# Properties

 Label 7623.2.a.cd Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} + (\beta_1 + 2) q^{4} + (\beta_{2} + \beta_1 - 1) q^{5} + q^{7} + (2 \beta_1 + 1) q^{8}+O(q^{10})$$ q + (b2 + 1) * q^2 + (b1 + 2) * q^4 + (b2 + b1 - 1) * q^5 + q^7 + (2*b1 + 1) * q^8 $$q + (\beta_{2} + 1) q^{2} + (\beta_1 + 2) q^{4} + (\beta_{2} + \beta_1 - 1) q^{5} + q^{7} + (2 \beta_1 + 1) q^{8} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{10} + ( - \beta_{2} + 3 \beta_1 + 1) q^{13} + (\beta_{2} + 1) q^{14} + (\beta_{2} + 2 \beta_1 - 1) q^{16} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{17} + (\beta_{2} - \beta_1 + 3) q^{19} + (3 \beta_{2} + 2 \beta_1 + 2) q^{20} + ( - 2 \beta_{2} - 4) q^{23} + ( - 3 \beta_{2} + \beta_1 + 4) q^{25} + (2 \beta_{2} + 5 \beta_1 + 1) q^{26} + (\beta_1 + 2) q^{28} + (\beta_{2} - 3 \beta_1 - 1) q^{29} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{31} + ( - 2 \beta_{2} + \beta_1 + 2) q^{32} + (4 \beta_{2} + 6 \beta_1) q^{34} + (\beta_{2} + \beta_1 - 1) q^{35} + (3 \beta_{2} - \beta_1 + 1) q^{37} + (2 \beta_{2} - \beta_1 + 5) q^{38} + (3 \beta_{2} + \beta_1 + 7) q^{40} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{41} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{43} + ( - 2 \beta_{2} - 2 \beta_1 - 10) q^{46} + ( - 3 \beta_{2} - \beta_1 - 1) q^{47} + q^{49} + (7 \beta_{2} - \beta_1 - 4) q^{50} + (\beta_{2} + 6 \beta_1 + 10) q^{52} + 2 \beta_1 q^{53} + (2 \beta_1 + 1) q^{56} + ( - 2 \beta_{2} - 5 \beta_1 - 1) q^{58} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{59} + 2 q^{61} + (2 \beta_{2} - 12) q^{62} + (2 \beta_{2} - 4 \beta_1 - 1) q^{64} + (7 \beta_{2} - \beta_1 + 7) q^{65} + (\beta_{2} - 5 \beta_1 - 1) q^{67} + (8 \beta_1 + 14) q^{68} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{70} + ( - 4 \beta_1 + 4) q^{71} + (\beta_{2} - 3 \beta_1 + 7) q^{73} + ( - 2 \beta_{2} + \beta_1 + 9) q^{74} + (\beta_{2} + 2 \beta_1 + 4) q^{76} + (4 \beta_1 - 4) q^{79} + ( - 2 \beta_{2} + \beta_1 + 13) q^{80} + (6 \beta_{2} + 2 \beta_1) q^{82} + ( - 6 \beta_1 + 2) q^{83} + (12 \beta_{2} - 2 \beta_1 + 6) q^{85} + (6 \beta_{2} - 10 \beta_1 - 6) q^{86} + ( - 4 \beta_{2} - 10) q^{89} + ( - \beta_{2} + 3 \beta_1 + 1) q^{91} + ( - 4 \beta_{2} - 6 \beta_1 - 10) q^{92} + (2 \beta_{2} - 5 \beta_1 - 11) q^{94} + ( - \beta_{2} + 5 \beta_1 - 3) q^{95} + (4 \beta_{2} + 2 \beta_1) q^{97} + (\beta_{2} + 1) q^{98}+O(q^{100})$$ q + (b2 + 1) * q^2 + (b1 + 2) * q^4 + (b2 + b1 - 1) * q^5 + q^7 + (2*b1 + 1) * q^8 + (-2*b2 + 3*b1 + 3) * q^10 + (-b2 + 3*b1 + 1) * q^13 + (b2 + 1) * q^14 + (b2 + 2*b1 - 1) * q^16 + (-2*b2 + 4*b1 + 2) * q^17 + (b2 - b1 + 3) * q^19 + (3*b2 + 2*b1 + 2) * q^20 + (-2*b2 - 4) * q^23 + (-3*b2 + b1 + 4) * q^25 + (2*b2 + 5*b1 + 1) * q^26 + (b1 + 2) * q^28 + (b2 - 3*b1 - 1) * q^29 + (-4*b2 + 2*b1 - 2) * q^31 + (-2*b2 + b1 + 2) * q^32 + (4*b2 + 6*b1) * q^34 + (b2 + b1 - 1) * q^35 + (3*b2 - b1 + 1) * q^37 + (2*b2 - b1 + 5) * q^38 + (3*b2 + b1 + 7) * q^40 + (-2*b2 + 2*b1 + 4) * q^41 + (-2*b2 - 4*b1 + 4) * q^43 + (-2*b2 - 2*b1 - 10) * q^46 + (-3*b2 - b1 - 1) * q^47 + q^49 + (7*b2 - b1 - 4) * q^50 + (b2 + 6*b1 + 10) * q^52 + 2*b1 * q^53 + (2*b1 + 1) * q^56 + (-2*b2 - 5*b1 - 1) * q^58 + (-3*b2 + 3*b1 - 1) * q^59 + 2 * q^61 + (2*b2 - 12) * q^62 + (2*b2 - 4*b1 - 1) * q^64 + (7*b2 - b1 + 7) * q^65 + (b2 - 5*b1 - 1) * q^67 + (8*b1 + 14) * q^68 + (-2*b2 + 3*b1 + 3) * q^70 + (-4*b1 + 4) * q^71 + (b2 - 3*b1 + 7) * q^73 + (-2*b2 + b1 + 9) * q^74 + (b2 + 2*b1 + 4) * q^76 + (4*b1 - 4) * q^79 + (-2*b2 + b1 + 13) * q^80 + (6*b2 + 2*b1) * q^82 + (-6*b1 + 2) * q^83 + (12*b2 - 2*b1 + 6) * q^85 + (6*b2 - 10*b1 - 6) * q^86 + (-4*b2 - 10) * q^89 + (-b2 + 3*b1 + 1) * q^91 + (-4*b2 - 6*b1 - 10) * q^92 + (2*b2 - 5*b1 - 11) * q^94 + (-b2 + 5*b1 - 3) * q^95 + (4*b2 + 2*b1) * q^97 + (b2 + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 + 3 * q^7 + 3 * q^8 $$3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{10} + 4 q^{13} + 2 q^{14} - 4 q^{16} + 8 q^{17} + 8 q^{19} + 3 q^{20} - 10 q^{23} + 15 q^{25} + q^{26} + 6 q^{28} - 4 q^{29} - 2 q^{31} + 8 q^{32} - 4 q^{34} - 4 q^{35} + 13 q^{38} + 18 q^{40} + 14 q^{41} + 14 q^{43} - 28 q^{46} + 3 q^{49} - 19 q^{50} + 29 q^{52} + 3 q^{56} - q^{58} + 6 q^{61} - 38 q^{62} - 5 q^{64} + 14 q^{65} - 4 q^{67} + 42 q^{68} + 11 q^{70} + 12 q^{71} + 20 q^{73} + 29 q^{74} + 11 q^{76} - 12 q^{79} + 41 q^{80} - 6 q^{82} + 6 q^{83} + 6 q^{85} - 24 q^{86} - 26 q^{89} + 4 q^{91} - 26 q^{92} - 35 q^{94} - 8 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100})$$ 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 + 3 * q^7 + 3 * q^8 + 11 * q^10 + 4 * q^13 + 2 * q^14 - 4 * q^16 + 8 * q^17 + 8 * q^19 + 3 * q^20 - 10 * q^23 + 15 * q^25 + q^26 + 6 * q^28 - 4 * q^29 - 2 * q^31 + 8 * q^32 - 4 * q^34 - 4 * q^35 + 13 * q^38 + 18 * q^40 + 14 * q^41 + 14 * q^43 - 28 * q^46 + 3 * q^49 - 19 * q^50 + 29 * q^52 + 3 * q^56 - q^58 + 6 * q^61 - 38 * q^62 - 5 * q^64 + 14 * q^65 - 4 * q^67 + 42 * q^68 + 11 * q^70 + 12 * q^71 + 20 * q^73 + 29 * q^74 + 11 * q^76 - 12 * q^79 + 41 * q^80 - 6 * q^82 + 6 * q^83 + 6 * q^85 - 24 * q^86 - 26 * q^89 + 4 * q^91 - 26 * q^92 - 35 * q^94 - 8 * q^95 - 4 * q^97 + 2 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3

## 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
 −0.254102 −1.86081 2.11491
−1.93543 0 1.74590 −4.18953 0 1.00000 0.491797 0 8.10856
1.2 1.46260 0 0.139194 −2.39821 0 1.00000 −2.72161 0 −3.50761
1.3 2.47283 0 4.11491 2.58774 0 1.00000 5.22982 0 6.39905
 $$n$$: 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.cd 3
3.b odd 2 1 2541.2.a.bg 3
11.b odd 2 1 693.2.a.l 3
33.d even 2 1 231.2.a.e 3
77.b even 2 1 4851.2.a.bi 3
132.d odd 2 1 3696.2.a.bo 3
165.d even 2 1 5775.2.a.bp 3
231.h odd 2 1 1617.2.a.t 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 33.d even 2 1
693.2.a.l 3 11.b odd 2 1
1617.2.a.t 3 231.h odd 2 1
2541.2.a.bg 3 3.b odd 2 1
3696.2.a.bo 3 132.d odd 2 1
4851.2.a.bi 3 77.b even 2 1
5775.2.a.bp 3 165.d even 2 1
7623.2.a.cd 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2}^{3} - 2T_{2}^{2} - 4T_{2} + 7$$ T2^3 - 2*T2^2 - 4*T2 + 7 $$T_{5}^{3} + 4T_{5}^{2} - 7T_{5} - 26$$ T5^3 + 4*T5^2 - 7*T5 - 26 $$T_{13}^{3} - 4T_{13}^{2} - 27T_{13} + 94$$ T13^3 - 4*T13^2 - 27*T13 + 94

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 2 T^{2} - 4 T + 7$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 4 T^{2} - 7 T - 26$$
$7$ $$(T - 1)^{3}$$
$11$ $$T^{3}$$
$13$ $$T^{3} - 4 T^{2} - 27 T + 94$$
$17$ $$T^{3} - 8 T^{2} - 40 T + 328$$
$19$ $$T^{3} - 8 T^{2} + 15 T - 4$$
$23$ $$T^{3} + 10 T^{2} + 12 T - 64$$
$29$ $$T^{3} + 4 T^{2} - 27 T - 94$$
$31$ $$T^{3} + 2 T^{2} - 76 T - 256$$
$37$ $$T^{3} - 43T + 106$$
$41$ $$T^{3} - 14 T^{2} + 40 T + 32$$
$43$ $$T^{3} - 14 T^{2} - 44 T + 848$$
$47$ $$T^{3} - 61T - 32$$
$53$ $$T^{3} - 16T - 8$$
$59$ $$T^{3} - 57T + 52$$
$61$ $$(T - 2)^{3}$$
$67$ $$T^{3} + 4 T^{2} - 85 T - 236$$
$71$ $$T^{3} - 12 T^{2} - 16 T + 256$$
$73$ $$T^{3} - 20 T^{2} + 101 T - 134$$
$79$ $$T^{3} + 12 T^{2} - 16 T - 256$$
$83$ $$T^{3} - 6 T^{2} - 132 T + 496$$
$89$ $$T^{3} + 26 T^{2} + 140 T - 328$$
$97$ $$T^{3} + 4 T^{2} - 120 T - 232$$