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Properties

Label 36.4.a.a.1.1
Level $36$
Weight $4$
Character 36.1
Self dual yes
Analytic conductor $2.124$
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] = [36,4,Mod(1,36)] 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("36.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 36.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.12406876021\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 36.1

$q$-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)
 
\(f(q)\) \(=\) \(q+18.0000 q^{5} +8.00000 q^{7} -36.0000 q^{11} -10.0000 q^{13} -18.0000 q^{17} -100.000 q^{19} -72.0000 q^{23} +199.000 q^{25} +234.000 q^{29} -16.0000 q^{31} +144.000 q^{35} -226.000 q^{37} -90.0000 q^{41} +452.000 q^{43} -432.000 q^{47} -279.000 q^{49} -414.000 q^{53} -648.000 q^{55} +684.000 q^{59} +422.000 q^{61} -180.000 q^{65} +332.000 q^{67} +360.000 q^{71} +26.0000 q^{73} -288.000 q^{77} +512.000 q^{79} +1188.00 q^{83} -324.000 q^{85} +630.000 q^{89} -80.0000 q^{91} -1800.00 q^{95} -1054.00 q^{97} +O(q^{100})\)

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
\(5\) 18.0000 1.60997 0.804984 − 0.593296i \(-0.202174\pi\)
0.804984 + 0.593296i \(0.202174\pi\)
\(6\) 0 0
\(7\) 8.00000 0.431959 0.215980 − 0.976398i \(-0.430705\pi\)
0.215980 + 0.976398i \(0.430705\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 − 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 0 0
\(13\) −10.0000 −0.213346 −0.106673 − 0.994294i \(-0.534020\pi\)
−0.106673 + 0.994294i \(0.534020\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −18.0000 −0.256802 −0.128401 − 0.991722i \(-0.540985\pi\)
−0.128401 + 0.991722i \(0.540985\pi\)
\(18\) 0 0
\(19\) −100.000 −1.20745 −0.603726 − 0.797192i \(-0.706318\pi\)
−0.603726 + 0.797192i \(0.706318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −72.0000 −0.652741 −0.326370 − 0.945242i \(-0.605826\pi\)
−0.326370 + 0.945242i \(0.605826\pi\)
\(24\) 0 0
\(25\) 199.000 1.59200
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 234.000 1.49837 0.749185 − 0.662361i \(-0.230446\pi\)
0.749185 + 0.662361i \(0.230446\pi\)
\(30\) 0 0
\(31\) −16.0000 −0.0926995 −0.0463498 − 0.998925i \(-0.514759\pi\)
−0.0463498 + 0.998925i \(0.514759\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 144.000 0.695441
\(36\) 0 0
\(37\) −226.000 −1.00417 −0.502083 − 0.864819i \(-0.667433\pi\)
−0.502083 + 0.864819i \(0.667433\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −90.0000 −0.342820 −0.171410 − 0.985200i \(-0.554832\pi\)
−0.171410 + 0.985200i \(0.554832\pi\)
\(42\) 0 0
\(43\) 452.000 1.60301 0.801504 − 0.597989i \(-0.204033\pi\)
0.801504 + 0.597989i \(0.204033\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −432.000 −1.34072 −0.670358 − 0.742038i \(-0.733860\pi\)
−0.670358 + 0.742038i \(0.733860\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −414.000 −1.07297 −0.536484 − 0.843911i \(-0.680248\pi\)
−0.536484 + 0.843911i \(0.680248\pi\)
\(54\) 0 0
\(55\) −648.000 −1.58866
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 684.000 1.50931 0.754654 − 0.656123i \(-0.227805\pi\)
0.754654 + 0.656123i \(0.227805\pi\)
\(60\) 0 0
\(61\) 422.000 0.885763 0.442882 − 0.896580i \(-0.353956\pi\)
0.442882 + 0.896580i \(0.353956\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −180.000 −0.343481
\(66\) 0 0
\(67\) 332.000 0.605377 0.302688 − 0.953090i \(-0.402116\pi\)
