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Properties

Label 8092.2.a.y
Level $8092$
Weight $2$
Character orbit 8092.a
Self dual yes
Analytic conductor $64.615$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8092,2,Mod(1,8092)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8092, 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("8092.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 8092 = 2^{2} \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8092.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,-8,0,-8,0,-20,0,28,0,-16] 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: \(64.6149453156\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 12 x^{18} + 240 x^{17} - 224 x^{16} - 2776 x^{15} + 5324 x^{14} + 15280 x^{13} + \cdots + 544 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 476)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{12} q^{5} - q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{14} - 1) q^{11} + (\beta_{15} + \beta_{5}) q^{13} + ( - \beta_{18} - \beta_{15} + \beta_{11} + \cdots + 1) q^{15}+ \cdots + (2 \beta_{19} + \beta_{17} - 2 \beta_{15} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 8 q^{3} - 8 q^{5} - 20 q^{7} + 28 q^{9} - 16 q^{11} + 8 q^{13} + 8 q^{15} + 8 q^{19} + 8 q^{21} - 40 q^{23} + 36 q^{25} - 32 q^{27} - 16 q^{29} - 16 q^{31} + 8 q^{33} + 8 q^{35} - 24 q^{37} - 16 q^{39}+ \cdots - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} - 12 x^{18} + 240 x^{17} - 224 x^{16} - 2776 x^{15} + 5324 x^{14} + 15280 x^{13} + \cdots + 544 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!90 \nu^{19} + \cdots - 16\!\cdots\!08 ) / 31\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 677873709331 \nu^{19} + 5102087233004 \nu^{18} + 10836688208710 \nu^{17} + \cdots + 11\!\cdots\!36 ) / 79189455252092 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 61\!\cdots\!89 \nu^{19} + \cdots - 91\!\cdots\!20 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18\!\cdots\!91 \nu^{19} + \cdots + 25\!\cdots\!04 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 31\!\cdots\!87 \nu^{19} + \cdots + 43\!\cdots\!32 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2106103986294 \nu^{19} - 16170958181021 \nu^{18} - 30375335068532 \nu^{17} + \cdots - 30\!\cdots\!48 ) / 79189455252092 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 38\!\cdots\!33 \nu^{19} + \cdots + 43\!\cdots\!22 ) / 31\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15\!\cdots\!77 \nu^{19} + \cdots + 18\!\cdots\!80 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 86\!\cdots\!96 \nu^{19} + \cdots - 10\!\cdots\!24 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!39 \nu^{19} + \cdots + 12\!\cdots\!08 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!81 \nu^{19} + \cdots - 12\!\cdots\!36 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 58\!\cdots\!70 \nu^{19} + \cdots + 66\!\cdots\!34 ) / 31\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 13\!\cdots\!64 \nu^{19} + \cdots + 15\!\cdots\!40 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 30\!\cdots\!15 \nu^{19} + \cdots + 30\!\cdots\!80 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 17\!\cdots\!25 \nu^{19} + \cdots - 21\!\cdots\!12 ) / 62\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 57\!\cdots\!01 \nu^{19} + \cdots - 64\!\cdots\!04 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 39\!\cdots\!49 \nu^{19} + \cdots + 43\!\cdots\!56 ) / 62\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{17} - \beta_{16} - \beta_{15} + \beta_{14} - \beta_{10} - 2 \beta_{8} - \beta_{5} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{18} + \beta_{17} - \beta_{16} - 2 \beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13 \beta_{19} + 13 \beta_{17} - 12 \beta_{16} - 15 \beta_{15} + 11 \beta_{14} + \beta_{13} + 3 \beta_{12} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{19} - 16 \beta_{18} + 17 \beta_{17} - 16 \beta_{16} - 34 \beta_{15} + 12 \beta_{14} + \cdots + 249 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 149 \beta_{19} - 8 \beta_{18} + 150 \beta_{17} - 130 \beta_{16} - 190 \beta_{15} + 104 \beta_{14} + \cdots + 374 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 98 \beta_{19} - 197 \beta_{18} + 237 \beta_{17} - 212 \beta_{16} - 454 \beta_{15} + 107 \beta_{14} + \cdots + 2315 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1674 \beta_{19} - 184 \beta_{18} + 1703 \beta_{17} - 1398 \beta_{16} - 2278 \beta_{15} + 950 \beta_{14} + \cdots + 4192 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1689 \beta_{19} - 2219 \beta_{18} + 3125 \beta_{17} - 2635 \beta_{16} - 5641 \beta_{15} + 848 \beta_{14} + \cdots + 22547 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 18860 \beta_{19} - 2823 \beta_{18} + 19435 \beta_{17} - 15160 \beta_{16} - 26649 \beta_{15} + \cdots + 45833 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 25261 \beta_{19} - 24006 \beta_{18} + 40204 \beta_{17} - 31827 \beta_{16} - 68149 \beta_{15} + \cdots + 226419 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 214056 \beta_{19} - 36593 \beta_{18} + 223648 \beta_{17} - 166153 \beta_{16} - 307880 \beta_{15} + \cdots + 497975 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 350495 \beta_{19} - 254230 \beta_{18} + 509034 \beta_{17} - 378998 \beta_{16} - 812391 \beta_{15} + \cdots + 2324091 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2446330 \beta_{19} - 434783 \beta_{18} + 2591349 \beta_{17} - 1839000 \beta_{16} - 3534588 \beta_{15} + \cdots + 5419104 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 4645946 \beta_{19} - 2659694 \beta_{18} + 6363879 \beta_{17} - 4479716 \beta_{16} - 9612990 \beta_{15} + \cdots + 24260024 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 28107944 \beta_{19} - 4917302 \beta_{18} + 30168087 \beta_{17} - 20525988 \beta_{16} - 40467114 \beta_{15} + \cdots + 59262972 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 59744183 \beta_{19} - 27627981 \beta_{18} + 78715291 \beta_{17} - 52737141 \beta_{16} - 113235567 \beta_{15} + \cdots + 256705001 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 324210315 \beta_{19} - 54024077 \beta_{18} + 352263720 \beta_{17} - 230719715 \beta_{16} + \cdots + 652092797 \) 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
3.41333
3.17150
3.07559
2.77836
2.76359
1.95750
1.67151
1.28083
0.715651
0.514204
0.484555
0.411306
−0.0812843
−0.948608
−1.20917
−2.05667
−2.09856
−2.12475
−2.57822
−3.14067
0 −3.41333 0 −2.69478 0 −1.00000 0 8.65080 0
1.2 0 −3.17150 0 −0.276099 0 −1.00000 0 7.05840 0
1.3 0 −3.07559 0 −3.74303 0 −1.00000 0 6.45926 0
1.4 0 −2.77836 0 3.43607 0 −1.00000 0 4.71928 0
1.5 0 −2.76359 0 1.73772 0 −1.00000 0 4.63745 0
1.6 0 −1.95750 0 −2.23360 0 −1.00000 0 0.831805 0
1.7 0 −1.67151 0 0.623960 0 −1.00000 0 −0.206045 0
1.8 0 −1.28083 0 3.03609 0 −1.00000 0 −1.35947 0
1.9 0 −0.715651 0 −3.10967 0 −1.00000 0 −2.48784 0
1.10 0 −0.514204 0 −4.35990 0 −1.00000 0 −2.73559 0
1.11 0 −0.484555 0 0.320543 0 −1.00000 0 −2.76521 0
1.12 0 −0.411306 0 0.830159 0 −1.00000 0 −2.83083 0
1.13 0 0.0812843 0 −3.83104 0 −1.00000 0 −2.99339 0
1.14 0 0.948608 0 3.07520 0 −1.00000 0 −2.10014 0
1.15 0 1.20917 0 3.53097 0 −1.00000 0 −1.53791 0
1.16 0 2.05667 0 −0.461780 0 −1.00000 0 1.22989 0
1.17 0 2.09856 0 −3.70341 0 −1.00000 0 1.40397 0
1.18 0 2.12475 0 −0.748276 0 −1.00000 0 1.51455 0
1.19 0 2.57822 0 1.78639 0 −1.00000 0 3.64723 0
1.20 0 3.14067 0 −1.21552 0 −1.00000 0 6.86379 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(17\) \( -1 \)

