# Properties

 Label 7600.2.a.g 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 190) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{7} - 2q^{9} + O(q^{10})$$ $$q - q^{3} - q^{7} - 2q^{9} + 3q^{13} + 7q^{17} + q^{19} + q^{21} - 5q^{23} + 5q^{27} - 5q^{29} - 10q^{31} - 2q^{37} - 3q^{39} + 2q^{41} + 6q^{43} - 6q^{49} - 7q^{51} - 9q^{53} - q^{57} + 7q^{59} - 4q^{61} + 2q^{63} + 7q^{67} + 5q^{69} + 9q^{73} + 10q^{79} + q^{81} - 2q^{83} + 5q^{87} - 10q^{89} - 3q^{91} + 10q^{93} + 18q^{97} + O(q^{100})$$

## 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 −1.00000 0 0 0 −1.00000 0 −2.00000 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.g 1
4.b odd 2 1 950.2.a.e 1
5.b even 2 1 1520.2.a.g 1
12.b even 2 1 8550.2.a.l 1
20.d odd 2 1 190.2.a.a 1
20.e even 4 2 950.2.b.d 2
40.e odd 2 1 6080.2.a.r 1
40.f even 2 1 6080.2.a.i 1
60.h even 2 1 1710.2.a.r 1
140.c even 2 1 9310.2.a.i 1
380.d even 2 1 3610.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.a 1 20.d odd 2 1
950.2.a.e 1 4.b odd 2 1
950.2.b.d 2 20.e even 4 2
1520.2.a.g 1 5.b even 2 1
1710.2.a.r 1 60.h even 2 1
3610.2.a.h 1 380.d even 2 1
6080.2.a.i 1 40.f even 2 1
6080.2.a.r 1 40.e odd 2 1
7600.2.a.g 1 1.a even 1 1 trivial
8550.2.a.l 1 12.b even 2 1
9310.2.a.i 1 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7600))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$T$$
$13$ $$-3 + T$$
$17$ $$-7 + T$$
$19$ $$-1 + T$$
$23$ $$5 + T$$
$29$ $$5 + T$$
$31$ $$10 + T$$
$37$ $$2 + T$$
$41$ $$-2 + T$$
$43$ $$-6 + T$$
$47$ $$T$$
$53$ $$9 + T$$
$59$ $$-7 + T$$
$61$ $$4 + T$$
$67$ $$-7 + T$$
$71$ $$T$$
$73$ $$-9 + T$$
$79$ $$-10 + T$$
$83$ $$2 + T$$
$89$ $$10 + T$$
$97$ $$-18 + T$$