LMFDB - Embedded newform 2070.2.a.z.1.3

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			1.3
Root			$-3.11903$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.50973 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.50973 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.33763 q^{11} -3.72833 q^{13} -4.50973 q^{14} +1.00000 q^{16} -1.11903 q^{17} +4.50973 q^{19} +1.00000 q^{20} +4.33763 q^{22} +1.00000 q^{23} +1.00000 q^{25} +3.72833 q^{26} +4.50973 q^{28} +8.23805 q^{29} +1.72833 q^{31} -1.00000 q^{32} +1.11903 q^{34} +4.50973 q^{35} -0.781399 q^{37} -4.50973 q^{38} -1.00000 q^{40} -3.90043 q^{41} +8.00000 q^{43} -4.33763 q^{44} -1.00000 q^{46} +11.4567 q^{47} +13.3376 q^{49} -1.00000 q^{50} -3.72833 q^{52} +6.00000 q^{53} -4.33763 q^{55} -4.50973 q^{56} -8.23805 q^{58} +2.23805 q^{59} +3.55623 q^{61} -1.72833 q^{62} +1.00000 q^{64} -3.72833 q^{65} +2.43720 q^{67} -1.11903 q^{68} -4.50973 q^{70} -7.11903 q^{71} -9.45665 q^{73} +0.781399 q^{74} +4.50973 q^{76} -19.5615 q^{77} -14.9133 q^{79} +1.00000 q^{80} +3.90043 q^{82} -2.78140 q^{83} -1.11903 q^{85} -8.00000 q^{86} +4.33763 q^{88} +7.69471 q^{89} -16.8137 q^{91} +1.00000 q^{92} -11.4567 q^{94} +4.50973 q^{95} -0.642920 q^{97} -13.3376 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 3 q^{11} - q^{13} - 3 q^{14} + 3 q^{16} + 7 q^{17} + 3 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{23} + 3 q^{25} + q^{26} + 3 q^{28} + 4 q^{29} - 5 q^{31} - 3 q^{32} - 7 q^{34} + 3 q^{35} - 2 q^{37} - 3 q^{38} - 3 q^{40} - q^{41} + 24 q^{43} - 3 q^{44} - 3 q^{46} + 14 q^{47} + 30 q^{49} - 3 q^{50} - q^{52} + 18 q^{53} - 3 q^{55} - 3 q^{56} - 4 q^{58} - 14 q^{59} + q^{61} + 5 q^{62} + 3 q^{64} - q^{65} + 8 q^{67} + 7 q^{68} - 3 q^{70} - 11 q^{71} - 8 q^{73} + 2 q^{74} + 3 q^{76} + 24 q^{77} - 4 q^{79} + 3 q^{80} + q^{82} - 8 q^{83} + 7 q^{85} - 24 q^{86} + 3 q^{88} - 18 q^{89} + q^{91} + 3 q^{92} - 14 q^{94} + 3 q^{95} - 33 q^{97} - 30 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^10 - 3 * q^11 - q^13 - 3 * q^14 + 3 * q^16 + 7 * q^17 + 3 * q^19 + 3 * q^20 + 3 * q^22 + 3 * q^23 + 3 * q^25 + q^26 + 3 * q^28 + 4 * q^29 - 5 * q^31 - 3 * q^32 - 7 * q^34 + 3 * q^35 - 2 * q^37 - 3 * q^38 - 3 * q^40 - q^41 + 24 * q^43 - 3 * q^44 - 3 * q^46 + 14 * q^47 + 30 * q^49 - 3 * q^50 - q^52 + 18 * q^53 - 3 * q^55 - 3 * q^56 - 4 * q^58 - 14 * q^59 + q^61 + 5 * q^62 + 3 * q^64 - q^65 + 8 * q^67 + 7 * q^68 - 3 * q^70 - 11 * q^71 - 8 * q^73 + 2 * q^74 + 3 * q^76 + 24 * q^77 - 4 * q^79 + 3 * q^80 + q^82 - 8 * q^83 + 7 * q^85 - 24 * q^86 + 3 * q^88 - 18 * q^89 + q^91 + 3 * q^92 - 14 * q^94 + 3 * q^95 - 33 * q^97 - 30 * 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
$7$	4.50973	1.70452	0.852258	−	0.523122i	$-0.175233\pi$
$7$	4.50973	1.70452	0.852258	+	0.523122i	$0.175233\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−4.33763	−1.30784	−0.653922	−	0.756562i	$-0.726878\pi$
$11$	−4.33763	−1.30784	−0.653922	+	0.756562i	$0.726878\pi$
$12$	0	0
$13$	−3.72833	−1.03405	−0.517026	−	0.855970i	$-0.672961\pi$
$13$	−3.72833	−1.03405	−0.517026	+	0.855970i	$0.672961\pi$
$14$	−4.50973	−1.20527
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.11903	−0.271404	−0.135702	−	0.990750i	$-0.543329\pi$
$17$	−1.11903	−0.271404	−0.135702	+	0.990750i	$0.543329\pi$
$18$	0	0
$19$	4.50973	1.03460	0.517301	−	0.855803i	$-0.326937\pi$
$19$	4.50973	1.03460	0.517301	+	0.855803i	$0.326937\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	4.33763	0.924785
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	3.72833	0.731185
$27$	0	0
$28$	4.50973	0.852258
$29$	8.23805	1.52977	0.764884	−	0.644168i	$-0.222796\pi$
$29$	8.23805	1.52977	0.764884	+	0.644168i	$0.222796\pi$
$30$	0	0
$31$	1.72833	0.310417	0.155208	−	0.987882i	$-0.450395\pi$
$31$	1.72833	0.310417	0.155208	+	0.987882i	$0.450395\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.11903	0.191911
$35$	4.50973	0.762283
$36$	0	0
$37$	−0.781399	−0.128461	−0.0642306	−	0.997935i	$-0.520459\pi$
$37$	−0.781399	−0.128461	−0.0642306	+	0.997935i	$0.520459\pi$
$38$	−4.50973	−0.731574
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−3.90043	−0.609144	−0.304572	−	0.952489i	$-0.598513\pi$
$41$	−3.90043	−0.609144	−0.304572	+	0.952489i	$0.598513\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−4.33763	−0.653922
$45$	0	0
$46$	−1.00000	−0.147442
$47$	11.4567	1.67112	0.835562	−	0.549396i	$-0.185142\pi$
$47$	11.4567	1.67112	0.835562	+	0.549396i	$0.185142\pi$
$48$	0	0
$49$	13.3376	1.90538
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.72833	−0.517026
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−4.33763	−0.584886
$56$	−4.50973	−0.602637
$57$	0	0
$58$	−8.23805	−1.08171
$59$	2.23805	0.291370	0.145685	−	0.989331i	$-0.453461\pi$
$59$	2.23805	0.291370	0.145685	+	0.989331i	$0.453461\pi$
$60$	0	0
$61$	3.55623	0.455329	0.227664	−	0.973740i	$-0.426891\pi$
$61$	3.55623	0.455329	0.227664	+	0.973740i	$0.426891\pi$
$62$	−1.72833	−0.219498
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.72833	−0.462442
$66$	0	0
$67$	2.43720	0.297752	0.148876	−	0.988856i	$-0.452435\pi$
