# Properties

 Label 7623.2.a.bs 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: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 847) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta + 1) q^{4} + ( - 2 \beta + 1) q^{5} + q^{7} + 3 q^{8}+O(q^{10})$$ q + b * q^2 + (b + 1) * q^4 + (-2*b + 1) * q^5 + q^7 + 3 * q^8 $$q + \beta q^{2} + (\beta + 1) q^{4} + ( - 2 \beta + 1) q^{5} + q^{7} + 3 q^{8} + ( - \beta - 6) q^{10} + ( - 2 \beta - 2) q^{13} + \beta q^{14} + (\beta - 2) q^{16} + ( - \beta + 5) q^{17} + 3 q^{19} + ( - 3 \beta - 5) q^{20} + ( - \beta + 5) q^{23} + 8 q^{25} + ( - 4 \beta - 6) q^{26} + (\beta + 1) q^{28} + ( - \beta + 7) q^{29} + q^{31} + ( - \beta - 3) q^{32} + (4 \beta - 3) q^{34} + ( - 2 \beta + 1) q^{35} + ( - 4 \beta + 4) q^{37} + 3 \beta q^{38} + ( - 6 \beta + 3) q^{40} + 7 q^{41} + (\beta - 4) q^{43} + (4 \beta - 3) q^{46} + (3 \beta - 5) q^{47} + q^{49} + 8 \beta q^{50} + ( - 6 \beta - 8) q^{52} + (3 \beta + 6) q^{53} + 3 q^{56} + (6 \beta - 3) q^{58} + (\beta - 9) q^{59} + (\beta + 2) q^{61} + \beta q^{62} + ( - 6 \beta + 1) q^{64} + (6 \beta + 10) q^{65} + (5 \beta - 3) q^{67} + (3 \beta + 2) q^{68} + ( - \beta - 6) q^{70} + (\beta + 2) q^{71} + 5 q^{73} - 12 q^{74} + (3 \beta + 3) q^{76} + (\beta + 6) q^{79} + (3 \beta - 8) q^{80} + 7 \beta q^{82} + 3 q^{83} + ( - 9 \beta + 11) q^{85} + ( - 3 \beta + 3) q^{86} + (9 \beta - 6) q^{89} + ( - 2 \beta - 2) q^{91} + (3 \beta + 2) q^{92} + ( - 2 \beta + 9) q^{94} + ( - 6 \beta + 3) q^{95} + ( - 2 \beta + 1) q^{97} + \beta q^{98}+O(q^{100})$$ q + b * q^2 + (b + 1) * q^4 + (-2*b + 1) * q^5 + q^7 + 3 * q^8 + (-b - 6) * q^10 + (-2*b - 2) * q^13 + b * q^14 + (b - 2) * q^16 + (-b + 5) * q^17 + 3 * q^19 + (-3*b - 5) * q^20 + (-b + 5) * q^23 + 8 * q^25 + (-4*b - 6) * q^26 + (b + 1) * q^28 + (-b + 7) * q^29 + q^31 + (-b - 3) * q^32 + (4*b - 3) * q^34 + (-2*b + 1) * q^35 + (-4*b + 4) * q^37 + 3*b * q^38 + (-6*b + 3) * q^40 + 7 * q^41 + (b - 4) * q^43 + (4*b - 3) * q^46 + (3*b - 5) * q^47 + q^49 + 8*b * q^50 + (-6*b - 8) * q^52 + (3*b + 6) * q^53 + 3 * q^56 + (6*b - 3) * q^58 + (b - 9) * q^59 + (b + 2) * q^61 + b * q^62 + (-6*b + 1) * q^64 + (6*b + 10) * q^65 + (5*b - 3) * q^67 + (3*b + 2) * q^68 + (-b - 6) * q^70 + (b + 2) * q^71 + 5 * q^73 - 12 * q^74 + (3*b + 3) * q^76 + (b + 6) * q^79 + (3*b - 8) * q^80 + 7*b * q^82 + 3 * q^83 + (-9*b + 11) * q^85 + (-3*b + 3) * q^86 + (9*b - 6) * q^89 + (-2*b - 2) * q^91 + (3*b + 2) * q^92 + (-2*b + 9) * q^94 + (-6*b + 3) * q^95 + (-2*b + 1) * q^97 + b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 3 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + 3 * q^4 + 2 * q^7 + 6 * q^8 $$2 q + q^{2} + 3 q^{4} + 2 q^{7} + 6 q^{8} - 13 q^{10} - 6 q^{13} + q^{14} - 3 q^{16} + 9 q^{17} + 6 q^{19} - 13 q^{20} + 9 q^{23} + 16 q^{25} - 16 q^{26} + 3 q^{28} + 13 q^{29} + 2 q^{31} - 7 q^{32} - 2 q^{34} + 4 q^{37} + 3 q^{38} + 14 q^{41} - 7 q^{43} - 2 q^{46} - 7 q^{47} + 2 q^{49} + 8 q^{50} - 22 q^{52} + 15 q^{53} + 6 q^{56} - 17 q^{59} + 5 q^{61} + q^{62} - 4 q^{64} + 26 q^{65} - q^{67} + 7 q^{68} - 13 q^{70} + 5 q^{71} + 10 q^{73} - 24 q^{74} + 9 q^{76} + 13 q^{79} - 13 q^{80} + 7 q^{82} + 6 q^{83} + 13 q^{85} + 3 q^{86} - 3 q^{89} - 6 q^{91} + 7 q^{92} + 16 q^{94} + q^{98}+O(q^{100})$$ 2 * q + q^2 + 3 * q^4 + 2 * q^7 + 6 * q^8 - 13 * q^10 - 6 * q^13 + q^14 - 3 * q^16 + 9 * q^17 + 6 * q^19 - 13 * q^20 + 9 * q^23 + 16 * q^25 - 16 * q^26 + 3 * q^28 + 13 * q^29 + 2 * q^31 - 7 * q^32 - 2 * q^34 + 4 * q^37 + 3 * q^38 + 14 * q^41 - 7 * q^43 - 2 * q^46 - 7 * q^47 + 2 * q^49 + 8 * q^50 - 22 * q^52 + 15 * q^53 + 6 * q^56 - 17 * q^59 + 5 * q^61 + q^62 - 4 * q^64 + 26 * q^65 - q^67 + 7 * q^68 - 13 * q^70 + 5 * q^71 + 10 * q^73 - 24 * q^74 + 9 * q^76 + 13 * q^79 - 13 * q^80 + 7 * q^82 + 6 * q^83 + 13 * q^85 + 3 * q^86 - 3 * q^89 - 6 * q^91 + 7 * q^92 + 16 * q^94 + 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
 −1.30278 2.30278
−1.30278 0 −0.302776 3.60555 0 1.00000 3.00000 0 −4.69722
1.2 2.30278 0 3.30278 −3.60555 0 1.00000 3.00000 0 −8.30278
 $$n$$: 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.bs 2
3.b odd 2 1 847.2.a.e 2
11.b odd 2 1 7623.2.a.bc 2
21.c even 2 1 5929.2.a.k 2
33.d even 2 1 847.2.a.g yes 2
33.f even 10 4 847.2.f.o 8
33.h odd 10 4 847.2.f.r 8
231.h odd 2 1 5929.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.e 2 3.b odd 2 1
847.2.a.g yes 2 33.d even 2 1
847.2.f.o 8 33.f even 10 4
847.2.f.r 8 33.h odd 10 4
5929.2.a.k 2 21.c even 2 1
5929.2.a.p 2 231.h odd 2 1
7623.2.a.bc 2 11.b odd 2 1
7623.2.a.bs 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{2} - T_{2} - 3$$ T2^2 - T2 - 3 $$T_{5}^{2} - 13$$ T5^2 - 13 $$T_{13}^{2} + 6T_{13} - 4$$ T13^2 + 6*T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 3$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 13$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T - 4$$
$17$ $$T^{2} - 9T + 17$$
$19$ $$(T - 3)^{2}$$
$23$ $$T^{2} - 9T + 17$$
$29$ $$T^{2} - 13T + 39$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} - 4T - 48$$
$41$ $$(T - 7)^{2}$$
$43$ $$T^{2} + 7T + 9$$
$47$ $$T^{2} + 7T - 17$$
$53$ $$T^{2} - 15T + 27$$
$59$ $$T^{2} + 17T + 69$$
$61$ $$T^{2} - 5T + 3$$
$67$ $$T^{2} + T - 81$$
$71$ $$T^{2} - 5T + 3$$
$73$ $$(T - 5)^{2}$$
$79$ $$T^{2} - 13T + 39$$
$83$ $$(T - 3)^{2}$$
$89$ $$T^{2} + 3T - 261$$
$97$ $$T^{2} - 13$$