LMFDB - Embedded newform 4842.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4842,2,Mod(1,4842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4842, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4842 = 2 \cdot 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4842.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:	$38.6635646587$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.38197 q^{5} +0.236068 q^{7} -1.00000 q^{8} -1.38197 q^{10} +4.23607 q^{11} -1.00000 q^{13} -0.236068 q^{14} +1.00000 q^{16} +4.23607 q^{17} +2.00000 q^{19} +1.38197 q^{20} -4.23607 q^{22} +3.76393 q^{23} -3.09017 q^{25} +1.00000 q^{26} +0.236068 q^{28} +5.85410 q^{29} -1.47214 q^{31} -1.00000 q^{32} -4.23607 q^{34} +0.326238 q^{35} +4.70820 q^{37} -2.00000 q^{38} -1.38197 q^{40} +5.70820 q^{41} -2.23607 q^{43} +4.23607 q^{44} -3.76393 q^{46} +0.145898 q^{47} -6.94427 q^{49} +3.09017 q^{50} -1.00000 q^{52} +5.85410 q^{55} -0.236068 q^{56} -5.85410 q^{58} -13.5623 q^{59} +3.47214 q^{61} +1.47214 q^{62} +1.00000 q^{64} -1.38197 q^{65} +6.23607 q^{67} +4.23607 q^{68} -0.326238 q^{70} +3.70820 q^{71} +2.70820 q^{73} -4.70820 q^{74} +2.00000 q^{76} +1.00000 q^{77} +10.7984 q^{79} +1.38197 q^{80} -5.70820 q^{82} +0.763932 q^{83} +5.85410 q^{85} +2.23607 q^{86} -4.23607 q^{88} -14.7984 q^{89} -0.236068 q^{91} +3.76393 q^{92} -0.145898 q^{94} +2.76393 q^{95} -18.6525 q^{97} +6.94427 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 5 q^{5} - 4 q^{7} - 2 q^{8} - 5 q^{10} + 4 q^{11} - 2 q^{13} + 4 q^{14} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 12 q^{23} + 5 q^{25} + 2 q^{26} - 4 q^{28} + 5 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 5 * q^5 - 4 * q^7 - 2 * q^8 - 5 * q^10 + 4 * q^11 - 2 * q^13 + 4 * q^14 + 2 * q^16 + 4 * q^17 + 4 * q^19 + 5 * q^20 - 4 * q^22 + 12 * q^23 + 5 * q^25 + 2 * q^26 - 4 * q^28 + 5 * q^29 + 6 * q^31 - 2 * q^32 - 4 * q^34 - 15 * q^35 - 4 * q^37 - 4 * q^38 - 5 * q^40 - 2 * q^41 + 4 * q^44 - 12 * q^46 + 7 * q^47 + 4 * q^49 - 5 * q^50 - 2 * q^52 + 5 * q^55 + 4 * q^56 - 5 * q^58 - 7 * q^59 - 2 * q^61 - 6 * q^62 + 2 * q^64 - 5 * q^65 + 8 * q^67 + 4 * q^68 + 15 * q^70 - 6 * q^71 - 8 * q^73 + 4 * q^74 + 4 * q^76 + 2 * q^77 - 3 * q^79 + 5 * q^80 + 2 * q^82 + 6 * q^83 + 5 * q^85 - 4 * q^88 - 5 * q^89 + 4 * q^91 + 12 * q^92 - 7 * q^94 + 10 * q^95 - 6 * q^97 - 4 * q^98

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
$5$	1.38197	0.618034	0.309017	−	0.951057i	$-0.400000\pi$
$5$	1.38197	0.618034	0.309017	+	0.951057i	$0.400000\pi$
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.38197	−0.437016
$11$	4.23607	1.27722	0.638611	−	0.769529i	$-0.279509\pi$
$11$	4.23607	1.27722	0.638611	+	0.769529i	$0.279509\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−0.236068	−0.0630918
$15$	0	0
$16$	1.00000	0.250000
$17$	4.23607	1.02740	0.513699	−	0.857971i	$-0.328275\pi$
$17$	4.23607	1.02740	0.513699	+	0.857971i	$0.328275\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	1.38197	0.309017
$21$	0	0
$22$	−4.23607	−0.903133
$23$	3.76393	0.784834	0.392417	−	0.919787i	$-0.371639\pi$
$23$	3.76393	0.784834	0.392417	+	0.919787i	$0.371639\pi$
$24$	0	0
$25$	−3.09017	−0.618034
$26$	1.00000	0.196116
$27$	0	0
$28$	0.236068	0.0446127
$29$	5.85410	1.08708	0.543540	−	0.839383i	$-0.317084\pi$
$29$	5.85410	1.08708	0.543540	+	0.839383i	$0.317084\pi$
$30$	0	0
$31$	−1.47214	−0.264403	−0.132202	−	0.991223i	$-0.542205\pi$
$31$	−1.47214	−0.264403	−0.132202	+	0.991223i	$0.542205\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.23607	−0.726480
$35$	0.326238	0.0551443
$36$	0	0
$37$	4.70820	0.774024	0.387012	−	0.922075i	$-0.373507\pi$
$37$	4.70820	0.774024	0.387012	+	0.922075i	$0.373507\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	−1.38197	−0.218508
$41$	5.70820	0.891472	0.445736	−	0.895165i	$-0.352942\pi$
$41$	5.70820	0.891472	0.445736	+	0.895165i	$0.352942\pi$
$42$	0	0
$43$	−2.23607	−0.340997	−0.170499	−	0.985358i	$-0.554538\pi$
$43$	−2.23607	−0.340997	−0.170499	+	0.985358i	$0.554538\pi$
$44$	4.23607	0.638611
$45$	0	0
$46$	−3.76393	−0.554962
$47$	0.145898	0.0212814	0.0106407	−	0.999943i	$-0.496613\pi$
$47$	0.145898	0.0212814	0.0106407	+	0.999943i	$0.496613\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	3.09017	0.437016
$51$	0	0
$52$	−1.00000	−0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	5.85410	0.789367
$56$	−0.236068	−0.0315459
$57$	0	0
$58$	−5.85410	−0.768681
$59$	−13.5623	−1.76566	−0.882831	−	0.469691i	$-0.844365\pi$
$59$	−13.5623	−1.76566	−0.882831	+	0.469691i	$0.844365\pi$
$60$	0	0
$61$	3.47214	0.444561	0.222281	−	0.974983i	$-0.428650\pi$
$61$	3.47214	0.444561	0.222281	+	0.974983i	$0.428650\pi$
$62$	1.47214	0.186961
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.38197	−0.171412
$66$	0	0
$67$	6.23607	0.761857	0.380928	−	0.924605i	$-0.375604\pi$
$67$	6.23607	0.761857	0.380928	+	0.924605i	$0.375604\pi$
$68$	4.23607	0.513699
$69$	0	0
$70$	−0.326238	−0.0389929
$71$	3.70820	0.440083	0.220041	−	0.975491i	$-0.429381\pi$