$67$	2.43720	0.297752	0.148876	+	0.988856i	$0.452435\pi$
$68$	−1.11903	−0.135702
$69$	0	0
$70$	−4.50973	−0.539015
$71$	−7.11903	−0.844873	−0.422437	−	0.906393i	$-0.638825\pi$
$71$	−7.11903	−0.844873	−0.422437	+	0.906393i	$0.638825\pi$
$72$	0	0
$73$	−9.45665	−1.10682	−0.553409	−	0.832910i	$-0.686673\pi$
$73$	−9.45665	−1.10682	−0.553409	+	0.832910i	$0.686673\pi$
$74$	0.781399	0.0908357
$75$	0	0
$76$	4.50973	0.517301
$77$	−19.5615	−2.22924
$78$	0	0
$79$	−14.9133	−1.67788	−0.838939	−	0.544225i	$-0.816824\pi$
$79$	−14.9133	−1.67788	−0.838939	+	0.544225i	$0.816824\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	3.90043	0.430730
$83$	−2.78140	−0.305298	−0.152649	−	0.988280i	$-0.548780\pi$
$83$	−2.78140	−0.305298	−0.152649	+	0.988280i	$0.548780\pi$
$84$	0	0
$85$	−1.11903	−0.121375
$86$	−8.00000	−0.862662
$87$	0	0
$88$	4.33763	0.462393
$89$	7.69471	0.815637	0.407819	−	0.913063i	$-0.366290\pi$
$89$	7.69471	0.815637	0.407819	+	0.913063i	$0.366290\pi$
$90$	0	0
$91$	−16.8137	−1.76256
$92$	1.00000	0.104257
$93$	0	0
$94$	−11.4567	−1.18166
$95$	4.50973	0.462688
$96$	0	0
$97$	−0.642920	−0.0652786	−0.0326393	−	0.999467i	$-0.510391\pi$
$97$	−0.642920	−0.0652786	−0.0326393	+	0.999467i	$0.510391\pi$
$98$	−13.3376	−1.34730
$99$	0	0
$100$	1.00000	0.100000
$101$	8.23805	0.819717	0.409858	−	0.912149i	$-0.365578\pi$
$101$	8.23805	0.819717	0.409858	+	0.912149i	$0.365578\pi$
$102$	0	0
$103$	12.3376	1.21566	0.607831	−	0.794066i	$-0.292040\pi$
$103$	12.3376	1.21566	0.607831	+	0.794066i	$0.292040\pi$
$104$	3.72833	0.365593
$105$	0	0
$106$	−6.00000	−0.582772
$107$	15.9328	1.54028	0.770139	−	0.637876i	$-0.220187\pi$
$107$	15.9328	1.54028	0.770139	+	0.637876i	$0.220187\pi$
$108$	0	0
$109$	−1.49027	−0.142742	−0.0713712	−	0.997450i	$-0.522737\pi$
$109$	−1.49027	−0.142742	−0.0713712	+	0.997450i	$0.522737\pi$
$110$	4.33763	0.413577
$111$	0	0
$112$	4.50973	0.426129
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	8.23805	0.764884
$117$	0	0
$118$	−2.23805	−0.206030
$119$	−5.04650	−0.462612
$120$	0	0
$121$	7.81502	0.710456
$122$	−3.55623	−0.321966
$123$	0	0
$124$	1.72833	0.155208
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.675256	−0.0599193	−0.0299597	−	0.999551i	$-0.509538\pi$
$127$	−0.675256	−0.0599193	−0.0299597	+	0.999551i	$0.509538\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	3.72833	0.326996
$131$	13.6947	1.19651	0.598256	−	0.801305i	$-0.295861\pi$
$131$	13.6947	1.19651	0.598256	+	0.801305i	$0.295861\pi$
$132$	0	0
$133$	20.3376	1.76350
$134$	−2.43720	−0.210542
$135$	0	0
$136$	1.11903	0.0959557
$137$	−7.52918	−0.643261	−0.321631	−	0.946865i	$-0.604231\pi$
$137$	−7.52918	−0.643261	−0.321631	+	0.946865i	$0.604231\pi$
$138$	0	0
$139$	4.67526	0.396550	0.198275	−	0.980146i	$-0.436466\pi$
$139$	4.67526	0.396550	0.198275	+	0.980146i	$0.436466\pi$
$140$	4.50973	0.381141
$141$	0	0
$142$	7.11903	0.597415
$143$	16.1721	1.35238
$144$	0	0
$145$	8.23805	0.684133
$146$	9.45665	0.782638
$147$	0	0
$148$	−0.781399	−0.0642306
$149$	−7.52918	−0.616814	−0.308407	−	0.951254i	$-0.599796\pi$
$149$	−7.52918	−0.616814	−0.308407	+	0.951254i	$0.599796\pi$
$150$	0	0
$151$	−13.3571	−1.08698	−0.543492	−	0.839414i	$-0.682898\pi$
$151$	−13.3571	−1.08698	−0.543492	+	0.839414i	$0.682898\pi$
$152$	−4.50973	−0.365787
$153$	0	0
$154$	19.5615	1.57631
$155$	1.72833	0.138823
$156$	0	0
$157$	16.2381	1.29594	0.647969	−	0.761667i	$-0.275619\pi$
$157$	16.2381	1.29594	0.647969	+	0.761667i	$0.275619\pi$
$158$	14.9133	1.18644
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	4.50973	0.355416
$162$	0	0
$163$	−3.29112	−0.257781	−0.128890	−	0.991659i	$-0.541142\pi$
$163$	−3.29112	−0.257781	−0.128890	+	0.991659i	$0.541142\pi$
$164$	−3.90043	−0.304572
$165$	0	0
$166$	2.78140	0.215878
$167$	−22.9133	−1.77309	−0.886543	−	0.462647i	$-0.846900\pi$
$167$	−22.9133	−1.77309	−0.886543	+	0.462647i	$0.846900\pi$
$168$	0	0
$169$	0.900425	0.0692635
$170$	1.11903	0.0858254
$171$	0	0
$172$	8.00000	0.609994
$173$	−0.575681	−0.0437683	−0.0218841	−	0.999761i	$-0.506966\pi$
$173$	−0.575681	−0.0437683	−0.0218841	+	0.999761i	$0.506966\pi$
$174$	0	0
$175$	4.50973	0.340903
$176$	−4.33763	−0.326961
$177$	0	0
$178$	−7.69471	−0.576743
$179$	−5.01945	−0.375171	−0.187586	−	0.982248i	$-0.560066\pi$
$179$	−5.01945	−0.375171	−0.187586	+	0.982248i	$0.560066\pi$
$180$	0	0
$181$	−11.5292	−0.856957	−0.428479	−	0.903552i	$-0.640950\pi$
$181$	−11.5292	−0.856957	−0.428479	+	0.903552i	$0.640950\pi$
$182$	16.8137	1.24632
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−0.781399	−0.0574496
$186$	0	0
$187$	4.85392	0.354954
$188$	11.4567	0.835562
$189$	0	0
$190$	−4.50973	−0.327170
$191$	18.7142	1.35411	0.677055	−	0.735933i	$-0.263256\pi$
$191$	18.7142	1.35411	0.677055	+	0.735933i	$0.263256\pi$
$192$	0	0
$193$	23.4956	1.69125	0.845624	−	0.533780i	$-0.179229\pi$
$193$	23.4956	1.69125	0.845624	+	0.533780i	$0.179229\pi$
$194$	0.642920	0.0461590
$195$	0	0
$196$	13.3376	0.952688
$197$	18.1385	1.29231	0.646157	−	0.763205i	$-0.276375\pi$
$197$	18.1385	1.29231	0.646157	+	0.763205i	$0.276375\pi$
$198$	0	0
$199$	−23.2575	−1.64868	−0.824340	−	0.566094i	$-0.808454\pi$
