Properties

Label 9.32.a.a
Level $9$
Weight $32$
Character orbit 9.a
Self dual yes
Analytic conductor $54.789$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.7894195371\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 4573872\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{18295489}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -19980 - \beta ) q^{2} + ( 886267168 + 39960 \beta ) q^{4} + ( 9695609010 + 1418560 \beta ) q^{5} + ( 15128763788600 - 71928864 \beta ) q^{7} + ( -80077529352960 + 462815680 \beta ) q^{8} +O(q^{10})\) \( q +(-19980 - \beta) q^{2} +(886267168 + 39960 \beta) q^{4} +(9695609010 + 1418560 \beta) q^{5} +(15128763788600 - 71928864 \beta) q^{7} +(-80077529352960 + 462815680 \beta) q^{8} +(-3930986106140760 - 38038437810 \beta) q^{10} +(3891176872559388 - 29000909200 \beta) q^{11} +(37354476525130310 - 4661016429888 \beta) q^{13} +(-112772481922620576 - 13691625085880 \beta) q^{14} +(-1522606456842450944 - 14982974507520 \beta) q^{16} +(-8612303914493544690 - 45085348093056 \beta) q^{17} +(-6185281664011082020 - 157805792764560 \beta) q^{19} +(\)\(15\!\cdots\!80\)\( + 1644659689877680 \beta) q^{20} +(-1341356516498345040 - 3311738706743388 \beta) q^{22} +(-\)\(94\!\cdots\!40\)\( + 18557618179251808 \beta) q^{23} +(\)\(73\!\cdots\!75\)\( + 27507606234451200 \beta) q^{25} +(\)\(11\!\cdots\!08\)\( + 55772631744031930 \beta) q^{26} +(\)\(58\!\cdots\!60\)\( + 540797210397718848 \beta) q^{28} +(-\)\(64\!\cdots\!70\)\( - 234192357384448960 \beta) q^{29} +(\)\(62\!\cdots\!32\)\( + 2412621171020361600 \beta) q^{31} +(\)\(24\!\cdots\!20\)\( + 828077182664699904 \beta) q^{32} +(\)\(29\!\cdots\!96\)\( + 9513109169392803570 \beta) q^{34} +(-\)\(12\!\cdots\!40\)\( + 20763665018078951360 \beta) q^{35} +(-\)\(41\!\cdots\!30\)\( + 32224244113578511296 \beta) q^{37} +(\)\(53\!\cdots\!60\)\( + 9338241403446990820 \beta) q^{38} +(\)\(95\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( \beta) q^{40} +(-\)\(43\!\cdots\!42\)\( - \)\(18\!\cdots\!00\)\( \beta) q^{41} +(-\)\(91\!\cdots\!00\)\( - \)\(34\!\cdots\!48\)\( \beta) q^{43} +(\)\(39\!\cdots\!84\)\( + \)\(12\!\cdots\!80\)\( \beta) q^{44} +(-\)\(29\!\cdots\!28\)\( + \)\(57\!\cdots\!00\)\( \beta) q^{46} +(-\)\(47\!\cdots\!60\)\( - \)\(51\!\cdots\!16\)\( \beta) q^{47} +(\)\(84\!\cdots\!93\)\( - \)\(21\!\cdots\!00\)\( \beta) q^{49} +(-\)\(87\!\cdots\!00\)\( - \)\(12\!\cdots\!75\)\( \beta) q^{50} +(-\)\(45\!\cdots\!00\)\( - \)\(26\!\cdots\!84\)\( \beta) q^{52} +(-\)\(97\!\cdots\!30\)\( + \)\(12\!\cdots\!68\)\( \beta) q^{53} +(-\)\(70\!\cdots\!20\)\( + \)\(52\!\cdots\!80\)\( \beta) q^{55} +(-\)\(12\!\cdots\!20\)\( + \)\(12\!\cdots\!40\)\( \beta) q^{56} +(\)\(19\!\cdots\!60\)\( + \)\(68\!\cdots\!70\)\( \beta) q^{58} +(\)\(99\!\cdots\!60\)\( + \)\(44\!\cdots\!80\)\( \beta) q^{59} +(-\)\(60\!\cdots\!38\)\( + \)\(11\!\cdots\!00\)\( \beta) q^{61} +(-\)\(76\!\cdots\!60\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{62} +(-\)\(37\!\cdots\!52\)\( - \)\(22\!