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Properties

Label 5184.2.a.bh.1.1
Level $5184$
Weight $2$
Character 5184.1
Self dual yes
Analytic conductor $41.394$
Analytic rank $1$
Dimension $2$
CM discriminant -4
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,-4,0,0,0,0,0,0,0,-6,0,0,0,8] 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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2592)
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) 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)
 
\(f(q)\) \(=\) \(q-3.73205 q^{5} -6.46410 q^{13} +5.73205 q^{17} +8.92820 q^{25} +10.6603 q^{29} +9.39230 q^{37} +8.00000 q^{41} -7.00000 q^{49} -4.00000 q^{53} -15.3923 q^{61} +24.1244 q^{65} -16.8564 q^{73} -21.3923 q^{85} -0.660254 q^{89} -18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 6 q^{13} + 8 q^{17} + 4 q^{25} + 4 q^{29} - 2 q^{37} + 16 q^{41} - 14 q^{49} - 8 q^{53} - 10 q^{61} + 24 q^{65} - 6 q^{73} - 22 q^{85} + 16 q^{89} - 36 q^{97}+O(q^{100}) \) 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
\(5\) −3.73205 −1.66902 −0.834512 − 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −6.46410 −1.79282 −0.896410 − 0.443227i \(-0.853834\pi\)
−0.896410 + 0.443227i \(0.853834\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.73205 1.39023 0.695113 − 0.718900i \(-0.255354\pi\)
0.695113 + 0.718900i \(0.255354\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 8.92820 1.78564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.6603 1.97956 0.989780 − 0.142605i \(-0.0455477\pi\)
0.989780 + 0.142605i \(0.0455477\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.39230 1.54409 0.772043 − 0.635571i \(-0.219235\pi\)
0.772043 + 0.635571i \(0.219235\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −15.3923 −1.97078 −0.985391 − 0.170305i \(-0.945525\pi\)
−0.985391 + 0.170305i \(0.945525\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 24.1244 2.99226
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −16.8564 −1.97289 −0.986447 − 0.164083i \(-0.947534\pi\)
−0.986447 + 0.164083i \(0.947534\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −21.3923 −2.32032
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.660254 −0.0699868 −0.0349934 − 0.999388i \(-0.511141\pi\)
−0.0349934 + 0.999388i \(0.511141\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 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.bh.1.1 2
3.2 odd 2 5184.2.a.ca.1.2 2
4.3 odd 2 CM 5184.2.a.bh.1.1 2
8.3 odd 2 2592.2.a.t.1.2 yes 2
8.5 even 2 2592.2.a.t.1.2 yes 2
12.11 even 2 5184.2.a.ca.1.2 2
24.5 odd 2 2592.2.a.i.1.1 ✓ 2
24.11 even 2 2592.2.a.i.1.1 ✓ 2
72.5 odd 6 2592.2.i.bf.865.2 4
72.11 even 6 2592.2.i.bf.1729.2 4
72.13 even 6 2592.2.i.y.865.1 4
72.29 odd 6 2592.2.i.bf.1729.2 4
72.43 odd 6 2592.2.i.y.1729.1 4
72.59 even 6 2592.2.i.bf.865.2 4
72.61 even 6 2592.2.i.y.1729.1 4
72.67 odd 6 2592.2.i.y.865.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2592.2.a.i.1.1 ✓ 2 24.5 odd 2
2592.2.a.i.1.1 ✓ 2 24.11 even 2
2592.2.a.t.1.2 yes 2 8.3 odd 2
2592.2.a.t.1.2 yes 2 8.5 even 2
2592.2.i.y.865.1 4 72.13 even 6
2592.2.i.y.865.1 4 72.67 odd 6
2592.2.i.y.1729.1 4 72.43 odd 6
2592.2.i.y.1729.1 4 72.61 even 6
2592.2.i.bf.865.2 4 72.5 odd 6
2592.2.i.bf.865.2 4 72.59 even 6
2592.2.i.bf.1729.2 4 72.11 even 6
2592.2.i.bf.1729.2 4 72.29 odd 6
5184.2.a.bh.1.1 2 1.1 even 1 trivial
5184.2.a.bh.1.1 2 4.3 odd 2 CM
5184.2.a.ca.1.2 2 3.2 odd 2
5184.2.a.ca.1.2 2 12.11 even 2