# Properties

 Label 7623.2.a.w 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: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

## 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
−1.00000 0 −1.00000 −3.85410 0 −1.00000 3.00000 0 3.85410
1.2 −1.00000 0 −1.00000 2.85410 0 −1.00000 3.00000 0 −2.85410
 $$n$$: 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.w 2
3.b odd 2 1 2541.2.a.bd 2
11.b odd 2 1 7623.2.a.bu 2
11.d odd 10 2 693.2.m.c 4
33.d even 2 1 2541.2.a.n 2
33.f even 10 2 231.2.j.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.c 4 33.f even 10 2
693.2.m.c 4 11.d odd 10 2
2541.2.a.n 2 33.d even 2 1
2541.2.a.bd 2 3.b odd 2 1
7623.2.a.w 2 1.a even 1 1 trivial
7623.2.a.bu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2} + 1$$ $$T_{5}^{2} + T_{5} - 11$$ $$T_{13}^{2} + 2 T_{13} - 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + 2 T^{2} )^{2}$$
$3$ 
$5$ $$1 + T - T^{2} + 5 T^{3} + 25 T^{4}$$
$7$ $$( 1 + T )^{2}$$
$11$ 
$13$ $$1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}$$
$17$ $$1 - 9 T + 43 T^{2} - 153 T^{3} + 289 T^{4}$$
$19$ $$1 + 3 T + 39 T^{2} + 57 T^{3} + 361 T^{4}$$
$23$ $$1 + 5 T + 21 T^{2} + 115 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 2 T + 29 T^{2} )^{2}$$
$31$ $$1 - T + T^{2} - 31 T^{3} + 961 T^{4}$$
$37$ $$1 + 13 T + 85 T^{2} + 481 T^{3} + 1369 T^{4}$$
$41$ $$1 + 15 T + 127 T^{2} + 615 T^{3} + 1681 T^{4}$$
$43$ $$1 + 2 T + 82 T^{2} + 86 T^{3} + 1849 T^{4}$$
$47$ $$( 1 + 2 T + 47 T^{2} )^{2}$$
$53$ $$1 + 2 T - 18 T^{2} + 106 T^{3} + 2809 T^{4}$$
$59$ $$1 + 18 T + 194 T^{2} + 1062 T^{3} + 3481 T^{4}$$
$61$ $$1 + 42 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 8 T + 67 T^{2} )^{2}$$
$71$ $$1 + 62 T^{2} + 5041 T^{4}$$
$73$ $$1 - 6 T + 150 T^{2} - 438 T^{3} + 5329 T^{4}$$
$79$ $$( 1 - 14 T + 79 T^{2} )^{2}$$
$83$ $$1 + 12 T + 122 T^{2} + 996 T^{3} + 6889 T^{4}$$
$89$ $$1 - 9 T + 167 T^{2} - 801 T^{3} + 7921 T^{4}$$
$97$ $$1 + 16 T + 238 T^{2} + 1552 T^{3} + 9409 T^{4}$$