# Properties

 Label 7605.2.a.br Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 845) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + ( - \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + b2 * q^4 + q^5 + (-b2 + b1 + 1) * q^7 + (-b2 + 2*b1 - 1) * q^8 $$q - \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + ( - \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} + 2 \beta_1 - 1) q^{8} - \beta_1 q^{10} + ( - \beta_{2} + 2 \beta_1 - 2) q^{11} + ( - \beta_1 - 1) q^{14} + ( - 3 \beta_{2} + \beta_1 - 3) q^{16} + (3 \beta_{2} - 3 \beta_1 + 3) q^{17} + ( - \beta_{2} - 3 \beta_1 - 2) q^{19} + \beta_{2} q^{20} + ( - \beta_{2} + 2 \beta_1 - 3) q^{22} + (5 \beta_1 - 1) q^{23} + q^{25} + (3 \beta_{2} - \beta_1) q^{28} + ( - 3 \beta_{2} - 3 \beta_1 + 2) q^{29} + (5 \beta_{2} - \beta_1 + 1) q^{31} + (4 \beta_{2} - \beta_1 + 3) q^{32} + ( - 3 \beta_1 + 3) q^{34} + ( - \beta_{2} + \beta_1 + 1) q^{35} + ( - 2 \beta_1 - 1) q^{37} + (4 \beta_{2} + 2 \beta_1 + 7) q^{38} + ( - \beta_{2} + 2 \beta_1 - 1) q^{40} + (4 \beta_{2} - 3 \beta_1 + 9) q^{41} + ( - \beta_{2} - 8) q^{43} + (\beta_{2} - \beta_1 + 1) q^{44} + ( - 5 \beta_{2} + \beta_1 - 10) q^{46} + (4 \beta_{2} - 3 \beta_1 - 3) q^{47} + ( - 4 \beta_{2} + 3 \beta_1 - 5) q^{49} - \beta_1 q^{50} + (\beta_{2} + 1) q^{53} + ( - \beta_{2} + 2 \beta_1 - 2) q^{55} + ( - 2 \beta_{2} + 2 \beta_1 + 1) q^{56} + (6 \beta_{2} - 2 \beta_1 + 9) q^{58} + (2 \beta_{2} - 3) q^{59} + ( - \beta_{2} - 6) q^{61} + ( - 4 \beta_{2} - \beta_1 - 3) q^{62} + (3 \beta_{2} - 5 \beta_1 + 4) q^{64} + (6 \beta_{2} - 3) q^{67} + ( - 3 \beta_{2} + 3 \beta_1) q^{68} + ( - \beta_1 - 1) q^{70} + ( - 4 \beta_{2} - 3 \beta_1 + 4) q^{71} + ( - 2 \beta_{2} + 10 \beta_1) q^{73} + (2 \beta_{2} + \beta_1 + 4) q^{74} + ( - 4 \beta_{2} - \beta_1 - 4) q^{76} + ( - \beta_{2} + \beta_1) q^{77} + (3 \beta_{2} + 2 \beta_1 - 12) q^{79} + ( - 3 \beta_{2} + \beta_1 - 3) q^{80} + ( - \beta_{2} - 9 \beta_1 + 2) q^{82} + ( - 2 \beta_{2} + 5 \beta_1 - 12) q^{83} + (3 \beta_{2} - 3 \beta_1 + 3) q^{85} + (\beta_{2} + 8 \beta_1 + 1) q^{86} + (2 \beta_{2} - 5 \beta_1 + 7) q^{88} + ( - 8 \beta_{2} - 2 \beta_1 - 3) q^{89} + (4 \beta_{2} + 5) q^{92} + ( - \beta_{2} + 3 \beta_1 + 2) q^{94} + ( - \beta_{2} - 3 \beta_1 - 2) q^{95} + (5 \beta_{2} - \beta_1 + 6) q^{97} + (\beta_{2} + 5 \beta_1 - 2) q^{98}+O(q^{100})$$ q - b1 * q^2 + b2 * q^4 + q^5 + (-b2 + b1 + 1) * q^7 + (-b2 + 2*b1 - 