Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6624,2,Mod(1,6624)] 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("6624.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6624 = 2^{5} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6624.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-6,0,4,0,0,0,2,0,2,0,0,0,-4,0,-6,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(52.8929062989\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.13768.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 736)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.53744\) of defining polynomial
Character \(\chi\) \(=\) 6624.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.78819 q^{5} -2.28669 q^{7} -0.788195 q^{11} +5.18787 q^{13} -4.28669 q^{17} +3.07489 q^{19} -1.00000 q^{23} +2.77403 q^{25} -3.61149 q^{29} +9.24860 q^{31} +6.37575 q^{35} +2.78819 q^{37} +2.76426 q^{41} -12.8772 q^{43} +7.05096 q^{47} -1.77103 q^{49} -0.727471 q^{53} +2.19764 q^{55} +12.1781 q^{59} +3.27253 q^{61} -14.4648 q^{65} +3.78519 q^{67} -3.89818 q^{71} +8.56662 q^{73} +1.80236 q^{77} -7.95214 q^{79} -1.01417 q^{83} +11.9521 q^{85} +4.37575 q^{89} -11.8631 q^{91} -8.57339 q^{95} -17.4335 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} + 4 q^{7} + 2 q^{11} + 2 q^{13} - 4 q^{17} - 6 q^{19} - 4 q^{23} + 12 q^{25} - 6 q^{29} + 6 q^{31} - 12 q^{35} + 6 q^{37} - 18 q^{41} - 22 q^{43} - 14 q^{47} + 8 q^{49} - 10 q^{53} + 20 q^{55}+ \cdots - 4 q^{97}+O(q^{100}) \) 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
\(5\) −2.78819 −1.24692 −0.623459 − 0.781856i \(-0.714273\pi\)
−0.623459 + 0.781856i \(0.714273\pi\)
\(6\) 0 0
\(7\) −2.28669 −0.864289 −0.432145 − 0.901804i \(-0.642243\pi\)
−0.432145 + 0.901804i \(0.642243\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.788195 −0.237650 −0.118825 − 0.992915i \(-0.537913\pi\)
−0.118825 + 0.992915i \(0.537913\pi\)
\(12\) 0 0
\(13\) 5.18787 1.43886 0.719429 − 0.694566i \(-0.244404\pi\)
0.719429 + 0.694566i \(0.244404\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.28669 −1.03968 −0.519838 − 0.854265i \(-0.674008\pi\)
−0.519838 + 0.854265i \(0.674008\pi\)
\(18\) 0 0
\(19\) 3.07489 0.705428 0.352714 − 0.935731i \(-0.385259\pi\)
0.352714 + 0.935731i \(0.385259\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 2.77403 0.554806
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.61149 −0.670636 −0.335318 − 0.942105i \(-0.608844\pi\)
−0.335318 + 0.942105i \(0.608844\pi\)
\(30\) 0 0
\(31\) 9.24860 1.66110 0.830549 − 0.556946i \(-0.188027\pi\)
0.830549 + 0.556946i \(0.188027\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.37575 1.07770
\(36\) 0 0
\(37\) 2.78819 0.458376 0.229188 − 0.973382i \(-0.426393\pi\)
0.229188 + 0.973382i \(0.426393\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.76426 0.431705 0.215853 − 0.976426i \(-0.430747\pi\)
0.215853 + 0.976426i \(0.430747\pi\)
\(42\) 0 0
\(43\) −12.8772 −1.96376 −0.981881 − 0.189498i \(-0.939314\pi\)
−0.981881 + 0.189498i \(0.939314\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.05096 1.02849 0.514244 − 0.857644i \(-0.328073\pi\)
0.514244 + 0.857644i \(0.328073\pi\)
\(48\) 0 0
\(49\) −1.77103 −0.253004
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.727471 −0.0999258 −0.0499629 − 0.998751i \(-0.515910\pi\)
−0.0499629 + 0.998751i \(0.515910\pi\)
\(54\) 0 0
\(55\) 2.19764 0.296330
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.1781 1.58545 0.792727 − 0.609576i \(-0.208660\pi\)
0.792727 + 0.609576i \(0.208660\pi\)
\(60\) 0 0
\(61\) 3.27253 0.419004 0.209502 − 0.977808i \(-0.432816\pi\)
0.209502 + 0.977808i \(0.432816\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −14.4648 −1.79414
\(66\) 0 0
\(67\) 3.78519 0.462435 0.231218 − 0.972902i \(-0.425729\pi\)
0.231218 + 0.972902i \(0.425729\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.89818 −0.462629 −0.231314 − 0.972879i \(-0.574303\pi\)
−0.231314 + 0.972879i \(0.574303\pi\)
\(72\) 0 0
\(73\) 8.56662 1.00265 0.501324 − 0.865260i \(-0.332847\pi\)
0.501324 + 0.865260i \(0.332847\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.80236 0.205398
\(78\) 0 0
\(79\) −7.95214 −0.894685 −0.447343 − 0.894363i \(-0.647630\pi\)
−0.447343 + 0.894363i \(0.647630\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.01417 −0.111319 −0.0556596 − 0.998450i \(-0.517726\pi\)
−0.0556596 + 0.998450i \(0.517726\pi\)
\(84\) 0 0
\(85\) 11.9521 1.29639
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.37575 0.463828 0.231914 − 0.972736i \(-0.425501\pi\)
0.231914 + 0.972736i \(0.425501\pi\)
\(90\) 0 0
\(91\) −11.8631 −1.24359
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.57339 −0.879611
\(96\) 0 0
\(97\) −17.4335 −1.77010 −0.885050 − 0.465495i \(-0.845876\pi\)
−0.885050 + 0.465495i \(0.845876\pi\)
\(98\) 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 6624.2.a.bf.1.2 4
3.2 odd 2 736.2.a.g.1.3 ✓ 4
4.3 odd 2 6624.2.a.be.1.2 4
12.11 even 2 736.2.a.h.1.2 yes 4
24.5 odd 2 1472.2.a.z.1.2 4
24.11 even 2 1472.2.a.y.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.g.1.3 ✓ 4 3.2 odd 2
736.2.a.h.1.2 yes 4 12.11 even 2
1472.2.a.y.1.3 4 24.11 even 2
1472.2.a.z.1.2 4 24.5 odd 2
6624.2.a.be.1.2 4 4.3 odd 2
6624.2.a.bf.1.2 4 1.1 even 1 trivial