Properties

Label 21.2.a.a
Level $21$
Weight $2$
Character orbit 21.a
Self dual yes
Analytic conductor $0.168$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.167685844245\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} - q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} - q^{7} + 3 q^{8} + q^{9} + 2 q^{10} + 4 q^{11} - q^{12} - 2 q^{13} + q^{14} - 2 q^{15} - q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + 2 q^{20} - q^{21} - 4 q^{22} + 3 q^{24} - q^{25} + 2 q^{26} + q^{27} + q^{28} - 2 q^{29} + 2 q^{30} - 5 q^{32} + 4 q^{33} + 6 q^{34} + 2 q^{35} - q^{36} + 6 q^{37} - 4 q^{38} - 2 q^{39} - 6 q^{40} + 2 q^{41} + q^{42} - 4 q^{43} - 4 q^{44} - 2 q^{45} - q^{48} + q^{49} + q^{50} - 6 q^{51} + 2 q^{52} + 6 q^{53} - q^{54} - 8 q^{55} - 3 q^{56} + 4 q^{57} + 2 q^{58} + 12 q^{59} + 2 q^{60} - 2 q^{61} - q^{63} + 7 q^{64} + 4 q^{65} - 4 q^{66} + 4 q^{67} + 6 q^{68} - 2 q^{70} + 3 q^{72} - 6 q^{73} - 6 q^{74} - q^{75} - 4 q^{76} - 4 q^{77} + 2 q^{78} - 16 q^{79} + 2 q^{80} + q^{81} - 2 q^{82} - 12 q^{83} + q^{84} + 12 q^{85} + 4 q^{86} - 2 q^{87} + 12 q^{88} - 14 q^{89} + 2 q^{90} + 2 q^{91} - 8 q^{95} - 5 q^{96} + 18 q^{97} - q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 1.00000 −1.00000 −2.00000 −1.00000 −1.00000 3.00000 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.2.a.a 1
3.b odd 2 1 63.2.a.a 1
4.b odd 2 1 336.2.a.a 1
5.b even 2 1 525.2.a.d 1
5.c odd 4 2 525.2.d.a 2
7.b odd 2 1 147.2.a.a 1
7.c even 3 2 147.2.e.b 2
7.d odd 6 2 147.2.e.c 2
8.b even 2 1 1344.2.a.g 1
8.d odd 2 1 1344.2.a.s 1
9.c even 3 2 567.2.f.g 2
9.d odd 6 2 567.2.f.b 2
11.b odd 2 1 2541.2.a.j 1
12.b even 2 1 1008.2.a.l 1
13.b even 2 1 3549.2.a.c 1
15.d odd 2 1 1575.2.a.c 1
15.e even 4 2 1575.2.d.a 2
16.e even 4 2 5376.2.c.r 2
16.f odd 4 2 5376.2.c.l 2
17.b even 2 1 6069.2.a.b 1
19.b odd 2 1 7581.2.a.d 1
20.d odd 2 1 8400.2.a.bn 1
21.c even 2 1 441.2.a.f 1
21.g even 6 2 441.2.e.b 2
21.h odd 6 2 441.2.e.a 2
24.f even 2 1 4032.2.a.k 1
24.h odd 2 1 4032.2.a.h 1
28.d even 2 1 2352.2.a.v 1
28.f even 6 2 2352.2.q.e 2
28.g odd 6 2 2352.2.q.x 2
33.d even 2 1 7623.2.a.g 1
35.c odd 2 1 3675.2.a.n 1
56.e even 2 1 9408.2.a.m 1
56.h odd 2 1 9408.2.a.bv 1
84.h odd 2 1 7056.2.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 1.a even 1 1 trivial
63.2.a.a 1 3.b odd 2 1
147.2.a.a 1 7.b odd 2 1
147.2.e.b 2 7.c even 3 2
147.2.e.c 2 7.d odd 6 2
336.2.a.a 1 4.b odd 2 1
441.2.a.f 1 21.c even 2 1
441.2.e.a 2 21.h odd 6 2
441.2.e.b 2 21.g even 6 2
525.2.a.d 1 5.b even 2 1
525.2.d.a 2 5.c odd 4 2
567.2.f.b 2 9.d odd 6 2
567.2.f.g 2 9.c even 3 2
1008.2.a.l 1 12.b even 2 1
1344.2.a.g 1 8.b even 2 1
1344.2.a.s 1 8.d odd 2 1
1575.2.a.c 1 15.d odd 2 1
1575.2.d.a 2 15.e even 4 2
2352.2.a.v 1 28.d even 2 1
2352.2.q.e 2 28.f even 6 2
2352.2.q.x 2 28.g odd 6 2
2541.2.a.j 1 11.b odd 2 1
3549.2.a.c 1 13.b even 2 1
3675.2.a.n 1 35.c odd 2 1
4032.2.a.h 1 24.h odd 2 1
4032.2.a.k 1 24.f even 2 1
5376.2.c.l 2 16.f odd 4 2
5376.2.c.r 2 16.e even 4 2
6069.2.a.b 1 17.b even 2 1
7056.2.a.p 1 84.h odd 2 1
7581.2.a.d 1 19.b odd 2 1
7623.2.a.g 1 33.d even 2 1
8400.2.a.bn 1 20.d odd 2 1
9408.2.a.m 1 56.e even 2 1
9408.2.a.bv 1 56.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T + 14 \) Copy content Toggle raw display
$97$ \( T - 18 \) Copy content Toggle raw display
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