Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5184,2,Mod(1,5184)] 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("5184.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,4,0,8,0,0,0,0] 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: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) 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 2592)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.18890\) of defining polynomial
Character \(\chi\) \(=\) 5184.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.64575 q^{5} -0.913701 q^{7} +5.58258 q^{11} -2.58258 q^{13} -1.73205 q^{17} +7.84190 q^{19} +1.58258 q^{23} +2.00000 q^{25} +0.818350 q^{29} -1.82740 q^{31} +2.41742 q^{35} +6.58258 q^{37} -8.75560 q^{41} -7.84190 q^{43} -8.00000 q^{47} -6.16515 q^{49} -8.75560 q^{53} -14.7701 q^{55} +8.00000 q^{59} +10.5826 q^{61} +6.83285 q^{65} -7.84190 q^{67} -9.58258 q^{71} +12.1652 q^{73} -5.10080 q^{77} +14.7701 q^{79} -3.16515 q^{83} +4.58258 q^{85} -8.85095 q^{89} +2.35970 q^{91} -20.7477 q^{95} +13.1652 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{11} + 8 q^{13} - 12 q^{23} + 8 q^{25} + 28 q^{35} + 8 q^{37} - 32 q^{47} + 12 q^{49} + 32 q^{59} + 24 q^{61} - 20 q^{71} + 12 q^{73} + 24 q^{83} - 28 q^{95} + 16 q^{97}+O(q^{100}) \) 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
\(5\) −2.64575 −1.18322 −0.591608 − 0.806226i \(-0.701507\pi\)
−0.591608 + 0.806226i \(0.701507\pi\)
\(6\) 0 0
\(7\) −0.913701 −0.345346 −0.172673 − 0.984979i \(-0.555240\pi\)
−0.172673 + 0.984979i \(0.555240\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.58258 1.68321 0.841605 − 0.540094i \(-0.181611\pi\)
0.841605 + 0.540094i \(0.181611\pi\)
\(12\) 0 0
\(13\) −2.58258 −0.716278 −0.358139 − 0.933668i \(-0.616589\pi\)
−0.358139 + 0.933668i \(0.616589\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.73205 −0.420084 −0.210042 − 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 0 0
\(19\) 7.84190 1.79906 0.899528 − 0.436863i \(-0.143911\pi\)
0.899528 + 0.436863i \(0.143911\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.58258 0.329990 0.164995 − 0.986294i \(-0.447239\pi\)
0.164995 + 0.986294i \(0.447239\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.818350 0.151964 0.0759819 − 0.997109i \(-0.475791\pi\)
0.0759819 + 0.997109i \(0.475791\pi\)
\(30\) 0 0
\(31\) −1.82740 −0.328211 −0.164105 − 0.986443i \(-0.552474\pi\)
−0.164105 + 0.986443i \(0.552474\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.41742 0.408619
\(36\) 0 0
\(37\) 6.58258 1.08217 0.541084 − 0.840968i \(-0.318014\pi\)
0.541084 + 0.840968i \(0.318014\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.75560 −1.36740 −0.683698 − 0.729765i \(-0.739629\pi\)
−0.683698 + 0.729765i \(0.739629\pi\)
\(42\) 0 0
\(43\) −7.84190 −1.19588 −0.597940 − 0.801541i \(-0.704014\pi\)
−0.597940 + 0.801541i \(0.704014\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −6.16515 −0.880736
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.75560 −1.20267 −0.601337 − 0.798995i \(-0.705365\pi\)
−0.601337 + 0.798995i \(0.705365\pi\)
\(54\) 0 0
\(55\) −14.7701 −1.99160
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 10.5826 1.35496 0.677480 − 0.735541i \(-0.263072\pi\)
0.677480 + 0.735541i \(0.263072\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.83285 0.847511
\(66\) 0 0
\(67\) −7.84190 −0.958041 −0.479021 − 0.877804i \(-0.659008\pi\)
−0.479021 + 0.877804i \(0.659008\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.58258 −1.13724 −0.568621 − 0.822599i \(-0.692523\pi\)
−0.568621 + 0.822599i \(0.692523\pi\)
\(72\) 0 0
\(73\) 12.1652 1.42382 0.711912 − 0.702269i \(-0.247830\pi\)
0.711912 + 0.702269i \(0.247830\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.10080 −0.581290
\(78\) 0 0
\(79\) 14.7701 1.66177 0.830883 − 0.556447i \(-0.187836\pi\)
0.830883 + 0.556447i \(0.187836\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.16515 −0.347421 −0.173710 − 0.984797i \(-0.555576\pi\)
−0.173710 + 0.984797i \(0.555576\pi\)
\(84\) 0 0
\(85\) 4.58258 0.497050
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.85095 −0.938199 −0.469100 − 0.883145i \(-0.655421\pi\)
−0.469100 + 0.883145i \(0.655421\pi\)
\(90\) 0 0
\(91\) 2.35970 0.247364
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −20.7477 −2.12867
\(96\) 0 0
\(97\) 13.1652 1.33672 0.668359 − 0.743839i \(-0.266997\pi\)
0.668359 + 0.743839i \(0.266997\pi\)
\(98\) 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 5184.2.a.ce.1.2 4
3.2 odd 2 5184.2.a.cd.1.4 4
4.3 odd 2 5184.2.a.cd.1.1 4
8.3 odd 2 2592.2.a.w.1.3 yes 4
8.5 even 2 2592.2.a.v.1.4 yes 4
12.11 even 2 inner 5184.2.a.ce.1.3 4
24.5 odd 2 2592.2.a.w.1.2 yes 4
24.11 even 2 2592.2.a.v.1.1 ✓ 4
72.5 odd 6 2592.2.i.bg.865.3 8
72.11 even 6 2592.2.i.bh.1729.4 8
72.13 even 6 2592.2.i.bh.865.1 8
72.29 odd 6 2592.2.i.bg.1729.3 8
72.43 odd 6 2592.2.i.bg.1729.2 8
72.59 even 6 2592.2.i.bh.865.4 8
72.61 even 6 2592.2.i.bh.1729.1 8
72.67 odd 6 2592.2.i.bg.865.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2592.2.a.v.1.1 ✓ 4 24.11 even 2
2592.2.a.v.1.4 yes 4 8.5 even 2
2592.2.a.w.1.2 yes 4 24.5 odd 2
2592.2.a.w.1.3 yes 4 8.3 odd 2
2592.2.i.bg.865.2 8 72.67 odd 6
2592.2.i.bg.865.3 8 72.5 odd 6
2592.2.i.bg.1729.2 8 72.43 odd 6
2592.2.i.bg.1729.3 8 72.29 odd 6
2592.2.i.bh.865.1 8 72.13 even 6
2592.2.i.bh.865.4 8 72.59 even 6
2592.2.i.bh.1729.1 8 72.61 even 6
2592.2.i.bh.1729.4 8 72.11 even 6
5184.2.a.cd.1.1 4 4.3 odd 2
5184.2.a.cd.1.4 4 3.2 odd 2
5184.2.a.ce.1.2 4 1.1 even 1 trivial
5184.2.a.ce.1.3 4 12.11 even 2 inner