Properties

Label 21.36.g.a
Level $21$
Weight $36$
Character orbit 21.g
Analytic conductor $162.950$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(162.949774331\)
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.

\(f(q)\) \(=\) \( q + ( -129140163 - 129140163 \zeta_{6} ) q^{3} + ( -34359738368 + 34359738368 \zeta_{6} ) q^{4} + ( 107883136881134 - 662291479140141 \zeta_{6} ) q^{7} + 50031545098999707 \zeta_{6} q^{9} +O(q^{10})\) \( q +(-129140163 - 129140163 \zeta_{6}) q^{3} +(-34359738368 + 34359738368 \zeta_{6}) q^{4} +(107883136881134 - 662291479140141 \zeta_{6}) q^{7} +50031545098999707 \zeta_{6} q^{9} +(8874444426961747968 - 4437222213480873984 \zeta_{6}) q^{12} +(28978182513508262987 - 57956365027016525974 \zeta_{6}) q^{13} -\)\(11\!\cdots\!24\)\( \zeta_{6} q^{16} +(-\)\(19\!\cdots\!14\)\( + \)\(96\!\cdots\!57\)\( \zeta_{6}) q^{19} +(-\)\(99\!\cdots\!25\)\( + \)\(15\!\cdots\!24\)\( \zeta_{6}) q^{21} +(\)\(29\!\cdots\!25\)\( - \)\(29\!\cdots\!25\)\( \zeta_{6}) q^{25} +(\)\(64\!\cdots\!41\)\( - \)\(12\!\cdots\!82\)\( \zeta_{6}) q^{27} +(\)\(19\!\cdots\!76\)\( + \)\(37\!\cdots\!12\)\( \zeta_{6}) q^{28} +(\)\(33\!\cdots\!75\)\( + \)\(33\!\cdots\!75\)\( \zeta_{6}) q^{31} -\)\(17\!\cdots\!76\)\( q^{36} +\)\(14\!\cdots\!23\)\( \zeta_{6} q^{37} +(-\)\(11\!\cdots\!43\)\( + \)\(11\!\cdots\!43\)\( \zeta_{6}) q^{39} +\)\(69\!\cdots\!25\)\( q^{43} +(-\)\(15\!\cdots\!12\)\( + \)\(30\!\cdots\!24\)\( \zeta_{6}) q^{48} +(-\)\(42\!\cdots\!25\)\( + \)\(29\!\cdots\!93\)\( \zeta_{6}) q^{49} +(\)\(99\!\cdots\!16\)\( + \)\(99\!\cdots\!16\)\( \zeta_{6}) q^{52} +\)\(37\!\cdots\!73\)\( q^{57} +(\)\(14\!\cdots\!48\)\( - \)\(74\!\cdots\!24\)\( \zeta_{6}) q^{61} +(\)\(33\!\cdots\!87\)\( - \)\(27\!\cdots\!49\)\( \zeta_{6}) q^{63} +\)\(40\!\cdots\!32\)\( q^{64} +(\)\(11\!\cdots\!23\)\( - \)\(11\!\cdots\!23\)\( \zeta_{6}) q^{67} +(\)\(27\!\cdots\!63\)\( + \)\(27\!\cdots\!63\)\( \zeta_{6}) q^{73} +(-\)\(75\!\cdots\!50\)\( + \)\(37\!\cdots\!75\)\( \zeta_{6}) q^{75} +(\)\(33\!\cdots\!76\)\( - \)\(66\!\cdots\!52\)\( \zeta_{6}) q^{76} +\)\(13\!\cdots\!07\)\( \zeta_{6} q^{79} +(-\)\(25\!\cdots\!49\)\( + \)\(25\!\cdots\!49\)\( \zeta_{6}) q^{81} +(-\)\(19\!\cdots\!32\)\( - \)\(34\!\cdots\!00\)\( \zeta_{6}) q^{84} +(-\)\(35\!\cdots\!76\)\( + \)\(12\!\cdots\!51\)\( \zeta_{6}) q^{91} -\)\(12\!\cdots\!75\)\( \zeta_{6} q^{93} +(\)\(55\!\cdots\!84\)\( - \)\(11\!\cdots\!68\)\( \zeta_{6}) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 387420489q^{3} - 34359738368q^{4} - 446525205377873q^{7} + 50031545098999707q^{9} + O(q^{10}) \) \( 2q - 387420489q^{3} - 34359738368q^{4} - 446525205377873q^{7} + 50031545098999707q^{9} + 13311666640442621952q^{12} - \)\(11\!\cdots\!24\)\(q^{16} - \)\(29\!\cdots\!71\)\(q^{19} - \)\(41\!\cdots\!26\)\(q^{21} + \)\(29\!\cdots\!25\)\(q^{25} + \)\(41\!\cdots\!64\)\(q^{28} + \)\(99\!\cdots\!25\)\(q^{31} - \)\(34\!\cdots\!52\)\(q^{36} + \)\(14\!\cdots\!23\)\(q^{37} - \)\(11\!\cdots\!43\)\(q^{39} + \)\(13\!\cdots\!50\)\(q^{43} - \)\(55\!\cdots\!57\)\(q^{49} + \)\(29\!\cdots\!48\)\(q^{52} + \)\(74\!\cdots\!46\)\(q^{57} + \)\(22\!\cdots\!72\)\(q^{61} + \)\(38\!\cdots\!25\)\(q^{63} + \)\(81\!\cdots\!64\)\(q^{64} + \)\(11\!\cdots\!23\)\(q^{67} + \)\(83\!\cdots\!89\)\(q^{73} - \)\(11\!\cdots\!25\)\(q^{75} + \)\(13\!\cdots\!07\)\(q^{79} - \)\(25\!\cdots\!49\)\(q^{81} - \)\(73\!\cdots\!64\)\(q^{84} - \)\(57\!\cdots\!01\)\(q^{91} - \)\(12\!\cdots\!75\)\(q^{93} + O(q^{100}) \)

