# Properties

 Label 7600.2.a.cd 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: 3.3.568.1 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 190) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( 1 + \beta_{1} ) q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( 1 + \beta_{1} ) q^{7} + ( 2 + \beta_{2} ) q^{9} + ( \beta_{1} + \beta_{2} ) q^{11} + ( -3 - \beta_{2} ) q^{13} + ( 1 + \beta_{2} ) q^{17} - q^{19} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{21} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{23} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{27} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{29} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{31} + ( -2 - \beta_{1} + \beta_{2} ) q^{33} + ( -4 - 2 \beta_{1} ) q^{37} + ( -5 + 3 \beta_{1} - 2 \beta_{2} ) q^{39} + ( 3 \beta_{1} + \beta_{2} ) q^{41} + ( 6 + \beta_{1} + \beta_{2} ) q^{43} + ( -4 + 2 \beta_{2} ) q^{47} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{49} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{51} + ( -5 + \beta_{2} ) q^{53} + ( -1 + \beta_{1} ) q^{57} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{59} + ( -10 - \beta_{1} - \beta_{2} ) q^{61} + 2 \beta_{1} q^{63} + ( 1 - 3 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -9 - 5 \beta_{2} ) q^{69} + ( 4 - 5 \beta_{1} - \beta_{2} ) q^{71} + ( -5 + 2 \beta_{1} - 3 \beta_{2} ) q^{73} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{77} + ( 2 - 6 \beta_{1} ) q^{79} + ( -5 - 2 \beta_{1} ) q^{81} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{83} + ( -1 + 5 \beta_{1} - 4 \beta_{2} ) q^{87} + ( -4 - \beta_{1} + \beta_{2} ) q^{89} + ( -1 - 3 \beta_{1} ) q^{91} + ( -4 + 3 \beta_{1} - 5 \beta_{2} ) q^{93} + ( -2 + 4 \beta_{2} ) q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 2q^{3} + 4q^{7} + 5q^{9} + O(q^{10})$$ $$3q + 2q^{3} + 4q^{7} + 5q^{9} - 8q^{13} + 2q^{17} - 3q^{19} - 10q^{21} + 2q^{27} - 8q^{29} - 4q^{31} - 8q^{33} - 14q^{37} - 10q^{39} + 2q^{41} + 18q^{43} - 14q^{47} - 3q^{49} + 6q^{51} - 16q^{53} - 2q^{57} - 2q^{59} - 30q^{61} + 2q^{63} + 2q^{67} - 22q^{69} + 8q^{71} - 10q^{73} + 8q^{77} - 17q^{81} - 6q^{83} + 6q^{87} - 14q^{89} - 6q^{91} - 4q^{93} - 10q^{97} + 12q^{99} + O(q^{100})$$

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 −2.12489 0 0 0 4.12489 0 1.51514 0
1.2 0 1.36333 0 0 0 0.636672 0 −1.14134 0
1.3 0 2.76156 0 0 0 −0.761557 0 4.62620 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.cd 3
4.b odd 2 1 950.2.a.i 3
5.b even 2 1 7600.2.a.bi 3
5.c odd 4 2 1520.2.d.j 6
12.b even 2 1 8550.2.a.cl 3
20.d odd 2 1 950.2.a.n 3
20.e even 4 2 190.2.b.b 6
60.h even 2 1 8550.2.a.ck 3
60.l odd 4 2 1710.2.d.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 20.e even 4 2
950.2.a.i 3 4.b odd 2 1
950.2.a.n 3 20.d odd 2 1
1520.2.d.j 6 5.c odd 4 2
1710.2.d.d 6 60.l odd 4 2
7600.2.a.bi 3 5.b even 2 1
7600.2.a.cd 3 1.a even 1 1 trivial
8550.2.a.ck 3 60.h even 2 1
8550.2.a.cl 3 12.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7600))$$:

 $$T_{3}^{3} - 2 T_{3}^{2} - 5 T_{3} + 8$$ $$T_{7}^{3} - 4 T_{7}^{2} - T_{7} + 2$$ $$T_{11}^{3} - 10 T_{11} + 8$$ $$T_{13}^{3} + 8 T_{13}^{2} + 13 T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$8 - 5 T - 2 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$2 - T - 4 T^{2} + T^{3}$$
$11$ $$8 - 10 T + T^{3}$$
$13$ $$-2 + 13 T + 8 T^{2} + T^{3}$$
$17$ $$4 - 7 T - 2 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$122 - 49 T + T^{3}$$
$29$ $$-410 - 51 T + 8 T^{2} + T^{3}$$
$31$ $$-232 - 62 T + 4 T^{2} + T^{3}$$
$37$ $$16 + 40 T + 14 T^{2} + T^{3}$$
$41$ $$-100 - 50 T - 2 T^{2} + T^{3}$$
$43$ $$-148 + 98 T - 18 T^{2} + T^{3}$$
$47$ $$-64 + 32 T + 14 T^{2} + T^{3}$$
$53$ $$106 + 77 T + 16 T^{2} + T^{3}$$
$59$ $$-80 - 29 T + 2 T^{2} + T^{3}$$
$61$ $$892 + 290 T + 30 T^{2} + T^{3}$$
$67$ $$64 - 61 T - 2 T^{2} + T^{3}$$
$71$ $$1016 - 122 T - 8 T^{2} + T^{3}$$
$73$ $$164 - 95 T + 10 T^{2} + T^{3}$$
$79$ $$880 - 228 T + T^{3}$$
$83$ $$-8 - 28 T + 6 T^{2} + T^{3}$$
$89$ $$-20 + 46 T + 14 T^{2} + T^{3}$$
$97$ $$-488 - 100 T + 10 T^{2} + T^{3}$$