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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5184,2,Mod(2591,5184)] 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("5184.2591"); 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, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8] 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: \(16\)
Coefficient field: 16.0.7465802011608416256.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.5
Root \(-1.37379 + 0.335728i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2591
Dual form 5184.2.f.c.2591.8

$q$-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)
 
\(f(q)\) \(=\) \(q-1.27582 q^{5} -2.37228i q^{7} -3.42703i q^{11} -2.20979i q^{13} -2.37686i q^{17} -3.72601 q^{19} -7.57301 q^{23} -3.37228 q^{25} +5.57825 q^{29} +2.37228i q^{31} +3.02661i q^{35} -11.8716i q^{37} +5.19615i q^{41} +10.3554 q^{43} -7.57301 q^{47} +1.37228 q^{49} -8.60485 q^{53} +4.37228i q^{55} +8.53032i q^{59} +9.66181i q^{61} +2.81929i q^{65} +1.51622 q^{67} -10.3923 q^{71} -0.627719 q^{73} -8.12989 q^{77} +5.11684i q^{79} -12.4323i q^{83} +3.03245i q^{85} +15.1460i q^{89} -5.24224 q^{91} +4.75372 q^{95} +7.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{25} - 24 q^{49} - 56 q^{73} + 32 q^{97}+O(q^{100}) \) 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
\(5\) −1.27582 −0.570564 −0.285282 − 0.958444i \(-0.592087\pi\)
−0.285282 + 0.958444i \(0.592087\pi\)
\(6\) 0 0
\(7\) − 2.37228i − 0.896638i −0.893874 − 0.448319i \(-0.852023\pi\)
0.893874 − 0.448319i \(-0.147977\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.42703i − 1.03329i −0.856200 − 0.516645i \(-0.827181\pi\)
0.856200 − 0.516645i \(-0.172819\pi\)
\(12\) 0 0
\(13\) − 2.20979i − 0.612884i −0.951889 − 0.306442i \(-0.900861\pi\)
0.951889 − 0.306442i \(-0.0991386\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.37686i − 0.576473i −0.957559 − 0.288237i \(-0.906931\pi\)
0.957559 − 0.288237i \(-0.0930690\pi\)
\(18\) 0 0
\(19\) −3.72601 −0.854805 −0.427403 − 0.904061i \(-0.640571\pi\)
−0.427403 + 0.904061i \(0.640571\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.57301 −1.57908 −0.789541 − 0.613698i \(-0.789681\pi\)
−0.789541 + 0.613698i \(0.789681\pi\)
\(24\) 0 0
\(25\) −3.37228 −0.674456
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.57825 1.03585 0.517927 − 0.855425i \(-0.326704\pi\)
0.517927 + 0.855425i \(0.326704\pi\)
\(30\) 0 0
\(31\) 2.37228i 0.426074i 0.977044 + 0.213037i \(0.0683355\pi\)
−0.977044 + 0.213037i \(0.931664\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.02661i 0.511590i
\(36\) 0 0
\(37\) − 11.8716i − 1.95168i −0.218491 − 0.975839i \(-0.570113\pi\)
0.218491 − 0.975839i \(-0.429887\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.19615i 0.811503i 0.913984 + 0.405751i \(0.132990\pi\)
−0.913984 + 0.405751i \(0.867010\pi\)
\(42\) 0 0
\(43\) 10.3554 1.57918 0.789590 − 0.613635i \(-0.210293\pi\)
0.789590 + 0.613635i \(0.210293\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.57301 −1.10464 −0.552319 − 0.833633i \(-0.686257\pi\)
−0.552319 + 0.833633i \(0.686257\pi\)
\(48\) 0 0
\(49\) 1.37228 0.196040
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.60485 −1.18197 −0.590984 − 0.806683i \(-0.701260\pi\)
