# Properties

 Label 7600.2.a.bl 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: 3.3.321.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1900) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + ( 2 - \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + ( 2 - \beta_{1} ) q^{9} -\beta_{2} q^{11} + ( 1 + \beta_{1} + \beta_{2} ) q^{13} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{17} - q^{19} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{21} + ( -5 - \beta_{1} ) q^{23} + ( 2 \beta_{1} + \beta_{2} ) q^{27} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{29} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{31} + ( 4 - \beta_{1} - \beta_{2} ) q^{33} + ( -4 + \beta_{2} ) q^{37} + ( -4 - \beta_{1} ) q^{39} -3 \beta_{1} q^{41} + ( -5 - 2 \beta_{2} ) q^{43} + ( -1 - 5 \beta_{1} + \beta_{2} ) q^{47} + ( 6 + 3 \beta_{1} ) q^{49} + ( 5 - 5 \beta_{1} - 2 \beta_{2} ) q^{51} + ( 1 - 4 \beta_{1} ) q^{53} + ( 1 + \beta_{2} ) q^{57} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -3 + 5 \beta_{1} + \beta_{2} ) q^{61} + ( -7 + 2 \beta_{1} ) q^{63} + ( -4 - 4 \beta_{1} + 3 \beta_{2} ) q^{67} + ( 4 + 2 \beta_{1} + 5 \beta_{2} ) q^{69} + ( 1 + 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 4 + \beta_{1} - 2 \beta_{2} ) q^{73} + ( -2 - \beta_{1} + 3 \beta_{2} ) q^{77} + ( 1 + 5 \beta_{2} ) q^{79} + ( -8 + \beta_{2} ) q^{81} + ( -10 + \beta_{1} ) q^{83} + ( -8 + 5 \beta_{1} - \beta_{2} ) q^{87} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 6 + 5 \beta_{1} ) q^{91} + ( 5 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 1 + \beta_{1} + 5 \beta_{2} ) q^{97} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 2q^{3} - 2q^{7} + 5q^{9} + O(q^{10})$$ $$3q - 2q^{3} - 2q^{7} + 5q^{9} + q^{11} + 3q^{13} + 6q^{17} - 3q^{19} - 8q^{21} - 16q^{23} + q^{27} + 3q^{29} + q^{31} + 12q^{33} - 13q^{37} - 13q^{39} - 3q^{41} - 13q^{43} - 9q^{47} + 21q^{49} + 12q^{51} - q^{53} + 2q^{57} + 6q^{59} - 5q^{61} - 19q^{63} - 19q^{67} + 9q^{69} + 3q^{71} + 15q^{73} - 10q^{77} - 2q^{79} - 25q^{81} - 29q^{83} - 18q^{87} + 14q^{89} + 23q^{91} + 7q^{93} - q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 1$$:

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

## 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
 −1.69963 2.46050 0.239123
0 −2.58836 0 0 0 −2.81089 0 3.69963 0
1.2 0 −1.59358 0 0 0 4.51459 0 −0.460505 0
1.3 0 2.18194 0 0 0 −3.70370 0 1.76088 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.bl 3
4.b odd 2 1 1900.2.a.i yes 3
5.b even 2 1 7600.2.a.ca 3
20.d odd 2 1 1900.2.a.g 3
20.e even 4 2 1900.2.c.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.g 3 20.d odd 2 1
1900.2.a.i yes 3 4.b odd 2 1
1900.2.c.f 6 20.e even 4 2
7600.2.a.bl 3 1.a even 1 1 trivial
7600.2.a.ca 3 5.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7600))$$:

 $$T_{3}^{3} + 2 T_{3}^{2} - 5 T_{3} - 9$$ $$T_{7}^{3} + 2 T_{7}^{2} - 19 T_{7} - 47$$ $$T_{11}^{3} - T_{11}^{2} - 6 T_{11} - 3$$ $$T_{13}^{3} - 3 T_{13}^{2} - 6 T_{13} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-9 - 5 T + 2 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-47 - 19 T + 2 T^{2} + T^{3}$$
$11$ $$-3 - 6 T - T^{2} + T^{3}$$
$13$ $$7 - 6 T - 3 T^{2} + T^{3}$$
$17$ $$99 - 15 T - 6 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$129 + 81 T + 16 T^{2} + T^{3}$$
$29$ $$-27 - 24 T - 3 T^{2} + T^{3}$$
$31$ $$109 - 40 T - T^{2} + T^{3}$$
$37$ $$59 + 50 T + 13 T^{2} + T^{3}$$
$41$ $$-27 - 36 T + 3 T^{2} + T^{3}$$
$43$ $$-69 + 31 T + 13 T^{2} + T^{3}$$
$47$ $$-621 - 96 T + 9 T^{2} + T^{3}$$
$53$ $$3 - 69 T + T^{2} + T^{3}$$
$59$ $$72 - 24 T - 6 T^{2} + T^{3}$$
$61$ $$-489 - 98 T + 5 T^{2} + T^{3}$$
$67$ $$-1323 - 26 T + 19 T^{2} + T^{3}$$
$71$ $$27 - 33 T - 3 T^{2} + T^{3}$$
$73$ $$49 + 42 T - 15 T^{2} + T^{3}$$
$79$ $$529 - 157 T + 2 T^{2} + T^{3}$$
$83$ $$861 + 276 T + 29 T^{2} + T^{3}$$
$89$ $$489 - 9 T - 14 T^{2} + T^{3}$$
$97$ $$683 - 154 T + T^{2} + T^{3}$$