# Properties

 Label 9633.2.a.h Level $9633$ Weight $2$ Character orbit 9633.a Self dual yes Analytic conductor $76.920$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9633 = 3 \cdot 13^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9633.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$76.9198922671$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 - q^4 + 2 * q^5 - q^6 + 3 * q^8 + q^9 $$q - q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} + 3 q^{8} + q^{9} - 2 q^{10} - q^{12} + 2 q^{15} - q^{16} - 6 q^{17} - q^{18} + q^{19} - 2 q^{20} + 4 q^{23} + 3 q^{24} - q^{25} + q^{27} + 2 q^{29} - 2 q^{30} - 8 q^{31} - 5 q^{32} + 6 q^{34} - q^{36} + 10 q^{37} - q^{38} + 6 q^{40} + 2 q^{41} - 4 q^{43} + 2 q^{45} - 4 q^{46} - 12 q^{47} - q^{48} - 7 q^{49} + q^{50} - 6 q^{51} - 6 q^{53} - q^{54} + q^{57} - 2 q^{58} + 12 q^{59} - 2 q^{60} - 2 q^{61} + 8 q^{62} + 7 q^{64} + 4 q^{67} + 6 q^{68} + 4 q^{69} + 3 q^{72} - 10 q^{73} - 10 q^{74} - q^{75} - q^{76} - 2 q^{80} + q^{81} - 2 q^{82} - 16 q^{83} - 12 q^{85} + 4 q^{86} + 2 q^{87} + 2 q^{89} - 2 q^{90} - 4 q^{92} - 8 q^{93} + 12 q^{94} + 2 q^{95} - 5 q^{96} - 10 q^{97} + 7 q^{98}+O(q^{100})$$ q - q^2 + q^3 - q^4 + 2 * q^5 - q^6 + 3 * q^8 + q^9 - 2 * q^10 - q^12 + 2 * q^15 - q^16 - 6 * q^17 - q^18 + q^19 - 2 * q^20 + 4 * q^23 + 3 * q^24 - q^25 + q^27 + 2 * q^29 - 2 * q^30 - 8 * q^31 - 5 * q^32 + 6 * q^34 - q^36 + 10 * q^37 - q^38 + 6 * q^40 + 2 * q^41 - 4 * q^43 + 2 * q^45 - 4 * q^46 - 12 * q^47 - q^48 - 7 * q^49 + q^50 - 6 * q^51 - 6 * q^53 - q^54 + q^57 - 2 * q^58 + 12 * q^59 - 2 * q^60 - 2 * q^61 + 8 * q^62 + 7 * q^64 + 4 * q^67 + 6 * q^68 + 4 * q^69 + 3 * q^72 - 10 * q^73 - 10 * q^74 - q^75 - q^76 - 2 * q^80 + q^81 - 2 * q^82 - 16 * q^83 - 12 * q^85 + 4 * q^86 + 2 * q^87 + 2 * q^89 - 2 * q^90 - 4 * q^92 - 8 * q^93 + 12 * q^94 + 2 * q^95 - 5 * q^96 - 10 * q^97 + 7 * 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
−1.00000 1.00000 −1.00000 2.00000 −1.00000 0 3.00000 1.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$1$$
$$19$$ $$-1$$

## 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 9633.2.a.h 1
13.b even 2 1 57.2.a.c 1
39.d odd 2 1 171.2.a.a 1
52.b odd 2 1 912.2.a.b 1
65.d even 2 1 1425.2.a.a 1
65.h odd 4 2 1425.2.c.g 2
91.b odd 2 1 2793.2.a.i 1
104.e even 2 1 3648.2.a.o 1
104.h odd 2 1 3648.2.a.bf 1
143.d odd 2 1 6897.2.a.a 1
156.h even 2 1 2736.2.a.s 1
195.e odd 2 1 4275.2.a.m 1
247.d odd 2 1 1083.2.a.a 1
273.g even 2 1 8379.2.a.e 1
741.d even 2 1 3249.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.c 1 13.b even 2 1
171.2.a.a 1 39.d odd 2 1
912.2.a.b 1 52.b odd 2 1
1083.2.a.a 1 247.d odd 2 1
1425.2.a.a 1 65.d even 2 1
1425.2.c.g 2 65.h odd 4 2
2736.2.a.s 1 156.h even 2 1
2793.2.a.i 1 91.b odd 2 1
3249.2.a.g 1 741.d even 2 1
3648.2.a.o 1 104.e even 2 1
3648.2.a.bf 1 104.h odd 2 1
4275.2.a.m 1 195.e odd 2 1
6897.2.a.a 1 143.d odd 2 1
8379.2.a.e 1 273.g even 2 1
9633.2.a.h 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9633))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 2$$ T5 - 2 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T - 2$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 6$$
$19$ $$T - 1$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T + 8$$
$37$ $$T - 10$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T + 12$$
$53$ $$T + 6$$
$59$ $$T - 12$$
$61$ $$T + 2$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T + 10$$
$79$ $$T$$
$83$ $$T + 16$$
$89$ $$T - 2$$
$97$ $$T + 10$$