Properties

Label 1.22.a.a
Level $1$
Weight $22$
Character orbit 1.a
Self dual yes
Analytic conductor $2.795$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.79477344287\)
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

\(f(q)\) \(=\) \( q - 288 q^{2} - 128844 q^{3} - 2014208 q^{4} + 21640950 q^{5} + 37107072 q^{6} - 768078808 q^{7} + 1184071680 q^{8} + 6140423133 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 288 q^{2} - 128844 q^{3} - 2014208 q^{4} + 21640950 q^{5} + 37107072 q^{6} - 768078808 q^{7} + 1184071680 q^{8} + 6140423133 q^{9} - 6232593600 q^{10} - 94724929188 q^{11} + 259518615552 q^{12} - 80621789794 q^{13} + 221206696704 q^{14} - 2788306561800 q^{15} + 3883087691776 q^{16} + 3052282930002 q^{17} - 1768441862304 q^{18} - 7920788351740 q^{19} - 43589374617600 q^{20} + 98962345937952 q^{21} + 27280779606144 q^{22} - 73845437470344 q^{23} - 152560531537920 q^{24} - 8506441300625 q^{25} + 23219075460672 q^{26} + 556597069939080 q^{27} + 15\!\cdots\!64 q^{28}+ \cdots - 58\!\cdots\!04 q^{99}+O(q^{100}) \) 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
−288.000 −128844. −2.01421e6 2.16410e7 3.71071e7 −7.68079e8 1.18407e9 6.14042e9 −6.23259e9
\(n\): 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 1.22.a.a 1
3.b odd 2 1 9.22.a.c 1
4.b odd 2 1 16.22.a.c 1
5.b even 2 1 25.22.a.a 1
5.c odd 4 2 25.22.b.a 2
7.b odd 2 1 49.22.a.a 1
8.b even 2 1 64.22.a.g 1
8.d odd 2 1 64.22.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.22.a.a 1 1.a even 1 1 trivial
9.22.a.c 1 3.b odd 2 1
16.22.a.c 1 4.b odd 2 1
25.22.a.a 1 5.b even 2 1
25.22.b.a 2 5.c odd 4 2
49.22.a.a 1 7.b odd 2 1
64.22.a.a 1 8.d odd 2 1
64.22.a.g 1 8.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 288 \) Copy content Toggle raw display
$3$ \( T + 128844 \) Copy content Toggle raw display
$5$ \( T - 21640950 \) Copy content Toggle raw display
$7$ \( T + 768078808 \) Copy content Toggle raw display
$11$ \( T + 94724929188 \) Copy content Toggle raw display
$13$ \( T + 80621789794 \) Copy content Toggle raw display
$17$ \( T - 3052282930002 \) Copy content Toggle raw display
$19$ \( T + 7920788351740 \) Copy content Toggle raw display
$23$ \( T + 73845437470344 \) Copy content Toggle raw display
$29$ \( T + 4253031736469010 \) Copy content Toggle raw display
$31$ \( T - 1900541176310432 \) Copy content Toggle raw display
$37$ \( T - 22\!\cdots\!22 \) Copy content Toggle raw display
$41$ \( T + 20\!\cdots\!58 \) Copy content Toggle raw display
$43$ \( T + 19\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T - 14\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T - 20\!\cdots\!06 \) Copy content Toggle raw display
$59$ \( T + 59\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T - 61\!\cdots\!62 \) Copy content Toggle raw display
$67$ \( T - 16\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T + 56\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T + 43\!\cdots\!94 \) Copy content Toggle raw display
$79$ \( T + 51\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T - 48\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T + 50\!\cdots\!30 \) Copy content Toggle raw display
$97$ \( T - 80\!\cdots\!82 \) Copy content Toggle raw display
show more
show less