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Properties

Label 1161.4.a.h
Level $1161$
Weight $4$
Character orbit 1161.a
Self dual yes
Analytic conductor $68.501$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1161,4,Mod(1,1161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1161, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1161.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1161 = 3^{3} \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1161.a (trivial)

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: yes
Analytic conductor: \(68.5012175167\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 114 q^{4} + 64 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 114 q^{4} + 64 q^{7} + 70 q^{10} - 14 q^{13} + 442 q^{16} + 348 q^{19} + 128 q^{22} + 618 q^{25} + 1164 q^{28} + 910 q^{31} + 972 q^{34} + 1068 q^{37} + 1742 q^{40} + 1032 q^{43} + 2226 q^{46} + 2356 q^{49} + 1978 q^{52} + 1246 q^{55} + 3240 q^{58} + 1458 q^{61} + 2858 q^{64} + 3650 q^{67} + 994 q^{70} + 948 q^{73} + 6812 q^{76} + 1582 q^{79} + 658 q^{82} + 5218 q^{85} + 3824 q^{88} + 882 q^{91} + 7274 q^{94} - 348 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.58634 0 23.2072 −3.48607 0 8.84963 −84.9528 0 19.4744
1.2 −5.00434 0 17.0434 1.05866 0 25.2687 −45.2564 0 −5.29791
1.3 −4.72764 0 14.3505 −7.78828 0 −26.5515 −30.0230 0 36.8202
1.4 −4.54841 0 12.6880 −22.1452 0 17.7299 −21.3231 0 100.726
1.5 −4.02623 0 8.21051 17.2051 0 35.4447 −0.847579 0 −69.2715
1.6 −3.52815 0 4.44786 12.1868 0 −30.9618 12.5325 0 −42.9969
1.7 −3.15292 0 1.94091 14.7909 0 −11.1292 19.1038 0 −46.6345
1.8 −2.79324 0 −0.197819 −1.33169 0 9.98795 22.8985 0 3.71974
1.9 −1.88109 0 −4.46150 −16.7765 0 −6.29617 23.4412 0 31.5581
1.10 −1.46463 0 −5.85486 −11.9297 0 22.3590 20.2922 0 17.4726
1.11 −1.15091 0 −6.67540 3.46710 0 −22.1468 16.8901 0 −3.99032
1.12 −0.548685 0 −7.69894 11.9914 0 9.44550 8.61378 0 −6.57948
1.13 0.548685 0 −7.69894 −11.9914 0 9.44550 −8.61378 0 −6.57948
1.14 1.15091 0 −6.67540 −3.46710 0 −22.1468 −16.8901 0 −3.99032
1.15 1.46463 0 −5.85486 11.9297 0 22.3590 −20.2922 0 17.4726
1.16 1.88109 0 −4.46150 16.7765 0 −6.29617 −23.4412 0 31.5581
1.17 2.79324 0 −0.197819 1.33169 0 9.98795 −22.8985 0 3.71974
1.18 3.15292 0 1.94091 −14.7909 0 −11.1292 −19.1038 0 −46.6345
1.19 3.52815 0 4.44786 −12.1868 0 −30.9618 −12.5325 0 −42.9969
1.20 4.02623 0 8.21051 −17.2051 0 35.4447 0.847579 0 −69.2715
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(43\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1161.4.a.h ✓ 24
3.b odd 2 1 inner 1161.4.a.h ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.4.a.h ✓ 24 1.a even 1 1 trivial
1161.4.a.h ✓ 24 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 153 T_{2}^{22} + 10142 T_{2}^{20} - 382474 T_{2}^{18} + 9059533 T_{2}^{16} + \cdots + 17119641600 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1161))\). Copy content Toggle raw display