# Properties

 Label 363.3.b.j Level $363$ Weight $3$ Character orbit 363.b Analytic conductor $9.891$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Defining polynomial: $$x^{6} + 21 x^{4} + 111 x^{2} + 47$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{3} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{4} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{5} + ( -2 + \beta_{2} - \beta_{4} ) q^{6} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{7} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{8} + ( -3 - 2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 - \beta_{3} ) q^{3} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{4} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{5} + ( -2 + \beta_{2} - \beta_{4} ) q^{6} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{7} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{8} + ( -3 - 2 \beta_{1} - \beta_{3} - \beta_{5} ) q^{9} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{10} + ( 2 - \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{12} + ( -7 + \beta_{3} + \beta_{4} ) q^{13} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} + \beta_{4} ) q^{14} + ( 5 + 7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{5} ) q^{15} + ( 5 + 2 \beta_{3} + 2 \beta_{4} ) q^{16} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{17} + ( 15 - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{18} + ( 14 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} + ( 4 + 7 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} + 2 \beta_{4} ) q^{20} + ( -4 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{21} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{23} + ( 9 + 7 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{24} + ( -18 - \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{25} + ( 2 - 8 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + \beta_{4} + \beta_{5} ) q^{26} + ( -6 - 5 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{27} + ( 22 - 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{28} + ( -4 + 3 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} - 2 \beta_{4} ) q^{29} + ( -25 + 12 \beta_{1} - 5 \beta_{3} + 6 \beta_{5} ) q^{30} + ( 4 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{31} + ( 4 - 5 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{32} + ( 11 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{34} + ( 6 - 8 \beta_{1} - 9 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} ) q^{35} + ( 9 + 21 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} ) q^{36} + ( -7 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} ) q^{37} + ( 2 + 16 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{38} + ( -16 + 5 \beta_{1} + 2 \beta_{2} + 7 \beta_{3} + \beta_{4} + \beta_{5} ) q^{39} + ( -49 - 5 \beta_{1} + 5 \beta_{2} - 11 \beta_{3} - 6 \beta_{4} + 5 \beta_{5} ) q^{40} + ( -2 - 19 \beta_{1} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{41} + ( -31 - 13 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{42} + ( -26 + 5 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} ) q^{43} + ( -21 + 5 \beta_{1} - 5 \beta_{2} - 7 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} ) q^{45} + ( -10 - 4 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{46} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{47} + ( -13 + 10 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{48} + ( 23 - 5 \beta_{1} + 5 \beta_{2} - 11 \beta_{3} - 6 \beta_{4} + 5 \beta_{5} ) q^{49} + ( -4 - 19 \beta_{1} + 7 \beta_{2} + 13 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{50} + ( 28 - 8 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} + \beta_{4} + 11 \beta_{5} ) q^{51} + ( 19 + 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} ) q^{52} + ( 22 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} - 7 \beta_{5} ) q^{53} + ( 27 + 5 \beta_{1} - 11 \beta_{2} - 16 \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{54} + ( 2 + 4 \beta_{2} + \beta_{3} + \beta_{4} - 