$199$	−23.2575	−1.64868	−0.824340	+	0.566094i	$0.808454\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−8.23805	−0.579627
$203$	37.1514	2.60751
$204$	0	0
$205$	−3.90043	−0.272418
$206$	−12.3376	−0.859603
$207$	0	0
$208$	−3.72833	−0.258513
$209$	−19.5615	−1.35310
$210$	0	0
$211$	4.34420	0.299067	0.149533	−	0.988757i	$-0.452223\pi$
$211$	4.34420	0.299067	0.149533	+	0.988757i	$0.452223\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−15.9328	−1.08914
$215$	8.00000	0.545595
$216$	0	0
$217$	7.79428	0.529110
$218$	1.49027	0.100934
$219$	0	0
$220$	−4.33763	−0.292443
$221$	4.17210	0.280646
$222$	0	0
$223$	12.4761	0.835462	0.417731	−	0.908571i	$-0.362825\pi$
$223$	12.4761	0.835462	0.417731	+	0.908571i	$0.362825\pi$
$224$	−4.50973	−0.301319
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−15.9328	−1.05749	−0.528747	−	0.848779i	$-0.677338\pi$
$227$	−15.9328	−1.05749	−0.528747	+	0.848779i	$0.677338\pi$
$228$	0	0
$229$	−3.56280	−0.235436	−0.117718	−	0.993047i	$-0.537558\pi$
$229$	−3.56280	−0.235436	−0.117718	+	0.993047i	$0.537558\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−8.23805	−0.540855
$233$	27.4956	1.80129	0.900647	−	0.434552i	$-0.143093\pi$
$233$	27.4956	1.80129	0.900647	+	0.434552i	$0.143093\pi$
$234$	0	0
$235$	11.4567	0.747350
$236$	2.23805	0.145685
$237$	0	0
$238$	5.04650	0.327116
$239$	−10.0389	−0.649363	−0.324681	−	0.945823i	$-0.605257\pi$
$239$	−10.0389	−0.649363	−0.324681	+	0.945823i	$0.605257\pi$
$240$	0	0
$241$	−23.6947	−1.52631	−0.763155	−	0.646215i	$-0.776351\pi$
$241$	−23.6947	−1.52631	−0.763155	+	0.646215i	$0.776351\pi$
$242$	−7.81502	−0.502368
$243$	0	0
$244$	3.55623	0.227664
$245$	13.3376	0.852110
$246$	0	0
$247$	−16.8137	−1.06983
$248$	−1.72833	−0.109749
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−12.4425	−0.785363	−0.392681	−	0.919675i	$-0.628452\pi$
$251$	−12.4425	−0.785363	−0.392681	+	0.919675i	$0.628452\pi$
$252$	0	0
$253$	−4.33763	−0.272704
$254$	0.675256	0.0423693
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.45665	−0.340377	−0.170188	−	0.985412i	$-0.554438\pi$
$257$	−5.45665	−0.340377	−0.170188	+	0.985412i	$0.554438\pi$
$258$	0	0
$259$	−3.52389	−0.218964
$260$	−3.72833	−0.231221
$261$	0	0
$262$	−13.6947	−0.846062
$263$	−0.138479	−0.00853895	−0.00426948	−	0.999991i	$-0.501359\pi$
$263$	−0.138479	−0.00853895	−0.00426948	+	0.999991i	$0.501359\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−20.3376	−1.24698
$267$	0	0
$268$	2.43720	0.148876
$269$	−14.6753	−0.894766	−0.447383	−	0.894342i	$-0.647644\pi$
$269$	−14.6753	−0.894766	−0.447383	+	0.894342i	$0.647644\pi$
$270$	0	0
$271$	−8.31058	−0.504832	−0.252416	−	0.967619i	$-0.581225\pi$
$271$	−8.31058	−0.504832	−0.252416	+	0.967619i	$0.581225\pi$
$272$	−1.11903	−0.0678510
$273$	0	0
$274$	7.52918	0.454854
$275$	−4.33763	−0.261569
$276$	0	0
$277$	12.9133	0.775886	0.387943	−	0.921683i	$-0.373186\pi$
$277$	12.9133	0.775886	0.387943	+	0.921683i	$0.373186\pi$
$278$	−4.67526	−0.280403
$279$	0	0
$280$	−4.50973	−0.269508
$281$	−2.67526	−0.159592	−0.0797962	−	0.996811i	$-0.525427\pi$
$281$	−2.67526	−0.159592	−0.0797962	+	0.996811i	$0.525427\pi$
$282$	0	0
$283$	0.742495	0.0441367	0.0220684	−	0.999756i	$-0.492975\pi$
$283$	0.742495	0.0441367	0.0220684	+	0.999756i	$0.492975\pi$
$284$	−7.11903	−0.422437
$285$	0	0
$286$	−16.1721	−0.956276
$287$	−17.5898	−1.03830
$288$	0	0
$289$	−15.7478	−0.926340
$290$	−8.23805	−0.483755
$291$	0	0
$292$	−9.45665	−0.553409
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	2.23805	0.130305
$296$	0.781399	0.0454179
$297$	0	0
$298$	7.52918	0.436154
$299$	−3.72833	−0.215615
$300$	0	0
$301$	36.0778	2.07949
$302$	13.3571	0.768614
$303$	0	0
$304$	4.50973	0.258651
$305$	3.55623	0.203629
$306$	0	0
$307$	30.5084	1.74121	0.870604	−	0.491984i	$-0.163728\pi$
$307$	30.5084	1.74121	0.870604	+	0.491984i	$0.163728\pi$
$308$	−19.5615	−1.11462
$309$	0	0
$310$	−1.72833	−0.0981624
$311$	−5.56280	−0.315437	−0.157719	−	0.987484i	$-0.550414\pi$
$311$	−5.56280	−0.315437	−0.157719	+	0.987484i	$0.550414\pi$
$312$	0	0
$313$	4.07252	0.230193	0.115096	−	0.993354i	$-0.463282\pi$
$313$	4.07252	0.230193	0.115096	+	0.993354i	$0.463282\pi$
$314$	−16.2381	−0.916366
$315$	0	0
$316$	−14.9133	−0.838939
$317$	6.16553	0.346291	0.173145	−	0.984896i	$-0.444607\pi$
$317$	6.16553	0.346291	0.173145	+	0.984896i	$0.444607\pi$
$318$	0	0
$319$	−35.7336	−2.00070
$320$	1.00000	0.0559017
$321$	0	0
$322$	−4.50973	−0.251317
$323$	−5.04650	−0.280795
$324$	0	0
$325$	−3.72833	−0.206810
$326$	3.29112	0.182279
$327$	0	0
$328$	3.90043	0.215365
$329$	51.6664	2.84846
$330$	0	0
$331$	27.5886	1.51640	0.758202	−	0.652019i	$-0.226078\pi$
$331$	27.5886	1.51640	0.758202	+	0.652019i	$0.226078\pi$
$332$	−2.78140	−0.152649
$333$	0	0
$334$	22.9133	1.25376
$335$	2.43720	0.133159
$336$	0	0
$337$	−17.4230	−0.949093	−0.474547	−	0.880230i	$-0.657388\pi$
$337$	−17.4230	−0.949093	−0.474547	+	0.880230i	$0.657388\pi$
$338$	−0.900425	−0.0489767
$339$	0	0
$340$	−1.11903	−0.0606877
$341$	−7.49684	−0.405977
$342$	0	0
$343$	28.5810	1.54323
$344$	−8.00000	−0.431331
$345$	0	0
$346$	0.575681	0.0309488
$347$	−4.88097	−0.262024	−0.131012	−	0.991381i	$-0.541823\pi$
