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Properties

Label 1260.2.bh.e
Level $1260$
Weight $2$
Character orbit 1260.bh
Analytic conductor $10.061$
Analytic rank $0$
Dimension $30$
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: \(30\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 4 q^{3} + 30 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 4 q^{3} + 30 q^{5} + 6 q^{13} - 4 q^{15} - 3 q^{17} - 12 q^{19} - 11 q^{21} + 30 q^{25} - 19 q^{27} + 3 q^{29} - 24 q^{31} + q^{33} - 9 q^{37} + 36 q^{39} - 3 q^{41} + 18 q^{43} + 18 q^{47} - 12 q^{49} - 35 q^{51} + 6 q^{53} + 3 q^{57} + 3 q^{59} + 15 q^{61} - 4 q^{63} + 6 q^{65} + 21 q^{67} + 4 q^{69} + 24 q^{73} - 4 q^{75} + 39 q^{77} - 12 q^{79} - 28 q^{81} + 9 q^{83} - 3 q^{85} + 31 q^{87} - 3 q^{91} + 40 q^{93} - 12 q^{95} - 78 q^{97} - 9 q^{99}+O(q^{100}) \) 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 −1.73191 − 0.0224713i 0 1.00000 0 −0.756432 − 2.53531i 0 2.99899 + 0.0778363i 0
941.2 0 −1.72644 − 0.139354i 0 1.00000 0 2.56409 + 0.652252i 0 2.96116 + 0.481171i 0
941.3 0 −1.65852 − 0.499298i 0 1.00000 0 −2.40534 + 1.10198i 0 2.50140 + 1.65619i 0
941.4 0 −1.29427 + 1.15103i 0 1.00000 0 2.63628 + 0.223706i 0 0.350268 − 2.97948i 0
941.5 0 −1.02760 + 1.39429i 0 1.00000 0 −0.797731 − 2.52262i 0 −0.888065 − 2.86554i 0
941.6 0 −1.00784 − 1.40863i 0 1.00000 0 1.37369 − 2.26119i 0 −0.968501 + 2.83937i 0
941.7 0 −0.598046 − 1.62553i 0 1.00000 0 0.337441 + 2.62414i 0 −2.28468 + 1.94428i 0
941.8 0 −0.544333 + 1.64429i 0 1.00000 0 −2.37752 + 1.16077i 0 −2.40740 − 1.79009i 0
941.9 0 0.489075 + 1.66157i 0 1.00000 0 0.642758 + 2.56649i 0 −2.52161 + 1.62526i 0
941.10 0 0.729336 − 1.57101i 0 1.00000 0 −1.28338 − 2.31364i 0 −1.93614 − 2.29159i 0
941.11 0 0.819173 + 1.52609i 0 1.00000 0 1.50341 + 2.17710i 0 −1.65791 + 2.50027i 0
941.12 0 1.15327 − 1.29227i 0 1.00000 0 −2.00024 + 1.73177i 0 −0.339920 − 2.98068i 0
941.13 0 1.16602 − 1.28078i 0 1.00000 0 0.795818 + 2.52323i 0 −0.280807 − 2.98683i 0
941.14 0 1.53412 + 0.804029i 0 1.00000 0 −2.60097 − 0.484730i 0 1.70707 + 2.46696i 0
941.15 0 1.69796 − 0.341950i 0 1.00000 0 2.36812 − 1.17984i 0 2.76614 − 1.16123i 0
1181.1 0 −1.73191 + 0.0224713i 0 1.00000 0 −0.756432 + 2.53531i 0 2.99899 − 0.0778363i 0
1181.2 0 −1.72644 + 0.139354i 0 1.00000 0 2.56409 − 0.652252i 0 2.96116 − 0.481171i 0
1181.3 0 −1.65852 + 0.499298i 0 1.00000 0 −2.40534 − 1.10198i 0 2.50140 − 1.65619i 0
1181.4 0 −1.29427 − 1.15103i 0 1.00000 0 2.63628 − 0.223706i 0 0.350268 + 2.97948i 0
1181.5 0 −1.02760 − 1.39429i 0 1.00000 0 −0.797731 + 2.52262i 0 −0.888065 + 2.86554i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 941.15
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.e ✓ 30
3.b odd 2 1 3780.2.bh.e 30
7.d odd 6 1 1260.2.cu.e yes 30
9.c even 3 1 3780.2.cu.e 30
9.d odd 6 1 1260.2.cu.e yes 30
21.g even 6 1 3780.2.cu.e 30
63.k odd 6 1 3780.2.bh.e 30
63.s even 6 1 inner 1260.2.bh.e ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.bh.e ✓ 30 1.a even 1 1 trivial
1260.2.bh.e ✓ 30 63.s even 6 1 inner
1260.2.cu.e yes 30 7.d odd 6 1
1260.2.cu.e yes 30 9.d odd 6 1
3780.2.bh.e 30 3.b odd 2 1
3780.2.bh.e 30 63.k odd 6 1
3780.2.cu.e 30 9.c even 3 1
3780.2.cu.e 30 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 \(S_{2}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{30} + 183 T_{11}^{28} + 14358 T_{11}^{26} + 635846 T_{11}^{24} + 17628693 T_{11}^{22} + \cdots + 66918059712 \) Copy content Toggle raw display
\( T_{13}^{30} - 6 T_{13}^{29} - 99 T_{13}^{28} + 666 T_{13}^{27} + 6222 T_{13}^{26} + \cdots + 117031476966732 \) Copy content Toggle raw display