Properties

Label 21.30.g.a
Level $21$
Weight $30$
Character orbit 21.g
Analytic conductor $111.884$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(111.883889004\)
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.

\(f(q)\) \(=\) \( q + ( 4782969 + 4782969 \zeta_{6} ) q^{3} + ( -536870912 + 536870912 \zeta_{6} ) q^{4} + ( -1992649800898 + 1488174302247 \zeta_{6} ) q^{7} + 68630377364883 \zeta_{6} q^{9} +O(q^{10})\) \( q +(4782969 + 4782969 \zeta_{6}) q^{3} +(-536870912 + 536870912 \zeta_{6}) q^{4} +(-1992649800898 + 1488174302247 \zeta_{6}) q^{7} +68630377364883 \zeta_{6} q^{9} +(-5135673858195456 + 2567836929097728 \zeta_{6}) q^{12} +(11187825231754739 - 22375650463509478 \zeta_{6}) q^{13} -288230376151711744 \zeta_{6} q^{16} +(-2616406980667028826 + 1308203490333514413 \zeta_{6}) q^{19} +(-16648673779795337505 + 4705000882936756524 \zeta_{6}) q^{21} +(\)\(18\!\cdots\!25\)\( - \)\(18\!\cdots\!25\)\( \zeta_{6}) q^{25} +(-\)\(32\!\cdots\!27\)\( + \)\(65\!\cdots\!54\)\( \zeta_{6}) q^{27} +(\)\(27\!\cdots\!12\)\( - \)\(10\!\cdots\!76\)\( \zeta_{6}) q^{28} +(-\)\(31\!\cdots\!45\)\( - \)\(31\!\cdots\!45\)\( \zeta_{6}) q^{31} -\)\(36\!\cdots\!96\)\( q^{36} -\)\(75\!\cdots\!61\)\( \zeta_{6} q^{37} +(\)\(16\!\cdots\!73\)\( - \)\(16\!\cdots\!73\)\( \zeta_{6}) q^{39} +\)\(91\!\cdots\!45\)\( q^{43} +(\)\(13\!\cdots\!36\)\( - \)\(27\!\cdots\!72\)\( \zeta_{6}) q^{48} +(\)\(17\!\cdots\!95\)\( - \)\(37\!\cdots\!03\)\( \zeta_{6}) q^{49} +(\)\(60\!\cdots\!68\)\( + \)\(60\!\cdots\!68\)\( \zeta_{6}) q^{52} -\)\(18\!\cdots\!91\)\( q^{57} +(-\)\(12\!\cdots\!72\)\( + \)\(61\!\cdots\!36\)\( \zeta_{6}) q^{61} +(-\)\(10\!\cdots\!01\)\( - \)\(34\!\cdots\!33\)\( \zeta_{6}) q^{63} +\)\(15\!\cdots\!28\)\( q^{64} +(-\)\(35\!\cdots\!81\)\( + \)\(35\!\cdots\!81\)\( \zeta_{6}) q^{67} +(\)\(11\!\cdots\!31\)\( + \)\(11\!\cdots\!31\)\( \zeta_{6}) q^{73} +(\)\(17\!\cdots\!50\)\( - \)\(89\!\cdots\!25\)\( \zeta_{6}) q^{75} +(\)\(70\!\cdots\!56\)\( - \)\(14\!\cdots\!12\)\( \zeta_{6}) q^{76} -\)\(63\!\cdots\!97\)\( \zeta_{6} q^{79} +(-\)\(47\!\cdots\!89\)\( + \)\(47\!\cdots\!89\)\( \zeta_{6}) q^{81} +(\)\(64\!\cdots\!72\)\( - \)\(89\!\cdots\!60\)\( \zeta_{6}) q^{84} +(\)\(11\!\cdots\!44\)\( + \)\(27\!\cdots\!11\)\( \zeta_{6}) q^{91} -\)\(45\!\cdots\!15\)\( \zeta_{6} q^{93} +(-\)\(44\!\cdots\!48\)\( + \)\(88\!\cdots\!96\)\( \zeta_{6}) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14348907q^{3} - 536870912q^{4} - 2497125299549q^{7} + 68630377364883q^{9} + O(q^{10}) \) \( 2q + 14348907q^{3} - 536870912q^{4} - 2497125299549q^{7} + 68630377364883q^{9} - 7703510787293184q^{12} - 288230376151711744q^{16} - 3924610471000543239q^{19} - 28592346676653918486q^{21} + \)\(18\!\cdots\!25\)\(q^{25} - \)\(52\!\cdots\!52\)\(q^{28} - \)\(94\!\cdots\!35\)\(q^{31} - \)\(73\!\cdots\!92\)\(q^{36} - \)\(75\!\cdots\!61\)\(q^{37} + \)\(16\!\cdots\!73\)\(q^{39} + \)\(18\!\cdots\!90\)\(q^{43} - \)\(20\!\cdots\!13\)\(q^{49} + \)\(18\!\cdots\!04\)\(q^{52} - \)\(37\!\cdots\!82\)\(q^{57} - \)\(18\!\cdots\!08\)\(q^{61} - \)\(23\!\cdots\!35\)\(q^{63} + \)\(30\!\cdots\!56\)\(q^{64} - \)\(35\!\cdots\!81\)\(q^{67} + \)\(34\!\cdots\!93\)\(q^{73} + \)\(26\!\cdots\!75\)\(q^{75} - \)\(63\!\cdots\!97\)\(q^{79} - \)\(47\!\cdots\!89\)\(q^{81} + \)\(38\!\cdots\!84\)\(q^{84} + \)\(49\!\cdots\!99\)\(q^{91} - \)\(45\!\cdots\!15\)\(q^{93} + O(q^{100}) \)

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 7.17445e6 4.14217e6i −2.68435e8 4.64944e8i 0 0 −1.24856e12 1.28880e12i 0 3.43152e13 5.94357e13i 0
17.1 0 7.17445e6 + 4.14217e6i −2.68435e8 + 4.64944e8i 0 0 −1.24856e12 + 1.28880e12i 0 3.43152e13 + 5.94357e13i 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 21.30.g.a 2
3.b odd 2 1 CM 21.30.g.a 2
7.d odd 6 1 inner 21.30.g.a 2
21.g even 6 1 inner 21.30.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.30.g.a 2 1.a even 1 1 trivial
21.30.g.a 2 3.b odd 2 1 CM
21.30.g.a 2 7.d odd 6 1 inner
21.30.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 \(S_{30}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 68630377364883 - 14348907 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(32\!\cdots\!07\)\( + 2497125299549 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(37\!\cdots\!63\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(51\!\cdots\!07\)\( + 3924610471000543239 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(29\!\cdots\!75\)\( + \)\(94\!\cdots\!35\)\( T + T^{2} \)
$37$ \( \)\(56\!\cdots\!21\)\( + \)\(75\!\cdots\!61\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -\)\(91\!\cdots\!45\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(11\!\cdots\!88\)\( + \)\(18\!\cdots\!08\)\( T + T^{2} \)
$67$ \( \)\(12\!\cdots\!61\)\( + \)\(35\!\cdots\!81\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(39\!\cdots\!83\)\( - \)\(34\!\cdots\!93\)\( T + T^{2} \)
$79$ \( \)\(40\!\cdots\!09\)\( + \)\(63\!\cdots\!97\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(58\!\cdots\!12\)\( + T^{2} \)
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