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Properties

Label 3381.2.a.bk
Level $3381$
Weight $2$
Character orbit 3381.a
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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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] = [3381,2,Mod(1,3381)] 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("3381.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3381, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,4,-10,8,-4,-4,0,12,10,-8,2] 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: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 108x^{3} - 33x^{2} - 36x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1) q^{4} + \beta_{7} q^{5} - \beta_1 q^{6} + ( - \beta_{9} - \beta_{8} + \beta_{4} + \cdots + 1) q^{8} + q^{9} + (\beta_{9} + \beta_{8} - \beta_1 - 1) q^{10}+ \cdots + (\beta_{8} - \beta_{5} + \cdots + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} - 10 q^{3} + 8 q^{4} - 4 q^{5} - 4 q^{6} + 12 q^{8} + 10 q^{9} - 8 q^{10} + 2 q^{11} - 8 q^{12} + 4 q^{15} + 4 q^{16} - 12 q^{17} + 4 q^{18} - 26 q^{19} - 24 q^{20} - 8 q^{22} - 10 q^{23}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 108x^{3} - 33x^{2} - 36x + 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 22\nu^{4} + 12\nu^{3} - 37\nu^{2} - 9\nu + 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{9} - 3\nu^{8} - 9\nu^{7} + 28\nu^{6} + 27\nu^{5} - 82\nu^{4} - 40\nu^{3} + 91\nu^{2} + 26\nu - 28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{9} + 3\nu^{8} + 10\nu^{7} - 31\nu^{6} - 33\nu^{5} + 103\nu^{4} + 45\nu^{3} - 123\nu^{2} - 26\nu + 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\nu^{9} + 7\nu^{8} + 15\nu^{7} - 63\nu^{6} - 30\nu^{5} + 173\nu^{4} + 30\nu^{3} - 176\nu^{2} - 25\nu + 46 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 3\nu^{9} - 11\nu^{8} - 21\nu^{7} + 98\nu^{6} + 34\nu^{5} - 266\nu^{4} - 26\nu^{3} + 271\nu^{2} + 30\nu - 70 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -4\nu^{9} + 14\nu^{8} + 31\nu^{7} - 128\nu^{6} - 69\nu^{5} + 362\nu^{4} + 83\nu^{3} - 385\nu^{2} - 67\nu + 108 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 5\nu^{9} - 17\nu^{8} - 41\nu^{7} + 159\nu^{6} + 103\nu^{5} - 466\nu^{4} - 136\nu^{3} + 514\nu^{2} + 106\nu - 149 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - \beta_{8} - \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} + 6\beta_{2} + 7\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{9} - 8\beta_{8} - \beta_{7} - 2\beta_{6} + 2\beta_{5} + 9\beta_{4} - 10\beta_{3} + 8\beta_{2} + 28\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{9} - 10 \beta_{8} - 8 \beta_{7} - 18 \beta_{6} + 10 \beta_{5} + 13 \beta_{4} - 22 \beta_{3} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 55 \beta_{9} - 52 \beta_{8} - 9 \beta_{7} - 24 \beta_{6} + 22 \beta_{5} + 68 \beta_{4} - 79 \beta_{3} + \cdots + 53 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 91 \beta_{9} - 75 \beta_{8} - 50 \beta_{7} - 131 \beta_{6} + 79 \beta_{5} + 122 \beta_{4} + \cdots + 230 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 366 \beta_{9} - 319 \beta_{8} - 62 \beta_{7} - 215 \beta_{6} + 183 \beta_{5} + 494 \beta_{4} + \cdots + 345 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.23864
−1.28424
−1.11439
−0.881528
0.507824
0.533756
1.46930
1.94871
2.42151
2.63770
−2.23864 −1.00000 3.01150 −2.46287 2.23864 0 −2.26439 1.00000 5.51348
