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Properties

Label 6069.2.a.be
Level $6069$
Weight $2$
Character orbit 6069.a
Self dual yes
Analytic conductor $48.461$
Analytic rank $0$
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] = [6069,2,Mod(1,6069)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6069, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) 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("6069.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6069.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,4,10,12,-6,4,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.4612089867\)
Analytic rank: \(0\)
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: \( x^{10} - 4x^{9} - 8x^{8} + 44x^{7} + 5x^{6} - 144x^{5} + 48x^{4} + 160x^{3} - 44x^{2} - 64x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 357)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{9} - 1) q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{7} + \beta_{6} - \beta_{4} + \cdots + 1) q^{8} + q^{9} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{10}+ \cdots + ( - \beta_{9} + \beta_{7} - \beta_{4} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} + 10 q^{3} + 12 q^{4} - 6 q^{5} + 4 q^{6} - 10 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{11} + 12 q^{12} + 6 q^{13} - 4 q^{14} - 6 q^{15} + 20 q^{16} + 4 q^{18} + 6 q^{19} - 16 q^{20} - 10 q^{21}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} - 8x^{8} + 44x^{7} + 5x^{6} - 144x^{5} + 48x^{4} + 160x^{3} - 44x^{2} - 64x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{9} - \nu^{8} - 36\nu^{7} + 17\nu^{6} + 218\nu^{5} - 86\nu^{4} - 502\nu^{3} + 114\nu^{2} + 344\nu + 42 ) / 22 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{9} + 20\nu^{8} + 71\nu^{7} - 208\nu^{6} - 202\nu^{5} + 598\nu^{4} + 206\nu^{3} - 432\nu^{2} - 170\nu - 4 ) / 22 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8 \nu^{9} - 15 \nu^{8} - 100 \nu^{7} + 145 \nu^{6} + 410 \nu^{5} - 344 \nu^{4} - 644 \nu^{3} + \cdots + 102 ) / 22 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{9} - 21\nu^{8} - 107\nu^{7} + 225\nu^{6} + 398\nu^{5} - 684\nu^{4} - 510\nu^{3} + 590\nu^{2} + 184\nu - 64 ) / 22 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{9} + 20\nu^{8} + 71\nu^{7} - 208\nu^{6} - 191\nu^{5} + 598\nu^{4} + 118\nu^{3} - 454\nu^{2} - 60\nu + 40 ) / 11 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15 \nu^{9} + 46 \nu^{8} + 149 \nu^{7} - 474 \nu^{6} - 392 \nu^{5} + 1316 \nu^{4} + 278 \nu^{3} + \cdots - 40 ) / 22 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -6\nu^{9} + 14\nu^{8} + 75\nu^{7} - 150\nu^{6} - 313\nu^{5} + 456\nu^{4} + 527\nu^{3} - 397\nu^{2} - 339\nu + 6 ) / 11 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} + 7\beta_{2} + 8\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} + 8\beta_{6} - 10\beta_{4} - 8\beta_{3} + 2\beta_{2} + 32\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{9} - 9 \beta_{8} + 10 \beta_{7} + 11 \beta_{6} - \beta_{5} - 2 \beta_{4} + 10 \beta_{3} + \cdots + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3\beta_{9} + 66\beta_{7} + 57\beta_{6} - 79\beta_{4} - 53\beta_{3} + 27\beta_{2} + 221\beta _1 + 92 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 82 \beta_{9} - 63 \beta_{8} + 80 \beta_{7} + 93 \beta_{6} - 9 \beta_{5} - 34 \beta_{4} + 78 \beta_{3} + \cdots + 670 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 53 \beta_{9} + 2 \beta_{8} + 456 \beta_{7} + 401 \beta_{6} + 4 \beta_{5} - 583 \beta_{4} - 325 \beta_{3} + \cdots + 798 \) 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
−2.46803
−1.80873
−0.901740
−0.661473
−0.0320370
1.23743
1.51109
1.71520
2.65813
2.75017
−2.46803 1.00000 4.09119 −1.62599 −2.46803 −1.00000 −5.16112 1.00000 4.01300
1.2 −1.80873 1.00000 1.27151 0.230597 −1.80873 −1.00000 1.31765 1.00000 −0.417087
1.3 −0.901740 1.00000 −1.18686 0.631060 −0.901740 −1.00000 2.87372 1.00000 −0.569052
1.4 −0.661473 1.00000 −1.56245 −2.35286 −0.661473 −1.00000 2.35647 1.00000 1.55635
1.5 −0.0320370 1.00000 −1.99897 −2.49420 −0.0320370 −1.00000 0.128115 1.00000 0.0799067
1.6 1.23743 1.00000 −0.468767 1.81608 1.23743 −1.00000 −3.05493 1.00000 2.24727
1.7 1.51109 1.00000 0.283389 −2.33464 1.51109 −1.00000 −2.59395 1.00000 −3.52785
1.8 1.71520 1.00000 0.941909 3.54799 1.71520 −1.00000 −1.81484 1.00000 6.08552
1.9 2.65813 1.00000 5.06565 0.737332 2.65813 −1.00000 8.14889 1.00000 1.95992
1.10 2.75017 1.00000 5.56342 −4.15538 2.75017 −1.00000 9.79999 1.00000 −11.4280
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( +1 \)
\(17\) \( +1 \)

