# Properties

 Label 9522.2.a.bq Level $9522$ Weight $2$ Character orbit 9522.a Self dual yes Analytic conductor $76.034$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9522 = 2 \cdot 3^{2} \cdot 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9522.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$76.0335528047$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\Q(\zeta_{22})^+$$ Defining polynomial: $$x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1$$ x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 138) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta_{4} - \beta_{3} - 1) q^{5} + (\beta_{4} - \beta_{3} + \beta_{2} + 2) q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b4 - b3 - 1) * q^5 + (b4 - b3 + b2 + 2) * q^7 - q^8 $$q - q^{2} + q^{4} + (\beta_{4} - \beta_{3} - 1) q^{5} + (\beta_{4} - \beta_{3} + \beta_{2} + 2) q^{7} - q^{8} + ( - \beta_{4} + \beta_{3} + 1) q^{10} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{11} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{13} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{14} + q^{16} + (\beta_{4} + \beta_{3} + \beta_1 - 2) q^{17} + (\beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{19} + (\beta_{4} - \beta_{3} - 1) q^{20} + (\beta_{3} - 2 \beta_{2} + 2) q^{22} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{25} + (2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{26} + (\beta_{4} - \beta_{3} + \beta_{2} + 2) q^{28} + ( - 5 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \beta_1 - 1) q^{29} + (\beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 1) q^{31} - q^{32} + ( - \beta_{4} - \beta_{3} - \beta_1 + 2) q^{34} + (\beta_{3} - 3 \beta_{2} - 1) q^{35} + ( - \beta_{4} + \beta_1 + 2) q^{37} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{38} + ( - \beta_{4} + \beta_{3} + 1) q^{40} + ( - \beta_{4} + 3 \beta_{2} - 3 \beta_1 + 3) q^{41} + (3 \beta_{4} + \beta_{3} + 4 \beta_{2} + 2) q^{43} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{44} + (4 \beta_{4} + \beta_{3} - \beta_{2} - 4 \beta_1 + 6) q^{47} + (2 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{49} + (\beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{50} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{52} + ( - 2 \beta_{4} + 4 \beta_{3} - 8 \beta_{2} + 2 \beta_1 - 5) q^{53} + ( - 6 \beta_{4} + 8 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 1) q^{55} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{56} + (5 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{58} + ( - \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - \beta_1 + 4) q^{59} + (2 \beta_{4} - 8 \beta_{3} + 3 \beta_{2} - 7 \beta_1 + 5) q^{61} + ( - \beta_{4} + 4 \beta_{3} - 3 \beta_{2} + 5 \beta_1 - 1) q^{62} + q^{64} + ( - 2 \beta_{3} + 4 \beta_{2} - \beta_1 - 1) q^{65} + (6 \beta_{4} - \beta_{3} - \beta_{2} - 6 \beta_1 + 5) q^{67} + (\beta_{4} + \beta_{3} + \beta_1 - 2) q^{68} + ( - \beta_{3} + 3 \beta_{2} + 1) q^{70} + ( - 4 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 7 \beta_1) q^{71} + ( - \beta_{4} - 3 \beta_{3} - 7 \beta_{2} + 4 \beta_1 - 1) q^{73} + (\beta_{4} - \beta_1 - 2) q^{74} + (\beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{76} + ( - 5 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{77} + ( - 3 \beta_{4} + 5 \beta_{3} - 4 \beta_1) q^{79} + (\beta_{4} - \beta_{3} - 1) q^{80} + (\beta_{4} - 3 \beta_{2} + 3 \beta_1 - 3) q^{82} + (5 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} - 7 \beta_1) q^{83} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{85} + ( - 3 \beta_{4} - \beta_{3} - 4 \beta_{2} - 2) q^{86} + (\beta_{3} - 2 \beta_{2} + 2) q^{88} + ( - 4 \beta_{4} - 3 \beta_{3} - 1) q^{89} + ( - 3 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 5) q^{91} + ( - 4 \beta_{4} - \beta_{3} + \beta_{2} + 4 \beta_1 - 6) q^{94} + (4 \beta_{4} - 3 \beta_{3} + \beta_{2} - 6 \beta_1 + 5) q^{95} + ( - 11 \beta_{4} + 5 \beta_{3} - 7 \beta_{2} + 8 \beta_1 - 8) q^{97} + ( - 2 \beta_{4} - \beta_{2} - \beta_1 + 2) q^{98}+O(q^{100})$$ q - q^2 + q^4 + (b4 - b3 - 1) * q^5 + (b4 - b3 + b2 + 2) * q^7 - q^8 + (-b4 + b3 + 1) * q^10 + (-b3 + 2*b2 - 2) * q^11 + (-2*b4 + 2*b3 - b2 + b1 - 2) * q^13 + (-b4 + b3 - b2 - 2) * q^14 + q^16 + (b4 + b3 + b1 - 2) * q^17 + (b4 - b3 - b2 + 2*b1 + 2) * q^19 + (b4 - b3 - 1) * q^20 + (b3 - 2*b2 + 2) * q^22 + (-b4 + 2*b3 - b2 - b1 - 1) * q^25 + (2*b4 - 2*b3 + b2 - b1 + 2) * q^26 + (b4 - b3 + b2 + 2) * q^28 + (-5*b4 + 2*b3 - 4*b2 + b1 - 1) * q^29 + (b4 - 4*b3 + 3*b2 - 5*b1 + 1) * q^31 - q^32 + (-b4 - b3 - b1 + 2) * q^34 + (b3 - 3*b2 - 1) * q^35 + (-b4 + b1 + 2) * q^37 + (-b4 + b3 + b2 - 2*b1 - 2) * q^38 + (-b4 + b3 + 1) * q^40 + (-b4 + 3*b2 - 3*b1 + 3) * q^41 + (3*b4 + b3 + 4*b2 + 2) * q^43 + (-b3 + 2*b2 - 2) * q^44 + (4*b4 + b3 - b2 - 4*b1 + 6) * q^47 + (2*b4 + b2 + b1 - 2) * q^49 + (b4 - 2*b3 + b2 + b1 + 1) * q^50 + (-2*b4 + 2*b3 - b2 + b1 - 2) * q^52 + (-2*b4 + 4*b3 - 8*b2 + 2*b1 - 5) * q^53 + (-6*b4 + 8*b3 - 5*b2 + 2*b1 - 1) * q^55 + (-b4 + b3 - b2 - 2) * q^56 + (5*b4 - 2*b3 + 4*b2 - b1 + 1) * q^58 + (-b4 - 2*b3 + 4*b2 - b1 + 4) * q^59 + (2*b4 - 8*b3 + 3*b2 - 7*b1 + 5) * q^61 + (-b4 + 4*b3 - 3*b2 + 5*b1 - 1) * q^62 + q^64 + (-2*b3 + 4*b2 - b1 - 1) * q^65 + (6*b4 - b3 - b2 - 6*b1 + 5) * q^67 + (b4 + b3 + b1 - 2) * q^68 + (-b3 + 3*b2 + 1) * q^70 + (-4*b4 + 6*b3 - 2*b2 + 7*b1) * q^71 + (-b4 - 3*b3 - 7*b2 + 4*b1 - 1) * q^73 + (b4 - b1 - 2) * q^74 + (b4 - b3 - b2 + 2*b1 + 2) * q^76 + (-5*b4 + 6*b3 - 2*b2 + 2*b1 - 4) * q^77 + (-3*b4 + 5*b3 - 4*b1) * q^79 + (b4 - b3 - 1) * q^80 + (b4 - 3*b2 + 3*b1 - 3) * q^82 + (5*b4 - 7*b3 + 5*b2 - 7*b1) * q^83 + (-2*b4 - b3 + b2 - 3*b1 + 4) * q^85 + (-3*b4 - b3 - 4*b2 - 2) * q^86 + (b3 - 2*b2 + 2) * q^88 + (-4*b4 - 3*b3 - 1) * q^89 + (-3*b4 + b3 + b2 + b1 - 5) * q^91 + (-4*b4 - b3 + b2 + 4*b1 - 6) * q^94 + (4*b4 - 3*b3 + b2 - 6*b1 + 5) * q^95 + (-11*b4 + 5*b3 - 7*b2 + 8*b1 - 8) * q^97 + (-2*b4 - b2 - b1 + 2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8}+O(q^{10})$$ 5 * q - 5 * q^2 + 5 * q^4 - 7 * q^5 + 7 * q^7 - 5 * q^8 $$5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8} + 7 q^{10} - 13 q^{11} - 4 q^{13} - 7 q^{14} + 5 q^{16} - 9 q^{17} + 11 q^{19} - 7 q^{20} + 13 q^{22} - 2 q^{25} + 4 q^{26} + 7 q^{28} + 7 q^{29} - 8 q^{31} - 5 q^{32} + 9 q^{34} - q^{35} + 12 q^{37} - 11 q^{38} + 7 q^{40} + 10 q^{41} + 4 q^{43} - 13 q^{44} + 24 q^{47} - 12 q^{49} + 2 q^{50} - 4 q^{52} - 9 q^{53} + 16 q^{55} - 7 q^{56} - 7 q^{58} + 14 q^{59} + 5 q^{61} + 8 q^{62} + 5 q^{64} - 12 q^{65} + 13 q^{67} - 9 q^{68} + q^{70} + 19 q^{71} + 4 q^{73} - 12 q^{74} + 11 q^{76} - 5 q^{77} + 4 q^{79} - 7 q^{80} - 10 q^{82} - 24 q^{83} + 17 q^{85} - 4 q^{86} + 13 q^{88} - 4 q^{89} - 21 q^{91} - 24 q^{94} + 11 q^{95} - 9 q^{97} + 12 q^{98}+O(q^{100})$$ 5 * q - 5 * q^2 + 5 * q^4 - 7 * q^5 + 7 * q^7 - 5 * q^8 + 7 * q^10 - 13 * q^11 - 4 * q^13 - 7 * q^14 + 5 * q^16 - 9 * q^17 + 11 * q^19 - 7 * q^20 + 13 * q^22 - 2 * q^25 + 4 * q^26 + 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 + 9 * q^34 - q^35 + 12 * q^37 - 11 * q^38 + 7 * q^40 + 10 * q^41 + 4 * q^43 - 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 - 9 * q^53 + 16 * q^55 - 7 * q^56 - 7 * q^58 + 14 * q^59 + 5 * q^61 + 8 * q^62 + 5 * q^64 - 12 * q^65 + 13 * q^67 - 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 - 12 * q^74 + 11 * q^76 - 5 * q^77 + 4 * q^79 - 7 * q^80 - 10 * q^82 - 24 * q^83 + 17 * q^85 - 4 * q^86 + 13 * q^88 - 4 * q^89 - 21 * q^91 - 24 * q^94 + 11 * q^95 - 9 * q^97 + 12 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{22} + \zeta_{22}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 4\nu^{2} + 2$$ v^4 - 4*v^2 + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 4\beta_{2} + 6$$ b4 + 4*b2 + 6

## 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
 −0.830830 −1.68251 1.91899 1.30972 0.284630
−1.00000 0 1.00000 −3.20362 0 −1.51334 −1.00000 0 3.20362
