# Properties

 Label 378.2.a.b Level $378$ Weight $2$ Character orbit 378.a Self dual yes Analytic conductor $3.018$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 3 q^{5} + q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - 3 * q^5 + q^7 - q^8 $$q - q^{2} + q^{4} - 3 q^{5} + q^{7} - q^{8} + 3 q^{10} + 3 q^{11} - 4 q^{13} - q^{14} + q^{16} - 6 q^{17} - 7 q^{19} - 3 q^{20} - 3 q^{22} - 3 q^{23} + 4 q^{25} + 4 q^{26} + q^{28} + 5 q^{31} - q^{32} + 6 q^{34} - 3 q^{35} - 7 q^{37} + 7 q^{38} + 3 q^{40} - 9 q^{41} - 10 q^{43} + 3 q^{44} + 3 q^{46} + 6 q^{47} + q^{49} - 4 q^{50} - 4 q^{52} + 12 q^{53} - 9 q^{55} - q^{56} - 6 q^{59} + 8 q^{61} - 5 q^{62} + q^{64} + 12 q^{65} - 4 q^{67} - 6 q^{68} + 3 q^{70} + 9 q^{71} + 2 q^{73} + 7 q^{74} - 7 q^{76} + 3 q^{77} - 10 q^{79} - 3 q^{80} + 9 q^{82} + 18 q^{85} + 10 q^{86} - 3 q^{88} + 15 q^{89} - 4 q^{91} - 3 q^{92} - 6 q^{94} + 21 q^{95} + 8 q^{97} - q^{98}+O(q^{100})$$ q - q^2 + q^4 - 3 * q^5 + q^7 - q^8 + 3 * q^10 + 3 * q^11 - 4 * q^13 - q^14 + q^16 - 6 * q^17 - 7 * q^19 - 3 * q^20 - 3 * q^22 - 3 * q^23 + 4 * q^25 + 4 * q^26 + q^28 + 5 * q^31 - q^32 + 6 * q^34 - 3 * q^35 - 7 * q^37 + 7 * q^38 + 3 * q^40 - 9 * q^41 - 10 * q^43 + 3 * q^44 + 3 * q^46 + 6 * q^47 + q^49 - 4 * q^50 - 4 * q^52 + 12 * q^53 - 9 * q^55 - q^56 - 6 * q^59 + 8 * q^61 - 5 * q^62 + q^64 + 12 * q^65 - 4 * q^67 - 6 * q^68 + 3 * q^70 + 9 * q^71 + 2 * q^73 + 7 * q^74 - 7 * q^76 + 3 * q^77 - 10 * q^79 - 3 * q^80 + 9 * q^82 + 18 * q^85 + 10 * q^86 - 3 * q^88 + 15 * q^89 - 4 * q^91 - 3 * q^92 - 6 * q^94 + 21 * q^95 + 8 * 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 −3.00000 0 1.00000 −1.00000 0 3.00000
 $$n$$: 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.b 1
3.b odd 2 1 378.2.a.g yes 1
4.b odd 2 1 3024.2.a.c 1
5.b even 2 1 9450.2.a.cu 1
7.b odd 2 1 2646.2.a.n 1
9.c even 3 2 1134.2.f.o 2
9.d odd 6 2 1134.2.f.b 2
12.b even 2 1 3024.2.a.bb 1
15.d odd 2 1 9450.2.a.h 1
21.c even 2 1 2646.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.b 1 1.a even 1 1 trivial
378.2.a.g yes 1 3.b odd 2 1
1134.2.f.b 2 9.d odd 6 2
1134.2.f.o 2 9.c even 3 2
2646.2.a.n 1 7.b odd 2 1
2646.2.a.q 1 21.c even 2 1
3024.2.a.c 1 4.b odd 2 1
3024.2.a.bb 1 12.b even 2 1
9450.2.a.h 1 15.d odd 2 1
9450.2.a.cu 1 5.b even 2 1

## Hecke kernels

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

 $$T_{5} + 3$$ T5 + 3 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T - 1$$
$11$ $$T - 3$$
$13$ $$T + 4$$
$17$ $$T + 6$$
$19$ $$T + 7$$
$23$ $$T + 3$$
$29$ $$T$$
$31$ $$T - 5$$
$37$ $$T + 7$$
$41$ $$T + 9$$
$43$ $$T + 10$$
$47$ $$T - 6$$
$53$ $$T - 12$$
$59$ $$T + 6$$
$61$ $$T - 8$$
$67$ $$T + 4$$
$71$ $$T - 9$$
$73$ $$T - 2$$
$79$ $$T + 10$$
$83$ $$T$$
$89$ $$T - 15$$
$97$ $$T - 8$$