LMFDB - Embedded newform 4842.2.a.q.1.8

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 33x^{6} + 352x^{4} - 18x^{3} - 1229x^{2} + 178x + 108 $ x^8 - 33x^6 + 352x^4 - 18x^3 - 1229x^2 + 178*x + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1614)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-3.72348$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.72348 q^{5} -3.69765 q^{7} +1.00000 q^{8} +3.72348 q^{10} -2.16938 q^{11} -1.78302 q^{13} -3.69765 q^{14} +1.00000 q^{16} +4.84900 q^{17} +6.84350 q^{19} +3.72348 q^{20} -2.16938 q^{22} -0.00549183 q^{23} +8.86432 q^{25} -1.78302 q^{26} -3.69765 q^{28} +8.82870 q^{29} -0.462161 q^{31} +1.00000 q^{32} +4.84900 q^{34} -13.7681 q^{35} -7.69250 q^{37} +6.84350 q^{38} +3.72348 q^{40} +1.22462 q^{41} +3.90914 q^{43} -2.16938 q^{44} -0.00549183 q^{46} -5.57907 q^{47} +6.67264 q^{49} +8.86432 q^{50} -1.78302 q^{52} -4.05603 q^{53} -8.07763 q^{55} -3.69765 q^{56} +8.82870 q^{58} -11.0037 q^{59} +12.9778 q^{61} -0.462161 q^{62} +1.00000 q^{64} -6.63904 q^{65} +11.0632 q^{67} +4.84900 q^{68} -13.7681 q^{70} -3.48703 q^{71} +14.0047 q^{73} -7.69250 q^{74} +6.84350 q^{76} +8.02160 q^{77} +8.36670 q^{79} +3.72348 q^{80} +1.22462 q^{82} -1.08042 q^{83} +18.0551 q^{85} +3.90914 q^{86} -2.16938 q^{88} -7.03265 q^{89} +6.59299 q^{91} -0.00549183 q^{92} -5.57907 q^{94} +25.4817 q^{95} +2.47385 q^{97} +6.67264 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} + 9 q^{7} + 8 q^{8} - 8 q^{11} - 3 q^{13} + 9 q^{14} + 8 q^{16} - 8 q^{17} + 18 q^{19} - 8 q^{22} + 10 q^{23} + 26 q^{25} - 3 q^{26} + 9 q^{28} + 9 q^{29} + 18 q^{31} + 8 q^{32}+ \cdots + 29 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 + 9 * q^7 + 8 * q^8 - 8 * q^11 - 3 * q^13 + 9 * q^14 + 8 * q^16 - 8 * q^17 + 18 * q^19 - 8 * q^22 + 10 * q^23 + 26 * q^25 - 3 * q^26 + 9 * q^28 + 9 * q^29 + 18 * q^31 + 8 * q^32 - 8 * q^34 - 9 * q^35 + 22 * q^37 + 18 * q^38 + q^41 + 14 * q^43 - 8 * q^44 + 10 * q^46 - 6 * q^47 + 29 * q^49 + 26 * q^50 - 3 * q^52 + 3 * q^53 + 17 * q^55 + 9 * q^56 + 9 * q^58 - q^59 + 18 * q^61 + 18 * q^62 + 8 * q^64 + 14 * q^65 + 30 * q^67 - 8 * q^68 - 9 * q^70 + 5 * q^71 + 15 * q^73 + 22 * q^74 + 18 * q^76 + 20 * q^77 + 21 * q^79 + q^82 + 11 * q^83 + 18 * q^85 + 14 * q^86 - 8 * q^88 - 22 * q^89 + 53 * q^91 + 10 * q^92 - 6 * q^94 + 9 * q^95 + 13 * q^97 + 29 * 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
$5$	3.72348	1.66519	0.832596	−	0.553881i	$-0.186854\pi$
$5$	3.72348	1.66519	0.832596	+	0.553881i	$0.186854\pi$
$6$	0	0
$7$	−3.69765	−1.39758	−0.698791	−	0.715326i	$-0.746278\pi$
$7$	−3.69765	−1.39758	−0.698791	+	0.715326i	$0.746278\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.72348	1.17747
$11$	−2.16938	−0.654091	−0.327046	−	0.945009i	$-0.606053\pi$
$11$	−2.16938	−0.654091	−0.327046	+	0.945009i	$0.606053\pi$
$12$	0	0
$13$	−1.78302	−0.494521	−0.247260	−	0.968949i	$-0.579530\pi$
$13$	−1.78302	−0.494521	−0.247260	+	0.968949i	$0.579530\pi$
$14$	−3.69765	−0.988239
$15$	0	0
$16$	1.00000	0.250000
$17$	4.84900	1.17605	0.588027	−	0.808841i	$-0.299905\pi$
$17$	4.84900	1.17605	0.588027	+	0.808841i	$0.299905\pi$
$18$	0	0
$19$	6.84350	1.57001	0.785004	−	0.619491i	$-0.212661\pi$
$19$	6.84350	1.57001	0.785004	+	0.619491i	$0.212661\pi$
$20$	3.72348	0.832596
$21$	0	0
$22$	−2.16938	−0.462512
$23$	−0.00549183	−0.00114513	−0.000572563	−	1.00000i	$-0.500182\pi$
$23$	−0.00549183	−0.00114513	−0.000572563		1.00000i	$0.500182\pi$
$24$	0	0
$25$	8.86432	1.77286
$26$	−1.78302	−0.349679
$27$	0	0
$28$	−3.69765	−0.698791
$29$	8.82870	1.63945	0.819725	−	0.572758i	$-0.194126\pi$
$29$	8.82870	1.63945	0.819725	+	0.572758i	$0.194126\pi$
$30$	0	0
$31$	−0.462161	−0.0830066	−0.0415033	−	0.999138i	$-0.513215\pi$
$31$	−0.462161	−0.0830066	−0.0415033	+	0.999138i	$0.513215\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.84900	0.831596
$35$	−13.7681	−2.32724
$36$	0	0
$37$	−7.69250	−1.26464	−0.632320	−	0.774708i	$-0.717897\pi$
$37$	−7.69250	−1.26464	−0.632320	+	0.774708i	$0.717897\pi$
$38$	6.84350	1.11016
$39$	0	0
$40$	3.72348	0.588734
$41$	1.22462	0.191254	0.0956269	−	0.995417i	$-0.469514\pi$
$41$	1.22462	0.191254	0.0956269	+	0.995417i	$0.469514\pi$
$42$	0	0
$43$	3.90914	0.596139	0.298069	−	0.954544i	$-0.403657\pi$
$43$	3.90914	0.596139	0.298069	+	0.954544i	$0.403657\pi$
$44$	−2.16938	−0.327046
$45$	0	0
$46$	−0.00549183	−0.000809726 0
$47$	−5.57907	−0.813791	−0.406896	−	0.913475i	$-0.633389\pi$
$47$	−5.57907	−0.813791	−0.406896	+	0.913475i	$0.633389\pi$
$48$	0	0
$49$	6.67264	0.953234
$50$	8.86432	1.25360
$51$	0	0
$52$	−1.78302	−0.247260
$53$	−4.05603	−0.557139	−0.278569	−	0.960416i	$-0.589860\pi$
$53$	−4.05603	−0.557139	−0.278569	+	0.960416i	$0.589860\pi$
$54$	0	0
$55$	−8.07763	−1.08919
$56$	−3.69765	−0.494120
$57$	0	0
$58$	8.82870	1.15927
$59$	−11.0037	−1.43256	−0.716282	−	0.697811i	$-0.754158\pi$
$59$	−11.0037	−1.43256	−0.716282	+	0.697811i	$0.754158\pi$
$60$	0	0
$61$	12.9778	1.66164	0.830818	−	0.556544i	$-0.187873\pi$
$61$	12.9778	1.66164	0.830818	+	0.556544i	$0.187873\pi$