$347$	−4.88097	−0.262024	−0.131012	+	0.991381i	$0.541823\pi$
$348$	0	0
$349$	24.0389	1.28677	0.643387	−	0.765542i	$-0.277529\pi$
$349$	24.0389	1.28677	0.643387	+	0.765542i	$0.277529\pi$
$350$	−4.50973	−0.241055
$351$	0	0
$352$	4.33763	0.231196
$353$	14.3442	0.763464	0.381732	−	0.924273i	$-0.375328\pi$
$353$	14.3442	0.763464	0.381732	+	0.924273i	$0.375328\pi$
$354$	0	0
$355$	−7.11903	−0.377839
$356$	7.69471	0.407819
$357$	0	0
$358$	5.01945	0.265286
$359$	−26.7814	−1.41347	−0.706734	−	0.707479i	$-0.749832\pi$
$359$	−26.7814	−1.41347	−0.706734	+	0.707479i	$0.749832\pi$
$360$	0	0
$361$	1.33763	0.0704015
$362$	11.5292	0.605960
$363$	0	0
$364$	−16.8137	−0.881279
$365$	−9.45665	−0.494984
$366$	0	0
$367$	−20.4761	−1.06884	−0.534422	−	0.845218i	$-0.679471\pi$
$367$	−20.4761	−1.06884	−0.534422	+	0.845218i	$0.679471\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	0.781399	0.0406230
$371$	27.0584	1.40480
$372$	0	0
$373$	−3.89386	−0.201616	−0.100808	−	0.994906i	$-0.532143\pi$
$373$	−3.89386	−0.201616	−0.100808	+	0.994906i	$0.532143\pi$
$374$	−4.85392	−0.250990
$375$	0	0
$376$	−11.4567	−0.590832
$377$	−30.7142	−1.58186
$378$	0	0
$379$	−30.3765	−1.56034	−0.780169	−	0.625569i	$-0.784867\pi$
$379$	−30.3765	−1.56034	−0.780169	+	0.625569i	$0.784867\pi$
$380$	4.50973	0.231344
$381$	0	0
$382$	−18.7142	−0.957500
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−19.5615	−0.996947
$386$	−23.4956	−1.19589
$387$	0	0
$388$	−0.642920	−0.0326393
$389$	18.6818	0.947206	0.473603	−	0.880738i	$-0.342953\pi$
$389$	18.6818	0.947206	0.473603	+	0.880738i	$0.342953\pi$
$390$	0	0
$391$	−1.11903	−0.0565916
$392$	−13.3376	−0.673652
$393$	0	0
$394$	−18.1385	−0.913803
$395$	−14.9133	−0.750370
$396$	0	0
$397$	−28.5757	−1.43417	−0.717086	−	0.696985i	$-0.754525\pi$
$397$	−28.5757	−1.43417	−0.717086	+	0.696985i	$0.754525\pi$
$398$	23.2575	1.16579
$399$	0	0
$400$	1.00000	0.0500000
$401$	−12.1061	−0.604552	−0.302276	−	0.953220i	$-0.597746\pi$
$401$	−12.1061	−0.604552	−0.302276	+	0.953220i	$0.597746\pi$
$402$	0	0
$403$	−6.44377	−0.320987
$404$	8.23805	0.409858
$405$	0	0
$406$	−37.1514	−1.84379
$407$	3.38942	0.168007
$408$	0	0
$409$	25.2911	1.25057	0.625283	−	0.780398i	$-0.284984\pi$
$409$	25.2911	1.25057	0.625283	+	0.780398i	$0.284984\pi$
$410$	3.90043	0.192628
$411$	0	0
$412$	12.3376	0.607831
$413$	10.0930	0.496644
$414$	0	0
$415$	−2.78140	−0.136533
$416$	3.72833	0.182796
$417$	0	0
$418$	19.5615	0.956785
$419$	17.3505	0.847628	0.423814	−	0.905749i	$-0.360691\pi$
$419$	17.3505	0.847628	0.423814	+	0.905749i	$0.360691\pi$
$420$	0	0
$421$	21.4230	1.04409	0.522047	−	0.852916i	$-0.325168\pi$
$421$	21.4230	1.04409	0.522047	+	0.852916i	$0.325168\pi$
$422$	−4.34420	−0.211472
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−1.11903	−0.0542808
$426$	0	0
$427$	16.0376	0.776115
$428$	15.9328	0.770139
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−22.5822	−1.08775	−0.543874	−	0.839167i	$-0.683043\pi$
$431$	−22.5822	−1.08775	−0.543874	+	0.839167i	$0.683043\pi$
$432$	0	0
$433$	1.01417	0.0487378	0.0243689	−	0.999703i	$-0.492242\pi$
$433$	1.01417	0.0487378	0.0243689	+	0.999703i	$0.492242\pi$
$434$	−7.79428	−0.374138
$435$	0	0
$436$	−1.49027	−0.0713712
$437$	4.50973	0.215729
$438$	0	0
$439$	−26.7478	−1.27660	−0.638301	−	0.769787i	$-0.720362\pi$
$439$	−26.7478	−1.27660	−0.638301	+	0.769787i	$0.720362\pi$
$440$	4.33763	0.206788
$441$	0	0
$442$	−4.17210	−0.198446
$443$	−10.2044	−0.484827	−0.242414	−	0.970173i	$-0.577939\pi$
$443$	−10.2044	−0.484827	−0.242414	+	0.970173i	$0.577939\pi$
$444$	0	0
$445$	7.69471	0.364764
$446$	−12.4761	−0.590761
$447$	0	0
$448$	4.50973	0.213065
$449$	−38.7867	−1.83046	−0.915228	−	0.402936i	$-0.867990\pi$
$449$	−38.7867	−1.83046	−0.915228	+	0.402936i	$0.867990\pi$
$450$	0	0
$451$	16.9186	0.796665
$452$	6.00000	0.282216
$453$	0	0
$454$	15.9328	0.747762
$455$	−16.8137	−0.788240
$456$	0	0
$457$	34.9522	1.63500	0.817498	−	0.575932i	$-0.195361\pi$
$457$	34.9522	1.63500	0.817498	+	0.575932i	$0.195361\pi$
$458$	3.56280	0.166479
$459$	0	0
$460$	1.00000	0.0466252
$461$	−16.3700	−0.762425	−0.381213	−	0.924487i	$-0.624493\pi$
$461$	−16.3700	−0.762425	−0.381213	+	0.924487i	$0.624493\pi$
$462$	0	0
$463$	29.2186	1.35790	0.678952	−	0.734183i	$-0.262435\pi$
$463$	29.2186	1.35790	0.678952	+	0.734183i	$0.262435\pi$
$464$	8.23805	0.382442
$465$	0	0
$466$	−27.4956	−1.27371
$467$	−24.2770	−1.12340	−0.561702	−	0.827340i	$-0.689853\pi$
$467$	−24.2770	−1.12340	−0.561702	+	0.827340i	$0.689853\pi$
$468$	0	0
$469$	10.9911	0.507523
$470$	−11.4567	−0.528456
$471$	0	0
$472$	−2.23805	−0.103015
$473$	−34.7010	−1.59555
$474$	0	0
$475$	4.50973	0.206920
$476$	−5.04650	−0.231306
$477$	0	0
$478$	10.0389	0.459169
$479$	−24.6080	−1.12437	−0.562185	−	0.827012i	$-0.690039\pi$
$479$	−24.6080	−1.12437	−0.562185	+	0.827012i	$0.690039\pi$
$480$	0	0
$481$	2.91331	0.132835
$482$	23.6947	1.07926
$483$	0	0
$484$	7.81502	0.355228
$485$	−0.642920	−0.0291935
$486$	0	0
$487$	−30.2381	−1.37022	−0.685108	−	0.728441i	$-0.740245\pi$
$487$	−30.2381	−1.37022	−0.685108	+	0.728441i	$0.740245\pi$
$488$	−3.55623	−0.160983
$489$	0	0
$490$	−13.3376	−0.602533