$71$	3.70820	0.440083	0.220041	+	0.975491i	$0.429381\pi$
$72$	0	0
$73$	2.70820	0.316971	0.158486	−	0.987361i	$-0.449339\pi$
$73$	2.70820	0.316971	0.158486	+	0.987361i	$0.449339\pi$
$74$	−4.70820	−0.547318
$75$	0	0
$76$	2.00000	0.229416
$77$	1.00000	0.113961
$78$	0	0
$79$	10.7984	1.21491	0.607456	−	0.794353i	$-0.292190\pi$
$79$	10.7984	1.21491	0.607456	+	0.794353i	$0.292190\pi$
$80$	1.38197	0.154508
$81$	0	0
$82$	−5.70820	−0.630366
$83$	0.763932	0.0838524	0.0419262	−	0.999121i	$-0.486651\pi$
$83$	0.763932	0.0838524	0.0419262	+	0.999121i	$0.486651\pi$
$84$	0	0
$85$	5.85410	0.634967
$86$	2.23607	0.241121
$87$	0	0
$88$	−4.23607	−0.451566
$89$	−14.7984	−1.56862	−0.784312	−	0.620366i	$-0.786984\pi$
$89$	−14.7984	−1.56862	−0.784312	+	0.620366i	$0.786984\pi$
$90$	0	0
$91$	−0.236068	−0.0247466
$92$	3.76393	0.392417
$93$	0	0
$94$	−0.145898	−0.0150482
$95$	2.76393	0.283573
$96$	0	0
$97$	−18.6525	−1.89387	−0.946936	−	0.321422i	$-0.895839\pi$
$97$	−18.6525	−1.89387	−0.946936	+	0.321422i	$0.895839\pi$
$98$	6.94427	0.701477
$99$	0	0
$100$	−3.09017	−0.309017
$101$	−6.09017	−0.605995	−0.302997	−	0.952991i	$-0.597987\pi$
$101$	−6.09017	−0.605995	−0.302997	+	0.952991i	$0.597987\pi$
$102$	0	0
$103$	4.14590	0.408507	0.204254	−	0.978918i	$-0.434523\pi$
$103$	4.14590	0.408507	0.204254	+	0.978918i	$0.434523\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	0	0
$107$	7.76393	0.750568	0.375284	−	0.926910i	$-0.377545\pi$
$107$	7.76393	0.750568	0.375284	+	0.926910i	$0.377545\pi$
$108$	0	0
$109$	8.70820	0.834095	0.417047	−	0.908885i	$-0.363065\pi$
$109$	8.70820	0.834095	0.417047	+	0.908885i	$0.363065\pi$
$110$	−5.85410	−0.558167
$111$	0	0
$112$	0.236068	0.0223063
$113$	17.6180	1.65737	0.828683	−	0.559719i	$-0.189091\pi$
$113$	17.6180	1.65737	0.828683	+	0.559719i	$0.189091\pi$
$114$	0	0
$115$	5.20163	0.485054
$116$	5.85410	0.543540
$117$	0	0
$118$	13.5623	1.24851
$119$	1.00000	0.0916698
$120$	0	0
$121$	6.94427	0.631297
$122$	−3.47214	−0.314352
$123$	0	0
$124$	−1.47214	−0.132202
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−3.79837	−0.337051	−0.168526	−	0.985697i	$-0.553901\pi$
$127$	−3.79837	−0.337051	−0.168526	+	0.985697i	$0.553901\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.38197	0.121206
$131$	−7.23607	−0.632218	−0.316109	−	0.948723i	$-0.602377\pi$
$131$	−7.23607	−0.632218	−0.316109	+	0.948723i	$0.602377\pi$
$132$	0	0
$133$	0.472136	0.0409394
$134$	−6.23607	−0.538714
$135$	0	0
$136$	−4.23607	−0.363240
$137$	13.3820	1.14330	0.571649	−	0.820498i	$-0.306304\pi$
$137$	13.3820	1.14330	0.571649	+	0.820498i	$0.306304\pi$
$138$	0	0
$139$	−9.56231	−0.811064	−0.405532	−	0.914081i	$-0.632914\pi$
$139$	−9.56231	−0.811064	−0.405532	+	0.914081i	$0.632914\pi$
$140$	0.326238	0.0275721
$141$	0	0
$142$	−3.70820	−0.311186
$143$	−4.23607	−0.354238
$144$	0	0
$145$	8.09017	0.671852
$146$	−2.70820	−0.224133
$147$	0	0
$148$	4.70820	0.387012
$149$	15.3262	1.25557	0.627787	−	0.778385i	$-0.283961\pi$
$149$	15.3262	1.25557	0.627787	+	0.778385i	$0.283961\pi$
$150$	0	0
$151$	7.94427	0.646496	0.323248	−	0.946314i	$-0.395225\pi$
$151$	7.94427	0.646496	0.323248	+	0.946314i	$0.395225\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−2.03444	−0.163410
$156$	0	0
$157$	−8.65248	−0.690543	−0.345271	−	0.938503i	$-0.612213\pi$
$157$	−8.65248	−0.690543	−0.345271	+	0.938503i	$0.612213\pi$
$158$	−10.7984	−0.859072
$159$	0	0
$160$	−1.38197	−0.109254
$161$	0.888544	0.0700271
$162$	0	0
$163$	1.67376	0.131099	0.0655496	−	0.997849i	$-0.479120\pi$
$163$	1.67376	0.131099	0.0655496	+	0.997849i	$0.479120\pi$
$164$	5.70820	0.445736
$165$	0	0
$166$	−0.763932	−0.0592926
$167$	−1.38197	−0.106940	−0.0534699	−	0.998569i	$-0.517028\pi$
$167$	−1.38197	−0.106940	−0.0534699	+	0.998569i	$0.517028\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−5.85410	−0.448989
$171$	0	0
$172$	−2.23607	−0.170499
$173$	−1.47214	−0.111924	−0.0559622	−	0.998433i	$-0.517823\pi$
$173$	−1.47214	−0.111924	−0.0559622	+	0.998433i	$0.517823\pi$
$174$	0	0
$175$	−0.729490	−0.0551443
$176$	4.23607	0.319306
$177$	0	0
$178$	14.7984	1.10919
$179$	−5.29180	−0.395527	−0.197764	−	0.980250i	$-0.563368\pi$
$179$	−5.29180	−0.395527	−0.197764	+	0.980250i	$0.563368\pi$
$180$	0	0
$181$	−7.41641	−0.551257	−0.275629	−	0.961264i	$-0.588886\pi$
$181$	−7.41641	−0.551257	−0.275629	+	0.961264i	$0.588886\pi$
$182$	0.236068	0.0174985
$183$	0	0
$184$	−3.76393	−0.277481
$185$	6.50658	0.478373
$186$	0	0
$187$	17.9443	1.31222
$188$	0.145898	0.0106407
$189$	0	0
$190$	−2.76393	−0.200517
$191$	22.9443	1.66019	0.830095	−	0.557623i	$-0.188286\pi$
$191$	22.9443	1.66019	0.830095	+	0.557623i	$0.188286\pi$
$192$	0	0
$193$	−3.76393	−0.270934	−0.135467	−	0.990782i	$-0.543253\pi$
$193$	−3.76393	−0.270934	−0.135467	+	0.990782i	$0.543253\pi$
$194$	18.6525	1.33917
$195$	0	0
$196$	−6.94427	−0.496019
$197$	24.6180	1.75396	0.876981	−	0.480525i	$-0.159554\pi$
$197$	24.6180	1.75396	0.876981	+	0.480525i	$0.159554\pi$
$198$	0	0
$199$	−7.61803	−0.540028	−0.270014	−	0.962856i	$-0.587028\pi$
