# Properties

 Label 363.3.b.i Level $363$ Weight $3$ Character orbit 363.b Analytic conductor $9.891$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{3} ) q^{3} -3 q^{4} + \beta_{3} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{6} + \beta_{1} q^{7} + \beta_{2} q^{8} + ( -7 - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -1 + \beta_{3} ) q^{3} -3 q^{4} + \beta_{3} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{6} + \beta_{1} q^{7} + \beta_{2} q^{8} + ( -7 - 2 \beta_{3} ) q^{9} -\beta_{1} q^{10} + ( 3 - 3 \beta_{3} ) q^{12} -3 \beta_{1} q^{13} + 7 \beta_{3} q^{14} + ( -8 - \beta_{3} ) q^{15} -19 q^{16} + 8 \beta_{2} q^{17} + ( 2 \beta_{1} - 7 \beta_{2} ) q^{18} -2 \beta_{1} q^{19} -3 \beta_{3} q^{20} + ( -\beta_{1} + 8 \beta_{2} ) q^{21} -11 \beta_{3} q^{23} + ( -\beta_{1} - \beta_{2} ) q^{24} + 17 q^{25} -21 \beta_{3} q^{26} + ( 23 - 5 \beta_{3} ) q^{27} -3 \beta_{1} q^{28} + 8 \beta_{2} q^{29} + ( \beta_{1} - 8 \beta_{2} ) q^{30} + 30 q^{31} -15 \beta_{2} q^{32} -56 q^{34} + 8 \beta_{2} q^{35} + ( 21 + 6 \beta_{3} ) q^{36} -10 q^{37} -14 \beta_{3} q^{38} + ( 3 \beta_{1} - 24 \beta_{2} ) q^{39} -\beta_{1} q^{40} + 16 \beta_{2} q^{41} + ( -56 - 7 \beta_{3} ) q^{42} + 2 \beta_{1} q^{43} + ( 16 - 7 \beta_{3} ) q^{45} + 11 \beta_{1} q^{46} + 13 \beta_{3} q^{47} + ( 19 - 19 \beta_{3} ) q^{48} + 7 q^{49} + 17 \beta_{2} q^{50} + ( -8 \beta_{1} - 8 \beta_{2} ) q^{51} + 9 \beta_{1} q^{52} -15 \beta_{3} q^{53} + ( 5 \beta_{1} + 23 \beta_{2} ) q^{54} + 7 \beta_{3} q^{56} + ( 2 \beta_{1} - 16 \beta_{2} ) q^{57} -56 q^{58} + 12 \beta_{3} q^{59} + ( 24 + 3 \beta_{3} ) q^{60} + 13 \beta_{1} q^{61} + 30 \beta_{2} q^{62} + ( -7 \beta_{1} - 16 \beta_{2} ) q^{63} + 29 q^{64} -24 \beta_{2} q^{65} -42 q^{67} -24 \beta_{2} q^{68} + ( 88 + 11 \beta_{3} ) q^{69} -56 q^{70} + 23 \beta_{3} q^{71} + ( 2 \beta_{1} - 7 \beta_{2} ) q^{72} -10 \beta_{1} q^{73} -10 \beta_{2} q^{74} + ( -17 + 17 \beta_{3} ) q^{75} + 6 \beta_{1} q^{76} + ( 168 + 21 \beta_{3} ) q^{78} -3 \beta_{1} q^{79} -19 \beta_{3} q^{80} + ( 17 + 28 \beta_{3} ) q^{81} -112 q^{82} + 8 \beta_{2} q^{83} + ( 3 \beta_{1} - 24 \beta_{2} ) q^{84} -8 \beta_{1} q^{85} + 14 \beta_{3} q^{86} + ( -8 \beta_{1} - 8 \beta_{2} ) q^{87} -22 \beta_{3} q^{89} + ( 7 \beta_{1} + 16 \beta_{2} ) q^{90} -168 q^{91} + 33 \beta_{3} q^{92} + ( -30 + 30 \beta_{3} ) q^{93} -13 \beta_{1} q^{94} -16 \beta_{2} q^{95} + ( 15 \beta_{1} + 15 \beta_{2} ) q^{96} + 74 q^{97} + 7 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 12 q^{4} - 28 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} - 12 q^{4} - 28 q^{9} + 12 q^{12} - 32 q^{15} - 76 q^{16} + 68 q^{25} + 92 q^{27} + 120 q^{31} - 224 q^{34} + 84 q^{36} - 40 q^{37} - 