Properties

Label 9.38.a.a
Level $9$
Weight $38$
Character orbit 9.a
Self dual yes
Analytic conductor $78.043$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.0426343121\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 15934380\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
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 \(\beta = 48\sqrt{63737521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 97200 - \beta ) q^{2} + ( 18860134912 - 194400 \beta ) q^{4} + ( -2764792192950 + 28866400 \beta ) q^{5} + ( -1724221976743000 + 9650004336 \beta ) q^{7} + ( 17022021521817600 + 99683138560 \beta ) q^{8} +O(q^{10})\) \( q +(97200 - \beta) q^{2} +(18860134912 - 194400 \beta) q^{4} +(-2764792192950 + 28866400 \beta) q^{5} +(-1724221976743000 + 9650004336 \beta) q^{7} +(17022021521817600 + 99683138560 \beta) q^{8} +(-4507804677506637600 + 5570606272950 \beta) q^{10} +(13367018177424269028 + 32186306471000 \beta) q^{11} +(\)\(26\!\cdots\!50\)\( + 724443400725408 \beta) q^{13} +(-\)\(15\!\cdots\!24\)\( + 2662202398202200 \beta) q^{14} +(-\)\(15\!\cdots\!04\)\( + 19385312101171200 \beta) q^{16} +(\)\(44\!\cdots\!50\)\( - 24633489938571456 \beta) q^{17} +(\)\(18\!\cdots\!60\)\( + 1292976182287157400 \beta) q^{19} +(-\)\(87\!\cdots\!00\)\( + 1081899800733236800 \beta) q^{20} +(-\)\(34\!\cdots\!00\)\( - 10238509188443069028 \beta) q^{22} +(\)\(13\!\cdots\!00\)\( - 34024622863285905488 \beta) q^{23} +(\)\(57\!\cdots\!75\)\( - \)\(15\!\cdots\!00\)\( \beta) q^{25} +(-\)\(80\!\cdots\!72\)\( - \)\(19\!\cdots\!50\)\( \beta) q^{26} +(-\)\(30\!\cdots\!00\)\( + \)\(51\!\cdots\!32\)\( \beta) q^{28} +(\)\(63\!\cdots\!10\)\( + \)\(23\!\cdots\!00\)\( \beta) q^{29} +(\)\(13\!\cdots\!12\)\( + \)\(64\!\cdots\!00\)\( \beta) q^{31} +(-\)\(67\!\cdots\!00\)\( + \)\(37\!\cdots\!84\)\( \beta) q^{32} +(\)\(79\!\cdots\!04\)\( - \)\(47\!\cdots\!50\)\( \beta) q^{34} +(\)\(45\!\cdots\!00\)\( - \)\(76\!\cdots\!00\)\( \beta) q^{35} +(-\)\(34\!\cdots\!50\)\( - \)\(25\!\cdots\!04\)\( \beta) q^{37} +(-\)\(17\!\cdots\!00\)\( - \)\(60\!\cdots\!60\)\( \beta) q^{38} +(\)\(37\!\cdots\!00\)\( + \)\(21\!\cdots\!00\)\( \beta) q^{40} +(\)\(63\!\cdots\!18\)\( - \)\(58\!\cdots\!00\)\( \beta) q^{41} +(-\)\(12\!\cdots\!00\)\( - \)\(42\!\cdots\!52\)\( \beta) q^{43} +(-\)\(66\!\cdots\!64\)\( - \)\(19\!\cdots\!00\)\( \beta) q^{44} +(\)\(62\!\cdots\!92\)\( - \)\(16\!\cdots\!00\)\( \beta) q^{46} +(-\)\(21\!\cdots\!00\)\( - \)\(38\!\cdots\!16\)\( \beta) q^{47} +(-\)\(19\!\cdots\!43\)\( - \)\(33\!\cdots\!00\)\( \beta) q^{49} +(\)\(29\!\cdots\!00\)\( - \)\(72\!\cdots\!75\)\( \beta) q^{50} +(-\)\(15\!\cdots\!00\)\( - \)\(37\!\cdots\!04\)\( \beta) q^{52} +(-\)\(79\!\cdots\!50\)\( + \)\(11\!\cdots\!72\)\( \beta) q^{53} +(\)\(99\!\cdots\!00\)\( + \)\(29\!\cdots\!00\)\( \beta) q^{55} +(\)\(11\!\cdots\!40\)\( - \)\(76\!\cdots\!00\)\( \beta) q^{56} +(-\)\(27\!\cdots\!00\)\( - \)\(41\!\cdots\!10\)\( \beta) q^{58} +(\)\(11\!\cdots\!20\)\( + \)\(48\!\cdots\!00\)\( \beta) q^{59} +(\)\(50\!\cdots\!22\)\( + \)\(28\!\cdots\!00\)\( \beta) q^{61} +(-\)\(93\!\cdots\!00\)\( + \)\(49\!\cdots\!88\)\( \beta) q^{62} +(\)\(93\!\cdots\!32\)\( + \)\(44\!\cdots\!00\)\( \beta) q^{64} +(\)\(23\!\cdots\!00\)\( + \)\(56\!\cdots\!00\)\( \beta) q^{65} +(-\)\(54\!\cdots\!00\)\( + \)\(78\!\cdots\!56\)\( \beta) q^{67} +(\)\(15\!\cdots\!00\)\( - \)\(91\!\cdots\!72\)\( \beta) q^{68} +(\)\(15\!\cdots\!00\)\( - \)\(53\!