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Properties

Label 1155.2.l.f
Level $1155$
Weight $2$
Character orbit 1155.l
Analytic conductor $9.223$
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] = [1155,2,Mod(1121,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.1121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.l (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: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{2} - 4 q^{3} + 44 q^{4} + 4 q^{6} + 12 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{2} - 4 q^{3} + 44 q^{4} + 4 q^{6} + 12 q^{8} - 2 q^{9} + 6 q^{11} + 8 q^{12} + 2 q^{15} + 68 q^{16} - 40 q^{18} + 2 q^{21} - 4 q^{22} + 12 q^{24} - 40 q^{25} + 32 q^{27} + 24 q^{29} - 8 q^{31} - 52 q^{32} + 28 q^{33} + 32 q^{34} - 40 q^{35} - 48 q^{37} - 66 q^{39} + 16 q^{41} - 32 q^{44} + 32 q^{48} - 40 q^{49} - 4 q^{50} + 14 q^{51} + 72 q^{54} + 8 q^{55} + 24 q^{57} - 80 q^{58} + 24 q^{60} - 48 q^{62} + 76 q^{64} - 4 q^{65} - 48 q^{67} + 32 q^{68} - 20 q^{69} - 4 q^{70} - 128 q^{72} + 4 q^{75} + 8 q^{77} - 28 q^{78} + 50 q^{81} - 32 q^{82} + 24 q^{83} + 24 q^{84} + 32 q^{87} - 16 q^{88} - 8 q^{90} - 4 q^{91} - 20 q^{93} + 12 q^{95} - 12 q^{96} + 100 q^{97} - 4 q^{98} + 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1 −2.72154 −1.44946 − 0.948193i 5.40675 − 1.00000i 3.94475 + 2.58054i − 1.00000i −9.27160 1.20186 + 2.74873i 2.72154i
1121.2 −2.72154 −1.44946 + 0.948193i 5.40675 1.00000i 3.94475 − 2.58054i 1.00000i −9.27160 1.20186 − 2.74873i − 2.72154i
1121.3 −2.50847 1.71957 − 0.207558i 4.29243 1.00000i −4.31349 + 0.520654i 1.00000i −5.75050 2.91384 − 0.713822i − 2.50847i
1121.4 −2.50847 1.71957 + 0.207558i 4.29243 − 1.00000i −4.31349 − 0.520654i − 1.00000i −5.75050 2.91384 + 0.713822i 2.50847i
1121.5 −2.45082 0.877309 + 1.49343i 4.00650 − 1.00000i −2.15012 − 3.66012i − 1.00000i −4.91756 −1.46066 + 2.62040i 2.45082i
1121.6 −2.45082 0.877309 − 1.49343i 4.00650 1.00000i −2.15012 + 3.66012i 1.00000i −4.91756 −1.46066 − 2.62040i − 2.45082i
1121.7 −1.78181 −1.68629 − 0.395519i 1.17485 1.00000i 3.00464 + 0.704739i 1.00000i 1.47026 2.68713 + 1.33392i − 1.78181i
1121.8 −1.78181 −1.68629 + 0.395519i 1.17485 − 1.00000i 3.00464 − 0.704739i − 1.00000i 1.47026 2.68713 − 1.33392i 1.78181i
1121.9 −1.41625 −1.72991 + 0.0861016i 0.00575714 − 1.00000i 2.44998 − 0.121941i − 1.00000i 2.82434 2.98517 − 0.297896i 1.41625i
1121.10 −1.41625 −1.72991 − 0.0861016i 0.00575714 1.00000i 2.44998 + 0.121941i 1.00000i 2.82434 2.98517 + 0.297896i − 1.41625i
1121.11 −1.30517 −0.259419 − 1.71251i −0.296539 1.00000i 0.338586 + 2.23512i 1.00000i 2.99737 −2.86540 + 0.888518i − 1.30517i
1121.12 −1.30517 −0.259419 + 1.71251i −0.296539 − 1.00000i 0.338586 − 2.23512i − 1.00000i 2.99737 −2.86540 − 0.888518i 1.30517i
1121.13 −0.892777 0.497839 − 1.65896i −1.20295 − 1.00000i −0.444460 + 1.48108i − 1.00000i 2.85952 −2.50431 − 1.65179i 0.892777i
1121.14 −0.892777 0.497839 + 1.65896i −1.20295 1.00000i −0.444460 − 1.48108i 1.00000i 2.85952 −2.50431 + 1.65179i − 0.892777i
1121.15 −0.490604 1.73069 + 0.0687106i −1.75931 − 1.00000i −0.849081 − 0.0337096i − 1.00000i 1.84433 2.99056 + 0.237833i 0.490604i
1121.16 −0.490604 1.73069 − 0.0687106i −1.75931 1.00000i −0.849081 + 0.0337096i 1.00000i 1.84433 2.99056 − 0.237833i − 0.490604i
1121.17 −0.371904 −0.305927 − 1.70482i −1.86169 − 1.00000i 0.113775 + 0.634029i − 1.00000i 1.43618 −2.81282 + 1.04310i 0.371904i
1121.18 −0.371904 −0.305927 + 1.70482i −1.86169 1.00000i 0.113775 − 0.634029i 1.00000i 1.43618 −2.81282 − 1.04310i − 0.371904i
1121.19 −0.261999 −0.733531 − 1.56905i −1.93136 1.00000i 0.192184 + 0.411090i 1.00000i 1.03001 −1.92386 + 2.30190i − 0.261999i
1121.20 −0.261999 −0.733531 + 1.56905i −1.93136 − 1.00000i 0.192184 − 0.411090i − 1.00000i 1.03001 −1.92386 − 2.30190i 0.261999i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1121.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.l.f yes 40
3.b odd 2 1 1155.2.l.e ✓ 40
11.b odd 2 1 1155.2.l.e ✓ 40
33.d even 2 1 inner 1155.2.l.f yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.l.e ✓ 40 3.b odd 2 1
1155.2.l.e ✓ 40 11.b odd 2 1
1155.2.l.f yes 40 1.a even 1 1 trivial
1155.2.l.f yes 40 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{20} - 2 T_{2}^{19} - 29 T_{2}^{18} + 56 T_{2}^{17} + 347 T_{2}^{16} - 634 T_{2}^{15} - 2237 T_{2}^{14} + \cdots - 32 \) Copy content Toggle raw display
\( T_{17}^{20} - 121 T_{17}^{18} - 96 T_{17}^{17} + 5899 T_{17}^{16} + 9008 T_{17}^{15} - 146479 T_{17}^{14} + \cdots - 18091520 \) Copy content Toggle raw display