.65685	−1.10050
$78$	0	0
$79$	2.82843	0.318223	0.159111	−	0.987261i	$-0.449137\pi$
$79$	2.82843	0.318223	0.159111	+	0.987261i	$0.449137\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	5.65685	0.624695
$83$	6.82843	0.749517	0.374759	−	0.927122i	$-0.377726\pi$
$83$	6.82843	0.749517	0.374759	+	0.927122i	$0.377726\pi$
$84$	0	0
$85$	4.82843	0.523716
$86$	8.48528	0.914991
$87$	0	0
$88$	2.00000	0.213201
$89$	5.65685	0.599625	0.299813	−	0.953998i	$-0.403076\pi$
$89$	5.65685	0.599625	0.299813	+	0.953998i	$0.403076\pi$
$90$	0	0
$91$	−13.6569	−1.43163
$92$	−1.00000	−0.104257
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.82843	0.896391	0.448195	−	0.893936i	$-0.352067\pi$
$97$	8.82843	0.896391	0.448195	+	0.893936i	$0.352067\pi$
$98$	−16.3137	−1.64793
$99$	0	0
$100$	1.00000	0.100000
$101$	6.48528	0.645310	0.322655	−	0.946517i	$-0.395425\pi$
$101$	6.48528	0.645310	0.322655	+	0.946517i	$0.395425\pi$
$102$	0	0
$103$	−6.48528	−0.639014	−0.319507	−	0.947584i	$-0.603517\pi$
$103$	−6.48528	−0.639014	−0.319507	+	0.947584i	$0.603517\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	−9.31371	−0.904627
$107$	−1.17157	−0.113260	−0.0566301	−	0.998395i	$-0.518036\pi$
$107$	−1.17157	−0.113260	−0.0566301	+	0.998395i	$0.518036\pi$
$108$	0	0
$109$	−14.4853	−1.38744	−0.693719	−	0.720246i	$-0.744029\pi$
$109$	−14.4853	−1.38744	−0.693719	+	0.720246i	$0.744029\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	4.82843	0.456243
$113$	−12.1421	−1.14224	−0.571118	−	0.820868i	$-0.693490\pi$
$113$	−12.1421	−1.14224	−0.571118	+	0.820868i	$0.693490\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0.828427	0.0769175
$117$	0	0
$118$	10.0000	0.920575
$119$	−23.3137	−2.13716
$120$	0	0
$121$	−7.00000	−0.636364
$122$	3.17157	0.287141
$123$	0	0
$124$	−9.65685	−0.867211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.8284	−1.13834	−0.569169	−	0.822220i	$-0.692735\pi$
$127$	−12.8284	−1.13834	−0.569169	+	0.822220i	$0.692735\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.82843	−0.248069
$131$	−17.3137	−1.51271	−0.756353	−	0.654164i	$-0.773021\pi$
$131$	−17.3137	−1.51271	−0.756353	+	0.654164i	$0.773021\pi$
$132$	0	0
$133$	0	0
$134$	−1.17157	−0.101208
$135$	0	0
$136$	4.82843	0.414034
$137$	−16.8284	−1.43775	−0.718875	−	0.695140i	$-0.755343\pi$
$137$	−16.8284	−1.43775	−0.718875	+	0.695140i	$0.755343\pi$
$138$	0	0
$139$	−6.34315	−0.538019	−0.269009	−	0.963138i	$-0.586696\pi$
$139$	−6.34315	−0.538019	−0.269009	+	0.963138i	$0.586696\pi$
$140$	−4.82843	−0.408077
$141$	0	0
$142$	−2.82843	−0.237356
$143$	5.65685	0.473050
$144$	0	0
$145$	−0.828427	−0.0687971
$146$	−9.31371	−0.770808
$147$	0	0
$148$	−6.82843	−0.561293
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	16.9706	1.38104	0.690522	−	0.723311i	$-0.257381\pi$
$151$	16.9706	1.38104	0.690522	+	0.723311i	$0.257381\pi$
$152$	0	0
$153$	0	0
$154$	9.65685	0.778171
$155$	9.65685	0.775657
$156$	0	0
$157$	8.48528	0.677199	0.338600	−	0.940931i	$-0.390047\pi$
$157$	8.48528	0.677199	0.338600	+	0.940931i	$0.390047\pi$
$158$	−2.82843	−0.225018
$159$	0	0
$160$	1.00000	0.0790569
$161$	−4.82843	−0.380533
$162$	0	0
$163$	−1.65685	−0.129775	−0.0648874	−	0.997893i	$-0.520669\pi$
$163$	−1.65685	−0.129775	−0.0648874	+	0.997893i	$0.520669\pi$
$164$	−5.65685	−0.441726
$165$	0	0
$166$	−6.82843	−0.529989
$167$	24.9706	1.93228	0.966140	−	0.258018i	$-0.0830694\pi$
$167$	24.9706	1.93228	0.966140	+	0.258018i	$0.0830694\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	−4.82843	−0.370323
$171$	0	0
$172$	−8.48528	−0.646997
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	4.82843	0.364995
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−5.65685	−0.423999
$179$	−23.6569	−1.76820	−0.884098	−	0.467301i	$-0.845226\pi$
$179$	−23.6569	−1.76820	−0.884098	+	0.467301i	$0.845226\pi$
$180$	0	0
$181$	−5.51472	−0.409906	−0.204953	−	0.978772i	$-0.565704\pi$
$181$	−5.51472	−0.409906	−0.204953	+	0.978772i	$0.565704\pi$
$182$	13.6569	1.01231
$183$	0	0
$184$	1.00000	0.0737210
$185$	6.82843	0.502036
$186$	0	0
$187$	9.65685	0.706179
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.9706	−0.938517	−0.469258	−	0.883061i	$-0.655479\pi$
