Properties

Label 9.36.a.a
Level $9$
Weight $36$
Character orbit 9.a
Self dual yes
Analytic conductor $69.836$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.8356175703\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2196841}) \)
Defining polynomial: \(x^{2} - x - 549210\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7 \)
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 \(\beta = 168\sqrt{2196841}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 30456 - \beta ) q^{2} + ( 28571469952 - 60912 \beta ) q^{4} + ( 666889748370 + 1210960 \beta ) q^{5} + ( -600756391476472 + 142131024 \beta ) q^{7} + ( 3600478240192512 + 3933132544 \beta ) q^{8} +O(q^{10})\) \( q +(30456 - \beta) q^{2} +(28571469952 - 60912 \beta) q^{4} +(666889748370 + 1210960 \beta) q^{5} +(-600756391476472 + 142131024 \beta) q^{7} +(3600478240192512 + 3933132544 \beta) q^{8} +(-54773134183051920 - 630008750610 \beta) q^{10} +(737221926110660316 + 3560156783456 \beta) q^{11} +(15002606829102839414 - 161668794040416 \beta) q^{13} +(-27109277558313104448 + 605085133943416 \beta) q^{14} +(-\)\(11\!\cdots\!60\)\( - 1387770371960832 \beta) q^{16} +(-\)\(30\!\cdots\!66\)\( - 5140788294161760 \beta) q^{17} +(-\)\(80\!\cdots\!32\)\( + 101300863898712096 \beta) q^{19} +(\)\(14\!\cdots\!60\)\( - 6022681099639520 \beta) q^{20} +(-\)\(19\!\cdots\!08\)\( - 628793791113724380 \beta) q^{22} +(-\)\(24\!\cdots\!36\)\( + 3053203101283215712 \beta) q^{23} +(-\)\(23\!\cdots\!25\)\( + 1615153619372270400 \beta) q^{25} +(\)\(10\!\cdots\!28\)\( - 19926391620397749110 \beta) q^{26} +(-\)\(17\!\cdots\!36\)\( + 40654165599077853312 \beta) q^{28} +(\)\(39\!\cdots\!74\)\( - 22123136384176647952 \beta) q^{29} +(-\)\(60\!\cdots\!24\)\( - \)\(15\!\cdots\!92\)\( \beta) q^{31} +(-\)\(71\!\cdots\!88\)\( + \)\(93\!\cdots\!76\)\( \beta) q^{32} +(\)\(22\!\cdots\!44\)\( + \)\(29\!\cdots\!06\)\( \beta) q^{34} +(-\)\(38\!\cdots\!80\)\( - \)\(63\!\cdots\!40\)\( \beta) q^{35} +(\)\(83\!\cdots\!58\)\( + \)\(15\!\cdots\!04\)\( \beta) q^{37} +(-\)\(65\!\cdots\!56\)\( + \)\(11\!\cdots\!08\)\( \beta) q^{38} +(\)\(26\!\cdots\!00\)\( + \)\(69\!\cdots\!00\)\( \beta) q^{40} +(\)\(16\!\cdots\!94\)\( + \)\(49\!\cdots\!32\)\( \beta) q^{41} +(\)\(50\!\cdots\!60\)\( + \)\(25\!\cdots\!56\)\( \beta) q^{43} +(\)\(76\!\cdots\!84\)\( + \)\(56\!\cdots\!20\)\( \beta) q^{44} +(-\)\(19\!\cdots\!24\)\( + \)\(33\!\cdots\!08\)\( \beta) q^{46} +(\)\(26\!\cdots\!60\)\( - \)\(10\!\cdots\!88\)\( \beta) q^{47} +(-\)\(16\!\cdots\!75\)\( - \)\(17\!\cdots\!56\)\( \beta) q^{49} +(-\)\(17\!\cdots\!00\)\( + \)\(24\!\cdots\!25\)\( \beta) q^{50} +(\)\(10\!