6 \beta_{5} ) q^{56} + ( 6 \beta_{1} - 15 \beta_{3} + 3 \beta_{5} ) q^{57} + ( -9 + 3 \beta_{1} - 3 \beta_{2} + 17 \beta_{3} + 14 \beta_{4} - 3 \beta_{5} ) q^{58} + ( -7 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} - 7 \beta_{5} ) q^{59} + ( -92 - 26 \beta_{1} + 11 \beta_{2} - 12 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{60} + ( 44 + 4 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{61} + ( 6 + 9 \beta_{1} - 13 \beta_{2} - 22 \beta_{3} + 3 \beta_{4} + 7 \beta_{5} ) q^{62} + ( -21 + 22 \beta_{1} + 5 \beta_{2} + 13 \beta_{3} - 2 \beta_{4} - 13 \beta_{5} ) q^{63} + ( 49 + 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{64} + ( 2 - 15 \beta_{1} - 6 \beta_{2} - 9 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{65} + ( 4 - 8 \beta_{1} + 8 \beta_{2} - 17 \beta_{3} - 9 \beta_{4} + 8 \beta_{5} ) q^{67} + ( -6 - \beta_{1} - 6 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 12 \beta_{5} ) q^{68} + ( 22 - 8 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 11 \beta_{5} ) q^{69} + ( 56 + 11 \beta_{1} - 11 \beta_{2} + 2 \beta_{3} - 9 \beta_{4} - 11 \beta_{5} ) q^{70} + ( -11 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{71} + ( -51 + 5 \beta_{1} - \beta_{2} - 15 \beta_{3} - 5 \beta_{4} + 10 \beta_{5} ) q^{72} + ( 12 + 17 \beta_{1} - 17 \beta_{2} + 19 \beta_{3} + 2 \beta_{4} - 17 \beta_{5} ) q^{73} + ( 6 - 16 \beta_{1} - 7 \beta_{2} - 16 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{74} + ( 5 - 11 \beta_{1} - 2 \beta_{2} + 19 \beta_{3} - \beta_{4} - 4 \beta_{5} ) q^{75} + ( -68 - 19 \beta_{1} + 19 \beta_{2} - 26 \beta_{3} - 7 \beta_{4} + 19 \beta_{5} ) q^{76} + ( -28 - 20 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} + 5 \beta_{5} ) q^{78} + ( 16 \beta_{1} - 16 \beta_{2} + 13 \beta_{3} - 3 \beta_{4} - 16 \beta_{5} ) q^{79} + ( 4 - 30 \beta_{1} + 7 \beta_{2} + \beta_{3} + 2 \beta_{4} - 11 \beta_{5} ) q^{80} + ( -45 + 5 \beta_{1} - 5 \beta_{2} - \beta_{3} + 2 \beta_{4} + 13 \beta_{5} ) q^{81} + ( 133 + 18 \beta_{1} - 18 \beta_{2} + 21 \beta_{3} + 3 \beta_{4} - 18 \beta_{5} ) q^{82} + ( 6 + 27 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - 9 \beta_{5} ) q^{83} + ( 64 - 18 \beta_{1} - 19 \beta_{3} - 9 \beta_{5} ) q^{84} + ( 55 - 20 \beta_{1} + 20 \beta_{2} - 17 \beta_{3} + 3 \beta_{4} + 20 \beta_{5} ) q^{85} + ( -4 - 9 \beta_{1} + \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{86} + ( 72 + 26 \beta_{1} - \beta_{2} + 12 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} ) q^{87} + ( 2 - 12 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{89} + ( -51 - 22 \beta_{1} + 30 \beta_{2} + 19 \beta_{3} - 12 \beta_{4} + 7 \beta_{5} ) q^{90} + ( 50 + 8 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} - 13 \beta_{4} - 8 \beta_{5} ) q^{91} + ( -6 - 19 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} + 9 \beta_{5} ) q^{92} + ( -47 + 19 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - \beta_{4} + 11 \beta_{5} ) q^{93} + ( 20 + 5 \beta_{1} - 5 \beta_{2} + 12 \beta_{3} + 7 \beta_{4} - 5 \beta_{5} ) q^{94} + ( -2 - 22 \beta_{1} + 16 \beta_{2} + 19 \beta_{3} - \beta_{4} - 14 \beta_{5} ) q^{95} + ( -58 - 12 \beta_{1} - 7 \beta_{2} - 6 \beta_{3} + 7 \beta_{4} - 6 \beta_{5} ) q^{96} + ( 21 - 25 \beta_{1} + 25 \beta_{2} - 23 \beta_{3} + 2 \beta_{4} + 25 \beta_{5} ) q^{97} + ( -12 + 14 \beta_{1} + 23 \beta_{2} + 41 \beta_{3} - 6 \beta_{4} - 11 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{3} - 18 q^{4} - 10 q^{6} - 22 q^{9} + O(q^{10})$$ $$6 q + 4 q^{3} - 18 q^{4} - 10 q^{6} - 22 q^{9} + 18 q^{10} + 14 q^{12} - 42 q^{13} + 28 q^{15} + 30 q^{16} + 94 q^{18} + 84 q^{19} - 28 q^{21} + 48 q^{24} - 108 q^{25} - 38 q^{27} + 132 q^{28} - 148 q^{30} + 66 q^{34} + 46 q^{36} - 42 q^{37} - 82 q^{39} - 294 q^{40} - 206 q^{42} - 156 q^{43} - 118 q^{45} - 60 q^{46} - 88 q^{48} + 138 q^{49} + 182 q^{51} + 114 q^{52} + 140 q^{54} - 24 q^{57} - 54 q^{58} - 562 q^{60} + 264 q^{61} - 122 q^{63} + 294 q^{64} + 24 q^{67} + 152 q^{69} + 336 