# Properties

 Label 363.3.b.h Level $363$ Weight $3$ Character orbit 363.b Analytic conductor $9.891$ Analytic rank $0$ Dimension $4$ CM no 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: no Analytic conductor: $$9.89103359628$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 33) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( -1 + \beta_{2} + \beta_{3} ) q^{3} + ( 1 + \beta_{1} + \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{6} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{7} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{8} + ( -3 - 5 \beta_{1} - 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{2} + ( -1 + \beta_{2} + \beta_{3} ) q^{3} + ( 1 + \beta_{1} + \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{6} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{7} + ( -3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{8} + ( -3 - 5 \beta_{1} - 3 \beta_{2} ) q^{9} + 2 q^{10} + ( 3 - 5 \beta_{1} + 3 \beta_{3} ) q^{12} + ( 4 + 4 \beta_{1} + 4 \beta_{3} ) q^{13} + ( 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{14} + ( 4 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{15} + ( -3 + 5 \beta_{1} + 5 \beta_{3} ) q^{16} + ( 2 \beta_{1} + 16 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -8 + \beta_{1} - 7 \beta_{2} + 5 \beta_{3} ) q^{18} + ( 6 - 6 \beta_{1} - 6 \beta_{3} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{20} + ( 8 - 10 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{21} + ( -\beta_{1} + 16 \beta_{2} + \beta_{3} ) q^{23} + ( -8 - \beta_{1} + 5 \beta_{2} - 7 \beta_{3} ) q^{24} + ( 21 - \beta_{1} - \beta_{3} ) q^{25} + ( 4 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} ) q^{26} + ( -5 + 16 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} ) q^{27} + 16 q^{28} + ( 24 \beta_{1} + 4 \beta_{2} - 24 \beta_{3} ) q^{29} + ( -2 + 2 \beta_{2} + 2 \beta_{3} ) q^{30} + ( 14 + 5 \beta_{1} + 5 \beta_{3} ) q^{31} + ( \beta_{1} + 19 \beta_{2} - \beta_{3} ) q^{32} + ( 20 - 16 \beta_{1} - 16 \beta_{3} ) q^{34} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{35} + ( -23 + \beta_{1} + 8 \beta_{2} - 7 \beta_{3} ) q^{36} + ( -26 - 7 \beta_{1} - 7 \beta_{3} ) q^{37} + ( -18 \beta_{1} - 30 \beta_{2} + 18 \beta_{3} ) q^{38} + ( 12 - 20 \beta_{1} + 12 \beta_{3} ) q^{39} + ( 10 + 2 \beta_{1} + 2 \beta_{3} ) q^{40} + ( -4 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{41} + ( -16 - 12 \beta_{1} - 20 \beta_{2} + 16 \beta_{3} ) q^{42} + ( -26 - 4 \beta_{1} - 4 \beta_{3} ) q^{43} + ( 4 - 6 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{45} + ( 14 - 16 \beta_{1} - 16 \beta_{3} ) q^{46} + ( -22 \beta_{1} - 14 \beta_{2} + 22 \beta_{3} ) q^{47} + ( 23 - 25 \beta_{1} - 8 \beta_{2} + 7 \beta_{3} ) q^{48} + ( -17 - 4 \beta_{1} - 4 \beta_{3} ) q^{49} + ( -23 \beta_{1} - 25 \beta_{2} + 23 \beta_{3} ) q^{50} + ( -56 - 30 \beta_{1} - 34 \beta_{2} + 20 \beta_{3} ) q^{51} + ( 36 + 4 \beta_{1} + 4 \beta_{3} ) q^{52} + ( 30 \beta_{1} + 38 \beta_{2} - 30 \beta_{3} ) q^{53} + ( 40 - 3 \beta_{1} + 5 \beta_{2} - 13 \beta_{3} ) q^{54} + 16 \beta_{2} q^{56} + ( -30 + 30 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} ) q^{57} + ( 52 - 4 \beta_{1} - 4 \beta_{3} ) q^{58} + ( 23 \beta_{1} - 2 \beta_{2} - 23 \beta_{3} ) q^{59} + ( 16 + 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{60} + ( -4 - 20 \beta_{1} - 20 \beta_{3} ) q^{61} + ( -4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{62} + ( -40 + 12 \beta_{1} + 22 \beta_{2} - 14 \beta_{3} ) q^{63} + ( 9 + \beta_{1} + \beta_{3} ) q^{64} + ( 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{65} + ( -6 + 17 \beta_{1} + 17 \beta_{3} ) q^{67} + ( -44 \beta_{1} - 20 \beta_{2} + 44 \beta_{3} ) q^{68} + ( -68 - 33 \beta_{1} - 31 \beta_{2} + 14 \beta_{3} ) q^{69} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{70} + ( 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{71} + ( -16 + 13 \beta_{1} - 17 \beta_{2} - 5 \beta_{3} ) q^{72} + ( 74 + 6 \beta_{1} + 6 \beta_{3} ) q^{73} + ( 12 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} ) q^{74} + ( -25 + 5 \beta_{1} + 22 \beta_{2} + 19 \beta_{3} ) q^{75} + ( -42 + 6 \beta_{1} + 6 \beta_{3} ) q^{76} + ( -32 - 20 \beta_{1} - 28 \beta_{2} + 20 \beta_{3} ) q^{78} + ( 32 + 26 \beta_{1} + 26 \beta_{3} ) q^{79} + ( 2 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} ) q^{80} + ( 37 - 37 \beta_{2} + 35 \beta_{3} ) q^{81} + ( 4 - 12 \beta_{1} - 12 \beta_{3} ) q^{82} + ( 30 \beta_{1} + 24 \beta_{2} - 30 \beta_{3} ) q^{83} + ( -16 + 16 \beta_{2} + 16 \beta_{3} ) q^{84} + ( 24 + 14 \beta_{1} + 14 \beta_{3} ) q^{85} + ( 18 \beta_{1} + 10 \beta_{2} - 18 \beta_{3} ) q^{86} + ( 80 + 16 \beta_{1} - 32 \beta_{2} + 52 \beta_{3} ) q^{87} + ( 39 \beta_{1} - 26 \beta_{2} - 39 \beta_{3} ) q^{89} + ( -6 - 10 \beta_{1} - 6 \beta_{2} ) q^{90} + 64 q^{91} + ( -50 \beta_{1} - 14 \beta_{2} + 50 \beta_{3} ) q^{92} + ( 6 - 25 \beta_{1} + 9 \beta_{2} + 24 \beta_{3} ) q^{93} + ( -58 + 14 \beta_{1} + 14 \beta_{3} ) q^{94} + 12 \beta_{2} q^{95} + ( -72 - 37 \beta_{1} - 39 \beta_{2} + 21 \beta_{3} ) q^{96} + ( 38 - 33 \beta_{1} - 33 \beta_{3} ) q^{97} + ( 9 \beta_{1} + \beta_{2} - 9 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} - 7 q^{9} + O(q^{10})$$ $$4 q - 5 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} - 7 q^{9} + 8 q^{10} + 14 q^{12} + 8 q^{13} + 13 q^{15} - 22 q^{16} - 38 q^{18} + 36 q^{19} + 38 q^{21} - 24 q^{24} + 86 q^{25} - 20 q^{27} + 64 q^{28} - 10 q^{30} + 46 q^{31} + 112 q^{34} - 86 q^{36} - 90 q^{37} + 56 q^{39} + 36 q^{40} - 68 q^{42} - 96 q^{43} + 17 q^{45} + 88 q^{46} + 110 q^{48} - 60 q^{49} - 214 q^{51} + 136 q^{52} + 176 q^{54} - 144 q^{57} + 