# Properties

 Label 7605.2.a.bg Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7605 = 3^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7605.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$60.7262307372$$ 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 65) 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 $$q + \beta q^{2} + (\beta + 1) q^{4} + q^{5} - q^{7} + 3 q^{8} + \beta q^{10} + ( - 2 \beta + 3) q^{11} - \beta q^{14} + (\beta - 2) q^{16} + ( - 2 \beta - 3) q^{17} + ( - 2 \beta - 1) q^{19} + (\beta + 1) q^{20} + (\beta - 6) q^{22} + 3 q^{23} + q^{25} + ( - \beta - 1) q^{28} + (4 \beta - 3) q^{29} - 4 q^{31} + ( - \beta - 3) q^{32} + ( - 5 \beta - 6) q^{34} - q^{35} + (2 \beta - 1) q^{37} + ( - 3 \beta - 6) q^{38} + 3 q^{40} - 3 q^{41} + ( - 4 \beta - 1) q^{43} + ( - \beta - 3) q^{44} + 3 \beta q^{46} - 4 \beta q^{47} - 6 q^{49} + \beta q^{50} + (4 \beta - 6) q^{53} + ( - 2 \beta + 3) q^{55} - 3 q^{56} + (\beta + 12) q^{58} + (6 \beta - 3) q^{59} - q^{61} - 4 \beta q^{62} + ( - 6 \beta + 1) q^{64} - 7 q^{67} + ( - 7 \beta - 9) q^{68} - \beta q^{70} + ( - 6 \beta + 9) q^{71} + (4 \beta - 10) q^{73} + (\beta + 6) q^{74} + ( - 5 \beta - 7) q^{76} + (2 \beta - 3) q^{77} + (4 \beta - 4) q^{79} + (\beta - 2) q^{80} - 3 \beta q^{82} + 4 \beta q^{83} + ( - 2 \beta - 3) q^{85} + ( - 5 \beta - 12) q^{86} + ( - 6 \beta + 9) q^{88} + (4 \beta - 3) q^{89} + (3 \beta + 3) q^{92} + ( - 4 \beta - 12) q^{94} + ( - 2 \beta - 1) q^{95} + (2 \beta - 13) q^{97} - 6 \beta q^{98} +O(q^{100})$$ q + b * q^2 + (b + 1) * q^4 + q^5 - q^7 + 3 * q^8 + b * q^10 + (-2*b + 3) * q^11 - b * q^14 + (b - 2) * q^16 + (-2*b - 3) * q^17 + (-2*b - 1) * q^19 + (b + 1) * q^20 + (b - 6) * q^22 + 3 * q^23 + q^25 + (-b - 1) * q^28 + (4*b - 3) * q^29 - 4 * q^31 + (-b - 3) * q^32 + (-5*b - 6) * q^34 - q^35 + (2*b - 1) * q^37 + (-3*b - 6) * q^38 + 3 * q^40 - 3 * q^41 + (-4*b - 1) * q^43 + (-b - 3) * q^44 + 3*b * q^46 - 4*b * q^47 - 6 * q^49 + b * q^50 + (4*b - 6) * q^53 + (-2*b + 3) * q^55 - 3 * q^56 + (b + 12) * q^58 + (6*b - 3) * q^59 - q^61 - 4*b * q^62 + (-6*b + 1) * q^64 - 7 * q^67 + (-7*b - 9) * q^68 - b * q^70 + (-6*b + 9) * q^71 + (4*b - 10) * q^73 + (b + 6) * q^74 + (-5*b - 7) * q^76 + (2*b - 3) * q^77 + (4*b - 4) * q^79 + (b - 2) * q^80 - 3*b * q^82 + 4*b * q^83 + (-2*b - 3) * q^85 + (-5*b - 12) * q^86 + (-6*b + 9) * q^88 + (4*b - 3) * q^89 + (3*b + 3) * q^92 + (-4*b - 12) * q^94 + (-2*b - 1) * q^95 + (2*b - 13) * q^97 - 6*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 $$2 q + q^{2} + 3 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + q^{10} + 4 q^{11} - q^{14} - 3 q^{16} - 8 q^{17} - 4 q^{19} + 3 q^{20} - 11 q^{22} + 6 q^{23} + 2 q^{25} - 3 q^{28} - 2 q^{29} - 8 q^{31} - 7 q^{32} - 17 q^{34} - 2 q^{35} - 15 q^{38} + 6 q^{40} - 6 q^{41} - 6 q^{43} - 7 q^{44} + 3 q^{46} - 4 q^{47} - 12 