# Properties

 Label 7600.2.a.bu Level $7600$ Weight $2$ Character orbit 7600.a Self dual yes Analytic conductor $60.686$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3800) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 0.713538 −1.91223 2.19869
0 −2.49086 0 0 0 2.49086 0 3.20440 0
1.2 0 0.656620 0 0 0 −0.656620 0 −2.56885 0
1.3 0 1.83424 0 0 0 −1.83424 0 0.364448 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.bu 3
4.b odd 2 1 3800.2.a.v yes 3
5.b even 2 1 7600.2.a.bt 3
20.d odd 2 1 3800.2.a.u 3
20.e even 4 2 3800.2.d.m 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.2.a.u 3 20.d odd 2 1
3800.2.a.v yes 3 4.b odd 2 1
3800.2.d.m 6 20.e even 4 2
7600.2.a.bt 3 5.b even 2 1
7600.2.a.bu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7600))$$:

 $$T_{3}^{3} - 5 T_{3} + 3$$ $$T_{7}^{3} - 5 T_{7} - 3$$ $$T_{11}^{3} - 3 T_{11}^{2} - 2 T_{11} + 1$$ $$T_{13}^{3} - T_{13}^{2} - 4 T_{13} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$3 - 5 T + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-3 - 5 T + T^{3}$$
$11$ $$1 - 2 T - 3 T^{2} + T^{3}$$
$13$ $$3 - 4 T - T^{2} + T^{3}$$
$17$ $$-45 - 19 T + 2 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$5 - 7 T - 2 T^{2} + T^{3}$$
$29$ $$49 - 78 T + T^{2} + T^{3}$$
$31$ $$81 - 36 T - 3 T^{2} + T^{3}$$
$37$ $$-45 - 24 T + T^{2} + T^{3}$$
$41$ $$-175 - 20 T + 9 T^{2} + T^{3}$$
$43$ $$-113 - 41 T + T^{2} + T^{3}$$
$47$ $$-9 + 30 T + 13 T^{2} + T^{3}$$
$53$ $$-9 + 7 T + 9 T^{2} + T^{3}$$
$59$ $$8 + 16 T - 10 T^{2} + T^{3}$$
$61$ $$-305 + 82 T + 21 T^{2} + T^{3}$$
$67$ $$25 + 36 T + 13 T^{2} + T^{3}$$
$71$ $$-281 - 81 T + T^{2} + T^{3}$$
$73$ $$-1075 - 42 T + 17 T^{2} + T^{3}$$
$79$ $$301 + 55 T - 20 T^{2} + T^{3}$$
$83$ $$109 - 118 T - T^{2} + T^{3}$$
$89$ $$-355 - 119 T - 4 T^{2} + T^{3}$$
$97$ $$-113 + 36 T + 15 T^{2} + T^{3}$$