# Properties

 Label 9.24.a.a Level $9$ Weight $24$ Character orbit 9.a Self dual yes Analytic conductor $30.168$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 1128 q^{2} - 7116224 q^{4} + 48863730 q^{5} - 1723688680 q^{7} + 17489450496 q^{8}+O(q^{10})$$ q - 1128 * q^2 - 7116224 * q^4 + 48863730 * q^5 - 1723688680 * q^7 + 17489450496 * q^8 $$q - 1128 q^{2} - 7116224 q^{4} + 48863730 q^{5} - 1723688680 q^{7} + 17489450496 q^{8} - 55118287440 q^{10} + 1428263180124 q^{11} - 8220964044826 q^{13} + 1944320831040 q^{14} + 39967113416704 q^{16} + 5989210330446 q^{17} + 680005481275676 q^{19} - 347725248155520 q^{20} - 16\!\cdots\!72 q^{22}+ \cdots + 27\!\cdots\!04 q^{98}+O(q^{100})$$ q - 1128 * q^2 - 7116224 * q^4 + 48863730 * q^5 - 1723688680 * q^7 + 17489450496 * q^8 - 55118287440 * q^10 + 1428263180124 * q^11 - 8220964044826 * q^13 + 1944320831040 * q^14 + 39967113416704 * q^16 + 5989210330446 * q^17 + 680005481275676 * q^19 - 347725248155520 * q^20 - 1611080867179872 * q^22 - 15440648191080 * q^23 - 9533264845565225 * q^25 + 9273247442563728 * q^26 + 12266154753144320 * q^28 - 115094192813324022 * q^29 - 90829724501108800 * q^31 - 191795048280391680 * q^32 - 6755829252743088 * q^34 - 84225858263576400 * q^35 - 1297873386623227570 * q^37 - 767046182878962528 * q^38 + 854599786884910080 * q^40 - 5214036225478655130 * q^41 - 2410434516296794108 * q^43 - 10163840720714731776 * q^44 + 17417051159538240 * q^46 + 23132669525900803824 * q^47 - 24397644674520773943 * q^49 + 10753522745797573800 * q^50 + 58502221638927857024 * q^52 + 44512631945276522850 * q^53 + 69790266402520502520 * q^55 - 30146367839375585280 * q^56 + 129826249493429496816 * q^58 + 323974479000840790476 * q^59 - 199406203121599312522 * q^61 + 102455929237250726400 * q^62 - 118923632883988692992 * q^64 - 401706967426085560980 * q^65 - 646392500721161158996 * q^67 - 42620562294567755904 * q^68 + 95006768121314179200 * q^70 - 3551461551813260928312 * q^71 + 3353187900182300778170 * q^73 + 1464001180111000698960 * q^74 - 4839071325985516167424 * q^76 - 2461881075640539796320 * q^77 - 6872134095241809038320 * q^79 + 1952942238873201745920 * q^80 + 5881432862339922986640 * q^82 + 1169769717495414820644 * q^83 + 292655156500124123580 * q^85 + 2718970134382783753824 * q^86 + 24979538184038229141504 * q^88 + 23457212631337905637974 * q^89 + 14170382662753588769680 * q^91 + 109879111232920081920 * q^92 - 26093651225216106713472 * q^94 + 33227604235574687631480 * q^95 - 30603881563463466110686 * q^97 + 27520543192859433007704 * q^98

## 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
 0
−1128.00 0 −7.11622e6 4.88637e7 0 −1.72369e9 1.74895e10 0 −5.51183e10
 $$n$$: 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

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 9.24.a.a 1
3.b odd 2 1 3.24.a.a 1
12.b even 2 1 48.24.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.24.a.a 1 3.b odd 2 1
9.24.a.a 1 1.a even 1 1 trivial
48.24.a.a 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1128$$ acting on $$S_{24}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1128$$
$3$ $$T$$
$5$ $$T - 48863730$$
$7$ $$T + 1723688680$$
$11$ $$T - 1428263180124$$
$13$ $$T + 8220964044826$$
$17$ $$T - 5989210330446$$
$19$ $$T - 680005481275676$$
$23$ $$T + 15440648191080$$
$29$ $$T + 11\!\cdots\!22$$
$31$ $$T + 90\!\cdots\!00$$
$37$ $$T + 12\!\cdots\!70$$
$41$ $$T + 52\!\cdots\!30$$
$43$ $$T + 24\!\cdots\!08$$
$47$ $$T - 23\!\cdots\!24$$
$53$ $$T - 44\!\cdots\!50$$
$59$ $$T - 32\!\cdots\!76$$
$61$ $$T + 19\!\cdots\!22$$
$67$ $$T + 64\!\cdots\!96$$
$71$ $$T + 35\!\cdots\!12$$
$73$ $$T - 33\!\cdots\!70$$
$79$ $$T + 68\!\cdots\!20$$
$83$ $$T - 11\!\cdots\!44$$
$89$ $$T - 23\!\cdots\!74$$
$97$ $$T + 30\!\cdots\!86$$