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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [83,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 83.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14] 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: \(13.3118570445\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 284 x^{12} + 192 x^{11} + 30530 x^{10} - 13538 x^{9} - 1578896 x^{8} + \cdots - 5316092640 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{7} - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 10) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_1 - 12) q^{5} + (\beta_{11} + \beta_{9} - 2 \beta_{7} + \cdots - 2) q^{6} + ( - \beta_{12} + 2 \beta_{11} + \cdots - 18) q^{7}+ \cdots + (287 \beta_{13} - 369 \beta_{12} + \cdots - 49927) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 13 q^{2} - 17 q^{3} + 133 q^{4} - 162 q^{5} - 15 q^{6} - 247 q^{7} - 609 q^{8} + 679 q^{9} - 519 q^{10} - 413 q^{11} - 1193 q^{12} - 2942 q^{13} - 1176 q^{14} - 1990 q^{15} + 437 q^{16} + 69 q^{17}+ \cdots - 708674 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 284 x^{12} + 192 x^{11} + 30530 x^{10} - 13538 x^{9} - 1578896 x^{8} + \cdots - 5316092640 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 41 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!49 \nu^{13} + \cdots + 69\!\cdots\!32 ) / 35\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53\!\cdots\!11 \nu^{13} + \cdots - 41\!\cdots\!24 ) / 11\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 75\!\cdots\!57 \nu^{13} + \cdots + 26\!\cdots\!16 ) / 13\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22\!\cdots\!77 \nu^{13} + \cdots - 52\!\cdots\!16 ) / 35\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41\!\cdots\!65 \nu^{13} + \cdots + 72\!\cdots\!16 ) / 53\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!07 \nu^{13} + \cdots - 92\!\cdots\!88 ) / 17\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16\!\cdots\!89 \nu^{13} + \cdots - 11\!\cdots\!80 ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27\!\cdots\!55 \nu^{13} + \cdots + 23\!\cdots\!76 ) / 13\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\!\cdots\!93 \nu^{13} + \cdots + 48\!\cdots\!64 ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!33 \nu^{13} + \cdots + 11\!\cdots\!88 ) / 35\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!93 \nu^{13} + \cdots + 27\!\cdots\!40 ) / 35\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 41 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{12} - \beta_{10} + \beta_{8} - 5\beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 69\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 11 \beta_{13} - 2 \beta_{12} + 12 \beta_{10} - 17 \beta_{9} - 9 \beta_{8} - 16 \beta_{7} + \cdots + 2841 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16 \beta_{13} + 286 \beta_{12} - 32 \beta_{11} - 159 \beta_{10} - 16 \beta_{9} + 107 \beta_{8} + \cdots + 2873 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1655 \beta_{13} - 94 \beta_{12} - 136 \beta_{11} + 2038 \beta_{10} - 2645 \beta_{9} - 1315 \beta_{8} + \cdots + 238707 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3088 \beta_{13} + 34098 \beta_{12} - 7480 \beta_{11} - 18369 \beta_{10} - 3768 \beta_{9} + 9529 \beta_{8} + \cdots + 425511 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 191215 \beta_{13} + 12350 \beta_{12} - 33376 \beta_{11} + 257184 \beta_{10} - 324141 \beta_{9} + \cdots + 22154997 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 401680 \beta_{13} + 3836390 \beta_{12} - 1142096 \beta_{11} - 1901139 \beta_{10} - 630624 \beta_{9} + \cdots + 57462109 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 20409839 \beta_{13} + 4054154 \beta_{12} - 5597560 \beta_{11} + 29203418 \beta_{10} + \cdots + 2184706719 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 43752280 \beta_{13} + 421424234 \beta_{12} - 148838888 \beta_{11} - 186396405 \beta_{10} + \cdots + 7386872891 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2120160071 \beta_{13} + 738362854 \beta_{12} - 800865552 \beta_{11} + 3160753284 \beta_{10} + \cdots + 223843393137 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4245090248 \beta_{13} + 45818187262 \beta_{12} - 18002957568 \beta_{11} - 17663862695 \beta_{10} + \cdots + 920913002337 \) 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
−9.89861
−7.82567
−7.59628
−5.40721
−4.24412
−2.84427
−1.73240
2.06614
3.00995
3.37236
6.26780
6.81995
8.38852
10.6238
−10.8986 12.4064 86.7797 −11.4967 −135.212 −56.6099 −597.022 −89.0824 125.299
1.2 −8.82567 15.4543 45.8924 22.0967 −136.395 −147.296 −122.610 −4.16317 −195.018
1.3 −8.59628 −21.9338 41.8960 −35.2526 188.549 −100.663 −85.0684 238.092 303.041
1.4 −6.40721 −6.52552 9.05235 −41.9805 41.8104 192.545 147.030 −200.418 268.978
1.5 −5.24412 −31.0765 −4.49922 −20.2301 162.969 133.375 191.406 722.746 106.089
1.6 −3.84427 13.5859 −17.2216 5.27843 −52.2280 10.6328 189.221 −58.4220 −20.2917
1.7 −2.73240 −18.5197 −24.5340 65.7005 50.6034 44.4378 154.474 99.9805 −179.520
1.8 1.06614 15.4216 −30.8634 −60.4801 16.4415 200.715 −67.0210 −5.17440 −64.4801
1.9 2.00995 7.39835 −27.9601 68.0447 14.8703 −236.195 −120.517 −188.264 136.767
1.10 2.37236 26.6591 −26.3719 −57.2562 63.2451 −196.988 −138.479 467.707 −135.833
1.11 5.26780 −16.6408 −4.25031 48.0519 −87.6606 144.632 −190.959 33.9177 253.127
1.12 5.81995 0.955026 1.87186 −10.2725 5.55821 −78.5130 −175.344 −242.088 −59.7852
1.13 7.38852 4.94050 22.5902 −104.763 36.5029 −32.1631 −69.5244 −218.592 −774.046
1.14 9.62382 −19.1248 60.6180 −29.4401 −184.054 −124.910 275.414 122.760 −283.326
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.14
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 83.6.a.a ✓ 14
3.b odd 2 1 747.6.a.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.6.a.a ✓ 14 1.a even 1 1 trivial
747.6.a.a 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 13 T_{2}^{13} - 206 T_{2}^{12} - 2930 T_{2}^{11} + 14613 T_{2}^{10} + 241129 T_{2}^{9} + \cdots - 3234172928 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(83))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots - 3234172928 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 980051407160400 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 94\!\cdots\!60 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 43\!\cdots\!35 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 92\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 29\!\cdots\!35 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 42\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 18\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 15\!\cdots\!45 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 26\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 79\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 58\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 86\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 96\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 42\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 18\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 93\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T + 6889)^{14} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 40\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
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