# Properties

 Label 7605.2.a.co Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ Analytic rank $0$ Dimension $5$ 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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.3352656.1 Defining polynomial: $$x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 6x - 6$$ x^5 - 2*x^4 - 6*x^3 + 10*x^2 + 6*x - 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_{4} q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 1) * q^4 + q^5 + b4 * q^7 + (b3 + b2 + 1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_{4} q^{7} + (\beta_{3} + \beta_{2} + 1) q^{8} + \beta_1 q^{10} + (\beta_{4} + \beta_1 + 1) q^{11} + (\beta_{4} + \beta_{3} - 1) q^{14} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{16} + (2 \beta_{2} - \beta_1) q^{17} + (\beta_{2} - 1) q^{19} + (\beta_{2} + 1) q^{20} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{22} + ( - \beta_{3} - \beta_{2} - 1) q^{23} + q^{25} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{28} + (\beta_{4} - \beta_{3} + 3) q^{29} + (\beta_{4} - \beta_{3} - \beta_1 + 2) q^{31} + (2 \beta_{4} + 2 \beta_{2} + 2) q^{32} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{34} + \beta_{4} q^{35} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{37} + (\beta_{3} + \beta_{2} + 1) q^{38} + (\beta_{3} + \beta_{2} + 1) q^{40} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} + 1) q^{41} + ( - \beta_{2} + 3 \beta_1 + 2) q^{43} + (2 \beta_{3} + 4 \beta_{2} + 2) q^{44} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{46} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{47} + ( - 2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 1) q^{49} + \beta_1 q^{50} + (2 \beta_{4} - 2 \beta_{2} - 4) q^{53} + (\beta_{4} + \beta_1 + 1) q^{55} + ( - \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{56} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{58} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{59} + ( - \beta_{4} + \beta_{3} + \beta_1 + 2) q^{61} + (\beta_{3} - 3 \beta_{2} + 3 \beta_1 - 5) q^{62} + (2 \beta_{3} + 2 \beta_1) q^{64} + ( - \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{67} + (2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 9) q^{68} + (\beta_{4} + \beta_{3} - 1) q^{70} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{71} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 - 2) q^{73} + ( - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4) q^{74} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 4) q^{76} + (2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 7) q^{77} + (2 \beta_{3} + \beta_{2} - 4 \beta_1 - 2) q^{79} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{80} + ( - 2 \beta_{4} + \beta_{3} + 4 \beta_1 + 2) q^{82} + ( - 3 \beta_{3} + \beta_{2} + 3) q^{83} + (2 \beta_{2} - \beta_1) q^{85} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 8) q^{86} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1 + 2) q^{88} + ( - \beta_{3} - 3 \beta_1 + 2) q^{89} + ( - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 4) q^{92} + ( - 3 