Properties

Label 40.11.e.b.19.1
Level $40$
Weight $11$
Character 40.19
Self dual yes
Analytic conductor $25.414$
Analytic rank $0$
Dimension $1$
CM discriminant -40
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] = [40,11,Mod(19,40)] 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("40.19"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 40.e (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: \(25.4142901069\)
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 \(\chi\) \(=\) 40.19

$q$-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)
 
\(f(q)\) \(=\) \(q+32.0000 q^{2} +1024.00 q^{4} -3125.00 q^{5} -26886.0 q^{7} +32768.0 q^{8} +59049.0 q^{9} -100000. q^{10} +321102. q^{11} +682086. q^{13} -860352. q^{14} +1.04858e6 q^{16} +1.88957e6 q^{18} -1.28880e6 q^{19} -3.20000e6 q^{20} +1.02753e7 q^{22} +1.77219e6 q^{23} +9.76562e6 q^{25} +2.18268e7 q^{26} -2.75313e7 q^{28} +3.35544e7 q^{32} +8.40188e7 q^{35} +6.04662e7 q^{36} -1.12653e8 q^{37} -4.12417e7 q^{38} -1.02400e8 q^{40} -611598. q^{41} +3.28808e8 q^{44} -1.84528e8 q^{45} +5.67100e7 q^{46} +2.22706e8 q^{47} +4.40382e8 q^{49} +3.12500e8 q^{50} +6.98456e8 q^{52} +5.52886e8 q^{53} -1.00344e9 q^{55} -8.81000e8 q^{56} -6.46632e8 q^{59} -1.58759e9 q^{63} +1.07374e9 q^{64} -2.13152e9 q^{65} +2.68860e9 q^{70} +1.93492e9 q^{72} -3.60488e9 q^{74} -1.31973e9 q^{76} -8.63315e9 q^{77} -3.27680e9 q^{80} +3.48678e9 q^{81} -1.95711e7 q^{82} +1.05219e10 q^{88} +5.41771e9 q^{89} -5.90490e9 q^{90} -1.83386e10 q^{91} +1.81472e9 q^{92} +7.12658e9 q^{94} +4.02751e9 q^{95} +1.40922e10 q^{98} +1.89608e10 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1024.00 1.00000
\(5\) −3125.00 −1.00000
\(6\) 0 0
\(7\) −26886.0 −1.59969 −0.799845 − 0.600206i \(-0.795085\pi\)
−0.799845 + 0.600206i \(0.795085\pi\)
\(8\) 32768.0 1.00000
\(9\) 59049.0 1.00000
\(10\) −100000. −1.00000
\(11\) 321102. 1.99379 0.996895 − 0.0787374i \(-0.0250889\pi\)
0.996895 + 0.0787374i \(0.0250889\pi\)
\(12\) 0 0
\(13\) 682086. 1.83706 0.918528 − 0.395356i \(-0.129379\pi\)
0.918528 + 0.395356i \(0.129379\pi\)
\(14\) −860352. −1.59969
\(15\) 0 0
\(16\) 1.04858e6 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.88957e6 1.00000
\(19\) −1.28880e6 −0.520497 −0.260248 − 0.965542i \(-0.583804\pi\)
−0.260248 + 0.965542i \(0.583804\pi\)
\(20\) −3.20000e6 −1.00000
\(21\) 0 0
\(22\) 1.02753e7 1.99379
\(23\) 1.77219e6 0.275341 0.137670 − 0.990478i \(-0.456039\pi\)
0.137670 + 0.990478i \(0.456039\pi\)
\(24\) 0 0
\(25\) 9.76562e6 1.00000
\(26\) 2.18268e7 1.83706
\(27\) 0 0
\(28\) −2.75313e7 −1.59969
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 3.35544e7 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 8.40188e7 1.59969
\(36\) 6.04662e7 1.00000
\(37\) −1.12653e8 −1.62455 −0.812274 − 0.583276i \(-0.801771\pi\)
−0.812274 + 0.583276i \(0.801771\pi\)
\(38\) −4.12417e7 −0.520497
\(39\) 0 0
\(40\) −1.02400e8 −1.00000
\(41\) −611598. −0.00527894 −0.00263947 − 0.999997i \(-0.500840\pi\)
−0.00263947 + 0.999997i \(0.500840\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 3.28808e8 1.99379
\(45\) −1.84528e8 −1.00000
\(46\) 5.67100e7 0.275341
\(47\) 2.22706e8 0.971050 0.485525 − 0.874223i \(-0.338628\pi\)
0.485525 + 0.874223i \(0.338628\pi\)
\(48\) 0 0
\(49\) 4.40382e8 1.55901
\(50\) 3.12500e8 1.00000
\(51\) 0 0
\(52\) 6.98456e8 1.83706
\(53\) 5.52886e8 1.32208 0.661038 − 0.750352i \(-0.270116\pi\)
0.661038 + 0.750352i \(0.270116\pi\)
\(54\) 0 0
\(55\) −1.00344e9 −1.99379
\(56\) −8.81000e8 −1.59969
\(57\) 0 0
\(58\) 0 0
\(59\) −6.46632e8 −0.904477 −0.452238 − 0.891897i \(-0.649374\pi\)
−0.452238 + 0.891897i \(0.649374\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −1.58759e9 −1.59969
\(64\) 1.07374e9 1.00000
\(65\) −2.13152e9 −1.83706
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 2.68860e9 1.59969
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.93492e9 1.00000
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −3.60488e9 −1.62455
\(75\) 0 0
\(76\) −1.31973e9 −0.520497
\(77\) −8.63315e9 −3.18945
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −3.27680e9 −1.00000
\(81\) 3.48678e9 1.00000
\(82\) −1.95711e7 −0.00527894
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 1.05219e10 1.99379
\(89\) 5.41771e9 0.970211 0.485105 − 0.874456i \(-0.338781\pi\)
0.485105 + 0.874456i \(0.338781\pi\)
\(90\) −5.90490e9 −1.00000
\(91\) −1.83386e10 −2.93872
\(92\) 1.81472e9 0.275341
\(93\) 0 0
\(94\) 7.12658e9 0.971050
\(95\) 4.02751e9 0.520497
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1.40922e10 1.55901
\(99\) 1.89608e10 1.99379
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 40.11.e.b.19.1 yes 1
4.3 odd 2 160.11.e.a.79.1 1
5.4 even 2 40.11.e.a.19.1 ✓ 1
8.3 odd 2 40.11.e.a.19.1 ✓ 1
8.5 even 2 160.11.e.b.79.1 1
20.19 odd 2 160.11.e.b.79.1 1
40.19 odd 2 CM 40.11.e.b.19.1 yes 1
40.29 even 2 160.11.e.a.79.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.e.a.19.1 ✓ 1 5.4 even 2
40.11.e.a.19.1 ✓ 1 8.3 odd 2
40.11.e.b.19.1 yes 1 1.1 even 1 trivial
40.11.e.b.19.1 yes 1 40.19 odd 2 CM
160.11.e.a.79.1 1 4.3 odd 2
160.11.e.a.79.1 1 40.29 even 2
160.11.e.b.79.1 1 8.5 even 2
160.11.e.b.79.1 1 20.19 odd 2