$491$	−12.3311	−0.556493	−0.278246	−	0.960510i	$-0.589753\pi$
$491$	−12.3311	−0.556493	−0.278246	+	0.960510i	$0.589753\pi$
$492$	0	0
$493$	−9.21860	−0.415185
$494$	16.8137	0.756486
$495$	0	0
$496$	1.72833	0.0776042
$497$	−32.1049	−1.44010
$498$	0	0
$499$	−26.9133	−1.20481	−0.602403	−	0.798192i	$-0.705790\pi$
$499$	−26.9133	−1.20481	−0.602403	+	0.798192i	$0.705790\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	12.4425	0.555335
$503$	−20.5097	−0.914483	−0.457242	−	0.889342i	$-0.651163\pi$
$503$	−20.5097	−0.914483	−0.457242	+	0.889342i	$0.651163\pi$
$504$	0	0
$505$	8.23805	0.366589
$506$	4.33763	0.192831
$507$	0	0
$508$	−0.675256	−0.0299597
$509$	36.7142	1.62733	0.813663	−	0.581336i	$-0.197470\pi$
$509$	36.7142	1.62733	0.813663	+	0.581336i	$0.197470\pi$
$510$	0	0
$511$	−42.6469	−1.88659
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	5.45665	0.240683
$515$	12.3376	0.543661
$516$	0	0
$517$	−49.6947	−2.18557
$518$	3.52389	0.154831
$519$	0	0
$520$	3.72833	0.163498
$521$	4.91331	0.215256	0.107628	−	0.994191i	$-0.465674\pi$
$521$	4.91331	0.215256	0.107628	+	0.994191i	$0.465674\pi$
$522$	0	0
$523$	−0.344196	−0.0150506	−0.00752531	−	0.999972i	$-0.502395\pi$
$523$	−0.344196	−0.0150506	−0.00752531	+	0.999972i	$0.502395\pi$
$524$	13.6947	0.598256
$525$	0	0
$526$	0.138479	0.00603795
$527$	−1.93404	−0.0842483
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	0	0
$532$	20.3376	0.881748
$533$	14.5421	0.629887
$534$	0	0
$535$	15.9328	0.688833
$536$	−2.43720	−0.105271
$537$	0	0
$538$	14.6753	0.632695
$539$	−57.8537	−2.49193
$540$	0	0
$541$	−6.13191	−0.263631	−0.131816	−	0.991274i	$-0.542081\pi$
$541$	−6.13191	−0.263631	−0.131816	+	0.991274i	$0.542081\pi$
$542$	8.31058	0.356970
$543$	0	0
$544$	1.11903	0.0479779
$545$	−1.49027	−0.0638363
$546$	0	0
$547$	−9.18498	−0.392721	−0.196361	−	0.980532i	$-0.562912\pi$
$547$	−9.18498	−0.392721	−0.196361	+	0.980532i	$0.562912\pi$
$548$	−7.52918	−0.321631
$549$	0	0
$550$	4.33763	0.184957
$551$	37.1514	1.58270
$552$	0	0
$553$	−67.2549	−2.85997
$554$	−12.9133	−0.548634
$555$	0	0
$556$	4.67526	0.198275
$557$	4.30529	0.182421	0.0912105	−	0.995832i	$-0.470926\pi$
$557$	4.30529	0.182421	0.0912105	+	0.995832i	$0.470926\pi$
$558$	0	0
$559$	−29.8266	−1.26153
$560$	4.50973	0.190571
$561$	0	0
$562$	2.67526	0.112849
$563$	−11.1256	−0.468888	−0.234444	−	0.972130i	$-0.575327\pi$
$563$	−11.1256	−0.468888	−0.234444	+	0.972130i	$0.575327\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	−0.742495	−0.0312094
$567$	0	0
$568$	7.11903	0.298708
$569$	16.0389	0.672386	0.336193	−	0.941793i	$-0.390861\pi$
$569$	16.0389	0.672386	0.336193	+	0.941793i	$0.390861\pi$
$570$	0	0
$571$	17.9004	0.749109	0.374555	−	0.927205i	$-0.377796\pi$
$571$	17.9004	0.749109	0.374555	+	0.927205i	$0.377796\pi$
$572$	16.1721	0.676189
$573$	0	0
$574$	17.5898	0.734186
$575$	1.00000	0.0417029
$576$	0	0
$577$	−9.12559	−0.379903	−0.189952	−	0.981793i	$-0.560833\pi$
$577$	−9.12559	−0.379903	−0.189952	+	0.981793i	$0.560833\pi$
$578$	15.7478	0.655021
$579$	0	0
$580$	8.23805	0.342067
$581$	−12.5433	−0.520386
$582$	0	0
$583$	−26.0258	−1.07788
$584$	9.45665	0.391319
$585$	0	0
$586$	−6.00000	−0.247858
$587$	33.6340	1.38823	0.694113	−	0.719866i	$-0.255797\pi$
$587$	33.6340	1.38823	0.694113	+	0.719866i	$0.255797\pi$
$588$	0	0
$589$	7.79428	0.321158
$590$	−2.23805	−0.0921392
$591$	0	0
$592$	−0.781399	−0.0321153
$593$	17.4567	0.716859	0.358429	−	0.933557i	$-0.383312\pi$
$593$	17.4567	0.716859	0.358429	+	0.933557i	$0.383312\pi$
$594$	0	0
$595$	−5.04650	−0.206886
$596$	−7.52918	−0.308407
$597$	0	0
$598$	3.72833	0.152463
$599$	11.5951	0.473764	0.236882	−	0.971538i	$-0.423874\pi$
$599$	11.5951	0.473764	0.236882	+	0.971538i	$0.423874\pi$
$600$	0	0
$601$	−31.6611	−1.29148	−0.645741	−	0.763556i	$-0.723452\pi$
$601$	−31.6611	−1.29148	−0.645741	+	0.763556i	$0.723452\pi$
$602$	−36.0778	−1.47042
$603$	0	0
$604$	−13.3571	−0.543492
$605$	7.81502	0.317726
$606$	0	0
$607$	−36.0778	−1.46435	−0.732177	−	0.681115i	$-0.761495\pi$
$607$	−36.0778	−1.46435	−0.732177	+	0.681115i	$0.761495\pi$
$608$	−4.50973	−0.182894
$609$	0	0
$610$	−3.55623	−0.143988
$611$	−42.7142	−1.72803
$612$	0	0
$613$	−32.0389	−1.29404	−0.647020	−	0.762473i	$-0.723985\pi$
$613$	−32.0389	−1.29404	−0.647020	+	0.762473i	$0.723985\pi$
$614$	−30.5084	−1.23122
$615$	0	0
$616$	19.5615	0.788156
$617$	13.3960	0.539302	0.269651	−	0.962958i	$-0.413092\pi$
$617$	13.3960	0.539302	0.269651	+	0.962958i	$0.413092\pi$
$618$	0	0
$619$	37.1309	1.49242	0.746208	−	0.665713i	$-0.231872\pi$
$619$	37.1309	1.49242	0.746208	+	0.665713i	$0.231872\pi$
$620$	1.72833	0.0694113
$621$	0	0
$622$	5.56280	0.223048
$623$	34.7010	1.39027
$624$	0	0
$625$	1.00000	0.0400000
$626$	−4.07252	−0.162771
$627$	0	0
$628$	16.2381	0.647969
$629$	0.874406	0.0348648
$630$	0	0
$631$	11.1125	0.442380	0.221190	−	0.975231i	$-0.429006\pi$
$631$	11.1125	0.442380	0.221190	+	0.975231i	$0.429006\pi$
$632$	14.9133	0.593220
$633$	0	0
$634$	−6.16553	−0.244864
$635$	−0.675256	−0.0267967
$636$	0	0
$637$	−49.7270	−1.97026
$638$	35.7336	1.41471
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−12.3831	−0.489103	−0.244552	−	0.969636i	$-0.578641\pi$
