# Properties

 Label 7623.2.a.bi Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
−1.73205 0 1.00000 −3.46410 0 −1.00000 1.73205 0 6.00000
1.2 1.73205 0 1.00000 3.46410 0 −1.00000 −1.73205 0 6.00000
 $$n$$: 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

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.bi 2
3.b odd 2 1 inner 7623.2.a.bi 2
11.b odd 2 1 63.2.a.b 2
33.d even 2 1 63.2.a.b 2
44.c even 2 1 1008.2.a.n 2
55.d odd 2 1 1575.2.a.q 2
55.e even 4 2 1575.2.d.i 4
77.b even 2 1 441.2.a.g 2
77.h odd 6 2 441.2.e.j 4
77.i even 6 2 441.2.e.i 4
88.b odd 2 1 4032.2.a.bt 2
88.g even 2 1 4032.2.a.bq 2
99.g even 6 2 567.2.f.j 4
99.h odd 6 2 567.2.f.j 4
132.d odd 2 1 1008.2.a.n 2
165.d even 2 1 1575.2.a.q 2
165.l odd 4 2 1575.2.d.i 4
231.h odd 2 1 441.2.a.g 2
231.k odd 6 2 441.2.e.i 4
231.l even 6 2 441.2.e.j 4
264.m even 2 1 4032.2.a.bt 2
264.p odd 2 1 4032.2.a.bq 2
308.g odd 2 1 7056.2.a.cm 2
924.n even 2 1 7056.2.a.cm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 11.b odd 2 1
63.2.a.b 2 33.d even 2 1
441.2.a.g 2 77.b even 2 1
441.2.a.g 2 231.h odd 2 1
441.2.e.i 4 77.i even 6 2
441.2.e.i 4 231.k odd 6 2
441.2.e.j 4 77.h odd 6 2
441.2.e.j 4 231.l even 6 2
567.2.f.j 4 99.g even 6 2
567.2.f.j 4 99.h odd 6 2
1008.2.a.n 2 44.c even 2 1
1008.2.a.n 2 132.d odd 2 1
1575.2.a.q 2 55.d odd 2 1
1575.2.a.q 2 165.d even 2 1
1575.2.d.i 4 55.e even 4 2
1575.2.d.i 4 165.l odd 4 2
4032.2.a.bq 2 88.g even 2 1
4032.2.a.bq 2 264.p odd 2 1
4032.2.a.bt 2 88.b odd 2 1
4032.2.a.bt 2 264.m even 2 1
7056.2.a.cm 2 308.g odd 2 1
7056.2.a.cm 2 924.n even 2 1
7623.2.a.bi 2 1.a even 1 1 trivial
7623.2.a.bi 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{2} - 3$$ $$T_{5}^{2} - 12$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-12 + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$-12 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$-108 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-48 + T^{2}$$
$59$ $$-48 + T^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$-108 + T^{2}$$
$73$ $$( 14 + T )^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$-12 + T^{2}$$
$97$ $$( -14 + T )^{2}$$