Properties

 Label 7644.2.a.k Level $7644$ Weight $2$ Character orbit 7644.a Self dual yes Analytic conductor $61.038$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7644.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$61.0376473051$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 4q^{5} + q^{9} + O(q^{10})$$ $$q + q^{3} + 4q^{5} + q^{9} - 4q^{11} - q^{13} + 4q^{15} - 2q^{17} + 2q^{19} + 11q^{25} + q^{27} - 6q^{29} + 10q^{31} - 4q^{33} + 10q^{37} - q^{39} - 8q^{41} + 4q^{43} + 4q^{45} + 4q^{47} - 2q^{51} - 10q^{53} - 16q^{55} + 2q^{57} + 8q^{59} + 14q^{61} - 4q^{65} + 2q^{67} + 16q^{71} + 10q^{73} + 11q^{75} - 16q^{79} + q^{81} - 8q^{85} - 6q^{87} + 4q^{89} + 10q^{93} + 8q^{95} + 2q^{97} - 4q^{99} + O(q^{100})$$

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 1.00000 0 4.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$1$$

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 7644.2.a.k 1
7.b odd 2 1 156.2.a.a 1
21.c even 2 1 468.2.a.d 1
28.d even 2 1 624.2.a.e 1
35.c odd 2 1 3900.2.a.m 1
35.f even 4 2 3900.2.h.b 2
56.e even 2 1 2496.2.a.o 1
56.h odd 2 1 2496.2.a.bc 1
63.l odd 6 2 4212.2.i.l 2
63.o even 6 2 4212.2.i.b 2
84.h odd 2 1 1872.2.a.s 1
91.b odd 2 1 2028.2.a.c 1
91.i even 4 2 2028.2.b.a 2
91.n odd 6 2 2028.2.i.e 2
91.t odd 6 2 2028.2.i.g 2
91.bc even 12 4 2028.2.q.h 4
168.e odd 2 1 7488.2.a.d 1
168.i even 2 1 7488.2.a.c 1
273.g even 2 1 6084.2.a.b 1
273.o odd 4 2 6084.2.b.j 2
364.h even 2 1 8112.2.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.a 1 7.b odd 2 1
468.2.a.d 1 21.c even 2 1
624.2.a.e 1 28.d even 2 1
1872.2.a.s 1 84.h odd 2 1
2028.2.a.c 1 91.b odd 2 1
2028.2.b.a 2 91.i even 4 2
2028.2.i.e 2 91.n odd 6 2
2028.2.i.g 2 91.t odd 6 2
2028.2.q.h 4 91.bc even 12 4
2496.2.a.o 1 56.e even 2 1
2496.2.a.bc 1 56.h odd 2 1
3900.2.a.m 1 35.c odd 2 1
3900.2.h.b 2 35.f even 4 2
4212.2.i.b 2 63.o even 6 2
4212.2.i.l 2 63.l odd 6 2
6084.2.a.b 1 273.g even 2 1
6084.2.b.j 2 273.o odd 4 2
7488.2.a.c 1 168.i even 2 1
7488.2.a.d 1 168.e odd 2 1
7644.2.a.k 1 1.a even 1 1 trivial
8112.2.a.bi 1 364.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7644))$$:

 $$T_{5} - 4$$ $$T_{11} + 4$$ $$T_{17} + 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$-4 + T$$
$7$ $$T$$
$11$ $$4 + T$$
$13$ $$1 + T$$
$17$ $$2 + T$$
$19$ $$-2 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$-10 + T$$
$37$ $$-10 + T$$
$41$ $$8 + T$$
$43$ $$-4 + T$$
$47$ $$-4 + T$$
$53$ $$10 + T$$
$59$ $$-8 + T$$
$61$ $$-14 + T$$
$67$ $$-2 + T$$
$71$ $$-16 + T$$
$73$ $$-10 + T$$
$79$ $$16 + T$$
$83$ $$T$$
$89$ $$-4 + T$$
$97$ $$-2 + T$$