Properties

Label 9.34.a.b
Level $9$
Weight $34$
Character orbit 9.a
Self dual yes
Analytic conductor $62.085$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.0845459929\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 589050\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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 \(\beta = 72\sqrt{2356201}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 60840 - \beta ) q^{2} + ( 7326116992 - 121680 \beta ) q^{4} + ( 90530768250 - 1004000 \beta ) q^{5} + ( -33576540033400 - 801594864 \beta ) q^{7} + ( 1409375292549120 - 6139193600 \beta ) q^{8} +O(q^{10})\) \( q +(60840 - \beta) q^{2} +(7326116992 - 121680 \beta) q^{4} +(90530768250 - 1004000 \beta) q^{5} +(-33576540033400 - 801594864 \beta) q^{7} +(1409375292549120 - 6139193600 \beta) q^{8} +(17771296108266000 - 151614128250 \beta) q^{10} +(-66935907720957132 - 1346611783000 \beta) q^{11} +(-1490805239129721970 + 3738595861728 \beta) q^{13} +(7748320631234170176 - 15192491492360 \beta) q^{14} +(97802989555947175936 - 737660590018560 \beta) q^{16} +(39680574630587762670 - 2662090686805056 \beta) q^{17} +(-\)\(68\!\cdots\!00\)\( - 4884703150768920 \beta) q^{19} +(\)\(21\!\cdots\!00\)\( - 18371205340628000 \beta) q^{20} +(\)\(12\!\cdots\!20\)\( - 14991953156762868 \beta) q^{22} +(-\)\(13\!\cdots\!20\)\( + 164629195362887632 \beta) q^{23} +(-\)\(95\!\cdots\!25\)\( - 181785782646000000 \beta) q^{25} +(-\)\(13\!\cdots\!52\)\( + 1718261411357253490 \beta) q^{26} +(\)\(94\!\cdots\!80\)\( - 1786984362586217088 \beta) q^{28} +(\)\(83\!\cdots\!50\)\( - 6085145360184173920 \beta) q^{29} +(-\)\(31\!\cdots\!08\)\( - 32916497064471576000 \beta) q^{31} +(\)\(28\!\cdots\!40\)\( - 89946988381051355136 \beta) q^{32} +(\)\(34\!\cdots\!04\)\( - \)\(20\!\cdots\!10\)\( \beta) q^{34} +(\)\(67\!\cdots\!00\)\( - 38858152669640668000 \beta) q^{35} +(-\)\(52\!\cdots\!10\)\( - \)\(20\!\cdots\!04\)\( \beta) q^{37} +(\)\(18\!\cdots\!80\)\( + \)\(38\!\cdots\!00\)\( \beta) q^{38} +(\)\(20\!\cdots\!00\)\( - \)\(19\!\cdots\!00\)\( \beta) q^{40} +(-\)\(13\!\cdots\!22\)\( - \)\(32\!\cdots\!00\)\( \beta) q^{41} +(\)\(78\!\cdots\!00\)\( - \)\(88\!\cdots\!52\)\( \beta) q^{43} +(\)\(15\!\cdots\!56\)\( - \)\(17\!\cdots\!40\)\( \beta) q^{44} +(-\)\(28\!\cdots\!88\)\( + \)\(23\!\cdots\!00\)\( \beta) q^{46} +(-\)\(27\!\cdots\!20\)\( + \)\(49\!\cdots\!84\)\( \beta) q^{47} +(\)\(12\!\cdots\!57\)\( + \)\(53\!\cdots\!00\)\( \beta) q^{49} +(-\)\(36\!\cdots\!00\)\( + \)\(84\!\cdots\!25\)\( \beta) q^{50} +(-\)\(16\!\cdots\!00\)\( + \)\(20\!\cdots\!76\)\( \beta) q^{52} +(\)\(13\!\cdots\!10\)\( + \)\(20\!\cdots\!12\)\( \beta) q^{53} +(\)\(10\!\cdots\!00\)\( - \)\(54\!\cdots\!00\)\( \beta) q^{55} +(\)\(12\!\cdots\!00\)\( - \)\(92\!\cdots\!80\)\( \beta) q^{56} +(\)\(12\!\cdots\!80\)\( - \)\(12\!\cdots\!50\)\( \beta) q^{58} +(\)\(15\!\cdots\!00\)\( + \)\(31\!\cdots\!60\)\( \beta) q^{59} +(-\)\(28\!\cdots\!18\)\( + \)\(63\!\cdots\!00\)\( \beta) q^{61} +(\)\(21\!\cdots\!80\)\( + \)\(11\!\cdots\!08\)\( \beta) q^{62} +(\)\(43\!