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 8092.2.a.y 20
17.b even 2 1 8092.2.a.z 20
17.e odd 16 2 476.2.u.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.u.a ✓ 40 17.e odd 16 2
8092.2.a.y 20 1.a even 1 1 trivial
8092.2.a.z 20 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8092))\):

\( T_{3}^{20} + 8 T_{3}^{19} - 12 T_{3}^{18} - 240 T_{3}^{17} - 224 T_{3}^{16} + 2776 T_{3}^{15} + \cdots + 544 \) Copy content Toggle raw display
\( T_{5}^{20} + 8 T_{5}^{19} - 36 T_{5}^{18} - 408 T_{5}^{17} + 306 T_{5}^{16} + 8440 T_{5}^{15} + \cdots - 29342 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 8 T^{19} + \cdots + 544 \) Copy content Toggle raw display
$5$ \( T^{20} + 8 T^{19} + \cdots - 29342 \) Copy content Toggle raw display
$7$ \( (T + 1)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 326861312 \) Copy content Toggle raw display
$13$ \( T^{20} - 8 T^{19} + \cdots - 17546176 \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 16120103936 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 587472896 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 152441824 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 267964288 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 261656653517856 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 332362533982 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 113632022759488 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 8042970112 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 159595962191356 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 2220114853888 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 61926659435074 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 16\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 26\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 15\!\cdots\!38 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 294304909312 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 332909802164224 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 12\!\cdots\!82 \) Copy content Toggle raw display
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