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 about

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

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

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 −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:

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

Atkin-Lehner signs

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

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T \)
$3$ \( 1 + 29 T + 243 T^{2} \)
$5$ \( 1 + 31 T + 3125 T^{2} \)
$7$ \( 1 + 230 T + 16807 T^{2} \)
$11$ \( 1 - 121 T \)
$13$ \( 1 - 112 T + 371293 T^{2} \)
$17$ \( 1 + 1142 T + 1419857 T^{2} \)
$19$ \( 1 + 612 T + 2476099 T^{2} \)
$23$ \( 1 + 1941 T + 6436343 T^{2} \)
$29$ \( 1 - 1192 T + 20511149 T^{2} \)
$31$ \( 1 + 1037 T + 28629151 T^{2} \)
$37$ \( 1 - 8083 T + 69343957 T^{2} \)
$41$ \( 1 + 10444 T + 115856201 T^{2} \)
$43$ \( 1 - 58 T + 147008443 T^{2} \)
$47$ \( 1 - 8656 T + 229345007 T^{2} \)
$53$ \( 1 + 20318 T + 418195493 T^{2} \)
$59$ \( 1 + 21351 T + 714924299 T^{2} \)
$61$ \( 1 - 47044 T + 844596301 T^{2} \)
$67$ \( 1 - 48093 T + 1350125107 T^{2} \)
$71$ \( 1 + 24967 T + 1804229351 T^{2} \)
$73$ \( 1 + 42288 T + 2073071593 T^{2} \)
$79$ \( 1 + 72410 T + 3077056399 T^{2} \)
$83$ \( 1 + 15806 T + 3939040643 T^{2} \)
$89$ \( 1 + 114761 T + 5584059449 T^{2} \)
$97$ \( 1 + 5159 T + 8587340257 T^{2} \)
show more
show less