# Properties

 Label 7623.2.a.x 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

# Learn more about

## 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: $$\Q(\sqrt{3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2541) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( 2 - 2 \beta ) q^{4} + ( 2 - \beta ) q^{5} - q^{7} + ( -6 + 2 \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( 2 - 2 \beta ) q^{4} + ( 2 - \beta ) q^{5} - q^{7} + ( -6 + 2 \beta ) q^{8} + ( -5 + 3 \beta ) q^{10} + ( 1 + \beta ) q^{13} + ( 1 - \beta ) q^{14} + ( 8 - 4 \beta ) q^{16} + ( 2 + \beta ) q^{17} + ( 3 - 3 \beta ) q^{19} + ( 10 - 6 \beta ) q^{20} + ( -5 + \beta ) q^{23} + ( 2 - 4 \beta ) q^{25} + 2 q^{26} + ( -2 + 2 \beta ) q^{28} + ( -3 + \beta ) q^{29} + ( 4 + 2 \beta ) q^{31} + ( -8 + 8 \beta ) q^{32} + ( 1 + \beta ) q^{34} + ( -2 + \beta ) q^{35} + ( -6 - 2 \beta ) q^{37} + ( -12 + 6 \beta ) q^{38} + ( -18 + 10 \beta ) q^{40} -4 q^{41} + ( -3 + 2 \beta ) q^{43} + ( 8 - 6 \beta ) q^{46} + ( 4 + 3 \beta ) q^{47} + q^{49} + ( -14 + 6 \beta ) q^{50} -4 q^{52} + ( -6 + 2 \beta ) q^{53} + ( 6 - 2 \beta ) q^{56} + ( 6 - 4 \beta ) q^{58} + ( -2 + 7 \beta ) q^{59} + ( 3 + 3 \beta ) q^{61} + ( 2 + 2 \beta ) q^{62} + ( 16 - 8 \beta ) q^{64} + ( -1 + \beta ) q^{65} -7 q^{67} + ( -2 - 2 \beta ) q^{68} + ( 5 - 3 \beta ) q^{70} + ( 3 - 7 \beta ) q^{71} + ( -5 - \beta ) q^{73} -4 \beta q^{74} + ( 24 - 12 \beta ) q^{76} + ( -4 + 2 \beta ) q^{79} + ( 28 - 16 \beta ) q^{80} + ( 4 - 4 \beta ) q^{82} + ( -6 + \beta ) q^{83} + q^{85} + ( 9 - 5 \beta ) q^{86} + ( 6 + 5 \beta ) q^{89} + ( -1 - \beta ) q^{91} + ( -16 + 12 \beta ) q^{92} + ( 5 + \beta ) q^{94} + ( 15 - 9 \beta ) q^{95} + ( 5 - \beta ) q^{97} + ( -1 + \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 4q^{4} + 4q^{5} - 2q^{7} - 12q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 4q^{4} + 4q^{5} - 2q^{7} - 12q^{8} - 10q^{10} + 2q^{13} + 2q^{14} + 16q^{16} + 4q^{17} + 6q^{19} + 20q^{20} - 10q^{23} + 4q^{25} + 4q^{26} - 4q^{28} - 6q^{29} + 8q^{31} - 16q^{32} + 2q^{34} - 4q^{35} - 12q^{37} - 24q^{38} - 36q^{40} - 8q^{41} - 6q^{43} + 16q^{46} + 8q^{47} + 2q^{49} - 28q^{50} - 8q^{52} - 12q^{53} + 12q^{56} + 12q^{58} - 4q^{59} + 6q^{61} + 4q^{62} + 32q^{64} - 2q^{65} - 14q^{67} - 4q^{68} + 10q^{70} + 6q^{71} - 10q^{73} + 48q^{76} - 8q^{79} + 56q^{80} + 8q^{82} - 12q^{83} + 2q^{85} + 18q^{86} + 12q^{89} - 2q^{91} - 32q^{92} + 10q^{94} + 30q^{95} + 10q^{97} - 2q^{98} + O(q^{100})$$

## 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
 −1.73205 1.73205
−2.73205 0 5.46410 3.73205 0 −1.00000 −9.46410 0 −10.1962
1.2 0.732051 0 −1.46410 0.267949 0 −1.00000 −2.53590 0 0.196152
 $$n$$: 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.x 2
3.b odd 2 1 2541.2.a.be yes 2
11.b odd 2 1 7623.2.a.bv 2
33.d even 2 1 2541.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.2.a.o 2 33.d even 2 1
2541.2.a.be yes 2 3.b odd 2 1
7623.2.a.x 2 1.a even 1 1 trivial
7623.2.a.bv 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4}$$
$3$ 
$5$ $$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ $$( 1 + T )^{2}$$
$11$ 
$13$ $$1 - 2 T + 24 T^{2} - 26 T^{3} + 169 T^{4}$$
$17$ $$1 - 4 T + 35 T^{2} - 68 T^{3} + 289 T^{4}$$
$19$ $$1 - 6 T + 20 T^{2} - 114 T^{3} + 361 T^{4}$$
$23$ $$1 + 10 T + 68 T^{2} + 230 T^{3} + 529 T^{4}$$
$29$ $$1 + 6 T + 64 T^{2} + 174 T^{3} + 841 T^{4}$$
$31$ $$1 - 8 T + 66 T^{2} - 248 T^{3} + 961 T^{4}$$
$37$ $$1 + 12 T + 98 T^{2} + 444 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 4 T + 41 T^{2} )^{2}$$
$43$ $$1 + 6 T + 83 T^{2} + 258 T^{3} + 1849 T^{4}$$
$47$ $$1 - 8 T + 83 T^{2} - 376 T^{3} + 2209 T^{4}$$
$53$ $$1 + 12 T + 130 T^{2} + 636 T^{3} + 2809 T^{4}$$
$59$ $$1 + 4 T - 25 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 6 T + 104 T^{2} - 366 T^{3} + 3721 T^{4}$$
$67$ $$( 1 + 7 T + 67 T^{2} )^{2}$$
$71$ $$1 - 6 T + 4 T^{2} - 426 T^{3} + 5041 T^{4}$$
$73$ $$1 + 10 T + 168 T^{2} + 730 T^{3} + 5329 T^{4}$$
$79$ $$1 + 8 T + 162 T^{2} + 632 T^{3} + 6241 T^{4}$$
$83$ $$1 + 12 T + 199 T^{2} + 996 T^{3} + 6889 T^{4}$$
$89$ $$1 - 12 T + 139 T^{2} - 1068 T^{3} + 7921 T^{4}$$
$97$ $$1 - 10 T + 216 T^{2} - 970 T^{3} + 9409 T^{4}$$
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