Properties

Label 1161.2.f.c
Level $1161$
Weight $2$
Character orbit 1161.f
Analytic conductor $9.271$
Analytic rank $0$
Dimension $38$
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: \(38\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 4 q^{2} - 22 q^{4} + 9 q^{5} - 7 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 4 q^{2} - 22 q^{4} + 9 q^{5} - 7 q^{7} - 24 q^{8} - 14 q^{10} + 5 q^{11} + 5 q^{13} + 17 q^{14} - 24 q^{16} - 42 q^{17} - 8 q^{19} + 21 q^{20} + 20 q^{22} + 22 q^{23} - 10 q^{25} - 34 q^{26} - 2 q^{28} + 30 q^{29} + 5 q^{31} + 48 q^{32} + 6 q^{34} - 106 q^{35} - 2 q^{37} + 21 q^{38} - 16 q^{40} + 29 q^{41} - 19 q^{43} - 58 q^{44} + 32 q^{47} + 10 q^{49} - 11 q^{50} - q^{52} - 76 q^{53} + 4 q^{55} + 46 q^{56} - 30 q^{58} + 30 q^{59} + 10 q^{61} - 50 q^{62} + 28 q^{64} + 8 q^{65} - 3 q^{67} + 47 q^{68} - 56 q^{70} - 42 q^{71} + 16 q^{73} + 28 q^{74} + 36 q^{76} + 49 q^{77} - 4 q^{79} - 140 q^{80} - 8 q^{82} + 29 q^{83} + 4 q^{85} + 4 q^{86} + 47 q^{88} - 108 q^{89} + 8 q^{91} + 12 q^{92} + 23 q^{94} + 33 q^{95} + 4 q^{97} - 98 q^{98}+O(q^{100}) \) 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 −1.27673 2.21136i 0 −2.26008 + 3.91458i 1.39974 2.42442i 0 0.0769910 + 0.133352i 6.43513 0 −7.14837
388.2 −1.12806 1.95386i 0 −1.54504 + 2.67608i 0.722147 1.25080i 0 −0.444392 0.769710i 2.45933 0 −3.25850
388.3 −1.09981 1.90493i 0 −1.41918 + 2.45809i −0.829860 + 1.43736i 0 1.66958 + 2.89180i 1.84408 0 3.65077
388.4 −0.887526 1.53724i 0 −0.575406 + 0.996632i 1.84963 3.20365i 0 0.0448503 + 0.0776831i −1.50735 0 −6.56638
388.5 −0.880687 1.52540i 0 −0.551220 + 0.954742i −0.398628 + 0.690444i 0 0.958880 + 1.66083i −1.58094 0 1.40427
388.6 −0.649321 1.12466i 0 0.156764 0.271524i −1.38166 + 2.39311i 0 −0.516159 0.894013i −3.00445 0 3.58856
388.7 −0.326572 0.565640i 0 0.786701 1.36261i 1.96622 3.40559i 0 −2.10690 3.64925i −2.33395 0 −2.56845
388.8 −0.152464 0.264076i 0 0.953509 1.65153i 1.13542 1.96661i 0 1.25241 + 2.16924i −1.19136 0 −0.692446
388.9 −0.0180090 0.0311926i 0 0.999351 1.73093i −0.638135 + 1.10528i 0 −1.94045 3.36095i −0.144026 0 0.0459688
388.10 0.100298 + 0.173722i 0 0.979881 1.69720i 0.316846 0.548794i 0 0.192248 + 0.332984i 0.794314 0 0.127116
388.11 0.449049 + 0.777775i 0 0.596711 1.03353i −0.818891 + 1.41836i 0 0.930316 + 1.61135i 2.86800 0 −1.47089
388.12 0.451326 + 0.781720i 0 0.592609 1.02643i −0.0415173 + 0.0719100i 0 −1.41989 2.45931i 2.87515 0 −0.0749514
388.13 0.651400 + 1.12826i 0 0.151357 0.262158i 1.84461 3.19495i 0 −1.71786 2.97543i 2.99997 0 4.80630
388.14 0.724508 + 1.25489i 0 −0.0498244 + 0.0862984i −1.28186 + 2.22025i 0 0.350586 + 0.607232i 2.75364 0 −3.71487
388.15 0.987453 + 1.71032i 0 −0.950127 + 1.64567i −1.66142 + 2.87766i 0 2.32616 + 4.02902i 0.196989 0 −6.56230
388.16 1.09093 + 1.88955i 0 −1.38028 + 2.39071i 0.845931 1.46520i 0 −1.08430 1.87806i −1.65943 0 3.69142
388.17 1.26912 + 2.19819i 0 −2.22135 + 3.84749i 0.946835 1.63997i 0 0.412403 + 0.714304i −6.20016 0 4.80660
388.18 1.33284 + 2.30855i 0 −2.55293 + 4.42180i −0.659842 + 1.14288i 0 −0.548434 0.949916i −8.27921 0 −3.51785
388.19 1.36225 + 2.35949i 0 −2.71146 + 4.69638i 1.18444 2.05150i 0 −1.93605 3.35333i −9.32575 0 6.45400
775.1 −1.27673 + 2.21136i 0 −2.26008 3.91458i 1.39974 + 2.42442i 0 0.0769910 0.133352i 6.43513 0 −7.14837
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 388.19
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.c 38
3.b odd 2 1 387.2.f.c 38
9.c even 3 1 inner 1161.2.f.c 38
9.c even 3 1 3483.2.a.r 19
9.d odd 6 1 387.2.f.c 38
9.d odd 6 1 3483.2.a.s 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.2.f.c 38 3.b odd 2 1
387.2.f.c 38 9.d odd 6 1
1161.2.f.c 38 1.a even 1 1 trivial
1161.2.f.c 38 9.c even 3 1 inner
3483.2.a.r 19 9.c even 3 1
3483.2.a.s 19 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{38} - 4 T_{2}^{37} + 38 T_{2}^{36} - 112 T_{2}^{35} + 703 T_{2}^{34} - 1772 T_{2}^{33} + 8664 T_{2}^{32} - 18846 T_{2}^{31} + 77334 T_{2}^{30} - 148750 T_{2}^{29} + 529864 T_{2}^{28} - 903317 T_{2}^{27} + 2841518 T_{2}^{26} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(1161, [\chi])\). Copy content Toggle raw display