# Properties

 Label 7623.2.a.bx Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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.

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

## 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
 −0.618034 1.61803
0.381966 0 −1.85410 −1.00000 0 1.00000 −1.47214 0 −0.381966
1.2 2.61803 0 4.85410 −1.00000 0 1.00000 7.47214 0 −2.61803
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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.bx 2
3.b odd 2 1 847.2.a.d 2
11.b odd 2 1 7623.2.a.t 2
21.c even 2 1 5929.2.a.i 2
33.d even 2 1 847.2.a.h yes 2
33.f even 10 2 847.2.f.c 4
33.f even 10 2 847.2.f.j 4
33.h odd 10 2 847.2.f.d 4
33.h odd 10 2 847.2.f.l 4
231.h odd 2 1 5929.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.d 2 3.b odd 2 1
847.2.a.h yes 2 33.d even 2 1
847.2.f.c 4 33.f even 10 2
847.2.f.d 4 33.h odd 10 2
847.2.f.j 4 33.f even 10 2
847.2.f.l 4 33.h odd 10 2
5929.2.a.i 2 21.c even 2 1
5929.2.a.s 2 231.h odd 2 1
7623.2.a.t 2 11.b odd 2 1
7623.2.a.bx 2 1.a even 1 1 trivial

## 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

$p$ $F_p(T)$
$2$ $$1 - 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$-4 - 2 T + T^{2}$$
$17$ $$-25 - 5 T + T^{2}$$
$19$ $$11 + 8 T + T^{2}$$
$23$ $$-31 - T + T^{2}$$
$29$ $$11 - 7 T + T^{2}$$
$31$ $$-1 + 4 T + T^{2}$$
$37$ $$-16 - 4 T + T^{2}$$
$41$ $$-125 + T^{2}$$
$43$ $$-95 + 5 T + T^{2}$$
$47$ $$29 - 11 T + T^{2}$$
$53$ $$11 - 7 T + T^{2}$$
$59$ $$-1 + 11 T + T^{2}$$
$61$ $$41 + 13 T + T^{2}$$
$67$ $$-61 + T + T^{2}$$
$71$ $$79 - 21 T + T^{2}$$
$73$ $$139 + 24 T + T^{2}$$
$79$ $$55 - 15 T + T^{2}$$
$83$ $$-29 - 8 T + T^{2}$$
$89$ $$1 + 7 T + T^{2}$$
$97$ $$( -7 + T )^{2}$$
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