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Properties

Label 5184.2.d.h.2593.2
Level $5184$
Weight $2$
Character 5184.2593
Analytic conductor $41.394$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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(2593,5184)] 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("5184.2593"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-40,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2593.2
Root \(-1.93649 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2593
Dual form 5184.2.d.h.2593.4

$q$-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)
 
\(f(q)\) \(=\) \(q-3.87298i q^{5} +3.87298 q^{7} +3.00000i q^{11} +3.87298i q^{13} -2.00000i q^{19} +3.87298 q^{23} -10.0000 q^{25} -3.87298i q^{29} +3.87298 q^{31} -15.0000i q^{35} +7.74597i q^{37} +9.00000 q^{41} -7.00000i q^{43} +3.87298 q^{47} +8.00000 q^{49} -7.74597i q^{53} +11.6190 q^{55} -3.00000i q^{59} +11.6190i q^{61} +15.0000 q^{65} +11.0000i q^{67} +7.74597 q^{71} +8.00000 q^{73} +11.6190i q^{77} -3.87298 q^{79} -3.00000i q^{83} -12.0000 q^{89} +15.0000i q^{91} -7.74597 q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{25} + 36 q^{41} + 32 q^{49} + 60 q^{65} + 32 q^{73} - 48 q^{89} + 4 q^{97}+O(q^{100}) \) 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
\(5\) − 3.87298i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(6\) 0 0
\(7\) 3.87298 1.46385 0.731925 − 0.681385i \(-0.238622\pi\)
0.731925 + 0.681385i \(0.238622\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) 3.87298i 1.07417i 0.843527 + 0.537086i \(0.180475\pi\)
−0.843527 + 0.537086i \(0.819525\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) − 2.00000i − 0.458831i −0.973329 − 0.229416i \(-0.926318\pi\)
0.973329 − 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.87298 0.807573 0.403786 − 0.914853i \(-0.367694\pi\)
0.403786 + 0.914853i \(0.367694\pi\)
\(24\) 0 0
\(25\) −10.0000 −2.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.87298i − 0.719195i −0.933108 − 0.359597i \(-0.882914\pi\)
0.933108 − 0.359597i \(-0.117086\pi\)
\(30\) 0 0
\(31\) 3.87298 0.695608 0.347804 − 0.937567i \(-0.386927\pi\)
0.347804 + 0.937567i \(0.386927\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 15.0000i − 2.53546i
\(36\) 0 0
\(37\) 7.74597i 1.27343i 0.771100 + 0.636715i \(0.219707\pi\)
−0.771100 + 0.636715i \(0.780293\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) − 7.00000i − 1.06749i −0.845645 − 0.533745i \(-0.820784\pi\)
0.845645 − 0.533745i \(-0.179216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.87298 0.564933 0.282466 − 0.959277i \(-0.408847\pi\)
0.282466 + 0.959277i \(0.408847\pi\)
\(48\) 0 0
\(49\) 8.00000 1.14286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 7.74597i − 1.06399i −0.846747 − 0.531995i \(-0.821442\pi\)
0.846747 − 0.531995i \(-0.178558\pi\)
\(54\) 0 0
\(55\) 11.6190 1.56670
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 3.00000i − 0.390567i −0.980747 − 0.195283i \(-0.937437\pi\)
0.980747 − 0.195283i \(-0.0625627\pi\)
\(60\) 0 0
\(61\) 11.6190i 1.48765i 0.668372 + 0.743827i \(0.266991\pi\)
−0.668372 + 0.743827i \(0.733009\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.0000 1.86052
\(66\) 0 0
\(67\) 11.0000i 1.34386i 0.740613 + 0.671932i \(0.234535\pi\)
−0.740613 + 0.671932i \(0.765465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.74597 0.919277 0.459639 − 0.888106i \(-0.347979\pi\)
0.459639 + 0.888106i \(0.347979\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.6190i 1.32410i
\(78\) 0 0
\(79\) −3.87298 −0.435745 −0.217872 − 0.975977i \(-0.569912\pi\)
−0.217872 + 0.975977i \(0.569912\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 3.00000i − 0.329293i −0.986353 − 0.164646i \(-0.947352\pi\)
0.986353 − 0.164646i \(-0.0526483\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 15.0000i 1.57243i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.74597 −0.794719
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 − 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 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.d.h.2593.2 4
3.2 odd 2 5184.2.d.g.2593.4 4
4.3 odd 2 inner 5184.2.d.h.2593.1 4
8.3 odd 2 inner 5184.2.d.h.2593.3 4
8.5 even 2 inner 5184.2.d.h.2593.4 4
9.2 odd 6 1728.2.r.d.1441.1 8
9.4 even 3 576.2.r.d.97.1 ✓ 8
9.5 odd 6 1728.2.r.d.289.3 8
9.7 even 3 576.2.r.d.481.4 yes 8
12.11 even 2 5184.2.d.g.2593.3 4
24.5 odd 2 5184.2.d.g.2593.2 4
24.11 even 2 5184.2.d.g.2593.1 4
36.7 odd 6 576.2.r.d.481.2 yes 8
36.11 even 6 1728.2.r.d.1441.2 8
36.23 even 6 1728.2.r.d.289.4 8
36.31 odd 6 576.2.r.d.97.3 yes 8
72.5 odd 6 1728.2.r.d.289.1 8
72.11 even 6 1728.2.r.d.1441.4 8
72.13 even 6 576.2.r.d.97.4 yes 8
72.29 odd 6 1728.2.r.d.1441.3 8
72.43 odd 6 576.2.r.d.481.3 yes 8
72.59 even 6 1728.2.r.d.289.2 8
72.61 even 6 576.2.r.d.481.1 yes 8
72.67 odd 6 576.2.r.d.97.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.d.97.1 ✓ 8 9.4 even 3
576.2.r.d.97.2 yes 8 72.67 odd 6
576.2.r.d.97.3 yes 8 36.31 odd 6
576.2.r.d.97.4 yes 8 72.13 even 6
576.2.r.d.481.1 yes 8 72.61 even 6
576.2.r.d.481.2 yes 8 36.7 odd 6
576.2.r.d.481.3 yes 8 72.43 odd 6
576.2.r.d.481.4 yes 8 9.7 even 3
1728.2.r.d.289.1 8 72.5 odd 6
1728.2.r.d.289.2 8 72.59 even 6
1728.2.r.d.289.3 8 9.5 odd 6
1728.2.r.d.289.4 8 36.23 even 6
1728.2.r.d.1441.1 8 9.2 odd 6
1728.2.r.d.1441.2 8 36.11 even 6
1728.2.r.d.1441.3 8 72.29 odd 6
1728.2.r.d.1441.4 8 72.11 even 6
5184.2.d.g.2593.1 4 24.11 even 2
5184.2.d.g.2593.2 4 24.5 odd 2
5184.2.d.g.2593.3 4 12.11 even 2
5184.2.d.g.2593.4 4 3.2 odd 2
5184.2.d.h.2593.1 4 4.3 odd 2 inner
5184.2.d.h.2593.2 4 1.1 even 1 trivial
5184.2.d.h.2593.3 4 8.3 odd 2 inner
5184.2.d.h.2593.4 4 8.5 even 2 inner