# Properties

 Label 525.4.d.n Level $525$ Weight $4$ Character orbit 525.d Analytic conductor $30.976$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 48 x^{6} + 668 x^{4} + 2217 x^{2} + 2116$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -3 \beta_{3} q^{3} + ( -4 + \beta_{2} ) q^{4} + 3 \beta_{4} q^{6} + 7 \beta_{3} q^{7} + ( -4 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -3 \beta_{3} q^{3} + ( -4 + \beta_{2} ) q^{4} + 3 \beta_{4} q^{6} + 7 \beta_{3} q^{7} + ( -4 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{8} -9 q^{9} + ( 5 + 3 \beta_{2} + 5 \beta_{4} - \beta_{7} ) q^{11} + ( 12 \beta_{3} - 3 \beta_{6} ) q^{12} + ( -3 \beta_{1} - 2 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{13} -7 \beta_{4} q^{14} + ( 18 + 3 \beta_{4} ) q^{16} + ( -7 \beta_{1} + 24 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} ) q^{17} -9 \beta_{1} q^{18} + ( -18 + 4 \beta_{2} - 6 \beta_{4} ) q^{19} + 21 q^{21} + ( -18 \beta_{1} + 56 \beta_{3} + 3 \beta_{5} - 9 \beta_{6} ) q^{22} + ( -10 \beta_{1} - 25 \beta_{3} + 2 \beta_{5} + 6 \beta_{6} ) q^{23} + ( -6 - 12 \beta_{4} + 3 \beta_{7} ) q^{24} + ( 32 + 9 \beta_{2} + 7 \beta_{4} - \beta_{7} ) q^{26} + 27 \beta_{3} q^{27} + ( -28 \beta_{3} + 7 \beta_{6} ) q^{28} + ( 59 - 30 \beta_{4} - 4 \beta_{7} ) q^{29} + ( 90 - 3 \beta_{2} - 9 \beta_{4} + 9 \beta_{7} ) q^{31} + ( -14 \beta_{1} + 20 \beta_{3} + 8 \beta_{5} - 3 \beta_{6} ) q^{32} + ( -15 \beta_{1} - 15 \beta_{3} + 3 \beta_{5} - 9 \beta_{6} ) q^{33} + ( 72 + 5 \beta_{2} - 51 \beta_{4} + 3 \beta_{7} ) q^{34} + ( 36 - 9 \beta_{2} ) q^{36} + ( -69 \beta_{1} + 99 \beta_{3} - 3 \beta_{5} + 9 \beta_{6} ) q^{37} + ( -50 \beta_{1} - 80 \beta_{3} + 4 \beta_{5} + 6 \beta_{6} ) q^{38} + ( -6 + 3 \beta_{2} - 9 \beta_{4} - 9 \beta_{7} ) q^{39} + ( 96 + 15 \beta_{2} - 7 \beta_{4} + 3 \beta_{7} ) q^{41} + 21 \beta_{1} q^{42} + ( 6 \beta_{1} - 115 \beta_{3} - 28 \beta_{6} ) q^{43} + ( 244 - 6 \beta_{2} - 85 \beta_{4} + \beta_{7} ) q^{44} + ( 136 - 18 \beta_{2} + 75 \beta_{4} - 6 \beta_{7} ) q^{46} + ( -30 \beta_{1} + 14 \beta_{3} - 4 \beta_{5} + 36 \beta_{6} ) q^{47} + ( -9 \beta_{1} - 54 \beta_{3} ) q^{48} -49 q^{49} + ( 72 - 9 \beta_{2} - 21 \beta_{4} - 9 \beta_{7} ) q^{51} + ( -63 \beta_{1} + 52 \beta_{3} - 15 \beta_{5} - 3 \beta_{6} ) q^{52} + ( -17 \beta_{1} - 222 \beta_{3} - 15 \beta_{5} - 3 \beta_{6} ) q^{53} -27 \beta_{4} q^{54} + ( 14 + 28 \beta_{4} - 7 \beta_{7} ) q^{56} + ( 18 \beta_{1} + 54 \beta_{3} - 12 \beta_{6} ) q^{57} + ( 63 \beta_{1} - 352 \beta_{3} + 14 \beta_{6} ) q^{58} + ( 218 - 27 \beta_{2} - 3 \beta_{4} + 17 \beta_{7} ) q^{59} + ( 168 - 3 \beta_{2} + 87 \beta_{4} - 15 \beta_{7} ) q^{61} + ( 105 \beta_{1} - 120 \beta_{3} - 3 \beta_{5} + 45 \beta_{6} ) q^{62} -63 \beta_{3} q^{63} + ( 322 - 46 \beta_{2} - 12 \beta_{4} + 3 \beta_{7} ) q^{64} + ( 168 - 27 \beta_{2} - 54 \beta_{4} + 9 \beta_{7} ) q^{66} + ( -105 \beta_{1} - 119 \beta_{3} + 27 \beta_{5} - 17 \beta_{6} ) q^{67} + ( -27 \beta_{1} - 436 \beta_{3} - 19 \beta_{5} + 39 \beta_{6} ) q^{68} + ( -75 + 18 \beta_{2} - 30 \beta_{4} + 6 \beta_{7} ) q^{69} + ( -167 + 21 \beta_{2} - 45 \beta_{4} + 13 \beta_{7} ) q^{71} + ( 36 \beta_{1} + 18 \beta_{3} - 9 \beta_{5} ) q^{72} + ( -162 \beta_{1} - 304 \beta_{3} + 12 \beta_{5} - 68 \beta_{6} ) q^{73} + ( 840 - 57 \beta_{2} - 30 \beta_{4} - 9 \beta_{7} ) q^{74} + ( 476 - 34 \beta_{2} + 84 \beta_{4} - 6 \beta_{7} ) q^{76} + ( 35 \beta_{1} + 35 \beta_{3} - 7 \beta_{5} + 21 \beta_{6} ) q^{77} + ( -21 \beta_{1} - 96 \beta_{3} + 3 \beta_{5} - 27 \beta_{6} ) q^{78} + ( -87 + 37 \beta_{2} + 153 \beta_{4} - 3 \beta_{7} ) q^{79} + 81 q^{81} + ( -27 \beta_{1} - 120 \beta_{3} + 15 \beta_{5} + 19 \beta_{6} ) q^{82} + ( 51 \beta_{1} - 370 \beta_{3} + 29 \beta_{5} + 33 \beta_{6} ) q^{83} + ( -84 + 21 \beta_{2} ) q^{84} + ( -128 + 6 \beta_{2} - 109 \beta_{4} + 28 \beta_{7} ) q^{86} + ( 90 \beta_{1} - 177 \beta_{3} + 12 \beta_{5} ) q^{87} + ( 147 \beta_{1} - 562 \beta_{3} + 18 \beta_{5} + 17 \beta_{6} ) q^{88} + ( 46 + 78 \beta_{2} + 140 \beta_{4} - 14 \beta_{7} ) q^{89} + ( 14 - 7 \beta_{2} + 21 \beta_{4} + 21 \beta_{7} ) q^{91} + ( 206 \beta_{1} + 748 \beta_{3} - 2 \beta_{5} - 51 \beta_{6} ) q^{92} + ( 27 \beta_{1} - 270 \beta_{3} - 27 \beta_{5} + 9 \beta_{6} ) q^{93} + ( 424 - 14 \beta_{2} + 270 \beta_{4} - 36 \beta_{7} ) q^{94} + ( 60 - 9 \beta_{2} - 42 \beta_{4} + 24 \beta_{7} ) q^{96} + ( -168 \beta_{1} - 102 \beta_{3} - 36 \beta_{5} - 28 \beta_{6} ) q^{97} -49 \beta_{1} q^{98} + ( -45 - 27 \beta_{2} - 45 \beta_{4} + 9 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 32q^{4} - 72q^{9} + O(q^{10})$$ $$8q - 32q^{4} - 72q^{9} + 42q^{11} + 144q^{16} - 144q^{19} + 168q^{21} - 54q^{24} + 258q^{26} + 480q^{29} + 702q^{31} + 570q^{34} + 288q^{36} - 30q^{39} + 762q^{41} + 1950q^{44} + 1100q^{46} - 392q^{49} + 594q^{51} + 126q^{56} + 1710q^{59} + 1374q^{61} + 2570q^{64} + 1326q^{66} - 612q^{69} - 1362q^{71} + 6738q^{74} + 3820q^{76} - 690q^{79} + 648q^{81} - 672q^{84} - 1080q^{86} + 396q^{89} + 70q^{91} + 3464q^{94} + 432q^{96} - 378q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 48 x^{6} + 668 x^{4} + 2217 x^{2} + 2116$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 12$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 48 \nu^{5} + 622 \nu^{3} + 1113 \nu$$$$)/138$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 24 \nu^{2} + 46$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 48 \nu^{5} + 691 \nu^{3} + 2493 \nu$$$$)/69$$ $$\beta_{6}$$ $$=$$ $$($$$$6 \nu^{7} + 265 \nu^{5} + 3180 \nu^{3} + 5620 \nu$$$$)/69$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 44 \nu^{4} + 526 \nu^{2} + 926$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 12$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - 2 \beta_{3} - 20 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{4} - 24 \beta_{2} + 242$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{6} - 24 \beta_{5} + 84 \beta_{3} + 434 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{7} - 132 \beta_{4} + 530 \beta_{2} - 5262$$ $$\nu^{7}$$ $$=$$ $$144 \beta_{6} + 530 \beta_{5} - 2650 \beta_{3} - 9505 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 − 4.75345i − 4.60171i − 1.52801i − 1.37627i 1.37627i 1.52801i 4.60171i 4.75345i