\cdots\!00\)\( \beta) q^{70} +(\)\(37\!\cdots\!08\)\( - \)\(17\!\cdots\!00\)\( \beta) q^{71} +(\)\(98\!\cdots\!50\)\( + \)\(11\!\cdots\!88\)\( \beta) q^{73} +(\)\(33\!\cdots\!36\)\( + \)\(96\!\cdots\!50\)\( \beta) q^{74} +(-\)\(33\!\cdots\!80\)\( - \)\(11\!\cdots\!00\)\( \beta) q^{76} +(\)\(22\!\cdots\!00\)\( + \)\(73\!\cdots\!08\)\( \beta) q^{77} +(\)\(13\!\cdots\!40\)\( - \)\(40\!\cdots\!00\)\( \beta) q^{79} +(\)\(12\!\cdots\!00\)\( - \)\(50\!\cdots\!00\)\( \beta) q^{80} +(\)\(14\!\cdots\!00\)\( - \)\(68\!\cdots\!18\)\( \beta) q^{82} +(\)\(23\!\cdots\!00\)\( + \)\(23\!\cdots\!32\)\( \beta) q^{83} +(-\)\(22\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( \beta) q^{85} +(\)\(50\!\cdots\!68\)\( + \)\(86\!\cdots\!00\)\( \beta) q^{86} +(\)\(69\!\cdots\!00\)\( + \)\(18\!\cdots\!80\)\( \beta) q^{88} +(\)\(66\!\cdots\!30\)\( + \)\(23\!\cdots\!00\)\( \beta) q^{89} +(\)\(56\!\cdots\!92\)\( + \)\(13\!\cdots\!00\)\( \beta) q^{91} +(\)\(12\!\cdots\!00\)\( - \)\(31\!\cdots\!56\)\( \beta) q^{92} +(\)\(35\!\cdots\!44\)\( + \)\(17\!\cdots\!00\)\( \beta) q^{94} +(\)\(49\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( \beta) q^{95} +(\)\(30\!\cdots\!50\)\( - \)\(10\!\cdots\!84\)\( \beta) q^{97} +(\)\(47\!\cdots\!00\)\( - \)\(13\!\cdots\!57\)\( \beta) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 194400q^{2} + 37720269824q^{4} - 5529584385900q^{5} - 3448443953486000q^{7} + 34044043043635200q^{8} + O(q^{10}) \) \( 2q + 194400q^{2} + 37720269824q^{4} - 5529584385900q^{5} - 3448443953486000q^{7} + 34044043043635200q^{8} - 9015609355013275200q^{10} + 26734036354848538056q^{11} + \)\(53\!\cdots\!00\)\(q^{13} - \)\(31\!\cdots\!48\)\(q^{14} - \)\(31\!\cdots\!08\)\(q^{16} + \)\(89\!\cdots\!00\)\(q^{17} + \)\(37\!\cdots\!20\)\(q^{19} - \)\(17\!\cdots\!00\)\(q^{20} - \)\(68\!\cdots\!00\)\(q^{22} + \)\(26\!\cdots\!00\)\(q^{23} + \)\(11\!\cdots\!50\)\(q^{25} - \)\(16\!\cdots\!44\)\(q^{26} - \)\(61\!\cdots\!00\)\(q^{28} + \)\(12\!\cdots\!20\)\(q^{29} + \)\(26\!\cdots\!24\)\(q^{31} - \)\(13\!\cdots\!00\)\(q^{32} + \)\(15\!\cdots\!08\)\(q^{34} + \)\(91\!\cdots\!00\)\(q^{35} - \)\(68\!\cdots\!00\)\(q^{37} - \)\(34\!\cdots\!00\)\(q^{38} + \)\(75\!\cdots\!00\)\(q^{40} + \)\(12\!\cdots\!36\)\(q^{41} - \)\(25\!\cdots\!00\)\(q^{43} - \)\(13\!\cdots\!28\)\(q^{44} + \)\(12\!\cdots\!84\)\(q^{46} - \)\(42\!\cdots\!00\)\(q^{47} - \)\(38\!\cdots\!86\)\(q^{49} + \)\(58\!\cdots\!00\)\(q^{50} - \)\(31\!\cdots\!00\)\(q^{52} - \)\(15\!\cdots\!00\)\(q^{53} + \)\(19\!\cdots\!00\)\(q^{55} + \)\(22\!\cdots\!80\)\(q^{56} - \)\(55\!\cdots\!00\)\(q^{58} + \)\(23\!\cdots\!40\)\(q^{59} + \)\(10\!\cdots\!44\)\(q^{61} - \)\(18\!\cdots\!00\)\(q^{62} + \)\(18\!\cdots\!64\)\(q^{64} + \)\(46\!\cdots\!00\)\(q^{65} - \)\(10\!\cdots\!00\)\(q^{67} + \)\(30\!\cdots\!00\)\(q^{68} + \)\(31\!\cdots\!00\)\(q^{70} + \)\(74\!\cdots\!16\)\(q^{71} + \)\(19\!\cdots\!00\)\(q^{73} + \)\(67\!\cdots\!72\)\(q^{74} - \)\(66\!\cdots\!60\)\(q^{76} + \)\(45\!\cdots\!00\)\(q^{77} + \)\(27\!\cdots\!80\)\(q^{79} + \)\(25\!\cdots\!00\)\(q^{80} + \)\(29\!\cdots\!00\)\(q^{82} + \)\(47\!\cdots\!00\)\(q^{83} - \)\(45\!\cdots\!00\)\(q^{85} + \)\(10\!\cdots\!36\)\(q^{86} + \)\(13\!\cdots\!00\)\(q^{88} + \)\(13\!\cdots\!60\)\(q^{89} + \)\(11\!\cdots\!84\)\(q^{91} + \)\(24\!\cdots\!00\)\(q^{92} + \)\(70\!\cdots\!88\)\(q^{94} + \)\(99\!\cdots\!00\)\(q^{95} + \)\(60\!\cdots\!00\)\(q^{97} + \)\(94\!\cdots\!00\)\(q^{98} + O(q^{100}) \)

