# Properties

 Label 7623.2.a.t 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 847) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{2} + 3 \beta q^{4} - q^{5} - q^{7} + ( -1 - 4 \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{2} + 3 \beta q^{4} - q^{5} - q^{7} + ( -1 - 4 \beta ) q^{8} + ( 1 + \beta ) q^{10} -2 \beta q^{13} + ( 1 + \beta ) q^{14} + ( 5 + 3 \beta ) q^{16} -5 \beta q^{17} + ( 3 + 2 \beta ) q^{19} -3 \beta q^{20} + ( -2 + 5 \beta ) q^{23} -4 q^{25} + ( 2 + 4 \beta ) q^{26} -3 \beta q^{28} + ( -4 + \beta ) q^{29} + ( -3 + 2 \beta ) q^{31} + ( -6 - 3 \beta ) q^{32} + ( 5 + 10 \beta ) q^{34} + q^{35} + ( 4 - 4 \beta ) q^{37} + ( -5 - 7 \beta ) q^{38} + ( 1 + 4 \beta ) q^{40} + ( -5 + 10 \beta ) q^{41} + ( 7 - 9 \beta ) q^{43} + ( -3 - 8 \beta ) q^{46} + ( 6 - \beta ) q^{47} + q^{49} + ( 4 + 4 \beta ) q^{50} + ( -6 - 6 \beta ) q^{52} + ( 3 + \beta ) q^{53} + ( 1 + 4 \beta ) q^{56} + ( 3 + 2 \beta ) q^{58} + ( -8 + 5 \beta ) q^{59} + ( 7 - \beta ) q^{61} + ( 1 - \beta ) q^{62} + ( -1 + 6 \beta ) q^{64} + 2 \beta q^{65} + ( -4 + 7 \beta ) q^{67} + ( -15 - 15 \beta ) q^{68} + ( -1 - \beta ) q^{70} + ( 13 - 5 \beta ) q^{71} + ( 13 - 2 \beta ) q^{73} + 4 \beta q^{74} + ( 6 + 15 \beta ) q^{76} + ( -7 - \beta ) q^{79} + ( -5 - 3 \beta ) q^{80} + ( -5 - 15 \beta ) q^{82} + ( -1 - 6 \beta ) q^{83} + 5 \beta q^{85} + ( 2 + 11 \beta ) q^{86} + ( -5 + 3 \beta ) q^{89} + 2 \beta q^{91} + ( 15 + 9 \beta ) q^{92} + ( -5 - 4 \beta ) q^{94} + ( -3 - 2 \beta ) q^{95} + 7 q^{97} + ( -1 - \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} + 3q^{4} - 2q^{5} - 2q^{7} - 6q^{8} + O(q^{10})$$ $$2q - 3q^{2} + 3q^{4} - 2q^{5} - 2q^{7} - 6q^{8} + 3q^{10} - 2q^{13} + 3q^{14} + 13q^{16} - 5q^{17} + 8q^{19} - 3q^{20} + q^{23} - 8q^{25} + 8q^{26} - 3q^{28} - 7q^{29} - 4q^{31} - 15q^{32} + 20q^{34} + 2q^{35} + 4q^{37} - 17q^{38} + 6q^{40} + 5q^{43} - 14q^{46} + 11q^{47} + 2q^{49} + 12q^{50} - 18q^{52} + 7q^{53} + 6q^{56} + 8q^{58} - 11q^{59} + 13q^{61} + q^{62} + 4q^{64} + 2q^{65} - q^{67} - 45q^{68} - 3q^{70} + 21q^{71} + 24q^{73} + 4q^{74} + 27q^{76} - 15q^{79} - 13q^{80} - 25q^{82} - 8q^{83} + 5q^{85} + 15q^{86} - 7q^{89} + 2q^{91} + 39q^{92} - 14q^{94} - 8q^{95} + 14q^{97} - 3q^{98} + O(q^{100})$$

## 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
−2.61803 0 4.85410 −1.00000 0 −1.00000 −7.47214 0 2.61803
1.2 −0.381966 0 −1.85410 −1.00000 0 −1.00000 1.47214 0 0.381966
 $$n$$: 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.t 2
3.b odd 2 1 847.2.a.h yes 2
11.b odd 2 1 7623.2.a.bx 2
21.c even 2 1 5929.2.a.s 2
33.d even 2 1 847.2.a.d 2
33.f even 10 2 847.2.f.d 4
33.f even 10 2 847.2.f.l 4
33.h odd 10 2 847.2.f.c 4
33.h odd 10 2 847.2.f.j 4
231.h odd 2 1 5929.2.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.d 2 33.d even 2 1
847.2.a.h yes 2 3.b odd 2 1
847.2.f.c 4 33.h odd 10 2
847.2.f.d 4 33.f even 10 2
847.2.f.j 4 33.h odd 10 2
847.2.f.l 4 33.f even 10 2
5929.2.a.i 2 231.h odd 2 1
5929.2.a.s 2 21.c even 2 1
7623.2.a.t 2 1.a even 1 1 trivial
7623.2.a.bx 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{2} + 3 T_{2} + 1$$ $$T_{5} + 1$$ $$T_{13}^{2} + 2 T_{13} - 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$3$ 
$5$ $$( 1 + T + 5 T^{2} )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ 
$13$ $$1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}$$
$17$ $$1 + 5 T + 9 T^{2} + 85 T^{3} + 289 T^{4}$$
$19$ $$1 - 8 T + 49 T^{2} - 152 T^{3} + 361 T^{4}$$
$23$ $$1 - T + 15 T^{2} - 23 T^{3} + 529 T^{4}$$
$29$ $$1 + 7 T + 69 T^{2} + 203 T^{3} + 841 T^{4}$$
$31$ $$1 + 4 T + 61 T^{2} + 124 T^{3} + 961 T^{4}$$
$37$ $$1 - 4 T + 58 T^{2} - 148 T^{3} + 1369 T^{4}$$
$41$ $$1 - 43 T^{2} + 1681 T^{4}$$
$43$ $$1 - 5 T - 9 T^{2} - 215 T^{3} + 1849 T^{4}$$
$47$ $$1 - 11 T + 123 T^{2} - 517 T^{3} + 2209 T^{4}$$
$53$ $$1 - 7 T + 117 T^{2} - 371 T^{3} + 2809 T^{4}$$
$59$ $$1 + 11 T + 117 T^{2} + 649 T^{3} + 3481 T^{4}$$
$61$ $$1 - 13 T + 163 T^{2} - 793 T^{3} + 3721 T^{4}$$
$67$ $$1 + T + 73 T^{2} + 67 T^{3} + 4489 T^{4}$$
$71$ $$1 - 21 T + 221 T^{2} - 1491 T^{3} + 5041 T^{4}$$
$73$ $$1 - 24 T + 285 T^{2} - 1752 T^{3} + 5329 T^{4}$$
$79$ $$1 + 15 T + 213 T^{2} + 1185 T^{3} + 6241 T^{4}$$
$83$ $$1 + 8 T + 137 T^{2} + 664 T^{3} + 6889 T^{4}$$
$89$ $$1 + 7 T + 179 T^{2} + 623 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 7 T + 97 T^{2} )^{2}$$