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Properties

Label 1260.2.l.d
Level $1260$
Weight $2$
Character orbit 1260.l
Analytic conductor $10.061$
Analytic rank $0$
Dimension $32$
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] = [1260,2,Mod(1079,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.1079");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{4} + 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{4} + 32 q^{7} - 12 q^{10} - 4 q^{16} - 4 q^{25} + 4 q^{28} + 8 q^{34} - 8 q^{40} + 40 q^{43} - 8 q^{46} + 32 q^{49} + 24 q^{52} + 64 q^{55} + 96 q^{58} - 16 q^{61} + 4 q^{64} + 8 q^{67} - 12 q^{70} + 16 q^{82} + 36 q^{85} - 104 q^{88} + 120 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1079.1 −1.38248 − 0.297922i 0 1.82249 + 0.823740i 2.23419 − 0.0915374i 0 1.00000 −2.27413 − 1.68176i 0 −3.11599 − 0.539066i
1079.2 −1.38248 + 0.297922i 0 1.82249 − 0.823740i 2.23419 + 0.0915374i 0 1.00000 −2.27413 + 1.68176i 0 −3.11599 + 0.539066i
1079.3 −1.34103 − 0.449028i 0 1.59675 + 1.20432i −0.229115 + 2.22430i 0 1.00000 −1.60052 − 2.33202i 0 1.30602 − 2.87998i
1079.4 −1.34103 + 0.449028i 0 1.59675 − 1.20432i −0.229115 − 2.22430i 0 1.00000 −1.60052 + 2.33202i 0 1.30602 + 2.87998i
1079.5 −1.33264 − 0.473355i 0 1.55187 + 1.26163i −2.09968 + 0.768989i 0 1.00000 −1.47089 − 2.41588i 0 3.16213 − 0.0308930i
1079.6 −1.33264 + 0.473355i 0 1.55187 − 1.26163i −2.09968 − 0.768989i 0 1.00000 −1.47089 + 2.41588i 0 3.16213 + 0.0308930i
1079.7 −1.12734 − 0.853873i 0 0.541801 + 1.92521i 0.418506 − 2.19655i 0 1.00000 1.03309 − 2.63301i 0 −2.34738 + 2.11892i
1079.8 −1.12734 + 0.853873i 0 0.541801 − 1.92521i 0.418506 + 2.19655i 0 1.00000 1.03309 + 2.63301i 0 −2.34738 − 2.11892i
1079.9 −1.00192 − 0.998080i 0 0.00767087 + 1.99999i 1.15189 − 1.91654i 0 1.00000 1.98846 − 2.01147i 0 −3.06697 + 0.770535i
1079.10 −1.00192 + 0.998080i 0 0.00767087 − 1.99999i 1.15189 + 1.91654i 0 1.00000 1.98846 + 2.01147i 0 −3.06697 − 0.770535i
1079.11 −0.619294 − 1.27141i 0 −1.23295 + 1.57475i 2.22003 + 0.267359i 0 1.00000 2.76570 + 0.592350i 0 −1.03493 − 2.98813i
1079.12 −0.619294 + 1.27141i 0 −1.23295 − 1.57475i 2.22003 − 0.267359i 0 1.00000 2.76570 − 0.592350i 0 −1.03493 + 2.98813i
1079.13 −0.442953 − 1.34305i 0 −1.60759 + 1.18982i −0.588805 + 2.15715i 0 1.00000 2.31007 + 1.63204i 0 3.15798 − 0.164720i
1079.14 −0.442953 + 1.34305i 0 −1.60759 − 1.18982i −0.588805 − 2.15715i 0 1.00000 2.31007 − 1.63204i 0 3.15798 + 0.164720i
1079.15 −0.399975 − 1.35647i 0 −1.68004 + 1.08511i −1.80832 − 1.31529i 0 1.00000 2.14390 + 1.84491i 0 −1.06087 + 2.97902i
1079.16 −0.399975 + 1.35647i 0 −1.68004 − 1.08511i −1.80832 + 1.31529i 0 1.00000 2.14390 − 1.84491i 0 −1.06087 − 2.97902i
1079.17 0.399975 − 1.35647i 0 −1.68004 − 1.08511i 1.80832 − 1.31529i 0 1.00000 −2.14390 + 1.84491i 0 −1.06087 − 2.97902i
1079.18 0.399975 + 1.35647i 0 −1.68004 + 1.08511i 1.80832 + 1.31529i 0 1.00000 −2.14390 − 1.84491i 0 −1.06087 + 2.97902i
1079.19 0.442953 − 1.34305i 0 −1.60759 − 1.18982i 0.588805 + 2.15715i 0 1.00000 −2.31007 + 1.63204i 0 3.15798 + 0.164720i
1079.20 0.442953 + 1.34305i 0 −1.60759 + 1.18982i 0.588805 − 2.15715i 0 1.00000 −2.31007 − 1.63204i 0 3.15798 − 0.164720i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1079.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.l.d yes 32
3.b odd 2 1 inner 1260.2.l.d yes 32
4.b odd 2 1 1260.2.l.c ✓ 32
5.b even 2 1 1260.2.l.c ✓ 32
12.b even 2 1 1260.2.l.c ✓ 32
15.d odd 2 1 1260.2.l.c ✓ 32
20.d odd 2 1 inner 1260.2.l.d yes 32
60.h even 2 1 inner 1260.2.l.d yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.l.c ✓ 32 4.b odd 2 1
1260.2.l.c ✓ 32 5.b even 2 1
1260.2.l.c ✓ 32 12.b even 2 1
1260.2.l.c ✓ 32 15.d odd 2 1
1260.2.l.d yes 32 1.a even 1 1 trivial
1260.2.l.d yes 32 3.b odd 2 1 inner
1260.2.l.d yes 32 20.d odd 2 1 inner
1260.2.l.d yes 32 60.h 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 \(S_{2}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{16} - 92 T_{11}^{14} + 2952 T_{11}^{12} - 40448 T_{11}^{10} + 262288 T_{11}^{8} - 811584 T_{11}^{6} + \cdots + 98304 \) Copy content Toggle raw display
\( T_{43}^{8} - 10 T_{43}^{7} - 130 T_{43}^{6} + 972 T_{43}^{5} + 6752 T_{43}^{4} - 28672 T_{43}^{3} + \cdots + 1212416 \) Copy content Toggle raw display