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
3992.29
−3991.29
−286012. 0 −5.56362e10 8.29715e12 0 1.97377e15 5.52218e16 0 −2.37308e18
1.2 480412. 0 9.33565e10 −1.38267e13 0 −5.42222e15 −2.11777e16 0 −6.64253e18
\(n\): 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 9.38.a.a 2
3.b odd 2 1 1.38.a.a 2
12.b even 2 1 16.38.a.b 2
15.d odd 2 1 25.38.a.a 2
15.e even 4 2 25.38.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.38.a.a 2 3.b odd 2 1
9.38.a.a 2 1.a even 1 1 trivial
16.38.a.b 2 12.b even 2 1
25.38.a.a 2 15.d odd 2 1
25.38.b.a 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 194400 T_{2} - 137403408384 \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 194400 T + 137474498560 T^{2} - 26718132554956800 T^{3} + \)\(18\!\cdots\!84\)\( T^{4} \)
$3$ 1
$5$ \( 1 + 5529584385900 T + \)\(30\!\cdots\!50\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + 3448443953486000 T + \)\(26\!\cdots\!50\)\( T^{2} + \)\(64\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 26734036354848538056 T + \)\(70\!\cdots\!26\)\( T^{2} - \)\(90\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \)
$13$ \( 1 - \)\(53\!\cdots\!00\)\( T + \)\(32\!\cdots\!90\)\( T^{2} - \)\(87\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \)
$17$ \( 1 - \)\(89\!\cdots\!00\)\( T + \)\(86\!\cdots\!30\)\( T^{2} - \)\(30\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \)
$19$ \( 1 - \)\(37\!\cdots\!20\)\( T + \)\(20\!\cdots\!78\)\( T^{2} - \)\(76\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 - \)\(26\!\cdots\!00\)\( T + \)\(48\!\cdots\!10\)\( T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!09\)\( T^{4} \)
$29$ \( 1 - \)\(12\!\cdots\!20\)\( T + \)\(21\!\cdots\!18\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 - \)\(26\!\cdots\!24\)\( T + \)\(24\!\cdots\!66\)\( T^{2} - \)\(39\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 + \)\(68\!\cdots\!00\)\( T + \)\(13\!\cdots\!90\)\( T^{2} + \)\(71\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 - \)\(12\!\cdots\!36\)\( T + \)\(12\!\cdots\!86\)\( T^{2} - \)\(59\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 + \)\(25\!\cdots\!00\)\( T + \)\(44\!\cdots\!50\)\( T^{2} + \)\(70\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(42\!\cdots\!00\)\( T + \)\(14\!\cdots\!70\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(16\!\cdots\!70\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 - \)\(23\!\cdots\!40\)\( T + \)\(64\!\cdots\!38\)\( T^{2} - \)\(79\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 - \)\(10\!\cdots\!44\)\( T + \)\(10\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(94\!\cdots\!30\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 - \)\(74\!\cdots\!16\)\( T + \)\(58\!\cdots\!46\)\( T^{2} - \)\(23\!\cdots\!56\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 - \)\(19\!\cdots\!00\)\( T + \)\(18\!\cdots\!10\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 - \)\(27\!\cdots\!80\)\( T + \)\(50\!\cdots\!18\)\( T^{2} - \)\(44\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 - \)\(47\!\cdots\!00\)\( T + \)\(24\!\cdots\!30\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 - \)\(13\!\cdots\!60\)\( T + \)\(23\!\cdots\!58\)\( T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 - \)\(60\!\cdots\!00\)\( T + \)\(57\!\cdots\!70\)\( T^{2} - \)\(19\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \)
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