# Properties

 Label 7623.2.a.by 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + 3 \beta q^{4} - \beta q^{5} - q^{7} + (4 \beta + 1) q^{8} +O(q^{10})$$ q + (b + 1) * q^2 + 3*b * q^4 - b * q^5 - q^7 + (4*b + 1) * q^8 $$q + (\beta + 1) q^{2} + 3 \beta q^{4} - \beta q^{5} - q^{7} + (4 \beta + 1) q^{8} + ( - 2 \beta - 1) q^{10} + ( - 4 \beta + 1) q^{13} + ( - \beta - 1) q^{14} + (3 \beta + 5) q^{16} - 3 q^{17} - 4 q^{19} + ( - 3 \beta - 3) q^{20} + (2 \beta + 4) q^{23} + (\beta - 4) q^{25} + ( - 7 \beta - 3) q^{26} - 3 \beta q^{28} + 3 q^{29} + (2 \beta - 3) q^{31} + (3 \beta + 6) q^{32} + ( - 3 \beta - 3) q^{34} + \beta q^{35} + ( - 4 \beta - 2) q^{37} + ( - 4 \beta - 4) q^{38} + ( - 5 \beta - 4) q^{40} + ( - 5 \beta + 5) q^{41} + 9 q^{43} + (8 \beta + 6) q^{46} + ( - 5 \beta - 2) q^{47} + q^{49} + ( - 2 \beta - 3) q^{50} + ( - 9 \beta - 12) q^{52} + ( - 5 \beta + 4) q^{53} + ( - 4 \beta - 1) q^{56} + (3 \beta + 3) q^{58} + ( - \beta - 11) q^{59} + (6 \beta - 3) q^{61} + (\beta - 1) q^{62} + (6 \beta - 1) q^{64} + (3 \beta + 4) q^{65} + ( - 6 \beta + 6) q^{67} - 9 \beta q^{68} + (2 \beta + 1) q^{70} + ( - 2 \beta - 9) q^{71} + (3 \beta - 1) q^{73} + ( - 10 \beta - 6) q^{74} - 12 \beta q^{76} + ( - 3 \beta - 2) q^{79} + ( - 8 \beta - 3) q^{80} - 5 \beta q^{82} + (8 \beta - 10) q^{83} + 3 \beta q^{85} + (9 \beta + 9) q^{86} + (7 \beta - 14) q^{89} + (4 \beta - 1) q^{91} + (18 \beta + 6) q^{92} + ( - 12 \beta - 7) q^{94} + 4 \beta q^{95} + ( - 12 \beta + 3) q^{97} + (\beta + 1) q^{98} +O(q^{100})$$ q + (b + 1) * q^2 + 3*b * q^4 - b * q^5 - q^7 + (4*b + 1) * q^8 + (-2*b - 1) * q^10 + (-4*b + 1) * q^13 + (-b - 1) * q^14 + (3*b + 5) * q^16 - 3 * q^17 - 4 * q^19 + (-3*b - 3) * q^20 + (2*b + 4) * q^23 + (b - 4) * q^25 + (-7*b - 3) * q^26 - 3*b * q^28 + 3 * q^29 + (2*b - 3) * q^31 + (3*b + 6) * q^32 + (-3*b - 3) * q^34 + b * q^35 + (-4*b - 2) * q^37 + (-4*b - 4) * q^38 + (-5*b - 4) * q^40 + (-5*b + 5) * q^41 + 9 * q^43 + (8*b + 6) * q^46 + (-5*b - 2) * q^47 + q^49 + (-2*b - 3) * q^50 + (-9*b - 12) * q^52 + (-5*b + 4) * q^53 + (-4*b - 1) * q^56 + (3*b + 3) * q^58 + (-b - 11) * q^59 + (6*b - 3) * q^61 + (b - 1) * q^62 + (6*b - 1) * q^64 + (3*b + 4) * q^65 + (-6*b + 6) * q^67 - 9*b * q^68 + (2*b + 1) * q^70 + (-2*b - 9) * q^71 + (3*b - 1) * q^73 + (-10*b - 6) * q^74 - 12*b * q^76 + (-3*b - 2) * q^79 + (-8*b - 3) * q^80 - 5*b * q^82 + (8*b - 10) * q^83 + 3*b * q^85 + (9*b + 9) * q^86 + (7*b - 14) * q^89 + (4*b - 1) * q^91 + (18*b + 6) * q^92 + (-12*b - 7) * q^94 + 4*b * q^95 + (-12*b + 3) * q^97 + (b + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 3 q^{4} - q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 3 * q^4 - q^5 - 2 * q^7 + 6 * q^8 $$2 q + 3 q^{2} + 3 q^{4} - q^{5} - 2 q^{7} + 6 q^{8} - 4 q^{10} - 2 q^{13} - 3 q^{14} + 13 q^{16} - 6 q^{17} - 8 q^{19} - 9 q^{20} + 10 q^{23} - 7 q^{25} - 13 q^{26} - 3 q^{28} + 6 q^{29} - 4 q^{31} + 15 q^{32} - 9 q^{34} + q^{35} - 8 q^{37} - 12 q^{38} - 13 q^{40} + 5 q^{41} + 18 q^{43} + 20 q^{46} - 9 q^{47} + 2 q^{49} - 8 q^{50} - 33 q^{52} + 3 q^{53} - 6 q^{56} + 9 q^{58} - 23 q^{59} - q^{62} + 4 q^{64} + 11 q^{65} + 6 q^{67} - 9 q^{68} + 4 q^{70} - 20 q^{71} + q^{73} - 22 q^{74} - 12 q^{76} - 7 q^{79} - 14 q^{80} - 5 q^{82} - 12 q^{83} + 3 q^{85} + 27 q^{86} - 21 q^{89} + 2 q^{91} + 30 q^{92} - 26 q^{94} + 4 q^{95} - 6 q^{97} + 3 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 + 3 * q^4 - q^5 - 2 * q^7 + 6 * q^8 - 4 * q^10 - 2 * q^13 - 3 * q^14 + 13 * q^16 - 6 * q^17 - 8 * q^19 - 9 * q^20 + 10 * q^23 - 7 * q^25 - 13 * q^26 - 3 * q^28 + 6 * q^29 - 4 * q^31 + 15 * q^32 - 9 * q^34 + q^35 - 8 * q^37 - 12 * q^38 - 13 * q^40 + 5 * q^41 + 18 * q^43 + 20 * q^46 - 9 * q^47 + 2 * q^49 - 8 * q^50 - 33 * q^52 + 3 * q^53 - 6 * q^56 + 9 * q^58 - 23 * q^59 - q^62 + 4 * q^64 + 11 * q^65 + 6 * q^67 - 9 * q^68 + 4 * q^70 - 20 * q^71 + q^73 - 22 * q^74 - 12 * q^76 - 7 * q^79 - 14 * q^80 - 5 * q^82 - 12 * q^83 + 3 * q^85 + 27 * q^86 - 21 * q^89 + 2 * q^91 + 30 * q^92 - 26 * q^94 + 4 * q^95 - 6 * 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
 −0.618034 1.61803
0.381966 0 −1.85410 0.618034 0 −1.00000 −1.47214 0 0.236068
1.2 2.61803 0 4.85410 −1.61803 0 −1.00000 7.47214 0 −4.23607
 $$n$$: 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.by 2
3.b odd 2 1 2541.2.a.m 2
11.b odd 2 1 7623.2.a.u 2
11.d odd 10 2 693.2.m.e 4
33.d even 2 1 2541.2.a.bf 2
33.f even 10 2 231.2.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.a 4 33.f even 10 2
693.2.m.e 4 11.d odd 10 2
2541.2.a.m 2 3.b odd 2 1
2541.2.a.bf 2 33.d even 2 1
7623.2.a.u 2 11.b odd 2 1
7623.2.a.by 2 1.a even 1 1 trivial

## 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 $$T_{5}^{2} + T_{5} - 1$$ T5^2 + T5 - 1 $$T_{13}^{2} + 2T_{13} - 19$$ T13^2 + 2*T13 - 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T - 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 2T - 19$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} - 10T + 20$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} + 4T - 1$$
$37$ $$T^{2} + 8T - 4$$
$41$ $$T^{2} - 5T - 25$$
$43$ $$(T - 9)^{2}$$
$47$ $$T^{2} + 9T - 11$$
$53$ $$T^{2} - 3T - 29$$
$59$ $$T^{2} + 23T + 131$$
$61$ $$T^{2} - 45$$
$67$ $$T^{2} - 6T - 36$$
$71$ $$T^{2} + 20T + 95$$
$73$ $$T^{2} - T - 11$$
$79$ $$T^{2} + 7T + 1$$
$83$ $$T^{2} + 12T - 44$$
$89$ $$T^{2} + 21T + 49$$
$97$ $$T^{2} + 6T - 171$$