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Properties

Label 5184.2.a.bn.1.2
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,-2,0,2,0,0,0,-2,0,-2,0,0,0,0] 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: \(\Q(\sqrt{6}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 288)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.44949\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +3.44949 q^{7} +1.44949 q^{11} -5.89898 q^{13} -4.89898 q^{17} -4.00000 q^{19} +5.44949 q^{23} -4.00000 q^{25} +0.101021 q^{29} +2.55051 q^{31} -3.44949 q^{35} +0.898979 q^{37} +11.8990 q^{41} +2.34847 q^{43} +6.34847 q^{47} +4.89898 q^{49} +8.89898 q^{53} -1.44949 q^{55} +14.3485 q^{59} +7.89898 q^{61} +5.89898 q^{65} -12.3485 q^{67} -7.79796 q^{71} -4.89898 q^{73} +5.00000 q^{77} +13.4495 q^{79} -0.550510 q^{83} +4.89898 q^{85} +12.8990 q^{89} -20.3485 q^{91} +4.00000 q^{95} -3.89898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{11} - 2 q^{13} - 8 q^{19} + 6 q^{23} - 8 q^{25} + 10 q^{29} + 10 q^{31} - 2 q^{35} - 8 q^{37} + 14 q^{41} - 10 q^{43} - 2 q^{47} + 8 q^{53} + 2 q^{55} + 14 q^{59} + 6 q^{61}+ \cdots + 2 q^{97}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 3.44949 1.30378 0.651892 − 0.758312i \(-0.273975\pi\)
0.651892 + 0.758312i \(0.273975\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.44949 0.437038 0.218519 − 0.975833i \(-0.429878\pi\)
0.218519 + 0.975833i \(0.429878\pi\)
\(12\) 0 0
\(13\) −5.89898 −1.63608 −0.818041 − 0.575160i \(-0.804940\pi\)
−0.818041 + 0.575160i \(0.804940\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.89898 −1.18818 −0.594089 − 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.44949 1.13630 0.568149 − 0.822926i \(-0.307660\pi\)
0.568149 + 0.822926i \(0.307660\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.101021 0.0187590 0.00937952 − 0.999956i \(-0.497014\pi\)
0.00937952 + 0.999956i \(0.497014\pi\)
\(30\) 0 0
\(31\) 2.55051 0.458085 0.229043 − 0.973416i \(-0.426440\pi\)
0.229043 + 0.973416i \(0.426440\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.44949 −0.583070
\(36\) 0 0
\(37\) 0.898979 0.147791 0.0738957 − 0.997266i \(-0.476457\pi\)
0.0738957 + 0.997266i \(0.476457\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.8990 1.85831 0.929154 − 0.369692i \(-0.120537\pi\)
0.929154 + 0.369692i \(0.120537\pi\)
\(42\) 0 0
\(43\) 2.34847 0.358138 0.179069 − 0.983836i \(-0.442691\pi\)
0.179069 + 0.983836i \(0.442691\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.34847 0.926019 0.463010 − 0.886353i \(-0.346770\pi\)
0.463010 + 0.886353i \(0.346770\pi\)
\(48\) 0 0
\(49\) 4.89898 0.699854
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.89898 1.22237 0.611184 − 0.791488i \(-0.290693\pi\)
0.611184 + 0.791488i \(0.290693\pi\)
\(54\) 0 0
\(55\) −1.44949 −0.195449
