# Properties

 Label 7605.2.a.bh 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta + 2) q^{4} + q^{5} + (\beta + 1) q^{7} + (\beta + 4) q^{8}+O(q^{10})$$ q + b * q^2 + (b + 2) * q^4 + q^5 + (b + 1) * q^7 + (b + 4) * q^8 $$q + \beta q^{2} + (\beta + 2) q^{4} + q^{5} + (\beta + 1) q^{7} + (\beta + 4) q^{8} + \beta q^{10} + (\beta + 3) q^{11} + (2 \beta + 4) q^{14} + 3 \beta q^{16} + (\beta - 3) q^{17} + (2 \beta - 4) q^{19} + (\beta + 2) q^{20} + (4 \beta + 4) q^{22} + (\beta + 3) q^{23} + q^{25} + (4 \beta + 6) q^{28} + ( - 4 \beta + 2) q^{29} - 6 q^{31} + (\beta + 4) q^{32} + ( - 2 \beta + 4) q^{34} + (\beta + 1) q^{35} + ( - \beta - 3) q^{37} + ( - 2 \beta + 8) q^{38} + (\beta + 4) q^{40} + ( - \beta + 3) q^{41} + ( - 2 \beta - 2) q^{43} + (6 \beta + 10) q^{44} + (4 \beta + 4) q^{46} - 4 q^{47} + (3 \beta - 2) q^{49} + \beta q^{50} + (5 \beta - 3) q^{53} + (\beta + 3) q^{55} + (6 \beta + 8) q^{56} + ( - 2 \beta - 16) q^{58} + 4 q^{59} + (\beta - 11) q^{61} - 6 \beta q^{62} + ( - \beta + 4) q^{64} + ( - 2 \beta + 8) q^{67} - 2 q^{68} + (2 \beta + 4) q^{70} + ( - 3 \beta - 1) q^{71} + ( - 2 \beta - 6) q^{73} + ( - 4 \beta - 4) q^{74} + 2 \beta q^{76} + (5 \beta + 7) q^{77} + ( - \beta - 3) q^{79} + 3 \beta q^{80} + (2 \beta - 4) q^{82} + ( - 6 \beta + 2) q^{83} + (\beta - 3) q^{85} + ( - 4 \beta - 8) q^{86} + (8 \beta + 16) q^{88} + ( - 3 \beta + 9) q^{89} + (6 \beta + 10) q^{92} - 4 \beta q^{94} + (2 \beta - 4) q^{95} + ( - 7 \beta + 3) q^{97} + (\beta + 12) q^{98} +O(q^{100})$$ q + b * q^2 + (b + 2) * q^4 + q^5 + (b + 1) * q^7 + (b + 4) * q^8 + b * q^10 + (b + 3) * q^11 + (2*b + 4) * q^14 + 3*b * q^16 + (b - 3) * q^17 + (2*b - 4) * q^19 + (b + 2) * q^20 + (4*b + 4) * q^22 + (b + 3) * q^23 + q^25 + (4*b + 6) * q^28 + (-4*b + 2) * q^29 - 6 * q^31 + (b + 4) * q^32 + (-2*b + 4) * q^34 + (b + 1) * q^35 + (-b - 3) * q^37 + (-2*b + 8) * q^38 + (b + 4) * q^40 + (-b + 3) * q^41 + (-2*b - 2) * q^43 + (6*b + 10) * q^44 + (4*b + 4) * q^46 - 4 * q^47 + (3*b - 2) * q^49 + b * q^50 + (5*b - 3) * q^53 + (b + 3) * q^55 + (6*b + 8) * q^56 + (-2*b - 16) * q^58 + 4 * q^59 + (b - 11) * q^61 - 6*b * q^62 + (-b + 4) * q^64 + (-2*b + 8) * q^67 - 2 * q^68 + (2*b + 4) * q^70 + (-3*b - 1) * q^71 + (-2*b - 6) * q^73 + (-4*b - 4) * q^74 + 2*b * q^76 + (5*b + 7) * q^77 + (-b - 3) * q^79 + 3*b * q^80 + (2*b - 4) * q^82 + (-6*b + 2) * q^83 + (b - 3) * q^85 + (-4*b - 8) * q^86 + (8*b + 16) * q^88 + (-3*b + 9) * q^89 + (6*b + 10) * q^92 - 4*b * q^94 + (2*b - 4) * q^95 + (-7*b + 3) * q^97 + (b + 12) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10})$$ 2 * q + q^2 + 5 * q^4 + 2 * q^5 + 3 * q^7 + 9 * q^8 $$2 q + q^{2} + 5 q^{4} + 2 q^{5} + 