# Properties

 Label 528.5.j.a Level $528$ Weight $5$ Character orbit 528.j Analytic conductor $54.579$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 528.j (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$54.5793405083$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 102 x^{6} + 2913 x^{4} + 23292 x^{2} + 41364$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{3} + ( -5 + \beta_{3} + \beta_{6} ) q^{5} + ( -3 \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} ) q^{7} + 27 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{3} + ( -5 + \beta_{3} + \beta_{6} ) q^{5} + ( -3 \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} ) q^{7} + 27 q^{9} + ( -4 + 2 \beta_{1} + \beta_{2} - 4 \beta_{3} - 7 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{11} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{5} - 2 \beta_{7} ) q^{13} + ( -15 - 3 \beta_{1} + 5 \beta_{3} + 6 \beta_{4} + 3 \beta_{6} ) q^{15} + ( 13 \beta_{1} - 8 \beta_{2} - 5 \beta_{5} + \beta_{7} ) q^{17} + ( -2 \beta_{1} + 9 \beta_{2} + 2 \beta_{5} + 5 \beta_{7} ) q^{19} + ( 7 \beta_{1} + 2 \beta_{2} - 6 \beta_{5} + 3 \beta_{7} ) q^{21} + ( -77 + \beta_{1} - 29 \beta_{3} - 2 \beta_{4} + 25 \beta_{6} ) q^{23} + ( -279 + 3 \beta_{1} - 48 \beta_{3} - 6 \beta_{4} - 12 \beta_{6} ) q^{25} -27 \beta_{3} q^{27} + ( -66 \beta_{1} + 16 \beta_{2} - 11 \beta_{5} + 13 \beta_{7} ) q^{29} + ( -338 + 9 \beta_{1} + 96 \beta_{3} - 18 \beta_{4} - 12 \beta_{6} ) q^{31} + ( 138 - 8 \beta_{1} + 8 \beta_{2} - 10 \beta_{3} + 15 \beta_{4} + 9 \beta_{5} - 24 \beta_{6} ) q^{33} + ( -5 \beta_{1} + 14 \beta_{2} - 16 \beta_{5} + 8 \beta_{7} ) q^{35} + ( 670 - 17 \beta_{1} + 76 \beta_{3} + 34 \beta_{4} - 16 \beta_{6} ) q^{37} + ( -32 \beta_{1} - 4 \beta_{2} + 3 \beta_{5} + 12 \beta_{7} ) q^{39} + ( -8 \beta_{1} - 84 \beta_{2} + 3 \beta_{5} - 3 \beta_{7} ) q^{41} + ( 70 \beta_{1} - 31 \beta_{2} + 10 \beta_{5} - 27 \beta_{7} ) q^{43} + ( -135 + 27 \beta_{3} + 27 \beta_{6} ) q^{45} + ( -43 + 78 \beta_{1} - 19 \beta_{3} - 156 \beta_{4} - 19 \beta_{6} ) q^{47} + ( -799 + 110 \beta_{1} + 218 \beta_{3} - 220 \beta_{4} - 110 \beta_{6} ) q^{49} + ( 54 \beta_{1} - 45 \beta_{2} - 18 \beta_{5} + 27 \beta_{7} ) q^{51} + ( 489 - 50 \beta_{1} - 453 \beta_{3} + 100 \beta_{4} - 93 \beta_{6} ) q^{53} + ( -486 + 42 \beta_{1} - 40 \beta_{2} + 292 \beta_{3} + 52 \beta_{4} + 9 \beta_{5} + 26 \beta_{6} - 16 \beta_{7} ) q^{55} + ( -111 \beta_{1} + 12 \beta_{2} - 9 \beta_{5} - 27 \beta_{7} ) q^{57} + ( 2156 + 45 \beta_{1} - 112 \beta_{3} - 90 \beta_{4} - 154 \beta_{6} ) q^{59} + ( 289 \beta_{1} - 38 \beta_{2} - 63 \beta_{5} + 28 \beta_{7} ) q^{61} + ( -81 \beta_{1} - 27 \beta_{2} - 27 \beta_{5} + 27 \beta_{7} ) q^{63} + ( -71 \beta_{1} - 52 \beta_{2} - 10 \beta_{5} + 2 \beta_{7} ) q^{65} + ( 544 - 171 \beta_{1} - 510 \beta_{3} + 342 \beta_{4} - 174 \beta_{6} ) q^{67} + ( 1095 - 78 \beta_{1} + 73 \beta_{3} + 156 \beta_{4} + 69 \beta_{6} ) q^{69} + ( 1737 - 160 \beta_{1} - 747 \beta_{3} + 320 \beta_{4} - 171 \beta_{6} ) q^{71} + ( 286 \beta_{1} - 16 \beta_{5} + 62 \beta_{7} ) q^{73} + ( 1188 + 27 \beta_{1} + 