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 −1.93710e8 + 1.11839e8i −1.71799e10 2.97564e10i 0 0 −2.23263e14 + 5.73561e14i 0 2.50158e16 4.33286e16i 0
17.1 0 −1.93710e8 1.11839e8i −1.71799e10 + 2.97564e10i 0 0 −2.23263e14 5.73561e14i 0 2.50158e16 + 4.33286e16i 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 21.36.g.a 2
3.b odd 2 1 CM 21.36.g.a 2
7.d odd 6 1 inner 21.36.g.a 2
21.g even 6 1 inner 21.36.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.36.g.a 2 1.a even 1 1 trivial
21.36.g.a 2 3.b odd 2 1 CM
21.36.g.a 2 7.d odd 6 1 inner
21.36.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 \(S_{36}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 50031545098999707 + 387420489 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(37\!\cdots\!43\)\( + 446525205377873 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(25\!\cdots\!07\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(28\!\cdots\!47\)\( + \)\(29\!\cdots\!71\)\( T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(32\!\cdots\!75\)\( - \)\(99\!\cdots\!25\)\( T + T^{2} \)
$37$ \( \)\(20\!\cdots\!29\)\( - \)\(14\!\cdots\!23\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -\)\(69\!\cdots\!25\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(16\!\cdots\!28\)\( - \)\(22\!\cdots\!72\)\( T + T^{2} \)
$67$ \( \)\(12\!\cdots\!29\)\( - \)\(11\!\cdots\!23\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(23\!\cdots\!07\)\( - \)\(83\!\cdots\!89\)\( T + T^{2} \)
$79$ \( \)\(18\!\cdots\!49\)\( - \)\(13\!\cdots\!07\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(93\!\cdots\!68\)\( + T^{2} \)
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