# Properties

 Label 363.2.e.m Level $363$ Weight $2$ Character orbit 363.e Analytic conductor $2.899$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 363.e (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.89856959337$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.324000000.3 Defining polynomial: $$x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81$$ x^8 + 3*x^6 + 9*x^4 + 27*x^2 + 81 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{2} - \beta_{6} q^{3} + \beta_{4} q^{4} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 3) q^{5} - \beta_{3} q^{6} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{7} - \beta_1 q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q + b7 * q^2 - b6 * q^3 + b4 * q^4 + (3*b6 + 3*b4 + 3*b2 + 3) * q^5 - b3 * q^6 + (2*b7 + 2*b5 + 2*b3 + 2*b1) * q^7 - b1 * q^8 + b2 * q^9 $$q + \beta_{7} q^{2} - \beta_{6} q^{3} + \beta_{4} q^{4} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 3) q^{5} - \beta_{3} q^{6} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{7} - \beta_1 q^{8} + \beta_{2} q^{9} - 3 \beta_{5} q^{10} - q^{12} - \beta_{7} q^{13} - 6 \beta_{6} q^{14} + 3 \beta_{4} q^{15} + (5 \beta_{6} + 5 \beta_{4} + 5 \beta_{2} + 5) q^{16} + \beta_{3} q^{17} + ( - \beta_{7} - \beta_{5} - \beta_{3} - \beta_1) q^{18} + 4 \beta_1 q^{19} - 3 \beta_{2} q^{20} + 2 \beta_{5} q^{21} - 6 q^{23} + \beta_{7} q^{24} + 4 \beta_{6} q^{25} - 3 \beta_{4} q^{26} + (\beta_{6} + \beta_{4} + \beta_{2} + 1) q^{27} - 2 \beta_{3} q^{28} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{29} + 3 \beta_1 q^{30} + 4 \beta_{2} q^{31} - 3 \beta_{5} q^{32} + 3 q^{34} + 6 \beta_{7} q^{35} + \beta_{6} q^{36} - 11 \beta_{4} q^{37} + ( - 12 \beta_{6} - 12 \beta_{4} - 12 \beta_{2} - 12) q^{38} + \beta_{3} q^{39} + ( - 3 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 3 \beta_1) q^{40} - \beta_1 q^{41} + 6 \beta_{2} q^{42} + 2 \beta_{5} q^{43} - 3 q^{45} - 6 \beta_{7} q^{46} + 5 \beta_{4} q^{48} + ( - 5 \beta_{6} - 5 \beta_{4} - 5 \beta_{2} - 5) q^{49} + 4 \beta_{3} q^{50} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{51} - \beta_1 q^{52} - 9 \beta_{2} q^{53} - \beta_{5} q^{54} + 6 q^{56} - 4 \beta_{7} q^{57} - 3 \beta_{6} q^{58} - 6 \beta_{4} q^{59} + ( - 3 \beta_{6} - 3 \beta_{4} - 3 \beta_{2} - 3) q^{60} + ( - 4 \beta_{7} - 4 \beta_{5} - 4 \beta_{3} - 4 \beta_1) q^{62} - 2 \beta_1 q^{63} + \beta_{2} q^{64} + 3 \beta_{5} q^{65} - 2 q^{67} + \beta_{7} q^{68} + 6 \beta_{6} q^{69} + 18 \beta_{4} q^{70} + (6 \beta_{6} + 6 \beta_{4} + 6 \beta_{2} + 