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Properties

Label 1274.2.g.p
Level $1274$
Weight $2$
Character orbit 1274.g
Analytic conductor $10.173$
Analytic rank $0$
Dimension $10$
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 = [10,-5,-1,-5,4,-1,0,10,-4,-2,-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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.23207289578928.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 3x^{8} + 13x^{7} + x^{6} - 39x^{5} + 3x^{4} + 117x^{3} - 81x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{2} + \beta_{6} q^{3} + \beta_{3} q^{4} - \beta_{8} q^{5} + \beta_{5} q^{6} + q^{8} + (\beta_{3} + \beta_{2}) q^{9} - \beta_{4} q^{10} + ( - \beta_{9} + \beta_{7} - \beta_{3} + \cdots - 1) q^{11}+ \cdots + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{2} - q^{3} - 5 q^{4} + 4 q^{5} - q^{6} + 10 q^{8} - 4 q^{9} - 2 q^{10} - 4 q^{11} + 2 q^{12} - 6 q^{13} + 3 q^{15} - 5 q^{16} - 3 q^{17} + 8 q^{18} - 5 q^{19} - 2 q^{20} - 4 q^{22} + 4 q^{23}+ \cdots + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 3x^{8} + 13x^{7} + x^{6} - 39x^{5} + 3x^{4} + 117x^{3} - 81x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{9} + \nu^{8} + 12\nu^{7} - 17\nu^{6} - 41\nu^{5} + 75\nu^{4} + 111\nu^{3} - 162\nu^{2} - 189\nu + 405 ) / 81 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{9} + 8\nu^{8} - 42\nu^{7} - 7\nu^{6} + 131\nu^{5} + 21\nu^{4} - 354\nu^{3} - 36\nu^{2} + 837\nu - 243 ) / 81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{9} - 2\nu^{8} - 45\nu^{7} + 28\nu^{6} + 136\nu^{5} - 99\nu^{4} - 393\nu^{3} + 342\nu^{2} + 675\nu - 648 ) / 81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{9} - 5\nu^{8} + 30\nu^{7} + 31\nu^{6} - 92\nu^{5} - 96\nu^{4} + 273\nu^{3} + 243\nu^{2} - 459\nu - 729 ) / 81 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{8} - 2\nu^{7} + 8\nu^{6} + 5\nu^{5} - 26\nu^{4} - 13\nu^{3} + 78\nu^{2} + 33\nu - 171 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{9} - 20\nu^{8} - 15\nu^{7} + 88\nu^{6} + 37\nu^{5} - 276\nu^{4} - 87\nu^{3} + 783\nu^{2} - 108\nu - 972 ) / 81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8\nu^{9} + 4\nu^{8} + 57\nu^{7} - 59\nu^{6} - 164\nu^{5} + 174\nu^{4} + 480\nu^{3} - 567\nu^{2} - 891\nu + 1215 ) / 81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{9} + 31 \nu^{8} + 36 \nu^{7} - 173 \nu^{6} - 83 \nu^{5} + 549 \nu^{4} + 201 \nu^{3} + \cdots + 2592 ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23 \nu^{9} - 19 \nu^{8} - 105 \nu^{7} + 101 \nu^{6} + 320 \nu^{5} - 312 \nu^{4} - 885 \nu^{3} + \cdots - 972 ) / 81 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{7} - 2\beta_{6} + 2\beta_{5} - \beta_{2} + \beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} - \beta_{8} + 4\beta_{7} + 3\beta_{5} - 2\beta_{4} - 5\beta_{3} + 5\beta_{2} + \beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{9} + 2\beta_{8} - 3\beta_{7} - \beta_{6} + \beta_{5} + 4\beta_{4} - 5\beta_{3} + 6\beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8 \beta_{9} + 6 \beta_{8} + 8 \beta_{7} + 5 \beta_{6} + \beta_{5} + 15 \beta_{4} - 9 \beta_{3} + \cdots - 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5 \beta_{9} + 4 \beta_{8} - 25 \beta_{7} + 3 \beta_{6} - 12 \beta_{5} + 14 \beta_{4} - 31 \beta_{3} + \cdots + 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 24 \beta_{9} + 43 \beta_{8} - 15 \beta_{7} + 16 \beta_{6} + 26 \beta_{5} + 35 \beta_{4} - 34 \beta_{3} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 19 \beta_{9} - 30 \beta_{8} - 26 \beta_{7} - 80 \beta_{6} - 13 \beta_{5} + 6 \beta_{4} - 63 \beta_{3} + \cdots + 108 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 34 \beta_{9} + 11 \beta_{8} - 26 \beta_{7} - 42 \beta_{6} + 84 \beta_{5} - 80 \beta_{4} - 158 \beta_{3} + \cdots - 211 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(885\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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
−1.73100 + 0.0603688i
1.16412 − 1.28251i
−1.24530 + 1.20384i
1.67396 + 0.444815i
1.13823 + 1.30554i
−1.73100 − 0.0603688i
1.16412 + 1.28251i
−1.24530 − 1.20384i
1.67396 − 0.444815i
1.13823 − 1.30554i
−0.500000 + 0.866025i −1.31322 + 2.27456i −0.500000 − 0.866025i 1.83556 −1.31322 − 2.27456i 0 1.00000 −1.94908 − 3.37591i −0.917780 + 1.58964i
295.2 −0.500000 + 0.866025i −1.02863 + 1.78164i −0.500000 − 0.866025i −3.38549 −1.02863 − 1.78164i 0 1.00000 −0.616155 − 1.06721i 1.69274 − 2.93192i
295.3 −0.500000 + 0.866025i −0.0800993 + 0.138736i −0.500000 − 0.866025i 3.33041 −0.0800993 − 0.138736i 0 1.00000 1.48717 + 2.57585i −1.66520 + 2.88422i
295.4 −0.500000 + 0.866025i 0.722201 − 1.25089i −0.500000 − 0.866025i −0.903518 0.722201 + 1.25089i 0 1.00000 0.456853 + 0.791292i 0.451759 − 0.782469i
295.5 −0.500000 + 0.866025i 1.19975 − 2.07802i −0.500000 − 0.866025i 1.12304 1.19975 + 2.07802i 0 1.00000 −1.37878 − 2.38812i −0.561520 + 0.972581i
393.1 −0.500000 − 0.866025i −1.31322 − 2.27456i −0.500000 + 0.866025i 1.83556 −1.31322 + 2.27456i 0 1.00000 −1.94908 + 3.37591i −0.917780 − 1.58964i
393.2 −0.500000 − 0.866025i −1.02863 − 1.78164i −0.500000 + 0.866025i −3.38549 −1.02863 + 1.78164i 0 1.00000 −0.616155 + 1.06721i 1.69274 + 2.93192i
393.3 −0.500000 − 0.866025i −0.0800993 − 0.138736i −0.500000 + 0.866025i 3.33041 −0.0800993 + 0.138736i 0 1.00000 1.48717 − 2.57585i −1.66520 − 2.88422i
393.4 −0.500000 − 0.866025i 0.722201 + 1.25089i −0.500000 + 0.866025i −0.903518 0.722201 − 1.25089i 0 1.00000 0.456853 − 0.791292i 0.451759 + 0.782469i
393.5 −0.500000 − 0.866025i 1.19975 + 2.07802i −0.500000 + 0.866025i 1.12304 1.19975 − 2.07802i 0 1.00000 −1.37878 + 2.38812i −0.561520 − 0.972581i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.5
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.p 10
7.b odd 2 1 1274.2.g.q 10
7.c even 3 1 182.2.e.d ✓ 10
7.c even 3 1 182.2.h.d yes 10
7.d odd 6 1 1274.2.e.s 10
7.d odd 6 1 1274.2.h.s 10
13.c even 3 1 inner 1274.2.g.p 10
21.h odd 6 1 1638.2.m.j 10
21.h odd 6 1 1638.2.p.k 10
91.g even 3 1 182.2.e.d ✓ 10
91.h even 3 1 182.2.h.d yes 10
91.m odd 6 1 1274.2.e.s 10
91.n odd 6 1 1274.2.g.q 10
91.v odd 6 1 1274.2.h.s 10
273.s odd 6 1 1638.2.p.k 10
273.bm odd 6 1 1638.2.m.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.e.d ✓ 10 7.c even 3 1
182.2.e.d ✓ 10 91.g even 3 1
182.2.h.d yes 10 7.c even 3 1
182.2.h.d yes 10 91.h even 3 1
1274.2.e.s 10 7.d odd 6 1
1274.2.e.s 10 91.m odd 6 1
1274.2.g.p 10 1.a even 1 1 trivial
1274.2.g.p 10 13.c even 3 1 inner
1274.2.g.q 10 7.b odd 2 1
1274.2.g.q 10 91.n odd 6 1
1274.2.h.s 10 7.d odd 6 1
1274.2.h.s 10 91.v odd 6 1
1638.2.m.j 10 21.h odd 6 1
1638.2.m.j 10 273.bm odd 6 1
1638.2.p.k 10 21.h odd 6 1
1638.2.p.k 10 273.s odd 6 1

