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Properties

Label 144.8.a.i.1.1
Level $144$
Weight $8$
Character 144.1
Self dual yes
Analytic conductor $44.983$
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] = [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 = [1,0,0,0,210] 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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

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)
 
\(f(q)\) \(=\) \(q+210.000 q^{5} -1016.00 q^{7} +1092.00 q^{11} +1382.00 q^{13} -14706.0 q^{17} +39940.0 q^{19} +68712.0 q^{23} -34025.0 q^{25} +102570. q^{29} -227552. q^{31} -213360. q^{35} +160526. q^{37} -10842.0 q^{41} +630748. q^{43} +472656. q^{47} +208713. q^{49} +1.49402e6 q^{53} +229320. q^{55} +2.64066e6 q^{59} +827702. q^{61} +290220. q^{65} +126004. q^{67} -1.41473e6 q^{71} +980282. q^{73} -1.10947e6 q^{77} +3.56680e6 q^{79} +5.67289e6 q^{83} -3.08826e6 q^{85} +1.19512e7 q^{89} -1.40411e6 q^{91} +8.38740e6 q^{95} +8.68215e6 q^{97} +O(q^{100})\)

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
\(5\) 210.000 0.751319 0.375659 − 0.926758i \(-0.377416\pi\)
0.375659 + 0.926758i \(0.377416\pi\)
\(6\) 0 0
\(7\) −1016.00 −1.11957 −0.559784 − 0.828638i \(-0.689116\pi\)
−0.559784 + 0.828638i \(0.689116\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1092.00 0.247371 0.123685 − 0.992321i \(-0.460529\pi\)
0.123685 + 0.992321i \(0.460529\pi\)
\(12\) 0 0
\(13\) 1382.00 0.174464 0.0872321 − 0.996188i \(-0.472198\pi\)
0.0872321 + 0.996188i \(0.472198\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −14706.0 −0.725978 −0.362989 − 0.931793i \(-0.618244\pi\)
−0.362989 + 0.931793i \(0.618244\pi\)
\(18\) 0 0
\(19\) 39940.0 1.33589 0.667945 − 0.744211i \(-0.267174\pi\)
0.667945 + 0.744211i \(0.267174\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 68712.0 1.17757 0.588783 − 0.808291i \(-0.299607\pi\)
0.588783 + 0.808291i \(0.299607\pi\)
\(24\) 0 0
\(25\) −34025.0 −0.435520
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 102570. 0.780957 0.390479 − 0.920612i \(-0.372310\pi\)
0.390479 + 0.920612i \(0.372310\pi\)
\(30\) 0 0
\(31\) −227552. −1.37188 −0.685938 − 0.727660i \(-0.740608\pi\)
−0.685938 + 0.727660i \(0.740608\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −213360. −0.841153
\(36\) 0 0
\(37\) 160526. 0.521002 0.260501 − 0.965474i \(-0.416112\pi\)
0.260501 + 0.965474i \(0.416112\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10842.0 −0.0245678 −0.0122839 − 0.999925i \(-0.503910\pi\)
−0.0122839 + 0.999925i \(0.503910\pi\)
\(42\) 0 0
\(43\) 630748. 1.20981 0.604904 − 0.796299i \(-0.293212\pi\)
0.604904 + 0.796299i \(0.293212\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 472656. 0.664053 0.332026 − 0.943270i \(-0.392268\pi\)
0.332026 + 0.943270i \(0.392268\pi\)
\(48\) 0 0
\(49\) 208713. 0.253433
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.49402e6 1.37845 0.689224 − 0.724548i \(-0.257952\pi\)
0.689224 + 0.724548i \(0.257952\pi\)
\(54\) 0 0
\(55\) 229320. 0.185854
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.64066e6 1.67390 0.836952 − 0.547277i \(-0.184335\pi\)
0.836952 + 0.547277i \(0.184335\pi\)
\(60\) 0 0
\(61\) 827702. 0.466895 0.233448 − 0.972369i \(-0.424999\pi\)