1) * q^8 - b1 * q^10 + (-b2 + 2*b1 - 2) * q^11 + (-b1 - 1) * q^14 + (-3*b2 + b1 - 3) * q^16 + (3*b2 - 3*b1 + 3) * q^17 + (-b2 - 3*b1 - 2) * q^19 + b2 * q^20 + (-b2 + 2*b1 - 3) * q^22 + (5*b1 - 1) * q^23 + q^25 + (3*b2 - b1) * q^28 + (-3*b2 - 3*b1 + 2) * q^29 + (5*b2 - b1 + 1) * q^31 + (4*b2 - b1 + 3) * q^32 + (-3*b1 + 3) * q^34 + (-b2 + b1 + 1) * q^35 + (-2*b1 - 1) * q^37 + (4*b2 + 2*b1 + 7) * q^38 + (-b2 + 2*b1 - 1) * q^40 + (4*b2 - 3*b1 + 9) * q^41 + (-b2 - 8) * q^43 + (b2 - b1 + 1) * q^44 + (-5*b2 + b1 - 10) * q^46 + (4*b2 - 3*b1 - 3) * q^47 + (-4*b2 + 3*b1 - 5) * q^49 - b1 * q^50 + (b2 + 1) * q^53 + (-b2 + 2*b1 - 2) * q^55 + (-2*b2 + 2*b1 + 1) * q^56 + (6*b2 - 2*b1 + 9) * q^58 + (2*b2 - 3) * q^59 + (-b2 - 6) * q^61 + (-4*b2 - b1 - 3) * q^62 + (3*b2 - 5*b1 + 4) * q^64 + (6*b2 - 3) * q^67 + (-3*b2 + 3*b1) * q^68 + (-b1 - 1) * q^70 + (-4*b2 - 3*b1 + 4) * q^71 + (-2*b2 + 10*b1) * q^73 + (2*b2 + b1 + 4) * q^74 + (-4*b2 - b1 - 4) * q^76 + (-b2 + b1) * q^77 + (3*b2 + 2*b1 - 12) * q^79 + (-3*b2 + b1 - 3) * q^80 + (-b2 - 9*b1 + 2) * q^82 + (-2*b2 + 5*b1 - 12) * q^83 + (3*b2 - 3*b1 + 3) * q^85 + (b2 + 8*b1 + 1) * q^86 + (2*b2 - 5*b1 + 7) * q^88 + (-8*b2 - 2*b1 - 3) * q^89 + (4*b2 + 5) * q^92 + (-b2 + 3*b1 + 2) * q^94 + (-b2 - 3*b1 - 2) * q^95 + (5*b2 - b1 + 6) * q^97 + (b2 + 5*b1 - 2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - q^{4} + 3 q^{5} + 5 q^{7}+O(q^{10})$$ 3 * q - q^2 - q^4 + 3 * q^5 + 5 * q^7 $$3 q - q^{2} - q^{4} + 3 q^{5} + 5 q^{7} - q^{10} - 3 q^{11} - 4 q^{14} - 5 q^{16} + 3 q^{17} - 8 q^{19} - q^{20} - 6 q^{22} + 2 q^{23} + 3 q^{25} - 4 q^{28} + 6 q^{29} - 3 q^{31} + 4 q^{32} + 6 q^{34} + 5 q^{35} - 5 q^{37} + 19 q^{38} + 20 q^{41} - 23 q^{43} + q^{44} - 24 q^{46} - 16 q^{47} - 8 q^{49} - q^{50} + 2 q^{53} - 3 q^{55} + 7 q^{56} + 19 q^{58} - 11 q^{59} - 17 q^{61} - 6 q^{62} + 4 q^{64} - 15 q^{67} + 6 q^{68} - 4 q^{70} + 13 q^{71} + 12 q^{73} + 11 q^{74} - 9 q^{76} + 2 q^{77} - 37 q^{79} - 5 q^{80} - 2 q^{82} - 29 q^{83} + 3 q^{85} + 10 q^{86} + 14 q^{88} - 3 q^{89} + 11 q^{92} + 10 q^{94} - 8 q^{95} + 12 q^{97} - 2 q^{98}+O(q^{100})$$ 3 * q - q^2 - q^4 + 3 * q^5 + 5 * q^7 - q^10 - 3 * q^11 - 4 * q^14 - 5 * q^16 + 3 * q^17 - 8 * q^19 - q^20 - 6 * q^22 + 2 * q^23 + 3 * q^25 - 4 * q^28 + 6 * q^29 - 3 * q^31 + 4 * q^32 + 6 * q^34 + 5 * q^35 - 5 * q^37 + 19 * q^38 + 20 * q^41 - 23 * q^43 + q^44 - 24 * q^46 - 16 * q^47 - 8 * q^49 - q^50 + 2 * q^53 - 3 * q^55 + 7 * q^56 + 19 * q^58 - 11 * q^59 - 17 * q^61 - 6 * q^62 + 4 * q^64 - 15 * q^67 + 6 * q^68 - 4 * q^70 + 13 * q^71 + 12 * q^73 + 11 * q^74 - 9 * q^76 + 2 * q^77 - 37 * q^79 - 5 * q^80 - 2 * q^82 - 29 * q^83 + 3 * q^85 + 10 * q^86 + 14 * q^88 - 3 * q^89 + 11 * q^92 + 10 * q^94 - 8 * q^95 + 12 * q^97 - 2 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