$62$	−0.462161	−0.0586945
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.63904	−0.823472
$66$	0	0
$67$	11.0632	1.35158	0.675790	−	0.737094i	$-0.263803\pi$
$67$	11.0632	1.35158	0.675790	+	0.737094i	$0.263803\pi$
$68$	4.84900	0.588027
$69$	0	0
$70$	−13.7681	−1.64561
$71$	−3.48703	−0.413834	−0.206917	−	0.978358i	$-0.566343\pi$
$71$	−3.48703	−0.413834	−0.206917	+	0.978358i	$0.566343\pi$
$72$	0	0
$73$	14.0047	1.63912	0.819561	−	0.572992i	$-0.194218\pi$
$73$	14.0047	1.63912	0.819561	+	0.572992i	$0.194218\pi$
$74$	−7.69250	−0.894235
$75$	0	0
$76$	6.84350	0.785004
$77$	8.02160	0.914146
$78$	0	0
$79$	8.36670	0.941328	0.470664	−	0.882313i	$-0.344014\pi$
$79$	8.36670	0.941328	0.470664	+	0.882313i	$0.344014\pi$
$80$	3.72348	0.416298
$81$	0	0
$82$	1.22462	0.135237
$83$	−1.08042	−0.118591	−0.0592957	−	0.998240i	$-0.518886\pi$
$83$	−1.08042	−0.118591	−0.0592957	+	0.998240i	$0.518886\pi$
$84$	0	0
$85$	18.0551	1.95836
$86$	3.90914	0.421534
$87$	0	0
$88$	−2.16938	−0.231256
$89$	−7.03265	−0.745460	−0.372730	−	0.927940i	$-0.621578\pi$
$89$	−7.03265	−0.745460	−0.372730	+	0.927940i	$0.621578\pi$
$90$	0	0
$91$	6.59299	0.691133
$92$	−0.00549183	−0.000572563 0
$93$	0	0
$94$	−5.57907	−0.575437
$95$	25.4817	2.61436
$96$	0	0
$97$	2.47385	0.251181	0.125591	−	0.992082i	$-0.459917\pi$
$97$	2.47385	0.251181	0.125591	+	0.992082i	$0.459917\pi$
$98$	6.67264	0.674038
$99$	0	0
$100$	8.86432	0.886432
$101$	6.48513	0.645294	0.322647	−	0.946519i	$-0.395427\pi$
$101$	6.48513	0.645294	0.322647	+	0.946519i	$0.395427\pi$
$102$	0	0
$103$	−4.89650	−0.482466	−0.241233	−	0.970467i	$-0.577552\pi$
$103$	−4.89650	−0.482466	−0.241233	+	0.970467i	$0.577552\pi$
$104$	−1.78302	−0.174839
$105$	0	0
$106$	−4.05603	−0.393957
$107$	8.77191	0.848013	0.424006	−	0.905659i	$-0.360623\pi$
$107$	8.77191	0.848013	0.424006	+	0.905659i	$0.360623\pi$
$108$	0	0
$109$	12.5009	1.19737	0.598686	−	0.800984i	$-0.295690\pi$
$109$	12.5009	1.19737	0.598686	+	0.800984i	$0.295690\pi$
$110$	−8.07763	−0.770172
$111$	0	0
$112$	−3.69765	−0.349395
$113$	3.42504	0.322201	0.161100	−	0.986938i	$-0.448496\pi$
$113$	3.42504	0.322201	0.161100	+	0.986938i	$0.448496\pi$
$114$	0	0
$115$	−0.0204487	−0.00190685
$116$	8.82870	0.819725
$117$	0	0
$118$	−11.0037	−1.01298
$119$	−17.9299	−1.64363
$120$	0	0
$121$	−6.29381	−0.572165
$122$	12.9778	1.17495
$123$	0	0
$124$	−0.462161	−0.0415033
$125$	14.3887	1.28697
$126$	0	0
$127$	10.5276	0.934171	0.467086	−	0.884212i	$-0.345304\pi$
$127$	10.5276	0.934171	0.467086	+	0.884212i	$0.345304\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−6.63904	−0.582282
$131$	−7.26762	−0.634975	−0.317488	−	0.948262i	$-0.602839\pi$
$131$	−7.26762	−0.634975	−0.317488	+	0.948262i	$0.602839\pi$
$132$	0	0
$133$	−25.3049	−2.19421
$134$	11.0632	0.955711
$135$	0	0
$136$	4.84900	0.415798
$137$	20.9183	1.78717	0.893587	−	0.448890i	$-0.148180\pi$
$137$	20.9183	1.78717	0.893587	+	0.448890i	$0.148180\pi$
$138$	0	0
$139$	−14.2845	−1.21160	−0.605799	−	0.795618i	$-0.707146\pi$
$139$	−14.2845	−1.21160	−0.605799	+	0.795618i	$0.707146\pi$
$140$	−13.7681	−1.16362
$141$	0	0
$142$	−3.48703	−0.292625
$143$	3.86804	0.323462
$144$	0	0
$145$	32.8735	2.73000
$146$	14.0047	1.15903
$147$	0	0
$148$	−7.69250	−0.632320
$149$	−1.74043	−0.142582	−0.0712908	−	0.997456i	$-0.522712\pi$
$149$	−1.74043	−0.142582	−0.0712908	+	0.997456i	$0.522712\pi$
$150$	0	0
$151$	7.47417	0.608239	0.304120	−	0.952634i	$-0.401638\pi$
$151$	7.47417	0.608239	0.304120	+	0.952634i	$0.401638\pi$
$152$	6.84350	0.555082
$153$	0	0
$154$	8.02160	0.646399
$155$	−1.72085	−0.138222
$156$	0	0
$157$	−2.01958	−0.161180	−0.0805900	−	0.996747i	$-0.525680\pi$
$157$	−2.01958	−0.161180	−0.0805900	+	0.996747i	$0.525680\pi$
$158$	8.36670	0.665619
$159$	0	0
$160$	3.72348	0.294367
$161$	0.0203069	0.00160041
$162$	0	0
$163$	9.21937	0.722117	0.361058	−	0.932543i	$-0.382416\pi$
$163$	9.21937	0.722117	0.361058	+	0.932543i	$0.382416\pi$
$164$	1.22462	0.0956269
$165$	0	0
$166$	−1.08042	−0.0838568
$167$	16.3353	1.26406	0.632030	−	0.774944i	$-0.282222\pi$
$167$	16.3353	1.26406	0.632030	+	0.774944i	$0.282222\pi$
$168$	0	0
$169$	−9.82084	−0.755449
$170$	18.0551	1.38477
$171$	0	0
$172$	3.90914	0.298069
$173$	1.93443	0.147072	0.0735359	−	0.997293i	$-0.476572\pi$
$173$	1.93443	0.147072	0.0735359	+	0.997293i	$0.476572\pi$
$174$	0	0
$175$	−32.7772	−2.47772
$176$	−2.16938	−0.163523
$177$	0	0
$178$	−7.03265	−0.527120
$179$	−6.75844	−0.505149	−0.252575	−	0.967577i	$-0.581277\pi$
$179$	−6.75844	−0.505149	−0.252575	+	0.967577i	$0.581277\pi$
$180$	0	0
$181$	−3.88998	−0.289140	−0.144570	−	0.989495i	$-0.546180\pi$
$181$	−3.88998	−0.289140	−0.144570	+	0.989495i	$0.546180\pi$
$182$	6.59299	0.488705
$183$	0	0
$184$	−0.00549183	−0.000404863 0
$185$	−28.6429	−2.10587
$186$	0	0
$187$	−10.5193	−0.769247
$188$	−5.57907	−0.406896
$189$	0	0
$190$	25.4817	1.84863
$191$	−22.3826	−1.61955	−0.809775	−	0.586740i	$-0.800411\pi$
$191$	−22.3826	−1.61955	−0.809775	+	0.586740i	$0.800411\pi$
$192$	0	0
$193$	−18.6907	−1.34539	−0.672695	−	0.739920i	$-0.734863\pi$