−0.590984 + 0.806683i \(0.701260\pi\)
\(54\) 0 0
\(55\) 4.37228i 0.589558i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.53032i 1.11055i 0.831666 + 0.555276i \(0.187388\pi\)
−0.831666 + 0.555276i \(0.812612\pi\)
\(60\) 0 0
\(61\) 9.66181i 1.23707i 0.785758 + 0.618534i \(0.212273\pi\)
−0.785758 + 0.618534i \(0.787727\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.81929i 0.349690i
\(66\) 0 0
\(67\) 1.51622 0.185236 0.0926181 − 0.995702i \(-0.470476\pi\)
0.0926181 + 0.995702i \(0.470476\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 − 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) −0.627719 −0.0734689 −0.0367345 − 0.999325i \(-0.511696\pi\)
−0.0367345 + 0.999325i \(0.511696\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.12989 −0.926487
\(78\) 0 0
\(79\) 5.11684i 0.575690i 0.957677 + 0.287845i \(0.0929388\pi\)
−0.957677 + 0.287845i \(0.907061\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 12.4323i − 1.36462i −0.731061 − 0.682312i \(-0.760975\pi\)
0.731061 − 0.682312i \(-0.239025\pi\)
\(84\) 0 0
\(85\) 3.03245i 0.328915i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.1460i 1.60548i 0.596332 + 0.802738i \(0.296624\pi\)
−0.596332 + 0.802738i \(0.703376\pi\)
\(90\) 0 0
\(91\) −5.24224 −0.549536
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.75372 0.487722
\(96\) 0 0
\(97\) 7.74456 0.786341 0.393171 − 0.919466i \(-0.371378\pi\)
0.393171 + 0.919466i \(0.371378\pi\)
\(98\) 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.f.c.2591.5 16
3.2 odd 2 inner 5184.2.f.c.2591.9 16
4.3 odd 2 inner 5184.2.f.c.2591.7 16
8.3 odd 2 inner 5184.2.f.c.2591.12 16
8.5 even 2 inner 5184.2.f.c.2591.10 16
9.2 odd 6 576.2.p.b.95.7 yes 16
9.4 even 3 576.2.p.b.479.8 yes 16
9.5 odd 6 1728.2.p.b.1439.4 16
9.7 even 3 1728.2.p.b.287.5 16
12.11 even 2 inner 5184.2.f.c.2591.11 16
24.5 odd 2 inner 5184.2.f.c.2591.6 16
24.11 even 2 inner 5184.2.f.c.2591.8 16
36.7 odd 6 1728.2.p.b.287.6 16
36.11 even 6 576.2.p.b.95.1 ✓ 16
36.23 even 6 1728.2.p.b.1439.3 16
36.31 odd 6 576.2.p.b.479.2 yes 16
72.5 odd 6 1728.2.p.b.1439.6 16
72.11 even 6 576.2.p.b.95.8 yes 16
72.13 even 6 576.2.p.b.479.1 yes 16
72.29 odd 6 576.2.p.b.95.2 yes 16
72.43 odd 6 1728.2.p.b.287.4 16
72.59 even 6 1728.2.p.b.1439.5 16
72.61 even 6 1728.2.p.b.287.3 16
72.67 odd 6 576.2.p.b.479.7 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.p.b.95.1 ✓ 16 36.11 even 6
576.2.p.b.95.2 yes 16 72.29 odd 6
576.2.p.b.95.7 yes 16 9.2 odd 6
576.2.p.b.95.8 yes 16 72.11 even 6
576.2.p.b.479.1 yes 16 72.13 even 6
576.2.p.b.479.2 yes 16 36.31 odd 6
576.2.p.b.479.7 yes 16 72.67 odd 6
576.2.p.b.479.8 yes 16 9.4 even 3
1728.2.p.b.287.3 16 72.61 even 6
1728.2.p.b.287.4 16 72.43 odd 6
1728.2.p.b.287.5 16 9.7 even 3
1728.2.p.b.287.6 16 36.7 odd 6
1728.2.p.b.1439.3 16 36.23 even 6
1728.2.p.b.1439.4 16 9.5 odd 6
1728.2.p.b.1439.5 16 72.59 even 6
1728.2.p.b.1439.6 16 72.5 odd 6
5184.2.f.c.2591.5 16 1.1 even 1 trivial
5184.2.f.c.2591.6 16 24.5 odd 2 inner
5184.2.f.c.2591.7 16 4.3 odd 2 inner
5184.2.f.c.2591.8 16 24.11 even 2 inner
5184.2.f.c.2591.9 16 3.2 odd 2 inner
5184.2.f.c.2591.10 16 8.5 even 2 inner
5184.2.f.c.2591.11 16 12.11 even 2 inner
5184.2.f.c.2591.12 16 8.3 odd 2 inner