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 6069.2.a.be 10
17.b even 2 1 6069.2.a.bd 10
17.d even 8 2 357.2.k.b ✓ 20
51.g odd 8 2 1071.2.n.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.k.b ✓ 20 17.d even 8 2
1071.2.n.b 20 51.g odd 8 2
6069.2.a.bd 10 17.b even 2 1
6069.2.a.be 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\):

\( T_{2}^{10} - 4T_{2}^{9} - 8T_{2}^{8} + 44T_{2}^{7} + 5T_{2}^{6} - 144T_{2}^{5} + 48T_{2}^{4} + 160T_{2}^{3} - 44T_{2}^{2} - 64T_{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{10} + 6 T_{5}^{9} - 9 T_{5}^{8} - 104 T_{5}^{7} - 79 T_{5}^{6} + 402 T_{5}^{5} + 441 T_{5}^{4} + \cdots - 64 \) Copy content Toggle raw display
\( T_{11}^{10} + 2 T_{11}^{9} - 63 T_{11}^{8} - 112 T_{11}^{7} + 1255 T_{11}^{6} + 1334 T_{11}^{5} + \cdots - 10084 \) Copy content Toggle raw display
\( T_{23}^{10} - 18 T_{23}^{9} + 29 T_{23}^{8} + 1116 T_{23}^{7} - 6257 T_{23}^{6} - 2122 T_{23}^{5} + \cdots - 484 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 4 T^{9} + \cdots - 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 6 T^{9} + \cdots - 64 \) Copy content Toggle raw display
$7$ \( (T + 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + 2 T^{9} + \cdots - 10084 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + \cdots - 91664 \) Copy content Toggle raw display
$17$ \( T^{10} \) Copy content Toggle raw display
$19$ \( T^{10} - 6 T^{9} + \cdots + 22972 \) Copy content Toggle raw display
$23$ \( T^{10} - 18 T^{9} + \cdots - 484 \) Copy content Toggle raw display
$29$ \( T^{10} + 12 T^{9} + \cdots - 5696 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + \cdots - 2336 \) Copy content Toggle raw display
$37$ \( T^{10} - 24 T^{9} + \cdots + 424456 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 833719792 \) Copy content Toggle raw display
$43$ \( T^{10} - 6 T^{9} + \cdots - 448 \) Copy content Toggle raw display
$47$ \( T^{10} - 16 T^{9} + \cdots - 9831296 \) Copy content Toggle raw display
$53$ \( T^{10} + 12 T^{9} + \cdots + 730592 \) Copy content Toggle raw display
$59$ \( T^{10} - 20 T^{9} + \cdots + 1360000 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 821767712 \) Copy content Toggle raw display
$67$ \( T^{10} - 20 T^{9} + \cdots + 4614400 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 101889904 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 424877056 \) Copy content Toggle raw display
$79$ \( T^{10} - 20 T^{9} + \cdots + 10713088 \) Copy content Toggle raw display
$83$ \( T^{10} - 36 T^{9} + \cdots - 6275584 \) Copy content Toggle raw display
$89$ \( T^{10} - 32 T^{9} + \cdots - 4681216 \) Copy content Toggle raw display
$97$ \( T^{10} - 36 T^{9} + \cdots - 310400 \) Copy content Toggle raw display
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