Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [276,2,Mod(47,276)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(276, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("276.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 276 = 2^{2} \cdot 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 276.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(2.20387109579\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | −1.33491 | − | 0.466911i | −1.38398 | − | 1.04144i | 1.56399 | + | 1.24657i | 0.588622i | 1.36124 | + | 2.03643i | − | 0.538069i | −1.50575 | − | 2.39431i | 0.830813 | + | 2.88266i | 0.274834 | − | 0.785760i | |||
47.2 | −1.33491 | + | 0.466911i | −1.38398 | + | 1.04144i | 1.56399 | − | 1.24657i | − | 0.588622i | 1.36124 | − | 2.03643i | 0.538069i | −1.50575 | + | 2.39431i | 0.830813 | − | 2.88266i | 0.274834 | + | 0.785760i | |||
47.3 | −1.27564 | − | 0.610520i | 0.436111 | − | 1.67625i | 1.25453 | + | 1.55761i | − | 3.87509i | −1.57971 | + | 1.87204i | 0.468259i | −0.649379 | − | 2.75287i | −2.61961 | − | 1.46206i | −2.36582 | + | 4.94323i | |||
47.4 | −1.27564 | + | 0.610520i | 0.436111 | + | 1.67625i | 1.25453 | − | 1.55761i | 3.87509i | −1.57971 | − | 1.87204i | − | 0.468259i | −0.649379 | + | 2.75287i | −2.61961 | + | 1.46206i | −2.36582 | − | 4.94323i | |||
47.5 | −1.04620 | − | 0.951558i | 1.08652 | + | 1.34888i | 0.189073 | + | 1.99104i | − | 0.289949i | 0.146820 | − | 2.44509i | 1.62347i | 1.69678 | − | 2.26295i | −0.638952 | + | 2.93117i | −0.275904 | + | 0.303345i | |||
47.6 | −1.04620 | + | 0.951558i | 1.08652 | − | 1.34888i | 0.189073 | − | 1.99104i | 0.289949i | 0.146820 | + | 2.44509i | − | 1.62347i | 1.69678 | + | 2.26295i | −0.638952 | − | 2.93117i | −0.275904 | − | 0.303345i | |||
47.7 | −0.725653 | − | 1.21385i | −0.173813 | − | 1.72331i | −0.946854 | + | 1.76167i | 1.59003i | −1.96571 | + | 1.46151i | − | 4.71351i | 2.82548 | − | 0.129022i | −2.93958 | + | 0.599066i | 1.93005 | − | 1.15381i | |||
47.8 | −0.725653 | + | 1.21385i | −0.173813 | + | 1.72331i | −0.946854 | − | 1.76167i | − | 1.59003i | −1.96571 | − | 1.46151i | 4.71351i | 2.82548 | + | 0.129022i | −2.93958 | − | 0.599066i | 1.93005 | + | 1.15381i | |||
47.9 | −0.456308 | − | 1.33857i | −0.784130 | + | 1.54439i | −1.58357 | + | 1.22161i | − | 1.42397i | 2.42509 | + | 0.344899i | − | 1.12562i | 2.35780 | + | 1.56229i | −1.77028 | − | 2.42200i | −1.90608 | + | 0.649767i | ||
47.10 | −0.456308 | + | 1.33857i | −0.784130 | − | 1.54439i | −1.58357 | − | 1.22161i | 1.42397i | 2.42509 | − | 0.344899i | 1.12562i | 2.35780 | − | 1.56229i | −1.77028 | + | 2.42200i | −1.90608 | − | 0.649767i | ||||
47.11 | −0.133777 | − | 1.40787i | 1.73013 | + | 0.0815319i | −1.96421 | + | 0.376681i | 3.66772i | −0.116665 | − | 2.44671i | 2.47568i | 0.793084 | + | 2.71496i | 2.98671 | + | 0.282122i | 5.16368 | − | 0.490656i | ||||
