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Properties

Label 4752.2.a.x.1.2
Level $4752$
Weight $2$
Character 4752.1
Self dual yes
Analytic conductor $37.945$
Analytic rank $1$
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] = [4752,2,Mod(1,4752)] 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("4752.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4752, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4752 = 2^{4} \cdot 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4752.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,-2,0,0] 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: \(37.9449110405\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2376)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 4752.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.64575 q^{7} -1.00000 q^{11} -0.645751 q^{17} -5.29150 q^{19} +2.29150 q^{23} -5.00000 q^{25} +1.35425 q^{29} -9.29150 q^{31} -8.29150 q^{37} +9.93725 q^{41} +0.645751 q^{43} -6.29150 q^{47} +4.00000 q^{53} -7.00000 q^{59} -10.5830 q^{61} -2.70850 q^{67} -14.5830 q^{71} +6.58301 q^{73} -2.64575 q^{77} -3.93725 q^{79} -8.00000 q^{83} +13.2915 q^{89} +7.58301 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{11} + 4 q^{17} - 6 q^{23} - 10 q^{25} + 8 q^{29} - 8 q^{31} - 6 q^{37} + 4 q^{41} - 4 q^{43} - 2 q^{47} + 8 q^{53} - 14 q^{59} - 16 q^{67} - 8 q^{71} - 8 q^{73} + 8 q^{79} - 16 q^{83} + 16 q^{89}+ \cdots - 6 q^{97}+O(q^{100}) \) 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
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 2.64575 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.645751 −0.156618 −0.0783088 − 0.996929i \(-0.524952\pi\)
−0.0783088 + 0.996929i \(0.524952\pi\)
\(18\) 0 0
\(19\) −5.29150 −1.21395 −0.606977 − 0.794719i \(-0.707618\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.29150 0.477811 0.238906 − 0.971043i \(-0.423211\pi\)
0.238906 + 0.971043i \(0.423211\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.35425 0.251478 0.125739 − 0.992063i \(-0.459870\pi\)
0.125739 + 0.992063i \(0.459870\pi\)
\(30\) 0 0
\(31\) −9.29150 −1.66880 −0.834402 − 0.551157i \(-0.814187\pi\)
−0.834402 + 0.551157i \(0.814187\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.29150 −1.36311 −0.681557 − 0.731765i \(-0.738697\pi\)
−0.681557 + 0.731765i \(0.738697\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.93725 1.55194 0.775969 − 0.630771i \(-0.217261\pi\)
0.775969 + 0.630771i \(0.217261\pi\)
\(42\) 0 0
\(43\) 0.645751 0.0984762 0.0492381 − 0.998787i \(-0.484321\pi\)
0.0492381 + 0.998787i \(0.484321\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.29150 −0.917710 −0.458855 − 0.888511i \(-0.651740\pi\)
−0.458855 + 0.888511i \(0.651740\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.00000 −0.911322 −0.455661 − 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 0 0
\(61\) −10.5830 −1.35501 −0.677507 − 0.735516i \(-0.736940\pi\)
−0.677507 + 0.735516i \(0.736940\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.70850 −0.330896 −0.165448 − 0.986219i \(-0.552907\pi\)
−0.165448 + 0.986219i \(0.552907\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.5830 −1.73068 −0.865342 − 0.501182i \(-0.832899\pi\)
−0.865342 + 0.501182i \(0.832899\pi\)
\(72\) 0 0
\(73\) 6.58301 0.770482 0.385241 − 0.922816i \(-0.374118\pi\)
0.385241 + 0.922816i \(0.374118\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.64575 −0.301511
\(78\) 0 0
\(79\) −3.93725 −0.442976 −0.221488 − 0.975163i \(-0.571091\pi\)
−0.221488 + 0.975163i \(0.571091\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.2915 1.40890 0.704448 − 0.709755i \(-0.251195\pi\)
0.704448 + 0.709755i \(0.251195\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.58301 0.769938 0.384969 − 0.922930i \(-0.374212\pi\)
0.384969 + 0.922930i \(0.374212\pi\)
\(98\) 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 4752.2.a.x.1.2 2
3.2 odd 2 4752.2.a.y.1.2 2
4.3 odd 2 2376.2.a.j.1.1 yes 2
12.11 even 2 2376.2.a.i.1.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2376.2.a.i.1.1 ✓ 2 12.11 even 2
2376.2.a.j.1.1 yes 2 4.3 odd 2
4752.2.a.x.1.2 2 1.1 even 1 trivial
4752.2.a.y.1.2 2 3.2 odd 2