$191$	−12.9706	−0.938517	−0.469258	+	0.883061i	$0.655479\pi$
$192$	0	0
$193$	11.6569	0.839079	0.419539	−	0.907737i	$-0.362192\pi$
$193$	11.6569	0.839079	0.419539	+	0.907737i	$0.362192\pi$
$194$	−8.82843	−0.633844
$195$	0	0
$196$	16.3137	1.16526
$197$	−26.9706	−1.92157	−0.960787	−	0.277289i	$-0.910564\pi$
$197$	−26.9706	−1.92157	−0.960787	+	0.277289i	$0.910564\pi$
$198$	0	0
$199$	−12.4853	−0.885058	−0.442529	−	0.896754i	$-0.645919\pi$
$199$	−12.4853	−0.885058	−0.442529	+	0.896754i	$0.645919\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−6.48528	−0.456303
$203$	4.00000	0.280745
$204$	0	0
$205$	5.65685	0.395092
$206$	6.48528	0.451851
$207$	0	0
$208$	−2.82843	−0.196116
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	9.31371	0.639668
$213$	0	0
$214$	1.17157	0.0800871
$215$	8.48528	0.578691
$216$	0	0
$217$	−46.6274	−3.16528
$218$	14.4853	0.981067
$219$	0	0
$220$	2.00000	0.134840
$221$	13.6569	0.918659
$222$	0	0
$223$	20.1421	1.34882	0.674409	−	0.738358i	$-0.264399\pi$
$223$	20.1421	1.34882	0.674409	+	0.738358i	$0.264399\pi$
$224$	−4.82843	−0.322613
$225$	0	0
$226$	12.1421	0.807683
$227$	−2.82843	−0.187729	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	−2.82843	−0.187729	−0.0938647	+	0.995585i	$0.529922\pi$
$228$	0	0
$229$	−10.4853	−0.692887	−0.346443	−	0.938071i	$-0.612611\pi$
$229$	−10.4853	−0.692887	−0.346443	+	0.938071i	$0.612611\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−0.828427	−0.0543889
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	0	0
$236$	−10.0000	−0.650945
$237$	0	0
$238$	23.3137	1.51120
$239$	−5.17157	−0.334521	−0.167261	−	0.985913i	$-0.553492\pi$
$239$	−5.17157	−0.334521	−0.167261	+	0.985913i	$0.553492\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−3.17157	−0.203039
$245$	−16.3137	−1.04224
$246$	0	0
$247$	0	0
$248$	9.65685	0.613211
$249$	0	0
$250$	1.00000	0.0632456
$251$	26.9706	1.70237	0.851183	−	0.524868i	$-0.175885\pi$
$251$	26.9706	1.70237	0.851183	+	0.524868i	$0.175885\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	12.8284	0.804927
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.31371	0.580973	0.290487	−	0.956879i	$-0.406183\pi$
$257$	9.31371	0.580973	0.290487	+	0.956879i	$0.406183\pi$
$258$	0	0
$259$	−32.9706	−2.04869
$260$	2.82843	0.175412
$261$	0	0
$262$	17.3137	1.06964
$263$	19.3137	1.19093	0.595467	−	0.803380i	$-0.296967\pi$
$263$	19.3137	1.19093	0.595467	+	0.803380i	$0.296967\pi$
$264$	0	0
$265$	−9.31371	−0.572137
$266$	0	0
$267$	0	0
$268$	1.17157	0.0715652
$269$	20.1421	1.22809	0.614044	−	0.789272i	$-0.289542\pi$
$269$	20.1421	1.22809	0.614044	+	0.789272i	$0.289542\pi$
$270$	0	0
$271$	−5.65685	−0.343629	−0.171815	−	0.985129i	$-0.554963\pi$
$271$	−5.65685	−0.343629	−0.171815	+	0.985129i	$0.554963\pi$
$272$	−4.82843	−0.292766
$273$	0	0
$274$	16.8284	1.01664
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−28.4853	−1.71151	−0.855757	−	0.517377i	$-0.826908\pi$
$277$	−28.4853	−1.71151	−0.855757	+	0.517377i	$0.826908\pi$
$278$	6.34315	0.380437
$279$	0	0
$280$	4.82843	0.288554
$281$	2.34315	0.139780	0.0698902	−	0.997555i	$-0.477735\pi$
$281$	2.34315	0.139780	0.0698902	+	0.997555i	$0.477735\pi$
$282$	0	0
$283$	26.1421	1.55399	0.776994	−	0.629508i	$-0.216743\pi$
$283$	26.1421	1.55399	0.776994	+	0.629508i	$0.216743\pi$
$284$	2.82843	0.167836
$285$	0	0
$286$	−5.65685	−0.334497
$287$	−27.3137	−1.61228
$288$	0	0
$289$	6.31371	0.371395
$290$	0.828427	0.0486469
$291$	0	0
$292$	9.31371	0.545044
$293$	7.65685	0.447318	0.223659	−	0.974667i	$-0.428200\pi$
$293$	7.65685	0.447318	0.223659	+	0.974667i	$0.428200\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	6.82843	0.396894
$297$	0	0
$298$	6.00000	0.347571
$299$	2.82843	0.163572
$300$	0	0
$301$	−40.9706	−2.36150
$302$	−16.9706	−0.976546
$303$	0	0
$304$	0	0
$305$	3.17157	0.181604
$306$	0	0
$307$	3.31371	0.189123	0.0945617	−	0.995519i	$-0.469855\pi$
$307$	3.31371	0.189123	0.0945617	+	0.995519i	$0.469855\pi$
$308$	−9.65685	−0.550250
$309$	0	0
$310$	−9.65685	−0.548472
$311$	−0.485281	−0.0275178	−0.0137589	−	0.999905i	$-0.504380\pi$
$311$	−0.485281	−0.0275178	−0.0137589	+	0.999905i	$0.504380\pi$
$312$	0	0
$313$	−12.1421	−0.686314	−0.343157	−	0.939278i	$-0.611496\pi$