1.2 −1.00000 0 1.00000 −2.59435 0 1.23648 −1.00000 0 2.59435
1.3 −1.00000 0 1.00000 −1.47889 0 3.20362 −1.00000 0 1.47889
1.4 −1.00000 0 1.00000 −1.23648 0 1.47889 −1.00000 0 1.23648
1.5 −1.00000 0 1.00000 1.51334 0 2.59435 −1.00000 0 −1.51334
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$23$$ $$-1$$

## 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 9522.2.a.bq 5
3.b odd 2 1 3174.2.a.bd 5
23.b odd 2 1 9522.2.a.bt 5
23.d odd 22 2 414.2.i.d 10
69.c even 2 1 3174.2.a.bc 5
69.g even 22 2 138.2.e.a 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.a 10 69.g even 22 2
414.2.i.d 10 23.d odd 22 2
3174.2.a.bc 5 69.c even 2 1
3174.2.a.bd 5 3.b odd 2 1
9522.2.a.bq 5 1.a even 1 1 trivial
9522.2.a.bt 5 23.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9522))$$:

 $$T_{5}^{5} + 7T_{5}^{4} + 13T_{5}^{3} - 6T_{5}^{2} - 35T_{5} - 23$$ T5^5 + 7*T5^4 + 13*T5^3 - 6*T5^2 - 35*T5 - 23 $$T_{7}^{5} - 7T_{7}^{4} + 13T_{7}^{3} + 6T_{7}^{2} - 35T_{7} + 23$$ T7^5 - 7*T7^4 + 13*T7^3 + 6*T7^2 - 35*T7 + 23 $$T_{11}^{5} + 13T_{11}^{4} + 50T_{11}^{3} + 53T_{11}^{2} + 15T_{11} - 1$$ T11^5 + 13*T11^4 + 50*T11^3 + 53*T11^2 + 15*T11 - 1 $$T_{29}^{5} - 7T_{29}^{4} - 75T_{29}^{3} + 413T_{29}^{2} + 1164T_{29} - 1693$$ T29^5 - 7*T29^4 - 75*T29^3 + 413*T29^2 + 1164*T29 - 1693

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{5}$$
$3$ $$T^{5}$$
$5$ $$T^{5} + 7 T^{4} + 13 T^{3} - 6 T^{2} + \cdots - 23$$
$7$ $$T^{5} - 7 T^{4} + 13 T^{3} + 6 T^{2} + \cdots + 23$$
$11$ $$T^{5} + 13 T^{4} + 50 T^{3} + 53 T^{2} + \cdots - 1$$
$13$ $$T^{5} + 4 T^{4} - 9 T^{3} - 27 T^{2} + \cdots + 1$$
$17$ $$T^{5} + 9 T^{4} + 17 T^{3} - 31 T^{2} + \cdots - 43$$
$19$ $$T^{5} - 11 T^{4} + 11 T^{3} + \cdots - 253$$
$23$ $$T^{5}$$
$29$ $$T^{5} - 7 T^{4} - 75 T^{3} + \cdots - 1693$$
$31$ $$T^{5} + 8 T^{4} - 69 T^{3} - 733 T^{2} + \cdots - 947$$
$37$ $$T^{5} - 12 T^{4} + 51 T^{3} - 96 T^{2} + \cdots - 23$$
$41$ $$T^{5} - 10 T^{4} - 37 T^{3} + 195 T^{2} + \cdots + 23$$
$43$ $$T^{5} - 4 T^{4} - 97 T^{3} - 149 T^{2} + \cdots + 439$$
$47$ $$T^{5} - 24 T^{4} + 83 T^{3} + \cdots + 10649$$
$53$ $$T^{5} + 9 T^{4} - 170 T^{3} + \cdots + 19009$$
$59$ $$T^{5} - 14 T^{4} - 3 T^{3} + \cdots - 5633$$
$61$ $$T^{5} - 5 T^{4} - 243 T^{3} + \cdots - 52933$$
$67$ $$T^{5} - 13 T^{4} - 181 T^{3} + \cdots - 42481$$
$71$ $$T^{5} - 19 T^{4} - 36 T^{3} + \cdots - 38609$$
$73$ $$T^{5} - 4 T^{4} - 317 T^{3} + \cdots + 15377$$
$79$ $$T^{5} - 4 T^{4} - 251 T^{3} + \cdots + 49169$$
$83$ $$T^{5} + 24 T^{4} + 50 T^{3} + \cdots + 25673$$
$89$ $$T^{5} + 4 T^{4} - 130 T^{3} + \cdots + 9637$$
$97$ $$T^{5} + 9 T^{4} - 335 T^{3} + \cdots + 149381$$