# Properties

 Label 363.2.e.e Level $363$ Weight $2$ Character orbit 363.e Analytic conductor $2.899$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} + \zeta_{10} q^{6} - 4 \zeta_{10}^{3} q^{7} - 3 \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10})$$ q + (z^3 - z^2 + z - 1) * q^2 - z^2 * q^3 + z^3 * q^4 + 2*z * q^5 + z * q^6 - 4*z^3 * q^7 - 3*z^2 * q^8 + (z^3 - z^2 + z - 1) * q^9 $$q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{2} - \zeta_{10}^{2} q^{3} + \zeta_{10}^{3} q^{4} + 2 \zeta_{10} q^{5} + \zeta_{10} q^{6} - 4 \zeta_{10}^{3} q^{7} - 3 \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} - 2 q^{10} + q^{12} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{13} + 4 \zeta_{10}^{2} q^{14} - 2 \zeta_{10}^{3} q^{15} + \zeta_{10} q^{16} + 2 \zeta_{10} q^{17} - \zeta_{10}^{3} q^{18} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{20} - 4 q^{21} + 8 q^{23} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{24} - \zeta_{10}^{2} q^{25} + 2 \zeta_{10}^{3} q^{26} + \zeta_{10} q^{27} + 4 \zeta_{10} q^{28} + 6 \zeta_{10}^{3} q^{29} + 2 \zeta_{10}^{2} q^{30} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{31} + 5 q^{32} - 2 q^{34} + ( - 8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 8) q^{35} - \zeta_{10}^{2} q^{36} - 6 \zeta_{10}^{3} q^{37} - 2 \zeta_{10} q^{39} - 6 \zeta_{10}^{3} q^{40} - 2 \zeta_{10}^{2} q^{41} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{42} - 2 q^{45} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 8) q^{46} + 8 \zeta_{10}^{2} q^{47} - \zeta_{10}^{3} q^{48} - 9 \zeta_{10} q^{49} + \zeta_{10} q^{50} - 2 \zeta_{10}^{3} q^{51} + 2 \zeta_{10}^{2} q^{52} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{53} - q^{54} - 12 q^{56} - 6 \zeta_{10}^{2} q^{58} + 4 \zeta_{10}^{3} q^{59} + 2 \zeta_{10} q^{60} - 6 \zeta_{10} q^{61} + 8 \zeta_{10}^{3} q^{62} + 4 \zeta_{10}^{2} q^{63} + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 7) q^{64} + 4 q^{65} - 4 q^{67} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{68} - 8 \zeta_{10}^{2} q^{69} + 8 \zeta_{10}^{3} q^{70} + 3 \zeta_{10} q^{72} + 14 \zeta_{10}^{3} q^{73} + 6 \zeta_{10}^{2} q^{74} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{75} + 2 q^{78} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{79} + 2 \zeta_{10}^{2} q^{80} - \zeta_{10}^{3} q^{81} + 2 \zeta_{10} q^{82} - 12 \zeta_{10} q^{83} - 4 \zeta_{10}^{3} q^{84} + 4 \zeta_{10}^{2} q^{85} + 6 q^{87} - 6 q^{89} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{90} - 8 \zeta_{10}^{2} q^{91} + 8 \zeta_{10}^{3} q^{92} - 8 \zeta_{10} q^{93} - 8 \zeta_{10} q^{94} - 5 \zeta_{10}^{2} q^{96} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{97} + 9 q^{98} +O(q^{100})$$ q + (z^3 - z^2 + z - 1) * q^2 - z^2 * q^3 + z^3 * q^4 + 2*z * q^5 + z * q^6 - 4*z^3 * q^7 - 3*z^2 * q^8 + (z^3 - z^2 + z - 1) * q^9 - 2 * q^10 + q^12 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^13 + 4*z^2 * q^14 - 2*z^3 * q^15 + z * q^16 + 2*z * q^17 - z^3 * q^18 + (2*z^3 - 2*z^2 + 2*z - 2) * q^20 - 4 * q^21 + 8 * q^23 + (3*z^3 - 3*z^2 + 3*z - 3) * q^24 - z^2 * q^25 + 2*z^3 * q^26 + z * q^27 + 4*z * q^28 + 6*z^3 * q^29 + 2*z^2 * q^30 + (-8*z^3 + 8*z^2 - 8*z + 8) * q^31 + 5 * q^32 - 2 * q^34 + (-8*z^3 + 