267 \beta_{3} - 54 \beta_{4} - 54 \beta_{6} ) q^{75} + ( 2837 + 301 \beta_{1} + 64 \beta_{2} - 421 \beta_{3} + 148 \beta_{4} + 49 \beta_{5} + 143 \beta_{6} - 71 \beta_{7} ) q^{77} + ( -303 \beta_{1} + 191 \beta_{2} - 91 \beta_{5} - 95 \beta_{7} ) q^{79} + 729 q^{81} + ( 196 \beta_{1} - 240 \beta_{2} - 150 \beta_{5} + 48 \beta_{7} ) q^{83} + ( -566 \beta_{1} + 216 \beta_{2} - 4 \beta_{5} - 10 \beta_{7} ) q^{85} + ( -245 \beta_{1} + 113 \beta_{2} - 72 \beta_{5} + 27 \beta_{7} ) q^{87} + ( 2044 + 78 \beta_{1} + 946 \beta_{3} - 156 \beta_{4} - 206 \beta_{6} ) q^{89} + ( 2560 - 151 \beta_{1} + 554 \beta_{3} + 302 \beta_{4} - 182 \beta_{6} ) q^{91} + ( -2628 + 9 \beta_{1} + 302 \beta_{3} - 18 \beta_{4} - 90 \beta_{6} ) q^{93} + ( 334 \beta_{1} - 158 \beta_{2} - 26 \beta_{5} + 82 \beta_{7} ) q^{95} + ( 938 - 246 \beta_{1} - 1098 \beta_{3} + 492 \beta_{4} + 30 \beta_{6} ) q^{97} + ( -108 + 54 \beta_{1} + 27 \beta_{2} - 108 \beta_{3} - 189 \beta_{4} + 27 \beta_{5} - 27 \beta_{6} - 54 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 36q^{5} + 216q^{9} + O(q^{10})$$ $$8q - 36q^{5} + 216q^{9} - 36q^{11} - 108q^{15} - 516q^{23} - 2280q^{25} - 2752q^{31} + 1008q^{33} + 5296q^{37} - 972q^{45} - 420q^{47} - 6832q^{49} + 3540q^{53} - 3784q^{55} + 16632q^{59} + 3656q^{67} + 9036q^{69} + 13212q^{71} + 9288q^{75} + 23268q^{77} + 5832q^{81} + 15528q^{89} + 19752q^{91} - 21384q^{93} + 7624q^{97} - 972q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 102 x^{6} + 2913 x^{4} + 23292 x^{2} + 41364$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} - 444 \nu^{5} - 9669 \nu^{3} - 41250 \nu$$$$)/3216$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{6} - 444 \nu^{4} - 9669 \nu^{2} - 38034$$$$)/3216$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{6} + 320 \nu^{4} + 8535 \nu^{2} + 1608 \nu + 31182$$$$)/1608$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} + 320 \nu^{5} + 10143 \nu^{3} + 93894 \nu$$$$)/1608$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 196 \nu^{4} + 10617 \nu^{2} + 107946$$$$)/3216$$ $$\beta_{7}$$ $$=$$ $$($$$$-2 \nu^{7} - 191 \nu^{5} - 4551 \nu^{3} - 16500 \nu$$$$)/402$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + \beta_{1} - 52$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} + 2 \beta_{5} - 4 \beta_{2} - 43 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-102 \beta_{6} + 162 \beta_{4} + 174 \beta_{3} - 81 \beta_{1} + 2340$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-162 \beta_{7} - 102 \beta_{5} + 396 \beta_{2} + 2193 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$5190 \beta_{6} - 10518 \beta_{4} - 12870 \beta_{3} + 5259 \beta_{1} - 122448$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$10518 \beta_{7} + 5190 \beta_{5} - 28716 \beta_{2} - 119835 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$133$$ $$145$$ $$353$$ $$463$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
241.1
 7.70102i − 7.70102i 3.00247i − 3.00247i − 5.58567i 5.58567i 1.57474i − 1.57474i
0 −5.19615 0 −12.5296 0 63.6540i 0 27.0000 0
241.2 0 −5.19615 0 −12.5296 0 63.6540i 0 27.0000 0
