Properties

Label 21.15.h.a
Level $21$
Weight $15$
Character orbit 21.h
Analytic conductor $26.109$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.1090833119\)
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.

\(f(q)\) \(=\) \( q + 2187 \zeta_{6} q^{3} -16384 \zeta_{6} q^{4} + ( 950035 - 438987 \zeta_{6} ) q^{7} + ( -4782969 + 4782969 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + 2187 \zeta_{6} q^{3} -16384 \zeta_{6} q^{4} + ( 950035 - 438987 \zeta_{6} ) q^{7} + ( -4782969 + 4782969 \zeta_{6} ) q^{9} + ( 35831808 - 35831808 \zeta_{6} ) q^{12} -121460953 q^{13} + ( -268435456 + 268435456 \zeta_{6} ) q^{16} + ( 1581620029 - 1581620029 \zeta_{6} ) q^{19} + ( 960064569 + 1117661976 \zeta_{6} ) q^{21} -6103515625 \zeta_{6} q^{25} -10460353203 q^{27} + ( -7192363008 - 8373010432 \zeta_{6} ) q^{28} -54792115547 \zeta_{6} q^{31} + 78364164096 q^{36} + ( 77194104697 - 77194104697 \zeta_{6} ) q^{37} -265635104211 \zeta_{6} q^{39} -375951067189 q^{43} -587068342272 q^{48} + ( 709856915056 - 641396442921 \zeta_{6} ) q^{49} + 1990016253952 \zeta_{6} q^{52} + 3459003003423 q^{57} + ( -6214599846074 + 6214599846074 \zeta_{6} ) q^{61} + ( -2444326741512 + 4543987953915 \zeta_{6} ) q^{63} + 4398046511104 q^{64} -7623508989083 \zeta_{6} q^{67} + 20565100535713 \zeta_{6} q^{73} + ( 13348388671875 - 13348388671875 \zeta_{6} ) q^{75} -25913262555136 q^{76} + ( 17998954613629 - 17998954613629 \zeta_{6} ) q^{79} -22876792454961 \zeta_{6} q^{81} + ( 18311773814784 - 34041471713280 \zeta_{6} ) q^{84} + ( -115392156483355 + 53319779374611 \zeta_{6} ) q^{91} + ( 119830356701289 - 119830356701289 \zeta_{6} ) q^{93} -11651694897022 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2187q^{3} - 16384q^{4} + 1461083q^{7} - 4782969q^{9} + O(q^{10}) \) \( 2q + 2187q^{3} - 16384q^{4} + 1461083q^{7} - 4782969q^{9} + 35831808q^{12} - 242921906q^{13} - 268435456q^{16} + 1581620029q^{19} + 3037791114q^{21} - 6103515625q^{25} - 20920706406q^{27} - 22757736448q^{28} - 54792115547q^{31} + 156728328192q^{36} + 77194104697q^{37} - 265635104211q^{39} - 751902134378q^{43} - 1174136684544q^{48} + 778317387191q^{49} + 1990016253952q^{52} + 6918006006846q^{57} - 6214599846074q^{61} - 344665529109q^{63} + 8796093022208q^{64} - 7623508989083q^{67} + 20565100535713q^{73} + 13348388671875q^{75} - 51826525110272q^{76} + 17998954613629q^{79} - 22876792454961q^{81} + 2582075916288q^{84} - 177464533592099q^{91} + 119830356701289q^{93} - 23303389794044q^{97} + O(q^{100}) \)

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 1093.50 + 1894.00i −8192.00 14189.0i 0 0 730542. 380174.i 0 −2.39148e6 + 4.14217e6i 0
11.1 0 1093.50 1894.00i −8192.00 + 14189.0i 0 0 730542. + 380174.i 0 −2.39148e6 4.14217e6i 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 21.15.h.a 2
3.b odd 2 1 CM 21.15.h.a 2
7.c even 3 1 inner 21.15.h.a 2
21.h odd 6 1 inner 21.15.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.15.h.a 2 1.a even 1 1 trivial
21.15.h.a 2 3.b odd 2 1 CM
21.15.h.a 2 7.c even 3 1 inner
21.15.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 \(S_{15}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4782969 - 2187 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 678223072849 - 1461083 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 121460953 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 2501521916133960841 - 1581620029 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(30\!\cdots\!09\)\( + 54792115547 T + T^{2} \)
$37$ \( \)\(59\!\cdots\!09\)\( - 77194104697 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 375951067189 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(38\!\cdots\!76\)\( + 6214599846074 T + T^{2} \)
$67$ \( \)\(58\!\cdots\!89\)\( + 7623508989083 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(42\!\cdots\!69\)\( - 20565100535713 T + T^{2} \)
$79$ \( \)\(32\!\cdots\!41\)\( - 17998954613629 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 11651694897022 + T )^{2} \)
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