# Properties

 Label 363.3.b.e Level $363$ Weight $3$ Character orbit 363.b Self dual yes Analytic conductor $9.891$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 4 q^{4} -5 \beta q^{7} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + 4 q^{4} -5 \beta q^{7} + 9 q^{9} + 12 q^{12} + 8 \beta q^{13} + 16 q^{16} + 21 \beta q^{19} -15 \beta q^{21} + 25 q^{25} + 27 q^{27} -20 \beta q^{28} -59 q^{31} + 36 q^{36} -47 q^{37} + 24 \beta q^{39} -48 \beta q^{43} + 48 q^{48} + 26 q^{49} + 32 \beta q^{52} + 63 \beta q^{57} -9 \beta q^{61} -45 \beta q^{63} + 64 q^{64} + 13 q^{67} + 17 \beta q^{73} + 75 q^{75} + 84 \beta q^{76} -91 \beta q^{79} + 81 q^{81} -60 \beta q^{84} -120 q^{91} -177 q^{93} -169 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 8 q^{4} + 18 q^{9} + O(q^{10})$$ $$2 q + 6 q^{3} + 8 q^{4} + 18 q^{9} + 24 q^{12} + 32 q^{16} + 50 q^{25} + 54 q^{27} - 118 q^{31} + 72 q^{36} - 94 q^{37} + 96 q^{48} + 52 q^{49} + 128 q^{64} + 26 q^{67} + 150 q^{75} + 162 q^{81} - 240 q^{91} - 354 q^{93} - 338 q^{97} + O(q^{100})$$

## 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
 1.73205 −1.73205
0 3.00000 4.00000 0 0 −8.66025 0 9.00000 0
122.2 0 3.00000 4.00000 0 0 8.66025 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 $$\Q(\sqrt{-3})$$
11.b odd 2 1 inner
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.e 2
3.b odd 2 1 CM 363.3.b.e 2
11.b odd 2 1 inner 363.3.b.e 2
11.c even 5 4 363.3.h.c 8
11.d odd 10 4 363.3.h.c 8
33.d even 2 1 inner 363.3.b.e 2
33.f even 10 4 363.3.h.c 8
33.h odd 10 4 363.3.h.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.e 2 1.a even 1 1 trivial
363.3.b.e 2 3.b odd 2 1 CM
363.3.b.e 2 11.b odd 2 1 inner
363.3.b.e 2 33.d even 2 1 inner
363.3.h.c 8 11.c even 5 4
363.3.h.c 8 11.d odd 10 4
363.3.h.c 8 33.f even 10 4
363.3.h.c 8 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 $$S_{3}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}$$ $$T_{5}$$ $$T_{7}^{2} - 75$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-75 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-192 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$-1323 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 59 + T )^{2}$$
$37$ $$( 47 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$-6912 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$-243 + T^{2}$$
$67$ $$( -13 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$-867 + T^{2}$$
$79$ $$-24843 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( 169 + T )^{2}$$