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Properties

Label 18.14.a.e.1.1
Level $18$
Weight $14$
Character 18.1
Self dual yes
Analytic conductor $19.302$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,14,Mod(1,18)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("18.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,64,0,4096,15936] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(19.3015672113\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 18.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+64.0000 q^{2} +4096.00 q^{4} +15936.0 q^{5} +98252.0 q^{7} +262144. q^{8} +1.01990e6 q^{10} -1.63046e6 q^{11} +2.23163e7 q^{13} +6.28813e6 q^{14} +1.67772e7 q^{16} +1.22938e8 q^{17} -7.42730e7 q^{19} +6.52739e7 q^{20} -1.04350e8 q^{22} +1.06951e9 q^{23} -9.66747e8 q^{25} +1.42824e9 q^{26} +4.02440e8 q^{28} +5.60709e9 q^{29} +2.16203e9 q^{31} +1.07374e9 q^{32} +7.86803e9 q^{34} +1.56574e9 q^{35} -5.95945e9 q^{37} -4.75347e9 q^{38} +4.17753e9 q^{40} -2.16769e10 q^{41} -6.11010e10 q^{43} -6.67838e9 q^{44} +6.84486e10 q^{46} -1.36472e11 q^{47} -8.72356e10 q^{49} -6.18718e10 q^{50} +9.14076e10 q^{52} -5.57016e8 q^{53} -2.59831e10 q^{55} +2.57562e10 q^{56} +3.58854e11 q^{58} +3.02212e11 q^{59} -1.90535e11 q^{61} +1.38370e11 q^{62} +6.87195e10 q^{64} +3.55633e11 q^{65} -9.18343e11 q^{67} +5.03554e11 q^{68} +1.00208e11 q^{70} +1.08659e12 q^{71} -7.72759e10 q^{73} -3.81405e11 q^{74} -3.04222e11 q^{76} -1.60196e11 q^{77} +2.62436e12 q^{79} +2.67362e11 q^{80} -1.38732e12 q^{82} -3.26961e12 q^{83} +1.95914e12 q^{85} -3.91047e12 q^{86} -4.27416e11 q^{88} -5.92223e12 q^{89} +2.19262e12 q^{91} +4.38071e12 q^{92} -8.73420e12 q^{94} -1.18361e12 q^{95} +5.34013e12 q^{97} -5.58308e12 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 64.0000 0.707107
\(3\) 0 0
\(4\) 4096.00 0.500000
\(5\) 15936.0 0.456115 0.228057 − 0.973648i \(-0.426763\pi\)
0.228057 + 0.973648i \(0.426763\pi\)
\(6\) 0 0
\(7\) 98252.0 0.315649 0.157824 − 0.987467i \(-0.449552\pi\)
0.157824 + 0.987467i \(0.449552\pi\)
\(8\) 262144. 0.353553
\(9\) 0 0
\(10\) 1.01990e6 0.322522
\(11\) −1.63046e6 −0.277497 −0.138749 − 0.990328i \(-0.544308\pi\)
−0.138749 + 0.990328i \(0.544308\pi\)
\(12\) 0 0
\(13\) 2.23163e7 1.28230 0.641151 − 0.767415i \(-0.278457\pi\)
0.641151 + 0.767415i \(0.278457\pi\)
\(14\) 6.28813e6 0.223197
\(15\) 0 0
\(16\) 1.67772e7 0.250000
\(17\) 1.22938e8 1.23529 0.617644 − 0.786458i \(-0.288087\pi\)
0.617644 + 0.786458i \(0.288087\pi\)
\(18\) 0 0
\(19\) −7.42730e7 −0.362187 −0.181093 − 0.983466i \(-0.557964\pi\)
−0.181093 + 0.983466i \(0.557964\pi\)
\(20\) 6.52739e7 0.228057
\(21\) 0 0
\(22\) −1.04350e8 −0.196220
\(23\) 1.06951e9 1.50645 0.753223 − 0.657765i \(-0.228498\pi\)
0.753223 + 0.657765i \(0.228498\pi\)
\(24\) 0 0
\(25\) −9.66747e8 −0.791959
\(26\) 1.42824e9 0.906725
\(27\) 0 0
\(28\) 4.02440e8 0.157824
\(29\) 5.60709e9 1.75045 0.875227 − 0.483713i \(-0.160712\pi\)
0.875227 + 0.483713i \(0.160712\pi\)
\(30\) 0 0
