Properties

Label 21.41.h.a
Level $21$
Weight $41$
Character orbit 21.h
Analytic conductor $212.819$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 41 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(212.818798913\)
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.

\(f(q)\) \(=\) \( q -3486784401 \zeta_{6} q^{3} -1099511627776 \zeta_{6} q^{4} + ( 64308113171822575 - 89295597483222351 \zeta_{6} ) q^{7} + ( -12157665459056928801 + 12157665459056928801 \zeta_{6} ) q^{9} +O(q^{10})\) \( q -3486784401 \zeta_{6} q^{3} -1099511627776 \zeta_{6} q^{4} +(64308113171822575 - 89295597483222351 \zeta_{6}) q^{7} +(-12157665459056928801 + 12157665459056928801 \zeta_{6}) q^{9} +(-\)\(38\!\cdots\!76\)\( + \)\(38\!\cdots\!76\)\( \zeta_{6}) q^{12} -\)\(36\!\cdots\!01\)\( q^{13} +(-\)\(12\!\cdots\!76\)\( + \)\(12\!\cdots\!76\)\( \zeta_{6}) q^{16} +(-\)\(68\!\cdots\!99\)\( + \)\(68\!\cdots\!99\)\( \zeta_{6}) q^{19} +(-\)\(31\!\cdots\!51\)\( + \)\(87\!\cdots\!76\)\( \zeta_{6}) q^{21} -\)\(90\!\cdots\!25\)\( \zeta_{6} q^{25} +\)\(42\!\cdots\!01\)\( q^{27} +(-\)\(98\!\cdots\!76\)\( + \)\(27\!\cdots\!76\)\( \zeta_{6}) q^{28} +\)\(62\!\cdots\!73\)\( \zeta_{6} q^{31} +\)\(13\!\cdots\!76\)\( q^{36} +(\)\(45\!\cdots\!01\)\( - \)\(45\!\cdots\!01\)\( \zeta_{6}) q^{37} +\)\(12\!\cdots\!01\)\( \zeta_{6} q^{39} -\)\(93\!\cdots\!73\)\( q^{43} +\)\(42\!\cdots\!76\)\( q^{48} +(-\)\(38\!\cdots\!76\)\( - \)\(35\!\cdots\!49\)\( \zeta_{6}) q^{49} +\)\(40\!\cdots\!76\)\( \zeta_{6} q^{52} +\)\(24\!\cdots\!99\)\( q^{57} +(-\)\(90\!\cdots\!74\)\( + \)\(90\!\cdots\!74\)\( \zeta_{6}) q^{61} +(\)\(30\!\cdots\!76\)\( + \)\(78\!\cdots\!75\)\( \zeta_{6}) q^{63} +\)\(13\!\cdots\!76\)\( q^{64} +\)\(24\!\cdots\!01\)\( \zeta_{6} q^{67} +\)\(14\!\cdots\!01\)\( \zeta_{6} q^{73} +(-\)\(31\!\cdots\!25\)\( + \)\(31\!\cdots\!25\)\( \zeta_{6}) q^{75} +\)\(75\!\cdots\!24\)\( q^{76} +(-\)\(31\!\cdots\!99\)\( + \)\(31\!\cdots\!99\)\( \zeta_{6}) q^{79} -\)\(14\!\cdots\!01\)\( \zeta_{6} q^{81} +(\)\(95\!\cdots\!76\)\( + \)\(24\!\cdots\!00\)\( \zeta_{6}) q^{84} +(-\)\(23\!\cdots\!75\)\( + \)\(32\!\cdots\!51\)\( \zeta_{6}) q^{91} +(\)\(21\!\cdots\!73\)\( - \)\(21\!\cdots\!73\)\( \zeta_{6}) q^{93} +\)\(10\!\cdots\!74\)\( q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3486784401q^{3} - 1099511627776q^{4} + 39320628860422799q^{7} - 12157665459056928801q^{9} + O(q^{10}) \) \( 2q - 3486784401q^{3} - 1099511627776q^{4} + 39320628860422799q^{7} - 12157665459056928801q^{9} - \)\(38\!\cdots\!76\)\(q^{12} - \)\(73\!\cdots\!02\)\(q^{13} - \)\(12\!\cdots\!76\)\(q^{16} - \)\(68\!\cdots\!99\)\(q^{19} - \)\(53\!\cdots\!26\)\(q^{21} - \)\(90\!\cdots\!25\)\(q^{25} + \)\(84\!\cdots\!02\)\(q^{27} - \)\(16\!\cdots\!76\)\(q^{28} + \)\(62\!\cdots\!73\)\(q^{31} + \)\(26\!\cdots\!52\)\(q^{36} + \)\(45\!\cdots\!01\)\(q^{37} + \)\(12\!\cdots\!01\)\(q^{39} - \)\(18\!\cdots\!46\)\(q^{43} + \)\(84\!\cdots\!52\)\(q^{48} - \)\(11\!\cdots\!01\)\(q^{49} + \)\(40\!\cdots\!76\)\(q^{52} + \)\(48\!\cdots\!98\)\(q^{57} - \)\(90\!\cdots\!74\)\(q^{61} + \)\(13\!\cdots\!27\)\(q^{63} + \)\(26\!\cdots\!52\)\(q^{64} + \)\(24\!\cdots\!01\)\(q^{67} + \)\(14\!\cdots\!01\)\(q^{73} - \)\(31\!\cdots\!25\)\(q^{75} + \)\(15\!\cdots\!48\)\(q^{76} - \)\(31\!\cdots\!99\)\(q^{79} - \)\(14\!\cdots\!01\)\(q^{81} + \)\(43\!\cdots\!52\)\(q^{84} - \)\(14\!\cdots\!99\)\(q^{91} + \)\(21\!\cdots\!73\)\(q^{93} + \)\(21\!\cdots\!48\)\(q^{97} + O(q^{100}) \)

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 −1.74339e9 3.01964e9i −5.49756e11 9.52205e11i 0 0 1.96603e16 7.73323e16i 0 −6.07883e18 + 1.05288e19i 0
11.1 0 −1.74339e9 + 3.01964e9i −5.49756e11 + 9.52205e11i 0 0 1.96603e16 + 7.73323e16i 0 −6.07883e18 1.05288e19i 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 21.41.h.a 2
3.b odd 2 1 CM 21.41.h.a 2
7.c even 3 1 inner 21.41.h.a 2
21.h odd 6 1 inner 21.41.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.41.h.a 2 1.a even 1 1 trivial
21.41.h.a 2 3.b odd 2 1 CM
21.41.h.a 2 7.c even 3 1 inner
21.41.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 \(S_{41}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 12157665459056928801 + 3486784401 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(63\!\cdots\!01\)\( - 39320628860422799 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( \)\(36\!\cdots\!01\)\( + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(47\!\cdots\!01\)\( + \)\(68\!\cdots\!99\)\( T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(39\!\cdots\!29\)\( - \)\(62\!\cdots\!73\)\( T + T^{2} \)
$37$ \( \)\(20\!\cdots\!01\)\( - \)\(45\!\cdots\!01\)\( T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( \)\(93\!\cdots\!73\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(82\!\cdots\!76\)\( + \)\(90\!\cdots\!74\)\( T + T^{2} \)
$67$ \( \)\(58\!\cdots\!01\)\( - \)\(24\!\cdots\!01\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(20\!\cdots\!01\)\( - \)\(14\!\cdots\!01\)\( T + T^{2} \)
$79$ \( \)\(99\!\cdots\!01\)\( + \)\(31\!\cdots\!99\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -\)\(10\!\cdots\!74\)\( + T )^{2} \)
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