# Properties

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

# 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 585) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} + q^{5} - \beta q^{7} +O(q^{10})$$ q - 2 * q^2 + 2 * q^4 + q^5 - b * q^7 $$q - 2 q^{2} + 2 q^{4} + q^{5} - \beta q^{7} - 2 q^{10} - 3 q^{11} + 2 \beta q^{14} - 4 q^{16} + \beta q^{17} + 2 \beta q^{19} + 2 q^{20} + 6 q^{22} - \beta q^{23} + q^{25} - 2 \beta q^{28} + 2 \beta q^{29} - 2 \beta q^{31} + 8 q^{32} - 2 \beta q^{34} - \beta q^{35} - \beta q^{37} - 4 \beta q^{38} - 11 q^{41} + 4 q^{43} - 6 q^{44} + 2 \beta q^{46} + 4 q^{47} + 6 q^{49} - 2 q^{50} - 3 \beta q^{53} - 3 q^{55} - 4 \beta q^{58} - 12 q^{59} + 13 q^{61} + 4 \beta q^{62} - 8 q^{64} + 2 \beta q^{68} + 2 \beta q^{70} + 5 q^{71} + 2 \beta q^{73} + 2 \beta q^{74} + 4 \beta q^{76} + 3 \beta q^{77} + 13 q^{79} - 4 q^{80} + 22 q^{82} - 6 q^{83} + \beta q^{85} - 8 q^{86} + 3 q^{89} - 2 \beta q^{92} - 8 q^{94} + 2 \beta q^{95} + \beta q^{97} - 12 q^{98} +O(q^{100})$$ q - 2 * q^2 + 2 * q^4 + q^5 - b * q^7 - 2 * q^10 - 3 * q^11 + 2*b * q^14 - 4 * q^16 + b * q^17 + 2*b * q^19 + 2 * q^20 + 6 * q^22 - b * q^23 + q^25 - 2*b * q^28 + 2*b * q^29 - 2*b * q^31 + 8 * q^32 - 2*b * q^34 - b * q^35 - b * q^37 - 4*b * q^38 - 11 * q^41 + 4 * q^43 - 6 * q^44 + 2*b * q^46 + 4 * q^47 + 6 * q^49 - 2 * q^50 - 3*b * q^53 - 3 * q^55 - 4*b * q^58 - 12 * q^59 + 13 * q^61 + 4*b * q^62 - 8 * q^64 + 2*b * q^68 + 2*b * q^70 + 5 * q^71 + 2*b * q^73 + 2*b * q^74 + 4*b * q^76 + 3*b * q^77 + 13 * q^79 - 4 * q^80 + 22 * q^82 - 6 * q^83 + b * q^85 - 8 * q^86 + 3 * q^89 - 2*b * q^92 - 8 * q^94 + 2*b * q^95 + b * q^97 - 12 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 4 q^{4} + 2 q^{5}+O(q^{10})$$ 2 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 $$2 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{10} - 6 q^{11} - 8 q^{16} + 4 q^{20} + 12 q^{22} + 2 q^{25} + 16 q^{32} - 22 q^{41} + 8 q^{43} - 12 q^{44} + 8 q^{47} + 12 q^{49} - 4 q^{50} - 6 q^{55} - 24 q^{59} + 26 q^{61} - 16 q^{64} + 10 q^{71} + 26 q^{79} - 8 q^{80} + 44 q^{82} - 12 q^{83} - 16 q^{86} + 6 q^{89} - 16 q^{94} - 24 q^{98}+O(q^{100})$$ 2 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^10 - 6 * q^11 - 8 * q^16 + 4 * q^20 + 12 * q^22 + 2 * q^25 + 16 * q^32 - 22 * q^41 + 8 * q^43 - 12 * q^44 + 8 * q^47 + 12 * q^49 - 4 * q^50 - 6 * q^55 - 24 * q^59 + 26 * q^61 - 16 * q^64 + 10 * q^71 + 26 * q^79 - 8 * q^80 + 44 * q^82 - 12 * q^83 - 16 * q^86 + 6 * q^89 - 16 * q^94 - 24 * 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
−2.00000 0 2.00000 1.00000 0 −3.60555 0 0 −2.00000
1.2 −2.00000 0 2.00000 1.00000 0 3.60555 0 0 −2.00000
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7605.2.a.w 2
3.b odd 2 1 7605.2.a.bl 2
13.b even 2 1 7605.2.a.bl 2
13.d odd 4 2 585.2.b.f 4
39.d odd 2 1 inner 7605.2.a.w 2
39.f even 4 2 585.2.b.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.b.f 4 13.d odd 4 2
585.2.b.f 4 39.f even 4 2
7605.2.a.w 2 1.a even 1 1 trivial
7605.2.a.w 2 39.d odd 2 1 inner
7605.2.a.bl 2 3.b odd 2 1
7605.2.a.bl 2 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{7}^{2} - 13$$ T7^2 - 13 $$T_{11} + 3$$ T11 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 13$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 13$$
$19$ $$T^{2} - 52$$
$23$ $$T^{2} - 13$$
$29$ $$T^{2} - 52$$
$31$ $$T^{2} - 52$$
$37$ $$T^{2} - 13$$
$41$ $$(T + 11)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$(T - 4)^{2}$$
$53$ $$T^{2} - 117$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 13)^{2}$$
$67$ $$T^{2}$$
$71$ $$(T - 5)^{2}$$
$73$ $$T^{2} - 52$$
$79$ $$(T - 13)^{2}$$
$83$ $$(T + 6)^{2}$$
$89$ $$(T - 3)^{2}$$
$97$ $$T^{2} - 13$$