# Properties

 Label 7623.2.a.be Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2541) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta + 2) q^{4} - q^{5} + q^{7} + ( - \beta - 4) q^{8} +O(q^{10})$$ q - b * q^2 + (b + 2) * q^4 - q^5 + q^7 + (-b - 4) * q^8 $$q - \beta q^{2} + (\beta + 2) q^{4} - q^{5} + q^{7} + ( - \beta - 4) q^{8} + \beta q^{10} + ( - \beta + 4) q^{13} - \beta q^{14} + 3 \beta q^{16} + (2 \beta - 1) q^{17} + 6 q^{19} + ( - \beta - 2) q^{20} + 4 q^{23} - 4 q^{25} + ( - 3 \beta + 4) q^{26} + (\beta + 2) q^{28} + (3 \beta - 2) q^{29} + ( - 4 \beta + 2) q^{31} + ( - \beta - 4) q^{32} + ( - \beta - 8) q^{34} - q^{35} + (3 \beta + 2) q^{37} - 6 \beta q^{38} + (\beta + 4) q^{40} + ( - \beta + 6) q^{41} + (3 \beta - 1) q^{43} - 4 \beta q^{46} + ( - \beta - 5) q^{47} + q^{49} + 4 \beta q^{50} + (\beta + 4) q^{52} + 5 \beta q^{53} + ( - \beta - 4) q^{56} + ( - \beta - 12) q^{58} + ( - \beta + 11) q^{59} + (6 \beta - 4) q^{61} + (2 \beta + 16) q^{62} + ( - \beta + 4) q^{64} + (\beta - 4) q^{65} + ( - 5 \beta + 1) q^{67} + (5 \beta + 6) q^{68} + \beta q^{70} + 2 \beta q^{71} + ( - 2 \beta + 4) q^{73} + ( - 5 \beta - 12) q^{74} + (6 \beta + 12) q^{76} - 2 \beta q^{79} - 3 \beta q^{80} + ( - 5 \beta + 4) q^{82} + ( - 5 \beta + 1) q^{83} + ( - 2 \beta + 1) q^{85} + ( - 2 \beta - 12) q^{86} + ( - 4 \beta + 5) q^{89} + ( - \beta + 4) q^{91} + (4 \beta + 8) q^{92} + (6 \beta + 4) q^{94} - 6 q^{95} + (3 \beta + 6) q^{97} - \beta q^{98} +O(q^{100})$$ q - b * q^2 + (b + 2) * q^4 - q^5 + q^7 + (-b - 4) * q^8 + b * q^10 + (-b + 4) * q^13 - b * q^14 + 3*b * q^16 + (2*b - 1) * q^17 + 6 * q^19 + (-b - 2) * q^20 + 4 * q^23 - 4 * q^25 + (-3*b + 4) * q^26 + (b + 2) * q^28 + (3*b - 2) * q^29 + (-4*b + 2) * q^31 + (-b - 4) * q^32 + (-b - 8) * q^34 - q^35 + (3*b + 2) * q^37 - 6*b * q^38 + (b + 4) * q^40 + (-b + 6) * q^41 + (3*b - 1) * q^43 - 4*b * q^46 + (-b - 5) * q^47 + q^49 + 4*b * q^50 + (b + 4) * q^52 + 5*b * q^53 + (-b - 4) * q^56 + (-b - 12) * q^58 + (-b + 11) * q^59 + (6*b - 4) * q^61 + (2*b + 16) * q^62 + (-b + 4) * q^64 + (b - 4) * q^65 + (-5*b + 1) * q^67 + (5*b + 6) * q^68 + b * q^70 + 2*b * q^71 + (-2*b + 4) * q^73 + (-5*b - 12) * q^74 + (6*b + 12) * q^76 - 2*b * q^79 - 3*b * q^80 + (-5*b + 4) * q^82 + (-5*b + 1) * q^83 + (-2*b + 1) * q^85 + (-2*b - 12) * q^86 + (-4*b + 5) * q^89 + (-b + 4) * q^91 + (4*b + 8) * q^92 + (6*b + 4) * q^94 - 6 * q^95 + (3*b + 6) * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 5 q^{4} - 2 q^{5} + 2 q^{7} - 9 q^{8}+O(q^{10})$$ 2 * q - q^2 + 5 * q^4 - 2 * q^5 + 