Properties

Label 144.8.a.e.1.1
Level $144$
Weight $8$
Character 144.1
Self dual yes
Analytic conductor $44.983$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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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] = [144,8,Mod(1,144)] 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("144.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0] 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: \(44.9834436697\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 144.1

$q$-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)
 
\(f(q)\) \(=\) \(q+508.000 q^{7} -14614.0 q^{13} +57448.0 q^{19} -78125.0 q^{25} -178916. q^{31} +279710. q^{37} -1.03522e6 q^{43} -565479. q^{49} -3.53555e6 q^{61} +385072. q^{67} -6.27481e6 q^{73} -8.76304e6 q^{79} -7.42391e6 q^{91} +1.22452e7 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 508.000 0.559784 0.279892 − 0.960031i \(-0.409701\pi\)
0.279892 + 0.960031i \(0.409701\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −14614.0 −1.84488 −0.922438 − 0.386144i \(-0.873807\pi\)
−0.922438 + 0.386144i \(0.873807\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 57448.0 1.92149 0.960743 − 0.277439i \(-0.0894857\pi\)
0.960743 + 0.277439i \(0.0894857\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −78125.0 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −178916. −1.07866 −0.539328 − 0.842096i \(-0.681322\pi\)
−0.539328 + 0.842096i \(0.681322\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 279710. 0.907825 0.453912 − 0.891046i \(-0.350028\pi\)
0.453912 + 0.891046i \(0.350028\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −1.03522e6 −1.98561 −0.992807 − 0.119727i \(-0.961798\pi\)
−0.992807 + 0.119727i \(0.961798\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −565479. −0.686642
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −3.53555e6 −1.99435 −0.997177 − 0.0750923i \(-0.976075\pi\)
−0.997177 + 0.0750923i \(0.976075\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 385072. 0.156416 0.0782078 − 0.996937i \(-0.475080\pi\)
0.0782078 + 0.996937i \(0.475080\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.27481e6 −1.88786 −0.943932 − 0.330141i \(-0.892904\pi\)
−0.943932 + 0.330141i \(0.892904\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.76304e6 −1.99968 −0.999839 − 0.0179303i \(-0.994292\pi\)
−0.999839 + 0.0179303i \(0.994292\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −7.42391e6 −1.03273
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.22452e7 1.36227 0.681137 − 0.732156i \(-0.261486\pi\)
0.681137 + 0.732156i \(0.261486\pi\)
\(98\) 0 0
\(99\) 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 144.8.a.e.1.1 1
3.2 odd 2 CM 144.8.a.e.1.1 1
4.3 odd 2 36.8.a.b.1.1 ✓ 1
8.3 odd 2 576.8.a.q.1.1 1
8.5 even 2 576.8.a.r.1.1 1
12.11 even 2 36.8.a.b.1.1 ✓ 1
24.5 odd 2 576.8.a.r.1.1 1
24.11 even 2 576.8.a.q.1.1 1
36.7 odd 6 324.8.e.c.109.1 2
36.11 even 6 324.8.e.c.109.1 2
36.23 even 6 324.8.e.c.217.1 2
36.31 odd 6 324.8.e.c.217.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.8.a.b.1.1 ✓ 1 4.3 odd 2
36.8.a.b.1.1 ✓ 1 12.11 even 2
144.8.a.e.1.1 1 1.1 even 1 trivial
144.8.a.e.1.1 1 3.2 odd 2 CM
324.8.e.c.109.1 2 36.7 odd 6
324.8.e.c.109.1 2 36.11 even 6
324.8.e.c.217.1 2 36.23 even 6
324.8.e.c.217.1 2 36.31 odd 6
576.8.a.q.1.1 1 8.3 odd 2
576.8.a.q.1.1 1 24.11 even 2
576.8.a.r.1.1 1 8.5 even 2
576.8.a.r.1.1 1 24.5 odd 2