Properties

Label 21.39.h.a
Level $21$
Weight $39$
Character orbit 21.h
Analytic conductor $192.073$
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: \( k \) \(=\) \( 39 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(192.073284970\)
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.

\(f(q)\) \(=\) \( q + 1162261467 \zeta_{6} q^{3} -274877906944 \zeta_{6} q^{4} + ( -1387488121601405 - 10641641639802267 \zeta_{6} ) q^{7} + ( -1350851717672992089 + 1350851717672992089 \zeta_{6} ) q^{9} +O(q^{10})\) \( q +1162261467 \zeta_{6} q^{3} -274877906944 \zeta_{6} q^{4} +(-1387488121601405 - 10641641639802267 \zeta_{6}) q^{7} +(-1350851717672992089 + 1350851717672992089 \zeta_{6}) q^{9} +(\)\(31\!\cdots\!48\)\( - \)\(31\!\cdots\!48\)\( \zeta_{6}) q^{12} -\)\(28\!\cdots\!13\)\( q^{13} +(-\)\(75\!\cdots\!36\)\( + \)\(75\!\cdots\!36\)\( \zeta_{6}) q^{16} +(-\)\(25\!\cdots\!91\)\( + \)\(25\!\cdots\!91\)\( \zeta_{6}) q^{19} +(\)\(12\!\cdots\!89\)\( - \)\(13\!\cdots\!24\)\( \zeta_{6}) q^{21} -\)\(36\!\cdots\!25\)\( \zeta_{6} q^{25} -\)\(15\!\cdots\!63\)\( q^{27} +(-\)\(29\!\cdots\!48\)\( + \)\(33\!\cdots\!68\)\( \zeta_{6}) q^{28} +\)\(33\!\cdots\!33\)\( \zeta_{6} q^{31} +\)\(37\!\cdots\!16\)\( q^{36} +(-\)\(12\!\cdots\!63\)\( + \)\(12\!\cdots\!63\)\( \zeta_{6}) q^{37} -\)\(33\!\cdots\!71\)\( \zeta_{6} q^{39} +\)\(14\!\cdots\!11\)\( q^{43} -\)\(87\!\cdots\!12\)\( q^{48} +(-\)\(11\!\cdots\!64\)\( + \)\(14\!\cdots\!59\)\( \zeta_{6}) q^{49} +\)\(78\!\cdots\!72\)\( \zeta_{6} q^{52} -\)\(29\!\cdots\!97\)\( q^{57} +(-\)\(30\!\cdots\!74\)\( + \)\(30\!\cdots\!74\)\( \zeta_{6}) q^{61} +(\)\(16\!\cdots\!08\)\( - \)\(18\!\cdots\!45\)\( \zeta_{6}) q^{63} +\)\(20\!\cdots\!84\)\( q^{64} -\)\(95\!\cdots\!23\)\( \zeta_{6} q^{67} -\)\(47\!\cdots\!67\)\( \zeta_{6} q^{73} +(\)\(42\!\cdots\!75\)\( - \)\(42\!\cdots\!75\)\( \zeta_{6}) q^{75} +\)\(70\!\cdots\!04\)\( q^{76} +(\)\(22\!\cdots\!09\)\( - \)\(22\!\cdots\!09\)\( \zeta_{6}) q^{79} -\)\(18\!\cdots\!21\)\( \zeta_{6} q^{81} +(-\)\(38\!\cdots\!56\)\( + \)\(44\!\cdots\!40\)\( \zeta_{6}) q^{84} +(\)\(39\!\cdots\!65\)\( + \)\(30\!\cdots\!71\)\( \zeta_{6}) q^{91} +(-\)\(38\!\cdots\!11\)\( + \)\(38\!\cdots\!11\)\( \zeta_{6}) q^{93} -\)\(21\!\cdots\!62\)\( q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 1162261467q^{3} - 274877906944q^{4} - 13416617883005077q^{7} - 1350851717672992089q^{9} + O(q^{10}) \) \( 2q + 1162261467q^{3} - 274877906944q^{4} - 13416617883005077q^{7} - 1350851717672992089q^{9} + \)\(31\!\cdots\!48\)\(q^{12} - \)\(57\!\cdots\!26\)\(q^{13} - \)\(75\!\cdots\!36\)\(q^{16} - \)\(25\!\cdots\!91\)\(q^{19} + \)\(10\!\cdots\!54\)\(q^{21} - \)\(36\!\cdots\!25\)\(q^{25} - \)\(31\!\cdots\!26\)\(q^{27} - \)\(25\!\cdots\!28\)\(q^{28} + \)\(33\!\cdots\!33\)\(q^{31} + \)\(74\!\cdots\!32\)\(q^{36} - \)\(12\!\cdots\!63\)\(q^{37} - \)\(33\!\cdots\!71\)\(q^{39} + \)\(28\!\cdots\!22\)\(q^{43} - \)\(17\!\cdots\!24\)\(q^{48} - \)\(79\!\cdots\!69\)\(q^{49} + \)\(78\!\cdots\!72\)\(q^{52} - \)\(59\!\cdots\!94\)\(q^{57} - \)\(30\!\cdots\!74\)\(q^{61} + \)\(30\!\cdots\!71\)\(q^{63} + \)\(41\!\cdots\!68\)\(q^{64} - \)\(95\!\cdots\!23\)\(q^{67} - \)\(47\!\cdots\!67\)\(q^{73} + \)\(42\!\cdots\!75\)\(q^{75} + \)\(14\!\cdots\!08\)\(q^{76} + \)\(22\!\cdots\!09\)\(q^{79} - \)\(18\!\cdots\!21\)\(q^{81} - \)\(72\!\cdots\!72\)\(q^{84} + \)\(38\!\cdots\!01\)\(q^{91} - \)\(38\!\cdots\!11\)\(q^{93} - \)\(43\!\cdots\!24\)\(q^{97} + O(q^{100}) \)

