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Properties

Label 82.6.c.a
Level $82$
Weight $6$
Character orbit 82.c
Analytic conductor $13.151$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [82,6,Mod(9,82)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(82, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("82.9"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 82 = 2 \cdot 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 82.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: no
Analytic conductor: \(13.1514732247\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
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: \( x^{16} - 2 x^{15} + 2 x^{14} + 3744 x^{13} + 591440 x^{12} + 1165004 x^{11} + 3495880 x^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - 4 \beta_{2} q^{2} + \beta_{6} q^{3} - 16 q^{4} + (\beta_{3} - 3 \beta_{2}) q^{5} - 4 \beta_1 q^{6} + (\beta_{7} + \beta_{6} - 6 \beta_{2} - 6) q^{7} + 64 \beta_{2} q^{8} + ( - \beta_{11} - 3 \beta_{6} + \cdots - 3 \beta_1) q^{9}+ \cdots + (142 \beta_{13} - 142 \beta_{12} + \cdots + 15938) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} - 256 q^{4} - 8 q^{6} - 100 q^{7} - 160 q^{10} - 418 q^{11} + 32 q^{12} - 194 q^{13} - 400 q^{14} + 1576 q^{15} + 4096 q^{16} + 2508 q^{17} + 3888 q^{18} + 1458 q^{19} - 1672 q^{22} + 11312 q^{23}+ \cdots + 266182 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 3744 x^{13} + 591440 x^{12} + 1165004 x^{11} + 3495880 x^{10} + \cdots + 22\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 75\!\cdots\!16 \nu^{15} + \cdots + 44\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 57\!\cdots\!03 \nu^{15} + \cdots - 28\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 36\!\cdots\!21 \nu^{15} + \cdots - 47\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 35\!\cdots\!06 \nu^{15} + \cdots - 14\!\cdots\!00 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32\!\cdots\!71 \nu^{15} + \cdots - 32\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\!\cdots\!81 \nu^{15} + \cdots - 32\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!63 \nu^{15} + \cdots - 43\!\cdots\!00 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20\!\cdots\!93 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25\!\cdots\!22 \nu^{15} + \cdots + 96\!\cdots\!00 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60\!\cdots\!97 \nu^{15} + \cdots + 45\!\cdots\!00 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!32 \nu^{15} + \cdots - 31\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 87\!\cdots\!29 \nu^{15} + \cdots - 75\!\cdots\!00 ) / 27\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19\!\cdots\!67 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 79\!\cdots\!67 \nu^{15} + \cdots - 26\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 3\beta_{6} + \beta_{3} - 304\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} - 5 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + 3 \beta_{11} - 3 \beta_{9} + 4 \beta_{8} + \cdots - 643 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{15} - 14 \beta_{14} + 13 \beta_{12} - 4 \beta_{10} - 589 \beta_{9} - 338 \beta_{8} + \cdots - 149135 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 1752 \beta_{13} + 1752 \beta_{12} - 2709 \beta_{11} + 1383 \beta_{10} - 2709 \beta_{9} + \cdots - 703299 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3258 \beta_{15} + 16713 \beta_{14} - 20958 \beta_{13} - 323989 \beta_{11} + 3258 \beta_{10} + \cdots - 2116683 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 974258 \beta_{15} + 1904618 \beta_{14} - 1226384 \beta_{13} - 1226384 \beta_{12} - 2025780 \beta_{11} + \cdots + 561152590 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6289220 \beta_{15} + 14850344 \beta_{14} - 21836146 \beta_{12} - 6289220 \beta_{10} + 178116310 \beta_{9} + \cdots + 44771506394 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 824917146 \beta_{13} - 824917146 \beta_{12} + 1404298848 \beta_{11} - 676677570 \beta_{10} + \cdots + 400825592328 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6468655266 \beta_{15} - 11558203848 \beta_{14} + 18808216194 \beta_{13} + 98943272860 \beta_{11} + \cdots + 992794025940 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 464200726190 \beta_{15} - 680882622920 \beta_{14} + 550795127810 \beta_{13} + 550795127810 \beta_{12} + \cdots - 271748926776610 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5500824670250 \beta_{15} - 8372048408810 \beta_{14} + 14672804543140 \beta_{12} + \cdots - 14\!\cdots\!74 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 367110406541616 \beta_{13} + 367110406541616 \beta_{12} - 600976224553434 \beta_{11} + \cdots - 17\!\cdots\!54 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 42\!\cdots\!88 \beta_{15} + \cdots - 41\!\cdots\!68 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 21\!\cdots\!04 \beta_{15} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/82\mathbb{Z}\right)^\times\).