$199$	−7.61803	−0.540028	−0.270014	+	0.962856i	$0.587028\pi$
$200$	3.09017	0.218508
$201$	0	0
$202$	6.09017	0.428503
$203$	1.38197	0.0969950
$204$	0	0
$205$	7.88854	0.550960
$206$	−4.14590	−0.288858
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	8.47214	0.586030
$210$	0	0
$211$	−5.85410	−0.403013	−0.201506	−	0.979487i	$-0.564584\pi$
$211$	−5.85410	−0.403013	−0.201506	+	0.979487i	$0.564584\pi$
$212$	0	0
$213$	0	0
$214$	−7.76393	−0.530731
$215$	−3.09017	−0.210748
$216$	0	0
$217$	−0.347524	−0.0235915
$218$	−8.70820	−0.589794
$219$	0	0
$220$	5.85410	0.394683
$221$	−4.23607	−0.284949
$222$	0	0
$223$	10.3262	0.691496	0.345748	−	0.938327i	$-0.387625\pi$
$223$	10.3262	0.691496	0.345748	+	0.938327i	$0.387625\pi$
$224$	−0.236068	−0.0157730
$225$	0	0
$226$	−17.6180	−1.17193
$227$	−5.03444	−0.334148	−0.167074	−	0.985944i	$-0.553432\pi$
$227$	−5.03444	−0.334148	−0.167074	+	0.985944i	$0.553432\pi$
$228$	0	0
$229$	2.70820	0.178963	0.0894816	−	0.995988i	$-0.471479\pi$
$229$	2.70820	0.178963	0.0894816	+	0.995988i	$0.471479\pi$
$230$	−5.20163	−0.342985
$231$	0	0
$232$	−5.85410	−0.384341
$233$	0.236068	0.0154653	0.00773266	−	0.999970i	$-0.497539\pi$
$233$	0.236068	0.0154653	0.00773266	+	0.999970i	$0.497539\pi$
$234$	0	0
$235$	0.201626	0.0131526
$236$	−13.5623	−0.882831
$237$	0	0
$238$	−1.00000	−0.0648204
$239$	−14.2705	−0.923083	−0.461541	−	0.887119i	$-0.652703\pi$
$239$	−14.2705	−0.923083	−0.461541	+	0.887119i	$0.652703\pi$
$240$	0	0
$241$	8.70820	0.560945	0.280472	−	0.959862i	$-0.409509\pi$
$241$	8.70820	0.560945	0.280472	+	0.959862i	$0.409509\pi$
$242$	−6.94427	−0.446395
$243$	0	0
$244$	3.47214	0.222281
$245$	−9.59675	−0.613114
$246$	0	0
$247$	−2.00000	−0.127257
$248$	1.47214	0.0934807
$249$	0	0
$250$	11.1803	0.707107
$251$	−12.9098	−0.814861	−0.407431	−	0.913236i	$-0.633575\pi$
$251$	−12.9098	−0.814861	−0.407431	+	0.913236i	$0.633575\pi$
$252$	0	0
$253$	15.9443	1.00241
$254$	3.79837	0.238331
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.47214	−0.216586	−0.108293	−	0.994119i	$-0.534538\pi$
$257$	−3.47214	−0.216586	−0.108293	+	0.994119i	$0.534538\pi$
$258$	0	0
$259$	1.11146	0.0690625
$260$	−1.38197	−0.0857059
$261$	0	0
$262$	7.23607	0.447046
$263$	10.2705	0.633307	0.316653	−	0.948541i	$-0.397441\pi$
$263$	10.2705	0.633307	0.316653	+	0.948541i	$0.397441\pi$
$264$	0	0
$265$	0	0
$266$	−0.472136	−0.0289485
$267$	0	0
$268$	6.23607	0.380928
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	−16.1246	−0.979500	−0.489750	−	0.871863i	$-0.662912\pi$
$271$	−16.1246	−0.979500	−0.489750	+	0.871863i	$0.662912\pi$
$272$	4.23607	0.256849
$273$	0	0
$274$	−13.3820	−0.808434
$275$	−13.0902	−0.789367
$276$	0	0
$277$	−23.6180	−1.41907	−0.709535	−	0.704670i	$-0.751095\pi$
$277$	−23.6180	−1.41907	−0.709535	+	0.704670i	$0.751095\pi$
$278$	9.56231	0.573509
$279$	0	0
$280$	−0.326238	−0.0194964
$281$	6.81966	0.406827	0.203413	−	0.979093i	$-0.434796\pi$
$281$	6.81966	0.406827	0.203413	+	0.979093i	$0.434796\pi$
$282$	0	0
$283$	10.2361	0.608471	0.304236	−	0.952597i	$-0.401599\pi$
$283$	10.2361	0.608471	0.304236	+	0.952597i	$0.401599\pi$
$284$	3.70820	0.220041
$285$	0	0
$286$	4.23607	0.250484
$287$	1.34752	0.0795418
$288$	0	0
$289$	0.944272	0.0555454
$290$	−8.09017	−0.475071
$291$	0	0
$292$	2.70820	0.158486
$293$	−16.4721	−0.962312	−0.481156	−	0.876635i	$-0.659783\pi$
$293$	−16.4721	−0.962312	−0.481156	+	0.876635i	$0.659783\pi$
$294$	0	0
$295$	−18.7426	−1.09124
$296$	−4.70820	−0.273659
$297$	0	0
$298$	−15.3262	−0.887825
$299$	−3.76393	−0.217674
$300$	0	0
$301$	−0.527864	−0.0304256
$302$	−7.94427	−0.457141
$303$	0	0
$304$	2.00000	0.114708
$305$	4.79837	0.274754
$306$	0	0
$307$	−8.12461	−0.463696	−0.231848	−	0.972752i	$-0.574477\pi$
$307$	−8.12461	−0.463696	−0.231848	+	0.972752i	$0.574477\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	2.03444	0.115549
$311$	12.7639	0.723776	0.361888	−	0.932222i	$-0.382132\pi$
$311$	12.7639	0.723776	0.361888	+	0.932222i	$0.382132\pi$
$312$	0	0
$313$	19.8541	1.12222	0.561110	−	0.827741i	$-0.310374\pi$
$313$	19.8541	1.12222	0.561110	+	0.827741i	$0.310374\pi$
$314$	8.65248	0.488287
$315$	0	0
$316$	10.7984	0.607456
$317$	25.7639	1.44705	0.723523	−	0.690300i	$-0.242521\pi$
$317$	25.7639	1.44705	0.723523	+	0.690300i	$0.242521\pi$
$318$	0	0
$319$	24.7984	1.38844
$320$	1.38197	0.0772542
$321$	0	0
$322$	−0.888544	−0.0495166
$323$	8.47214	0.471402
$324$	0	0
$325$	3.09017	0.171412
$326$	−1.67376	−0.0927011
$327$	0	0
$328$	−5.70820	−0.315183
$329$	0.0344419	0.00189884
$330$	0	0
$331$	23.1803	1.27411	0.637053	−	0.770820i	$-0.280153\pi$
$331$	23.1803	1.27411	0.637053	+	0.770820i	$0.280153\pi$
$332$	0.763932	0.0419262
$333$	0	0
$334$	1.38197	0.0756178
$335$	8.61803	0.470853
$336$	0	0
$337$	18.9787	1.03384	0.516918	−	0.856035i	$-0.327079\pi$
$337$	18.9787	1.03384	0.516918	+	0.856035i	$0.327079\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	5.85410	0.317483
$341$	−6.23607	−0.337702
$342$	0	0
$343$	−3.29180	−0.177740