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 3) q^{94} + (\beta_{2} - 1) q^{95} + ( - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{97} + ( - 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 7) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 1) * q^4 + q^5 + b4 * q^7 + (b3 + b2 + 1) * q^8 + b1 * q^10 + (b4 + b1 + 1) * q^11 + (b4 + b3 - 1) * q^14 + (b4 + b3 + b2 + b1) * q^16 + (2*b2 - b1) * q^17 + (b2 - 1) * q^19 + (b2 + 1) * q^20 + (b4 + b3 + b2 + b1 + 2) * q^22 + (-b3 - b2 - 1) * q^23 + q^25 + (b3 + 2*b2 - 2*b1) * q^28 + (b4 - b3 + 3) * q^29 + (b4 - b3 - b1 + 2) * q^31 + (2*b4 + 2*b2 + 2) * q^32 + (2*b3 + b2 + 2*b1 - 1) * q^34 + b4 * q^35 + (-b4 - b3 - b2 - b1) * q^37 + (b3 + b2 + 1) * q^38 + (b3 + b2 + 1) * q^40 + (-b4 - b3 + 2*b2 + 1) * q^41 + (-b2 + 3*b1 + 2) * q^43 + (2*b3 + 4*b2 + 2) * q^44 + (-b4 - b3 - 3*b2 - b1 - 2) * q^46 + (-2*b4 - b3 - b2 - b1 + 3) * q^47 + (-2*b3 - 3*b2 + 4*b1 + 1) * q^49 + b1 * q^50 + (2*b4 - 2*b2 - 4) * q^53 + (b4 + b1 + 1) * q^55 + (-b4 + 2*b2 + b1 - 1) * q^56 + (b3 - 2*b2 + 4*b1 - 2) * q^58 + (-b4 + b3 - 2*b2 + 3) * q^59 + (-b4 + b3 + b1 + 2) * q^61 + (b3 - 3*b2 + 3*b1 - 5) * q^62 + (2*b3 + 2*b1) * q^64 + (-b4 + b3 - 2*b1 + 2) * q^67 + (2*b4 + b3 + 3*b2 + 9) * q^68 + (b4 + b3 - 1) * q^70 + (-b3 - 2*b2 + b1 + 2) * q^71 + (-b3 + b2 + 5*b1 - 2) * q^73 + (-2*b4 - 2*b3 - 4*b2 - 4) * q^74 + (b4 + b3 + b2 + b1 + 4) * q^76 + (2*b4 - b3 - 3*b2 + 4*b1 + 7) * q^77 + (2*b3 + b2 - 4*b1 - 2) * q^79 + (b4 + b3 + b2 + b1) * q^80 + (-2*b4 + b3 + 4*b1 + 2) * q^82 + (-3*b3 + b2 + 3) * q^83 + (2*b2 - b1) * q^85 + (-b3 + 2*b2 + b1 + 8) * q^86 + (2*b3 + 6*b2 + 2*b1 + 2) * q^88 + (-b3 - 3*b1 + 2) * q^89 + (-2*b4 - 2*b3 - 4*b2 - 4*b1 - 4) * q^92 + (-3*b4 - 3*b3 - 4*b2 + 3*b1 - 3) * q^94 + (b2 - 1) * q^95 + (-2*b4 - 2*b3 - b2 + b1 - 2) * q^97 + (-2*b4 - 3*b3 - 3*b2 + 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + q^{7} + 6 q^{8}+O(q^{10})$$ 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 + q^7 + 6 * q^8 $$5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{11} - 4 q^{14} + 4 q^{16} - 4 q^{19} + 6 q^{20} + 14 q^{22} - 6 q^{23} + 5 q^{25} - 2 q^{28} + 16 q^{29} + 9 q^{31} + 14 q^{32} + q^{35} - 4 q^{37} + 6 q^{38} + 6 q^{40} + 6 q^{41} + 15 q^{43} + 14 q^{44} - 16 q^{46} + 10 q^{47} + 10 q^{49} + 2 q^{50} - 20 q^{53} + 8 q^{55} - 2 q^{56} - 4 q^{58} + 12 q^{59} + 11 q^{61} - 22 q^{62} + 4 q^{64} + 5 q^{67} + 50 q^{68} - 4 q^{70} + 10 q^{71} + q^{73} - 26 q^{74} + 24 q^{76} + 42 q^{77} - 17 q^{79} + 4 q^{80} + 16 q^{82} + 16 q^{83} + 44 q^{86} + 20 q^{88} + 4 q^{89} - 34 q^{92} - 16 q^{94} - 4 q^{95} - 11 q^{97} + 30 q^{98}+O(q^{100})$$ 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 + q^7 + 6 * q^8 + 2 * q^10 + 8 * q^11 - 4 * q^14 + 4 * q^16 - 4 * q^19 + 6 * q^20 + 14 * q^22 - 6 * q^23 + 5 * q^25 - 2 * q^28 + 16 * q^29 + 9 * q^31 + 14 * q^32 + q^35 - 4 * q^37 + 6 * q^38 + 6 * q^40 + 6 * q^41 + 15 * q^43 + 14 * q^44 - 16 * q^46 + 10 * q^47 + 10 * q^49 + 2 * q^50 - 20 * q^53 + 8 * q^55 - 2 * q^56 - 4 * q^58 + 12 * q^59 + 11 * q^61 - 22 * q^62 + 4 * q^64 + 5 * q^67 + 50 * q^68 - 4 * q^70 + 10 * q^71 + q^73 - 26 * q^74 + 24 * q^76 + 42 * q^77 - 17 * q^79 + 4 * q^80 + 16 * q^82 + 16 * q^83 + 44 * q^86 + 20 * q^88 + 4 * q^89 - 34 * q^92 - 16 * q^94 - 4 * q^95 - 11 * q^97 + 30 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 6x - 6$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu + 2$$ v^3 - v^2 - 4*v + 2 $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 5$$ v^4 - v^3 - 6*v^2 + 3*v + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 1$$ b3 + b2 + 4*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 14$$ b4 + b3 + 7*b2 + b1 + 14

## 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
 −2.12283 −0.946366 0.626791 1.81031 2.63209
−2.12283 0 2.50640 1.00000 0 1.46707 −1.07500 0 −2.12283
1.2 −0.946366 0 −1.10439 1.00000 0 −1.56306 2.93789 0 −0.946366
1.3 0.626791 0 −1.60713 1.00000 0 4.43127 −2.26092 0 0.626791
1.4 1.81031 0 1.27724 1.00000 0 −4.42503 −1.30843 0 1.81031
1.5 2.63209 0 4.92789 1.00000 0 1.08975 7.70645 0 2.63209
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.co 5
3.b odd 2 1 7605.2.a.cm 5
13.b even 2 1 7605.2.a.cl 5
13.c even 3 2 585.2.j.h 10
39.d odd 2 1 7605.2.a.cn 5
39.i odd 6 2 585.2.j.i yes 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.j.h 10 13.c even 3 2
585.2.j.i yes 10 39.i odd 6 2
7605.2.a.cl 5 13.b even 2 1
7605.2.a.cm 5 3.b odd 2 1
7605.2.a.cn 5 39.d odd 2 1
7605.2.a.co 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{5} - 2T_{2}^{4} - 6T_{2}^{3} + 10T_{2}^{2} + 6T_{2} - 6$$ T2^5 - 2*T2^4 - 6*T2^3 + 10*T2^2 + 6*T2 - 6 $$T_{7}^{5} - T_{7}^{4} - 22T_{7}^{3} + 22T_{7}^{2} + 47T_{7} - 49$$ T7^5 - T7^4 - 22*T7^3 + 22*T7^2 + 47*T7 - 49 $$T_{11}^{5} - 8T_{11}^{4} + 64T_{11}^{2} + 48T_{11} - 24$$ T11^5 - 8*T11^4 + 64*T11^2 + 48*T11 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 2 T^{4} - 6 T^{3} + 10 T^{2} + \cdots - 6$$
$3$ $$T^{5}$$
$5$ $$(T - 1)^{5}$$
$7$ $$T^{5} - T^{4} - 22 T^{3} + 22 T^{2} + \cdots - 49$$
$11$ $$T^{5} - 8 T^{4} + 64 T^{2} + 48 T - 24$$
$13$ $$T^{5}$$
$17$ $$T^{5} - 50 T^{3} - 14 T^{2} + \cdots + 642$$
$19$ $$T^{5} + 4 T^{4} - 8 T^{3} - 36 T^{2} + \cdots + 12$$
$23$ $$T^{5} + 6 T^{4} - 20 T^{3} - 36 T^{2} + \cdots - 72$$
$29$ $$T^{5} - 16 T^{4} + 58 T^{3} + \cdots + 258$$
$31$ $$T^{5} - 9 T^{4} - 30 T^{3} + 178 T^{2} + \cdots + 603$$
$37$ $$T^{5} + 4 T^{4} - 64 T^{3} + 56 T^{2} + \cdots - 144$$
$41$ $$T^{5} - 6 T^{4} - 98 T^{3} + \cdots - 11898$$
$43$ $$T^{5} - 15 T^{4} + 30 T^{3} + \cdots + 2059$$
$47$ $$T^{5} - 10 T^{4} - 90 T^{3} + \cdots - 7434$$
$53$ $$T^{5} + 20 T^{4} - 1872 T^{2} + \cdots - 15552$$
$59$ $$T^{5} - 12 T^{4} - 42 T^{3} + \cdots - 2754$$
$61$ $$T^{5} - 11 T^{4} - 14 T^{3} + \cdots - 3023$$
$67$ $$T^{5} - 5 T^{4} - 44 T^{3} + 4 T^{2} + \cdots - 17$$
$71$ $$T^{5} - 10 T^{4} - 22 T^{3} + \cdots + 162$$
$73$ $$T^{5} - T^{4} - 236 T^{3} + 144 T^{2} + \cdots + 9693$$
$79$ $$T^{5} + 17 T^{4} - 26 T^{3} + \cdots + 2349$$
$83$ $$T^{5} - 16 T^{4} - 116 T^{3} + \cdots - 6264$$
$89$ $$T^{5} - 4 T^{4} - 102 T^{3} + 78 T^{2} + \cdots - 54$$
$97$ $$T^{5} + 11 T^{4} - 130 T^{3} + \cdots - 7679$$