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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7942,2,Mod(1,7942)] 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("7942.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7942, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-12,6,12,-6,-6,-6] 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: \(63.4171892853\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + x^{10} + 48 x^{9} - 47 x^{8} - 138 x^{7} + 150 x^{6} + 172 x^{5} - 139 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta_1 + 1) q^{3} + q^{4} + \beta_{11} q^{5} + (\beta_1 - 1) q^{6} + (\beta_{11} - \beta_{6} + \cdots - \beta_{2}) q^{7} - q^{8} + (\beta_{4} + \beta_{3} - \beta_1) q^{9} - \beta_{11} q^{10}+ \cdots + (\beta_{4} + \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 6 q^{3} + 12 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} - 12 q^{8} - 2 q^{9} + 6 q^{10} + 12 q^{11} + 6 q^{12} + 26 q^{13} + 6 q^{14} - 16 q^{15} + 12 q^{16} + 2 q^{18} - 6 q^{20} + 10 q^{21}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + x^{10} + 48 x^{9} - 47 x^{8} - 138 x^{7} + 150 x^{6} + 172 x^{5} - 139 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7 \nu^{11} + 31 \nu^{10} + 85 \nu^{9} - 491 \nu^{8} - 154 \nu^{7} + 2340 \nu^{6} - 475 \nu^{5} + \cdots - 170 ) / 101 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13 \nu^{11} + 72 \nu^{10} + 28 \nu^{9} - 681 \nu^{8} + 421 \nu^{7} + 2369 \nu^{6} - 1921 \nu^{5} + \cdots - 359 ) / 101 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13 \nu^{11} - 72 \nu^{10} - 28 \nu^{9} + 681 \nu^{8} - 421 \nu^{7} - 2369 \nu^{6} + 1921 \nu^{5} + \cdots + 157 ) / 101 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 24 \nu^{11} + 164 \nu^{10} - 127 \nu^{9} - 1164 \nu^{8} + 1795 \nu^{7} + 2944 \nu^{6} - 5077 \nu^{5} + \cdots - 150 ) / 101 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 80 \nu^{11} - 513 \nu^{10} + 255 \nu^{9} + 3880 \nu^{8} - 5108 \nu^{7} - 10150 \nu^{6} + 15442 \nu^{5} + \cdots + 399 ) / 101 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( - \nu^{11} + 6 \nu^{10} - \nu^{9} - 48 \nu^{8} + 47 \nu^{7} + 138 \nu^{6} - 150 \nu^{5} - 172 \nu^{4} + \cdots - 6 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 150 \nu^{11} - 924 \nu^{10} + 314 \nu^{9} + 7073 \nu^{8} - 8214 \nu^{7} - 18905 \nu^{6} + 25444 \nu^{5} + \cdots + 786 ) / 101 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 249 \nu^{11} + 1550 \nu^{10} - 598 \nu^{9} - 11824 \nu^{8} + 14419 \nu^{7} + 31049 \nu^{6} + \cdots - 1228 ) / 101 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 279 \nu^{11} - 1755 \nu^{10} + 782 \nu^{9} + 13178 \nu^{8} - 16789 \nu^{7} - 34022 \nu^{6} + \cdots + 1062 ) / 101 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 380 \nu^{11} - 2361 \nu^{10} + 883 \nu^{9} + 18026 \nu^{8} - 21536 \nu^{7} - 47960 \nu^{6} + \cdots + 1870 ) / 101 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{7} + 2\beta_{4} + 2\beta_{3} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} - 3\beta_{10} + 2\beta_{8} + 3\beta_{7} + \beta_{6} + 9\beta_{4} + 9\beta_{3} + 6\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{11} - 12 \beta_{10} + \beta_{9} + 6 \beta_{8} + 13 \beta_{7} + 3 \beta_{6} + \beta_{5} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27 \beta_{11} - 37 \beta_{10} + 4 \beta_{9} + 27 \beta_{8} + 40 \beta_{7} + 15 \beta_{6} + 5 \beta_{5} + \cdots + 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 96 \beta_{11} - 121 \beta_{10} + 20 \beta_{9} + 85 \beta_{8} + 136 \beta_{7} + 49 \beta_{6} + 23 \beta_{5} + \cdots + 204 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 283 \beta_{11} - 370 \beta_{10} + 72 \beta_{9} + 294 \beta_{8} + 419 \beta_{7} + 180 \beta_{6} + \cdots + 642 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 907 \beta_{11} - 1146 \beta_{10} + 269 \beta_{9} + 923 \beta_{8} + 1326 \beta_{7} + 584 \beta_{6} + \cdots + 1818 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2734 \beta_{11} - 3476 \beta_{10} + 909 \beta_{9} + 2957 \beta_{8} + 4060 \beta_{7} + 1940 \beta_{6} + \cdots + 5500 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8447 \beta_{11} - 10567 \beta_{10} + 3071 \beta_{9} + 9184 \beta_{8} + 12507 \beta_{7} + 6167 \beta_{6} + \cdots + 16106 \) 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
