Properties

Label 22.10.a.c
Level $22$
Weight $10$
Character orbit 22.a
Self dual yes
Analytic conductor $11.331$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.3307883956\)
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

\(f(q)\) \(=\) \( q + 16 q^{2} + 201 q^{3} + 256 q^{4} + 2349 q^{5} + 3216 q^{6} - 8806 q^{7} + 4096 q^{8} + 20718 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{2} + 201 q^{3} + 256 q^{4} + 2349 q^{5} + 3216 q^{6} - 8806 q^{7} + 4096 q^{8} + 20718 q^{9} + 37584 q^{10} - 14641 q^{11} + 51456 q^{12} - 131068 q^{13} - 140896 q^{14} + 472149 q^{15} + 65536 q^{16} - 55698 q^{17} + 331488 q^{18} + 1041824 q^{19} + 601344 q^{20} - 1770006 q^{21} - 234256 q^{22} - 662139 q^{23} + 823296 q^{24} + 3564676 q^{25} - 2097088 q^{26} + 208035 q^{27} - 2254336 q^{28} - 4819344 q^{29} + 7554384 q^{30} - 180115 q^{31} + 1048576 q^{32} - 2942841 q^{33} - 891168 q^{34} - 20685294 q^{35} + 5303808 q^{36} - 7803025 q^{37} + 16669184 q^{38} - 26344668 q^{39} + 9621504 q^{40} - 5927736 q^{41} - 28320096 q^{42} - 5929162 q^{43} - 3748096 q^{44} + 48666582 q^{45} - 10594224 q^{46} + 61576176 q^{47} + 13172736 q^{48} + 37192029 q^{49} + 57034816 q^{50} - 11195298 q^{51} - 33553408 q^{52} + 7349514 q^{53} + 3328560 q^{54} - 34391709 q^{55} - 36069376 q^{56} + 209406624 q^{57} - 77109504 q^{58} - 113901909 q^{59} + 120870144 q^{60} - 13814260 q^{61} - 2881840 q^{62} - 182442708 q^{63} + 16777216 q^{64} - 307878732 q^{65} - 47085456 q^{66} + 309980903 q^{67} - 14258688 q^{68} - 133089939 q^{69} - 330964704 q^{70} + 42752631 q^{71} + 84860928 q^{72} + 142018340 q^{73} - 124848400 q^{74} + 716499876 q^{75} + 266706944 q^{76} + 128928646 q^{77} - 421514688 q^{78} - 325376446 q^{79} + 153944064 q^{80} - 365977359 q^{81} - 94843776 q^{82} + 253502934 q^{83} - 453121536 q^{84} - 130834602 q^{85} - 94866592 q^{86} - 968688144 q^{87} - 59969536 q^{88} - 994227705 q^{89} + 778665312 q^{90} + 1154184808 q^{91} - 169507584 q^{92} - 36203115 q^{93} + 985218816 q^{94} + 2447244576 q^{95} + 210763776 q^{96} - 352091047 q^{97} + 595072464 q^{98} - 303332238 q^{99}+O(q^{100}) \) 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
16.0000 201.000 256.000 2349.00 3216.00 −8806.00 4096.00 20718.0 37584.0
\(n\): 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 22.10.a.c 1
3.b odd 2 1 198.10.a.a 1
4.b odd 2 1 176.10.a.a 1
11.b odd 2 1 242.10.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.10.a.c 1 1.a even 1 1 trivial
176.10.a.a 1 4.b odd 2 1
198.10.a.a 1 3.b odd 2 1
242.10.a.d 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 201 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(22))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T - 201 \) Copy content Toggle raw display
$5$ \( T - 2349 \) Copy content Toggle raw display
$7$ \( T + 8806 \) Copy content Toggle raw display
$11$ \( T + 14641 \) Copy content Toggle raw display
$13$ \( T + 131068 \) Copy content Toggle raw display
$17$ \( T + 55698 \) Copy content Toggle raw display
$19$ \( T - 1041824 \) Copy content Toggle raw display
$23$ \( T + 662139 \) Copy content Toggle raw display
$29$ \( T + 4819344 \) Copy content Toggle raw display
$31$ \( T + 180115 \) Copy content Toggle raw display
$37$ \( T + 7803025 \) Copy content Toggle raw display
$41$ \( T + 5927736 \) Copy content Toggle raw display
$43$ \( T + 5929162 \) Copy content Toggle raw display
$47$ \( T - 61576176 \) Copy content Toggle raw display
$53$ \( T - 7349514 \) Copy content Toggle raw display
$59$ \( T + 113901909 \) Copy content Toggle raw display
$61$ \( T + 13814260 \) Copy content Toggle raw display
$67$ \( T - 309980903 \) Copy content Toggle raw display
$71$ \( T - 42752631 \) Copy content Toggle raw display
$73$ \( T - 142018340 \) Copy content Toggle raw display
$79$ \( T + 325376446 \) Copy content Toggle raw display
$83$ \( T - 253502934 \) Copy content Toggle raw display
$89$ \( T + 994227705 \) Copy content Toggle raw display
$97$ \( T + 352091047 \) Copy content Toggle raw display
show more
show less