241.3 0 −5.19615 0 8.72578 0 1.45810i 0 27.0000 0
241.4 0 −5.19615 0 8.72578 0 1.45810i 0 27.0000 0
241.5 0 5.19615 0 −29.7487 0 12.9420i 0 27.0000 0
241.6 0 5.19615 0 −29.7487 0 12.9420i 0 27.0000 0
241.7 0 5.19615 0 15.5526 0 93.8006i 0 27.0000 0
241.8 0 5.19615 0 15.5526 0 93.8006i 0 27.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 241.8 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.5.j.a 8
4.b odd 2 1 33.5.c.a 8
11.b odd 2 1 inner 528.5.j.a 8
12.b even 2 1 99.5.c.c 8
44.c even 2 1 33.5.c.a 8
132.d odd 2 1 99.5.c.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.5.c.a 8 4.b odd 2 1
33.5.c.a 8 44.c even 2 1
99.5.c.c 8 12.b even 2 1
99.5.c.c 8 132.d odd 2 1
528.5.j.a 8 1.a even 1 1 trivial
528.5.j.a 8 11.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 18 T_{5}^{3} - 518 T_{5}^{2} - 3312 T_{5} + 50584$$ acting on $$S_{5}^{\mathrm{new}}(528, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -27 + T^{2} )^{4}$$
$5$ $$( 50584 - 3312 T - 518 T^{2} + 18 T^{3} + T^{4} )^{2}$$
$7$ $$12695273424 + 6051633840 T^{2} + 37830264 T^{4} + 13020 T^{6} + T^{8}$$
$11$ $$45949729863572161 + 112983421561956 T - 1032352370896 T^{2} + 16417538940 T^{3} + 160159230 T^{4} + 1121340 T^{5} - 4816 T^{6} + 36 T^{7} + T^{8}$$
$13$ $$1164901648483584 + 2284214269248 T^{2} + 1142069460 T^{4} + 66480 T^{6} + T^{8}$$
$17$ $$5763713931173964096 + 770067814668192 T^{2} + 27408756564 T^{4} + 326700 T^{6} + T^{8}$$
$19$ $$1148548870541318976 + 433340712892512 T^{2} + 42562524276 T^{4} + 467172 T^{6} + T^{8}$$
$23$ $$( 2066510872 - 154896312 T - 458558 T^{2} + 258 T^{3} + T^{4} )^{2}$$
$29$ $$32\!\cdots\!84$$$$+ 668949031877815296 T^{2} + 3572518482852 T^{4} + 3652884 T^{6} + T^{8}$$
$31$ $$( 6236194624 - 171603616 T - 14412 T^{2} + 1376 T^{3} + T^{4} )^{2}$$
$37$ $$( 26585618752 - 398290592 T + 1748388 T^{2} - 2648 T^{3} + T^{4} )^{2}$$
$41$ $$10\!\cdots\!76$$$$+$$$$19\!\cdots\!24$$$$T^{2} + 107439635858532 T^{4} + 18633108 T^{6} + T^{8}$$
$43$ $$25\!\cdots\!84$$$$+ 27608350007521242720 T^{2} + 26793750661620 T^{4} + 8964516 T^{6} + T^{8}$$
$47$ $$( 12403328125408 - 291605376 T - 7278002 T^{2} + 210 T^{3} + T^{4} )^{2}$$
$53$ $$( -15377555198312 + 31141326432 T - 14617718 T^{2} - 1770 T^{3} + T^{4} )^{2}$$
$59$ $$( -39877809797408 + 30362839632 T + 7759660 T^{2} - 8316 T^{3} + T^{4} )^{2}$$
$61$ $$11\!\cdots\!36$$$$+$$$$94\!\cdots\!12$$$$T^{2} + 1388503615694388 T^{4} + 65474352 T^{6} + T^{8}$$
$67$ $$( -192515994273728 + 221754664832 T - 53637708 T^{2} - 1828 T^{3} + T^{4} )^{2}$$
$71$ $$( -855560872809632 + 445427720832 T - 46867514 T^{2} - 6606 T^{3} + T^{4} )^{2}$$
$73$ $$16\!\cdots\!84$$$$+$$$$86\!\cdots\!80$$$$T^{2} + 1203338659893120 T^{4} + 62146512 T^{6} + T^{8}$$
$79$ $$28\!\cdots\!84$$$$+$$$$16\!\cdots\!20$$$$T^{2} + 35826082579997880 T^{4} + 318678684 T^{6} + T^{8}$$
$83$ $$76\!\cdots\!44$$$$+$$$$30\!\cdots\!48$$$$T^{2} + 16776000917143872 T^{4} + 244092288 T^{6} + T^{8}$$
$89$ $$( -1270313049471344 + 705522821424 T - 69427640 T^{2} - 7764 T^{3} + T^{4} )^{2}$$
$97$ $$( -466619407059968 + 494162140288 T - 121159848 T^{2} - 3812 T^{3} + T^{4} )^{2}$$