$344$	2.23607	0.120561
$345$	0	0
$346$	1.47214	0.0791425
$347$	31.3607	1.68353	0.841765	−	0.539845i	$-0.181517\pi$
$347$	31.3607	1.68353	0.841765	+	0.539845i	$0.181517\pi$
$348$	0	0
$349$	−12.2705	−0.656825	−0.328413	−	0.944534i	$-0.606514\pi$
$349$	−12.2705	−0.656825	−0.328413	+	0.944534i	$0.606514\pi$
$350$	0.729490	0.0389929
$351$	0	0
$352$	−4.23607	−0.225783
$353$	3.14590	0.167439	0.0837196	−	0.996489i	$-0.473320\pi$
$353$	3.14590	0.167439	0.0837196	+	0.996489i	$0.473320\pi$
$354$	0	0
$355$	5.12461	0.271986
$356$	−14.7984	−0.784312
$357$	0	0
$358$	5.29180	0.279680
$359$	−0.944272	−0.0498368	−0.0249184	−	0.999689i	$-0.507933\pi$
$359$	−0.944272	−0.0498368	−0.0249184	+	0.999689i	$0.507933\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	7.41641	0.389798
$363$	0	0
$364$	−0.236068	−0.0123733
$365$	3.74265	0.195899
$366$	0	0
$367$	−33.7984	−1.76426	−0.882130	−	0.471005i	$-0.843891\pi$
$367$	−33.7984	−1.76426	−0.882130	+	0.471005i	$0.843891\pi$
$368$	3.76393	0.196209
$369$	0	0
$370$	−6.50658	−0.338261
$371$	0	0
$372$	0	0
$373$	−10.9443	−0.566673	−0.283336	−	0.959021i	$-0.591441\pi$
$373$	−10.9443	−0.566673	−0.283336	+	0.959021i	$0.591441\pi$
$374$	−17.9443	−0.927876
$375$	0	0
$376$	−0.145898	−0.00752412
$377$	−5.85410	−0.301502
$378$	0	0
$379$	12.8541	0.660271	0.330135	−	0.943934i	$-0.392906\pi$
$379$	12.8541	0.660271	0.330135	+	0.943934i	$0.392906\pi$
$380$	2.76393	0.141787
$381$	0	0
$382$	−22.9443	−1.17393
$383$	19.6180	1.00243	0.501217	−	0.865321i	$-0.332886\pi$
$383$	19.6180	1.00243	0.501217	+	0.865321i	$0.332886\pi$
$384$	0	0
$385$	1.38197	0.0704315
$386$	3.76393	0.191579
$387$	0	0
$388$	−18.6525	−0.946936
$389$	−11.0557	−0.560548	−0.280274	−	0.959920i	$-0.590425\pi$
$389$	−11.0557	−0.560548	−0.280274	+	0.959920i	$0.590425\pi$
$390$	0	0
$391$	15.9443	0.806336
$392$	6.94427	0.350739
$393$	0	0
$394$	−24.6180	−1.24024
$395$	14.9230	0.750857
$396$	0	0
$397$	−0.729490	−0.0366121	−0.0183060	−	0.999832i	$-0.505827\pi$
$397$	−0.729490	−0.0366121	−0.0183060	+	0.999832i	$0.505827\pi$
$398$	7.61803	0.381858
$399$	0	0
$400$	−3.09017	−0.154508
$401$	−29.2705	−1.46170	−0.730850	−	0.682538i	$-0.760876\pi$
$401$	−29.2705	−1.46170	−0.730850	+	0.682538i	$0.760876\pi$
$402$	0	0
$403$	1.47214	0.0733323
$404$	−6.09017	−0.302997
$405$	0	0
$406$	−1.38197	−0.0685858
$407$	19.9443	0.988601
$408$	0	0
$409$	19.6525	0.971752	0.485876	−	0.874028i	$-0.338501\pi$
$409$	19.6525	0.971752	0.485876	+	0.874028i	$0.338501\pi$
$410$	−7.88854	−0.389587
$411$	0	0
$412$	4.14590	0.204254
$413$	−3.20163	−0.157542
$414$	0	0
$415$	1.05573	0.0518237
$416$	1.00000	0.0490290
$417$	0	0
$418$	−8.47214	−0.414386
$419$	37.5623	1.83504	0.917519	−	0.397691i	$-0.130188\pi$
$419$	37.5623	1.83504	0.917519	+	0.397691i	$0.130188\pi$
$420$	0	0
$421$	2.52786	0.123201	0.0616003	−	0.998101i	$-0.480380\pi$
$421$	2.52786	0.123201	0.0616003	+	0.998101i	$0.480380\pi$
$422$	5.85410	0.284973
$423$	0	0
$424$	0	0
$425$	−13.0902	−0.634967
$426$	0	0
$427$	0.819660	0.0396661
$428$	7.76393	0.375284
$429$	0	0
$430$	3.09017	0.149021
$431$	6.52786	0.314436	0.157218	−	0.987564i	$-0.449747\pi$
$431$	6.52786	0.314436	0.157218	+	0.987564i	$0.449747\pi$
$432$	0	0
$433$	36.7984	1.76842	0.884208	−	0.467092i	$-0.154698\pi$
$433$	36.7984	1.76842	0.884208	+	0.467092i	$0.154698\pi$
$434$	0.347524	0.0166817
$435$	0	0
$436$	8.70820	0.417047
$437$	7.52786	0.360107
$438$	0	0
$439$	25.8541	1.23395	0.616974	−	0.786983i	$-0.288358\pi$
$439$	25.8541	1.23395	0.616974	+	0.786983i	$0.288358\pi$
$440$	−5.85410	−0.279083
$441$	0	0
$442$	4.23607	0.201489
$443$	−6.58359	−0.312796	−0.156398	−	0.987694i	$-0.549988\pi$
$443$	−6.58359	−0.312796	−0.156398	+	0.987694i	$0.549988\pi$
$444$	0	0
$445$	−20.4508	−0.969463
$446$	−10.3262	−0.488962
$447$	0	0
$448$	0.236068	0.0111532
$449$	−14.0902	−0.664956	−0.332478	−	0.943111i	$-0.607885\pi$
$449$	−14.0902	−0.664956	−0.332478	+	0.943111i	$0.607885\pi$
$450$	0	0
$451$	24.1803	1.13861
$452$	17.6180	0.828683
$453$	0	0
$454$	5.03444	0.236278
$455$	−0.326238	−0.0152943
$456$	0	0
$457$	2.20163	0.102988	0.0514939	−	0.998673i	$-0.483602\pi$
$457$	2.20163	0.102988	0.0514939	+	0.998673i	$0.483602\pi$
$458$	−2.70820	−0.126546
$459$	0	0
$460$	5.20163	0.242527
$461$	19.1246	0.890722	0.445361	−	0.895351i	$-0.353075\pi$
$461$	19.1246	0.890722	0.445361	+	0.895351i	$0.353075\pi$
$462$	0	0
$463$	−8.90983	−0.414075	−0.207037	−	0.978333i	$-0.566382\pi$
$463$	−8.90983	−0.414075	−0.207037	+	0.978333i	$0.566382\pi$
$464$	5.85410	0.271770
$465$	0	0
$466$	−0.236068	−0.0109356
$467$	33.4164	1.54633	0.773163	−	0.634207i	$-0.218673\pi$
$467$	33.4164	1.54633	0.773163	+	0.634207i	$0.218673\pi$
$468$	0	0
$469$	1.47214	0.0679769
$470$	−0.201626	−0.00930032
$471$	0	0
$472$	13.5623	0.624256
$473$	−9.47214	−0.435529
$474$	0	0
$475$	−6.18034	−0.283573
$476$	1.00000	0.0458349
$477$	0	0
$478$	14.2705	0.652718
$479$	−12.5967	−0.575560	−0.287780	−	0.957697i	$-0.592917\pi$
$479$	−12.5967	−0.575560	−0.287780	+	0.957697i	$0.592917\pi$
$480$	0	0