6) q^{71} - \beta_{3} q^{72} + ( - 4 \beta_{7} - 4 \beta_{5} - 4 \beta_{3} - 4 \beta_1) q^{73} - 11 \beta_1 q^{74} - 4 \beta_{2} q^{75} + 4 \beta_{5} q^{76} + 3 q^{78} + 15 \beta_{6} q^{80} + \beta_{4} q^{81} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 3) q^{82} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{84} - 3 \beta_1 q^{85} + 6 \beta_{2} q^{86} + \beta_{5} q^{87} + 9 q^{89} - 3 \beta_{7} q^{90} + 6 \beta_{6} q^{91} - 6 \beta_{4} q^{92} + (4 \beta_{6} + 4 \beta_{4} + 4 \beta_{2} + 4) q^{93} + (12 \beta_{7} + 12 \beta_{5} + 12 \beta_{3} + 12 \beta_1) q^{95} + 3 \beta_1 q^{96} - 7 \beta_{2} q^{97} + 5 \beta_{5} q^{98}+O(q^{100})$$ q + b7 * q^2 - b6 * q^3 + b4 * q^4 + (3*b6 + 3*b4 + 3*b2 + 3) * q^5 - b3 * q^6 + (2*b7 + 2*b5 + 2*b3 + 2*b1) * q^7 - b1 * q^8 + b2 * q^9 - 3*b5 * q^10 - q^12 - b7 * q^13 - 6*b6 * q^14 + 3*b4 * q^15 + (5*b6 + 5*b4 + 5*b2 + 5) * q^16 + b3 * q^17 + (-b7 - b5 - b3 - b1) * q^18 + 4*b1 * q^19 - 3*b2 * q^20 + 2*b5 * q^21 - 6 * q^23 + b7 * q^24 + 4*b6 * q^25 - 3*b4 * q^26 + (b6 + b4 + b2 + 1) * q^27 - 2*b3 * q^28 + (b7 + b5 + b3 + b1) * q^29 + 3*b1 * q^30 + 4*b2 * q^31 - 3*b5 * q^32 + 3 * q^34 + 6*b7 * q^35 + b6 * q^36 - 11*b4 * q^37 + (-12*b6 - 12*b4 - 12*b2 - 12) * q^38 + b3 * q^39 + (-3*b7 - 3*b5 - 3*b3 - 3*b1) * q^40 - b1 * q^41 + 6*b2 * q^42 + 2*b5 * q^43 - 3 * q^45 - 6*b7 * q^46 + 5*b4 * q^48 + (-5*b6 - 5*b4 - 5*b2 - 5) * q^49 + 4*b3 * q^50 + (b7 + b5 + b3 + b1) * q^51 - b1 * q^52 - 9*b2 * q^53 - b5 * q^54 + 6 * q^56 - 4*b7 * q^57 - 3*b6 * q^58 - 6*b4 * q^59 + (-3*b6 - 3*b4 - 3*b2 - 3) * q^60 + (-4*b7 - 4*b5 - 4*b3 - 4*b1) * q^62 - 2*b1 * q^63 + b2 * q^64 + 3*b5 * q^65 - 2 * q^67 + b7 * q^68 + 6*b6 * q^69 + 18*b4 * q^70 + (6*b6 + 6*b4 + 6*b2 + 6) * q^71 - b3 * q^72 + (-4*b7 - 4*b5 - 4*b3 - 4*b1) * q^73 - 11*b1 * q^74 - 4*b2 * q^75 + 4*b5 * q^76 + 3 * q^78 + 15*b6 * q^80 + b4 * q^81 + (3*b6 + 3*b4 + 3*b2 + 3) * q^82 + (-2*b7 - 2*b5 - 2*b3 - 2*b1) * q^84 - 3*b1 * q^85 + 6*b2 * q^86 + b5 * q^87 + 9 * q^89 - 3*b7 * q^90 + 6*b6 * q^91 - 6*b4 * q^92 + (4*b6 + 4*b4 + 4*b2 + 4) * q^93 + (12*b7 + 12*b5 + 12*b3 + 12*b1) * q^95 + 3*b1 * q^96 - 7*b2 * q^97 + 5*b5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{3} - 2 q^{4} + 6 q^{5} - 2 q^{9}+O(q^{10})$$ 8 * q + 2 * q^3 - 2 * q^4 + 6 * q^5 - 2 * q^9 $$8 q + 2 q^{3} - 2 q^{4} + 6 q^{5} - 2 q^{9} - 8 q^{12} + 12 q^{14} - 6 q^{15} + 10 q^{16} + 6 q^{20} - 48 q^{23} - 8 q^{25} + 6 q^{26} + 2 q^{27} - 8 q^{31} + 24 q^{34} - 2 q^{36} + 22 q^{37} - 24 q^{38} - 12 q^{42} - 24 q^{45} - 10 q^{48} - 10 q^{49} + 18 q^{53} + 