# Properties

 Label 7600.2.a.d Level $7600$ Weight $2$ Character orbit 7600.a Self dual yes Analytic conductor $60.686$ Analytic rank $1$ 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: $$1$$ 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

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

## 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 −2.00000 0 0 0 0 0 1.00000 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.d 1
4.b odd 2 1 3800.2.a.h 1
5.b even 2 1 1520.2.a.i 1
20.d odd 2 1 760.2.a.a 1
20.e even 4 2 3800.2.d.b 2
40.e odd 2 1 6080.2.a.t 1
40.f even 2 1 6080.2.a.d 1
60.h even 2 1 6840.2.a.i 1

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-4 + T$$
$13$ $$4 + T$$
$17$ $$-2 + T$$
$19$ $$1 + T$$
$23$ $$4 + T$$
$29$ $$6 + T$$
$31$ $$-8 + T$$
$37$ $$-4 + T$$
$41$ $$2 + T$$
$43$ $$-4 + T$$
$47$ $$8 + T$$
$53$ $$T$$
$59$ $$-8 + T$$
$61$ $$-2 + T$$
$67$ $$14 + T$$
$71$ $$-8 + T$$
$73$ $$6 + T$$
$79$ $$-4 + T$$
$83$ $$-16 + T$$
$89$ $$18 + T$$
$97$ $$-4 + T$$