Properties

Label 1155.2.k.b
Level $1155$
Weight $2$
Character orbit 1155.k
Analytic conductor $9.223$
Analytic rank $0$
Dimension $48$
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] = [1155,2,Mod(769,1155)]
 
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([0, 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("1155.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.k (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: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{3} + 48 q^{4} - 4 q^{5} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{3} + 48 q^{4} - 4 q^{5} + 48 q^{9} + 48 q^{12} - 4 q^{15} + 40 q^{16} - 18 q^{20} + 20 q^{25} + 48 q^{27} + 48 q^{36} - 20 q^{38} - 16 q^{44} - 4 q^{45} + 8 q^{47} + 40 q^{48} + 24 q^{49} - 8 q^{55} - 8 q^{56} - 18 q^{60} - 4 q^{64} - 14 q^{70} - 32 q^{71} + 20 q^{75} - 32 q^{77} - 46 q^{80} + 48 q^{81} - 32 q^{82} - 16 q^{86} + 68 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1 −2.66471 1.00000 5.10065 −1.18527 1.89608i −2.66471 2.09935 + 1.61019i −8.26233 1.00000 3.15840 + 5.05250i
769.2 −2.66471 1.00000 5.10065 −1.18527 + 1.89608i −2.66471 2.09935 1.61019i −8.26233 1.00000 3.15840 5.05250i
769.3 −2.52930 1.00000 4.39737 −1.63381 1.52665i −2.52930 −2.48590 0.905718i −6.06368 1.00000 4.13240 + 3.86136i
769.4 −2.52930 1.00000 4.39737 −1.63381 + 1.52665i −2.52930 −2.48590 + 0.905718i −6.06368 1.00000 4.13240 3.86136i
769.5 −2.43104 1.00000 3.90997 2.21714 + 0.290295i −2.43104 1.43522 + 2.22265i −4.64322 1.00000 −5.38997 0.705719i
769.6 −2.43104 1.00000 3.90997 2.21714 0.290295i −2.43104 1.43522 2.22265i −4.64322 1.00000 −5.38997 + 0.705719i
769.7 −2.15215 1.00000 2.63174 1.03706 1.98104i −2.15215 −2.36665 1.18277i −1.35961 1.00000 −2.23191 + 4.26348i
769.8 −2.15215 1.00000 2.63174 1.03706 + 1.98104i −2.15215 −2.36665 + 1.18277i −1.35961 1.00000 −2.23191 4.26348i
769.9 −1.83264 1.00000 1.35857 0.538446 2.17027i −1.83264 −0.996474 + 2.45093i 1.17551 1.00000 −0.986778 + 3.97732i
769.10 −1.83264 1.00000 1.35857 0.538446 + 2.17027i −1.83264 −0.996474 2.45093i 1.17551 1.00000 −0.986778 3.97732i
769.11 −1.69546 1.00000 0.874594 −2.22547 0.217459i −1.69546 0.698105 2.55199i 1.90808 1.00000 3.77320 + 0.368694i
769.12 −1.69546 1.00000 0.874594 −2.22547 + 0.217459i −1.69546 0.698105 + 2.55199i 1.90808 1.00000 3.77320 0.368694i
769.13 −1.63529 1.00000 0.674169 −1.84877 1.25780i −1.63529 1.79271 1.94582i 2.16812 1.00000 3.02327 + 2.05687i
769.14 −1.63529 1.00000 0.674169 −1.84877 + 1.25780i −1.63529 1.79271 + 1.94582i 2.16812 1.00000 3.02327 2.05687i
769.15 −1.17317 1.00000 −0.623669 2.20931 0.344884i −1.17317 −1.71739 + 2.01260i 3.07801 1.00000 −2.59190 + 0.404608i
769.16 −1.17317 1.00000 −0.623669 2.20931 + 0.344884i −1.17317 −1.71739 2.01260i 3.07801 1.00000 −2.59190 0.404608i
769.17 −1.12050 1.00000 −0.744486 1.27295 + 1.83837i −1.12050 2.54893 0.709189i 3.07519 1.00000 −1.42634 2.05989i
769.18 −1.12050 1.00000 −0.744486 1.27295 1.83837i −1.12050 2.54893 + 0.709189i 3.07519 1.00000 −1.42634 + 2.05989i
769.19 −0.567201 1.00000 −1.67828 −2.10264 0.760858i −0.567201 −2.63586 + 0.228555i 2.08633 1.00000 1.19262 + 0.431559i
769.20 −0.567201 1.00000 −1.67828 −2.10264 + 0.760858i −0.567201 −2.63586 0.228555i 2.08633 1.00000 1.19262 0.431559i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
35.c odd 2 1 inner
385.h 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.k.b yes 48
5.b even 2 1 1155.2.k.a 48
7.b odd 2 1 1155.2.k.a 48
11.b odd 2 1 inner 1155.2.k.b yes 48
35.c odd 2 1 inner 1155.2.k.b yes 48
55.d odd 2 1 1155.2.k.a 48
77.b even 2 1 1155.2.k.a 48
385.h even 2 1 inner 1155.2.k.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.k.a 48 5.b even 2 1
1155.2.k.a 48 7.b odd 2 1
1155.2.k.a 48 55.d odd 2 1
1155.2.k.a 48 77.b even 2 1
1155.2.k.b yes 48 1.a even 1 1 trivial
1155.2.k.b yes 48 11.b odd 2 1 inner
1155.2.k.b yes 48 35.c odd 2 1 inner
1155.2.k.b yes 48 385.h even 2 1 inner