# Properties

 Label 7605.2.a.bn Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2535) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{2} + (\beta_{2} + \beta_1) q^{4} - q^{5} + (2 \beta_1 - 2) q^{7} + (\beta_{2} - 2 \beta_1) q^{8}+O(q^{10})$$ q + (-b2 - 1) * q^2 + (b2 + b1) * q^4 - q^5 + (2*b1 - 2) * q^7 + (b2 - 2*b1) * q^8 $$q + ( - \beta_{2} - 1) q^{2} + (\beta_{2} + \beta_1) q^{4} - q^{5} + (2 \beta_1 - 2) q^{7} + (\beta_{2} - 2 \beta_1) q^{8} + (\beta_{2} + 1) q^{10} + 2 q^{11} - 2 \beta_1 q^{14} + ( - \beta_1 + 1) q^{16} + ( - 3 \beta_{2} + \beta_1 - 3) q^{17} + (\beta_{2} - 2 \beta_1) q^{19} + ( - \beta_{2} - \beta_1) q^{20} + ( - 2 \beta_{2} - 2) q^{22} + ( - 2 \beta_{2} + \beta_1 - 4) q^{23} + q^{25} + (2 \beta_{2} - 2 \beta_1 + 6) q^{28} + (4 \beta_{2} - 2 \beta_1) q^{29} + ( - 3 \beta_{2} + 5 \beta_1 - 1) q^{31} + ( - 2 \beta_{2} + 5 \beta_1) q^{32} + (2 \beta_{2} + 2 \beta_1 + 5) q^{34} + ( - 2 \beta_1 + 2) q^{35} + (4 \beta_{2} - 4 \beta_1 + 8) q^{37} + (2 \beta_{2} + \beta_1 + 1) q^{38} + ( - \beta_{2} + 2 \beta_1) q^{40} + ( - 4 \beta_{2} - 6) q^{41} + ( - 2 \beta_{2} + 6 \beta_1 + 4) q^{43} + (2 \beta_{2} + 2 \beta_1) q^{44} + (3 \beta_{2} + \beta_1 + 5) q^{46} + (3 \beta_{2} - 4 \beta_1 + 8) q^{47} + (4 \beta_{2} - 8 \beta_1 + 5) q^{49} + ( - \beta_{2} - 1) q^{50} + (5 \beta_{2} + 2 \beta_1 - 4) q^{53} - 2 q^{55} + ( - 4 \beta_{2} + 4 \beta_1 - 6) q^{56} + (2 \beta_{2} - 2 \beta_1 - 2) q^{58} + (8 \beta_1 - 2) q^{59} + ( - 7 \beta_{2} - \beta_1 - 3) q^{61} + ( - 4 \beta_{2} - 2 \beta_1 - 1) q^{62} + ( - 5 \beta_{2} - \beta_1 - 5) q^{64} + ( - 2 \beta_{2} - 4 \beta_1 + 8) q^{67} + ( - \beta_{2} - 6 \beta_1 - 3) q^{68} + 2 \beta_1 q^{70} + 12 q^{71} + (6 \beta_{2} - 8 \beta_1 + 4) q^{73} + ( - 4 \beta_{2} - 8) q^{74} + ( - 4 \beta_{2} + \beta_1 - 4) q^{76} + (4 \beta_1 - 4) q^{77} + (2 \beta_{2} - 9 \beta_1 + 2) q^{79} + (\beta_1 - 1) q^{80} + (6 \beta_{2} + 4 \beta_1 + 10) q^{82} + (\beta_{2} - \beta_1 + 3) q^{83} + (3 \beta_{2} - \beta_1 + 3) q^{85} + ( - 10 \beta_{2} - 4 \beta_1 - 8) q^{86} + (2 \beta_{2} - 4 \beta_1) q^{88} + 4 \beta_{2} q^{89} + ( - 2 \beta_{2} - 6 \beta_1 - 1) q^{92} + ( - 4 \beta_{2} + \beta_1 - 7) q^{94} + ( - \beta_{2} + 2 \beta_1) q^{95} + (4 \beta_{2} - 4 \beta_1 - 2) q^{97} + (3 \beta_{2} + 4 \beta_1 - 1) q^{98}+O(q^{100})$$ q + (-b2 - 1) * q^2 + (b2 + b1) * q^4 - q^5 + (2*b1 - 2) * q^7 + (b2 - 2*b1) * q^8 + (b2 + 1) * q^10 + 2 * q^11 - 2*b1 * q^14 + (-b1 + 1) * q^16 + (-3*b2 + b1 - 3) * q^17 + (b2 - 2*b1) * q^19 + (-b2 - b1) * q^20 + (-2*b2 - 2) * q^22 + (-2*b2 + b1 - 4) * q^23 + q^25 + (2*b2 - 2*b1 + 6) * q^28 + (4*b2 - 2*b1) * q^29 + (-3*b2 + 5*b1 - 1) * q^31 + (-2*b2 + 5*b1) * q^32 + (2*b2 + 2*b1 + 5) * q^34 + (-2*b1 + 2) * q^35 + (4*b2 - 4*b1 + 8) * q^37 + (2*b2 + b1 + 1) * q^38 + (-b2 + 2*b1) * q^40 + (-4*b2 - 6) * q^41 + (-2*b2 + 6*b1 + 4) * q^43 + (2*b2 + 2*b1) * q^44 + (3*b2 + b1 + 5) * q^46 + (3*b2 - 4*b1 + 8) * q^47 + (4*b2 - 8*b1 + 5) * q^49 + (-b2 - 1) * q^50 + (5*b2 + 2*b1 - 4) * q^53 - 2 * q^55 + (-4*b2 + 4*b1 - 6) * q^56 + (2*b2 - 2*b1 - 2) * q^58 + (8*b1 - 2) * q^59 + (-7*b2 - b1 - 3) * q^61 + (-4*b2 - 2*b1 - 1) * q^62 + (-5*b2 - b1 - 5) * q^64 + (-2*b2 - 4*b1 + 8) * q^67 + (-b2 - 6*b1 - 3) * q^68 + 2*b1 * q^70 + 12 * q^71 + (6*b2 - 8*b1 + 4) * q^73 + (-4*b2 - 8) * q^74 + (-4*b2 + b1 - 4) * q^76 + (4*b1 - 4) * q^77 + (2*b2 - 9*b1 + 2) * q^79 + (b1 - 1) * q^80 + (6*b2 + 4*b1 + 10) * q^82 + (b2 - b1 + 3) * q^83 + (3*b2 - b1 + 3) * q^85 + (-10*b2 - 4*b1 - 8) * q^86 + (2*b2 - 4*b1) * q^88 + 4*b2 * q^89 + (-2*b2 - 6*b1 - 1) * q^92 + (-4*b2 + b1 - 7) * q^94 + (-b2 + 2*b1) * q^95 + (4*b2 - 4*b1 - 2) * q^97 + (3*b2 + 4*b1 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 3 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - 2 * q^2 - 3 * q^5 - 4 * q^7 - 3 * q^8 $$3 q - 2 q^{2} - 3 q^{5} - 4 q^{7} - 3 q^{8} + 2 q^{10} + 6 q^{11} - 2 q^{14} + 2 q^{16} - 5 q^{17} - 3 q^{19} - 4 q^{22} - 9 q^{23} + 3 q^{25} + 14 q^{28} - 6 q^{29} + 5 q^{31} + 7 q^{32} + 15 q^{34} + 4 q^{35} + 16 q^{37} + 2 q^{38} + 3 q^{40} - 14 q^{41} + 20 q^{43} + 13 q^{46} + 17 q^{47} + 3 q^{49} - 2 q^{50} - 15 q^{53} - 6 q^{55} - 10 q^{56} - 10 q^{58} + 2 q^{59} - 3 q^{61} - q^{62} - 11 q^{64} + 22 q^{67} - 14 q^{68} + 2 q^{70} + 36 q^{71} - 2 q^{73} - 20 q^{74} - 7 q^{76} - 8 q^{77} - 5 q^{79} - 2 q^{80} + 28 q^{82} + 7 q^{83} + 5 q^{85} - 18 q^{86} - 6 q^{88} - 4 q^{89} - 7 q^{92} - 16 q^{94} + 3 q^{95} - 14 q^{97} - 2 q^{98}+O(q^{100})$$ 3 * q - 2 * q^2 - 3 * q^5 - 4 * q^7 - 3 * q^8 + 2 * q^10 + 6 * q^11 - 2 * q^14 + 2 * q^16 - 5 * q^17 - 3 * q^19 - 4 * q^22 - 9 * q^23 + 3 * q^25 + 14 * q^28 - 6 * q^29 + 5 * q^31 + 7 * q^32 + 15 * q^34 + 4 * q^35 + 16 * q^37 + 2 * q^38 + 3 * q^40 - 14 * q^41 + 20 * q^43 + 13 * q^46 + 17 * q^47 + 3 * q^49 - 2 * q^50 - 15 * q^53 - 6 * q^55 - 10 * q^56 - 10 * q^58 + 2 * q^59 - 3 * q^61 - q^62 - 11 * q^64 + 22 * q^67 - 14 * q^68 + 2 * q^70 + 36 * q^71 - 2 * q^73 - 20 * q^74 - 7 * q^76 - 8 * q^77 - 5 * q^79 - 2 * q^80 + 28 * q^82 + 7 * q^83 + 5 * q^85 - 18 * q^86 - 6 * q^88 - 4 * q^89 - 7 * q^92 - 16 * q^94 + 3 * q^95 - 14 * q^97 - 2 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

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

## 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
 1.80194 −1.24698 0.445042
−2.24698 0 3.04892 −1.00000 0 1.60388 −2.35690 0 2.24698
1.2 −0.554958 0 −1.69202 −1.00000 0 −4.49396 2.04892 0 0.554958
1.3 0.801938 0 −1.35690 −1.00000 0 −1.10992 −2.69202 0 −0.801938
 $$n$$: 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.bn 3
3.b odd 2 1 2535.2.a.bg yes 3
13.b even 2 1 7605.2.a.cd 3
39.d odd 2 1 2535.2.a.u 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2535.2.a.u 3 39.d odd 2 1
2535.2.a.bg yes 3 3.b odd 2 1
7605.2.a.bn 3 1.a even 1 1 trivial
7605.2.a.cd 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{3} + 2T_{2}^{2} - T_{2} - 1$$ T2^3 + 2*T2^2 - T2 - 1 $$T_{7}^{3} + 4T_{7}^{2} - 4T_{7} - 8$$ T7^3 + 4*T7^2 - 4*T7 - 8 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2T^{2} - T - 1$$
$3$ $$T^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 4 T^{2} - 4 T - 8$$
$11$ $$(T - 2)^{3}$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 5 T^{2} - 8 T - 41$$
$19$ $$T^{3} + 3 T^{2} - 4 T - 13$$
$23$ $$T^{3} + 9 T^{2} + 20 T - 1$$
$29$ $$T^{3} + 6 T^{2} - 16 T + 8$$
$31$ $$T^{3} - 5 T^{2} - 36 T + 167$$
$37$ $$T^{3} - 16 T^{2} + 48 T + 64$$
$41$ $$T^{3} + 14 T^{2} + 28 T - 56$$
$43$ $$T^{3} - 20 T^{2} + 68 T + 328$$
$47$ $$T^{3} - 17 T^{2} + 66 T - 43$$
$53$ $$T^{3} + 15 T^{2} - 16 T - 617$$
$59$ $$T^{3} - 2 T^{2} - 148 T + 232$$
$61$ $$T^{3} + 3 T^{2} - 130 T + 169$$
$67$ $$T^{3} - 22 T^{2} + 96 T + 232$$
$71$ $$(T - 12)^{3}$$
$73$ $$T^{3} + 2 T^{2} - 120 T - 344$$
$79$ $$T^{3} + 5 T^{2} - 148 T - 811$$
$83$ $$T^{3} - 7 T^{2} + 14 T - 7$$
$89$ $$T^{3} + 4 T^{2} - 32 T - 64$$
$97$ $$T^{3} + 14 T^{2} + 28 T - 56$$