Properties

Label 9.40.a.c
Level $9$
Weight $40$
Character orbit 9.a
Self dual yes
Analytic conductor $86.706$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 40 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(86.7055962508\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - 3876249523 x - 18467420411022\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6}\cdot 5\cdot 7\cdot 13 \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -177858 + \beta_{1} ) q^{2} + ( 319147551244 - 227081 \beta_{1} + \beta_{2} ) q^{4} + ( 17793991314810 - 12092638 \beta_{1} - 54 \beta_{2} ) q^{5} + ( -525770892074176 - 12596523938 \beta_{1} + 56758 \beta_{2} ) q^{7} + ( -149112778455452232 + 182657330678 \beta_{1} - 533574 \beta_{2} ) q^{8} +O(q^{10})\) \( q +(-177858 + \beta_{1}) q^{2} +(319147551244 - 227081 \beta_{1} + \beta_{2}) q^{4} +(17793991314810 - 12092638 \beta_{1} - 54 \beta_{2}) q^{5} +(-525770892074176 - 12596523938 \beta_{1} + 56758 \beta_{2}) q^{7} +(-149112778455452232 + 182657330678 \beta_{1} - 533574 \beta_{2}) q^{8} +(-13289593742193286260 - 3323523968902 \beta_{1} + 4457984 \beta_{2}) q^{10} +(\)\(17\!\cdots\!80\)\( - 2617390166972 \beta_{1} + 42093972 \beta_{2}) q^{11} +(\)\(18\!\cdots\!30\)\( - 3296990601565580 \beta_{1} + 2302171012 \beta_{2}) q^{13} +(-\)\(10\!\cdots\!84\)\( + 22915977673056320 \beta_{1} - 29992453632 \beta_{2}) q^{14} +(\)\(40\!\cdots\!12\)\( - 247808314916500164 \beta_{1} - 203561787228 \beta_{2}) q^{16} +(-\)\(24\!\cdots\!86\)\( + 1432579830459091908 \beta_{1} - 73008839916 \beta_{2}) q^{17} +(-\)\(35\!\cdots\!08\)\( - 3709293048070314324 \beta_{1} + 14019280007196 \beta_{2}) q^{19} +(-\)\(10\!\cdots\!20\)\( - 4685500071252208714 \beta_{1} + 24996949090938 \beta_{2}) q^{20} +(-\)\(33\!\cdots\!08\)\( + \)\(19\!\cdots\!84\)\( \beta_{1} - 15518897927168 \beta_{2}) q^{22} +(-\)\(13\!\cdots\!88\)\( - 71070379474158455740 \beta_{1} - 864912659360364 \beta_{2}) q^{23} +(-\)\(39\!\cdots\!25\)\( - 37236871224450578840 \beta_{1} - 3116005674041720 \beta_{2}) q^{25} +(-\)\(30\!\cdots\!92\)\( + \)\(29\!\cdots\!78\)\( \beta_{1} - 4002589901546496 \beta_{2}) q^{26} +(\)\(21\!\cdots\!52\)\( - \)\(16\!\cdots\!88\)\( \beta_{1} + 905414279433792 \beta_{2}) q^{28} +(-\)\(24\!\cdots\!54\)\( + \)\(60\!\cdots\!54\)\( \beta_{1} - 10883586973384674 \beta_{2}) q^{29} +(-\)\(12\!\cdots\!52\)\( + \)\(13\!\cdots\!26\)\( \beta_{1} - 51025473408711938 \beta_{2}) q^{31} +(-\)\(12\!\cdots\!04\)\( - \)\(16\!\cdots\!32\)\( \beta_{1} + 107917356575846952 \beta_{2}) q^{32} +(\)\(12\!