4.75345i 3.00000i −14.5953 0 14.2604 7.00000i 31.3504i −9.00000 0
274.2 4.60171i 3.00000i −13.1758 0 −13.8051 7.00000i 23.8174i −9.00000 0
274.3 1.52801i 3.00000i 5.66519 0 −4.58402 7.00000i 20.8805i −9.00000 0
274.4 1.37627i 3.00000i 6.10588 0 4.12881 7.00000i 19.4135i −9.00000 0
274.5 1.37627i 3.00000i 6.10588 0 4.12881 7.00000i 19.4135i −9.00000 0
274.6 1.52801i 3.00000i 5.66519 0 −4.58402 7.00000i 20.8805i −9.00000 0
274.7 4.60171i 3.00000i −13.1758 0 −13.8051 7.00000i 23.8174i −9.00000 0
274.8 4.75345i 3.00000i −14.5953 0 14.2604 7.00000i 31.3504i −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 274.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.4.d.n 8
5.b even 2 1 inner 525.4.d.n 8
5.c odd 4 1 525.4.a.t 4
5.c odd 4 1 525.4.a.u yes 4
15.e even 4 1 1575.4.a.bj 4
15.e even 4 1 1575.4.a.bk 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.t 4 5.c odd 4 1
525.4.a.u yes 4 5.c odd 4 1
525.4.d.n 8 1.a even 1 1 trivial
525.4.d.n 8 5.b even 2 1 inner
1575.4.a.bj 4 15.e even 4 1
1575.4.a.bk 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{8} + 48 T_{2}^{6} + 668 T_{2}^{4} + 2217 T_{2}^{2} + 2116$$ $$T_{11}^{4} - 21 T_{11}^{3} - 2643 T_{11}^{2} + 61617 T_{11} - 304010$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 16 T^{2} + 156 T^{4} - 1751 T^{6} + 18356 T^{8} - 112064 T^{10} + 638976 T^{12} - 4194304 T^{14} + 16777216 T^{16}$$
$3$ $$( 1 + 9 T^{2} )^{4}$$
$5$ 1
$7$ $$( 1 + 49 T^{2} )^{4}$$
$11$ $$( 1 - 21 T + 2681 T^{2} - 22236 T^{3} + 3289690 T^{4} - 29596116 T^{5} + 4749555041 T^{6} - 49516901511 T^{7} + 3138428376721 T^{8} )^{2}$$
$13$ $$1 - 3671 T^{2} + 2135358 T^{4} - 10866464017 T^{6} + 58692802872482 T^{8} - 52450346315431753 T^{10} + 49749752450970783198 T^{12} -$$$$41\!\cdots\!59$$$$T^{14} +$$$$54\!\cdots\!61$$$$T^{16}$$
$17$ $$1 - 16795 T^{2} + 189484074 T^{4} - 1396068939605 T^{6} + 8048224142687690 T^{8} - 33697710358472520245 T^{10} +$$$$11\!\cdots\!14$$$$T^{12} -$$$$23\!\cdots\!55$$$$T^{14} +$$$$33\!\cdots\!21$$$$T^{16}$$
$19$ $$( 1 + 72 T + 25164 T^{2} + 1424592 T^{3} + 252125750 T^{4} + 9771276528 T^{5} + 1183862549484 T^{6} + 23233514240088 T^{7} + 2213314919066161 T^{8} )^{2}$$
$23$ $$1 - 66652 T^{2} + 2057185554 T^{4} - 40083083332832 T^{6} + 562578923508162875 T^{8} -$$$$59\!\cdots\!48$$$$T^{10} +$$$$45\!\cdots\!34$$$$T^{12} -$$$$21\!\cdots\!88$$$$T^{14} +$$$$48\!\cdots\!41$$$$T^{16}$$
$29$ $$( 1 - 240 T + 86430 T^{2} - 14441664 T^{3} + 3156579983 T^{4} - 352217743296 T^{5} + 51410579634030 T^{6} - 3481715034208560 T^{7} + 353814783205469041 T^{8} )^{2}$$
$31$ $$( 1 - 351 T + 102010 T^{2} - 21726711 T^{3} + 4440563682 T^{4} - 647260447401 T^{5} + 90534250498810 T^{6} - 9280307378395521 T^{7} + 787662783788549761 T^{8} )^{2}$$
