# Properties

 Label 7605.2.a.bk 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (2 \beta + 2) q^{4} + q^{5} + ( - 2 \beta + 1) q^{7} + (2 \beta + 6) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (2*b + 2) * q^4 + q^5 + (-2*b + 1) * q^7 + (2*b + 6) * q^8 $$q + (\beta + 1) q^{2} + (2 \beta + 2) q^{4} + q^{5} + ( - 2 \beta + 1) q^{7} + (2 \beta + 6) q^{8} + (\beta + 1) q^{10} - 2 \beta q^{11} + ( - \beta - 5) q^{14} + (4 \beta + 8) q^{16} + ( - \beta + 5) q^{17} + (2 \beta - 2) q^{19} + (2 \beta + 2) q^{20} + ( - 2 \beta - 6) q^{22} + (2 \beta + 4) q^{23} + q^{25} + ( - 2 \beta - 10) q^{28} + ( - \beta + 1) q^{29} + (3 \beta + 2) q^{31} + (8 \beta + 8) q^{32} + (4 \beta + 2) q^{34} + ( - 2 \beta + 1) q^{35} + 4 q^{37} + 4 q^{38} + (2 \beta + 6) q^{40} + (\beta + 7) q^{41} + ( - \beta - 2) q^{43} + ( - 4 \beta - 12) q^{44} + (6 \beta + 10) q^{46} + ( - 3 \beta - 5) q^{47} + ( - 4 \beta + 6) q^{49} + (\beta + 1) q^{50} - 4 \beta q^{53} - 2 \beta q^{55} + ( - 10 \beta - 6) q^{56} - 2 q^{58} + (\beta + 9) q^{59} + ( - 2 \beta + 1) q^{61} + (5 \beta + 11) q^{62} + (8 \beta + 16) q^{64} + ( - 2 \beta + 9) q^{67} + (8 \beta + 4) q^{68} + ( - \beta - 5) q^{70} + ( - \beta + 11) q^{71} + (6 \beta - 5) q^{73} + (4 \beta + 4) q^{74} + 8 q^{76} + ( - 2 \beta + 12) q^{77} + ( - 4 \beta - 5) q^{79} + (4 \beta + 8) q^{80} + (8 \beta + 10) q^{82} + (2 \beta + 6) q^{83} + ( - \beta + 5) q^{85} + ( - 3 \beta - 5) q^{86} + ( - 12 \beta - 12) q^{88} + (\beta + 3) q^{89} + (12 \beta + 20) q^{92} + ( - 8 \beta - 14) q^{94} + (2 \beta - 2) q^{95} + (2 \beta - 13) q^{97} + (2 \beta - 6) q^{98} +O(q^{100})$$ q + (b + 1) * q^2 + (2*b + 2) * q^4 + q^5 + (-2*b + 1) * q^7 + (2*b + 6) * q^8 + (b + 1) * q^10 - 2*b * q^11 + (-b - 5) * q^14 + (4*b + 8) * q^16 + (-b + 5) * q^17 + (2*b - 2) * q^19 + (2*b + 2) * q^20 + (-2*b - 6) * q^22 + (2*b + 4) * q^23 + q^25 + (-2*b - 10) * q^28 + (-b + 1) * q^29 + (3*b + 2) * q^31 + (8*b + 8) * q^32 + (4*b + 2) * q^34 + (-2*b + 1) * q^35 + 4 * q^37 + 4 * q^38 + (2*b + 6) * q^40 + (b + 7) * q^41 + (-b - 2) * q^43 + (-4*b - 12) * q^44 + (6*b + 10) * q^46 + (-3*b - 5) * q^47 + (-4*b + 6) * q^49 + (b + 1) * q^50 - 4*b * q^53 - 2*b * q^55 + (-10*b - 6) * q^56 - 2 * q^58 + (b + 9) * q^59 + (-2*b + 1) * q^61 + (5*b + 11) * q^62 + (8*b + 16) * q^64 + (-2*b + 9) * q^67 + (8*b + 4) * q^68 + (-b - 5) * q^70 + (-b + 11) * q^71 + (6*b - 5) * q^73 + (4*b + 4) * q^74 + 8 * q^76 + (-2*b + 12) * q^77 + (-4*b - 5) * q^79 + (4*b + 8) * q^80 + (8*b + 10) * q^82 + (2*b + 6) * q^83 + (-b + 5) * q^85 + (-3*b - 5) * q^86 + (-12*b - 12) * q^88 + (b + 3) * q^89 + (12*b + 20) * q^92 + (-8*b - 14) * q^94 + (2*b - 2) * q^95 + (2*b - 13) * q^97 + (2*b - 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 12 * q^8 $$2 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} + 12 q^{8} + 2 q^{10} - 