\cdots\!48\)\( - \)\(34\!\cdots\!14\)\( \beta_{1} + 1454956528831466496 \beta_{2}) q^{34} +(-\)\(91\!\cdots\!40\)\( - \)\(42\!\cdots\!08\)\( \beta_{1} + 2423830555879147836 \beta_{2}) q^{35} +(\)\(96\!\cdots\!74\)\( - \)\(33\!\cdots\!92\)\( \beta_{1} - 3950416425949998144 \beta_{2}) q^{37} +(-\)\(24\!\cdots\!64\)\( + \)\(22\!\cdots\!08\)\( \beta_{1} - 8006104235315837952 \beta_{2}) q^{38} +(\)\(51\!\cdots\!00\)\( + \)\(19\!\cdots\!20\)\( \beta_{1} - 14797692611200750940 \beta_{2}) q^{40} +(\)\(46\!\cdots\!42\)\( - \)\(18\!\cdots\!28\)\( \beta_{1} - 41049028078685916228 \beta_{2}) q^{41} +(-\)\(80\!\cdots\!76\)\( - \)\(20\!\cdots\!56\)\( \beta_{1} + 42956960985072135700 \beta_{2}) q^{43} +(\)\(71\!\cdots\!04\)\( - \)\(48\!\cdots\!16\)\( \beta_{1} + \)\(17\!\cdots\!72\)\( \beta_{2}) q^{44} +(-\)\(34\!\cdots\!52\)\( - \)\(48\!\cdots\!44\)\( \beta_{1} + \)\(19\!\cdots\!12\)\( \beta_{2}) q^{46} +(\)\(36\!\cdots\!68\)\( + \)\(27\!\cdots\!32\)\( \beta_{1} - \)\(15\!\cdots\!28\)\( \beta_{2}) q^{47} +(\)\(30\!\cdots\!81\)\( - \)\(70\!\cdots\!80\)\( \beta_{1} - \)\(10\!\cdots\!68\)\( \beta_{2}) q^{49} +(\)\(39\!\cdots\!50\)\( - \)\(16\!\cdots\!85\)\( \beta_{1} + \)\(91\!\cdots\!20\)\( \beta_{2}) q^{50} +(\)\(19\!\cdots\!96\)\( - \)\(30\!\cdots\!10\)\( \beta_{1} + \)\(28\!\cdots\!50\)\( \beta_{2}) q^{52} +(-\)\(20\!\cdots\!78\)\( - \)\(33\!\cdots\!18\)\( \beta_{1} - \)\(29\!\cdots\!98\)\( \beta_{2}) q^{53} +(\)\(24\!\cdots\!80\)\( - \)\(22\!\cdots\!24\)\( \beta_{1} - \)\(78\!\cdots\!92\)\( \beta_{2}) q^{55} +(-\)\(12\!\cdots\!00\)\( + \)\(99\!\cdots\!44\)\( \beta_{1} - \)\(50\!\cdots\!28\)\( \beta_{2}) q^{56} +(\)\(93\!\cdots\!28\)\( - \)\(28\!\cdots\!62\)\( \beta_{1} + \)\(93\!\cdots\!36\)\( \beta_{2}) q^{58} +(\)\(25\!\cdots\!52\)\( - \)\(10\!\cdots\!28\)\( \beta_{1} - \)\(27\!\cdots\!56\)\( \beta_{2}) q^{59} +(-\)\(23\!\cdots\!66\)\( + \)\(74\!\cdots\!00\)\( \beta_{1} + \)\(14\!\cdots\!48\)\( \beta_{2}) q^{61} +(\)\(11\!\cdots\!72\)\( - \)\(28\!\cdots\!92\)\( \beta_{1} + \)\(14\!\cdots\!60\)\( \beta_{2}) q^{62} +(-\)\(11\!\cdots\!52\)\( + \)\(61\!\cdots\!32\)\( \beta_{1} - \)\(87\!\cdots\!04\)\( \beta_{2}) q^{64} +(\)\(23\!\cdots\!20\)\( - \)\(87\!\cdots\!96\)\( \beta_{1} + \)\(12\!\cdots\!32\)\( \beta_{2}) q^{65} +(-\)\(17\!\cdots\!24\)\( + \)\(39\!\cdots\!84\)\( \beta_{1} + \)\(35\!\cdots\!32\)\( \beta_{2}) q^{67} +(-\)\(37\!\cdots\!64\)\( + \)\(10\!\cdots\!30\)\( \beta_{1} - \)\(74\!\cdots\!34\)\( \beta_{2}) q^{68} +(\)\(12\!\cdots\!40\)\( + \)\(63\!\cdots\!68\)\( \beta_{1} - \)\(78\!