Properties

Label 21.6.a.a
Level 21
Weight 6
Character orbit 21.a
Self dual yes
Analytic conductor 3.368
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.36806021607\)
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

\(f(q)\) \(=\) \( q - 6q^{2} - 9q^{3} + 4q^{4} + 78q^{5} + 54q^{6} + 49q^{7} + 168q^{8} + 81q^{9} + O(q^{10}) \) \( q - 6q^{2} - 9q^{3} + 4q^{4} + 78q^{5} + 54q^{6} + 49q^{7} + 168q^{8} + 81q^{9} - 468q^{10} + 444q^{11} - 36q^{12} - 442q^{13} - 294q^{14} - 702q^{15} - 1136q^{16} - 126q^{17} - 486q^{18} + 2684q^{19} + 312q^{20} - 441q^{21} - 2664q^{22} + 4200q^{23} - 1512q^{24} + 2959q^{25} + 2652q^{26} - 729q^{27} + 196q^{28} - 5442q^{29} + 4212q^{30} + 80q^{31} + 1440q^{32} - 3996q^{33} + 756q^{34} + 3822q^{35} + 324q^{36} - 5434q^{37} - 16104q^{38} + 3978q^{39} + 13104q^{40} + 7962q^{41} + 2646q^{42} - 11524q^{43} + 1776q^{44} + 6318q^{45} - 25200q^{46} - 13920q^{47} + 10224q^{48} + 2401q^{49} - 17754q^{50} + 1134q^{51} - 1768q^{52} - 9594q^{53} + 4374q^{54} + 34632q^{55} + 8232q^{56} - 24156q^{57} + 32652q^{58} + 27492q^{59} - 2808q^{60} + 49478q^{61} - 480q^{62} + 3969q^{63} + 27712q^{64} - 34476q^{65} + 23976q^{66} - 59356q^{67} - 504q^{68} - 37800q^{69} - 22932q^{70} + 32040q^{71} + 13608q^{72} - 61846q^{73} + 32604q^{74} - 26631q^{75} + 10736q^{76} + 21756q^{77} - 23868q^{78} - 65776q^{79} - 88608q^{80} + 6561q^{81} - 47772q^{82} + 40188q^{83} - 1764q^{84} - 9828q^{85} + 69144q^{86} + 48978q^{87} + 74592q^{88} - 7974q^{89} - 37908q^{90} - 21658q^{91} + 16800q^{92} - 720q^{93} + 83520q^{94} + 209352q^{95} - 12960q^{96} - 143662q^{97} - 14406q^{98} + 35964q^{99} + O(q^{100}) \)

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
−6.00000 −9.00000 4.00000 78.0000 54.0000 49.0000 168.000 81.0000 −468.000
\(n\): 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 21.6.a.a 1
3.b odd 2 1 63.6.a.d 1
4.b odd 2 1 336.6.a.r 1
5.b even 2 1 525.6.a.d 1
5.c odd 4 2 525.6.d.b 2
7.b odd 2 1 147.6.a.b 1
7.c even 3 2 147.6.e.j 2
7.d odd 6 2 147.6.e.i 2
12.b even 2 1 1008.6.a.c 1
21.c even 2 1 441.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.a 1 1.a even 1 1 trivial
63.6.a.d 1 3.b odd 2 1
147.6.a.b 1 7.b odd 2 1
147.6.e.i 2 7.d odd 6 2
147.6.e.j 2 7.c even 3 2
336.6.a.r 1 4.b odd 2 1
441.6.a.j 1 21.c even 2 1
525.6.a.d 1 5.b even 2 1
525.6.d.b 2 5.c odd 4 2
1008.6.a.c 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 32 T^{2} \)
$3$ \( 1 + 9 T \)
$5$ \( 1 - 78 T + 3125 T^{2} \)
$7$ \( 1 - 49 T \)
$11$ \( 1 - 444 T + 161051 T^{2} \)
$13$ \( 1 + 442 T + 371293 T^{2} \)
$17$ \( 1 + 126 T + 1419857 T^{2} \)
$19$ \( 1 - 2684 T + 2476099 T^{2} \)
$23$ \( 1 - 4200 T + 6436343 T^{2} \)
$29$ \( 1 + 5442 T + 20511149 T^{2} \)
$31$ \( 1 - 80 T + 28629151 T^{2} \)
$37$ \( 1 + 5434 T + 69343957 T^{2} \)
$41$ \( 1 - 7962 T + 115856201 T^{2} \)
$43$ \( 1 + 11524 T + 147008443 T^{2} \)
$47$ \( 1 + 13920 T + 229345007 T^{2} \)
$53$ \( 1 + 9594 T + 418195493 T^{2} \)
$59$ \( 1 - 27492 T + 714924299 T^{2} \)
$61$ \( 1 - 49478 T + 844596301 T^{2} \)
$67$ \( 1 + 59356 T + 1350125107 T^{2} \)
$71$ \( 1 - 32040 T + 1804229351 T^{2} \)
$73$ \( 1 + 61846 T + 2073071593 T^{2} \)
$79$ \( 1 + 65776 T + 3077056399 T^{2} \)
$83$ \( 1 - 40188 T + 3939040643 T^{2} \)
$89$ \( 1 + 7974 T + 5584059449 T^{2} \)
$97$ \( 1 + 143662 T + 8587340257 T^{2} \)
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