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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7792,2,Mod(1,7792)] 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("7792.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7792 = 2^{4} \cdot 487 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7792.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,4,0,9,0,-1] 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: \(62.2194332550\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7 x^{15} - 5 x^{14} + 131 x^{13} - 132 x^{12} - 977 x^{11} + 1666 x^{10} + 3671 x^{9} + \cdots - 3825 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 487)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + (\beta_{11} + \beta_{3} + 1) q^{5} + ( - \beta_{15} + \beta_{14} + \beta_{5}) q^{7} + ( - \beta_{13} + \beta_{9} - \beta_{3} + 1) q^{9} + (\beta_{6} + \beta_{5}) q^{11} + ( - \beta_{15} + \beta_{14} - \beta_{13} + \cdots + 1) q^{13}+ \cdots + (\beta_{15} - 3 \beta_{14} - \beta_{13} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} + 9 q^{5} - q^{7} + 22 q^{9} - q^{11} + 4 q^{13} + 20 q^{15} + 38 q^{17} - 7 q^{19} - 31 q^{21} - 8 q^{23} + 35 q^{25} + 4 q^{27} + 6 q^{29} + 10 q^{31} + 40 q^{33} - 8 q^{35} - 8 q^{37}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7 x^{15} - 5 x^{14} + 131 x^{13} - 132 x^{12} - 977 x^{11} + 1666 x^{10} + 3671 x^{9} + \cdots - 3825 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4 \nu^{15} + 19 \nu^{14} + 47 \nu^{13} - 308 \nu^{12} - 165 \nu^{11} + 2009 \nu^{10} + 35 \nu^{9} + \cdots - 1851 ) / 21 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13 \nu^{15} - 81 \nu^{14} - 25 \nu^{13} + 1043 \nu^{12} - 1331 \nu^{11} - 4326 \nu^{10} + 10178 \nu^{9} + \cdots - 13075 ) / 35 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7 \nu^{15} + 49 \nu^{14} + 5 \nu^{13} - 707 \nu^{12} + 939 \nu^{11} + 3839 \nu^{10} - 7567 \nu^{9} + \cdots - 465 ) / 15 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7 \nu^{15} - 49 \nu^{14} - 5 \nu^{13} + 707 \nu^{12} - 939 \nu^{11} - 3839 \nu^{10} + 7567 \nu^{9} + \cdots + 555 ) / 15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2 \nu^{15} + \nu^{14} - 125 \nu^{13} + 287 \nu^{12} + 1416 \nu^{11} - 4564 \nu^{10} - 6118 \nu^{9} + \cdots + 21145 ) / 35 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16 \nu^{15} + 132 \nu^{14} - 120 \nu^{13} - 1666 \nu^{12} + 4037 \nu^{11} + 6727 \nu^{10} + \cdots + 11895 ) / 35 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8 \nu^{15} - 206 \nu^{14} + 935 \nu^{13} + 1393 \nu^{12} - 14811 \nu^{11} + 7994 \nu^{10} + \cdots - 89370 ) / 105 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19 \nu^{15} + 62 \nu^{14} - 1310 \nu^{13} + 1484 \nu^{12} + 16182 \nu^{11} - 31598 \nu^{10} + \cdots + 144195 ) / 105 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 149 \nu^{15} - 923 \nu^{14} - 895 \nu^{13} + 14749 \nu^{12} - 8538 \nu^{11} - 94528 \nu^{10} + \cdots + 104865 ) / 105 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2 \nu^{15} - 16 \nu^{14} + 220 \nu^{13} - 232 \nu^{12} - 2811 \nu^{11} + 5884 \nu^{10} + \cdots - 31035 ) / 15 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2 \nu^{15} + 16 \nu^{14} - 220 \nu^{13} + 232 \nu^{12} + 2811 \nu^{11} - 5884 \nu^{10} - 13888 \nu^{9} + \cdots + 31065 ) / 15 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 4 \nu^{15} + 38 \nu^{14} - 65 \nu^{13} - 419 \nu^{12} + 1508 \nu^{11} + 1003 \nu^{10} - 9539 \nu^{9} + \cdots + 7390 ) / 5 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 151 \nu^{15} + 922 \nu^{14} + 1160 \nu^{13} - 15701 \nu^{12} + 5547 \nu^{11} + 109592 \nu^{10} + \cdots - 184005 ) / 105 \) 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
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{6} + \beta_{5} + 7\beta_{2} + 8\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - \beta_{14} + 9 \beta_{13} + 7 \beta_{12} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{15} - \beta_{14} + 14 \beta_{13} + 10 \beta_{12} + \beta_{11} + \beta_{10} + 3 \beta_{9} + \cdots + 67 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{15} - 10 \beta_{14} + 70 \beta_{13} + 43 \beta_{12} + 2 \beta_{11} + 10 \beta_{10} + 16 \beta_{9} + \cdots + 81 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 32 \beta_{15} - 13 \beta_{14} + 139 \beta_{13} + 75 \beta_{12} + 16 \beta_{11} + 14 \beta_{10} + \cdots + 380 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 132 \beta_{15} - 74 \beta_{14} + 532 \beta_{13} + 260 \beta_{12} + 