0.233448 + 0.972369i \(0.424999\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 290220. 0.131078
\(66\) 0 0
\(67\) 126004. 0.0511826 0.0255913 − 0.999672i \(-0.491853\pi\)
0.0255913 + 0.999672i \(0.491853\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.41473e6 −0.469104 −0.234552 − 0.972104i \(-0.575362\pi\)
−0.234552 + 0.972104i \(0.575362\pi\)
\(72\) 0 0
\(73\) 980282. 0.294931 0.147466 − 0.989067i \(-0.452888\pi\)
0.147466 + 0.989067i \(0.452888\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.10947e6 −0.276948
\(78\) 0 0
\(79\) 3.56680e6 0.813924 0.406962 − 0.913445i \(-0.366588\pi\)
0.406962 + 0.913445i \(0.366588\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.67289e6 1.08901 0.544504 − 0.838758i \(-0.316718\pi\)
0.544504 + 0.838758i \(0.316718\pi\)
\(84\) 0 0
\(85\) −3.08826e6 −0.545441
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.19512e7 1.79699 0.898496 − 0.438982i \(-0.144661\pi\)
0.898496 + 0.438982i \(0.144661\pi\)
\(90\) 0 0
\(91\) −1.40411e6 −0.195325
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.38740e6 1.00368
\(96\) 0 0
\(97\) 8.68215e6 0.965886 0.482943 − 0.875652i \(-0.339568\pi\)
0.482943 + 0.875652i \(0.339568\pi\)
\(98\) 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.i.1.1 1
3.2 odd 2 16.8.a.b.1.1 1
4.3 odd 2 18.8.a.b.1.1 1
8.3 odd 2 576.8.a.g.1.1 1
8.5 even 2 576.8.a.f.1.1 1
12.11 even 2 2.8.a.a.1.1 ✓ 1
15.2 even 4 400.8.c.j.49.2 2
15.8 even 4 400.8.c.j.49.1 2
15.14 odd 2 400.8.a.l.1.1 1
20.3 even 4 450.8.c.g.199.1 2
20.7 even 4 450.8.c.g.199.2 2
20.19 odd 2 450.8.a.c.1.1 1
24.5 odd 2 64.8.a.e.1.1 1
24.11 even 2 64.8.a.c.1.1 1
36.7 odd 6 162.8.c.a.109.1 2
36.11 even 6 162.8.c.l.109.1 2
36.23 even 6 162.8.c.l.55.1 2
36.31 odd 6 162.8.c.a.55.1 2
48.5 odd 4 256.8.b.f.129.2 2
48.11 even 4 256.8.b.b.129.1 2
48.29 odd 4 256.8.b.f.129.1 2
48.35 even 4 256.8.b.b.129.2 2
60.23 odd 4 50.8.b.c.49.2 2
60.47 odd 4 50.8.b.c.49.1 2
60.59 even 2 50.8.a.g.1.1 1
84.11 even 6 98.8.c.d.79.1 2
84.23 even 6 98.8.c.d.67.1 2
84.47 odd 6 98.8.c.e.67.1 2
84.59 odd 6 98.8.c.e.79.1 2
84.83 odd 2 98.8.a.a.1.1 1
132.131 odd 2 242.8.a.e.1.1 1
156.47 odd 4 338.8.b.d.337.1 2
156.83 odd 4 338.8.b.d.337.2 2
156.155 even 2 338.8.a.d.1.1 1
204.203 even 2 578.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 12.11 even 2
16.8.a.b.1.1 1 3.2 odd 2
18.8.a.b.1.1 1 4.3 odd 2
50.8.a.g.1.1 1 60.59 even 2
50.8.b.c.49.1 2 60.47 odd 4
50.8.b.c.49.2 2 60.23 odd 4
64.8.a.c.1.1 1 24.11 even 2
64.8.a.e.1.1 1 24.5 odd 2
98.8.a.a.1.1 1 84.83 odd 2
98.8.c.d.67.1 2 84.23 even 6
98.8.c.d.79.1 2 84.11 even 6
98.8.c.e.67.1 2 84.47 odd 6
98.8.c.e.79.1 2 84.59 odd 6
144.8.a.i.1.1 1 1.1 even 1 trivial
162.8.c.a.55.1 2 36.31 odd 6
162.8.c.a.109.1 2 36.7 odd 6
162.8.c.l.55.1 2 36.23 even 6
162.8.c.l.109.1 2 36.11 even 6
242.8.a.e.1.1 1 132.131 odd 2
256.8.b.b.129.1 2 48.11 even 4
256.8.b.b.129.2 2 48.35 even 4
256.8.b.f.129.1 2 48.29 odd 4
256.8.b.f.129.2 2 48.5 odd 4
338.8.a.d.1.1 1 156.155 even 2
338.8.b.d.337.1 2 156.47 odd 4
338.8.b.d.337.2 2 156.83 odd 4
400.8.a.l.1.1 1 15.14 odd 2
400.8.c.j.49.1 2 15.8 even 4
400.8.c.j.49.2 2 15.2 even 4
450.8.a.c.1.1 1 20.19 odd 2
450.8.c.g.199.1 2 20.3 even 4
450.8.c.g.199.2 2 20.7 even 4
576.8.a.f.1.1 1 8.5 even 2
576.8.a.g.1.1 1 8.3 odd 2
578.8.a.b.1.1 1 204.203 even 2