Properties

 Label 7605.2.a.bs 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: 3.3.564.1 Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + (\beta_{2} + 1) q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + 2) * q^4 + q^5 + (b2 + 1) * q^7 + (-b2 - b1 - 1) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + (\beta_{2} + 1) q^{7} + ( - \beta_{2} - \beta_1 - 1) q^{8} - \beta_1 q^{10} + (\beta_{2} + \beta_1 - 2) q^{11} + ( - \beta_{2} - 2 \beta_1 - 1) q^{14} + (2 \beta_1 + 1) q^{16} + (2 \beta_{2} + 2) q^{17} + ( - \beta_{2} + \beta_1 + 2) q^{19} + (\beta_{2} + 2) q^{20} + ( - 2 \beta_{2} + \beta_1 - 5) q^{22} + ( - \beta_1 + 3) q^{23} + q^{25} + (\beta_{2} + 2 \beta_1 + 7) q^{28} + (\beta_{2} + 1) q^{29} + (\beta_{2} - \beta_1 + 4) q^{31} + (\beta_1 - 6) q^{32} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{34} + (\beta_{2} + 1) q^{35} + (\beta_{2} + 7) q^{37} + ( - \beta_1 - 3) q^{38} + ( - \beta_{2} - \beta_1 - 1) q^{40} + (2 \beta_{2} - 2 \beta_1 + 2) q^{41} + (2 \beta_{2} - \beta_1 - 3) q^{43} + ( - \beta_{2} + 5 \beta_1 + 2) q^{44} + (\beta_{2} - 3 \beta_1 + 4) q^{46} + (\beta_{2} + 1) q^{47} + (2 \beta_1 - 1) q^{49} - \beta_1 q^{50} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{53} + (\beta_{2} + \beta_1 - 2) q^{55} + ( - \beta_{2} - 4 \beta_1 - 7) q^{56} + ( - \beta_{2} - 2 \beta_1 - 1) q^{58} + (\beta_{2} + \beta_1 + 4) q^{59} + (\beta_{2} - 2 \beta_1 - 1) q^{61} + ( - 5 \beta_1 + 3) q^{62} + ( - \beta_{2} + 2 \beta_1 - 6) q^{64} + (\beta_{2} - 2 \beta_1 + 7) q^{67} + (2 \beta_{2} + 4 \beta_1 + 14) q^{68} + ( - \beta_{2} - 2 \beta_1 - 1) q^{70} + (\beta_{2} - \beta_1 - 2) q^{71} + ( - \beta_{2} + 4 \beta_1 + 5) q^{73} + ( - \beta_{2} - 8 \beta_1 - 1) q^{74} + (3 \beta_{2} + \beta_1) q^{76} + ( - 2 \beta_{2} + 4 \beta_1 + 4) q^{77} + (2 \beta_{2} - 4 \beta_1 + 6) q^{79} + (2 \beta_1 + 1) q^{80} + ( - 4 \beta_1 + 6) q^{82} + (\beta_{2} - 2 \beta_1 - 5) q^{83} + (2 \beta_{2} + 2) q^{85} + ( - \beta_{2} + \beta_1 + 2) q^{86} + ( - 3 \beta_1 - 9) q^{88} + (2 \beta_{2} - 4 \beta_1 + 2) q^{89} + (2 \beta_{2} - 3 \beta_1 + 5) q^{92} + ( - \beta_{2} - 2 \beta_1 - 1) q^{94} + ( - \beta_{2} + \beta_1 + 2) q^{95} + (2 \beta_{2} - 4) q^{97} + ( - 2 \beta_{2} + \beta_1 - 8) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b2 + 2) * q^4 + q^5 + (b2 + 1) * q^7 + (-b2 - b1 - 1) * q^8 - b1 * q^10 + (b2 + b1 - 2) * q^11 + (-b2 - 2*b1 - 1) * q^14 + (2*b1 + 1) * q^16 + (2*b2 + 2) * q^17 + (-b2 + b1 + 2) * q^19 + (b2 + 2) * q^20 + (-2*b2 + b1 - 5) * q^22 + (-b1 + 3) * q^23 + q^25 + (b2 + 2*b1 + 7) * q^28 + (b2 + 1) * q^29 + (b2 - b1 + 4) * q^31 + (b1 - 6) * q^32 + (-2*b2 - 4*b1 - 2) * q^34 + (b2 + 1) * q^35 + (b2 + 7) * q^37 + (-b1 - 3) * q^38 + (-b2 - b1 - 1) * q^40 + (2*b2 - 2*b1 + 2) * q^41 + (2*b2 - b1 - 3) * q^43 + (-b2 + 5*b1 + 2) * q^44 + (b2 - 3*b1 + 4) * q^46 + (b2 + 1) * q^47 + (2*b1 - 1) * q^49 - b1 * q^50 + (-2*b2 - 2*b1 - 2) * q^53 + (b2 + b1 - 2) * q^55 + (-b2 - 4*b1 - 7) * q^56 + (-b2 - 2*b1 - 1) * q^58 + (b2 + b1 + 4) * q^59 + (b2 - 2*b1 - 1) * q^61 + (-5*b1 + 3) * q^62 + (-b2 + 2*b1 - 6) * q^64 + (b2 - 2*b1 + 7) * q^67 + (2*b2 + 4*b1 + 14) * q^68 + (-b2 - 2*b1 - 1) * q^70 + (b2 - b1 - 2) * q^71 + (-b2 + 4*b1 + 5) * q^73 + (-b2 - 8*b1 - 1) * q^74 + (3*b2 + b1) * q^76 + (-2*b2 + 4*b1 + 4) * q^77 + (2*b2 - 4*b1 + 6) * q^79 + (2*b1 + 1) * q^80 + (-4*b1 + 6) * q^82 + (b2 - 2*b1 - 5) * q^83 + (2*b2 + 2) * q^85 + (-b2 + b1 + 2) * q^86 + (-3*b1 - 9) * q^88 + (2*b2 - 4*b1 + 2) * q^89 + (2*b2 - 3*b1 + 5) * q^92 + (-b2 - 2*b1 - 1) * q^94 + (-b2 + b1 + 2) * q^95 + (2*b2 - 4) * q^97 + (-2*b2 + b1 - 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 5 q^{4} + 3 q^{5} + 2 q^{7} - 3 q^{8}+O(q^{10})$$ 3 * q - q^2 + 5 * q^4 + 3 * q^5 + 2 * q^7 - 3 * q^8 $$3 q - q^{2} + 5 q^{4} + 3 q^{5} + 2 q^{7} - 3 q^{8} - q^{10} - 6 q^{11} - 4 q^{14} + 5 q^{16} + 4 q^{17} + 8 q^{19} + 5 q^{20} - 12 q^{22} + 8 q^{23} + 3 q^{25} + 22 q^{28} + 2 q^{29} + 10 q^{31} - 17 q^{32} - 8 q^{34} + 2 q^{35} + 20 q^{37} - 10 q^{38} - 3 q^{40} + 2 q^{41} - 12 q^{43} + 12 q^{44} + 8 q^{46} + 2 q^{47} - q^{49} - q^{50} - 6 q^{53} - 6 q^{55} - 24 q^{56} - 4 q^{58} + 12 q^{59} - 6 q^{61} + 4 q^{62} - 15 q^{64} + 18 q^{67} + 44 q^{68} - 4 q^{70} - 8 q^{71} + 20 q^{73} - 10 q^{74} - 2 q^{76} + 18 q^{77} + 12 q^{79} + 5 q^{80} + 14 q^{82} - 18 q^{83} + 4 q^{85} + 8 q^{86} - 30 q^{88} + 10 q^{92} - 4 q^{94} + 8 q^{95} - 14 q^{97} - 21 q^{98}+O(q^{100})$$ 3 * q - q^2 + 5 * q^4 + 3 * q^5 + 2 * q^7 - 3 * q^8 - q^10 - 6 * q^11 - 4 * q^14 + 5 * q^16 + 4 * q^17 + 8 * q^19 + 5 * q^20 - 12 * q^22 + 8 * q^23 + 3 * q^25 + 22 * q^28 + 2 * q^29 + 10 * q^31 - 17 * q^32 - 8 * q^34 + 2 * q^35 + 20 * q^37 - 10 * q^38 - 3 * q^40 + 2 * q^41 - 12 * q^43 + 12 * q^44 + 8 * q^46 + 2 * q^47 - q^49 - q^50 - 6 * q^53 - 6 * q^55 - 24 * q^56 - 4 * q^58 + 12 * q^59 - 6 * q^61 + 4 * q^62 - 15 * q^64 + 18 * q^67 + 44 * q^68 - 4 * q^70 - 8 * q^71 + 20 * q^73 - 10 * q^74 - 2 * q^76 + 18 * q^77 + 12 * q^79 + 5 * q^80 + 14 * q^82 - 18 * q^83 + 4 * q^85 + 8 * q^86 - 30 * q^88 + 10 * q^92 - 4 * q^94 + 8 * q^95 - 14 * q^97 - 21 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4

