↑

Properties

Label 11.53.b.a.10.1
Level $11$
Weight $53$
Character 11.10
Self dual yes
Analytic conductor $188.379$
Analytic rank $0$
Dimension $1$
CM discriminant -11
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [11,53,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, 53); 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, 53, names="a")
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 53 \)
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: \(188.379295253\)
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)
 
\(f(q)\) \(=\) \(q-4.50892e12 q^{3} +4.50360e15 q^{4} +2.56039e18 q^{5} +1.38693e25 q^{9} +1.19182e27 q^{11} -2.03064e28 q^{12} -1.15446e31 q^{15} +2.02824e31 q^{16} +1.15310e34 q^{20} +4.48364e35 q^{23} +4.33515e36 q^{25} -3.34028e37 q^{27} +7.23932e38 q^{31} -5.37381e39 q^{33} +6.24616e40 q^{36} +1.06070e41 q^{37} +5.36747e42 q^{44} +3.55107e43 q^{45} +2.56855e43 q^{47} -9.14517e43 q^{48} +8.81248e43 q^{49} -1.34968e45 q^{53} +3.05152e45 q^{55} +6.93259e45 q^{59} -5.19922e46 q^{60} +9.13439e46 q^{64} -5.00329e47 q^{67} -2.02164e48 q^{69} -2.69589e48 q^{71} -1.95469e49 q^{75} +5.19309e49 q^{80} +6.10003e49 q^{81} +7.24276e50 q^{89} +2.01925e51 q^{92} -3.26415e51 q^{93} -6.69771e51 q^{97} +1.65296e52 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −4.50892e12 −1.77386 −0.886931 − 0.461903i \(-0.847167\pi\)
−0.886931 + 0.461903i \(0.847167\pi\)
\(4\) 4.50360e15 1.00000
\(5\) 2.56039e18 1.71825 0.859125 − 0.511767i \(-0.171009\pi\)
0.859125 + 0.511767i \(0.171009\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.38693e25 2.14658
\(10\) 0 0
\(11\) 1.19182e27 1.00000
\(12\) −2.03064e28 −1.77386
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −1.15446e31 −3.04794
\(16\) 2.02824e31 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.15310e34 1.71825
\(21\) 0 0
\(22\) 0 0
\(23\) 4.48364e35 1.76485 0.882424 − 0.470455i \(-0.155910\pi\)
0.882424 + 0.470455i \(0.155910\pi\)
\(24\) 0 0
\(25\) 4.33515e36 1.95238
\(26\) 0 0
\(27\) −3.34028e37 −2.03388
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 7.23932e38 1.21421 0.607105 − 0.794622i \(-0.292331\pi\)
0.607105 + 0.794622i \(0.292331\pi\)
\(32\) 0 0
\(33\) −5.37381e39 −1.77386
\(34\) 0 0
\(35\) 0 0
\(36\) 6.24616e40 2.14658
\(37\) 1.06070e41 1.78792 0.893960 − 0.448147i \(-0.147916\pi\)
0.893960 + 0.448147i \(0.147916\pi\)
\(38\) 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 \(\pi\)
\(44\) 5.36747e42 1.00000
\(45\) 3.55107e43 3.68837
\(46\) 0 0
\(47\) 2.56855e43 0.861278 0.430639 − 0.902524i \(-0.358288\pi\)
0.430639 + 0.902524i \(0.358288\pi\)
\(48\) −9.14517e43 −1.77386
\(49\) 8.81248e43 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.34968e45 −1.99094 −0.995471 − 0.0950618i \(-0.969695\pi\)
−0.995471 + 0.0950618i \(0.969695\pi\)
\(54\) 0 0
\(55\) 3.05152e45 1.71825
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.93259e45 0.629135 0.314567 − 0.949235i \(-0.398141\pi\)
0.314567 + 0.949235i \(0.398141\pi\)
\(60\) −5.19922e46 −3.04794
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 9.13439e46 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.00329e47 −1.66460 −0.832302 − 0.554322i \(-0.812978\pi\)
−0.832302 + 0.554322i \(0.812978\pi\)
\(68\) 0 0
\(69\) −2.02164e48 −3.13060
\(70\) 0 0
\(71\) −2.69589e48 −1.98603 −0.993013 − 0.118003i \(-0.962351\pi\)
−0.993013 + 0.118003i \(0.962351\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −1.95469e49 −3.46325
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 5.19309e49 1.71825
\(81\) 6.10003e49 1.46124
\(82\) 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
\(89\) 7.24276e50 1.49887 0.749435 − 0.662078i \(-0.230325\pi\)
0.749435 + 0.662078i \(0.230325\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.01925e51 1.76485
\(93\) −3.26415e51 −2.15384
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.69771e51 −1.47864 −0.739318 − 0.673356i \(-0.764852\pi\)
−0.739318 + 0.673356i \(0.764852\pi\)
\(98\) 0 0
\(99\) 1.65296e52 2.14658
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 11.53.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.53.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.53.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.53.b.a.10.1 ✓ 1 11.10 odd 2 CM