Properties

Label 308.2.y.b
Level 308308
Weight 22
Character orbit 308.y
Analytic conductor 2.4592.459
Analytic rank 00
Dimension 4848
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [308,2,Mod(9,308)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(308, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([0, 10, 18])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("308.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 308=22711 308 = 2^{2} \cdot 7 \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 308.y (of order 1515, degree 88, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 2.459392382262.45939238226
Analytic rank: 00
Dimension: 4848
Relative dimension: 66 over Q(ζ15)\Q(\zeta_{15})
Twist minimal: yes
Sato-Tate group: SU(2)[C15]\mathrm{SU}(2)[C_{15}]

qq-expansion

The algebraic qq-expansion of this newform has not been computed, but we have computed the trace expansion.

Tr(f)(q)=\operatorname{Tr}(f)(q) = 48qq3+7q57q7+13q95q118q1348q15+19q17+13q1922q2130q237q25+26q2712q29+3q3128q33+37q35+19q37++140q99+O(q100) 48 q - q^{3} + 7 q^{5} - 7 q^{7} + 13 q^{9} - 5 q^{11} - 8 q^{13} - 48 q^{15} + 19 q^{17} + 13 q^{19} - 22 q^{21} - 30 q^{23} - 7 q^{25} + 26 q^{27} - 12 q^{29} + 3 q^{31} - 28 q^{33} + 37 q^{35} + 19 q^{37}+ \cdots + 140 q^{99}+O(q^{100}) Copy content Toggle raw display

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\iota_m(a_n) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
9.1 0 −2.83413 1.26184i 0 1.32307 + 0.281228i 0 1.68941 + 2.03615i 0 4.43267 + 4.92298i 0
9.2 0 −1.59083 0.708282i 0 3.53897 + 0.752231i 0 −1.44841 2.21407i 0 0.0216755 + 0.0240731i 0
9.3 0 −0.416782 0.185563i 0 −0.887136 0.188567i 0 −2.58171 0.578578i 0 −1.86812 2.07476i 0
9.4 0 0.298336 + 0.132828i 0 −3.45675 0.734755i 0 2.29997 1.30772i 0 −1.93603 2.15018i 0
9.5 0 2.27473 + 1.01277i 0 −1.34506 0.285901i 0 1.38806 + 2.25240i 0 2.14129 + 2.37814i 0
9.6 0 2.61762 + 1.16544i 0 1.05782 + 0.224846i 0 −2.30739 1.29459i 0 3.48627 + 3.87190i 0
25.1 0 −3.17936 0.675793i 0 −0.0774019 + 0.736430i 0 −2.58614 + 0.558460i 0 6.91098 + 3.07697i 0
25.2 0 −1.94235 0.412859i 0 −0.0302385 + 0.287700i 0 2.64441 + 0.0843530i 0 0.861626 + 0.383621i 0
25.3 0 −1.23333 0.262153i 0 −0.367897 + 3.50030i 0 0.745250 2.53862i 0 −1.28825 0.573565i 0
25.4 0 −0.876178 0.186237i 0 0.461396 4.38989i 0 −2.16649 + 1.51865i 0 −2.00763 0.893856i 0
25.5 0 1.54096 + 0.327542i 0 −0.0703085 + 0.668941i 0 1.10028 + 2.40611i 0 −0.473354 0.210751i 0
25.6 0 3.12943 + 0.665181i 0 −0.358340 + 3.40938i 0 −2.48911 0.896852i 0 6.61023 + 2.94307i 0
37.1 0 −3.17936 + 0.675793i 0 −0.0774019 0.736430i 0 −2.58614 0.558460i 0 6.91098 3.07697i 0
37.2 0 −1.94235 + 0.412859i 0 −0.0302385 0.287700i 0 2.64441 0.0843530i 0 0.861626 0.383621i 0
37.3 0 −1.23333 + 0.262153i 0 −0.367897 3.50030i 0 0.745250 + 2.53862i 0 −1.28825 + 0.573565i 0
37.4 0 −0.876178 + 0.186237i 0 0.461396 + 4.38989i 0 −2.16649 1.51865i 0 −2.00763 + 0.893856i 0
37.5 0 1.54096 0.327542i 0 −0.0703085 0.668941i 0 1.10028 2.40611i 0 −0.473354 + 0.210751i 0
37.6 0 3.12943 0.665181i 0 −0.358340 3.40938i 0 −2.48911 + 0.896852i 0 6.61023 2.94307i 0
53.1 0 −0.299509 2.84964i 0 −0.723630 + 0.803673i 0 −1.94425 1.79440i 0 −5.09630 + 1.08325i 0
53.2 0 −0.260276 2.47636i 0 0.920128 1.02191i 0 2.57109 + 0.624089i 0 −3.13018 + 0.665340i 0
See all 48 embeddings
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 308.2.y.b 48
7.c even 3 1 inner 308.2.y.b 48
11.c even 5 1 inner 308.2.y.b 48
77.m even 15 1 inner 308.2.y.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.y.b 48 1.a even 1 1 trivial
308.2.y.b 48 7.c even 3 1 inner
308.2.y.b 48 11.c even 5 1 inner
308.2.y.b 48 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T348+T34715T3468T345+73T34424T343++1982119441 T_{3}^{48} + T_{3}^{47} - 15 T_{3}^{46} - 8 T_{3}^{45} + 73 T_{3}^{44} - 24 T_{3}^{43} + \cdots + 1982119441 acting on S2new(308,[χ])S_{2}^{\mathrm{new}}(308, [\chi]). Copy content Toggle raw display