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

## 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
 1.80194 0.445042 −1.24698
−1.80194 0 1.24698 1.00000 0 1.55496 1.35690 0 −1.80194
1.2 −0.445042 0 −1.80194 1.00000 0 3.24698 1.69202 0 −0.445042
1.3 1.24698 0 −0.445042 1.00000 0 0.198062 −3.04892 0 1.24698
 $$n$$: 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.br 3
3.b odd 2 1 845.2.a.j yes 3
13.b even 2 1 7605.2.a.by 3
15.d odd 2 1 4225.2.a.bd 3
39.d odd 2 1 845.2.a.h 3
39.f even 4 2 845.2.c.f 6
39.h odd 6 2 845.2.e.l 6
39.i odd 6 2 845.2.e.j 6
39.k even 12 4 845.2.m.i 12
195.e odd 2 1 4225.2.a.bf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.a.h 3 39.d odd 2 1
845.2.a.j yes 3 3.b odd 2 1
845.2.c.f 6 39.f even 4 2
845.2.e.j 6 39.i odd 6 2
845.2.e.l 6 39.h odd 6 2
845.2.m.i 12 39.k even 12 4
4225.2.a.bd 3 15.d odd 2 1
4225.2.a.bf 3 195.e odd 2 1
7605.2.a.br 3 1.a even 1 1 trivial
7605.2.a.by 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{3} + T_{2}^{2} - 2T_{2} - 1$$ T2^3 + T2^2 - 2*T2 - 1 $$T_{7}^{3} - 5T_{7}^{2} + 6T_{7} - 1$$ T7^3 - 5*T7^2 + 6*T7 - 1 $$T_{11}^{3} + 3T_{11}^{2} - 4T_{11} + 1$$ T11^3 + 3*T11^2 - 4*T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 2T - 1$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 5 T^{2} + 6 T - 1$$
$11$ $$T^{3} + 3 T^{2} - 4 T + 1$$
$13$ $$T^{3}$$
$17$ $$T^{3} - 3 T^{2} - 18 T + 27$$
$19$ $$T^{3} + 8 T^{2} - 9 T - 29$$
$23$ $$T^{3} - 2 T^{2} - 57 T + 71$$
$29$ $$T^{3} - 6 T^{2} - 51 T + 307$$
$31$ $$T^{3} + 3 T^{2} - 46 T + 1$$
$37$ $$T^{3} + 5T^{2} - T - 13$$
$41$ $$T^{3} - 20 T^{2} + 103 T - 43$$
$43$ $$T^{3} + 23 T^{2} + 174 T + 433$$
$47$ $$T^{3} + 16 T^{2} + 55 T + 41$$
$53$ $$T^{3} - 2T^{2} - T + 1$$
$59$ $$T^{3} + 11 T^{2} + 31 T + 13$$
$61$ $$T^{3} + 17 T^{2} + 94 T + 169$$
$67$ $$T^{3} + 15 T^{2} - 9 T - 351$$
$71$ $$T^{3} - 13 T^{2} - 30 T + 601$$
$73$ $$T^{3} - 12 T^{2} - 148 T + 1448$$
$79$ $$T^{3} + 37 T^{2} + 412 T + 1217$$
$83$ $$T^{3} + 29 T^{2} + 236 T + 587$$
$89$ $$T^{3} + 3 T^{2} - 193 T + 533$$
$97$ $$T^{3} - 12 T^{2} - T + 181$$