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Properties

Label 297.3.m.b
Level $297$
Weight $3$
Character orbit 297.m
Analytic conductor $8.093$
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] = [297,3,Mod(26,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.m (of order \(10\), degree \(4\), 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: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{4} + 2 q^{7} - 68 q^{10} - 118 q^{13} - 140 q^{16} - 40 q^{19} + 134 q^{22} + 206 q^{25} + 174 q^{28} + 64 q^{31} - 288 q^{34} - 186 q^{37} + 316 q^{40} + 96 q^{43} + 164 q^{46} - 36 q^{49} - 322 q^{52} - 24 q^{55} + 238 q^{58} + 324 q^{61} + 484 q^{64} - 196 q^{67} - 478 q^{70} + 322 q^{73} - 108 q^{76} + 66 q^{79} - 294 q^{82} - 630 q^{85} - 366 q^{88} + 44 q^{91} - 380 q^{94} + 576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −2.22109 + 3.05707i 0 −3.17637 − 9.77587i 2.26833 + 3.12208i 0 0.526472 + 1.62031i 22.5654 + 7.33193i 0 −14.5826
26.2 −1.34303 + 1.84852i 0 −0.377237 − 1.16102i −0.168100 − 0.231370i 0 3.79451 + 11.6783i −6.03947 − 1.96234i 0 0.653455
26.3 −0.202176 + 0.278271i 0 1.19951 + 3.69171i −3.54597 − 4.88061i 0 −0.157859 − 0.485840i −2.57832 − 0.837746i 0 2.07504
26.4 0.202176 − 0.278271i 0 1.19951 + 3.69171i 3.54597 + 4.88061i 0 −0.157859 − 0.485840i 2.57832 + 0.837746i 0 2.07504
26.5 1.34303 − 1.84852i 0 −0.377237 − 1.16102i 0.168100 + 0.231370i 0 3.79451 + 11.6783i 6.03947 + 1.96234i 0 0.653455
26.6 2.22109 − 3.05707i 0 −3.17637 − 9.77587i −2.26833 − 3.12208i 0 0.526472 + 1.62031i −22.5654 − 7.33193i 0 −14.5826
53.1 −3.23014 − 1.04954i 0 6.09621 + 4.42916i −2.92344 + 0.949885i 0 0.764086 + 0.555141i −7.05771 − 9.71411i 0 10.4401
53.2 −2.19785 − 0.714123i 0 1.08448 + 0.787924i 8.34039 − 2.70996i 0 4.05799 + 2.94830i 3.61252 + 4.97220i 0 −20.2661
53.3 −0.676614 − 0.219845i 0 −2.82659 − 2.05364i −6.25651 + 2.03286i 0 −8.48519 − 6.16485i 3.13371 + 4.31318i 0 4.68016
53.4 0.676614 + 0.219845i 0 −2.82659 − 2.05364i 6.25651 − 2.03286i 0 −8.48519 − 6.16485i −3.13371 − 4.31318i 0 4.68016
53.5 2.19785 + 0.714123i 0 1.08448 + 0.787924i −8.34039 + 2.70996i 0 4.05799 + 2.94830i −3.61252 − 4.97220i 0 −20.2661
53.6 3.23014 + 1.04954i 0 6.09621 + 4.42916i 2.92344 − 0.949885i 0 0.764086 + 0.555141i 7.05771 + 9.71411i 0 10.4401
80.1 −2.22109 − 3.05707i 0 −3.17637 + 9.77587i 2.26833 − 3.12208i 0 0.526472 − 1.62031i 22.5654 − 7.33193i 0 −14.5826
80.2 −1.34303 − 1.84852i 0 −0.377237 + 1.16102i −0.168100 + 0.231370i 0 3.79451 − 11.6783i −6.03947 + 1.96234i 0 0.653455
80.3 −0.202176 − 0.278271i 0 1.19951 − 3.69171i −3.54597 + 4.88061i 0 −0.157859 + 0.485840i −2.57832 + 0.837746i 0 2.07504
80.4 0.202176 + 0.278271i 0 1.19951 − 3.69171i 3.54597 − 4.88061i 0 −0.157859 + 0.485840i 2.57832 − 0.837746i 0 2.07504
80.5 1.34303 + 1.84852i 0 −0.377237 + 1.16102i 0.168100 − 0.231370i 0 3.79451 − 11.6783i 6.03947 − 1.96234i 0 0.653455
80.6 2.22109 + 3.05707i 0 −3.17637 + 9.77587i −2.26833 + 3.12208i 0 0.526472 − 1.62031i −22.5654 + 7.33193i 0 −14.5826
269.1 −3.23014 + 1.04954i 0 6.09621 − 4.42916i −2.92344 − 0.949885i 0 0.764086 − 0.555141i −7.05771 + 9.71411i 0 10.4401
269.2 −2.19785 + 0.714123i 0 1.08448 − 0.787924i 8.34039 + 2.70996i 0 4.05799 − 2.94830i 3.61252 − 4.97220i 0 −20.2661
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.m.b ✓ 24
3.b odd 2 1 inner 297.3.m.b ✓ 24
11.c even 5 1 inner 297.3.m.b ✓ 24
33.h odd 10 1 inner 297.3.m.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.3.m.b ✓ 24 1.a even 1 1 trivial
297.3.m.b ✓ 24 3.b odd 2 1 inner
297.3.m.b ✓ 24 11.c even 5 1 inner
297.3.m.b ✓ 24 33.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 16 T_{2}^{22} + 265 T_{2}^{20} - 4174 T_{2}^{18} + 51394 T_{2}^{16} - 289574 T_{2}^{14} + \cdots + 75625 \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display