Properties

 Label 378.2.a.e Level $378$ Weight $2$ Character orbit 378.a Self dual yes Analytic conductor $3.018$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + q^{7} + q^{8} + 5q^{13} + q^{14} + q^{16} + 3q^{17} + 2q^{19} - 9q^{23} - 5q^{25} + 5q^{26} + q^{28} - 3q^{29} + 5q^{31} + q^{32} + 3q^{34} + 2q^{37} + 2q^{38} - 6q^{41} - q^{43} - 9q^{46} - 6q^{47} + q^{49} - 5q^{50} + 5q^{52} + 3q^{53} + q^{56} - 3q^{58} - 3q^{59} - 10q^{61} + 5q^{62} + q^{64} - 13q^{67} + 3q^{68} + 9q^{71} + 2q^{73} + 2q^{74} + 2q^{76} - 10q^{79} - 6q^{82} - 12q^{83} - q^{86} + 15q^{89} + 5q^{91} - 9q^{92} - 6q^{94} + 8q^{97} + q^{98} + O(q^{100})$$

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

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 378.2.a.e yes 1
3.b odd 2 1 378.2.a.d 1
4.b odd 2 1 3024.2.a.p 1
5.b even 2 1 9450.2.a.l 1
7.b odd 2 1 2646.2.a.y 1
9.c even 3 2 1134.2.f.e 2
9.d odd 6 2 1134.2.f.k 2
12.b even 2 1 3024.2.a.o 1
15.d odd 2 1 9450.2.a.cl 1
21.c even 2 1 2646.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.d 1 3.b odd 2 1
378.2.a.e yes 1 1.a even 1 1 trivial
1134.2.f.e 2 9.c even 3 2
1134.2.f.k 2 9.d odd 6 2
2646.2.a.f 1 21.c even 2 1
2646.2.a.y 1 7.b odd 2 1
3024.2.a.o 1 12.b even 2 1
3024.2.a.p 1 4.b odd 2 1
9450.2.a.l 1 5.b even 2 1
9450.2.a.cl 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(378))$$:

 $$T_{5}$$ $$T_{17} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$-5 + T$$
$17$ $$-3 + T$$
$19$ $$-2 + T$$
$23$ $$9 + T$$
$29$ $$3 + T$$
$31$ $$-5 + T$$
$37$ $$-2 + T$$
$41$ $$6 + T$$
$43$ $$1 + T$$
$47$ $$6 + T$$
$53$ $$-3 + T$$
$59$ $$3 + T$$
$61$ $$10 + T$$
$67$ $$13 + T$$
$71$ $$-9 + T$$
$73$ $$-2 + T$$
$79$ $$10 + T$$
$83$ $$12 + T$$
$89$ $$-15 + T$$
$97$ $$-8 + T$$