q^{49} + q^{50} - 8 q^{53} + 4 q^{55} - 6 q^{56} + 25 q^{58} - 2 q^{61} - 4 q^{62} - 4 q^{64} - 14 q^{67} - 25 q^{68} - q^{70} + 12 q^{71} - 16 q^{73} + 13 q^{74} - 19 q^{76} - 4 q^{77} - 4 q^{79} - 3 q^{80} - 3 q^{82} + 4 q^{83} - 8 q^{85} - 29 q^{86} + 12 q^{88} - 2 q^{89} + 9 q^{92} - 28 q^{94} - 4 q^{95} - 24 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q + q^2 + 3 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + q^10 + 4 * q^11 - q^14 - 3 * q^16 - 8 * q^17 - 4 * q^19 + 3 * q^20 - 11 * q^22 + 6 * q^23 + 2 * q^25 - 3 * q^28 - 2 * q^29 - 8 * q^31 - 7 * q^32 - 17 * q^34 - 2 * q^35 - 15 * q^38 + 6 * q^40 - 6 * q^41 - 6 * q^43 - 7 * q^44 + 3 * q^46 - 4 * q^47 - 12 * q^49 + q^50 - 8 * q^53 + 4 * q^55 - 6 * q^56 + 25 * q^58 - 2 * q^61 - 4 * q^62 - 4 * q^64 - 14 * q^67 - 25 * q^68 - q^70 + 12 * q^71 - 16 * q^73 + 13 * q^74 - 19 * q^76 - 4 * q^77 - 4 * q^79 - 3 * q^80 - 3 * q^82 + 4 * q^83 - 8 * q^85 - 29 * q^86 + 12 * q^88 - 2 * q^89 + 9 * q^92 - 28 * q^94 - 4 * q^95 - 24 * q^97 - 6 * 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
 −1.30278 2.30278
−1.30278 0 −0.302776 1.00000 0 −1.00000 3.00000 0 −1.30278
1.2 2.30278 0 3.30278 1.00000 0 −1.00000 3.00000 0 2.30278
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

## 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 7605.2.a.bg 2
3.b odd 2 1 845.2.a.c 2
13.b even 2 1 7605.2.a.bb 2
13.c even 3 2 585.2.j.d 4
15.d odd 2 1 4225.2.a.x 2
39.d odd 2 1 845.2.a.f 2
39.f even 4 2 845.2.c.d 4
39.h odd 6 2 845.2.e.d 4
39.i odd 6 2 65.2.e.b 4
39.k even 12 4 845.2.m.d 8
156.p even 6 2 1040.2.q.o 4
195.e odd 2 1 4225.2.a.t 2
195.x odd 6 2 325.2.e.a 4
195.bl even 12 4 325.2.o.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.b 4 39.i odd 6 2
325.2.e.a 4 195.x odd 6 2
325.2.o.b 8 195.bl even 12 4
585.2.j.d 4 13.c even 3 2
845.2.a.c 2 3.b odd 2 1
845.2.a.f 2 39.d odd 2 1
845.2.c.d 4 39.f even 4 2
845.2.e.d 4 39.h odd 6 2
845.2.m.d 8 39.k even 12 4
1040.2.q.o 4 156.p even 6 2
4225.2.a.t 2 195.e odd 2 1
4225.2.a.x 2 15.d odd 2 1
7605.2.a.bb 2 13.b even 2 1
7605.2.a.bg 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{2} - T_{2} - 3$$ T2^2 - T2 - 3 $$T_{7} + 1$$ T7 + 1 $$T_{11}^{2} - 4T_{11} - 9$$ T11^2 - 4*T11 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 3$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 4T - 9$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 8T + 3$$
$19$ $$T^{2} + 4T - 9$$
$23$ $$(T - 3)^{2}$$
$29$ $$T^{2} + 2T - 51$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} - 13$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 6T - 43$$
$47$ $$T^{2} + 4T - 48$$
$53$ $$T^{2} + 8T - 36$$
$59$ $$T^{2} - 117$$
$61$ $$(T + 1)^{2}$$
$67$ $$(T + 7)^{2}$$
$71$ $$T^{2} - 12T - 81$$
$73$ $$T^{2} + 16T + 12$$
$79$ $$T^{2} + 4T - 48$$
$83$ $$T^{2} - 4T - 48$$
$89$ $$T^{2} + 2T - 51$$
$97$ $$T^{2} + 24T + 131$$