Properties

Label 21.11.h.a
Level $21$
Weight $11$
Character orbit 21.h
Analytic conductor $13.343$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.3425023061\)
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.

\(f(q)\) \(=\) \( q + 243 \zeta_{6} q^{3} -1024 \zeta_{6} q^{4} + ( -3725 + 18357 \zeta_{6} ) q^{7} + ( -59049 + 59049 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + 243 \zeta_{6} q^{3} -1024 \zeta_{6} q^{4} + ( -3725 + 18357 \zeta_{6} ) q^{7} + ( -59049 + 59049 \zeta_{6} ) q^{9} + ( 248832 - 248832 \zeta_{6} ) q^{12} -560257 q^{13} + ( -1048576 + 1048576 \zeta_{6} ) q^{16} + ( -4926251 + 4926251 \zeta_{6} ) q^{19} + ( -4460751 + 3555576 \zeta_{6} ) q^{21} -9765625 \zeta_{6} q^{25} -14348907 q^{27} + ( 18797568 - 14983168 \zeta_{6} ) q^{28} + 49843573 \zeta_{6} q^{31} + 60466176 q^{36} + ( 94318993 - 94318993 \zeta_{6} ) q^{37} -136142451 \zeta_{6} q^{39} + 211108739 q^{43} -254803968 q^{48} + ( -323103824 + 200219799 \zeta_{6} ) q^{49} + 573703168 \zeta_{6} q^{52} -1197078993 q^{57} + ( 197224726 - 197224726 \zeta_{6} ) q^{61} + ( -864004968 - 219957525 \zeta_{6} ) q^{63} + 1073741824 q^{64} + 1260882493 \zeta_{6} q^{67} -1957684943 \zeta_{6} q^{73} + ( 2373046875 - 2373046875 \zeta_{6} ) q^{75} + 5044481024 q^{76} + ( -2100881651 + 2100881651 \zeta_{6} ) q^{79} -3486784401 \zeta_{6} q^{81} + ( 3640909824 + 926899200 \zeta_{6} ) q^{84} + ( 2086957325 - 10284637749 \zeta_{6} ) q^{91} + ( -12111988239 + 12111988239 \zeta_{6} ) q^{93} + 884916482 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 243q^{3} - 1024q^{4} + 10907q^{7} - 59049q^{9} + O(q^{10}) \) \( 2q + 243q^{3} - 1024q^{4} + 10907q^{7} - 59049q^{9} + 248832q^{12} - 1120514q^{13} - 1048576q^{16} - 4926251q^{19} - 5365926q^{21} - 9765625q^{25} - 28697814q^{27} + 22611968q^{28} + 49843573q^{31} + 120932352q^{36} + 94318993q^{37} - 136142451q^{39} + 422217478q^{43} - 509607936q^{48} - 445987849q^{49} + 573703168q^{52} - 2394157986q^{57} + 197224726q^{61} - 1947967461q^{63} + 2147483648q^{64} + 1260882493q^{67} - 1957684943q^{73} + 2373046875q^{75} + 10088962048q^{76} - 2100881651q^{79} - 3486784401q^{81} + 8208718848q^{84} - 6110723099q^{91} - 12111988239q^{93} + 1769832964q^{97} + O(q^{100}) \)

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
0 121.500 + 210.444i −512.000 886.810i 0 0 5453.50 + 15897.6i 0 −29524.5 + 51137.9i 0
11.1 0 121.500 210.444i −512.000 + 886.810i 0 0 5453.50 15897.6i 0 −29524.5 51137.9i 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 21.11.h.a 2
3.b odd 2 1 CM 21.11.h.a 2
7.c even 3 1 inner 21.11.h.a 2
21.h odd 6 1 inner 21.11.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.11.h.a 2 1.a even 1 1 trivial
21.11.h.a 2 3.b odd 2 1 CM
21.11.h.a 2 7.c even 3 1 inner
21.11.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 \(S_{11}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 59049 - 243 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 282475249 - 10907 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 560257 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 24267948915001 + 4926251 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 2484381769406329 - 49843573 T + T^{2} \)
$37$ \( 8896072440534049 - 94318993 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -211108739 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 38897592545775076 - 197224726 T + T^{2} \)
$67$ \( 1589824661153895049 - 1260882493 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 3832530336048913249 + 1957684943 T + T^{2} \)
$79$ \( 4413703711508485801 + 2100881651 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -884916482 + T )^{2} \)
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