# Properties

 Label 7623.2.a.v Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $0$ 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: $$0$$ 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: $$1$$ Twist minimal: no (minimal twist has level 693) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + 3 \beta q^{4} + q^{5} - q^{7} + ( - 4 \beta - 1) q^{8}+O(q^{10})$$ q + (-b - 1) * q^2 + 3*b * q^4 + q^5 - q^7 + (-4*b - 1) * q^8 $$q + ( - \beta - 1) q^{2} + 3 \beta q^{4} + q^{5} - q^{7} + ( - 4 \beta - 1) q^{8} + ( - \beta - 1) q^{10} + (2 \beta + 1) q^{13} + (\beta + 1) q^{14} + (3 \beta + 5) q^{16} + (4 \beta - 4) q^{17} + (4 \beta - 1) q^{19} + 3 \beta q^{20} + ( - 4 \beta + 6) q^{23} - 4 q^{25} + ( - 5 \beta - 3) q^{26} - 3 \beta q^{28} + ( - 2 \beta - 3) q^{29} + (4 \beta + 2) q^{31} + ( - 3 \beta - 6) q^{32} - 4 \beta q^{34} - q^{35} + (4 \beta - 3) q^{37} + ( - 7 \beta - 3) q^{38} + ( - 4 \beta - 1) q^{40} + (4 \beta - 2) q^{41} + 4 q^{43} + (2 \beta - 2) q^{46} + (2 \beta + 1) q^{47} + q^{49} + (4 \beta + 4) q^{50} + (9 \beta + 6) q^{52} + (4 \beta + 8) q^{53} + (4 \beta + 1) q^{56} + (7 \beta + 5) q^{58} + ( - 10 \beta + 9) q^{59} + (4 \beta - 6) q^{61} + ( - 10 \beta - 6) q^{62} + (6 \beta - 1) q^{64} + (2 \beta + 1) q^{65} + ( - 10 \beta + 3) q^{67} + 12 q^{68} + (\beta + 1) q^{70} + (4 \beta - 2) q^{71} + (2 \beta + 11) q^{73} + ( - 5 \beta - 1) q^{74} + (9 \beta + 12) q^{76} + (4 \beta - 2) q^{79} + (3 \beta + 5) q^{80} + ( - 6 \beta - 2) q^{82} + ( - 12 \beta + 4) q^{83} + (4 \beta - 4) q^{85} + ( - 4 \beta - 4) q^{86} - 10 q^{89} + ( - 2 \beta - 1) q^{91} + (6 \beta - 12) q^{92} + ( - 5 \beta - 3) q^{94} + (4 \beta - 1) q^{95} - 8 q^{97} + ( - \beta - 1) q^{98} +O(q^{100})$$ q + (-b - 1) * q^2 + 3*b * q^4 + q^5 - q^7 + (-4*b - 1) * q^8 + (-b - 1) * q^10 + (2*b + 1) * q^13 + (b + 1) * q^14 + (3*b + 5) * q^16 + (4*b - 4) * q^17 + (4*b - 1) * q^19 + 3*b * q^20 + (-4*b + 6) * q^23 - 4 * q^25 + (-5*b - 3) * q^26 - 3*b * q^28 + (-2*b - 3) * q^29 + (4*b + 2) * q^31 + (-3*b - 6) * q^32 - 4*b * q^34 - q^35 + (4*b - 3) * q^37 + (-7*b - 3) * q^38 + (-4*b - 1) * q^40 + (4*b - 2) * q^41 + 4 * q^43 + (2*b - 2) * q^46 + (2*b + 1) * q^47 + q^49 + (4*b + 4) * q^50 + (9*b + 6) * q^52 + (4*b + 8) * q^53 + (4*b + 1) * q^56 + (7*b + 5) * q^58 + (-10*b + 9) * q^59 + (4*b - 6) * q^61 + (-10*b - 6) * q^62 + (6*b - 1) * q^64 + (2*b + 1) * q^65 + (-10*b + 3) * q^67 + 12 * q^68 + (b + 1) * q^70 + (4*b - 2) * q^71 + (2*b + 11) * q^73 + (-5*b - 1) * q^74 + (9*b + 12) * q^76 + (4*b - 2) * q^79 + (3*b + 5) * q^80 + (-6*b - 2) * q^82 + (-12*b + 4) * q^83 + (4*b - 4) * q^85 + (-4*b - 4) * q^86 - 10 * q^89 + (-2*b - 1) * q^91 + (6*b - 12) * q^92 + (-5*b - 3) * q^94 + (4*b - 1) * q^95 - 8 * q^97 + (-b - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 3 q^{4} + 2 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - 3 * q^2 + 3 * q^4 + 2 * q^5 - 2 * q^7 - 6 * q^8 $$2 q - 3 q^{2} + 3 q^{4} + 2 q^{5} - 2 q^{7} - 6 q^{8} - 3 q^{10} + 4 q^{13} + 3 q^{14} + 13 q^{16} - 4 q^{17} + 2 q^{19} + 3 q^{20} + 8 q^{23} - 8 q^{25} - 11 q^{26} - 3 q^{28} - 8 q^{29} + 8 q^{31} - 15 q^{32} - 4 q^{34} - 2 q^{35} - 2 q^{37} - 13 q^{38} - 6 q^{40} + 8 q^{43} - 2 q^{46} + 4 q^{47} + 2 q^{49} + 12 q^{50} + 21 q^{52} + 20 q^{53} + 6 q^{56} + 17 q^{58} + 8 q^{59} - 8 q^{61} - 22 q^{62} + 4 q^{64} + 4 q^{65} - 4 q^{67} + 24 q^{68} + 3 q^{70} + 24 q^{73} - 7 q^{74} + 33 q^{76} + 13 q^{80} - 10 q^{82} - 4 q^{83} - 4 q^{85} - 12 q^{86} - 20 q^{89} - 4 q^{91} - 18 q^{92} - 11 q^{94} + 2 q^{95} - 16 q^{97} - 3 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 + 3 * q^4 + 2 * q^5 - 2 * q^7 - 6 * q^8 - 3 * q^10 + 4 * q^13 + 3 * q^14 + 13 * q^16 - 4 * q^17 + 2 * q^19 + 3 * q^20 + 8 * q^23 - 8 * q^25 - 11 * q^26 - 3 * q^28 - 8 * q^29 + 8 * q^31 - 15 * q^32 - 4 * q^34 - 2 * q^35 - 2 * q^37 - 13 * q^38 - 6 * q^40 + 8 * q^43 - 2 * q^46 + 4 * q^47 + 2 * q^49 + 12 * q^50 + 21 * q^52 + 20 * q^53 + 6 * q^56 + 17 * q^58 + 8 * q^59 - 8 * q^61 - 22 * q^62 + 4 * q^64 + 4 * q^65 - 4 * q^67 + 24 * q^68 + 3 * q^70 + 24 * q^73 - 7 * q^74 + 33 * q^76 + 13 * q^80 - 10 * q^82 - 4 * q^83 - 4 * q^85 - 12 * q^86 - 20 * q^89 - 4 * q^91 - 18 * q^92 - 11 * q^94 + 2 * q^95 - 16 * q^97 - 3 * q^98

## 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
−2.61803 0 4.85410 1.00000 0 −1.00000 −7.47214 0 −2.61803
1.2 −0.381966 0 −1.85410 1.00000 0 −1.00000 1.47214 0 −0.381966
 $$n$$: 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.v 2
3.b odd 2 1 7623.2.a.bw 2
11.b odd 2 1 693.2.a.k yes 2
33.d even 2 1 693.2.a.e 2
77.b even 2 1 4851.2.a.bf 2
231.h odd 2 1 4851.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.a.e 2 33.d even 2 1
693.2.a.k yes 2 11.b odd 2 1
4851.2.a.u 2 231.h odd 2 1
4851.2.a.bf 2 77.b even 2 1
7623.2.a.v 2 1.a even 1 1 trivial
7623.2.a.bw 2 3.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 3T_{2} + 1$$ T2^2 + 3*T2 + 1 $$T_{5} - 1$$ T5 - 1 $$T_{13}^{2} - 4T_{13} - 1$$ T13^2 - 4*T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 1$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T - 1$$
$17$ $$T^{2} + 4T - 16$$
$19$ $$T^{2} - 2T - 19$$
$23$ $$T^{2} - 8T - 4$$
$29$ $$T^{2} + 8T + 11$$
$31$ $$T^{2} - 8T - 4$$
$37$ $$T^{2} + 2T - 19$$
$41$ $$T^{2} - 20$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 4T - 1$$
$53$ $$T^{2} - 20T + 80$$
$59$ $$T^{2} - 8T - 109$$
$61$ $$T^{2} + 8T - 4$$
$67$ $$T^{2} + 4T - 121$$
$71$ $$T^{2} - 20$$
$73$ $$T^{2} - 24T + 139$$
$79$ $$T^{2} - 20$$
$83$ $$T^{2} + 4T - 176$$
$89$ $$(T + 10)^{2}$$
$97$ $$(T + 8)^{2}$$