Properties

Label 1122.2.h.g
Level $1122$
Weight $2$
Character orbit 1122.h
Analytic conductor $8.959$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1122,2,Mod(1121,1122)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1122.1121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1122.h (of order \(2\), degree \(1\), 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: \(8.95921510679\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 28 q^{2} + 28 q^{4} - 28 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 28 q^{2} + 28 q^{4} - 28 q^{8} + 2 q^{9} - 8 q^{15} + 28 q^{16} - 24 q^{17} - 2 q^{18} + 36 q^{21} + 80 q^{25} + 8 q^{30} - 28 q^{32} - 10 q^{33} + 24 q^{34} + 2 q^{36} - 36 q^{42} - 24 q^{49} - 80 q^{50} + 26 q^{51} - 4 q^{55} - 8 q^{60} + 28 q^{64} + 10 q^{66} - 40 q^{67} - 24 q^{68} - 4 q^{69} - 2 q^{72} + 4 q^{77} + 14 q^{81} + 12 q^{83} + 36 q^{84} + 48 q^{87} + 56 q^{93} + 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1 −1.00000 −1.63364 0.575506i 1.00000 2.19878 1.63364 + 0.575506i −4.54668 −1.00000 2.33759 + 1.88034i −2.19878
1121.2 −1.00000 −1.63364 + 0.575506i 1.00000 2.19878 1.63364 0.575506i −4.54668 −1.00000 2.33759 1.88034i −2.19878
1121.3 −1.00000 −1.60123 0.660342i 1.00000 −1.88112 1.60123 + 0.660342i −2.39717 −1.00000 2.12790 + 2.11472i 1.88112
1121.4 −1.00000 −1.60123 + 0.660342i 1.00000 −1.88112 1.60123 0.660342i −2.39717 −1.00000 2.12790 2.11472i 1.88112
1121.5 −1.00000 −1.57348 0.723996i 1.00000 3.66340 1.57348 + 0.723996i 1.80951 −1.00000 1.95166 + 2.27838i −3.66340
1121.6 −1.00000 −1.57348 + 0.723996i 1.00000 3.66340 1.57348 0.723996i 1.80951 −1.00000 1.95166 2.27838i −3.66340
1121.7 −1.00000 −1.46442 0.924915i 1.00000 −3.53846 1.46442 + 0.924915i −1.65251 −1.00000 1.28906 + 2.70893i 3.53846
1121.8 −1.00000 −1.46442 + 0.924915i 1.00000 −3.53846 1.46442 0.924915i −1.65251 −1.00000 1.28906 2.70893i 3.53846
1121.9 −1.00000 −0.744620 1.56382i 1.00000 −2.04802 0.744620 + 1.56382i 2.83134 −1.00000 −1.89108 + 2.32891i 2.04802
1121.10 −1.00000 −0.744620 + 1.56382i 1.00000 −2.04802 0.744620 1.56382i 2.83134 −1.00000 −1.89108 2.32891i 2.04802
1121.11 −1.00000 −0.524078 1.65086i 1.00000 3.81008 0.524078 + 1.65086i 0.269313 −1.00000 −2.45068 + 1.73036i −3.81008
1121.12 −1.00000 −0.524078 + 1.65086i 1.00000 3.81008 0.524078 1.65086i 0.269313 −1.00000 −2.45068 1.73036i −3.81008
1121.13 −1.00000 −0.260343 1.71237i 1.00000 1.40515 0.260343 + 1.71237i −1.57706 −1.00000 −2.86444 + 0.891608i −1.40515
1121.14 −1.00000 −0.260343 + 1.71237i 1.00000 1.40515 0.260343 1.71237i −1.57706 −1.00000 −2.86444 0.891608i −1.40515
1121.15 −1.00000 0.260343 1.71237i 1.00000 −1.40515 −0.260343 + 1.71237i 1.57706 −1.00000 −2.86444 0.891608i 1.40515
1121.16 −1.00000 0.260343 + 1.71237i 1.00000 −1.40515 −0.260343 1.71237i 1.57706 −1.00000 −2.86444 + 0.891608i 1.40515
1121.17 −1.00000 0.524078 1.65086i 1.00000 −3.81008 −0.524078 + 1.65086i −0.269313 −1.00000 −2.45068 1.73036i 3.81008
1121.18 −1.00000 0.524078 + 1.65086i 1.00000 −3.81008 −0.524078 1.65086i −0.269313 −1.00000 −2.45068 + 1.73036i 3.81008
1121.19 −1.00000 0.744620 1.56382i 1.00000 2.04802 −0.744620 + 1.56382i −2.83134 −1.00000 −1.89108 2.32891i −2.04802
1121.20 −1.00000 0.744620 + 1.56382i 1.00000 2.04802 −0.744620 1.56382i −2.83134 −1.00000 −1.89108 + 2.32891i −2.04802
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1121.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
33.d even 2 1 inner
561.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1122.2.h.g 28
3.b odd 2 1 1122.2.h.h yes 28
11.b odd 2 1 1122.2.h.h yes 28
17.b even 2 1 inner 1122.2.h.g 28
33.d even 2 1 inner 1122.2.h.g 28
51.c odd 2 1 1122.2.h.h yes 28
187.b odd 2 1 1122.2.h.h yes 28
561.h even 2 1 inner 1122.2.h.g 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1122.2.h.g 28 1.a even 1 1 trivial
1122.2.h.g 28 17.b even 2 1 inner
1122.2.h.g 28 33.d even 2 1 inner
1122.2.h.g 28 561.h even 2 1 inner
1122.2.h.h yes 28 3.b odd 2 1
1122.2.h.h yes 28 11.b odd 2 1
1122.2.h.h yes 28 51.c odd 2 1
1122.2.h.h yes 28 187.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1122, [\chi])\):

\( T_{5}^{14} - 55T_{5}^{12} + 1210T_{5}^{10} - 13651T_{5}^{8} + 84648T_{5}^{6} - 288904T_{5}^{4} + 503744T_{5}^{2} - 345600 \) Copy content Toggle raw display
\( T_{7}^{14} - 43T_{7}^{12} + 650T_{7}^{10} - 4651T_{7}^{8} + 17080T_{7}^{6} - 31304T_{7}^{4} + 23360T_{7}^{2} - 1536 \) Copy content Toggle raw display
\( T_{23}^{14} - 160 T_{23}^{12} + 8616 T_{23}^{10} - 217024 T_{23}^{8} + 2785664 T_{23}^{6} - 17878528 T_{23}^{4} + 51064832 T_{23}^{2} - 52002816 \) Copy content Toggle raw display
\( T_{83}^{7} - 3T_{83}^{6} - 246T_{83}^{5} - 841T_{83}^{4} + 9288T_{83}^{3} + 65152T_{83}^{2} + 141184T_{83} + 100608 \) Copy content Toggle raw display