# Properties

 Label 9522.2.a.bs Level $9522$ Weight $2$ Character orbit 9522.a Self dual yes Analytic conductor $76.034$ Analytic rank $0$ 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: $$0$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b4 + b2 - 2*b1 + 1) * q^5 + (-b4 - b2 - b1 + 2) * q^7 - q^8 $$q - q^{2} + q^{4} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{7} - q^{8} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 - 1) q^{10} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{11} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{13} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{14} + q^{16} + ( - 3 \beta_{3} - \beta_1 + 1) q^{17} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{19} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{20} + (3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{22} + ( - 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{25} + (\beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{26} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{28} + ( - 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{29} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} - 4) q^{31} - q^{32} + (3 \beta_{3} + \beta_1 - 1) q^{34} + (3 \beta_{4} + 4 \beta_{2} - 7 \beta_1 + 5) q^{35} + (\beta_{4} - 3 \beta_{3} - \beta_1 + 3) q^{37} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{38} + ( - \beta_{4} - \beta_{2} + 2 \beta_1 - 1) q^{40} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 4) q^{41} + ( - 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 4) q^{43} + ( - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{44} + (\beta_{4} + 2 \beta_{3} - 4 \beta_{2} - \beta_1) q^{47} + ( - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{49} + (3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{50} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{52} + ( - 4 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 6 \beta_1 - 3) q^{53} + (4 \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 7) q^{55} + (\beta_{4} + \beta_{2} + \beta_1 - 2) q^{56} + (3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{58} + (3 \beta_{4} - 7 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 3) q^{59} + (7 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{61} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 4) q^{62} + q^{64} + (3 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + \beta_1 - 6) q^{65} + ( - \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 5 \beta_1 + 3) q^{67} + ( - 3 \beta_{3} - \beta_1 + 1) q^{68} + ( - 3 \beta_{4} - 4 \beta_{2} + 7 \beta_1 - 5) q^{70} + ( - 5 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{71} + ( - \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 1) q^{73} + ( - \beta_{4} + 3 \beta_{3} + \beta_1 - 3) q^{74} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{76} + ( - 6 \beta_{4} + 8 \beta_{3} - 5 \beta_{2} + 10 \beta_1 - 8) q^{77} + ( - \beta_{4} + 3 \beta_{2} + 2 \beta_1 + 8) q^{79} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{80} + (\beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 4) q^{82} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{83} + (6 \beta_{4} - \beta_{3} + 5 \beta_{2} - 6 \beta_1 + 1) q^{85} + (3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 