\cdots\!56\)\( - \)\(55\!\cdots\!00\)\( \beta) q^{52} +(\)\(15\!\cdots\!34\)\( - \)\(99\!\cdots\!88\)\( \beta) q^{53} +(\)\(75\!\cdots\!60\)\( + \)\(32\!\cdots\!80\)\( \beta) q^{55} +(-\)\(21\!\cdots\!60\)\( - \)\(18\!\cdots\!80\)\( \beta) q^{56} +(\)\(25\!\cdots\!12\)\( - \)\(40\!\cdots\!86\)\( \beta) q^{58} +(-\)\(41\!\cdots\!32\)\( - \)\(66\!\cdots\!64\)\( \beta) q^{59} +(-\)\(20\!\cdots\!70\)\( - \)\(26\!\cdots\!24\)\( \beta) q^{61} +(\)\(76\!\cdots\!84\)\( + \)\(55\!\cdots\!72\)\( \beta) q^{62} +(-\)\(22\!\cdots\!32\)\( + \)\(14\!\cdots\!20\)\( \beta) q^{64} +(-\)\(21\!\cdots\!60\)\( - \)\(89\!\cdots\!80\)\( \beta) q^{65} +(\)\(48\!\cdots\!56\)\( + \)\(28\!\cdots\!40\)\( \beta) q^{67} +(-\)\(68\!\cdots\!52\)\( + \)\(41\!\cdots\!72\)\( \beta) q^{68} +(\)\(27\!\cdots\!80\)\( + \)\(37\!\cdots\!40\)\( \beta) q^{70} +(-\)\(22\!\cdots\!12\)\( + \)\(10\!\cdots\!80\)\( \beta) q^{71} +(-\)\(68\!\cdots\!54\)\( - \)\(82\!\cdots\!92\)\( \beta) q^{73} +(-\)\(90\!\cdots\!88\)\( - \)\(37\!\cdots\!34\)\( \beta) q^{74} +(-\)\(61\!\cdots\!32\)\( + \)\(33\!\cdots\!76\)\( \beta) q^{76} +(-\)\(41\!\cdots\!56\)\( - \)\(20\!\cdots\!48\)\( \beta) q^{77} +(-\)\(53\!\cdots\!40\)\( - \)\(64\!\cdots\!20\)\( \beta) q^{79} +(-\)\(84\!\cdots\!80\)\( - \)\(22\!\cdots\!40\)\( \beta) q^{80} +(-\)\(25\!\cdots\!24\)\( - \)\(14\!\cdots\!02\)\( \beta) q^{82} +(\)\(27\!\cdots\!72\)\( - \)\(21\!\cdots\!32\)\( \beta) q^{83} +(-\)\(24\!\cdots\!20\)\( - \)\(71\!\cdots\!60\)\( \beta) q^{85} +(-\)\(15\!\cdots\!44\)\( + \)\(27\!\cdots\!76\)\( \beta) q^{86} +(\)\(35\!\cdots\!68\)\( + \)\(15\!\cdots\!76\)\( \beta) q^{88} +(-\)\(12\!\cdots\!18\)\( - \)\(35\!\cdots\!36\)\( \beta) q^{89} +(-\)\(10\!\cdots\!64\)\( + \)\(99\!\cdots\!88\)\( \beta) q^{91} +(-\)\(18\!\cdots\!68\)\( + \)\(10\!\cdots\!56\)\( \beta) q^{92} +(\)\(72\!\cdots\!52\)\( - \)\(29\!\cdots\!88\)\( \beta) q^{94} +(\)\(22\!\cdots\!00\)\( + \)\(57\!\cdots\!00\)\( \beta) q^{95} +(\)\(41\!\cdots\!26\)\( + \)\(60\!\cdots\!00\)\( \beta) q^{97} +(\)\(10\!\cdots\!04\)\( + \)\(11\!\cdots\!39\)\( \beta) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 60912q^{2} + 57142939904q^{4} + 1333779496740q^{5} - 1201512782952944q^{7} + 7200956480385024q^{8} + O(q^{10}) \) \( 2q + 60912q^{2} + 57142939904q^{4} + 1333779496740q^{5} - 1201512782952944q^{7} + 7200956480385024q^{8} - 109546268366103840q^{10} + 1474443852221320632q^{11} + 30005213658205678828q^{13} - 54218555116626208896q^{14} - \)\(22\!\cdots\!20\)\(q^{16} - \)\(61\!\cdots\!32\)\(q^{17} - \)\(16\!\cdots\!64\)\(q^{19} + \)\(28\!\cdots\!20\)\(q^{20} - \)\(39\!\cdots\!16\)\(q^{22} - \)\(49\!\cdots\!72\)\(q^{23} - \)\(47\!