Properties

 Label 7623.2.a.bd 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: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta + 1) q^{4} + q^{5} + q^{7} - 3 q^{8} +O(q^{10})$$ q - b * q^2 + (b + 1) * q^4 + q^5 + q^7 - 3 * q^8 $$q - \beta q^{2} + (\beta + 1) q^{4} + q^{5} + q^{7} - 3 q^{8} - \beta q^{10} + ( - 2 \beta + 1) q^{13} - \beta q^{14} + (\beta - 2) q^{16} - 4 q^{17} - 3 q^{19} + (\beta + 1) q^{20} + 2 q^{23} - 4 q^{25} + (\beta + 6) q^{26} + (\beta + 1) q^{28} + (2 \beta + 1) q^{29} - 2 q^{31} + (\beta + 3) q^{32} + 4 \beta q^{34} + q^{35} + ( - 4 \beta + 1) q^{37} + 3 \beta q^{38} - 3 q^{40} + (4 \beta - 2) q^{41} + (4 \beta - 4) q^{43} - 2 \beta q^{46} + ( - 2 \beta + 7) q^{47} + q^{49} + 4 \beta q^{50} + ( - 3 \beta - 5) q^{52} - 3 q^{56} + ( - 3 \beta - 6) q^{58} + (2 \beta + 3) q^{59} + (4 \beta + 2) q^{61} + 2 \beta q^{62} + ( - 6 \beta + 1) q^{64} + ( - 2 \beta + 1) q^{65} + (2 \beta - 3) q^{67} + ( - 4 \beta - 4) q^{68} - \beta q^{70} + (4 \beta + 2) q^{71} + (6 \beta - 1) q^{73} + (3 \beta + 12) q^{74} + ( - 3 \beta - 3) q^{76} + (4 \beta - 6) q^{79} + (\beta - 2) q^{80} + ( - 2 \beta - 12) q^{82} - 12 q^{83} - 4 q^{85} - 12 q^{86} + 6 q^{89} + ( - 2 \beta + 1) q^{91} + (2 \beta + 2) q^{92} + ( - 5 \beta + 6) q^{94} - 3 q^{95} + (4 \beta - 8) q^{97} - \beta q^{98} +O(q^{100})$$ q - b * q^2 + (b + 1) * q^4 + q^5 + q^7 - 3 * q^8 - b * q^10 + (-2*b + 1) * q^13 - b * q^14 + (b - 2) * q^16 - 4 * q^17 - 3 * q^19 + (b + 1) * q^20 + 2 * q^23 - 4 * q^25 + (b + 6) * q^26 + (b + 1) * q^28 + (2*b + 1) * q^29 - 2 * q^31 + (b + 3) * q^32 + 4*b * q^34 + q^35 + (-4*b + 1) * q^37 + 3*b * q^38 - 3 * q^40 + (4*b - 2) * q^41 + (4*b - 4) * q^43 - 2*b * q^46 + (-2*b + 7) * q^47 + q^49 + 4*b * q^50 + (-3*b - 5) * q^52 - 3 * q^56 + (-3*b - 6) * q^58 + (2*b + 3) * q^59 + (4*b + 2) * q^61 + 2*b * q^62 + (-6*b + 1) * q^64 + (-2*b + 1) * q^65 + (2*b - 3) * q^67 + (-4*b - 4) * q^68 - b * q^70 + (4*b + 2) * q^71 + (6*b - 1) * q^73 + (3*b + 12) * q^74 + (-3*b - 3) * q^76 + (4*b - 6) * q^79 + (b - 2) * q^80 + (-2*b - 12) * q^82 - 12 * q^83 - 4 * q^85 - 12 * q^86 + 6 * q^89 + (-2*b + 1) * q^91 + (2*b + 2) * q^92 + (-5*b + 6) * q^94 - 3 * q^95 + (4*b - 8) * q^97 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + 3 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8 $$2 q - q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8} - q^{10} - q^{14} - 3 q^{16} - 8 q^{17} - 6 q^{19} + 3 q^{20} + 4 q^{23} - 8 q^{25} + 13 q^{26} + 3 q^{28} + 4 q^{29} - 4 q^{31} + 7 q^{32} + 4 q^{34} + 2 q^{35} - 2 q^{37} + 3 q^{38} - 6 q^{40} - 4 q^{43} - 2 q^{46} + 12 q^{47} + 2 q^{49} + 4 q^{50} - 13 q^{52} - 6 q^{56} - 15 q^{58} + 8 q^{59} + 8 q^{61} + 2 q^{62} - 4 q^{64} - 4 q^{67} - 12 q^{68} - q^{70} + 8 q^{71} + 4 q^{73} + 27 q^{74} - 9 q^{76} - 8 q^{79} - 3 q^{80} - 26 q^{82} - 24 q^{83} - 8 q^{85} - 24 q^{86} + 12 q^{89} + 6 q^{92} + 7 q^{94} - 6 q^{95} - 12 q^{97} - q^{98}+O(q^{100})$$ 2 * q - q^2 + 3 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8 - q^10 - q^14 - 3 * q^16 - 8 * q^17 - 6 * q^19 + 3 * q^20 + 4 * q^23 - 8 * q^25 + 13 * q^26 + 3 * q^28 + 4 * q^29 - 4 * q^31 + 7 * q^32 + 4 * q^34 + 2 * q^35 - 2 * q^37 + 3 * q^38 - 6 * q^40 - 4 * q^43 - 2 * q^46 + 12 * q^47 + 2 * q^49 + 4 * q^50 - 13 * q^52 - 6 * q^56 - 15 * q^58 + 8 * q^59 + 8 * q^61 + 2 * q^62 - 4 * q^64 - 4 * q^67 - 12 * q^68 - q^70 + 8 * q^71 + 4 * q^73 + 27 * q^74 - 9 * q^76 - 8 * q^79 - 3 * q^80 - 26 * q^82 - 24 * q^83 - 8 * q^85 - 24 * q^86 + 12 * q^89 + 6 * q^92 + 7 * q^94 - 6 * q^95 - 12 * 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
 2.30278 −1.30278
−2.30278 0 3.30278 1.00000 0 1.00000 −3.00000 0 −2.30278
1.2 1.30278 0 −0.302776 1.00000 0 1.00000 −3.00000 0 1.30278
 $$n$$: 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.bd 2
3.b odd 2 1 7623.2.a.br 2
11.b odd 2 1 693.2.a.i yes 2
33.d even 2 1 693.2.a.g 2
77.b even 2 1 4851.2.a.z 2
231.h odd 2 1 4851.2.a.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.a.g 2 33.d even 2 1
693.2.a.i yes 2 11.b odd 2 1
4851.2.a.x 2 231.h odd 2 1
4851.2.a.z 2 77.b even 2 1
7623.2.a.bd 2 1.a even 1 1 trivial
7623.2.a.br 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 3$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 13$$
$17$ $$(T + 4)^{2}$$
$19$ $$(T + 3)^{2}$$
$23$ $$(T - 2)^{2}$$
$29$ $$T^{2} - 4T - 9$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 2T - 51$$
$41$ $$T^{2} - 52$$
$43$ $$T^{2} + 4T - 48$$
$47$ $$T^{2} - 12T + 23$$
$53$ $$T^{2}$$
$59$ $$T^{2} - 8T + 3$$
$61$ $$T^{2} - 8T - 36$$
$67$ $$T^{2} + 4T - 9$$
$71$ $$T^{2} - 8T - 36$$
$73$ $$T^{2} - 4T - 113$$
$79$ $$T^{2} + 8T - 36$$
$83$ $$(T + 12)^{2}$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 12T - 16$$