Properties

Label 21.12.g.a
Level $21$
Weight $12$
Character orbit 21.g
Analytic conductor $16.135$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.1352067918\)
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.

\(f(q)\) \(=\) \( q + ( -243 - 243 \zeta_{6} ) q^{3} + ( -2048 + 2048 \zeta_{6} ) q^{4} + ( -51346 + 25539 \zeta_{6} ) q^{7} + 177147 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -243 - 243 \zeta_{6} ) q^{3} + ( -2048 + 2048 \zeta_{6} ) q^{4} + ( -51346 + 25539 \zeta_{6} ) q^{7} + 177147 \zeta_{6} q^{9} + ( 995328 - 497664 \zeta_{6} ) q^{12} + ( 704747 - 1409494 \zeta_{6} ) q^{13} -4194304 \zeta_{6} q^{16} + ( 16824846 - 8412423 \zeta_{6} ) q^{19} + ( 18683055 + 65124 \zeta_{6} ) q^{21} + ( 48828125 - 48828125 \zeta_{6} ) q^{25} + ( 43046721 - 86093442 \zeta_{6} ) q^{27} + ( 52852736 - 105156608 \zeta_{6} ) q^{28} + ( 67664195 + 67664195 \zeta_{6} ) q^{31} -362797056 q^{36} + 663545123 \zeta_{6} q^{37} + ( -513760563 + 513760563 \zeta_{6} ) q^{39} -1768358135 q^{43} + ( -1019215872 + 2038431744 \zeta_{6} ) q^{48} + ( 1984171195 - 1970410467 \zeta_{6} ) q^{49} + ( 1443321856 + 1443321856 \zeta_{6} ) q^{52} -6132656367 q^{57} + ( 14349096968 - 7174548484 \zeta_{6} ) q^{61} + ( -4524157233 - 4571632629 \zeta_{6} ) q^{63} + 8589934592 q^{64} + ( 21410042863 - 21410042863 \zeta_{6} ) q^{67} + ( 1424119743 + 1424119743 \zeta_{6} ) q^{73} + ( -23730468750 + 11865234375 \zeta_{6} ) q^{75} + ( -17228642304 + 34457284608 \zeta_{6} ) q^{76} -21410392133 \zeta_{6} q^{79} + ( -31381059609 + 31381059609 \zeta_{6} ) q^{81} + ( -38396270592 + 38262896640 \zeta_{6} ) q^{84} + ( -188872196 + 54373345291 \zeta_{6} ) q^{91} -49327198155 \zeta_{6} q^{93} + ( 72858114904 - 145716229808 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 729q^{3} - 2048q^{4} - 77153q^{7} + 177147q^{9} + O(q^{10}) \) \( 2q - 729q^{3} - 2048q^{4} - 77153q^{7} + 177147q^{9} + 1492992q^{12} - 4194304q^{16} + 25237269q^{19} + 37431234q^{21} + 48828125q^{25} + 548864q^{28} + 202992585q^{31} - 725594112q^{36} + 663545123q^{37} - 513760563q^{39} - 3536716270q^{43} + 1997931923q^{49} + 4329965568q^{52} - 12265312734q^{57} + 21523645452q^{61} - 13619947095q^{63} + 17179869184q^{64} + 21410042863q^{67} + 4272359229q^{73} - 35595703125q^{75} - 21410392133q^{79} - 31381059609q^{81} - 38529644544q^{84} + 53995600899q^{91} - 49327198155q^{93} + O(q^{100}) \)

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 −364.500 + 210.444i −1024.00 1773.62i 0 0 −38576.5 22117.4i 0 88573.5 153414.i 0
17.1 0 −364.500 210.444i −1024.00 + 1773.62i 0 0 −38576.5 + 22117.4i 0 88573.5 + 153414.i 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 21.12.g.a 2
3.b odd 2 1 CM 21.12.g.a 2
7.d odd 6 1 inner 21.12.g.a 2
21.g even 6 1 inner 21.12.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.12.g.a 2 1.a even 1 1 trivial
21.12.g.a 2 3.b odd 2 1 CM
21.12.g.a 2 7.d odd 6 1 inner
21.12.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 \(S_{12}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 177147 + 729 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1977326743 + 77153 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 1490005002027 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 212306582192787 - 25237269 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 13735329854994075 - 202992585 T + T^{2} \)
$37$ \( 440292130257085129 - 663545123 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 1768358135 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(15\!\cdots\!68\)\( - 21523645452 T + T^{2} \)
$67$ \( \)\(45\!\cdots\!69\)\( - 21410042863 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 6084351127207158147 - 4272359229 T + T^{2} \)
$79$ \( \)\(45\!\cdots\!89\)\( + 21410392133 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(15\!\cdots\!48\)\( + T^{2} \)
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