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Properties

Label 11.35.b.a.10.1
Level $11$
Weight $35$
Character 11.10
Self dual yes
Analytic conductor $80.548$
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] = [11,35,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, 35); 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, 35, names="a")
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 35 \)
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: \(80.5482194470\)
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)
 
\(f(q)\) \(=\) \(q+2.22601e8 q^{3} +1.71799e10 q^{4} -1.51259e12 q^{5} +3.28739e16 q^{9} -5.05447e17 q^{11} +3.82425e18 q^{12} -3.36705e20 q^{15} +2.95148e20 q^{16} -2.59862e22 q^{20} +2.38723e23 q^{23} +1.70587e24 q^{25} +3.60540e24 q^{27} +4.47511e25 q^{31} -1.12513e26 q^{33} +5.64769e26 q^{36} +3.76162e26 q^{37} -8.68351e27 q^{44} -4.97249e28 q^{45} -5.84263e27 q^{47} +6.57001e28 q^{48} +5.41170e28 q^{49} +1.22959e29 q^{53} +7.64537e29 q^{55} +1.11875e30 q^{59} -5.78454e30 q^{60} +5.07060e30 q^{64} -2.15629e31 q^{67} +5.31399e31 q^{69} -5.80548e31 q^{71} +3.79727e32 q^{75} -4.46439e32 q^{80} +2.54321e32 q^{81} +1.00442e33 q^{89} +4.10123e33 q^{92} +9.96164e33 q^{93} +7.90479e33 q^{97} -1.66160e34 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) 2.22601e8 1.72371 0.861857 − 0.507151i \(-0.169301\pi\)
0.861857 + 0.507151i \(0.169301\pi\)
\(4\) 1.71799e10 1.00000
\(5\) −1.51259e12 −1.98259 −0.991294 − 0.131666i \(-0.957967\pi\)
−0.991294 + 0.131666i \(0.957967\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 3.28739e16 1.97119
\(10\) 0 0
\(11\) −5.05447e17 −1.00000
\(12\) 3.82425e18 1.72371
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −3.36705e20 −3.41742
\(16\) 2.95148e20 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −2.59862e22 −1.98259
\(21\) 0 0
\(22\) 0 0
\(23\) 2.38723e23 1.69247 0.846234 − 0.532811i \(-0.178864\pi\)
0.846234 + 0.532811i \(0.178864\pi\)
\(24\) 0 0
\(25\) 1.70587e24 2.93066
\(26\) 0 0
\(27\) 3.60540e24 1.67405
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 4.47511e25 1.98452 0.992260 − 0.124179i \(-0.0396298\pi\)
0.992260 + 0.124179i \(0.0396298\pi\)
\(32\) 0 0
\(33\) −1.12513e26 −1.72371
\(34\) 0 0
\(35\) 0 0
\(36\) 5.64769e26 1.97119
\(37\) 3.76162e26 0.824035 0.412017 − 0.911176i \(-0.364824\pi\)
0.412017 + 0.911176i \(0.364824\pi\)
\(38\) 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 \(\pi\)
\(44\) −8.68351e27 −1.00000
\(45\) −4.97249e28 −3.90806
\(46\) 0 0
\(47\) −5.84263e27 −0.219253 −0.109626 − 0.993973i \(-0.534965\pi\)
−0.109626 + 0.993973i \(0.534965\pi\)
\(48\) 6.57001e28 1.72371
\(49\) 5.41170e28 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.22959e29 0.598508 0.299254 − 0.954173i \(-0.403262\pi\)
0.299254 + 0.954173i \(0.403262\pi\)
\(54\) 0 0
\(55\) 7.64537e29 1.98259
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.11875e30 0.879529 0.439764 − 0.898113i \(-0.355062\pi\)
0.439764 + 0.898113i \(0.355062\pi\)
\(60\) −5.78454e30 −3.41742
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 5.07060e30 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.15629e31 −1.95180 −0.975900 − 0.218217i \(-0.929976\pi\)
−0.975900 + 0.218217i \(0.929976\pi\)
\(68\) 0 0
\(69\) 5.31399e31 2.91733
\(70\) 0 0
\(71\) −5.80548e31 −1.96086 −0.980429 − 0.196872i \(-0.936922\pi\)
−0.980429 + 0.196872i \(0.936922\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 3.79727e32 5.05161
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −4.46439e32 −1.98259
\(81\) 2.54321e32 0.914401
\(82\) 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
\(89\) 1.00442e33 0.728259 0.364129 − 0.931348i \(-0.381367\pi\)
0.364129 + 0.931348i \(0.381367\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.10123e33 1.69247
\(93\) 9.96164e33 3.42074
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.90479e33 1.32669 0.663347 − 0.748312i \(-0.269135\pi\)
0.663347 + 0.748312i \(0.269135\pi\)
\(98\) 0 0
\(99\) −1.66160e34 −1.97119
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 11.35.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.35.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.35.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.35.b.a.10.1 ✓ 1 11.10 odd 2 CM