# Properties

 Label 7605.2.a.bm 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 2535) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_1 + 4) q^{4} + q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (3 \beta_1 - 5) q^{8}+O(q^{10})$$ q + (b2 + b1 - 1) * q^2 + (-b1 + 4) * q^4 + q^5 + (-b2 - b1) * q^7 + (3*b1 - 5) * q^8 $$q + (\beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_1 + 4) q^{4} + q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (3 \beta_1 - 5) q^{8} + (\beta_{2} + \beta_1 - 1) q^{10} + (2 \beta_{2} - 2 \beta_1 + 3) q^{11} + ( - \beta_{2} - 5) q^{14} + (\beta_{2} - 6 \beta_1 + 6) q^{16} + (2 \beta_{2} + 2 \beta_1 - 2) q^{17} + ( - 4 \beta_{2} + 3 \beta_1 - 3) q^{19} + ( - \beta_1 + 4) q^{20} + ( - 3 \beta_{2} + 7 \beta_1 - 5) q^{22} + ( - 5 \beta_{2} + 4 \beta_1 - 4) q^{23} + q^{25} + ( - 2 \beta_{2} - 4 \beta_1 + 3) q^{28} + (2 \beta_{2} - \beta_1 - 4) q^{29} + (2 \beta_{2} - 6 \beta_1 - 1) q^{31} + ( - 7 \beta_{2} + 7 \beta_1 - 12) q^{32} + ( - 2 \beta_1 + 12) q^{34} + ( - \beta_{2} - \beta_1) q^{35} + (2 \beta_{2} + \beta_1 + 6) q^{37} + (7 \beta_{2} - 10 \beta_1 + 4) q^{38} + (3 \beta_1 - 5) q^{40} + (3 \beta_{2} - 5 \beta_1 + 6) q^{41} + ( - 3 \beta_{2} + \beta_1 - 10) q^{43} + (8 \beta_{2} - 11 \beta_1 + 14) q^{44} + (9 \beta_{2} - 13 \beta_1 + 6) q^{46} + ( - \beta_{2} + \beta_1 - 4) q^{47} + (2 \beta_{2} + \beta_1 - 2) q^{49} + (\beta_{2} + \beta_1 - 1) q^{50} + (5 \beta_{2} - 9 \beta_1 + 1) q^{53} + (2 \beta_{2} - 2 \beta_1 + 3) q^{55} + ( - \beta_{2} + 5 \beta_1 - 9) q^{56} + ( - 8 \beta_{2} - \beta_1 + 5) q^{58} + ( - 7 \beta_{2} + 7 \beta_1 - 5) q^{59} + ( - \beta_{2} + 7) q^{61} + ( - 15 \beta_{2} + 7 \beta_1 - 13) q^{62} + (7 \beta_{2} - 14 \beta_1 + 7) q^{64} + ( - 7 \beta_{2} + 4 \beta_1 - 9) q^{67} + (4 \beta_{2} + 10 \beta_1 - 14) q^{68} + ( - \beta_{2} - 5) q^{70} + ( - 4 \beta_{2} + 3 \beta_1 + 7) q^{71} + (6 \beta_{2} + \beta_1 + 3) q^{73} + (6 \beta_{2} + 7 \beta_1 + 1) q^{74} + ( - 15 \beta_{2} + 15 \beta_1 - 14) q^{76} + (\beta_{2} - 5 \beta_1 + 2) q^{77} + ( - 7 \beta_{2} + 5 \beta_1 - 2) q^{79} + (\beta_{2} - 6 \beta_1 + 6) q^{80} + ( - 7 \beta_{2} + 14 \beta_1 - 15) q^{82} + ( - 11 \beta_1 + 2) q^{83} + (2 \beta_{2} + 2 \beta_1 - 2) q^{85} + ( - 5 \beta_{2} - 14 \beta_1 + 7) q^{86} + ( - 10 \beta_{2} + 19 \beta_1 - 21) q^{88} + ( - 2 \beta_{2} + 3 \beta_1 - 1) q^{89} + ( - 19 \beta_{2} + 20 \beta_1 - 19) q^{92} + ( - \beta_{2} - 