# Properties

 Label 7623.2.a.br Level 7623 Weight 2 Character orbit 7623.a Self dual yes Analytic conductor 60.870 Analytic rank 1 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 693) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 1 + \beta ) q^{4} - q^{5} + q^{7} + 3 q^{8} +O(q^{10})$$ $$q + \beta q^{2} + ( 1 + \beta ) q^{4} - q^{5} + q^{7} + 3 q^{8} -\beta q^{10} + ( 1 - 2 \beta ) q^{13} + \beta q^{14} + ( -2 + \beta ) q^{16} + 4 q^{17} -3 q^{19} + ( -1 - \beta ) q^{20} -2 q^{23} -4 q^{25} + ( -6 - \beta ) q^{26} + ( 1 + \beta ) q^{28} + ( -1 - 2 \beta ) q^{29} -2 q^{31} + ( -3 - \beta ) q^{32} + 4 \beta q^{34} - q^{35} + ( 1 - 4 \beta ) q^{37} -3 \beta q^{38} -3 q^{40} + ( 2 - 4 \beta ) q^{41} + ( -4 + 4 \beta ) q^{43} -2 \beta q^{46} + ( -7 + 2 \beta ) q^{47} + q^{49} -4 \beta q^{50} + ( -5 - 3 \beta ) q^{52} + 3 q^{56} + ( -6 - 3 \beta ) q^{58} + ( -3 - 2 \beta ) q^{59} + ( 2 + 4 \beta ) q^{61} -2 \beta q^{62} + ( 1 - 6 \beta ) q^{64} + ( -1 + 2 \beta ) q^{65} + ( -3 + 2 \beta ) q^{67} + ( 4 + 4 \beta ) q^{68} -\beta q^{70} + ( -2 - 4 \beta ) q^{71} + ( -1 + 6 \beta ) q^{73} + ( -12 - 3 \beta ) q^{74} + ( -3 - 3 \beta ) q^{76} + ( -6 + 4 \beta ) q^{79} + ( 2 - \beta ) q^{80} + ( -12 - 2 \beta ) q^{82} + 12 q^{83} -4 q^{85} + 12 q^{86} -6 q^{89} + ( 1 - 2 \beta ) q^{91} + ( -2 - 2 \beta ) q^{92} + ( 6 - 5 \beta ) q^{94} + 3 q^{95} + ( -8 + 4 \beta ) q^{97} + \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 3q^{4} - 2q^{5} + 2q^{7} + 6q^{8} + O(q^{10})$$ $$2q + q^{2} + 3q^{4} - 2q^{5} + 2q^{7} + 6q^{8} - q^{10} + q^{14} - 3q^{16} + 8q^{17} - 6q^{19} - 3q^{20} - 4q^{23} - 8q^{25} - 13q^{26} + 3q^{28} - 4q^{29} - 4q^{31} - 7q^{32} + 4q^{34} - 2q^{35} - 2q^{37} - 3q^{38} - 6q^{40} - 4q^{43} - 2q^{46} - 12q^{47} + 2q^{49} - 4q^{50} - 13q^{52} + 6q^{56} - 15q^{58} - 8q^{59} + 8q^{61} - 2q^{62} - 4q^{64} - 4q^{67} + 12q^{68} - q^{70} - 8q^{71} + 4q^{73} - 27q^{74} - 9q^{76} - 8q^{79} + 3q^{80} - 26q^{82} + 24q^{83} - 8q^{85} + 24q^{86} - 12q^{89} - 6q^{92} + 7q^{94} + 6q^{95} - 12q^{97} + q^{98} + O(q^{100})$$

## 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 −1.00000 0 1.00000 3.00000 0 1.30278
1.2 2.30278 0 3.30278 −1.00000 0 1.00000 3.00000 0 −2.30278
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.br 2
3.b odd 2 1 7623.2.a.bd 2
11.b odd 2 1 693.2.a.g 2
33.d even 2 1 693.2.a.i yes 2
77.b even 2 1 4851.2.a.x 2
231.h odd 2 1 4851.2.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.a.g 2 11.b odd 2 1
693.2.a.i yes 2 33.d even 2 1
4851.2.a.x 2 77.b even 2 1
4851.2.a.z 2 231.h odd 2 1
7623.2.a.bd 2 3.b odd 2 1
7623.2.a.br 2 1.a even 1 1 trivial

## Atkin-Lehner signs

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

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 3$$ $$T_{5} + 1$$ $$T_{13}^{2} - 13$$