\(n\) \(47\)
\(\chi(n)\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
17.8600 − 17.8600i
16.8295 − 16.8295i
6.56256 − 6.56256i
2.52920 − 2.52920i
−2.67676 + 2.67676i
−9.71571 + 9.71571i
−15.0753 + 15.0753i
−15.3135 + 15.3135i
17.8600 + 17.8600i
16.8295 + 16.8295i
6.56256 + 6.56256i
2.52920 + 2.52920i
−2.67676 − 2.67676i
−9.71571 − 9.71571i
−15.0753 − 15.0753i
−15.3135 − 15.3135i
− 4.00000i −17.8600 − 17.8600i −16.0000 − 7.25919i −71.4400 + 71.4400i −148.544 − 148.544i 64.0000i 394.959i −29.0368
9.2 − 4.00000i −16.8295 − 16.8295i −16.0000 26.2061i −67.3178 + 67.3178i 133.421 + 133.421i 64.0000i 323.461i 104.824
9.3 − 4.00000i −6.56256 − 6.56256i −16.0000 − 80.1361i −26.2502 + 26.2502i −46.4912 − 46.4912i 64.0000i − 156.866i −320.544
9.4 − 4.00000i −2.52920 − 2.52920i −16.0000 50.8239i −10.1168 + 10.1168i −34.3737 − 34.3737i 64.0000i − 230.206i 203.295
9.5 − 4.00000i 2.67676 + 2.67676i −16.0000 85.7530i 10.7070 − 10.7070i −17.2810 − 17.2810i 64.0000i − 228.670i 343.012
9.6 − 4.00000i 9.71571 + 9.71571i −16.0000 − 65.0493i 38.8628 − 38.8628i 173.630 + 173.630i 64.0000i − 54.2101i −260.197
9.7 − 4.00000i 15.0753 + 15.0753i −16.0000 29.0716i 60.3011 − 60.3011i 1.29408 + 1.29408i 64.0000i 211.528i 116.286
9.8 − 4.00000i 15.3135 + 15.3135i −16.0000 − 59.4099i 61.2538 − 61.2538i −111.655 − 111.655i 64.0000i 226.004i −237.640
73.1 4.00000i −17.8600 + 17.8600i −16.0000 7.25919i −71.4400 − 71.4400i −148.544 + 148.544i − 64.0000i − 394.959i −29.0368
73.2 4.00000i −16.8295 + 16.8295i −16.0000 − 26.2061i −67.3178 − 67.3178i 133.421 − 133.421i − 64.0000i − 323.461i 104.824
73.3 4.00000i −6.56256 + 6.56256i −16.0000 80.1361i −26.2502 − 26.2502i −46.4912 + 46.4912i − 64.0000i 156.866i −320.544
73.4 4.00000i −2.52920 + 2.52920i −16.0000 − 50.8239i −10.1168 − 10.1168i −34.3737 + 34.3737i − 64.0000i 230.206i 203.295
73.5 4.00000i 2.67676 − 2.67676i −16.0000 − 85.7530i 10.7070 + 10.7070i −17.2810 + 17.2810i − 64.0000i 228.670i 343.012
73.6 4.00000i 9.71571 − 9.71571i −16.0000 65.0493i 38.8628 + 38.8628i 173.630 − 173.630i − 64.0000i 54.2101i −260.197
73.7 4.00000i 15.0753 − 15.0753i −16.0000 − 29.0716i 60.3011 + 60.3011i 1.29408 − 1.29408i − 64.0000i − 211.528i 116.286
73.8 4.00000i 15.3135 − 15.3135i −16.0000 59.4099i 61.2538 + 61.2538i −111.655 + 111.655i − 64.0000i − 226.004i −237.640
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 82.6.c.a ✓ 16
41.c even 4 1 inner 82.6.c.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
82.6.c.a ✓ 16 1.a even 1 1 trivial
82.6.c.a ✓ 16 41.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 2 T_{3}^{15} + 2 T_{3}^{14} - 3744 T_{3}^{13} + 591440 T_{3}^{12} - 1165004 T_{3}^{11} + \cdots + 22\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(82, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots - 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 80\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 35\!\cdots\!20)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 32\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 10\!\cdots\!60)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 56\!\cdots\!48)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
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