3.01246
2.55692
2.37041
1.85058
1.26387
0.410422
−0.162801
−0.220619
−0.443244
−1.24800
−1.65830
−1.73171
−1.00000 −2.01246 1.00000 1.68755 2.01246 −3.48754 −1.00000 1.05001 −1.68755
1.2 −1.00000 −1.55692 1.00000 −0.954341 1.55692 0.269753 −1.00000 −0.576007 0.954341
1.3 −1.00000 −1.37041 1.00000 −0.107181 1.37041 −0.953307 −1.00000 −1.12197 0.107181
1.4 −1.00000 −0.850582 1.00000 3.18154 0.850582 −2.86948 −1.00000 −2.27651 −3.18154
1.5 −1.00000 −0.263871 1.00000 −2.13699 0.263871 2.93250 −1.00000 −2.93037 2.13699
1.6 −1.00000 0.589578 1.00000 −3.50950 −0.589578 −2.22346 −1.00000 −2.65240 3.50950
1.7 −1.00000 1.16280 1.00000 2.05915 −1.16280 1.61923 −1.00000 −1.64789 −2.05915
1.8 −1.00000 1.22062 1.00000 −2.91567 −1.22062 −2.48215 −1.00000 −1.51009 2.91567
1.9 −1.00000 1.44324 1.00000 0.168593 −1.44324 −1.52457 −1.00000 −0.917047 −0.168593
1.10 −1.00000 2.24800 1.00000 −0.981787 −2.24800 4.63571 −1.00000 2.05349 0.981787
1.11 −1.00000 2.65830 1.00000 1.24304 −2.65830 1.34865 −1.00000 4.06656 −1.24304
1.12 −1.00000 2.73171 1.00000 −3.73440 −2.73171 −3.26533 −1.00000 4.46223 3.73440
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(11\) \( -1 \)
\(19\) \( +1 \)

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 7942.2.a.bu ✓ 12
19.b odd 2 1 7942.2.a.bv yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7942.2.a.bu ✓ 12 1.a even 1 1 trivial
7942.2.a.bv yes 12 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{12} - 6 T_{3}^{11} + T_{3}^{10} + 52 T_{3}^{9} - 65 T_{3}^{8} - 146 T_{3}^{7} + 262 T_{3}^{6} + \cdots - 19 \) Copy content Toggle raw display
\( T_{5}^{12} + 6 T_{5}^{11} - 12 T_{5}^{10} - 116 T_{5}^{9} - 11 T_{5}^{8} + 718 T_{5}^{7} + 412 T_{5}^{6} + \cdots - 19 \) Copy content Toggle raw display
\( T_{13}^{12} - 26 T_{13}^{11} + 238 T_{13}^{10} - 626 T_{13}^{9} - 3367 T_{13}^{8} + 22068 T_{13}^{7} + \cdots - 14039 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 6 T^{11} + \cdots - 19 \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots - 19 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + \cdots - 2099 \) Copy content Toggle raw display
$11$ \( (T - 1)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 26 T^{11} + \cdots - 14039 \) Copy content Toggle raw display
$17$ \( T^{12} - 101 T^{10} + \cdots + 67241 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} - 4 T^{11} + \cdots + 779741 \) Copy content Toggle raw display
$29$ \( T^{12} - 24 T^{11} + \cdots - 581879 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 371736479 \) Copy content Toggle raw display
$37$ \( T^{12} - 8 T^{11} + \cdots - 34252279 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 830751079 \) Copy content Toggle raw display
$43$ \( T^{12} + 32 T^{11} + \cdots - 55491259 \) Copy content Toggle raw display
$47$ \( T^{12} + 2 T^{11} + \cdots + 7560841 \) Copy content Toggle raw display
$53$ \( T^{12} - 28 T^{11} + \cdots - 59875559 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 5254326481 \) Copy content Toggle raw display
$61$ \( T^{12} - 363 T^{10} + \cdots + 12707561 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 18632171741 \) Copy content Toggle raw display
$71$ \( T^{12} - 4 T^{11} + \cdots + 7382021 \) Copy content Toggle raw display
$73$ \( T^{12} + 22 T^{11} + \cdots + 6570181 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 260545559 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 70919562121 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 447105176501 \) Copy content Toggle raw display
$97$ \( T^{12} - 20 T^{11} + \cdots - 12591559 \) Copy content Toggle raw display
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