Properties

Label 9.44.a.b
Level $9$
Weight $44$
Character orbit 9.a
Self dual yes
Analytic conductor $105.399$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 44 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(105.399355811\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - 11258260111 x - 264759545317170\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{5}\cdot 5\cdot 7 \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 736648 + \beta_{1} ) q^{2} + ( 3274206140608 + 810680 \beta_{1} - 136 \beta_{2} ) q^{4} + ( -178401793591390 + 23668880 \beta_{1} + 9948 \beta_{2} ) q^{5} + ( 100657141888492952 - 411883648456 \beta_{1} - 20265742 \beta_{2} ) q^{7} + ( 5276954637929116160 - 137106139840 \beta_{1} - 300552384 \beta_{2} ) q^{8} +O(q^{10})\) \( q +(736648 + \beta_{1}) q^{2} +(3274206140608 + 810680 \beta_{1} - 136 \beta_{2}) q^{4} +(-178401793591390 + 23668880 \beta_{1} + 9948 \beta_{2}) q^{5} +(100657141888492952 - 411883648456 \beta_{1} - 20265742 \beta_{2}) q^{7} +(5276954637929116160 - 137106139840 \beta_{1} - 300552384 \beta_{2}) q^{8} +(\)\(14\!\cdots\!40\)\( - 566140399707230 \beta_{1} + 10700910592 \beta_{2}) q^{10} +(-\)\(88\!\cdots\!32\)\( + 4605449192710300 \beta_{1} - 45747233895 \beta_{2}) q^{11} +(\)\(89\!\cdots\!94\)\( - 6721472103363472 \beta_{1} + 7602640063460 \beta_{2}) q^{13} +(-\)\(46\!\cdots\!76\)\( + 863622683114558360 \beta_{1} + 27659052976128 \beta_{2}) q^{14} +(-\)\(26\!\cdots\!64\)\( + 9903419018930019840 \beta_{1} + 794362954993152 \beta_{2}) q^{16} +(\)\(13\!\cdots\!98\)\( + 4610357498452581216 \beta_{1} + 5127718083059496 \beta_{2}) q^{17} +(\)\(52\!\cdots\!00\)\( + \)\(85\!\cdots\!20\)\( \beta_{1} - 35769349521761799 \beta_{2}) q^{19} +(-\)\(48\!\cdots\!20\)\( - \)\(52\!\cdots\!60\)\( \beta_{1} + 4464959933862384 \beta_{2}) q^{20} +(\)\(46\!\cdots\!64\)\( - \)\(67\!\cdots\!32\)\( \beta_{1} - 690353547697454080 \beta_{2}) q^{22} +(\)\(40\!\cdots\!16\)\( - \)\(54\!\cdots\!88\)\( \beta_{1} - 625371170988949302 \beta_{2}) q^{23} +(-\)\(77\!\cdots\!25\)\( - \)\(83\!\cdots\!00\)\( \beta_{1} + 561020137038857840 \beta_{2}) q^{25} +(\)\(57\!\cdots\!48\)\( + \)\(59\!\cdots\!90\)\( \beta_{1} + 11552220751814725632 \beta_{2}) q^{26} +(\)\(56\!\cdots\!56\)\( - \)\(20\!\cdots\!08\)\( \beta_{1} + 99508983996076431168 \beta_{2}) q^{28} +(-\)\(19\!\cdots\!50\)\( - \)\(17\!\cdots\!80\)\( \beta_{1} + \)\(33\!\cdots\!96\)\( \beta_{2}) q^{29} +(-\)\(85\!\cdots\!08\)\( + \)\(11\!