Properties

Label 21.9.h.a
Level $21$
Weight $9$
Character orbit 21.h
Analytic conductor $8.555$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.55495081128\)
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.

\(f(q)\) \(=\) \( q -81 \zeta_{6} q^{3} -256 \zeta_{6} q^{4} + ( -1265 + 2769 \zeta_{6} ) q^{7} + ( -6561 + 6561 \zeta_{6} ) q^{9} +O(q^{10})\) \( q -81 \zeta_{6} q^{3} -256 \zeta_{6} q^{4} + ( -1265 + 2769 \zeta_{6} ) q^{7} + ( -6561 + 6561 \zeta_{6} ) q^{9} + ( -20736 + 20736 \zeta_{6} ) q^{12} -20641 q^{13} + ( -65536 + 65536 \zeta_{6} ) q^{16} + ( -100559 + 100559 \zeta_{6} ) q^{19} + ( 224289 - 121824 \zeta_{6} ) q^{21} -390625 \zeta_{6} q^{25} + 531441 q^{27} + ( 708864 - 385024 \zeta_{6} ) q^{28} -1225967 \zeta_{6} q^{31} + 1679616 q^{36} + ( -2964959 + 2964959 \zeta_{6} ) q^{37} + 1671921 \zeta_{6} q^{39} -6837073 q^{43} + 5308416 q^{48} + ( -6067136 + 661791 \zeta_{6} ) q^{49} + 5284096 \zeta_{6} q^{52} + 8145279 q^{57} + ( 23826526 - 23826526 \zeta_{6} ) q^{61} + ( -9867744 - 8299665 \zeta_{6} ) q^{63} + 16777216 q^{64} -37296239 \zeta_{6} q^{67} + 55236481 \zeta_{6} q^{73} + ( -31640625 + 31640625 \zeta_{6} ) q^{75} + 25743104 q^{76} + ( -74894159 + 74894159 \zeta_{6} ) q^{79} -43046721 \zeta_{6} q^{81} + ( -31186944 - 26231040 \zeta_{6} ) q^{84} + ( 26110865 - 57154929 \zeta_{6} ) q^{91} + ( -99303327 + 99303327 \zeta_{6} ) q^{93} + 176908034 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 81q^{3} - 256q^{4} + 239q^{7} - 6561q^{9} + O(q^{10}) \) \( 2q - 81q^{3} - 256q^{4} + 239q^{7} - 6561q^{9} - 20736q^{12} - 41282q^{13} - 65536q^{16} - 100559q^{19} + 326754q^{21} - 390625q^{25} + 1062882q^{27} + 1032704q^{28} - 1225967q^{31} + 3359232q^{36} - 2964959q^{37} + 1671921q^{39} - 13674146q^{43} + 10616832q^{48} - 11472481q^{49} + 5284096q^{52} + 16290558q^{57} + 23826526q^{61} - 28035153q^{63} + 33554432q^{64} - 37296239q^{67} + 55236481q^{73} - 31640625q^{75} + 51486208q^{76} - 74894159q^{79} - 43046721q^{81} - 88604928q^{84} - 4933199q^{91} - 99303327q^{93} + 353816068q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −40.5000 70.1481i −128.000 221.703i 0 0 119.500 + 2398.02i 0 −3280.50 + 5681.99i 0
11.1 0 −40.5000 + 70.1481i −128.000 + 221.703i 0 0 119.500 2398.02i 0 −3280.50 5681.99i 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.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.9.h.a 2
3.b odd 2 1 CM 21.9.h.a 2
7.c even 3 1 inner 21.9.h.a 2
21.h odd 6 1 inner 21.9.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.9.h.a 2 1.a even 1 1 trivial
21.9.h.a 2 3.b odd 2 1 CM
21.9.h.a 2 7.c even 3 1 inner
21.9.h.a 2 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{9}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 6561 + 81 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 5764801 - 239 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 20641 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 10112112481 + 100559 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 1502995085089 + 1225967 T + T^{2} \)
$37$ \( 8790981871681 + 2964959 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 6837073 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 567703341228676 - 23826526 T + T^{2} \)
$67$ \( 1391009443545121 + 37296239 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 3051068833263361 - 55236481 T + T^{2} \)
$79$ \( 5609135052317281 + 74894159 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -176908034 + T )^{2} \)
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