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Properties

Label 1260.2.bh.d
Level $1260$
Weight $2$
Character orbit 1260.bh
Analytic conductor $10.061$
Analytic rank $0$
Dimension $28$
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] = [1260,2,Mod(941,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 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("1260.941");
 
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.bh (of order \(6\), degree \(2\), 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: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{3} - 28 q^{5} - 7 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{3} - 28 q^{5} - 7 q^{7} + 9 q^{9} - 18 q^{13} - q^{15} + 3 q^{17} - 6 q^{19} - 3 q^{21} + 28 q^{25} + q^{27} - 18 q^{29} - 12 q^{31} + 17 q^{33} + 7 q^{35} + 7 q^{37} - 12 q^{39} - 18 q^{41} - 17 q^{43} - 9 q^{45} + 21 q^{47} + 7 q^{49} + 43 q^{51} - 24 q^{53} - 33 q^{57} + 21 q^{59} - 21 q^{61} + 11 q^{63} + 18 q^{65} - 7 q^{67} - 10 q^{69} - 12 q^{73} + q^{75} + 21 q^{77} + 8 q^{79} - 19 q^{81} - 18 q^{83} - 3 q^{85} - 31 q^{87} + 6 q^{89} - 21 q^{91} + 16 q^{93} + 6 q^{95} + 18 q^{97} + 9 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
941.1 0 −1.68675 + 0.393557i 0 −1.00000 0 −0.518243 + 2.59450i 0 2.69023 − 1.32766i 0
941.2 0 −1.66724 + 0.469359i 0 −1.00000 0 1.83003 − 1.91076i 0 2.55940 − 1.56507i 0
941.3 0 −1.65049 − 0.525238i 0 −1.00000 0 −1.71672 − 2.01317i 0 2.44825 + 1.73380i 0
941.4 0 −1.10002 − 1.33790i 0 −1.00000 0 −1.22654 + 2.34427i 0 −0.579927 + 2.94341i 0
941.5 0 −0.926114 + 1.46366i 0 −1.00000 0 −2.15676 − 1.53245i 0 −1.28463 − 2.71104i 0
941.6 0 −0.785272 + 1.54381i 0 −1.00000 0 1.81891 + 1.92134i 0 −1.76670 − 2.42462i 0
941.7 0 −0.324432 − 1.70139i 0 −1.00000 0 0.287831 − 2.63005i 0 −2.78949 + 1.10397i 0
941.8 0 0.274228 − 1.71020i 0 −1.00000 0 −2.64575 + 0.00406263i 0 −2.84960 − 0.937973i 0
941.9 0 0.980031 + 1.42812i 0 −1.00000 0 −2.26312 + 1.37051i 0 −1.07908 + 2.79921i 0
941.10 0 1.32477 − 1.11579i 0 −1.00000 0 2.45126 − 0.995648i 0 0.510019 − 2.95633i 0
941.11 0 1.33109 − 1.10824i 0 −1.00000 0 −1.80850 + 1.93114i 0 0.543616 − 2.95034i 0
941.12 0 1.33205 + 1.10708i 0 −1.00000 0 2.61911 − 0.374550i 0 0.548736 + 2.94939i 0
941.13 0 1.67795 + 0.429499i 0 −1.00000 0 1.69785 + 2.02911i 0 2.63106 + 1.44136i 0
941.14 0 1.72019 − 0.202359i 0 −1.00000 0 −1.86937 − 1.87229i 0 2.91810 − 0.696192i 0
1181.1 0 −1.68675 − 0.393557i 0 −1.00000 0 −0.518243 − 2.59450i 0 2.69023 + 1.32766i 0
1181.2 0 −1.66724 − 0.469359i 0 −1.00000 0 1.83003 + 1.91076i 0 2.55940 + 1.56507i 0
1181.3 0 −1.65049 + 0.525238i 0 −1.00000 0 −1.71672 + 2.01317i 0 2.44825 − 1.73380i 0
1181.4 0 −1.10002 + 1.33790i 0 −1.00000 0 −1.22654 − 2.34427i 0 −0.579927 − 2.94341i 0
1181.5 0 −0.926114 − 1.46366i 0 −1.00000 0 −2.15676 + 1.53245i 0 −1.28463 + 2.71104i 0
1181.6 0 −0.785272 − 1.54381i 0 −1.00000 0 1.81891 − 1.92134i 0 −1.76670 + 2.42462i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 941.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.bh.d ✓ 28
3.b odd 2 1 3780.2.bh.d 28
7.d odd 6 1 1260.2.cu.d yes 28
9.c even 3 1 3780.2.cu.d 28
9.d odd 6 1 1260.2.cu.d yes 28
21.g even 6 1 3780.2.cu.d 28
63.k odd 6 1 3780.2.bh.d 28
63.s even 6 1 inner 1260.2.bh.d ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.bh.d ✓ 28 1.a even 1 1 trivial
1260.2.bh.d ✓ 28 63.s even 6 1 inner
1260.2.cu.d yes 28 7.d odd 6 1
1260.2.cu.d yes 28 9.d odd 6 1
3780.2.bh.d 28 3.b odd 2 1
3780.2.bh.d 28 63.k odd 6 1
3780.2.cu.d 28 9.c even 3 1
3780.2.cu.d 28 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{28} + 171 T_{11}^{26} + 12714 T_{11}^{24} + 542222 T_{11}^{22} + 14717277 T_{11}^{20} + \cdots + 6123375504 \) Copy content Toggle raw display
\( T_{13}^{28} + 18 T_{13}^{27} + 84 T_{13}^{26} - 432 T_{13}^{25} - 4131 T_{13}^{24} + 9969 T_{13}^{23} + \cdots + 1016064 \) Copy content Toggle raw display