1.2 −1.28424 −1.00000 −0.350734 1.91754 1.28424 0 3.01890 1.00000 −2.46257
1.3 −1.11439 −1.00000 −0.758127 −1.17161 1.11439 0 3.07364 1.00000 1.30563
1.4 −0.881528 −1.00000 −1.22291 2.92694 0.881528 0 2.84108 1.00000 −2.58018
1.5 0.507824 −1.00000 −1.74212 −3.36946 −0.507824 0 −1.90033 1.00000 −1.71109
1.6 0.533756 −1.00000 −1.71510 1.09368 −0.533756 0 −1.98296 1.00000 0.583758
1.7 1.46930 −1.00000 0.158829 2.14842 −1.46930 0 −2.70522 1.00000 3.15666
1.8 1.94871 −1.00000 1.79746 −1.40111 −1.94871 0 −0.394699 1.00000 −2.73035
1.9 2.42151 −1.00000 3.86372 −2.93909 −2.42151 0 4.51302 1.00000 −7.11705
1.10 2.63770 −1.00000 4.95748 −0.742420 −2.63770 0 7.80096 1.00000 −1.95828
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( +1 \)
\(23\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3381.2.a.bk ✓ 10
7.b odd 2 1 3381.2.a.bl yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3381.2.a.bk ✓ 10 1.a even 1 1 trivial
3381.2.a.bl yes 10 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3381))\):

\( T_{2}^{10} - 4T_{2}^{9} - 6T_{2}^{8} + 36T_{2}^{7} + T_{2}^{6} - 100T_{2}^{5} + 26T_{2}^{4} + 108T_{2}^{3} - 33T_{2}^{2} - 36T_{2} + 14 \) Copy content Toggle raw display
\( T_{5}^{10} + 4 T_{5}^{9} - 16 T_{5}^{8} - 72 T_{5}^{7} + 71 T_{5}^{6} + 428 T_{5}^{5} - 24 T_{5}^{4} + \cdots + 392 \) Copy content Toggle raw display
\( T_{11}^{10} - 2 T_{11}^{9} - 57 T_{11}^{8} + 140 T_{11}^{7} + 1038 T_{11}^{6} - 2936 T_{11}^{5} + \cdots - 12953 \) Copy content Toggle raw display
\( T_{13}^{10} - 52 T_{13}^{8} - 96 T_{13}^{7} + 747 T_{13}^{6} + 2584 T_{13}^{5} - 350 T_{13}^{4} + \cdots + 5598 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 4 T^{9} + \cdots + 14 \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 4 T^{9} + \cdots + 392 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} + \cdots - 12953 \) Copy content Toggle raw display
$13$ \( T^{10} - 52 T^{8} + \cdots + 5598 \) Copy content Toggle raw display
$17$ \( T^{10} + 12 T^{9} + \cdots - 226 \) Copy content Toggle raw display
$19$ \( T^{10} + 26 T^{9} + \cdots - 24113 \) Copy content Toggle raw display
$23$ \( (T + 1)^{10} \) Copy content Toggle raw display
$29$ \( T^{10} - 16 T^{9} + \cdots - 486482 \) Copy content Toggle raw display
$31$ \( T^{10} + 12 T^{9} + \cdots + 8641252 \) Copy content Toggle raw display
$37$ \( T^{10} + 8 T^{9} + \cdots + 679266 \) Copy content Toggle raw display
$41$ \( T^{10} + 10 T^{9} + \cdots + 1815793 \) Copy content Toggle raw display
$43$ \( T^{10} + 4 T^{9} + \cdots - 14455202 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 112744688 \) Copy content Toggle raw display
$53$ \( T^{10} - 14 T^{9} + \cdots + 5454391 \) Copy content Toggle raw display
$59$ \( T^{10} + 38 T^{9} + \cdots + 42698601 \) Copy content Toggle raw display
$61$ \( T^{10} + 14 T^{9} + \cdots + 141967 \) Copy content Toggle raw display
$67$ \( T^{10} - 268 T^{8} + \cdots - 376888 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 472303438 \) Copy content Toggle raw display
$73$ \( T^{10} + 8 T^{9} + \cdots - 2744 \) Copy content Toggle raw display
$79$ \( T^{10} + 16 T^{9} + \cdots + 537938 \) Copy content Toggle raw display
$83$ \( T^{10} + 28 T^{9} + \cdots + 11428 \) Copy content Toggle raw display
$89$ \( T^{10} + 32 T^{9} + \cdots + 1212644 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 608495324 \) Copy content Toggle raw display
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