\cdots\!56\)\( \beta_{2}) q^{70} +(-\)\(28\!\cdots\!92\)\( - \)\(73\!\cdots\!04\)\( \beta_{1} + \)\(74\!\cdots\!88\)\( \beta_{2}) q^{71} +(\)\(21\!\cdots\!38\)\( + \)\(25\!\cdots\!92\)\( \beta_{1} + \)\(69\!\cdots\!00\)\( \beta_{2}) q^{73} +(-\)\(29\!\cdots\!04\)\( - \)\(45\!\cdots\!06\)\( \beta_{1} - \)\(21\!\cdots\!00\)\( \beta_{2}) q^{74} +(\)\(42\!\cdots\!12\)\( - \)\(37\!\cdots\!04\)\( \beta_{1} - \)\(29\!\cdots\!04\)\( \beta_{2}) q^{76} +(\)\(73\!\cdots\!12\)\( - \)\(26\!\cdots\!32\)\( \beta_{1} + \)\(91\!\cdots\!40\)\( \beta_{2}) q^{77} +(-\)\(54\!\cdots\!00\)\( - \)\(11\!\cdots\!74\)\( \beta_{1} + \)\(11\!\cdots\!38\)\( \beta_{2}) q^{79} +(\)\(62\!\cdots\!60\)\( + \)\(17\!\cdots\!12\)\( \beta_{1} - \)\(72\!\cdots\!04\)\( \beta_{2}) q^{80} +(-\)\(16\!\cdots\!12\)\( - \)\(10\!\cdots\!66\)\( \beta_{1} - \)\(59\!\cdots\!24\)\( \beta_{2}) q^{82} +(-\)\(19\!\cdots\!16\)\( - \)\(12\!\cdots\!88\)\( \beta_{1} + \)\(40\!\cdots\!00\)\( \beta_{2}) q^{83} +(-\)\(17\!\cdots\!60\)\( - \)\(47\!\cdots\!32\)\( \beta_{1} + \)\(26\!\cdots\!44\)\( \beta_{2}) q^{85} +(-\)\(28\!\cdots\!00\)\( - \)\(62\!\cdots\!88\)\( \beta_{1} - \)\(33\!\cdots\!56\)\( \beta_{2}) q^{86} +(-\)\(34\!\cdots\!88\)\( + \)\(37\!\cdots\!28\)\( \beta_{1} - \)\(93\!\cdots\!28\)\( \beta_{2}) q^{88} +(-\)\(60\!\cdots\!62\)\( + \)\(73\!\cdots\!36\)\( \beta_{1} - \)\(13\!\cdots\!40\)\( \beta_{2}) q^{89} +(\)\(77\!\cdots\!08\)\( - \)\(10\!\cdots\!60\)\( \beta_{1} + \)\(11\!\cdots\!64\)\( \beta_{2}) q^{91} +(-\)\(32\!\cdots\!44\)\( + \)\(10\!\cdots\!88\)\( \beta_{1} - \)\(67\!\cdots\!28\)\( \beta_{2}) q^{92} +(-\)\(41\!\cdots\!40\)\( + \)\(29\!\cdots\!80\)\( \beta_{1} + \)\(75\!\cdots\!36\)\( \beta_{2}) q^{94} +(-\)\(28\!\cdots\!00\)\( + \)\(34\!\cdots\!60\)\( \beta_{1} + \)\(78\!\cdots\!80\)\( \beta_{2}) q^{95} +(-\)\(33\!\cdots\!14\)\( + \)\(51\!\cdots\!24\)\( \beta_{1} + \)\(22\!\cdots\!12\)\( \beta_{2}) q^{97} +(-\)\(64\!\cdots\!70\)\( - \)\(64\!\cdots\!91\)\( \beta_{1} - \)\(39\!\cdots\!56\)\( \beta_{2}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 533574q^{2} + 957442653732q^{4} + 53381973944430q^{5} - 1577312676222528q^{7} - 447338335366356696q^{8} + O(q^{10}) \) \( 3q - 533574q^{2} + 957442653732q^{4} + 53381973944430q^{5} - 1577312676222528q^{7} - 447338335366356696q^{8} - 39868781226579858780q^{10} + \)\(53\!\cdots\!40\)\(q^{11} + \)\(54\!\cdots\!90\)\(q^{13} - \)\(31\!\cdots\!52\)\(q^{14} + \)\(12\!\cdots\!36\)\(q^{16} - \)\(72\!\cdots\!58\)\(q^{17} - \)\(10\!\cdots\!24\)\(q^{19} - \)\(30\!\cdots\!60\)\(q^{20} - \)\(10\!