# Properties

 Label 7623.2.a.ba 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta - 1) q^{4} + (\beta + 1) q^{5} + q^{7} + (2 \beta - 1) q^{8} +O(q^{10})$$ q - b * q^2 + (b - 1) * q^4 + (b + 1) * q^5 + q^7 + (2*b - 1) * q^8 $$q - \beta q^{2} + (\beta - 1) q^{4} + (\beta + 1) q^{5} + q^{7} + (2 \beta - 1) q^{8} + ( - 2 \beta - 1) q^{10} - q^{13} - \beta q^{14} - 3 \beta q^{16} + ( - 2 \beta + 3) q^{17} + \beta q^{20} + ( - 2 \beta + 2) q^{23} + (3 \beta - 3) q^{25} + \beta q^{26} + (\beta - 1) q^{28} + (6 \beta - 3) q^{29} + (2 \beta - 9) q^{31} + ( - \beta + 5) q^{32} + ( - \beta + 2) q^{34} + (\beta + 1) q^{35} + ( - 8 \beta + 2) q^{37} + (3 \beta + 1) q^{40} + (5 \beta - 2) q^{41} - q^{43} + 2 q^{46} + ( - 7 \beta + 3) q^{47} + q^{49} - 3 q^{50} + ( - \beta + 1) q^{52} + (\beta - 7) q^{53} + (2 \beta - 1) q^{56} + ( - 3 \beta - 6) q^{58} + (3 \beta + 6) q^{59} + ( - 2 \beta - 7) q^{61} + (7 \beta - 2) q^{62} + (2 \beta + 1) q^{64} + ( - \beta - 1) q^{65} + (2 \beta - 8) q^{67} + (3 \beta - 5) q^{68} + ( - 2 \beta - 1) q^{70} + ( - 4 \beta + 5) q^{71} + ( - 9 \beta + 6) q^{73} + (6 \beta + 8) q^{74} + ( - 3 \beta + 9) q^{79} + ( - 6 \beta - 3) q^{80} + ( - 3 \beta - 5) q^{82} + 6 q^{83} + ( - \beta + 1) q^{85} + \beta q^{86} + (\beta + 7) q^{89} - q^{91} + (2 \beta - 4) q^{92} + (4 \beta + 7) q^{94} - 17 q^{97} - \beta q^{98} +O(q^{100})$$ q - b * q^2 + (b - 1) * q^4 + (b + 1) * q^5 + q^7 + (2*b - 1) * q^8 + (-2*b - 1) * q^10 - q^13 - b * q^14 - 3*b * q^16 + (-2*b + 3) * q^17 + b * q^20 + (-2*b + 2) * q^23 + (3*b - 3) * q^25 + b * q^26 + (b - 1) * q^28 + (6*b - 3) * q^29 + (2*b - 9) * q^31 + (-b + 5) * q^32 + (-b + 2) * q^34 + (b + 1) * q^35 + (-8*b + 2) * q^37 + (3*b + 1) * q^40 + (5*b - 2) * q^41 - q^43 + 2 * q^46 + (-7*b + 3) * q^47 + q^49 - 3 * q^50 + (-b + 1) * q^52 + (b - 7) * q^53 + (2*b - 1) * q^56 + (-3*b - 6) * q^58 + (3*b + 6) * q^59 + (-2*b - 7) * q^61 + (7*b - 2) * q^62 + (2*b + 1) * q^64 + (-b - 1) * q^65 + (2*b - 8) * q^67 + (3*b - 5) * q^68 + (-2*b - 1) * q^70 + (-4*b + 5) * q^71 + (-9*b + 6) * q^73 + (6*b + 8) * q^74 + (-3*b + 9) * q^79 + (-6*b - 3) * q^80 + (-3*b - 5) * q^82 + 6 * q^83 + (-b + 1) * q^85 + b * q^86 + (b + 7) * q^89 - q^91 + (2*b - 4) * q^92 + (4*b + 7) * q^94 - 17 * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 3 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q - q^2 - q^4 + 3 * q^5 + 2 * q^7 $$2 q - q^{2} - q^{4} + 3 q^{5} + 2 q^{7} - 4 q^{10} - 2 q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + q^{20} + 2 q^{23} - 3 q^{25} + q^{26} - q^{28} - 16 q^{31} + 9 q^{32} + 3 q^{34} + 3 q^{35} - 4 q^{37} + 5 q^{40} + q^{41} - 2 q^{43} + 4 q^{46} - q^{47} + 2 q^{49} - 6 q^{50} + q^{52} - 13 q^{53} - 15 q^{58} + 15 q^{59} - 16 q^{61} + 3 q^{62} + 4 q^{64} - 3 q^{65} - 14 q^{67} - 7 q^{68} - 4 q^{70} + 6 q^{71} + 3 q^{73} + 22 q^{74} + 15 q^{79} - 12 q^{80} - 13 q^{82} + 12 q^{83} + q^{85} + q^{86} + 15 q^{89} - 2 q^{91} - 6 q^{92} + 18 q^{94} - 34 q^{97} - q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 + 3 * q^5 + 2 * q^7 - 4 * q^10 - 2 * q^13 - q^14 - 3 * q^16 + 4 * q^17 + q^20 + 2 * q^23 - 3 * q^25 + q^26 - q^28 - 16 * q^31 + 9 * q^32 + 3 * q^34 + 3 * q^35 - 4 * q^37 + 5 * q^40 + q^41 - 2 * q^43 + 4 * q^46 - q^47 + 2 * q^49 - 6 * q^50 + q^52 - 13 * q^53 - 15 * q^58 + 15 * q^59 - 16 * q^61 + 3 * q^62 + 4 * q^64 - 3 * q^65 - 14 * q^67 - 7 * q^68 - 4 * q^70 + 6 * q^71 + 3 * q^73 + 22 * q^74 + 15 * q^79 - 12 * q^80 - 13 * q^82 + 12 * q^83 + q^85 + q^86 + 15 * q^89 - 2 * q^91 - 6 * q^92 + 18 * q^94 - 34 * q^97 - 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
−1.61803 0 0.618034 2.61803 0 1.00000 2.23607 0 −4.23607
1.2 0.618034 0 −1.61803 0.381966 0 1.00000 −2.23607 0 0.236068
 $$n$$: 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.ba 2
3.b odd 2 1 2541.2.a.bb 2
11.b odd 2 1 7623.2.a.bp 2
11.c even 5 2 693.2.m.b 4
33.d even 2 1 2541.2.a.s 2
33.h odd 10 2 231.2.j.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.d 4 33.h odd 10 2
693.2.m.b 4 11.c even 5 2
2541.2.a.s 2 33.d even 2 1
2541.2.a.bb 2 3.b odd 2 1
7623.2.a.ba 2 1.a even 1 1 trivial
7623.2.a.bp 2 11.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 1$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 4T - 1$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 2T - 4$$
$29$ $$T^{2} - 45$$
$31$ $$T^{2} + 16T + 59$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} - T - 31$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + T - 61$$
$53$ $$T^{2} + 13T + 41$$
$59$ $$T^{2} - 15T + 45$$
$61$ $$T^{2} + 16T + 59$$
$67$ $$T^{2} + 14T + 44$$
$71$ $$T^{2} - 6T - 11$$
$73$ $$T^{2} - 3T - 99$$
$79$ $$T^{2} - 15T + 45$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 15T + 55$$
$97$ $$(T + 17)^{2}$$