Properties

Label 21.4.a.a
Level $21$
Weight $4$
Character orbit 21.a
Self dual yes
Analytic conductor $1.239$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.23904011012\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 3 q^{2} - 3 q^{3} + q^{4} - 18 q^{5} + 9 q^{6} + 7 q^{7} + 21 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} - 3 q^{3} + q^{4} - 18 q^{5} + 9 q^{6} + 7 q^{7} + 21 q^{8} + 9 q^{9} + 54 q^{10} - 36 q^{11} - 3 q^{12} - 34 q^{13} - 21 q^{14} + 54 q^{15} - 71 q^{16} + 42 q^{17} - 27 q^{18} - 124 q^{19} - 18 q^{20} - 21 q^{21} + 108 q^{22} - 63 q^{24} + 199 q^{25} + 102 q^{26} - 27 q^{27} + 7 q^{28} + 102 q^{29} - 162 q^{30} - 160 q^{31} + 45 q^{32} + 108 q^{33} - 126 q^{34} - 126 q^{35} + 9 q^{36} + 398 q^{37} + 372 q^{38} + 102 q^{39} - 378 q^{40} - 318 q^{41} + 63 q^{42} - 268 q^{43} - 36 q^{44} - 162 q^{45} + 240 q^{47} + 213 q^{48} + 49 q^{49} - 597 q^{50} - 126 q^{51} - 34 q^{52} - 498 q^{53} + 81 q^{54} + 648 q^{55} + 147 q^{56} + 372 q^{57} - 306 q^{58} - 132 q^{59} + 54 q^{60} + 398 q^{61} + 480 q^{62} + 63 q^{63} + 433 q^{64} + 612 q^{65} - 324 q^{66} + 92 q^{67} + 42 q^{68} + 378 q^{70} - 720 q^{71} + 189 q^{72} - 502 q^{73} - 1194 q^{74} - 597 q^{75} - 124 q^{76} - 252 q^{77} - 306 q^{78} - 1024 q^{79} + 1278 q^{80} + 81 q^{81} + 954 q^{82} - 204 q^{83} - 21 q^{84} - 756 q^{85} + 804 q^{86} - 306 q^{87} - 756 q^{88} + 354 q^{89} + 486 q^{90} - 238 q^{91} + 480 q^{93} - 720 q^{94} + 2232 q^{95} - 135 q^{96} - 286 q^{97} - 147 q^{98} - 324 q^{99}+O(q^{100}) \) 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
−3.00000 −3.00000 1.00000 −18.0000 9.00000 7.00000 21.0000 9.00000 54.0000
\(n\): 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 21.4.a.a 1
3.b odd 2 1 63.4.a.c 1
4.b odd 2 1 336.4.a.f 1
5.b even 2 1 525.4.a.g 1
5.c odd 4 2 525.4.d.c 2
7.b odd 2 1 147.4.a.c 1
7.c even 3 2 147.4.e.i 2
7.d odd 6 2 147.4.e.g 2
8.b even 2 1 1344.4.a.ba 1
8.d odd 2 1 1344.4.a.n 1
12.b even 2 1 1008.4.a.v 1
15.d odd 2 1 1575.4.a.b 1
21.c even 2 1 441.4.a.j 1
21.g even 6 2 441.4.e.d 2
21.h odd 6 2 441.4.e.b 2
28.d even 2 1 2352.4.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 1.a even 1 1 trivial
63.4.a.c 1 3.b odd 2 1
147.4.a.c 1 7.b odd 2 1
147.4.e.g 2 7.d odd 6 2
147.4.e.i 2 7.c even 3 2
336.4.a.f 1 4.b odd 2 1
441.4.a.j 1 21.c even 2 1
441.4.e.b 2 21.h odd 6 2
441.4.e.d 2 21.g even 6 2
525.4.a.g 1 5.b even 2 1
525.4.d.c 2 5.c odd 4 2
1008.4.a.v 1 12.b even 2 1
1344.4.a.n 1 8.d odd 2 1
1344.4.a.ba 1 8.b even 2 1
1575.4.a.b 1 15.d odd 2 1
2352.4.a.r 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(21))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 3 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 18 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 36 \) Copy content Toggle raw display
$13$ \( T + 34 \) Copy content Toggle raw display
$17$ \( T - 42 \) Copy content Toggle raw display
$19$ \( T + 124 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 102 \) Copy content Toggle raw display
$31$ \( T + 160 \) Copy content Toggle raw display
$37$ \( T - 398 \) Copy content Toggle raw display
$41$ \( T + 318 \) Copy content Toggle raw display
$43$ \( T + 268 \) Copy content Toggle raw display
$47$ \( T - 240 \) Copy content Toggle raw display
$53$ \( T + 498 \) Copy content Toggle raw display
$59$ \( T + 132 \) Copy content Toggle raw display
$61$ \( T - 398 \) Copy content Toggle raw display
$67$ \( T - 92 \) Copy content Toggle raw display
$71$ \( T + 720 \) Copy content Toggle raw display
$73$ \( T + 502 \) Copy content Toggle raw display
$79$ \( T + 1024 \) Copy content Toggle raw display
$83$ \( T + 204 \) Copy content Toggle raw display
$89$ \( T - 354 \) Copy content Toggle raw display
$97$ \( T + 286 \) Copy content Toggle raw display
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