Properties

Label 21.28.c.a
Level $21$
Weight $28$
Character orbit 21.c
Analytic conductor $96.990$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(96.9896707160\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1594323 - 3188646 \zeta_{6} ) q^{3} + 134217728 q^{4} + ( 173476059101 + 120968265618 \zeta_{6} ) q^{7} -7625597484987 q^{9} +O(q^{10})\) \( q +(1594323 - 3188646 \zeta_{6}) q^{3} +134217728 q^{4} +(173476059101 + 120968265618 \zeta_{6}) q^{7} -7625597484987 q^{9} +(213986410758144 - 427972821516288 \zeta_{6}) q^{12} +(-331524223037316 + 663048446074632 \zeta_{6}) q^{13} +18014398509481984 q^{16} +(-211512059159899590 + 423024118319799180 \zeta_{6}) q^{19} +(662301847263856851 - 746016230093053860 \zeta_{6}) q^{21} -7450580596923828125 q^{25} +(-12157665459056928801 + 24315330918113857602 \zeta_{6}) q^{27} +(23283562514929942528 + 16236085771348475904 \zeta_{6}) q^{28} +(-\)\(13\!\cdots\!90\)\( + \)\(27\!\cdots\!80\)\( \zeta_{6}) q^{31} -\)\(10\!\cdots\!36\)\( q^{36} -\)\(23\!\cdots\!90\)\( q^{37} +\)\(15\!\cdots\!04\)\( q^{39} -\)\(20\!\cdots\!20\)\( q^{43} +(\)\(28\!\cdots\!32\)\( - \)\(57\!\cdots\!64\)\( \zeta_{6}) q^{48} +(\)\(15\!\cdots\!77\)\( + \)\(56\!\cdots\!60\)\( \zeta_{6}) q^{49} +(-\)\(44\!\cdots\!48\)\( + \)\(88\!\cdots\!96\)\( \zeta_{6}) q^{52} +\)\(10\!\cdots\!10\)\( q^{57} +(\)\(23\!\cdots\!40\)\( - \)\(46\!\cdots\!80\)\( \zeta_{6}) q^{61} +(-\)\(13\!\cdots\!87\)\( - \)\(92\!\cdots\!66\)\( \zeta_{6}) q^{63} +\)\(24\!\cdots\!52\)\( q^{64} -\)\(77\!\cdots\!20\)\( q^{67} +(\)\(64\!\cdots\!36\)\( - \)\(12\!\cdots\!72\)\( \zeta_{6}) q^{73} +(-\)\(11\!\cdots\!75\)\( + \)\(23\!\cdots\!50\)\( \zeta_{6}) q^{75} +(-\)\(28\!\cdots\!20\)\( + \)\(56\!\cdots\!40\)\( \zeta_{6}) q^{76} -\)\(12\!\cdots\!56\)\( q^{79} +\)\(58\!\cdots\!69\)\( q^{81} +(\)\(88\!\cdots\!28\)\( - \)\(10\!\cdots\!80\)\( \zeta_{6}) q^{84} +(-\)\(13\!\cdots\!92\)\( + \)\(15\!\cdots\!20\)\( \zeta_{6}) q^{91} +\)\(66\!\cdots\!10\)\( q^{93} +(-\)\(61\!\cdots\!72\)\( + \)\(12\!\cdots\!44\)\( \zeta_{6}) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 268435456q^{4} + 467920383820q^{7} - 15251194969974q^{9} + O(q^{10}) \) \( 2q + 268435456q^{4} + 467920383820q^{7} - 15251194969974q^{9} + 36028797018963968q^{16} + 578587464434659842q^{21} - 14901161193847656250q^{25} + 62803210801208360960q^{28} - \)\(20\!\cdots\!72\)\(q^{36} - \)\(47\!\cdots\!80\)\(q^{37} + \)\(31\!\cdots\!08\)\(q^{39} - \)\(40\!\cdots\!40\)\(q^{43} + \)\(87\!\cdots\!14\)\(q^{49} + \)\(20\!\cdots\!20\)\(q^{57} - \)\(35\!\cdots\!40\)\(q^{63} + \)\(48\!\cdots\!04\)\(q^{64} - \)\(15\!\cdots\!40\)\(q^{67} - \)\(25\!\cdots\!12\)\(q^{79} + \)\(11\!\cdots\!38\)\(q^{81} + \)\(77\!\cdots\!76\)\(q^{84} - \)\(12\!\cdots\!64\)\(q^{91} + \)\(13\!\cdots\!20\)\(q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1
0.500000 + 0.866025i
0.500000 0.866025i
0 2.76145e6i 1.34218e8 0 0 2.33960e11 + 1.04762e11i 0 −7.62560e12 0
20.2 0 2.76145e6i 1.34218e8 0 0 2.33960e11 1.04762e11i 0 −7.62560e12 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.28.c.a 2
3.b odd 2 1 CM 21.28.c.a 2
7.b odd 2 1 inner 21.28.c.a 2
21.c even 2 1 inner 21.28.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.28.c.a 2 1.a even 1 1 trivial
21.28.c.a 2 3.b odd 2 1 CM
21.28.c.a 2 7.b odd 2 1 inner
21.28.c.a 2 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{28}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 7625597484987 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(65\!\cdots\!43\)\( - 467920383820 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(32\!\cdots\!68\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(13\!\cdots\!00\)\( + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(57\!\cdots\!00\)\( + T^{2} \)
$37$ \( ( \)\(23\!\cdots\!90\)\( + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( \)\(20\!\cdots\!20\)\( + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(16\!\cdots\!00\)\( + T^{2} \)
$67$ \( ( \)\(77\!\cdots\!20\)\( + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(12\!\cdots\!88\)\( + T^{2} \)
$79$ \( ( \)\(12\!\cdots\!56\)\( + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(11\!\cdots\!52\)\( + T^{2} \)
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