# Properties

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

# 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: $$8$$ Coefficient field: 8.8.6988960000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 693) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -\beta_{1} - \beta_{7} ) q^{5} + q^{7} + ( \beta_{1} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -\beta_{1} - \beta_{7} ) q^{5} + q^{7} + ( \beta_{1} + \beta_{5} + \beta_{6} ) q^{8} + ( -1 - \beta_{2} + \beta_{3} ) q^{10} + ( -\beta_{3} - \beta_{4} ) q^{13} + \beta_{1} q^{14} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{5} + \beta_{7} ) q^{17} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{19} + ( -2 \beta_{1} - \beta_{6} + \beta_{7} ) q^{20} + ( 3 \beta_{1} + \beta_{6} + \beta_{7} ) q^{23} + ( -2 - \beta_{3} ) q^{25} + ( -\beta_{1} - 3 \beta_{5} + 2 \beta_{7} ) q^{26} + \beta_{2} q^{28} + ( -3 \beta_{1} - 3 \beta_{6} ) q^{29} + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{31} + ( -4 \beta_{1} - 3 \beta_{6} - 2 \beta_{7} ) q^{32} + ( -3 - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{34} + ( -\beta_{1} - \beta_{7} ) q^{35} + ( -5 - \beta_{2} + \beta_{4} ) q^{37} + ( 5 \beta_{1} + 6 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{38} + ( -2 - 3 \beta_{3} ) q^{40} + ( 4 \beta_{1} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{41} + ( -2 - 3 \beta_{3} - 3 \beta_{4} ) q^{43} + ( 4 + 3 \beta_{2} - \beta_{3} ) q^{46} + ( 5 \beta_{5} - \beta_{6} ) q^{47} + q^{49} + ( -2 \beta_{1} - \beta_{5} + \beta_{7} ) q^{50} + ( -1 - \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{52} + ( 2 \beta_{1} - \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{53} + ( \beta_{1} + \beta_{5} + \beta_{6} ) q^{56} + ( -3 - 3 \beta_{2} ) q^{58} + ( \beta_{1} - 3 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{59} + ( -4 + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{61} + ( -4 \beta_{1} - 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{62} + ( -3 - 2 \beta_{2} - 2 \beta_{4} ) q^{64} + ( 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{65} + ( -6 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{67} + ( -7 \beta_{1} - 5 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{68} + ( -1 - \beta_{2} + \beta_{3} ) q^{70} + ( -3 \beta_{1} - 4 \beta_{5} + \beta_{6} ) q^{71} + ( -3 - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{73} + ( -7 \beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{74} + ( 6 + 3 \beta_{2} + 7 \beta_{3} + 2 \beta_{4} ) q^{76} + ( 1 - 3 \beta_{2} + 7 \beta_{3} + \beta_{4} ) q^{79} + ( 2 \beta_{1} - 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{80} + ( 4 + 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{82} + ( -2 \beta_{1} + 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{83} + ( 2 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{85} + ( -5 \beta_{1} - 9 \beta_{5} + 6 \beta_{7} ) q^{86} + ( \beta_{1} - \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{89} + ( -\beta_{3} - \beta_{4} ) q^{91} + ( 7 \beta_{1} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{92} + ( -4 + 5 \beta_{3} + 5 \beta_{4} ) q^{94} + ( -2 \beta_{1} - 5 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{95} + ( -2 - 4 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{4} + 8q^{7} + O(q^{10})$$ $$8q + 2q^{4} + 8q^{7} - 14q^{10} - 2q^{16} + 6q^{19} - 12q^{25} + 2q^{28} - 6q^{31} - 24q^{34} - 38q^{37} - 4q^{40} - 16q^{43} + 42q^{46} + 8q^{49} - 2q^{52} - 30q^{58} - 28q^{61} - 36q^{64} - 36q^{67} - 14q^{70} - 14q^{73} + 34q^{76} - 22q^{79} + 36q^{82} + 18q^{85} - 32q^{94} - 48q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9 x^{6} + 22 x^{4} - 11 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 8 \nu^{4} + 16 \nu^{2} - 5$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} + 10 \nu^{4} - 26 \nu^{2} + 9$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$-\nu^{7} + 9 \nu^{5} - 21 \nu^{3} + 6 \nu$$ $$\beta_{6}$$ $$=$$ $$\nu^{7} - 9 \nu^{5} + 22 \nu^{3} - 11 \nu$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{7} + 26 \nu^{5} - 58 \nu^{3} + 17 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5 \beta_{2} + 8$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{7} + 5 \beta_{6} + 8 \beta_{5} + 24 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{4} + 10 \beta_{3} + 24 \beta_{2} + 37$$ $$\nu^{7}$$ $$=$$ $$-18 \beta_{7} + 24 \beta_{6} + 50 \beta_{5} + 117 \beta_{1}$$

## 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
 −2.25947 −1.80692 −0.716111 −0.342036 0.342036 0.716111 1.80692 2.25947
−2.25947 0 3.10522 1.54336 0 1.00000 −2.49721 0 −3.48718
1.2 −1.80692 0 1.26498 2.14896 0 1.00000 1.32813 0 −3.88301
1.3 −0.716111 0 −1.48718 −1.54336 0 1.00000 2.49721 0 1.10522
1.4 −0.342036 0 −1.88301 2.14896 0 1.00000 1.32813 0 −0.735023
1.5 0.342036 0 −1.88301 −2.14896 0 1.00000 −1.32813 0 −0.735023
1.6 0.716111 0 −1.48718 1.54336 0 1.00000 −2.49721 0 1.10522
1.7 1.80692 0 1.26498 −2.14896 0 1.00000 −1.32813 0 −3.88301
1.8 2.25947 0 3.10522 −1.54336 0 1.00000 2.49721 0 −3.48718
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.cv 8
3.b odd 2 1 inner 7623.2.a.cv 8
11.b odd 2 1 7623.2.a.cu 8
11.d odd 10 2 693.2.m.h 16
33.d even 2 1 7623.2.a.cu 8
33.f even 10 2 693.2.m.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.m.h 16 11.d odd 10 2
693.2.m.h 16 33.f even 10 2
7623.2.a.cu 8 11.b odd 2 1
7623.2.a.cu 8 33.d even 2 1
7623.2.a.cv 8 1.a even 1 1 trivial
7623.2.a.cv 8 3.b odd 2 1 inner

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

 $$T_{2}^{8} - 9 T_{2}^{6} + 22 T_{2}^{4} - 11 T_{2}^{2} + 1$$ $$T_{5}^{4} - 7 T_{5}^{2} + 11$$ $$T_{13}^{4} - 11 T_{13}^{2} - 10 T_{13} + 4$$