\cdots\!24\)\(q^{22} - \)\(41\!\cdots\!64\)\(q^{23} - \)\(11\!\cdots\!75\)\(q^{25} - \)\(92\!\cdots\!76\)\(q^{26} + \)\(64\!\cdots\!56\)\(q^{28} - \)\(72\!\cdots\!62\)\(q^{29} - \)\(38\!\cdots\!56\)\(q^{31} - \)\(37\!\cdots\!12\)\(q^{32} + \)\(37\!\cdots\!44\)\(q^{34} - \)\(27\!\cdots\!20\)\(q^{35} + \)\(29\!\cdots\!22\)\(q^{37} - \)\(74\!\cdots\!92\)\(q^{38} + \)\(15\!\cdots\!00\)\(q^{40} + \)\(13\!\cdots\!26\)\(q^{41} - \)\(24\!\cdots\!28\)\(q^{43} + \)\(21\!\cdots\!12\)\(q^{44} - \)\(10\!\cdots\!56\)\(q^{46} + \)\(10\!\cdots\!04\)\(q^{47} + \)\(92\!\cdots\!43\)\(q^{49} + \)\(11\!\cdots\!50\)\(q^{50} + \)\(59\!\cdots\!88\)\(q^{52} - \)\(62\!\cdots\!34\)\(q^{53} + \)\(72\!\cdots\!40\)\(q^{55} - \)\(36\!\cdots\!00\)\(q^{56} + \)\(28\!\cdots\!84\)\(q^{58} + \)\(75\!\cdots\!56\)\(q^{59} - \)\(71\!\cdots\!98\)\(q^{61} + \)\(33\!\cdots\!16\)\(q^{62} - \)\(35\!\cdots\!56\)\(q^{64} + \)\(71\!\cdots\!60\)\(q^{65} - \)\(51\!\cdots\!72\)\(q^{67} - \)\(11\!\cdots\!92\)\(q^{68} + \)\(38\!\cdots\!20\)\(q^{70} - \)\(84\!\cdots\!76\)\(q^{71} + \)\(63\!\cdots\!14\)\(q^{73} - \)\(89\!\cdots\!12\)\(q^{74} + \)\(12\!\cdots\!36\)\(q^{76} + \)\(22\!\cdots\!36\)\(q^{77} - \)\(16\!\cdots\!00\)\(q^{79} + \)\(18\!\cdots\!80\)\(q^{80} - \)\(48\!\cdots\!36\)\(q^{82} - \)\(59\!\cdots\!48\)\(q^{83} - \)\(52\!\cdots\!80\)\(q^{85} - \)\(85\!\cdots\!00\)\(q^{86} - \)\(10\!\cdots\!64\)\(q^{88} - \)\(18\!\cdots\!86\)\(q^{89} + \)\(23\!\cdots\!24\)\(q^{91} - \)\(96\!\cdots\!32\)\(q^{92} - \)\(12\!\cdots\!20\)\(q^{94} - \)\(84\!\cdots\!00\)\(q^{95} - \)\(99\!\cdots\!42\)\(q^{97} - \)\(19\!\cdots\!10\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 3876249523 x - 18467420411022\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 18 \nu \)
\(\beta_{2}\)\(=\)\( 324 \nu^{2} - 2315430 \nu - 837269896968 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/18\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 128635 \beta_{1} + 837269896968\)\()/324\)

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
−59724.7
−4792.65
64517.4
−1.25290e6 0 1.02001e12 6.13018e12 0 3.89397e16 −5.89182e17 0 −7.68052e18
1.2 −264126. 0 −4.79993e11 6.30487e13 0 −4.59086e16 2.71983e17 0 −1.66528e19
1.3 983454. 0 4.17427e11 −1.57969e13 0 5.39162e15 −1.30140e17 0 −1.55355e19
\(n\): 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 9.40.a.c 3
3.b odd 2 1 3.40.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.40.a.b 3 3.b odd 2 1
