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Properties

Label 1274.2.n.k
Level $1274$
Weight $2$
Character orbit 1274.n
Analytic conductor $10.173$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1274,2,Mod(753,1274)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1274.753"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1274, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,4,4,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1729412175\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} + \beta_{2}) q^{3} + \beta_{2} q^{4} + \beta_{6} q^{5} + ( - \beta_{6} + \beta_{5} - \beta_{3}) q^{6} - \beta_{3} q^{8} + ( - 2 \beta_{7} + 2 \beta_{4}) q^{9} - \beta_{4} q^{10}+ \cdots + (2 \beta_{6} - 2 \beta_{5} + 12 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{4} - 4 q^{12} + 24 q^{13} - 4 q^{16} + 24 q^{17} - 8 q^{22} + 12 q^{23} - 12 q^{25} - 8 q^{26} - 8 q^{27} - 48 q^{29} + 8 q^{30} - 16 q^{38} + 12 q^{39} + 64 q^{43} - 8 q^{48} - 24 q^{51}+ \cdots - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(197\) \(885\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
753.1
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.866025 + 0.500000i −0.207107 + 0.358719i 0.500000 − 0.866025i −1.22474 + 0.707107i − 0.414214i 0 1.00000i 1.41421 + 2.44949i 0.707107 − 1.22474i
753.2 −0.866025 + 0.500000i 1.20711 − 2.09077i 0.500000 − 0.866025i 1.22474 − 0.707107i 2.41421i 0 1.00000i −1.41421 − 2.44949i −0.707107 + 1.22474i
753.3 0.866025 − 0.500000i −0.207107 + 0.358719i 0.500000 − 0.866025i 1.22474 − 0.707107i 0.414214i 0 − 1.00000i 1.41421 + 2.44949i 0.707107 − 1.22474i
753.4 0.866025 − 0.500000i 1.20711 − 2.09077i 0.500000 − 0.866025i −1.22474 + 0.707107i − 2.41421i 0 − 1.00000i −1.41421 − 2.44949i −0.707107 + 1.22474i
961.1 −0.866025 − 0.500000i −0.207107 − 0.358719i 0.500000 + 0.866025i −1.22474 − 0.707107i 0.414214i 0 − 1.00000i 1.41421 − 2.44949i 0.707107 + 1.22474i
961.2 −0.866025 − 0.500000i 1.20711 + 2.09077i 0.500000 + 0.866025i 1.22474 + 0.707107i − 2.41421i 0 − 1.00000i −1.41421 + 2.44949i −0.707107 − 1.22474i
961.3 0.866025 + 0.500000i −0.207107 − 0.358719i 0.500000 + 0.866025i 1.22474 + 0.707107i − 0.414214i 0 1.00000i 1.41421 − 2.44949i 0.707107 + 1.22474i
961.4 0.866025 + 0.500000i 1.20711 + 2.09077i 0.500000 + 0.866025i −1.22474 − 0.707107i 2.41421i 0 1.00000i −1.41421 + 2.44949i −0.707107 − 1.22474i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 753.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1274.2.n.k 8
7.b odd 2 1 1274.2.n.h 8
7.c even 3 1 1274.2.d.g ✓ 4
7.c even 3 1 inner 1274.2.n.k 8
7.d odd 6 1 1274.2.d.j yes 4
7.d odd 6 1 1274.2.n.h 8
13.b even 2 1 inner 1274.2.n.k 8
91.b odd 2 1 1274.2.n.h 8
91.r even 6 1 1274.2.d.g ✓ 4
91.r even 6 1 inner 1274.2.n.k 8
91.s odd 6 1 1274.2.d.j yes 4
91.s odd 6 1 1274.2.n.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1274.2.d.g ✓ 4 7.c even 3 1
1274.2.d.g ✓ 4 91.r even 6 1
1274.2.d.j yes 4 7.d odd 6 1
1274.2.d.j yes 4 91.s odd 6 1
1274.2.n.h 8 7.b odd 2 1
1274.2.n.h 8 7.d odd 6 1
1274.2.n.h 8 91.b odd 2 1
1274.2.n.h 8 91.s odd 6 1
1274.2.n.k 8 1.a even 1 1 trivial
1274.2.n.k 8 7.c even 3 1 inner
1274.2.n.k 8 13.b even 2 1 inner
1274.2.n.k 8 91.r even 6 1 inner

Hecke kernels

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

\( T_{3}^{4} - 2T_{3}^{3} + 5T_{3}^{2} + 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 2T_{5}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} + 5 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 38 T^{6} + \cdots + 83521 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 13)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T + 36)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} - 68 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{3} + 29 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 34)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} - 38 T^{6} + \cdots + 83521 \) Copy content Toggle raw display
$37$ \( T^{8} - 82 T^{6} + \cdots + 279841 \) Copy content Toggle raw display
$41$ \( (T^{2} + 81)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 16 T + 46)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 22 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 108 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$61$ \( (T^{2} - 3 T + 9)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 214 T^{6} + \cdots + 62742241 \) Copy content Toggle raw display
$71$ \( (T^{4} + 164 T^{2} + 2116)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 194 T^{6} + \cdots + 4879681 \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} + \cdots + 25921)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 76 T^{2} + 1156)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 128 T^{2} + 16384)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 146 T^{2} + 5041)^{2} \) Copy content Toggle raw display
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