Properties

Label 21.33.h.a
Level $21$
Weight $33$
Character orbit 21.h
Analytic conductor $136.220$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(136.219975799\)
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.

\(f(q)\) \(=\) \( q -43046721 \zeta_{6} q^{3} -4294967296 \zeta_{6} q^{4} + ( 36372171914815 - 7592384673471 \zeta_{6} ) q^{7} + ( -1853020188851841 + 1853020188851841 \zeta_{6} ) q^{9} +O(q^{10})\) \( q -43046721 \zeta_{6} q^{3} -4294967296 \zeta_{6} q^{4} +(36372171914815 - 7592384673471 \zeta_{6}) q^{7} +(-1853020188851841 + 1853020188851841 \zeta_{6}) q^{9} +(-184884258895036416 + 184884258895036416 \zeta_{6}) q^{12} +122181402203975039 q^{13} +(-18446744073709551616 + 18446744073709551616 \zeta_{6}) q^{16} +(7821153231347348161 - 7821153231347348161 \zeta_{6}) q^{19} +(-\)\(32\!\cdots\!91\)\( - \)\(12\!\cdots\!24\)\( \zeta_{6}) q^{21} -\)\(23\!\cdots\!25\)\( \zeta_{6} q^{25} +\)\(79\!\cdots\!61\)\( q^{27} +(-\)\(32\!\cdots\!16\)\( - \)\(12\!\cdots\!24\)\( \zeta_{6}) q^{28} +\)\(14\!\cdots\!13\)\( \zeta_{6} q^{31} +\)\(79\!\cdots\!36\)\( q^{36} +(\)\(21\!\cdots\!61\)\( - \)\(21\!\cdots\!61\)\( \zeta_{6}) q^{37} -\)\(52\!\cdots\!19\)\( \zeta_{6} q^{39} +\)\(27\!\cdots\!27\)\( q^{43} +\)\(79\!\cdots\!36\)\( q^{48} +(\)\(12\!\cdots\!84\)\( - \)\(49\!\cdots\!89\)\( \zeta_{6}) q^{49} -\)\(52\!\cdots\!44\)\( \zeta_{6} q^{52} -\)\(33\!\cdots\!81\)\( q^{57} +(\)\(39\!\cdots\!26\)\( - \)\(39\!\cdots\!26\)\( \zeta_{6}) q^{61} +(-\)\(53\!\cdots\!04\)\( + \)\(67\!\cdots\!15\)\( \zeta_{6}) q^{63} +\)\(79\!\cdots\!36\)\( q^{64} -\)\(53\!\cdots\!59\)\( \zeta_{6} q^{67} -\)\(76\!\cdots\!79\)\( \zeta_{6} q^{73} +(-\)\(10\!\cdots\!25\)\( + \)\(10\!\cdots\!25\)\( \zeta_{6}) q^{75} -\)\(33\!\cdots\!56\)\( q^{76} +(-\)\(20\!\cdots\!39\)\( + \)\(20\!\cdots\!39\)\( \zeta_{6}) q^{79} -\)\(34\!\cdots\!81\)\( \zeta_{6} q^{81} +(-\)\(53\!\cdots\!04\)\( + \)\(67\!\cdots\!40\)\( \zeta_{6}) q^{84} +(\)\(44\!\cdots\!85\)\( - \)\(92\!\cdots\!69\)\( \zeta_{6}) q^{91} +(\)\(60\!\cdots\!73\)\( - \)\(60\!\cdots\!73\)\( \zeta_{6}) q^{93} +\)\(12\!\cdots\!14\)\( q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 43046721q^{3} - 4294967296q^{4} + 65151959156159q^{7} - 1853020188851841q^{9} + O(q^{10}) \) \( 2q - 43046721q^{3} - 4294967296q^{4} + 65151959156159q^{7} - 1853020188851841q^{9} - 184884258895036416q^{12} + 244362804407950078q^{13} - 18446744073709551616q^{16} + 7821153231347348161q^{19} - \)\(18\!\cdots\!06\)\(q^{21} - \)\(23\!\cdots\!25\)\(q^{25} + \)\(15\!\cdots\!22\)\(q^{27} - \)\(18\!\cdots\!56\)\(q^{28} + \)\(14\!\cdots\!13\)\(q^{31} + \)\(15\!\cdots\!72\)\(q^{36} + \)\(21\!\cdots\!61\)\(q^{37} - \)\(52\!\cdots\!19\)\(q^{39} + \)\(54\!\cdots\!54\)\(q^{43} + \)\(15\!\cdots\!72\)\(q^{48} + \)\(20\!\cdots\!79\)\(q^{49} - \)\(52\!\cdots\!44\)\(q^{52} - \)\(67\!\cdots\!62\)\(q^{57} + \)\(39\!\cdots\!26\)\(q^{61} - \)\(39\!\cdots\!93\)\(q^{63} + \)\(15\!\cdots\!72\)\(q^{64} - \)\(53\!\cdots\!59\)\(q^{67} - \)\(76\!\cdots\!79\)\(q^{73} - \)\(10\!\cdots\!25\)\(q^{75} - \)\(67\!\cdots\!12\)\(q^{76} - \)\(20\!\cdots\!39\)\(q^{79} - \)\(34\!\cdots\!81\)\(q^{81} - \)\(39\!\cdots\!68\)\(q^{84} + \)\(79\!\cdots\!01\)\(q^{91} + \)\(60\!\cdots\!73\)\(q^{93} + \)\(24\!\cdots\!28\)\(q^{97} + O(q^{100}) \)

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 −2.15234e7 3.72796e7i −2.14748e9 3.71955e9i 0 0 3.25760e13 6.57520e12i 0 −9.26510e14 + 1.60476e15i 0
11.1 0 −2.15234e7 + 3.72796e7i −2.14748e9 + 3.71955e9i 0 0 3.25760e13 + 6.57520e12i 0 −9.26510e14 1.60476e15i 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 21.33.h.a 2
3.b odd 2 1 CM 21.33.h.a 2
7.c even 3 1 inner 21.33.h.a 2
21.h odd 6 1 inner 21.33.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.33.h.a 2 1.a even 1 1 trivial
21.33.h.a 2 3.b odd 2 1 CM
21.33.h.a 2 7.c even 3 1 inner
21.33.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 \(S_{33}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1853020188851841 + 43046721 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(11\!\cdots\!01\)\( - 65151959156159 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -122181402203975039 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(61\!\cdots\!21\)\( - 7821153231347348161 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(19\!\cdots\!69\)\( - \)\(14\!\cdots\!13\)\( T + T^{2} \)
$37$ \( \)\(46\!\cdots\!21\)\( - \)\(21\!\cdots\!61\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -\)\(27\!\cdots\!27\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(15\!\cdots\!76\)\( - \)\(39\!\cdots\!26\)\( T + T^{2} \)
$67$ \( \)\(28\!\cdots\!81\)\( + \)\(53\!\cdots\!59\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(58\!\cdots\!41\)\( + \)\(76\!\cdots\!79\)\( T + T^{2} \)
$79$ \( \)\(41\!\cdots\!21\)\( + \)\(20\!\cdots\!39\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -\)\(12\!\cdots\!14\)\( + T )^{2} \)
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