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Properties

Label 980.2.g.b
Level $980$
Weight $2$
Character orbit 980.g
Analytic conductor $7.825$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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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] = [980,2,Mod(391,980)] 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("980.391"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.g (of order \(2\), degree \(1\), 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: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{2} - 8 q^{4} + 8 q^{8} + 48 q^{9} - 40 q^{16} + 40 q^{18} - 32 q^{22} - 48 q^{25} + 64 q^{29} + 8 q^{32} - 40 q^{36} + 64 q^{37} - 32 q^{46} - 8 q^{50} - 64 q^{53} - 48 q^{58} - 8 q^{64} - 72 q^{72}+ \cdots - 112 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
391.1 −1.27833 − 0.604870i −1.35351 1.26826 + 1.54645i 1.00000i 1.73023 + 0.818697i 0 −0.685859 − 2.74401i −1.16802 0.604870 − 1.27833i
391.2 −1.27833 − 0.604870i 1.35351 1.26826 + 1.54645i − 1.00000i −1.73023 − 0.818697i 0 −0.685859 − 2.74401i −1.16802 −0.604870 + 1.27833i
391.3 −1.27833 + 0.604870i −1.35351 1.26826 − 1.54645i − 1.00000i 1.73023 − 0.818697i 0 −0.685859 + 2.74401i −1.16802 0.604870 + 1.27833i
391.4 −1.27833 + 0.604870i 1.35351 1.26826 − 1.54645i 1.00000i −1.73023 + 0.818697i 0 −0.685859 + 2.74401i −1.16802 −0.604870 − 1.27833i
391.5 −1.04820 − 0.949355i −1.29812 0.197452 + 1.99023i − 1.00000i 1.36069 + 1.23237i 0 1.68246 − 2.27361i −1.31490 −0.949355 + 1.04820i
391.6 −1.04820 − 0.949355i 1.29812 0.197452 + 1.99023i 1.00000i −1.36069 − 1.23237i 0 1.68246 − 2.27361i −1.31490 0.949355 − 1.04820i
391.7 −1.04820 + 0.949355i −1.29812 0.197452 − 1.99023i 1.00000i 1.36069 − 1.23237i 0 1.68246 + 2.27361i −1.31490 −0.949355 − 1.04820i
391.8 −1.04820 + 0.949355i 1.29812 0.197452 − 1.99023i − 1.00000i −1.36069 + 1.23237i 0 1.68246 + 2.27361i −1.31490 0.949355 + 1.04820i
391.9 −1.01402 − 0.985786i −2.46214 0.0564536 + 1.99920i − 1.00000i 2.49664 + 2.42714i 0 1.91354 − 2.08287i 3.06212 −0.985786 + 1.01402i
391.10 −1.01402 − 0.985786i 2.46214 0.0564536 + 1.99920i 1.00000i −2.49664 − 2.42714i 0 1.91354 − 2.08287i 3.06212 0.985786 − 1.01402i
391.11 −1.01402 + 0.985786i −2.46214 0.0564536 − 1.99920i 1.00000i 2.49664 − 2.42714i 0 1.91354 + 2.08287i 3.06212 −0.985786 − 1.01402i
391.12 −1.01402 + 0.985786i 2.46214 0.0564536 − 1.99920i − 1.00000i −2.49664 + 2.42714i 0 1.91354 + 2.08287i 3.06212 0.985786 + 1.01402i
391.13 −0.560755 − 1.29829i −0.396131 −1.37111 + 1.45604i − 1.00000i 0.222133 + 0.514293i 0 2.65922 + 0.963609i −2.84308 −1.29829 + 0.560755i
391.14 −0.560755 − 1.29829i 0.396131 −1.37111 + 1.45604i 1.00000i −0.222133 − 0.514293i 0 2.65922 + 0.963609i −2.84308 1.29829 − 0.560755i
391.15 −0.560755 + 1.29829i −0.396131 −1.37111 − 1.45604i 1.00000i 0.222133 − 0.514293i 0 2.65922 − 0.963609i −2.84308 −1.29829 − 0.560755i
391.16 −0.560755 + 1.29829i 0.396131 −1.37111 − 1.45604i − 1.00000i −0.222133 + 0.514293i 0 2.65922 − 0.963609i −2.84308 1.29829 + 0.560755i
391.17 −0.281497 − 1.38591i −1.31255 −1.84152 + 0.780262i 1.00000i 0.369479 + 1.81908i 0 1.59976 + 2.33255i −1.27722 1.38591 − 0.281497i
391.18 −0.281497 − 1.38591i 1.31255 −1.84152 + 0.780262i − 1.00000i −0.369479 − 1.81908i 0 1.59976 + 2.33255i −1.27722 −1.38591 + 0.281497i
391.19 −0.281497 + 1.38591i −1.31255 −1.84152 − 0.780262i − 1.00000i 0.369479 − 1.81908i 0 1.59976 − 2.33255i −1.27722 1.38591 + 0.281497i
391.20 −0.281497 + 1.38591i 1.31255 −1.84152 − 0.780262i 1.00000i −0.369479 + 1.81908i 0 1.59976 − 2.33255i −1.27722 −1.38591 − 0.281497i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.g.b ✓ 48
4.b odd 2 1 inner 980.2.g.b ✓ 48
7.b odd 2 1 inner 980.2.g.b ✓ 48
7.c even 3 2 980.2.o.g 96
7.d odd 6 2 980.2.o.g 96
28.d even 2 1 inner 980.2.g.b ✓ 48
28.f even 6 2 980.2.o.g 96
28.g odd 6 2 980.2.o.g 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.g.b ✓ 48 1.a even 1 1 trivial
980.2.g.b ✓ 48 4.b odd 2 1 inner
980.2.g.b ✓ 48 7.b odd 2 1 inner
980.2.g.b ✓ 48 28.d even 2 1 inner
980.2.o.g 96 7.c even 3 2
980.2.o.g 96 7.d odd 6 2
980.2.o.g 96 28.f even 6 2
980.2.o.g 96 28.g odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 48 T_{3}^{22} + 984 T_{3}^{20} - 11304 T_{3}^{18} + 80202 T_{3}^{16} - 365048 T_{3}^{14} + \cdots + 1348 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\). Copy content Toggle raw display