\cdots\!00\)\( \beta_{1} + \)\(11\!\cdots\!60\)\( \beta_{2}) q^{31} +(\)\(48\!\cdots\!28\)\( - \)\(55\!\cdots\!64\)\( \beta_{1} + \)\(24\!\cdots\!60\)\( \beta_{2}) q^{32} +(\)\(15\!\cdots\!96\)\( - \)\(64\!\cdots\!90\)\( \beta_{1} + \)\(65\!\cdots\!68\)\( \beta_{2}) q^{34} +(-\)\(79\!\cdots\!40\)\( + \)\(22\!\cdots\!80\)\( \beta_{1} - \)\(49\!\cdots\!12\)\( \beta_{2}) q^{35} +(-\)\(77\!\cdots\!98\)\( - \)\(43\!\cdots\!36\)\( \beta_{1} + \)\(15\!\cdots\!36\)\( \beta_{2}) q^{37} +(\)\(10\!\cdots\!40\)\( + \)\(19\!\cdots\!40\)\( \beta_{1} - \)\(16\!\cdots\!56\)\( \beta_{2}) q^{38} +(-\)\(10\!\cdots\!00\)\( - \)\(86\!\cdots\!00\)\( \beta_{1} - \)\(16\!\cdots\!00\)\( \beta_{2}) q^{40} +(-\)\(85\!\cdots\!22\)\( - \)\(87\!\cdots\!00\)\( \beta_{1} - \)\(76\!\cdots\!60\)\( \beta_{2}) q^{41} +(\)\(80\!\cdots\!64\)\( - \)\(12\!\cdots\!92\)\( \beta_{1} + \)\(59\!\cdots\!91\)\( \beta_{2}) q^{43} +(\)\(34\!\cdots\!44\)\( + \)\(32\!\cdots\!40\)\( \beta_{1} + \)\(35\!\cdots\!92\)\( \beta_{2}) q^{44} +(-\)\(63\!\cdots\!88\)\( + \)\(20\!\cdots\!00\)\( \beta_{1} + \)\(65\!\cdots\!40\)\( \beta_{2}) q^{46} +(-\)\(10\!\cdots\!52\)\( - \)\(31\!\cdots\!04\)\( \beta_{1} + \)\(83\!\cdots\!28\)\( \beta_{2}) q^{47} +(\)\(11\!\cdots\!93\)\( - \)\(77\!\cdots\!00\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2}) q^{49} +(-\)\(66\!\cdots\!00\)\( - \)\(79\!\cdots\!25\)\( \beta_{1} + \)\(19\!\cdots\!60\)\( \beta_{2}) q^{50} +(-\)\(57\!\cdots\!68\)\( + \)\(23\!\cdots\!04\)\( \beta_{1} - \)\(13\!\cdots\!72\)\( \beta_{2}) q^{52} +(-\)\(50\!\cdots\!54\)\( - \)\(47\!\cdots\!68\)\( \beta_{1} + \)\(30\!\cdots\!12\)\( \beta_{2}) q^{53} +(\)\(13\!\cdots\!80\)\( - \)\(29\!\cdots\!60\)\( \beta_{1} - \)\(83\!\cdots\!86\)\( \beta_{2}) q^{55} +(\)\(21\!\cdots\!00\)\( - \)\(60\!\cdots\!80\)\( \beta_{1} + \)\(17\!\cdots\!16\)\( \beta_{2}) q^{56} +(-\)\(34\!\cdots\!60\)\( - \)\(32\!\cdots\!10\)\( \beta_{1} + \)\(71\!\cdots\!24\)\( \beta_{2}) q^{58} +(-\)\(75\!\cdots\!00\)\( + \)\(62\!\cdots\!40\)\( \beta_{1} - \)\(12\!\cdots\!03\)\( \beta_{2}) q^{59} +(-\)\(31\!\cdots\!18\)\( - \)\(68\!\cdots\!00\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2}) q^{61} +(\)\(69\!\cdots\!16\)\( - \)\(12\!\cdots\!08\)\( \beta_{1} + \)\(68\!\cdots\!40\)\( \beta_{2}) q^{62} +(-\)\(37\!\cdots\!12\)\( - \)\(13\!\cdots\!40\)\( \beta_{1} + \)\(39\!\cdots\!28\)\( \beta_{2}) q^{64} +(\)\(90\!\cdots\!20\)\( + \)\(32\!\cdots\!60\)\( \beta_{1} + \)\(10\!\cdots\!76\)\( \beta_{2}) q^{65} +(-\)\(24\!