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Properties

Label 49.2.a.a.1.1
Level $49$
Weight $2$
Character 49.1
Self dual yes
Analytic conductor $0.391$
Analytic rank $0$
Dimension $1$
CM discriminant -7
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] = [49,2,Mod(1,49)] 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("49.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(0.391266969904\)
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: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} -3.00000 q^{8} -3.00000 q^{9} +4.00000 q^{11} -1.00000 q^{16} -3.00000 q^{18} +4.00000 q^{22} +8.00000 q^{23} -5.00000 q^{25} +2.00000 q^{29} +5.00000 q^{32} +3.00000 q^{36} -6.00000 q^{37} -12.0000 q^{43} -4.00000 q^{44} +8.00000 q^{46} -5.00000 q^{50} -10.0000 q^{53} +2.00000 q^{58} +7.00000 q^{64} +4.00000 q^{67} +16.0000 q^{71} +9.00000 q^{72} -6.00000 q^{74} +8.00000 q^{79} +9.00000 q^{81} -12.0000 q^{86} -12.0000 q^{88} -8.00000 q^{92} -12.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −3.00000 −1.06066
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −3.00000 −0.707107
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 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
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 − 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 9.00000 1.06066
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) −12.0000 −1.27920
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
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 49.2.a.a.1.1 ✓ 1
3.2 odd 2 441.2.a.c.1.1 1
4.3 odd 2 784.2.a.f.1.1 1
5.2 odd 4 1225.2.b.c.99.2 2
5.3 odd 4 1225.2.b.c.99.1 2
5.4 even 2 1225.2.a.c.1.1 1
7.2 even 3 49.2.c.a.18.1 2
7.3 odd 6 49.2.c.a.30.1 2
7.4 even 3 49.2.c.a.30.1 2
7.5 odd 6 49.2.c.a.18.1 2
7.6 odd 2 CM 49.2.a.a.1.1 ✓ 1
8.3 odd 2 3136.2.a.o.1.1 1
8.5 even 2 3136.2.a.n.1.1 1
11.10 odd 2 5929.2.a.c.1.1 1
12.11 even 2 7056.2.a.bg.1.1 1
13.12 even 2 8281.2.a.d.1.1 1
21.2 odd 6 441.2.e.d.361.1 2
21.5 even 6 441.2.e.d.361.1 2
21.11 odd 6 441.2.e.d.226.1 2
21.17 even 6 441.2.e.d.226.1 2
21.20 even 2 441.2.a.c.1.1 1
28.3 even 6 784.2.i.f.177.1 2
28.11 odd 6 784.2.i.f.177.1 2
28.19 even 6 784.2.i.f.753.1 2
28.23 odd 6 784.2.i.f.753.1 2
28.27 even 2 784.2.a.f.1.1 1
35.13 even 4 1225.2.b.c.99.1 2
35.27 even 4 1225.2.b.c.99.2 2
35.34 odd 2 1225.2.a.c.1.1 1
56.13 odd 2 3136.2.a.n.1.1 1
56.27 even 2 3136.2.a.o.1.1 1
77.76 even 2 5929.2.a.c.1.1 1
84.83 odd 2 7056.2.a.bg.1.1 1
91.90 odd 2 8281.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
49.2.a.a.1.1 ✓ 1 7.6 odd 2 CM
49.2.c.a.18.1 2 7.2 even 3
49.2.c.a.18.1 2 7.5 odd 6
49.2.c.a.30.1 2 7.3 odd 6
49.2.c.a.30.1 2 7.4 even 3
441.2.a.c.1.1 1 3.2 odd 2
441.2.a.c.1.1 1 21.20 even 2
441.2.e.d.226.1 2 21.11 odd 6
441.2.e.d.226.1 2 21.17 even 6
441.2.e.d.361.1 2 21.2 odd 6
441.2.e.d.361.1 2 21.5 even 6
784.2.a.f.1.1 1 4.3 odd 2
784.2.a.f.1.1 1 28.27 even 2
784.2.i.f.177.1 2 28.3 even 6
784.2.i.f.177.1 2 28.11 odd 6
784.2.i.f.753.1 2 28.19 even 6
784.2.i.f.753.1 2 28.23 odd 6
1225.2.a.c.1.1 1 5.4 even 2
1225.2.a.c.1.1 1 35.34 odd 2
1225.2.b.c.99.1 2 5.3 odd 4
1225.2.b.c.99.1 2 35.13 even 4
1225.2.b.c.99.2 2 5.2 odd 4
1225.2.b.c.99.2 2 35.27 even 4
3136.2.a.n.1.1 1 8.5 even 2
3136.2.a.n.1.1 1 56.13 odd 2
3136.2.a.o.1.1 1 8.3 odd 2
3136.2.a.o.1.1 1 56.27 even 2
5929.2.a.c.1.1 1 11.10 odd 2
5929.2.a.c.1.1 1 77.76 even 2
7056.2.a.bg.1.1 1 12.11 even 2
7056.2.a.bg.1.1 1 84.83 odd 2
8281.2.a.d.1.1 1 13.12 even 2
8281.2.a.d.1.1 1 91.90 odd 2