$313$	−12.1421	−0.686314	−0.343157	+	0.939278i	$0.611496\pi$
$314$	−8.48528	−0.478852
$315$	0	0
$316$	2.82843	0.159111
$317$	33.3137	1.87108	0.935542	−	0.353215i	$-0.114912\pi$
$317$	33.3137	1.87108	0.935542	+	0.353215i	$0.114912\pi$
$318$	0	0
$319$	−1.65685	−0.0927660
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	4.82843	0.269078
$323$	0	0
$324$	0	0
$325$	−2.82843	−0.156893
$326$	1.65685	0.0917647
$327$	0	0
$328$	5.65685	0.312348
$329$	0	0
$330$	0	0
$331$	−26.6274	−1.46358	−0.731788	−	0.681533i	$-0.761314\pi$
$331$	−26.6274	−1.46358	−0.731788	+	0.681533i	$0.761314\pi$
$332$	6.82843	0.374759
$333$	0	0
$334$	−24.9706	−1.36633
$335$	−1.17157	−0.0640099
$336$	0	0
$337$	21.7990	1.18747	0.593733	−	0.804662i	$-0.297653\pi$
$337$	21.7990	1.18747	0.593733	+	0.804662i	$0.297653\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	4.82843	0.261858
$341$	19.3137	1.04590
$342$	0	0
$343$	44.9706	2.42818
$344$	8.48528	0.457496
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−5.65685	−0.303676	−0.151838	−	0.988405i	$-0.548519\pi$
$347$	−5.65685	−0.303676	−0.151838	+	0.988405i	$0.548519\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−4.82843	−0.258090
$351$	0	0
$352$	2.00000	0.106600
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−2.82843	−0.150117
$356$	5.65685	0.299813
$357$	0	0
$358$	23.6569	1.25030
$359$	−27.3137	−1.44156	−0.720781	−	0.693163i	$-0.756217\pi$
$359$	−27.3137	−1.44156	−0.720781	+	0.693163i	$0.756217\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	5.51472	0.289847
$363$	0	0
$364$	−13.6569	−0.715814
$365$	−9.31371	−0.487502
$366$	0	0
$367$	−0.828427	−0.0432435	−0.0216218	−	0.999766i	$-0.506883\pi$
$367$	−0.828427	−0.0432435	−0.0216218	+	0.999766i	$0.506883\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−6.82843	−0.354993
$371$	44.9706	2.33476
$372$	0	0
$373$	21.1716	1.09622	0.548111	−	0.836405i	$-0.315347\pi$
$373$	21.1716	1.09622	0.548111	+	0.836405i	$0.315347\pi$
$374$	−9.65685	−0.499344
$375$	0	0
$376$	0	0
$377$	−2.34315	−0.120678
$378$	0	0
$379$	−13.6569	−0.701505	−0.350753	−	0.936468i	$-0.614074\pi$
$379$	−13.6569	−0.701505	−0.350753	+	0.936468i	$0.614074\pi$
$380$	0	0
$381$	0	0
$382$	12.9706	0.663632
$383$	29.6569	1.51539	0.757697	−	0.652606i	$-0.226324\pi$
$383$	29.6569	1.51539	0.757697	+	0.652606i	$0.226324\pi$
$384$	0	0
$385$	9.65685	0.492159
$386$	−11.6569	−0.593318
$387$	0	0
$388$	8.82843	0.448195
$389$	−28.6274	−1.45147	−0.725734	−	0.687976i	$-0.758500\pi$
$389$	−28.6274	−1.45147	−0.725734	+	0.687976i	$0.758500\pi$
$390$	0	0
$391$	4.82843	0.244184
$392$	−16.3137	−0.823967
$393$	0	0
$394$	26.9706	1.35876
$395$	−2.82843	−0.142314
$396$	0	0
$397$	−22.8284	−1.14573	−0.572863	−	0.819651i	$-0.694167\pi$
$397$	−22.8284	−1.14573	−0.572863	+	0.819651i	$0.694167\pi$
$398$	12.4853	0.625831
$399$	0	0
$400$	1.00000	0.0500000
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	0	0
$403$	27.3137	1.36059
$404$	6.48528	0.322655
$405$	0	0
$406$	−4.00000	−0.198517
$407$	13.6569	0.676945
$408$	0	0
$409$	−17.3137	−0.856108	−0.428054	−	0.903753i	$-0.640801\pi$
$409$	−17.3137	−0.856108	−0.428054	+	0.903753i	$0.640801\pi$
$410$	−5.65685	−0.279372
$411$	0	0
$412$	−6.48528	−0.319507
$413$	−48.2843	−2.37591
$414$	0	0
$415$	−6.82843	−0.335194
$416$	2.82843	0.138675
$417$	0	0
$418$	0	0
$419$	−3.65685	−0.178649	−0.0893245	−	0.996003i	$-0.528471\pi$
$419$	−3.65685	−0.178649	−0.0893245	+	0.996003i	$0.528471\pi$
$420$	0	0
$421$	20.1421	0.981668	0.490834	−	0.871253i	$-0.336692\pi$
$421$	20.1421	0.981668	0.490834	+	0.871253i	$0.336692\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	−9.31371	−0.452314
$425$	−4.82843	−0.234213
$426$	0	0
$427$	−15.3137	−0.741082
$428$	−1.17157	−0.0566301
$429$	0	0
$430$	−8.48528	−0.409197
$431$	−14.6274	−0.704578	−0.352289	−	0.935891i	$-0.614597\pi$
$431$	−14.6274	−0.704578	−0.352289	+	0.935891i	$0.614597\pi$
$432$	0	0
$433$	−20.1421	−0.967969	−0.483985	−	0.875076i	$-0.660811\pi$
$433$	−20.1421	−0.967969	−0.483985	+	0.875076i	$0.660811\pi$
$434$	46.6274	2.23819
$435$	0	0
$436$	−14.4853	−0.693719
$437$	0	0
$438$	0	0
$439$	30.6274	1.46177	0.730883	−	0.682502i	$-0.239108\pi$
