Properties

Label 21.6.a.d
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

\(f(q)\) \(=\) \( q + 10q^{2} + 9q^{3} + 68q^{4} - 106q^{5} + 90q^{6} - 49q^{7} + 360q^{8} + 81q^{9} + O(q^{10}) \) \( q + 10q^{2} + 9q^{3} + 68q^{4} - 106q^{5} + 90q^{6} - 49q^{7} + 360q^{8} + 81q^{9} - 1060q^{10} + 92q^{11} + 612q^{12} + 670q^{13} - 490q^{14} - 954q^{15} + 1424q^{16} - 222q^{17} + 810q^{18} - 908q^{19} - 7208q^{20} - 441q^{21} + 920q^{22} - 1176q^{23} + 3240q^{24} + 8111q^{25} + 6700q^{26} + 729q^{27} - 3332q^{28} + 1118q^{29} - 9540q^{30} + 3696q^{31} + 2720q^{32} + 828q^{33} - 2220q^{34} + 5194q^{35} + 5508q^{36} + 4182q^{37} - 9080q^{38} + 6030q^{39} - 38160q^{40} - 6662q^{41} - 4410q^{42} - 3700q^{43} + 6256q^{44} - 8586q^{45} - 11760q^{46} - 7056q^{47} + 12816q^{48} + 2401q^{49} + 81110q^{50} - 1998q^{51} + 45560q^{52} - 37578q^{53} + 7290q^{54} - 9752q^{55} - 17640q^{56} - 8172q^{57} + 11180q^{58} + 32700q^{59} - 64872q^{60} - 10802q^{61} + 36960q^{62} - 3969q^{63} - 18368q^{64} - 71020q^{65} + 8280q^{66} + 64996q^{67} - 15096q^{68} - 10584q^{69} + 51940q^{70} - 61320q^{71} + 29160q^{72} + 38922q^{73} + 41820q^{74} + 72999q^{75} - 61744q^{76} - 4508q^{77} + 60300q^{78} - 88096q^{79} - 150944q^{80} + 6561q^{81} - 66620q^{82} + 71892q^{83} - 29988q^{84} + 23532q^{85} - 37000q^{86} + 10062q^{87} + 33120q^{88} + 111818q^{89} - 85860q^{90} - 32830q^{91} - 79968q^{92} + 33264q^{93} - 70560q^{94} + 96248q^{95} + 24480q^{96} - 150846q^{97} + 24010q^{98} + 7452q^{99} + O(q^{100}) \)

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
10.0000 9.00000 68.0000 −106.000 90.0000 −49.0000 360.000 81.0000 −1060.00
\(n\): 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 21.6.a.d 1
3.b odd 2 1 63.6.a.a 1
4.b odd 2 1 336.6.a.a 1
5.b even 2 1 525.6.a.a 1
5.c odd 4 2 525.6.d.a 2
7.b odd 2 1 147.6.a.g 1
7.c even 3 2 147.6.e.a 2
7.d odd 6 2 147.6.e.b 2
12.b even 2 1 1008.6.a.bc 1
21.c even 2 1 441.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.d 1 1.a even 1 1 trivial
63.6.a.a 1 3.b odd 2 1
147.6.a.g 1 7.b odd 2 1
147.6.e.a 2 7.c even 3 2
147.6.e.b 2 7.d odd 6 2
336.6.a.a 1 4.b odd 2 1
441.6.a.b 1 21.c even 2 1
525.6.a.a 1 5.b even 2 1
525.6.d.a 2 5.c odd 4 2
1008.6.a.bc 1 12.b 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 \( T_{2} - 10 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 10 T + 32 T^{2} \)
$3$ \( 1 - 9 T \)
$5$ \( 1 + 106 T + 3125 T^{2} \)
$7$ \( 1 + 49 T \)
$11$ \( 1 - 92 T + 161051 T^{2} \)
$13$ \( 1 - 670 T + 371293 T^{2} \)
$17$ \( 1 + 222 T + 1419857 T^{2} \)
$19$ \( 1 + 908 T + 2476099 T^{2} \)
$23$ \( 1 + 1176 T + 6436343 T^{2} \)
$29$ \( 1 - 1118 T + 20511149 T^{2} \)
$31$ \( 1 - 3696 T + 28629151 T^{2} \)
$37$ \( 1 - 4182 T + 69343957 T^{2} \)
$41$ \( 1 + 6662 T + 115856201 T^{2} \)
$43$ \( 1 + 3700 T + 147008443 T^{2} \)
$47$ \( 1 + 7056 T + 229345007 T^{2} \)
$53$ \( 1 + 37578 T + 418195493 T^{2} \)
$59$ \( 1 - 32700 T + 714924299 T^{2} \)
$61$ \( 1 + 10802 T + 844596301 T^{2} \)
$67$ \( 1 - 64996 T + 1350125107 T^{2} \)
$71$ \( 1 + 61320 T + 1804229351 T^{2} \)
$73$ \( 1 - 38922 T + 2073071593 T^{2} \)
$79$ \( 1 + 88096 T + 3077056399 T^{2} \)
$83$ \( 1 - 71892 T + 3939040643 T^{2} \)
$89$ \( 1 - 111818 T + 5584059449 T^{2} \)
$97$ \( 1 + 150846 T + 8587340257 T^{2} \)
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