Properties

Label 21.18.g.a
Level $21$
Weight $18$
Character orbit 21.g
Analytic conductor $38.477$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(38.4766383424\)
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.

\(f(q)\) \(=\) \( q + ( 6561 + 6561 \zeta_{6} ) q^{3} + ( -131072 + 131072 \zeta_{6} ) q^{4} + ( 7390718 - 17539617 \zeta_{6} ) q^{7} + 129140163 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 6561 + 6561 \zeta_{6} ) q^{3} + ( -131072 + 131072 \zeta_{6} ) q^{4} + ( 7390718 - 17539617 \zeta_{6} ) q^{7} + 129140163 \zeta_{6} q^{9} + ( -1719926784 + 859963392 \zeta_{6} ) q^{12} + ( -2824265149 + 5648530298 \zeta_{6} ) q^{13} -17179869184 \zeta_{6} q^{16} + ( -169289986506 + 84644993253 \zeta_{6} ) q^{19} + ( 163567927935 - 181664353476 \zeta_{6} ) q^{21} + ( 762939453125 - 762939453125 \zeta_{6} ) q^{25} + ( -847288609443 + 1694577218886 \zeta_{6} ) q^{27} + ( 1330236489728 + 968716189696 \zeta_{6} ) q^{28} + ( -2915506365985 - 2915506365985 \zeta_{6} ) q^{31} -16926659444736 q^{36} -42101902631989 \zeta_{6} q^{37} + ( -55590010927767 + 55590010927767 \zeta_{6} ) q^{39} -54897704698835 q^{43} + ( 112717121716224 - 225434243432448 \zeta_{6} ) q^{48} + ( -253015451951165 + 48377438356677 \zeta_{6} ) q^{49} + ( -370182081609728 - 370182081609728 \zeta_{6} ) q^{52} -1666067402198799 q^{57} + ( -3454795454866312 + 1727397727433156 \zeta_{6} ) q^{61} + ( 2265068998337571 - 1310630471130537 \zeta_{6} ) q^{63} + 2251799813685248 q^{64} + ( 269339188788571 - 269339188788571 \zeta_{6} ) q^{67} + ( 4996068505776639 + 4996068505776639 \zeta_{6} ) q^{73} + ( 10011291503906250 - 5005645751953125 \zeta_{6} ) q^{75} + ( 11094588555657216 - 22189177111314432 \zeta_{6} ) q^{76} + 26467234205310223 \zeta_{6} q^{79} + ( -16677181699666569 + 16677181699666569 \zeta_{6} ) q^{81} + ( 2371934688509952 + 21439175450296320 \zeta_{6} ) q^{84} + ( 78199710766328884 - 7789834472933969 \zeta_{6} ) q^{91} -57385911801682755 \zeta_{6} q^{93} + ( 57267491945336968 - 114534983890673936 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 19683q^{3} - 131072q^{4} - 2758181q^{7} + 129140163q^{9} + O(q^{10}) \) \( 2q + 19683q^{3} - 131072q^{4} - 2758181q^{7} + 129140163q^{9} - 2579890176q^{12} - 17179869184q^{16} - 253934979759q^{19} + 145471502394q^{21} + 762939453125q^{25} + 3629189169152q^{28} - 8746519097955q^{31} - 33853318889472q^{36} - 42101902631989q^{37} - 55590010927767q^{39} - 109795409397670q^{43} - 457653465545653q^{49} - 1110546244829184q^{52} - 3332134804397598q^{57} - 5182193182299468q^{61} + 3219507525544605q^{63} + 4503599627370496q^{64} + 269339188788571q^{67} + 14988205517329917q^{73} + 15016937255859375q^{75} + 26467234205310223q^{79} - 16677181699666569q^{81} + 26183044827316224q^{84} + 148609587059723799q^{91} - 57385911801682755q^{93} + O(q^{100}) \)

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 9841.50 5681.99i −65536.0 113512.i 0 0 −1.37909e6 + 1.51898e7i 0 6.45701e7 1.11839e8i 0
17.1 0 9841.50 + 5681.99i −65536.0 + 113512.i 0 0 −1.37909e6 1.51898e7i 0 6.45701e7 + 1.11839e8i 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 21.18.g.a 2
3.b odd 2 1 CM 21.18.g.a 2
7.d odd 6 1 inner 21.18.g.a 2
21.g even 6 1 inner 21.18.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.18.g.a 2 1.a even 1 1 trivial
21.18.g.a 2 3.b odd 2 1 CM
21.18.g.a 2 7.d odd 6 1 inner
21.18.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 \(S_{18}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 129140163 - 19683 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 232630513987207 + 2758181 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 23929420895567976603 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( \)\(21\!\cdots\!27\)\( + 253934979759 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( \)\(25\!\cdots\!75\)\( + 8746519097955 T + T^{2} \)
$37$ \( \)\(17\!\cdots\!21\)\( + 42101902631989 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 54897704698835 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(89\!\cdots\!08\)\( + 5182193182299468 T + T^{2} \)
$67$ \( \)\(72\!\cdots\!41\)\( - 269339188788571 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(74\!\cdots\!63\)\( - 14988205517329917 T + T^{2} \)
$79$ \( \)\(70\!\cdots\!29\)\( - 26467234205310223 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(98\!\cdots\!72\)\( + T^{2} \)
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