Properties

Label 21.16.c.a
Level $21$
Weight $16$
Character orbit 21.c
Analytic conductor $29.966$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9656360710\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2187 - 4374 \zeta_{6} ) q^{3} + 32768 q^{4} + ( 583109 - 2411118 \zeta_{6} ) q^{7} -14348907 q^{9} +O(q^{10})\) \( q + ( 2187 - 4374 \zeta_{6} ) q^{3} + 32768 q^{4} + ( 583109 - 2411118 \zeta_{6} ) q^{7} -14348907 q^{9} + ( 71663616 - 143327232 \zeta_{6} ) q^{12} + ( 124527276 - 249054552 \zeta_{6} ) q^{13} + 1073741824 q^{16} + ( -688418550 + 1376837100 \zeta_{6} ) q^{19} + ( -9270970749 + 2722596300 \zeta_{6} ) q^{21} -30517578125 q^{25} + ( -31381059609 + 62762119218 \zeta_{6} ) q^{27} + ( 19107315712 - 79007514624 \zeta_{6} ) q^{28} + ( 126756171450 - 253512342900 \zeta_{6} ) q^{31} -470184984576 q^{36} -1090158909950 q^{37} -817023457836 q^{39} -1440654152600 q^{43} + ( 2348273369088 - 4696546738176 \zeta_{6} ) q^{48} + ( -5473473904043 + 3001600798200 \zeta_{6} ) q^{49} + ( 4080509779968 - 8161019559936 \zeta_{6} ) q^{52} + 4516714106550 q^{57} + ( -16233489614700 + 32466979229400 \zeta_{6} ) q^{61} + ( -8366976811863 + 34596907948026 \zeta_{6} ) q^{63} + 35184372088832 q^{64} + 99059017336400 q^{67} + ( 108866703280824 - 217733406561648 \zeta_{6} ) q^{73} + ( -66741943359375 + 133483886718750 \zeta_{6} ) q^{75} + ( -22558099046400 + 45116198092800 \zeta_{6} ) q^{76} -88692309079036 q^{79} + 205891132094649 q^{81} + ( -303791169503232 + 89214035558400 \zeta_{6} ) q^{84} + ( -527886937928052 + 155024005892400 \zeta_{6} ) q^{91} -831647240883450 q^{93} + ( 697991654583432 - 1395983309166864 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 65536q^{4} - 1244900q^{7} - 28697814q^{9} + O(q^{10}) \) \( 2q + 65536q^{4} - 1244900q^{7} - 28697814q^{9} + 2147483648q^{16} - 15819345198q^{21} - 61035156250q^{25} - 40792883200q^{28} - 940369969152q^{36} - 2180317819900q^{37} - 1634046915672q^{39} - 2881308305200q^{43} - 7945347009886q^{49} + 9033428213100q^{57} + 17862954324300q^{63} + 70368744177664q^{64} + 198118034672800q^{67} - 177384618158072q^{79} + 411782264189298q^{81} - 518368303448064q^{84} - 900749869963704q^{91} - 1663294481766900q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1
0.500000 + 0.866025i
0.500000 0.866025i
0 3788.00i 32768.0 0 0 −622450. 2.08809e6i 0 −1.43489e7 0
20.2 0 3788.00i 32768.0 0 0 −622450. + 2.08809e6i 0 −1.43489e7 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.16.c.a 2
3.b odd 2 1 CM 21.16.c.a 2
7.b odd 2 1 inner 21.16.c.a 2
21.c even 2 1 inner 21.16.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.16.c.a 2 1.a even 1 1 trivial
21.16.c.a 2 3.b odd 2 1 CM
21.16.c.a 2 7.b odd 2 1 inner
21.16.c.a 2 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{16}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 14348907 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4747561509943 + 1244900 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 46521127403940528 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 1421760299952307500 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(48\!\cdots\!00\)\( + T^{2} \)
$37$ \( ( 1090158909950 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 1440654152600 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(79\!\cdots\!00\)\( + T^{2} \)
$67$ \( ( -99059017336400 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(35\!\cdots\!28\)\( + T^{2} \)
$79$ \( ( 88692309079036 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(14\!\cdots\!72\)\( + T^{2} \)
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