# Properties

 Label 363.2.f.c Level $363$ Weight $2$ Character orbit 363.f Analytic conductor $2.899$ Analytic rank $0$ Dimension $8$ CM discriminant -11 Inner twists $16$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.f (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.89856959337$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.228765625.1 Defining polynomial: $$x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + x^{4} + 15 x^{3} - 18 x^{2} - 27 x + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -2 \beta_{7} q^{4} + ( -\beta_{4} - 2 \beta_{5} ) q^{5} + ( 2 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -2 \beta_{7} q^{4} + ( -\beta_{4} - 2 \beta_{5} ) q^{5} + ( 2 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{9} + ( -2 + 2 \beta_{6} ) q^{12} + ( -\beta_{2} + 5 \beta_{7} ) q^{15} -4 \beta_{4} q^{16} + ( -2 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{20} + ( -1 + 2 \beta_{6} ) q^{23} + 6 \beta_{3} q^{25} + ( 3 \beta_{4} - 2 \beta_{5} ) q^{27} + ( -5 + 5 \beta_{3} + 5 \beta_{4} - 5 \beta_{7} ) q^{31} + ( 2 \beta_{1} - 6 \beta_{3} ) q^{36} -7 \beta_{7} q^{37} + ( 8 - 5 \beta_{6} ) q^{45} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{47} + ( 4 \beta_{2} + 4 \beta_{7} ) q^{48} -7 \beta_{4} q^{49} + ( 4 - 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} ) q^{53} + ( -2 \beta_{2} - \beta_{7} ) q^{59} + ( 12 \beta_{4} + 2 \beta_{5} ) q^{60} + ( 8 - 8 \beta_{3} - 8 \beta_{4} + 8 \beta_{7} ) q^{64} -13 q^{67} + ( \beta_{1} - 6 \beta_{3} ) q^{69} + ( 5 \beta_{4} + 10 \beta_{5} ) q^{71} + ( 6 - 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{75} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{80} + ( -5 \beta_{2} + \beta_{7} ) q^{81} + ( 5 - 10 \beta_{6} ) q^{89} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{92} + 5 \beta_{5} q^{93} + ( -17 + 17 \beta_{3} + 17 \beta_{4} - 17 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - q^{3} + 4 q^{4} + 5 q^{9} + O(q^{10})$$ $$8 q - q^{3} + 4 q^{4} + 5 q^{9} - 8 q^{12} - 11 q^{15} - 8 q^{16} + 12 q^{25} + 8 q^{27} - 10 q^{31} - 10 q^{36} + 14 q^{37} + 44 q^{45} - 4 q^{48} - 14 q^{49} + 22 q^{60} + 16 q^{64} - 104 q^{67} - 11 q^{69} + 6 q^{75} - 7 q^{81} - 5 q^{93} - 34 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + x^{4} + 15 x^{3} - 18 x^{2} - 27 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + 2 \nu^{5} - 5 \nu^{4} - \nu^{3} - 15 \nu^{2} + 18 \nu + 27$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 16 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{7} - 4 \nu^{6} + 10 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} - 36 \nu^{2} - 54 \nu + 162$$$$)/27$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{6} + 5 \nu^{5} + \nu^{4} + 2 \nu^{3} - 18 \nu^{2} - 27 \nu + 81$$$$)/9$$ $$\beta_{6}$$ $$=$$ $$\nu^{5} + 16$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 5 \nu^{6} + 10 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} - 75 \nu^{2} + 90 \nu + 135$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - 5 \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$\beta_{6} - 16$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{3} - 16 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-35 \beta_{7} - 13 \beta_{6} + 13 \beta_{5} + 48 \beta_{4} + 48 \beta_{3} + 13 \beta_{2} - 13 \beta_{1} - 35$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$-\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 1.42264 + 0.987975i −1.73166 − 0.0369185i 1.37924 − 1.04771i −0.570223 + 1.63550i 1.37924 + 1.04771i −0.570223 − 1.63550i 1.42264 − 0.987975i −1.73166 + 0.0369185i
0 −1.42264 0.987975i −0.618034 1.90211i −1.94946 2.68321i 0 0 0 1.04781 + 2.81107i 0
161.2 0 1.73166 + 0.0369185i −0.618034 1.90211i 1.94946 + 2.68321i 0 0 0 2.99727 + 0.127860i 0