47.12 | −0.133777 | + | 1.40787i | 1.73013 | − | 0.0815319i | −1.96421 | − | 0.376681i | − | 3.66772i | −0.116665 | + | 2.44671i | − | 2.47568i | 0.793084 | − | 2.71496i | 2.98671 | − | 0.282122i | 5.16368 | + | 0.490656i | ||
47.13 | 0.469504 | − | 1.33400i | −1.64481 | + | 0.542779i | −1.55913 | − | 1.25264i | − | 2.34886i | −0.0481753 | + | 2.44902i | 0.796368i | −2.40305 | + | 1.49177i | 2.41078 | − | 1.78553i | −3.13338 | − | 1.10280i | |||
47.14 | 0.469504 | + | 1.33400i | −1.64481 | − | 0.542779i | −1.55913 | + | 1.25264i | 2.34886i | −0.0481753 | − | 2.44902i | − | 0.796368i | −2.40305 | − | 1.49177i | 2.41078 | + | 1.78553i | −3.13338 | + | 1.10280i | |||
47.15 | 0.565709 | − | 1.29614i | 1.40231 | + | 1.01662i | −1.35995 | − | 1.46647i | − | 0.662888i | 2.11098 | − | 1.24248i | − | 4.60297i | −2.67009 | + | 0.933083i | 0.932961 | + | 2.85124i | −0.859194 | − | 0.375001i | ||
47.16 | 0.565709 | + | 1.29614i | 1.40231 | − | 1.01662i | −1.35995 | + | 1.46647i | 0.662888i | 2.11098 | + | 1.24248i | 4.60297i | −2.67009 | − | 0.933083i | 0.932961 | − | 2.85124i | −0.859194 | + | 0.375001i | ||||
47.17 | 1.15961 | − | 0.809507i | 1.20859 | − | 1.24069i | 0.689397 | − | 1.87743i | 3.47795i | 0.397138 | − | 2.41708i | − | 0.968466i | −0.720358 | − | 2.73516i | −0.0786444 | − | 2.99897i | 2.81542 | + | 4.03307i | |||
47.18 | 1.15961 | + | 0.809507i | 1.20859 | + | 1.24069i | 0.689397 | + | 1.87743i | − | 3.47795i | 0.397138 | + | 2.41708i | 0.968466i | −0.720358 | + | 2.73516i | −0.0786444 | + | 2.99897i | 2.81542 | − | 4.03307i | |||
47.19 | 1.37110 | − | 0.346521i | −1.70706 | + | 0.293173i | 1.75985 | − | 0.950233i | 2.20059i | −2.23896 | + | 0.993503i | 3.71987i | 2.08365 | − | 1.91269i | 2.82810 | − | 1.00093i | 0.762550 | + | 3.01723i | ||||
47.20 | 1.37110 | + | 0.346521i | −1.70706 | − | 0.293173i | 1.75985 | + | 0.950233i | − | 2.20059i | −2.23896 | − | 0.993503i | − | 3.71987i | 2.08365 | + | 1.91269i | 2.82810 | + | 1.00093i | 0.762550 | − | 3.01723i | ||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 276.2.c.b | yes | 22 |
3.b | odd | 2 | 1 | 276.2.c.a | ✓ | 22 | |
4.b | odd | 2 | 1 | 276.2.c.a | ✓ | 22 | |
12.b | even | 2 | 1 | inner | 276.2.c.b | yes | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
276.2.c.a | ✓ | 22 | 3.b | odd | 2 | 1 | |
276.2.c.a | ✓ | 22 | 4.b | odd | 2 | 1 | |
276.2.c.b | yes | 22 | 1.a | even | 1 | 1 | trivial |
276.2.c.b | yes | 22 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{11} - 58 T_{11}^{9} + 20 T_{11}^{8} + 972 T_{11}^{7} - 928 T_{11}^{6} - 5512 T_{11}^{5} + 8272 T_{11}^{4} + 5344 T_{11}^{3} - 10496 T_{11}^{2} - 1280 T_{11} + 3072 \)
acting on \(S_{2}^{\mathrm{new}}(276, [\chi])\).