\cdots\!48\)\( - \)\(23\!\cdots\!16\)\( \beta_{1} + \)\(15\!\cdots\!09\)\( \beta_{2}) q^{67} +(-\)\(18\!\cdots\!56\)\( - \)\(14\!\cdots\!12\)\( \beta_{1} - \)\(27\!\cdots\!76\)\( \beta_{2}) q^{68} +(\)\(19\!\cdots\!40\)\( - \)\(58\!\cdots\!80\)\( \beta_{1} - \)\(37\!\cdots\!48\)\( \beta_{2}) q^{70} +(\)\(61\!\cdots\!88\)\( + \)\(47\!\cdots\!00\)\( \beta_{1} - \)\(32\!\cdots\!50\)\( \beta_{2}) q^{71} +(-\)\(67\!\cdots\!66\)\( + \)\(18\!\cdots\!88\)\( \beta_{1} - \)\(43\!\cdots\!48\)\( \beta_{2}) q^{73} +(-\)\(55\!\cdots\!36\)\( - \)\(14\!\cdots\!50\)\( \beta_{1} + \)\(80\!\cdots\!00\)\( \beta_{2}) q^{74} +(\)\(25\!\cdots\!00\)\( + \)\(93\!\cdots\!60\)\( \beta_{1} - \)\(18\!\cdots\!32\)\( \beta_{2}) q^{76} +(-\)\(19\!\cdots\!64\)\( + \)\(84\!\cdots\!92\)\( \beta_{1} + \)\(42\!\cdots\!24\)\( \beta_{2}) q^{77} +(\)\(50\!\cdots\!00\)\( + \)\(26\!\cdots\!80\)\( \beta_{1} + \)\(39\!\cdots\!84\)\( \beta_{2}) q^{79} +(\)\(33\!\cdots\!60\)\( - \)\(56\!\cdots\!20\)\( \beta_{1} - \)\(50\!\cdots\!72\)\( \beta_{2}) q^{80} +(-\)\(10\!\cdots\!56\)\( + \)\(20\!\cdots\!78\)\( \beta_{1} + \)\(10\!\cdots\!60\)\( \beta_{2}) q^{82} +(\)\(29\!\cdots\!76\)\( + \)\(22\!\cdots\!52\)\( \beta_{1} - \)\(22\!\cdots\!29\)\( \beta_{2}) q^{83} +(\)\(14\!\cdots\!40\)\( + \)\(55\!\cdots\!20\)\( \beta_{1} + \)\(25\!\cdots\!52\)\( \beta_{2}) q^{85} +(-\)\(83\!\cdots\!32\)\( + \)\(55\!\cdots\!20\)\( \beta_{1} + \)\(25\!\cdots\!36\)\( \beta_{2}) q^{86} +(-\)\(85\!\cdots\!20\)\( + \)\(82\!\cdots\!80\)\( \beta_{1} + \)\(21\!\cdots\!88\)\( \beta_{2}) q^{88} +(-\)\(67\!\cdots\!50\)\( + \)\(45\!\cdots\!60\)\( \beta_{1} - \)\(10\!\cdots\!92\)\( \beta_{2}) q^{89} +(-\)\(38\!\cdots\!28\)\( - \)\(28\!\cdots\!20\)\( \beta_{1} - \)\(25\!\cdots\!56\)\( \beta_{2}) q^{91} +(-\)\(22\!\cdots\!52\)\( - \)\(40\!\cdots\!84\)\( \beta_{1} + \)\(11\!\cdots\!76\)\( \beta_{2}) q^{92} +(-\)\(44\!\cdots\!44\)\( - \)\(43\!\cdots\!80\)\( \beta_{1} + \)\(16\!\cdots\!36\)\( \beta_{2}) q^{94} +(-\)\(10\!\cdots\!00\)\( - \)\(52\!\cdots\!00\)\( \beta_{1} - \)\(42\!\cdots\!50\)\( \beta_{2}) q^{95} +(-\)\(12\!\cdots\!98\)\( - \)\(11\!\cdots\!96\)\( \beta_{1} + \)\(70\!\cdots\!32\)\( \beta_{2}) q^{97} +(-\)\(80\!\cdots\!36\)\( + \)\(10\!\cdots\!93\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 2209944q^{2} + 9822618421824q^{4} - 535205380774170q^{5} + 301971425665478856q^{7} + 15830863913787348480q^{8} + O(q^{10}) \) \( 3q + 2209944q^{2} + 9822618421824q^{4} - 535205380774170q^{5} + 301971425665478856q^{7} + 15830863913787348480q^{8} + \)\(42\!