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Properties

Label 4788.2.a.k.1.2
Level $4788$
Weight $2$
Character 4788.1
Self dual yes
Analytic conductor $38.232$
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] = [4788,2,Mod(1,4788)] 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("4788.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4788, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4788.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,-2,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.2323724878\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1596)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 4788.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.64575 q^{5} -1.00000 q^{7} -5.29150 q^{11} -0.354249 q^{17} +1.00000 q^{19} -5.29150 q^{23} +8.29150 q^{25} +9.64575 q^{29} -6.00000 q^{31} -3.64575 q^{35} +5.29150 q^{37} +3.29150 q^{41} -1.29150 q^{43} +4.35425 q^{47} +1.00000 q^{49} +13.6458 q^{53} -19.2915 q^{55} +8.00000 q^{59} +9.29150 q^{61} +7.29150 q^{67} +8.93725 q^{71} -2.00000 q^{73} +5.29150 q^{77} -3.29150 q^{79} +14.9373 q^{83} -1.29150 q^{85} +3.29150 q^{89} +3.64575 q^{95} +4.58301 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{7} - 6 q^{17} + 2 q^{19} + 6 q^{25} + 14 q^{29} - 12 q^{31} - 2 q^{35} - 4 q^{41} + 8 q^{43} + 14 q^{47} + 2 q^{49} + 22 q^{53} - 28 q^{55} + 16 q^{59} + 8 q^{61} + 4 q^{67} + 2 q^{71}+ \cdots - 12 q^{97}+O(q^{100}) \) 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
\(5\) 3.64575 1.63043 0.815215 − 0.579159i \(-0.196619\pi\)
0.815215 + 0.579159i \(0.196619\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.29150 −1.59545 −0.797724 − 0.603023i \(-0.793963\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.354249 −0.0859179 −0.0429590 − 0.999077i \(-0.513678\pi\)
−0.0429590 + 0.999077i \(0.513678\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.29150 −1.10335 −0.551677 − 0.834058i \(-0.686012\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) 0 0
\(25\) 8.29150 1.65830
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.64575 1.79117 0.895586 − 0.444889i \(-0.146757\pi\)
0.895586 + 0.444889i \(0.146757\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.64575 −0.616244
\(36\) 0 0
\(37\) 5.29150 0.869918 0.434959 − 0.900450i \(-0.356763\pi\)
0.434959 + 0.900450i \(0.356763\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.29150 0.514046 0.257023 − 0.966405i \(-0.417258\pi\)
0.257023 + 0.966405i \(0.417258\pi\)
\(42\) 0 0
\(43\) −1.29150 −0.196952 −0.0984762 − 0.995139i \(-0.531397\pi\)
−0.0984762 + 0.995139i \(0.531397\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.35425 0.635132 0.317566 − 0.948236i \(-0.397134\pi\)
0.317566 + 0.948236i \(0.397134\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.6458 1.87439 0.937194 − 0.348808i \(-0.113414\pi\)
0.937194 + 0.348808i \(0.113414\pi\)
\(54\) 0 0
\(55\) −19.2915 −2.60127
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 9.29150 1.18966 0.594828 − 0.803853i \(-0.297220\pi\)
0.594828 + 0.803853i \(0.297220\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.29150 0.890799 0.445399 − 0.895332i \(-0.353062\pi\)
0.445399 + 0.895332i \(0.353062\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.93725 1.06066 0.530328 − 0.847792i \(-0.322069\pi\)
0.530328 + 0.847792i \(0.322069\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.29150 0.603023
\(78\) 0 0
\(79\) −3.29150 −0.370323 −0.185161 − 0.982708i \(-0.559281\pi\)
−0.185161 + 0.982708i \(0.559281\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.9373 1.63958 0.819788 − 0.572667i \(-0.194091\pi\)
0.819788 + 0.572667i \(0.194091\pi\)
\(84\) 0 0
\(85\) −1.29150 −0.140083
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.29150 0.348899 0.174449 − 0.984666i \(-0.444185\pi\)
0.174449 + 0.984666i \(0.444185\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.64575 0.374046
\(96\) 0 0
\(97\) 4.58301 0.465334 0.232667 − 0.972556i \(-0.425255\pi\)
0.232667 + 0.972556i \(0.425255\pi\)
\(98\) 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 4788.2.a.k.1.2 2
3.2 odd 2 1596.2.a.f.1.1 ✓ 2
12.11 even 2 6384.2.a.bo.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1596.2.a.f.1.1 ✓ 2 3.2 odd 2
4788.2.a.k.1.2 2 1.1 even 1 trivial
6384.2.a.bo.1.1 2 12.11 even 2