Properties

Label 21.27.h.a
Level $21$
Weight $27$
Character orbit 21.h
Analytic conductor $89.942$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 27 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(89.9415133532\)
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.

\(f(q)\) \(=\) \( q + 1594323 \zeta_{6} q^{3} -67108864 \zeta_{6} q^{4} + ( -76625836565 + 108909244077 \zeta_{6} ) q^{7} + ( -2541865828329 + 2541865828329 \zeta_{6} ) q^{9} +O(q^{10})\) \( q +1594323 \zeta_{6} q^{3} -67108864 \zeta_{6} q^{4} +(-76625836565 + 108909244077 \zeta_{6}) q^{7} +(-2541865828329 + 2541865828329 \zeta_{6}) q^{9} +(106993205379072 - 106993205379072 \zeta_{6}) q^{12} +605583307798583 q^{13} +(-4503599627370496 + 4503599627370496 \zeta_{6}) q^{16} +(52636841466889669 - 52636841466889669 \zeta_{6}) q^{19} +(-173636512744574871 + 51470179114754376 \zeta_{6}) q^{21} -1490116119384765625 \zeta_{6} q^{25} -4052555153018976267 q^{27} +(7308775649106198528 - 2166502804179386368 \zeta_{6}) q^{28} +18810497092058970493 \zeta_{6} q^{31} +\)\(17\!\cdots\!56\)\( q^{36} +(-\)\(26\!\cdots\!67\)\( + \)\(26\!\cdots\!67\)\( \zeta_{6}) q^{37} +\)\(96\!\cdots\!09\)\( \zeta_{6} q^{39} +\)\(25\!\cdots\!39\)\( q^{43} -\)\(71\!\cdots\!08\)\( q^{48} +(-\)\(59\!\cdots\!04\)\( - \)\(48\!\cdots\!81\)\( \zeta_{6}) q^{49} -\)\(40\!\cdots\!12\)\( \zeta_{6} q^{52} +\)\(83\!\cdots\!87\)\( q^{57} +(-\)\(26\!\cdots\!74\)\( + \)\(26\!\cdots\!74\)\( \zeta_{6}) q^{61} +(-\)\(82\!\cdots\!48\)\( - \)\(19\!\cdots\!85\)\( \zeta_{6}) q^{63} +\)\(30\!\cdots\!44\)\( q^{64} +\)\(10\!\cdots\!53\)\( \zeta_{6} q^{67} -\)\(73\!\cdots\!23\)\( \zeta_{6} q^{73} +(\)\(23\!\cdots\!75\)\( - \)\(23\!\cdots\!75\)\( \zeta_{6}) q^{75} -\)\(35\!\cdots\!16\)\( q^{76} +(-\)\(73\!\cdots\!31\)\( + \)\(73\!\cdots\!31\)\( \zeta_{6}) q^{79} -\)\(64\!\cdots\!41\)\( \zeta_{6} q^{81} +(\)\(34\!\cdots\!64\)\( + \)\(81\!\cdots\!80\)\( \zeta_{6}) q^{84} +(-\)\(46\!\cdots\!95\)\( + \)\(65\!\cdots\!91\)\( \zeta_{6}) q^{91} +(-\)\(29\!\cdots\!39\)\( + \)\(29\!\cdots\!39\)\( \zeta_{6}) q^{93} +\)\(17\!\cdots\!42\)\( q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 1594323q^{3} - 67108864q^{4} - 44342429053q^{7} - 2541865828329q^{9} + O(q^{10}) \) \( 2q + 1594323q^{3} - 67108864q^{4} - 44342429053q^{7} - 2541865828329q^{9} + 106993205379072q^{12} + 1211166615597166q^{13} - 4503599627370496q^{16} + 52636841466889669q^{19} - 295802846374395366q^{21} - 1490116119384765625q^{25} - 8105110306037952534q^{27} + 12451048494033010688q^{28} + 18810497092058970493q^{31} + \)\(34\!\cdots\!12\)\(q^{36} - \)\(26\!\cdots\!67\)\(q^{37} + \)\(96\!\cdots\!09\)\(q^{39} + \)\(50\!\cdots\!78\)\(q^{43} - \)\(14\!\cdots\!16\)\(q^{48} - \)\(16\!\cdots\!89\)\(q^{49} - \)\(40\!\cdots\!12\)\(q^{52} + \)\(16\!\cdots\!74\)\(q^{57} - \)\(26\!\cdots\!74\)\(q^{61} - \)\(35\!\cdots\!81\)\(q^{63} + \)\(60\!\cdots\!88\)\(q^{64} + \)\(10\!\cdots\!53\)\(q^{67} - \)\(73\!\cdots\!23\)\(q^{73} + \)\(23\!\cdots\!75\)\(q^{75} - \)\(70\!\cdots\!32\)\(q^{76} - \)\(73\!\cdots\!31\)\(q^{79} - \)\(64\!\cdots\!41\)\(q^{81} + \)\(15\!\cdots\!08\)\(q^{84} - \)\(26\!\cdots\!99\)\(q^{91} - \)\(29\!\cdots\!39\)\(q^{93} + \)\(35\!\cdots\!84\)\(q^{97} + O(q^{100}) \)

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 797162. + 1.38072e6i −3.35544e7 5.81180e7i 0 0 −2.21712e10 + 9.43182e10i 0 −1.27093e12 + 2.20132e12i 0
11.1 0 797162. 1.38072e6i −3.35544e7 + 5.81180e7i 0 0 −2.21712e10 9.43182e10i 0 −1.27093e12 2.20132e12i 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 21.27.h.a 2
3.b odd 2 1 CM 21.27.h.a 2
7.c even 3 1 inner 21.27.h.a 2
21.h odd 6 1 inner 21.27.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.27.h.a 2 1.a even 1 1 trivial
21.27.h.a 2 3.b odd 2 1 CM
21.27.h.a 2 7.c even 3 1 inner
21.27.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 \(S_{27}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 2541865828329 - 1594323 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(93\!\cdots\!49\)\( + 44342429053 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -605583307798583 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(27\!\cdots\!61\)\( - 52636841466889669 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(35\!\cdots\!49\)\( - 18810497092058970493 T + T^{2} \)
$37$ \( \)\(68\!\cdots\!89\)\( + \)\(26\!\cdots\!67\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -\)\(25\!\cdots\!39\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(69\!\cdots\!76\)\( + \)\(26\!\cdots\!74\)\( T + T^{2} \)
$67$ \( \)\(10\!\cdots\!09\)\( - \)\(10\!\cdots\!53\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(53\!\cdots\!29\)\( + \)\(73\!\cdots\!23\)\( T + T^{2} \)
$79$ \( \)\(53\!\cdots\!61\)\( + \)\(73\!\cdots\!31\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -\)\(17\!\cdots\!42\)\( + T )^{2} \)
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