Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5328,2,Mod(1,5328)] 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("5328.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5328.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-1,0,2,0,0,0,-5,0,1,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(42.5442941969\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5328.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.85410 q^{5} -1.23607 q^{7} -3.61803 q^{11} +3.85410 q^{13} -4.47214 q^{17} +4.47214 q^{19} -3.85410 q^{23} +3.14590 q^{25} -6.32624 q^{29} -9.61803 q^{31} -3.52786 q^{35} -1.00000 q^{37} -7.38197 q^{41} +0.763932 q^{43} +3.23607 q^{47} -5.47214 q^{49} +8.47214 q^{53} -10.3262 q^{55} -9.23607 q^{59} +8.38197 q^{61} +11.0000 q^{65} +10.0902 q^{67} -14.9443 q^{71} -4.09017 q^{73} +4.47214 q^{77} -11.5623 q^{79} -5.52786 q^{83} -12.7639 q^{85} +10.4721 q^{89} -4.76393 q^{91} +12.7639 q^{95} +8.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 2 q^{7} - 5 q^{11} + q^{13} - q^{23} + 13 q^{25} + 3 q^{29} - 17 q^{31} - 16 q^{35} - 2 q^{37} - 17 q^{41} + 6 q^{43} + 2 q^{47} - 2 q^{49} + 8 q^{53} - 5 q^{55} - 14 q^{59} + 19 q^{61}+ \cdots + 8 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.85410 1.27639 0.638197 − 0.769873i \(-0.279681\pi\)
0.638197 + 0.769873i \(0.279681\pi\)
\(6\) 0 0
\(7\) −1.23607 −0.467190 −0.233595 − 0.972334i \(-0.575049\pi\)
−0.233595 + 0.972334i \(0.575049\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.61803 −1.09088 −0.545439 − 0.838150i \(-0.683637\pi\)
−0.545439 + 0.838150i \(0.683637\pi\)
\(12\) 0 0
\(13\) 3.85410 1.06894 0.534468 − 0.845189i \(-0.320512\pi\)
0.534468 + 0.845189i \(0.320512\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 − 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 4.47214 1.02598 0.512989 − 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.85410 −0.803636 −0.401818 − 0.915720i \(-0.631622\pi\)
−0.401818 + 0.915720i \(0.631622\pi\)
\(24\) 0 0
\(25\) 3.14590 0.629180
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.32624 −1.17475 −0.587376 − 0.809314i \(-0.699839\pi\)
−0.587376 + 0.809314i \(0.699839\pi\)
\(30\) 0 0
\(31\) −9.61803 −1.72745 −0.863725 − 0.503964i \(-0.831875\pi\)
−0.863725 + 0.503964i \(0.831875\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.52786 −0.596318
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −7.38197 −1.15287 −0.576435 − 0.817143i \(-0.695556\pi\)
−0.576435 + 0.817143i \(0.695556\pi\)
\(42\) 0 0
\(43\) 0.763932 0.116499 0.0582493 − 0.998302i \(-0.481448\pi\)
0.0582493 + 0.998302i \(0.481448\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.23607 0.472029 0.236015 − 0.971750i \(-0.424159\pi\)
0.236015 + 0.971750i \(0.424159\pi\)
\(48\) 0 0
\(49\) −5.47214 −0.781734
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.47214 1.16374 0.581869 − 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) −10.3262 −1.39239
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.23607 −1.20243 −0.601217 − 0.799086i \(-0.705317\pi\)
−0.601217 + 0.799086i \(0.705317\pi\)
\(60\) 0 0
\(61\) 8.38197 1.07320 0.536600 − 0.843836i \(-0.319708\pi\)
0.536600 + 0.843836i \(0.319708\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 11.0000 1.36438
\(66\) 0 0
\(67\) 10.0902 1.23271 0.616355 − 0.787468i \(-0.288609\pi\)
0.616355 + 0.787468i \(0.288609\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.9443 −1.77356 −0.886779 − 0.462193i \(-0.847063\pi\)
−0.886779 + 0.462193i \(0.847063\pi\)
\(72\) 0 0
\(73\) −4.09017 −0.478718 −0.239359 − 0.970931i \(-0.576937\pi\)
−0.239359 + 0.970931i \(0.576937\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.47214 0.509647
\(78\) 0 0
\(79\) −11.5623 −1.30086 −0.650431 − 0.759566i \(-0.725411\pi\)
−0.650431 + 0.759566i \(0.725411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.52786 −0.606762 −0.303381 − 0.952869i \(-0.598115\pi\)
−0.303381 + 0.952869i \(0.598115\pi\)
\(84\) 0 0
\(85\) −12.7639 −1.38444
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.4721 1.11004 0.555022 − 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) −4.76393 −0.499396
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.7639 1.30955
\(96\) 0 0
\(97\) 8.47214 0.860215 0.430108 − 0.902778i \(-0.358476\pi\)
0.430108 + 0.902778i \(0.358476\pi\)
\(98\) 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 5328.2.a.bc.1.2 2
3.2 odd 2 592.2.a.g.1.1 2
4.3 odd 2 666.2.a.i.1.2 2
12.11 even 2 74.2.a.b.1.2 ✓ 2
24.5 odd 2 2368.2.a.u.1.2 2
24.11 even 2 2368.2.a.y.1.1 2
60.23 odd 4 1850.2.b.j.149.2 4
60.47 odd 4 1850.2.b.j.149.3 4
60.59 even 2 1850.2.a.t.1.1 2
84.83 odd 2 3626.2.a.s.1.1 2
132.131 odd 2 8954.2.a.j.1.2 2
444.443 even 2 2738.2.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.b.1.2 ✓ 2 12.11 even 2
592.2.a.g.1.1 2 3.2 odd 2
666.2.a.i.1.2 2 4.3 odd 2
1850.2.a.t.1.1 2 60.59 even 2
1850.2.b.j.149.2 4 60.23 odd 4
1850.2.b.j.149.3 4 60.47 odd 4
2368.2.a.u.1.2 2 24.5 odd 2
2368.2.a.y.1.1 2 24.11 even 2
2738.2.a.g.1.2 2 444.443 even 2
3626.2.a.s.1.1 2 84.83 odd 2
5328.2.a.bc.1.2 2 1.1 even 1 trivial
8954.2.a.j.1.2 2 132.131 odd 2