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 5.81131e8 + 1.00655e9i −1.37439e11 2.38051e11i 0 0 −6.70831e15 9.21593e15i 0 −6.75426e17 + 1.16987e18i 0
11.1 0 5.81131e8 1.00655e9i −1.37439e11 + 2.38051e11i 0 0 −6.70831e15 + 9.21593e15i 0 −6.75426e17 1.16987e18i 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 21.39.h.a 2
3.b odd 2 1 CM 21.39.h.a 2
7.c even 3 1 inner 21.39.h.a 2
21.h odd 6 1 inner 21.39.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.39.h.a 2 1.a even 1 1 trivial
21.39.h.a 2 3.b odd 2 1 CM
21.39.h.a 2 7.c even 3 1 inner
21.39.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 \(S_{39}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1350851717672992089 - 1162261467 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(12\!\cdots\!49\)\( + 13416617883005077 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( \)\(28\!\cdots\!13\)\( + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(65\!\cdots\!81\)\( + \)\(25\!\cdots\!91\)\( T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(11\!\cdots\!89\)\( - \)\(33\!\cdots\!33\)\( T + T^{2} \)
$37$ \( \)\(15\!\cdots\!69\)\( + \)\(12\!\cdots\!63\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -\)\(14\!\cdots\!11\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(92\!\cdots\!76\)\( + \)\(30\!\cdots\!74\)\( T + T^{2} \)
$67$ \( \)\(91\!\cdots\!29\)\( + \)\(95\!\cdots\!23\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(22\!\cdots\!89\)\( + \)\(47\!\cdots\!67\)\( T + T^{2} \)
$79$ \( \)\(48\!\cdots\!81\)\( - \)\(22\!\cdots\!09\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( \)\(21\!\cdots\!62\)\( + T )^{2} \)
show more
show less