Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2496,4,Mod(1,2496)] 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("2496.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2496.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-6,0,4,0,-20,0,18,0,60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(147.268767374\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 2496.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +7.29150 q^{5} +5.87451 q^{7} +9.00000 q^{9} +51.1660 q^{11} +13.0000 q^{13} -21.8745 q^{15} -73.7490 q^{17} -59.9555 q^{19} -17.6235 q^{21} +69.8301 q^{23} -71.8340 q^{25} -27.0000 q^{27} +294.826 q^{29} -334.450 q^{31} -153.498 q^{33} +42.8340 q^{35} -261.409 q^{37} -39.0000 q^{39} +222.701 q^{41} -79.2470 q^{43} +65.6235 q^{45} -584.405 q^{47} -308.490 q^{49} +221.247 q^{51} -465.158 q^{53} +373.077 q^{55} +179.867 q^{57} +530.089 q^{59} -548.332 q^{61} +52.8706 q^{63} +94.7895 q^{65} +384.959 q^{67} -209.490 q^{69} -307.004 q^{71} -844.891 q^{73} +215.502 q^{75} +300.575 q^{77} +30.1699 q^{79} +81.0000 q^{81} -19.5633 q^{83} -537.741 q^{85} -884.478 q^{87} -513.017 q^{89} +76.3686 q^{91} +1003.35 q^{93} -437.166 q^{95} +787.903 q^{97} +460.494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 4 q^{5} - 20 q^{7} + 18 q^{9} + 60 q^{11} + 26 q^{13} - 12 q^{15} - 84 q^{17} + 60 q^{19} + 60 q^{21} - 72 q^{23} - 186 q^{25} - 54 q^{27} + 124 q^{29} - 108 q^{31} - 180 q^{33} + 128 q^{35}+ \cdots + 540 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 7.29150 0.652172 0.326086 − 0.945340i \(-0.394270\pi\)
0.326086 + 0.945340i \(0.394270\pi\)
\(6\) 0 0
\(7\) 5.87451 0.317194 0.158597 − 0.987343i \(-0.449303\pi\)
0.158597 + 0.987343i \(0.449303\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 51.1660 1.40247 0.701233 − 0.712932i \(-0.252633\pi\)
0.701233 + 0.712932i \(0.252633\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) −21.8745 −0.376532
\(16\) 0 0
\(17\) −73.7490 −1.05216 −0.526081 − 0.850434i \(-0.676339\pi\)
−0.526081 + 0.850434i \(0.676339\pi\)
\(18\) 0 0
\(19\) −59.9555 −0.723934 −0.361967 − 0.932191i \(-0.617895\pi\)
−0.361967 + 0.932191i \(0.617895\pi\)
\(20\) 0 0
\(21\) −17.6235 −0.183132
\(22\) 0 0
\(23\) 69.8301 0.633068 0.316534 − 0.948581i \(-0.397481\pi\)
0.316534 + 0.948581i \(0.397481\pi\)
\(24\) 0 0
\(25\) −71.8340 −0.574672
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 294.826 1.88786 0.943928 − 0.330151i \(-0.107100\pi\)
0.943928 + 0.330151i \(0.107100\pi\)
\(30\) 0 0
\(31\) −334.450 −1.93771 −0.968854 − 0.247634i \(-0.920347\pi\)
−0.968854 + 0.247634i \(0.920347\pi\)
\(32\) 0 0
\(33\) −153.498 −0.809714
\(34\) 0 0
