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Properties

Label 4752.2.a.bc.1.1
Level $4752$
Weight $2$
Character 4752.1
Self dual yes
Analytic conductor $37.945$
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] = [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 = [2,0,0,0,0,0,2,0,0,0,2,0,-2] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 1.1
Root \(-1.41421\) 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)
 
\(f(q)\) \(=\) \(q-1.41421 q^{5} -0.414214 q^{7} +1.00000 q^{11} -2.41421 q^{13} +6.24264 q^{17} +5.24264 q^{19} -5.65685 q^{23} -3.00000 q^{25} +0.585786 q^{29} -3.65685 q^{31} +0.585786 q^{35} -3.82843 q^{37} +2.82843 q^{41} +11.6569 q^{43} +8.24264 q^{47} -6.82843 q^{49} -5.89949 q^{53} -1.41421 q^{55} +2.58579 q^{59} -7.58579 q^{61} +3.41421 q^{65} -8.31371 q^{67} +4.00000 q^{71} +10.8995 q^{73} -0.414214 q^{77} -0.0710678 q^{79} +9.65685 q^{83} -8.82843 q^{85} -0.242641 q^{89} +1.00000 q^{91} -7.41421 q^{95} -14.6569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} + 2 q^{11} - 2 q^{13} + 4 q^{17} + 2 q^{19} - 6 q^{25} + 4 q^{29} + 4 q^{31} + 4 q^{35} - 2 q^{37} + 12 q^{43} + 8 q^{47} - 8 q^{49} + 8 q^{53} + 8 q^{59} - 18 q^{61} + 4 q^{65} + 6 q^{67}+ \cdots - 18 q^{97}+O(q^{100}) \) 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
\(5\) −1.41421 −0.632456 −0.316228 − 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) −0.414214 −0.156558 −0.0782790 − 0.996931i \(-0.524942\pi\)
−0.0782790 + 0.996931i \(0.524942\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.41421 −0.669582 −0.334791 − 0.942292i \(-0.608666\pi\)
−0.334791 + 0.942292i \(0.608666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.24264 1.51406 0.757031 − 0.653379i \(-0.226649\pi\)
0.757031 + 0.653379i \(0.226649\pi\)
\(18\) 0 0
\(19\) 5.24264 1.20274 0.601372 − 0.798969i \(-0.294621\pi\)
0.601372 + 0.798969i \(0.294621\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.65685 −1.17954 −0.589768 − 0.807573i \(-0.700781\pi\)
−0.589768 + 0.807573i \(0.700781\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.585786 0.108778 0.0543889 − 0.998520i \(-0.482679\pi\)
0.0543889 + 0.998520i \(0.482679\pi\)
\(30\) 0 0
\(31\) −3.65685 −0.656790 −0.328395 − 0.944540i \(-0.606508\pi\)
−0.328395 + 0.944540i \(0.606508\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.585786 0.0990160
\(36\) 0 0
\(37\) −3.82843 −0.629390 −0.314695 − 0.949193i \(-0.601902\pi\)
−0.314695 + 0.949193i \(0.601902\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.82843 0.441726 0.220863 − 0.975305i \(-0.429113\pi\)
0.220863 + 0.975305i \(0.429113\pi\)
\(42\) 0 0
\(43\) 11.6569 1.77765 0.888827 − 0.458243i \(-0.151521\pi\)
0.888827 + 0.458243i \(0.151521\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.24264 1.20231 0.601156 − 0.799131i \(-0.294707\pi\)
0.601156 + 0.799131i \(0.294707\pi\)
\(48\) 0 0
\(49\) −6.82843 −0.975490
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.89949 −0.810358 −0.405179 − 0.914237i \(-0.632791\pi\)
−0.405179 + 0.914237i \(0.632791\pi\)
\(54\) 0 0
\(55\) −1.41421 −0.190693
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.58579 0.336641 0.168320 − 0.985732i \(-0.446166\pi\)
0.168320 + 0.985732i \(0.446166\pi\)
\(60\) 0 0
\(61\) −7.58579 −0.971260 −0.485630 − 0.874164i \(-0.661410\pi\)
−0.485630 + 0.874164i \(0.661410\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.41421 0.423481
\(66\) 0 0
\(67\) −8.31371 −1.01568 −0.507841 − 0.861451i \(-0.669556\pi\)
−0.507841 + 0.861451i \(0.669556\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 10.8995 1.27569 0.637845 − 0.770165i \(-0.279826\pi\)
0.637845 + 0.770165i \(0.279826\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.414214 −0.0472040
\(78\) 0 0
\(79\) −0.0710678 −0.00799575 −0.00399788 − 0.999992i \(-0.501273\pi\)
−0.00399788 + 0.999992i \(0.501273\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.65685 1.05998 0.529989 − 0.848005i \(-0.322196\pi\)
0.529989 + 0.848005i \(0.322196\pi\)
\(84\) 0 0
\(85\) −8.82843 −0.957577
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.242641 −0.0257199 −0.0128599 − 0.999917i \(-0.504094\pi\)
−0.0128599 + 0.999917i \(0.504094\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.41421 −0.760682
\(96\) 0 0
\(97\) −14.6569 −1.48818 −0.744089 − 0.668080i \(-0.767116\pi\)
−0.744089 + 0.668080i \(0.767116\pi\)
\(98\) 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.bc.1.1 2
3.2 odd 2 4752.2.a.ba.1.2 2
4.3 odd 2 2376.2.a.f.1.1 ✓ 2
12.11 even 2 2376.2.a.h.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2376.2.a.f.1.1 ✓ 2 4.3 odd 2
2376.2.a.h.1.2 yes 2 12.11 even 2
4752.2.a.ba.1.2 2 3.2 odd 2
4752.2.a.bc.1.1 2 1.1 even 1 trivial