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Properties

Label 1260.2.cu.e
Level $1260$
Weight $2$
Character orbit 1260.cu
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(101,1260)]
 
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, 1, 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("1260.101");
 
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.cu (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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{3} + 15 q^{5} + 6 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{3} + 15 q^{5} + 6 q^{7} - 6 q^{9} - 3 q^{11} - 6 q^{13} - 4 q^{15} + 3 q^{17} - 12 q^{19} + 12 q^{21} - 3 q^{23} - 15 q^{25} + 19 q^{27} + 3 q^{29} - 19 q^{33} - 9 q^{37} - 33 q^{39} + 3 q^{41} + 18 q^{43} - 6 q^{45} + 36 q^{47} - 6 q^{49} + 34 q^{51} - 6 q^{53} + 3 q^{57} + 6 q^{59} + 26 q^{63} - 42 q^{67} - 4 q^{69} + 24 q^{73} - 2 q^{75} + 33 q^{77} + 24 q^{79} + 2 q^{81} - 9 q^{83} - 3 q^{85} - 25 q^{87} - 3 q^{91} - 23 q^{93} + 78 q^{97} - 9 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 0 −1.72384 − 0.168501i 0 0.500000 − 0.866025i 0 −2.64509 − 0.0590591i 0 2.94321 + 0.580937i 0
101.2 0 −1.70677 + 0.294841i 0 0.500000 − 0.866025i 0 2.10386 − 1.60430i 0 2.82614 − 1.00645i 0
101.3 0 −1.26167 − 1.18668i 0 0.500000 − 0.866025i 0 2.15701 + 1.53210i 0 0.183605 + 2.99438i 0
101.4 0 −0.995866 + 1.41713i 0 0.500000 − 0.866025i 0 −1.36198 + 2.26826i 0 −1.01650 − 2.82254i 0
101.5 0 −0.983902 − 1.42546i 0 0.500000 − 0.866025i 0 −0.717180 − 2.54669i 0 −1.06387 + 2.80503i 0
101.6 0 −0.885413 − 1.48864i 0 0.500000 − 0.866025i 0 −1.81743 + 1.92275i 0 −1.43209 + 2.63612i 0
101.7 0 −0.542501 + 1.64490i 0 0.500000 − 0.866025i 0 2.49988 + 0.866374i 0 −2.41138 − 1.78472i 0
101.8 0 −0.526181 + 1.65019i 0 0.500000 − 0.866025i 0 1.78727 − 1.95081i 0 −2.44627 − 1.73660i 0
101.9 0 0.349685 − 1.69638i 0 0.500000 − 0.866025i 0 −1.12440 − 2.39494i 0 −2.75544 − 1.18640i 0
101.10 0 0.552843 + 1.64145i 0 0.500000 − 0.866025i 0 −2.20583 − 1.46093i 0 −2.38873 + 1.81493i 0
101.11 0 0.693685 − 1.58707i 0 0.500000 − 0.866025i 0 −1.78579 + 1.95217i 0 −2.03760 − 2.20186i 0
101.12 0 1.15183 − 1.29355i 0 0.500000 − 0.866025i 0 2.19402 + 1.47861i 0 −0.346558 − 2.97992i 0
101.13 0 1.46337 + 0.926576i 0 0.500000 − 0.866025i 0 0.880696 + 2.49487i 0 1.28291 + 2.71185i 0
101.14 0 1.68350 − 0.407232i 0 0.500000 − 0.866025i 0 1.90127 − 1.83989i 0 2.66832 − 1.37115i 0
101.15 0 1.73122 − 0.0536207i 0 0.500000 − 0.866025i 0 1.13371 − 2.39054i 0 2.99425 − 0.185658i 0
761.1 0 −1.72384 + 0.168501i 0 0.500000 + 0.866025i 0 −2.64509 + 0.0590591i 0 2.94321 − 0.580937i 0
761.2 0 −1.70677 − 0.294841i 0 0.500000 + 0.866025i 0 2.10386 + 1.60430i 0 2.82614 + 1.00645i 0
761.3 0 −1.26167 + 1.18668i 0 0.500000 + 0.866025i 0 2.15701 − 1.53210i 0 0.183605 − 2.99438i 0
761.4 0 −0.995866 − 1.41713i 0 0.500000 + 0.866025i 0 −1.36198 − 2.26826i 0 −1.01650 + 2.82254i 0
761.5 0 −0.983902 + 1.42546i 0 0.500000 + 0.866025i 0 −0.717180 + 2.54669i 0 −1.06387 − 2.80503i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i 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.cu.e yes 30
3.b odd 2 1 3780.2.cu.e 30
7.d odd 6 1 1260.2.bh.e ✓ 30
9.c even 3 1 3780.2.bh.e 30
9.d odd 6 1 1260.2.bh.e ✓ 30
21.g even 6 1 3780.2.bh.e 30
63.i even 6 1 inner 1260.2.cu.e yes 30
63.t odd 6 1 3780.2.cu.e 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.bh.e ✓ 30 7.d odd 6 1
1260.2.bh.e ✓ 30 9.d odd 6 1
1260.2.cu.e yes 30 1.a even 1 1 trivial
1260.2.cu.e yes 30 63.i even 6 1 inner
3780.2.bh.e 30 9.c even 3 1
3780.2.bh.e 30 21.g even 6 1
3780.2.cu.e 30 3.b odd 2 1
3780.2.cu.e 30 63.t odd 6 1

Hecke kernels

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

\( T_{11}^{30} + 3 T_{11}^{29} - 87 T_{11}^{28} - 270 T_{11}^{27} + 4971 T_{11}^{26} + 14727 T_{11}^{25} + \cdots + 66918059712 \) 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