Properties

Label 21.10.c.a
Level $21$
Weight $10$
Character orbit 21.c
Analytic conductor $10.816$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.8157525594\)
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: \( 2\cdot 3^{3} \)
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.

\(f(q)\) \(=\) \( q + ( 81 - 162 \zeta_{6} ) q^{3} + 512 q^{4} + ( 6803 - 1026 \zeta_{6} ) q^{7} -19683 q^{9} +O(q^{10})\) \( q + ( 81 - 162 \zeta_{6} ) q^{3} + 512 q^{4} + ( 6803 - 1026 \zeta_{6} ) q^{7} -19683 q^{9} + ( 41472 - 82944 \zeta_{6} ) q^{12} + ( 97308 - 194616 \zeta_{6} ) q^{13} + 262144 q^{16} + ( 335070 - 670140 \zeta_{6} ) q^{19} + ( 384831 - 1018980 \zeta_{6} ) q^{21} -1953125 q^{25} + ( -1594323 + 3188646 \zeta_{6} ) q^{27} + ( 3483136 - 525312 \zeta_{6} ) q^{28} + ( -5856570 + 11713140 \zeta_{6} ) q^{31} -10077696 q^{36} + 15384490 q^{37} -23645844 q^{39} + 16577080 q^{43} + ( 21233664 - 42467328 \zeta_{6} ) q^{48} + ( 45228133 - 12907080 \zeta_{6} ) q^{49} + ( 49821696 - 99643392 \zeta_{6} ) q^{52} -81422010 q^{57} + ( -104682780 + 209365560 \zeta_{6} ) q^{61} + ( -133903449 + 20194758 \zeta_{6} ) q^{63} + 134217728 q^{64} + 112542320 q^{67} + ( -221849928 + 443699856 \zeta_{6} ) q^{73} + ( -158203125 + 316406250 \zeta_{6} ) q^{75} + ( 171555840 - 343111680 \zeta_{6} ) q^{76} -616732324 q^{79} + 387420489 q^{81} + ( 197033472 - 521717760 \zeta_{6} ) q^{84} + ( 462310308 - 1224134640 \zeta_{6} ) q^{91} + 1423146510 q^{93} + ( 678131784 - 1356263568 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 1024q^{4} + 12580q^{7} - 39366q^{9} + O(q^{10}) \) \( 2q + 1024q^{4} + 12580q^{7} - 39366q^{9} + 524288q^{16} - 249318q^{21} - 3906250q^{25} + 6440960q^{28} - 20155392q^{36} + 30768980q^{37} - 47291688q^{39} + 33154160q^{43} + 77549186q^{49} - 162844020q^{57} - 247612140q^{63} + 268435456q^{64} + 225084640q^{67} - 1233464648q^{79} + 774840978q^{81} - 127650816q^{84} - 299514024q^{91} + 2846293020q^{93} + O(q^{100}) \)

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
0 140.296i 512.000 0 0 6290.00 888.542i 0 −19683.0 0
20.2 0 140.296i 512.000 0 0 6290.00 + 888.542i 0 −19683.0 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 21.10.c.a 2
3.b odd 2 1 CM 21.10.c.a 2
7.b odd 2 1 inner 21.10.c.a 2
21.c even 2 1 inner 21.10.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.10.c.a 2 1.a even 1 1 trivial
21.10.c.a 2 3.b odd 2 1 CM
21.10.c.a 2 7.b odd 2 1 inner
21.10.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 \(S_{10}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 19683 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 40353607 - 12580 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 28406540592 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 336815714700 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 102898236494700 + T^{2} \)
$37$ \( ( -15384490 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -16577080 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 32875453285585200 + T^{2} \)
$67$ \( ( -112542320 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( 147652171660815552 + T^{2} \)
$79$ \( ( 616732324 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 1379588149413067968 + T^{2} \)
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