39 \beta_{11} + 78 \beta_{10} + \cdots + 607 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 355 \beta_{15} - 114 \beta_{14} + 1216 \beta_{13} + 513 \beta_{12} + 182 \beta_{11} + 137 \beta_{10} + \cdots + 2293 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1220 \beta_{15} - 490 \beta_{14} + 4044 \beta_{13} + 1583 \beta_{12} + 486 \beta_{11} + 572 \beta_{10} + \cdots + 4382 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3403 \beta_{15} - 848 \beta_{14} + 10031 \beta_{13} + 3385 \beta_{12} + 1801 \beta_{11} + 1173 \beta_{10} + \cdots + 14463 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 10708 \beta_{15} - 3074 \beta_{14} + 30857 \beta_{13} + 9748 \beta_{12} + 5008 \beta_{11} + 4148 \beta_{10} + \cdots + 31076 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 30280 \beta_{15} - 5779 \beta_{14} + 80244 \beta_{13} + 21998 \beta_{12} + 16518 \beta_{11} + \cdots + 94309 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 90924 \beta_{15} - 18712 \beta_{14} + 236118 \beta_{13} + 60667 \beta_{12} + 46690 \beta_{11} + \cdots + 218861 \) 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
1.76531
−1.22075
2.56721
−2.07454
1.72465
2.49691
−1.03114
2.41686
−1.80312
−0.672138
1.44349
−2.10546
2.75110
2.17402
−2.10279
0.670372
0 −2.88221 0 −2.07593 0 −1.23587 0 5.30713 0
1.2 0 −2.87556 0 −3.27010 0 −0.627241 0 5.26883 0
1.3 0 −2.37003 0 −1.19418 0 4.36698 0 2.61703 0
1.4 0 −2.32654 0 1.98734 0 3.05697 0 2.41281 0
1.5 0 −1.53488 0 4.08045 0 4.44601 0 −0.644129 0
1.6 0 −0.498388 0 0.0587589 0 0.265202 0 −2.75161 0
1.7 0 −0.439542 0 4.17391 0 −2.25691 0 −2.80680 0
1.8 0 0.523806 0 −1.14731 0 −5.17054 0 −2.72563 0
1.9 0 0.869745 0 −0.139670 0 −2.94775 0 −2.24354 0
1.10 0 1.46680 0 −3.64004 0 2.34760 0 −0.848487 0
1.11 0 1.48569 0 3.49515 0 −3.42362 0 −0.792727 0
1.12 0 1.54239 0 −0.121792 0 3.20952 0 −0.621033 0
1.13 0 1.76693 0 4.06573 0 0.976032 0 0.122026 0
1.14 0 2.90748 0 0.701983 0 −0.0299776 0 5.45345 0
1.15 0 3.16730 0 3.71978 0 −3.38749 0 7.03181 0
1.16 0 3.19701 0 −1.69409 0 −0.588925 0 7.22087 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(487\) \( +1 \)

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 7792.2.a.t 16
4.b odd 2 1 487.2.a.d ✓ 16
12.b even 2 1 4383.2.a.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
487.2.a.d ✓ 16 4.b odd 2 1
4383.2.a.f 16 12.b even 2 1
7792.2.a.t 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7792))\):

\( T_{3}^{16} - 4 T_{3}^{15} - 27 T_{3}^{14} + 120 T_{3}^{13} + 254 T_{3}^{12} - 1398 T_{3}^{11} + \cdots - 1224 \) Copy content Toggle raw display
\( T_{5}^{16} - 9 T_{5}^{15} - 17 T_{5}^{14} + 326 T_{5}^{13} - 155 T_{5}^{12} - 4490 T_{5}^{11} + \cdots + 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{15} + \cdots - 1224 \) Copy content Toggle raw display
$5$ \( T^{16} - 9 T^{15} + \cdots + 72 \) Copy content Toggle raw display
$7$ \( T^{16} + T^{15} + \cdots - 632 \) Copy content Toggle raw display
$11$ \( T^{16} + T^{15} + \cdots - 50616 \) Copy content Toggle raw display
$13$ \( T^{16} - 4 T^{15} + \cdots - 26544200 \) Copy content Toggle raw display
$17$ \( T^{16} - 38 T^{15} + \cdots - 3212856 \) Copy content Toggle raw display
$19$ \( T^{16} + 7 T^{15} + \cdots + 57166587 \) Copy content Toggle raw display
$23$ \( T^{16} + 8 T^{15} + \cdots - 33132600 \) Copy content Toggle raw display
$29$ \( T^{16} - 6 T^{15} + \cdots - 17301 \) Copy content Toggle raw display
$31$ \( T^{16} - 10 T^{15} + \cdots - 3147649 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots - 476550392568 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 10411698483 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 1074595624 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 1443891048 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 60642302967 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 216910227192 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 38879870388355 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 1094760181 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 32941340595 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 454215094299 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 10950003598648 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 247711920327 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 1291850272305 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 19558912908721 \) Copy content Toggle raw display
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