Properties

Label 21.26.g.a
Level $21$
Weight $26$
Character orbit 21.g
Analytic conductor $83.159$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(83.1593237900\)
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.

\(f(q)\) \(=\) \( q + ( -531441 - 531441 \zeta_{6} ) q^{3} + ( -33554432 + 33554432 \zeta_{6} ) q^{4} + ( -35728842238 - 1722717363 \zeta_{6} ) q^{7} + 847288609443 \zeta_{6} q^{9} +O(q^{10})\) \( q +(-531441 - 531441 \zeta_{6}) q^{3} +(-33554432 + 33554432 \zeta_{6}) q^{4} +(-35728842238 - 1722717363 \zeta_{6}) q^{7} +847288609443 \zeta_{6} q^{9} +(35664401793024 - 17832200896512 \zeta_{6}) q^{12} +(21081468058559 - 42162936117118 \zeta_{6}) q^{13} -1125899906842624 \zeta_{6} q^{16} +(5642266816761114 - 2821133408380557 \zeta_{6}) q^{19} +(18072249009694875 + 20818816924025124 \zeta_{6}) q^{21} +(298023223876953125 - 298023223876953125 \zeta_{6}) q^{25} +(450283905890997363 - 900567811781994726 \zeta_{6}) q^{27} +(1256665809925701632 - 1198861007313698816 \zeta_{6}) q^{28} +(-1420701470158304875 - 1420701470158304875 \zeta_{6}) q^{31} -28430288029929701376 q^{36} -70792710091786388161 \zeta_{6} q^{37} +(-33610669399525960557 + 33610669399525960557 \zeta_{6}) q^{39} +\)\(18\!\cdots\!75\)\( q^{43} +(-\)\(59\!\cdots\!84\)\( + \)\(11\!\cdots\!68\)\( \zeta_{6}) q^{48} +(\)\(12\!\cdots\!75\)\( + \)\(12\!\cdots\!57\)\( \zeta_{6}) q^{49} +(\)\(70\!\cdots\!88\)\( + \)\(70\!\cdots\!88\)\( \zeta_{6}) q^{52} -\)\(44\!\cdots\!11\)\( q^{57} +(-\)\(42\!\cdots\!52\)\( + \)\(21\!\cdots\!76\)\( \zeta_{6}) q^{61} +(\)\(14\!\cdots\!09\)\( - \)\(31\!\cdots\!43\)\( \zeta_{6}) q^{63} +\)\(37\!\cdots\!68\)\( q^{64} +(\)\(55\!\cdots\!89\)\( - \)\(55\!\cdots\!89\)\( \zeta_{6}) q^{67} +(\)\(97\!\cdots\!91\)\( + \)\(97\!\cdots\!91\)\( \zeta_{6}) q^{73} +(-\)\(31\!\cdots\!50\)\( + \)\(15\!\cdots\!25\)\( \zeta_{6}) q^{75} +(-\)\(94\!\cdots\!24\)\( + \)\(18\!\cdots\!48\)\( \zeta_{6}) q^{76} +\)\(10\!\cdots\!93\)\( \zeta_{6} q^{79} +(-\)\(71\!\cdots\!49\)\( + \)\(71\!\cdots\!49\)\( \zeta_{6}) q^{81} +(-\)\(13\!\cdots\!68\)\( + \)\(60\!\cdots\!00\)\( \zeta_{6}) q^{84} +(-\)\(82\!\cdots\!76\)\( + \)\(15\!\cdots\!01\)\( \zeta_{6}) q^{91} +\)\(22\!\cdots\!25\)\( \zeta_{6} q^{93} +(-\)\(48\!\cdots\!88\)\( + \)\(96\!\cdots\!76\)\( \zeta_{6}) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 1594323q^{3} - 33554432q^{4} - 73180401839q^{7} + 847288609443q^{9} + O(q^{10}) \) \( 2q - 1594323q^{3} - 33554432q^{4} - 73180401839q^{7} + 847288609443q^{9} + 53496602689536q^{12} - 1125899906842624q^{16} + 8463400225141671q^{19} + 56963314943414874q^{21} + 298023223876953125q^{25} + 1314470612537704448q^{28} - 4262104410474914625q^{31} - 56860576059859402752q^{36} - 70792710091786388161q^{37} - 33610669399525960557q^{39} + \)\(36\!\cdots\!50\)\(q^{43} + \)\(26\!\cdots\!07\)\(q^{49} + \)\(21\!\cdots\!64\)\(q^{52} - \)\(89\!\cdots\!22\)\(q^{57} - \)\(63\!\cdots\!28\)\(q^{61} - \)\(28\!\cdots\!25\)\(q^{63} + \)\(75\!\cdots\!36\)\(q^{64} + \)\(55\!\cdots\!89\)\(q^{67} + \)\(29\!\cdots\!73\)\(q^{73} - \)\(47\!\cdots\!75\)\(q^{75} + \)\(10\!\cdots\!93\)\(q^{79} - \)\(71\!\cdots\!49\)\(q^{81} - \)\(20\!\cdots\!36\)\(q^{84} - \)\(10\!\cdots\!51\)\(q^{91} + \)\(22\!\cdots\!25\)\(q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −797161. + 460241.i −1.67772e7 2.90590e7i 0 0 −3.65902e10 + 1.49192e9i 0 4.23644e11 7.33773e11i 0
17.1 0 −797161. 460241.i −1.67772e7 + 2.90590e7i 0 0 −3.65902e10 1.49192e9i 0 4.23644e11 + 7.33773e11i 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.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.26.g.a 2
3.b odd 2 1 CM 21.26.g.a 2
7.d odd 6 1 inner 21.26.g.a 2
21.g even 6 1 inner 21.26.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.26.g.a 2 1.a even 1 1 trivial
21.26.g.a 2 3.b odd 2 1 CM
21.26.g.a 2 7.d odd 6 1 inner
21.26.g.a 2 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{26}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 847288609443 + 1594323 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(13\!\cdots\!07\)\( + 73180401839 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(13\!\cdots\!43\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(23\!\cdots\!47\)\( - 8463400225141671 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(60\!\cdots\!75\)\( + 4262104410474914625 T + T^{2} \)
$37$ \( \)\(50\!\cdots\!21\)\( + 70792710091786388161 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -\)\(18\!\cdots\!75\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(13\!\cdots\!28\)\( + \)\(63\!\cdots\!28\)\( T + T^{2} \)
$67$ \( \)\(31\!\cdots\!21\)\( - \)\(55\!\cdots\!89\)\( T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(28\!\cdots\!43\)\( - \)\(29\!\cdots\!73\)\( T + T^{2} \)
$79$ \( \)\(10\!\cdots\!49\)\( - \)\(10\!\cdots\!93\)\( T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(69\!\cdots\!32\)\( + T^{2} \)
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