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
 2.51414 0.571993 −2.08613
−2.51414 0 4.32088 1.00000 0 3.32088 −5.83502 0 −2.51414
1.2 −0.571993 0 −1.67282 1.00000 0 −2.67282 2.10083 0 −0.571993
1.3 2.08613 0 2.35194 1.00000 0 1.35194 0.734191 0 2.08613
 $$n$$: 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.bs 3
3.b odd 2 1 845.2.a.k 3
13.b even 2 1 7605.2.a.cc 3
13.d odd 4 2 585.2.b.g 6
15.d odd 2 1 4225.2.a.bc 3
39.d odd 2 1 845.2.a.i 3
39.f even 4 2 65.2.c.a 6
39.h odd 6 2 845.2.e.k 6
39.i odd 6 2 845.2.e.i 6
39.k even 12 4 845.2.m.h 12
156.l odd 4 2 1040.2.k.d 6
195.e odd 2 1 4225.2.a.be 3
195.j odd 4 2 325.2.d.f 6
195.n even 4 2 325.2.c.g 6
195.u odd 4 2 325.2.d.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.c.a 6 39.f even 4 2
325.2.c.g 6 195.n even 4 2
325.2.d.e 6 195.u odd 4 2
325.2.d.f 6 195.j odd 4 2
585.2.b.g 6 13.d odd 4 2
845.2.a.i 3 39.d odd 2 1
845.2.a.k 3 3.b odd 2 1
845.2.e.i 6 39.i odd 6 2
845.2.e.k 6 39.h odd 6 2
845.2.m.h 12 39.k even 12 4
1040.2.k.d 6 156.l odd 4 2
4225.2.a.bc 3 15.d odd 2 1
4225.2.a.be 3 195.e odd 2 1
7605.2.a.bs 3 1.a even 1 1 trivial
7605.2.a.cc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{3} + T_{2}^{2} - 5T_{2} - 3$$ T2^3 + T2^2 - 5*T2 - 3 $$T_{7}^{3} - 2T_{7}^{2} - 8T_{7} + 12$$ T7^3 - 2*T7^2 - 8*T7 + 12 $$T_{11}^{3} + 6T_{11}^{2} - 6T_{11} - 54$$ T11^3 + 6*T11^2 - 6*T11 - 54

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 5T - 3$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 2 T^{2} - 8 T + 12$$
$11$ $$T^{3} + 6 T^{2} - 6 T - 54$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 4 T^{2} - 32 T + 96$$
$19$ $$T^{3} - 8 T^{2} + 10 T + 6$$
$23$ $$T^{3} - 8 T^{2} + 16 T - 6$$
$29$ $$T^{3} - 2 T^{2} - 8 T + 12$$
$31$ $$T^{3} - 10 T^{2} + 22 T + 6$$
$37$ $$T^{3} - 20 T^{2} + 124 T - 228$$
$41$ $$T^{3} - 2 T^{2} - 44 T + 72$$
$43$ $$T^{3} + 12 T^{2} + 12 T + 2$$
$47$ $$T^{3} - 2 T^{2} - 8 T + 12$$
$53$ $$T^{3} + 6 T^{2} - 60 T + 72$$
$59$ $$T^{3} - 12 T^{2} + 30 T - 18$$
$61$ $$T^{3} + 6 T^{2} - 12 T - 76$$
$67$ $$T^{3} - 18 T^{2} + 84 T - 108$$
$71$ $$T^{3} + 8 T^{2} + 10 T - 6$$
$73$ $$T^{3} - 20 T^{2} + 52 T + 516$$
$79$ $$T^{3} - 12 T^{2} - 48 T + 32$$
$83$ $$T^{3} + 18 T^{2} + 84 T + 36$$
$89$ $$T^{3} - 96T - 288$$
$97$ $$T^{3} + 14 T^{2} + 28 T - 24$$