$193$	−18.6907	−1.34539	−0.672695	+	0.739920i	$0.734863\pi$
$194$	2.47385	0.177612
$195$	0	0
$196$	6.67264	0.476617
$197$	−3.01518	−0.214823	−0.107411	−	0.994215i	$-0.534256\pi$
$197$	−3.01518	−0.214823	−0.107411	+	0.994215i	$0.534256\pi$
$198$	0	0
$199$	10.3713	0.735204	0.367602	−	0.929983i	$-0.380179\pi$
$199$	10.3713	0.735204	0.367602	+	0.929983i	$0.380179\pi$
$200$	8.86432	0.626802
$201$	0	0
$202$	6.48513	0.456292
$203$	−32.6455	−2.29126
$204$	0	0
$205$	4.55985	0.318474
$206$	−4.89650	−0.341155
$207$	0	0
$208$	−1.78302	−0.123630
$209$	−14.8461	−1.02693
$210$	0	0
$211$	−24.5601	−1.69079	−0.845395	−	0.534141i	$-0.820635\pi$
$211$	−24.5601	−1.69079	−0.845395	+	0.534141i	$0.820635\pi$
$212$	−4.05603	−0.278569
$213$	0	0
$214$	8.77191	0.599636
$215$	14.5556	0.992685
$216$	0	0
$217$	1.70891	0.116008
$218$	12.5009	0.846670
$219$	0	0
$220$	−8.07763	−0.544594
$221$	−8.64586	−0.581583
$222$	0	0
$223$	24.7890	1.66000	0.829999	−	0.557766i	$-0.188341\pi$
$223$	24.7890	1.66000	0.829999	+	0.557766i	$0.188341\pi$
$224$	−3.69765	−0.247060
$225$	0	0
$226$	3.42504	0.227830
$227$	10.2299	0.678980	0.339490	−	0.940610i	$-0.389746\pi$
$227$	10.2299	0.678980	0.339490	+	0.940610i	$0.389746\pi$
$228$	0	0
$229$	15.9880	1.05652	0.528258	−	0.849084i	$-0.322845\pi$
$229$	15.9880	1.05652	0.528258	+	0.849084i	$0.322845\pi$
$230$	−0.0204487	−0.00134835
$231$	0	0
$232$	8.82870	0.579633
$233$	−13.0383	−0.854169	−0.427085	−	0.904212i	$-0.640459\pi$
$233$	−13.0383	−0.854169	−0.427085	+	0.904212i	$0.640459\pi$
$234$	0	0
$235$	−20.7736	−1.35512
$236$	−11.0037	−0.716282
$237$	0	0
$238$	−17.9299	−1.16222
$239$	−21.8996	−1.41657	−0.708285	−	0.705927i	$-0.750531\pi$
$239$	−21.8996	−1.41657	−0.708285	+	0.705927i	$0.750531\pi$
$240$	0	0
$241$	−27.1588	−1.74945	−0.874726	−	0.484618i	$-0.838959\pi$
$241$	−27.1588	−1.74945	−0.874726	+	0.484618i	$0.838959\pi$
$242$	−6.29381	−0.404581
$243$	0	0
$244$	12.9778	0.830818
$245$	24.8455	1.58732
$246$	0	0
$247$	−12.2021	−0.776401
$248$	−0.462161	−0.0293473
$249$	0	0
$250$	14.3887	0.910022
$251$	−5.86677	−0.370307	−0.185154	−	0.982710i	$-0.559278\pi$
$251$	−5.86677	−0.370307	−0.185154	+	0.982710i	$0.559278\pi$
$252$	0	0
$253$	0.0119138	0.000749016 0
$254$	10.5276	0.660559
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.8923	1.49036	0.745180	−	0.666864i	$-0.232364\pi$
$257$	23.8923	1.49036	0.745180	+	0.666864i	$0.232364\pi$
$258$	0	0
$259$	28.4442	1.76744
$260$	−6.63904	−0.411736
$261$	0	0
$262$	−7.26762	−0.448995
$263$	19.3456	1.19290	0.596452	−	0.802649i	$-0.296577\pi$
$263$	19.3456	1.19290	0.596452	+	0.802649i	$0.296577\pi$
$264$	0	0
$265$	−15.1026	−0.927743
$266$	−25.3049	−1.55154
$267$	0	0
$268$	11.0632	0.675790
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	−21.6428	−1.31470	−0.657352	−	0.753583i	$-0.728324\pi$
$271$	−21.6428	−1.31470	−0.657352	+	0.753583i	$0.728324\pi$
$272$	4.84900	0.294014
$273$	0	0
$274$	20.9183	1.26372
$275$	−19.2300	−1.15961
$276$	0	0
$277$	23.8215	1.43129	0.715647	−	0.698462i	$-0.246132\pi$
$277$	23.8215	1.43129	0.715647	+	0.698462i	$0.246132\pi$
$278$	−14.2845	−0.856729
$279$	0	0
$280$	−13.7681	−0.822804
$281$	−24.9995	−1.49134	−0.745672	−	0.666313i	$-0.767871\pi$
$281$	−24.9995	−1.49134	−0.745672	+	0.666313i	$0.767871\pi$
$282$	0	0
$283$	−12.8336	−0.762879	−0.381439	−	0.924394i	$-0.624572\pi$
$283$	−12.8336	−0.762879	−0.381439	+	0.924394i	$0.624572\pi$
$284$	−3.48703	−0.206917
$285$	0	0
$286$	3.86804	0.228722
$287$	−4.52822	−0.267293
$288$	0	0
$289$	6.51277	0.383104
$290$	32.8735	1.93040
$291$	0	0
$292$	14.0047	0.819561
$293$	−2.32243	−0.135678	−0.0678388	−	0.997696i	$-0.521610\pi$
$293$	−2.32243	−0.135678	−0.0678388	+	0.997696i	$0.521610\pi$
$294$	0	0
$295$	−40.9722	−2.38549
$296$	−7.69250	−0.447118
$297$	0	0
$298$	−1.74043	−0.100820
$299$	0.00979204	0.000566288 0
$300$	0	0
$301$	−14.4547	−0.833152
$302$	7.47417	0.430090
$303$	0	0
$304$	6.84350	0.392502
$305$	48.3226	2.76694
$306$	0	0
$307$	18.6636	1.06519	0.532595	−	0.846370i	$-0.321217\pi$
$307$	18.6636	1.06519	0.532595	+	0.846370i	$0.321217\pi$
$308$	8.02160	0.457073
$309$	0	0
$310$	−1.72085	−0.0977376
$311$	19.4077	1.10051	0.550256	−	0.834996i	$-0.314530\pi$
$311$	19.4077	1.10051	0.550256	+	0.834996i	$0.314530\pi$
$312$	0	0
$313$	9.22747	0.521567	0.260784	−	0.965397i	$-0.416019\pi$
$313$	9.22747	0.521567	0.260784	+	0.965397i	$0.416019\pi$
$314$	−2.01958	−0.113971
$315$	0	0
$316$	8.36670	0.470664
$317$	−26.7021	−1.49974	−0.749869	−	0.661586i	$-0.769884\pi$
$317$	−26.7021	−1.49974	−0.749869	+	0.661586i	$0.769884\pi$
$318$	0	0
$319$	−19.1528	−1.07235
$320$	3.72348	0.208149
$321$	0	0
$322$	0.0203069	0.00113166
$323$	33.1841	1.84641
$324$	0	0
$325$	−15.8053	−0.876718
$326$	9.21937	0.510614
$327$	0	0
$328$	1.22462	0.0676184
$329$	20.6295	1.13734
$330$	0	0
$331$	−18.6211	−1.02351	−0.511753	−	0.859133i	$-0.671004\pi$
$331$	−18.6211	−1.02351	−0.511753	+	0.859133i	$0.671004\pi$
$332$	−1.08042	−0.0592957
$333$	0	0
$334$	16.3353	0.893826
$335$	41.1935	2.25064
$336$	0	0
$337$	−22.3970	−1.22004	−0.610021	−	0.792385i	$-0.708839\pi$
