# Properties

 Label 7605.2.a.ba Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta - 1) q^{4} - q^{5} + ( - 2 \beta + 3) q^{7} + (2 \beta - 1) q^{8} +O(q^{10})$$ q - b * q^2 + (b - 1) * q^4 - q^5 + (-2*b + 3) * q^7 + (2*b - 1) * q^8 $$q - \beta q^{2} + (\beta - 1) q^{4} - q^{5} + ( - 2 \beta + 3) q^{7} + (2 \beta - 1) q^{8} + \beta q^{10} + ( - 2 \beta - 1) q^{11} + ( - \beta + 2) q^{14} - 3 \beta q^{16} + ( - 4 \beta + 1) q^{17} + ( - 2 \beta + 3) q^{19} + ( - \beta + 1) q^{20} + (3 \beta + 2) q^{22} + ( - 2 \beta - 5) q^{23} + q^{25} + (3 \beta - 5) q^{28} + ( - 4 \beta + 5) q^{29} + ( - \beta + 5) q^{32} + (3 \beta + 4) q^{34} + (2 \beta - 3) q^{35} + 3 q^{37} + ( - \beta + 2) q^{38} + ( - 2 \beta + 1) q^{40} + (8 \beta - 7) q^{41} + (2 \beta - 5) q^{43} + ( - \beta - 1) q^{44} + (7 \beta + 2) q^{46} - 8 \beta q^{47} + ( - 8 \beta + 6) q^{49} - \beta q^{50} - 6 q^{53} + (2 \beta + 1) q^{55} + (4 \beta - 7) q^{56} + ( - \beta + 4) q^{58} + ( - 6 \beta - 3) q^{59} + ( - 12 \beta + 7) q^{61} + (2 \beta + 1) q^{64} + (6 \beta + 1) q^{67} + (\beta - 5) q^{68} + (\beta - 2) q^{70} + ( - 2 \beta + 5) q^{71} - 6 q^{73} - 3 \beta q^{74} + (3 \beta - 5) q^{76} + q^{77} + 3 \beta q^{80} + ( - \beta - 8) q^{82} + (8 \beta - 4) q^{83} + (4 \beta - 1) q^{85} + (3 \beta - 2) q^{86} + ( - 4 \beta - 3) q^{88} + 9 q^{89} + ( - 5 \beta + 3) q^{92} + (8 \beta + 8) q^{94} + (2 \beta - 3) q^{95} + (4 \beta - 1) q^{97} + (2 \beta + 8) q^{98} +O(q^{100})$$ q - b * q^2 + (b - 1) * q^4 - q^5 + (-2*b + 3) * q^7 + (2*b - 1) * q^8 + b * q^10 + (-2*b - 1) * q^11 + (-b + 2) * q^14 - 3*b * q^16 + (-4*b + 1) * q^17 + (-2*b + 3) * q^19 + (-b + 1) * q^20 + (3*b + 2) * q^22 + (-2*b - 5) * q^23 + q^25 + (3*b - 5) * q^28 + (-4*b + 5) * q^29 + (-b + 5) * q^32 + (3*b + 4) * q^34 + (2*b - 3) * q^35 + 3 * q^37 + (-b + 2) * q^38 + (-2*b + 1) * q^40 + (8*b - 7) * q^41 + (2*b - 5) * q^43 + (-b - 1) * q^44 + (7*b + 2) * q^46 - 8*b * q^47 + (-8*b + 6) * q^49 - b * q^50 - 6 * q^53 + (2*b + 1) * q^55 + (4*b - 7) * q^56 + (-b + 4) * q^58 + (-6*b - 3) * q^59 + (-12*b + 7) * q^61 + (2*b + 1) * q^64 + (6*b + 1) * q^67 + (b - 5) * q^68 + (b - 2) * q^70 + (-2*b + 5) * q^71 - 6 * q^73 - 3*b * q^74 + (3*b - 5) * q^76 + q^77 + 3*b * q^80 + (-b - 8) * q^82 + (8*b - 4) * q^83 + (4*b - 1) * q^85 + (3*b - 2) * q^86 + (-4*b - 3) * q^88 + 9 * q^89 + (-5*b + 3) * q^92 + (8*b + 8) * q^94 + (2*b - 3) * q^95 + (4*b - 1) * q^97 + (2*b + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 2 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q - q^2 - q^4 - 2 * q^5 + 4 * q^7 $$2 q - q^{2} - q^{4} - 2 q^{5} + 4 q^{7} + q^{10} - 4 q^{11} + 3 q^{14} - 3 q^{16} - 2 q^{17} + 4 q^{19} + q^{20} + 