Properties

Label 20.11.d.a.19.1
Level $20$
Weight $11$
Character 20.19
Self dual yes
Analytic conductor $12.707$
Analytic rank $0$
Dimension $1$
CM discriminant -20
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] = [20,11,Mod(19,20)] 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("20.19"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.7071450535\)
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 19.1
Character \(\chi\) \(=\) 20.19

$q$-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)
 
\(f(q)\) \(=\) \(q-32.0000 q^{2} -236.000 q^{3} +1024.00 q^{4} -3125.00 q^{5} +7552.00 q^{6} -33364.0 q^{7} -32768.0 q^{8} -3353.00 q^{9} +100000. q^{10} -241664. q^{12} +1.06765e6 q^{14} +737500. q^{15} +1.04858e6 q^{16} +107296. q^{18} -3.20000e6 q^{20} +7.87390e6 q^{21} +1.16956e6 q^{23} +7.73325e6 q^{24} +9.76562e6 q^{25} +1.47269e7 q^{27} -3.41647e7 q^{28} -3.81797e7 q^{29} -2.36000e7 q^{30} -3.35544e7 q^{32} +1.04262e8 q^{35} -3.43347e6 q^{36} +1.02400e8 q^{40} -2.11028e8 q^{41} -2.51965e8 q^{42} +2.23663e8 q^{43} +1.04781e7 q^{45} -3.74260e7 q^{46} -9.68878e7 q^{47} -2.47464e8 q^{48} +8.30681e8 q^{49} -3.12500e8 q^{50} -4.71260e8 q^{54} +1.09327e9 q^{56} +1.22175e9 q^{58} +7.55200e8 q^{60} -1.04159e9 q^{61} +1.11869e8 q^{63} +1.07374e9 q^{64} -2.34324e9 q^{67} -2.76017e8 q^{69} -3.33640e9 q^{70} +1.09871e8 q^{72} -2.30469e9 q^{75} -3.27680e9 q^{80} -3.27755e9 q^{81} +6.75290e9 q^{82} -5.44916e9 q^{83} +8.06288e9 q^{84} -7.15723e9 q^{86} +9.01041e9 q^{87} +1.11182e10 q^{89} -3.35300e8 q^{90} +1.19763e9 q^{92} +3.10041e9 q^{94} +7.91885e9 q^{96} -2.65818e10 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −32.0000 −1.00000
\(3\) −236.000 −0.971193 −0.485597 − 0.874183i \(-0.661398\pi\)
−0.485597 + 0.874183i \(0.661398\pi\)
\(4\) 1024.00 1.00000
\(5\) −3125.00 −1.00000
\(6\) 7552.00 0.971193
\(7\) −33364.0 −1.98513 −0.992563 − 0.121735i \(-0.961154\pi\)
−0.992563 + 0.121735i \(0.961154\pi\)
\(8\) −32768.0 −1.00000
\(9\) −3353.00 −0.0567833
\(10\) 100000. 1.00000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −241664. −0.971193
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 1.06765e6 1.98513
\(15\) 737500. 0.971193
\(16\) 1.04858e6 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 107296. 0.0567833
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −3.20000e6 −1.00000
\(21\) 7.87390e6 1.92794
\(22\) 0 0
\(23\) 1.16956e6 0.181713 0.0908563 − 0.995864i \(-0.471040\pi\)
0.0908563 + 0.995864i \(0.471040\pi\)
\(24\) 7.73325e6 0.971193
\(25\) 9.76562e6 1.00000
\(26\) 0 0
\(27\) 1.47269e7 1.02634
\(28\) −3.41647e7 −1.98513
\(29\) −3.81797e7 −1.86141 −0.930706 − 0.365768i \(-0.880806\pi\)
−0.930706 + 0.365768i \(0.880806\pi\)
\(30\) −2.36000e7 −0.971193
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −3.35544e7 −1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 1.04262e8 1.98513
\(36\) −3.43347e6 −0.0567833
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.02400e8 1.00000
\(41\) −2.11028e8 −1.82147 −0.910733 − 0.412996i \(-0.864482\pi\)
−0.910733 + 0.412996i \(0.864482\pi\)
\(42\) −2.51965e8 −1.92794
\(43\) 2.23663e8 1.52143 0.760716 − 0.649085i \(-0.224848\pi\)
0.760716 + 0.649085i \(0.224848\pi\)
\(44\) 0 0
\(45\) 1.04781e7 0.0567833
\(46\) −3.74260e7 −0.181713
\(47\) −9.68878e7 −0.422454 −0.211227 − 0.977437i \(-0.567746\pi\)
−0.211227 + 0.977437i \(0.567746\pi\)
\(48\) −2.47464e8 −0.971193
\(49\) 8.30681e8 2.94072
\(50\) −3.12500e8 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −4.71260e8 −1.02634
\(55\) 0 0
\(56\) 1.09327e9 1.98513
\(57\) 0 0
\(58\) 1.22175e9 1.86141
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 7.55200e8 0.971193
\(61\) −1.04159e9 −1.23324 −0.616621 − 0.787260i \(-0.711499\pi\)
−0.616621 + 0.787260i \(0.711499\pi\)
\(62\) 0 0
\(63\) 1.11869e8 0.112722
\(64\) 1.07374e9 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.34324e9 −1.73558 −0.867788 − 0.496935i \(-0.834459\pi\)
−0.867788 + 0.496935i \(0.834459\pi\)
\(68\) 0 0
\(69\) −2.76017e8 −0.176478
\(70\) −3.33640e9 −1.98513
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.09871e8 0.0567833
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −2.30469e9 −0.971193
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −3.27680e9 −1.00000
\(81\) −3.27755e9 −0.939992
\(82\) 6.75290e9 1.82147
\(83\) −5.44916e9 −1.38337 −0.691686 − 0.722198i \(-0.743132\pi\)
−0.691686 + 0.722198i \(0.743132\pi\)
\(84\) 8.06288e9 1.92794
\(85\) 0 0
\(86\) −7.15723e9 −1.52143
\(87\) 9.01041e9 1.80779
\(88\) 0 0
\(89\) 1.11182e10 1.99106 0.995529 − 0.0944520i \(-0.0301099\pi\)
0.995529 + 0.0944520i \(0.0301099\pi\)
\(90\) −3.35300e8 −0.0567833
\(91\) 0 0
\(92\) 1.19763e9 0.181713
\(93\) 0 0
\(94\) 3.10041e9 0.422454
\(95\) 0 0
\(96\) 7.91885e9 0.971193
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −2.65818e10 −2.94072
\(99\) 0 0
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 20.11.d.a.19.1 ✓ 1
4.3 odd 2 20.11.d.b.19.1 yes 1
5.2 odd 4 100.11.b.c.51.1 2
5.3 odd 4 100.11.b.c.51.2 2
5.4 even 2 20.11.d.b.19.1 yes 1
20.3 even 4 100.11.b.c.51.1 2
20.7 even 4 100.11.b.c.51.2 2
20.19 odd 2 CM 20.11.d.a.19.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.11.d.a.19.1 ✓ 1 1.1 even 1 trivial
20.11.d.a.19.1 ✓ 1 20.19 odd 2 CM
20.11.d.b.19.1 yes 1 4.3 odd 2
20.11.d.b.19.1 yes 1 5.4 even 2
100.11.b.c.51.1 2 5.2 odd 4
100.11.b.c.51.1 2 20.3 even 4
100.11.b.c.51.2 2 5.3 odd 4
100.11.b.c.51.2 2 20.7 even 4