10 q^{14} + 16 q^{16} + 10 q^{17} - 4 q^{19} + 4 q^{20} - 12 q^{22} + 8 q^{23} + 2 q^{25} - 20 q^{28} + 2 q^{29} + 4 q^{31} + 16 q^{32} + 4 q^{34} + 2 q^{35} + 8 q^{37} + 8 q^{38} + 12 q^{40} + 14 q^{41} - 4 q^{43} - 24 q^{44} + 20 q^{46} - 10 q^{47} + 12 q^{49} + 2 q^{50} - 12 q^{56} - 4 q^{58} + 18 q^{59} + 2 q^{61} + 22 q^{62} + 32 q^{64} + 18 q^{67} + 8 q^{68} - 10 q^{70} + 22 q^{71} - 10 q^{73} + 8 q^{74} + 16 q^{76} + 24 q^{77} - 10 q^{79} + 16 q^{80} + 20 q^{82} + 12 q^{83} + 10 q^{85} - 10 q^{86} - 24 q^{88} + 6 q^{89} + 40 q^{92} - 28 q^{94} - 4 q^{95} - 26 q^{97} - 12 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 + 12 * q^8 + 2 * q^10 - 10 * q^14 + 16 * q^16 + 10 * q^17 - 4 * q^19 + 4 * q^20 - 12 * q^22 + 8 * q^23 + 2 * q^25 - 20 * q^28 + 2 * q^29 + 4 * q^31 + 16 * q^32 + 4 * q^34 + 2 * q^35 + 8 * q^37 + 8 * q^38 + 12 * q^40 + 14 * q^41 - 4 * q^43 - 24 * q^44 + 20 * q^46 - 10 * q^47 + 12 * q^49 + 2 * q^50 - 12 * q^56 - 4 * q^58 + 18 * q^59 + 2 * q^61 + 22 * q^62 + 32 * q^64 + 18 * q^67 + 8 * q^68 - 10 * q^70 + 22 * q^71 - 10 * q^73 + 8 * q^74 + 16 * q^76 + 24 * q^77 - 10 * q^79 + 16 * q^80 + 20 * q^82 + 12 * q^83 + 10 * q^85 - 10 * q^86 - 24 * q^88 + 6 * q^89 + 40 * q^92 - 28 * q^94 - 4 * q^95 - 26 * q^97 - 12 * 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
 −1.73205 1.73205
−0.732051 0 −1.46410 1.00000 0 4.46410 2.53590 0 −0.732051
1.2 2.73205 0 5.46410 1.00000 0 −2.46410 9.46410 0 2.73205
 $$n$$: 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.bk 2
3.b odd 2 1 2535.2.a.n 2
13.b even 2 1 7605.2.a.y 2
13.f odd 12 2 585.2.bu.a 4
39.d odd 2 1 2535.2.a.s 2
39.k even 12 2 195.2.bb.a 4
195.bc odd 12 2 975.2.w.a 4
195.bh even 12 2 975.2.bc.h 4
195.bn odd 12 2 975.2.w.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bb.a 4 39.k even 12 2
585.2.bu.a 4 13.f odd 12 2
975.2.w.a 4 195.bc odd 12 2
975.2.w.f 4 195.bn odd 12 2
975.2.bc.h 4 195.bh even 12 2
2535.2.a.n 2 3.b odd 2 1
2535.2.a.s 2 39.d odd 2 1
7605.2.a.y 2 13.b even 2 1
7605.2.a.bk 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{2} - 2T_{2} - 2$$ T2^2 - 2*T2 - 2 $$T_{7}^{2} - 2T_{7} - 11$$ T7^2 - 2*T7 - 11 $$T_{11}^{2} - 12$$ T11^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 2$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 2T - 11$$
$11$ $$T^{2} - 12$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 10T + 22$$
$19$ $$T^{2} + 4T - 8$$
$23$ $$T^{2} - 8T + 4$$
$29$ $$T^{2} - 2T - 2$$
$31$ $$T^{2} - 4T - 23$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} - 14T + 46$$
$43$ $$T^{2} + 4T + 1$$
$47$ $$T^{2} + 10T - 2$$
$53$ $$T^{2} - 48$$
$59$ $$T^{2} - 18T + 78$$
$61$ $$T^{2} - 2T - 11$$
$67$ $$T^{2} - 18T + 69$$
$71$ $$T^{2} - 22T + 118$$
$73$ $$T^{2} + 10T - 83$$
$79$ $$T^{2} + 10T - 23$$
$83$ $$T^{2} - 12T + 24$$
$89$ $$T^{2} - 6T + 6$$
$97$ $$T^{2} + 26T + 157$$