# Properties

 Label 7623.2.a.bl 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−2.23607 0 3.00000 2.00000 0 −1.00000 −2.23607 0 −4.47214
1.2 2.23607 0 3.00000 2.00000 0 −1.00000 2.23607 0 4.47214
 $$n$$: 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.bl 2
3.b odd 2 1 847.2.a.f 2
11.b odd 2 1 693.2.a.h 2
21.c even 2 1 5929.2.a.m 2
33.d even 2 1 77.2.a.d 2
33.f even 10 2 847.2.f.a 4
33.f even 10 2 847.2.f.n 4
33.h odd 10 2 847.2.f.b 4
33.h odd 10 2 847.2.f.m 4
77.b even 2 1 4851.2.a.y 2
132.d odd 2 1 1232.2.a.m 2
165.d even 2 1 1925.2.a.r 2
165.l odd 4 2 1925.2.b.h 4
231.h odd 2 1 539.2.a.f 2
231.k odd 6 2 539.2.e.j 4
231.l even 6 2 539.2.e.i 4
264.m even 2 1 4928.2.a.bm 2
264.p odd 2 1 4928.2.a.bv 2
924.n even 2 1 8624.2.a.ce 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 33.d even 2 1
539.2.a.f 2 231.h odd 2 1
539.2.e.i 4 231.l even 6 2
539.2.e.j 4 231.k odd 6 2
693.2.a.h 2 11.b odd 2 1
847.2.a.f 2 3.b odd 2 1
847.2.f.a 4 33.f even 10 2
847.2.f.b 4 33.h odd 10 2
847.2.f.m 4 33.h odd 10 2
847.2.f.n 4 33.f even 10 2
1232.2.a.m 2 132.d odd 2 1
1925.2.a.r 2 165.d even 2 1
1925.2.b.h 4 165.l odd 4 2
4851.2.a.y 2 77.b even 2 1
4928.2.a.bm 2 264.m even 2 1
4928.2.a.bv 2 264.p odd 2 1
5929.2.a.m 2 21.c even 2 1
7623.2.a.bl 2 1.a even 1 1 trivial
8624.2.a.ce 2 924.n even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{5} - 2$$ T5 - 2 $$T_{13}^{2} + 2T_{13} - 4$$ T13^2 + 2*T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5$$
$3$ $$T^{2}$$
$5$ $$(T - 2)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 2T - 4$$
$17$ $$T^{2} + 2T - 4$$
$19$ $$T^{2} + 4T - 16$$
$23$ $$T^{2} - 4T - 16$$
$29$ $$T^{2} - 8T - 4$$
$31$ $$T^{2} + 10T + 20$$
$37$ $$T^{2} + 8T - 4$$
$41$ $$T^{2} + 18T + 76$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} + 10T + 20$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} + 2T - 4$$
$61$ $$T^{2} - 10T + 20$$
$67$ $$T^{2} - 20T + 80$$
$71$ $$T^{2} - 12T + 16$$
$73$ $$T^{2} - 6T + 4$$
$79$ $$T^{2} - 80$$
$83$ $$T^{2} - 4T - 176$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} - 8T - 164$$