# Properties

 Label 7605.2.a.bi Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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## 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 585) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta + 2) q^{4} + q^{5} + ( - \beta + 3) q^{7} + (\beta + 4) q^{8}+O(q^{10})$$ q + b * q^2 + (b + 2) * q^4 + q^5 + (-b + 3) * q^7 + (b + 4) * q^8 $$q + \beta q^{2} + (\beta + 2) q^{4} + q^{5} + ( - \beta + 3) q^{7} + (\beta + 4) q^{8} + \beta q^{10} + ( - \beta + 1) q^{11} + (2 \beta - 4) q^{14} + 3 \beta q^{16} + ( - \beta + 1) q^{17} + 2 \beta q^{19} + (\beta + 2) q^{20} - 4 q^{22} + (\beta - 5) q^{23} + q^{25} + 2 q^{28} + (2 \beta + 2) q^{29} - 6 q^{31} + (\beta + 4) q^{32} - 4 q^{34} + ( - \beta + 3) q^{35} + (3 \beta + 3) q^{37} + (2 \beta + 8) q^{38} + (\beta + 4) q^{40} + (\beta + 1) q^{41} + (2 \beta - 2) q^{43} + ( - 2 \beta - 2) q^{44} + ( - 4 \beta + 4) q^{46} + (2 \beta + 6) q^{47} + ( - 5 \beta + 6) q^{49} + \beta q^{50} + (3 \beta - 3) q^{53} + ( - \beta + 1) q^{55} + ( - 2 \beta + 8) q^{56} + (4 \beta + 8) q^{58} + 12 q^{59} + ( - 3 \beta + 1) q^{61} - 6 \beta q^{62} + ( - \beta + 4) q^{64} + (6 \beta - 4) q^{67} + ( - 2 \beta - 2) q^{68} + (2 \beta - 4) q^{70} + ( - \beta + 13) q^{71} + 6 q^{73} + (6 \beta + 12) q^{74} + (6 \beta + 8) q^{76} + ( - 3 \beta + 7) q^{77} + (3 \beta - 3) q^{79} + 3 \beta q^{80} + (2 \beta + 4) q^{82} + ( - 8 \beta + 4) q^{83} + ( - \beta + 1) q^{85} + 8 q^{86} - 4 \beta q^{88} + (3 \beta + 3) q^{89} + ( - 2 \beta - 6) q^{92} + (8 \beta + 8) q^{94} + 2 \beta q^{95} + ( - 7 \beta + 1) q^{97} + (\beta - 20) q^{98} +O(q^{100})$$ q + b * q^2 + (b + 2) * q^4 + q^5 + (-b + 3) * q^7 + (b + 4) * q^8 + b * q^10 + (-b + 1) * q^11 + (2*b - 4) * q^14 + 3*b * q^16 + (-b + 1) * q^17 + 2*b * q^19 + (b + 2) * q^20 - 4 * q^22 + (b - 5) * q^23 + q^25 + 2 * q^28 + (2*b + 2) * q^29 - 6 * q^31 + (b + 4) * q^32 - 4 * q^34 + (-b + 3) * q^35 + (3*b + 3) * q^37 + (2*b + 8) * q^38 + (b + 4) * q^40 + (b + 1) * q^41 + (2*b - 2) * q^43 + (-2*b - 2) * q^44 + (-4*b + 4) * q^46 + (2*b + 6) * q^47 + (-5*b + 6) * q^49 + b * q^50 + (3*b - 3) * q^53 + (-b + 1) * q^55 + (-2*b + 8) * q^56 + (4*b + 8) * q^58 + 12 * q^59 + (-3*b + 1) * q^61 - 6*b * q^62 + (-b + 4) * q^64 + (6*b - 4) * q^67 + (-2*b - 2) * q^68 + (2*b - 4) * q^70 + (-b + 13) * q^71 + 6 * q^73 + (6*b + 12) * q^74 + (6*b + 8) * q^76 + (-3*b + 7) * q^77 + (3*b - 3) * q^79 + 3*b * q^80 + (2*b + 4) * q^82 + (-8*b + 4) * q^83 + (-b + 1) * q^85 + 8 * q^86 - 4*b * q^88 + (3*b + 3) * q^89 + (-2*b - 6) * q^92 + (8*b + 8) * q^94 + 2*b * q^95 + (-7*b + 1) * q^97 + (b - 20) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 5 q^{4} + 2 q^{5} + 5 q^{7} + 9 q^{8}+O(q^{10})$$ 2 * q + q^2 + 5 * q^4 + 2 * q^5 + 5 * q^7 + 9 * q^8 $$2 q + q^{2} + 5 q^{4} + 