# Properties

 Label 7623.2.a.bz Level 7623 Weight 2 Character orbit 7623.a Self dual yes Analytic conductor 60.870 Analytic rank 1 Dimension 3 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: $$3$$ Coefficient field: 3.3.568.1 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 3 + \beta_{2} ) q^{4} + \beta_{2} q^{5} + q^{7} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 3 + \beta_{2} ) q^{4} + \beta_{2} q^{5} + q^{7} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{8} + ( -2 - 2 \beta_{2} ) q^{10} + ( 2 + \beta_{1} - \beta_{2} ) q^{13} + ( -1 + \beta_{1} ) q^{14} + ( 5 - 2 \beta_{1} + 3 \beta_{2} ) q^{16} + ( -2 - \beta_{1} + \beta_{2} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{19} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{20} + ( -2 + \beta_{2} ) q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{25} + ( 4 + 3 \beta_{1} + 3 \beta_{2} ) q^{26} + ( 3 + \beta_{2} ) q^{28} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -4 - \beta_{1} ) q^{31} + ( -13 + \beta_{1} - 4 \beta_{2} ) q^{32} + ( -4 - 3 \beta_{1} - 3 \beta_{2} ) q^{34} + \beta_{2} q^{35} + ( -6 + 2 \beta_{1} + \beta_{2} ) q^{37} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{38} + ( -14 + 4 \beta_{1} - 2 \beta_{2} ) q^{40} + ( -6 + \beta_{1} - \beta_{2} ) q^{41} + ( 2 + 2 \beta_{2} ) q^{43} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{46} + ( -2 + \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( -7 - \beta_{1} ) q^{50} + ( -2 + 5 \beta_{1} - \beta_{2} ) q^{52} + ( 4 + 2 \beta_{2} ) q^{53} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{56} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{58} -\beta_{1} q^{59} + ( -6 + \beta_{1} - \beta_{2} ) q^{61} + ( -5 \beta_{1} - \beta_{2} ) q^{62} + ( 15 - 8 \beta_{1} + 3 \beta_{2} ) q^{64} + ( -8 + 2 \beta_{1} + 2 \beta_{2} ) q^{65} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{67} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{68} + ( -2 - 2 \beta_{2} ) q^{70} + ( -2 - 4 \beta_{1} - 5 \beta_{2} ) q^{71} + ( 6 - \beta_{1} + \beta_{2} ) q^{73} + ( 12 - 4 \beta_{1} ) q^{74} + ( -8 - 2 \beta_{1} - 2 \beta_{2} ) q^{76} + ( 10 + 2 \beta_{2} ) q^{79} + ( 22 - 6 \beta_{1} + 4 \beta_{2} ) q^{80} + ( 12 - 5 \beta_{1} + 3 \beta_{2} ) q^{82} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{83} + ( 8 - 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{86} + ( 8 - 2 \beta_{1} + \beta_{2} ) q^{89} + ( 2 + \beta_{1} - \beta_{2} ) q^{91} -2 \beta_{1} q^{92} + ( 8 - \beta_{1} + 3 \beta_{2} ) q^{94} + ( -8 + 4 \beta_{1} + 4 \beta_{2} ) q^{95} + ( -4 - 2 \beta_{1} - 3 \beta_{2} ) q^{97} + ( -1 + \beta_{1} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 2q^{2} + 8q^{4} - q^{5} + 3q^{7} - 6q^{8} + O(q^{10})$$ $$3q - 2q^{2} + 8q^{4} - q^{5} + 3q^{7} - 6q^{8} - 4q^{10} + 8q^{13} - 2q^{14} + 10q^{16} - 8q^{17} + 14q^{20} - 7q^{23} + 2q^{25} + 12q^{26} + 8q^{28} - 13q^{31} - 34q^{32} - 12q^{34} - q^{35} - 17q^{37} - 16q^{38} - 36q^{40} - 16q^{41} + 4q^{43} - 4q^{47} + 3q^{49} - 22q^{50} + 10q^{53} - 6q^{56} - 16q^{58} - q^{59} - 16q^{61} - 4q^{62} + 34q^{64} - 24q^{65} - 3q^{67} - 4q^{70} - 5q^{71} + 16q^{73} + 32q^{74} - 24q^{76} + 28q^{79} + 56q^{80} + 28q^{82} - 8q^{83} + 24q^{85} - 12q^{86} + 21q^{89} + 8q^{91} - 2q^{92} + 20q^{94} - 24q^{95} - 11q^{97} - 2q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

