Properties

Label 1161.2.f.d
Level $1161$
Weight $2$
Character orbit 1161.f
Analytic conductor $9.271$
Analytic rank $0$
Dimension $40$
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] = [1161,2,Mod(388,1161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1161, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1161.388");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1161 = 3^{3} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1161.f (of order \(3\), degree \(2\), not 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: \(9.27063167467\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 387)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 6 q^{2} - 22 q^{4} - 17 q^{5} - 3 q^{7} + 30 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 6 q^{2} - 22 q^{4} - 17 q^{5} - 3 q^{7} + 30 q^{8} - 4 q^{10} - 10 q^{11} + q^{13} - 10 q^{14} - 22 q^{16} + 40 q^{17} + 16 q^{19} - 30 q^{20} - 15 q^{22} - 19 q^{23} - 19 q^{25} + 50 q^{26} - 6 q^{28} - 25 q^{29} + 11 q^{31} - 36 q^{32} - 9 q^{34} + 18 q^{37} - 28 q^{38} - 12 q^{40} - 12 q^{41} + 20 q^{43} + 10 q^{44} + 8 q^{46} - 38 q^{47} - 37 q^{49} - 36 q^{50} + 8 q^{52} + 138 q^{53} - 18 q^{55} - 30 q^{56} + 27 q^{58} - 31 q^{59} - 19 q^{61} + 64 q^{62} + 22 q^{64} - 47 q^{65} - 9 q^{67} - 68 q^{68} + 6 q^{70} + 42 q^{71} - 4 q^{73} + 16 q^{74} - 37 q^{76} - 85 q^{77} + 4 q^{79} + 122 q^{80} + 2 q^{82} - 19 q^{83} + 6 q^{85} + 6 q^{86} - 60 q^{88} + 108 q^{89} - 6 q^{91} - 85 q^{92} + 19 q^{94} + 11 q^{95} - 2 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
388.1 −1.38785 2.40383i 0 −2.85226 + 4.94026i −1.14436 + 1.98209i 0 2.15274 + 3.72866i 10.2826 0 6.35282
388.2 −1.28103 2.21881i 0 −2.28209 + 3.95269i 1.16256 2.01361i 0 −0.00975820 0.0169017i 6.56959 0 −5.95711
388.3 −1.18355 2.04996i 0 −1.80157 + 3.12041i 0.867534 1.50261i 0 −2.31405 4.00804i 3.79478 0 −4.10707
388.4 −1.16730 2.02182i 0 −1.72517 + 2.98809i −1.89009 + 3.27373i 0 −0.823076 1.42561i 3.38597 0 8.82518
388.5 −1.16214 2.01289i 0 −1.70115 + 2.94647i −1.73647 + 3.00766i 0 −0.946105 1.63870i 3.25933 0 8.07210
388.6 −0.856434 1.48339i 0 −0.466959 + 0.808796i −0.461242 + 0.798894i 0 −1.56362 2.70827i −1.82606 0 1.58009
388.7 −0.774348 1.34121i 0 −0.199230 + 0.345076i 0.0851621 0.147505i 0 0.931330 + 1.61311i −2.48030 0 −0.263781
388.8 −0.535034 0.926706i 0 0.427478 0.740413i 1.71565 2.97160i 0 0.0278005 + 0.0481519i −3.05500 0 −3.67173
388.9 −0.352536 0.610611i 0 0.751436 1.30153i 0.464078 0.803807i 0 2.55550 + 4.42626i −2.46978 0 −0.654418
388.10 −0.342086 0.592511i 0 0.765954 1.32667i −0.741129 + 1.28367i 0 −1.90727 3.30349i −2.41643 0 1.01412
388.11 −0.254302 0.440465i 0 0.870661 1.50803i −1.90300 + 3.29610i 0 0.340171 + 0.589193i −1.90285 0 1.93575
388.12 0.0942911 + 0.163317i 0 0.982218 1.70125i −1.89854 + 3.28836i 0 2.00490 + 3.47259i 0.747622 0 −0.716061
388.13 0.110277 + 0.191005i 0 0.975678 1.68992i 0.797629 1.38153i 0 −0.709781 1.22938i 0.871488 0 0.351841
388.14 0.391509 + 0.678114i 0 0.693441 1.20108i −1.47053 + 2.54704i 0 −1.24831 2.16213i 2.65199 0 −2.30291
388.15 0.562737 + 0.974689i 0 0.366654 0.635063i 1.19873 2.07626i 0 0.174845 + 0.302841i 3.07627 0 2.69827
388.16 0.875101 + 1.51572i 0 −0.531605 + 0.920767i −1.34854 + 2.33575i 0 −1.33599 2.31400i 1.63957 0 −4.72045
388.17 0.911404 + 1.57860i 0 −0.661315 + 1.14543i −0.216442 + 0.374888i 0 0.345529 + 0.598473i 1.23472 0 −0.789063
388.18 0.988037 + 1.71133i 0 −0.952435 + 1.64967i −0.717842 + 1.24334i 0 −2.37609 4.11551i 0.187983 0 −2.83702
388.19 1.04491 + 1.80984i 0 −1.18368 + 2.05019i 0.136674 0.236726i 0 1.21012 + 2.09599i −0.767710 0 0.571248
388.20 1.31834 + 2.28344i 0 −2.47606 + 4.28866i −1.39983 + 2.42457i 0 1.99111 + 3.44870i −7.78382 0 −7.38182
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 388.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1161.2.f.d 40
3.b odd 2 1 387.2.f.d 40
9.c even 3 1 inner 1161.2.f.d 40
9.c even 3 1 3483.2.a.u 20
9.d odd 6 1 387.2.f.d 40
9.d odd 6 1 3483.2.a.t 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.2.f.d 40 3.b odd 2 1
387.2.f.d 40 9.d odd 6 1
1161.2.f.d 40 1.a even 1 1 trivial
1161.2.f.d 40 9.c even 3 1 inner
3483.2.a.t 20 9.d odd 6 1
3483.2.a.u 20 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 6 T_{2}^{39} + 49 T_{2}^{38} + 196 T_{2}^{37} + 1015 T_{2}^{36} + 3316 T_{2}^{35} + 13489 T_{2}^{34} + 37496 T_{2}^{33} + 127462 T_{2}^{32} + 308874 T_{2}^{31} + 912872 T_{2}^{30} + 1954589 T_{2}^{29} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(1161, [\chi])\). Copy content Toggle raw display