# Properties

 Label 528.2.y.f Level 528 Weight 2 Character orbit 528.y Analytic conductor 4.216 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 528.y (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.21610122672$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{10}^{2} q^{3} + ( -1 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{5} + 3 \zeta_{10}^{3} q^{7} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q -\zeta_{10}^{2} q^{3} + ( -1 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{5} + 3 \zeta_{10}^{3} q^{7} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + ( -4 + 3 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{11} + ( -5 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{13} + ( -1 + \zeta_{10} - \zeta_{10}^{3} ) q^{15} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{17} + ( 3 \zeta_{10} - 4 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{19} + 3 q^{21} + ( 3 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{23} + ( -3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{25} + \zeta_{10} q^{27} + ( -4 + 4 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{29} + ( -2 - \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{31} + ( 2 \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{33} + ( -3 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{35} + ( -2 + 2 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{37} + ( 2 + 3 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{39} + ( 8 \zeta_{10} - 7 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{41} + ( -3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{43} + ( -1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{45} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{47} -2 \zeta_{10} q^{49} + ( 1 - \zeta_{10} ) q^{51} + ( 8 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{53} + ( -5 \zeta_{10} + 6 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{55} + ( -1 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{57} + ( 7 - 7 \zeta_{10} - \zeta_{10}^{3} ) q^{59} + ( 3 - 6 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{61} -3 \zeta_{10}^{2} q^{63} + ( -3 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{65} + ( -5 - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{67} + ( -4 \zeta_{10} + \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{69} + ( 9 + 9 \zeta_{10}^{2} ) q^{71} + ( 2 - 2 \zeta_{10} ) q^{73} + ( -3 + 3 \zeta_{10}^{3} ) q^{75} + ( -3 + 3 \zeta_{10} - 9 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{77} + ( 3 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{79} -\zeta_{10}^{3} q^{81} + ( -6 - 3 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{83} + ( 2 \zeta_{10} - 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{85} + ( 2 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{87} + ( 7 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{89} + ( -6 \zeta_{10} - 9 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{91} + ( 3 - \zeta_{10} + 3 \zeta_{10}^{2} ) q^{93} + ( -7 + 7 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{95} + ( -7 - 6 \zeta_{10} + 6 \zeta_{10}^{2} + 7 \zeta_{10}^{3} ) q^{97} + ( 1 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{3} - q^{5} + 3q^{7} - q^{9} + O(q^{10})$$ $$4q + q^{3} - q^{5} + 3q^{7} - q^{9} - 9q^{11} - 9q^{13} - 4q^{15} + 2q^{17} + 10q^{19} + 12q^{21} + 4q^{23} - 6q^{25} + q^{27} - 10q^{29} - 8q^{31} - q^{33} + 3q^{35} - 3q^{37} + 9q^{39} + 23q^{41} - 16q^{43} - 6q^{45} + 3q^{47} - 2q^{49} + 3q^{51} + 6q^{53} - 14q^{55} + 5q^{57} + 20q^{59} + 3q^{61} + 3q^{63} - 14q^{65} - 2q^{67} - 9q^{69} + 27q^{71} + 6q^{73} - 9q^{75} - 3q^{77} - 5q^{79} - q^{81} - 21q^{83} + 7q^{85} + 20q^{89} - 3q^{91} + 8q^{93} - 25q^{95} - 33q^{97} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$$.

