Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5520,2,Mod(1,5520)] 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("5520.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,3,0,-1,0,3,0,0,0,4,0,-3,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +3.58774 q^{7} +1.00000 q^{9} +0.715853 q^{13} -1.00000 q^{15} -1.58774 q^{17} -3.58774 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.58774 q^{29} -4.87189 q^{31} +3.58774 q^{35} +8.15604 q^{37} -0.715853 q^{39} +4.30359 q^{41} -8.45963 q^{43} +1.00000 q^{45} +7.17548 q^{47} +5.87189 q^{49} +1.58774 q^{51} +8.30359 q^{53} +2.30359 q^{59} +6.45963 q^{61} +3.58774 q^{63} +0.715853 q^{65} +6.15604 q^{67} +1.00000 q^{69} +1.01945 q^{71} -5.17548 q^{73} -1.00000 q^{75} -9.89134 q^{79} +1.00000 q^{81} -5.01945 q^{83} -1.58774 q^{85} -5.58774 q^{87} +15.7438 q^{89} +2.56829 q^{91} +4.87189 q^{93} -6.45963 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 4 q^{13} - 3 q^{15} + 7 q^{17} + q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 5 q^{29} - q^{31} - q^{35} + 9 q^{37} - 4 q^{39} + 3 q^{41} + 3 q^{45} - 2 q^{47}+ \cdots + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.58774 1.35604 0.678019 − 0.735044i \(-0.262839\pi\)
0.678019 + 0.735044i \(0.262839\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0.715853 0.198542 0.0992709 − 0.995060i \(-0.468349\pi\)
0.0992709 + 0.995060i \(0.468349\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.58774 −0.385084 −0.192542 − 0.981289i \(-0.561673\pi\)
−0.192542 + 0.981289i \(0.561673\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −3.58774 −0.782909
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.58774 1.03762 0.518809 − 0.854890i \(-0.326376\pi\)
0.518809 + 0.854890i \(0.326376\pi\)
\(30\) 0 0
\(31\) −4.87189 −0.875017 −0.437509 − 0.899214i \(-0.644139\pi\)
−0.437509 + 0.899214i \(0.644139\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.58774 0.606439
\(36\) 0 0
\(37\) 8.15604 1.34084 0.670422 − 0.741980i \(-0.266113\pi\)
0.670422 + 0.741980i \(0.266113\pi\)
\(38\) 0 0
\(39\) −0.715853 −0.114628
\(40\) 0 0
\(41\) 4.30359 0.672108 0.336054 − 0.941843i \(-0.390907\pi\)
0.336054 + 0.941843i \(0.390907\pi\)
\(42\) 0 0
\(43\) −8.45963 −1.29008 −0.645041 − 0.764148i \(-0.723160\pi\)
−0.645041 + 0.764148i \(0.723160\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 7.17548 1.04665 0.523326 − 0.852133i \(-0.324691\pi\)
0.523326 + 0.852133i \(0.324691\pi\)
\(48\) 0 0
\(49\) 5.87189 0.838841
\(50\) 0 0
\(51\) 1.58774 0.222328
\(52\) 0 0
\(53\) 8.30359 1.14059 0.570293 − 0.821441i \(-0.306830\pi\)
0.570293 + 0.821441i \(0.306830\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.30359 0.299902 0.149951 − 0.988693i \(-0.452088\pi\)
0.149951 + 0.988693i \(0.452088\pi\)
\(60\) 0 0
\(61\) 6.45963 0.827071 0.413535 − 0.910488i \(-0.364294\pi\)
0.413535 + 0.910488i \(0.364294\pi\)
\(62\) 0 0
\(63\) 3.58774 0.452013
\(64\) 0 0
\(65\) 0.715853 0.0887906
\(66\) 0 0
\(67\) 6.15604 0.752079 0.376040 − 0.926604i \(-0.377286\pi\)
0.376040 + 0.926604i \(0.377286\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 1.01945 0.120986 0.0604930 − 0.998169i \(-0.480733\pi\)
0.0604930 + 0.998169i \(0.480733\pi\)
\(72\) 0 0
\(73\) −5.17548 −0.605744 −0.302872 − 0.953031i \(-0.597946\pi\)
−0.302872 + 0.953031i \(0.597946\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −9.89134 −1.11286 −0.556431 − 0.830894i \(-0.687830\pi\)
−0.556431 + 0.830894i \(0.687830\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.01945 −0.550956 −0.275478 − 0.961307i \(-0.588836\pi\)
−0.275478 + 0.961307i \(0.588836\pi\)
\(84\) 0 0
\(85\) −1.58774 −0.172215
\(86\) 0 0
\(87\) −5.58774 −0.599069
\(88\) 0 0
\(89\) 15.7438 1.66884 0.834419 − 0.551131i \(-0.185804\pi\)
0.834419 + 0.551131i \(0.185804\pi\)
\(90\) 0 0
\(91\) 2.56829 0.269230
\(92\) 0 0
\(93\) 4.87189 0.505191
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.45963 −0.655876 −0.327938 − 0.944699i \(-0.606354\pi\)
−0.327938 + 0.944699i \(0.606354\pi\)
\(98\) 0 0
\(99\) 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 5520.2.a.ca.1.3 3
4.3 odd 2 2760.2.a.t.1.1 ✓ 3
12.11 even 2 8280.2.a.bh.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.t.1.1 ✓ 3 4.3 odd 2
5520.2.a.ca.1.3 3 1.1 even 1 trivial
8280.2.a.bh.1.1 3 12.11 even 2