# Properties

 Label 363.3.b.a Level $363$ Weight $3$ Character orbit 363.b Self dual yes Analytic conductor $9.891$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 363.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.89103359628$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 4 q^{4} - 11 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 4 * q^4 - 11 * q^7 + 9 * q^9 $$q - 3 q^{3} + 4 q^{4} - 11 q^{7} + 9 q^{9} - 12 q^{12} + 22 q^{13} + 16 q^{16} - 11 q^{19} + 33 q^{21} + 25 q^{25} - 27 q^{27} - 44 q^{28} + 59 q^{31} + 36 q^{36} + 47 q^{37} - 66 q^{39} + 22 q^{43} - 48 q^{48} + 72 q^{49} + 88 q^{52} + 33 q^{57} + 121 q^{61} - 99 q^{63} + 64 q^{64} - 13 q^{67} - 143 q^{73} - 75 q^{75} - 44 q^{76} - 11 q^{79} + 81 q^{81} + 132 q^{84} - 242 q^{91} - 177 q^{93} - 169 q^{97}+O(q^{100})$$ q - 3 * q^3 + 4 * q^4 - 11 * q^7 + 9 * q^9 - 12 * q^12 + 22 * q^13 + 16 * q^16 - 11 * q^19 + 33 * q^21 + 25 * q^25 - 27 * q^27 - 44 * q^28 + 59 * q^31 + 36 * q^36 + 47 * q^37 - 66 * q^39 + 22 * q^43 - 48 * q^48 + 72 * q^49 + 88 * q^52 + 33 * q^57 + 121 * q^61 - 99 * q^63 + 64 * q^64 - 13 * q^67 - 143 * q^73 - 75 * q^75 - 44 * q^76 - 11 * q^79 + 81 * q^81 + 132 * q^84 - 242 * q^91 - 177 * q^93 - 169 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
122.1
 0
0 −3.00000 4.00000 0 0 −11.0000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.a 1
3.b odd 2 1 CM 363.3.b.a 1
11.b odd 2 1 363.3.b.b yes 1
11.c even 5 4 363.3.h.b 4
11.d odd 10 4 363.3.h.a 4
33.d even 2 1 363.3.b.b yes 1
33.f even 10 4 363.3.h.a 4
33.h odd 10 4 363.3.h.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.a 1 1.a even 1 1 trivial
363.3.b.a 1 3.b odd 2 1 CM
363.3.b.b yes 1 11.b odd 2 1
363.3.b.b yes 1 33.d even 2 1
363.3.h.a 4 11.d odd 10 4
363.3.h.a 4 33.f even 10 4
363.3.h.b 4 11.c even 5 4
363.3.h.b 4 33.h odd 10 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7} + 11$$ T7 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T$$
$7$ $$T + 11$$
$11$ $$T$$
$13$ $$T - 22$$
$17$ $$T$$
$19$ $$T + 11$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T - 59$$
$37$ $$T - 47$$
$41$ $$T$$
$43$ $$T - 22$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 121$$
$67$ $$T + 13$$
$71$ $$T$$
$73$ $$T + 143$$
$79$ $$T + 11$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 169$$