9.40.a.c 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 533574 T_{2}^{2} - \)\(11\!\cdots\!60\)\( T_{2} - \)\(32\!\cdots\!08\)\( \) acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -325448454458769408 - 1161004440960 T + 533574 T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( \)\(61\!\cdots\!00\)\( - \)\(70\!\cdots\!00\)\( T - 53381973944430 T^{2} + T^{3} \)
$7$ \( \)\(96\!\cdots\!40\)\( - \)\(18\!\cdots\!44\)\( T + 1577312676222528 T^{2} + T^{3} \)
$11$ \( -\)\(53\!\cdots\!36\)\( + \)\(93\!\cdots\!48\)\( T - \)\(53\!\cdots\!40\)\( T^{2} + T^{3} \)
$13$ \( \)\(16\!\cdots\!36\)\( - \)\(63\!\cdots\!92\)\( T - \)\(54\!\cdots\!90\)\( T^{2} + T^{3} \)
$17$ \( -\)\(77\!\cdots\!44\)\( - \)\(24\!\cdots\!12\)\( T + \)\(72\!\cdots\!58\)\( T^{2} + T^{3} \)
$19$ \( -\)\(28\!\cdots\!00\)\( - \)\(78\!\cdots\!40\)\( T + \)\(10\!\cdots\!24\)\( T^{2} + T^{3} \)
$23$ \( -\)\(14\!\cdots\!40\)\( - \)\(32\!\cdots\!96\)\( T + \)\(41\!\cdots\!64\)\( T^{2} + T^{3} \)
$29$ \( \)\(12\!\cdots\!00\)\( + \)\(16\!\cdots\!80\)\( T + \)\(72\!\cdots\!62\)\( T^{2} + T^{3} \)
$31$ \( \)\(65\!\cdots\!00\)\( - \)\(23\!\cdots\!16\)\( T + \)\(38\!\cdots\!56\)\( T^{2} + T^{3} \)
$37$ \( \)\(53\!\cdots\!80\)\( - \)\(19\!\cdots\!84\)\( T - \)\(29\!\cdots\!22\)\( T^{2} + T^{3} \)
$41$ \( \)\(72\!\cdots\!00\)\( - \)\(12\!\cdots\!56\)\( T - \)\(13\!\cdots\!26\)\( T^{2} + T^{3} \)
$43$ \( \)\(39\!\cdots\!68\)\( + \)\(17\!\cdots\!56\)\( T + \)\(24\!\cdots\!28\)\( T^{2} + T^{3} \)
$47$ \( -\)\(42\!\cdots\!16\)\( + \)\(38\!\cdots\!04\)\( T - \)\(10\!\cdots\!04\)\( T^{2} + T^{3} \)
$53$ \( \)\(61\!\cdots\!80\)\( - \)\(51\!\cdots\!96\)\( T + \)\(62\!\cdots\!34\)\( T^{2} + T^{3} \)
$59$ \( \)\(56\!\cdots\!00\)\( - \)\(50\!\cdots\!80\)\( T - \)\(75\!\cdots\!56\)\( T^{2} + T^{3} \)
$61$ \( -\)\(27\!\cdots\!52\)\( - \)\(53\!\cdots\!24\)\( T + \)\(71\!\cdots\!98\)\( T^{2} + T^{3} \)
$67$ \( -\)\(88\!\cdots\!76\)\( - \)\(16\!\cdots\!72\)\( T + \)\(51\!\cdots\!72\)\( T^{2} + T^{3} \)
$71$ \( -\)\(61\!\cdots\!12\)\( - \)\(72\!\cdots\!08\)\( T + \)\(84\!\cdots\!76\)\( T^{2} + T^{3} \)
$73$ \( -\)\(42\!\cdots\!60\)\( - \)\(81\!\cdots\!96\)\( T - \)\(63\!\cdots\!14\)\( T^{2} + T^{3} \)
$79$ \( -\)\(23\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( T + \)\(16\!\cdots\!00\)\( T^{2} + T^{3} \)
$83$ \( \)\(76\!\cdots\!76\)\( + \)\(11\!\cdots\!80\)\( T + \)\(59\!\cdots\!48\)\( T^{2} + T^{3} \)
$89$ \( -\)\(15\!\cdots\!00\)\( + \)\(41\!\cdots\!20\)\( T + \)\(18\!\cdots\!86\)\( T^{2} + T^{3} \)
$97$ \( -\)\(15\!\cdots\!56\)\( - \)\(29\!\cdots\!12\)\( T + \)\(99\!\cdots\!42\)\( T^{2} + T^{3} \)
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