4) q^{86} + (3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{88} + (\beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 1) q^{89} + ( - 6 \beta_{4} + 7 \beta_{3} - 12 \beta_{2} + 8 \beta_1) q^{91} + ( - \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \beta_1) q^{94} + (\beta_{4} + 2 \beta_{3} - 3 \beta_1 + 1) q^{95} + ( - 3 \beta_{4} - \beta_{2} - 7 \beta_1 + 4) q^{97} + (3 \beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{98}+O(q^{100})$$ q - q^2 + q^4 + (b4 + b2 - 2*b1 + 1) * q^5 + (-b4 - b2 - b1 + 2) * q^7 - q^8 + (-b4 - b2 + 2*b1 - 1) * q^10 + (-3*b4 + 2*b3 - 2*b2 + 2*b1 - 4) * q^11 + (-b4 + b3 - 3*b2 + 2*b1 + 1) * q^13 + (b4 + b2 + b1 - 2) * q^14 + q^16 + (-3*b3 - b1 + 1) * q^17 + (-b4 + 2*b3 - b2 + b1 + 2) * q^19 + (b4 + b2 - 2*b1 + 1) * q^20 + (3*b4 - 2*b3 + 2*b2 - 2*b1 + 4) * q^22 + (-3*b4 - 3*b3 + 2*b2 - 2*b1 + 2) * q^25 + (b4 - b3 + 3*b2 - 2*b1 - 1) * q^26 + (-b4 - b2 - b1 + 2) * q^28 + (-3*b4 + 3*b3 - 2*b2 + b1 - 2) * q^29 + (-b4 - b3 - 2*b2 - 4) * q^31 - q^32 + (3*b3 + b1 - 1) * q^34 + (3*b4 + 4*b2 - 7*b1 + 5) * q^35 + (b4 - 3*b3 - b1 + 3) * q^37 + (b4 - 2*b3 + b2 - b1 - 2) * q^38 + (-b4 - b2 + 2*b1 - 1) * q^40 + (-b4 - 3*b3 + 2*b2 + 4) * q^41 + (-3*b4 - 2*b3 + 3*b2 + 4) * q^43 + (-3*b4 + 2*b3 - 2*b2 + 2*b1 - 4) * q^44 + (b4 + 2*b3 - 4*b2 - b1) * q^47 + (-3*b4 + 3*b3 - b2 - 2*b1 + 3) * q^49 + (3*b4 + 3*b3 - 2*b2 + 2*b1 - 2) * q^50 + (-b4 + b3 - 3*b2 + 2*b1 + 1) * q^52 + (-4*b4 + 2*b3 - 4*b2 + 6*b1 - 3) * q^53 + (4*b3 - 4*b2 + 5*b1 - 7) * q^55 + (b4 + b2 + b1 - 2) * q^56 + (3*b4 - 3*b3 + 2*b2 - b1 + 2) * q^58 + (3*b4 - 7*b3 + 2*b2 - 5*b1 + 3) * q^59 + (7*b4 - b3 + b2 - 2*b1 + 2) * q^61 + (b4 + b3 + 2*b2 + 4) * q^62 + q^64 + (3*b4 + 5*b3 - 3*b2 + b1 - 6) * q^65 + (-b4 + 4*b3 - 4*b2 + 5*b1 + 3) * q^67 + (-3*b3 - b1 + 1) * q^68 + (-3*b4 - 4*b2 + 7*b1 - 5) * q^70 + (-5*b4 - b3 - 3*b2 - b1 + 1) * q^71 + (-b4 + 4*b3 + 3*b2 - 5*b1 - 1) * q^73 + (-b4 + 3*b3 + b1 - 3) * q^74 + (-b4 + 2*b3 - b2 + b1 + 2) * q^76 + (-6*b4 + 8*b3 - 5*b2 + 10*b1 - 8) * q^77 + (-b4 + 3*b2 + 2*b1 + 8) * q^79 + (b4 + b2 - 2*b1 + 1) * q^80 + (b4 + 3*b3 - 2*b2 - 4) * q^82 + (-b3 + 2*b2 - 1) * q^83 + (6*b4 - b3 + 5*b2 - 6*b1 + 1) * q^85 + (3*b4 + 2*b3 - 3*b2 - 4) * q^86 + (3*b4 - 2*b3 + 2*b2 - 2*b1 + 4) * q^88 + (b4 - 4*b3 - 2*b2 + 6*b1 - 1) * q^89 + (-6*b4 + 7*b3 - 12*b2 + 8*b1) * q^91 + (-b4 - 2*b3 + 4*b2 + b1) * q^94 + (b4 + 2*b3 - 3*b1 + 1) * q^95 + (-3*b4 - b2 - 7*b1 + 4) * q^97 + (3*b4 - 3*b3 + b2 + 2*b1 - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{2} + 5 q^{4} + q^{5} + 11 q^{7} - 5 q^{8}+O(q^{10})$$ 5 * q - 5 * q^2 + 5 * q^4 + q^5 + 11 * q^7 - 5 * q^8 $$5 q - 5 q^{2} + 5 q^{4} + q^{5} + 11 q^{7} - 5 q^{8} - q^{10} - 11 q^{11} + 12 q^{13} - 11 q^{14} + 5 q^{16} + q^{17} + 15 q^{19} + q^{20} + 11 q^{22} + 6 q^{25} - 12 q^{26} + 11 q^{28} - q^{29} - 18 q^{31} - 5 q^{32} - q^{34} + 11 q^{35} + 10 q^{37} - 15 q^{38} - q^{40} + 16 q^{41} + 18 q^{43} - 11 q^{44} + 4 q^{47} + 20 q^{49} - 6 q^{50} + 12 q^{52} + q^{53} - 22 q^{55} - 11 q^{56} + q^{58} - 2 q^{59} - q^{61} + 18 q^{62} + 5 q^{64} - 24 q^{65} + 29 q^{67} + q^{68} - 11 q^{70} + 11 q^{71} - 8 q^{73} - 10 q^{74} + 15 q^{76} - 11 q^{77} + 40 q^{79} + q^{80} - 16 q^{82} - 8 q^{83} - 13 q^{85} - 18 q^{86} + 11 q^{88} - 2 q^{89} + 33 q^{91} - 4 q^{94} + 3 q^{95} + 17 q^{97} - 20 q^{98}+O(q^{100})$$ 5 * q - 5 * q^2 + 5 * q^4 + q^5 + 11 * q^7 - 5 * q^8 - q^10 - 11 * q^11 + 12 * q^13 - 11 * q^14 + 5 * q^16 + q^17 + 15 * q^19 + q^20 + 11 * q^22 + 6 * q^25 - 12 * q^26 + 11 * q^28 - q^29 - 18 * q^31 - 5 * q^32 - q^34 + 11 * q^35 + 10 * q^37 - 15 * q^38 - q^40 + 16 * q^41 + 18 * q^43 - 11 * q^44 + 4 * q^47 + 20 * q^49 - 6 * q^50 + 12 * q^52 + q^53 - 22 * q^55 - 11 * q^56 + q^58 - 2 * q^59 - q^61 + 18 * q^62 + 5 * q^64 - 24 * q^65 + 29 * q^67 + q^68 - 11 * q^70 + 11 * q^71 - 8 * q^73 - 10 * q^74 + 15 * q^76 - 11 * q^77 + 40 * q^79 + q^80 - 16 * q^82 - 8 * q^83 - 13 * q^85 - 18 * q^86 + 11 * q^88 - 2 * q^89 + 33 * q^91 - 4 * q^94 + 3 * q^95 + 17 * q^97 - 20 * 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
 1.30972 1.91899 0.284630 −0.830830 −1.68251
−1.00000 0 1.00000 −3.82306 0 2.89389 −1.00000 0 3.82306
1.2 −1.00000 0 1.00000 −0.324635 0 −2.43232 −1.00000 0 0.324635
1.3 −1.00000 0 1.00000 0.194262 0 1.95185 −1.00000 0 −0.194262
1.4 −1.00000 0 1.00000 1.06731 0 4.42518 −1.00000 0 −1.06731
1.5 −1.00000 0 1.00000 3.88612 0 4.16140 −1.00000 0 −3.88612
 $$n$$: 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.bs 5
3.b odd 2 1 3174.2.a.ba 5
23.b odd 2 1 9522.2.a.br 5
23.c even 11 2 414.2.i.e 10
69.c even 2 1 3174.2.a.bb 5
69.h odd 22 2 138.2.e.b 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.e.b 10 69.h odd 22 2
414.2.i.e 10 23.c even 11 2
3174.2.a.ba 5 3.b odd 2 1
3174.2.a.bb 5 69.c even 2 1
9522.2.a.br 5 23.b odd 2 1
9522.2.a.bs 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9522))$$:

 $$T_{5}^{5} - T_{5}^{4} - 15T_{5}^{3} + 14T_{5}^{2} + 3T_{5} - 1$$ T5^5 - T5^4 - 15*T5^3 + 14*T5^2 + 3*T5 - 1 $$T_{7}^{5} - 11T_{7}^{4} + 33T_{7}^{3} + 22T_{7}^{2} - 231T_{7} + 253$$ T7^5 - 11*T7^4 + 33*T7^3 + 22*T7^2 - 231*T7 + 253 $$T_{11}^{5} + 11T_{11}^{4} + 22T_{11}^{3} - 77T_{11}^{2} - 143T_{11} + 253$$ T11^5 + 11*T11^4 + 22*T11^3 - 77*T11^2 - 143*T11 + 253 $$T_{29}^{5} + T_{29}^{4} - 37T_{29}^{3} - 69T_{29}^{2} + 14T_{29} + 23$$ T29^5 + T29^4 - 37*T29^3 - 69*T29^2 + 14*T29 + 23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{5}$$
$3$ $$T^{5}$$
$5$ $$T^{5} - T^{4} - 15 T^{3} + 14 T^{2} + \cdots - 1$$
$7$ $$T^{5} - 11 T^{4} + 33 T^{3} + \cdots + 253$$
$11$ $$T^{5} + 11 T^{4} + 22 T^{3} + \cdots + 253$$
$13$ $$T^{5} - 12 T^{4} + 29 T^{3} + \cdots + 109$$
$17$ $$T^{5} - T^{4} - 37 T^{3} + 47 T^{2} + \cdots - 529$$
$19$ $$T^{5} - 15 T^{4} + 79 T^{3} - 182 T^{2} + \cdots - 67$$
$23$ $$T^{5}$$
$29$ $$T^{5} + T^{4} - 37 T^{3} - 69 T^{2} + \cdots + 23$$
$31$ $$T^{5} + 18 T^{4} + 101 T^{3} + \cdots - 23$$
$37$ $$T^{5} - 10 T^{4} + 7 T^{3} + 118 T^{2} + \cdots - 373$$
$41$ $$T^{5} - 16 T^{4} + 43 T^{3} + \cdots + 659$$
$43$ $$T^{5} - 18 T^{4} + 13 T^{3} + \cdots - 7127$$
$47$ $$T^{5} - 4 T^{4} - 97 T^{3} + \cdots + 4817$$
$53$ $$T^{5} - T^{4} - 114 T^{3} + 146 T^{2} + \cdots - 529$$
$59$ $$T^{5} + 2 T^{4} - 159 T^{3} + \cdots - 11309$$
$61$ $$T^{5} + T^{4} - 169 T^{3} - 102 T^{2} + \cdots + 3389$$
$67$ $$T^{5} - 29 T^{4} + 233 T^{3} + \cdots + 32429$$
$71$ $$T^{5} - 11 T^{4} - 110 T^{3} + \cdots - 5093$$
$73$ $$T^{5} + 8 T^{4} - 245 T^{3} + \cdots + 17621$$
$79$ $$T^{5} - 40 T^{4} + 563 T^{3} + \cdots - 4817$$
$83$ $$T^{5} + 8 T^{4} + 8 T^{3} - 29 T^{2} + \cdots - 1$$
$89$ $$T^{5} + 2 T^{4} - 302 T^{3} + \cdots - 55177$$
$97$ $$T^{5} - 17 T^{4} - 199 T^{3} + \cdots - 3013$$