Properties

Label 21.23.h.a
Level $21$
Weight $23$
Character orbit 21.h
Analytic conductor $64.409$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 23 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.4085613168\)
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.

\(f(q)\) \(=\) \( q + 177147 \zeta_{6} q^{3} -4194304 \zeta_{6} q^{4} + ( 1984171195 - 1970410467 \zeta_{6} ) q^{7} + ( -31381059609 + 31381059609 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + 177147 \zeta_{6} q^{3} -4194304 \zeta_{6} q^{4} + ( 1984171195 - 1970410467 \zeta_{6} ) q^{7} + ( -31381059609 + 31381059609 \zeta_{6} ) q^{9} + ( 743008370688 - 743008370688 \zeta_{6} ) q^{12} + 2094315786047 q^{13} + ( -17592186044416 + 17592186044416 \zeta_{6} ) q^{16} + ( -20673935603651 + 20673935603651 \zeta_{6} ) q^{19} + ( 349052302997649 + 2437671683016 \zeta_{6} ) q^{21} -2384185791015625 \zeta_{6} q^{25} -5559060566555523 q^{27} + ( -8264500503379968 - 57716676493312 \zeta_{6} ) q^{28} -37081623937815587 \zeta_{6} q^{31} + 131621703842267136 q^{36} + ( -84456886698164303 + 84456886698164303 \zeta_{6} ) q^{37} + 371001758550867909 \zeta_{6} q^{39} + 1268503014678232811 q^{43} -3116402981210161152 q^{48} + ( 54417922604569936 - 3936745973432638041 \zeta_{6} ) q^{49} -8784197078680076288 \zeta_{6} q^{52} -3662325670379963697 q^{57} + ( 67394602624928417446 - 67394602624928417446 \zeta_{6} ) q^{61} + ( -431826225631235352 + 62265394544755762755 \zeta_{6} ) q^{63} + 73786976294838206464 q^{64} -214129669585561202603 \zeta_{6} q^{67} -621369020009512258607 \zeta_{6} q^{73} + ( 422351360321044921875 - 422351360321044921875 \zeta_{6} ) q^{75} + 86712770798135803904 q^{76} + ( 1037582729766213568069 - 1037582729766213568069 \zeta_{6} ) q^{79} -984770902183611232881 \zeta_{6} q^{81} + ( 10224336090760740864 - 1474255806763011932160 \zeta_{6} ) q^{84} + ( 4155481055908240316165 - 4126661746030341353949 \zeta_{6} ) q^{91} + ( 6568898435712217790289 - 6568898435712217790289 \zeta_{6} ) q^{93} -1618886660331792534142 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 177147q^{3} - 4194304q^{4} + 1997931923q^{7} - 31381059609q^{9} + O(q^{10}) \) \( 2q + 177147q^{3} - 4194304q^{4} + 1997931923q^{7} - 31381059609q^{9} + 743008370688q^{12} + 4188631572094q^{13} - 17592186044416q^{16} - 20673935603651q^{19} + 700542277678314q^{21} - 2384185791015625q^{25} - 11118121133111046q^{27} - 16586717683253248q^{28} - 37081623937815587q^{31} + 263243407684534272q^{36} - 84456886698164303q^{37} + 371001758550867909q^{39} + 2537006029356465622q^{43} - 6232805962420322304q^{48} - 3827910128223498169q^{49} - 8784197078680076288q^{52} - 7324651340759927394q^{57} + 67394602624928417446q^{61} + 61401742093493292051q^{63} + 147573952589676412928q^{64} - 214129669585561202603q^{67} - 621369020009512258607q^{73} + 422351360321044921875q^{75} + 173425541596271607808q^{76} + 1037582729766213568069q^{79} - 984770902183611232881q^{81} - 1453807134581490450432q^{84} + 4184300365786139278381q^{91} + 6568898435712217790289q^{93} - 3237773320663585068284q^{97} + O(q^{100}) \)

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 88573.5 + 153414.i −2.09715e6 3.63237e6i 0 0 9.98966e8 1.70643e9i 0 −1.56905e10 + 2.71768e10i 0
11.1 0 88573.5 153414.i −2.09715e6 + 3.63237e6i 0 0 9.98966e8 + 1.70643e9i 0 −1.56905e10 2.71768e10i 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 21.23.h.a 2
3.b odd 2 1 CM 21.23.h.a 2
7.c even 3 1 inner 21.23.h.a 2
21.h odd 6 1 inner 21.23.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.23.h.a 2 1.a even 1 1 trivial
21.23.h.a 2 3.b odd 2 1 CM
21.23.h.a 2 7.c even 3 1 inner
21.23.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 \(S_{23}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 31381059609 - 177147 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 3909821048582988049 - 1997931923 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -2094315786047 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(42\!\cdots\!01\)\( + 20673935603651 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(13\!\cdots\!69\)\( + 37081623937815587 T + T^{2} \)
$37$ \( \)\(71\!\cdots\!09\)\( + 84456886698164303 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -1268503014678232811 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(45\!\cdots\!16\)\( - 67394602624928417446 T + T^{2} \)
$67$ \( \)\(45\!\cdots\!09\)\( + \)\(21\!\cdots\!03\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(38\!\cdots\!49\)\( + \)\(62\!\cdots\!07\)\( T + T^{2} \)
$79$ \( \)\(10\!\cdots\!61\)\( - \)\(10\!\cdots\!69\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( \)\(16\!\cdots\!42\)\( + T )^{2} \)
show more
show less