# Properties

 Label 7600.2.a.ba Level $7600$ Weight $2$ Character orbit 7600.a Self dual yes Analytic conductor $60.686$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 2 \beta q^{7} - q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 2 \beta q^{7} - q^{9} + ( -2 - 2 \beta ) q^{11} + ( -2 + \beta ) q^{13} + ( -2 + 2 \beta ) q^{17} + q^{19} + 4 q^{21} + 4 q^{23} -4 \beta q^{27} + ( 2 + 2 \beta ) q^{29} + ( -4 - 2 \beta ) q^{33} + ( -6 + 3 \beta ) q^{37} + ( 2 - 2 \beta ) q^{39} + ( 2 + 2 \beta ) q^{41} -2 \beta q^{43} + 6 \beta q^{47} + q^{49} + ( 4 - 2 \beta ) q^{51} + ( -6 + 5 \beta ) q^{53} + \beta q^{57} + 2 \beta q^{59} + ( 4 + 4 \beta ) q^{61} -2 \beta q^{63} + ( -8 + \beta ) q^{67} + 4 \beta q^{69} + ( -4 + 8 \beta ) q^{71} + ( 2 + 10 \beta ) q^{73} + ( -8 - 4 \beta ) q^{77} + ( 12 + 2 \beta ) q^{79} -5 q^{81} + 8 q^{83} + ( 4 + 2 \beta ) q^{87} + ( -6 - 2 \beta ) q^{89} + ( 4 - 4 \beta ) q^{91} + ( -2 - 3 \beta ) q^{97} + ( 2 + 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 4q^{11} - 4q^{13} - 4q^{17} + 2q^{19} + 8q^{21} + 8q^{23} + 4q^{29} - 8q^{33} - 12q^{37} + 4q^{39} + 4q^{41} + 2q^{49} + 8q^{51} - 12q^{53} + 8q^{61} - 16q^{67} - 8q^{71} + 4q^{73} - 16q^{77} + 24q^{79} - 10q^{81} + 16q^{83} + 8q^{87} - 12q^{89} + 8q^{91} - 4q^{97} + 4q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −1.41421 0 0 0 −2.82843 0 −1.00000 0
1.2 0 1.41421 0 0 0 2.82843 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.ba 2
4.b odd 2 1 3800.2.a.n 2
5.b even 2 1 1520.2.a.m 2
20.d odd 2 1 760.2.a.f 2
20.e even 4 2 3800.2.d.i 4
40.e odd 2 1 6080.2.a.bf 2
40.f even 2 1 6080.2.a.bg 2
60.h even 2 1 6840.2.a.z 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-8 + T^{2}$$
$11$ $$-4 + 4 T + T^{2}$$
$13$ $$2 + 4 T + T^{2}$$
$17$ $$-4 + 4 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$-4 - 4 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$18 + 12 T + T^{2}$$
$41$ $$-4 - 4 T + T^{2}$$
$43$ $$-8 + T^{2}$$
$47$ $$-72 + T^{2}$$
$53$ $$-14 + 12 T + T^{2}$$
$59$ $$-8 + T^{2}$$
$61$ $$-16 - 8 T + T^{2}$$
$67$ $$62 + 16 T + T^{2}$$
$71$ $$-112 + 8 T + T^{2}$$
$73$ $$-196 - 4 T + T^{2}$$
$79$ $$136 - 24 T + T^{2}$$
$83$ $$( -8 + T )^{2}$$
$89$ $$28 + 12 T + T^{2}$$
$97$ $$-14 + 4 T + T^{2}$$