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Properties

Label 21.8.e.a
Level $21$
Weight $8$
Character orbit 21.e
Analytic conductor $6.560$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [21,8,Mod(4,21)] 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("21.4"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(6.56008553517\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 362x^{6} - 1072x^{5} + 36131x^{4} - 70480x^{3} + 698554x^{2} - 663492x + 2868273 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + 27 \beta_{4} q^{3} + ( - \beta_{7} + 2 \beta_{6} + \cdots - 2 \beta_1) q^{4} + ( - 2 \beta_{7} - 9 \beta_{6} + \cdots - 46) q^{5} - 27 \beta_1 q^{6} + (14 \beta_{6} + 273 \beta_{4} + \cdots + 161) q^{7}+ \cdots + ( - 29889 \beta_{3} + 7290 \beta_{2} + \cdots - 928746) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 108 q^{3} - 101 q^{4} - 196 q^{5} - 54 q^{6} + 154 q^{7} - 2694 q^{8} - 2916 q^{9} + 5185 q^{10} + 5210 q^{11} - 2727 q^{12} - 7588 q^{13} - 29932 q^{14} + 10584 q^{15} + 21055 q^{16} + 11436 q^{17}+ \cdots - 7596180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 362x^{6} - 1072x^{5} + 36131x^{4} - 70480x^{3} + 698554x^{2} - 663492x + 2868273 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 3\nu^{5} + 334\nu^{4} - 663\nu^{3} + 29038\nu^{2} - 28707\nu + 287577 ) / 6912 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 3\nu^{5} + 238\nu^{4} - 471\nu^{3} + 1486\nu^{2} - 1251\nu - 797319 ) / 6912 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} - 670\nu^{4} + 1335\nu^{3} - 89182\nu^{2} + 88515\nu - 851625 ) / 6912 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} - 2882\nu^{5} + 7135\nu^{4} - 272324\nu^{3} + 401365\nu^{2} - 3032736\nu - 444825 ) / 3789072 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 477 \nu^{7} - 6055 \nu^{6} + 191574 \nu^{5} - 3380155 \nu^{4} + 25170582 \nu^{3} + \cdots - 4290009840 ) / 60625152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 541 \nu^{7} + 6279 \nu^{6} - 214630 \nu^{5} + 1963707 \nu^{4} - 24402118 \nu^{3} + \cdots + 1509692976 ) / 60625152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2357 \nu^{7} - 8876 \nu^{6} + 804941 \nu^{5} - 2136137 \nu^{4} + 71969297 \nu^{3} + \cdots + 119000043 ) / 8660736 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} - \beta _1 + 10 ) / 21 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + 2\beta_{5} - \beta_{4} + 3\beta_{3} - 14\beta_{2} + 17\beta _1 - 1861 ) / 21 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 14\beta_{7} - 994\beta_{6} - 308\beta_{5} + 4858\beta_{4} + 160\beta_{3} - 14\beta_{2} + 524\beta _1 - 367 ) / 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 28 \beta_{7} - 1990 \beta_{6} - 618 \beta_{5} + 9717 \beta_{4} - 827 \beta_{3} + 2478 \beta_{2} + \cdots + 298912 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6482 \beta_{7} + 176108 \beta_{6} + 52026 \beta_{5} - 1385973 \beta_{4} - 28863 \beta_{3} + \cdots + 46809 ) / 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 19516 \beta_{7} + 533300 \beta_{6} + 157624 \beta_{5} - 4182212 \beta_{4} + 179888 \beta_{3} + \cdots - 51651831 ) / 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1815240 \beta_{7} - 30172774 \beta_{6} - 8915230 \beta_{5} + 318339239 \beta_{4} + 5373047 \beta_{3} + \cdots - 16885466 ) / 21 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.500000 + 4.06331i
0.500000 − 13.6728i
0.500000 + 12.0633i
0.500000 − 2.45385i
0.500000 − 4.06331i
0.500000 + 13.6728i
0.500000 − 12.0633i
0.500000 + 2.45385i
−7.61660 − 13.1923i −13.5000 + 23.3827i −52.0252 + 90.1103i 122.567 + 212.292i 411.296 906.017 + 51.7322i −364.830 −364.500 − 631.333i 1867.09 − 3233.89i
4.2 −3.40109 − 5.89086i −13.5000 + 23.3827i 40.8652 − 70.7806i −6.28050 − 10.8781i 183.659 −894.339 + 153.953i −1426.62 −364.500 − 631.333i −42.7211 + 73.9951i
4.3 2.81424 + 4.87440i −13.5000 + 23.3827i 48.1601 − 83.4158i −251.471 − 435.561i −151.969 854.822 + 304.667i 1262.58 −364.500 − 631.333i 1415.40 − 2451.54i
4.4 8.70345 + 15.0748i −13.5000 + 23.3827i −87.5001 + 151.555i 37.1845 + 64.4054i −469.986 −789.501 + 447.472i −818.128 −364.500 − 631.333i −647.267 + 1121.10i
16.1 −7.61660 + 13.1923i −13.5000 − 23.3827i −52.0252 − 90.1103i 122.567 − 212.292i 411.296 906.017 − 51.7322i −364.830 −364.500 + 631.333i 1867.09 + 3233.89i
16.2 −3.40109 + 5.89086i −13.5000 − 23.3827i 40.8652 + 70.7806i −6.28050 + 10.8781i 183.659 −894.339 − 153.953i −1426.62 −364.500 + 631.333i −42.7211 − 73.9951i
16.3 2.81424 − 4.87440i −13.5000 − 23.3827i 48.1601 + 83.4158i −251.471 + 435.561i −151.969 854.822 − 304.667i 1262.58 −364.500 + 631.333i 1415.40 + 2451.54i
16.4 8.70345 − 15.0748i −13.5000 − 23.3827i −87.5001 − 151.555i 37.1845 − 64.4054i −469.986 −789.501 − 447.472i −818.128 −364.500 + 631.333i −647.267 − 1121.10i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.8.e.a ✓ 8
3.b odd 2 1 63.8.e.c 8
7.b odd 2 1 147.8.e.k 8
7.c even 3 1 inner 21.8.e.a ✓ 8
7.c even 3 1 147.8.a.i 4
7.d odd 6 1 147.8.a.h 4
7.d odd 6 1 147.8.e.k 8
21.g even 6 1 441.8.a.r 4
21.h odd 6 1 63.8.e.c 8
21.h odd 6 1 441.8.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.8.e.a ✓ 8 1.a even 1 1 trivial
21.8.e.a ✓ 8 7.c even 3 1 inner
63.8.e.c 8 3.b odd 2 1
63.8.e.c 8 21.h odd 6 1
147.8.a.h 4 7.d odd 6 1
147.8.a.i 4 7.c even 3 1
147.8.e.k 8 7.b odd 2 1
147.8.e.k 8 7.d odd 6 1
441.8.a.q 4 21.h odd 6 1
441.8.a.r 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} + 307T_{2}^{6} + 762T_{2}^{5} + 83256T_{2}^{4} + 90072T_{2}^{3} + 3158496T_{2}^{2} - 2314656T_{2} + 103063104 \) acting on \(S_{8}^{\mathrm{new}}(21, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + \cdots + 103063104 \) Copy content Toggle raw display
$3$ \( (T^{2} + 27 T + 729)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 45\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 32\!\cdots\!12)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 66\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 72\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 65\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 40\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 81\!\cdots\!40)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 27\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 64\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 21\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 35\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 20\!\cdots\!88)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 98\!\cdots\!56)^{2} \) Copy content Toggle raw display
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