$481$	−4.70820	−0.214676
$482$	−8.70820	−0.396648
$483$	0	0
$484$	6.94427	0.315649
$485$	−25.7771	−1.17048
$486$	0	0
$487$	38.8328	1.75968	0.879841	−	0.475267i	$-0.157649\pi$
$487$	38.8328	1.75968	0.879841	+	0.475267i	$0.157649\pi$
$488$	−3.47214	−0.157176
$489$	0	0
$490$	9.59675	0.433537
$491$	−11.2361	−0.507077	−0.253538	−	0.967325i	$-0.581594\pi$
$491$	−11.2361	−0.507077	−0.253538	+	0.967325i	$0.581594\pi$
$492$	0	0
$493$	24.7984	1.11686
$494$	2.00000	0.0899843
$495$	0	0
$496$	−1.47214	−0.0661009
$497$	0.875388	0.0392665
$498$	0	0
$499$	−5.94427	−0.266102	−0.133051	−	0.991109i	$-0.542477\pi$
$499$	−5.94427	−0.266102	−0.133051	+	0.991109i	$0.542477\pi$
$500$	−11.1803	−0.500000
$501$	0	0
$502$	12.9098	0.576194
$503$	10.3607	0.461960	0.230980	−	0.972959i	$-0.425807\pi$
$503$	10.3607	0.461960	0.230980	+	0.972959i	$0.425807\pi$
$504$	0	0
$505$	−8.41641	−0.374525
$506$	−15.9443	−0.708809
$507$	0	0
$508$	−3.79837	−0.168526
$509$	−42.3262	−1.87608	−0.938039	−	0.346530i	$-0.887360\pi$
$509$	−42.3262	−1.87608	−0.938039	+	0.346530i	$0.887360\pi$
$510$	0	0
$511$	0.639320	0.0282819
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	3.47214	0.153149
$515$	5.72949	0.252472
$516$	0	0
$517$	0.618034	0.0271811
$518$	−1.11146	−0.0488346
$519$	0	0
$520$	1.38197	0.0606032
$521$	−3.58359	−0.157000	−0.0785000	−	0.996914i	$-0.525013\pi$
$521$	−3.58359	−0.157000	−0.0785000	+	0.996914i	$0.525013\pi$
$522$	0	0
$523$	25.3820	1.10988	0.554938	−	0.831892i	$-0.312742\pi$
$523$	25.3820	1.10988	0.554938	+	0.831892i	$0.312742\pi$
$524$	−7.23607	−0.316109
$525$	0	0
$526$	−10.2705	−0.447816
$527$	−6.23607	−0.271647
$528$	0	0
$529$	−8.83282	−0.384035
$530$	0	0
$531$	0	0
$532$	0.472136	0.0204697
$533$	−5.70820	−0.247250
$534$	0	0
$535$	10.7295	0.463876
$536$	−6.23607	−0.269357
$537$	0	0
$538$	1.00000	0.0431131
$539$	−29.4164	−1.26705
$540$	0	0
$541$	11.5279	0.495622	0.247811	−	0.968808i	$-0.420289\pi$
$541$	11.5279	0.495622	0.247811	+	0.968808i	$0.420289\pi$
$542$	16.1246	0.692611
$543$	0	0
$544$	−4.23607	−0.181620
$545$	12.0344	0.515499
$546$	0	0
$547$	27.7984	1.18857	0.594286	−	0.804254i	$-0.297435\pi$
$547$	27.7984	1.18857	0.594286	+	0.804254i	$0.297435\pi$
$548$	13.3820	0.571649
$549$	0	0
$550$	13.0902	0.558167
$551$	11.7082	0.498786
$552$	0	0
$553$	2.54915	0.108401
$554$	23.6180	1.00343
$555$	0	0
$556$	−9.56231	−0.405532
$557$	−41.7082	−1.76723	−0.883617	−	0.468211i	$-0.844899\pi$
$557$	−41.7082	−1.76723	−0.883617	+	0.468211i	$0.844899\pi$
$558$	0	0
$559$	2.23607	0.0945756
$560$	0.326238	0.0137861
$561$	0	0
$562$	−6.81966	−0.287670
$563$	−20.7082	−0.872747	−0.436373	−	0.899766i	$-0.643737\pi$
$563$	−20.7082	−0.872747	−0.436373	+	0.899766i	$0.643737\pi$
$564$	0	0
$565$	24.3475	1.02431
$566$	−10.2361	−0.430254
$567$	0	0
$568$	−3.70820	−0.155593
$569$	17.0902	0.716457	0.358229	−	0.933634i	$-0.383381\pi$
$569$	17.0902	0.716457	0.358229	+	0.933634i	$0.383381\pi$
$570$	0	0
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	−4.23607	−0.177119
$573$	0	0
$574$	−1.34752	−0.0562446
$575$	−11.6312	−0.485054
$576$	0	0
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	−0.944272	−0.0392765
$579$	0	0
$580$	8.09017	0.335926
$581$	0.180340	0.00748176
$582$	0	0
$583$	0	0
$584$	−2.70820	−0.112066
$585$	0	0
$586$	16.4721	0.680458
$587$	26.3262	1.08660	0.543300	−	0.839539i	$-0.317175\pi$
$587$	26.3262	1.08660	0.543300	+	0.839539i	$0.317175\pi$
$588$	0	0
$589$	−2.94427	−0.121317
$590$	18.7426	0.771623
$591$	0	0
$592$	4.70820	0.193506
$593$	27.4164	1.12586	0.562928	−	0.826506i	$-0.309675\pi$
$593$	27.4164	1.12586	0.562928	+	0.826506i	$0.309675\pi$
$594$	0	0
$595$	1.38197	0.0566551
$596$	15.3262	0.627787
$597$	0	0
$598$	3.76393	0.153919
$599$	−33.5967	−1.37273	−0.686363	−	0.727259i	$-0.740794\pi$
$599$	−33.5967	−1.37273	−0.686363	+	0.727259i	$0.740794\pi$
$600$	0	0
$601$	34.6869	1.41491	0.707454	−	0.706759i	$-0.249843\pi$
$601$	34.6869	1.41491	0.707454	+	0.706759i	$0.249843\pi$
$602$	0.527864	0.0215141
$603$	0	0
$604$	7.94427	0.323248
$605$	9.59675	0.390163
$606$	0	0
$607$	−48.5755	−1.97162	−0.985809	−	0.167873i	$-0.946310\pi$
$607$	−48.5755	−1.97162	−0.985809	+	0.167873i	$0.946310\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	−4.79837	−0.194280
$611$	−0.145898	−0.00590240
$612$	0	0
$613$	−35.4721	−1.43271	−0.716353	−	0.697738i	$-0.754190\pi$
$613$	−35.4721	−1.43271	−0.716353	+	0.697738i	$0.754190\pi$
$614$	8.12461	0.327883
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	33.1591	1.33493	0.667467	−	0.744640i	$-0.267379\pi$
$617$	33.1591	1.33493	0.667467	+	0.744640i	$0.267379\pi$
$618$	0	0
$619$	12.8328	0.515794	0.257897	−	0.966172i	$-0.416970\pi$
$619$	12.8328	0.515794	0.257897	+	0.966172i	$0.416970\pi$
$620$	−2.03444	−0.0817052
$621$	0	0
$622$	−12.7639	−0.511787
$623$	−3.49342	−0.139961
$624$	0	0
$625$	0	0
$626$	−19.8541	−0.793530
$627$	0	0
$628$	−8.65248	−0.345271
$629$	19.9443	0.795230
$630$	0	0
$631$	29.7984	1.18625	0.593127	−	0.805109i	$-0.297893\pi$
