# Properties

 Label 7623.2.a.ck Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7623 = 3^{2} \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7623.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$60.8699614608$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{7})$$ Defining polynomial: $$x^{4} - 8x^{2} + 9$$ x^4 - 8*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{3} q^{5} + q^{7} - 2 \beta_1 q^{8}+O(q^{10})$$ q + b1 * q^2 - b3 * q^5 + q^7 - 2*b1 * q^8 $$q + \beta_1 q^{2} - \beta_{3} q^{5} + q^{7} - 2 \beta_1 q^{8} - \beta_{2} q^{10} + ( - \beta_{2} - 2) q^{13} + \beta_1 q^{14} - 4 q^{16} + (\beta_{3} + 2 \beta_1) q^{17} + (\beta_{2} + 2) q^{19} + (2 \beta_{3} - \beta_1) q^{23} + 2 q^{25} + ( - 2 \beta_{3} - 2 \beta_1) q^{26} + (2 \beta_{3} - \beta_1) q^{29} + (2 \beta_{2} - 2) q^{31} + (\beta_{2} + 4) q^{34} - \beta_{3} q^{35} + (2 \beta_{2} - 4) q^{37} + (2 \beta_{3} + 2 \beta_1) q^{38} + 2 \beta_{2} q^{40} - 4 \beta_1 q^{41} - q^{43} + (2 \beta_{2} - 2) q^{46} + (\beta_{3} + 2 \beta_1) q^{47} + q^{49} + 2 \beta_1 q^{50} + ( - 4 \beta_{3} - 2 \beta_1) q^{53} - 2 \beta_1 q^{56} + (2 \beta_{2} - 2) q^{58} + ( - \beta_{3} - 4 \beta_1) q^{59} + (\beta_{2} - 2) q^{61} + (4 \beta_{3} - 2 \beta_1) q^{62} + 8 q^{64} + (2 \beta_{3} + 7 \beta_1) q^{65} + ( - 2 \beta_{2} + 1) q^{67} - \beta_{2} q^{70} + \beta_1 q^{71} + ( - \beta_{2} - 12) q^{73} + (4 \beta_{3} - 4 \beta_1) q^{74} + ( - 2 \beta_{2} - 6) q^{79} + 4 \beta_{3} q^{80} - 8 q^{82} + (\beta_{3} + 6 \beta_1) q^{83} + ( - 2 \beta_{2} - 7) q^{85} - \beta_1 q^{86} + (\beta_{3} - 6 \beta_1) q^{89} + ( - \beta_{2} - 2) q^{91} + (\beta_{2} + 4) q^{94} + ( - 2 \beta_{3} - 7 \beta_1) q^{95} + ( - 3 \beta_{2} + 4) q^{97} + \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 - b3 * q^5 + q^7 - 2*b1 * q^8 - b2 * q^10 + (-b2 - 2) * q^13 + b1 * q^14 - 4 * q^16 + (b3 + 2*b1) * q^17 + (b2 + 2) * q^19 + (2*b3 - b1) * q^23 + 2 * q^25 + (-2*b3 - 2*b1) * q^26 + (2*b3 - b1) * q^29 + (2*b2 - 2) * q^31 + (b2 + 4) * q^34 - b3 * q^35 + (2*b2 - 4) * q^37 + (2*b3 + 2*b1) * q^38 + 2*b2 * q^40 - 4*b1 * q^41 - q^43 + (2*b2 - 2) * q^46 + (b3 + 2*b1) * q^47 + q^49 + 2*b1 * q^50 + (-4*b3 - 2*b1) * q^53 - 2*b1 * q^56 + (2*b2 - 2) * q^58 + (-b3 - 4*b1) * q^59 + (b2 - 2) * q^61 + (4*b3 - 2*b1) * q^62 + 8 * q^64 + (2*b3 + 7*b1) * q^65 + (-2*b2 + 1) * q^67 - b2 * q^70 + b1 * q^71 + (-b2 - 12) * q^73 + (4*b3 - 4*b1) * q^74 + (-2*b2 - 6) * q^79 + 4*b3 * q^80 - 8 * q^82 + (b3 + 6*b1) * q^83 + (-2*b2 - 7) * q^85 - b1 * q^86 + (b3 - 6*b1) * q^89 + (-b2 - 