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Properties

Label 5184.2.a.s.1.1
Level $5184$
Weight $2$
Character 5184.1
Self dual yes
Analytic conductor $41.394$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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(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 = [1,0,0,0,1,0,-3,0,0,0,-5,0,5,0,0,0,-2] 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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -3.00000 q^{7} -5.00000 q^{11} +5.00000 q^{13} -2.00000 q^{17} +4.00000 q^{19} -1.00000 q^{23} -4.00000 q^{25} +9.00000 q^{29} -1.00000 q^{31} -3.00000 q^{35} +6.00000 q^{37} +3.00000 q^{41} -1.00000 q^{43} -3.00000 q^{47} +2.00000 q^{49} -2.00000 q^{53} -5.00000 q^{55} -11.0000 q^{59} -7.00000 q^{61} +5.00000 q^{65} +1.00000 q^{67} +4.00000 q^{71} -2.00000 q^{73} +15.0000 q^{77} +1.00000 q^{79} -1.00000 q^{83} -2.00000 q^{85} -18.0000 q^{89} -15.0000 q^{91} +4.00000 q^{95} -13.0000 q^{97} +O(q^{100})\)

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
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 − 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.0000 −1.43208 −0.716039 − 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.00000 0.620174
\(66\) 0 0
\(67\) 1.00000 0.122169 0.0610847 − 0.998133i \(-0.480544\pi\)
0.0610847 + 0.998133i \(0.480544\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.0000 1.70941
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 − 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.00000 −0.109764 −0.0548821 − 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) −15.0000 −1.57243
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 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.s.1.1 1
3.2 odd 2 5184.2.a.i.1.1 1
4.3 odd 2 5184.2.a.x.1.1 1
8.3 odd 2 1296.2.a.e.1.1 1
8.5 even 2 648.2.a.a.1.1 1
9.2 odd 6 1728.2.i.h.577.1 2
9.4 even 3 576.2.i.d.385.1 2
9.5 odd 6 1728.2.i.h.1153.1 2
9.7 even 3 576.2.i.d.193.1 2
12.11 even 2 5184.2.a.n.1.1 1
24.5 odd 2 648.2.a.c.1.1 1
24.11 even 2 1296.2.a.i.1.1 1
36.7 odd 6 576.2.i.c.193.1 2
36.11 even 6 1728.2.i.g.577.1 2
36.23 even 6 1728.2.i.g.1153.1 2
36.31 odd 6 576.2.i.c.385.1 2
72.5 odd 6 216.2.i.a.73.1 2
72.11 even 6 432.2.i.a.145.1 2
72.13 even 6 72.2.i.a.25.1 ✓ 2
72.29 odd 6 216.2.i.a.145.1 2
72.43 odd 6 144.2.i.b.49.1 2
72.59 even 6 432.2.i.a.289.1 2
72.61 even 6 72.2.i.a.49.1 yes 2
72.67 odd 6 144.2.i.b.97.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.i.a.25.1 ✓ 2 72.13 even 6
72.2.i.a.49.1 yes 2 72.61 even 6
144.2.i.b.49.1 2 72.43 odd 6
144.2.i.b.97.1 2 72.67 odd 6
216.2.i.a.73.1 2 72.5 odd 6
216.2.i.a.145.1 2 72.29 odd 6
432.2.i.a.145.1 2 72.11 even 6
432.2.i.a.289.1 2 72.59 even 6
576.2.i.c.193.1 2 36.7 odd 6
576.2.i.c.385.1 2 36.31 odd 6
576.2.i.d.193.1 2 9.7 even 3
576.2.i.d.385.1 2 9.4 even 3
648.2.a.a.1.1 1 8.5 even 2
648.2.a.c.1.1 1 24.5 odd 2
1296.2.a.e.1.1 1 8.3 odd 2
1296.2.a.i.1.1 1 24.11 even 2
1728.2.i.g.577.1 2 36.11 even 6
1728.2.i.g.1153.1 2 36.23 even 6
1728.2.i.h.577.1 2 9.2 odd 6
1728.2.i.h.1153.1 2 9.5 odd 6
5184.2.a.i.1.1 1 3.2 odd 2
5184.2.a.n.1.1 1 12.11 even 2
5184.2.a.s.1.1 1 1.1 even 1 trivial
5184.2.a.x.1.1 1 4.3 odd 2