$631$	29.7984	1.18625	0.593127	+	0.805109i	$0.297893\pi$
$632$	−10.7984	−0.429536
$633$	0	0
$634$	−25.7639	−1.02322
$635$	−5.24922	−0.208309
$636$	0	0
$637$	6.94427	0.275142
$638$	−24.7984	−0.981777
$639$	0	0
$640$	−1.38197	−0.0546270
$641$	19.1803	0.757578	0.378789	−	0.925483i	$-0.376341\pi$
$641$	19.1803	0.757578	0.378789	+	0.925483i	$0.376341\pi$
$642$	0	0
$643$	−10.8541	−0.428044	−0.214022	−	0.976829i	$-0.568656\pi$
$643$	−10.8541	−0.428044	−0.214022	+	0.976829i	$0.568656\pi$
$644$	0.888544	0.0350135
$645$	0	0
$646$	−8.47214	−0.333332
$647$	25.1803	0.989941	0.494971	−	0.868910i	$-0.335179\pi$
$647$	25.1803	0.989941	0.494971	+	0.868910i	$0.335179\pi$
$648$	0	0
$649$	−57.4508	−2.25514
$650$	−3.09017	−0.121206
$651$	0	0
$652$	1.67376	0.0655496
$653$	−39.5410	−1.54736	−0.773680	−	0.633577i	$-0.781586\pi$
$653$	−39.5410	−1.54736	−0.773680	+	0.633577i	$0.781586\pi$
$654$	0	0
$655$	−10.0000	−0.390732
$656$	5.70820	0.222868
$657$	0	0
$658$	−0.0344419	−0.00134268
$659$	15.6738	0.610563	0.305282	−	0.952262i	$-0.401249\pi$
$659$	15.6738	0.610563	0.305282	+	0.952262i	$0.401249\pi$
$660$	0	0
$661$	−14.1459	−0.550212	−0.275106	−	0.961414i	$-0.588713\pi$
$661$	−14.1459	−0.550212	−0.275106	+	0.961414i	$0.588713\pi$
$662$	−23.1803	−0.900929
$663$	0	0
$664$	−0.763932	−0.0296463
$665$	0.652476	0.0253019
$666$	0	0
$667$	22.0344	0.853177
$668$	−1.38197	−0.0534699
$669$	0	0
$670$	−8.61803	−0.332944
$671$	14.7082	0.567804
$672$	0	0
$673$	−2.70820	−0.104394	−0.0521968	−	0.998637i	$-0.516622\pi$
$673$	−2.70820	−0.104394	−0.0521968	+	0.998637i	$0.516622\pi$
$674$	−18.9787	−0.731033
$675$	0	0
$676$	−12.0000	−0.461538
$677$	37.5623	1.44364	0.721818	−	0.692083i	$-0.243307\pi$
$677$	37.5623	1.44364	0.721818	+	0.692083i	$0.243307\pi$
$678$	0	0
$679$	−4.40325	−0.168981
$680$	−5.85410	−0.224495
$681$	0	0
$682$	6.23607	0.238791
$683$	16.7639	0.641454	0.320727	−	0.947172i	$-0.396073\pi$
$683$	16.7639	0.641454	0.320727	+	0.947172i	$0.396073\pi$
$684$	0	0
$685$	18.4934	0.706597
$686$	3.29180	0.125681
$687$	0	0
$688$	−2.23607	−0.0852493
$689$	0	0
$690$	0	0
$691$	25.1246	0.955785	0.477893	−	0.878418i	$-0.341401\pi$
$691$	25.1246	0.955785	0.477893	+	0.878418i	$0.341401\pi$
$692$	−1.47214	−0.0559622
$693$	0	0
$694$	−31.3607	−1.19044
$695$	−13.2148	−0.501265
$696$	0	0
$697$	24.1803	0.915896
$698$	12.2705	0.464446
$699$	0	0
$700$	−0.729490	−0.0275721
$701$	−35.9787	−1.35890	−0.679449	−	0.733723i	$-0.737781\pi$
$701$	−35.9787	−1.35890	−0.679449	+	0.733723i	$0.737781\pi$
$702$	0	0
$703$	9.41641	0.355147
$704$	4.23607	0.159653
$705$	0	0
$706$	−3.14590	−0.118397
$707$	−1.43769	−0.0540701
$708$	0	0
$709$	−20.0557	−0.753209	−0.376604	−	0.926374i	$-0.622908\pi$
$709$	−20.0557	−0.753209	−0.376604	+	0.926374i	$0.622908\pi$
$710$	−5.12461	−0.192323
$711$	0	0
$712$	14.7984	0.554593
$713$	−5.54102	−0.207513
$714$	0	0
$715$	−5.85410	−0.218931
$716$	−5.29180	−0.197764
$717$	0	0
$718$	0.944272	0.0352399
$719$	−19.8197	−0.739149	−0.369574	−	0.929201i	$-0.620496\pi$
$719$	−19.8197	−0.739149	−0.369574	+	0.929201i	$0.620496\pi$
$720$	0	0
$721$	0.978714	0.0364492
$722$	15.0000	0.558242
$723$	0	0
$724$	−7.41641	−0.275629
$725$	−18.0902	−0.671852
$726$	0	0
$727$	−31.2918	−1.16055	−0.580274	−	0.814421i	$-0.697055\pi$
$727$	−31.2918	−1.16055	−0.580274	+	0.814421i	$0.697055\pi$
$728$	0.236068	0.00874926
$729$	0	0
$730$	−3.74265	−0.138522
$731$	−9.47214	−0.350340
$732$	0	0
$733$	11.3607	0.419616	0.209808	−	0.977743i	$-0.432716\pi$
$733$	11.3607	0.419616	0.209808	+	0.977743i	$0.432716\pi$
$734$	33.7984	1.24752
$735$	0	0
$736$	−3.76393	−0.138740
$737$	26.4164	0.973061
$738$	0	0
$739$	5.96556	0.219447	0.109723	−	0.993962i	$-0.465004\pi$
$739$	5.96556	0.219447	0.109723	+	0.993962i	$0.465004\pi$
$740$	6.50658	0.239187
$741$	0	0
$742$	0	0
$743$	33.7984	1.23994	0.619971	−	0.784625i	$-0.287144\pi$
$743$	33.7984	1.23994	0.619971	+	0.784625i	$0.287144\pi$
$744$	0	0
$745$	21.1803	0.775988
$746$	10.9443	0.400698
$747$	0	0
$748$	17.9443	0.656108
$749$	1.83282	0.0669696
$750$	0	0
$751$	−42.8673	−1.56425	−0.782124	−	0.623123i	$-0.785864\pi$
$751$	−42.8673	−1.56425	−0.782124	+	0.623123i	$0.785864\pi$
$752$	0.145898	0.00532035
$753$	0	0
$754$	5.85410	0.213194
$755$	10.9787	0.399556
$756$	0	0
$757$	−21.0557	−0.765283	−0.382642	−	0.923897i	$-0.624986\pi$
$757$	−21.0557	−0.765283	−0.382642	+	0.923897i	$0.624986\pi$
$758$	−12.8541	−0.466882
$759$	0	0
$760$	−2.76393	−0.100258
$761$	−25.4164	−0.921344	−0.460672	−	0.887570i	$-0.652392\pi$
$761$	−25.4164	−0.921344	−0.460672	+	0.887570i	$0.652392\pi$
$762$	0	0
$763$	2.05573	0.0744224
$764$	22.9443	0.830095
$765$	0	0
$766$	−19.6180	−0.708828
$767$	13.5623	0.489706
$768$	0	0
$769$	−17.6738	−0.637332	−0.318666	−	0.947867i	$-0.603235\pi$
$769$	−17.6738	−0.637332	−0.318666	+	0.947867i	$0.603235\pi$
$770$	−1.38197	−0.0498026
$771$	0	0
$772$	−3.76393	−0.135467
$773$	1.59675	0.0574310	0.0287155	−	0.999588i	$-0.490858\pi$
$773$	1.59675	0.0574310	0.0287155	+	0.999588i	$0.490858\pi$
$774$	0	0
$775$	4.54915	0.163410
