Properties

Label 21.29.h.a
Level $21$
Weight $29$
Character orbit 21.h
Analytic conductor $104.304$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 29 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(104.303795921\)
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.

\(f(q)\) \(=\) \( q -4782969 \zeta_{6} q^{3} -268435456 \zeta_{6} q^{4} + ( 68460472135 + 641396442921 \zeta_{6} ) q^{7} + ( -22876792454961 + 22876792454961 \zeta_{6} ) q^{9} +O(q^{10})\) \( q -4782969 \zeta_{6} q^{3} -268435456 \zeta_{6} q^{4} +(68460472135 + 641396442921 \zeta_{6}) q^{7} +(-22876792454961 + 22876792454961 \zeta_{6}) q^{9} +(-1283918464548864 + 1283918464548864 \zeta_{6}) q^{12} +6878010332269631 q^{13} +(-72057594037927936 + 72057594037927936 \zeta_{6}) q^{16} +(-903508544568192599 + 903508544568192599 \zeta_{6}) q^{19} +(3067779303201412449 - 3395223619148481264 \zeta_{6}) q^{21} -37252902984619140625 \zeta_{6} q^{25} +\)\(10\!\cdots\!09\)\( q^{27} +(\)\(17\!\cdots\!76\)\( - \)\(19\!\cdots\!36\)\( \zeta_{6}) q^{28} -\)\(14\!\cdots\!67\)\( \zeta_{6} q^{31} +\)\(61\!\cdots\!16\)\( q^{36} +(\)\(12\!\cdots\!69\)\( - \)\(12\!\cdots\!69\)\( \zeta_{6}) q^{37} -\)\(32\!\cdots\!39\)\( \zeta_{6} q^{39} -\)\(64\!\cdots\!77\)\( q^{43} +\)\(34\!\cdots\!84\)\( q^{48} +(-\)\(40\!\cdots\!16\)\( + \)\(49\!\cdots\!11\)\( \zeta_{6}) q^{49} -\)\(18\!\cdots\!36\)\( \zeta_{6} q^{52} +\)\(43\!\cdots\!31\)\( q^{57} +(-\)\(18\!\cdots\!94\)\( + \)\(18\!\cdots\!94\)\( \zeta_{6}) q^{61} +(-\)\(16\!\cdots\!16\)\( + \)\(15\!\cdots\!35\)\( \zeta_{6}) q^{63} +\)\(19\!\cdots\!16\)\( q^{64} +\)\(15\!\cdots\!69\)\( \zeta_{6} q^{67} -\)\(17\!\cdots\!51\)\( \zeta_{6} q^{73} +(-\)\(17\!\cdots\!25\)\( + \)\(17\!\cdots\!25\)\( \zeta_{6}) q^{75} +\)\(24\!\cdots\!44\)\( q^{76} +(\)\(41\!\cdots\!21\)\( - \)\(41\!\cdots\!21\)\( \zeta_{6}) q^{79} -\)\(52\!\cdots\!21\)\( \zeta_{6} q^{81} +(-\)\(91\!\cdots\!84\)\( + \)\(87\!\cdots\!40\)\( \zeta_{6}) q^{84} +(\)\(47\!\cdots\!85\)\( + \)\(44\!\cdots\!51\)\( \zeta_{6}) q^{91} +(-\)\(71\!\cdots\!23\)\( + \)\(71\!\cdots\!23\)\( \zeta_{6}) q^{93} -\)\(12\!\cdots\!54\)\( q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4782969q^{3} - 268435456q^{4} + 778317387191q^{7} - 22876792454961q^{9} + O(q^{10}) \) \( 2q - 4782969q^{3} - 268435456q^{4} + 778317387191q^{7} - 22876792454961q^{9} - 1283918464548864q^{12} + 13756020664539262q^{13} - 72057594037927936q^{16} - 903508544568192599q^{19} + 2740334987254343634q^{21} - 37252902984619140625q^{25} + \)\(21\!\cdots\!18\)\(q^{27} + \)\(15\!\cdots\!16\)\(q^{28} - \)\(14\!\cdots\!67\)\(q^{31} + \)\(12\!\cdots\!32\)\(q^{36} + \)\(12\!\cdots\!69\)\(q^{37} - \)\(32\!\cdots\!39\)\(q^{39} - \)\(12\!\cdots\!54\)\(q^{43} + \)\(68\!\cdots\!68\)\(q^{48} - \)\(31\!\cdots\!21\)\(q^{49} - \)\(18\!\cdots\!36\)\(q^{52} + \)\(86\!\cdots\!62\)\(q^{57} - \)\(18\!\cdots\!94\)\(q^{61} - \)\(30\!\cdots\!97\)\(q^{63} + \)\(38\!\cdots\!32\)\(q^{64} + \)\(15\!\cdots\!69\)\(q^{67} - \)\(17\!\cdots\!51\)\(q^{73} - \)\(17\!\cdots\!25\)\(q^{75} + \)\(48\!\cdots\!88\)\(q^{76} + \)\(41\!\cdots\!21\)\(q^{79} - \)\(52\!\cdots\!21\)\(q^{81} - \)\(17\!\cdots\!28\)\(q^{84} + \)\(53\!\cdots\!21\)\(q^{91} - \)\(71\!\cdots\!23\)\(q^{93} - \)\(25\!\cdots\!08\)\(q^{97} + O(q^{100}) \)

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 −2.39148e6 4.14217e6i −1.34218e8 2.32472e8i 0 0 3.89159e11 + 5.55466e11i 0 −1.14384e13 + 1.98119e13i 0
11.1 0 −2.39148e6 + 4.14217e6i −1.34218e8 + 2.32472e8i 0 0 3.89159e11 5.55466e11i 0 −1.14384e13 1.98119e13i 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 21.29.h.a 2
3.b odd 2 1 CM 21.29.h.a 2
7.c even 3 1 inner 21.29.h.a 2
21.h odd 6 1 inner 21.29.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.29.h.a 2 1.a even 1 1 trivial
21.29.h.a 2 3.b odd 2 1 CM
21.29.h.a 2 7.c even 3 1 inner
21.29.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 \(S_{29}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 22876792454961 + 4782969 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(45\!\cdots\!01\)\( - 778317387191 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -6878010332269631 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(81\!\cdots\!01\)\( + 903508544568192599 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(22\!\cdots\!89\)\( + \)\(14\!\cdots\!67\)\( T + T^{2} \)
$37$ \( \)\(14\!\cdots\!61\)\( - \)\(12\!\cdots\!69\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( \)\(64\!\cdots\!77\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(35\!\cdots\!36\)\( + \)\(18\!\cdots\!94\)\( T + T^{2} \)
$67$ \( \)\(23\!\cdots\!61\)\( - \)\(15\!\cdots\!69\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(31\!\cdots\!01\)\( + \)\(17\!\cdots\!51\)\( T + T^{2} \)
$79$ \( \)\(17\!\cdots\!41\)\( - \)\(41\!\cdots\!21\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( \)\(12\!\cdots\!54\)\( + T )^{2} \)
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