# Properties

 Label 7623.2.a.bw 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: $$\Q(\sqrt{5})$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + 3 \beta q^{4} - q^{5} - q^{7} + ( 1 + 4 \beta ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + 3 \beta q^{4} - q^{5} - q^{7} + ( 1 + 4 \beta ) q^{8} + ( -1 - \beta ) q^{10} + ( 1 + 2 \beta ) q^{13} + ( -1 - \beta ) q^{14} + ( 5 + 3 \beta ) q^{16} + ( 4 - 4 \beta ) q^{17} + ( -1 + 4 \beta ) q^{19} -3 \beta q^{20} + ( -6 + 4 \beta ) q^{23} -4 q^{25} + ( 3 + 5 \beta ) q^{26} -3 \beta q^{28} + ( 3 + 2 \beta ) q^{29} + ( 2 + 4 \beta ) q^{31} + ( 6 + 3 \beta ) q^{32} -4 \beta q^{34} + q^{35} + ( -3 + 4 \beta ) q^{37} + ( 3 + 7 \beta ) q^{38} + ( -1 - 4 \beta ) q^{40} + ( 2 - 4 \beta ) q^{41} + 4 q^{43} + ( -2 + 2 \beta ) q^{46} + ( -1 - 2 \beta ) q^{47} + q^{49} + ( -4 - 4 \beta ) q^{50} + ( 6 + 9 \beta ) q^{52} + ( -8 - 4 \beta ) q^{53} + ( -1 - 4 \beta ) q^{56} + ( 5 + 7 \beta ) q^{58} + ( -9 + 10 \beta ) q^{59} + ( -6 + 4 \beta ) q^{61} + ( 6 + 10 \beta ) q^{62} + ( -1 + 6 \beta ) q^{64} + ( -1 - 2 \beta ) q^{65} + ( 3 - 10 \beta ) q^{67} -12 q^{68} + ( 1 + \beta ) q^{70} + ( 2 - 4 \beta ) q^{71} + ( 11 + 2 \beta ) q^{73} + ( 1 + 5 \beta ) q^{74} + ( 12 + 9 \beta ) q^{76} + ( -2 + 4 \beta ) q^{79} + ( -5 - 3 \beta ) q^{80} + ( -2 - 6 \beta ) q^{82} + ( -4 + 12 \beta ) q^{83} + ( -4 + 4 \beta ) q^{85} + ( 4 + 4 \beta ) q^{86} + 10 q^{89} + ( -1 - 2 \beta ) q^{91} + ( 12 - 6 \beta ) q^{92} + ( -3 - 5 \beta ) q^{94} + ( 1 - 4 \beta ) q^{95} -8 q^{97} + ( 1 + \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{2} + 3q^{4} - 2q^{5} - 2q^{7} + 6q^{8} + O(q^{10})$$ $$2q + 3q^{2} + 3q^{4} - 2q^{5} - 2q^{7} + 6q^{8} - 3q^{10} + 4q^{13} - 3q^{14} + 13q^{16} + 4q^{17} + 2q^{19} - 3q^{20} - 8q^{23} - 8q^{25} + 11q^{26} - 3q^{28} + 8q^{29} + 8q^{31} + 15q^{32} - 4q^{34} + 2q^{35} - 2q^{37} + 13q^{38} - 6q^{40} + 8q^{43} - 2q^{46} - 4q^{47} + 2q^{49} - 12q^{50} + 21q^{52} - 20q^{53} - 6q^{56} + 17q^{58} - 8q^{59} - 8q^{61} + 22q^{62} + 4q^{64} - 4q^{65} - 4q^{67} - 24q^{68} + 3q^{70} + 24q^{73} + 7q^{74} + 33q^{76} - 13q^{80} - 10q^{82} + 4q^{83} - 4q^{85} + 12q^{86} + 20q^{89} - 4q^{91} + 18q^{92} - 11q^{94} - 2q^{95} - 16q^{97} + 3q^{98} + O(q^{100})$$

## 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
 −0.618034 1.61803
0.381966 0 −1.85410 −1.00000 0 −1.00000 −1.47214 0 −0.381966
1.2 2.61803 0 4.85410 −1.00000 0 −1.00000 7.47214 0 −2.61803
 $$n$$: 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.bw 2
3.b odd 2 1 7623.2.a.v 2
11.b odd 2 1 693.2.a.e 2
33.d even 2 1 693.2.a.k yes 2
77.b even 2 1 4851.2.a.u 2
231.h odd 2 1 4851.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.a.e 2 11.b odd 2 1
693.2.a.k yes 2 33.d even 2 1
4851.2.a.u 2 77.b even 2 1
4851.2.a.bf 2 231.h odd 2 1
7623.2.a.v 2 3.b odd 2 1
7623.2.a.bw 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{2} - 3 T_{2} + 1$$ $$T_{5} + 1$$ $$T_{13}^{2} - 4 T_{13} - 1$$