# Properties

 Label 484.2.e.a Level $484$ Weight $2$ Character orbit 484.e Analytic conductor $3.865$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$484 = 2^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 484.e (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.86475945783$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{10}^{2} q^{3} + 3 \zeta_{10} q^{5} - 2 \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} +O(q^{10})$$ q + z^2 * q^3 + 3*z * q^5 - 2*z^3 * q^7 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^9 $$q + \zeta_{10}^{2} q^{3} + 3 \zeta_{10} q^{5} - 2 \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{9} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{13} + 3 \zeta_{10}^{3} q^{15} - 6 \zeta_{10} q^{17} + 8 \zeta_{10}^{2} q^{19} + 2 q^{21} - 3 q^{23} + 4 \zeta_{10}^{2} q^{25} + 5 \zeta_{10} q^{27} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{31} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{35} + \zeta_{10}^{3} q^{37} + 4 \zeta_{10} q^{39} - 10 q^{43} + 6 q^{45} + 3 \zeta_{10} q^{49} - 6 \zeta_{10}^{3} q^{51} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 6) q^{53} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 8) q^{57} - 3 \zeta_{10}^{3} q^{59} + 4 \zeta_{10} q^{61} - 4 \zeta_{10}^{2} q^{63} + 12 q^{65} - q^{67} - 3 \zeta_{10}^{2} q^{69} - 15 \zeta_{10} q^{71} + 4 \zeta_{10}^{3} q^{73} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{75} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{79} - \zeta_{10}^{3} q^{81} - 6 \zeta_{10} q^{83} - 18 \zeta_{10}^{2} q^{85} - 9 q^{89} - 8 \zeta_{10}^{2} q^{91} - 5 \zeta_{10} q^{93} + 24 \zeta_{10}^{3} q^{95} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 7) q^{97} +O(q^{100})$$ q + z^2 * q^3 + 3*z * q^5 - 2*z^3 * q^7 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^9 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^13 + 3*z^3 * q^15 - 6*z * q^17 + 8*z^2 * q^19 + 2 * q^21 - 3 * q^23 + 4*z^2 * q^25 + 5*z * q^27 + (5*z^3 - 5*z^2 + 5*z - 5) * q^31 + (-6*z^3 + 6*z^2 - 6*z + 6) * q^35 + z^3 * q^37 + 4*z * q^39 - 10 * q^43 + 6 * q^45 + 3*z * q^49 - 6*z^3 * q^51 + (-6*z^3 + 6*z^2 - 6*z + 6) * q^53 + (8*z^3 - 8*z^2 + 8*z - 8) * q^57 - 3*z^3 * q^59 + 4*z * q^61 - 4*z^2 * q^63 + 12 * q^65 - q^67 - 3*z^2 * q^69 - 15*z * q^71 + 4*z^3 * q^73 + (4*z^3 - 4*z^2 + 4*z - 4) * q^75 + (2*z^3 - 2*z^2 + 2*z - 2) * q^79 - z^3 * q^81 - 6*z * q^83 - 18*z^2 * q^85 - 9 * q^89 - 8*z^2 * q^91 - 5*z * q^93 + 24*z^3 * q^95 + (-7*z^3 + 7*z^2 - 7*z + 7) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{3} + 3 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 4 * q - q^3 + 3 * q^5 - 2 * q^7 + 2 * q^9 $$4 q - q^{3} + 3 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{13} + 3 q^{15} - 6 q^{17} - 8 q^{19} + 8 q^{21} - 12 q^{23} - 4 q^{25} + 5 q^{27} - 5 q^{31} + 6 q^{35} + q^{37} + 4 q^{39} - 40 q^{43} + 24 q^{45} + 3 q^{49} - 6 q^{51} + 6 q^{53} - 8 q^{57} - 3 q^{59} + 4 q^{61} + 4 q^{63} + 48 q^{65} - 4 q^{67} + 3 q^{69} - 15 q^{71} + 4 q^{73} - 4 q^{75} - 2 q^{79} - q^{81} - 6 q^{83} + 18 q^{85} - 36 q^{89} + 8 q^{91} - 5 q^{93} + 24 q^{95} + 7 q^{97}+O(q^{100})$$ 4 * q - q^3 + 3 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^13 + 3 * q^15 - 6 * q^17 - 8 * q^19 + 8 * q^21 - 12 * q^23 - 4 * q^25 + 5 * q^27 - 5 * q^31 + 6 * q^35 + q^37 + 4 * q^39 - 40 * q^43 + 24 * q^45 + 3 * q^49 - 6 * q^51 + 6 * q^53 - 8 * q^57 - 3 * q^59 + 4 * q^61 + 4 * q^63 + 48 * q^65 - 4 * q^67 + 3 * q^69 - 15 * q^71 + 4 * q^73 - 4 * q^75 - 2 * q^79 - q^81 - 6 * q^83 + 18 * q^85 - 36 * q^89 + 8 * q^91 - 5 * q^93 + 24 * q^95 + 7 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/484\mathbb{Z}\right)^\times$$.

