# Properties

 Label 7605.2.a.bt Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{2} + 2 \beta_1 + 1) q^{7} + 2 q^{8}+O(q^{10})$$ q - b2 * q^2 + (-b2 - b1 + 1) * q^4 - q^5 + (-b2 + 2*b1 + 1) * q^7 + 2 * q^8 $$q - \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{2} + 2 \beta_1 + 1) q^{7} + 2 q^{8} + \beta_{2} q^{10} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{11} + ( - 2 \beta_{2} - 3 \beta_1 + 5) q^{14} + (2 \beta_1 - 2) q^{16} + ( - \beta_{2} - 2 \beta_1 + 2) q^{17} + (\beta_{2} - 3 \beta_1 + 1) q^{19} + (\beta_{2} + \beta_1 - 1) q^{20} + ( - 4 \beta_{2} + 4) q^{22} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{23} + q^{25} + ( - 5 \beta_{2} - 3 \beta_1 + 1) q^{28} + (3 \beta_1 - 3) q^{29} + ( - \beta_{2} - 3 \beta_1 + 4) q^{31} + (2 \beta_{2} - 2 \beta_1 - 2) q^{32} + ( - 3 \beta_{2} + \beta_1 + 1) q^{34} + (\beta_{2} - 2 \beta_1 - 1) q^{35} + ( - 4 \beta_1 + 6) q^{37} + (4 \beta_1 - 6) q^{38} - 2 q^{40} + (2 \beta_{2} + \beta_1 - 7) q^{41} + (2 \beta_{2} + 3 \beta_1 - 6) q^{43} + ( - 4 \beta_{2} + 8) q^{44} + ( - 2 \beta_{2} - 2 \beta_1 + 10) q^{46} - \beta_{2} q^{47} + (\beta_{2} + 3 \beta_1 + 9) q^{49} - \beta_{2} q^{50} + (2 \beta_{2} + 4 \beta_1 + 4) q^{53} + (2 \beta_{2} + 2 \beta_1 - 2) q^{55} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{56} + (3 \beta_{2} - 3 \beta_1 + 3) q^{58} + (6 \beta_{2} + \beta_1 + 3) q^{59} + ( - \beta_{2} - 3 \beta_1 + 4) q^{61} + ( - 5 \beta_{2} + 2 \beta_1) q^{62} + (4 \beta_{2} - 4) q^{64} + (5 \beta_1 + 2) q^{67} + ( - 2 \beta_{2} + 6) q^{68} + (2 \beta_{2} + 3 \beta_1 - 5) q^{70} + ( - 2 \beta_{2} + \beta_1 + 1) q^{71} + (\beta_{2} + 5) q^{73} + ( - 6 \beta_{2} + 4 \beta_1 - 4) q^{74} + (4 \beta_{2} + 2 \beta_1 + 2) q^{76} + ( - 10 \beta_{2} - 6 \beta_1 + 2) q^{77} + ( - 3 \beta_{2} - \beta_1 + 6) q^{79} + ( - 2 \beta_1 + 2) q^{80} + (9 \beta_{2} + \beta_1 - 5) q^{82} + 2 \beta_1 q^{83} + (\beta_{2} + 2 \beta_1 - 2) q^{85} + (8 \beta_{2} - \beta_1 - 3) q^{86} + ( - 4 \beta_{2} - 4 \beta_1 + 4) q^{88} + (4 \beta_{2} - \beta_1 - 7) q^{89} + ( - 4 \beta_{2} + 4 \beta_1 + 8) q^{92} + ( - \beta_{2} - \beta_1 + 3) q^{94} + ( - \beta_{2} + 3 \beta_1 - 1) q^{95} + (2 \beta_{2} + 7 \beta_1 - 2) q^{97} + ( - 8 \beta_{2} - 2 \beta_1) q^{98}+O(q^{100})$$ q - b2 * q^2 + (-b2 - b1 + 1) * q^4 - q^5 + (-b2 + 