$439$	30.6274	1.46177	0.730883	+	0.682502i	$0.239108\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	−13.6569	−0.649590
$443$	−11.3137	−0.537531	−0.268765	−	0.963206i	$-0.586616\pi$
$443$	−11.3137	−0.537531	−0.268765	+	0.963206i	$0.586616\pi$
$444$	0	0
$445$	−5.65685	−0.268161
$446$	−20.1421	−0.953758
$447$	0	0
$448$	4.82843	0.228122
$449$	36.9706	1.74475	0.872374	−	0.488838i	$-0.162579\pi$
$449$	36.9706	1.74475	0.872374	+	0.488838i	$0.162579\pi$
$450$	0	0
$451$	11.3137	0.532742
$452$	−12.1421	−0.571118
$453$	0	0
$454$	2.82843	0.132745
$455$	13.6569	0.640243
$456$	0	0
$457$	−4.82843	−0.225864	−0.112932	−	0.993603i	$-0.536024\pi$
$457$	−4.82843	−0.225864	−0.112932	+	0.993603i	$0.536024\pi$
$458$	10.4853	0.489945
$459$	0	0
$460$	1.00000	0.0466252
$461$	−4.14214	−0.192918	−0.0964592	−	0.995337i	$-0.530752\pi$
$461$	−4.14214	−0.192918	−0.0964592	+	0.995337i	$0.530752\pi$
$462$	0	0
$463$	34.4853	1.60267	0.801333	−	0.598218i	$-0.204124\pi$
$463$	34.4853	1.60267	0.801333	+	0.598218i	$0.204124\pi$
$464$	0.828427	0.0384588
$465$	0	0
$466$	−2.00000	−0.0926482
$467$	−3.51472	−0.162642	−0.0813209	−	0.996688i	$-0.525914\pi$
$467$	−3.51472	−0.162642	−0.0813209	+	0.996688i	$0.525914\pi$
$468$	0	0
$469$	5.65685	0.261209
$470$	0	0
$471$	0	0
$472$	10.0000	0.460287
$473$	16.9706	0.780307
$474$	0	0
$475$	0	0
$476$	−23.3137	−1.06858
$477$	0	0
$478$	5.17157	0.236542
$479$	25.6569	1.17229	0.586146	−	0.810206i	$-0.300645\pi$
$479$	25.6569	1.17229	0.586146	+	0.810206i	$0.300645\pi$
$480$	0	0
$481$	19.3137	0.880629
$482$	18.0000	0.819878
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−8.82843	−0.400878
$486$	0	0
$487$	33.1127	1.50048	0.750240	−	0.661166i	$-0.229938\pi$
$487$	33.1127	1.50048	0.750240	+	0.661166i	$0.229938\pi$
$488$	3.17157	0.143570
$489$	0	0
$490$	16.3137	0.736978
$491$	37.3137	1.68394	0.841972	−	0.539521i	$-0.181395\pi$
$491$	37.3137	1.68394	0.841972	+	0.539521i	$0.181395\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	−9.65685	−0.433606
$497$	13.6569	0.612594
$498$	0	0
$499$	14.3431	0.642087	0.321044	−	0.947064i	$-0.395966\pi$
$499$	14.3431	0.642087	0.321044	+	0.947064i	$0.395966\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−26.9706	−1.20376
$503$	−22.6274	−1.00891	−0.504453	−	0.863439i	$-0.668306\pi$
$503$	−22.6274	−1.00891	−0.504453	+	0.863439i	$0.668306\pi$
$504$	0	0
$505$	−6.48528	−0.288591
$506$	−2.00000	−0.0889108
$507$	0	0
$508$	−12.8284	−0.569169
$509$	20.1421	0.892784	0.446392	−	0.894837i	$-0.352709\pi$
$509$	20.1421	0.892784	0.446392	+	0.894837i	$0.352709\pi$
$510$	0	0
$511$	44.9706	1.98938
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−9.31371	−0.410810
$515$	6.48528	0.285776
$516$	0	0
$517$	0	0
$518$	32.9706	1.44864
$519$	0	0
$520$	−2.82843	−0.124035
$521$	19.3137	0.846149	0.423074	−	0.906095i	$-0.360951\pi$
$521$	19.3137	0.846149	0.423074	+	0.906095i	$0.360951\pi$
$522$	0	0
$523$	19.5147	0.853319	0.426660	−	0.904412i	$-0.359690\pi$
$523$	19.5147	0.853319	0.426660	+	0.904412i	$0.359690\pi$
$524$	−17.3137	−0.756353
$525$	0	0
$526$	−19.3137	−0.842118
$527$	46.6274	2.03112
$528$	0	0
$529$	1.00000	0.0434783
$530$	9.31371	0.404562
$531$	0	0
$532$	0	0
$533$	16.0000	0.693037
$534$	0	0
$535$	1.17157	0.0506515
$536$	−1.17157	−0.0506042
$537$	0	0
$538$	−20.1421	−0.868389
$539$	−32.6274	−1.40536
$540$	0	0
$541$	−38.9706	−1.67548	−0.837738	−	0.546073i	$-0.816122\pi$
$541$	−38.9706	−1.67548	−0.837738	+	0.546073i	$0.816122\pi$
$542$	5.65685	0.242983
$543$	0	0
$544$	4.82843	0.207017
$545$	14.4853	0.620481
$546$	0	0
$547$	−22.6274	−0.967478	−0.483739	−	0.875212i	$-0.660722\pi$
$547$	−22.6274	−0.967478	−0.483739	+	0.875212i	$0.660722\pi$
$548$	−16.8284	−0.718875
$549$	0	0
$550$	2.00000	0.0852803
$551$	0	0
$552$	0	0
$553$	13.6569	0.580749
$554$	28.4853	1.21022
$555$	0	0
$556$	−6.34315	−0.269009
$557$	−26.9706	−1.14278	−0.571390	−	0.820679i	$-0.693596\pi$
$557$	−26.9706	−1.14278	−0.571390	+	0.820679i	$0.693596\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	−4.82843	−0.204038
$561$	0	0
$562$	−2.34315	−0.0988396
$563$	−18.8284	−0.793524	−0.396762	−	0.917922i	$-0.629866\pi$
$563$	−18.8284	−0.793524	−0.396762	+	0.917922i	$0.629866\pi$
$564$	0	0
