# Properties

 Label 363.3.h.a Level $363$ Weight $3$ Character orbit 363.h Analytic conductor $9.891$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.89103359628$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{10}^{3} q^{3} + 4 \zeta_{10}^{2} q^{4} + 11 \zeta_{10}^{2} q^{7} - 9 \zeta_{10} q^{9} +O(q^{10})$$ q + 3*z^3 * q^3 + 4*z^2 * q^4 + 11*z^2 * q^7 - 9*z * q^9 $$q + 3 \zeta_{10}^{3} q^{3} + 4 \zeta_{10}^{2} q^{4} + 11 \zeta_{10}^{2} q^{7} - 9 \zeta_{10} q^{9} - 12 q^{12} + 22 \zeta_{10} q^{13} + (16 \zeta_{10}^{3} - 16 \zeta_{10}^{2} + 16 \zeta_{10} - 16) q^{16} - 11 \zeta_{10}^{3} q^{19} - 33 q^{21} - 25 \zeta_{10}^{3} q^{25} + ( - 27 \zeta_{10}^{3} + 27 \zeta_{10}^{2} - 27 \zeta_{10} + 27) q^{27} + (44 \zeta_{10}^{3} - 44 \zeta_{10}^{2} + 44 \zeta_{10} - 44) q^{28} - 59 \zeta_{10} q^{31} - 36 \zeta_{10}^{3} q^{36} + 47 \zeta_{10}^{2} q^{37} + (66 \zeta_{10}^{3} - 66 \zeta_{10}^{2} + 66 \zeta_{10} - 66) q^{39} - 22 q^{43} - 48 \zeta_{10}^{2} q^{48} + (72 \zeta_{10}^{3} - 72 \zeta_{10}^{2} + 72 \zeta_{10} - 72) q^{49} + 88 \zeta_{10}^{3} q^{52} + 33 \zeta_{10} q^{57} + ( - 121 \zeta_{10}^{3} + 121 \zeta_{10}^{2} - 121 \zeta_{10} + 121) q^{61} - 99 \zeta_{10}^{3} q^{63} - 64 \zeta_{10} q^{64} - 13 q^{67} + 143 \zeta_{10}^{2} q^{73} + 75 \zeta_{10} q^{75} + 44 q^{76} - 11 \zeta_{10} q^{79} + 81 \zeta_{10}^{2} q^{81} - 132 \zeta_{10}^{2} q^{84} + 242 \zeta_{10}^{3} q^{91} + ( - 177 \zeta_{10}^{3} + 177 \zeta_{10}^{2} - 177 \zeta_{10} + 177) q^{93} + 169 \zeta_{10} q^{97} +O(q^{100})$$ q + 3*z^3 * q^3 + 4*z^2 * q^4 + 11*z^2 * q^7 - 9*z * q^9 - 12 * q^12 + 22*z * q^13 + (16*z^3 - 16*z^2 + 16*z - 16) * q^16 - 11*z^3 * q^19 - 33 * q^21 - 25*z^3 * q^25 + (-27*z^3 + 27*z^2 - 27*z + 27) * q^27 + (44*z^3 - 44*z^2 + 44*z - 44) * q^28 - 59*z * q^31 - 36*z^3 * q^36 + 47*z^2 * q^37 + (66*z^3 - 66*z^2 + 66*z - 66) * q^39 - 22 * q^43 - 48*z^2 * q^48 + (72*z^3 - 72*z^2 + 72*z - 72) * q^49 + 88*z^3 * q^52 + 33*z * q^57 + (-121*z^3 + 121*z^2 - 121*z + 121) * q^61 - 99*z^3 * q^63 - 64*z * q^64 - 13 * q^67 + 143*z^2 * q^73 + 75*z * q^75 + 44 * q^76 - 11*z * q^79 + 81*z^2 * q^81 - 132*z^2 * q^84 + 242*z^3 * q^91 + (-177*z^3 + 177*z^2 - 177*z + 177) * q^93 + 169*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{3} - 4 q^{4} - 11 q^{7} - 9 q^{9}+O(q^{10})$$ 4 * q + 3 * q^3 - 4 * q^4 - 11 * q^7 - 9 * q^9 $$4 q + 3 q^{3} - 4 q^{4} - 11 q^{7} - 9 q^{9} - 48 q^{12} + 22 q^{13} - 16 q^{16} - 11 q^{19} - 132 q^{21} - 25 q^{25} + 27 q^{27} - 44 q^{28} - 59 q^{31} - 36 q^{36} - 47 q^{37} - 66 q^{39} - 88 q^{43} + 48 q^{48} - 72 q^{49} + 88 q^{52} + 33 q^{57} + 121 q^{61} - 99 q^{63} - 64 q^{64} - 52 q^{67} - 143 q^{73} + 75 q^{75} + 176 q^{76} - 11 q^{79} - 81 q^{81} + 132 q^{84} + 242 q^{91} + 177 q^{93} + 169 q^{97}+O(q^{100})$$ 4 * q + 3 * q^3 - 4 * q^4 - 11 * q^7 - 9 * q^9 - 48 * q^12 + 22 * q^13 - 16 * q^16 - 11 * q^19 - 132 * q^21 - 25 * q^25 + 27 * q^27 - 44 * q^28 - 59 * q^31 - 36 * q^36 - 47 * q^37 - 66 * q^39 - 88 * q^43 + 48 * q^48 - 72 * q^49 + 88 * q^52 + 33 * q^57 + 121 * q^61 - 99 * q^63 - 64 * q^64 - 52 * q^67 - 143 * q^73 + 75 * q^75 + 176 * q^76 - 11 * q^79 - 81 * q^81 + 132 * q^84 + 242 * q^91 + 177 * q^93 + 169 * q^97