\cdots\!50\)\(q^{25} + \)\(20\!\cdots\!56\)\(q^{26} - \)\(35\!\cdots\!72\)\(q^{28} + \)\(78\!\cdots\!48\)\(q^{29} - \)\(12\!\cdots\!48\)\(q^{31} - \)\(14\!\cdots\!76\)\(q^{32} + \)\(44\!\cdots\!88\)\(q^{34} - \)\(77\!\cdots\!60\)\(q^{35} + \)\(16\!\cdots\!16\)\(q^{37} - \)\(13\!\cdots\!12\)\(q^{38} + \)\(53\!\cdots\!00\)\(q^{40} + \)\(32\!\cdots\!88\)\(q^{41} + \)\(10\!\cdots\!20\)\(q^{43} + \)\(15\!\cdots\!68\)\(q^{44} - \)\(39\!\cdots\!48\)\(q^{46} + \)\(52\!\cdots\!20\)\(q^{47} - \)\(33\!\cdots\!50\)\(q^{49} - \)\(34\!\cdots\!00\)\(q^{50} + \)\(20\!\cdots\!12\)\(q^{52} + \)\(31\!\cdots\!68\)\(q^{53} + \)\(15\!\cdots\!20\)\(q^{55} - \)\(42\!\cdots\!20\)\(q^{56} + \)\(51\!\cdots\!24\)\(q^{58} - \)\(82\!\cdots\!64\)\(q^{59} - \)\(40\!\cdots\!40\)\(q^{61} + \)\(15\!\cdots\!68\)\(q^{62} - \)\(44\!\cdots\!64\)\(q^{64} - \)\(42\!\cdots\!20\)\(q^{65} + \)\(96\!\cdots\!12\)\(q^{67} - \)\(13\!\cdots\!04\)\(q^{68} + \)\(54\!\cdots\!60\)\(q^{70} - \)\(44\!\cdots\!24\)\(q^{71} - \)\(13\!\cdots\!08\)\(q^{73} - \)\(18\!\cdots\!76\)\(q^{74} - \)\(12\!\cdots\!64\)\(q^{76} - \)\(82\!\cdots\!12\)\(q^{77} - \)\(10\!\cdots\!80\)\(q^{79} - \)\(16\!\cdots\!60\)\(q^{80} - \)\(51\!\cdots\!48\)\(q^{82} + \)\(55\!\cdots\!44\)\(q^{83} - \)\(48\!\cdots\!40\)\(q^{85} - \)\(31\!\cdots\!88\)\(q^{86} + \)\(70\!\cdots\!36\)\(q^{88} - \)\(24\!\cdots\!36\)\(q^{89} - \)\(20\!\cdots\!28\)\(q^{91} - \)\(37\!\cdots\!36\)\(q^{92} + \)\(14\!\cdots\!04\)\(q^{94} + \)\(45\!\cdots\!00\)\(q^{95} + \)\(83\!\cdots\!52\)\(q^{97} + \)\(20\!\cdots\!08\)\(q^{98} + O(q^{100}) \)

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
741.587
−740.587
−218549. 0 1.34041e10 9.68425e11 0 −5.65365e14 4.57985e15 0 −2.11649e17
1.2 279461. 0 4.37389e10 3.65354e11 0 −6.36148e14 2.62111e15 0 1.02102e17
\(n\): 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 9.36.a.a 2
3.b odd 2 1 3.36.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.36.a.a 2 3.b odd 2 1
9.36.a.a 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 60912 T_{2} - 61076072448 \) acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 60912 T + 7643404288 T^{2} - 2092920383471616 T^{3} + \)\(11\!\cdots\!24\)\( T^{4} \)
$3$ 1
$5$ \( 1 - 1333779496740 T + \)\(61\!\cdots\!50\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + 1201512782952944 T + \)\(11\!\cdots\!86\)\( T^{2} + \)\(45\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 1474443852221320632 T + \)\(53\!\cdots\!34\)\( T^{2} - \)\(41\!\cdots\!32\)\( T^{3} + \)\(78\!\cdots\!01\)\( T^{4} \)
$13$ \( 1 - 30005213658205678828 T + \)\(55\!\cdots\!06\)\( T^{2} - \)\(29\!\cdots\!96\)\( T^{3} + \)\(94\!\cdots\!49\)\( T^{4} \)
