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Properties

Label 5184.2.a.bw.1.2
Level $5184$
Weight $2$
Character 5184.1
Self dual yes
Analytic conductor $41.394$
Analytic rank $0$
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 = [2,0,0,0,2,0,0,0,0,0,0,0,6,0,0,0,2] 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: \(0\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 2592)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.2
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)
 
\(f(q)\) \(=\) \(q+4.46410 q^{5} -0.464102 q^{13} +7.92820 q^{17} +14.9282 q^{25} +8.46410 q^{29} -11.3923 q^{37} -10.0000 q^{41} -7.00000 q^{49} +14.0000 q^{53} +5.39230 q^{61} -2.07180 q^{65} -10.8564 q^{73} +35.3923 q^{85} -8.85641 q^{89} +18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 6 q^{13} + 2 q^{17} + 16 q^{25} + 10 q^{29} - 2 q^{37} - 20 q^{41} - 14 q^{49} + 28 q^{53} - 10 q^{61} - 18 q^{65} + 6 q^{73} + 50 q^{85} + 10 q^{89} + 36 q^{97}+O(q^{100}) \) 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
\(5\) 4.46410 1.99641 0.998203 − 0.0599153i \(-0.0190830\pi\)
0.998203 + 0.0599153i \(0.0190830\pi\)
\(6\) 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
\(13\) −0.464102 −0.128719 −0.0643593 − 0.997927i \(-0.520500\pi\)
−0.0643593 + 0.997927i \(0.520500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.92820 1.92287 0.961436 − 0.275029i \(-0.0886875\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 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
\(25\) 14.9282 2.98564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.46410 1.57174 0.785872 − 0.618389i \(-0.212214\pi\)
0.785872 + 0.618389i \(0.212214\pi\)
\(30\) 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
\(37\) −11.3923 −1.87288 −0.936442 − 0.350823i \(-0.885902\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 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
\(53\) 14.0000 1.92305 0.961524 − 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 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
\(61\) 5.39230 0.690414 0.345207 − 0.938527i \(-0.387809\pi\)
0.345207 + 0.938527i \(0.387809\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.07180 −0.256975
\(66\) 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
\(73\) −10.8564 −1.27065 −0.635323 − 0.772246i \(-0.719133\pi\)
−0.635323 + 0.772246i \(0.719133\pi\)
\(74\) 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
\(85\) 35.3923 3.83883
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.85641 −0.938777 −0.469389 − 0.882992i \(-0.655526\pi\)
−0.469389 + 0.882992i \(0.655526\pi\)
\(90\) 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 \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 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.bw.1.2 2
3.2 odd 2 5184.2.a.bm.1.1 2
4.3 odd 2 CM 5184.2.a.bw.1.2 2
8.3 odd 2 2592.2.a.k.1.1 ✓ 2
8.5 even 2 2592.2.a.k.1.1 ✓ 2
12.11 even 2 5184.2.a.bm.1.1 2
24.5 odd 2 2592.2.a.q.1.2 yes 2
24.11 even 2 2592.2.a.q.1.2 yes 2
72.5 odd 6 2592.2.i.ba.865.1 4
72.11 even 6 2592.2.i.ba.1729.1 4
72.13 even 6 2592.2.i.be.865.2 4
72.29 odd 6 2592.2.i.ba.1729.1 4
72.43 odd 6 2592.2.i.be.1729.2 4
72.59 even 6 2592.2.i.ba.865.1 4
72.61 even 6 2592.2.i.be.1729.2 4
72.67 odd 6 2592.2.i.be.865.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2592.2.a.k.1.1 ✓ 2 8.3 odd 2
2592.2.a.k.1.1 ✓ 2 8.5 even 2
2592.2.a.q.1.2 yes 2 24.5 odd 2
2592.2.a.q.1.2 yes 2 24.11 even 2
2592.2.i.ba.865.1 4 72.5 odd 6
2592.2.i.ba.865.1 4 72.59 even 6
2592.2.i.ba.1729.1 4 72.11 even 6
2592.2.i.ba.1729.1 4 72.29 odd 6
2592.2.i.be.865.2 4 72.13 even 6
2592.2.i.be.865.2 4 72.67 odd 6
2592.2.i.be.1729.2 4 72.43 odd 6
2592.2.i.be.1729.2 4 72.61 even 6
5184.2.a.bm.1.1 2 3.2 odd 2
5184.2.a.bm.1.1 2 12.11 even 2
5184.2.a.bw.1.2 2 1.1 even 1 trivial
5184.2.a.bw.1.2 2 4.3 odd 2 CM