2 q^{5} + 5 q^{7} + 9 q^{8} + q^{10} + q^{11} - 6 q^{14} + 3 q^{16} + q^{17} + 2 q^{19} + 5 q^{20} - 8 q^{22} - 9 q^{23} + 2 q^{25} + 4 q^{28} + 6 q^{29} - 12 q^{31} + 9 q^{32} - 8 q^{34} + 5 q^{35} + 9 q^{37} + 18 q^{38} + 9 q^{40} + 3 q^{41} - 2 q^{43} - 6 q^{44} + 4 q^{46} + 14 q^{47} + 7 q^{49} + q^{50} - 3 q^{53} + q^{55} + 14 q^{56} + 20 q^{58} + 24 q^{59} - q^{61} - 6 q^{62} + 7 q^{64} - 2 q^{67} - 6 q^{68} - 6 q^{70} + 25 q^{71} + 12 q^{73} + 30 q^{74} + 22 q^{76} + 11 q^{77} - 3 q^{79} + 3 q^{80} + 10 q^{82} + q^{85} + 16 q^{86} - 4 q^{88} + 9 q^{89} - 14 q^{92} + 24 q^{94} + 2 q^{95} - 5 q^{97} - 39 q^{98}+O(q^{100})$$ 2 * q + q^2 + 5 * q^4 + 2 * q^5 + 5 * q^7 + 9 * q^8 + q^10 + q^11 - 6 * q^14 + 3 * q^16 + q^17 + 2 * q^19 + 5 * q^20 - 8 * q^22 - 9 * q^23 + 2 * q^25 + 4 * q^28 + 6 * q^29 - 12 * q^31 + 9 * q^32 - 8 * q^34 + 5 * q^35 + 9 * q^37 + 18 * q^38 + 9 * q^40 + 3 * q^41 - 2 * q^43 - 6 * q^44 + 4 * q^46 + 14 * q^47 + 7 * q^49 + q^50 - 3 * q^53 + q^55 + 14 * q^56 + 20 * q^58 + 24 * q^59 - q^61 - 6 * q^62 + 7 * q^64 - 2 * q^67 - 6 * q^68 - 6 * q^70 + 25 * q^71 + 12 * q^73 + 30 * q^74 + 22 * q^76 + 11 * q^77 - 3 * q^79 + 3 * q^80 + 10 * q^82 + q^85 + 16 * q^86 - 4 * q^88 + 9 * q^89 - 14 * q^92 + 24 * q^94 + 2 * q^95 - 5 * q^97 - 39 * 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 4.56155 2.43845 0 −1.56155
1.2 2.56155 0 4.56155 1.00000 0 0.438447 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.bi 2
3.b odd 2 1 7605.2.a.bd 2
13.b even 2 1 585.2.a.j 2
39.d odd 2 1 585.2.a.l yes 2
52.b odd 2 1 9360.2.a.cl 2
65.d even 2 1 2925.2.a.bc 2
65.h odd 4 2 2925.2.c.o 4
156.h even 2 1 9360.2.a.cw 2
195.e odd 2 1 2925.2.a.x 2
195.s even 4 2 2925.2.c.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.a.j 2 13.b even 2 1
585.2.a.l yes 2 39.d odd 2 1
2925.2.a.x 2 195.e odd 2 1
2925.2.a.bc 2 65.d even 2 1
2925.2.c.o 4 65.h odd 4 2
2925.2.c.p 4 195.s even 4 2
7605.2.a.bd 2 3.b odd 2 1
7605.2.a.bi 2 1.a even 1 1 trivial
9360.2.a.cl 2 52.b odd 2 1
9360.2.a.cw 2 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{2} - T_{2} - 4$$ T2^2 - T2 - 4 $$T_{7}^{2} - 5T_{7} + 2$$ T7^2 - 5*T7 + 2 $$T_{11}^{2} - T_{11} - 4$$ T11^2 - T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 5T + 2$$
$11$ $$T^{2} - T - 4$$
$13$ $$T^{2}$$
$17$ $$T^{2} - T - 4$$
$19$ $$T^{2} - 2T - 16$$
$23$ $$T^{2} + 9T + 16$$
$29$ $$T^{2} - 6T - 8$$
$31$ $$(T + 6)^{2}$$
$37$ $$T^{2} - 9T - 18$$
$41$ $$T^{2} - 3T - 2$$
$43$ $$T^{2} + 2T - 16$$
$47$ $$T^{2} - 14T + 32$$
$53$ $$T^{2} + 3T - 36$$
$59$ $$(T - 12)^{2}$$
$61$ $$T^{2} + T - 38$$
$67$ $$T^{2} + 2T - 152$$
$71$ $$T^{2} - 25T + 152$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} + 3T - 36$$
$83$ $$T^{2} - 272$$
$89$ $$T^{2} - 9T - 18$$
$97$ $$T^{2} + 5T - 202$$
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