$337$	−22.3970	−1.22004	−0.610021	+	0.792385i	$0.708839\pi$
$338$	−9.82084	−0.534183
$339$	0	0
$340$	18.0551	0.979178
$341$	1.00260	0.0542939
$342$	0	0
$343$	1.21046	0.0653588
$344$	3.90914	0.210767
$345$	0	0
$346$	1.93443	0.103996
$347$	−8.34172	−0.447807	−0.223903	−	0.974611i	$-0.571880\pi$
$347$	−8.34172	−0.447807	−0.223903	+	0.974611i	$0.571880\pi$
$348$	0	0
$349$	18.1802	0.973166	0.486583	−	0.873634i	$-0.338243\pi$
$349$	18.1802	0.973166	0.486583	+	0.873634i	$0.338243\pi$
$350$	−32.7772	−1.75201
$351$	0	0
$352$	−2.16938	−0.115628
$353$	−6.16109	−0.327922	−0.163961	−	0.986467i	$-0.552427\pi$
$353$	−6.16109	−0.327922	−0.163961	+	0.986467i	$0.552427\pi$
$354$	0	0
$355$	−12.9839	−0.689113
$356$	−7.03265	−0.372730
$357$	0	0
$358$	−6.75844	−0.357195
$359$	−1.80036	−0.0950191	−0.0475096	−	0.998871i	$-0.515128\pi$
$359$	−1.80036	−0.0950191	−0.0475096	+	0.998871i	$0.515128\pi$
$360$	0	0
$361$	27.8336	1.46492
$362$	−3.88998	−0.204453
$363$	0	0
$364$	6.59299	0.345567
$365$	52.1461	2.72945
$366$	0	0
$367$	8.75146	0.456823	0.228411	−	0.973565i	$-0.426647\pi$
$367$	8.75146	0.456823	0.228411	+	0.973565i	$0.426647\pi$
$368$	−0.00549183	−0.000286281 0
$369$	0	0
$370$	−28.6429	−1.48907
$371$	14.9978	0.778647
$372$	0	0
$373$	−17.7211	−0.917564	−0.458782	−	0.888549i	$-0.651714\pi$
$373$	−17.7211	−0.917564	−0.458782	+	0.888549i	$0.651714\pi$
$374$	−10.5193	−0.543940
$375$	0	0
$376$	−5.57907	−0.287719
$377$	−15.7418	−0.810742
$378$	0	0
$379$	−36.4211	−1.87082	−0.935412	−	0.353559i	$-0.884971\pi$
$379$	−36.4211	−1.87082	−0.935412	+	0.353559i	$0.884971\pi$
$380$	25.4817	1.30718
$381$	0	0
$382$	−22.3826	−1.14520
$383$	−32.9376	−1.68303	−0.841515	−	0.540233i	$-0.818336\pi$
$383$	−32.9376	−1.68303	−0.841515	+	0.540233i	$0.818336\pi$
$384$	0	0
$385$	29.8683	1.52223
$386$	−18.6907	−0.951334
$387$	0	0
$388$	2.47385	0.125591
$389$	−8.36791	−0.424270	−0.212135	−	0.977240i	$-0.568042\pi$
$389$	−8.36791	−0.424270	−0.212135	+	0.977240i	$0.568042\pi$
$390$	0	0
$391$	−0.0266298	−0.00134673
$392$	6.67264	0.337019
$393$	0	0
$394$	−3.01518	−0.151903
$395$	31.1533	1.56749
$396$	0	0
$397$	22.6515	1.13685	0.568423	−	0.822737i	$-0.307554\pi$
$397$	22.6515	1.13685	0.568423	+	0.822737i	$0.307554\pi$
$398$	10.3713	0.519868
$399$	0	0
$400$	8.86432	0.443216
$401$	5.29174	0.264257	0.132128	−	0.991233i	$-0.457819\pi$
$401$	5.29174	0.264257	0.132128	+	0.991233i	$0.457819\pi$
$402$	0	0
$403$	0.824042	0.0410485
$404$	6.48513	0.322647
$405$	0	0
$406$	−32.6455	−1.62017
$407$	16.6879	0.827190
$408$	0	0
$409$	−15.0093	−0.742163	−0.371082	−	0.928600i	$-0.621013\pi$
$409$	−15.0093	−0.742163	−0.371082	+	0.928600i	$0.621013\pi$
$410$	4.55985	0.225195
$411$	0	0
$412$	−4.89650	−0.241233
$413$	40.6880	2.00213
$414$	0	0
$415$	−4.02292	−0.197478
$416$	−1.78302	−0.0874197
$417$	0	0
$418$	−14.8461	−0.726148
$419$	−37.8003	−1.84666	−0.923332	−	0.384002i	$-0.874545\pi$
$419$	−37.8003	−1.84666	−0.923332	+	0.384002i	$0.874545\pi$
$420$	0	0
$421$	29.6392	1.44453	0.722263	−	0.691618i	$-0.243102\pi$
$421$	29.6392	1.44453	0.722263	+	0.691618i	$0.243102\pi$
$422$	−24.5601	−1.19557
$423$	0	0
$424$	−4.05603	−0.196978
$425$	42.9830	2.08498
$426$	0	0
$427$	−47.9874	−2.32227
$428$	8.77191	0.424006
$429$	0	0
$430$	14.5556	0.701934
$431$	3.66080	0.176335	0.0881673	−	0.996106i	$-0.471899\pi$
$431$	3.66080	0.176335	0.0881673	+	0.996106i	$0.471899\pi$
$432$	0	0
$433$	14.0744	0.676374	0.338187	−	0.941079i	$-0.390186\pi$
$433$	14.0744	0.676374	0.338187	+	0.941079i	$0.390186\pi$
$434$	1.70891	0.0820304
$435$	0	0
$436$	12.5009	0.598686
$437$	−0.0375833	−0.00179785
$438$	0	0
$439$	−15.3969	−0.734853	−0.367426	−	0.930053i	$-0.619761\pi$
$439$	−15.3969	−0.734853	−0.367426	+	0.930053i	$0.619761\pi$
$440$	−8.07763	−0.385086
$441$	0	0
$442$	−8.64586	−0.411241
$443$	17.8950	0.850215	0.425108	−	0.905143i	$-0.360236\pi$
$443$	17.8950	0.850215	0.425108	+	0.905143i	$0.360236\pi$
$444$	0	0
$445$	−26.1860	−1.24133
$446$	24.7890	1.17380
$447$	0	0
$448$	−3.69765	−0.174698
$449$	−23.2268	−1.09614	−0.548071	−	0.836432i	$-0.684638\pi$
$449$	−23.2268	−1.09614	−0.548071	+	0.836432i	$0.684638\pi$
$450$	0	0
$451$	−2.65666	−0.125097
$452$	3.42504	0.161100
$453$	0	0
$454$	10.2299	0.480111
$455$	24.5489	1.15087
$456$	0	0
$457$	−0.553089	−0.0258724	−0.0129362	−	0.999916i	$-0.504118\pi$
$457$	−0.553089	−0.0258724	−0.0129362	+	0.999916i	$0.504118\pi$
$458$	15.9880	0.747070
$459$	0	0
$460$	−0.0204487	−0.000953426 0
$461$	−26.4326	−1.23109	−0.615544	−	0.788103i	$-0.711064\pi$
$461$	−26.4326	−1.23109	−0.615544	+	0.788103i	$0.711064\pi$
$462$	0	0
$463$	3.39737	0.157889	0.0789446	−	0.996879i	$-0.474845\pi$
$463$	3.39737	0.157889	0.0789446	+	0.996879i	$0.474845\pi$
$464$	8.82870	0.409862
$465$	0	0
$466$	−13.0383	−0.603989
$467$	−14.5812	−0.674739	−0.337370	−	0.941372i	$-0.609537\pi$
$467$	−14.5812	−0.674739	−0.337370	+	0.941372i	$0.609537\pi$
$468$	0	0
$469$	−40.9077	−1.88894
$470$	−20.7736	−0.958213
$471$	0	0
$472$	−11.0037	−0.506488
$473$	−8.48040	−0.389929
$474$	0	0
