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(5183,5184)] 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("5184.5183"); 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([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.5780865024.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 9x^{4} - 16x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 324)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5183.6
Root \(1.03295 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 5184.5183
Dual form 5184.2.c.k.5183.3

$q$-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)
 
\(f(q)\) \(=\) \(q+0.517638i q^{5} +2.92163i q^{7} -5.64415 q^{11} +4.46410 q^{13} -2.31079i q^{17} -5.06040i q^{19} +1.51234 q^{23} +4.73205 q^{25} +4.76028i q^{29} -7.98203i q^{31} -1.51234 q^{35} -0.267949 q^{37} -8.10634i q^{41} -2.92163i q^{43} -4.13180 q^{47} -1.53590 q^{49} +4.52004i q^{53} -2.92163i q^{55} -4.13180 q^{59} +2.26795 q^{61} +2.31079i q^{65} +2.92163i q^{67} +9.77595 q^{71} +4.66025 q^{73} -16.4901i q^{77} -10.9037i q^{79} +11.2883 q^{83} +1.19615 q^{85} +13.5230i q^{89} +13.0424i q^{91} +2.61946 q^{95} -0.535898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{13} + 24 q^{25} - 16 q^{37} - 40 q^{49} + 32 q^{61} - 32 q^{73} - 32 q^{85} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) 0.517638i 0.231495i 0.993279 + 0.115747i \(0.0369263\pi\)
−0.993279 + 0.115747i \(0.963074\pi\)
\(6\) 0 0
\(7\) 2.92163i 1.10427i 0.833754 + 0.552135i \(0.186187\pi\)
−0.833754 + 0.552135i \(0.813813\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.64415 −1.70177 −0.850887 − 0.525348i \(-0.823935\pi\)
−0.850887 + 0.525348i \(0.823935\pi\)
\(12\) 0 0
\(13\) 4.46410 1.23812 0.619060 − 0.785344i \(-0.287514\pi\)
0.619060 + 0.785344i \(0.287514\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.31079i − 0.560449i −0.959935 − 0.280224i \(-0.909591\pi\)
0.959935 − 0.280224i \(-0.0904089\pi\)
\(18\) 0 0
\(19\) − 5.06040i − 1.16094i −0.814283 − 0.580468i \(-0.802870\pi\)
0.814283 − 0.580468i \(-0.197130\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.51234 0.315346 0.157673 − 0.987491i \(-0.449601\pi\)
0.157673 + 0.987491i \(0.449601\pi\)
\(24\) 0 0
\(25\) 4.73205 0.946410
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.76028i 0.883962i 0.897025 + 0.441981i \(0.145724\pi\)
−0.897025 + 0.441981i \(0.854276\pi\)
\(30\) 0 0
\(31\) − 7.98203i − 1.43362i −0.697271 − 0.716808i \(-0.745603\pi\)
0.697271 − 0.716808i \(-0.254397\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.51234 −0.255633
\(36\) 0 0
\(37\) −0.267949 −0.0440506 −0.0220253 − 0.999757i \(-0.507011\pi\)
−0.0220253 + 0.999757i \(0.507011\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 8.10634i − 1.26600i −0.774153 − 0.632999i \(-0.781824\pi\)
0.774153 − 0.632999i \(-0.218176\pi\)
\(42\) 0 0
\(43\) − 2.92163i − 0.445544i −0.974871 − 0.222772i \(-0.928489\pi\)
0.974871 − 0.222772i \(-0.0715105\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.13180 −0.602685 −0.301343 − 0.953516i \(-0.597435\pi\)
−0.301343 + 0.953516i \(0.597435\pi\)
\(48\) 0 0
\(49\) −1.53590 −0.219414
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.52004i 0.620876i 0.950594 + 0.310438i \(0.100476\pi\)
−0.950594 + 0.310438i \(0.899524\pi\)
\(54\) 0 0
\(55\) − 2.92163i − 0.393952i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.13180 −0.537915 −0.268957 − 0.963152i \(-0.586679\pi\)
−0.268957 + 0.963152i \(0.586679\pi\)
\(60\) 0 0
\(61\) 2.26795 0.290381 0.145191 − 0.989404i \(-0.453620\pi\)
0.145191 + 0.989404i \(0.453620\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.31079i 0.286618i
\(66\) 0 0
\(67\) 2.92163i 0.356933i 0.983946 + 0.178467i \(0.0571137\pi\)
−0.983946 + 0.178467i \(0.942886\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.77595 1.16019 0.580096 − 0.814548i \(-0.303015\pi\)
0.580096 + 0.814548i \(0.303015\pi\)
\(72\) 0 0
\(73\) 4.66025 0.545441 0.272721 − 0.962093i \(-0.412076\pi\)
0.272721 + 0.962093i \(0.412076\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 16.4901i − 1.87922i
\(78\) 0 0
\(79\) − 10.9037i − 1.22676i −0.789789 − 0.613379i \(-0.789810\pi\)
0.789789 − 0.613379i \(-0.210190\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.2883 1.23905 0.619526 − 0.784976i \(-0.287325\pi\)
0.619526 + 0.784976i \(0.287325\pi\)
\(84\) 0 0
\(85\) 1.19615 0.129741
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.5230i 1.43343i 0.697366 + 0.716716i \(0.254355\pi\)
−0.697366 + 0.716716i \(0.745645\pi\)
\(90\) 0 0
\(91\) 13.0424i 1.36722i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.61946 0.268751
\(96\) 0 0
\(97\) −0.535898 −0.0544122 −0.0272061 − 0.999630i \(-0.508661\pi\)
−0.0272061 + 0.999630i \(0.508661\pi\)
\(98\) 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.c.k.5183.6 8
3.2 odd 2 inner 5184.2.c.k.5183.4 8
4.3 odd 2 inner 5184.2.c.k.5183.5 8
8.3 odd 2 324.2.b.c.323.4 yes 8
8.5 even 2 324.2.b.c.323.6 yes 8
12.11 even 2 inner 5184.2.c.k.5183.3 8
24.5 odd 2 324.2.b.c.323.3 ✓ 8
24.11 even 2 324.2.b.c.323.5 yes 8
72.5 odd 6 324.2.h.f.107.8 16
72.11 even 6 324.2.h.f.215.1 16
72.13 even 6 324.2.h.f.107.1 16
72.29 odd 6 324.2.h.f.215.4 16
72.43 odd 6 324.2.h.f.215.8 16
72.59 even 6 324.2.h.f.107.5 16
72.61 even 6 324.2.h.f.215.5 16
72.67 odd 6 324.2.h.f.107.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.b.c.323.3 ✓ 8 24.5 odd 2
324.2.b.c.323.4 yes 8 8.3 odd 2
324.2.b.c.323.5 yes 8 24.11 even 2
324.2.b.c.323.6 yes 8 8.5 even 2
324.2.h.f.107.1 16 72.13 even 6
324.2.h.f.107.4 16 72.67 odd 6
324.2.h.f.107.5 16 72.59 even 6
324.2.h.f.107.8 16 72.5 odd 6
324.2.h.f.215.1 16 72.11 even 6
324.2.h.f.215.4 16 72.29 odd 6
324.2.h.f.215.5 16 72.61 even 6
324.2.h.f.215.8 16 72.43 odd 6
5184.2.c.k.5183.3 8 12.11 even 2 inner
5184.2.c.k.5183.4 8 3.2 odd 2 inner
5184.2.c.k.5183.5 8 4.3 odd 2 inner
5184.2.c.k.5183.6 8 1.1 even 1 trivial