Properties

Label 21.42.g.a
Level $21$
Weight $42$
Character orbit 21.g
Analytic conductor $223.591$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(223.590507958\)
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.

\(f(q)\) \(=\) \( q + ( 3486784401 + 3486784401 \zeta_{6} ) q^{3} + ( -2199023255552 + 2199023255552 \zeta_{6} ) q^{4} + ( 203578679277844478 + 14333144549023023 \zeta_{6} ) q^{7} + 36472996377170786403 \zeta_{6} q^{9} +O(q^{10})\) \( q +(3486784401 + 3486784401 \zeta_{6}) q^{3} +(-2199023255552 + 2199023255552 \zeta_{6}) q^{4} +(203578679277844478 + 14333144549023023 \zeta_{6}) q^{7} +36472996377170786403 \zeta_{6} q^{9} +(-\)\(15\!\cdots\!04\)\( + \)\(76\!\cdots\!52\)\( \zeta_{6}) q^{12} +(-\)\(39\!\cdots\!29\)\( + \)\(78\!\cdots\!58\)\( \zeta_{6}) q^{13} -\)\(48\!\cdots\!04\)\( \zeta_{6} q^{16} +(\)\(32\!\cdots\!54\)\( - \)\(16\!\cdots\!27\)\( \zeta_{6}) q^{19} +(\)\(65\!\cdots\!55\)\( + \)\(80\!\cdots\!24\)\( \zeta_{6}) q^{21} +(\)\(45\!\cdots\!25\)\( - \)\(45\!\cdots\!25\)\( \zeta_{6}) q^{25} +(-\)\(12\!\cdots\!03\)\( + \)\(25\!\cdots\!06\)\( \zeta_{6}) q^{27} +(-\)\(47\!\cdots\!52\)\( + \)\(44\!\cdots\!56\)\( \zeta_{6}) q^{28} +(\)\(39\!\cdots\!95\)\( + \)\(39\!\cdots\!95\)\( \zeta_{6}) q^{31} -\)\(80\!\cdots\!56\)\( q^{36} +\)\(64\!\cdots\!11\)\( \zeta_{6} q^{37} +(-\)\(41\!\cdots\!87\)\( + \)\(41\!\cdots\!87\)\( \zeta_{6}) q^{39} +\)\(19\!\cdots\!45\)\( q^{43} +(\)\(16\!\cdots\!04\)\( - \)\(33\!\cdots\!08\)\( \zeta_{6}) q^{48} +(\)\(41\!\cdots\!55\)\( + \)\(60\!\cdots\!17\)\( \zeta_{6}) q^{49} +(-\)\(86\!\cdots\!08\)\( - \)\(86\!\cdots\!08\)\( \zeta_{6}) q^{52} +\)\(17\!\cdots\!81\)\( q^{57} +(\)\(78\!\cdots\!68\)\( - \)\(39\!\cdots\!84\)\( \zeta_{6}) q^{61} +(-\)\(52\!\cdots\!69\)\( + \)\(79\!\cdots\!03\)\( \zeta_{6}) q^{63} +\)\(10\!\cdots\!08\)\( q^{64} +(\)\(50\!\cdots\!91\)\( - \)\(50\!\cdots\!91\)\( \zeta_{6}) q^{67} +(\)\(47\!\cdots\!99\)\( + \)\(47\!\cdots\!99\)\( \zeta_{6}) q^{73} +(\)\(31\!\cdots\!50\)\( - \)\(15\!\cdots\!25\)\( \zeta_{6}) q^{75} +(-\)\(36\!\cdots\!04\)\( + \)\(72\!\cdots\!08\)\( \zeta_{6}) q^{76} +\)\(12\!\cdots\!83\)\( \zeta_{6} q^{79} +(-\)\(13\!\cdots\!09\)\( + \)\(13\!\cdots\!09\)\( \zeta_{6}) q^{81} +(-\)\(32\!\cdots\!08\)\( + \)\(14\!\cdots\!60\)\( \zeta_{6}) q^{84} +(-\)\(91\!\cdots\!96\)\( + \)\(16\!\cdots\!91\)\( \zeta_{6}) q^{91} +\)\(41\!\cdots\!85\)\( \zeta_{6} q^{93} +(\)\(33\!\cdots\!28\)\( - \)\(67\!\cdots\!56\)\( \zeta_{6}) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10460353203q^{3} - 2199023255552q^{4} + 421490503104711979q^{7} + 36472996377170786403q^{9} + O(q^{10}) \) \( 2q + 10460353203q^{3} - 2199023255552q^{4} + 421490503104711979q^{7} + 36472996377170786403q^{9} - \)\(23\!\cdots\!56\)\(q^{12} - \)\(48\!\cdots\!04\)\(q^{16} + \)\(49\!\cdots\!81\)\(q^{19} + \)\(21\!\cdots\!34\)\(q^{21} + \)\(45\!\cdots\!25\)\(q^{25} - \)\(51\!\cdots\!48\)\(q^{28} + \)\(11\!\cdots\!85\)\(q^{31} - \)\(16\!\cdots\!12\)\(q^{36} + \)\(64\!\cdots\!11\)\(q^{37} - \)\(41\!\cdots\!87\)\(q^{39} + \)\(39\!\cdots\!90\)\(q^{43} + \)\(88\!\cdots\!27\)\(q^{49} - \)\(25\!\cdots\!24\)\(q^{52} + \)\(34\!\cdots\!62\)\(q^{57} + \)\(11\!\cdots\!52\)\(q^{61} + \)\(69\!\cdots\!65\)\(q^{63} + \)\(21\!\cdots\!16\)\(q^{64} + \)\(50\!\cdots\!91\)\(q^{67} + \)\(14\!\cdots\!97\)\(q^{73} + \)\(47\!\cdots\!75\)\(q^{75} + \)\(12\!\cdots\!83\)\(q^{79} - \)\(13\!\cdots\!09\)\(q^{81} - \)\(50\!\cdots\!56\)\(q^{84} - \)\(16\!\cdots\!01\)\(q^{91} + \)\(41\!\cdots\!85\)\(q^{93} + O(q^{100}) \)