48 q^{56} + 6 q^{58} + 12 q^{59} - 6 q^{60} - 2 q^{64} - 16 q^{67} - 12 q^{69} - 36 q^{70} + 12 q^{71} + 8 q^{75} + 24 q^{78} - 30 q^{80} - 2 q^{81} + 6 q^{82} - 12 q^{86} + 72 q^{89} - 12 q^{91} + 12 q^{92} + 8 q^{93} + 14 q^{97}+O(q^{100})$$ 8 * q + 2 * q^3 - 2 * q^4 + 6 * q^5 - 2 * q^9 - 8 * q^12 + 12 * q^14 - 6 * q^15 + 10 * q^16 + 6 * q^20 - 48 * q^23 - 8 * q^25 + 6 * q^26 + 2 * q^27 - 8 * q^31 + 24 * q^34 - 2 * q^36 + 22 * q^37 - 24 * q^38 - 12 * q^42 - 24 * q^45 - 10 * q^48 - 10 * q^49 + 18 * q^53 + 48 * q^56 + 6 * q^58 + 12 * q^59 - 6 * q^60 - 2 * q^64 - 16 * q^67 - 12 * q^69 - 36 * q^70 + 12 * q^71 + 8 * q^75 + 24 * q^78 - 30 * q^80 - 2 * q^81 + 6 * q^82 - 12 * q^86 + 72 * q^89 - 12 * q^91 + 12 * q^92 + 8 * q^93 + 14 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3 $$\beta_{4}$$ $$=$$ $$( \nu^{4} ) / 9$$ (v^4) / 9 $$\beta_{5}$$ $$=$$ $$( \nu^{5} ) / 9$$ (v^5) / 9 $$\beta_{6}$$ $$=$$ $$( \nu^{6} ) / 27$$ (v^6) / 27 $$\beta_{7}$$ $$=$$ $$( \nu^{7} ) / 27$$ (v^7) / 27
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3 $$\nu^{4}$$ $$=$$ $$9\beta_{4}$$ 9*b4 $$\nu^{5}$$ $$=$$ $$9\beta_{5}$$ 9*b5 $$\nu^{6}$$ $$=$$ $$27\beta_{6}$$ 27*b6 $$\nu^{7}$$ $$=$$ $$27\beta_{7}$$ 27*b7

## Character values

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

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 0.535233 + 1.64728i −0.535233 − 1.64728i 1.40126 − 1.01807i −1.40126 + 1.01807i 1.40126 + 1.01807i −1.40126 − 1.01807i 0.535233 − 1.64728i −0.535233 + 1.64728i
−1.40126 + 1.01807i −0.309017 0.951057i 0.309017 0.951057i 2.42705 + 1.76336i 1.40126 + 1.01807i −1.07047 + 3.29456i −0.535233 1.64728i −0.809017 + 0.587785i −5.19615
124.2 1.40126 1.01807i −0.309017 0.951057i 0.309017 0.951057i 2.42705 + 1.76336i −1.40126 1.01807i 1.07047 3.29456i 0.535233 + 1.64728i −0.809017 + 0.587785i 5.19615
130.1 −0.535233 + 1.64728i 0.809017 0.587785i −0.809017 0.587785i −0.927051 2.85317i 0.535233 + 1.64728i −2.80252 2.03615i −1.40126 + 1.01807i 0.309017 0.951057i 5.19615
130.2 0.535233 1.64728i 0.809017 0.587785i −0.809017 0.587785i −0.927051 2.85317i −0.535233 1.64728i 2.80252 + 2.03615i 1.40126 1.01807i 0.309017 0.951057i −5.19615
148.1 −0.535233 1.64728i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.927051 + 2.85317i 0.535233 1.64728i −2.80252 + 2.03615i −1.40126 1.01807i 0.309017 + 0.951057i 5.19615
148.2 0.535233 + 1.64728i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.927051 + 2.85317i −0.535233 + 1.64728i 2.80252 2.03615i 1.40126 + 1.01807i 0.309017 + 0.951057i −5.19615