8*z^2 - 8*z + 8) * q^35 - z^2 * q^36 - 6*z^3 * q^37 - 2*z * q^39 - 6*z^3 * q^40 - 2*z^2 * q^41 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^42 - 2 * q^45 + (8*z^3 - 8*z^2 + 8*z - 8) * q^46 + 8*z^2 * q^47 - z^3 * q^48 - 9*z * q^49 + z * q^50 - 2*z^3 * q^51 + 2*z^2 * q^52 + (6*z^3 - 6*z^2 + 6*z - 6) * q^53 - q^54 - 12 * q^56 - 6*z^2 * q^58 + 4*z^3 * q^59 + 2*z * q^60 - 6*z * q^61 + 8*z^3 * q^62 + 4*z^2 * q^63 + (7*z^3 - 7*z^2 + 7*z - 7) * q^64 + 4 * q^65 - 4 * q^67 + (2*z^3 - 2*z^2 + 2*z - 2) * q^68 - 8*z^2 * q^69 + 8*z^3 * q^70 + 3*z * q^72 + 14*z^3 * q^73 + 6*z^2 * q^74 + (z^3 - z^2 + z - 1) * q^75 + 2 * q^78 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^79 + 2*z^2 * q^80 - z^3 * q^81 + 2*z * q^82 - 12*z * q^83 - 4*z^3 * q^84 + 4*z^2 * q^85 + 6 * q^87 - 6 * q^89 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^90 - 8*z^2 * q^91 + 8*z^3 * q^92 - 8*z * q^93 - 8*z * q^94 - 5*z^2 * q^96 + (2*z^3 - 2*z^2 + 2*z - 2) * q^97 + 9 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 3 q^{8} - q^{9}+O(q^{10})$$ 4 * q - q^2 + q^3 + q^4 + 2 * q^5 + q^6 - 4 * q^7 + 3 * q^8 - q^9 $$4 q - q^{2} + q^{3} + q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 3 q^{8} - q^{9} - 8 q^{10} + 4 q^{12} + 2 q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 2 q^{20} - 16 q^{21} + 32 q^{23} - 3 q^{24} + q^{25} + 2 q^{26} + q^{27} + 4 q^{28} + 6 q^{29} - 2 q^{30} + 8 q^{31} + 20 q^{32} - 8 q^{34} + 8 q^{35} + q^{36} - 6 q^{37} - 2 q^{39} - 6 q^{40} + 2 q^{41} + 4 q^{42} - 8 q^{45} - 8 q^{46} - 8 q^{47} - q^{48} - 9 q^{49} + q^{50} - 2 q^{51} - 2 q^{52} - 6 q^{53} - 4 q^{54} - 48 q^{56} + 6 q^{58} + 4 q^{59} + 2 q^{60} - 6 q^{61} + 8 q^{62} - 4 q^{63} - 7 q^{64} + 16 q^{65} - 16 q^{67} - 2 q^{68} + 8 q^{69} + 8 q^{70} + 3 q^{72} + 14 q^{73} - 6 q^{74} - q^{75} + 8 q^{78} + 4 q^{79} - 2 q^{80} - q^{81} + 2 q^{82} - 12 q^{83} - 4 q^{84} - 4 q^{85} + 24 q^{87} - 24 q^{89} + 2 q^{90} + 8 q^{91} + 8 q^{92} - 8 q^{93} - 8 q^{94} + 5 q^{96} - 2 q^{97} + 36 q^{98}+O(q^{100})$$ 4 * q - q^2 + q^3 + q^4 + 2 * q^5 + q^6 - 4 * q^7 + 3 * q^8 - q^9 - 8 * q^10 + 4 * q^12 + 2 * q^13 - 4 * q^14 - 2 * q^15 + q^16 + 2 * q^17 - q^18 - 2 * q^20 - 16 * q^21 + 32 * q^23 - 3 * q^24 + q^25 + 2 * q^26 + q^27 + 4 * q^28 + 6 * q^29 - 2 * q^30 + 8 * q^31 + 20 * q^32 - 8 * q^34 + 8 * q^35 + q^36 - 6 * q^37 - 2 * q^39 - 6 * q^40 + 2 * q^41 + 4 * q^42 - 8 * q^45 - 8 * q^46 - 8 * q^47 - q^48 - 9 * q^49 + q^50 - 2 * q^51 - 2 * q^52 - 6 * q^53 - 4 * q^54 - 48 * q^56 + 6 * q^58 + 4 * q^59 + 2 * q^60 - 6 * q^61 + 8 * q^62 - 4 * q^63 - 7 * q^64 + 16 * q^65 - 16 * q^67 - 2 * q^68 + 8 * q^69 + 8 * q^70 + 3 * q^72 + 14 * q^73 - 6 * q^74 - q^75 + 8 * q^78 + 4 * q^79 - 2 * q^80 - q^81 + 2 * q^82 - 12 * q^83 - 4 * q^84 - 4 * q^85 + 24 * q^87 - 24 * q^89 + 2 * q^90 + 8 * q^91 + 8 * q^92 - 8 * q^93 - 8 * q^94 + 5 * q^96 - 2 * q^97 + 36 * q^98

## Character values

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

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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
 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i
−0.809017 + 0.587785i −0.309017 0.951057i −0.309017 + 0.951057i 1.61803 + 1.17557i 0.809017 + 0.587785i 1.23607 3.80423i −0.927051 2.85317i −0.809017 + 0.587785i −2.00000