Hecke kernels

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

\( T_{3}^{10} + T_{3}^{9} + 10T_{3}^{8} + 3T_{3}^{7} + 69T_{3}^{6} + 21T_{3}^{5} + 195T_{3}^{4} - 54T_{3}^{3} + 342T_{3}^{2} + 54T_{3} + 9 \) Copy content Toggle raw display
\( T_{5}^{5} - 2T_{5}^{4} - 12T_{5}^{3} + 25T_{5}^{2} + 7T_{5} - 21 \) Copy content Toggle raw display
\( T_{11}^{10} + 4 T_{11}^{9} + 40 T_{11}^{8} + 34 T_{11}^{7} + 727 T_{11}^{6} + 679 T_{11}^{5} + 6877 T_{11}^{4} + \cdots + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( (T^{5} - 2 T^{4} - 12 T^{3} + \cdots - 21)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 4 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{10} + 6 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} + 3 T^{9} + \cdots + 6718464 \) Copy content Toggle raw display
$19$ \( T^{10} + 5 T^{9} + \cdots + 50253921 \) Copy content Toggle raw display
$23$ \( T^{10} - 4 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{10} - T^{9} + \cdots + 103041 \) Copy content Toggle raw display
$31$ \( (T^{5} - 21 T^{4} + \cdots + 5677)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} - 10 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( T^{10} + 9 T^{9} + \cdots + 7733961 \) Copy content Toggle raw display
$43$ \( T^{10} - 9 T^{9} + \cdots + 6538249 \) Copy content Toggle raw display
$47$ \( (T^{5} - 13 T^{4} + \cdots + 3087)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} - 15 T^{4} + \cdots - 2439)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - T^{9} + \cdots + 52012944 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 493417369 \) Copy content Toggle raw display
$67$ \( T^{10} + 220 T^{8} + \cdots + 22118209 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 1069878681 \) Copy content Toggle raw display
$73$ \( (T^{5} - 4 T^{4} + \cdots - 6461)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - T^{4} - 185 T^{3} + \cdots - 257)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} - 48 T^{4} + \cdots - 20412)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 9 T^{9} + \cdots + 44836416 \) Copy content Toggle raw display
$97$ \( T^{10} + 19 T^{9} + \cdots + 42523441 \) Copy content Toggle raw display
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