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.3485 1.86801 0.934006 − 0.357258i \(-0.116288\pi\)
0.934006 + 0.357258i \(0.116288\pi\)
\(60\) 0 0
\(61\) 7.89898 1.01136 0.505680 − 0.862721i \(-0.331242\pi\)
0.505680 + 0.862721i \(0.331242\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.89898 0.731678
\(66\) 0 0
\(67\) −12.3485 −1.50861 −0.754303 − 0.656527i \(-0.772025\pi\)
−0.754303 + 0.656527i \(0.772025\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7.79796 −0.925447 −0.462724 − 0.886503i \(-0.653128\pi\)
−0.462724 + 0.886503i \(0.653128\pi\)
\(72\) 0 0
\(73\) −4.89898 −0.573382 −0.286691 − 0.958023i \(-0.592555\pi\)
−0.286691 + 0.958023i \(0.592555\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) 13.4495 1.51319 0.756593 − 0.653886i \(-0.226863\pi\)
0.756593 + 0.653886i \(0.226863\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.550510 −0.0604264 −0.0302132 − 0.999543i \(-0.509619\pi\)
−0.0302132 + 0.999543i \(0.509619\pi\)
\(84\) 0 0
\(85\) 4.89898 0.531369
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.8990 1.36729 0.683645 − 0.729815i \(-0.260394\pi\)
0.683645 + 0.729815i \(0.260394\pi\)
\(90\) 0 0
\(91\) −20.3485 −2.13310
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −3.89898 −0.395881 −0.197941 − 0.980214i \(-0.563425\pi\)
−0.197941 + 0.980214i \(0.563425\pi\)
\(98\) 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.bn.1.2 2
3.2 odd 2 5184.2.a.by.1.2 2
4.3 odd 2 5184.2.a.bj.1.1 2
8.3 odd 2 2592.2.a.o.1.1 2
8.5 even 2 2592.2.a.s.1.2 2
9.2 odd 6 576.2.i.i.193.2 4
9.4 even 3 1728.2.i.k.1153.1 4
9.5 odd 6 576.2.i.i.385.1 4
9.7 even 3 1728.2.i.k.577.1 4
12.11 even 2 5184.2.a.bu.1.1 2
24.5 odd 2 2592.2.a.n.1.2 2
24.11 even 2 2592.2.a.j.1.1 2
36.7 odd 6 1728.2.i.m.577.2 4
36.11 even 6 576.2.i.m.193.1 4
36.23 even 6 576.2.i.m.385.2 4
36.31 odd 6 1728.2.i.m.1153.2 4
72.5 odd 6 288.2.i.e.97.2 yes 4
72.11 even 6 288.2.i.c.193.2 yes 4
72.13 even 6 864.2.i.c.289.1 4
72.29 odd 6 288.2.i.e.193.1 yes 4
72.43 odd 6 864.2.i.e.577.2 4
72.59 even 6 288.2.i.c.97.1 ✓ 4
72.61 even 6 864.2.i.c.577.1 4
72.67 odd 6 864.2.i.e.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.c.97.1 ✓ 4 72.59 even 6
288.2.i.c.193.2 yes 4 72.11 even 6
288.2.i.e.97.2 yes 4 72.5 odd 6
288.2.i.e.193.1 yes 4 72.29 odd 6
576.2.i.i.193.2 4 9.2 odd 6
576.2.i.i.385.1 4 9.5 odd 6
576.2.i.m.193.1 4 36.11 even 6
576.2.i.m.385.2 4 36.23 even 6
864.2.i.c.289.1 4 72.13 even 6
864.2.i.c.577.1 4 72.61 even 6
864.2.i.e.289.2 4 72.67 odd 6
864.2.i.e.577.2 4 72.43 odd 6
1728.2.i.k.577.1 4 9.7 even 3
1728.2.i.k.1153.1 4 9.4 even 3
1728.2.i.m.577.2 4 36.7 odd 6
1728.2.i.m.1153.2 4 36.31 odd 6
2592.2.a.j.1.1 2 24.11 even 2
2592.2.a.n.1.2 2 24.5 odd 2
2592.2.a.o.1.1 2 8.3 odd 2
2592.2.a.s.1.2 2 8.5 even 2
5184.2.a.bj.1.1 2 4.3 odd 2
5184.2.a.bn.1.2 2 1.1 even 1 trivial
5184.2.a.bu.1.1 2 12.11 even 2
5184.2.a.by.1.2 2 3.2 odd 2