Properties

Label 9.8.a.b
Level 9
Weight 8
Character orbit 9.a
Self dual yes
Analytic conductor 2.811
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.81146522936\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Defining polynomial: \(x^{2} - 10\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 232 q^{4} -16 \beta q^{5} + 260 q^{7} + 104 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 232 q^{4} -16 \beta q^{5} + 260 q^{7} + 104 \beta q^{8} -5760 q^{10} -320 \beta q^{11} + 6890 q^{13} + 260 \beta q^{14} + 7744 q^{16} + 1248 \beta q^{17} + 33176 q^{19} -3712 \beta q^{20} -115200 q^{22} + 1664 \beta q^{23} + 14035 q^{25} + 6890 \beta q^{26} + 60320 q^{28} -7280 \beta q^{29} + 1508 q^{31} -5568 \beta q^{32} + 449280 q^{34} -4160 \beta q^{35} -380770 q^{37} + 33176 \beta q^{38} -599040 q^{40} + 4640 \beta q^{41} + 7640 q^{43} -74240 \beta q^{44} + 599040 q^{46} + 29824 \beta q^{47} -755943 q^{49} + 14035 \beta q^{50} + 1598480 q^{52} + 54288 \beta q^{53} + 1843200 q^{55} + 27040 \beta q^{56} -2620800 q^{58} -142720 \beta q^{59} -988858 q^{61} + 1508 \beta q^{62} -2995712 q^{64} -110240 \beta q^{65} + 3857360 q^{67} + 289536 \beta q^{68} -1497600 q^{70} + 222720 \beta q^{71} -2004730 q^{73} -380770 \beta q^{74} + 7696832 q^{76} -83200 \beta q^{77} + 2699684 q^{79} -123904 \beta q^{80} + 1670400 q^{82} + 142912 \beta q^{83} -7188480 q^{85} + 7640 \beta q^{86} -11980800 q^{88} + 408000 \beta q^{89} + 1791400 q^{91} + 386048 \beta q^{92} + 10736640 q^{94} -530816 \beta q^{95} -12957490 q^{97} -755943 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 464q^{4} + 520q^{7} + O(q^{10}) \) \( 2q + 464q^{4} + 520q^{7} - 11520q^{10} + 13780q^{13} + 15488q^{16} + 66352q^{19} - 230400q^{22} + 28070q^{25} + 120640q^{28} + 3016q^{31} + 898560q^{34} - 761540q^{37} - 1198080q^{40} + 15280q^{43} + 1198080q^{46} - 1511886q^{49} + 3196960q^{52} + 3686400q^{55} - 5241600q^{58} - 1977716q^{61} - 5991424q^{64} + 7714720q^{67} - 2995200q^{70} - 4009460q^{73} + 15393664q^{76} + 5399368q^{79} + 3340800q^{82} - 14376960q^{85} - 23961600q^{88} + 3582800q^{91} + 21473280q^{94} - 25914980q^{97} + O(q^{100}) \)

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
−3.16228
3.16228
−18.9737 0 232.000 303.579 0 260.000 −1973.26 0 −5760.00
1.2 18.9737 0 232.000 −303.579 0 260.000 1973.26 0 −5760.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.8.a.b 2
3.b odd 2 1 inner 9.8.a.b 2
4.b odd 2 1 144.8.a.m 2
5.b even 2 1 225.8.a.q 2
5.c odd 4 2 225.8.b.k 4
7.b odd 2 1 441.8.a.k 2
8.b even 2 1 576.8.a.bj 2
8.d odd 2 1 576.8.a.bi 2
9.c even 3 2 81.8.c.f 4
9.d odd 6 2 81.8.c.f 4
12.b even 2 1 144.8.a.m 2
15.d odd 2 1 225.8.a.q 2
15.e even 4 2 225.8.b.k 4
21.c even 2 1 441.8.a.k 2
24.f even 2 1 576.8.a.bi 2
24.h odd 2 1 576.8.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.8.a.b 2 1.a even 1 1 trivial
9.8.a.b 2 3.b odd 2 1 inner
81.8.c.f 4 9.c even 3 2
81.8.c.f 4 9.d odd 6 2
144.8.a.m 2 4.b odd 2 1
144.8.a.m 2 12.b even 2 1
225.8.a.q 2 5.b even 2 1
225.8.a.q 2 15.d odd 2 1
225.8.b.k 4 5.c odd 4 2
225.8.b.k 4 15.e even 4 2
441.8.a.k 2 7.b odd 2 1
441.8.a.k 2 21.c even 2 1
576.8.a.bi 2 8.d odd 2 1
576.8.a.bi 2 24.f even 2 1
576.8.a.bj 2 8.b even 2 1
576.8.a.bj 2 24.h odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 360 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 104 T^{2} + 16384 T^{4} \)
$3$ 1
$5$ \( 1 + 64090 T^{2} + 6103515625 T^{4} \)
$7$ \( ( 1 - 260 T + 823543 T^{2} )^{2} \)
$11$ \( 1 + 2110342 T^{2} + 379749833583241 T^{4} \)
$13$ \( ( 1 - 6890 T + 62748517 T^{2} )^{2} \)
$17$ \( 1 + 259975906 T^{2} + 168377826559400929 T^{4} \)
$19$ \( ( 1 - 33176 T + 893871739 T^{2} )^{2} \)
$23$ \( 1 + 5812848334 T^{2} + 11592836324538749809 T^{4} \)
$29$ \( 1 + 15420328618 T^{2} + \)\(29\!\cdots\!81\)\( T^{4} \)
$31$ \( ( 1 - 1508 T + 27512614111 T^{2} )^{2} \)
$37$ \( ( 1 + 380770 T + 94931877133 T^{2} )^{2} \)
$41$ \( 1 + 381757891762 T^{2} + \)\(37\!\cdots\!61\)\( T^{4} \)
$43$ \( ( 1 - 7640 T + 271818611107 T^{2} )^{2} \)
$47$ \( 1 + 693036689566 T^{2} + \)\(25\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 + 1288434979834 T^{2} + \)\(13\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 - 2355536454362 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} \)
$61$ \( ( 1 + 988858 T + 3142742836021 T^{2} )^{2} \)
$67$ \( ( 1 - 3857360 T + 6060711605323 T^{2} )^{2} \)
$71$ \( 1 + 332728892782 T^{2} + \)\(82\!\cdots\!81\)\( T^{4} \)
$73$ \( ( 1 + 2004730 T + 11047398519097 T^{2} )^{2} \)
$79$ \( ( 1 - 2699684 T + 19203908986159 T^{2} )^{2} \)
$83$ \( 1 + 46919519671414 T^{2} + \)\(73\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 + 28535629791058 T^{2} + \)\(19\!\cdots\!41\)\( T^{4} \)
$97$ \( ( 1 + 12957490 T + 80798284478113 T^{2} )^{2} \)
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