Properties

Label 21.24.g.a
Level $21$
Weight $24$
Character orbit 21.g
Analytic conductor $70.393$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(70.3928478425\)
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.

\(f(q)\) \(=\) \( q + ( -177147 - 177147 \zeta_{6} ) q^{3} + ( -8388608 + 8388608 \zeta_{6} ) q^{4} + ( 3927060206 + 2011692651 \zeta_{6} ) q^{7} + 94143178827 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -177147 - 177147 \zeta_{6} ) q^{3} + ( -8388608 + 8388608 \zeta_{6} ) q^{4} + ( 3927060206 + 2011692651 \zeta_{6} ) q^{7} + 94143178827 \zeta_{6} q^{9} + ( 2972033482752 - 1486016741376 \zeta_{6} ) q^{12} + ( -4830757327717 + 9661514655434 \zeta_{6} ) q^{13} -70368744177664 \zeta_{6} q^{16} + ( 999886934447166 - 499943467223583 \zeta_{6} ) q^{19} + ( -339301616265585 - 1408397570405676 \zeta_{6} ) q^{21} + ( 11920928955078125 - 11920928955078125 \zeta_{6} ) q^{25} + ( 16677181699666569 - 33354363399333138 \zeta_{6} ) q^{27} + ( -49817869726253056 + 32942568660533248 \zeta_{6} ) q^{28} + ( -162917117739084565 - 162917117739084565 \zeta_{6} ) q^{31} -789730223053602816 q^{36} + 2053267080544957427 \zeta_{6} q^{37} + ( 2567262504999250197 - 2567262504999250197 \zeta_{6} ) q^{39} + 5063598036403118305 q^{43} + ( -12465611924840644608 + 24931223849681289216 \zeta_{6} ) q^{48} + ( 11374894539461354635 + 19846983634976900013 \zeta_{6} ) q^{49} + ( -40523329565345447936 - 40523329565345447936 \zeta_{6} ) q^{52} -265690456164768173103 q^{57} + ( 714638362101378972008 - 357319181050689486004 \zeta_{6} ) q^{61} + ( -189387140988054700377 + 559093072225908158739 \zeta_{6} ) q^{63} + 590295810358705651712 q^{64} + ( -1890156096633289486793 + 1890156096633289486793 \zeta_{6} ) q^{67} + ( 2972303387757082416447 + 2972303387757082416447 \zeta_{6} ) q^{73} + ( -4223513603210449218750 + 2111756801605224609375 \zeta_{6} ) q^{75} + ( -4193829768699486142464 + 8387659537398972284928 \zeta_{6} ) q^{76} -211180652241182634413 \zeta_{6} q^{79} + ( -8862938119652501095929 + 8862938119652501095929 \zeta_{6} ) q^{81} + ( 14660763378904033394688 - 2846268252618416455680 \zeta_{6} ) q^{84} + ( -38406672896385706545236 + 47659348747973350567171 \zeta_{6} ) q^{91} + 86580835968376840308165 \zeta_{6} q^{93} + ( -63921290897278826857544 + 127842581794557653715088 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 531441q^{3} - 8388608q^{4} + 9865813063q^{7} + 94143178827q^{9} + O(q^{10}) \) \( 2q - 531441q^{3} - 8388608q^{4} + 9865813063q^{7} + 94143178827q^{9} + 4458050224128q^{12} - 70368744177664q^{16} + 1499830401670749q^{19} - 2087000802936846q^{21} + 11920928955078125q^{25} - 66693170791972864q^{28} - 488751353217253695q^{31} - 1579460446107205632q^{36} + 2053267080544957427q^{37} + 2567262504999250197q^{39} + 10127196072806236610q^{43} + 42596772713899609283q^{49} - 121569988696036343808q^{52} - 531380912329536346206q^{57} + 1071957543152068458012q^{61} + 180318790249798757985q^{63} + 1180591620717411303424q^{64} - 1890156096633289486793q^{67} + 8916910163271247249341q^{73} - 6335270404815673828125q^{75} - 211180652241182634413q^{79} - 8862938119652501095929q^{81} + 26475258505189650333696q^{84} - 29153997044798062523301q^{91} + 86580835968376840308165q^{93} + O(q^{100}) \)

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 −265720. + 153414.i −4.19430e6 7.26475e6i 0 0 4.93291e9 1.74218e9i 0 4.70716e10 8.15304e10i 0
17.1 0 −265720. 153414.i −4.19430e6 + 7.26475e6i 0 0 4.93291e9 + 1.74218e9i 0 4.70716e10 + 8.15304e10i 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 21.24.g.a 2
3.b odd 2 1 CM 21.24.g.a 2
7.d odd 6 1 inner 21.24.g.a 2
21.g even 6 1 inner 21.24.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.24.g.a 2 1.a even 1 1 trivial
21.24.g.a 2 3.b odd 2 1 CM
21.24.g.a 2 7.d odd 6 1 inner
21.24.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 \(S_{24}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 94143178827 + 531441 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 27368747340080916343 - 9865813063 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(70\!\cdots\!67\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(74\!\cdots\!67\)\( - 1499830401670749 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(79\!\cdots\!75\)\( + 488751353217253695 T + T^{2} \)
$37$ \( \)\(42\!\cdots\!29\)\( - 2053267080544957427 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -5063598036403118305 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(38\!\cdots\!48\)\( - \)\(10\!\cdots\!12\)\( T + T^{2} \)
$67$ \( \)\(35\!\cdots\!49\)\( + \)\(18\!\cdots\!93\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(26\!\cdots\!27\)\( - \)\(89\!\cdots\!41\)\( T + T^{2} \)
$79$ \( \)\(44\!\cdots\!69\)\( + \)\(21\!\cdots\!13\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(12\!\cdots\!08\)\( + T^{2} \)
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