2 * q^7 - 9 * q^8 $$2 q - q^{2} + 5 q^{4} - 2 q^{5} + 2 q^{7} - 9 q^{8} + q^{10} + 7 q^{13} - q^{14} + 3 q^{16} + 12 q^{19} - 5 q^{20} + 8 q^{23} - 8 q^{25} + 5 q^{26} + 5 q^{28} - q^{29} - 9 q^{32} - 17 q^{34} - 2 q^{35} + 7 q^{37} - 6 q^{38} + 9 q^{40} + 11 q^{41} + q^{43} - 4 q^{46} - 11 q^{47} + 2 q^{49} + 4 q^{50} + 9 q^{52} + 5 q^{53} - 9 q^{56} - 25 q^{58} + 21 q^{59} - 2 q^{61} + 34 q^{62} + 7 q^{64} - 7 q^{65} - 3 q^{67} + 17 q^{68} + q^{70} + 2 q^{71} + 6 q^{73} - 29 q^{74} + 30 q^{76} - 2 q^{79} - 3 q^{80} + 3 q^{82} - 3 q^{83} - 26 q^{86} + 6 q^{89} + 7 q^{91} + 20 q^{92} + 14 q^{94} - 12 q^{95} + 15 q^{97} - q^{98}+O(q^{100})$$ 2 * q - q^2 + 5 * q^4 - 2 * q^5 + 2 * q^7 - 9 * q^8 + q^10 + 7 * q^13 - q^14 + 3 * q^16 + 12 * q^19 - 5 * q^20 + 8 * q^23 - 8 * q^25 + 5 * q^26 + 5 * q^28 - q^29 - 9 * q^32 - 17 * q^34 - 2 * q^35 + 7 * q^37 - 6 * q^38 + 9 * q^40 + 11 * q^41 + q^43 - 4 * q^46 - 11 * q^47 + 2 * q^49 + 4 * q^50 + 9 * q^52 + 5 * q^53 - 9 * q^56 - 25 * q^58 + 21 * q^59 - 2 * q^61 + 34 * q^62 + 7 * q^64 - 7 * q^65 - 3 * q^67 + 17 * q^68 + q^70 + 2 * q^71 + 6 * q^73 - 29 * q^74 + 30 * q^76 - 2 * q^79 - 3 * q^80 + 3 * q^82 - 3 * q^83 - 26 * q^86 + 6 * q^89 + 7 * q^91 + 20 * q^92 + 14 * q^94 - 12 * q^95 + 15 * q^97 - 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
 2.56155 −1.56155
−2.56155 0 4.56155 −1.00000 0 1.00000 −6.56155 0 2.56155
1.2 1.56155 0 0.438447 −1.00000 0 1.00000 −2.43845 0 −1.56155
 $$n$$: 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

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 7623.2.a.be 2
3.b odd 2 1 2541.2.a.bc yes 2
11.b odd 2 1 7623.2.a.bt 2
33.d even 2 1 2541.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.u 2 33.d even 2 1
2541.2.a.bc yes 2 3.b odd 2 1
7623.2.a.be 2 1.a even 1 1 trivial
7623.2.a.bt 2 11.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{2} + T_{2} - 4$$ T2^2 + T2 - 4 $$T_{5} + 1$$ T5 + 1 $$T_{13}^{2} - 7T_{13} + 8$$ T13^2 - 7*T13 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 4$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 7T + 8$$
$17$ $$T^{2} - 17$$
$19$ $$(T - 6)^{2}$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} + T - 38$$
$31$ $$T^{2} - 68$$
$37$ $$T^{2} - 7T - 26$$
$41$ $$T^{2} - 11T + 26$$
$43$ $$T^{2} - T - 38$$
$47$ $$T^{2} + 11T + 26$$
$53$ $$T^{2} - 5T - 100$$
$59$ $$T^{2} - 21T + 106$$
$61$ $$T^{2} + 2T - 152$$
$67$ $$T^{2} + 3T - 104$$
$71$ $$T^{2} - 2T - 16$$
$73$ $$T^{2} - 6T - 8$$
$79$ $$T^{2} + 2T - 16$$
$83$ $$T^{2} + 3T - 104$$
$89$ $$T^{2} - 6T - 59$$
$97$ $$T^{2} - 15T + 18$$