$641$	−12.3831	−0.489103	−0.244552	+	0.969636i	$0.578641\pi$
$642$	0	0
$643$	−37.4956	−1.47868	−0.739340	−	0.673332i	$-0.764862\pi$
$643$	−37.4956	−1.47868	−0.739340	+	0.673332i	$0.764862\pi$
$644$	4.50973	0.177708
$645$	0	0
$646$	5.04650	0.198552
$647$	−14.5691	−0.572771	−0.286385	−	0.958114i	$-0.592454\pi$
$647$	−14.5691	−0.572771	−0.286385	+	0.958114i	$0.592454\pi$
$648$	0	0
$649$	−9.70784	−0.381066
$650$	3.72833	0.146237
$651$	0	0
$652$	−3.29112	−0.128890
$653$	4.41672	0.172840	0.0864198	−	0.996259i	$-0.472457\pi$
$653$	4.41672	0.172840	0.0864198	+	0.996259i	$0.472457\pi$
$654$	0	0
$655$	13.6947	0.535097
$656$	−3.90043	−0.152286
$657$	0	0
$658$	−51.6664	−2.01416
$659$	31.8655	1.24130	0.620652	−	0.784086i	$-0.286868\pi$
$659$	31.8655	1.24130	0.620652	+	0.784086i	$0.286868\pi$
$660$	0	0
$661$	33.1190	1.28818	0.644090	−	0.764949i	$-0.277236\pi$
$661$	33.1190	1.28818	0.644090	+	0.764949i	$0.277236\pi$
$662$	−27.5886	−1.07226
$663$	0	0
$664$	2.78140	0.107939
$665$	20.3376	0.788659
$666$	0	0
$667$	8.23805	0.318979
$668$	−22.9133	−0.886543
$669$	0	0
$670$	−2.43720	−0.0941574
$671$	−15.4256	−0.595499
$672$	0	0
$673$	19.3505	0.745907	0.372954	−	0.927850i	$-0.378345\pi$
$673$	19.3505	0.745907	0.372954	+	0.927850i	$0.378345\pi$
$674$	17.4230	0.671110
$675$	0	0
$676$	0.900425	0.0346317
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	−2.89939	−0.111268
$680$	1.11903	0.0429127
$681$	0	0
$682$	7.49684	0.287069
$683$	−38.3495	−1.46740	−0.733701	−	0.679472i	$-0.762209\pi$
$683$	−38.3495	−1.46740	−0.733701	+	0.679472i	$0.762209\pi$
$684$	0	0
$685$	−7.52918	−0.287675
$686$	−28.5810	−1.09123
$687$	0	0
$688$	8.00000	0.304997
$689$	−22.3700	−0.852228
$690$	0	0
$691$	−21.0195	−0.799618	−0.399809	−	0.916599i	$-0.630923\pi$
$691$	−21.0195	−0.799618	−0.399809	+	0.916599i	$0.630923\pi$
$692$	−0.575681	−0.0218841
$693$	0	0
$694$	4.88097	0.185279
$695$	4.67526	0.177343
$696$	0	0
$697$	4.36468	0.165324
$698$	−24.0389	−0.909886
$699$	0	0
$700$	4.50973	0.170452
$701$	−19.3169	−0.729589	−0.364794	−	0.931088i	$-0.618861\pi$
$701$	−19.3169	−0.729589	−0.364794	+	0.931088i	$0.618861\pi$
$702$	0	0
$703$	−3.52389	−0.132906
$704$	−4.33763	−0.163481
$705$	0	0
$706$	−14.3442	−0.539851
$707$	37.1514	1.39722
$708$	0	0
$709$	12.2315	0.459363	0.229682	−	0.973266i	$-0.426232\pi$
$709$	12.2315	0.459363	0.229682	+	0.973266i	$0.426232\pi$
$710$	7.11903	0.267172
$711$	0	0
$712$	−7.69471	−0.288371
$713$	1.72833	0.0647264
$714$	0	0
$715$	16.1721	0.604802
$716$	−5.01945	−0.187586
$717$	0	0
$718$	26.7814	0.999473
$719$	40.6416	1.51568	0.757839	−	0.652442i	$-0.226255\pi$
$719$	40.6416	1.51568	0.757839	+	0.652442i	$0.226255\pi$
$720$	0	0
$721$	55.6393	2.07212
$722$	−1.33763	−0.0497814
$723$	0	0
$724$	−11.5292	−0.428479
$725$	8.23805	0.305954
$726$	0	0
$727$	−23.4501	−0.869716	−0.434858	−	0.900499i	$-0.643201\pi$
$727$	−23.4501	−0.869716	−0.434858	+	0.900499i	$0.643201\pi$
$728$	16.8137	0.623158
$729$	0	0
$730$	9.45665	0.350006
$731$	−8.95221	−0.331110
$732$	0	0
$733$	−12.5150	−0.462252	−0.231126	−	0.972924i	$-0.574241\pi$
$733$	−12.5150	−0.462252	−0.231126	+	0.972924i	$0.574241\pi$
$734$	20.4761	0.755787
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−10.5717	−0.389413
$738$	0	0
$739$	−21.3505	−0.785391	−0.392696	−	0.919668i	$-0.628457\pi$
$739$	−21.3505	−0.785391	−0.392696	+	0.919668i	$0.628457\pi$
$740$	−0.781399	−0.0287248
$741$	0	0
$742$	−27.0584	−0.993343
$743$	−24.9858	−0.916641	−0.458321	−	0.888787i	$-0.651549\pi$
$743$	−24.9858	−0.916641	−0.458321	+	0.888787i	$0.651549\pi$
$744$	0	0
$745$	−7.52918	−0.275848
$746$	3.89386	0.142564
$747$	0	0
$748$	4.85392	0.177477
$749$	71.8524	2.62543
$750$	0	0
$751$	−33.6275	−1.22708	−0.613542	−	0.789662i	$-0.710256\pi$
$751$	−33.6275	−1.22708	−0.613542	+	0.789662i	$0.710256\pi$
$752$	11.4567	0.417781
$753$	0	0
$754$	30.7142	1.11854
$755$	−13.3571	−0.486114
$756$	0	0
$757$	−37.1230	−1.34926	−0.674630	−	0.738156i	$-0.735697\pi$
$757$	−37.1230	−1.34926	−0.674630	+	0.738156i	$0.735697\pi$
$758$	30.3765	1.10333
$759$	0	0
$760$	−4.50973	−0.163585
$761$	3.87337	0.140410	0.0702048	−	0.997533i	$-0.477635\pi$
$761$	3.87337	0.140410	0.0702048	+	0.997533i	$0.477635\pi$
$762$	0	0
$763$	−6.72073	−0.243307
$764$	18.7142	0.677055
$765$	0	0
$766$	0	0
$767$	−8.34420	−0.301291
$768$	0	0
$769$	23.1645	0.835333	0.417667	−	0.908600i	$-0.362848\pi$
$769$	23.1645	0.835333	0.417667	+	0.908600i	$0.362848\pi$
$770$	19.5615	0.704948
$771$	0	0
$772$	23.4956	0.845624
$773$	8.78140	0.315845	0.157922	−	0.987452i	$-0.449520\pi$
$773$	8.78140	0.315845	0.157922	+	0.987452i	$0.449520\pi$
$774$	0	0
$775$	1.72833	0.0620834
$776$	0.642920	0.0230795
$777$	0	0
$778$	−18.6818	−0.669776
$779$	−17.5898	−0.630222
$780$	0	0
$781$	30.8797	1.10496
$782$	1.11903	0.0400163
$783$	0	0
$784$	13.3376	0.476344
$785$	16.2381	0.579561
$786$	0	0
$787$	49.6275	1.76903	0.884514	−	0.466513i	$-0.154490\pi$
$787$	49.6275	1.76903	0.884514	+	0.466513i	$0.154490\pi$
$788$	18.1385	0.646157
$789$	0	0
$790$	14.9133	0.530592
$791$	27.0584	0.962084
$792$	0	0
$793$	−13.2588	−0.470833
$794$	28.5757	1.01411
$795$	0	0
