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Properties

Label 297.3.m.c
Level $297$
Weight $3$
Character orbit 297.m
Analytic conductor $8.093$
Analytic rank $0$
Dimension $32$
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: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 34 q^{4} + 8 q^{7} + 12 q^{10} + 20 q^{13} + 90 q^{16} + 80 q^{19} - 30 q^{22} + 10 q^{25} + 12 q^{28} - 110 q^{31} + 212 q^{34} - 702 q^{40} - 92 q^{43} - 190 q^{46} + 52 q^{49} + 374 q^{52} - 350 q^{55}+ \cdots - 564 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −1.91915 + 2.64148i 0 −2.05823 − 6.33458i −5.62765 − 7.74580i 0 1.38002 + 4.24727i 8.26173 + 2.68440i 0 31.2607
26.2 −1.70641 + 2.34868i 0 −1.36837 − 4.21142i 1.87123 + 2.57553i 0 −2.03023 − 6.24840i 1.18212 + 0.384095i 0 −9.24219
26.3 −0.932126 + 1.28296i 0 0.458937 + 1.41246i 3.97855 + 5.47600i 0 2.86444 + 8.81585i −8.27277 − 2.68799i 0 −10.7340
26.4 −0.432865 + 0.595788i 0 1.06848 + 3.28843i −2.00643 − 2.76162i 0 −1.21424 − 3.73704i −5.22328 − 1.69715i 0 2.51386
26.5 0.432865 − 0.595788i 0 1.06848 + 3.28843i 2.00643 + 2.76162i 0 −1.21424 − 3.73704i 5.22328 + 1.69715i 0 2.51386
26.6 0.932126 − 1.28296i 0 0.458937 + 1.41246i −3.97855 − 5.47600i 0 2.86444 + 8.81585i 8.27277 + 2.68799i 0 −10.7340
26.7 1.70641 − 2.34868i 0 −1.36837 − 4.21142i −1.87123 − 2.57553i 0 −2.03023 − 6.24840i −1.18212 − 0.384095i 0 −9.24219
26.8 1.91915 − 2.64148i 0 −2.05823 − 6.33458i 5.62765 + 7.74580i 0 1.38002 + 4.24727i −8.26173 − 2.68440i 0 31.2607
53.1 −3.74988 − 1.21841i 0 9.34099 + 6.78663i 6.49312 − 2.10974i 0 0.851148 + 0.618395i −17.4885 − 24.0708i 0 −26.9189
53.2 −2.68763 − 0.873263i 0 3.22469 + 2.34287i −7.76821 + 2.52404i 0 4.13431 + 3.00376i 0.0233650 + 0.0321592i 0 23.0822
53.3 −2.05552 − 0.667878i 0 0.543019 + 0.394526i 2.08499 − 0.677453i 0 −8.15858 − 5.92755i 4.22883 + 5.82048i 0 −4.73818
53.4 −0.767275 − 0.249303i 0 −2.70951 − 1.96857i 2.62117 − 0.851670i 0 4.17312 + 3.03195i 3.48498 + 4.79666i 0 −2.22348
53.5 0.767275 + 0.249303i 0 −2.70951 − 1.96857i −2.62117 + 0.851670i 0 4.17312 + 3.03195i −3.48498 − 4.79666i 0 −2.22348
53.6 2.05552 + 0.667878i 0 0.543019 + 0.394526i −2.08499 + 0.677453i 0 −8.15858 − 5.92755i −4.22883 − 5.82048i 0 −4.73818
53.7 2.68763 + 0.873263i 0 3.22469 + 2.34287i 7.76821 − 2.52404i 0 4.13431 + 3.00376i −0.0233650 − 0.0321592i 0 23.0822
53.8 3.74988 + 1.21841i 0 9.34099 + 6.78663i −6.49312 + 2.10974i 0 0.851148 + 0.618395i 17.4885 + 24.0708i 0 −26.9189
80.1 −1.91915 − 2.64148i 0 −2.05823 + 6.33458i −5.62765 + 7.74580i 0 1.38002 − 4.24727i 8.26173 − 2.68440i 0 31.2607
80.2 −1.70641 − 2.34868i 0 −1.36837 + 4.21142i 1.87123 − 2.57553i 0 −2.03023 + 6.24840i 1.18212 − 0.384095i 0 −9.24219
80.3 −0.932126 − 1.28296i 0 0.458937 − 1.41246i 3.97855 − 5.47600i 0 2.86444 − 8.81585i −8.27277 + 2.68799i 0 −10.7340
80.4 −0.432865 − 0.595788i 0 1.06848 − 3.28843i −2.00643 + 2.76162i 0 −1.21424 + 3.73704i −5.22328 + 1.69715i 0 2.51386
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.8
Significant digits:
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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.c ✓ 32
3.b odd 2 1 inner 297.3.m.c ✓ 32
11.c even 5 1 inner 297.3.m.c ✓ 32
33.h odd 10 1 inner 297.3.m.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.3.m.c ✓ 32 1.a even 1 1 trivial
297.3.m.c ✓ 32 3.b odd 2 1 inner
297.3.m.c ✓ 32 11.c even 5 1 inner
297.3.m.c ✓ 32 33.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 33 T_{2}^{30} + 598 T_{2}^{28} - 7840 T_{2}^{26} + 105464 T_{2}^{24} - 1043961 T_{2}^{22} + \cdots + 2139525025 \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display