 $$n$$ $$133$$ $$145$$ $$353$$ $$463$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.309017 + 0.951057i −0.309017 − 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i
0 0.809017 + 0.587785i 0 −0.809017 + 2.48990i 0 2.42705 1.76336i 0 0.309017 + 0.951057i 0
97.1 0 0.809017 0.587785i 0 −0.809017 2.48990i 0 2.42705 + 1.76336i 0 0.309017 0.951057i 0
289.1 0 −0.309017 0.951057i 0 0.309017 + 0.224514i 0 −0.927051 + 2.85317i 0 −0.809017 + 0.587785i 0
433.1 0 −0.309017 + 0.951057i 0 0.309017 0.224514i 0 −0.927051 2.85317i 0 −0.809017 0.587785i 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 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.y.f 4
4.b odd 2 1 33.2.e.a 4
11.c even 5 1 inner 528.2.y.f 4
11.c even 5 1 5808.2.a.bl 2
11.d odd 10 1 5808.2.a.bm 2
12.b even 2 1 99.2.f.b 4
20.d odd 2 1 825.2.n.f 4
20.e even 4 2 825.2.bx.b 8
36.f odd 6 2 891.2.n.d 8
36.h even 6 2 891.2.n.a 8
44.c even 2 1 363.2.e.j 4
44.g even 10 1 363.2.a.e 2
44.g even 10 2 363.2.e.c 4
44.g even 10 1 363.2.e.j 4
44.h odd 10 1 33.2.e.a 4
44.h odd 10 1 363.2.a.h 2
44.h odd 10 2 363.2.e.h 4
132.n odd 10 1 1089.2.a.s 2
132.o even 10 1 99.2.f.b 4
132.o even 10 1 1089.2.a.m 2
220.n odd 10 1 825.2.n.f 4
220.n odd 10 1 9075.2.a.x 2
220.o even 10 1 9075.2.a.bv 2
220.v even 20 2 825.2.bx.b 8
396.ba even 30 2 891.2.n.a 8
396.be odd 30 2 891.2.n.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.a 4 4.b odd 2 1
33.2.e.a 4 44.h odd 10 1
99.2.f.b 4 12.b even 2 1
99.2.f.b 4 132.o even 10 1
363.2.a.e 2 44.g even 10 1
363.2.a.h 2 44.h odd 10 1
363.2.e.c 4 44.g even 10 2
363.2.e.h 4 44.h odd 10 2
363.2.e.j 4 44.c even 2 1
363.2.e.j 4 44.g even 10 1
528.2.y.f 4 1.a even 1 1 trivial
528.2.y.f 4 11.c even 5 1 inner
825.2.n.f 4 20.d odd 2 1
825.2.n.f 4 220.n odd 10 1
825.2.bx.b 8 20.e even 4 2
825.2.bx.b 8 220.v even 20 2
891.2.n.a 8 36.h even 6 2
891.2.n.a 8 396.ba even 30 2
891.2.n.d 8 36.f odd 6 2
891.2.n.d 8 396.be odd 30 2
1089.2.a.m 2 132.o even 10 1
1089.2.a.s 2 132.n odd 10 1
5808.2.a.bl 2 11.c even 5 1
5808.2.a.bm 2 11.d odd 10 1
9075.2.a.x 2 220.n odd 10 1
9075.2.a.bv 2 220.o even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(528, [\chi])$$:

 $$T_{5}^{4} + T_{5}^{3} + 6 T_{5}^{2} - 4 T_{5} + 1$$ $$T_{7}^{4} - 3 T_{7}^{3} + 9 T_{7}^{2} - 27 T_{7} + 81$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 55 T^{5} + 25 T^{6} + 125 T^{7} + 625 T^{8}$$
$7$ $$1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 105 T^{5} + 98 T^{6} - 1029 T^{7} + 2401 T^{8}$$
$11$ $$1 + 9 T + 41 T^{2} + 99 T^{3} + 121 T^{4}$$
$13$ $$1 + 9 T + 18 T^{2} - 115 T^{3} - 789 T^{4} - 1495 T^{5} + 3042 T^{6} + 19773 T^{7} + 28561 T^{8}$$
$17$ $$1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 340 T^{5} - 3757 T^{6} - 9826 T^{7} + 83521 T^{8}$$
$19$ $$1 - 10 T + 21 T^{2} + 70 T^{3} - 469 T^{4} + 1330 T^{5} + 7581 T^{6} - 68590 T^{7} + 130321 T^{8}$$
$23$ $$( 1 - 2 T + 27 T^{2} - 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 5800 T^{5} + 26071 T^{6} + 243890 T^{7} + 707281 T^{8}$$
$31$ $$1 + 8 T + 3 T^{2} + 46 T^{3} + 1175 T^{4} + 1426 T^{5} + 2883 T^{6} + 238328 T^{7} + 923521 T^{8}$$
$37$ $$1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 5735 T^{5} - 24642 T^{6} + 151959 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 23 T + 208 T^{2} - 961 T^{3} + 3975 T^{4} - 39401 T^{5} + 349648 T^{6} - 1585183 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$1 - 3 T - 43 T^{2} + 45 T^{3} + 2116 T^{4} + 2115 T^{5} - 94987 T^{6} - 311469 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 6 T + 23 T^{2} + 120 T^{3} - 1319 T^{4} + 6360 T^{5} + 64607 T^{6} - 893262 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 20 T + 131 T^{2} - 530 T^{3} + 3851 T^{4} - 31270 T^{5} + 456011 T^{6} - 4107580 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 3 T - 7 T^{2} - 441 T^{3} + 4900 T^{4} - 26901 T^{5} - 26047 T^{6} - 680943 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 + T + 33 T^{2} + 67 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 - 27 T + 253 T^{2} - 819 T^{3} + 100 T^{4} - 58149 T^{5} + 1275373 T^{6} - 9663597 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 6 T - 57 T^{2} + 130 T^{3} + 4761 T^{4} + 9490 T^{5} - 303753 T^{6} - 2334102 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 5 T + 6 T^{2} + 715 T^{3} + 9821 T^{4} + 56485 T^{5} + 37446 T^{6} + 2465195 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 21 T + 88 T^{2} - 915 T^{3} - 13199 T^{4} - 75945 T^{5} + 606232 T^{6} + 12007527 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 10 T + 183 T^{2} - 890 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$1 + 33 T + 537 T^{2} + 6655 T^{3} + 71196 T^{4} + 645535 T^{5} + 5052633 T^{6} + 30118209 T^{7} + 88529281 T^{8}$$