$565$	12.1421	0.510823
$566$	−26.1421	−1.09884
$567$	0	0
$568$	−2.82843	−0.118678
$569$	32.0000	1.34151	0.670755	−	0.741679i	$-0.265970\pi$
$569$	32.0000	1.34151	0.670755	+	0.741679i	$0.265970\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	5.65685	0.236525
$573$	0	0
$574$	27.3137	1.14005
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−8.34315	−0.347330	−0.173665	−	0.984805i	$-0.555561\pi$
$577$	−8.34315	−0.347330	−0.173665	+	0.984805i	$0.555561\pi$
$578$	−6.31371	−0.262616
$579$	0	0
$580$	−0.828427	−0.0343986
$581$	32.9706	1.36785
$582$	0	0
$583$	−18.6274	−0.771469
$584$	−9.31371	−0.385404
$585$	0	0
$586$	−7.65685	−0.316302
$587$	−35.5980	−1.46929	−0.734643	−	0.678454i	$-0.762650\pi$
$587$	−35.5980	−1.46929	−0.734643	+	0.678454i	$0.762650\pi$
$588$	0	0
$589$	0	0
$590$	−10.0000	−0.411693
$591$	0	0
$592$	−6.82843	−0.280647
$593$	−21.3137	−0.875249	−0.437625	−	0.899158i	$-0.644180\pi$
$593$	−21.3137	−0.875249	−0.437625	+	0.899158i	$0.644180\pi$
$594$	0	0
$595$	23.3137	0.955769
$596$	−6.00000	−0.245770
$597$	0	0
$598$	−2.82843	−0.115663
$599$	−26.1421	−1.06814	−0.534069	−	0.845441i	$-0.679338\pi$
$599$	−26.1421	−1.06814	−0.534069	+	0.845441i	$0.679338\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	40.9706	1.66984
$603$	0	0
$604$	16.9706	0.690522
$605$	7.00000	0.284590
$606$	0	0
$607$	48.1421	1.95403	0.977015	−	0.213173i	$-0.0683797\pi$
$607$	48.1421	1.95403	0.977015	+	0.213173i	$0.0683797\pi$
$608$	0	0
$609$	0	0
$610$	−3.17157	−0.128413
$611$	0	0
$612$	0	0
$613$	−14.8284	−0.598915	−0.299457	−	0.954110i	$-0.596806\pi$
$613$	−14.8284	−0.598915	−0.299457	+	0.954110i	$0.596806\pi$
$614$	−3.31371	−0.133730
$615$	0	0
$616$	9.65685	0.389086
$617$	−37.7990	−1.52173	−0.760865	−	0.648910i	$-0.775225\pi$
$617$	−37.7990	−1.52173	−0.760865	+	0.648910i	$0.775225\pi$
$618$	0	0
$619$	−3.31371	−0.133189	−0.0665946	−	0.997780i	$-0.521213\pi$
$619$	−3.31371	−0.133189	−0.0665946	+	0.997780i	$0.521213\pi$
$620$	9.65685	0.387829
$621$	0	0
$622$	0.485281	0.0194580
$623$	27.3137	1.09430
$624$	0	0
$625$	1.00000	0.0400000
$626$	12.1421	0.485297
$627$	0	0
$628$	8.48528	0.338600
$629$	32.9706	1.31462
$630$	0	0
$631$	11.7990	0.469710	0.234855	−	0.972030i	$-0.424538\pi$
$631$	11.7990	0.469710	0.234855	+	0.972030i	$0.424538\pi$
$632$	−2.82843	−0.112509
$633$	0	0
$634$	−33.3137	−1.32306
$635$	12.8284	0.509081
$636$	0	0
$637$	−46.1421	−1.82822
$638$	1.65685	0.0655955
$639$	0	0
$640$	1.00000	0.0395285
$641$	9.65685	0.381423	0.190711	−	0.981646i	$-0.438921\pi$
$641$	9.65685	0.381423	0.190711	+	0.981646i	$0.438921\pi$
$642$	0	0
$643$	27.5147	1.08507	0.542537	−	0.840032i	$-0.317464\pi$
$643$	27.5147	1.08507	0.542537	+	0.840032i	$0.317464\pi$
$644$	−4.82843	−0.190267
$645$	0	0
$646$	0	0
$647$	24.9706	0.981694	0.490847	−	0.871246i	$-0.336687\pi$
$647$	24.9706	0.981694	0.490847	+	0.871246i	$0.336687\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	2.82843	0.110940
$651$	0	0
$652$	−1.65685	−0.0648874
$653$	−3.37258	−0.131979	−0.0659897	−	0.997820i	$-0.521020\pi$
$653$	−3.37258	−0.131979	−0.0659897	+	0.997820i	$0.521020\pi$
$654$	0	0
$655$	17.3137	0.676503
$656$	−5.65685	−0.220863
$657$	0	0
$658$	0	0
$659$	27.9411	1.08843	0.544216	−	0.838945i	$-0.316827\pi$
$659$	27.9411	1.08843	0.544216	+	0.838945i	$0.316827\pi$
$660$	0	0
$661$	−28.1421	−1.09460	−0.547301	−	0.836936i	$-0.684345\pi$
$661$	−28.1421	−1.09460	−0.547301	+	0.836936i	$0.684345\pi$
$662$	26.6274	1.03490
$663$	0	0
$664$	−6.82843	−0.264994
$665$	0	0
$666$	0	0
$667$	−0.828427	−0.0320768
$668$	24.9706	0.966140
$669$	0	0
$670$	1.17157	0.0452618
$671$	6.34315	0.244874
$672$	0	0
$673$	27.6569	1.06609	0.533047	−	0.846086i	$-0.321047\pi$
$673$	27.6569	1.06609	0.533047	+	0.846086i	$0.321047\pi$
$674$	−21.7990	−0.839666
$675$	0	0
$676$	−5.00000	−0.192308
$677$	5.31371	0.204222	0.102111	−	0.994773i	$-0.467440\pi$
$677$	5.31371	0.204222	0.102111	+	0.994773i	$0.467440\pi$
$678$	0	0
$679$	42.6274	1.63589
$680$	−4.82843	−0.185162
$681$	0	0
$682$	−19.3137	−0.739560
$683$	−44.9706	−1.72075	−0.860375	−	0.509661i	$-0.829771\pi$
$683$	−44.9706	−1.72075	−0.860375	+	0.509661i	$0.829771\pi$
$684$	0	0
$685$	16.8284	0.642981
