Properties

Label 22.6.a.c
Level $22$
Weight $6$
Character orbit 22.a
Self dual yes
Analytic conductor $3.528$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.52844403589\)
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

\(f(q)\) \(=\) \( q + 4 q^{2} - 29 q^{3} + 16 q^{4} - 31 q^{5} - 116 q^{6} - 230 q^{7} + 64 q^{8} + 598 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - 29 q^{3} + 16 q^{4} - 31 q^{5} - 116 q^{6} - 230 q^{7} + 64 q^{8} + 598 q^{9} - 124 q^{10} + 121 q^{11} - 464 q^{12} + 112 q^{13} - 920 q^{14} + 899 q^{15} + 256 q^{16} - 1142 q^{17} + 2392 q^{18} - 612 q^{19} - 496 q^{20} + 6670 q^{21} + 484 q^{22} - 1941 q^{23} - 1856 q^{24} - 2164 q^{25} + 448 q^{26} - 10295 q^{27} - 3680 q^{28} + 1192 q^{29} + 3596 q^{30} - 1037 q^{31} + 1024 q^{32} - 3509 q^{33} - 4568 q^{34} + 7130 q^{35} + 9568 q^{36} + 8083 q^{37} - 2448 q^{38} - 3248 q^{39} - 1984 q^{40} - 10444 q^{41} + 26680 q^{42} + 58 q^{43} + 1936 q^{44} - 18538 q^{45} - 7764 q^{46} + 8656 q^{47} - 7424 q^{48} + 36093 q^{49} - 8656 q^{50} + 33118 q^{51} + 1792 q^{52} - 20318 q^{53} - 41180 q^{54} - 3751 q^{55} - 14720 q^{56} + 17748 q^{57} + 4768 q^{58} - 21351 q^{59} + 14384 q^{60} + 47044 q^{61} - 4148 q^{62} - 137540 q^{63} + 4096 q^{64} - 3472 q^{65} - 14036 q^{66} + 48093 q^{67} - 18272 q^{68} + 56289 q^{69} + 28520 q^{70} - 24967 q^{71} + 38272 q^{72} - 42288 q^{73} + 32332 q^{74} + 62756 q^{75} - 9792 q^{76} - 27830 q^{77} - 12992 q^{78} - 72410 q^{79} - 7936 q^{80} + 153241 q^{81} - 41776 q^{82} - 15806 q^{83} + 106720 q^{84} + 35402 q^{85} + 232 q^{86} - 34568 q^{87} + 7744 q^{88} - 114761 q^{89} - 74152 q^{90} - 25760 q^{91} - 31056 q^{92} + 30073 q^{93} + 34624 q^{94} + 18972 q^{95} - 29696 q^{96} - 5159 q^{97} + 144372 q^{98} + 72358 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −29.0000 16.0000 −31.0000 −116.000 −230.000 64.0000 598.000 −124.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)

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 22.6.a.c 1
3.b odd 2 1 198.6.a.b 1
4.b odd 2 1 176.6.a.e 1
5.b even 2 1 550.6.a.c 1
5.c odd 4 2 550.6.b.a 2
7.b odd 2 1 1078.6.a.f 1
8.b even 2 1 704.6.a.j 1
8.d odd 2 1 704.6.a.a 1
11.b odd 2 1 242.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.c 1 1.a even 1 1 trivial
176.6.a.e 1 4.b odd 2 1
198.6.a.b 1 3.b odd 2 1
242.6.a.a 1 11.b odd 2 1
550.6.a.c 1 5.b even 2 1
550.6.b.a 2 5.c odd 4 2
704.6.a.a 1 8.d odd 2 1
704.6.a.j 1 8.b even 2 1
1078.6.a.f 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 29 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(22))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T + 29 \) Copy content Toggle raw display
$5$ \( T + 31 \) Copy content Toggle raw display
$7$ \( T + 230 \) Copy content Toggle raw display
$11$ \( T - 121 \) Copy content Toggle raw display
$13$ \( T - 112 \) Copy content Toggle raw display
$17$ \( T + 1142 \) Copy content Toggle raw display
$19$ \( T + 612 \) Copy content Toggle raw display
$23$ \( T + 1941 \) Copy content Toggle raw display
$29$ \( T - 1192 \) Copy content Toggle raw display
$31$ \( T + 1037 \) Copy content Toggle raw display
$37$ \( T - 8083 \) Copy content Toggle raw display
$41$ \( T + 10444 \) Copy content Toggle raw display
$43$ \( T - 58 \) Copy content Toggle raw display
$47$ \( T - 8656 \) Copy content Toggle raw display
$53$ \( T + 20318 \) Copy content Toggle raw display
$59$ \( T + 21351 \) Copy content Toggle raw display
$61$ \( T - 47044 \) Copy content Toggle raw display
$67$ \( T - 48093 \) Copy content Toggle raw display
$71$ \( T + 24967 \) Copy content Toggle raw display
$73$ \( T + 42288 \) Copy content Toggle raw display
$79$ \( T + 72410 \) Copy content Toggle raw display
$83$ \( T + 15806 \) Copy content Toggle raw display
$89$ \( T + 114761 \) Copy content Toggle raw display
$97$ \( T + 5159 \) Copy content Toggle raw display
show more
show less