# Properties

 Label 363.2.f.a Level $363$ Weight $2$ Character orbit 363.f 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.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: 8.0.64000000.1 Defining polynomial: $$x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16$$ x^8 - 2*x^6 + 4*x^4 - 8*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} + (\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{3} + \beta_{2} q^{4} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{5} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{6} + \beta_{7} q^{7} + ( - 3 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 3) q^{8} + (\beta_{6} + 2 \beta_1) q^{9}+O(q^{10})$$ q - b6 * q^2 + (b6 - b4 - b3 + b2 - 1) * q^3 + b2 * q^4 + (2*b7 - 2*b5 + 2*b3 - 2*b1) * q^5 + (b7 - b5 + b4 + b3 - b1) * q^6 + b7 * q^7 + (-3*b6 + 3*b4 - 3*b2 + 3) * q^8 + (b6 + 2*b1) * q^9 $$q - \beta_{6} q^{2} + (\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{3} + \beta_{2} q^{4} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{5} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{6} + \beta_{7} q^{7} + ( - 3 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 3) q^{8} + (\beta_{6} + 2 \beta_1) q^{9} + 2 \beta_{5} q^{10} + ( - \beta_{5} - 1) q^{12} + 3 \beta_1 q^{13} + \beta_{3} q^{14} + ( - 2 \beta_{7} + 4 \beta_{2}) q^{15} - \beta_{4} q^{16} + 6 \beta_{4} q^{17} + ( - 2 \beta_{7} + \beta_{2}) q^{18} + 3 \beta_{3} q^{19} - 2 \beta_1 q^{20} + ( - \beta_{5} + 2) q^{21} + (3 \beta_{6} - 3 \beta_1) q^{24} + ( - 3 \beta_{6} + 3 \beta_{4} - 3 \beta_{2} + 3) q^{25} - 3 \beta_{7} q^{26} + (\beta_{7} - \beta_{5} - 5 \beta_{4} + \beta_{3} - \beta_1) q^{27} + (\beta_{7} - \beta_{5} + \beta_{3} - \beta_1) q^{28} + 2 \beta_{2} q^{29} + ( - 4 \beta_{6} + 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 4) q^{30} + 2 \beta_{6} q^{31} + 5 q^{32} + 6 q^{34} - 4 \beta_{6} q^{35} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{36} - 8 \beta_{2} q^{37} + ( - 3 \beta_{7} + 3 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{38} + (3 \beta_{7} - 3 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 3 \beta_1) q^{39} + 6 \beta_{7} q^{40} + (6 \beta_{6} - 6 \beta_{4} + 6 \beta_{2} - 6) q^{41} + ( - 2 \beta_{6} - \beta_1) q^{42} - 3 \beta_{5} q^{43} + ( - 2 \beta_{5} - 8) q^{45} - 2 \beta_{3} q^{47} + (\beta_{7} + \beta_{2}) q^{48} + 5 \beta_{4} q^{49} - 3 \beta_{4} q^{50} + ( - 6 \beta_{7} - 6 \beta_{2}) q^{51} + 3 \beta_{3} q^{52} + 4 \beta_1 q^{53} + (\beta_{5} - 5) q^{54} + 3 \beta_{5} q^{56} + ( - 6 \beta_{6} - 3 \beta_1) q^{57} + ( - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{2} + 2) q^{58} + 8 \beta_{7} q^{59} + ( - 2 \beta_{7} + 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{60} + ( - 7 \beta_{7} + 7 \beta_{5} - 7 \beta_{3} + 7 \beta_1) q^{61} + 2 \beta_{2} q^{62} + (4 \beta_{6} - 4 \beta_{4} - \beta_{3} + 4 \beta_{2} - 4) q^{63} - 7 \beta_{6} q^{64} - 12 q^{65} + 2 q^{67} + 6 \beta_{6} q^{68} - 4 \beta_{2} q^{70} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{71} + ( - 6 \beta_{7} + 6 \beta_{5} + 3 \beta_{4} - 6 \beta_{3} + 6 \beta_1) q^{72} + \beta_{7} q^{73} + (8 \beta_{6} - 8 \beta_{4} + 8 \beta_{2} - 8) q^{74} + (3 \beta_{6} - 3 \beta_1) q^{75} + 3 \beta_{5} q^{76} + (3 \beta_{5} - 6) q^{78} + 3 \beta_1 q^{79} + 2 \beta_{3} q^{80} + (4 \beta_{7} + 7 \beta_{2}) q^{81} + 6 \beta_{4} q^{82} - 16 \beta_{4} q^{83} + ( - \beta_{7} + 2 \beta_{2}) q^{84} - 12 \beta_{3} q^{85} - 3 \beta_1 q^{86} + ( - 2 \beta_{5} - 2) q^{87} + (8 \beta_{6} - 2 \beta_1) q^{90} + (6 \beta_{6} - 6 \beta_{4} + 6 \beta_{2} - 6) q^{91} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{93} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{94} - 12 \beta_{2} q^{95} + (5 \beta_{6} - 5 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} - 5) q^{96} + 2 \beta_{6} q^{97} + 5 q^{98}+O(q^{100})$$ q - b6 * q^2 + (b6 - b4 - b3 + b2 - 1) * q^3 + b2 * q^4 + (2*b7 - 2*b5 + 2*b3 - 2*b1) * q^5 + (b7 - b5 + b4 + b3 - b1) * q^6 + b7 * q^7 + (-3*b6 + 3*b4 - 3*b2 + 3) * q^8 + (b6 + 2*b1) * q^9 + 2*b5 * q^10 + (-b5 - 1) * q^12 + 3*b1 * q^13 + b3 * q^14 + (-2*b7 + 4*b2) * q^15 - b4 * q^16 + 6*b4 * q^17 + (-2*b7 + b2) * q^18 + 3*b3 * q^19 - 2*b1 * q^20 + (-b5 + 2) * q^21 + (3*b6 - 3*b1) * q^24 + (-3*b6 + 3*b4 - 3*b2 + 3) * q^25 - 3*b7 * q^26 + (b7 - b5 - 5*b4 + b3 - b1) * q^27 + (b7 - b5 + b3 - b1) * q^28 + 2*b2 * q^29 + (-4*b6 + 4*b4 - 2*b3 - 4*b2 + 4) * q^30 + 2*b6 * q^31 + 5 * q^32 + 6 * q^34 - 4*b6 * q^35 + (b6 - b4 + 2*b3 + b2 - 1) * q^36 - 8*b2 * q^37 + (-3*b7 + 3*b5 - 3*b3 + 3*b1) * q^38 + (3*b7 - 3*b5 - 6*b4 + 3*b3 - 3*b1) * q^39 + 6*b7 * q^40 + (6*b6 - 6*b4 + 6*b2 - 6) * q^41 + (-2*b6 - b1) * q^42 - 3*b5 * q^43 + (-2*b5 - 8) * q^45 - 2*b3 * q^47 + (b7 + b2) * q^48 + 5*b4 * q^49 - 3*b4 * q^50 + (-6*b7 - 6*b2) * q^51 + 3*b3 * q^52 + 4*b1 * q^53 + (b5 - 5) * q^54 + 3*b5 * q^56 + (-6*b6 - 3*b1) * q^57 + (-2*b6 + 2*b4 - 2*b2 + 2) * q^58 + 8*b7 * q^59 + (-2*b7 + 2*b5 + 4*b4 - 2*b3 + 2*b1) * q^60 + (-7*b7 + 7*b5 - 7*b3 + 7*b1) * q^61 + 2*b2 * q^62 + (4*b6 - 4*b4 - b3 + 4*b2 - 4) * q^63 - 7*b6 * q^64 - 12 * q^65 + 2 * q^67 + 6*b6 * q^68 - 4*b2 * q^70 + (2*b7 - 2*b5 + 2*b3 - 2*b1) * q^71 + (-6*b7 + 6*b5 + 3*b4 - 6*b3 + 6*b1) * q^72 + b7 * q^73 + (8*b6 - 8*b4 + 8*b2 - 8) * q^74 + (3*b6 - 3*b1) * q^75 + 3*b5 * q^76 + (3*b5 - 6) * q^78 + 3*b1 * q^79 + 2*b3 * q^80 + (4*b7 + 7*b2) * q^81 + 6*b4 * q^82 - 16*b4 * q^83 + (-b7 + 2*b2) * q^84 - 12*b3 * q^85 - 3*b1 * q^86 + (-2*b5 - 2) * q^87 + (8*b6 - 2*b1) * q^90 + (6*b6 - 6*b4 + 6*b2 - 6) * q^91 + (-2*b7 + 2*b5 - 2*b4 - 2*b3 + 2*b1) * q^93 + (2*b7 - 2*b5 + 2*b3 - 2*b1) * q^94 - 12*b2 * q^95 + (5*b6 - 5*b4 - 5*b3 + 5*b2 - 5) * q^96 + 2*b6 * q^97 + 5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 8 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^8 + 2 * q^9 $$8 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{8} + 2 q^{9} - 8 q^{12} + 8 q^{15} + 2 q^{16} - 12 q^{17} + 2 q^{18} + 16 q^{21} + 6 q^{24} + 6 q^{25} + 10 q^{27} + 4 q^{29} + 8 q^{30} + 4 q^{31} + 40 q^{32} + 48 q^{34} - 8 q^{35} - 2 q^{36} - 16 q^{37} + 12 q^{39} - 12 q^{41} - 4 q^{42} - 64 q^{45} + 2 q^{48} - 10 q^{49} + 6 q^{50} - 12 q^{51} - 40 q^{54} - 12 q^{57} + 4 q^{58} - 8 q^{60} + 4 q^{62} - 8 q^{63} - 14 q^{64} - 96 q^{65} + 16 q^{67} + 12 q^{68} - 8 q^{70} - 6 q^{72} - 16 q^{74} + 6 q^{75} - 48 q^{78} + 14 q^{81} - 12 q^{82} + 32 q^{83} + 4 q^{84} - 16 q^{87} + 16 q^{90} - 12 q^{91} + 4 q^{93} - 24 q^{95} - 10 q^{96} + 4 q^{97} + 40 q^{98}+O(q^{100})$$ 8 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^8 + 2 * q^9 - 8 * q^12 + 8 * q^15 + 2 * q^16 - 12 * q^17 + 2 * q^18 + 16 * q^21 + 6 * q^24 + 6 * q^25 + 10 * q^27 + 4 * q^29 + 8 * q^30 + 4 * q^31 + 40 * q^32 + 48 * q^34 - 8 * q^35 - 2 * q^36 - 16 * q^37 + 12 * q^39 - 12 * q^41 - 4 * q^42 - 64 * q^45 + 2 * q^48 - 10 * q^49 + 6 * q^50 - 12 * q^51 - 40 * q^54 - 12 * q^57 + 4 * q^58 - 8 * q^60 + 4 * q^62 - 8 * q^63 - 14 * q^64 - 96 * q^65 + 16 * q^67 + 12 * q^68 - 8 * q^70 - 6 * q^72 - 16 * q^74 + 6 * q^75 - 48 * q^78 + 14 * q^81 - 12 * q^82 + 32 * q^83 + 4 * q^84 - 16 * q^87 + 16 * q^90 - 12 * q^91 + 4 * q^93 - 24 * q^95 - 10 * q^96 + 4 * q^97 + 40 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} ) / 4$$ (v^4) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{5} ) / 4$$ (v^5) / 4 $$\beta_{6}$$ $$=$$ $$( \nu^{6} ) / 8$$ (v^6) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{7} ) / 8$$ (v^7) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3 $$\nu^{4}$$ $$=$$ $$4\beta_{4}$$ 4*b4 $$\nu^{5}$$ $$=$$ $$4\beta_{5}$$ 4*b5 $$\nu^{6}$$ $$=$$ $$8\beta_{6}$$ 8*b6 $$\nu^{7}$$ $$=$$ $$8\beta_{7}$$ 8*b7

