# Properties

 Label 7600.2.a.bo Level $7600$ Weight $2$ Character orbit 7600.a Self dual yes Analytic conductor $60.686$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$60.6863055362$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -2 - \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -2 - \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{9} + ( 2 - 2 \beta_{1} ) q^{11} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{13} + ( 4 + \beta_{2} ) q^{17} + q^{19} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{21} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{23} + ( -6 - 2 \beta_{1} - \beta_{2} ) q^{27} -3 \beta_{2} q^{29} + ( 4 - 2 \beta_{2} ) q^{31} + ( 8 + 2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( -4 - \beta_{1} + \beta_{2} ) q^{37} + ( 4 \beta_{1} - \beta_{2} ) q^{39} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{49} + ( 2 - 4 \beta_{1} + \beta_{2} ) q^{51} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{53} -\beta_{1} q^{57} + ( 2 - \beta_{2} ) q^{59} + ( 8 - 4 \beta_{1} ) q^{61} + ( -4 - 2 \beta_{1} ) q^{63} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 10 + 6 \beta_{1} + 3 \beta_{2} ) q^{69} + ( 4 - 4 \beta_{1} ) q^{71} + ( 4 + 4 \beta_{1} + 3 \beta_{2} ) q^{73} + ( -8 + 4 \beta_{1} - 4 \beta_{2} ) q^{77} + ( -4 + 2 \beta_{1} + 4 \beta_{2} ) q^{79} + ( 3 + 4 \beta_{1} - 2 \beta_{2} ) q^{81} + ( -8 + 2 \beta_{2} ) q^{83} + ( -6 - 3 \beta_{2} ) q^{87} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 14 - 2 \beta_{1} + 3 \beta_{2} ) q^{91} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 4 - 7 \beta_{1} - \beta_{2} ) q^{97} + ( -10 - 6 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{3} - 5q^{7} + 4q^{9} + O(q^{10})$$ $$3q - q^{3} - 5q^{7} + 4q^{9} + 4q^{11} - 5q^{13} + 11q^{17} + 3q^{19} - 3q^{21} - 9q^{23} - 19q^{27} + 3q^{29} + 14q^{31} + 24q^{33} - 14q^{37} + 5q^{39} - 10q^{41} - 10q^{43} + 4q^{49} + q^{51} + 7q^{53} - q^{57} + 7q^{59} + 20q^{61} - 14q^{63} - q^{67} + 33q^{69} + 8q^{71} + 13q^{73} - 16q^{77} - 14q^{79} + 15q^{81} - 26q^{83} - 15q^{87} + 18q^{89} + 37q^{91} - 14q^{93} + 6q^{97} - 36q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

## 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
 3.12489 −0.363328 −1.76156
0 −3.12489 0 0 0 −1.51514 0 6.76491 0
1.2 0 0.363328 0 0 0 1.14134 0 −2.86799 0
1.3 0 1.76156 0 0 0 −4.62620 0 0.103084 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 7600.2.a.bo 3
4.b odd 2 1 3800.2.a.y 3
5.b even 2 1 1520.2.a.r 3
20.d odd 2 1 760.2.a.h 3
20.e even 4 2 3800.2.d.k 6
40.e odd 2 1 6080.2.a.bw 3
40.f even 2 1 6080.2.a.bs 3
60.h even 2 1 6840.2.a.bj 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.h 3 20.d odd 2 1
1520.2.a.r 3 5.b even 2 1
3800.2.a.y 3 4.b odd 2 1
3800.2.d.k 6 20.e even 4 2
6080.2.a.bs 3 40.f even 2 1
6080.2.a.bw 3 40.e odd 2 1
6840.2.a.bj 3 60.h even 2 1
7600.2.a.bo 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7600))$$:

 $$T_{3}^{3} + T_{3}^{2} - 6 T_{3} + 2$$ $$T_{7}^{3} + 5 T_{7}^{2} - 8$$ $$T_{11}^{3} - 4 T_{11}^{2} - 20 T_{11} + 64$$ $$T_{13}^{3} + 5 T_{13}^{2} - 22 T_{13} - 106$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$2 - 6 T + T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-8 + 5 T^{2} + T^{3}$$
$11$ $$64 - 20 T - 4 T^{2} + T^{3}$$
$13$ $$-106 - 22 T + 5 T^{2} + T^{3}$$
$17$ $$-20 + 32 T - 11 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$-160 - 16 T + 9 T^{2} + T^{3}$$
$29$ $$108 - 72 T - 3 T^{2} + T^{3}$$
$31$ $$64 + 32 T - 14 T^{2} + T^{3}$$
$37$ $$-20 + 46 T + 14 T^{2} + T^{3}$$
$41$ $$-472 - 44 T + 10 T^{2} + T^{3}$$
$43$ $$-848 - 88 T + 10 T^{2} + T^{3}$$
$47$ $$64 - 40 T + T^{3}$$
$53$ $$-2 - 14 T - 7 T^{2} + T^{3}$$
$59$ $$8 + 8 T - 7 T^{2} + T^{3}$$
$61$ $$640 + 32 T - 20 T^{2} + T^{3}$$
$67$ $$530 - 134 T + T^{2} + T^{3}$$
$71$ $$512 - 80 T - 8 T^{2} + T^{3}$$
$73$ $$500 - 64 T - 13 T^{2} + T^{3}$$
$79$ $$16 - 56 T + 14 T^{2} + T^{3}$$
$83$ $$352 + 192 T + 26 T^{2} + T^{3}$$
$89$ $$-40 + 68 T - 18 T^{2} + T^{3}$$
$97$ $$2308 - 274 T - 6 T^{2} + T^{3}$$