↑

Properties

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

Related objects

Downloads

Learn more

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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 7 q^{2} + 87 q^{4} - 37 q^{5} + 3 q^{7} - 27 q^{8}+O(q^{10}) \) 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 −5.47906 0 22.0200 −11.4538 0 10.4530 −76.8166 0 62.7559
1.2 −5.33353 0 20.4466 18.4445 0 3.33060 −66.3841 0 −98.3745
1.3 −4.64432 0 13.5697 −21.3481 0 −26.1614 −25.8675 0 99.1474
1.4 −4.29852 0 10.4773 −10.6921 0 30.3987 −10.6486 0 45.9602
1.5 −4.04904 0 8.39473 0.434917 0 6.63981 −1.59827 0 −1.76100
1.6 −3.00617 0 1.03707 19.5716 0 −13.2606 20.9318 0 −58.8357
1.7 −2.64180 0 −1.02090 7.88919 0 −8.92293 23.8314 0 −20.8417
1.8 −2.44668 0 −2.01374 −11.6462 0 20.4800 24.5005 0 28.4945
1.9 −2.42076 0 −2.13992 −3.48472 0 −36.7573 24.5463 0 8.43567
1.10 −0.632689 0 −7.59970 8.95671 0 29.4406 9.86976 0 −5.66682
1.11 −0.0632306 0 −7.99600 −15.4871 0 −15.3919 1.01144 0 0.979261
1.12 0.176254 0 −7.96893 12.6112 0 4.39568 −2.81459 0 2.22278
1.13 0.556217 0 −7.69062 −3.06650 0 −18.2705 −8.72739 0 −1.70564
1.14 0.861032 0 −7.25862 −22.1043 0 20.0309 −13.1382 0 −19.0325
1.15 1.96463 0 −4.14021 6.04897 0 24.6993 −23.8511 0 11.8840
1.16 2.03545 0 −3.85695 −0.414175 0 −13.3218 −24.1342 0 −0.843031
1.17 3.89674 0 7.18461 11.1651 0 1.83209 −3.17736 0 43.5075
1.18 3.90091 0 7.21713 −11.1921 0 13.4217 −3.05391 0 −43.6594
1.19 4.06492 0 8.52355 4.30497 0 −6.63046 2.12820 0 17.4993
1.20 5.10386 0 18.0494 −15.9312 0 10.4685 51.2909 0 −81.3106
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.c ✓ 21
3.b odd 2 1 1161.4.a.f yes 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.4.a.c ✓ 21 1.a even 1 1 trivial
1161.4.a.f yes 21 3.b odd 2 1

Hecke kernels

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