# Properties

 Label 363.3.b.g 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: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{3} + 3 q^{4} + ( -\beta_{1} - 6 \beta_{3} ) q^{5} + ( 1 + 2 \beta_{2} - 2 \beta_{3} ) q^{6} + ( 7 + 3 \beta_{2} ) q^{7} + 7 \beta_{3} q^{8} + ( -1 + 8 \beta_{1} + 4 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{3} + 3 q^{4} + ( -\beta_{1} - 6 \beta_{3} ) q^{5} + ( 1 + 2 \beta_{2} - 2 \beta_{3} ) q^{6} + ( 7 + 3 \beta_{2} ) q^{7} + 7 \beta_{3} q^{8} + ( -1 + 8 \beta_{1} + 4 \beta_{3} ) q^{9} + ( 6 + \beta_{2} ) q^{10} + ( -6 - 6 \beta_{1} - 3 \beta_{3} ) q^{12} + ( 12 + 2 \beta_{2} ) q^{13} + ( 3 \beta_{1} + 7 \beta_{3} ) q^{14} + ( -8 + 2 \beta_{1} - 11 \beta_{2} + 12 \beta_{3} ) q^{15} + 5 q^{16} -6 \beta_{3} q^{17} + ( -4 - 8 \beta_{2} - \beta_{3} ) q^{18} + ( 2 - 18 \beta_{2} ) q^{19} + ( -3 \beta_{1} - 18 \beta_{3} ) q^{20} + ( -14 - 11 \beta_{1} - 6 \beta_{2} - 13 \beta_{3} ) q^{21} + ( 4 \beta_{1} - 20 \beta_{3} ) q^{23} + ( 7 + 14 \beta_{2} - 14 \beta_{3} ) q^{24} + ( -12 - 11 \beta_{2} ) q^{25} + ( 2 \beta_{1} + 12 \beta_{3} ) q^{26} + ( 22 - 14 \beta_{1} - 7 \beta_{3} ) q^{27} + ( 21 + 9 \beta_{2} ) q^{28} + ( -33 \beta_{1} - 6 \beta_{3} ) q^{29} + ( -12 - 11 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} ) q^{30} + ( -13 + 11 \beta_{2} ) q^{31} + 33 \beta_{3} q^{32} + 6 q^{34} + ( -22 \beta_{1} - 45 \beta_{3} ) q^{35} + ( -3 + 24 \beta_{1} + 12 \beta_{3} ) q^{36} + ( -6 - 22 \beta_{2} ) q^{37} + ( -18 \beta_{1} + 2 \beta_{3} ) q^{38} + ( -24 - 22 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} ) q^{39} + ( 42 + 7 \beta_{2} ) q^{40} + ( -22 \beta_{1} + 8 \beta_{3} ) q^{41} + ( 13 - 6 \beta_{1} + 11 \beta_{2} - 14 \beta_{3} ) q^{42} + ( 30 - 6 \beta_{2} ) q^{43} + ( 32 + \beta_{1} + 44 \beta_{2} + 6 \beta_{3} ) q^{45} + ( 20 - 4 \beta_{2} ) q^{46} + ( -6 \beta_{1} - 14 \beta_{3} ) q^{47} + ( -10 - 10 \beta_{1} - 5 \beta_{3} ) q^{48} + ( 9 + 33 \beta_{2} ) q^{49} + ( -11 \beta_{1} - 12 \beta_{3} ) q^{50} + ( -6 - 12 \beta_{2} + 12 \beta_{3} ) q^{51} + ( 36 + 6 \beta_{2} ) q^{52} + ( 15 \beta_{1} + 13 \beta_{3} ) q^{53} + ( 7 + 14 \beta_{2} + 22 \beta_{3} ) q^{54} + ( 21 \beta_{1} + 49 \beta_{3} ) q^{56} + ( -4 - 22 \beta_{1} + 36 \beta_{2} + 34 \beta_{3} ) q^{57} + ( 6 + 33 \beta_{2} ) q^{58} + ( -5 \beta_{1} + 25 \beta_{3} ) q^{59} + ( -24 + 6 \beta_{1} - 33 \beta_{2} + 36 \beta_{3} ) q^{60} + ( 60 + 54 \beta_{2} ) q^{61} + ( 11 \beta_{1} - 13 \beta_{3} ) q^{62} + ( -7 + 44 \beta_{1} - 3 \beta_{2} + 52 \beta_{3} ) q^{63} -13 q^{64} + ( -22 \beta_{1} - 74 \beta_{3} ) q^{65} + ( -36 - 66 \beta_{2} ) q^{67} -18 \beta_{3} q^{68} + ( -12 - 8 \beta_{1} - 44 \beta_{2} + 40 \beta_{3} ) q^{69} + ( 45 + 22 \beta_{2} ) q^{70} + ( 10 \beta_{1} - 72 \beta_{3} ) q^{71} + ( -28 - 56 \beta_{2} - 7 \beta_{3} ) q^{72} + ( -28 + 21 \beta_{2} ) q^{73} + ( -22 \beta_{1} - 6 \beta_{3} ) q^{74} + ( 24 + 13 \beta_{1} + 22 \beta_{2} + 34 \beta_{3} ) q^{75} + ( 6 - 54 \beta_{2} ) q^{76} + ( 16 - 4 \beta_{1} + 22 \beta_{2} - 24 \beta_{3} ) q^{78} + ( -68 - 81 \beta_{2} ) q^{79} + ( -5 \beta_{1} - 30 \beta_{3} ) q^{80} + ( -79 - 16 \beta_{1} - 8 \beta_{3} ) q^{81} + ( -8 + 22 \beta_{2} ) q^{82} + ( 99 \beta_{1} + 50 \beta_{3} ) q^{83} + ( -42 - 33 \beta_{1} - 18 \beta_{2} - 39 \beta_{3} ) q^{84} + ( -36 - 6 \beta_{2} ) q^{85} + ( -6 \beta_{1} + 30 \beta_{3} ) q^{86} + ( -72 + 66 \beta_{1} + 21 \beta_{2} + 12 \beta_{3} ) q^{87} + ( -48 \beta_{1} - 68 \beta_{3} ) q^{89} + ( -6 + 44 \beta_{1} - \beta_{2} + 32 \beta_{3} ) q^{90} + ( 90 + 44 \beta_{2} ) q^{91} + ( 12 \beta_{1} - 60 \beta_{3} ) q^{92} + ( 26 + 37 \beta_{1} - 22 \beta_{2} - 9 \beta_{3} ) q^{93} + ( 14 + 6 \beta_{2} ) q^{94} + ( 88 \beta_{1} + 6 \beta_{3} ) q^{95} + ( 33 + 66 \beta_{2} - 66 \beta_{3} ) q^{96} + ( -60 - 99 \beta_{2} ) q^{97} + ( 33 \beta_{1} + 9 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{3} + 12 q^{4} + 22 q^{7} - 4 q^{9} + O(q^{10})$$ $$4 q - 8 q^{3} + 12 q^{4} + 22 q^{7} - 4 q^{9} + 22 q^{10} - 24 q^{12} + 44 q^{13} - 10 q^{15} + 20 q^{16} + 44 q^{19} - 44 q^{21} - 26 q^{25} + 88 q^{27} + 66 q^{28} - 44 q^{30} - 74 q^{31} + 24 q^{34} - 12 q^{36} + 20 q^{37} - 88 q^{39} + 154 q^{40} + 30 q^{42} + 132 q^{43} + 40 q^{45} + 88 q^{46} - 40 q^{48} - 30 q^{49} + 132 q^{52} - 88 q^{57} - 42 q^{58} - 30 q^{60} + 132 q^{61} - 22 q^{63} - 52 q^{64} - 12 q^{67} + 40 q^{69} + 136 q^{70} - 154 q^{73} + 52 q^{75} + 132 q^{76} + 20 q^{78} - 110 q^{79} - 316 q^{81} - 76 q^{82} - 132 q^{84} - 132 q^{85} - 330 q^{87} - 22 q^{90} + 272 q^{91} + 148 q^{93} + 44 q^{94} - 42 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2 \beta_{1}$$

