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Properties

Label 1161.4.a.f
Level $1161$
Weight $4$
Character orbit 1161.a
Self dual yes
Analytic conductor $68.501$
Analytic rank $0$
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: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 7 q^{2} + 87 q^{4} + 37 q^{5} + 3 q^{7} + 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 7 q^{2} + 87 q^{4} + 37 q^{5} + 3 q^{7} + 27 q^{8} - 9 q^{10} + 135 q^{11} - 66 q^{13} + 96 q^{14} + 543 q^{16} + 136 q^{17} - 66 q^{19} + 357 q^{20} - 51 q^{22} + 426 q^{23} + 546 q^{25} + 499 q^{26} - 336 q^{28} + 372 q^{29} - 159 q^{31} + 138 q^{32} + 12 q^{34} + 275 q^{35} + 282 q^{37} + 1543 q^{38} - 12 q^{40} + 846 q^{41} + 903 q^{43} + 2660 q^{44} + 504 q^{46} + 932 q^{47} + 726 q^{49} + 3138 q^{50} - 1707 q^{52} + 1083 q^{53} + 2073 q^{55} + 3749 q^{56} + 234 q^{58} + 812 q^{59} - 450 q^{61} + 2029 q^{62} + 2439 q^{64} + 2194 q^{65} - 804 q^{67} + 2288 q^{68} - 1404 q^{70} + 4656 q^{71} + 159 q^{73} + 3202 q^{74} - 1041 q^{76} + 2413 q^{77} - 2778 q^{79} + 3345 q^{80} + 2190 q^{82} + 551 q^{83} - 846 q^{85} + 301 q^{86} - 1674 q^{88} + 834 q^{89} + 4902 q^{91} + 3096 q^{92} + 2490 q^{94} + 4446 q^{95} + 69 q^{97} + 1377 q^{98}+O(q^{100}) \) 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 −5.45578 0 21.7655 −0.393043 0 −33.8741 −75.1015 0 2.14435
1.2 −5.10386 0 18.0494 15.9312 0 10.4685 −51.2909 0 −81.3106
1.3 −4.06492 0 8.52355 −4.30497 0 −6.63046 −2.12820 0 17.4993
1.4 −3.90091 0 7.21713 11.1921 0 13.4217 3.05391 0 −43.6594
1.5 −3.89674 0 7.18461 −11.1651 0 1.83209 3.17736 0 43.5075
1.6 −2.03545 0 −3.85695 0.414175 0 −13.3218 24.1342 0 −0.843031
1.7 −1.96463 0 −4.14021 −6.04897 0 24.6993 23.8511 0 11.8840
1.8 −0.861032 0 −7.25862 22.1043 0 20.0309 13.1382 0 −19.0325
1.9 −0.556217 0 −7.69062 3.06650 0 −18.2705 8.72739 0 −1.70564
1.10 −0.176254 0 −7.96893 −12.6112 0 4.39568 2.81459 0 2.22278
1.11 0.0632306 0 −7.99600 15.4871 0 −15.3919 −1.01144 0 0.979261
1.12 0.632689 0 −7.59970 −8.95671 0 29.4406 −9.86976 0 −5.66682
1.13 2.42076 0 −2.13992 3.48472 0 −36.7573 −24.5463 0 8.43567
1.14 2.44668 0 −2.01374 11.6462 0 20.4800 −24.5005 0 28.4945
1.15 2.64180 0 −1.02090 −7.88919 0 −8.92293 −23.8314 0 −20.8417
1.16 3.00617 0 1.03707 −19.5716 0 −13.2606 −20.9318 0 −58.8357
1.17 4.04904 0 8.39473 −0.434917 0 6.63981 1.59827 0 −1.76100
1.18 4.29852 0 10.4773 10.6921 0 30.3987 10.6486 0 45.9602
1.19 4.64432 0 13.5697 21.3481 0 −26.1614 25.8675 0 99.1474
1.20 5.33353 0 20.4466 −18.4445 0 3.33060 66.3841 0 −98.3745
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.f yes 21
3.b odd 2 1 1161.4.a.c ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.4.a.c ✓ 21 3.b odd 2 1
1161.4.a.f yes 21 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} - 7 T_{2}^{20} - 103 T_{2}^{19} + 789 T_{2}^{18} + 4099 T_{2}^{17} - 36316 T_{2}^{16} + \cdots - 2581632 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1161))\). Copy content Toggle raw display