\(31\) 2.16203e9 0.437533 0.218767 − 0.975777i \(-0.429797\pi\)
0.218767 + 0.975777i \(0.429797\pi\)
\(32\) 1.07374e9 0.176777
\(33\) 0 0
\(34\) 7.86803e9 0.873480
\(35\) 1.56574e9 0.143972
\(36\) 0 0
\(37\) −5.95945e9 −0.381852 −0.190926 − 0.981604i \(-0.561149\pi\)
−0.190926 + 0.981604i \(0.561149\pi\)
\(38\) −4.75347e9 −0.256105
\(39\) 0 0
\(40\) 4.17753e9 0.161261
\(41\) −2.16769e10 −0.712691 −0.356345 − 0.934354i \(-0.615977\pi\)
−0.356345 + 0.934354i \(0.615977\pi\)
\(42\) 0 0
\(43\) −6.11010e10 −1.47402 −0.737010 − 0.675881i \(-0.763763\pi\)
−0.737010 + 0.675881i \(0.763763\pi\)
\(44\) −6.67838e9 −0.138749
\(45\) 0 0
\(46\) 6.84486e10 1.06522
\(47\) −1.36472e11 −1.84675 −0.923373 − 0.383903i \(-0.874580\pi\)
−0.923373 + 0.383903i \(0.874580\pi\)
\(48\) 0 0
\(49\) −8.72356e10 −0.900366
\(50\) −6.18718e10 −0.560000
\(51\) 0 0
\(52\) 9.14076e10 0.641151
\(53\) −5.57016e8 −0.00345203 −0.00172601 − 0.999999i \(-0.500549\pi\)
−0.00172601 + 0.999999i \(0.500549\pi\)
\(54\) 0 0
\(55\) −2.59831e10 −0.126571
\(56\) 2.57562e10 0.111599
\(57\) 0 0
\(58\) 3.58854e11 1.23776
\(59\) 3.02212e11 0.932768 0.466384 − 0.884582i \(-0.345557\pi\)
0.466384 + 0.884582i \(0.345557\pi\)
\(60\) 0 0
\(61\) −1.90535e11 −0.473513 −0.236756 − 0.971569i \(-0.576084\pi\)
−0.236756 + 0.971569i \(0.576084\pi\)
\(62\) 1.38370e11 0.309383
\(63\) 0 0
\(64\) 6.87195e10 0.125000
\(65\) 3.55633e11 0.584877
\(66\) 0 0
\(67\) −9.18343e11 −1.24028 −0.620139 − 0.784492i \(-0.712924\pi\)
−0.620139 + 0.784492i \(0.712924\pi\)
\(68\) 5.03554e11 0.617644
\(69\) 0 0
\(70\) 1.00208e11 0.101804
\(71\) 1.08659e12 1.00667 0.503336 − 0.864091i \(-0.332106\pi\)
0.503336 + 0.864091i \(0.332106\pi\)
\(72\) 0 0
\(73\) −7.72759e10 −0.0597648 −0.0298824 − 0.999553i \(-0.509513\pi\)
−0.0298824 + 0.999553i \(0.509513\pi\)
\(74\) −3.81405e11 −0.270010
\(75\) 0 0
\(76\) −3.04222e11 −0.181093
\(77\) −1.60196e11 −0.0875917
\(78\) 0 0
\(79\) 2.62436e12 1.21464 0.607320 − 0.794457i \(-0.292244\pi\)
0.607320 + 0.794457i \(0.292244\pi\)
\(80\) 2.67362e11 0.114029
\(81\) 0 0
\(82\) −1.38732e12 −0.503948
\(83\) −3.26961e12 −1.09771 −0.548856 − 0.835917i \(-0.684936\pi\)
−0.548856 + 0.835917i \(0.684936\pi\)
\(84\) 0 0
\(85\) 1.95914e12 0.563433
\(86\) −3.91047e12 −1.04229
\(87\) 0 0
\(88\) −4.27416e11 −0.0981101
\(89\) −5.92223e12 −1.26314 −0.631568 − 0.775320i \(-0.717588\pi\)
−0.631568 + 0.775320i \(0.717588\pi\)
\(90\) 0 0
\(91\) 2.19262e12 0.404757
\(92\) 4.38071e12 0.753223
\(93\) 0 0
\(94\) −8.73420e12 −1.30585
\(95\) −1.18361e12 −0.165199
\(96\) 0 0
\(97\) 5.34013e12 0.650932 0.325466 − 0.945554i \(-0.394479\pi\)
0.325466 + 0.945554i \(0.394479\pi\)
\(98\) −5.58308e12 −0.636655
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 18.14.a.e.1.1 yes 1
3.2 odd 2 18.14.a.b.1.1 ✓ 1
4.3 odd 2 144.14.a.h.1.1 1
12.11 even 2 144.14.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.14.a.b.1.1 ✓ 1 3.2 odd 2
18.14.a.e.1.1 yes 1 1.1 even 1 trivial
144.14.a.c.1.1 1 12.11 even 2
144.14.a.h.1.1 1 4.3 odd 2