$686$	−44.9706	−1.71698
$687$	0	0
$688$	−8.48528	−0.323498
$689$	−26.3431	−1.00359
$690$	0	0
$691$	−30.3431	−1.15431	−0.577154	−	0.816635i	$-0.695837\pi$
$691$	−30.3431	−1.15431	−0.577154	+	0.816635i	$0.695837\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	5.65685	0.214731
$695$	6.34315	0.240609
$696$	0	0
$697$	27.3137	1.03458
$698$	14.0000	0.529908
$699$	0	0
$700$	4.82843	0.182497
$701$	−20.3431	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$701$	−20.3431	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$702$	0	0
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−18.0000	−0.677439
$707$	31.3137	1.17767
$708$	0	0
$709$	45.1127	1.69424	0.847121	−	0.531399i	$-0.178334\pi$
$709$	45.1127	1.69424	0.847121	+	0.531399i	$0.178334\pi$
$710$	2.82843	0.106149
$711$	0	0
$712$	−5.65685	−0.212000
$713$	9.65685	0.361652
$714$	0	0
$715$	−5.65685	−0.211554
$716$	−23.6569	−0.884098
$717$	0	0
$718$	27.3137	1.01934
$719$	12.4853	0.465622	0.232811	−	0.972522i	$-0.425208\pi$
$719$	12.4853	0.465622	0.232811	+	0.972522i	$0.425208\pi$
$720$	0	0
$721$	−31.3137	−1.16618
$722$	19.0000	0.707107
$723$	0	0
$724$	−5.51472	−0.204953
$725$	0.828427	0.0307670
$726$	0	0
$727$	−20.8284	−0.772484	−0.386242	−	0.922398i	$-0.626227\pi$
$727$	−20.8284	−0.772484	−0.386242	+	0.922398i	$0.626227\pi$
$728$	13.6569	0.506157
$729$	0	0
$730$	9.31371	0.344716
$731$	40.9706	1.51535
$732$	0	0
$733$	−12.4853	−0.461154	−0.230577	−	0.973054i	$-0.574061\pi$
$733$	−12.4853	−0.461154	−0.230577	+	0.973054i	$0.574061\pi$
$734$	0.828427	0.0305778
$735$	0	0
$736$	1.00000	0.0368605
$737$	−2.34315	−0.0863109
$738$	0	0
$739$	22.3431	0.821906	0.410953	−	0.911657i	$-0.365196\pi$
$739$	22.3431	0.821906	0.410953	+	0.911657i	$0.365196\pi$
$740$	6.82843	0.251018
$741$	0	0
$742$	−44.9706	−1.65092
$743$	−36.2843	−1.33114	−0.665570	−	0.746335i	$-0.731812\pi$
$743$	−36.2843	−1.33114	−0.665570	+	0.746335i	$0.731812\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−21.1716	−0.775146
$747$	0	0
$748$	9.65685	0.353090
$749$	−5.65685	−0.206697
$750$	0	0
$751$	−31.1127	−1.13532	−0.567659	−	0.823264i	$-0.692151\pi$
$751$	−31.1127	−1.13532	−0.567659	+	0.823264i	$0.692151\pi$
$752$	0	0
$753$	0	0
$754$	2.34315	0.0853323
$755$	−16.9706	−0.617622
$756$	0	0
$757$	−9.17157	−0.333346	−0.166673	−	0.986012i	$-0.553303\pi$
$757$	−9.17157	−0.333346	−0.166673	+	0.986012i	$0.553303\pi$
$758$	13.6569	0.496039
$759$	0	0
$760$	0	0
$761$	14.3431	0.519939	0.259969	−	0.965617i	$-0.416288\pi$
$761$	14.3431	0.519939	0.259969	+	0.965617i	$0.416288\pi$
$762$	0	0
$763$	−69.9411	−2.53204
$764$	−12.9706	−0.469258
$765$	0	0
$766$	−29.6569	−1.07155
$767$	28.2843	1.02129
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	−9.65685	−0.348009
$771$	0	0
$772$	11.6569	0.419539
$773$	−8.34315	−0.300082	−0.150041	−	0.988680i	$-0.547941\pi$
$773$	−8.34315	−0.300082	−0.150041	+	0.988680i	$0.547941\pi$
$774$	0	0
$775$	−9.65685	−0.346884
$776$	−8.82843	−0.316922
$777$	0	0
$778$	28.6274	1.02634
$779$	0	0
$780$	0	0
$781$	−5.65685	−0.202418
$782$	−4.82843	−0.172664
$783$	0	0
$784$	16.3137	0.582632
$785$	−8.48528	−0.302853
$786$	0	0
$787$	19.5147	0.695625	0.347812	−	0.937564i	$-0.386925\pi$
$787$	19.5147	0.695625	0.347812	+	0.937564i	$0.386925\pi$
$788$	−26.9706	−0.960787
$789$	0	0
$790$	2.82843	0.100631
$791$	−58.6274	−2.08455
$792$	0	0
$793$	8.97056	0.318554
$794$	22.8284	0.810151
$795$	0	0
$796$	−12.4853	−0.442529
$797$	−50.9706	−1.80547	−0.902735	−	0.430197i	$-0.858444\pi$
$797$	−50.9706	−1.80547	−0.902735	+	0.430197i	$0.858444\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−16.0000	−0.564980
$803$	−18.6274	−0.657347
$804$	0	0
$805$	4.82843	0.170180
$806$	−27.3137	−0.962084
$807$	0	0
$808$	−6.48528	−0.228151
$809$	−52.2843	−1.83822	−0.919109	−	0.394004i	$-0.871089\pi$
$809$	−52.2843	−1.83822	−0.919109	+	0.394004i	$0.871089\pi$
$810$	0	0
$811$	40.2843	1.41457	0.707286	−	0.706927i	$-0.249919\pi$
$811$	40.2843	1.41457	0.707286	+	0.706927i	$0.249919\pi$
$812$	4.00000	0.140372
$813$	0	0
$814$	−13.6569	−0.478672
$815$	1.65685	0.0580371
$816$	0	0
$817$	0	0
$818$	17.3137	0.605360
$819$	0	0
$820$	5.65685	0.197546
$821$	−3.85786	−0.134640	−0.0673202	−	0.997731i	$-0.521445\pi$
