# Properties

 Label 7623.2.a.cd Level 7623 Weight 2 Character orbit 7623.a Self dual yes Analytic conductor 60.870 Analytic rank 0 Dimension 3 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: $$3$$ Coefficient field: 3.3.229.1 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{2} + ( 2 + \beta_{1} ) q^{4} + ( -1 + \beta_{1} + \beta_{2} ) q^{5} + q^{7} + ( 1 + 2 \beta_{1} ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{2} + ( 2 + \beta_{1} ) q^{4} + ( -1 + \beta_{1} + \beta_{2} ) q^{5} + q^{7} + ( 1 + 2 \beta_{1} ) q^{8} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{10} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{13} + ( 1 + \beta_{2} ) q^{14} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{16} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 3 - \beta_{1} + \beta_{2} ) q^{19} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{20} + ( -4 - 2 \beta_{2} ) q^{23} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{25} + ( 1 + 5 \beta_{1} + 2 \beta_{2} ) q^{26} + ( 2 + \beta_{1} ) q^{28} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{31} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{32} + ( 6 \beta_{1} + 4 \beta_{2} ) q^{34} + ( -1 + \beta_{1} + \beta_{2} ) q^{35} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{37} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{38} + ( 7 + \beta_{1} + 3 \beta_{2} ) q^{40} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 4 - 4 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -10 - 2 \beta_{1} - 2 \beta_{2} ) q^{46} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{47} + q^{49} + ( -4 - \beta_{1} + 7 \beta_{2} ) q^{50} + ( 10 + 6 \beta_{1} + \beta_{2} ) q^{52} + 2 \beta_{1} q^{53} + ( 1 + 2 \beta_{1} ) q^{56} + ( -1 - 5 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -1 + 3 \beta_{1} - 3 \beta_{2} ) q^{59} + 2 q^{61} + ( -12 + 2 \beta_{2} ) q^{62} + ( -1 - 4 \beta_{1} + 2 \beta_{2} ) q^{64} + ( 7 - \beta_{1} + 7 \beta_{2} ) q^{65} + ( -1 - 5 \beta_{1} + \beta_{2} ) q^{67} + ( 14 + 8 \beta_{1} ) q^{68} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{70} + ( 4 - 4 \beta_{1} ) q^{71} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{73} + ( 9 + \beta_{1} - 2 \beta_{2} ) q^{74} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{76} + ( -4 + 4 \beta_{1} ) q^{79} + ( 13 + \beta_{1} - 2 \beta_{2} ) q^{80} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{82} + ( 2 - 6 \beta_{1} ) q^{83} + ( 6 - 2 \beta_{1} + 12 \beta_{2} ) q^{85} + ( -6 - 10 \beta_{1} + 6 \beta_{2} ) q^{86} + ( -10 - 4 \beta_{2} ) q^{89} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{91} + ( -10 - 6 \beta_{1} - 4 \beta_{2} ) q^{92} + ( -11 - 5 \beta_{1} + 2 \beta_{2} ) q^{94} + ( -3 + 5 \beta_{1} - \beta_{2} ) q^{95} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{97} + ( 1 + \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 2q^{2} + 6q^{4} - 4q^{5} + 3q^{7} + 3q^{8} + O(q^{10})$$ $$3q + 2q^{2} + 6q^{4} - 4q^{5} + 3q^{7} + 3q^{8} + 11q^{10} + 4q^{13} + 2q^{14} - 4q^{16} + 8q^{17} + 8q^{19} + 3q^{20} - 10q^{23} + 15q^{25} + q^{26} + 6q^{28} - 4q^{29} - 2q^{31} + 8q^{32} - 4q^{34} - 4q^{35} + 13q^{38} + 18q^{40} + 14q^{41} + 14q^{43} - 28q^{46} + 3q^{49} - 19q^{50} + 29q^{52} + 3q^{56} - q^{58} + 6q^{61} - 38q^{62} - 5q^{64} + 14q^{65} - 4q^{67} + 42q^{68} + 11q^{70} + 12q^{71} + 20q^{73} + 29q^{74} + 11q^{76} - 12q^{79} + 41q^{80} - 6q^{82} + 6q^{83} + 6q^{85} - 24q^{86} - 26q^{89} + 4q^{91} - 26q^{92} - 35q^{94} - 8q^{95} - 4q^{97} + 2q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 −0.254102 −1.86081 2.11491
−1.93543 0 1.74590 −4.18953 0 1.00000 0.491797 0 8.10856
1.2 1.46260 0 0.139194 −2.39821 0 1.00000 −2.72161 0 −3.50761
1.3 2.47283 0 4.11491 2.58774 0 1.00000 5.22982 0 6.39905
 $$n$$: 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.cd 3
3.b odd 2 1 2541.2.a.bg 3
11.b odd 2 1 693.2.a.l 3
33.d even 2 1 231.2.a.e 3
77.b even 2 1 4851.2.a.bi 3
132.d odd 2 1 3696.2.a.bo 3
165.d even 2 1 5775.2.a.bp 3
231.h odd 2 1 1617.2.a.t 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 33.d even 2 1
693.2.a.l 3 11.b odd 2 1
1617.2.a.t 3 231.h odd 2 1
2541.2.a.bg 3 3.b odd 2 1
3696.2.a.bo 3 132.d odd 2 1
4851.2.a.bi 3 77.b even 2 1
5775.2.a.bp 3 165.d even 2 1
7623.2.a.cd 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{3} - 2 T_{2}^{2} - 4 T_{2} + 7$$ $$T_{5}^{3} + 4 T_{5}^{2} - 7 T_{5} - 26$$ $$T_{13}^{3} - 4 T_{13}^{2} - 27 T_{13} + 94$$