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 5.23018e9 3.01964e9i −1.09951e12 1.90441e12i 0 0 2.10745e17 1.24129e16i 0 1.82365e19 3.15865e19i 0
17.1 0 5.23018e9 + 3.01964e9i −1.09951e12 + 1.90441e12i 0 0 2.10745e17 + 1.24129e16i 0 1.82365e19 + 3.15865e19i 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 21.42.g.a 2
3.b odd 2 1 CM 21.42.g.a 2
7.d odd 6 1 inner 21.42.g.a 2
21.g even 6 1 inner 21.42.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.42.g.a 2 1.a even 1 1 trivial
21.42.g.a 2 3.b odd 2 1 CM
21.42.g.a 2 7.d odd 6 1 inner
21.42.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 \(S_{42}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 36472996377170786403 - 10460353203 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(44\!\cdots\!07\)\( - 421490503104711979 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(46\!\cdots\!23\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(80\!\cdots\!87\)\( - \)\(49\!\cdots\!81\)\( T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(47\!\cdots\!75\)\( - \)\(11\!\cdots\!85\)\( T + T^{2} \)
$37$ \( \)\(41\!\cdots\!21\)\( - \)\(64\!\cdots\!11\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -\)\(19\!\cdots\!45\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(46\!\cdots\!68\)\( - \)\(11\!\cdots\!52\)\( T + T^{2} \)
$67$ \( \)\(25\!\cdots\!81\)\( - \)\(50\!\cdots\!91\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(66\!\cdots\!03\)\( - \)\(14\!\cdots\!97\)\( T + T^{2} \)
$79$ \( \)\(16\!\cdots\!89\)\( - \)\(12\!\cdots\!83\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(33\!\cdots\!52\)\( + T^{2} \)
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