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Properties

Label 1260.2.cu.d
Level $1260$
Weight $2$
Character orbit 1260.cu
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(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: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{3} - 14 q^{5} + 5 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{3} - 14 q^{5} + 5 q^{7} - 9 q^{9} - 9 q^{11} + 18 q^{13} - q^{15} - 3 q^{17} - 6 q^{19} + q^{21} + 3 q^{23} - 14 q^{25} - q^{27} - 18 q^{29} + 7 q^{33} - 7 q^{35} + 7 q^{37} + 9 q^{39} + 18 q^{41} - 17 q^{43} + 42 q^{47} + q^{49} - 2 q^{51} + 24 q^{53} - 33 q^{57} + 42 q^{59} + 26 q^{63} + 14 q^{67} + 10 q^{69} - 12 q^{73} + 2 q^{75} + 45 q^{77} - 16 q^{79} + 11 q^{81} + 18 q^{83} - 3 q^{85} - 56 q^{87} - 6 q^{89} - 21 q^{91} + 7 q^{93} - 18 q^{97} + 9 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 0 −1.70866 − 0.283695i 0 −0.500000 + 0.866025i 0 2.64347 − 0.109925i 0 2.83903 + 0.969475i 0
101.2 0 −1.63567 + 0.569731i 0 −0.500000 + 0.866025i 0 −2.42160 + 1.06575i 0 2.35081 − 1.86378i 0
101.3 0 −1.34397 + 1.09259i 0 −0.500000 + 0.866025i 0 1.32639 + 2.28925i 0 0.612490 − 2.93681i 0
101.4 0 −1.28012 − 1.16675i 0 −0.500000 + 0.866025i 0 −0.885100 + 2.49331i 0 0.277392 + 2.98715i 0
101.5 0 −0.502542 − 1.65754i 0 −0.500000 + 0.866025i 0 2.50602 − 0.848438i 0 −2.49490 + 1.66597i 0
101.6 0 −0.427145 − 1.67855i 0 −0.500000 + 0.866025i 0 −2.56978 − 0.629475i 0 −2.63509 + 1.43397i 0
101.7 0 −0.303920 + 1.70518i 0 −0.500000 + 0.866025i 0 −2.08789 − 1.62503i 0 −2.81527 − 1.03648i 0
101.8 0 −0.294216 + 1.70688i 0 −0.500000 + 0.866025i 0 2.57667 + 0.600639i 0 −2.82687 − 1.00438i 0
101.9 0 0.684846 + 1.59091i 0 −0.500000 + 0.866025i 0 −0.686766 + 2.55506i 0 −2.06197 + 2.17905i 0
101.10 0 0.804513 − 1.53387i 0 −0.500000 + 0.866025i 0 −0.248762 + 2.63403i 0 −1.70552 − 2.46804i 0
101.11 0 0.944342 − 1.45197i 0 −0.500000 + 0.866025i 0 0.754474 − 2.53590i 0 −1.21643 − 2.74231i 0
101.12 0 1.21093 + 1.23840i 0 −0.500000 + 0.866025i 0 0.908339 − 2.48494i 0 −0.0672783 + 2.99925i 0
101.13 0 1.62479 + 0.600051i 0 −0.500000 + 0.866025i 0 −1.63392 − 2.08094i 0 2.27988 + 1.94991i 0
101.14 0 1.72681 + 0.134669i 0 −0.500000 + 0.866025i 0 2.31845 + 1.27466i 0 2.96373 + 0.465095i 0
761.1 0 −1.70866 + 0.283695i 0 −0.500000 − 0.866025i 0 2.64347 + 0.109925i 0 2.83903 − 0.969475i 0
761.2 0 −1.63567 − 0.569731i 0 −0.500000 − 0.866025i 0 −2.42160 − 1.06575i 0 2.35081 + 1.86378i 0
761.3 0 −1.34397 − 1.09259i 0 −0.500000 − 0.866025i 0 1.32639 − 2.28925i 0 0.612490 + 2.93681i 0
761.4 0 −1.28012 + 1.16675i 0 −0.500000 − 0.866025i 0 −0.885100 − 2.49331i 0 0.277392 − 2.98715i 0
761.5 0 −0.502542 + 1.65754i 0 −0.500000 − 0.866025i 0 2.50602 + 0.848438i 0 −2.49490 − 1.66597i 0
761.6 0 −0.427145 + 1.67855i 0 −0.500000 − 0.866025i 0 −2.56978 + 0.629475i 0 −2.63509 − 1.43397i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.14
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.d yes 28
3.b odd 2 1 3780.2.cu.d 28
7.d odd 6 1 1260.2.bh.d ✓ 28
9.c even 3 1 3780.2.bh.d 28
9.d odd 6 1 1260.2.bh.d ✓ 28
21.g even 6 1 3780.2.bh.d 28
63.i even 6 1 inner 1260.2.cu.d yes 28
63.t odd 6 1 3780.2.cu.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.bh.d ✓ 28 7.d odd 6 1
1260.2.bh.d ✓ 28 9.d odd 6 1
1260.2.cu.d yes 28 1.a even 1 1 trivial
1260.2.cu.d yes 28 63.i even 6 1 inner
3780.2.bh.d 28 9.c even 3 1
3780.2.bh.d 28 21.g even 6 1
3780.2.cu.d 28 3.b odd 2 1
3780.2.cu.d 28 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 \(S_{2}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{28} + 9 T_{11}^{27} - 45 T_{11}^{26} - 648 T_{11}^{25} + 1419 T_{11}^{24} + 32169 T_{11}^{23} + \cdots + 6123375504 \) 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