\cdots\!12\)\( - \)\(19\!\cdots\!60\)\( \beta) q^{64} +(-\)\(18\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( \beta) q^{65} +(\)\(79\!\cdots\!80\)\( - \)\(28\!\cdots\!44\)\( \beta) q^{67} +(\)\(42\!\cdots\!60\)\( - \)\(24\!\cdots\!52\)\( \beta) q^{68} +(\)\(88\!\cdots\!00\)\( - \)\(91\!\cdots\!00\)\( \beta) q^{70} +(\)\(13\!\cdots\!88\)\( + \)\(18\!\cdots\!00\)\( \beta) q^{71} +(\)\(47\!\cdots\!70\)\( + \)\(59\!\cdots\!68\)\( \beta) q^{73} +(-\)\(72\!\cdots\!64\)\( + \)\(40\!\cdots\!50\)\( \beta) q^{74} +(\)\(22\!\cdots\!00\)\( + \)\(46\!\cdots\!60\)\( \beta) q^{76} +(\)\(15\!\cdots\!00\)\( + \)\(98\!\cdots\!48\)\( \beta) q^{77} +(-\)\(42\!\cdots\!00\)\( + \)\(11\!\cdots\!20\)\( \beta) q^{79} +(\)\(17\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( \beta) q^{80} +(\)\(31\!\cdots\!20\)\( - \)\(58\!\cdots\!78\)\( \beta) q^{82} +(-\)\(14\!\cdots\!60\)\( + \)\(24\!\cdots\!92\)\( \beta) q^{83} +(\)\(36\!\cdots\!00\)\( - \)\(28\!\cdots\!00\)\( \beta) q^{85} +(\)\(15\!\cdots\!68\)\( - \)\(13\!\cdots\!80\)\( \beta) q^{86} +(\)\(66\!\cdots\!60\)\( - \)\(14\!\cdots\!00\)\( \beta) q^{88} +(-\)\(66\!\cdots\!50\)\( - \)\(38\!\cdots\!60\)\( \beta) q^{89} +(\)\(13\!\cdots\!72\)\( + \)\(10\!\cdots\!80\)\( \beta) q^{91} +(-\)\(34\!\cdots\!80\)\( + \)\(27\!\cdots\!44\)\( \beta) q^{92} +(-\)\(22\!\cdots\!56\)\( + \)\(30\!\cdots\!80\)\( \beta) q^{94} +(-\)\(16\!\cdots\!00\)\( + \)\(24\!\cdots\!00\)\( \beta) q^{95} +(-\)\(18\!\cdots\!30\)\( - \)\(53\!\cdots\!84\)\( \beta) q^{97} +(-\)\(58\!\cdots\!20\)\( + \)\(20\!\cdots\!43\)\( \beta) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 121680q^{2} + 14652233984q^{4} + 181061536500q^{5} - 67153080066800q^{7} + 2818750585098240q^{8} + O(q^{10}) \) \( 2q + 121680q^{2} + 14652233984q^{4} + 181061536500q^{5} - 67153080066800q^{7} + 2818750585098240q^{8} + 35542592216532000q^{10} - 133871815441914264q^{11} - 2981610478259443940q^{13} + 15496641262468340352q^{14} + \)\(19\!\cdots\!72\)\(q^{16} + 79361149261175525340q^{17} - \)\(13\!\cdots\!00\)\(q^{19} + \)\(43\!\cdots\!00\)\(q^{20} + \)\(24\!\cdots\!40\)\(q^{22} - \)\(26\!\cdots\!40\)\(q^{23} - \)\(19\!\cdots\!50\)\(q^{25} - \)\(27\!\cdots\!04\)\(q^{26} + \)\(18\!\cdots\!60\)\(q^{28} + \)\(16\!\cdots\!00\)\(q^{29} - \)\(62\!\cdots\!16\)\(q^{31} + \)\(57\!\cdots\!80\)\(q^{32} + \)\(69\!\cdots\!08\)\(q^{34} + \)\(13\!\cdots\!00\)\(q^{35} - \)\(10\!\cdots\!20\)\(q^{37} + \)\(36\!\cdots\!60\)\(q^{38} + \)\(40\!\cdots\!00\)\(q^{40} - \)\(27\!\cdots\!44\)\(q^{41} + \)\(15\!\cdots\!00\)\(q^{43} + \)\(30\!\cdots\!12\)\(q^{44} - \)\(56\!\cdots\!76\)\(q^{46} - \)\(54\!\cdots\!40\)\(q^{47} + \)\(24\!\cdots\!14\)\(q^{49} - \)\(72\!\cdots\!00\)\(q^{50} - \)\(32\!\cdots\!00\)\(q^{52} + \)\(26\!\cdots\!20\)\(q^{53} + \)\(20\!\cdots\!00\)\(q^{55} + \)\(25\!\cdots\!00\)\(q^{56} + \)\(25\!\cdots\!60\)\(q^{58} + \)\(30\!\cdots\!00\)\(q^{59} - \)\(57\!\cdots\!36\)\(q^{61} + \)\(42\!\cdots\!60\)\(q^{62} + \)\(86\!\cdots\!24\)\(q^{64} - \)\(36\!\cdots\!00\)\(q^{65} + \)\(15\!\cdots\!60\)\(q^{67} + \)\(84\!\cdots\!20\)\(q^{68} + \)\(17\!