$475$	60.6630	2.78341
$476$	−17.9299	−0.821816
$477$	0	0
$478$	−21.8996	−1.00167
$479$	−41.3750	−1.89047	−0.945236	−	0.326386i	$-0.894169\pi$
$479$	−41.3750	−1.89047	−0.945236	+	0.326386i	$0.894169\pi$
$480$	0	0
$481$	13.7159	0.625390
$482$	−27.1588	−1.23705
$483$	0	0
$484$	−6.29381	−0.286082
$485$	9.21133	0.418265
$486$	0	0
$487$	27.7547	1.25769	0.628844	−	0.777532i	$-0.283529\pi$
$487$	27.7547	1.25769	0.628844	+	0.777532i	$0.283529\pi$
$488$	12.9778	0.587477
$489$	0	0
$490$	24.8455	1.12240
$491$	−28.7642	−1.29811	−0.649054	−	0.760742i	$-0.724835\pi$
$491$	−28.7642	−1.29811	−0.649054	+	0.760742i	$0.724835\pi$
$492$	0	0
$493$	42.8104	1.92808
$494$	−12.2021	−0.548999
$495$	0	0
$496$	−0.462161	−0.0207516
$497$	12.8938	0.578367
$498$	0	0
$499$	32.0457	1.43456	0.717281	−	0.696784i	$-0.245386\pi$
$499$	32.0457	1.43456	0.717281	+	0.696784i	$0.245386\pi$
$500$	14.3887	0.643483
$501$	0	0
$502$	−5.86677	−0.261847
$503$	−29.7110	−1.32475	−0.662373	−	0.749174i	$-0.730451\pi$
$503$	−29.7110	−1.32475	−0.662373	+	0.749174i	$0.730451\pi$
$504$	0	0
$505$	24.1473	1.07454
$506$	0.0119138	0.000529634 0
$507$	0	0
$508$	10.5276	0.467086
$509$	41.7898	1.85230	0.926150	−	0.377156i	$-0.123098\pi$
$509$	41.7898	1.85230	0.926150	+	0.377156i	$0.123098\pi$
$510$	0	0
$511$	−51.7844	−2.29081
$512$	1.00000	0.0441942
$513$	0	0
$514$	23.8923	1.05384
$515$	−18.2320	−0.803398
$516$	0	0
$517$	12.1031	0.532294
$518$	28.4442	1.24977
$519$	0	0
$520$	−6.63904	−0.291141
$521$	−1.80228	−0.0789594	−0.0394797	−	0.999220i	$-0.512570\pi$
$521$	−1.80228	−0.0789594	−0.0394797	+	0.999220i	$0.512570\pi$
$522$	0	0
$523$	−2.65984	−0.116307	−0.0581533	−	0.998308i	$-0.518521\pi$
$523$	−2.65984	−0.116307	−0.0581533	+	0.998308i	$0.518521\pi$
$524$	−7.26762	−0.317488
$525$	0	0
$526$	19.3456	0.843510
$527$	−2.24102	−0.0976202
$528$	0	0
$529$	−23.0000	−0.999999
$530$	−15.1026	−0.656013
$531$	0	0
$532$	−25.3049	−1.09711
$533$	−2.18352	−0.0945789
$534$	0	0
$535$	32.6621	1.41210
$536$	11.0632	0.477856
$537$	0	0
$538$	1.00000	0.0431131
$539$	−14.4755	−0.623502
$540$	0	0
$541$	−35.0400	−1.50649	−0.753243	−	0.657742i	$-0.771512\pi$
$541$	−35.0400	−1.50649	−0.753243	+	0.657742i	$0.771512\pi$
$542$	−21.6428	−0.929637
$543$	0	0
$544$	4.84900	0.207899
$545$	46.5470	1.99385
$546$	0	0
$547$	−40.2366	−1.72039	−0.860196	−	0.509964i	$-0.829659\pi$
$547$	−40.2366	−1.72039	−0.860196	+	0.509964i	$0.829659\pi$
$548$	20.9183	0.893587
$549$	0	0
$550$	−19.2300	−0.819971
$551$	60.4193	2.57395
$552$	0	0
$553$	−30.9372	−1.31558
$554$	23.8215	1.01208
$555$	0	0
$556$	−14.2845	−0.605799
$557$	13.5515	0.574197	0.287098	−	0.957901i	$-0.407309\pi$
$557$	13.5515	0.574197	0.287098	+	0.957901i	$0.407309\pi$
$558$	0	0
$559$	−6.97008	−0.294803
$560$	−13.7681	−0.581810
$561$	0	0
$562$	−24.9995	−1.05454
$563$	2.88000	0.121378	0.0606888	−	0.998157i	$-0.480670\pi$
$563$	2.88000	0.121378	0.0606888	+	0.998157i	$0.480670\pi$
$564$	0	0
$565$	12.7531	0.536526
$566$	−12.8336	−0.539437
$567$	0	0
$568$	−3.48703	−0.146313
$569$	−14.6555	−0.614392	−0.307196	−	0.951646i	$-0.599391\pi$
$569$	−14.6555	−0.614392	−0.307196	+	0.951646i	$0.599391\pi$
$570$	0	0
$571$	−27.8159	−1.16406	−0.582030	−	0.813167i	$-0.697742\pi$
$571$	−27.8159	−1.16406	−0.582030	+	0.813167i	$0.697742\pi$
$572$	3.86804	0.161731
$573$	0	0
$574$	−4.52822	−0.189004
$575$	−0.0486813	−0.00203015
$576$	0	0
$577$	14.1066	0.587266	0.293633	−	0.955918i	$-0.405136\pi$
$577$	14.1066	0.587266	0.293633	+	0.955918i	$0.405136\pi$
$578$	6.51277	0.270895
$579$	0	0
$580$	32.8735	1.36500
$581$	3.99502	0.165741
$582$	0	0
$583$	8.79905	0.364420
$584$	14.0047	0.579517
$585$	0	0
$586$	−2.32243	−0.0959385
$587$	22.0377	0.909594	0.454797	−	0.890595i	$-0.349712\pi$
$587$	22.0377	0.909594	0.454797	+	0.890595i	$0.349712\pi$
$588$	0	0
$589$	−3.16280	−0.130321
$590$	−40.9722	−1.68680
$591$	0	0
$592$	−7.69250	−0.316160
$593$	−33.5598	−1.37813	−0.689067	−	0.724698i	$-0.741979\pi$
$593$	−33.5598	−1.37813	−0.689067	+	0.724698i	$0.741979\pi$
$594$	0	0
$595$	−66.7617	−2.73696
$596$	−1.74043	−0.0712908
$597$	0	0
$598$	0.00979204	0.000400426 0
$599$	17.5125	0.715541	0.357770	−	0.933810i	$-0.383537\pi$
$599$	17.5125	0.715541	0.357770	+	0.933810i	$0.383537\pi$
$600$	0	0
$601$	29.5189	1.20410	0.602051	−	0.798458i	$-0.294350\pi$
$601$	29.5189	1.20410	0.602051	+	0.798458i	$0.294350\pi$
$602$	−14.4547	−0.589128
$603$	0	0
$604$	7.47417	0.304120
$605$	−23.4349	−0.952764
$606$	0	0
$607$	30.5165	1.23862	0.619312	−	0.785145i	$-0.287412\pi$
$607$	30.5165	1.23862	0.619312	+	0.785145i	$0.287412\pi$
$608$	6.84350	0.277541
$609$	0	0
$610$	48.3226	1.95652
$611$	9.94759	0.402437
$612$	0	0
$613$	−13.4372	−0.542725	−0.271363	−	0.962477i	$-0.587474\pi$
$613$	−13.4372	−0.542725	−0.271363	+	0.962477i	$0.587474\pi$
$614$	18.6636	0.753203
$615$	0	0
$616$	8.02160	0.323199
$617$	−48.4723	−1.95142	−0.975710	−	0.219067i	$-0.929699\pi$
$617$	−48.4723	−1.95142	−0.975710	+	0.219067i	$0.929699\pi$
$618$	0	0
$619$	−28.1728	−1.13236	−0.566180	−	0.824281i	$-0.691579\pi$
