# Properties

 Label 7600.2.a.t Level $7600$ Weight $2$ Character orbit 7600.a Self dual yes Analytic conductor $60.686$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} - q^{7} + 6q^{9} + O(q^{10})$$ $$q + 3q^{3} - q^{7} + 6q^{9} - 4q^{11} - q^{13} + 7q^{17} + q^{19} - 3q^{21} - 5q^{23} + 9q^{27} + 7q^{29} + 2q^{31} - 12q^{33} + 6q^{37} - 3q^{39} + 6q^{41} + 10q^{43} - 8q^{47} - 6q^{49} + 21q^{51} + 3q^{53} + 3q^{57} - 5q^{59} - 8q^{61} - 6q^{63} + 11q^{67} - 15q^{69} + 12q^{71} + 9q^{73} + 4q^{77} - 6q^{79} + 9q^{81} + 14q^{83} + 21q^{87} - 6q^{89} + q^{91} + 6q^{93} + 2q^{97} - 24q^{99} + O(q^{100})$$

## 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
0 3.00000 0 0 0 −1.00000 0 6.00000 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.t 1
4.b odd 2 1 3800.2.a.a 1
5.b even 2 1 1520.2.a.a 1
20.d odd 2 1 760.2.a.e 1
20.e even 4 2 3800.2.d.a 2
40.e odd 2 1 6080.2.a.a 1
40.f even 2 1 6080.2.a.w 1
60.h even 2 1 6840.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.e 1 20.d odd 2 1
1520.2.a.a 1 5.b even 2 1
3800.2.a.a 1 4.b odd 2 1
3800.2.d.a 2 20.e even 4 2
6080.2.a.a 1 40.e odd 2 1
6080.2.a.w 1 40.f even 2 1
6840.2.a.c 1 60.h even 2 1
7600.2.a.t 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7600))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$4 + T$$
$13$ $$1 + T$$
$17$ $$-7 + T$$
$19$ $$-1 + T$$
$23$ $$5 + T$$
$29$ $$-7 + T$$
$31$ $$-2 + T$$
$37$ $$-6 + T$$
$41$ $$-6 + T$$
$43$ $$-10 + T$$
$47$ $$8 + T$$
$53$ $$-3 + T$$
$59$ $$5 + T$$
$61$ $$8 + T$$
$67$ $$-11 + T$$
$71$ $$-12 + T$$
$73$ $$-9 + T$$
$79$ $$6 + T$$
$83$ $$-14 + T$$
$89$ $$6 + T$$
$97$ $$-2 + T$$