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

\(f(q)\) \(=\) \( q - 3q^{2} - 3q^{3} + q^{4} - 18q^{5} + 9q^{6} + 7q^{7} + 21q^{8} + 9q^{9} + O(q^{10}) \) \( q - 3q^{2} - 3q^{3} + q^{4} - 18q^{5} + 9q^{6} + 7q^{7} + 21q^{8} + 9q^{9} + 54q^{10} - 36q^{11} - 3q^{12} - 34q^{13} - 21q^{14} + 54q^{15} - 71q^{16} + 42q^{17} - 27q^{18} - 124q^{19} - 18q^{20} - 21q^{21} + 108q^{22} - 63q^{24} + 199q^{25} + 102q^{26} - 27q^{27} + 7q^{28} + 102q^{29} - 162q^{30} - 160q^{31} + 45q^{32} + 108q^{33} - 126q^{34} - 126q^{35} + 9q^{36} + 398q^{37} + 372q^{38} + 102q^{39} - 378q^{40} - 318q^{41} + 63q^{42} - 268q^{43} - 36q^{44} - 162q^{45} + 240q^{47} + 213q^{48} + 49q^{49} - 597q^{50} - 126q^{51} - 34q^{52} - 498q^{53} + 81q^{54} + 648q^{55} + 147q^{56} + 372q^{57} - 306q^{58} - 132q^{59} + 54q^{60} + 398q^{61} + 480q^{62} + 63q^{63} + 433q^{64} + 612q^{65} - 324q^{66} + 92q^{67} + 42q^{68} + 378q^{70} - 720q^{71} + 189q^{72} - 502q^{73} - 1194q^{74} - 597q^{75} - 124q^{76} - 252q^{77} - 306q^{78} - 1024q^{79} + 1278q^{80} + 81q^{81} + 954q^{82} - 204q^{83} - 21q^{84} - 756q^{85} + 804q^{86} - 306q^{87} - 756q^{88} + 354q^{89} + 486q^{90} - 238q^{91} + 480q^{93} - 720q^{94} + 2232q^{95} - 135q^{96} - 286q^{97} - 147q^{98} - 324q^{99} + O(q^{100}) \)

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:

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

Atkin-Lehner signs

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 8 T^{2} \)
$3$ \( 1 + 3 T \)
$5$ \( 1 + 18 T + 125 T^{2} \)
$7$ \( 1 - 7 T \)
$11$ \( 1 + 36 T + 1331 T^{2} \)
$13$ \( 1 + 34 T + 2197 T^{2} \)
$17$ \( 1 - 42 T + 4913 T^{2} \)
$19$ \( 1 + 124 T + 6859 T^{2} \)
$23$ \( 1 + 12167 T^{2} \)
$29$ \( 1 - 102 T + 24389 T^{2} \)
$31$ \( 1 + 160 T + 29791 T^{2} \)
$37$ \( 1 - 398 T + 50653 T^{2} \)
$41$ \( 1 + 318 T + 68921 T^{2} \)
$43$ \( 1 + 268 T + 79507 T^{2} \)
$47$ \( 1 - 240 T + 103823 T^{2} \)
$53$ \( 1 + 498 T + 148877 T^{2} \)
$59$ \( 1 + 132 T + 205379 T^{2} \)
$61$ \( 1 - 398 T + 226981 T^{2} \)
$67$ \( 1 - 92 T + 300763 T^{2} \)
$71$ \( 1 + 720 T + 357911 T^{2} \)
$73$ \( 1 + 502 T + 389017 T^{2} \)
$79$ \( 1 + 1024 T + 493039 T^{2} \)
$83$ \( 1 + 204 T + 571787 T^{2} \)
$89$ \( 1 - 354 T + 704969 T^{2} \)
$97$ \( 1 + 286 T + 912673 T^{2} \)
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