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Properties

Label 441.2.a.d.1.1
Level $441$
Weight $2$
Character 441.1
Self dual yes
Analytic conductor $3.521$
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] = [441,2,Mod(1,441)] 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("441.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,-2,0,0,0,0,0,0,0,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.52140272914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 441.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} -7.00000 q^{13} +4.00000 q^{16} -7.00000 q^{19} -5.00000 q^{25} -7.00000 q^{31} -1.00000 q^{37} +5.00000 q^{43} +14.0000 q^{52} +14.0000 q^{61} -8.00000 q^{64} +11.0000 q^{67} -7.00000 q^{73} +14.0000 q^{76} -13.0000 q^{79} +14.0000 q^{97} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −7.00000 −1.94145 −0.970725 − 0.240192i \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −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
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 − 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 14.0000 1.94145
\(53\) 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
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 11.0000 1.34386 0.671932 − 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 14.0000 1.60591
\(77\) 0 0
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 − 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 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
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 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 441.2.a.d.1.1 1
3.2 odd 2 CM 441.2.a.d.1.1 1
4.3 odd 2 7056.2.a.y.1.1 1
7.2 even 3 63.2.e.a.46.1 yes 2
7.3 odd 6 441.2.e.c.226.1 2
7.4 even 3 63.2.e.a.37.1 ✓ 2
7.5 odd 6 441.2.e.c.361.1 2
7.6 odd 2 441.2.a.e.1.1 1
12.11 even 2 7056.2.a.y.1.1 1
21.2 odd 6 63.2.e.a.46.1 yes 2
21.5 even 6 441.2.e.c.361.1 2
21.11 odd 6 63.2.e.a.37.1 ✓ 2
21.17 even 6 441.2.e.c.226.1 2
21.20 even 2 441.2.a.e.1.1 1
28.11 odd 6 1008.2.s.j.289.1 2
28.23 odd 6 1008.2.s.j.865.1 2
28.27 even 2 7056.2.a.bf.1.1 1
63.2 odd 6 567.2.g.d.109.1 2
63.4 even 3 567.2.g.d.541.1 2
63.11 odd 6 567.2.h.c.352.1 2
63.16 even 3 567.2.g.d.109.1 2
63.23 odd 6 567.2.h.c.298.1 2
63.25 even 3 567.2.h.c.352.1 2
63.32 odd 6 567.2.g.d.541.1 2
63.58 even 3 567.2.h.c.298.1 2
84.11 even 6 1008.2.s.j.289.1 2
84.23 even 6 1008.2.s.j.865.1 2
84.83 odd 2 7056.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.e.a.37.1 ✓ 2 7.4 even 3
63.2.e.a.37.1 ✓ 2 21.11 odd 6
63.2.e.a.46.1 yes 2 7.2 even 3
63.2.e.a.46.1 yes 2 21.2 odd 6
441.2.a.d.1.1 1 1.1 even 1 trivial
441.2.a.d.1.1 1 3.2 odd 2 CM
441.2.a.e.1.1 1 7.6 odd 2
441.2.a.e.1.1 1 21.20 even 2
441.2.e.c.226.1 2 7.3 odd 6
441.2.e.c.226.1 2 21.17 even 6
441.2.e.c.361.1 2 7.5 odd 6
441.2.e.c.361.1 2 21.5 even 6
567.2.g.d.109.1 2 63.2 odd 6
567.2.g.d.109.1 2 63.16 even 3
567.2.g.d.541.1 2 63.4 even 3
567.2.g.d.541.1 2 63.32 odd 6
567.2.h.c.298.1 2 63.23 odd 6
567.2.h.c.298.1 2 63.58 even 3
567.2.h.c.352.1 2 63.11 odd 6
567.2.h.c.352.1 2 63.25 even 3
1008.2.s.j.289.1 2 28.11 odd 6
1008.2.s.j.289.1 2 84.11 even 6
1008.2.s.j.865.1 2 28.23 odd 6
1008.2.s.j.865.1 2 84.23 even 6
7056.2.a.y.1.1 1 4.3 odd 2
7056.2.a.y.1.1 1 12.11 even 2
7056.2.a.bf.1.1 1 28.27 even 2
7056.2.a.bf.1.1 1 84.83 odd 2