Properties

Label 21.21.h.a
Level $21$
Weight $21$
Character orbit 21.h
Analytic conductor $53.238$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(53.2378906716\)
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.

\(f(q)\) \(=\) \( q -59049 \zeta_{6} q^{3} -1048576 \zeta_{6} q^{4} + ( -122884025 - 200219799 \zeta_{6} ) q^{7} + ( -3486784401 + 3486784401 \zeta_{6} ) q^{9} +O(q^{10})\) \( q -59049 \zeta_{6} q^{3} -1048576 \zeta_{6} q^{4} + ( -122884025 - 200219799 \zeta_{6} ) q^{7} + ( -3486784401 + 3486784401 \zeta_{6} ) q^{9} + ( -61917364224 + 61917364224 \zeta_{6} ) q^{12} + 38170922351 q^{13} + ( -1099511627776 + 1099511627776 \zeta_{6} ) q^{16} + ( -12005816399399 + 12005816399399 \zeta_{6} ) q^{19} + ( -11822778911151 + 19078957703376 \zeta_{6} ) q^{21} -95367431640625 \zeta_{6} q^{25} + 205891132094649 q^{27} + ( -209945675956224 + 338798915354624 \zeta_{6} ) q^{28} -845125195444727 \zeta_{6} q^{31} + 3656158440062976 q^{36} + ( 721096304301649 - 721096304301649 \zeta_{6} ) q^{37} -2253954793904199 \zeta_{6} q^{39} + 1343935055601623 q^{43} + 64925062108545024 q^{48} + ( -24987484311399776 + 89295597483222351 \zeta_{6} ) q^{49} -40025113075122176 \zeta_{6} q^{52} + 708931452568111551 q^{57} + ( 1387788230779990126 - 1387788230779990126 \zeta_{6} ) q^{61} + ( 1126593373426649424 - 428470101502094025 \zeta_{6} ) q^{63} + 1152921504606846976 q^{64} + 2055850947949627849 \zeta_{6} q^{67} + 4762721323358202049 \zeta_{6} q^{73} + ( -5631351470947265625 + 5631351470947265625 \zeta_{6} ) q^{75} + 12589010936816205824 q^{76} + ( 14522848453745208601 - 14522848453745208601 \zeta_{6} ) q^{79} -12157665459056928801 \zeta_{6} q^{81} + ( 20005737152775192576 - 7608654933236121600 \zeta_{6} ) q^{84} + ( -4690596576453342775 - 7642574400761827449 \zeta_{6} ) q^{91} + ( -49903797665815684623 + 49903797665815684623 \zeta_{6} ) q^{93} -146701748198870395774 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 59049q^{3} - 1048576q^{4} - 445987849q^{7} - 3486784401q^{9} + O(q^{10}) \) \( 2q - 59049q^{3} - 1048576q^{4} - 445987849q^{7} - 3486784401q^{9} - 61917364224q^{12} + 76341844702q^{13} - 1099511627776q^{16} - 12005816399399q^{19} - 4566600118926q^{21} - 95367431640625q^{25} + 411782264189298q^{27} - 81092436557824q^{28} - 845125195444727q^{31} + 7312316880125952q^{36} + 721096304301649q^{37} - 2253954793904199q^{39} + 2687870111203246q^{43} + 129850124217090048q^{48} + 39320628860422799q^{49} - 40025113075122176q^{52} + 1417862905136223102q^{57} + 1387788230779990126q^{61} + 1824716645351204823q^{63} + 2305843009213693952q^{64} + 2055850947949627849q^{67} + 4762721323358202049q^{73} - 5631351470947265625q^{75} + 25178021873632411648q^{76} + 14522848453745208601q^{79} - 12157665459056928801q^{81} + 32402819372314263552q^{84} - 17023767553668512999q^{91} - 49903797665815684623q^{93} - 293403496397740791548q^{97} + O(q^{100}) \)

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 −29524.5 51137.9i −524288. 908093.i 0 0 −2.22994e8 1.73395e8i 0 −1.74339e9 + 3.01964e9i 0
11.1 0 −29524.5 + 51137.9i −524288. + 908093.i 0 0 −2.22994e8 + 1.73395e8i 0 −1.74339e9 3.01964e9i 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 21.21.h.a 2
3.b odd 2 1 CM 21.21.h.a 2
7.c even 3 1 inner 21.21.h.a 2
21.h odd 6 1 inner 21.21.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.21.h.a 2 1.a even 1 1 trivial
21.21.h.a 2 3.b odd 2 1 CM
21.21.h.a 2 7.c even 3 1 inner
21.21.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 \(S_{21}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3486784401 + 59049 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 79792266297612001 + 445987849 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -38170922351 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(14\!\cdots\!01\)\( + 12005816399399 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(71\!\cdots\!29\)\( + 845125195444727 T + T^{2} \)
$37$ \( \)\(51\!\cdots\!01\)\( - 721096304301649 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -1343935055601623 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(19\!\cdots\!76\)\( - 1387788230779990126 T + T^{2} \)
$67$ \( \)\(42\!\cdots\!01\)\( - 2055850947949627849 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(22\!\cdots\!01\)\( - 4762721323358202049 T + T^{2} \)
$79$ \( \)\(21\!\cdots\!01\)\( - 14522848453745208601 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( \)\(14\!\cdots\!74\)\( + T )^{2} \)
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