Properties

Label 18.26.a.d.1.1
Level $18$
Weight $26$
Character 18.1
Self dual yes
Analytic conductor $71.279$
Analytic rank $1$
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] = [18,26,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, 26); 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, 26, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,4096,0,16777216,292754850] 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: \(71.2794203914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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)
 
\(f(q)\) \(=\) \(q+4096.00 q^{2} +1.67772e7 q^{4} +2.92755e8 q^{5} +3.58064e9 q^{7} +6.87195e10 q^{8} +1.19912e12 q^{10} -1.51116e13 q^{11} +1.22107e12 q^{13} +1.46663e13 q^{14} +2.81475e14 q^{16} -2.51825e15 q^{17} -7.99269e15 q^{19} +4.91161e15 q^{20} -6.18970e16 q^{22} +9.96456e16 q^{23} -2.12318e17 q^{25} +5.00151e15 q^{26} +6.00732e16 q^{28} +2.08067e18 q^{29} -4.93767e18 q^{31} +1.15292e18 q^{32} -1.03148e19 q^{34} +1.04825e18 q^{35} +1.98292e19 q^{37} -3.27381e19 q^{38} +2.01180e19 q^{40} -2.24696e20 q^{41} -7.22210e19 q^{43} -2.53530e20 q^{44} +4.08149e20 q^{46} -1.89872e20 q^{47} -1.32825e21 q^{49} -8.69654e20 q^{50} +2.04862e19 q^{52} +2.64568e21 q^{53} -4.42399e21 q^{55} +2.46060e20 q^{56} +8.52244e21 q^{58} +1.64546e22 q^{59} -3.55470e22 q^{61} -2.02247e22 q^{62} +4.72237e21 q^{64} +3.57475e20 q^{65} +1.06704e23 q^{67} -4.22492e22 q^{68} +4.29364e21 q^{70} -7.36720e22 q^{71} -2.62403e23 q^{73} +8.12202e22 q^{74} -1.34095e23 q^{76} -5.41092e22 q^{77} -1.00264e24 q^{79} +8.24032e22 q^{80} -9.20355e23 q^{82} -1.55859e24 q^{83} -7.37230e23 q^{85} -2.95817e23 q^{86} -1.03846e24 q^{88} -2.18167e24 q^{89} +4.37222e21 q^{91} +1.67178e24 q^{92} -7.77717e23 q^{94} -2.33990e24 q^{95} -4.40165e23 q^{97} -5.44050e24 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4096.00 0.707107
\(3\) 0 0
\(4\) 1.67772e7 0.500000
\(5\) 2.92755e8 0.536264 0.268132 − 0.963382i \(-0.413594\pi\)
0.268132 + 0.963382i \(0.413594\pi\)
\(6\) 0 0
\(7\) 3.58064e9 0.0977768 0.0488884 − 0.998804i \(-0.484432\pi\)
0.0488884 + 0.998804i \(0.484432\pi\)
\(8\) 6.87195e10 0.353553
\(9\) 0 0
\(10\) 1.19912e12 0.379196
\(11\) −1.51116e13 −1.45178 −0.725891 − 0.687810i \(-0.758572\pi\)
−0.725891 + 0.687810i \(0.758572\pi\)
\(12\) 0 0
\(13\) 1.22107e12 0.0145361 0.00726807 − 0.999974i \(-0.497686\pi\)
0.00726807 + 0.999974i \(0.497686\pi\)
\(14\) 1.46663e13 0.0691386
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −2.51825e15 −1.04830 −0.524152 − 0.851625i \(-0.675618\pi\)
−0.524152 + 0.851625i \(0.675618\pi\)
\(18\) 0 0
\(19\) −7.99269e15 −0.828463 −0.414232 − 0.910172i \(-0.635950\pi\)
−0.414232 + 0.910172i \(0.635950\pi\)
\(20\) 4.91161e15 0.268132
\(21\) 0 0
\(22\) −6.18970e16 −1.02656
\(23\) 9.96456e16 0.948114 0.474057 − 0.880494i \(-0.342789\pi\)
0.474057 + 0.880494i \(0.342789\pi\)
\(24\) 0 0
\(25\) −2.12318e17 −0.712420
\(26\) 5.00151e15 0.0102786
\(27\) 0 0
\(28\) 6.00732e16 0.0488884
\(29\) 2.08067e18 1.09202 0.546008 − 0.837780i \(-0.316147\pi\)
