Properties

 Label 7600.2.a.bz 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: 3.3.993.1 Defining polynomial: $$x^{3} - x^{2} - 6 x + 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 950) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( 1 - \beta_{2} ) q^{7} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( 1 - \beta_{2} ) q^{7} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{9} -2 \beta_{1} q^{11} + ( 2 - \beta_{1} + \beta_{2} ) q^{13} + ( -4 + \beta_{1} - \beta_{2} ) q^{17} - q^{19} + ( 2 + \beta_{1} ) q^{21} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{23} + ( 6 - 3 \beta_{1} + 2 \beta_{2} ) q^{27} + ( 2 + \beta_{1} - \beta_{2} ) q^{29} + ( -2 - 2 \beta_{1} ) q^{31} + ( 8 - 2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( 1 - 2 \beta_{1} ) q^{37} + ( 5 - 5 \beta_{1} + \beta_{2} ) q^{39} + ( -4 - 2 \beta_{1} ) q^{43} + ( 1 + 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 3 + 3 \beta_{1} - 3 \beta_{2} ) q^{49} + ( -7 + 7 \beta_{1} - \beta_{2} ) q^{51} + ( 5 - 4 \beta_{1} - \beta_{2} ) q^{53} + ( -1 + \beta_{1} ) q^{57} + ( -1 + \beta_{1} - 4 \beta_{2} ) q^{59} -2 \beta_{2} q^{61} + ( -5 - \beta_{1} + 2 \beta_{2} ) q^{63} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{67} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{69} + ( 4 - 4 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 5 - \beta_{1} - 2 \beta_{2} ) q^{73} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( -8 + 4 \beta_{1} ) q^{79} + ( 10 - 7 \beta_{1} ) q^{81} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{83} + ( -1 + \beta_{1} - \beta_{2} ) q^{87} + ( 6 - 4 \beta_{1} ) q^{89} + ( -6 - 2 \beta_{1} + \beta_{2} ) q^{91} + ( 6 + 2 \beta_{2} ) q^{93} + ( 8 + 6 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 14 - 8 \beta_{1} + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 2q^{3} + 2q^{7} + 5q^{9} + O(q^{10})$$ $$3q + 2q^{3} + 2q^{7} + 5q^{9} - 2q^{11} + 6q^{13} - 12q^{17} - 3q^{19} + 7q^{21} - 2q^{23} + 17q^{27} + 6q^{29} - 8q^{31} + 24q^{33} + q^{37} + 11q^{39} - 14q^{43} + 3q^{47} + 9q^{49} - 15q^{51} + 10q^{53} - 2q^{57} - 6q^{59} - 2q^{61} - 14q^{63} + 4q^{67} + 6q^{71} + 12q^{73} + 10q^{77} - 20q^{79} + 23q^{81} - 4q^{83} - 3q^{87} + 14q^{89} - 19q^{91} + 20q^{93} + 28q^{97} + 36q^{99} + O(q^{100})$$

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

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

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
 2.77339 0.480031 −2.25342
0 −1.77339 0 0 0 −2.69168 0 0.144903 0
1.2 0 0.519969 0 0 0 4.76957 0 −2.72963 0
1.3 0 3.25342 0 0 0 −0.0778929 0 7.58473 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.bz 3
4.b odd 2 1 950.2.a.j 3
5.b even 2 1 7600.2.a.bk 3
12.b even 2 1 8550.2.a.cp 3
20.d odd 2 1 950.2.a.l yes 3
20.e even 4 2 950.2.b.h 6
60.h even 2 1 8550.2.a.ci 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.j 3 4.b odd 2 1
950.2.a.l yes 3 20.d odd 2 1
950.2.b.h 6 20.e even 4 2
7600.2.a.bk 3 5.b even 2 1
7600.2.a.bz 3 1.a even 1 1 trivial
8550.2.a.ci 3 60.h even 2 1
8550.2.a.cp 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7600))$$:

 $$T_{3}^{3} - 2 T_{3}^{2} - 5 T_{3} + 3$$ $$T_{7}^{3} - 2 T_{7}^{2} - 13 T_{7} - 1$$ $$T_{11}^{3} + 2 T_{11}^{2} - 24 T_{11} - 24$$ $$T_{13}^{3} - 6 T_{13}^{2} - 3 T_{13} + 35$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$3 - 5 T - 2 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-1 - 13 T - 2 T^{2} + T^{3}$$
$11$ $$-24 - 24 T + 2 T^{2} + T^{3}$$
$13$ $$35 - 3 T - 6 T^{2} + T^{3}$$
$17$ $$-9 + 33 T + 12 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$111 - 51 T + 2 T^{2} + T^{3}$$
$29$ $$9 - 3 T - 6 T^{2} + T^{3}$$
$31$ $$-56 - 4 T + 8 T^{2} + T^{3}$$
$37$ $$1 - 25 T - T^{2} + T^{3}$$
$41$ $$T^{3}$$
$43$ $$-24 + 40 T + 14 T^{2} + T^{3}$$
$47$ $$-45 - 57 T - 3 T^{2} + T^{3}$$
$53$ $$867 - 105 T - 10 T^{2} + T^{3}$$
$59$ $$-1431 - 201 T + 6 T^{2} + T^{3}$$
$61$ $$-120 - 56 T + 2 T^{2} + T^{3}$$
$67$ $$75 - 23 T - 4 T^{2} + T^{3}$$
$71$ $$1512 - 192 T - 6 T^{2} + T^{3}$$
$73$ $$317 - 27 T - 12 T^{2} + T^{3}$$
$79$ $$-320 + 32 T + 20 T^{2} + T^{3}$$
$83$ $$168 - 108 T + 4 T^{2} + T^{3}$$
$89$ $$312 - 36 T - 14 T^{2} + T^{3}$$
$97$ $$2440 + 44 T - 28 T^{2} + T^{3}$$