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Properties

Label 297.3.h.b
Level $297$
Weight $3$
Character orbit 297.h
Analytic conductor $8.093$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [297,3,Mod(10,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.h (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: \(8.09266385150\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 46 q^{4} - 2 q^{5} + 23 q^{11} + 18 q^{14} - 26 q^{16} - 50 q^{20} - 21 q^{22} - 32 q^{23} + 30 q^{25} + 216 q^{26} + 50 q^{31} + 54 q^{34} - 16 q^{37} + 60 q^{38} + 50 q^{44} + 112 q^{47} + 166 q^{49}+ \cdots - 328 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −3.30671 + 1.90913i 0 5.28955 − 9.16177i 2.91356 − 5.04644i 0 5.33694 − 3.08128i 25.1207i 0 22.2495i
10.2 −3.03057 + 1.74970i 0 4.12292 − 7.14111i 0.198992 − 0.344665i 0 −2.03014 + 1.17210i 14.8579i 0 1.39271i
10.3 −2.67675 + 1.54542i 0 2.77667 − 4.80934i 1.89480 − 3.28189i 0 −5.27354 + 3.04468i 4.80116i 0 11.7131i
10.4 −2.41819 + 1.39614i 0 1.89841 − 3.28815i −3.95412 + 6.84874i 0 −11.3316 + 6.54229i − 0.567308i 0 − 22.0820i
10.5 −2.38735 + 1.37834i 0 1.79962 − 3.11703i −3.89274 + 6.74242i 0 6.72208 − 3.88100i − 1.10477i 0 − 21.4620i
10.6 −1.81513 + 1.04797i 0 0.196474 − 0.340304i −1.28105 + 2.21884i 0 4.60933 − 2.66120i − 7.56015i 0 − 5.37000i
10.7 −1.64469 + 0.949563i 0 −0.196660 + 0.340626i 3.05607 − 5.29326i 0 −4.13935 + 2.38985i − 8.34347i 0 11.6077i
10.8 −1.10231 + 0.636421i 0 −1.18994 + 2.06103i 1.84774 − 3.20038i 0 6.26503 − 3.61711i − 8.12057i 0 4.70376i
10.9 −0.981412 + 0.566618i 0 −1.35789 + 2.35193i −0.0827703 + 0.143362i 0 3.82724 − 2.20966i − 7.61056i 0 − 0.187597i
10.10 −0.491166 + 0.283575i 0 −1.83917 + 3.18554i −1.20048 + 2.07930i 0 −10.1718 + 5.87270i − 4.35477i 0 − 1.36171i
10.11 0.491166 − 0.283575i 0 −1.83917 + 3.18554i −1.20048 + 2.07930i 0 10.1718 − 5.87270i 4.35477i 0 1.36171i
10.12 0.981412 − 0.566618i 0 −1.35789 + 2.35193i −0.0827703 + 0.143362i 0 −3.82724 + 2.20966i 7.61056i 0 0.187597i
10.13 1.10231 − 0.636421i 0 −1.18994 + 2.06103i 1.84774 − 3.20038i 0 −6.26503 + 3.61711i 8.12057i 0 − 4.70376i
10.14 1.64469 − 0.949563i 0 −0.196660 + 0.340626i 3.05607 − 5.29326i 0 4.13935 − 2.38985i 8.34347i 0 − 11.6077i
10.15 1.81513 − 1.04797i 0 0.196474 − 0.340304i −1.28105 + 2.21884i 0 −4.60933 + 2.66120i 7.56015i 0 5.37000i
10.16 2.38735 − 1.37834i 0 1.79962 − 3.11703i −3.89274 + 6.74242i 0 −6.72208 + 3.88100i 1.10477i 0 21.4620i
10.17 2.41819 − 1.39614i 0 1.89841 − 3.28815i −3.95412 + 6.84874i 0 11.3316 − 6.54229i 0.567308i 0 22.0820i
10.18 2.67675 − 1.54542i 0 2.77667 − 4.80934i 1.89480 − 3.28189i 0 5.27354 − 3.04468i − 4.80116i 0 − 11.7131i
10.19 3.03057 − 1.74970i 0 4.12292 − 7.14111i 0.198992 − 0.344665i 0 2.03014 − 1.17210i − 14.8579i 0 − 1.39271i
10.20 3.30671 − 1.90913i 0 5.28955 − 9.16177i 2.91356 − 5.04644i 0 −5.33694 + 3.08128i − 25.1207i 0 − 22.2495i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.b odd 2 1 inner
99.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.h.b 40
3.b odd 2 1 99.3.h.b ✓ 40
9.c even 3 1 inner 297.3.h.b 40
9.c even 3 1 891.3.c.f 20
9.d odd 6 1 99.3.h.b ✓ 40
9.d odd 6 1 891.3.c.e 20
11.b odd 2 1 inner 297.3.h.b 40
33.d even 2 1 99.3.h.b ✓ 40
99.g even 6 1 99.3.h.b ✓ 40
99.g even 6 1 891.3.c.e 20
99.h odd 6 1 inner 297.3.h.b 40
99.h odd 6 1 891.3.c.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.h.b ✓ 40 3.b odd 2 1
99.3.h.b ✓ 40 9.d odd 6 1
99.3.h.b ✓ 40 33.d even 2 1
99.3.h.b ✓ 40 99.g even 6 1
297.3.h.b 40 1.a even 1 1 trivial
297.3.h.b 40 9.c even 3 1 inner
297.3.h.b 40 11.b odd 2 1 inner
297.3.h.b 40 99.h odd 6 1 inner
891.3.c.e 20 9.d odd 6 1
891.3.c.e 20 99.g even 6 1
891.3.c.f 20 9.c even 3 1
891.3.c.f 20 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} - 63 T_{2}^{38} + 2289 T_{2}^{36} - 56286 T_{2}^{34} + 1039872 T_{2}^{32} + \cdots + 1148217259401 \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display