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Properties

Label 444.2.a.a.1.1
Level $444$
Weight $2$
Character 444.1
Self dual yes
Analytic conductor $3.545$
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] = [444,2,Mod(1,444)] 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("444.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.54535784974\)
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 \(\chi\) \(=\) 444.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{9} +4.00000 q^{11} -2.00000 q^{13} +6.00000 q^{19} +8.00000 q^{23} -5.00000 q^{25} -1.00000 q^{27} +8.00000 q^{29} +6.00000 q^{31} -4.00000 q^{33} -1.00000 q^{37} +2.00000 q^{39} +2.00000 q^{41} -6.00000 q^{43} -7.00000 q^{49} +2.00000 q^{53} -6.00000 q^{57} +2.00000 q^{61} +8.00000 q^{67} -8.00000 q^{69} -6.00000 q^{73} +5.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -8.00000 q^{87} -12.0000 q^{89} -6.00000 q^{93} -10.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 5.00000 0.577350
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.00000 −0.857690
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 444.2.a.a.1.1 ✓ 1
3.2 odd 2 1332.2.a.b.1.1 1
4.3 odd 2 1776.2.a.i.1.1 1
8.3 odd 2 7104.2.a.i.1.1 1
8.5 even 2 7104.2.a.s.1.1 1
12.11 even 2 5328.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
444.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
1332.2.a.b.1.1 1 3.2 odd 2
1776.2.a.i.1.1 1 4.3 odd 2
5328.2.a.j.1.1 1 12.11 even 2
7104.2.a.i.1.1 1 8.3 odd 2
7104.2.a.s.1.1 1 8.5 even 2