Properties

 Label 9.26.a.a Level $9$ Weight $26$ Character orbit 9.a Self dual yes Analytic conductor $35.640$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$26$$ Character orbit: $$[\chi]$$ $$=$$ 9.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$35.6397101957$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 48 q^{2} - 33552128 q^{4} + 741989850 q^{5} + 39080597192 q^{7} - 3221114880 q^{8}+O(q^{10})$$ q + 48 * q^2 - 33552128 * q^4 + 741989850 * q^5 + 39080597192 * q^7 - 3221114880 * q^8 $$q + 48 q^{2} - 33552128 q^{4} + 741989850 q^{5} + 39080597192 q^{7} - 3221114880 q^{8} + 35615512800 q^{10} - 8419515299052 q^{11} - 81651045335314 q^{13} + 1875868665216 q^{14} + 11\!\cdots\!56 q^{16}+ \cdots + 89\!\cdots\!36 q^{98}+O(q^{100})$$ q + 48 * q^2 - 33552128 * q^4 + 741989850 * q^5 + 39080597192 * q^7 - 3221114880 * q^8 + 35615512800 * q^10 - 8419515299052 * q^11 - 81651045335314 * q^13 + 1875868665216 * q^14 + 1125667983917056 * q^16 + 2519900028948078 * q^17 - 6082056370308940 * q^19 - 24895338421900800 * q^20 - 404136734354496 * q^22 + 94995280296320424 * q^23 + 252525713626069375 * q^25 - 3919250176095072 * q^26 - 1311237199302424576 * q^28 + 271246959476737410 * q^29 + 4291666067521509152 * q^31 + 162114743433166848 * q^32 + 120955201389507744 * q^34 + 28997406448402501200 * q^35 + 20301484446109126982 * q^37 - 291938705774829120 * q^38 - 2390034546643968000 * q^40 + 183744249574071224598 * q^41 + 300901824185586335756 * q^43 + 282492655011750982656 * q^44 + 4559773454223380352 * q^46 + 924361048064704868688 * q^47 + 186224457219393384057 * q^49 + 12121234254051330000 * q^50 + 2739566324424258248192 * q^52 + 990292205554990470954 * q^53 - 6247194893816298622200 * q^55 - 125883093134437416960 * q^56 + 13019854054883395680 * q^58 - 13052569416454201837980 * q^59 + 9015451224701414617502 * q^61 + 205999971241032439296 * q^62 - 37763368313237157183488 * q^64 - 60584246880692834562900 * q^65 - 26689067808908579702428 * q^67 - 84548008318469618409984 * q^68 + 1391875509523320057600 * q^70 + 192390516186217637440248 * q^71 + 42404584838092453858826 * q^73 + 974471253413238095136 * q^74 + 204065933839820954424320 * q^76 - 329039685954132631461984 * q^77 - 271681055025772277197360 * q^79 + 835234218536418793881600 * q^80 + 8819723979555418780704 * q^82 + 931454457307013524361484 * q^83 + 1869740244494180053008300 * q^85 + 14443287560908144116288 * q^86 + 27120226012164047093760 * q^88 + 1763635518049807316502630 * q^89 - 3190971613055137006838288 * q^91 - 3187293803898020795062272 * q^92 + 44369330307105833697024 * q^94 - 4512824093897074844259000 * q^95 + 2829240869926872086187362 * q^97 + 8938773946530882434736 * 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
48.0000 0 −3.35521e7 7.41990e8 0 3.90806e10 −3.22111e9 0 3.56155e10
 $$n$$: 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 9.26.a.a 1
3.b odd 2 1 1.26.a.a 1
12.b even 2 1 16.26.a.b 1
15.d odd 2 1 25.26.a.a 1
15.e even 4 2 25.26.b.a 2
21.c even 2 1 49.26.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.26.a.a 1 3.b odd 2 1
9.26.a.a 1 1.a even 1 1 trivial
16.26.a.b 1 12.b even 2 1
25.26.a.a 1 15.d odd 2 1
25.26.b.a 2 15.e even 4 2
49.26.a.a 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 48$$ acting on $$S_{26}^{\mathrm{new}}(\Gamma_0(9))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 48$$
$3$ $$T$$
$5$ $$T - 741989850$$
$7$ $$T - 39080597192$$
$11$ $$T + 8419515299052$$
$13$ $$T + 81651045335314$$
$17$ $$T - 2519900028948078$$
$19$ $$T + 6082056370308940$$
$23$ $$T - 94\!\cdots\!24$$
$29$ $$T - 27\!\cdots\!10$$
$31$ $$T - 42\!\cdots\!52$$
$37$ $$T - 20\!\cdots\!82$$
$41$ $$T - 18\!\cdots\!98$$
$43$ $$T - 30\!\cdots\!56$$
$47$ $$T - 92\!\cdots\!88$$
$53$ $$T - 99\!\cdots\!54$$
$59$ $$T + 13\!\cdots\!80$$
$61$ $$T - 90\!\cdots\!02$$
$67$ $$T + 26\!\cdots\!28$$
$71$ $$T - 19\!\cdots\!48$$
$73$ $$T - 42\!\cdots\!26$$
$79$ $$T + 27\!\cdots\!60$$
$83$ $$T - 93\!\cdots\!84$$
$89$ $$T - 17\!\cdots\!30$$
$97$ $$T - 28\!\cdots\!62$$