215.1 0 −1.37924 + 1.04771i 1.61803 1.17557i 3.15430 + 1.02489i 0 0 0 0.804606 2.89009i 0
215.2 0 0.570223 1.63550i 1.61803 1.17557i −3.15430 1.02489i 0 0 0 −2.34969 1.86519i 0
233.1 0 −1.37924 1.04771i 1.61803 + 1.17557i 3.15430 1.02489i 0 0 0 0.804606 + 2.89009i 0
233.2 0 0.570223 + 1.63550i 1.61803 + 1.17557i −3.15430 + 1.02489i 0 0 0 −2.34969 + 1.86519i 0
239.1 0 −1.42264 + 0.987975i −0.618034 + 1.90211i −1.94946 + 2.68321i 0 0 0 1.04781 2.81107i 0
239.2 0 1.73166 0.0369185i −0.618034 + 1.90211i 1.94946 2.68321i 0 0 0 2.99727 0.127860i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.2 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
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.f.c 8
3.b odd 2 1 inner 363.2.f.c 8
11.b odd 2 1 CM 363.2.f.c 8
11.c even 5 1 33.2.d.a 2
11.c even 5 3 inner 363.2.f.c 8
11.d odd 10 1 33.2.d.a 2
11.d odd 10 3 inner 363.2.f.c 8
33.d even 2 1 inner 363.2.f.c 8
33.f even 10 1 33.2.d.a 2
33.f even 10 3 inner 363.2.f.c 8
33.h odd 10 1 33.2.d.a 2
33.h odd 10 3 inner 363.2.f.c 8
44.g even 10 1 528.2.b.a 2
44.h odd 10 1 528.2.b.a 2
55.h odd 10 1 825.2.f.a 2
55.j even 10 1 825.2.f.a 2
55.k odd 20 2 825.2.d.a 4
55.l even 20 2 825.2.d.a 4
88.k even 10 1 2112.2.b.f 2
88.l odd 10 1 2112.2.b.f 2
88.o even 10 1 2112.2.b.e 2
88.p odd 10 1 2112.2.b.e 2
99.m even 15 2 891.2.g.a 4
99.n odd 30 2 891.2.g.a 4
99.o odd 30 2 891.2.g.a 4
99.p even 30 2 891.2.g.a 4
132.n odd 10 1 528.2.b.a 2
132.o even 10 1 528.2.b.a 2
165.o odd 10 1 825.2.f.a 2
165.r even 10 1 825.2.f.a 2
165.u odd 20 2 825.2.d.a 4
165.v even 20 2 825.2.d.a 4
264.r odd 10 1 2112.2.b.f 2
264.t odd 10 1 2112.2.b.e 2
264.u even 10 1 2112.2.b.e 2
264.w even 10 1 2112.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a 2 11.c even 5 1
33.2.d.a 2 11.d odd 10 1
33.2.d.a 2 33.f even 10 1
33.2.d.a 2 33.h odd 10 1
363.2.f.c 8 1.a even 1 1 trivial
363.2.f.c 8 3.b odd 2 1 inner
363.2.f.c 8 11.b odd 2 1 CM
363.2.f.c 8 11.c even 5 3 inner
363.2.f.c 8 11.d odd 10 3 inner
363.2.f.c 8 33.d even 2 1 inner
363.2.f.c 8 33.f even 10 3 inner
363.2.f.c 8 33.h odd 10 3 inner
528.2.b.a 2 44.g even 10 1
528.2.b.a 2 44.h odd 10 1
528.2.b.a 2 132.n odd 10 1
528.2.b.a 2 132.o even 10 1
825.2.d.a 4 55.k odd 20 2
825.2.d.a 4 55.l even 20 2
825.2.d.a 4 165.u odd 20 2
825.2.d.a 4 165.v even 20 2
825.2.f.a 2 55.h odd 10 1
825.2.f.a 2 55.j even 10 1
825.2.f.a 2 165.o odd 10 1
825.2.f.a 2 165.r even 10 1
891.2.g.a 4 99.m even 15 2
891.2.g.a 4 99.n odd 30 2
891.2.g.a 4 99.o odd 30 2
891.2.g.a 4 99.p even 30 2
2112.2.b.e 2 88.o even 10 1
2112.2.b.e 2 88.p odd 10 1
2112.2.b.e 2 264.t odd 10 1
2112.2.b.e 2 264.u even 10 1
2112.2.b.f 2 88.k even 10 1
2112.2.b.f 2 88.l odd 10 1
2112.2.b.f 2 264.r odd 10 1
2112.2.b.f 2 264.w even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}$$ $$T_{5}^{8} - 11 T_{5}^{6} + 121 T_{5}^{4} - 1331 T_{5}^{2} + 14641$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$81 + 27 T - 18 T^{2} - 15 T^{3} + T^{4} - 5 T^{5} - 2 T^{6} + T^{7} + T^{8}$$
$5$ $$14641 - 1331 T^{2} + 121 T^{4} - 11 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( 11 + T^{2} )^{4}$$
$29$ $$T^{8}$$
$31$ $$( 625 + 125 T + 25 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$37$ $$( 2401 - 343 T + 49 T^{2} - 7 T^{3} + T^{4} )^{2}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$3748096 - 85184 T^{2} + 1936 T^{4} - 44 T^{6} + T^{8}$$
$53$ $$959512576 - 5451776 T^{2} + 30976 T^{4} - 176 T^{6} + T^{8}$$
$59$ $$14641 - 1331 T^{2} + 121 T^{4} - 11 T^{6} + T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 13 + T )^{8}$$
$71$ $$5719140625 - 20796875 T^{2} + 75625 T^{4} - 275 T^{6} + T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 275 + T^{2} )^{4}$$
$97$ $$( 83521 + 4913 T + 289 T^{2} + 17 T^{3} + T^{4} )^{2}$$