\cdots\!80\)\( \beta) q^{64} +(-\)\(17\!\cdots\!80\)\( + \)\(77\!\cdots\!20\)\( \beta) q^{65} +(-\)\(48\!\cdots\!60\)\( - \)\(88\!\cdots\!44\)\( \beta) q^{67} +(-\)\(12\!\cdots\!80\)\( - \)\(38\!\cdots\!08\)\( \beta) q^{68} +(-\)\(52\!\cdots\!60\)\( - \)\(29\!\cdots\!60\)\( \beta) q^{70} +(-\)\(27\!\cdots\!72\)\( + \)\(95\!\cdots\!00\)\( \beta) q^{71} +(\)\(31\!\cdots\!90\)\( + \)\(15\!\cdots\!92\)\( \beta) q^{73} +(-\)\(76\!\cdots\!36\)\( - \)\(22\!\cdots\!50\)\( \beta) q^{74} +(-\)\(22\!\cdots\!60\)\( - \)\(38\!\cdots\!80\)\( \beta) q^{76} +(\)\(64\!\cdots\!00\)\( - \)\(71\!\cdots\!32\)\( \beta) q^{77} +(-\)\(59\!\cdots\!80\)\( + \)\(46\!\cdots\!60\)\( \beta) q^{79} +(-\)\(70\!\cdots\!40\)\( - \)\(23\!\cdots\!40\)\( \beta) q^{80} +(\)\(57\!\cdots\!60\)\( + \)\(80\!\cdots\!42\)\( \beta) q^{82} +(\)\(13\!\cdots\!80\)\( + \)\(62\!\cdots\!28\)\( \beta) q^{83} +(-\)\(25\!\cdots\!60\)\( - \)\(12\!\cdots\!60\)\( \beta) q^{85} +(\)\(10\!\cdots\!68\)\( + \)\(16\!\cdots\!40\)\( \beta) q^{86} +(-\)\(34\!\cdots\!80\)\( + \)\(41\!\cdots\!40\)\( \beta) q^{88} +(\)\(10\!\cdots\!90\)\( - \)\(26\!\cdots\!80\)\( \beta) q^{89} +(\)\(14\!\cdots\!12\)\( - \)\(73\!\cdots\!40\)\( \beta) q^{91} +(\)\(11\!\cdots\!60\)\( - \)\(21\!\cdots\!56\)\( \beta) q^{92} +(\)\(23\!\cdots\!56\)\( + \)\(57\!\cdots\!40\)\( \beta) q^{94} +(-\)\(64\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( \beta) q^{95} +(-\)\(45\!\cdots\!90\)\( - \)\(16\!\cdots\!84\)\( \beta) q^{97} +(\)\(40\!\cdots\!60\)\( - \)\(41\!\cdots\!93\)\( \beta) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 39960q^{2} + 1772534336q^{4} + 19391218020q^{5} + 30257527577200q^{7} - 160155058705920q^{8} + O(q^{10}) \) \( 2q - 39960q^{2} + 1772534336q^{4} + 19391218020q^{5} + 30257527577200q^{7} - 160155058705920q^{8} - 7861972212281520q^{10} + 7782353745118776q^{11} + 74708953050260620q^{13} - 225544963845241152q^{14} - 3045212913684901888q^{16} - 17224607828987089380q^{17} - 12370563328022164040q^{19} + \)\(31\!\cdots\!60\)\(q^{20} - 2682713032996690080q^{22} - \)\(18\!\cdots\!80\)\(q^{23} + \)\(14\!\cdots\!50\)\(q^{25} + \)\(23\!\cdots\!16\)\(q^{26} + \)\(11\!\cdots\!20\)\(q^{28} - \)\(12\!\cdots\!40\)\(q^{29} + \)\(12\!\cdots\!64\)\(q^{31} + \)\(48\!\cdots\!40\)\(q^{32} + \)\(58\!\cdots\!92\)\(q^{34} - \)\(24\!\cdots\!80\)\(q^{35} - \)\(83\!\cdots\!60\)\(q^{37} + \)\(10\!\cdots\!20\)\(q^{38} + \)\(19\!\cdots\!00\)\(q^{40} - \)\(87\!\cdots\!84\)\(q^{41} - \)\(18\!\cdots\!00\)\(q^{43} + \)\(79\!\cdots\!68\)\(q^{44} - \)\(59\!\cdots\!56\)\(q^{46} - \)\(95\!\cdots\!20\)\(q^{47} + \)\(16\!\cdots\!86\)\(q^{49} - \)\(17\!\cdots\!00\)\(q^{50} - \)\(91\!\cdots\!00\)\(q^{52} - \)\(19\!\cdots\!60\)\(q^{53} - \)\(14\!\cdots\!40\)\(q^{55} - \)\(25\!\cdots\!40\)\(q^{56} + \)\(38\!\cdots\!20\)\(q^{58} + \)\(19\!\cdots\!20\)\(q^{59} - \)\(12\!\cdots\!76\)\(q^{61} - \)\(15\!\cdots\!20\)\(q^{62} - \)\(74\!\cdots\!04\)\(q^{64} - \)\(34\!\cdots\!60\)\(q^{65} - \)\(96\!\cdots\!20\)\(q^{67} - \)\(24\!