# Properties

 Label 7623.2.a.co Level 7623 Weight 2 Character orbit 7623.a Self dual yes Analytic conductor 60.870 Analytic rank 0 Dimension 4 CM no Inner twists 1

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## 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: $$4$$ Coefficient field: 4.4.2525.1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 3$$

## 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
 −1.46673 −0.777484 1.77748 2.46673
−1.46673 0 0.151302 −0.466732 0 1.00000 2.71154 0 0.684570
1.2 −0.777484 0 −1.39552 0.222516 0 1.00000 2.63996 0 −0.173002
1.3 1.77748 0 1.15945 2.77748 0 1.00000 −1.49406 0 4.93693
1.4 2.46673 0 4.08477 3.46673 0 1.00000 5.14256 0 8.55150
 $$n$$: 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.co 4
3.b odd 2 1 847.2.a.k 4
11.b odd 2 1 7623.2.a.ch 4
11.d odd 10 2 693.2.m.g 8
21.c even 2 1 5929.2.a.bb 4
33.d even 2 1 847.2.a.l 4
33.f even 10 2 77.2.f.a 8
33.f even 10 2 847.2.f.p 8
33.h odd 10 2 847.2.f.q 8
33.h odd 10 2 847.2.f.s 8
231.h odd 2 1 5929.2.a.bi 4
231.r odd 10 2 539.2.f.d 8
231.be even 30 4 539.2.q.c 16
231.bf odd 30 4 539.2.q.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 33.f even 10 2
539.2.f.d 8 231.r odd 10 2
539.2.q.b 16 231.bf odd 30 4
539.2.q.c 16 231.be even 30 4
693.2.m.g 8 11.d odd 10 2
847.2.a.k 4 3.b odd 2 1
847.2.a.l 4 33.d even 2 1
847.2.f.p 8 33.f even 10 2
847.2.f.q 8 33.h odd 10 2
847.2.f.s 8 33.h odd 10 2
5929.2.a.bb 4 21.c even 2 1
5929.2.a.bi 4 231.h odd 2 1
7623.2.a.ch 4 11.b odd 2 1
7623.2.a.co 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{4} - 2 T_{2}^{3} - 4 T_{2}^{2} + 5 T_{2} + 5$$ $$T_{5}^{4} - 6 T_{5}^{3} + 8 T_{5}^{2} + 3 T_{5} - 1$$ $$T_{13}^{4} - 32 T_{13}^{2} + 65 T_{13} - 29$$