Properties

Label 975.2.bb.l
Level $975$
Weight $2$
Character orbit 975.bb
Analytic conductor $7.785$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(724,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.724");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bb (of order \(6\), degree \(2\), not 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 16 q^{4} + 12 q^{9} - 2 q^{11} - 60 q^{14} - 40 q^{16} - 6 q^{19} - 4 q^{21} + 12 q^{24} + 18 q^{26} + 24 q^{29} + 48 q^{31} + 112 q^{34} - 16 q^{36} + 6 q^{39} + 16 q^{41} - 156 q^{44} - 10 q^{46} + 18 q^{49} - 32 q^{51} - 34 q^{56} - 4 q^{59} - 66 q^{61} - 144 q^{64} + 12 q^{66} + 2 q^{69} + 46 q^{71} + 118 q^{74} + 70 q^{76} - 44 q^{79} - 12 q^{81} - 6 q^{84} - 52 q^{86} + 16 q^{89} + 10 q^{91} + 16 q^{94} + 128 q^{96} - 4 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
724.1 −2.43531 1.40603i 0.866025 + 0.500000i 2.95384 + 5.11620i 0 −1.40603 2.43531i 0.577759 0.333570i 10.9886i 0.500000 + 0.866025i 0
724.2 −2.12585 1.22736i −0.866025 0.500000i 2.01283 + 3.48633i 0 1.22736 + 2.12585i 3.62470 2.09272i 4.97244i 0.500000 + 0.866025i 0
724.3 −1.54501 0.892010i −0.866025 0.500000i 0.591364 + 1.02427i 0 0.892010 + 1.54501i −2.03615 + 1.17557i 1.45803i 0.500000 + 0.866025i 0
724.4 −1.40830 0.813080i 0.866025 + 0.500000i 0.322200 + 0.558066i 0 −0.813080 1.40830i 2.67150 1.54239i 2.20443i 0.500000 + 0.866025i 0
724.5 −0.373368 0.215564i −0.866025 0.500000i −0.907064 1.57108i 0 0.215564 + 0.373368i 3.53303 2.03980i 1.64438i 0.500000 + 0.866025i 0
724.6 −0.200615 0.115825i 0.866025 + 0.500000i −0.973169 1.68558i 0 −0.115825 0.200615i 1.00629 0.580982i 0.914171i 0.500000 + 0.866025i 0
724.7 0.200615 + 0.115825i −0.866025 0.500000i −0.973169 1.68558i 0 −0.115825 0.200615i −1.00629 + 0.580982i 0.914171i 0.500000 + 0.866025i 0
724.8 0.373368 + 0.215564i 0.866025 + 0.500000i −0.907064 1.57108i 0 0.215564 + 0.373368i −3.53303 + 2.03980i 1.64438i 0.500000 + 0.866025i 0
724.9 1.40830 + 0.813080i −0.866025 0.500000i 0.322200 + 0.558066i 0 −0.813080 1.40830i −2.67150 + 1.54239i 2.20443i 0.500000 + 0.866025i 0
724.10 1.54501 + 0.892010i 0.866025 + 0.500000i 0.591364 + 1.02427i 0 0.892010 + 1.54501i 2.03615 1.17557i 1.45803i 0.500000 + 0.866025i 0
724.11 2.12585 + 1.22736i 0.866025 + 0.500000i 2.01283 + 3.48633i 0 1.22736 + 2.12585i −3.62470 + 2.09272i 4.97244i 0.500000 + 0.866025i 0
724.12 2.43531 + 1.40603i −0.866025 0.500000i 2.95384 + 5.11620i 0 −1.40603 2.43531i −0.577759 + 0.333570i 10.9886i 0.500000 + 0.866025i 0
874.1 −2.43531 + 1.40603i 0.866025 0.500000i 2.95384 5.11620i 0 −1.40603 + 2.43531i 0.577759 + 0.333570i 10.9886i 0.500000 0.866025i 0
874.2 −2.12585 + 1.22736i −0.866025 + 0.500000i 2.01283 3.48633i 0 1.22736 2.12585i 3.62470 + 2.09272i 4.97244i 0.500000 0.866025i 0
874.3 −1.54501 + 0.892010i −0.866025 + 0.500000i 0.591364 1.02427i 0 0.892010 1.54501i −2.03615 1.17557i 1.45803i 0.500000 0.866025i 0
874.4 −1.40830 + 0.813080i 0.866025 0.500000i 0.322200 0.558066i 0 −0.813080 + 1.40830i 2.67150 + 1.54239i 2.20443i 0.500000 0.866025i 0
874.5 −0.373368 + 0.215564i −0.866025 + 0.500000i −0.907064 + 1.57108i 0 0.215564 0.373368i 3.53303 + 2.03980i 1.64438i 0.500000 0.866025i 0
874.6 −0.200615 + 0.115825i 0.866025 0.500000i −0.973169 + 1.68558i 0 −0.115825 + 0.200615i 1.00629 + 0.580982i 0.914171i 0.500000 0.866025i 0
874.7 0.200615 0.115825i −0.866025 + 0.500000i −0.973169 + 1.68558i 0 −0.115825 + 0.200615i −1.00629 0.580982i 0.914171i 0.500000 0.866025i 0
874.8 0.373368 0.215564i 0.866025 0.500000i −0.907064 + 1.57108i 0 0.215564 0.373368i −3.53303 2.03980i 1.64438i 0.500000 0.866025i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 724.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bb.l 24
5.b even 2 1 inner 975.2.bb.l 24
5.c odd 4 1 975.2.i.n 12
5.c odd 4 1 975.2.i.p yes 12
13.c even 3 1 inner 975.2.bb.l 24
65.n even 6 1 inner 975.2.bb.l 24
65.q odd 12 1 975.2.i.n 12
65.q odd 12 1 975.2.i.p yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.i.n 12 5.c odd 4 1
975.2.i.n 12 65.q odd 12 1
975.2.i.p yes 12 5.c odd 4 1
975.2.i.p yes 12 65.q odd 12 1
975.2.bb.l 24 1.a even 1 1 trivial
975.2.bb.l 24 5.b even 2 1 inner
975.2.bb.l 24 13.c even 3 1 inner
975.2.bb.l 24 65.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{24} - 20 T_{2}^{22} + 258 T_{2}^{20} - 1984 T_{2}^{18} + 11107 T_{2}^{16} - 40996 T_{2}^{14} + 110618 T_{2}^{12} - 184396 T_{2}^{10} + 203641 T_{2}^{8} - 46276 T_{2}^{6} + 8012 T_{2}^{4} - 400 T_{2}^{2} + \cdots + 16 \) Copy content Toggle raw display
\( T_{7}^{24} - 51 T_{7}^{22} + 1654 T_{7}^{20} - 32791 T_{7}^{18} + 474454 T_{7}^{16} - 4624235 T_{7}^{14} + 33010249 T_{7}^{12} - 150245160 T_{7}^{10} + 475417008 T_{7}^{8} - 699292800 T_{7}^{6} + \cdots + 84934656 \) Copy content Toggle raw display