# Properties

 Label 7623.2.a.u 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: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{2} + 3 \beta q^{4} -\beta q^{5} + q^{7} + ( -1 - 4 \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{2} + 3 \beta q^{4} -\beta q^{5} + q^{7} + ( -1 - 4 \beta ) q^{8} + ( 1 + 2 \beta ) q^{10} + ( -1 + 4 \beta ) q^{13} + ( -1 - \beta ) q^{14} + ( 5 + 3 \beta ) q^{16} + 3 q^{17} + 4 q^{19} + ( -3 - 3 \beta ) q^{20} + ( 4 + 2 \beta ) q^{23} + ( -4 + \beta ) q^{25} + ( -3 - 7 \beta ) q^{26} + 3 \beta q^{28} -3 q^{29} + ( -3 + 2 \beta ) q^{31} + ( -6 - 3 \beta ) q^{32} + ( -3 - 3 \beta ) q^{34} -\beta q^{35} + ( -2 - 4 \beta ) q^{37} + ( -4 - 4 \beta ) q^{38} + ( 4 + 5 \beta ) q^{40} + ( -5 + 5 \beta ) q^{41} -9 q^{43} + ( -6 - 8 \beta ) q^{46} + ( -2 - 5 \beta ) q^{47} + q^{49} + ( 3 + 2 \beta ) q^{50} + ( 12 + 9 \beta ) q^{52} + ( 4 - 5 \beta ) q^{53} + ( -1 - 4 \beta ) q^{56} + ( 3 + 3 \beta ) q^{58} + ( -11 - \beta ) q^{59} + ( 3 - 6 \beta ) q^{61} + ( 1 - \beta ) q^{62} + ( -1 + 6 \beta ) q^{64} + ( -4 - 3 \beta ) q^{65} + ( 6 - 6 \beta ) q^{67} + 9 \beta q^{68} + ( 1 + 2 \beta ) q^{70} + ( -9 - 2 \beta ) q^{71} + ( 1 - 3 \beta ) q^{73} + ( 6 + 10 \beta ) q^{74} + 12 \beta q^{76} + ( 2 + 3 \beta ) q^{79} + ( -3 - 8 \beta ) q^{80} -5 \beta q^{82} + ( 10 - 8 \beta ) q^{83} -3 \beta q^{85} + ( 9 + 9 \beta ) q^{86} + ( -14 + 7 \beta ) q^{89} + ( -1 + 4 \beta ) q^{91} + ( 6 + 18 \beta ) q^{92} + ( 7 + 12 \beta ) q^{94} -4 \beta q^{95} + ( 3 - 12 \beta ) q^{97} + ( -1 - \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} + 3q^{4} - q^{5} + 2q^{7} - 6q^{8} + O(q^{10})$$ $$2q - 3q^{2} + 3q^{4} - q^{5} + 2q^{7} - 6q^{8} + 4q^{10} + 2q^{13} - 3q^{14} + 13q^{16} + 6q^{17} + 8q^{19} - 9q^{20} + 10q^{23} - 7q^{25} - 13q^{26} + 3q^{28} - 6q^{29} - 4q^{31} - 15q^{32} - 9q^{34} - q^{35} - 8q^{37} - 12q^{38} + 13q^{40} - 5q^{41} - 18q^{43} - 20q^{46} - 9q^{47} + 2q^{49} + 8q^{50} + 33q^{52} + 3q^{53} - 6q^{56} + 9q^{58} - 23q^{59} + q^{62} + 4q^{64} - 11q^{65} + 6q^{67} + 9q^{68} + 4q^{70} - 20q^{71} - q^{73} + 22q^{74} + 12q^{76} + 7q^{79} - 14q^{80} - 5q^{82} + 12q^{83} - 3q^{85} + 27q^{86} - 21q^{89} + 2q^{91} + 30q^{92} + 26q^{94} - 4q^{95} - 6q^{97} - 3q^{98} + O(q^{100})$$

## 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
 1.61803 −0.618034
−2.61803 0 4.85410 −1.61803 0 1.00000 −7.47214 0 4.23607
1.2 −0.381966 0 −1.85410 0.618034 0 1.00000 1.47214 0 −0.236068
 $$n$$: 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.u 2
3.b odd 2 1 2541.2.a.bf 2
11.b odd 2 1 7623.2.a.by 2
11.c even 5 2 693.2.m.e 4
33.d even 2 1 2541.2.a.m 2
33.h odd 10 2 231.2.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.a 4 33.h odd 10 2
693.2.m.e 4 11.c even 5 2
2541.2.a.m 2 33.d even 2 1
2541.2.a.bf 2 3.b odd 2 1
7623.2.a.u 2 1.a even 1 1 trivial
7623.2.a.by 2 11.b odd 2 1

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{2} + 3 T_{2} + 1$$ $$T_{5}^{2} + T_{5} - 1$$ $$T_{13}^{2} - 2 T_{13} - 19$$