\cdots\!20\)\(q^{10} - \)\(26\!\cdots\!96\)\(q^{11} + \)\(26\!\cdots\!82\)\(q^{13} - \)\(14\!\cdots\!28\)\(q^{14} - \)\(79\!\cdots\!92\)\(q^{16} + \)\(40\!\cdots\!94\)\(q^{17} + \)\(15\!\cdots\!00\)\(q^{19} - \)\(14\!\cdots\!60\)\(q^{20} + \)\(13\!\cdots\!92\)\(q^{22} + \)\(12\!\cdots\!48\)\(q^{23} - \)\(23\!\cdots\!75\)\(q^{25} + \)\(17\!\cdots\!44\)\(q^{26} + \)\(16\!\cdots\!68\)\(q^{28} - \)\(57\!\cdots\!50\)\(q^{29} - \)\(25\!\cdots\!24\)\(q^{31} + \)\(14\!\cdots\!84\)\(q^{32} + \)\(46\!\cdots\!88\)\(q^{34} - \)\(23\!\cdots\!20\)\(q^{35} - \)\(23\!\cdots\!94\)\(q^{37} + \)\(30\!\cdots\!20\)\(q^{38} - \)\(32\!\cdots\!00\)\(q^{40} - \)\(25\!\cdots\!66\)\(q^{41} + \)\(24\!\cdots\!92\)\(q^{43} + \)\(10\!\cdots\!32\)\(q^{44} - \)\(18\!\cdots\!64\)\(q^{46} - \)\(30\!\cdots\!56\)\(q^{47} + \)\(34\!\cdots\!79\)\(q^{49} - \)\(19\!\cdots\!00\)\(q^{50} - \)\(17\!\cdots\!04\)\(q^{52} - \)\(15\!\cdots\!62\)\(q^{53} + \)\(39\!\cdots\!40\)\(q^{55} + \)\(64\!\cdots\!00\)\(q^{56} - \)\(10\!\cdots\!80\)\(q^{58} - \)\(22\!\cdots\!00\)\(q^{59} - \)\(93\!\cdots\!54\)\(q^{61} + \)\(20\!\cdots\!48\)\(q^{62} - \)\(11\!\cdots\!36\)\(q^{64} + \)\(27\!\cdots\!60\)\(q^{65} - \)\(73\!\cdots\!44\)\(q^{67} - \)\(54\!\cdots\!68\)\(q^{68} + \)\(59\!\cdots\!20\)\(q^{70} + \)\(18\!\cdots\!64\)\(q^{71} - \)\(20\!\cdots\!98\)\(q^{73} - \)\(16\!\cdots\!08\)\(q^{74} + \)\(77\!\cdots\!00\)\(q^{76} - \)\(59\!\cdots\!92\)\(q^{77} + \)\(15\!\cdots\!00\)\(q^{79} + \)\(10\!\cdots\!80\)\(q^{80} - \)\(32\!\cdots\!68\)\(q^{82} + \)\(89\!\cdots\!28\)\(q^{83} + \)\(43\!\cdots\!20\)\(q^{85} - \)\(25\!\cdots\!96\)\(q^{86} - \)\(25\!\cdots\!60\)\(q^{88} - \)\(20\!\cdots\!50\)\(q^{89} - \)\(11\!\cdots\!84\)\(q^{91} - \)\(68\!\cdots\!56\)\(q^{92} - \)\(13\!\cdots\!32\)\(q^{94} - \)\(31\!\cdots\!00\)\(q^{95} - \)\(38\!\cdots\!94\)\(q^{97} - \)\(24\!\cdots\!08\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 11258260111 x - 264759545317170\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -6 \nu^{2} + 343074 \nu + 45033040444 \)\()/8369\)
\(\beta_{2}\)\(=\)\((\)\( 44520 \nu^{2} + 3528276360 \nu - 334145160094480 \)\()/8369\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 7420 \beta_{1}\)\()/725760\)
\(\nu^{2}\)\(=\)\((\)\(57179 \beta_{2} - 588046060 \beta_{1} + 5447196572106240\)\()/725760\)

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
−91450.2
116336.
