# Properties

 Label 7623.2.a.bp 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.

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

## 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
 −0.618034 1.61803
−0.618034 0 −1.61803 0.381966 0 −1.00000 2.23607 0 −0.236068
1.2 1.61803 0 0.618034 2.61803 0 −1.00000 −2.23607 0 4.23607
 $$n$$: 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.bp 2
3.b odd 2 1 2541.2.a.s 2
11.b odd 2 1 7623.2.a.ba 2
11.d odd 10 2 693.2.m.b 4
33.d even 2 1 2541.2.a.bb 2
33.f even 10 2 231.2.j.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.d 4 33.f even 10 2
693.2.m.b 4 11.d odd 10 2
2541.2.a.s 2 3.b odd 2 1
2541.2.a.bb 2 33.d even 2 1
7623.2.a.ba 2 11.b odd 2 1
7623.2.a.bp 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

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