Properties

Label 21.5.h.a
Level $21$
Weight $5$
Character orbit 21.h
Analytic conductor $2.171$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17076922476\)
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.

\(f(q)\) \(=\) \( q -9 \zeta_{6} q^{3} -16 \zeta_{6} q^{4} + ( 55 - 39 \zeta_{6} ) q^{7} + ( -81 + 81 \zeta_{6} ) q^{9} +O(q^{10})\) \( q -9 \zeta_{6} q^{3} -16 \zeta_{6} q^{4} + ( 55 - 39 \zeta_{6} ) q^{7} + ( -81 + 81 \zeta_{6} ) q^{9} + ( -144 + 144 \zeta_{6} ) q^{12} + 191 q^{13} + ( -256 + 256 \zeta_{6} ) q^{16} + ( 601 - 601 \zeta_{6} ) q^{19} + ( -351 - 144 \zeta_{6} ) q^{21} -625 \zeta_{6} q^{25} + 729 q^{27} + ( -624 - 256 \zeta_{6} ) q^{28} + 1753 \zeta_{6} q^{31} + 1296 q^{36} + ( -2591 + 2591 \zeta_{6} ) q^{37} -1719 \zeta_{6} q^{39} + 23 q^{43} + 2304 q^{48} + ( 1504 - 2769 \zeta_{6} ) q^{49} -3056 \zeta_{6} q^{52} -5409 q^{57} + ( 1966 - 1966 \zeta_{6} ) q^{61} + ( -1296 + 4455 \zeta_{6} ) q^{63} + 4096 q^{64} + 8809 \zeta_{6} q^{67} + 1249 \zeta_{6} q^{73} + ( -5625 + 5625 \zeta_{6} ) q^{75} -9616 q^{76} + ( 12361 - 12361 \zeta_{6} ) q^{79} -6561 \zeta_{6} q^{81} + ( -2304 + 7920 \zeta_{6} ) q^{84} + ( 10505 - 7449 \zeta_{6} ) q^{91} + ( 15777 - 15777 \zeta_{6} ) q^{93} -18814 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 9q^{3} - 16q^{4} + 71q^{7} - 81q^{9} + O(q^{10}) \) \( 2q - 9q^{3} - 16q^{4} + 71q^{7} - 81q^{9} - 144q^{12} + 382q^{13} - 256q^{16} + 601q^{19} - 846q^{21} - 625q^{25} + 1458q^{27} - 1504q^{28} + 1753q^{31} + 2592q^{36} - 2591q^{37} - 1719q^{39} + 46q^{43} + 4608q^{48} + 239q^{49} - 3056q^{52} - 10818q^{57} + 1966q^{61} + 1863q^{63} + 8192q^{64} + 8809q^{67} + 1249q^{73} - 5625q^{75} - 19232q^{76} + 12361q^{79} - 6561q^{81} + 3312q^{84} + 13561q^{91} + 15777q^{93} - 37628q^{97} + O(q^{100}) \)

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 −4.50000 7.79423i −8.00000 13.8564i 0 0 35.5000 33.7750i 0 −40.5000 + 70.1481i 0
11.1 0 −4.50000 + 7.79423i −8.00000 + 13.8564i 0 0 35.5000 + 33.7750i 0 −40.5000 70.1481i 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 21.5.h.a 2
3.b odd 2 1 CM 21.5.h.a 2
7.b odd 2 1 147.5.h.a 2
7.c even 3 1 inner 21.5.h.a 2
7.c even 3 1 147.5.b.b 1
7.d odd 6 1 147.5.b.a 1
7.d odd 6 1 147.5.h.a 2
21.c even 2 1 147.5.h.a 2
21.g even 6 1 147.5.b.a 1
21.g even 6 1 147.5.h.a 2
21.h odd 6 1 inner 21.5.h.a 2
21.h odd 6 1 147.5.b.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.h.a 2 1.a even 1 1 trivial
21.5.h.a 2 3.b odd 2 1 CM
21.5.h.a 2 7.c even 3 1 inner
21.5.h.a 2 21.h odd 6 1 inner
147.5.b.a 1 7.d odd 6 1
147.5.b.a 1 21.g even 6 1
147.5.b.b 1 7.c even 3 1
147.5.b.b 1 21.h odd 6 1
147.5.h.a 2 7.b odd 2 1
147.5.h.a 2 7.d odd 6 1
147.5.h.a 2 21.c even 2 1
147.5.h.a 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{5}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 81 + 9 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 2401 - 71 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -191 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 361201 - 601 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 3073009 - 1753 T + T^{2} \)
$37$ \( 6713281 + 2591 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -23 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 3865156 - 1966 T + T^{2} \)
$67$ \( 77598481 - 8809 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1560001 - 1249 T + T^{2} \)
$79$ \( 152794321 - 12361 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 18814 + T )^{2} \)
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