−24885.9
−3.62707e6 0 4.35955e12 −6.19839e14 0 2.58688e18 1.60917e19 0 2.24820e21
1.2 1.18359e6 0 −7.39521e12 6.39116e14 0 −1.72730e18 −1.91639e19 0 7.56451e20
1.3 4.65343e6 0 1.28583e13 −5.54482e14 0 −5.57604e17 1.89031e19 0 −2.58024e21
\(n\): 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 9.44.a.b 3
3.b odd 2 1 1.44.a.a 3
12.b even 2 1 16.44.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.44.a.a 3 3.b odd 2 1
9.44.a.b 3 1.a even 1 1 trivial
16.44.a.c 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 2209944 T_{2}^{2} - \)\(15\!\cdots\!56\)\( T_{2} + \)\(19\!\cdots\!64\)\( \) acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 19976984434430705664 - 15663522502656 T - 2209944 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( -\)\(21\!\cdots\!00\)\( - \)\(40\!\cdots\!00\)\( T + 535205380774170 T^{2} + T^{3} \)
$7$ \( -\)\(24\!\cdots\!84\)\( - \)\(49\!\cdots\!36\)\( T - 301971425665478856 T^{2} + T^{3} \)
$11$ \( -\)\(32\!\cdots\!32\)\( - \)\(14\!\cdots\!28\)\( T + \)\(26\!\cdots\!96\)\( T^{2} + T^{3} \)
$13$ \( -\)\(50\!\cdots\!72\)\( + \)\(20\!\cdots\!96\)\( T - \)\(26\!\cdots\!82\)\( T^{2} + T^{3} \)
$17$ \( -\)\(28\!\cdots\!76\)\( - \)\(75\!\cdots\!96\)\( T - \)\(40\!\cdots\!94\)\( T^{2} + T^{3} \)
$19$ \( -\)\(21\!\cdots\!00\)\( - \)\(18\!\cdots\!00\)\( T - \)\(15\!\cdots\!00\)\( T^{2} + T^{3} \)
$23$ \( -\)\(36\!\cdots\!48\)\( - \)\(54\!\cdots\!24\)\( T - \)\(12\!\cdots\!48\)\( T^{2} + T^{3} \)
$29$ \( -\)\(73\!\cdots\!00\)\( + \)\(46\!\cdots\!00\)\( T + \)\(57\!\cdots\!50\)\( T^{2} + T^{3} \)
$31$ \( -\)\(17\!\cdots\!88\)\( + \)\(13\!\cdots\!92\)\( T + \)\(25\!\cdots\!24\)\( T^{2} + T^{3} \)
$37$ \( \)\(56\!\cdots\!96\)\( - \)\(26\!\cdots\!16\)\( T + \)\(23\!\cdots\!94\)\( T^{2} + T^{3} \)
$41$ \( -\)\(27\!\cdots\!52\)\( - \)\(40\!\cdots\!48\)\( T + \)\(25\!\cdots\!66\)\( T^{2} + T^{3} \)
$43$ \( -\)\(49\!\cdots\!12\)\( + \)\(14\!\cdots\!36\)\( T - \)\(24\!\cdots\!92\)\( T^{2} + T^{3} \)
$47$ \( -\)\(86\!\cdots\!56\)\( - \)\(33\!\cdots\!76\)\( T + \)\(30\!\cdots\!56\)\( T^{2} + T^{3} \)
$53$ \( \)\(90\!\cdots\!92\)\( + \)\(66\!\cdots\!16\)\( T + \)\(15\!\cdots\!62\)\( T^{2} + T^{3} \)
$59$ \( -\)\(78\!\cdots\!00\)\( + \)\(87\!\cdots\!00\)\( T + \)\(22\!\cdots\!00\)\( T^{2} + T^{3} \)
$61$ \( -\)\(87\!\cdots\!68\)\( - \)\(87\!\cdots\!28\)\( T + \)\(93\!\cdots\!54\)\( T^{2} + T^{3} \)
$67$ \( \)\(42\!\cdots\!76\)\( - \)\(19\!\cdots\!96\)\( T + \)\(73\!\cdots\!44\)\( T^{2} + T^{3} \)
$71$ \( -\)\(18\!\cdots\!72\)\( + \)\(10\!\cdots\!32\)\( T - \)\(18\!\cdots\!64\)\( T^{2} + T^{3} \)
$73$ \( -\)\(31\!\cdots\!52\)\( + \)\(69\!\cdots\!76\)\( T + \)\(20\!\cdots\!98\)\( T^{2} + T^{3} \)
$79$ \( -\)\(26\!\cdots\!00\)\( - \)\(83\!\cdots\!00\)\( T - \)\(15\!\cdots\!00\)\( T^{2} + T^{3} \)
$83$ \( \)\(17\!\cdots\!32\)\( - \)\(30\!\cdots\!44\)\( T - \)\(89\!\cdots\!28\)\( T^{2} + T^{3} \)
$89$ \( \)\(10\!\cdots\!00\)\( - \)\(52\!\cdots\!00\)\( T + \)\(20\!\cdots\!50\)\( T^{2} + T^{3} \)
$97$ \( -\)\(26\!\cdots\!44\)\( - \)\(17\!\cdots\!76\)\( T + \)\(38\!\cdots\!94\)\( T^{2} + T^{3} \)
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