\cdots\!00\)\(q^{70} + \)\(26\!\cdots\!76\)\(q^{71} + \)\(94\!\cdots\!40\)\(q^{73} - \)\(14\!\cdots\!28\)\(q^{74} + \)\(45\!\cdots\!00\)\(q^{76} + \)\(30\!\cdots\!00\)\(q^{77} - \)\(85\!\cdots\!00\)\(q^{79} + \)\(35\!\cdots\!00\)\(q^{80} + \)\(62\!\cdots\!40\)\(q^{82} - \)\(29\!\cdots\!20\)\(q^{83} + \)\(72\!\cdots\!00\)\(q^{85} + \)\(31\!\cdots\!36\)\(q^{86} + \)\(13\!\cdots\!20\)\(q^{88} - \)\(13\!\cdots\!00\)\(q^{89} + \)\(26\!\cdots\!44\)\(q^{91} - \)\(68\!\cdots\!60\)\(q^{92} - \)\(45\!\cdots\!12\)\(q^{94} - \)\(33\!\cdots\!00\)\(q^{95} - \)\(36\!\cdots\!60\)\(q^{97} - \)\(11\!\cdots\!40\)\(q^{98} + O(q^{100}) \)

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
767.996
−766.996
−49679.4 0 −6.12189e9 −2.04307e10 0 −1.22168e14 7.30875e14 0 1.01499e15
1.2 171359. 0 2.07741e10 2.01492e11 0 5.50153e13 2.08788e15 0 3.45276e16
\(n\): 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 9.34.a.b 2
3.b odd 2 1 1.34.a.a 2
12.b even 2 1 16.34.a.b 2
15.d odd 2 1 25.34.a.a 2
15.e even 4 2 25.34.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.34.a.a 2 3.b odd 2 1
9.34.a.b 2 1.a even 1 1 trivial
16.34.a.b 2 12.b even 2 1
25.34.a.a 2 15.d odd 2 1
25.34.b.a 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 121680 T_{2} - 8513040384 \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -8513040384 - 121680 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -\)\(41\!\cdots\!00\)\( - 181061536500 T + T^{2} \)
$7$ \( -\)\(67\!\cdots\!64\)\( + 67153080066800 T + T^{2} \)
$11$ \( -\)\(17\!\cdots\!76\)\( + 133871815441914264 T + T^{2} \)
$13$ \( \)\(20\!\cdots\!44\)\( + 2981610478259443940 T + T^{2} \)
$17$ \( -\)\(84\!\cdots\!24\)\( - 79361149261175525340 T + T^{2} \)
$19$ \( \)\(17\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T + T^{2} \)
$23$ \( -\)\(15\!\cdots\!16\)\( + \)\(26\!\cdots\!40\)\( T + T^{2} \)
$29$ \( \)\(24\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( T + T^{2} \)
$31$ \( -\)\(35\!\cdots\!36\)\( + \)\(62\!\cdots\!16\)\( T + T^{2} \)
$37$ \( \)\(22\!\cdots\!56\)\( + \)\(10\!\cdots\!20\)\( T + T^{2} \)
$41$ \( -\)\(10\!\cdots\!16\)\( + \)\(27\!\cdots\!44\)\( T + T^{2} \)
$43$ \( -\)\(35\!\cdots\!36\)\( - \)\(15\!\cdots\!00\)\( T + T^{2} \)
$47$ \( \)\(70\!\cdots\!96\)\( + \)\(54\!\cdots\!40\)\( T + T^{2} \)
$53$ \( -\)\(33\!\cdots\!96\)\( - \)\(26\!\cdots\!20\)\( T + T^{2} \)
$59$ \( \)\(22\!\cdots\!00\)\( - \)\(30\!\cdots\!00\)\( T + T^{2} \)
$61$ \( -\)\(41\!\cdots\!76\)\( + \)\(57\!\cdots\!36\)\( T + T^{2} \)
$67$ \( \)\(54\!\cdots\!76\)\( - \)\(15\!\cdots\!60\)\( T + T^{2} \)
$71$ \( -\)\(26\!\cdots\!56\)\( - \)\(26\!\cdots\!76\)\( T + T^{2} \)
$73$ \( -\)\(42\!\cdots\!16\)\( - \)\(94\!\cdots\!40\)\( T + T^{2} \)
$79$ \( \)\(16\!\cdots\!00\)\( + \)\(85\!\cdots\!00\)\( T + T^{2} \)
$83$ \( -\)\(50\!\cdots\!76\)\( + \)\(29\!\cdots\!20\)\( T + T^{2} \)
$89$ \( \)\(43\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T + T^{2} \)
$97$ \( -\)\(31\!\cdots\!04\)\( + \)\(36\!\cdots\!60\)\( T + T^{2} \)
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