6 \beta_1 + 5) q^{94} + ( - 4 \beta_{2} + 3 \beta_1 - 3) q^{95} + (4 \beta_{2} + 2 \beta_1 - 1) q^{97} + ( - 2 \beta_{2} - \beta_1 + 9) q^{98}+O(q^{100})$$ q + (b2 + b1 - 1) * q^2 + (-b1 + 4) * q^4 + q^5 + (-b2 - b1) * q^7 + (3*b1 - 5) * q^8 + (b2 + b1 - 1) * q^10 + (2*b2 - 2*b1 + 3) * q^11 + (-b2 - 5) * q^14 + (b2 - 6*b1 + 6) * q^16 + (2*b2 + 2*b1 - 2) * q^17 + (-4*b2 + 3*b1 - 3) * q^19 + (-b1 + 4) * q^20 + (-3*b2 + 7*b1 - 5) * q^22 + (-5*b2 + 4*b1 - 4) * q^23 + q^25 + (-2*b2 - 4*b1 + 3) * q^28 + (2*b2 - b1 - 4) * q^29 + (2*b2 - 6*b1 - 1) * q^31 + (-7*b2 + 7*b1 - 12) * q^32 + (-2*b1 + 12) * q^34 + (-b2 - b1) * q^35 + (2*b2 + b1 + 6) * q^37 + (7*b2 - 10*b1 + 4) * q^38 + (3*b1 - 5) * q^40 + (3*b2 - 5*b1 + 6) * q^41 + (-3*b2 + b1 - 10) * q^43 + (8*b2 - 11*b1 + 14) * q^44 + (9*b2 - 13*b1 + 6) * q^46 + (-b2 + b1 - 4) * q^47 + (2*b2 + b1 - 2) * q^49 + (b2 + b1 - 1) * q^50 + (5*b2 - 9*b1 + 1) * q^53 + (2*b2 - 2*b1 + 3) * q^55 + (-b2 + 5*b1 - 9) * q^56 + (-8*b2 - b1 + 5) * q^58 + (-7*b2 + 7*b1 - 5) * q^59 + (-b2 + 7) * q^61 + (-15*b2 + 7*b1 - 13) * q^62 + (7*b2 - 14*b1 + 7) * q^64 + (-7*b2 + 4*b1 - 9) * q^67 + (4*b2 + 10*b1 - 14) * q^68 + (-b2 - 5) * q^70 + (-4*b2 + 3*b1 + 7) * q^71 + (6*b2 + b1 + 3) * q^73 + (6*b2 + 7*b1 + 1) * q^74 + (-15*b2 + 15*b1 - 14) * q^76 + (b2 - 5*b1 + 2) * q^77 + (-7*b2 + 5*b1 - 2) * q^79 + (b2 - 6*b1 + 6) * q^80 + (-7*b2 + 14*b1 - 15) * q^82 + (-11*b1 + 2) * q^83 + (2*b2 + 2*b1 - 2) * q^85 + (-5*b2 - 14*b1 + 7) * q^86 + (-10*b2 + 19*b1 - 21) * q^88 + (-2*b2 + 3*b1 - 1) * q^89 + (-19*b2 + 20*b1 - 19) * q^92 + (-b2 - 6*b1 + 5) * q^94 + (-4*b2 + 3*b1 - 3) * q^95 + (4*b2 + 2*b1 - 1) * q^97 + (-2*b2 - b1 + 9) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 11 q^{4} + 3 q^{5} - 12 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 11 * q^4 + 3 * q^5 - 12 * q^8 $$3 q - 3 q^{2} + 11 q^{4} + 3 q^{5} - 12 q^{8} - 3 q^{10} + 5 q^{11} - 14 q^{14} + 11 q^{16} - 6 q^{17} - 2 q^{19} + 11 q^{20} - 5 q^{22} - 3 q^{23} + 3 q^{25} + 7 q^{28} - 15 q^{29} - 11 q^{31} - 22 q^{32} + 34 q^{34} + 17 q^{37} - 5 q^{38} - 12 q^{40} + 10 q^{41} - 26 q^{43} + 23 q^{44} - 4 q^{46} - 10 q^{47} - 7 q^{49} - 3 q^{50} - 11 q^{53} + 5 q^{55} - 21 q^{56} + 22 q^{58} - q^{59} + 22 q^{61} - 17 q^{62} - 16 q^{67} - 36 q^{68} - 14 q^{70} + 28 q^{71} + 4 q^{73} + 4 q^{74} - 12 q^{76} + 6 q^{79} + 11 q^{80} - 24 q^{82} - 5 q^{83} - 6 q^{85} + 12 q^{86} - 34 q^{88} + 2 q^{89} - 18 q^{92} + 10 q^{94} - 2 