# Properties

 Label 363.3.b.c Level $363$ Weight $3$ Character orbit 363.b Analytic conductor $9.891$ Analytic rank $0$ Dimension $2$ CM discriminant -11 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: no Analytic conductor: $$9.89103359628$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{-11})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 3) q^{3} + 4 q^{4} + (6 \beta - 3) q^{5} + ( - 5 \beta + 6) q^{9}+O(q^{10})$$ q + (-b + 3) * q^3 + 4 * q^4 + (6*b - 3) * q^5 + (-5*b + 6) * q^9 $$q + ( - \beta + 3) q^{3} + 4 q^{4} + (6 \beta - 3) q^{5} + ( - 5 \beta + 6) q^{9} + ( - 4 \beta + 12) q^{12} + (15 \beta + 9) q^{15} + 16 q^{16} + (24 \beta - 12) q^{20} + (18 \beta - 9) q^{23} - 74 q^{25} + ( - 16 \beta + 3) q^{27} + 37 q^{31} + ( - 20 \beta + 24) q^{36} + 25 q^{37} + (21 \beta + 72) q^{45} + ( - 48 \beta + 24) q^{47} + ( - 16 \beta + 48) q^{48} - 49 q^{49} + ( - 48 \beta + 24) q^{53} + (30 \beta - 15) q^{59} + (60 \beta + 36) q^{60} + 64 q^{64} - 35 q^{67} + (45 \beta + 27) q^{69} + ( - 30 \beta + 15) q^{71} + (74 \beta - 222) q^{75} + (96 \beta - 48) q^{80} + ( - 35 \beta - 39) q^{81} + ( - 90 \beta + 45) q^{89} + (72 \beta - 36) q^{92} + ( - 37 \beta + 111) q^{93} + 95 q^{97}+O(q^{100})$$ q + (-b + 3) * q^3 + 4 * q^4 + (6*b - 3) * q^5 + (-5*b + 6) * q^9 + (-4*b + 12) * q^12 + (15*b + 9) * q^15 + 16 * q^16 + (24*b - 12) * q^20 + (18*b - 9) * q^23 - 74 * q^25 + (-16*b + 3) * q^27 + 37 * q^31 + (-20*b + 24) * q^36 + 25 * q^37 + (21*b + 72) * q^45 + (-48*b + 24) * q^47 + (-16*b + 48) * q^48 - 49 * q^49 + (-48*b + 24) * q^53 + (30*b - 15) * q^59 + (60*b + 36) * q^60 + 64 * q^64 - 35 * q^67 + (45*b + 27) * q^69 + (-30*b + 15) * q^71 + (74*b - 222) * q^75 + (96*b - 48) * q^80 + (-35*b - 39) * q^81 + (-90*b + 45) * q^89 + (72*b - 36) * q^92 + (-37*b + 111) * q^93 + 95 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{3} + 8 q^{4} + 7 q^{9}+O(q^{10})$$ 2 * q + 5 * q^3 + 8 * q^4 + 7 * q^9 $$2 q + 5 q^{3} + 8 q^{4} + 7 q^{9} + 20 q^{12} + 33 q^{15} + 32 q^{16} - 148 q^{25} - 10 q^{27} + 74 q^{31} + 28 q^{36} + 50 q^{37} + 165 q^{45} + 80 q^{48} - 98 q^{49} + 132 q^{60} + 128 q^{64} - 70 q^{67} + 99 q^{69} - 370 q^{75} - 113 q^{81} + 185 q^{93} + 190 q^{97}+O(q^{100})$$ 2 * q + 5 * q^3 + 8 * q^4 + 7 * q^9 + 20 * q^12 + 33 * q^15 + 32 * q^16 - 148 * q^25 - 10 * q^27 + 74 * q^31 + 28 * q^36 + 50 * q^37 + 165 * q^45 + 80 * q^48 - 98 * q^49 + 132 * q^60 + 128 * q^64 - 70 * q^67 + 99 * q^69 - 370 * q^75 - 113 * q^81 + 185 * q^93 + 190 * 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
 0.5 + 1.65831i 0.5 − 1.65831i
0 2.50000 1.65831i 4.00000 9.94987i 0 0 0 3.50000 8.29156i 0
122.2 0 2.50000 + 1.65831i 4.00000 9.94987i 0 0 0 3.50000 + 8.29156i 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
3.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.c 2
3.b odd 2 1 inner 363.3.b.c 2
11.b odd 2 1 CM 363.3.b.c 2
11.c even 5 4 363.3.h.f 8
11.d odd 10 4 363.3.h.f 8
33.d even 2 1 inner 363.3.b.c 2
33.f even 10 4 363.3.h.f 8
33.h odd 10 4 363.3.h.f 8

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

 $$T_{2}$$ T2 $$T_{5}^{2} + 99$$ T5^2 + 99 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 5T + 9$$
$5$ $$T^{2} + 99$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 891$$
$29$ $$T^{2}$$
$31$ $$(T - 37)^{2}$$
$37$ $$(T - 25)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 6336$$
$53$ $$T^{2} + 6336$$
$59$ $$T^{2} + 2475$$
$61$ $$T^{2}$$
$67$ $$(T + 35)^{2}$$
$71$ $$T^{2} + 2475$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 22275$$
$97$ $$(T - 95)^{2}$$