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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [83,4,Mod(1,83)] 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("83.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(83, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 83.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.89715853048\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 76 x^{11} + 374 x^{10} + 2129 x^{9} - 9943 x^{8} - 28670 x^{7} + 114750 x^{6} + \cdots - 6976 \) 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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{8} + 1) q^{3} + (\beta_{2} + 6) q^{4} + (\beta_{12} + \beta_{5} + \beta_1 + 2) q^{5} + ( - \beta_{12} + \beta_{8} + \cdots + \beta_1) q^{6} + (\beta_{7} - \beta_{6} + 4) q^{7}+ \cdots + ( - 18 \beta_{12} + 86 \beta_{11} + \cdots - 530) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 5 q^{2} + 7 q^{3} + 73 q^{4} + 33 q^{5} - 3 q^{6} + 52 q^{7} + 63 q^{8} + 178 q^{9} + 35 q^{10} + 7 q^{11} + 127 q^{12} + 237 q^{13} + 114 q^{14} + 104 q^{15} + 365 q^{16} - 12 q^{17} + 262 q^{18}+ \cdots - 8456 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 5 x^{12} - 76 x^{11} + 374 x^{10} + 2129 x^{9} - 9943 x^{8} - 28670 x^{7} + 114750 x^{6} + \cdots - 6976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 279707708 \nu^{12} + 32799653612 \nu^{11} - 247350621751 \nu^{10} - 1625978100588 \nu^{9} + \cdots - 715170660049408 ) / 50909819752144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3126316687 \nu^{12} - 921852208 \nu^{11} + 351747851852 \nu^{10} + \cdots + 149999827632064 ) / 50909819752144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23874100150 \nu^{12} - 194702439969 \nu^{11} - 1182075881643 \nu^{10} + \cdots + 12\!\cdots\!24 ) / 50909819752144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31958242378 \nu^{12} + 225365018255 \nu^{11} + 2015105742751 \nu^{10} + \cdots - 252710825961488 ) / 50909819752144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35391627155 \nu^{12} + 247284352608 \nu^{11} + 2149166771102 \nu^{10} + \cdots + 50379320943328 ) / 50909819752144 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38206728119 \nu^{12} + 271216957111 \nu^{11} + 2294159434744 \nu^{10} + \cdots - 331288340490208 ) / 50909819752144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 27081172078 \nu^{12} - 227446639806 \nu^{11} - 1344183692221 \nu^{10} + \cdots - 259956897375520 ) / 25454909876072 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 67895566266 \nu^{12} - 473198431316 \nu^{11} - 4144413598717 \nu^{10} + \cdots - 11\!\cdots\!96 ) / 50909819752144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 103680538656 \nu^{12} - 747204986625 \nu^{11} - 5997982781008 \nu^{10} + \cdots + 510318814633600 ) / 50909819752144 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 106872517630 \nu^{12} + 828186946772 \nu^{11} + 5958243168695 \nu^{10} + \cdots - 640456165136400 ) / 50909819752144 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} - \beta_{9} - \beta_{8} + 2\beta_{6} - \beta_{5} + \beta_{4} + 21\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{12} - 2 \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + \cdots + 304 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 28 \beta_{12} - 4 \beta_{11} - 4 \beta_{10} - 26 \beta_{9} - 52 \beta_{8} + \beta_{7} + 63 \beta_{6} + \cdots - 34 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{12} - 30 \beta_{11} - 42 \beta_{10} + 8 \beta_{9} - 98 \beta_{8} + 74 \beta_{7} - 60 \beta_{6} + \cdots + 7422 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 679 \beta_{12} - 160 \beta_{11} - 232 \beta_{10} - 583 \beta_{9} - 1927 \beta_{8} + 114 \beta_{7} + \cdots - 3146 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1155 \beta_{12} - 1760 \beta_{11} - 306 \beta_{10} + 1403 \beta_{9} - 4677 \beta_{8} + 3239 \beta_{7} + \cdots + 188696 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 16314 \beta_{12} - 4652 \beta_{11} - 9600 \beta_{10} - 12908 \beta_{9} - 63418 \beta_{8} + \cdots - 133094 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 60606 \beta_{12} - 71802 \beta_{11} + 17882 \beta_{10} + 68108 \beta_{9} - 174958 \beta_{8} + \cdots + 4890654 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 400045 \beta_{12} - 114668 \beta_{11} - 349556 \beta_{10} - 292185 \beta_{9} - 1968821 \beta_{8} + \cdots - 4638122 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2377915 \beta_{12} - 2530216 \beta_{11} + 1213538 \beta_{10} + 2572419 \beta_{9} - 5843261 \beta_{8} + \cdots + 128313476 \) 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
−5.33392
−5.06588
−2.61289
−2.59240
−1.74457
−1.65981
0.00424280
2.16563
2.90302
4.08231
4.61904
4.97334
5.26189
−5.33392 7.87257 20.4507 21.5535 −41.9917 9.97200 −66.4112 34.9774 −114.965
1.2 −5.06588 −0.527559 17.6632 −11.1157 2.67255 1.40455 −48.9526 −26.7217 56.3107
1.3 −2.61289 −8.27665 −1.17281 −9.66402 21.6260 −31.6072 23.9675 41.5030 25.2510
1.4 −2.59240 1.71328 −1.27948 17.6302 −4.44149 −24.8560 24.0561 −24.0647 −45.7044
1.5 −1.74457 −4.85948 −4.95647 −18.2878 8.47771 29.0420 22.6035 −3.38542 31.9043
1.6 −1.65981 7.37487 −5.24503 −0.102414 −12.2409 23.5648 21.9842 27.3887 0.169988
1.7 0.00424280 −7.83724 −7.99998 8.54661 −0.0332519 10.0055 −0.0678848 34.4223 0.0362616
1.8 2.16563 7.89819 −3.31003 11.8864 17.1046 −4.90538 −24.4934 35.3814 25.7415
1.9 2.90302 0.0552626 0.427512 11.7434 0.160428 34.5834 −21.9831 −26.9969 34.0912
1.10 4.08231 9.18464 8.66522 −18.8026 37.4945 8.58868 2.71565 57.3577 −76.7578
1.11 4.61904 3.20706 13.3356 8.23698 14.8135 −22.2043 24.6451 −16.7148 38.0469
1.12 4.97334 1.07884 16.7341 −3.52903 5.36546 7.39925 43.4375 −25.8361 −17.5510
1.13 5.26189 −9.88379 19.6875 14.9045 −52.0074 11.0128 61.4985 70.6892 78.4259
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(83\) \( +1 \)

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 83.4.a.b ✓ 13
3.b odd 2 1 747.4.a.f 13
4.b odd 2 1 1328.4.a.j 13
5.b even 2 1 2075.4.a.d 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.4.a.b ✓ 13 1.a even 1 1 trivial
747.4.a.f 13 3.b odd 2 1
1328.4.a.j 13 4.b odd 2 1
2075.4.a.d 13 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 5 T_{2}^{12} - 76 T_{2}^{11} + 374 T_{2}^{10} + 2129 T_{2}^{9} - 9943 T_{2}^{8} + \cdots - 6976 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(83))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 5 T^{12} + \cdots - 6976 \) Copy content Toggle raw display
$3$ \( T^{13} - 7 T^{12} + \cdots + 2267708 \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots - 742987149824 \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots - 198634180601741 \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 52\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 65\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 12\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots + 56\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 51\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 75\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 12\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 49\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 48\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T + 83)^{13} \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 32\!\cdots\!04 \) Copy content Toggle raw display
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