# Properties

 Label 378.2.a.f 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^5 - q^7 + q^8 $$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} + 5 q^{11} - q^{14} + q^{16} + 2 q^{17} - q^{19} + q^{20} + 5 q^{22} - q^{23} - 4 q^{25} - q^{28} + 4 q^{29} - 9 q^{31} + q^{32} + 2 q^{34} - q^{35} + 5 q^{37} - q^{38} + q^{40} - 9 q^{41} - 10 q^{43} + 5 q^{44} - q^{46} + 6 q^{47} + q^{49} - 4 q^{50} + 12 q^{53} + 5 q^{55} - q^{56} + 4 q^{58} - 14 q^{59} - 9 q^{62} + q^{64} - 8 q^{67} + 2 q^{68} - q^{70} - 13 q^{71} - 2 q^{73} + 5 q^{74} - q^{76} - 5 q^{77} + 6 q^{79} + q^{80} - 9 q^{82} - 4 q^{83} + 2 q^{85} - 10 q^{86} + 5 q^{88} - 9 q^{89} - q^{92} + 6 q^{94} - q^{95} + 16 q^{97} + q^{98}+O(q^{100})$$ q + q^2 + q^4 + q^5 - q^7 + q^8 + q^10 + 5 * q^11 - q^14 + q^16 + 2 * q^17 - q^19 + q^20 + 5 * q^22 - q^23 - 4 * q^25 - q^28 + 4 * q^29 - 9 * q^31 + q^32 + 2 * q^34 - q^35 + 5 * q^37 - q^38 + q^40 - 9 * q^41 - 10 * q^43 + 5 * q^44 - q^46 + 6 * q^47 + q^49 - 4 * q^50 + 12 * q^53 + 5 * q^55 - q^56 + 4 * q^58 - 14 * q^59 - 9 * q^62 + q^64 - 8 * q^67 + 2 * q^68 - q^70 - 13 * q^71 - 2 * q^73 + 5 * q^74 - q^76 - 5 * q^77 + 6 * q^79 + q^80 - 9 * q^82 - 4 * q^83 + 2 * q^85 - 10 * q^86 + 5 * q^88 - 9 * q^89 - q^92 + 6 * q^94 - q^95 + 16 * q^97 + q^98

## 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 1.00000 0 −1.00000 1.00000 0 1.00000
 $$n$$: 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.f yes 1
3.b odd 2 1 378.2.a.c 1
4.b odd 2 1 3024.2.a.t 1
5.b even 2 1 9450.2.a.bx 1
7.b odd 2 1 2646.2.a.v 1
9.c even 3 2 1134.2.f.c 2
9.d odd 6 2 1134.2.f.n 2
12.b even 2 1 3024.2.a.m 1
15.d odd 2 1 9450.2.a.dc 1
21.c even 2 1 2646.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.c 1 3.b odd 2 1
378.2.a.f yes 1 1.a even 1 1 trivial
1134.2.f.c 2 9.c even 3 2
1134.2.f.n 2 9.d odd 6 2
2646.2.a.i 1 21.c even 2 1
2646.2.a.v 1 7.b odd 2 1
3024.2.a.m 1 12.b even 2 1
3024.2.a.t 1 4.b odd 2 1
9450.2.a.bx 1 5.b even 2 1
9450.2.a.dc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(378))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T - 5$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T + 1$$
$23$ $$T + 1$$
$29$ $$T - 4$$
$31$ $$T + 9$$
$37$ $$T - 5$$
$41$ $$T + 9$$
$43$ $$T + 10$$
$47$ $$T - 6$$
$53$ $$T - 12$$
$59$ $$T + 14$$
$61$ $$T$$
$67$ $$T + 8$$
$71$ $$T + 13$$
$73$ $$T + 2$$
$79$ $$T - 6$$
$83$ $$T + 4$$
$89$ $$T + 9$$
$97$ $$T - 16$$