## Character values

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

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$\zeta_{10}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
245.1
 0.809017 − 0.587785i −0.309017 + 0.951057i −0.309017 − 0.951057i 0.809017 + 0.587785i
0 −0.927051 2.85317i 1.23607 3.80423i 0 0 3.39919 10.4616i 0 −7.28115 + 5.29007i 0
251.1 0 2.42705 1.76336i −3.23607 2.35114i 0 0 −8.89919 6.46564i 0 2.78115 8.55951i 0
269.1 0 2.42705 + 1.76336i −3.23607 + 2.35114i 0 0 −8.89919 + 6.46564i 0 2.78115 + 8.55951i 0
323.1 0 −0.927051 + 2.85317i 1.23607 + 3.80423i 0 0 3.39919 + 10.4616i 0 −7.28115 5.29007i 0
 $$n$$: 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 CM by $$\Q(\sqrt{-3})$$
11.c even 5 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 363.3.h.a 4
3.b odd 2 1 CM 363.3.h.a 4
11.b odd 2 1 363.3.h.b 4
11.c even 5 1 363.3.b.b yes 1
11.c even 5 3 inner 363.3.h.a 4
11.d odd 10 1 363.3.b.a 1
11.d odd 10 3 363.3.h.b 4
33.d even 2 1 363.3.h.b 4
33.f even 10 1 363.3.b.a 1
33.f even 10 3 363.3.h.b 4
33.h odd 10 1 363.3.b.b yes 1
33.h odd 10 3 inner 363.3.h.a 4

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

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{7}^{4} + 11T_{7}^{3} + 121T_{7}^{2} + 1331T_{7} + 14641$$ T7^4 + 11*T7^3 + 121*T7^2 + 1331*T7 + 14641

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 22 T^{3} + 484 T^{2} + \cdots + 234256$$
$17$ $$T^{4}$$
$19$ $$T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 59 T^{3} + 3481 T^{2} + \cdots + 12117361$$
$37$ $$T^{4} + 47 T^{3} + 2209 T^{2} + \cdots + 4879681$$
$41$ $$T^{4}$$
$43$ $$(T + 22)^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4} - 121 T^{3} + \cdots + 214358881$$
$67$ $$(T + 13)^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 143 T^{3} + \cdots + 418161601$$
$79$ $$T^{4} + 11 T^{3} + 121 T^{2} + \cdots + 14641$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} - 169 T^{3} + \cdots + 815730721$$