Properties

 Label 7600.2.a.l 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 95) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{7} - 3q^{9} + O(q^{10})$$ $$q + 2q^{7} - 3q^{9} + 4q^{11} + 2q^{13} - 4q^{17} - q^{19} - 6q^{23} - 6q^{29} + 4q^{31} + 10q^{37} - 10q^{41} + 2q^{43} - 6q^{47} - 3q^{49} - 10q^{53} + 2q^{61} - 6q^{63} + 8q^{67} - 4q^{71} - 4q^{73} + 8q^{77} - 4q^{79} + 9q^{81} - 18q^{83} - 2q^{89} + 4q^{91} - 6q^{97} - 12q^{99} + O(q^{100})$$

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 0 0 0 0 2.00000 0 −3.00000 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.l 1
4.b odd 2 1 475.2.a.c 1
5.b even 2 1 7600.2.a.i 1
5.c odd 4 2 1520.2.d.b 2
12.b even 2 1 4275.2.a.e 1
20.d odd 2 1 475.2.a.a 1
20.e even 4 2 95.2.b.a 2
60.h even 2 1 4275.2.a.p 1
60.l odd 4 2 855.2.c.b 2
76.d even 2 1 9025.2.a.c 1
380.d even 2 1 9025.2.a.h 1
380.j odd 4 2 1805.2.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.a 2 20.e even 4 2
475.2.a.a 1 20.d odd 2 1
475.2.a.c 1 4.b odd 2 1
855.2.c.b 2 60.l odd 4 2
1520.2.d.b 2 5.c odd 4 2
1805.2.b.c 2 380.j odd 4 2
4275.2.a.e 1 12.b even 2 1
4275.2.a.p 1 60.h even 2 1
7600.2.a.i 1 5.b even 2 1
7600.2.a.l 1 1.a even 1 1 trivial
9025.2.a.c 1 76.d even 2 1
9025.2.a.h 1 380.d even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-2 + T$$
$11$ $$-4 + T$$
$13$ $$-2 + T$$
$17$ $$4 + T$$
$19$ $$1 + T$$
$23$ $$6 + T$$
$29$ $$6 + T$$
$31$ $$-4 + T$$
$37$ $$-10 + T$$
$41$ $$10 + T$$
$43$ $$-2 + T$$
$47$ $$6 + T$$
$53$ $$10 + T$$
$59$ $$T$$
$61$ $$-2 + T$$
$67$ $$-8 + T$$
$71$ $$4 + T$$
$73$ $$4 + T$$
$79$ $$4 + T$$
$83$ $$18 + T$$
$89$ $$2 + T$$
$97$ $$6 + T$$