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Properties

Label 1274.2.g.l
Level $1274$
Weight $2$
Character orbit 1274.g
Analytic conductor $10.173$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1274,2,Mod(295,1274)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1274, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1274.295"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,0,-2,-8,0,0,4,0,4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1729412175\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_1 - 1) q^{4} + (\beta_{3} - 2) q^{5} + ( - \beta_{3} + \beta_{2}) q^{6} + q^{8} + ( - \beta_{2} + 2 \beta_1) q^{10} + (2 \beta_{2} + 2 \beta_1) q^{11} + \beta_{3} q^{12}+ \cdots + (4 \beta_{3} - 4 \beta_{2} + 10 \beta_1 - 10) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} - 8 q^{5} + 4 q^{8} + 4 q^{10} + 4 q^{11} + 4 q^{13} - 6 q^{15} - 2 q^{16} + 4 q^{20} + 4 q^{22} - 10 q^{23} + 8 q^{25} - 8 q^{26} - 4 q^{29} - 6 q^{30} - 16 q^{31} - 2 q^{32} + 12 q^{33}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
−0.500000 + 0.866025i −0.866025 + 1.50000i −0.500000 − 0.866025i −0.267949 −0.866025 − 1.50000i 0 1.00000 0 0.133975 − 0.232051i
295.2 −0.500000 + 0.866025i 0.866025 − 1.50000i −0.500000 − 0.866025i −3.73205 0.866025 + 1.50000i 0 1.00000 0 1.86603 − 3.23205i
393.1 −0.500000 − 0.866025i −0.866025 − 1.50000i −0.500000 + 0.866025i −0.267949 −0.866025 + 1.50000i 0 1.00000 0 0.133975 + 0.232051i
393.2 −0.500000 − 0.866025i 0.866025 + 1.50000i −0.500000 + 0.866025i −3.73205 0.866025 − 1.50000i 0 1.00000 0 1.86603 + 3.23205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1274.2.g.l 4
7.b odd 2 1 182.2.g.e ✓ 4
7.c even 3 1 1274.2.e.p 4
7.c even 3 1 1274.2.h.p 4
7.d odd 6 1 1274.2.e.o 4
7.d odd 6 1 1274.2.h.o 4
13.c even 3 1 inner 1274.2.g.l 4
21.c even 2 1 1638.2.r.v 4
28.d even 2 1 1456.2.s.l 4
91.g even 3 1 1274.2.e.p 4
91.h even 3 1 1274.2.h.p 4
91.m odd 6 1 1274.2.e.o 4
91.n odd 6 1 182.2.g.e ✓ 4
91.n odd 6 1 2366.2.a.t 2
91.t odd 6 1 2366.2.a.r 2
91.v odd 6 1 1274.2.h.o 4
91.bc even 12 2 2366.2.d.l 4
273.bn even 6 1 1638.2.r.v 4
364.v even 6 1 1456.2.s.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.e ✓ 4 7.b odd 2 1
182.2.g.e ✓ 4 91.n odd 6 1
1274.2.e.o 4 7.d odd 6 1
1274.2.e.o 4 91.m odd 6 1
1274.2.e.p 4 7.c even 3 1
1274.2.e.p 4 91.g even 3 1
1274.2.g.l 4 1.a even 1 1 trivial
1274.2.g.l 4 13.c even 3 1 inner
1274.2.h.o 4 7.d odd 6 1
1274.2.h.o 4 91.v odd 6 1
1274.2.h.p 4 7.c even 3 1
1274.2.h.p 4 91.h even 3 1
1456.2.s.l 4 28.d even 2 1
1456.2.s.l 4 364.v even 6 1
1638.2.r.v 4 21.c even 2 1
1638.2.r.v 4 273.bn even 6 1
2366.2.a.r 2 91.t odd 6 1
2366.2.a.t 2 91.n odd 6 1
2366.2.d.l 4 91.bc even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1274, [\chi])\):

\( T_{3}^{4} + 3T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{2} + 4T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} + 24T_{11}^{2} + 32T_{11} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$19$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 4 T - 44)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 16 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$73$ \( (T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + \cdots + 17424 \) Copy content Toggle raw display
$97$ \( T^{4} + 20 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
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