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Properties

Label 1274.2.e.b
Level $1274$
Weight $2$
Character orbit 1274.e
Analytic conductor $10.173$
Analytic rank $0$
Dimension $2$
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(165,1274)] 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([4, 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("1274.165"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-2,2,-3,2,0,-2,-1,3,6] 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - 2 \zeta_{6} q^{3} + q^{4} - 3 \zeta_{6} q^{5} + 2 \zeta_{6} q^{6} - q^{8} + (\zeta_{6} - 1) q^{9} + 3 \zeta_{6} q^{10} + 6 \zeta_{6} q^{11} - 2 \zeta_{6} q^{12} + ( - 3 \zeta_{6} - 1) q^{13} + \cdots - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} - 2 q^{8} - q^{9} + 3 q^{10} + 6 q^{11} - 2 q^{12} - 5 q^{13} - 6 q^{15} + 2 q^{16} + 6 q^{17} + q^{18} - 4 q^{19} - 3 q^{20} - 6 q^{22} - 12 q^{23}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(885\)
\(\chi(n)\) \(-\zeta_{6}\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−1.00000 −1.00000 − 1.73205i 1.00000 −1.50000 − 2.59808i 1.00000 + 1.73205i 0 −1.00000 −0.500000 + 0.866025i 1.50000 + 2.59808i
471.1 −1.00000 −1.00000 + 1.73205i 1.00000 −1.50000 + 2.59808i 1.00000 − 1.73205i 0 −1.00000 −0.500000 − 0.866025i 1.50000 − 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h 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.e.b 2
7.b odd 2 1 1274.2.e.k 2
7.c even 3 1 1274.2.g.c 2
7.c even 3 1 1274.2.h.l 2
7.d odd 6 1 182.2.g.c ✓ 2
7.d odd 6 1 1274.2.h.e 2
13.c even 3 1 1274.2.h.l 2
21.g even 6 1 1638.2.r.m 2
28.f even 6 1 1456.2.s.a 2
91.g even 3 1 1274.2.g.c 2
91.h even 3 1 inner 1274.2.e.b 2
91.l odd 6 1 2366.2.a.k 1
91.m odd 6 1 182.2.g.c ✓ 2
91.n odd 6 1 1274.2.h.e 2
91.v odd 6 1 1274.2.e.k 2
91.v odd 6 1 2366.2.a.a 1
91.ba even 12 2 2366.2.d.a 2
273.bf even 6 1 1638.2.r.m 2
364.br even 6 1 1456.2.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.c ✓ 2 7.d odd 6 1
182.2.g.c ✓ 2 91.m odd 6 1
1274.2.e.b 2 1.a even 1 1 trivial
1274.2.e.b 2 91.h even 3 1 inner
1274.2.e.k 2 7.b odd 2 1
1274.2.e.k 2 91.v odd 6 1
1274.2.g.c 2 7.c even 3 1
1274.2.g.c 2 91.g even 3 1
1274.2.h.e 2 7.d odd 6 1
1274.2.h.e 2 91.n odd 6 1
1274.2.h.l 2 7.c even 3 1
1274.2.h.l 2 13.c even 3 1
1456.2.s.a 2 28.f even 6 1
1456.2.s.a 2 364.br even 6 1
1638.2.r.m 2 21.g even 6 1
1638.2.r.m 2 273.bf even 6 1
2366.2.a.a 1 91.v odd 6 1
2366.2.a.k 1 91.l odd 6 1
2366.2.d.a 2 91.ba even 12 2

Hecke kernels

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

\( T_{3}^{2} + 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( (T - 18)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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