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Properties

Label 207.3.j.c
Level $207$
Weight $3$
Character orbit 207.j
Analytic conductor $5.640$
Analytic rank $0$
Dimension $80$
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] = [207,3,Mod(10,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 207.j (of order \(22\), degree \(10\), 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: \(5.64034147226\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 24 q^{4} - 16 q^{13} + 224 q^{16} + 66 q^{19} - 44 q^{25} - 528 q^{28} - 22 q^{31} - 242 q^{34} - 352 q^{37} - 110 q^{40} - 88 q^{43} + 336 q^{46} + 40 q^{49} + 530 q^{52} + 442 q^{55} - 618 q^{58} - 308 q^{61} - 312 q^{64} - 22 q^{67} - 400 q^{70} + 46 q^{73} + 2354 q^{76} + 1540 q^{79} + 1676 q^{82} + 1118 q^{85} + 528 q^{88} - 674 q^{94} - 792 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.20725 − 2.54730i 0 −1.04753 + 7.28575i −2.99886 − 1.36953i 0 −1.31584 + 4.48135i 9.52916 − 6.12402i 0 3.13061 + 10.6619i
10.2 −1.25712 − 1.45079i 0 0.0448082 − 0.311648i 1.56906 + 0.716567i 0 1.25321 − 4.26806i −6.96820 + 4.47819i 0 −0.932909 − 3.17720i
10.3 −0.718657 − 0.829374i 0 0.397865 − 2.76721i −4.86076 − 2.21983i 0 2.07875 − 7.07956i −6.27382 + 4.03194i 0 1.65214 + 5.62668i
10.4 −0.705934 − 0.814691i 0 0.403880 − 2.80905i 6.96870 + 3.18250i 0 −3.55376 + 12.1030i −6.20108 + 3.98519i 0 −2.32669 − 7.92397i
10.5 0.705934 + 0.814691i 0 0.403880 − 2.80905i −6.96870 − 3.18250i 0 −3.55376 + 12.1030i 6.20108 − 3.98519i 0 −2.32669 − 7.92397i
10.6 0.718657 + 0.829374i 0 0.397865 − 2.76721i 4.86076 + 2.21983i 0 2.07875 − 7.07956i 6.27382 − 4.03194i 0 1.65214 + 5.62668i
10.7 1.25712 + 1.45079i 0 0.0448082 − 0.311648i −1.56906 − 0.716567i 0 1.25321 − 4.26806i 6.96820 − 4.47819i 0 −0.932909 − 3.17720i
10.8 2.20725 + 2.54730i 0 −1.04753 + 7.28575i 2.99886 + 1.36953i 0 −1.31584 + 4.48135i −9.52916 + 6.12402i 0 3.13061 + 10.6619i
19.1 −1.38976 − 3.04316i 0 −4.70992 + 5.43554i 0.743426 − 1.15679i 0 −9.10473 + 1.30906i 10.2470 + 3.00879i 0 −4.55349 − 0.654693i
19.2 −1.17192 − 2.56615i 0 −2.59227 + 2.99164i −2.02261 + 3.14724i 0 13.5294 − 1.94524i −0.112326 − 0.0329818i 0 10.4466 + 1.50199i
19.3 −0.526428 − 1.15272i 0 1.56781 − 1.80935i −1.64209 + 2.55515i 0 −5.42892 + 0.780561i −7.77462 − 2.28283i 0 3.80981 + 0.547767i
19.4 −0.349287 − 0.764832i 0 2.15648 − 2.48871i 4.49156 − 6.98901i 0 4.21506 − 0.606035i −5.88370 − 1.72761i 0 −6.91426 − 0.994121i
19.5 0.349287 + 0.764832i 0 2.15648 − 2.48871i −4.49156 + 6.98901i 0 4.21506 − 0.606035i 5.88370 + 1.72761i 0 −6.91426 − 0.994121i
19.6 0.526428 + 1.15272i 0 1.56781 − 1.80935i 1.64209 − 2.55515i 0 −5.42892 + 0.780561i 7.77462 + 2.28283i 0 3.80981 + 0.547767i
19.7 1.17192 + 2.56615i 0 −2.59227 + 2.99164i 2.02261 − 3.14724i 0 13.5294 − 1.94524i 0.112326 + 0.0329818i 0 10.4466 + 1.50199i
19.8 1.38976 + 3.04316i 0 −4.70992 + 5.43554i −0.743426 + 1.15679i 0 −9.10473 + 1.30906i −10.2470 − 3.00879i 0 −4.55349 − 0.654693i
28.1 −3.74119 − 1.09851i 0 9.42477 + 6.05693i −0.516461 − 0.0742558i 0 −8.21158 + 3.75010i −18.3927 − 21.2263i 0 1.85061 + 0.845144i
28.2 −2.45804 − 0.721745i 0 2.15601 + 1.38559i −3.85616 − 0.554432i 0 7.70732 − 3.51982i 2.41099 + 2.78243i 0 9.07841 + 4.14597i
28.3 −2.08416 − 0.611965i 0 0.604218 + 0.388307i 9.37344 + 1.34770i 0 −3.14185 + 1.43483i 4.66816 + 5.38735i 0 −18.7110 − 8.54504i
28.4 −0.519179 − 0.152445i 0 −3.11871 − 2.00427i −1.82492 − 0.262384i 0 −1.75613 + 0.801999i 2.73100 + 3.15174i 0 0.907461 + 0.414423i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.3.j.c ✓ 80
3.b odd 2 1 inner 207.3.j.c ✓ 80
23.d odd 22 1 inner 207.3.j.c ✓ 80
69.g even 22 1 inner 207.3.j.c ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.3.j.c ✓ 80 1.a even 1 1 trivial
207.3.j.c ✓ 80 3.b odd 2 1 inner
207.3.j.c ✓ 80 23.d odd 22 1 inner
207.3.j.c ✓ 80 69.g even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 28 T_{2}^{78} + 274 T_{2}^{76} + 2364 T_{2}^{74} + 52391 T_{2}^{72} + 1043886 T_{2}^{70} + \cdots + 12\!\cdots\!81 \) acting on \(S_{3}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display