3 q^{7} + 9 q^{8} + q^{10} + 7 q^{11} + 10 q^{14} + 3 q^{16} - 5 q^{17} - 6 q^{19} + 5 q^{20} + 12 q^{22} + 7 q^{23} + 2 q^{25} + 16 q^{28} - 12 q^{31} + 9 q^{32} + 6 q^{34} + 3 q^{35} - 7 q^{37} + 14 q^{38} + 9 q^{40} + 5 q^{41} - 6 q^{43} + 26 q^{44} + 12 q^{46} - 8 q^{47} - q^{49} + q^{50} - q^{53} + 7 q^{55} + 22 q^{56} - 34 q^{58} + 8 q^{59} - 21 q^{61} - 6 q^{62} + 7 q^{64} + 14 q^{67} - 4 q^{68} + 10 q^{70} - 5 q^{71} - 14 q^{73} - 12 q^{74} + 2 q^{76} + 19 q^{77} - 7 q^{79} + 3 q^{80} - 6 q^{82} - 2 q^{83} - 5 q^{85} - 20 q^{86} + 40 q^{88} + 15 q^{89} + 26 q^{92} - 4 q^{94} - 6 q^{95} - q^{97} + 25 q^{98}+O(q^{100})$$ 2 * q + q^2 + 5 * q^4 + 2 * q^5 + 3 * q^7 + 9 * q^8 + q^10 + 7 * q^11 + 10 * q^14 + 3 * q^16 - 5 * q^17 - 6 * q^19 + 5 * q^20 + 12 * q^22 + 7 * q^23 + 2 * q^25 + 16 * q^28 - 12 * q^31 + 9 * q^32 + 6 * q^34 + 3 * q^35 - 7 * q^37 + 14 * q^38 + 9 * q^40 + 5 * q^41 - 6 * q^43 + 26 * q^44 + 12 * q^46 - 8 * q^47 - q^49 + q^50 - q^53 + 7 * q^55 + 22 * q^56 - 34 * q^58 + 8 * q^59 - 21 * q^61 - 6 * q^62 + 7 * q^64 + 14 * q^67 - 4 * q^68 + 10 * q^70 - 5 * q^71 - 14 * q^73 - 12 * q^74 + 2 * q^76 + 19 * q^77 - 7 * q^79 + 3 * q^80 - 6 * q^82 - 2 * q^83 - 5 * q^85 - 20 * q^86 + 40 * q^88 + 15 * q^89 + 26 * q^92 - 4 * q^94 - 6 * q^95 - q^97 + 25 * 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
 −1.56155 2.56155
−1.56155 0 0.438447 1.00000 0 −0.561553 2.43845 0 −1.56155
1.2 2.56155 0 4.56155 1.00000 0 3.56155 6.56155 0 2.56155
 $$n$$: 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.bh 2
3.b odd 2 1 2535.2.a.p 2
13.b even 2 1 7605.2.a.bc 2
13.d odd 4 2 585.2.b.e 4
39.d odd 2 1 2535.2.a.q 2
39.f even 4 2 195.2.b.c 4
156.l odd 4 2 3120.2.g.n 4
195.j odd 4 2 975.2.h.e 4
195.n even 4 2 975.2.b.f 4
195.u odd 4 2 975.2.h.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.b.c 4 39.f even 4 2
585.2.b.e 4 13.d odd 4 2
975.2.b.f 4 195.n even 4 2
975.2.h.e 4 195.j odd 4 2
975.2.h.g 4 195.u odd 4 2
2535.2.a.p 2 3.b odd 2 1
2535.2.a.q 2 39.d odd 2 1
3120.2.g.n 4 156.l odd 4 2
7605.2.a.bc 2 13.b even 2 1
7605.2.a.bh 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 3T - 2$$
$11$ $$T^{2} - 7T + 8$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 5T + 2$$
$19$ $$T^{2} + 6T - 8$$
$23$ $$T^{2} - 7T + 8$$
$29$ $$T^{2} - 68$$
$31$ $$(T + 6)^{2}$$
$37$ $$T^{2} + 7T + 8$$
$41$ $$T^{2} - 5T + 2$$
$43$ $$T^{2} + 6T - 8$$
$47$ $$(T + 4)^{2}$$
$53$ $$T^{2} + T - 106$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} + 21T + 106$$
$67$ $$T^{2} - 14T + 32$$
$71$ $$T^{2} + 5T - 32$$
$73$ $$T^{2} + 14T + 32$$
$79$ $$T^{2} + 7T + 8$$
$83$ $$T^{2} + 2T - 152$$
$89$ $$T^{2} - 15T + 18$$
$97$ $$T^{2} + T - 208$$