Properties

Label 9.22.a.e
Level $9$
Weight $22$
Character orbit 9.a
Self dual yes
Analytic conductor $25.153$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.1529609858\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{649}) \)
Defining polynomial: \( x^{2} - x - 162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 63\sqrt{649}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 333) q^{2} + (666 \beta + 589618) q^{4} + ( - 13568 \beta - 498438) q^{5} + ( - 182016 \beta + 339948056) q^{7} + (1285756 \beta - 1213527924) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 333) q^{2} + (666 \beta + 589618) q^{4} + ( - 13568 \beta - 498438) q^{5} + ( - 182016 \beta + 339948056) q^{7} + (1285756 \beta - 1213527924) q^{8} + (5016582 \beta + 35115533262) q^{10} + ( - 31263232 \beta - 109934561484) q^{11} + ( - 475236864 \beta - 24234454978) q^{13} + ( - 279336728 \beta + 355648853448) q^{14} + ( - 611332056 \beta - 4144368220280) q^{16} + ( - 1677304320 \beta + 5666764520718) q^{17} + ( - 22384332288 \beta + 5980292505812) q^{19} + ( - 8331896732 \beta - 23570290586412) q^{20} + (120345217740 \beta + 117138574281564) q^{22} + (73613782528 \beta + 73254195031752) q^{23} + (13525613568 \beta - 2393177123537) q^{25} + (182488330690 \beta + 12\!\cdots\!58) q^{26}+ \cdots + (39\!\cdots\!71 \beta + 43\!\cdots\!07) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 666 q^{2} + 1179236 q^{4} - 996876 q^{5} + 679896112 q^{7} - 2427055848 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 666 q^{2} + 1179236 q^{4} - 996876 q^{5} + 679896112 q^{7} - 2427055848 q^{8} + 70231066524 q^{10} - 219869122968 q^{11} - 48468909956 q^{13} + 711297706896 q^{14} - 8288736440560 q^{16} + 11333529041436 q^{17} + 11960585011624 q^{19} - 47140581172824 q^{20} + 234277148563128 q^{22} + 146508390063504 q^{23} - 4786354247074 q^{25} + 24\!\cdots\!16 q^{26}+ \cdots + 87\!\cdots\!14 q^{98}+O(q^{100}) \) 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
13.2377
−12.2377
−1937.96 0 1.65852e6 −2.22745e7 0 4.78205e7 8.50053e8 0 4.31669e10
1.2 1271.96 0 −479282. 2.12776e7 0 6.32076e8 −3.27711e9 0 2.70641e10
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

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 9.22.a.e 2
3.b odd 2 1 3.22.a.c 2
12.b even 2 1 48.22.a.g 2
15.d odd 2 1 75.22.a.d 2
15.e even 4 2 75.22.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.c 2 3.b odd 2 1
9.22.a.e 2 1.a even 1 1 trivial
48.22.a.g 2 12.b even 2 1
75.22.a.d 2 15.d odd 2 1
75.22.b.d 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 666T_{2} - 2464992 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 666 T - 2464992 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 473947100199900 \) Copy content Toggle raw display
$7$ \( T^{2} - 679896112 T + 30\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{2} + 219869122968 T + 95\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{2} + 48468909956 T - 58\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{2} - 11333529041436 T + 24\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{2} - 11960585011624 T - 12\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} - 146508390063504 T - 85\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 54\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 32\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 20\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 83\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 74\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 73\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 11\!\cdots\!40 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 33\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 17\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 27\!\cdots\!76 \) Copy content Toggle raw display
show more
show less