7 q^{22} - 12 q^{23} + 2 q^{25} - 7 q^{28} + 6 q^{29} + 9 q^{32} + 11 q^{34} - 4 q^{35} + 6 q^{37} + 3 q^{38} - 6 q^{41} - 8 q^{43} - 3 q^{44} + 11 q^{46} - 8 q^{47} + 4 q^{49} - q^{50} - 12 q^{53} + 4 q^{55} - 10 q^{56} + 7 q^{58} - 12 q^{59} + 2 q^{61} + 4 q^{64} + 8 q^{67} - 9 q^{68} - 3 q^{70} + 8 q^{71} - 12 q^{73} - 3 q^{74} - 7 q^{76} + 2 q^{77} + 3 q^{80} - 17 q^{82} + 2 q^{85} - q^{86} - 10 q^{88} + 18 q^{89} + q^{92} + 24 q^{94} - 4 q^{95} + 2 q^{97} + 18 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - 2 * q^5 + 4 * q^7 + q^10 - 4 * q^11 + 3 * q^14 - 3 * q^16 - 2 * q^17 + 4 * q^19 + q^20 + 7 * q^22 - 12 * q^23 + 2 * q^25 - 7 * q^28 + 6 * q^29 + 9 * q^32 + 11 * q^34 - 4 * q^35 + 6 * q^37 + 3 * q^38 - 6 * q^41 - 8 * q^43 - 3 * q^44 + 11 * q^46 - 8 * q^47 + 4 * q^49 - q^50 - 12 * q^53 + 4 * q^55 - 10 * q^56 + 7 * q^58 - 12 * q^59 + 2 * q^61 + 4 * q^64 + 8 * q^67 - 9 * q^68 - 3 * q^70 + 8 * q^71 - 12 * q^73 - 3 * q^74 - 7 * q^76 + 2 * q^77 + 3 * q^80 - 17 * q^82 + 2 * q^85 - q^86 - 10 * q^88 + 18 * q^89 + q^92 + 24 * q^94 - 4 * q^95 + 2 * q^97 + 18 * 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
 1.61803 −0.618034
−1.61803 0 0.618034 −1.00000 0 −0.236068 2.23607 0 1.61803
1.2 0.618034 0 −1.61803 −1.00000 0 4.23607 −2.23607 0 −0.618034
 $$n$$: 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.ba 2
3.b odd 2 1 845.2.a.e 2
13.b even 2 1 7605.2.a.bf 2
13.c even 3 2 585.2.j.e 4
15.d odd 2 1 4225.2.a.u 2
39.d odd 2 1 845.2.a.b 2
39.f even 4 2 845.2.c.c 4
39.h odd 6 2 845.2.e.g 4
39.i odd 6 2 65.2.e.a 4
39.k even 12 4 845.2.m.e 8
156.p even 6 2 1040.2.q.n 4
195.e odd 2 1 4225.2.a.y 2
195.x odd 6 2 325.2.e.b 4
195.bl even 12 4 325.2.o.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a 4 39.i odd 6 2
325.2.e.b 4 195.x odd 6 2
325.2.o.a 8 195.bl even 12 4
585.2.j.e 4 13.c even 3 2
845.2.a.b 2 39.d odd 2 1
845.2.a.e 2 3.b odd 2 1
845.2.c.c 4 39.f even 4 2
845.2.e.g 4 39.h odd 6 2
845.2.m.e 8 39.k even 12 4
1040.2.q.n 4 156.p even 6 2
4225.2.a.u 2 15.d odd 2 1
4225.2.a.y 2 195.e odd 2 1
7605.2.a.ba 2 1.a even 1 1 trivial
7605.2.a.bf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 4T - 1$$
$11$ $$T^{2} + 4T - 1$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 2T - 19$$
$19$ $$T^{2} - 4T - 1$$
$23$ $$T^{2} + 12T + 31$$
$29$ $$T^{2} - 6T - 11$$
$31$ $$T^{2}$$
$37$ $$(T - 3)^{2}$$
$41$ $$T^{2} + 6T - 71$$
$43$ $$T^{2} + 8T + 11$$
$47$ $$T^{2} + 8T - 64$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 12T - 9$$
$61$ $$T^{2} - 2T - 179$$
$67$ $$T^{2} - 8T - 29$$
$71$ $$T^{2} - 8T + 11$$
$73$ $$(T + 6)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 80$$
$89$ $$(T - 9)^{2}$$
$97$ $$T^{2} - 2T - 19$$