2*b1 + 1) * q^7 + 2 * q^8 + b2 * q^10 + (-2*b2 - 2*b1 + 2) * q^11 + (-2*b2 - 3*b1 + 5) * q^14 + (2*b1 - 2) * q^16 + (-b2 - 2*b1 + 2) * q^17 + (b2 - 3*b1 + 1) * q^19 + (b2 + b1 - 1) * q^20 + (-4*b2 + 4) * q^22 + (-4*b2 - 2*b1 - 2) * q^23 + q^25 + (-5*b2 - 3*b1 + 1) * q^28 + (3*b1 - 3) * q^29 + (-b2 - 3*b1 + 4) * q^31 + (2*b2 - 2*b1 - 2) * q^32 + (-3*b2 + b1 + 1) * q^34 + (b2 - 2*b1 - 1) * q^35 + (-4*b1 + 6) * q^37 + (4*b1 - 6) * q^38 - 2 * q^40 + (2*b2 + b1 - 7) * q^41 + (2*b2 + 3*b1 - 6) * q^43 + (-4*b2 + 8) * q^44 + (-2*b2 - 2*b1 + 10) * q^46 - b2 * q^47 + (b2 + 3*b1 + 9) * q^49 - b2 * q^50 + (2*b2 + 4*b1 + 4) * q^53 + (2*b2 + 2*b1 - 2) * q^55 + (-2*b2 + 4*b1 + 2) * q^56 + (3*b2 - 3*b1 + 3) * q^58 + (6*b2 + b1 + 3) * q^59 + (-b2 - 3*b1 + 4) * q^61 + (-5*b2 + 2*b1) * q^62 + (4*b2 - 4) * q^64 + (5*b1 + 2) * q^67 + (-2*b2 + 6) * q^68 + (2*b2 + 3*b1 - 5) * q^70 + (-2*b2 + b1 + 1) * q^71 + (b2 + 5) * q^73 + (-6*b2 + 4*b1 - 4) * q^74 + (4*b2 + 2*b1 + 2) * q^76 + (-10*b2 - 6*b1 + 2) * q^77 + (-3*b2 - b1 + 6) * q^79 + (-2*b1 + 2) * q^80 + (9*b2 + b1 - 5) * q^82 + 2*b1 * q^83 + (b2 + 2*b1 - 2) * q^85 + (8*b2 - b1 - 3) * q^86 + (-4*b2 - 4*b1 + 4) * q^88 + (4*b2 - b1 - 7) * q^89 + (-4*b2 + 4*b1 + 8) * q^92 + (-b2 - b1 + 3) * q^94 + (-b2 + 3*b1 - 1) * q^95 + (2*b2 + 7*b1 - 2) * q^97 + (-8*b2 - 2*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{4} - 3 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10})$$ 3 * q + 2 * q^4 - 3 * q^5 + 5 * q^7 + 6 * q^8 $$3 q + 2 q^{4} - 3 q^{5} + 5 q^{7} + 6 q^{8} + 4 q^{11} + 12 q^{14} - 4 q^{16} + 4 q^{17} - 2 q^{20} + 12 q^{22} - 8 q^{23} + 3 q^{25} - 6 q^{29} + 9 q^{31} - 8 q^{32} + 4 q^{34} - 5 q^{35} + 14 q^{37} - 14 q^{38} - 6 q^{40} - 20 q^{41} - 15 q^{43} + 24 q^{44} + 28 q^{46} + 30 q^{49} + 16 q^{53} - 4 q^{55} + 10 q^{56} + 6 q^{58} + 10 q^{59} + 9 q^{61} + 2 q^{62} - 12 q^{64} + 11 q^{67} + 18 q^{68} - 12 q^{70} + 4 q^{71} + 15 q^{73} - 8 q^{74} + 8 q^{76} + 17 q^{79} + 4 q^{80} - 14 q^{82} + 2 q^{83} - 4 q^{85} - 10 q^{86} + 8 q^{88} - 22 q^{89} + 28 q^{92} + 8 q^{94} + q^{97} - 2 q^{98}+O(q^{100})$$ 3 * q + 2 * q^4 - 3 * q^5 + 5 * q^7 + 6 * q^8 + 4 * q^11 + 12 * q^14 - 4 * q^16 + 4 * q^17 - 2 * q^20 + 12 * q^22 - 8 * q^23 + 3 * q^25 - 6 * q^29 + 9 * q^31 - 8 * q^32 + 4 * q^34 - 5 * q^35 + 14 * q^37 - 14 * q^38 - 6 * q^40 - 20 * q^41 - 15 * q^43 + 24 * q^44 + 28 * q^46 + 30 * q^49 + 16 * q^53 - 4 * q^55 + 10 * q^56 + 6 * q^58 + 10 * q^59 + 9 * q^61 + 2 * q^62 - 12 * q^64 + 11 * q^67 + 18 * q^68 - 12 * q^70 + 4 * q^71 + 15 * q^73 - 8 * q^74 + 8 * q^76 + 17 * q^79 + 4 * q^80 - 14 * q^82 + 2 * q^83 - 4 * q^85 - 10 * q^86 + 8 * q^88 - 22 * q^89 + 28 * q^92 + 8 * q^94 + q^97 - 2 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 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
 −1.48119 2.17009 0.311108
−1.67513 0 0.806063 −1.00000 0 −3.63752 2.00000 0 1.67513
1.2 −0.539189 0 −1.70928 −1.00000 0 4.80098 2.00000 0 0.539189
1.3 2.21432 0 2.90321 −1.00000 0 3.83654 2.00000 0 −2.21432
 $$n$$: 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.bt 3
3.b odd 2 1 2535.2.a.z 3
13.b even 2 1 7605.2.a.bu 3
13.e even 6 2 585.2.j.g 6
39.d odd 2 1 2535.2.a.y 3
39.h odd 6 2 195.2.i.e 6
195.y odd 6 2 975.2.i.m 6
195.bf even 12 4 975.2.bb.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.e 6 39.h odd 6 2
585.2.j.g 6 13.e even 6 2
975.2.i.m 6 195.y odd 6 2
975.2.bb.j 12 195.bf even 12 4
2535.2.a.y 3 39.d odd 2 1
2535.2.a.z 3 3.b odd 2 1
7605.2.a.bt 3 1.a even 1 1 trivial
7605.2.a.bu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{3} - 4T_{2} - 2$$ T2^3 - 4*T2 - 2 $$T_{7}^{3} - 5T_{7}^{2} - 13T_{7} + 67$$ T7^3 - 5*T7^2 - 13*T7 + 67 $$T_{11}^{3} - 4T_{11}^{2} - 16T_{11} + 32$$ T11^3 - 4*T11^2 - 16*T11 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 4T - 2$$
$3$ $$T^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - 5 T^{2} - 13 T + 67$$
$11$ $$T^{3} - 4 T^{2} - 16 T + 32$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 4 T^{2} - 8 T + 34$$
$19$ $$T^{3} - 40T - 76$$
$23$ $$T^{3} + 8 T^{2} - 40 T - 304$$
$29$ $$T^{3} + 6 T^{2} - 18 T - 54$$
$31$ $$T^{3} - 9 T^{2} - T + 109$$
$37$ $$T^{3} - 14 T^{2} + 12 T + 152$$
$41$ $$T^{3} + 20 T^{2} + 118 T + 214$$
$43$ $$T^{3} + 15 T^{2} + 41 T - 107$$
$47$ $$T^{3} - 4T - 2$$
$53$ $$T^{3} - 16 T^{2} + 32 T - 16$$
$59$ $$T^{3} - 10 T^{2} - 102 T + 970$$
$61$ $$T^{3} - 9 T^{2} - T + 109$$
$67$ $$T^{3} - 11 T^{2} - 43 T + 247$$
$71$ $$T^{3} - 4 T^{2} - 18 T + 46$$
$73$ $$T^{3} - 15 T^{2} + 71 T - 103$$
$79$ $$T^{3} - 17 T^{2} + 63 T - 67$$
$83$ $$T^{3} - 2 T^{2} - 12 T + 8$$
$89$ $$T^{3} + 22 T^{2} + 86 T - 134$$
$97$ $$T^{3} - T^{2} - 151 T - 547$$