0.302688 + 0.953090i \(0.402116\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 360.000 0.601748 0.300874 − 0.953664i \(-0.402722\pi\)
0.300874 + 0.953664i \(0.402722\pi\)
\(72\) 0 0
\(73\) 26.0000 0.0416859 0.0208429 − 0.999783i \(-0.493365\pi\)
0.0208429 + 0.999783i \(0.493365\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −288.000 −0.426242
\(78\) 0 0
\(79\) 512.000 0.729171 0.364585 − 0.931170i \(-0.381211\pi\)
0.364585 + 0.931170i \(0.381211\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1188.00 1.57108 0.785542 − 0.618809i \(-0.212384\pi\)
0.785542 + 0.618809i \(0.212384\pi\)
\(84\) 0 0
\(85\) −324.000 −0.413444
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 630.000 0.750336 0.375168 − 0.926957i \(-0.377585\pi\)
0.375168 + 0.926957i \(0.377585\pi\)
\(90\) 0 0
\(91\) −80.0000 −0.0921569
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1800.00 −1.94396
\(96\) 0 0
\(97\) −1054.00 −1.10327 −0.551637 − 0.834085i \(-0.685996\pi\)
−0.551637 + 0.834085i \(0.685996\pi\)
\(98\) 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 36.4.a.a.1.1 1
3.2 odd 2 12.4.a.a.1.1 ✓ 1
4.3 odd 2 144.4.a.g.1.1 1
5.2 odd 4 900.4.d.c.649.2 2
5.3 odd 4 900.4.d.c.649.1 2
5.4 even 2 900.4.a.g.1.1 1
7.2 even 3 1764.4.k.b.361.1 2
7.3 odd 6 1764.4.k.o.1549.1 2
7.4 even 3 1764.4.k.b.1549.1 2
7.5 odd 6 1764.4.k.o.361.1 2
7.6 odd 2 1764.4.a.b.1.1 1
8.3 odd 2 576.4.a.a.1.1 1
8.5 even 2 576.4.a.b.1.1 1
9.2 odd 6 324.4.e.h.109.1 2
9.4 even 3 324.4.e.a.217.1 2
9.5 odd 6 324.4.e.h.217.1 2
9.7 even 3 324.4.e.a.109.1 2
12.11 even 2 48.4.a.a.1.1 1
15.2 even 4 300.4.d.e.49.1 2
15.8 even 4 300.4.d.e.49.2 2
15.14 odd 2 300.4.a.b.1.1 1
21.2 odd 6 588.4.i.d.361.1 2
21.5 even 6 588.4.i.e.361.1 2
21.11 odd 6 588.4.i.d.373.1 2
21.17 even 6 588.4.i.e.373.1 2
21.20 even 2 588.4.a.c.1.1 1
24.5 odd 2 192.4.a.f.1.1 1
24.11 even 2 192.4.a.l.1.1 1
33.32 even 2 1452.4.a.d.1.1 1
39.5 even 4 2028.4.b.c.337.2 2
39.8 even 4 2028.4.b.c.337.1 2
39.38 odd 2 2028.4.a.c.1.1 1
48.5 odd 4 768.4.d.g.385.1 2
48.11 even 4 768.4.d.j.385.2 2
48.29 odd 4 768.4.d.g.385.2 2
48.35 even 4 768.4.d.j.385.1 2
60.23 odd 4 1200.4.f.d.49.1 2
60.47 odd 4 1200.4.f.d.49.2 2
60.59 even 2 1200.4.a.be.1.1 1
84.83 odd 2 2352.4.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.4.a.a.1.1 ✓ 1 3.2 odd 2
36.4.a.a.1.1 1 1.1 even 1 trivial
48.4.a.a.1.1 1 12.11 even 2
144.4.a.g.1.1 1 4.3 odd 2
192.4.a.f.1.1 1 24.5 odd 2
192.4.a.l.1.1 1 24.11 even 2
300.4.a.b.1.1 1 15.14 odd 2
300.4.d.e.49.1 2 15.2 even 4
300.4.d.e.49.2 2 15.8 even 4
324.4.e.a.109.1 2 9.7 even 3
324.4.e.a.217.1 2 9.4 even 3
324.4.e.h.109.1 2 9.2 odd 6
324.4.e.h.217.1 2 9.5 odd 6
576.4.a.a.1.1 1 8.3 odd 2
576.4.a.b.1.1 1 8.5 even 2
588.4.a.c.1.1 1 21.20 even 2
588.4.i.d.361.1 2 21.2 odd 6
588.4.i.d.373.1 2 21.11 odd 6
588.4.i.e.361.1 2 21.5 even 6
588.4.i.e.373.1 2 21.17 even 6
768.4.d.g.385.1 2 48.5 odd 4
768.4.d.g.385.2 2 48.29 odd 4
768.4.d.j.385.1 2 48.35 even 4
768.4.d.j.385.2 2 48.11 even 4
900.4.a.g.1.1 1 5.4 even 2
900.4.d.c.649.1 2 5.3 odd 4
900.4.d.c.649.2 2 5.2 odd 4
1200.4.a.be.1.1 1 60.59 even 2
1200.4.f.d.49.1 2 60.23 odd 4
1200.4.f.d.49.2 2 60.47 odd 4
1452.4.a.d.1.1 1 33.32 even 2
1764.4.a.b.1.1 1 7.6 odd 2
1764.4.k.b.361.1 2 7.2 even 3
1764.4.k.b.1549.1 2 7.4 even 3
1764.4.k.o.361.1 2 7.5 odd 6
1764.4.k.o.1549.1 2 7.3 odd 6
2028.4.a.c.1.1 1 39.38 odd 2
2028.4.b.c.337.1 2 39.8 even 4
2028.4.b.c.337.2 2 39.5 even 4
2352.4.a.bk.1.1 1 84.83 odd 2