202.1 −1.40126 1.01807i −0.309017 + 0.951057i 0.309017 + 0.951057i 2.42705 1.76336i 1.40126 1.01807i −1.07047 3.29456i −0.535233 + 1.64728i −0.809017 0.587785i −5.19615
202.2 1.40126 + 1.01807i −0.309017 + 0.951057i 0.309017 + 0.951057i 2.42705 1.76336i −1.40126 + 1.01807i 1.07047 + 3.29456i 0.535233 1.64728i −0.809017 0.587785i 5.19615
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 202.2 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 inner
11.c even 5 3 inner
11.d 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.e.m 8
11.b odd 2 1 inner 363.2.e.m 8
11.c even 5 1 363.2.a.f 2
11.c even 5 3 inner 363.2.e.m 8
11.d odd 10 1 363.2.a.f 2
11.d odd 10 3 inner 363.2.e.m 8
33.f even 10 1 1089.2.a.o 2
33.h odd 10 1 1089.2.a.o 2
44.g even 10 1 5808.2.a.ca 2
44.h odd 10 1 5808.2.a.ca 2
55.h odd 10 1 9075.2.a.bo 2
55.j even 10 1 9075.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.f 2 11.c even 5 1
363.2.a.f 2 11.d odd 10 1
363.2.e.m 8 1.a even 1 1 trivial
363.2.e.m 8 11.b odd 2 1 inner
363.2.e.m 8 11.c even 5 3 inner
363.2.e.m 8 11.d odd 10 3 inner
1089.2.a.o 2 33.f even 10 1
1089.2.a.o 2 33.h odd 10 1
5808.2.a.ca 2 44.g even 10 1
5808.2.a.ca 2 44.h odd 10 1
9075.2.a.bo 2 55.h odd 10 1
9075.2.a.bo 2 55.j even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 3T_{2}^{6} + 9T_{2}^{4} + 27T_{2}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(363, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81$$
$3$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$5$ $$(T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81)^{2}$$
$7$ $$T^{8} + 12 T^{6} + 144 T^{4} + \cdots + 20736$$
$11$ $$T^{8}$$
$13$ $$T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81$$
$17$ $$T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81$$
$19$ $$T^{8} + 48 T^{6} + 2304 T^{4} + \cdots + 5308416$$
$23$ $$(T + 6)^{8}$$
$29$ $$T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81$$
$31$ $$(T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256)^{2}$$
$37$ $$(T^{4} - 11 T^{3} + 121 T^{2} + \cdots + 14641)^{2}$$
$41$ $$T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81$$
$43$ $$(T^{2} - 12)^{4}$$
$47$ $$T^{8}$$
$53$ $$(T^{4} - 9 T^{3} + 81 T^{2} - 729 T + 6561)^{2}$$
$59$ $$(T^{4} - 6 T^{3} + 36 T^{2} - 216 T + 1296)^{2}$$
$61$ $$T^{8}$$
$67$ $$(T + 2)^{8}$$
$71$ $$(T^{4} - 6 T^{3} + 36 T^{2} - 216 T + 1296)^{2}$$
$73$ $$T^{8} + 48 T^{6} + 2304 T^{4} + \cdots + 5308416$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$(T - 9)^{8}$$
$97$ $$(T^{4} - 7 T^{3} + 49 T^{2} - 343 T + 2401)^{2}$$