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Properties

Label 1260.2.cx.c
Level $1260$
Weight $2$
Character orbit 1260.cx
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(41,1260)]
 
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, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.41");
 
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.cx (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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 5 q^{3} + 15 q^{5} + 6 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 5 q^{3} + 15 q^{5} + 6 q^{7} - 3 q^{9} + 3 q^{11} - 9 q^{13} - 4 q^{15} - 6 q^{17} - 12 q^{21} - 33 q^{23} - 15 q^{25} - 8 q^{27} + 21 q^{29} + 12 q^{31} - q^{33} + 3 q^{35} + 12 q^{37} + 3 q^{39} - 3 q^{41} - 12 q^{43} - 9 q^{45} + 6 q^{47} - 30 q^{49} + 7 q^{51} + 18 q^{57} - 6 q^{59} - 15 q^{61} + 20 q^{63} - 9 q^{65} - 9 q^{67} - 37 q^{69} + q^{75} + 9 q^{77} - 15 q^{79} + 17 q^{81} + 3 q^{83} - 3 q^{85} + 47 q^{87} - 30 q^{89} + 36 q^{91} - 14 q^{93} + 27 q^{97} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 −1.72037 − 0.200843i 0 0.500000 − 0.866025i 0 −0.0135086 + 2.64572i 0 2.91932 + 0.691046i 0
41.2 0 −1.71090 + 0.269884i 0 0.500000 − 0.866025i 0 2.64541 + 0.0422262i 0 2.85433 − 0.923486i 0
41.3 0 −1.57834 + 0.713331i 0 0.500000 − 0.866025i 0 −0.961927 − 2.46469i 0 1.98232 − 2.25176i 0
41.4 0 −1.34333 − 1.09337i 0 0.500000 − 0.866025i 0 0.760089 − 2.53422i 0 0.609094 + 2.93752i 0
41.5 0 −1.32129 − 1.11991i 0 0.500000 − 0.866025i 0 −1.27866 + 2.31625i 0 0.491598 + 2.95945i 0
41.6 0 −0.756600 + 1.55806i 0 0.500000 − 0.866025i 0 2.53732 + 0.749679i 0 −1.85511 − 2.35766i 0
41.7 0 −0.444104 + 1.67415i 0 0.500000 − 0.866025i 0 −2.06748 + 1.65092i 0 −2.60554 − 1.48699i 0
41.8 0 −0.439161 − 1.67545i 0 0.500000 − 0.866025i 0 2.51360 − 0.825724i 0 −2.61428 + 1.47158i 0
41.9 0 0.0956042 + 1.72941i 0 0.500000 − 0.866025i 0 −0.737797 − 2.54080i 0 −2.98172 + 0.330678i 0
41.10 0 0.444052 − 1.67416i 0 0.500000 − 0.866025i 0 −0.669097 − 2.55975i 0 −2.60563 − 1.48683i 0
41.11 0 0.634116 + 1.61180i 0 0.500000 − 0.866025i 0 1.14259 + 2.38631i 0 −2.19579 + 2.04414i 0
41.12 0 0.955577 − 1.44460i 0 0.500000 − 0.866025i 0 −1.65633 + 2.06315i 0 −1.17375 − 2.76085i 0
41.13 0 1.45676 − 0.936939i 0 0.500000 − 0.866025i 0 −2.61358 − 0.411352i 0 1.24429 − 2.72979i 0
41.14 0 1.56151 − 0.749467i 0 0.500000 − 0.866025i 0 2.00246 + 1.72920i 0 1.87660 − 2.34059i 0
41.15 0 1.66647 + 0.472082i 0 0.500000 − 0.866025i 0 1.39690 − 2.24692i 0 2.55428 + 1.57343i 0
461.1 0 −1.72037 + 0.200843i 0 0.500000 + 0.866025i 0 −0.0135086 − 2.64572i 0 2.91932 − 0.691046i 0
461.2 0 −1.71090 − 0.269884i 0 0.500000 + 0.866025i 0 2.64541 − 0.0422262i 0 2.85433 + 0.923486i 0
461.3 0 −1.57834 − 0.713331i 0 0.500000 + 0.866025i 0 −0.961927 + 2.46469i 0 1.98232 + 2.25176i 0
461.4 0 −1.34333 + 1.09337i 0 0.500000 + 0.866025i 0 0.760089 + 2.53422i 0 0.609094 − 2.93752i 0
461.5 0 −1.32129 + 1.11991i 0 0.500000 + 0.866025i 0 −1.27866 − 2.31625i 0 0.491598 − 2.95945i 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.o 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.cx.c ✓ 30
3.b odd 2 1 3780.2.cx.c 30
7.b odd 2 1 1260.2.cx.d yes 30
9.c even 3 1 3780.2.cx.d 30
9.d odd 6 1 1260.2.cx.d yes 30
21.c even 2 1 3780.2.cx.d 30
63.l odd 6 1 3780.2.cx.c 30
63.o even 6 1 inner 1260.2.cx.c ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.cx.c ✓ 30 1.a even 1 1 trivial
1260.2.cx.c ✓ 30 63.o even 6 1 inner
1260.2.cx.d yes 30 7.b odd 2 1
1260.2.cx.d yes 30 9.d odd 6 1
3780.2.cx.c 30 3.b odd 2 1
3780.2.cx.c 30 63.l odd 6 1
3780.2.cx.d 30 9.c even 3 1
3780.2.cx.d 30 21.c even 2 1

Hecke kernels

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

\( T_{11}^{30} - 3 T_{11}^{29} - 87 T_{11}^{28} + 270 T_{11}^{27} + 4851 T_{11}^{26} - 17805 T_{11}^{25} + \cdots + 1728 \) Copy content Toggle raw display
\( T_{13}^{30} + 9 T_{13}^{29} - 78 T_{13}^{28} - 945 T_{13}^{27} + 4524 T_{13}^{26} + 76335 T_{13}^{25} + \cdots + 2229977088 \) Copy content Toggle raw display