# Properties

 Label 7623.2.a.cw Level 7623 Weight 2 Character orbit 7623.a Self dual yes Analytic conductor 60.870 Analytic rank 1 Dimension 8 CM no Inner twists 1

# Learn more about

## 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{5} + \beta_{6} ) q^{4} + ( -2 - \beta_{2} - \beta_{3} ) q^{5} - q^{7} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{5} + \beta_{6} ) q^{4} + ( -2 - \beta_{2} - \beta_{3} ) q^{5} - q^{7} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{8} + ( -1 - 2 \beta_{1} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{10} + ( 1 - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{13} -\beta_{1} q^{14} + ( \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{16} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{17} + ( 2 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} ) q^{19} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{20} + ( \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{23} + ( 2 + \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{25} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{26} + ( -1 - \beta_{5} - \beta_{6} ) q^{28} + ( -\beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{29} + ( -\beta_{1} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + ( 3 + \beta_{2} + 2 \beta_{3} - \beta_{6} + \beta_{7} ) q^{32} + ( -2 - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{34} + ( 2 + \beta_{2} + \beta_{3} ) q^{35} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{7} ) q^{37} + ( 2 + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{38} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{40} + ( -1 - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{41} + ( 2 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{43} + ( 1 - 3 \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{46} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + q^{49} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 5 \beta_{5} + 4 \beta_{7} ) q^{50} + ( 4 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{52} + ( -4 - \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{53} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{56} + ( -2 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{58} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( -3 - 3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{61} + ( -3 - \beta_{2} + \beta_{3} + \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{62} + ( 1 + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{64} + ( 3 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{65} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{67} + ( 1 - 5 \beta_{1} + \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{68} + ( 1 + 2 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{70} + ( 1 + \beta_{1} + 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{71} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - 5 \beta_{5} + 2 \beta_{6} ) q^{73} + ( 4 + 2 \beta_{2} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{74} + ( -3 + 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{76} + ( -2 \beta_{1} + \beta_{3} + 4 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{79} + ( -3 - 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 10 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{80} + ( -3 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{82} + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{83} + ( 1 + 3 \beta_{1} + 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{85} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 5 \beta_{5} - \beta_{6} + \beta_{7} ) q^{86} + ( -4 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{89} + ( -1 + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{91} + ( -7 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{92} + ( 5 - 2 \beta_{2} - 4 \beta_{3} - \beta_{4} + 6 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{94} + ( -2 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 5 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{95} + ( 5 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + q^{2} + 7q^{4} - 10q^{5} - 8q^{7} + O(q^{10})$$ $$8q + q^{2} + 7q^{4} - 10q^{5} - 8q^{7} - 6q^{10} + 6q^{13} - q^{14} + q^{16} - 5q^{17} + 13q^{19} - 23q^{20} - 16q^{23} + 16q^{25} + 6q^{26} - 7q^{28} + 9q^{29} + 9q^{31} + 16q^{32} - 12q^{34} + 10q^{35} + 7q^{37} + 10q^{38} - 5q^{40} - 10q^{41} + 4q^{43} - 4q^{46} - 16q^{47} + 8q^{49} + 6q^{50} + 41q^{52} - 37q^{53} - 15q^{58} - q^{59} - 19q^{61} - 18q^{62} - 4q^{64} - 4q^{65} - 19q^{67} + 9q^{68} + 6q^{70} - 13q^{71} + 25q^{73} + 33q^{74} - 26q^{76} - 4q^{80} - 13q^{82} - 25q^{83} - 3q^{85} - 4q^{86} - 37q^{89} - 6q^{91} - 35q^{92} + 42q^{94} + 21q^{95} + 15q^{97} + q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 35 x^{4} - 30 x^{3} - 30 x^{2} + 30 x - 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} - 5 \nu^{5} - 30 \nu^{4} - 30 \nu^{3} + 130 \nu^{2} + 135 \nu - 140$$$$)/25$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} - 4 \nu^{6} + 35 \nu^{5} + 35 \nu^{4} - 165 \nu^{3} - 60 \nu^{2} + 180 \nu - 45$$$$)/25$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} + 6 \nu^{6} - 40 \nu^{5} - 65 \nu^{4} + 160 \nu^{3} + 165 \nu^{2} - 170 \nu - 20$$$$)/25$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{7} + 7 \nu^{6} + 95 \nu^{5} - 55 \nu^{4} - 280 \nu^{3} + 105 \nu^{2} + 210 \nu - 90$$$$)/25$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{7} - 7 \nu^{6} - 95 \nu^{5} + 55 \nu^{4} + 280 \nu^{3} - 80 \nu^{2} - 210 \nu + 15$$$$)/25$$ $$\beta_{7}$$ $$=$$ $$($$$$12 \nu^{7} - \nu^{6} - 135 \nu^{5} + 15 \nu^{4} + 440 \nu^{3} - 90 \nu^{2} - 405 \nu + 170$$$$)/25$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 6 \beta_{6} + 7 \beta_{5} - \beta_{4} + \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 7 \beta_{6} + 8 \beta_{5} + 8 \beta_{4} + 10 \beta_{3} - 7 \beta_{2} + 28 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{7} + 35 \beta_{6} + 47 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + \beta_{2} + 10 \beta_{1} + 77$$ $$\nu^{7}$$ $$=$$ $$13 \beta_{7} + 45 \beta_{6} + 56 \beta_{5} + 54 \beta_{4} + 76 \beta_{3} - 42 \beta_{2} + 165 \beta_{1} + 31$$