216 q^{58} + 56 q^{60} + 24 q^{61} - 158 q^{63} + 34 q^{64} - 58 q^{67} - 253 q^{69} - 8 q^{70} - 72 q^{72} + 284 q^{73} - 124 q^{75} - 180 q^{76} - 128 q^{78} + 76 q^{79} + 113 q^{81} + 40 q^{82} - 80 q^{84} + 68 q^{85} + 252 q^{87} - 14 q^{90} + 256 q^{91} + 25 q^{93} - 260 q^{94} - 272 q^{96} + 218 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} + 4 \nu - 9$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 2 \nu^{2} + 2 \nu + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} - 3 \beta_{2} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4$$

## 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
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
2.52434i −2.68614 + 1.33591i −2.37228 0.792287i 3.37228 + 6.78073i −6.74456 4.10891i 5.43070 7.17687i 2.00000
122.2 0.792287i 0.186141 2.99422i 3.37228 2.52434i −2.37228 0.147477i 4.74456 5.84096i −8.93070 1.11469i 2.00000
122.3 0.792287i 0.186141 + 2.99422i 3.37228 2.52434i −2.37228 + 0.147477i 4.74456 5.84096i −8.93070 + 1.11469i 2.00000
122.4 2.52434i −2.68614 1.33591i −2.37228 0.792287i 3.37228 6.78073i −6.74456 4.10891i 5.43070 + 7.17687i 2.00000
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.h 4
3.b odd 2 1 inner 363.3.b.h 4
11.b odd 2 1 33.3.b.b 4
11.c even 5 4 363.3.h.l 16
11.d odd 10 4 363.3.h.m 16
33.d even 2 1 33.3.b.b 4
33.f even 10 4 363.3.h.m 16
33.h odd 10 4 363.3.h.l 16
44.c even 2 1 528.3.i.d 4
132.d odd 2 1 528.3.i.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.b 4 11.b odd 2 1
33.3.b.b 4 33.d even 2 1
363.3.b.h 4 1.a even 1 1 trivial
363.3.b.h 4 3.b odd 2 1 inner
363.3.h.l 16 11.c even 5 4
363.3.h.l 16 33.h odd 10 4
363.3.h.m 16 11.d odd 10 4
363.3.h.m 16 33.f even 10 4
528.3.i.d 4 44.c even 2 1
528.3.i.d 4 132.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 7 T_{2}^{2} + 4$$ $$T_{5}^{4} + 7 T_{5}^{2} + 4$$ $$T_{7}^{2} + 2 T_{7} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 7 T^{2} + T^{4}$$
$3$ $$81 + 45 T + 16 T^{2} + 5 T^{3} + T^{4}$$
$5$ $$4 + 7 T^{2} + T^{4}$$
$7$ $$( -32 + 2 T + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( -128 - 4 T + T^{2} )^{2}$$
$17$ $$440896 + 1372 T^{2} + T^{4}$$
$19$ $$( -216 - 18 T + T^{2} )^{2}$$
$23$ $$662596 + 1639 T^{2} + T^{4}$$
$29$ $$1937664 + 3552 T^{2} + T^{4}$$
$31$ $$( -74 - 23 T + T^{2} )^{2}$$
$37$ $$( 102 + 45 T + T^{2} )^{2}$$
$41$ $$295936 + 1264 T^{2} + T^{4}$$
$43$ $$( 444 + 48 T + T^{2} )^{2}$$
$47$ $$1700416 + 2716 T^{2} + T^{4}$$
$53$ $$788544 + 8124 T^{2} + T^{4}$$
$59$ $$824464 + 4003 T^{2} + T^{4}$$
$61$ $$( -3264 - 12 T + T^{2} )^{2}$$
$67$ $$( -2174 + 29 T + T^{2} )^{2}$$
$71$ $$4356 + 231 T^{2} + T^{4}$$
$73$ $$( 4744 - 142 T + T^{2} )^{2}$$
$79$ $$( -5216 - 38 T + T^{2} )^{2}$$
$83$ $$4981824 + 5436 T^{2} + T^{4}$$
$89$ $$4112784 + 20787 T^{2} + T^{4}$$
$97$ $$( -6014 - 109 T + T^{2} )^{2}$$