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Properties

Label 11.29.b.a.10.1
Level $11$
Weight $29$
Character 11.10
Self dual yes
Analytic conductor $54.635$
Analytic rank $0$
Dimension $1$
CM discriminant -11
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] = [11,29,Mod(10,11)] 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("11.10"); S:= CuspForms(chi, 29); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(11, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 29, names="a")
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 29 \)
Character orbit: \([\chi]\) \(=\) 11.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.6353216730\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 10.1
Character \(\chi\) \(=\) 11.10

$q$-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)
 
\(f(q)\) \(=\) \(q-3.24071e6 q^{3} +2.68435e8 q^{4} -2.04659e9 q^{5} -1.23746e13 q^{9} +3.79750e14 q^{11} -8.69922e14 q^{12} +6.63242e15 q^{15} +7.20576e16 q^{16} -5.49378e17 q^{20} -2.07628e19 q^{23} -3.30644e19 q^{25} +1.14240e20 q^{27} +1.34827e21 q^{31} -1.23066e21 q^{33} -3.32177e21 q^{36} -2.01937e21 q^{37} +1.01938e23 q^{44} +2.53257e22 q^{45} -9.94560e20 q^{47} -2.33518e23 q^{48} +4.59987e23 q^{49} +2.15688e24 q^{53} -7.77193e23 q^{55} +1.21634e25 q^{59} +1.78038e24 q^{60} +1.93428e25 q^{64} +6.23198e25 q^{67} +6.72862e25 q^{69} +4.87024e25 q^{71} +1.07152e26 q^{75} -1.47473e26 q^{80} -8.71271e25 q^{81} +8.32986e26 q^{89} -5.57347e27 q^{92} -4.36936e27 q^{93} -8.30799e27 q^{97} -4.69924e27 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −3.24071e6 −0.677553 −0.338776 − 0.940867i \(-0.610013\pi\)
−0.338776 + 0.940867i \(0.610013\pi\)
\(4\) 2.68435e8 1.00000
\(5\) −2.04659e9 −0.335314 −0.167657 − 0.985845i \(-0.553620\pi\)
−0.167657 + 0.985845i \(0.553620\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −1.23746e13 −0.540922
\(10\) 0 0
\(11\) 3.79750e14 1.00000
\(12\) −8.69922e14 −0.677553
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 6.63242e15 0.227193
\(16\) 7.20576e16 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −5.49378e17 −0.335314
\(21\) 0 0
\(22\) 0 0
\(23\) −2.07628e19 −1.79100 −0.895501 − 0.445060i \(-0.853182\pi\)
−0.895501 + 0.445060i \(0.853182\pi\)
\(24\) 0 0
\(25\) −3.30644e19 −0.887565
\(26\) 0 0
\(27\) 1.14240e20 1.04406
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.34827e21 1.78120 0.890602 − 0.454783i \(-0.150283\pi\)
0.890602 + 0.454783i \(0.150283\pi\)
\(32\) 0 0
\(33\) −1.23066e21 −0.677553
\(34\) 0 0
\(35\) 0 0
\(36\) −3.32177e21 −0.540922
\(37\) −2.01937e21 −0.224075 −0.112037 − 0.993704i \(-0.535738\pi\)
−0.112037 + 0.993704i \(0.535738\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 1.01938e23 1.00000
\(45\) 2.53257e22 0.181379
\(46\) 0 0
\(47\) −9.94560e20 −0.00387491 −0.00193745 − 0.999998i \(-0.500617\pi\)
−0.00193745 + 0.999998i \(0.500617\pi\)
\(48\) −2.33518e23 −0.677553
\(49\) 4.59987e23 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.15688e24 1.56301 0.781507 − 0.623896i \(-0.214451\pi\)
0.781507 + 0.623896i \(0.214451\pi\)
\(54\) 0 0
\(55\) −7.77193e23 −0.335314
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.21634e25 1.96394 0.981970 − 0.189039i \(-0.0605372\pi\)
0.981970 + 0.189039i \(0.0605372\pi\)
\(60\) 1.78038e24 0.227193
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.93428e25 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.23198e25 1.69660 0.848298 − 0.529519i \(-0.177627\pi\)
0.848298 + 0.529519i \(0.177627\pi\)
\(68\) 0 0
\(69\) 6.72862e25 1.21350
\(70\) 0 0
\(71\) 4.87024e25 0.588753 0.294377 − 0.955689i \(-0.404888\pi\)
0.294377 + 0.955689i \(0.404888\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.07152e26 0.601372
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −1.47473e26 −0.335314
\(81\) −8.71271e25 −0.166480
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.32986e26 0.425772 0.212886 − 0.977077i \(-0.431714\pi\)
0.212886 + 0.977077i \(0.431714\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.57347e27 −1.79100
\(93\) −4.36936e27 −1.20686
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.30799e27 −1.27260 −0.636299 − 0.771442i \(-0.719536\pi\)
−0.636299 + 0.771442i \(0.719536\pi\)
\(98\) 0 0
\(99\) −4.69924e27 −0.540922
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 11.29.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.29.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.29.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.29.b.a.10.1 ✓ 1 11.10 odd 2 CM