$796$	−23.2575	−0.824340
$797$	−18.3311	−0.649319	−0.324660	−	0.945831i	$-0.605250\pi$
$797$	−18.3311	−0.649319	−0.324660	+	0.945831i	$0.605250\pi$
$798$	0	0
$799$	−12.8203	−0.453550
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	12.1061	0.427483
$803$	41.0195	1.44755
$804$	0	0
$805$	4.50973	0.158947
$806$	6.44377	0.226972
$807$	0	0
$808$	−8.23805	−0.289814
$809$	1.93933	0.0681832	0.0340916	−	0.999419i	$-0.489146\pi$
$809$	1.93933	0.0681832	0.0340916	+	0.999419i	$0.489146\pi$
$810$	0	0
$811$	−5.41775	−0.190243	−0.0951215	−	0.995466i	$-0.530324\pi$
$811$	−5.41775	−0.190243	−0.0951215	+	0.995466i	$0.530324\pi$
$812$	37.1514	1.30376
$813$	0	0
$814$	−3.38942	−0.118799
$815$	−3.29112	−0.115283
$816$	0	0
$817$	36.0778	1.26220
$818$	−25.2911	−0.884283
$819$	0	0
$820$	−3.90043	−0.136209
$821$	−37.8655	−1.32152	−0.660758	−	0.750599i	$-0.729765\pi$
$821$	−37.8655	−1.32152	−0.660758	+	0.750599i	$0.729765\pi$
$822$	0	0
$823$	43.7336	1.52446	0.762229	−	0.647308i	$-0.224105\pi$
$823$	43.7336	1.52446	0.762229	+	0.647308i	$0.224105\pi$
$824$	−12.3376	−0.429802
$825$	0	0
$826$	−10.0930	−0.351181
$827$	−28.1991	−0.980581	−0.490290	−	0.871559i	$-0.663109\pi$
$827$	−28.1991	−0.980581	−0.490290	+	0.871559i	$0.663109\pi$
$828$	0	0
$829$	1.12559	0.0390935	0.0195468	−	0.999809i	$-0.493778\pi$
$829$	1.12559	0.0390935	0.0195468	+	0.999809i	$0.493778\pi$
$830$	2.78140	0.0965438
$831$	0	0
$832$	−3.72833	−0.129256
$833$	−14.9252	−0.517126
$834$	0	0
$835$	−22.9133	−0.792948
$836$	−19.5615	−0.676549
$837$	0	0
$838$	−17.3505	−0.599364
$839$	−39.9328	−1.37863	−0.689316	−	0.724461i	$-0.742089\pi$
$839$	−39.9328	−1.37863	−0.689316	+	0.724461i	$0.742089\pi$
$840$	0	0
$841$	38.8655	1.34019
$842$	−21.4230	−0.738287
$843$	0	0
$844$	4.34420	0.149533
$845$	0.900425	0.0309756
$846$	0	0
$847$	35.2436	1.21098
$848$	6.00000	0.206041
$849$	0	0
$850$	1.11903	0.0383823
$851$	−0.781399	−0.0267860
$852$	0	0
$853$	47.9921	1.64322	0.821610	−	0.570050i	$-0.193076\pi$
$853$	47.9921	1.64322	0.821610	+	0.570050i	$0.193076\pi$
$854$	−16.0376	−0.548796
$855$	0	0
$856$	−15.9328	−0.544571
$857$	43.4283	1.48348	0.741742	−	0.670686i	$-0.234000\pi$
$857$	43.4283	1.48348	0.741742	+	0.670686i	$0.234000\pi$
$858$	0	0
$859$	32.5433	1.11036	0.555182	−	0.831729i	$-0.312648\pi$
$859$	32.5433	1.11036	0.555182	+	0.831729i	$0.312648\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	22.5822	0.769154
$863$	−46.1036	−1.56938	−0.784692	−	0.619886i	$-0.787179\pi$
$863$	−46.1036	−1.56938	−0.784692	+	0.619886i	$0.787179\pi$
$864$	0	0
$865$	−0.575681	−0.0195738
$866$	−1.01417	−0.0344628
$867$	0	0
$868$	7.79428	0.264555
$869$	64.6884	2.19440
$870$	0	0
$871$	−9.08669	−0.307891
$872$	1.49027	0.0504670
$873$	0	0
$874$	−4.50973	−0.152544
$875$	4.50973	0.152457
$876$	0	0
$877$	−24.0996	−0.813785	−0.406892	−	0.913476i	$-0.633388\pi$
$877$	−24.0996	−0.813785	−0.406892	+	0.913476i	$0.633388\pi$
$878$	26.7478	0.902694
$879$	0	0
$880$	−4.33763	−0.146221
$881$	−2.34420	−0.0789780	−0.0394890	−	0.999220i	$-0.512573\pi$
$881$	−2.34420	−0.0789780	−0.0394890	+	0.999220i	$0.512573\pi$
$882$	0	0
$883$	−41.0505	−1.38146	−0.690730	−	0.723113i	$-0.742711\pi$
$883$	−41.0505	−1.38146	−0.690730	+	0.723113i	$0.742711\pi$
$884$	4.17210	0.140323
$885$	0	0
$886$	10.2044	0.342825
$887$	−54.7788	−1.83929	−0.919647	−	0.392747i	$-0.871525\pi$
$887$	−54.7788	−1.83929	−0.919647	+	0.392747i	$0.871525\pi$
$888$	0	0
$889$	−3.04522	−0.102133
$890$	−7.69471	−0.257927
$891$	0	0
$892$	12.4761	0.417731
$893$	51.6664	1.72895
$894$	0	0
$895$	−5.01945	−0.167782
$896$	−4.50973	−0.150659
$897$	0	0
$898$	38.7867	1.29433
$899$	14.2381	0.474866
$900$	0	0
$901$	−6.71416	−0.223681
$902$	−16.9186	−0.563328
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−11.5292	−0.383243
$906$	0	0
$907$	−10.1061	−0.335569	−0.167784	−	0.985824i	$-0.553661\pi$
$907$	−10.1061	−0.335569	−0.167784	+	0.985824i	$0.553661\pi$
$908$	−15.9328	−0.528747
$909$	0	0
$910$	16.8137	0.557370
$911$	25.4178	0.842128	0.421064	−	0.907031i	$-0.361657\pi$
$911$	25.4178	0.842128	0.421064	+	0.907031i	$0.361657\pi$
$912$	0	0
$913$	12.0647	0.399282
$914$	−34.9522	−1.15612
$915$	0	0
$916$	−3.56280	−0.117718
$917$	61.7594	2.03947
$918$	0	0
$919$	23.6017	0.778548	0.389274	−	0.921122i	$-0.372726\pi$
$919$	23.6017	0.778548	0.389274	+	0.921122i	$0.372726\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	16.3700	0.539116
$923$	26.5421	0.873643
$924$	0	0
$925$	−0.781399	−0.0256922
$926$	−29.2186	−0.960183
$927$	0	0
$928$	−8.23805	−0.270427
$929$	7.08669	0.232507	0.116253	−	0.993220i	$-0.462912\pi$
$929$	7.08669	0.232507	0.116253	+	0.993220i	$0.462912\pi$
$930$	0	0
$931$	60.1490	1.97131
$932$	27.4956	0.900647
$933$	0	0
$934$	24.2770	0.794366
$935$	4.85392	0.158740
$936$	0	0
$937$	27.3169	0.892404	0.446202	−	0.894932i	$-0.352776\pi$
$937$	27.3169	0.892404	0.446202	+	0.894932i	$0.352776\pi$
$938$	−10.9911	−0.358873
$939$	0	0
$940$	11.4567	0.373675
$941$	−55.8979	−1.82222	−0.911109	−	0.412165i	$-0.864773\pi$
$941$	−55.8979	−1.82222	−0.911109	+	0.412165i	$0.864773\pi$
$942$	0	0
$943$	−3.90043	−0.127015
$944$	2.23805	0.0728424
$945$	0	0