$821$	−3.85786	−0.134640	−0.0673202	+	0.997731i	$0.521445\pi$
$822$	0	0
$823$	9.51472	0.331662	0.165831	−	0.986154i	$-0.446969\pi$
$823$	9.51472	0.331662	0.165831	+	0.986154i	$0.446969\pi$
$824$	6.48528	0.225925
$825$	0	0
$826$	48.2843	1.68002
$827$	1.45584	0.0506247	0.0253123	−	0.999680i	$-0.491942\pi$
$827$	1.45584	0.0506247	0.0253123	+	0.999680i	$0.491942\pi$
$828$	0	0
$829$	−43.6569	−1.51627	−0.758133	−	0.652100i	$-0.773888\pi$
$829$	−43.6569	−1.51627	−0.758133	+	0.652100i	$0.773888\pi$
$830$	6.82843	0.237018
$831$	0	0
$832$	−2.82843	−0.0980581
$833$	−78.7696	−2.72920
$834$	0	0
$835$	−24.9706	−0.864142
$836$	0	0
$837$	0	0
$838$	3.65685	0.126324
$839$	36.2843	1.25267	0.626336	−	0.779553i	$-0.284554\pi$
$839$	36.2843	1.25267	0.626336	+	0.779553i	$0.284554\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	−20.1421	−0.694144
$843$	0	0
$844$	−4.00000	−0.137686
$845$	5.00000	0.172005
$846$	0	0
$847$	−33.7990	−1.16135
$848$	9.31371	0.319834
$849$	0	0
$850$	4.82843	0.165614
$851$	6.82843	0.234075
$852$	0	0
$853$	19.1127	0.654406	0.327203	−	0.944954i	$-0.393894\pi$
$853$	19.1127	0.654406	0.327203	+	0.944954i	$0.393894\pi$
$854$	15.3137	0.524024
$855$	0	0
$856$	1.17157	0.0400435
$857$	−14.9706	−0.511385	−0.255692	−	0.966758i	$-0.582303\pi$
$857$	−14.9706	−0.511385	−0.255692	+	0.966758i	$0.582303\pi$
$858$	0	0
$859$	35.5980	1.21459	0.607294	−	0.794477i	$-0.292255\pi$
$859$	35.5980	1.21459	0.607294	+	0.794477i	$0.292255\pi$
$860$	8.48528	0.289346
$861$	0	0
$862$	14.6274	0.498212
$863$	37.2548	1.26817	0.634085	−	0.773264i	$-0.281377\pi$
$863$	37.2548	1.26817	0.634085	+	0.773264i	$0.281377\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	20.1421	0.684458
$867$	0	0
$868$	−46.6274	−1.58264
$869$	−5.65685	−0.191896
$870$	0	0
$871$	−3.31371	−0.112281
$872$	14.4853	0.490534
$873$	0	0
$874$	0	0
$875$	−4.82843	−0.163231
$876$	0	0
$877$	−11.7990	−0.398424	−0.199212	−	0.979956i	$-0.563838\pi$
$877$	−11.7990	−0.398424	−0.199212	+	0.979956i	$0.563838\pi$
$878$	−30.6274	−1.03363
$879$	0	0
$880$	2.00000	0.0674200
$881$	24.9706	0.841280	0.420640	−	0.907228i	$-0.361806\pi$
$881$	24.9706	0.841280	0.420640	+	0.907228i	$0.361806\pi$
$882$	0	0
$883$	11.0294	0.371170	0.185585	−	0.982628i	$-0.440582\pi$
$883$	11.0294	0.371170	0.185585	+	0.982628i	$0.440582\pi$
$884$	13.6569	0.459330
$885$	0	0
$886$	11.3137	0.380091
$887$	−56.9706	−1.91288	−0.956442	−	0.291922i	$-0.905705\pi$
$887$	−56.9706	−1.91288	−0.956442	+	0.291922i	$0.905705\pi$
$888$	0	0
$889$	−61.9411	−2.07744
$890$	5.65685	0.189618
$891$	0	0
$892$	20.1421	0.674409
$893$	0	0
$894$	0	0
$895$	23.6569	0.790761
$896$	−4.82843	−0.161306
$897$	0	0
$898$	−36.9706	−1.23372
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−44.9706	−1.49819
$902$	−11.3137	−0.376705
$903$	0	0
$904$	12.1421	0.403841
$905$	5.51472	0.183315
$906$	0	0
$907$	−0.485281	−0.0161135	−0.00805675	−	0.999968i	$-0.502565\pi$
$907$	−0.485281	−0.0161135	−0.00805675	+	0.999968i	$0.502565\pi$
$908$	−2.82843	−0.0938647
$909$	0	0
$910$	−13.6569	−0.452720
$911$	−10.3431	−0.342684	−0.171342	−	0.985212i	$-0.554810\pi$
$911$	−10.3431	−0.342684	−0.171342	+	0.985212i	$0.554810\pi$
$912$	0	0
$913$	−13.6569	−0.451976
$914$	4.82843	0.159710
$915$	0	0
$916$	−10.4853	−0.346443
$917$	−83.5980	−2.76065
$918$	0	0
$919$	41.4558	1.36750	0.683751	−	0.729715i	$-0.260347\pi$
$919$	41.4558	1.36750	0.683751	+	0.729715i	$0.260347\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	4.14214	0.136414
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−6.82843	−0.224517
$926$	−34.4853	−1.13326
$927$	0	0
$928$	−0.828427	−0.0271945
$929$	−19.5980	−0.642989	−0.321494	−	0.946911i	$-0.604185\pi$
$929$	−19.5980	−0.642989	−0.321494	+	0.946911i	$0.604185\pi$
$930$	0	0
$931$	0	0
$932$	2.00000	0.0655122
$933$	0	0
$934$	3.51472	0.115005
$935$	−9.65685	−0.315813
$936$	0	0
$937$	−28.8284	−0.941784	−0.470892	−	0.882191i	$-0.656068\pi$
$937$	−28.8284	−0.941784	−0.470892	+	0.882191i	$0.656068\pi$
$938$	−5.65685	−0.184703
$939$	0	0
$940$	0	0
$941$	−42.2843	−1.37843	−0.689214	−	0.724558i	$-0.742044\pi$
$941$	−42.2843	−1.37843	−0.689214	+	0.724558i	$0.742044\pi$
$942$	0	0