## 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
 −1.76156 −0.363328 3.12489
−2.76156 0 5.62620 2.62620 0 1.00000 −10.0140 0 −7.25240
1.2 −1.36333 0 −0.141336 −3.14134 0 1.00000 2.91934 0 4.28267
1.3 2.12489 0 2.51514 −0.484862 0 1.00000 1.09461 0 −1.03028
 $$n$$: 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.bz 3
3.b odd 2 1 847.2.a.j yes 3
11.b odd 2 1 7623.2.a.ce 3
21.c even 2 1 5929.2.a.y 3
33.d even 2 1 847.2.a.i 3
33.f even 10 4 847.2.f.u 12
33.h odd 10 4 847.2.f.t 12
231.h odd 2 1 5929.2.a.t 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.i 3 33.d even 2 1
847.2.a.j yes 3 3.b odd 2 1
847.2.f.t 12 33.h odd 10 4
847.2.f.u 12 33.f even 10 4
5929.2.a.t 3 231.h odd 2 1
5929.2.a.y 3 21.c even 2 1
7623.2.a.bz 3 1.a even 1 1 trivial
7623.2.a.ce 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{3} + 2 T_{2}^{2} - 5 T_{2} - 8$$ $$T_{5}^{3} + T_{5}^{2} - 8 T_{5} - 4$$ $$T_{13}^{3} - 8 T_{13}^{2} + 2 T_{13} + 64$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + T^{2} + 2 T^{4} + 8 T^{5} + 8 T^{6}$$
$3$ 
$5$ $$1 + T + 7 T^{2} + 6 T^{3} + 35 T^{4} + 25 T^{5} + 125 T^{6}$$
$7$ $$( 1 - T )^{3}$$
$11$ 
$13$ $$1 - 8 T + 41 T^{2} - 144 T^{3} + 533 T^{4} - 1352 T^{5} + 2197 T^{6}$$
$17$ $$1 + 8 T + 53 T^{2} + 208 T^{3} + 901 T^{4} + 2312 T^{5} + 4913 T^{6}$$
$19$ $$1 + 17 T^{2} - 64 T^{3} + 323 T^{4} + 6859 T^{6}$$
$23$ $$1 + 7 T + 77 T^{2} + 314 T^{3} + 1771 T^{4} + 3703 T^{5} + 12167 T^{6}$$
$29$ $$1 + 47 T^{2} - 64 T^{3} + 1363 T^{4} + 24389 T^{6}$$
$31$ $$1 + 13 T + 143 T^{2} + 864 T^{3} + 4433 T^{4} + 12493 T^{5} + 29791 T^{6}$$
$37$ $$1 + 17 T + 183 T^{2} + 1274 T^{3} + 6771 T^{4} + 23273 T^{5} + 50653 T^{6}$$
$41$ $$1 + 16 T + 189 T^{2} + 1392 T^{3} + 7749 T^{4} + 26896 T^{5} + 68921 T^{6}$$
$43$ $$1 - 4 T + 101 T^{2} - 312 T^{3} + 4343 T^{4} - 7396 T^{5} + 79507 T^{6}$$
$47$ $$1 + 4 T + 127 T^{2} + 384 T^{3} + 5969 T^{4} + 8836 T^{5} + 103823 T^{6}$$
$53$ $$1 - 10 T + 159 T^{2} - 996 T^{3} + 8427 T^{4} - 28090 T^{5} + 148877 T^{6}$$
$59$ $$1 + T + 171 T^{2} + 120 T^{3} + 10089 T^{4} + 3481 T^{5} + 205379 T^{6}$$
$61$ $$1 + 16 T + 249 T^{2} + 2032 T^{3} + 15189 T^{4} + 59536 T^{5} + 226981 T^{6}$$
$67$ $$1 + 3 T + 113 T^{2} - 22 T^{3} + 7571 T^{4} + 13467 T^{5} + 300763 T^{6}$$
$71$ $$1 + 5 T + 5 T^{2} - 770 T^{3} + 355 T^{4} + 25205 T^{5} + 357911 T^{6}$$
$73$ $$1 - 16 T + 285 T^{2} - 2416 T^{3} + 20805 T^{4} - 85264 T^{5} + 389017 T^{6}$$
$79$ $$1 - 28 T + 465 T^{2} - 4936 T^{3} + 36735 T^{4} - 174748 T^{5} + 493039 T^{6}$$
$83$ $$1 + 8 T + 193 T^{2} + 1392 T^{3} + 16019 T^{4} + 55112 T^{5} + 571787 T^{6}$$
$89$ $$1 - 21 T + 371 T^{2} - 3838 T^{3} + 33019 T^{4} - 166341 T^{5} + 704969 T^{6}$$
$97$ $$1 + 11 T + 259 T^{2} + 1682 T^{3} + 25123 T^{4} + 103499 T^{5} + 912673 T^{6}$$