# Properties

 Label 7623.2.a.bj 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \beta ) q^{2} + 3 q^{4} -\beta q^{5} - q^{7} + ( 1 - 2 \beta ) q^{8} +O(q^{10})$$ $$q + ( 1 - 2 \beta ) q^{2} + 3 q^{4} -\beta q^{5} - q^{7} + ( 1 - 2 \beta ) q^{8} + ( 2 + \beta ) q^{10} + ( -2 + 2 \beta ) q^{13} + ( -1 + 2 \beta ) q^{14} - q^{16} + ( 3 - 3 \beta ) q^{17} + ( -1 + 3 \beta ) q^{19} -3 \beta q^{20} + ( -5 - \beta ) q^{23} + ( -4 + \beta ) q^{25} + ( -6 + 2 \beta ) q^{26} -3 q^{28} -6 q^{29} + 5 \beta q^{31} + ( -3 + 6 \beta ) q^{32} + ( 9 - 3 \beta ) q^{34} + \beta q^{35} + ( 4 - \beta ) q^{37} + ( -7 - \beta ) q^{38} + ( 2 + \beta ) q^{40} + ( 8 + \beta ) q^{41} + ( -6 + 6 \beta ) q^{43} + ( -3 + 11 \beta ) q^{46} + ( -2 + 4 \beta ) q^{47} + q^{49} + ( -6 + 7 \beta ) q^{50} + ( -6 + 6 \beta ) q^{52} + ( -8 - 2 \beta ) q^{53} + ( -1 + 2 \beta ) q^{56} + ( -6 + 12 \beta ) q^{58} + ( -2 + 2 \beta ) q^{59} + ( -10 - 5 \beta ) q^{62} -13 q^{64} -2 q^{65} + ( 9 - 9 \beta ) q^{68} + ( -2 - \beta ) q^{70} -8 \beta q^{71} + ( -10 + 6 \beta ) q^{73} + ( 6 - 7 \beta ) q^{74} + ( -3 + 9 \beta ) q^{76} + 10 q^{79} + \beta q^{80} + ( 6 - 17 \beta ) q^{82} + ( 14 - 4 \beta ) q^{83} + 3 q^{85} + ( -18 + 6 \beta ) q^{86} + ( -5 + \beta ) q^{89} + ( 2 - 2 \beta ) q^{91} + ( -15 - 3 \beta ) q^{92} -10 q^{94} + ( -3 - 2 \beta ) q^{95} -6 q^{97} + ( 1 - 2 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{4} - q^{5} - 2q^{7} + O(q^{10})$$ $$2q + 6q^{4} - q^{5} - 2q^{7} + 5q^{10} - 2q^{13} - 2q^{16} + 3q^{17} + q^{19} - 3q^{20} - 11q^{23} - 7q^{25} - 10q^{26} - 6q^{28} - 12q^{29} + 5q^{31} + 15q^{34} + q^{35} + 7q^{37} - 15q^{38} + 5q^{40} + 17q^{41} - 6q^{43} + 5q^{46} + 2q^{49} - 5q^{50} - 6q^{52} - 18q^{53} - 2q^{59} - 25q^{62} - 26q^{64} - 4q^{65} + 9q^{68} - 5q^{70} - 8q^{71} - 14q^{73} + 5q^{74} + 3q^{76} + 20q^{79} + q^{80} - 5q^{82} + 24q^{83} + 6q^{85} - 30q^{86} - 9q^{89} + 2q^{91} - 33q^{92} - 20q^{94} - 8q^{95} - 12q^{97} + O(q^{100})$$

## 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
 1.61803 −0.618034
−2.23607 0 3.00000 −1.61803 0 −1.00000 −2.23607 0 3.61803
1.2 2.23607 0 3.00000 0.618034 0 −1.00000 2.23607 0 1.38197
 $$n$$: 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.bj 2
3.b odd 2 1 2541.2.a.v 2
11.b odd 2 1 7623.2.a.bk 2
11.d odd 10 2 693.2.m.a 4
33.d even 2 1 2541.2.a.w 2
33.f even 10 2 231.2.j.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.e 4 33.f even 10 2
693.2.m.a 4 11.d odd 10 2
2541.2.a.v 2 3.b odd 2 1
2541.2.a.w 2 33.d even 2 1
7623.2.a.bj 2 1.a even 1 1 trivial
7623.2.a.bk 2 11.b odd 2 1

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

 $$T_{2}^{2} - 5$$ $$T_{5}^{2} + T_{5} - 1$$ $$T_{13}^{2} + 2 T_{13} - 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + 4 T^{4}$$
$3$ 
$5$ $$1 + T + 9 T^{2} + 5 T^{3} + 25 T^{4}$$
$7$ $$( 1 + T )^{2}$$
$11$ 
$13$ $$1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}$$
$17$ $$1 - 3 T + 25 T^{2} - 51 T^{3} + 289 T^{4}$$
$19$ $$1 - T + 27 T^{2} - 19 T^{3} + 361 T^{4}$$
$23$ $$1 + 11 T + 75 T^{2} + 253 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$1 - 5 T + 37 T^{2} - 155 T^{3} + 961 T^{4}$$
$37$ $$1 - 7 T + 85 T^{2} - 259 T^{3} + 1369 T^{4}$$
$41$ $$1 - 17 T + 153 T^{2} - 697 T^{3} + 1681 T^{4}$$
$43$ $$1 + 6 T + 50 T^{2} + 258 T^{3} + 1849 T^{4}$$
$47$ $$1 + 74 T^{2} + 2209 T^{4}$$
$53$ $$1 + 18 T + 182 T^{2} + 954 T^{3} + 2809 T^{4}$$
$59$ $$1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4}$$
$61$ $$( 1 + 61 T^{2} )^{2}$$
$67$ $$( 1 + 67 T^{2} )^{2}$$
$71$ $$1 + 8 T + 78 T^{2} + 568 T^{3} + 5041 T^{4}$$
$73$ $$1 + 14 T + 150 T^{2} + 1022 T^{3} + 5329 T^{4}$$
$79$ $$( 1 - 10 T + 79 T^{2} )^{2}$$
$83$ $$1 - 24 T + 290 T^{2} - 1992 T^{3} + 6889 T^{4}$$
$89$ $$1 + 9 T + 197 T^{2} + 801 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 6 T + 97 T^{2} )^{2}$$