$946$	34.7010	1.12823
$947$	−37.5939	−1.22164	−0.610818	−	0.791771i	$-0.709159\pi$
$947$	−37.5939	−1.22164	−0.610818	+	0.791771i	$0.709159\pi$
$948$	0	0
$949$	35.2575	1.14451
$950$	−4.50973	−0.146315
$951$	0	0
$952$	5.04650	0.163558
$953$	29.3828	0.951804	0.475902	−	0.879498i	$-0.342122\pi$
$953$	29.3828	0.951804	0.475902	+	0.879498i	$0.342122\pi$
$954$	0	0
$955$	18.7142	0.605576
$956$	−10.0389	−0.324681
$957$	0	0
$958$	24.6080	0.795049
$959$	−33.9545	−1.09645
$960$	0	0
$961$	−28.0129	−0.903641
$962$	−2.91331	−0.0939289
$963$	0	0
$964$	−23.6947	−0.763155
$965$	23.4956	0.756349
$966$	0	0
$967$	−49.2292	−1.58310	−0.791552	−	0.611102i	$-0.790726\pi$
$967$	−49.2292	−1.58310	−0.791552	+	0.611102i	$0.790726\pi$
$968$	−7.81502	−0.251184
$969$	0	0
$970$	0.642920	0.0206429
$971$	−9.62347	−0.308832	−0.154416	−	0.988006i	$-0.549350\pi$
$971$	−9.62347	−0.308832	−0.154416	+	0.988006i	$0.549350\pi$
$972$	0	0
$973$	21.0841	0.675926
$974$	30.2381	0.968890
$975$	0	0
$976$	3.55623	0.113832
$977$	−18.9858	−0.607411	−0.303705	−	0.952766i	$-0.598224\pi$
$977$	−18.9858	−0.607411	−0.303705	+	0.952766i	$0.598224\pi$
$978$	0	0
$979$	−33.3768	−1.06673
$980$	13.3376	0.426055
$981$	0	0
$982$	12.3311	0.393500
$983$	−4.33763	−0.138349	−0.0691744	−	0.997605i	$-0.522037\pi$
$983$	−4.33763	−0.138349	−0.0691744	+	0.997605i	$0.522037\pi$
$984$	0	0
$985$	18.1385	0.577940
$986$	9.21860	0.293580
$987$	0	0
$988$	−16.8137	−0.534916
$989$	8.00000	0.254385
$990$	0	0
$991$	1.96766	0.0625049	0.0312524	−	0.999512i	$-0.490050\pi$
$991$	1.96766	0.0625049	0.0312524	+	0.999512i	$0.490050\pi$
$992$	−1.72833	−0.0548744
$993$	0	0
$994$	32.1049	1.01830
$995$	−23.2575	−0.737312
$996$	0	0
$997$	3.96110	0.125449	0.0627246	−	0.998031i	$-0.480021\pi$
$997$	3.96110	0.125449	0.0627246	+	0.998031i	$0.480021\pi$
$998$	26.9133	0.851926
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.z.1.3		3
3.2	odd	2		230.2.a.d.1.1	✓	3
12.11	even	2		1840.2.a.r.1.3		3
15.2	even	4		1150.2.b.j.599.6		6
15.8	even	4		1150.2.b.j.599.1		6
15.14	odd	2		1150.2.a.q.1.3		3
24.5	odd	2		7360.2.a.bz.1.3		3
24.11	even	2		7360.2.a.ce.1.1		3
60.59	even	2		9200.2.a.cf.1.1		3
69.68	even	2		5290.2.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.d.1.1	✓	3	3.2	odd	2
1150.2.a.q.1.3		3	15.14	odd	2
1150.2.b.j.599.1		6	15.8	even	4
1150.2.b.j.599.6		6	15.2	even	4
1840.2.a.r.1.3		3	12.11	even	2
2070.2.a.z.1.3		3	1.1	even	1	trivial
5290.2.a.r.1.1		3	69.68	even	2
7360.2.a.bz.1.3		3	24.5	odd	2
7360.2.a.ce.1.1		3	24.11	even	2
9200.2.a.cf.1.1		3	60.59	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.50973 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.50973 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.33763 q^{11} -3.72833 q^{13} -4.50973 q^{14} +1.00000 q^{16} -1.11903 q^{17} +4.50973 q^{19} +1.00000 q^{20} +4.33763 q^{22} +1.00000 q^{23} +1.00000 q^{25} +3.72833 q^{26} +4.50973 q^{28} +8.23805 q^{29} +1.72833 q^{31} -1.00000 q^{32} +1.11903 q^{34} +4.50973 q^{35} -0.781399 q^{37} -4.50973 q^{38} -1.00000 q^{40} -3.90043 q^{41} +8.00000 q^{43} -4.33763 q^{44} -1.00000 q^{46} +11.4567 q^{47} +13.3376 q^{49} -1.00000 q^{50} -3.72833 q^{52} +6.00000 q^{53} -4.33763 q^{55} -4.50973 q^{56} -8.23805 q^{58} +2.23805 q^{59} +3.55623 q^{61} -1.72833 q^{62} +1.00000 q^{64} -3.72833 q^{65} +2.43720 q^{67} -1.11903 q^{68} -4.50973 q^{70} -7.11903 q^{71} -9.45665 q^{73} +0.781399 q^{74} +4.50973 q^{76} -19.5615 q^{77} -14.9133 q^{79} +1.00000 q^{80} +3.90043 q^{82} -2.78140 q^{83} -1.11903 q^{85} -8.00000 q^{86} +4.33763 q^{88} +7.69471 q^{89} -16.8137 q^{91} +1.00000 q^{92} -11.4567 q^{94} +4.50973 q^{95} -0.642920 q^{97} -13.3376 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 3 q^{11} - q^{13} - 3 q^{14} + 3 q^{16} + 7 q^{17} + 3 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{23} + 3 q^{25} + q^{26} + 3 q^{28} + 4 q^{29} - 5 q^{31} - 3 q^{32} - 7 q^{34} + 3 q^{35} - 2 q^{37} - 3 q^{38} - 3 q^{40} - q^{41} + 24 q^{43} - 3 q^{44} - 3 q^{46} + 14 q^{47} + 30 q^{49} - 3 q^{50} - q^{52} + 18 q^{53} - 3 q^{55} - 3 q^{56} - 4 q^{58} - 14 q^{59} + q^{61} + 5 q^{62} + 3 q^{64} - q^{65} + 8 q^{67} + 7 q^{68} - 3 q^{70} - 11 q^{71} - 8 q^{73} + 2 q^{74} + 3 q^{76} + 24 q^{77} - 4 q^{79} + 3 q^{80} + q^{82} - 8 q^{83} + 7 q^{85} - 24 q^{86} + 3 q^{88} - 18 q^{89} + q^{91} + 3 q^{92} - 14 q^{94} + 3 q^{95} - 33 q^{97} - 30 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^10 - 3 * q^11 - q^13 - 3 * q^14 + 3 * q^16 + 7 * q^17 + 3 * q^19 + 3 * q^20 + 3 * q^22 + 3 * q^23 + 3 * q^25 + q^26 + 3 * q^28 + 4 * q^29 - 5 * q^31 - 3 * q^32 - 7 * q^34 + 3 * q^35 - 2 * q^37 - 3 * q^38 - 3 * q^40 - q^41 + 24 * q^43 - 3 * q^44 - 3 * q^46 + 14 * q^47 + 30 * q^49 - 3 * q^50 - q^52 + 18 * q^53 - 3 * q^55 - 3 * q^56 - 4 * q^58 - 14 * q^59 + q^61 + 5 * q^62 + 3 * q^64 - q^65 + 8 * q^67 + 7 * q^68 - 3 * q^70 - 11 * q^71 - 8 * q^73 + 2 * q^74 + 3 * q^76 + 24 * q^77 - 4 * q^79 + 3 * q^80 + q^82 - 8 * q^83 + 7 * q^85 - 24 * q^86 + 3 * q^88 - 18 * q^89 + q^91 + 3 * q^92 - 14 * q^94 + 3 * q^95 - 33 * q^97 - 30 * q^98

Introduction

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