Properties

Label 21.32.g.a
Level $21$
Weight $32$
Character orbit 21.g
Analytic conductor $127.842$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(127.841978920\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 14348907 + 14348907 \zeta_{6} ) q^{3} + ( -2147483648 + 2147483648 \zeta_{6} ) q^{4} + ( -1942145413486 + 13418820913929 \zeta_{6} ) q^{7} + 617673396283947 \zeta_{6} q^{9} +O(q^{10})\) \( q +(14348907 + 14348907 \zeta_{6}) q^{3} +(-2147483648 + 2147483648 \zeta_{6}) q^{4} +(-1942145413486 + 13418820913929 \zeta_{6}) q^{7} +617673396283947 \zeta_{6} q^{9} +(-61628086298345472 + 30814043149172736 \zeta_{6}) q^{12} +(-201292386637052593 + 402584773274105186 \zeta_{6}) q^{13} -4611686018427387904 \zeta_{6} q^{16} +(-\)\(12\!\cdots\!94\)\( + 61965614963055093447 \zeta_{6}) q^{19} +(-\)\(22\!\cdots\!05\)\( + \)\(35\!\cdots\!04\)\( \zeta_{6}) q^{21} +(\)\(46\!\cdots\!25\)\( - \)\(46\!\cdots\!25\)\( \zeta_{6}) q^{25} +(-\)\(88\!\cdots\!29\)\( + \)\(17\!\cdots\!58\)\( \zeta_{6}) q^{27} +(-\)\(24\!\cdots\!64\)\( - \)\(41\!\cdots\!28\)\( \zeta_{6}) q^{28} +(\)\(91\!\cdots\!05\)\( + \)\(91\!\cdots\!05\)\( \zeta_{6}) q^{31} -\)\(13\!\cdots\!56\)\( q^{36} +\)\(22\!\cdots\!43\)\( \zeta_{6} q^{37} +(-\)\(86\!\cdots\!53\)\( + \)\(86\!\cdots\!53\)\( \zeta_{6}) q^{39} -\)\(17\!\cdots\!65\)\( q^{43} +(\)\(66\!\cdots\!28\)\( - \)\(13\!\cdots\!56\)\( \zeta_{6}) q^{48} +(-\)\(17\!\cdots\!45\)\( + \)\(12\!\cdots\!53\)\( \zeta_{6}) q^{49} +(-\)\(43\!\cdots\!64\)\( - \)\(43\!\cdots\!64\)\( \zeta_{6}) q^{52} -\)\(26\!\cdots\!87\)\( q^{57} +(\)\(10\!\cdots\!08\)\( - \)\(54\!\cdots\!04\)\( \zeta_{6}) q^{61} +(-\)\(82\!\cdots\!63\)\( + \)\(70\!\cdots\!21\)\( \zeta_{6}) q^{63} +\)\(99\!\cdots\!92\)\( q^{64} +(-\)\(30\!\cdots\!27\)\( + \)\(30\!\cdots\!27\)\( \zeta_{6}) q^{67} +(\)\(76\!\cdots\!63\)\( + \)\(76\!\cdots\!63\)\( \zeta_{6}) q^{73} +(\)\(13\!\cdots\!50\)\( - \)\(66\!\cdots\!75\)\( \zeta_{6}) q^{75} +(\)\(13\!\cdots\!56\)\( - \)\(26\!\cdots\!12\)\( \zeta_{6}) q^{76} -\)\(15\!\cdots\!63\)\( \zeta_{6} q^{79} +(-\)\(38\!\cdots\!09\)\( + \)\(38\!\cdots\!09\)\( \zeta_{6}) q^{81} +(-\)\(29\!\cdots\!52\)\( - \)\(47\!\cdots\!40\)\( \zeta_{6}) q^{84} +(-\)\(50\!\cdots\!96\)\( + \)\(19\!\cdots\!01\)\( \zeta_{6}) q^{91} +\)\(39\!\cdots\!05\)\( \zeta_{6} q^{93} +(\)\(58\!\cdots\!44\)\( - \)\(11\!\cdots\!88\)\( \zeta_{6}) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 43046721q^{3} - 2147483648q^{4} + 9534530086957q^{7} + 617673396283947q^{9} + O(q^{10}) \) \( 2q + 43046721q^{3} - 2147483648q^{4} + 9534530086957q^{7} + 617673396283947q^{9} - 92442129447518208q^{12} - 4611686018427387904q^{16} - \)\(18\!\cdots\!41\)\(q^{19} - 83602991755761479406q^{21} + \)\(46\!\cdots\!25\)\(q^{25} - \)\(53\!\cdots\!56\)\(q^{28} + \)\(27\!\cdots\!15\)\(q^{31} - \)\(26\!\cdots\!12\)\(q^{36} + \)\(22\!\cdots\!43\)\(q^{37} - \)\(86\!\cdots\!53\)\(q^{39} - \)\(35\!\cdots\!30\)\(q^{43} - \)\(22\!\cdots\!37\)\(q^{49} - \)\(12\!\cdots\!92\)\(q^{52} - \)\(53\!\cdots\!74\)\(q^{57} + \)\(16\!\cdots\!12\)\(q^{61} - \)\(94\!\cdots\!05\)\(q^{63} + \)\(19\!\cdots\!84\)\(q^{64} - \)\(30\!\cdots\!27\)\(q^{67} + \)\(22\!\cdots\!89\)\(q^{73} + \)\(20\!\cdots\!25\)\(q^{75} - \)\(15\!\cdots\!63\)\(q^{79} - \)\(38\!\cdots\!09\)\(q^{81} - \)\(10\!\cdots\!44\)\(q^{84} - \)\(81\!\cdots\!91\)\(q^{91} + \)\(39\!\cdots\!05\)\(q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 0.866025i
0.500000 + 0.866025i
0 2.15234e7 1.24265e7i −1.07374e9 1.85978e9i 0 0 4.76727e12 1.16210e13i 0 3.08837e14 5.34921e14i 0
17.1 0 2.15234e7 + 1.24265e7i −1.07374e9 + 1.85978e9i 0 0 4.76727e12 + 1.16210e13i 0 3.08837e14 + 5.34921e14i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.32.g.a 2
3.b odd 2 1 CM 21.32.g.a 2
7.d odd 6 1 inner 21.32.g.a 2
21.g even 6 1 inner 21.32.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.32.g.a 2 1.a even 1 1 trivial
21.32.g.a 2 3.b odd 2 1 CM
21.32.g.a 2 7.d odd 6 1 inner
21.32.g.a 2 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{32}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 617673396283947 - 43046721 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(15\!\cdots\!43\)\( - 9534530086957 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(12\!\cdots\!47\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(11\!\cdots\!27\)\( + \)\(18\!\cdots\!41\)\( T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(25\!\cdots\!75\)\( - \)\(27\!\cdots\!15\)\( T + T^{2} \)
$37$ \( \)\(51\!\cdots\!49\)\( - \)\(22\!\cdots\!43\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( \)\(17\!\cdots\!65\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(87\!\cdots\!48\)\( - \)\(16\!\cdots\!12\)\( T + T^{2} \)
$67$ \( \)\(91\!\cdots\!29\)\( + \)\(30\!\cdots\!27\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(17\!\cdots\!07\)\( - \)\(22\!\cdots\!89\)\( T + T^{2} \)
$79$ \( \)\(22\!\cdots\!69\)\( + \)\(15\!\cdots\!63\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(10\!\cdots\!08\)\( + T^{2} \)
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