\cdots\!60\)\(q^{68} - \)\(10\!\cdots\!20\)\(q^{70} - \)\(55\!\cdots\!44\)\(q^{71} + \)\(62\!\cdots\!80\)\(q^{73} - \)\(15\!\cdots\!72\)\(q^{74} - \)\(44\!\cdots\!20\)\(q^{76} + \)\(12\!\cdots\!00\)\(q^{77} - \)\(11\!\cdots\!60\)\(q^{79} - \)\(14\!\cdots\!80\)\(q^{80} + \)\(11\!\cdots\!20\)\(q^{82} + \)\(26\!\cdots\!60\)\(q^{83} - \)\(50\!\cdots\!20\)\(q^{85} + \)\(21\!\cdots\!36\)\(q^{86} - \)\(69\!\cdots\!60\)\(q^{88} + \)\(21\!\cdots\!80\)\(q^{89} + \)\(28\!\cdots\!24\)\(q^{91} + \)\(22\!\cdots\!20\)\(q^{92} + \)\(46\!\cdots\!12\)\(q^{94} - \)\(12\!\cdots\!00\)\(q^{95} - \)\(90\!\cdots\!80\)\(q^{97} + \)\(80\!\cdots\!20\)\(q^{98} + O(q^{100}) \)

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
2139.16
−2138.16
−71307.9 0 2.93733e9 8.25073e10 0 1.14368e13 −5.63222e13 0 −5.88342e15
1.2 31347.9 0 −1.16479e9 −6.31161e10 0 1.88207e13 −1.03833e14 0 −1.97855e15
\(n\): 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 9.32.a.a 2
3.b odd 2 1 1.32.a.a 2
12.b even 2 1 16.32.a.b 2
15.d odd 2 1 25.32.a.a 2
15.e even 4 2 25.32.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.32.a.a 2 3.b odd 2 1
9.32.a.a 2 1.a even 1 1 trivial
16.32.a.b 2 12.b even 2 1
25.32.a.a 2 15.d odd 2 1
25.32.b.a 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 39960 T_{2} - 2235350016 \) acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2235350016 + 39960 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -\)\(52\!\cdots\!00\)\( - 19391218020 T + T^{2} \)
$7$ \( \)\(21\!\cdots\!64\)\( - 30257527577200 T + T^{2} \)
$11$ \( \)\(12\!\cdots\!44\)\( - 7782353745118776 T + T^{2} \)
$13$ \( -\)\(55\!\cdots\!04\)\( - 74708953050260620 T + T^{2} \)
$17$ \( \)\(68\!\cdots\!24\)\( + 17224607828987089380 T + T^{2} \)
$19$ \( -\)\(27\!\cdots\!00\)\( + 12370563328022164040 T + T^{2} \)
$23$ \( -\)\(73\!\cdots\!24\)\( + \)\(18\!\cdots\!80\)\( T + T^{2} \)
$29$ \( \)\(39\!\cdots\!00\)\( + \)\(12\!\cdots\!40\)\( T + T^{2} \)
$31$ \( -\)\(11\!\cdots\!76\)\( - \)\(12\!\cdots\!64\)\( T + T^{2} \)
$37$ \( -\)\(25\!\cdots\!56\)\( + \)\(83\!\cdots\!60\)\( T + T^{2} \)
$41$ \( -\)\(70\!\cdots\!36\)\( + \)\(87\!\cdots\!84\)\( T + T^{2} \)
$43$ \( -\)\(22\!\cdots\!64\)\( + \)\(18\!\cdots\!00\)\( T + T^{2} \)
$47$ \( \)\(15\!\cdots\!04\)\( + \)\(95\!\cdots\!20\)\( T + T^{2} \)
$53$ \( \)\(56\!\cdots\!16\)\( + \)\(19\!\cdots\!60\)\( T + T^{2} \)
$59$ \( -\)\(51\!\cdots\!00\)\( - \)\(19\!\cdots\!20\)\( T + T^{2} \)
$61$ \( \)\(36\!\cdots\!44\)\( + \)\(12\!\cdots\!76\)\( T + T^{2} \)
$67$ \( \)\(26\!\cdots\!24\)\( + \)\(96\!\cdots\!20\)\( T + T^{2} \)
$71$ \( -\)\(16\!\cdots\!16\)\( + \)\(55\!\cdots\!44\)\( T + T^{2} \)
$73$ \( \)\(90\!\cdots\!76\)\( - \)\(62\!\cdots\!80\)\( T + T^{2} \)
$79$ \( -\)\(53\!\cdots\!00\)\( + \)\(11\!\cdots\!60\)\( T + T^{2} \)
$83$ \( -\)\(86\!\cdots\!44\)\( - \)\(26\!\cdots\!60\)\( T + T^{2} \)
$89$ \( -\)\(62\!\cdots\!00\)\( - \)\(21\!\cdots\!80\)\( T + T^{2} \)
$97$ \( \)\(19\!\cdots\!04\)\( + \)\(90\!\cdots\!80\)\( T + T^{2} \)
show more
show less