q^{95} - 5 q^{97} + 28 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 11 * q^4 + 3 * q^5 - 12 * q^8 - 3 * q^10 + 5 * q^11 - 14 * q^14 + 11 * q^16 - 6 * q^17 - 2 * q^19 + 11 * q^20 - 5 * q^22 - 3 * q^23 + 3 * q^25 + 7 * q^28 - 15 * q^29 - 11 * q^31 - 22 * q^32 + 34 * q^34 + 17 * q^37 - 5 * q^38 - 12 * q^40 + 10 * q^41 - 26 * q^43 + 23 * q^44 - 4 * q^46 - 10 * q^47 - 7 * q^49 - 3 * q^50 - 11 * q^53 + 5 * q^55 - 21 * q^56 + 22 * q^58 - q^59 + 22 * q^61 - 17 * q^62 - 16 * q^67 - 36 * q^68 - 14 * q^70 + 28 * q^71 + 4 * q^73 + 4 * q^74 - 12 * q^76 + 6 * q^79 + 11 * q^80 - 24 * q^82 - 5 * q^83 - 6 * q^85 + 12 * q^86 - 34 * q^88 + 2 * q^89 - 18 * q^92 + 10 * q^94 - 2 * q^95 - 5 * q^97 + 28 * 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
 −1.24698 0.445042 1.80194
−2.69202 0 5.24698 1.00000 0 1.69202 −8.74094 0 −2.69202
1.2 −2.35690 0 3.55496 1.00000 0 1.35690 −3.66487 0 −2.35690
1.3 2.04892 0 2.19806 1.00000 0 −3.04892 0.405813 0 2.04892
 $$n$$: 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.bm 3
3.b odd 2 1 2535.2.a.bh yes 3
13.b even 2 1 7605.2.a.ce 3
39.d odd 2 1 2535.2.a.t 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2535.2.a.t 3 39.d odd 2 1
2535.2.a.bh yes 3 3.b odd 2 1
7605.2.a.bm 3 1.a even 1 1 trivial
7605.2.a.ce 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2}^{3} + 3T_{2}^{2} - 4T_{2} - 13$$ T2^3 + 3*T2^2 - 4*T2 - 13 $$T_{7}^{3} - 7T_{7} + 7$$ T7^3 - 7*T7 + 7 $$T_{11}^{3} - 5T_{11}^{2} - T_{11} + 13$$ T11^3 - 5*T11^2 - T11 + 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 3 T^{2} - 4 T - 13$$
$3$ $$T^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 7T + 7$$
$11$ $$T^{3} - 5T^{2} - T + 13$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 6 T^{2} - 16 T - 104$$
$19$ $$T^{3} + 2 T^{2} - 29 T - 71$$
$23$ $$T^{3} + 3 T^{2} - 46 T - 139$$
$29$ $$T^{3} + 15 T^{2} + 68 T + 97$$
$31$ $$T^{3} + 11 T^{2} - 25 T - 379$$
$37$ $$T^{3} - 17 T^{2} + 80 T - 113$$
$41$ $$T^{3} - 10 T^{2} - 11 T + 13$$
$43$ $$T^{3} + 26 T^{2} + 209 T + 491$$
$47$ $$T^{3} + 10 T^{2} + 31 T + 29$$
$53$ $$T^{3} + 11 T^{2} - 102 T - 1079$$
$59$ $$T^{3} + T^{2} - 114 T - 127$$
$61$ $$T^{3} - 22 T^{2} + 159 T - 377$$
$67$ $$T^{3} + 16 T^{2} - T - 617$$
$71$ $$T^{3} - 28 T^{2} + 231 T - 581$$
$73$ $$T^{3} - 4 T^{2} - 95 T - 83$$
$79$ $$T^{3} - 6 T^{2} - 79 T - 113$$
$83$ $$T^{3} + 5 T^{2} - 274 T - 811$$
$89$ $$T^{3} - 2 T^{2} - 15 T + 29$$
$97$ $$T^{3} + 5 T^{2} - 57 T - 293$$