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Properties

Label 5184.2.a.bk.1.2
Level $5184$
Weight $2$
Character 5184.1
Self dual yes
Analytic conductor $41.394$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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,-2,0,0,0,0,0,0,0,-6,0,0,0,10] 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: \( 2 \)
Twist minimal: no (minimal twist has level 2592)
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +4.89898 q^{7} -4.89898 q^{11} -3.00000 q^{13} +5.00000 q^{17} +4.89898 q^{19} +4.89898 q^{23} -4.00000 q^{25} -5.00000 q^{29} -4.89898 q^{35} +5.00000 q^{37} -2.00000 q^{41} +4.89898 q^{43} -9.79796 q^{47} +17.0000 q^{49} -2.00000 q^{53} +4.89898 q^{55} +9.79796 q^{59} +13.0000 q^{61} +3.00000 q^{65} +4.89898 q^{67} -4.89898 q^{71} +3.00000 q^{73} -24.0000 q^{77} -14.6969 q^{79} +9.79796 q^{83} -5.00000 q^{85} +13.0000 q^{89} -14.6969 q^{91} -4.89898 q^{95} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 6 q^{13} + 10 q^{17} - 8 q^{25} - 10 q^{29} + 10 q^{37} - 4 q^{41} + 34 q^{49} - 4 q^{53} + 26 q^{61} + 6 q^{65} + 6 q^{73} - 48 q^{77} - 10 q^{85} + 26 q^{89} - 12 q^{97}+O(q^{100}) \) 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
\(7\) 4.89898 1.85164 0.925820 − 0.377964i \(-0.123376\pi\)
0.925820 + 0.377964i \(0.123376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.89898 −1.47710 −0.738549 − 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 − 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 4.89898 1.12390 0.561951 − 0.827170i \(-0.310051\pi\)
0.561951 + 0.827170i \(0.310051\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.89898 1.02151 0.510754 − 0.859727i \(-0.329366\pi\)
0.510754 + 0.859727i \(0.329366\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.89898 −0.828079
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.89898 0.747087 0.373544 − 0.927613i \(-0.378143\pi\)
0.373544 + 0.927613i \(0.378143\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.79796 −1.42918 −0.714590 − 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 0 0
\(49\) 17.0000 2.42857
\(50\) 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
\(55\) 4.89898 0.660578
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.79796 1.27559 0.637793 − 0.770208i \(-0.279848\pi\)
0.637793 + 0.770208i \(0.279848\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 − 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 4.89898 0.598506 0.299253 − 0.954174i \(-0.403263\pi\)
0.299253 + 0.954174i \(0.403263\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.89898 −0.581402 −0.290701 − 0.956814i \(-0.593888\pi\)
−0.290701 + 0.956814i \(0.593888\pi\)
\(72\) 0 0
\(73\) 3.00000 0.351123 0.175562 − 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −24.0000 −2.73505
\(78\) 0 0
\(79\) −14.6969 −1.65353 −0.826767 − 0.562544i \(-0.809823\pi\)
−0.826767 + 0.562544i \(0.809823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.79796 1.07547 0.537733 − 0.843115i \(-0.319281\pi\)
0.537733 + 0.843115i \(0.319281\pi\)
\(84\) 0 0
\(85\) −5.00000 −0.542326
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.0000 1.37800 0.688999 − 0.724763i \(-0.258051\pi\)
0.688999 + 0.724763i \(0.258051\pi\)
\(90\) 0 0
\(91\) −14.6969 −1.54066
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.89898 −0.502625
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 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.bk.1.2 2
3.2 odd 2 5184.2.a.bv.1.2 2
4.3 odd 2 inner 5184.2.a.bk.1.1 2
8.3 odd 2 2592.2.a.r.1.1 yes 2
8.5 even 2 2592.2.a.r.1.2 yes 2
12.11 even 2 5184.2.a.bv.1.1 2
24.5 odd 2 2592.2.a.m.1.2 yes 2
24.11 even 2 2592.2.a.m.1.1 ✓ 2
72.5 odd 6 2592.2.i.bd.865.1 4
72.11 even 6 2592.2.i.bd.1729.2 4
72.13 even 6 2592.2.i.z.865.1 4
72.29 odd 6 2592.2.i.bd.1729.1 4
72.43 odd 6 2592.2.i.z.1729.2 4
72.59 even 6 2592.2.i.bd.865.2 4
72.61 even 6 2592.2.i.z.1729.1 4
72.67 odd 6 2592.2.i.z.865.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2592.2.a.m.1.1 ✓ 2 24.11 even 2
2592.2.a.m.1.2 yes 2 24.5 odd 2
2592.2.a.r.1.1 yes 2 8.3 odd 2
2592.2.a.r.1.2 yes 2 8.5 even 2
2592.2.i.z.865.1 4 72.13 even 6
2592.2.i.z.865.2 4 72.67 odd 6
2592.2.i.z.1729.1 4 72.61 even 6
2592.2.i.z.1729.2 4 72.43 odd 6
2592.2.i.bd.865.1 4 72.5 odd 6
2592.2.i.bd.865.2 4 72.59 even 6
2592.2.i.bd.1729.1 4 72.29 odd 6
2592.2.i.bd.1729.2 4 72.11 even 6
5184.2.a.bk.1.1 2 4.3 odd 2 inner
5184.2.a.bk.1.2 2 1.1 even 1 trivial
5184.2.a.bv.1.1 2 12.11 even 2
5184.2.a.bv.1.2 2 3.2 odd 2