Properties

Label 21.8.a.a
Level $21$
Weight $8$
Character orbit 21.a
Self dual yes
Analytic conductor $6.560$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.56008553517\)
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

\(f(q)\) \(=\) \( q + 2q^{2} + 27q^{3} - 124q^{4} - 278q^{5} + 54q^{6} - 343q^{7} - 504q^{8} + 729q^{9} + O(q^{10}) \) \( q + 2q^{2} + 27q^{3} - 124q^{4} - 278q^{5} + 54q^{6} - 343q^{7} - 504q^{8} + 729q^{9} - 556q^{10} - 4496q^{11} - 3348q^{12} - 7274q^{13} - 686q^{14} - 7506q^{15} + 14864q^{16} + 11382q^{17} + 1458q^{18} - 15884q^{19} + 34472q^{20} - 9261q^{21} - 8992q^{22} + 86100q^{23} - 13608q^{24} - 841q^{25} - 14548q^{26} + 19683q^{27} + 42532q^{28} + 40702q^{29} - 15012q^{30} - 44760q^{31} + 94240q^{32} - 121392q^{33} + 22764q^{34} + 95354q^{35} - 90396q^{36} - 580962q^{37} - 31768q^{38} - 196398q^{39} + 140112q^{40} - 171658q^{41} - 18522q^{42} - 741148q^{43} + 557504q^{44} - 202662q^{45} + 172200q^{46} + 1071720q^{47} + 401328q^{48} + 117649q^{49} - 1682q^{50} + 307314q^{51} + 901976q^{52} - 1721778q^{53} + 39366q^{54} + 1249888q^{55} + 172872q^{56} - 428868q^{57} + 81404q^{58} - 1557012q^{59} + 930744q^{60} + 2597998q^{61} - 89520q^{62} - 250047q^{63} - 1714112q^{64} + 2022172q^{65} - 242784q^{66} - 963548q^{67} - 1411368q^{68} + 2324700q^{69} + 190708q^{70} - 4063380q^{71} - 367416q^{72} - 5370222q^{73} - 1161924q^{74} - 22707q^{75} + 1969616q^{76} + 1542128q^{77} - 392796q^{78} + 4094936q^{79} - 4132192q^{80} + 531441q^{81} - 343316q^{82} - 1343124q^{83} + 1148364q^{84} - 3164196q^{85} - 1482296q^{86} + 1098954q^{87} + 2265984q^{88} + 9081574q^{89} - 405324q^{90} + 2494982q^{91} - 10676400q^{92} - 1208520q^{93} + 2143440q^{94} + 4415752q^{95} + 2544480q^{96} + 6487914q^{97} + 235298q^{98} - 3277584q^{99} + O(q^{100}) \)

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
2.00000 27.0000 −124.000 −278.000 54.0000 −343.000 −504.000 729.000 −556.000
\(n\): 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 21.8.a.a 1
3.b odd 2 1 63.8.a.a 1
4.b odd 2 1 336.8.a.b 1
5.b even 2 1 525.8.a.b 1
7.b odd 2 1 147.8.a.a 1
7.c even 3 2 147.8.e.c 2
7.d odd 6 2 147.8.e.d 2
21.c even 2 1 441.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.8.a.a 1 1.a even 1 1 trivial
63.8.a.a 1 3.b odd 2 1
147.8.a.a 1 7.b odd 2 1
147.8.e.c 2 7.c even 3 2
147.8.e.d 2 7.d odd 6 2
336.8.a.b 1 4.b odd 2 1
441.8.a.b 1 21.c even 2 1
525.8.a.b 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(21))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( -27 + T \)
$5$ \( 278 + T \)
$7$ \( 343 + T \)
$11$ \( 4496 + T \)
$13$ \( 7274 + T \)
$17$ \( -11382 + T \)
$19$ \( 15884 + T \)
$23$ \( -86100 + T \)
$29$ \( -40702 + T \)
$31$ \( 44760 + T \)
$37$ \( 580962 + T \)
$41$ \( 171658 + T \)
$43$ \( 741148 + T \)
$47$ \( -1071720 + T \)
$53$ \( 1721778 + T \)
$59$ \( 1557012 + T \)
$61$ \( -2597998 + T \)
$67$ \( 963548 + T \)
$71$ \( 4063380 + T \)
$73$ \( 5370222 + T \)
$79$ \( -4094936 + T \)
$83$ \( 1343124 + T \)
$89$ \( -9081574 + T \)
$97$ \( -6487914 + T \)
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