130.1 0.309017 0.951057i 0.809017 0.587785i 0.809017 + 0.587785i −0.618034 1.90211i −0.309017 0.951057i −3.23607 2.35114i 2.42705 1.76336i 0.309017 0.951057i −2.00000
148.1 0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 0.587785i −0.618034 + 1.90211i −0.309017 + 0.951057i −3.23607 + 2.35114i 2.42705 + 1.76336i 0.309017 + 0.951057i −2.00000
202.1 −0.809017 0.587785i −0.309017 + 0.951057i −0.309017 0.951057i 1.61803 1.17557i 0.809017 0.587785i 1.23607 + 3.80423i −0.927051 + 2.85317i −0.809017 0.587785i −2.00000
 $$n$$: 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
11.c even 5 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.e.e 4
11.b odd 2 1 363.2.e.g 4
11.c even 5 1 33.2.a.a 1
11.c even 5 3 inner 363.2.e.e 4
11.d odd 10 1 363.2.a.b 1
11.d odd 10 3 363.2.e.g 4
33.f even 10 1 1089.2.a.j 1
33.h odd 10 1 99.2.a.b 1
44.g even 10 1 5808.2.a.t 1
44.h odd 10 1 528.2.a.g 1
55.h odd 10 1 9075.2.a.q 1
55.j even 10 1 825.2.a.a 1
55.k odd 20 2 825.2.c.a 2
77.j odd 10 1 1617.2.a.j 1
88.l odd 10 1 2112.2.a.j 1
88.o even 10 1 2112.2.a.bb 1
99.m even 15 2 891.2.e.e 2
99.n odd 30 2 891.2.e.g 2
132.o even 10 1 1584.2.a.o 1
143.n even 10 1 5577.2.a.a 1
165.o odd 10 1 2475.2.a.g 1
165.v even 20 2 2475.2.c.d 2
187.j even 10 1 9537.2.a.m 1
231.u even 10 1 4851.2.a.b 1
264.t odd 10 1 6336.2.a.x 1
264.w even 10 1 6336.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 11.c even 5 1
99.2.a.b 1 33.h odd 10 1
363.2.a.b 1 11.d odd 10 1
363.2.e.e 4 1.a even 1 1 trivial
363.2.e.e 4 11.c even 5 3 inner
363.2.e.g 4 11.b odd 2 1
363.2.e.g 4 11.d odd 10 3
528.2.a.g 1 44.h odd 10 1
825.2.a.a 1 55.j even 10 1
825.2.c.a 2 55.k odd 20 2
891.2.e.e 2 99.m even 15 2
891.2.e.g 2 99.n odd 30 2
1089.2.a.j 1 33.f even 10 1
1584.2.a.o 1 132.o even 10 1
1617.2.a.j 1 77.j odd 10 1
2112.2.a.j 1 88.l odd 10 1
2112.2.a.bb 1 88.o even 10 1
2475.2.a.g 1 165.o odd 10 1
2475.2.c.d 2 165.v even 20 2
4851.2.a.b 1 231.u even 10 1
5577.2.a.a 1 143.n even 10 1
5808.2.a.t 1 44.g even 10 1
6336.2.a.n 1 264.w even 10 1
6336.2.a.x 1 264.t odd 10 1
9075.2.a.q 1 55.h odd 10 1
9537.2.a.m 1 187.j even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(363, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$3$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$5$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$7$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$17$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$19$ $$T^{4}$$
$23$ $$(T - 8)^{4}$$
$29$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$31$ $$T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$37$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$41$ $$T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$53$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$59$ $$T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256$$
$61$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$67$ $$(T + 4)^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 14 T^{3} + 196 T^{2} + \cdots + 38416$$
$79$ $$T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256$$
$83$ $$T^{4} + 12 T^{3} + 144 T^{2} + \cdots + 20736$$
$89$ $$(T + 6)^{4}$$
$97$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$