$619$	−28.1728	−1.13236	−0.566180	+	0.824281i	$0.691579\pi$
$620$	−1.72085	−0.0691109
$621$	0	0
$622$	19.4077	0.778180
$623$	26.0043	1.04184
$624$	0	0
$625$	9.25453	0.370181
$626$	9.22747	0.368804
$627$	0	0
$628$	−2.01958	−0.0805900
$629$	−37.3009	−1.48728
$630$	0	0
$631$	15.1117	0.601589	0.300794	−	0.953689i	$-0.402748\pi$
$631$	15.1117	0.601589	0.300794	+	0.953689i	$0.402748\pi$
$632$	8.36670	0.332810
$633$	0	0
$634$	−26.7021	−1.06047
$635$	39.1992	1.55557
$636$	0	0
$637$	−11.8974	−0.471394
$638$	−19.1528	−0.758266
$639$	0	0
$640$	3.72348	0.147184
$641$	−15.8943	−0.627786	−0.313893	−	0.949458i	$-0.601633\pi$
$641$	−15.8943	−0.627786	−0.313893	+	0.949458i	$0.601633\pi$
$642$	0	0
$643$	−35.0377	−1.38175	−0.690875	−	0.722974i	$-0.742775\pi$
$643$	−35.0377	−1.38175	−0.690875	+	0.722974i	$0.742775\pi$
$644$	0.0203069	0.000800203 0
$645$	0	0
$646$	33.1841	1.30561
$647$	−19.8670	−0.781054	−0.390527	−	0.920592i	$-0.627707\pi$
$647$	−19.8670	−0.781054	−0.390527	+	0.920592i	$0.627707\pi$
$648$	0	0
$649$	23.8712	0.937028
$650$	−15.8053	−0.619933
$651$	0	0
$652$	9.21937	0.361058
$653$	−9.98929	−0.390911	−0.195455	−	0.980713i	$-0.562619\pi$
$653$	−9.98929	−0.390911	−0.195455	+	0.980713i	$0.562619\pi$
$654$	0	0
$655$	−27.0609	−1.05736
$656$	1.22462	0.0478134
$657$	0	0
$658$	20.6295	0.804220
$659$	25.3834	0.988796	0.494398	−	0.869236i	$-0.335389\pi$
$659$	25.3834	0.988796	0.494398	+	0.869236i	$0.335389\pi$
$660$	0	0
$661$	−0.0919517	−0.00357651	−0.00178825	−	0.999998i	$-0.500569\pi$
$661$	−0.0919517	−0.00357651	−0.00178825	+	0.999998i	$0.500569\pi$
$662$	−18.6211	−0.723728
$663$	0	0
$664$	−1.08042	−0.0419284
$665$	−94.2224	−3.65379
$666$	0	0
$667$	−0.0484857	−0.00187737
$668$	16.3353	0.632030
$669$	0	0
$670$	41.1935	1.59144
$671$	−28.1537	−1.08686
$672$	0	0
$673$	20.7844	0.801181	0.400591	−	0.916257i	$-0.368805\pi$
$673$	20.7844	0.801181	0.400591	+	0.916257i	$0.368805\pi$
$674$	−22.3970	−0.862700
$675$	0	0
$676$	−9.82084	−0.377725
$677$	6.38739	0.245487	0.122744	−	0.992438i	$-0.460831\pi$
$677$	6.38739	0.245487	0.122744	+	0.992438i	$0.460831\pi$
$678$	0	0
$679$	−9.14743	−0.351046
$680$	18.0551	0.692383
$681$	0	0
$682$	1.00260	0.0383916
$683$	11.9773	0.458300	0.229150	−	0.973391i	$-0.426405\pi$
$683$	11.9773	0.458300	0.229150	+	0.973391i	$0.426405\pi$
$684$	0	0
$685$	77.8890	2.97599
$686$	1.21046	0.0462157
$687$	0	0
$688$	3.90914	0.149035
$689$	7.23198	0.275517
$690$	0	0
$691$	−14.6174	−0.556071	−0.278035	−	0.960571i	$-0.589683\pi$
$691$	−14.6174	−0.556071	−0.278035	+	0.960571i	$0.589683\pi$
$692$	1.93443	0.0735359
$693$	0	0
$694$	−8.34172	−0.316647
$695$	−53.1882	−2.01754
$696$	0	0
$697$	5.93818	0.224925
$698$	18.1802	0.688132
$699$	0	0
$700$	−32.7772	−1.23886
$701$	6.98645	0.263875	0.131937	−	0.991258i	$-0.457880\pi$
$701$	6.98645	0.263875	0.131937	+	0.991258i	$0.457880\pi$
$702$	0	0
$703$	−52.6437	−1.98549
$704$	−2.16938	−0.0817614
$705$	0	0
$706$	−6.16109	−0.231876
$707$	−23.9798	−0.901851
$708$	0	0
$709$	37.5538	1.41036	0.705182	−	0.709027i	$-0.250865\pi$
$709$	37.5538	1.41036	0.705182	+	0.709027i	$0.250865\pi$
$710$	−12.9839	−0.487277
$711$	0	0
$712$	−7.03265	−0.263560
$713$	0.00253811	9.50529e−5 0
$714$	0	0
$715$	14.4026	0.538626
$716$	−6.75844	−0.252575
$717$	0	0
$718$	−1.80036	−0.0671887
$719$	13.4033	0.499860	0.249930	−	0.968264i	$-0.419592\pi$
$719$	13.4033	0.499860	0.249930	+	0.968264i	$0.419592\pi$
$720$	0	0
$721$	18.1055	0.674286
$722$	27.8336	1.03586
$723$	0	0
$724$	−3.88998	−0.144570
$725$	78.2604	2.90652
$726$	0	0
$727$	21.7956	0.808356	0.404178	−	0.914680i	$-0.367558\pi$
$727$	21.7956	0.808356	0.404178	+	0.914680i	$0.367558\pi$
$728$	6.59299	0.244352
$729$	0	0
$730$	52.1461	1.93001
$731$	18.9554	0.701091
$732$	0	0
$733$	0.583668	0.0215583	0.0107791	−	0.999942i	$-0.496569\pi$
$733$	0.583668	0.0215583	0.0107791	+	0.999942i	$0.496569\pi$
$734$	8.75146	0.323022
$735$	0	0
$736$	−0.00549183	−0.000202431 0
$737$	−24.0001	−0.884057
$738$	0	0
$739$	33.8863	1.24653	0.623264	−	0.782011i	$-0.285806\pi$
$739$	33.8863	1.24653	0.623264	+	0.782011i	$0.285806\pi$
$740$	−28.6429	−1.05293
$741$	0	0
$742$	14.9978	0.550586
$743$	3.84358	0.141007	0.0705036	−	0.997512i	$-0.477539\pi$
$743$	3.84358	0.141007	0.0705036	+	0.997512i	$0.477539\pi$
$744$	0	0
$745$	−6.48046	−0.237426
$746$	−17.7211	−0.648816
$747$	0	0
$748$	−10.5193	−0.384623
$749$	−32.4355	−1.18517
$750$	0	0
$751$	−18.6019	−0.678791	−0.339396	−	0.940644i	$-0.610223\pi$
$751$	−18.6019	−0.678791	−0.339396	+	0.940644i	$0.610223\pi$
$752$	−5.57907	−0.203448
$753$	0	0
$754$	−15.7418	−0.573281
$755$	27.8299	1.01284
$756$	0	0
$757$	−43.1574	−1.56858	−0.784291	−	0.620393i	$-0.786973\pi$
$757$	−43.1574	−1.56858	−0.784291	+	0.620393i	$0.786973\pi$
$758$	−36.4211	−1.32287
$759$	0	0
$760$	25.4817	0.924317
$761$	−2.44728	−0.0887139	−0.0443570	−	0.999016i	$-0.514124\pi$
$761$	−2.44728	−0.0887139	−0.0443570	+	0.999016i	$0.514124\pi$
$762$	0	0
$763$	−46.2241	−1.67343
$764$	−22.3826	−0.809775
$765$	0	0
$766$	−32.9376	−1.19008
$767$	19.6199	0.708433
$768$	0	0
