Properties

Label 21.38.g.a
Level $21$
Weight $38$
Character orbit 21.g
Analytic conductor $182.099$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(182.099480062\)
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.

\(f(q)\) \(=\) \( q + ( -387420489 - 387420489 \zeta_{6} ) q^{3} + ( -137438953472 + 137438953472 \zeta_{6} ) q^{4} + ( 4758916148715458 - 3635106806344107 \zeta_{6} ) q^{7} + 450283905890997363 \zeta_{6} q^{9} +O(q^{10})\) \( q +(-387420489 - 387420489 \zeta_{6}) q^{3} +(-137438953472 + 137438953472 \zeta_{6}) q^{4} +(4758916148715458 - 3635106806344107 \zeta_{6}) q^{7} +450283905890997363 \zeta_{6} q^{9} +(\)\(10\!\cdots\!16\)\( - 53246666561770487808 \zeta_{6}) q^{12} +(-\)\(20\!\cdots\!89\)\( + \)\(40\!\cdots\!78\)\( \zeta_{6}) q^{13} -\)\(18\!\cdots\!84\)\( \zeta_{6} q^{16} +(\)\(10\!\cdots\!74\)\( - \)\(52\!\cdots\!37\)\( \zeta_{6}) q^{19} +(-\)\(32\!\cdots\!85\)\( + \)\(97\!\cdots\!84\)\( \zeta_{6}) q^{21} +(\)\(72\!\cdots\!25\)\( - \)\(72\!\cdots\!25\)\( \zeta_{6}) q^{25} +(\)\(17\!\cdots\!07\)\( - \)\(34\!\cdots\!14\)\( \zeta_{6}) q^{27} +(-\)\(15\!\cdots\!72\)\( + \)\(65\!\cdots\!76\)\( \zeta_{6}) q^{28} +(\)\(44\!\cdots\!85\)\( + \)\(44\!\cdots\!85\)\( \zeta_{6}) q^{31} -\)\(61\!\cdots\!36\)\( q^{36} -\)\(19\!\cdots\!09\)\( \zeta_{6} q^{37} +(\)\(23\!\cdots\!63\)\( - \)\(23\!\cdots\!63\)\( \zeta_{6}) q^{39} -\)\(31\!\cdots\!65\)\( q^{43} +(-\)\(73\!\cdots\!76\)\( + \)\(14\!\cdots\!52\)\( \zeta_{6}) q^{48} +(\)\(94\!\cdots\!15\)\( - \)\(21\!\cdots\!63\)\( \zeta_{6}) q^{49} +(-\)\(27\!\cdots\!08\)\( - \)\(27\!\cdots\!08\)\( \zeta_{6}) q^{52} -\)\(60\!\cdots\!79\)\( q^{57} +(-\)\(23\!\cdots\!32\)\( + \)\(11\!\cdots\!16\)\( \zeta_{6}) q^{61} +(\)\(16\!\cdots\!41\)\( + \)\(50\!\cdots\!13\)\( \zeta_{6}) q^{63} +\)\(25\!\cdots\!48\)\( q^{64} +(-\)\(10\!\cdots\!39\)\( + \)\(10\!\cdots\!39\)\( \zeta_{6}) q^{67} +(\)\(34\!\cdots\!59\)\( + \)\(34\!\cdots\!59\)\( \zeta_{6}) q^{73} +(-\)\(56\!\cdots\!50\)\( + \)\(28\!\cdots\!25\)\( \zeta_{6}) q^{75} +(-\)\(71\!\cdots\!64\)\( + \)\(14\!\cdots\!28\)\( \zeta_{6}) q^{76} -\)\(22\!\cdots\!67\)\( \zeta_{6} q^{79} +(-\)\(20\!\cdots\!69\)\( + \)\(20\!\cdots\!69\)\( \zeta_{6}) q^{81} +(\)\(31\!\cdots\!72\)\( - \)\(44\!\cdots\!20\)\( \zeta_{6}) q^{84} +(\)\(51\!\cdots\!84\)\( + \)\(11\!\cdots\!01\)\( \zeta_{6}) q^{91} -\)\(52\!\cdots\!95\)\( \zeta_{6} q^{93} +(\)\(36\!\cdots\!28\)\( - \)\(73\!\cdots\!56\)\( \zeta_{6}) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 1162261467q^{3} - 137438953472q^{4} + 5882725491086809q^{7} + 450283905890997363q^{9} + O(q^{10}) \) \( 2q - 1162261467q^{3} - 137438953472q^{4} + 5882725491086809q^{7} + 450283905890997363q^{9} + \)\(15\!\cdots\!24\)\(q^{12} - \)\(18\!\cdots\!84\)\(q^{16} + \)\(15\!\cdots\!11\)\(q^{19} - \)\(55\!\cdots\!86\)\(q^{21} + \)\(72\!\cdots\!25\)\(q^{25} + \)\(34\!\cdots\!32\)\(q^{28} + \)\(13\!\cdots\!55\)\(q^{31} - \)\(12\!\cdots\!72\)\(q^{36} - \)\(19\!\cdots\!09\)\(q^{37} + \)\(23\!\cdots\!63\)\(q^{39} - \)\(62\!\cdots\!30\)\(q^{43} - \)\(25\!\cdots\!33\)\(q^{49} - \)\(83\!\cdots\!24\)\(q^{52} - \)\(12\!\cdots\!58\)\(q^{57} - \)\(35\!\cdots\!48\)\(q^{61} + \)\(37\!\cdots\!95\)\(q^{63} + \)\(51\!\cdots\!96\)\(q^{64} - \)\(10\!\cdots\!39\)\(q^{67} + \)\(10\!\cdots\!77\)\(q^{73} - \)\(84\!\cdots\!75\)\(q^{75} - \)\(22\!\cdots\!67\)\(q^{79} - \)\(20\!\cdots\!69\)\(q^{81} + \)\(17\!\cdots\!24\)\(q^{84} + \)\(22\!\cdots\!69\)\(q^{91} - \)\(52\!\cdots\!95\)\(q^{93} + O(q^{100}) \)

