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Properties

Label 1274.2.g.m
Level $1274$
Weight $2$
Character orbit 1274.g
Analytic conductor $10.173$
Analytic rank $0$
Dimension $6$
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 = [6,3,-1,-3,0,1,0,-6,-8,0,-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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.4740147.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 10x^{4} + 25x^{2} + 3 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + \beta_{2}) q^{3} + (\beta_1 - 1) q^{4} + ( - \beta_{3} - \beta_{2}) q^{5} + \beta_{4} q^{6} - q^{8} + ( - \beta_{5} + 3 \beta_1 - 3) q^{9} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_{2}) q^{10}+ \cdots + ( - 2 \beta_{3} - 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - q^{3} - 3 q^{4} + q^{6} - 6 q^{8} - 8 q^{9} - 4 q^{11} + 2 q^{12} + 4 q^{13} - 9 q^{15} - 3 q^{16} - q^{17} - 16 q^{18} - 3 q^{19} + 4 q^{22} + 4 q^{23} + q^{24} + 30 q^{25} + 2 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 10x^{4} + 25x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 5\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} - 6\nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 8\nu^{3} - \nu^{2} + 15\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 6\nu^{2} - 12\nu + 3 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{5} - 10\beta_{4} - 5\beta_{3} - 5\beta_{2} + 6\beta _1 - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{3} - 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 50\beta_{5} + 56\beta_{4} + 25\beta_{3} + 28\beta_{2} - 48\beta _1 + 24 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.39248i
− 2.03709i
− 0.355387i
− 2.39248i
2.03709i
0.355387i
0.500000 − 0.866025i −1.36198 + 2.35902i −0.500000 − 0.866025i 4.14390 1.36198 + 2.35902i 0 −1.00000 −2.20997 − 3.82778i 2.07195 − 3.58872i
295.2 0.500000 − 0.866025i −0.574872 + 0.995707i −0.500000 − 0.866025i −3.52835 0.574872 + 0.995707i 0 −1.00000 0.839045 + 1.45327i −1.76417 + 3.05564i
295.3 0.500000 − 0.866025i 1.43685 − 2.48870i −0.500000 − 0.866025i −0.615549 −1.43685 − 2.48870i 0 −1.00000 −2.62908 − 4.55369i −0.307774 + 0.533081i
393.1 0.500000 + 0.866025i −1.36198 − 2.35902i −0.500000 + 0.866025i 4.14390 1.36198 − 2.35902i 0 −1.00000 −2.20997 + 3.82778i 2.07195 + 3.58872i
393.2 0.500000 + 0.866025i −0.574872 − 0.995707i −0.500000 + 0.866025i −3.52835 0.574872 − 0.995707i 0 −1.00000 0.839045 − 1.45327i −1.76417 − 3.05564i
393.3 0.500000 + 0.866025i 1.43685 + 2.48870i −0.500000 + 0.866025i −0.615549 −1.43685 + 2.48870i 0 −1.00000 −2.62908 + 4.55369i −0.307774 − 0.533081i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.3
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.m 6
7.b odd 2 1 1274.2.g.n 6
7.c even 3 1 1274.2.e.q 6
7.c even 3 1 1274.2.h.q 6
7.d odd 6 1 182.2.e.c ✓ 6
7.d odd 6 1 182.2.h.c yes 6
13.c even 3 1 inner 1274.2.g.m 6
21.g even 6 1 1638.2.m.f 6
21.g even 6 1 1638.2.p.f 6
91.g even 3 1 1274.2.e.q 6
91.h even 3 1 1274.2.h.q 6
91.m odd 6 1 182.2.e.c ✓ 6
91.n odd 6 1 1274.2.g.n 6
91.v odd 6 1 182.2.h.c yes 6
273.r even 6 1 1638.2.p.f 6
273.bf even 6 1 1638.2.m.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.e.c ✓ 6 7.d odd 6 1
182.2.e.c ✓ 6 91.m odd 6 1
182.2.h.c yes 6 7.d odd 6 1
182.2.h.c yes 6 91.v odd 6 1
1274.2.e.q 6 7.c even 3 1
1274.2.e.q 6 91.g even 3 1
1274.2.g.m 6 1.a even 1 1 trivial
1274.2.g.m 6 13.c even 3 1 inner
1274.2.g.n 6 7.b odd 2 1
1274.2.g.n 6 91.n odd 6 1
1274.2.h.q 6 7.c even 3 1
1274.2.h.q 6 91.h even 3 1
1638.2.m.f 6 21.g even 6 1
1638.2.m.f 6 273.bf even 6 1
1638.2.p.f 6 21.g even 6 1
1638.2.p.f 6 273.r even 6 1

Hecke kernels

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

\( T_{3}^{6} + T_{3}^{5} + 9T_{3}^{4} + 10T_{3}^{3} + 73T_{3}^{2} + 72T_{3} + 81 \) Copy content Toggle raw display
\( T_{5}^{3} - 15T_{5} - 9 \) Copy content Toggle raw display
\( T_{11}^{6} + 4T_{11}^{5} + 19T_{11}^{4} + 18T_{11}^{3} + 69T_{11}^{2} + 45T_{11} + 225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{3} - 15 T - 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 4 T^{5} + \cdots + 225 \) Copy content Toggle raw display
$13$ \( T^{6} - 4 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + T^{5} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$29$ \( T^{6} + T^{5} + \cdots + 31329 \) Copy content Toggle raw display
$31$ \( (T^{3} - 13 T^{2} + \cdots + 31)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 2 T^{5} + \cdots + 7921 \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} + 7 T^{5} + \cdots + 1521 \) Copy content Toggle raw display
$47$ \( (T^{3} - 13 T^{2} + \cdots - 39)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 13 T^{2} + \cdots + 39)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3 T + 9)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} - 11 T^{5} + \cdots + 51529 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$71$ \( T^{6} + 15 T^{5} + \cdots + 342225 \) Copy content Toggle raw display
$73$ \( (T^{3} - 22 T^{2} + \cdots - 191)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + T^{2} - 184 T - 13)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 14 T^{2} + \cdots + 225)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 17 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$97$ \( T^{6} + 27 T^{5} + \cdots + 49729 \) Copy content Toggle raw display
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