\(35\) 42.8340 0.206865
\(36\) 0 0
\(37\) −261.409 −1.16150 −0.580749 − 0.814083i \(-0.697240\pi\)
−0.580749 + 0.814083i \(0.697240\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 222.701 0.848293 0.424146 − 0.905594i \(-0.360574\pi\)
0.424146 + 0.905594i \(0.360574\pi\)
\(42\) 0 0
\(43\) −79.2470 −0.281048 −0.140524 − 0.990077i \(-0.544879\pi\)
−0.140524 + 0.990077i \(0.544879\pi\)
\(44\) 0 0
\(45\) 65.6235 0.217391
\(46\) 0 0
\(47\) −584.405 −1.81371 −0.906854 − 0.421445i \(-0.861523\pi\)
−0.906854 + 0.421445i \(0.861523\pi\)
\(48\) 0 0
\(49\) −308.490 −0.899388
\(50\) 0 0
\(51\) 221.247 0.607466
\(52\) 0 0
\(53\) −465.158 −1.20555 −0.602777 − 0.797910i \(-0.705939\pi\)
−0.602777 + 0.797910i \(0.705939\pi\)
\(54\) 0 0
\(55\) 373.077 0.914649
\(56\) 0 0
\(57\) 179.867 0.417963
\(58\) 0 0
\(59\) 530.089 1.16969 0.584845 − 0.811145i \(-0.301155\pi\)
0.584845 + 0.811145i \(0.301155\pi\)
\(60\) 0 0
\(61\) −548.332 −1.15093 −0.575465 − 0.817826i \(-0.695179\pi\)
−0.575465 + 0.817826i \(0.695179\pi\)
\(62\) 0 0
\(63\) 52.8706 0.105731
\(64\) 0 0
\(65\) 94.7895 0.180880
\(66\) 0 0
\(67\) 384.959 0.701945 0.350972 − 0.936386i \(-0.385851\pi\)
0.350972 + 0.936386i \(0.385851\pi\)
\(68\) 0 0
\(69\) −209.490 −0.365502
\(70\) 0 0
\(71\) −307.004 −0.513164 −0.256582 − 0.966522i \(-0.582596\pi\)
−0.256582 + 0.966522i \(0.582596\pi\)
\(72\) 0 0
\(73\) −844.891 −1.35462 −0.677309 − 0.735699i \(-0.736854\pi\)
−0.677309 + 0.735699i \(0.736854\pi\)
\(74\) 0 0
\(75\) 215.502 0.331787
\(76\) 0 0
\(77\) 300.575 0.444853
\(78\) 0 0
\(79\) 30.1699 0.0429669 0.0214834 − 0.999769i \(-0.493161\pi\)
0.0214834 + 0.999769i \(0.493161\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −19.5633 −0.0258717 −0.0129359 − 0.999916i \(-0.504118\pi\)
−0.0129359 + 0.999916i \(0.504118\pi\)
\(84\) 0 0
\(85\) −537.741 −0.686191
\(86\) 0 0
\(87\) −884.478 −1.08995
\(88\) 0 0
\(89\) −513.017 −0.611008 −0.305504 − 0.952191i \(-0.598825\pi\)
−0.305504 + 0.952191i \(0.598825\pi\)
\(90\) 0 0
\(91\) 76.3686 0.0879737
\(92\) 0 0
\(93\) 1003.35 1.11874
\(94\) 0 0
\(95\) −437.166 −0.472129
\(96\) 0 0
\(97\) 787.903 0.824737 0.412368 − 0.911017i \(-0.364702\pi\)
0.412368 + 0.911017i \(0.364702\pi\)
\(98\) 0 0
\(99\) 460.494 0.467489
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 2496.4.a.y.1.2 2
4.3 odd 2 2496.4.a.bh.1.2 2
8.3 odd 2 624.4.a.k.1.1 2
8.5 even 2 312.4.a.e.1.1 ✓ 2
24.5 odd 2 936.4.a.f.1.2 2
24.11 even 2 1872.4.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.e.1.1 ✓ 2 8.5 even 2
624.4.a.k.1.1 2 8.3 odd 2
936.4.a.f.1.2 2 24.5 odd 2
1872.4.a.bf.1.2 2 24.11 even 2
2496.4.a.y.1.2 2 1.1 even 1 trivial
2496.4.a.bh.1.2 2 4.3 odd 2