$37$ $$1 - 87569 T^{2} + 8318550313 T^{4} - 613716143379878 T^{6} + 29309622284809592422 T^{8} -$$$$15\!\cdots\!02$$$$T^{10} +$$$$54\!\cdots\!53$$$$T^{12} -$$$$14\!\cdots\!01$$$$T^{14} +$$$$43\!\cdots\!61$$$$T^{16}$$
$41$ $$( 1 - 381 T + 273692 T^{2} - 71526783 T^{3} + 28134293782 T^{4} - 4929697411143 T^{5} + 1300065529927772 T^{6} - 124732517004099141 T^{7} + 22563490300366186081 T^{8} )^{2}$$
$43$ $$1 - 271076 T^{2} + 44752340058 T^{4} - 5319114613093072 T^{6} +$$$$48\!\cdots\!07$$$$T^{8} -$$$$33\!\cdots\!28$$$$T^{10} +$$$$17\!\cdots\!58$$$$T^{12} -$$$$68\!\cdots\!24$$$$T^{14} +$$$$15\!\cdots\!01$$$$T^{16}$$
$47$ $$1 - 257496 T^{2} + 49366812892 T^{4} - 6931866867101352 T^{6} +$$$$84\!\cdots\!54$$$$T^{8} -$$$$74\!\cdots\!08$$$$T^{10} +$$$$57\!\cdots\!72$$$$T^{12} -$$$$32\!\cdots\!44$$$$T^{14} +$$$$13\!\cdots\!81$$$$T^{16}$$
$53$ $$1 - 641683 T^{2} + 172519035378 T^{4} - 26517744919429925 T^{6} +$$$$35\!\cdots\!22$$$$T^{8} -$$$$58\!\cdots\!25$$$$T^{10} +$$$$84\!\cdots\!98$$$$T^{12} -$$$$69\!\cdots\!87$$$$T^{14} +$$$$24\!\cdots\!81$$$$T^{16}$$
$59$ $$( 1 - 855 T + 767616 T^{2} - 344685519 T^{3} + 195821792246 T^{4} - 70791167206701 T^{5} + 32378452511369856 T^{6} - 7406861424949972845 T^{7} +$$$$17\!\cdots\!81$$$$T^{8} )^{2}$$
$61$ $$( 1 - 687 T + 715504 T^{2} - 385624593 T^{3} + 235308406830 T^{4} - 87529455743733 T^{5} + 36863033936792944 T^{6} - 8033878365777054867 T^{7} +$$$$26\!\cdots\!21$$$$T^{8} )^{2}$$
$67$ $$1 - 651893 T^{2} + 433059221049 T^{4} - 177336848417309398 T^{6} +$$$$61\!\cdots\!50$$$$T^{8} -$$$$16\!\cdots\!62$$$$T^{10} +$$$$35\!\cdots\!89$$$$T^{12} -$$$$48\!\cdots\!37$$$$T^{14} +$$$$66\!\cdots\!21$$$$T^{16}$$
$71$ $$( 1 + 681 T + 1310019 T^{2} + 668258544 T^{3} + 678493576820 T^{4} + 239177083741584 T^{5} + 167813805841904499 T^{6} + 31222828989263790111 T^{7} +$$$$16\!\cdots\!41$$$$T^{8} )^{2}$$
$73$ $$1 + 270040 T^{2} + 305634878940 T^{4} + 41673922681088936 T^{6} +$$$$45\!\cdots\!18$$$$T^{8} +$$$$63\!\cdots\!04$$$$T^{10} +$$$$69\!\cdots\!40$$$$T^{12} +$$$$93\!\cdots\!60$$$$T^{14} +$$$$52\!\cdots\!41$$$$T^{16}$$
$79$ $$( 1 + 345 T + 1235121 T^{2} + 721148448 T^{3} + 716751093026 T^{4} + 355554309653472 T^{5} + 300242421150553041 T^{6} + 41348800614003320055 T^{7} +$$$$59\!\cdots\!41$$$$T^{8} )^{2}$$
$83$ $$1 - 2098899 T^{2} + 2942941076302 T^{4} - 2619304253314086933 T^{6} +$$$$17\!\cdots\!54$$$$T^{8} -$$$$85\!\cdots\!77$$$$T^{10} +$$$$31\!\cdots\!22$$$$T^{12} -$$$$73\!\cdots\!91$$$$T^{14} +$$$$11\!\cdots\!21$$$$T^{16}$$
$89$ $$( 1 - 198 T + 1184884 T^{2} + 492558966 T^{3} + 650980125750 T^{4} + 347238801702054 T^{5} + 588865179959033524 T^{6} - 69370567934082071382 T^{7} +$$$$24\!\cdots\!21$$$$T^{8} )^{2}$$
$97$ $$1 - 3683064 T^{2} + 6611927715868 T^{4} - 7859364237946659528 T^{6} +$$$$75\!\cdots\!62$$$$T^{8} -$$$$65\!\cdots\!12$$$$T^{10} +$$$$45\!\cdots\!88$$$$T^{12} -$$$$21\!\cdots\!96$$$$T^{14} +$$$$48\!\cdots\!81$$$$T^{16}$$