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Properties

Label 1161.4.a.d
Level $1161$
Weight $4$
Character orbit 1161.a
Self dual yes
Analytic conductor $68.501$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - q^{2} + 87 q^{4} - 3 q^{5} + 3 q^{7} - 69 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q - q^{2} + 87 q^{4} - 3 q^{5} + 3 q^{7} - 69 q^{8} - 9 q^{10} - 129 q^{11} + 54 q^{13} - 184 q^{14} + 111 q^{16} - 68 q^{17} + 78 q^{19} - 123 q^{20} - 51 q^{22} - 586 q^{23} + 402 q^{25} - 125 q^{26} - 6 q^{28} - 556 q^{29} - 111 q^{31} - 758 q^{32} - 420 q^{34} - 1409 q^{35} + 330 q^{37} - 1067 q^{38} + 180 q^{40} - 678 q^{41} - 903 q^{43} - 1510 q^{44} - 1062 q^{46} - 1580 q^{47} + 1218 q^{49} - 2054 q^{50} + 717 q^{52} - 1069 q^{53} - 2259 q^{55} - 2823 q^{56} + 666 q^{58} - 2248 q^{59} - 162 q^{61} - 3211 q^{62} - 2025 q^{64} - 910 q^{65} - 216 q^{67} - 1868 q^{68} + 3636 q^{70} - 3088 q^{71} - 1605 q^{73} - 2516 q^{74} + 1263 q^{76} - 891 q^{77} + 2094 q^{79} - 5251 q^{80} - 966 q^{82} - 1993 q^{83} - 114 q^{85} + 43 q^{86} + 846 q^{88} - 2494 q^{89} - 2382 q^{91} - 8034 q^{92} - 1062 q^{94} - 4798 q^{95} + 2565 q^{97} - 3103 q^{98}+O(q^{100}) \) 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 −5.31637 0 20.2638 3.65414 0 20.8267 −65.1986 0 −19.4267
1.2 −5.00944 0 17.0945 −12.5119 0 −9.33198 −45.5584 0 62.6777
1.3 −4.84783 0 15.5015 −16.9421 0 29.1204 −36.3660 0 82.1327
1.4 −4.17639 0 9.44227 10.0927 0 −15.5108 −6.02347 0 −42.1510
1.5 −4.14895 0 9.21376 20.9870 0 −30.4510 −5.03581 0 −87.0737
1.6 −2.89508 0 0.381517 7.01872 0 27.0395 22.0562 0 −20.3198
1.7 −2.42400 0 −2.12424 −17.6279 0 19.4399 24.5411 0 42.7299
1.8 −1.99899 0 −4.00403 5.98333 0 10.1419 23.9960 0 −11.9606
1.9 −1.91123 0 −4.34718 10.3634 0 −23.3949 23.5984 0 −19.8069
1.10 −0.318560 0 −7.89852 −16.9346 0 −25.6773 5.06463 0 5.39468
1.11 0.300662 0 −7.90960 −9.08026 0 −6.92598 −4.78341 0 −2.73009
1.12 0.563204 0 −7.68280 0.559868 0 20.8351 −8.83262 0 0.315320
1.13 1.32933 0 −6.23288 9.83042 0 −9.59659 −18.9202 0 13.0679
1.14 1.88873 0 −4.43270 7.93489 0 11.0434 −23.4820 0 14.9869
1.15 2.68882 0 −0.770244 −17.0644 0 9.05110 −23.5816 0 −45.8830
1.16 3.22858 0 2.42373 20.5591 0 −5.01145 −18.0034 0 66.3767
1.17 3.98048 0 7.84421 11.2891 0 −0.169008 −0.620112 0 44.9358
1.18 4.08934 0 8.72270 −3.99157 0 −21.9460 2.95535 0 −16.3229
1.19 4.19298 0 9.58108 −10.8469 0 36.6605 6.62944 0 −45.4810
1.20 4.71525 0 14.2336 −3.77842 0 −4.57768 29.3931 0 −17.8162
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1161.4.a.d ✓ 21
3.b odd 2 1 1161.4.a.e yes 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.4.a.d ✓ 21 1.a even 1 1 trivial
1161.4.a.e yes 21 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} + T_{2}^{20} - 127 T_{2}^{19} - 99 T_{2}^{18} + 6871 T_{2}^{17} + 3944 T_{2}^{16} + \cdots - 115053696 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1161))\). Copy content Toggle raw display