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Properties

Label 5184.2.a.bs.1.1
Level $5184$
Weight $2$
Character 5184.1
Self dual yes
Analytic conductor $41.394$
Analytic rank $0$
Dimension $2$
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 = [2,0,0,0,1,0,-3,0,0,0,2,0,-5,0,0,0,-5] 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: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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
Root \(-2.37228\) 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)
 
\(f(q)\) \(=\) \(q-2.37228 q^{5} -4.37228 q^{7} +1.00000 q^{11} +0.372281 q^{13} -5.37228 q^{17} +0.627719 q^{19} +0.372281 q^{23} +0.627719 q^{25} +4.37228 q^{29} -6.37228 q^{31} +10.3723 q^{35} -8.74456 q^{37} -11.7446 q^{41} -1.74456 q^{43} -4.37228 q^{47} +12.1168 q^{49} -0.744563 q^{53} -2.37228 q^{55} +7.00000 q^{59} -2.37228 q^{61} -0.883156 q^{65} -3.74456 q^{67} +4.00000 q^{71} -12.1168 q^{73} -4.37228 q^{77} +6.37228 q^{79} +9.62772 q^{83} +12.7446 q^{85} +6.00000 q^{89} -1.62772 q^{91} -1.48913 q^{95} +1.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - 3 q^{7} + 2 q^{11} - 5 q^{13} - 5 q^{17} + 7 q^{19} - 5 q^{23} + 7 q^{25} + 3 q^{29} - 7 q^{31} + 15 q^{35} - 6 q^{37} - 12 q^{41} + 8 q^{43} - 3 q^{47} + 7 q^{49} + 10 q^{53} + q^{55} + 14 q^{59}+ \cdots - 8 q^{97}+O(q^{100}) \) 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
\(5\) −2.37228 −1.06092 −0.530458 − 0.847711i \(-0.677980\pi\)
−0.530458 + 0.847711i \(0.677980\pi\)
\(6\) 0 0
\(7\) −4.37228 −1.65257 −0.826284 − 0.563254i \(-0.809549\pi\)
−0.826284 + 0.563254i \(0.809549\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 0.372281 0.103252 0.0516261 − 0.998666i \(-0.483560\pi\)
0.0516261 + 0.998666i \(0.483560\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.37228 −1.30297 −0.651485 − 0.758662i \(-0.725854\pi\)
−0.651485 + 0.758662i \(0.725854\pi\)
\(18\) 0 0
\(19\) 0.627719 0.144009 0.0720043 − 0.997404i \(-0.477060\pi\)
0.0720043 + 0.997404i \(0.477060\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.372281 0.0776260 0.0388130 − 0.999246i \(-0.487642\pi\)
0.0388130 + 0.999246i \(0.487642\pi\)
\(24\) 0 0
\(25\) 0.627719 0.125544
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.37228 0.811912 0.405956 − 0.913893i \(-0.366939\pi\)
0.405956 + 0.913893i \(0.366939\pi\)
\(30\) 0 0
\(31\) −6.37228 −1.14450 −0.572248 − 0.820081i \(-0.693928\pi\)
−0.572248 + 0.820081i \(0.693928\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.3723 1.75324
\(36\) 0 0
\(37\) −8.74456 −1.43760 −0.718799 − 0.695218i \(-0.755308\pi\)
−0.718799 + 0.695218i \(0.755308\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.7446 −1.83419 −0.917096 − 0.398666i \(-0.869473\pi\)
−0.917096 + 0.398666i \(0.869473\pi\)
\(42\) 0 0
\(43\) −1.74456 −0.266043 −0.133022 − 0.991113i \(-0.542468\pi\)
−0.133022 + 0.991113i \(0.542468\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.37228 −0.637763 −0.318881 − 0.947795i \(-0.603307\pi\)