$776$	18.6525	0.669585
$777$	0	0
$778$	11.0557	0.396367
$779$	11.4164	0.409035
$780$	0	0
$781$	15.7082	0.562084
$782$	−15.9443	−0.570166
$783$	0	0
$784$	−6.94427	−0.248010
$785$	−11.9574	−0.426779
$786$	0	0
$787$	−33.6525	−1.19958	−0.599791	−	0.800157i	$-0.704749\pi$
$787$	−33.6525	−1.19958	−0.599791	+	0.800157i	$0.704749\pi$
$788$	24.6180	0.876981
$789$	0	0
$790$	−14.9230	−0.530936
$791$	4.15905	0.147879
$792$	0	0
$793$	−3.47214	−0.123299
$794$	0.729490	0.0258886
$795$	0	0
$796$	−7.61803	−0.270014
$797$	−37.7984	−1.33889	−0.669444	−	0.742863i	$-0.733467\pi$
$797$	−37.7984	−1.33889	−0.669444	+	0.742863i	$0.733467\pi$
$798$	0	0
$799$	0.618034	0.0218645
$800$	3.09017	0.109254
$801$	0	0
$802$	29.2705	1.03358
$803$	11.4721	0.404843
$804$	0	0
$805$	1.22794	0.0432791
$806$	−1.47214	−0.0518538
$807$	0	0
$808$	6.09017	0.214251
$809$	−24.4721	−0.860394	−0.430197	−	0.902735i	$-0.641556\pi$
$809$	−24.4721	−0.860394	−0.430197	+	0.902735i	$0.641556\pi$
$810$	0	0
$811$	1.59675	0.0560694	0.0280347	−	0.999607i	$-0.491075\pi$
$811$	1.59675	0.0560694	0.0280347	+	0.999607i	$0.491075\pi$
$812$	1.38197	0.0484975
$813$	0	0
$814$	−19.9443	−0.699046
$815$	2.31308	0.0810237
$816$	0	0
$817$	−4.47214	−0.156460
$818$	−19.6525	−0.687133
$819$	0	0
$820$	7.88854	0.275480
$821$	6.36068	0.221989	0.110995	−	0.993821i	$-0.464596\pi$
$821$	6.36068	0.221989	0.110995	+	0.993821i	$0.464596\pi$
$822$	0	0
$823$	−15.1115	−0.526752	−0.263376	−	0.964693i	$-0.584836\pi$
$823$	−15.1115	−0.526752	−0.263376	+	0.964693i	$0.584836\pi$
$824$	−4.14590	−0.144429
$825$	0	0
$826$	3.20163	0.111399
$827$	−4.74265	−0.164918	−0.0824590	−	0.996594i	$-0.526277\pi$
$827$	−4.74265	−0.164918	−0.0824590	+	0.996594i	$0.526277\pi$
$828$	0	0
$829$	19.2016	0.666900	0.333450	−	0.942768i	$-0.391787\pi$
$829$	19.2016	0.666900	0.333450	+	0.942768i	$0.391787\pi$
$830$	−1.05573	−0.0366449
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	−29.4164	−1.01922
$834$	0	0
$835$	−1.90983	−0.0660924
$836$	8.47214	0.293015
$837$	0	0
$838$	−37.5623	−1.29757
$839$	41.4508	1.43104	0.715521	−	0.698591i	$-0.246189\pi$
$839$	41.4508	1.43104	0.715521	+	0.698591i	$0.246189\pi$
$840$	0	0
$841$	5.27051	0.181742
$842$	−2.52786	−0.0871159
$843$	0	0
$844$	−5.85410	−0.201506
$845$	−16.5836	−0.570493
$846$	0	0
$847$	1.63932	0.0563277
$848$	0	0
$849$	0	0
$850$	13.0902	0.448989
$851$	17.7214	0.607480
$852$	0	0
$853$	−48.8115	−1.67127	−0.835637	−	0.549281i	$-0.814902\pi$
$853$	−48.8115	−1.67127	−0.835637	+	0.549281i	$0.814902\pi$
$854$	−0.819660	−0.0280482
$855$	0	0
$856$	−7.76393	−0.265366
$857$	−16.0689	−0.548903	−0.274451	−	0.961601i	$-0.588496\pi$
$857$	−16.0689	−0.548903	−0.274451	+	0.961601i	$0.588496\pi$
$858$	0	0
$859$	29.7426	1.01481	0.507403	−	0.861709i	$-0.330606\pi$
$859$	29.7426	1.01481	0.507403	+	0.861709i	$0.330606\pi$
$860$	−3.09017	−0.105374
$861$	0	0
$862$	−6.52786	−0.222340
$863$	−1.67376	−0.0569755	−0.0284878	−	0.999594i	$-0.509069\pi$
$863$	−1.67376	−0.0569755	−0.0284878	+	0.999594i	$0.509069\pi$
$864$	0	0
$865$	−2.03444	−0.0691731
$866$	−36.7984	−1.25046
$867$	0	0
$868$	−0.347524	−0.0117957
$869$	45.7426	1.55171
$870$	0	0
$871$	−6.23607	−0.211301
$872$	−8.70820	−0.294897
$873$	0	0
$874$	−7.52786	−0.254634
$875$	−2.63932	−0.0892253
$876$	0	0
$877$	−53.5755	−1.80911	−0.904557	−	0.426352i	$-0.859799\pi$
$877$	−53.5755	−1.80911	−0.904557	+	0.426352i	$0.859799\pi$
$878$	−25.8541	−0.872534
$879$	0	0
$880$	5.85410	0.197342
$881$	6.49342	0.218769	0.109384	−	0.994000i	$-0.465112\pi$
$881$	6.49342	0.218769	0.109384	+	0.994000i	$0.465112\pi$
$882$	0	0
$883$	48.1935	1.62184	0.810920	−	0.585157i	$-0.198967\pi$
$883$	48.1935	1.62184	0.810920	+	0.585157i	$0.198967\pi$
$884$	−4.23607	−0.142474
$885$	0	0
$886$	6.58359	0.221180
$887$	−39.8328	−1.33746	−0.668728	−	0.743508i	$-0.733161\pi$
$887$	−39.8328	−1.33746	−0.668728	+	0.743508i	$0.733161\pi$
$888$	0	0
$889$	−0.896674	−0.0300735
$890$	20.4508	0.685514
$891$	0	0
$892$	10.3262	0.345748
$893$	0.291796	0.00976458
$894$	0	0
$895$	−7.31308	−0.244449
$896$	−0.236068	−0.00788648
$897$	0	0
$898$	14.0902	0.470195
$899$	−8.61803	−0.287428
$900$	0	0
$901$	0	0
$902$	−24.1803	−0.805117
$903$	0	0
$904$	−17.6180	−0.585967
$905$	−10.2492	−0.340696
$906$	0	0
$907$	26.1246	0.867453	0.433727	−	0.901044i	$-0.357198\pi$
$907$	26.1246	0.867453	0.433727	+	0.901044i	$0.357198\pi$
$908$	−5.03444	−0.167074
$909$	0	0
$910$	0.326238	0.0108147
$911$	44.3394	1.46903	0.734515	−	0.678593i	$-0.237410\pi$
$911$	44.3394	1.46903	0.734515	+	0.678593i	$0.237410\pi$
$912$	0	0
$913$	3.23607	0.107098
$914$	−2.20163	−0.0728233
$915$	0	0
$916$	2.70820	0.0894816
$917$	−1.70820	−0.0564099
$918$	0	0
$919$	−36.6525	−1.20905	−0.604527	−	0.796585i	$-0.706638\pi$
$919$	−36.6525	−1.20905	−0.604527	+	0.796585i	$0.706638\pi$
$920$	−5.20163	−0.171493
$921$	0	0
$922$	−19.1246	−0.629836
$923$	−3.70820	−0.122057
$924$	0	0
$925$	−14.5492	−0.478373
$926$	8.90983	0.292795
$927$	0	0
$928$	−5.85410	−0.192170
$929$	−40.1803	−1.31827	−0.659137	−	0.752023i	$-0.729078\pi$