2) * q^91 + (b2 + 4) * q^94 + (-2*b3 - 7*b1) * q^95 + (-3*b2 + 4) * q^97 + b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{7}+O(q^{10})$$ 4 * q + 4 * q^7 $$4 q + 4 q^{7} - 8 q^{13} - 16 q^{16} + 8 q^{19} + 8 q^{25} - 8 q^{31} + 16 q^{34} - 16 q^{37} - 4 q^{43} - 8 q^{46} + 4 q^{49} - 8 q^{58} - 8 q^{61} + 32 q^{64} + 4 q^{67} - 48 q^{73} - 24 q^{79} - 32 q^{82} - 28 q^{85} - 8 q^{91} + 16 q^{94} + 16 q^{97}+O(q^{100})$$ 4 * q + 4 * q^7 - 8 * q^13 - 16 * q^16 + 8 * q^19 + 8 * q^25 - 8 * q^31 + 16 * q^34 - 16 * q^37 - 4 * q^43 - 8 * q^46 + 4 * q^49 - 8 * q^58 - 8 * q^61 + 32 * q^64 + 4 * q^67 - 48 * q^73 - 24 * q^79 - 32 * q^82 - 28 * q^85 - 8 * q^91 + 16 * q^94 + 16 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 8x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 5\nu ) / 3$$ (v^3 - 5*v) / 3 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 11\nu ) / 3$$ (-v^3 + 11*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4$$ b3 + 4 $$\nu^{3}$$ $$=$$ $$( 5\beta_{2} + 11\beta_1 ) / 2$$ (5*b2 + 11*b1) / 2

## 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
 −2.57794 1.16372 2.57794 −1.16372
−1.41421 0 0 −2.64575 0 1.00000 2.82843 0 3.74166
1.2 −1.41421 0 0 2.64575 0 1.00000 2.82843 0 −3.74166
1.3 1.41421 0 0 −2.64575 0 1.00000 −2.82843 0 −3.74166
1.4 1.41421 0 0 2.64575 0 1.00000 −2.82843 0 3.74166
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.ck yes 4
3.b odd 2 1 inner 7623.2.a.ck yes 4
11.b odd 2 1 7623.2.a.cj 4
33.d even 2 1 7623.2.a.cj 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7623.2.a.cj 4 11.b odd 2 1
7623.2.a.cj 4 33.d even 2 1
7623.2.a.ck yes 4 1.a even 1 1 trivial
7623.2.a.ck yes 4 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{2} - 2$$ T2^2 - 2 $$T_{5}^{2} - 7$$ T5^2 - 7 $$T_{13}^{2} + 4T_{13} - 10$$ T13^2 + 4*T13 - 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 7)^{2}$$
$7$ $$(T - 1)^{4}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 4 T - 10)^{2}$$
$17$ $$T^{4} - 30T^{2} + 1$$
$19$ $$(T^{2} - 4 T - 10)^{2}$$
$23$ $$T^{4} - 60T^{2} + 676$$
$29$ $$T^{4} - 60T^{2} + 676$$
$31$ $$(T^{2} + 4 T - 52)^{2}$$
$37$ $$(T^{2} + 8 T - 40)^{2}$$
$41$ $$(T^{2} - 32)^{2}$$
$43$ $$(T + 1)^{4}$$
$47$ $$T^{4} - 30T^{2} + 1$$
$53$ $$T^{4} - 240 T^{2} + 10816$$
$59$ $$T^{4} - 78T^{2} + 625$$
$61$ $$(T^{2} + 4 T - 10)^{2}$$
$67$ $$(T^{2} - 2 T - 55)^{2}$$
$71$ $$(T^{2} - 2)^{2}$$
$73$ $$(T^{2} + 24 T + 130)^{2}$$
$79$ $$(T^{2} + 12 T - 20)^{2}$$
$83$ $$T^{4} - 158T^{2} + 4225$$
$89$ $$T^{4} - 158T^{2} + 4225$$
$97$ $$(T^{2} - 8 T - 110)^{2}$$