Properties

Label 21.4.a.b
Level 21
Weight 4
Character orbit 21.a
Self dual yes
Analytic conductor 1.239
Analytic rank 0
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: \(0\)
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

\(f(q)\) \(=\) \( q + 4q^{2} - 3q^{3} + 8q^{4} - 4q^{5} - 12q^{6} - 7q^{7} + 9q^{9} + O(q^{10}) \) \( q + 4q^{2} - 3q^{3} + 8q^{4} - 4q^{5} - 12q^{6} - 7q^{7} + 9q^{9} - 16q^{10} + 62q^{11} - 24q^{12} - 62q^{13} - 28q^{14} + 12q^{15} - 64q^{16} + 84q^{17} + 36q^{18} + 100q^{19} - 32q^{20} + 21q^{21} + 248q^{22} - 42q^{23} - 109q^{25} - 248q^{26} - 27q^{27} - 56q^{28} - 10q^{29} + 48q^{30} - 48q^{31} - 256q^{32} - 186q^{33} + 336q^{34} + 28q^{35} + 72q^{36} - 246q^{37} + 400q^{38} + 186q^{39} - 248q^{41} + 84q^{42} + 68q^{43} + 496q^{44} - 36q^{45} - 168q^{46} + 324q^{47} + 192q^{48} + 49q^{49} - 436q^{50} - 252q^{51} - 496q^{52} + 258q^{53} - 108q^{54} - 248q^{55} - 300q^{57} - 40q^{58} + 120q^{59} + 96q^{60} + 622q^{61} - 192q^{62} - 63q^{63} - 512q^{64} + 248q^{65} - 744q^{66} + 904q^{67} + 672q^{68} + 126q^{69} + 112q^{70} - 678q^{71} - 642q^{73} - 984q^{74} + 327q^{75} + 800q^{76} - 434q^{77} + 744q^{78} + 740q^{79} + 256q^{80} + 81q^{81} - 992q^{82} + 468q^{83} + 168q^{84} - 336q^{85} + 272q^{86} + 30q^{87} + 200q^{89} - 144q^{90} + 434q^{91} - 336q^{92} + 144q^{93} + 1296q^{94} - 400q^{95} + 768q^{96} - 1266q^{97} + 196q^{98} + 558q^{99} + O(q^{100}) \)

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
4.00000 −3.00000 8.00000 −4.00000 −12.0000 −7.00000 0 9.00000 −16.0000
\(n\): 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.b 1
3.b odd 2 1 63.4.a.a 1
4.b odd 2 1 336.4.a.h 1
5.b even 2 1 525.4.a.b 1
5.c odd 4 2 525.4.d.b 2
7.b odd 2 1 147.4.a.g 1
7.c even 3 2 147.4.e.c 2
7.d odd 6 2 147.4.e.b 2
8.b even 2 1 1344.4.a.w 1
8.d odd 2 1 1344.4.a.i 1
12.b even 2 1 1008.4.a.m 1
15.d odd 2 1 1575.4.a.k 1
21.c even 2 1 441.4.a.b 1
21.g even 6 2 441.4.e.n 2
21.h odd 6 2 441.4.e.m 2
28.d even 2 1 2352.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 1.a even 1 1 trivial
63.4.a.a 1 3.b odd 2 1
147.4.a.g 1 7.b odd 2 1
147.4.e.b 2 7.d odd 6 2
147.4.e.c 2 7.c even 3 2
336.4.a.h 1 4.b odd 2 1
441.4.a.b 1 21.c even 2 1
441.4.e.m 2 21.h odd 6 2
441.4.e.n 2 21.g even 6 2
525.4.a.b 1 5.b even 2 1
525.4.d.b 2 5.c odd 4 2
1008.4.a.m 1 12.b even 2 1
1344.4.a.i 1 8.d odd 2 1
1344.4.a.w 1 8.b even 2 1
1575.4.a.k 1 15.d odd 2 1
2352.4.a.l 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 \( T_{2} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 8 T^{2} \)
$3$ \( 1 + 3 T \)
$5$ \( 1 + 4 T + 125 T^{2} \)
$7$ \( 1 + 7 T \)
$11$ \( 1 - 62 T + 1331 T^{2} \)
$13$ \( 1 + 62 T + 2197 T^{2} \)
$17$ \( 1 - 84 T + 4913 T^{2} \)
$19$ \( 1 - 100 T + 6859 T^{2} \)
$23$ \( 1 + 42 T + 12167 T^{2} \)
$29$ \( 1 + 10 T + 24389 T^{2} \)
$31$ \( 1 + 48 T + 29791 T^{2} \)
$37$ \( 1 + 246 T + 50653 T^{2} \)
$41$ \( 1 + 248 T + 68921 T^{2} \)
$43$ \( 1 - 68 T + 79507 T^{2} \)
$47$ \( 1 - 324 T + 103823 T^{2} \)
$53$ \( 1 - 258 T + 148877 T^{2} \)
$59$ \( 1 - 120 T + 205379 T^{2} \)
$61$ \( 1 - 622 T + 226981 T^{2} \)
$67$ \( 1 - 904 T + 300763 T^{2} \)
$71$ \( 1 + 678 T + 357911 T^{2} \)
$73$ \( 1 + 642 T + 389017 T^{2} \)
$79$ \( 1 - 740 T + 493039 T^{2} \)
$83$ \( 1 - 468 T + 571787 T^{2} \)
$89$ \( 1 - 200 T + 704969 T^{2} \)
$97$ \( 1 + 1266 T + 912673 T^{2} \)
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