Properties

Label 21.22.c.a
Level $21$
Weight $22$
Character orbit 21.c
Analytic conductor $58.690$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 21.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(58.6902423003\)
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: \( 2\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -59049 + 118098 \zeta_{6} ) q^{3} + 2097152 q^{4} + ( -846427447 + 568871874 \zeta_{6} ) q^{7} -10460353203 q^{9} +O(q^{10})\) \( q + ( -59049 + 118098 \zeta_{6} ) q^{3} + 2097152 q^{4} + ( -846427447 + 568871874 \zeta_{6} ) q^{7} -10460353203 q^{9} + ( -123834728448 + 247669456896 \zeta_{6} ) q^{12} + ( 532697931828 - 1065395863656 \zeta_{6} ) q^{13} + 4398046511104 q^{16} + ( 23048508442770 - 46097016885540 \zeta_{6} ) q^{19} + ( -17201936257749 - 66370073347980 \zeta_{6} ) q^{21} -476837158203125 q^{25} + ( 617673396283947 - 1235346792567894 \zeta_{6} ) q^{27} + ( -1775087013330944 + 1193010788302848 \zeta_{6} ) q^{28} + ( -725516191935270 + 1451032383870540 \zeta_{6} ) q^{31} -21936950640377856 q^{36} + 57776323439003290 q^{37} + 94365840529534716 q^{39} -265258444413820520 q^{43} + ( -259700248434180096 + 519400496868360192 \zeta_{6} ) q^{48} + ( 392824214006665933 - 639402326931579480 \zeta_{6} ) q^{49} + ( 1117148533128953856 - 2234297066257907712 \zeta_{6} ) q^{52} + 4082974125111377190 q^{57} + ( 1438373287720915020 - 2876746575441830040 \zeta_{6} ) q^{61} + ( 8853930056333562741 - 5950600729292512422 \zeta_{6} ) q^{63} + 9223372036854775808 q^{64} + 6944332370266921520 q^{67} + ( 35890654689431889192 - 71781309378863778384 \zeta_{6} ) q^{73} + ( 28156757354736328125 - 56313514709472656250 \zeta_{6} ) q^{75} + ( 48336225577771991040 - 96672451155543982080 \zeta_{6} ) q^{76} + 168068289047083465196 q^{79} + 109418989131512359209 q^{81} + ( -36075075026810830848 - 139188132061862952960 \zeta_{6} ) q^{84} + ( 155183591050483128228 + 598743430163789560560 \zeta_{6} ) q^{91} -128523016852757274690 q^{93} + ( 525465954789943160664 - 1050931909579886321328 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4194304q^{4} - 1123983020q^{7} - 20920706406q^{9} + O(q^{10}) \) \( 2q + 4194304q^{4} - 1123983020q^{7} - 20920706406q^{9} + 8796093022208q^{16} - 100773945863478q^{21} - 953674316406250q^{25} - 2357163238359040q^{28} - 43873901280755712q^{36} + 115552646878006580q^{37} + 188731681059069432q^{39} - 530516888827641040q^{43} + 146246101081752386q^{49} + 8165948250222754380q^{57} + 11757259383374613060q^{63} + 18446744073709551616q^{64} + 13888664740533843040q^{67} + 336136578094166930392q^{79} + 218837978263024718418q^{81} - 211338282115484614656q^{84} + 909110612264755817016q^{91} - 257046033705514549380q^{93} + O(q^{100}) \)

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 102276.i 2.09715e6 0 0 −5.61992e8 4.92657e8i 0 −1.04604e10 0
20.2 0 102276.i 2.09715e6 0 0 −5.61992e8 + 4.92657e8i 0 −1.04604e10 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 21.22.c.a 2
3.b odd 2 1 CM 21.22.c.a 2
7.b odd 2 1 inner 21.22.c.a 2
21.c even 2 1 inner 21.22.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.22.c.a 2 1.a even 1 1 trivial
21.22.c.a 2 3.b odd 2 1 CM
21.22.c.a 2 7.b odd 2 1 inner
21.22.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 \(S_{22}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 10460353203 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 558545864083284007 + 1123983020 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( \)\(85\!\cdots\!52\)\( + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(15\!\cdots\!00\)\( + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(15\!\cdots\!00\)\( + T^{2} \)
$37$ \( ( -57776323439003290 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 265258444413820520 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(62\!\cdots\!00\)\( + T^{2} \)
$67$ \( ( -6944332370266921520 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(38\!\cdots\!92\)\( + T^{2} \)
$79$ \( ( -\)\(16\!\cdots\!96\)\( + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(82\!\cdots\!88\)\( + T^{2} \)
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