−0.318881 + 0.947795i \(0.603307\pi\)
\(48\) 0 0
\(49\) 12.1168 1.73098
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.744563 −0.102274 −0.0511368 − 0.998692i \(-0.516284\pi\)
−0.0511368 + 0.998692i \(0.516284\pi\)
\(54\) 0 0
\(55\) −2.37228 −0.319878
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.00000 0.911322 0.455661 − 0.890153i \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) 0 0
\(61\) −2.37228 −0.303739 −0.151870 − 0.988401i \(-0.548529\pi\)
−0.151870 + 0.988401i \(0.548529\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.883156 −0.109542
\(66\) 0 0
\(67\) −3.74456 −0.457471 −0.228736 − 0.973489i \(-0.573459\pi\)
−0.228736 + 0.973489i \(0.573459\pi\)
\(68\) 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
\(73\) −12.1168 −1.41817 −0.709085 − 0.705123i \(-0.750892\pi\)
−0.709085 + 0.705123i \(0.750892\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.37228 −0.498268
\(78\) 0 0
\(79\) 6.37228 0.716938 0.358469 − 0.933542i \(-0.383299\pi\)
0.358469 + 0.933542i \(0.383299\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.62772 1.05678 0.528390 − 0.849002i \(-0.322796\pi\)
0.528390 + 0.849002i \(0.322796\pi\)
\(84\) 0 0
\(85\) 12.7446 1.38234
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −1.62772 −0.170631
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.48913 −0.152781
\(96\) 0 0
\(97\) 1.74456 0.177133 0.0885667 − 0.996070i \(-0.471771\pi\)
0.0885667 + 0.996070i \(0.471771\pi\)
\(98\) 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.bs.1.1 2
3.2 odd 2 5184.2.a.bo.1.2 2
4.3 odd 2 5184.2.a.bt.1.1 2
8.3 odd 2 648.2.a.f.1.2 2
8.5 even 2 1296.2.a.n.1.2 2
9.2 odd 6 1728.2.i.j.577.1 4
9.4 even 3 576.2.i.l.385.2 4
9.5 odd 6 1728.2.i.j.1153.1 4
9.7 even 3 576.2.i.l.193.2 4
12.11 even 2 5184.2.a.bp.1.2 2
24.5 odd 2 1296.2.a.p.1.1 2
24.11 even 2 648.2.a.g.1.1 2
36.7 odd 6 576.2.i.j.193.1 4
36.11 even 6 1728.2.i.i.577.1 4
36.23 even 6 1728.2.i.i.1153.1 4
36.31 odd 6 576.2.i.j.385.1 4
72.5 odd 6 432.2.i.d.289.2 4
72.11 even 6 216.2.i.b.145.2 4
72.13 even 6 144.2.i.d.97.1 4
72.29 odd 6 432.2.i.d.145.2 4
72.43 odd 6 72.2.i.b.49.2 yes 4
72.59 even 6 216.2.i.b.73.2 4
72.61 even 6 144.2.i.d.49.1 4
72.67 odd 6 72.2.i.b.25.2 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.i.b.25.2 ✓ 4 72.67 odd 6
72.2.i.b.49.2 yes 4 72.43 odd 6
144.2.i.d.49.1 4 72.61 even 6
144.2.i.d.97.1 4 72.13 even 6
216.2.i.b.73.2 4 72.59 even 6
216.2.i.b.145.2 4 72.11 even 6
432.2.i.d.145.2 4 72.29 odd 6
432.2.i.d.289.2 4 72.5 odd 6
576.2.i.j.193.1 4 36.7 odd 6
576.2.i.j.385.1 4 36.31 odd 6
576.2.i.l.193.2 4 9.7 even 3
576.2.i.l.385.2 4 9.4 even 3
648.2.a.f.1.2 2 8.3 odd 2
648.2.a.g.1.1 2 24.11 even 2
1296.2.a.n.1.2 2 8.5 even 2
1296.2.a.p.1.1 2 24.5 odd 2
1728.2.i.i.577.1 4 36.11 even 6
1728.2.i.i.1153.1 4 36.23 even 6
1728.2.i.j.577.1 4 9.2 odd 6
1728.2.i.j.1153.1 4 9.5 odd 6
5184.2.a.bo.1.2 2 3.2 odd 2
5184.2.a.bp.1.2 2 12.11 even 2
5184.2.a.bs.1.1 2 1.1 even 1 trivial
5184.2.a.bt.1.1 2 4.3 odd 2