## 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
 −2.43045 −1.70716 −1.40927 0.226211 0.669744 1.11447 1.98451 2.55194
−2.43045 0 3.90710 −1.25863 0 −1.00000 −4.63512 0 3.05904
1.2 −1.70716 0 0.914391 −4.06637 0 −1.00000 1.85331 0 6.94194
1.3 −1.40927 0 −0.0139645 1.83139 0 −1.00000 2.83822 0 −2.58091
1.4 0.226211 0 −1.94883 2.49552 0 −1.00000 −0.893270 0 0.564516
1.5 0.669744 0 −1.55144 −2.14378 0 −1.00000 −2.37856 0 −1.43578
1.6 1.11447 0 −0.757964 −3.45608 0 −1.00000 −3.07366 0 −3.85168
1.7 1.98451 0 1.93830 0.0269243 0 −1.00000 −0.122446 0 0.0534317
1.8 2.55194 0 4.51241 −3.42898 0 −1.00000 6.41153 0 −8.75055
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.cw 8
3.b odd 2 1 847.2.a.o 8
11.b odd 2 1 7623.2.a.ct 8
11.d odd 10 2 693.2.m.i 16
21.c even 2 1 5929.2.a.bs 8
33.d even 2 1 847.2.a.p 8
33.f even 10 2 77.2.f.b 16
33.f even 10 2 847.2.f.w 16
33.h odd 10 2 847.2.f.v 16
33.h odd 10 2 847.2.f.x 16
231.h odd 2 1 5929.2.a.bt 8
231.r odd 10 2 539.2.f.e 16
231.be even 30 4 539.2.q.g 32
231.bf odd 30 4 539.2.q.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 33.f even 10 2
539.2.f.e 16 231.r odd 10 2
539.2.q.f 32 231.bf odd 30 4
539.2.q.g 32 231.be even 30 4
693.2.m.i 16 11.d odd 10 2
847.2.a.o 8 3.b odd 2 1
847.2.a.p 8 33.d even 2 1
847.2.f.v 16 33.h odd 10 2
847.2.f.w 16 33.f even 10 2
847.2.f.x 16 33.h odd 10 2
5929.2.a.bs 8 21.c even 2 1
5929.2.a.bt 8 231.h odd 2 1
7623.2.a.ct 8 11.b odd 2 1
7623.2.a.cw 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{8} - \cdots$$ $$T_{5}^{8} + \cdots$$ $$T_{13}^{8} - \cdots$$