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 −5.81131e8 + 3.35516e8i −6.87195e10 1.19026e11i 0 0 2.94136e15 + 3.14809e15i 0 2.25142e17 3.89957e17i 0
17.1 0 −5.81131e8 3.35516e8i −6.87195e10 + 1.19026e11i 0 0 2.94136e15 3.14809e15i 0 2.25142e17 + 3.89957e17i 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 21.38.g.a 2
3.b odd 2 1 CM 21.38.g.a 2
7.d odd 6 1 inner 21.38.g.a 2
21.g even 6 1 inner 21.38.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.38.g.a 2 1.a even 1 1 trivial
21.38.g.a 2 3.b odd 2 1 CM
21.38.g.a 2 7.d odd 6 1 inner
21.38.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 \(S_{38}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 450283905890997363 + 1162261467 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(18\!\cdots\!07\)\( - 5882725491086809 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(12\!\cdots\!63\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(82\!\cdots\!07\)\( - \)\(15\!\cdots\!11\)\( T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(60\!\cdots\!75\)\( - \)\(13\!\cdots\!55\)\( T + T^{2} \)
$37$ \( \)\(39\!\cdots\!81\)\( + \)\(19\!\cdots\!09\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( \)\(31\!\cdots\!65\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(42\!\cdots\!68\)\( + \)\(35\!\cdots\!48\)\( T + T^{2} \)
$67$ \( \)\(10\!\cdots\!21\)\( + \)\(10\!\cdots\!39\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(34\!\cdots\!43\)\( - \)\(10\!\cdots\!77\)\( T + T^{2} \)
$79$ \( \)\(50\!\cdots\!89\)\( + \)\(22\!\cdots\!67\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(40\!\cdots\!52\)\( + T^{2} \)
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