q^{70} - 306 q^{72} + 72 q^{73} + 62 q^{75} - 408 q^{76} - 194 q^{78} - 250 q^{81} + 798 q^{82} + 328 q^{84} + 330 q^{85} + 462 q^{87} - 230 q^{90} + 300 q^{91} - 266 q^{93} + 120 q^{94} - 386 q^{96} + 126 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 21 x^{4} + 111 x^{2} + 47$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} + 11 \nu + 7$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 16 \nu^{3} + 10 \nu^{2} - 55 \nu + 1$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} + 5 \nu^{4} + 16 \nu^{3} + 62 \nu^{2} + 55 \nu + 65$$$$)/12$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 22 \nu^{3} + 16 \nu^{2} - 109 \nu + 43$$$$)/12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{3} + \beta_{2} - \beta_{1} - 7$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{3} + \beta_{2} - 10 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-12 \beta_{5} + 2 \beta_{4} + 14 \beta_{3} - 12 \beta_{2} + 12 \beta_{1} + 73$$ $$\nu^{5}$$ $$=$$ $$14 \beta_{5} + 2 \beta_{4} - 24 \beta_{3} - 18 \beta_{2} + 107 \beta_{1} + 4$$

## 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
 − 3.50091i − 2.87759i − 0.680517i 0.680517i 2.87759i 3.50091i
3.50091i −1.38788 2.65966i −8.25635 7.89925i −9.31122 + 4.85883i 1.81458 14.9011i −5.14758 + 7.38257i 27.6545
122.2 2.87759i 2.10316 + 2.13932i −4.28055 4.94788i 6.15610 6.05206i −11.1801 0.807314i −0.153394 + 8.99869i −14.2380
122.3 0.680517i 1.28471 2.71100i 3.53690 6.49002i −1.84488 0.874269i 9.36551 5.12898i −5.69902 6.96571i −4.41657
122.4 0.680517i 1.28471 + 2.71100i 3.53690 6.49002i −1.84488 + 0.874269i 9.36551 5.12898i −5.69902 + 6.96571i −4.41657
122.5 2.87759i 2.10316 2.13932i −4.28055 4.94788i 6.15610 + 6.05206i −11.1801 0.807314i −0.153394 8.99869i −14.2380
122.6 3.50091i −1.38788 + 2.65966i −8.25635 7.89925i −9.31122 4.85883i 1.81458 14.9011i −5.14758 7.38257i 27.6545
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 122.6 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.j 6
3.b odd 2 1 inner 363.3.b.j 6
11.b odd 2 1 363.3.b.k yes 6
11.c even 5 4 363.3.h.q 24
11.d odd 10 4 363.3.h.p 24
33.d even 2 1 363.3.b.k yes 6
33.f even 10 4 363.3.h.p 24
33.h odd 10 4 363.3.h.q 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.j 6 1.a even 1 1 trivial
363.3.b.j 6 3.b odd 2 1 inner
363.3.b.k yes 6 11.b odd 2 1
363.3.b.k yes 6 33.d even 2 1
363.3.h.p 24 11.d odd 10 4
363.3.h.p 24 33.f even 10 4
363.3.h.q 24 11.c even 5 4
363.3.h.q 24 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 $$S_{3}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}^{6} + 21 T_{2}^{4} + 111 T_{2}^{2} + 47$$ $$T_{5}^{6} + 129 T_{5}^{4} + 5187 T_{5}^{2} + 64343$$ $$T_{7}^{3} - 108 T_{7} + 190$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$47 + 111 T^{2} + 21 T^{4} + T^{6}$$
$3$ $$729 - 324 T + 171 T^{2} - 42 T^{3} + 19 T^{4} - 4 T^{5} + T^{6}$$
$5$ $$64343 + 5187 T^{2} + 129 T^{4} + T^{6}$$
$7$ $$( 190 - 108 T + T^{3} )^{2}$$
$11$ $$T^{6}$$
$13$ $$( -5 + 69 T + 21 T^{2} + T^{3} )^{2}$$
$17$ $$47 + 12003 T^{2} + 909 T^{4} + T^{6}$$
$19$ $$( -1458 + 468 T - 42 T^{2} + T^{3} )^{2}$$
$23$ $$2752508 + 98736 T^{2} + 900 T^{4} + T^{6}$$
$29$ $$593987175 + 3166407 T^{2} + 3669 T^{4} + T^{6}$$
$31$ $$( -18334 - 1350 T + T^{3} )^{2}$$
$37$ $$( 2115 - 663 T + 21 T^{2} + T^{3} )^{2}$$
$41$ $$11934438575 + 18292083 T^{2} + 8205 T^{4} + T^{6}$$
$43$ $$( -180 + 876 T + 78 T^{2} + T^{3} )^{2}$$
$47$ $$18183548 + 272436 T^{2} + 1080 T^{4} + T^{6}$$
$53$ $$16518150423 + 41246883 T^{2} + 13713 T^{4} + T^{6}$$
$59$ $$2211801200 + 12216288 T^{2} + 8232 T^{4} + T^{6}$$
$61$ $$( -54576 + 4872 T - 132 T^{2} + T^{3} )^{2}$$
$67$ $$( 259810 - 9006 T - 12 T^{2} + T^{3} )^{2}$$
$71$ $$1522800 + 1538928 T^{2} + 2868 T^{4} + T^{6}$$
$73$ $$( 150820 - 10488 T - 36 T^{2} + T^{3} )^{2}$$
$79$ $$( 356894 - 9630 T + T^{3} )^{2}$$
$83$ $$280649740572 + 142551576 T^{2} + 21600 T^{4} + T^{6}$$
$89$ $$1393457175 + 4881987 T^{2} + 4101 T^{4} + T^{6}$$
$97$ $$( -563825 - 21189 T - 63 T^{2} + T^{3} )^{2}$$