$929$	−40.1803	−1.31827	−0.659137	+	0.752023i	$0.729078\pi$
$930$	0	0
$931$	−13.8885	−0.455179
$932$	0.236068	0.00773266
$933$	0	0
$934$	−33.4164	−1.09342
$935$	24.7984	0.810994
$936$	0	0
$937$	−21.9098	−0.715763	−0.357881	−	0.933767i	$-0.616501\pi$
$937$	−21.9098	−0.715763	−0.357881	+	0.933767i	$0.616501\pi$
$938$	−1.47214	−0.0480669
$939$	0	0
$940$	0.201626	0.00657632
$941$	0.0212862	0.000693911 0	0.000346956	−	1.00000i	$-0.499890\pi$
$941$	0.0212862	0.000693911 0	0.000346956		1.00000i	$0.499890\pi$
$942$	0	0
$943$	21.4853	0.699657
$944$	−13.5623	−0.441415
$945$	0	0
$946$	9.47214	0.307966
$947$	−1.21478	−0.0394751	−0.0197376	−	0.999805i	$-0.506283\pi$
$947$	−1.21478	−0.0394751	−0.0197376	+	0.999805i	$0.506283\pi$
$948$	0	0
$949$	−2.70820	−0.0879120
$950$	6.18034	0.200517
$951$	0	0
$952$	−1.00000	−0.0324102
$953$	37.3050	1.20843	0.604213	−	0.796823i	$-0.293488\pi$
$953$	37.3050	1.20843	0.604213	+	0.796823i	$0.293488\pi$
$954$	0	0
$955$	31.7082	1.02605
$956$	−14.2705	−0.461541
$957$	0	0
$958$	12.5967	0.406982
$959$	3.15905	0.102011
$960$	0	0
$961$	−28.8328	−0.930091
$962$	4.70820	0.151799
$963$	0	0
$964$	8.70820	0.280472
$965$	−5.20163	−0.167446
$966$	0	0
$967$	−4.85410	−0.156097	−0.0780487	−	0.996950i	$-0.524869\pi$
$967$	−4.85410	−0.156097	−0.0780487	+	0.996950i	$0.524869\pi$
$968$	−6.94427	−0.223197
$969$	0	0
$970$	25.7771	0.827652
$971$	−30.6525	−0.983685	−0.491842	−	0.870684i	$-0.663676\pi$
$971$	−30.6525	−0.983685	−0.491842	+	0.870684i	$0.663676\pi$
$972$	0	0
$973$	−2.25735	−0.0723675
$974$	−38.8328	−1.24428
$975$	0	0
$976$	3.47214	0.111140
$977$	−37.8541	−1.21106	−0.605530	−	0.795822i	$-0.707039\pi$
$977$	−37.8541	−1.21106	−0.605530	+	0.795822i	$0.707039\pi$
$978$	0	0
$979$	−62.6869	−2.00348
$980$	−9.59675	−0.306557
$981$	0	0
$982$	11.2361	0.358557
$983$	51.5623	1.64458	0.822291	−	0.569067i	$-0.192696\pi$
$983$	51.5623	1.64458	0.822291	+	0.569067i	$0.192696\pi$
$984$	0	0
$985$	34.0213	1.08401
$986$	−24.7984	−0.789741
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	−8.41641	−0.267626
$990$	0	0
$991$	−18.6180	−0.591421	−0.295711	−	0.955278i	$-0.595556\pi$
$991$	−18.6180	−0.591421	−0.295711	+	0.955278i	$0.595556\pi$
$992$	1.47214	0.0467404
$993$	0	0
$994$	−0.875388	−0.0277656
$995$	−10.5279	−0.333756
$996$	0	0
$997$	−44.4164	−1.40668	−0.703341	−	0.710853i	$-0.748309\pi$
$997$	−44.4164	−1.40668	−0.703341	+	0.710853i	$0.748309\pi$
$998$	5.94427	0.188163
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.f.1.1		2
3.2	odd	2		538.2.a.a.1.1	✓	2
12.11	even	2		4304.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.a.1.1	✓	2	3.2	odd	2
4304.2.a.d.1.2		2	12.11	even	2
4842.2.a.f.1.1		2	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:	\( N \)	\(=\)	\( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4842.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.38197 q^{5} +0.236068 q^{7} -1.00000 q^{8} -1.38197 q^{10} +4.23607 q^{11} -1.00000 q^{13} -0.236068 q^{14} +1.00000 q^{16} +4.23607 q^{17} +2.00000 q^{19} +1.38197 q^{20} -4.23607 q^{22} +3.76393 q^{23} -3.09017 q^{25} +1.00000 q^{26} +0.236068 q^{28} +5.85410 q^{29} -1.47214 q^{31} -1.00000 q^{32} -4.23607 q^{34} +0.326238 q^{35} +4.70820 q^{37} -2.00000 q^{38} -1.38197 q^{40} +5.70820 q^{41} -2.23607 q^{43} +4.23607 q^{44} -3.76393 q^{46} +0.145898 q^{47} -6.94427 q^{49} +3.09017 q^{50} -1.00000 q^{52} +5.85410 q^{55} -0.236068 q^{56} -5.85410 q^{58} -13.5623 q^{59} +3.47214 q^{61} +1.47214 q^{62} +1.00000 q^{64} -1.38197 q^{65} +6.23607 q^{67} +4.23607 q^{68} -0.326238 q^{70} +3.70820 q^{71} +2.70820 q^{73} -4.70820 q^{74} +2.00000 q^{76} +1.00000 q^{77} +10.7984 q^{79} +1.38197 q^{80} -5.70820 q^{82} +0.763932 q^{83} +5.85410 q^{85} +2.23607 q^{86} -4.23607 q^{88} -14.7984 q^{89} -0.236068 q^{91} +3.76393 q^{92} -0.145898 q^{94} +2.76393 q^{95} -18.6525 q^{97} +6.94427 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 5 q^{5} - 4 q^{7} - 2 q^{8} - 5 q^{10} + 4 q^{11} - 2 q^{13} + 4 q^{14} + 2 q^{16} + 4 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 12 q^{23} + 5 q^{25} + 2 q^{26} - 4 q^{28} + 5 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 5 * q^5 - 4 * q^7 - 2 * q^8 - 5 * q^10 + 4 * q^11 - 2 * q^13 + 4 * q^14 + 2 * q^16 + 4 * q^17 + 4 * q^19 + 5 * q^20 - 4 * q^22 + 12 * q^23 + 5 * q^25 + 2 * q^26 - 4 * q^28 + 5 * q^29 + 6 * q^31 - 2 * q^32 - 4 * q^34 - 15 * q^35 - 4 * q^37 - 4 * q^38 - 5 * q^40 - 2 * q^41 + 4 * q^44 - 12 * q^46 + 7 * q^47 + 4 * q^49 - 5 * q^50 - 2 * q^52 + 5 * q^55 + 4 * q^56 - 5 * q^58 - 7 * q^59 - 2 * q^61 - 6 * q^62 + 2 * q^64 - 5 * q^65 + 8 * q^67 + 4 * q^68 + 15 * q^70 - 6 * q^71 - 8 * q^73 + 4 * q^74 + 4 * q^76 + 2 * q^77 - 3 * q^79 + 5 * q^80 + 2 * q^82 + 6 * q^83 + 5 * q^85 - 4 * q^88 - 5 * q^89 + 4 * q^91 + 12 * q^92 - 7 * q^94 + 10 * q^95 - 6 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

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Database

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