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Properties

Label 11.37.b.a.10.1
Level $11$
Weight $37$
Character 11.10
Self dual yes
Analytic conductor $90.300$
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] = [11,37,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, 37); 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, 37, names="a")
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 37 \)
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: \(90.3003845253\)
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)
 
\(f(q)\) \(=\) \(q-3.39284e8 q^{3} +6.87195e10 q^{4} +1.75580e12 q^{5} -3.49809e16 q^{9} +5.55992e18 q^{11} -2.33154e19 q^{12} -5.95714e20 q^{15} +4.72237e21 q^{16} +1.20657e23 q^{20} +6.42094e24 q^{23} -1.14691e25 q^{25} +6.27932e25 q^{27} -6.88585e26 q^{31} -1.88639e27 q^{33} -2.40387e27 q^{36} -3.36719e28 q^{37} +3.82075e29 q^{44} -6.14194e28 q^{45} +1.96230e30 q^{47} -1.60222e30 q^{48} +2.65173e30 q^{49} -1.99072e31 q^{53} +9.76209e30 q^{55} +3.02001e30 q^{59} -4.09372e31 q^{60} +3.24519e32 q^{64} -6.55191e31 q^{67} -2.17852e33 q^{69} +4.15062e33 q^{71} +3.89128e33 q^{75} +8.29152e33 q^{80} -1.60543e34 q^{81} +1.42998e35 q^{89} +4.41243e35 q^{92} +2.33626e35 q^{93} +1.12965e36 q^{97} -1.94491e35 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −3.39284e8 −0.875752 −0.437876 − 0.899035i \(-0.644269\pi\)
−0.437876 + 0.899035i \(0.644269\pi\)
\(4\) 6.87195e10 1.00000
\(5\) 1.75580e12 0.460272 0.230136 − 0.973159i \(-0.426083\pi\)
0.230136 + 0.973159i \(0.426083\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −3.49809e16 −0.233059
\(10\) 0 0
\(11\) 5.55992e18 1.00000
\(12\) −2.33154e19 −0.875752
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −5.95714e20 −0.403084
\(16\) 4.72237e21 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.20657e23 0.460272
\(21\) 0 0
\(22\) 0 0
\(23\) 6.42094e24 1.97924 0.989618 − 0.143726i \(-0.0459083\pi\)
0.989618 + 0.143726i \(0.0459083\pi\)
\(24\) 0 0
\(25\) −1.14691e25 −0.788150
\(26\) 0 0
\(27\) 6.27932e25 1.07985
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −6.88585e26 −0.985025 −0.492512 − 0.870305i \(-0.663921\pi\)
−0.492512 + 0.870305i \(0.663921\pi\)
\(32\) 0 0
\(33\) −1.88639e27 −0.875752
\(34\) 0 0
\(35\) 0 0
\(36\) −2.40387e27 −0.233059
\(37\) −3.36719e28 −1.99360 −0.996798 − 0.0799629i \(-0.974520\pi\)
−0.996798 + 0.0799629i \(0.974520\pi\)
\(38\) 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 \(\pi\)
\(44\) 3.82075e29 1.00000
\(45\) −6.14194e28 −0.107271
\(46\) 0 0
\(47\) 1.96230e30 1.56676 0.783382 − 0.621540i \(-0.213493\pi\)
0.783382 + 0.621540i \(0.213493\pi\)
\(48\) −1.60222e30 −0.875752
\(49\) 2.65173e30 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.99072e31 −1.82828 −0.914142 − 0.405395i \(-0.867134\pi\)
−0.914142 + 0.405395i \(0.867134\pi\)
\(54\) 0 0
\(55\) 9.76209e30 0.460272
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.02001e30 0.0402413 0.0201206 − 0.999798i \(-0.493595\pi\)
0.0201206 + 0.999798i \(0.493595\pi\)
\(60\) −4.09372e31 −0.403084
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 3.24519e32 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −6.55191e31 −0.0885159 −0.0442579 − 0.999020i \(-0.514092\pi\)
−0.0442579 + 0.999020i \(0.514092\pi\)
\(68\) 0 0
\(69\) −2.17852e33 −1.73332
\(70\) 0 0
\(71\) 4.15062e33 1.97453 0.987263 − 0.159097i \(-0.0508583\pi\)
0.987263 + 0.159097i \(0.0508583\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 3.89128e33 0.690224
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 8.29152e33 0.460272
\(81\) −1.60543e34 −0.712624
\(82\) 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
\(89\) 1.42998e35 1.16496 0.582480 − 0.812845i \(-0.302082\pi\)
0.582480 + 0.812845i \(0.302082\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.41243e35 1.97924
\(93\) 2.33626e35 0.862637
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.12965e36 1.95457 0.977285 − 0.211929i \(-0.0679744\pi\)
0.977285 + 0.211929i \(0.0679744\pi\)
\(98\) 0 0
\(99\) −1.94491e35 −0.233059
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 11.37.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.37.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.37.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.37.b.a.10.1 ✓ 1 11.10 odd 2 CM