## 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
 1.61803i − 0.618034i 0.618034i − 1.61803i
1.00000i −2.00000 2.23607i 3.00000 4.38197i −2.23607 + 2.00000i 2.14590 7.00000i −1.00000 + 8.94427i 4.38197
122.2 1.00000i −2.00000 + 2.23607i 3.00000 6.61803i 2.23607 + 2.00000i 8.85410 7.00000i −1.00000 8.94427i 6.61803
122.3 1.00000i −2.00000 2.23607i 3.00000 6.61803i 2.23607 2.00000i 8.85410 7.00000i −1.00000 + 8.94427i 6.61803
122.4 1.00000i −2.00000 + 2.23607i 3.00000 4.38197i −2.23607 2.00000i 2.14590 7.00000i −1.00000 8.94427i 4.38197
 $$n$$: 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.g 4
3.b odd 2 1 inner 363.3.b.g 4
11.b odd 2 1 363.3.b.f 4
11.c even 5 2 363.3.h.g 8
11.c even 5 2 363.3.h.i 8
11.d odd 10 2 33.3.h.a 8
11.d odd 10 2 363.3.h.h 8
33.d even 2 1 363.3.b.f 4
33.f even 10 2 33.3.h.a 8
33.f even 10 2 363.3.h.h 8
33.h odd 10 2 363.3.h.g 8
33.h odd 10 2 363.3.h.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.a 8 11.d odd 10 2
33.3.h.a 8 33.f even 10 2
363.3.b.f 4 11.b odd 2 1
363.3.b.f 4 33.d even 2 1
363.3.b.g 4 1.a even 1 1 trivial
363.3.b.g 4 3.b odd 2 1 inner
363.3.h.g 8 11.c even 5 2
363.3.h.g 8 33.h odd 10 2
363.3.h.h 8 11.d odd 10 2
363.3.h.h 8 33.f even 10 2
363.3.h.i 8 11.c even 5 2
363.3.h.i 8 33.h odd 10 2

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ $$T_{5}^{4} + 63 T_{5}^{2} + 841$$ $$T_{7}^{2} - 11 T_{7} + 19$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 9 + 4 T + T^{2} )^{2}$$
$5$ $$841 + 63 T^{2} + T^{4}$$
$7$ $$( 19 - 11 T + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( 116 - 22 T + T^{2} )^{2}$$
$17$ $$( 36 + T^{2} )^{2}$$
$19$ $$( -284 - 22 T + T^{2} )^{2}$$
$23$ $$215296 + 1008 T^{2} + T^{4}$$
$29$ $$1565001 + 2943 T^{2} + T^{4}$$
$31$ $$( 191 + 37 T + T^{2} )^{2}$$
$37$ $$( -580 - 10 T + T^{2} )^{2}$$
$41$ $$59536 + 1932 T^{2} + T^{4}$$
$43$ $$( 1044 - 66 T + T^{2} )^{2}$$
$47$ $$5776 + 332 T^{2} + T^{4}$$
$53$ $$63001 + 623 T^{2} + T^{4}$$
$59$ $$525625 + 1575 T^{2} + T^{4}$$
$61$ $$( -2556 - 66 T + T^{2} )^{2}$$
$67$ $$( -5436 + 6 T + T^{2} )^{2}$$
$71$ $$33686416 + 12108 T^{2} + T^{4}$$
$73$ $$( 931 + 77 T + T^{2} )^{2}$$
$79$ $$( -7445 + 55 T + T^{2} )^{2}$$
$83$ $$150087001 + 24503 T^{2} + T^{4}$$
$89$ $$891136 + 9632 T^{2} + T^{4}$$
$97$ $$( -12141 + 21 T + T^{2} )^{2}$$