$769$	50.6095	1.82503	0.912513	−	0.409048i	$-0.134139\pi$
$769$	50.6095	1.82503	0.912513	+	0.409048i	$0.134139\pi$
$770$	29.8683	1.07638
$771$	0	0
$772$	−18.6907	−0.672695
$773$	0.217769	0.00783260	0.00391630	−	0.999992i	$-0.498753\pi$
$773$	0.217769	0.00783260	0.00391630	+	0.999992i	$0.498753\pi$
$774$	0	0
$775$	−4.09674	−0.147159
$776$	2.47385	0.0888060
$777$	0	0
$778$	−8.36791	−0.300004
$779$	8.38070	0.300270
$780$	0	0
$781$	7.56468	0.270685
$782$	−0.0266298	−0.000952281 0
$783$	0	0
$784$	6.67264	0.238309
$785$	−7.51987	−0.268396
$786$	0	0
$787$	−11.9639	−0.426468	−0.213234	−	0.977001i	$-0.568400\pi$
$787$	−11.9639	−0.426468	−0.213234	+	0.977001i	$0.568400\pi$
$788$	−3.01518	−0.107411
$789$	0	0
$790$	31.1533	1.10838
$791$	−12.6646	−0.450302
$792$	0	0
$793$	−23.1397	−0.821713
$794$	22.6515	0.803871
$795$	0	0
$796$	10.3713	0.367602
$797$	−2.42030	−0.0857315	−0.0428658	−	0.999081i	$-0.513649\pi$
$797$	−2.42030	−0.0857315	−0.0428658	+	0.999081i	$0.513649\pi$
$798$	0	0
$799$	−27.0529	−0.957063
$800$	8.86432	0.313401
$801$	0	0
$802$	5.29174	0.186858
$803$	−30.3814	−1.07213
$804$	0	0
$805$	0.0756123	0.00266498
$806$	0.824042	0.0290256
$807$	0	0
$808$	6.48513	0.228146
$809$	−2.37249	−0.0834122	−0.0417061	−	0.999130i	$-0.513279\pi$
$809$	−2.37249	−0.0834122	−0.0417061	+	0.999130i	$0.513279\pi$
$810$	0	0
$811$	33.2511	1.16760	0.583802	−	0.811896i	$-0.301565\pi$
$811$	33.2511	1.16760	0.583802	+	0.811896i	$0.301565\pi$
$812$	−32.6455	−1.14563
$813$	0	0
$814$	16.6879	0.584911
$815$	34.3282	1.20246
$816$	0	0
$817$	26.7522	0.935942
$818$	−15.0093	−0.524789
$819$	0	0
$820$	4.55985	0.159237
$821$	56.6425	1.97684	0.988419	−	0.151752i	$-0.0484914\pi$
$821$	56.6425	1.97684	0.988419	+	0.151752i	$0.0484914\pi$
$822$	0	0
$823$	−15.2584	−0.531873	−0.265937	−	0.963991i	$-0.585681\pi$
$823$	−15.2584	−0.531873	−0.265937	+	0.963991i	$0.585681\pi$
$824$	−4.89650	−0.170577
$825$	0	0
$826$	40.6880	1.41572
$827$	−20.9250	−0.727632	−0.363816	−	0.931471i	$-0.618526\pi$
$827$	−20.9250	−0.727632	−0.363816	+	0.931471i	$0.618526\pi$
$828$	0	0
$829$	44.2289	1.53613	0.768066	−	0.640371i	$-0.221219\pi$
$829$	44.2289	1.53613	0.768066	+	0.640371i	$0.221219\pi$
$830$	−4.02292	−0.139638
$831$	0	0
$832$	−1.78302	−0.0618151
$833$	32.3556	1.12106
$834$	0	0
$835$	60.8240	2.10490
$836$	−14.8461	−0.513464
$837$	0	0
$838$	−37.8003	−1.30579
$839$	−18.8567	−0.651007	−0.325503	−	0.945541i	$-0.605534\pi$
$839$	−18.8567	−0.651007	−0.325503	+	0.945541i	$0.605534\pi$
$840$	0	0
$841$	48.9460	1.68779
$842$	29.6392	1.02143
$843$	0	0
$844$	−24.5601	−0.845395
$845$	−36.5677	−1.25797
$846$	0	0
$847$	23.2723	0.799647
$848$	−4.05603	−0.139285
$849$	0	0
$850$	42.9830	1.47431
$851$	0.0422459	0.00144817
$852$	0	0
$853$	46.6652	1.59779	0.798893	−	0.601474i	$-0.205419\pi$
$853$	46.6652	1.59779	0.798893	+	0.601474i	$0.205419\pi$
$854$	−47.9874	−1.64209
$855$	0	0
$856$	8.77191	0.299818
$857$	−48.3193	−1.65056	−0.825278	−	0.564726i	$-0.808982\pi$
$857$	−48.3193	−1.65056	−0.825278	+	0.564726i	$0.808982\pi$
$858$	0	0
$859$	−29.5664	−1.00879	−0.504396	−	0.863472i	$-0.668285\pi$
$859$	−29.5664	−1.00879	−0.504396	+	0.863472i	$0.668285\pi$
$860$	14.5556	0.496342
$861$	0	0
$862$	3.66080	0.124687
$863$	40.7507	1.38717	0.693585	−	0.720375i	$-0.256030\pi$
$863$	40.7507	1.38717	0.693585	+	0.720375i	$0.256030\pi$
$864$	0	0
$865$	7.20281	0.244903
$866$	14.0744	0.478269
$867$	0	0
$868$	1.70891	0.0580042
$869$	−18.1505	−0.615714
$870$	0	0
$871$	−19.7258	−0.668384
$872$	12.5009	0.423335
$873$	0	0
$874$	−0.0375833	−0.00127128
$875$	−53.2045	−1.79864
$876$	0	0
$877$	−38.1412	−1.28794	−0.643968	−	0.765053i	$-0.722713\pi$
$877$	−38.1412	−1.28794	−0.643968	+	0.765053i	$0.722713\pi$
$878$	−15.3969	−0.519619
$879$	0	0
$880$	−8.07763	−0.272297
$881$	−42.4945	−1.43168	−0.715839	−	0.698266i	$-0.753955\pi$
$881$	−42.4945	−1.43168	−0.715839	+	0.698266i	$0.753955\pi$
$882$	0	0
$883$	37.8722	1.27450	0.637250	−	0.770657i	$-0.280072\pi$
$883$	37.8722	1.27450	0.637250	+	0.770657i	$0.280072\pi$
$884$	−8.64586	−0.290792
$885$	0	0
$886$	17.8950	0.601193
$887$	−22.6121	−0.759242	−0.379621	−	0.925142i	$-0.623946\pi$
$887$	−22.6121	−0.759242	−0.379621	+	0.925142i	$0.623946\pi$
$888$	0	0
$889$	−38.9273	−1.30558
$890$	−26.1860	−0.877755
$891$	0	0
$892$	24.7890	0.829999
$893$	−38.1804	−1.27766
$894$	0	0
$895$	−25.1649	−0.841171
$896$	−3.69765	−0.123530
$897$	0	0
$898$	−23.2268	−0.775090
$899$	−4.08028	−0.136085
$900$	0	0
$901$	−19.6677	−0.655225
$902$	−2.65666	−0.0884572
$903$	0	0
$904$	3.42504	0.113915
$905$	−14.4843	−0.481474
$906$	0	0
$907$	33.0529	1.09750	0.548752	−	0.835985i	$-0.315103\pi$
$907$	33.0529	1.09750	0.548752	+	0.835985i	$0.315103\pi$
$908$	10.2299	0.339490
$909$	0	0
$910$	24.5489	0.813787
$911$	24.7088	0.818639	0.409319	−	0.912391i	$-0.365766\pi$
$911$	24.7088	0.818639	0.409319	+	0.912391i	$0.365766\pi$
$912$	0	0
$913$	2.34384	0.0775697
$914$	−0.553089	−0.0182946
$915$	0	0
$916$	15.9880	0.528258
$917$	26.8731	0.887429
$918$	0	0
$919$	−24.2103	−0.798623	−0.399311	−	0.916815i	$-0.630751\pi$
