Properties

Label 9.24.a.d
Level $9$
Weight $24$
Character orbit 9.a
Self dual yes
Analytic conductor $30.168$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.1683633611\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 29258 x^{2} + 97377280\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{10}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 4777492 + \beta_{3} ) q^{4} + ( 17070 \beta_{1} + \beta_{2} ) q^{5} + ( 2140359620 + 448 \beta_{3} ) q^{7} + ( 3564184 \beta_{1} + 200 \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 4777492 + \beta_{3} ) q^{4} + ( 17070 \beta_{1} + \beta_{2} ) q^{5} + ( 2140359620 + 448 \beta_{3} ) q^{7} + ( 3564184 \beta_{1} + 200 \beta_{2} ) q^{8} + ( 224744702400 + 47024 \beta_{3} ) q^{10} + ( -52068840 \beta_{1} - 6700 \beta_{2} ) q^{11} + ( -3383642499670 - 712448 \beta_{3} ) q^{13} + ( 5354894020 \beta_{1} + 89600 \beta_{2} ) q^{14} + ( 6849770431264 + 1166376 \beta_{3} ) q^{16} + ( 56834260348 \beta_{1} - 587150 \beta_{2} ) q^{17} + ( 205778476850456 + 35757696 \beta_{3} ) q^{19} + ( 418962471040 \beta_{1} + 1016192 \beta_{2} ) q^{20} + ( -685539369504000 - 252760640 \beta_{3} ) q^{22} + ( 515525395024 \beta_{1} + 14763800 \beta_{2} ) q^{23} + ( 6064335486675475 + 239366656 \beta_{3} ) q^{25} + ( -8495670634070 \beta_{1} - 142489600 \beta_{2} ) q^{26} + ( 52548376361353040 + 4280676036 \beta_{3} ) q^{28} + ( -9854914990910 \beta_{1} + 630733575 \beta_{2} ) q^{29} + ( 49791379648383908 - 21764383808 \beta_{3} ) q^{31} + ( -14679674271808 \beta_{1} - 1444446400 \beta_{2} ) q^{32} + ( 748285921901692800 + 39246769248 \beta_{3} ) q^{34} + ( 187695924518200 \beta_{1} + 455297220 \beta_{2} ) q^{35} + ( 1528578128356354910 - 10500608256 \beta_{3} ) q^{37} + ( 462350672959256 \beta_{1} + 7151539200 \beta_{2} ) q^{38} + ( 3630805946735961600 + 54935583616 \beta_{3} ) q^{40} + ( -784858875864220 \beta_{1} - 18217322850 \beta_{2} ) q^{41} + ( 2096109129722497880 - 589933098368 \beta_{3} ) q^{43} + ( -2062387701921280 \beta_{1} + 5651545600 \beta_{2} ) q^{44} + ( 6787449681956006400 + 957760260224 \beta_{3} ) q^{46} + ( -3121499288156656 \beta_{1} + 33704659000 \beta_{2} ) q^{47} + ( -3826982258097731943 + 1917762219520 \beta_{3} ) q^{49} + ( 7781863053472275 \beta_{1} + 47873331200 \beta_{2} ) q^{50} + ( -83470709594353107640 - 6787357120086 \beta_{3} ) q^{52} + ( 6648804893204178 \beta_{1} - 357950546625 \beta_{2} ) q^{53} + ( -106499640841076736000 + 1325846128640 \beta_{3} ) q^{55} + ( 38343404307139680 \beta_{1} + 104515930400 \beta_{2} ) q^{56} + ( -129751190218011096000 + 9038078514640 \beta_{3} ) q^{58} + ( -24469385199585840 \beta_{1} + 2135962067800 \beta_{2} ) q^{59} + ( -46908974320486263418 + 21337371081472 \beta_{3} ) q^{61} + ( -106374603489158492 \beta_{1} - 4352876761600 \beta_{2} ) q^{62} + ( -250733196266694509312 - 67730892782016 \beta_{3} ) q^{64} + ( -298146343466371700 \beta_{1} - 703911837270 \beta_{2} ) q^{65} + ( 554467225276943936720 - 20960113417472 \beta_{3} ) q^{67} + ( 553132934257551616 \beta_{1} + 12774725036800 \beta_{2} ) q^{68} + ( 2471223027420429408000 + 201333897446080 \beta_{3} ) q^{70} + ( -256949632766269760 \beta_{1} - 13593026376800 \beta_{2} ) q^{71} + ( 1389554758074421664390 - 241346655051264 \beta_{3} ) q^{73} + ( 1453233113937078110 \beta_{1} - 2100121651200 \beta_{2} ) q^{74} + ( 4361155751261926096352 + 376610583428888 \beta_{3} ) q^{76} + ( -923951940042896800 \beta_{1} + 2531602962000 \beta_{2} ) q^{77} + ( -390645815131573557916 - 380716817262656 \beta_{3} ) q^{79} + ( 510473303589934080 \beta_{1} + 2462680382464 \beta_{2} ) q^{80} + ( -10333519066976054832000 - 1330540564513120 \beta_{3} ) q^{82} + ( -4165449121937070968 \beta_{1} + 61106262577500 \beta_{2} ) q^{83} + ( 4465688514941465395200 + 3003335243645952 \beta_{3} ) q^{85} + ( -2136837830997412520 \beta_{1} - 117986619673600 \beta_{2} ) q^{86} + ( -21402885212884935936000 + 227208621770240 \beta_{3} ) q^{88} + ( -528574317491009800 \beta_{1} - 1168489865500 \beta_{2} ) q^{89} + ( -37395024085160747965400 - 3040766770401920 \beta_{3} ) q^{91} + ( 9335126424239787008 \beta_{1} + 67704321254400 \beta_{2} ) q^{92} + ( -41097992829729359961600 - 2111909932470656 \beta_{3} ) q^{94} + ( 15577667215400573520 \beta_{1} + 71283054885656 \beta_{2} ) q^{95} + ( 15422116373173492952270 + 1488781249113088 \beta_{3} ) q^{97} + ( 9933536995624124057 \beta_{1} + 383552443904000 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 19109968 q^{4} + 8561438480 q^{7} + O(q^{10}) \) \( 4 q + 19109968 q^{4} + 8561438480 q^{7} + 898978809600 q^{10} - 13534569998680 q^{13} + 27399081725056 q^{16} + 823113907401824 q^{19} - 2742157478016000 q^{22} + 24257341946701900 q^{25} + 210193505445412160 q^{28} + 199165518593535632 q^{31} + 2993143687606771200 q^{34} + 6114312513425419640 q^{37} + 14523223786943846400 q^{40} + 8384436518889991520 q^{43} + 27149798727824025600 q^{46} - 15307929032390927772 q^{49} - 333882838377412430560 q^{52} - 425998563364306944000 q^{55} - 519004760872044384000 q^{58} - 187635897281945053672 q^{61} - 1002932785066778037248 q^{64} + 2217868901107775746880 q^{67} + 9884892109681717632000 q^{70} + 5558219032297686657560 q^{73} + 17444623005047704385408 q^{76} - 1562583260526294231664 q^{79} - 41334076267904219328000 q^{82} + 17862754059765861580800 q^{85} - 85611540851539743744000 q^{88} - 149580096340642991861600 q^{91} - 164391971318917439846400 q^{94} + 61688465492693971809080 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 29258 x^{2} + 97377280\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 30 \nu \)
\(\beta_{2}\)\(=\)\( 135 \nu^{3} - 3051210 \nu \)
\(\beta_{3}\)\(=\)\( 900 \nu^{2} - 13166100 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/30\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 13166100\)\()/900\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 101707 \beta_{1}\)\()/135\)

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
−159.463
−61.8825
61.8825
159.463
−4783.90 0 1.44971e7 −1.42520e8 0 6.49474e9 −2.92224e10 0 6.81799e11
1.2 −1856.48 0 −4.94211e6 1.25135e8 0 −2.21402e9 2.47481e10 0 −2.32310e11
1.3 1856.48 0 −4.94211e6 −1.25135e8 0 −2.21402e9 −2.47481e10 0 −2.32310e11
1.4 4783.90 0 1.44971e7 1.42520e8 0 6.49474e9 2.92224e10 0 6.81799e11
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.24.a.d 4
3.b odd 2 1 inner 9.24.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.24.a.d 4 1.a even 1 1 trivial
9.24.a.d 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 26332200 T_{2}^{2} + \)\(78\!\cdots\!00\)\( \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 78875596800000 - 26332200 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( \)\(31\!\cdots\!00\)\( - 35970528883507200 T^{2} + T^{4} \)
$7$ \( ( -14379486476130095600 - 4280719240 T + T^{2} )^{2} \)
$11$ \( \)\(39\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( T^{2} + T^{4} \)
$13$ \( ( -\)\(36\!\cdots\!00\)\( + 6767284999340 T + T^{2} )^{2} \)
$17$ \( \)\(21\!\cdots\!00\)\( - \)\(94\!\cdots\!00\)\( T^{2} + T^{4} \)
$19$ \( ( -\)\(78\!\cdots\!64\)\( - 411556953700912 T + T^{2} )^{2} \)
$23$ \( \)\(20\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( T^{2} + T^{4} \)
$29$ \( \)\(10\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( T^{2} + T^{4} \)
$31$ \( ( -\)\(42\!\cdots\!36\)\( - 99582759296767816 T + T^{2} )^{2} \)
$37$ \( ( \)\(23\!\cdots\!00\)\( - 3057156256712709820 T + T^{2} )^{2} \)
$41$ \( \)\(46\!\cdots\!00\)\( - \)\(25\!\cdots\!00\)\( T^{2} + T^{4} \)
$43$ \( ( -\)\(28\!\cdots\!00\)\( - 4192218259444995760 T + T^{2} )^{2} \)
$47$ \( \)\(20\!\cdots\!00\)\( - \)\(28\!\cdots\!00\)\( T^{2} + T^{4} \)
$53$ \( \)\(47\!\cdots\!00\)\( - \)\(47\!\cdots\!00\)\( T^{2} + T^{4} \)
$59$ \( \)\(24\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( T^{2} + T^{4} \)
$61$ \( ( -\)\(40\!\cdots\!76\)\( + 93817948640972526836 T + T^{2} )^{2} \)
$67$ \( ( \)\(26\!\cdots\!00\)\( - \)\(11\!\cdots\!40\)\( T + T^{2} )^{2} \)
$71$ \( \)\(11\!\cdots\!00\)\( - \)\(69\!\cdots\!00\)\( T^{2} + T^{4} \)
$73$ \( ( -\)\(35\!\cdots\!00\)\( - \)\(27\!\cdots\!80\)\( T + T^{2} )^{2} \)
$79$ \( ( -\)\(13\!\cdots\!44\)\( + \)\(78\!\cdots\!32\)\( T + T^{2} )^{2} \)
$83$ \( \)\(78\!\cdots\!00\)\( - \)\(56\!\cdots\!00\)\( T^{2} + T^{4} \)
$89$ \( \)\(43\!\cdots\!00\)\( - \)\(73\!\cdots\!00\)\( T^{2} + T^{4} \)
$97$ \( ( \)\(28\!\cdots\!00\)\( - \)\(30\!\cdots\!40\)\( T + T^{2} )^{2} \)
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