# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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