$919$	−24.2103	−0.798623	−0.399311	+	0.916815i	$0.630751\pi$
$920$	−0.0204487	−0.000674174 0
$921$	0	0
$922$	−26.4326	−0.870510
$923$	6.21744	0.204650
$924$	0	0
$925$	−68.1888	−2.24203
$926$	3.39737	0.111645
$927$	0	0
$928$	8.82870	0.289816
$929$	−22.9565	−0.753179	−0.376590	−	0.926380i	$-0.622903\pi$
$929$	−22.9565	−0.753179	−0.376590	+	0.926380i	$0.622903\pi$
$930$	0	0
$931$	45.6642	1.49659
$932$	−13.0383	−0.427085
$933$	0	0
$934$	−14.5812	−0.477113
$935$	−39.1684	−1.28094
$936$	0	0
$937$	3.01896	0.0986253	0.0493126	−	0.998783i	$-0.484297\pi$
$937$	3.01896	0.0986253	0.0493126	+	0.998783i	$0.484297\pi$
$938$	−40.9077	−1.33568
$939$	0	0
$940$	−20.7736	−0.677559
$941$	19.2308	0.626906	0.313453	−	0.949604i	$-0.398514\pi$
$941$	19.2308	0.626906	0.313453	+	0.949604i	$0.398514\pi$
$942$	0	0
$943$	−0.00672541	−0.000219009 0
$944$	−11.0037	−0.358141
$945$	0	0
$946$	−8.48040	−0.275721
$947$	−19.2407	−0.625237	−0.312619	−	0.949879i	$-0.601206\pi$
$947$	−19.2407	−0.625237	−0.312619	+	0.949879i	$0.601206\pi$
$948$	0	0
$949$	−24.9706	−0.810579
$950$	60.6630	1.96817
$951$	0	0
$952$	−17.9299	−0.581112
$953$	3.75904	0.121767	0.0608837	−	0.998145i	$-0.480608\pi$
$953$	3.75904	0.121767	0.0608837	+	0.998145i	$0.480608\pi$
$954$	0	0
$955$	−83.3413	−2.69686
$956$	−21.8996	−0.708285
$957$	0	0
$958$	−41.3750	−1.33677
$959$	−77.3488	−2.49772
$960$	0	0
$961$	−30.7864	−0.993110
$962$	13.7159	0.442218
$963$	0	0
$964$	−27.1588	−0.874726
$965$	−69.5946	−2.24033
$966$	0	0
$967$	−4.34893	−0.139852	−0.0699261	−	0.997552i	$-0.522276\pi$
$967$	−4.34893	−0.139852	−0.0699261	+	0.997552i	$0.522276\pi$
$968$	−6.29381	−0.202291
$969$	0	0
$970$	9.21133	0.295758
$971$	7.50521	0.240854	0.120427	−	0.992722i	$-0.461574\pi$
$971$	7.50521	0.240854	0.120427	+	0.992722i	$0.461574\pi$
$972$	0	0
$973$	52.8192	1.69331
$974$	27.7547	0.889319
$975$	0	0
$976$	12.9778	0.415409
$977$	−9.96282	−0.318739	−0.159369	−	0.987219i	$-0.550946\pi$
$977$	−9.96282	−0.318739	−0.159369	+	0.987219i	$0.550946\pi$
$978$	0	0
$979$	15.2565	0.487599
$980$	24.8455	0.793659
$981$	0	0
$982$	−28.7642	−0.917902
$983$	0.659591	0.0210377	0.0105188	−	0.999945i	$-0.496652\pi$
$983$	0.659591	0.0210377	0.0105188	+	0.999945i	$0.496652\pi$
$984$	0	0
$985$	−11.2270	−0.357721
$986$	42.8104	1.36336
$987$	0	0
$988$	−12.2021	−0.388201
$989$	−0.0214683	−0.000682653 0
$990$	0	0
$991$	−53.1280	−1.68767	−0.843833	−	0.536605i	$-0.819706\pi$
$991$	−53.1280	−1.68767	−0.843833	+	0.536605i	$0.819706\pi$
$992$	−0.462161	−0.0146736
$993$	0	0
$994$	12.8938	0.408967
$995$	38.6175	1.22426
$996$	0	0
$997$	−13.2729	−0.420358	−0.210179	−	0.977663i	$-0.567405\pi$
$997$	−13.2729	−0.420358	−0.210179	+	0.977663i	$0.567405\pi$
$998$	32.0457	1.01439
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4842.2.a.q.1.8		8
3.2	odd	2		1614.2.a.i.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1614.2.a.i.1.1	✓	8	3.2	odd	2
4842.2.a.q.1.8		8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.72348 q^{5} -3.69765 q^{7} +1.00000 q^{8} +3.72348 q^{10} -2.16938 q^{11} -1.78302 q^{13} -3.69765 q^{14} +1.00000 q^{16} +4.84900 q^{17} +6.84350 q^{19} +3.72348 q^{20} -2.16938 q^{22} -0.00549183 q^{23} +8.86432 q^{25} -1.78302 q^{26} -3.69765 q^{28} +8.82870 q^{29} -0.462161 q^{31} +1.00000 q^{32} +4.84900 q^{34} -13.7681 q^{35} -7.69250 q^{37} +6.84350 q^{38} +3.72348 q^{40} +1.22462 q^{41} +3.90914 q^{43} -2.16938 q^{44} -0.00549183 q^{46} -5.57907 q^{47} +6.67264 q^{49} +8.86432 q^{50} -1.78302 q^{52} -4.05603 q^{53} -8.07763 q^{55} -3.69765 q^{56} +8.82870 q^{58} -11.0037 q^{59} +12.9778 q^{61} -0.462161 q^{62} +1.00000 q^{64} -6.63904 q^{65} +11.0632 q^{67} +4.84900 q^{68} -13.7681 q^{70} -3.48703 q^{71} +14.0047 q^{73} -7.69250 q^{74} +6.84350 q^{76} +8.02160 q^{77} +8.36670 q^{79} +3.72348 q^{80} +1.22462 q^{82} -1.08042 q^{83} +18.0551 q^{85} +3.90914 q^{86} -2.16938 q^{88} -7.03265 q^{89} +6.59299 q^{91} -0.00549183 q^{92} -5.57907 q^{94} +25.4817 q^{95} +2.47385 q^{97} +6.67264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 9 q^{7} + 8 q^{8} - 8 q^{11} - 3 q^{13} + 9 q^{14} + 8 q^{16} - 8 q^{17} + 18 q^{19} - 8 q^{22} + 10 q^{23} + 26 q^{25} - 3 q^{26} + 9 q^{28} + 9 q^{29} + 18 q^{31} + 8 q^{32}+ \cdots + 29 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 + 9 * q^7 + 8 * q^8 - 8 * q^11 - 3 * q^13 + 9 * q^14 + 8 * q^16 - 8 * q^17 + 18 * q^19 - 8 * q^22 + 10 * q^23 + 26 * q^25 - 3 * q^26 + 9 * q^28 + 9 * q^29 + 18 * q^31 + 8 * q^32 - 8 * q^34 - 9 * q^35 + 22 * q^37 + 18 * q^38 + q^41 + 14 * q^43 - 8 * q^44 + 10 * q^46 - 6 * q^47 + 29 * q^49 + 26 * q^50 - 3 * q^52 + 3 * q^53 + 17 * q^55 + 9 * q^56 + 9 * q^58 - q^59 + 18 * q^61 + 18 * q^62 + 8 * q^64 + 14 * q^65 + 30 * q^67 - 8 * q^68 - 9 * q^70 + 5 * q^71 + 15 * q^73 + 22 * q^74 + 18 * q^76 + 20 * q^77 + 21 * q^79 + q^82 + 11 * q^83 + 18 * q^85 + 14 * q^86 - 8 * q^88 - 22 * q^89 + 53 * q^91 + 10 * q^92 - 6 * q^94 + 9 * q^95 + 13 * q^97 + 29 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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