## Character values

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

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$

## 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
 −0.831254 + 1.14412i 0.831254 − 1.14412i 1.34500 − 0.437016i −1.34500 + 0.437016i 1.34500 + 0.437016i −1.34500 − 0.437016i −0.831254 − 1.14412i 0.831254 + 1.14412i
−0.809017 0.587785i −1.03598 1.38807i −0.309017 0.951057i 1.66251 + 2.28825i 0.0222369 + 1.73191i −1.34500 + 0.437016i −0.927051 + 2.85317i −0.853491 + 2.87603i 2.82843i
161.2 −0.809017 0.587785i 1.65401 0.514040i −0.309017 0.951057i −1.66251 2.28825i −1.64027 0.556338i 1.34500 0.437016i −0.927051 + 2.85317i 2.47152 1.70046i 2.82843i
215.1 0.309017 + 0.951057i −1.64027 + 0.556338i 0.809017 0.587785i −2.68999 0.874032i −1.03598 1.38807i −0.831254 1.14412i 2.42705 + 1.76336i 2.38098 1.82509i 2.82843i
215.2 0.309017 + 0.951057i 0.0222369 1.73191i 0.809017 0.587785i 2.68999 + 0.874032i 1.65401 0.514040i 0.831254 + 1.14412i 2.42705 + 1.76336i −2.99901 0.0770245i 2.82843i
233.1 0.309017 0.951057i −1.64027 0.556338i 0.809017 + 0.587785i −2.68999 + 0.874032i −1.03598 + 1.38807i −0.831254 + 1.14412i 2.42705 1.76336i 2.38098 + 1.82509i 2.82843i
233.2 0.309017 0.951057i 0.0222369 + 1.73191i 0.809017 + 0.587785i 2.68999 0.874032i 1.65401 + 0.514040i 0.831254 1.14412i 2.42705 1.76336i −2.99901 + 0.0770245i 2.82843i
239.1 −0.809017 + 0.587785i −1.03598 + 1.38807i −0.309017 + 0.951057i 1.66251 2.28825i 0.0222369 1.73191i −1.34500 0.437016i −0.927051 2.85317i −0.853491 2.87603i 2.82843i
239.2 −0.809017 + 0.587785i 1.65401 + 0.514040i −0.309017 + 0.951057i −1.66251 + 2.28825i −1.64027 + 0.556338i 1.34500 + 0.437016i −0.927051 2.85317i 2.47152 + 1.70046i 2.82843i
 $$n$$: 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.c even 5 3 inner
33.d even 2 1 inner
33.f even 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.a 8
3.b odd 2 1 363.2.f.f 8
11.b odd 2 1 363.2.f.f 8
11.c even 5 1 363.2.d.b yes 2
11.c even 5 3 inner 363.2.f.a 8
11.d odd 10 1 363.2.d.a 2
11.d odd 10 3 363.2.f.f 8
33.d even 2 1 inner 363.2.f.a 8
33.f even 10 1 363.2.d.b yes 2
33.f even 10 3 inner 363.2.f.a 8
33.h odd 10 1 363.2.d.a 2
33.h odd 10 3 363.2.f.f 8

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

## Hecke kernels

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

 $$T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1$$ T2^4 + T2^3 + T2^2 + T2 + 1 $$T_{5}^{8} - 8T_{5}^{6} + 64T_{5}^{4} - 512T_{5}^{2} + 4096$$ T5^8 - 8*T5^6 + 64*T5^4 - 512*T5^2 + 4096 $$T_{7}^{8} - 2T_{7}^{6} + 4T_{7}^{4} - 8T_{7}^{2} + 16$$ T7^8 - 2*T7^6 + 4*T7^4 - 8*T7^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$3$ $$T^{8} + 2 T^{7} + T^{6} - 4 T^{5} + \cdots + 81$$
$5$ $$T^{8} - 8 T^{6} + 64 T^{4} + \cdots + 4096$$
$7$ $$T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + \cdots + 16$$
$11$ $$T^{8}$$
$13$ $$T^{8} - 18 T^{6} + 324 T^{4} + \cdots + 104976$$
$17$ $$(T^{4} + 6 T^{3} + 36 T^{2} + 216 T + 1296)^{2}$$
$19$ $$T^{8} - 18 T^{6} + 324 T^{4} + \cdots + 104976$$
$23$ $$T^{8}$$
$29$ $$(T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2}$$
$31$ $$(T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2}$$
$37$ $$(T^{4} + 8 T^{3} + 64 T^{2} + 512 T + 4096)^{2}$$
$41$ $$(T^{4} + 6 T^{3} + 36 T^{2} + 216 T + 1296)^{2}$$
$43$ $$(T^{2} + 18)^{4}$$
$47$ $$T^{8} - 8 T^{6} + 64 T^{4} + \cdots + 4096$$
$53$ $$T^{8} - 32 T^{6} + 1024 T^{4} + \cdots + 1048576$$
$59$ $$T^{8} - 128 T^{6} + \cdots + 268435456$$
$61$ $$T^{8} - 98 T^{6} + 9604 T^{4} + \cdots + 92236816$$
$67$ $$(T - 2)^{8}$$
$71$ $$T^{8} - 8 T^{6} + 64 T^{4} + \cdots + 4096$$
$73$ $$T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + \cdots + 16$$
$79$ $$T^{8} - 18 T^{6} + 324 T^{4} + \cdots + 104976$$
$83$ $$(T^{4} - 16 T^{3} + 256 T^{2} + \cdots + 65536)^{2}$$
$89$ $$T^{8}$$
$97$ $$(T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2}$$