# Properties

 Label 468.2.a.d Level $468$ Weight $2$ Character orbit 468.a Self dual yes Analytic conductor $3.737$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$468 = 2^{2} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 468.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.73699881460$$ 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

 $$f(q)$$ $$=$$ $$q + 4q^{5} - 2q^{7} + O(q^{10})$$ $$q + 4q^{5} - 2q^{7} + 4q^{11} + q^{13} - 2q^{17} - 2q^{19} + 11q^{25} + 6q^{29} - 10q^{31} - 8q^{35} + 10q^{37} - 8q^{41} + 4q^{43} + 4q^{47} - 3q^{49} + 10q^{53} + 16q^{55} + 8q^{59} - 14q^{61} + 4q^{65} + 2q^{67} - 16q^{71} - 10q^{73} - 8q^{77} - 16q^{79} - 8q^{85} + 4q^{89} - 2q^{91} - 8q^{95} - 2q^{97} + O(q^{100})$$

## 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 0 0 4.00000 0 −2.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## 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 468.2.a.d 1
3.b odd 2 1 156.2.a.a 1
4.b odd 2 1 1872.2.a.s 1
8.b even 2 1 7488.2.a.c 1
8.d odd 2 1 7488.2.a.d 1
9.c even 3 2 4212.2.i.b 2
9.d odd 6 2 4212.2.i.l 2
12.b even 2 1 624.2.a.e 1
13.b even 2 1 6084.2.a.b 1
13.d odd 4 2 6084.2.b.j 2
15.d odd 2 1 3900.2.a.m 1
15.e even 4 2 3900.2.h.b 2
21.c even 2 1 7644.2.a.k 1
24.f even 2 1 2496.2.a.o 1
24.h odd 2 1 2496.2.a.bc 1
39.d odd 2 1 2028.2.a.c 1
39.f even 4 2 2028.2.b.a 2
39.h odd 6 2 2028.2.i.g 2
39.i odd 6 2 2028.2.i.e 2
39.k even 12 4 2028.2.q.h 4
156.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 3.b odd 2 1
468.2.a.d 1 1.a even 1 1 trivial
624.2.a.e 1 12.b even 2 1
1872.2.a.s 1 4.b odd 2 1
2028.2.a.c 1 39.d odd 2 1
2028.2.b.a 2 39.f even 4 2
2028.2.i.e 2 39.i odd 6 2
2028.2.i.g 2 39.h odd 6 2
2028.2.q.h 4 39.k even 12 4
2496.2.a.o 1 24.f even 2 1
2496.2.a.bc 1 24.h odd 2 1
3900.2.a.m 1 15.d odd 2 1
3900.2.h.b 2 15.e even 4 2
4212.2.i.b 2 9.c even 3 2
4212.2.i.l 2 9.d odd 6 2
6084.2.a.b 1 13.b even 2 1
6084.2.b.j 2 13.d odd 4 2
7488.2.a.c 1 8.b even 2 1
7488.2.a.d 1 8.d odd 2 1
7644.2.a.k 1 21.c even 2 1
8112.2.a.bi 1 156.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(468))$$:

 $$T_{5} - 4$$ $$T_{7} + 2$$

## Hecke characteristic polynomials

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