$943$	5.65685	0.184213
$944$	−10.0000	−0.325472
$945$	0	0
$946$	−16.9706	−0.551761
$947$	48.9706	1.59133	0.795665	−	0.605737i	$-0.207122\pi$
$947$	48.9706	1.59133	0.795665	+	0.605737i	$0.207122\pi$
$948$	0	0
$949$	−26.3431	−0.855135
$950$	0	0
$951$	0	0
$952$	23.3137	0.755602
$953$	17.1127	0.554335	0.277167	−	0.960822i	$-0.410604\pi$
$953$	17.1127	0.554335	0.277167	+	0.960822i	$0.410604\pi$
$954$	0	0
$955$	12.9706	0.419718
$956$	−5.17157	−0.167261
$957$	0	0
$958$	−25.6569	−0.828935
$959$	−81.2548	−2.62386
$960$	0	0
$961$	62.2548	2.00822
$962$	−19.3137	−0.622699
$963$	0	0
$964$	−18.0000	−0.579741
$965$	−11.6569	−0.375247
$966$	0	0
$967$	−11.4558	−0.368395	−0.184198	−	0.982889i	$-0.558969\pi$
$967$	−11.4558	−0.368395	−0.184198	+	0.982889i	$0.558969\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	8.82843	0.283464
$971$	35.9411	1.15341	0.576703	−	0.816954i	$-0.304339\pi$
$971$	35.9411	1.15341	0.576703	+	0.816954i	$0.304339\pi$
$972$	0	0
$973$	−30.6274	−0.981870
$974$	−33.1127	−1.06100
$975$	0	0
$976$	−3.17157	−0.101520
$977$	−50.4853	−1.61517	−0.807584	−	0.589753i	$-0.799225\pi$
$977$	−50.4853	−1.61517	−0.807584	+	0.589753i	$0.799225\pi$
$978$	0	0
$979$	−11.3137	−0.361588
$980$	−16.3137	−0.521122
$981$	0	0
$982$	−37.3137	−1.19073
$983$	15.0294	0.479365	0.239682	−	0.970851i	$-0.422957\pi$
$983$	15.0294	0.479365	0.239682	+	0.970851i	$0.422957\pi$
$984$	0	0
$985$	26.9706	0.859354
$986$	4.00000	0.127386
$987$	0	0
$988$	0	0
$989$	8.48528	0.269816
$990$	0	0
$991$	3.31371	0.105263	0.0526317	−	0.998614i	$-0.483239\pi$
$991$	3.31371	0.105263	0.0526317	+	0.998614i	$0.483239\pi$
$992$	9.65685	0.306605
$993$	0	0
$994$	−13.6569	−0.433169
$995$	12.4853	0.395810
$996$	0	0
$997$	44.7696	1.41787	0.708933	−	0.705276i	$-0.249177\pi$
$997$	44.7696	1.41787	0.708933	+	0.705276i	$0.249177\pi$
$998$	−14.3431	−0.454024
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.v.1.2	✓	2
3.2	odd	2		2070.2.a.y.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2070.2.a.v.1.2	✓	2	1.1	even	1	trivial
2070.2.a.y.1.2	yes	2	3.2	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.82843 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.82843 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} -2.82843 q^{13} -4.82843 q^{14} +1.00000 q^{16} -4.82843 q^{17} -1.00000 q^{20} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +2.82843 q^{26} +4.82843 q^{28} +0.828427 q^{29} -9.65685 q^{31} -1.00000 q^{32} +4.82843 q^{34} -4.82843 q^{35} -6.82843 q^{37} +1.00000 q^{40} -5.65685 q^{41} -8.48528 q^{43} -2.00000 q^{44} +1.00000 q^{46} +16.3137 q^{49} -1.00000 q^{50} -2.82843 q^{52} +9.31371 q^{53} +2.00000 q^{55} -4.82843 q^{56} -0.828427 q^{58} -10.0000 q^{59} -3.17157 q^{61} +9.65685 q^{62} +1.00000 q^{64} +2.82843 q^{65} +1.17157 q^{67} -4.82843 q^{68} +4.82843 q^{70} +2.82843 q^{71} +9.31371 q^{73} +6.82843 q^{74} -9.65685 q^{77} +2.82843 q^{79} -1.00000 q^{80} +5.65685 q^{82} +6.82843 q^{83} +4.82843 q^{85} +8.48528 q^{86} +2.00000 q^{88} +5.65685 q^{89} -13.6569 q^{91} -1.00000 q^{92} +8.82843 q^{97} -16.3137 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 2 q^{20} + 4 q^{22} - 2 q^{23} + 2 q^{25} + 4 q^{28} - 4 q^{29} - 8 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{35} - 8 q^{37} + 2 q^{40} - 4 q^{44} + 2 q^{46} + 10 q^{49} - 2 q^{50} - 4 q^{53} + 4 q^{55} - 4 q^{56} + 4 q^{58} - 20 q^{59} - 12 q^{61} + 8 q^{62} + 2 q^{64} + 8 q^{67} - 4 q^{68} + 4 q^{70} - 4 q^{73} + 8 q^{74} - 8 q^{77} - 2 q^{80} + 8 q^{83} + 4 q^{85} + 4 q^{88} - 16 q^{91} - 2 q^{92} + 12 q^{97} - 10 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 - 4 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^17 - 2 * q^20 + 4 * q^22 - 2 * q^23 + 2 * q^25 + 4 * q^28 - 4 * q^29 - 8 * q^31 - 2 * q^32 + 4 * q^34 - 4 * q^35 - 8 * q^37 + 2 * q^40 - 4 * q^44 + 2 * q^46 + 10 * q^49 - 2 * q^50 - 4 * q^53 + 4 * q^55 - 4 * q^56 + 4 * q^58 - 20 * q^59 - 12 * q^61 + 8 * q^62 + 2 * q^64 + 8 * q^67 - 4 * q^68 + 4 * q^70 - 4 * q^73 + 8 * q^74 - 8 * q^77 - 2 * q^80 + 8 * q^83 + 4 * q^85 + 4 * q^88 - 16 * q^91 - 2 * q^92 + 12 * q^97 - 10 * q^98

Introduction

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