224 q^{42} + 64 q^{45} + 76 q^{48} + 28 q^{49} - 224 q^{58} + 96 q^{60} + 116 q^{64} - 168 q^{67} + 352 q^{69} - 224 q^{70} - 68 q^{75} + 672 q^{78} + 68 q^{81} - 448 q^{82} - 672 q^{91} - 120 q^{93} + 296 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2} - 2 \nu + 8$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{3} - 6 \nu^{2} + 32 \nu - 15$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{3} - 6 \nu^{2} + 34 \nu - 16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + \beta_{1} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-13 \beta_{3} + 14 \beta_{2} + 3 \beta_{1} - 22$$$$)/4$$

## 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 − 0.0913379i 0.5 + 2.73709i 0.5 − 2.73709i 0.5 + 0.0913379i
2.64575i −1.00000 2.82843i −3.00000 2.82843i −7.48331 + 2.64575i 7.48331 2.64575i −7.00000 + 5.65685i −7.48331
122.2 2.64575i −1.00000 + 2.82843i −3.00000 2.82843i 7.48331 + 2.64575i −7.48331 2.64575i −7.00000 5.65685i 7.48331
122.3 2.64575i −1.00000 2.82843i −3.00000 2.82843i 7.48331 2.64575i −7.48331 2.64575i −7.00000 + 5.65685i 7.48331
122.4 2.64575i −1.00000 + 2.82843i −3.00000 2.82843i −7.48331 2.64575i 7.48331 2.64575i −7.00000 5.65685i −7.48331
 $$n$$: 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 inner
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.i 4
3.b odd 2 1 inner 363.3.b.i 4
11.b odd 2 1 inner 363.3.b.i 4
11.c even 5 4 363.3.h.k 16
11.d odd 10 4 363.3.h.k 16
33.d even 2 1 inner 363.3.b.i 4
33.f even 10 4 363.3.h.k 16
33.h odd 10 4 363.3.h.k 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.i 4 1.a even 1 1 trivial
363.3.b.i 4 3.b odd 2 1 inner
363.3.b.i 4 11.b odd 2 1 inner
363.3.b.i 4 33.d even 2 1 inner
363.3.h.k 16 11.c even 5 4
363.3.h.k 16 11.d odd 10 4
363.3.h.k 16 33.f even 10 4
363.3.h.k 16 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 $$S_{3}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}^{2} + 7$$ $$T_{5}^{2} + 8$$ $$T_{7}^{2} - 56$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 7 + T^{2} )^{2}$$
$3$ $$( 9 + 2 T + T^{2} )^{2}$$
$5$ $$( 8 + T^{2} )^{2}$$
$7$ $$( -56 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( -504 + T^{2} )^{2}$$
$17$ $$( 448 + T^{2} )^{2}$$
$19$ $$( -224 + T^{2} )^{2}$$
$23$ $$( 968 + T^{2} )^{2}$$
$29$ $$( 448 + T^{2} )^{2}$$
$31$ $$( -30 + T )^{4}$$
$37$ $$( 10 + T )^{4}$$
$41$ $$( 1792 + T^{2} )^{2}$$
$43$ $$( -224 + T^{2} )^{2}$$
$47$ $$( 1352 + T^{2} )^{2}$$
$53$ $$( 1800 + T^{2} )^{2}$$
$59$ $$( 1152 + T^{2} )^{2}$$
$61$ $$( -9464 + T^{2} )^{2}$$
$67$ $$( 42 + T )^{4}$$
$71$ $$( 4232 + T^{2} )^{2}$$
$73$ $$( -5600 + T^{2} )^{2}$$
$79$ $$( -504 + T^{2} )^{2}$$
$83$ $$( 448 + T^{2} )^{2}$$
$89$ $$( 3872 + T^{2} )^{2}$$
$97$ $$( -74 + T )^{4}$$