0.546008 + 0.837780i \(0.316147\pi\)
\(30\) 0 0
\(31\) −4.93767e18 −1.12590 −0.562952 − 0.826490i \(-0.690334\pi\)
−0.562952 + 0.826490i \(0.690334\pi\)
\(32\) 1.15292e18 0.176777
\(33\) 0 0
\(34\) −1.03148e19 −0.741263
\(35\) 1.04825e18 0.0524342
\(36\) 0 0
\(37\) 1.98292e19 0.495202 0.247601 − 0.968862i \(-0.420358\pi\)
0.247601 + 0.968862i \(0.420358\pi\)
\(38\) −3.27381e19 −0.585812
\(39\) 0 0
\(40\) 2.01180e19 0.189598
\(41\) −2.24696e20 −1.55524 −0.777620 − 0.628735i \(-0.783573\pi\)
−0.777620 + 0.628735i \(0.783573\pi\)
\(42\) 0 0
\(43\) −7.22210e19 −0.275618 −0.137809 − 0.990459i \(-0.544006\pi\)
−0.137809 + 0.990459i \(0.544006\pi\)
\(44\) −2.53530e20 −0.725891
\(45\) 0 0
\(46\) 4.08149e20 0.670418
\(47\) −1.89872e20 −0.238363 −0.119181 − 0.992872i \(-0.538027\pi\)
−0.119181 + 0.992872i \(0.538027\pi\)
\(48\) 0 0
\(49\) −1.32825e21 −0.990440
\(50\) −8.69654e20 −0.503757
\(51\) 0 0
\(52\) 2.04862e19 0.00726807
\(53\) 2.64568e21 0.739756 0.369878 − 0.929080i \(-0.379400\pi\)
0.369878 + 0.929080i \(0.379400\pi\)
\(54\) 0 0
\(55\) −4.42399e21 −0.778539
\(56\) 2.46060e20 0.0345693
\(57\) 0 0
\(58\) 8.52244e21 0.772172
\(59\) 1.64546e22 1.20403 0.602015 − 0.798485i \(-0.294365\pi\)
0.602015 + 0.798485i \(0.294365\pi\)
\(60\) 0 0
\(61\) −3.55470e22 −1.71467 −0.857333 − 0.514762i \(-0.827880\pi\)
−0.857333 + 0.514762i \(0.827880\pi\)
\(62\) −2.02247e22 −0.796134
\(63\) 0 0
\(64\) 4.72237e21 0.125000
\(65\) 3.57475e20 0.00779522
\(66\) 0 0
\(67\) 1.06704e23 1.59311 0.796553 − 0.604569i \(-0.206655\pi\)
0.796553 + 0.604569i \(0.206655\pi\)
\(68\) −4.22492e22 −0.524152
\(69\) 0 0
\(70\) 4.29364e21 0.0370766
\(71\) −7.36720e22 −0.532811 −0.266405 − 0.963861i \(-0.585836\pi\)
−0.266405 + 0.963861i \(0.585836\pi\)
\(72\) 0 0
\(73\) −2.62403e23 −1.34101 −0.670506 − 0.741904i \(-0.733923\pi\)
−0.670506 + 0.741904i \(0.733923\pi\)
\(74\) 8.12202e22 0.350161
\(75\) 0 0
\(76\) −1.34095e23 −0.414232
\(77\) −5.41092e22 −0.141950
\(78\) 0 0
\(79\) −1.00264e24 −1.90901 −0.954503 − 0.298201i \(-0.903613\pi\)
−0.954503 + 0.298201i \(0.903613\pi\)
\(80\) 8.24032e22 0.134066
\(81\) 0 0
\(82\) −9.20355e23 −1.09972
\(83\) −1.55859e24 −1.60050 −0.800249 − 0.599667i \(-0.795300\pi\)
−0.800249 + 0.599667i \(0.795300\pi\)
\(84\) 0 0
\(85\) −7.37230e23 −0.562169
\(86\) −2.95817e23 −0.194891
\(87\) 0 0
\(88\) −1.03846e24 −0.513282
\(89\) −2.18167e24 −0.936298 −0.468149 − 0.883650i \(-0.655079\pi\)
−0.468149 + 0.883650i \(0.655079\pi\)
\(90\) 0 0
\(91\) 4.37222e21 0.00142130
\(92\) 1.67178e24 0.474057
\(93\) 0 0
\(94\) −7.77717e23 −0.168548
\(95\) −2.33990e24 −0.444275
\(96\) 0 0
\(97\) −4.40165e23 −0.0644123 −0.0322062 − 0.999481i \(-0.510253\pi\)
−0.0322062 + 0.999481i \(0.510253\pi\)
\(98\) −5.44050e24 −0.700347
\(99\) 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 18.26.a.d.1.1 1
3.2 odd 2 6.26.a.a.1.1 ✓ 1
12.11 even 2 48.26.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.26.a.a.1.1 ✓ 1 3.2 odd 2
18.26.a.d.1.1 1 1.1 even 1 trivial
48.26.a.c.1.1 1 12.11 even 2