$17$ \( 1 + \)\(61\!\cdots\!32\)\( T + \)\(31\!\cdots\!42\)\( T^{2} + \)\(71\!\cdots\!76\)\( T^{3} + \)\(13\!\cdots\!49\)\( T^{4} \)
$19$ \( 1 + \)\(16\!\cdots\!64\)\( T + \)\(56\!\cdots\!78\)\( T^{2} + \)\(91\!\cdots\!36\)\( T^{3} + \)\(32\!\cdots\!01\)\( T^{4} \)
$23$ \( 1 + \)\(49\!\cdots\!72\)\( T + \)\(39\!\cdots\!14\)\( T^{2} + \)\(22\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!49\)\( T^{4} \)
$29$ \( 1 - \)\(78\!\cdots\!48\)\( T + \)\(45\!\cdots\!38\)\( T^{2} - \)\(12\!\cdots\!52\)\( T^{3} + \)\(23\!\cdots\!01\)\( T^{4} \)
$31$ \( 1 + \)\(12\!\cdots\!48\)\( T + \)\(33\!\cdots\!02\)\( T^{2} + \)\(19\!\cdots\!48\)\( T^{3} + \)\(24\!\cdots\!01\)\( T^{4} \)
$37$ \( 1 - \)\(16\!\cdots\!16\)\( T + \)\(20\!\cdots\!06\)\( T^{2} - \)\(12\!\cdots\!88\)\( T^{3} + \)\(59\!\cdots\!49\)\( T^{4} \)
$41$ \( 1 - \)\(32\!\cdots\!88\)\( T + \)\(68\!\cdots\!22\)\( T^{2} - \)\(92\!\cdots\!88\)\( T^{3} + \)\(78\!\cdots\!01\)\( T^{4} \)
$43$ \( 1 - \)\(10\!\cdots\!20\)\( T - \)\(10\!\cdots\!10\)\( T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - \)\(52\!\cdots\!20\)\( T + \)\(66\!\cdots\!90\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \)
$53$ \( 1 - \)\(31\!\cdots\!68\)\( T - \)\(16\!\cdots\!26\)\( T^{2} - \)\(69\!\cdots\!76\)\( T^{3} + \)\(50\!\cdots\!49\)\( T^{4} \)
$59$ \( 1 + \)\(82\!\cdots\!64\)\( T + \)\(20\!\cdots\!58\)\( T^{2} + \)\(78\!\cdots\!36\)\( T^{3} + \)\(91\!\cdots\!01\)\( T^{4} \)
$61$ \( 1 + \)\(40\!\cdots\!40\)\( T + \)\(98\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + \)\(93\!\cdots\!01\)\( T^{4} \)
$67$ \( 1 - \)\(96\!\cdots\!12\)\( T + \)\(11\!\cdots\!22\)\( T^{2} - \)\(78\!\cdots\!16\)\( T^{3} + \)\(66\!\cdots\!49\)\( T^{4} \)
$71$ \( 1 + \)\(44\!\cdots\!24\)\( T + \)\(99\!\cdots\!46\)\( T^{2} + \)\(27\!\cdots\!24\)\( T^{3} + \)\(38\!\cdots\!01\)\( T^{4} \)
$73$ \( 1 + \)\(13\!\cdots\!08\)\( T + \)\(29\!\cdots\!54\)\( T^{2} + \)\(22\!\cdots\!56\)\( T^{3} + \)\(27\!\cdots\!49\)\( T^{4} \)
$79$ \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(29\!\cdots\!98\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(68\!\cdots\!01\)\( T^{4} \)
$83$ \( 1 - \)\(55\!\cdots\!44\)\( T + \)\(36\!\cdots\!82\)\( T^{2} - \)\(80\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!49\)\( T^{4} \)
$89$ \( 1 + \)\(24\!\cdots\!36\)\( T + \)\(40\!\cdots\!58\)\( T^{2} + \)\(40\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!01\)\( T^{4} \)
$97$ \( 1 - \)\(83\!\cdots\!52\)\( T + \)\(66\!\cdots\!62\)\( T^{2} - \)\(28\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \)
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