 $$n$$ $$243$$ $$365$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 −0.309017 − 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i −0.309017 + 0.951057i
0 −0.809017 + 0.587785i 0 −0.927051 2.85317i 0 −1.61803 1.17557i 0 −0.618034 + 1.90211i 0
81.1 0 0.309017 0.951057i 0 2.42705 1.76336i 0 0.618034 + 1.90211i 0 1.61803 + 1.17557i 0
245.1 0 0.309017 + 0.951057i 0 2.42705 + 1.76336i 0 0.618034 1.90211i 0 1.61803 1.17557i 0
269.1 0 −0.809017 0.587785i 0 −0.927051 + 2.85317i 0 −1.61803 + 1.17557i 0 −0.618034 1.90211i 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
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 484.2.e.a 4
11.b odd 2 1 484.2.e.b 4
11.c even 5 1 44.2.a.a 1
11.c even 5 3 inner 484.2.e.a 4
11.d odd 10 1 484.2.a.a 1
11.d odd 10 3 484.2.e.b 4
33.f even 10 1 4356.2.a.j 1
33.h odd 10 1 396.2.a.c 1
44.g even 10 1 1936.2.a.c 1
44.h odd 10 1 176.2.a.a 1
55.j even 10 1 1100.2.a.b 1
55.k odd 20 2 1100.2.b.c 2
77.j odd 10 1 2156.2.a.a 1
77.m even 15 2 2156.2.i.b 2
77.p odd 30 2 2156.2.i.c 2
88.k even 10 1 7744.2.a.bc 1
88.l odd 10 1 704.2.a.i 1
88.o even 10 1 704.2.a.f 1
88.p odd 10 1 7744.2.a.m 1
99.m even 15 2 3564.2.i.j 2
99.n odd 30 2 3564.2.i.a 2
132.o even 10 1 1584.2.a.p 1
143.n even 10 1 7436.2.a.d 1
165.o odd 10 1 9900.2.a.h 1
165.v even 20 2 9900.2.c.g 2
176.v odd 20 2 2816.2.c.k 2
176.w even 20 2 2816.2.c.e 2
220.n odd 10 1 4400.2.a.v 1
220.v even 20 2 4400.2.b.k 2
264.t odd 10 1 6336.2.a.j 1
264.w even 10 1 6336.2.a.i 1
308.t even 10 1 8624.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 11.c even 5 1
176.2.a.a 1 44.h odd 10 1
396.2.a.c 1 33.h odd 10 1
484.2.a.a 1 11.d odd 10 1
484.2.e.a 4 1.a even 1 1 trivial
484.2.e.a 4 11.c even 5 3 inner
484.2.e.b 4 11.b odd 2 1
484.2.e.b 4 11.d odd 10 3
704.2.a.f 1 88.o even 10 1
704.2.a.i 1 88.l odd 10 1
1100.2.a.b 1 55.j even 10 1
1100.2.b.c 2 55.k odd 20 2
1584.2.a.p 1 132.o even 10 1
1936.2.a.c 1 44.g even 10 1
2156.2.a.a 1 77.j odd 10 1
2156.2.i.b 2 77.m even 15 2
2156.2.i.c 2 77.p odd 30 2
2816.2.c.e 2 176.w even 20 2
2816.2.c.k 2 176.v odd 20 2
3564.2.i.a 2 99.n odd 30 2
3564.2.i.j 2 99.m even 15 2
4356.2.a.j 1 33.f even 10 1
4400.2.a.v 1 220.n odd 10 1
4400.2.b.k 2 220.v even 20 2
6336.2.a.i 1 264.w even 10 1
6336.2.a.j 1 264.t odd 10 1
7436.2.a.d 1 143.n even 10 1
7744.2.a.m 1 88.p odd 10 1
7744.2.a.bc 1 88.k even 10 1
8624.2.a.w 1 308.t even 10 1
9900.2.a.h 1 165.o odd 10 1
9900.2.c.g 2 165.v even 20 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(484, [\chi])$$:

 $$T_{3}^{4} + T_{3}^{3} + T_{3}^{2} + T_{3} + 1$$ T3^4 + T3^3 + T3^2 + T3 + 1 $$T_{7}^{4} + 2T_{7}^{3} + 4T_{7}^{2} + 8T_{7} + 16$$ T7^4 + 2*T7^3 + 4*T7^2 + 8*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$5$ $$T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81$$
$7$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256$$
$17$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$19$ $$T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$23$ $$(T + 3)^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625$$
$37$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$41$ $$T^{4}$$
$43$ $$(T + 10)^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} - 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$59$ $$T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81$$
$61$ $$T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256$$
$67$ $$(T + 1)^{4}$$
$71$ $$T^{4} + 15 T^{3} + 225 T^{2} + \cdots + 50625$$
$73$ $$T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256$$
$79$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$83$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$89$ $$(T + 9)^{4}$$
$97$ $$T^{4} - 7 T^{3} + 49 T^{2} + \cdots + 2401$$