# Properties

 Label 525.4.a.t Level $525$ Weight $4$ Character orbit 525.a Self dual yes Analytic conductor $30.976$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.9760027530$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 24 x^{2} - 3 x + 46$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -3 q^{3} + ( 4 + \beta_{2} ) q^{4} + 3 \beta_{1} q^{6} -7 q^{7} + ( -2 - 4 \beta_{1} - \beta_{3} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} -3 q^{3} + ( 4 + \beta_{2} ) q^{4} + 3 \beta_{1} q^{6} -7 q^{7} + ( -2 - 4 \beta_{1} - \beta_{3} ) q^{8} + 9 q^{9} + ( 5 + 5 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{11} + ( -12 - 3 \beta_{2} ) q^{12} + ( -2 - 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{13} + 7 \beta_{1} q^{14} + ( 18 + 3 \beta_{1} ) q^{16} + ( -24 + 7 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{17} -9 \beta_{1} q^{18} + ( 18 + 6 \beta_{1} + 4 \beta_{2} ) q^{19} + 21 q^{21} + ( -56 + 18 \beta_{1} - 9 \beta_{2} + 3 \beta_{3} ) q^{22} + ( -25 - 10 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{23} + ( 6 + 12 \beta_{1} + 3 \beta_{3} ) q^{24} + ( 32 + 7 \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{26} -27 q^{27} + ( -28 - 7 \beta_{2} ) q^{28} + ( -59 + 30 \beta_{1} - 4 \beta_{3} ) q^{29} + ( 90 - 9 \beta_{1} + 3 \beta_{2} - 9 \beta_{3} ) q^{31} + ( -20 + 14 \beta_{1} - 3 \beta_{2} + 8 \beta_{3} ) q^{32} + ( -15 - 15 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{33} + ( -72 + 51 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{34} + ( 36 + 9 \beta_{2} ) q^{36} + ( -99 + 69 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{37} + ( -80 - 50 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{38} + ( 6 + 9 \beta_{1} + 3 \beta_{2} - 9 \beta_{3} ) q^{39} + ( 96 - 7 \beta_{1} - 15 \beta_{2} - 3 \beta_{3} ) q^{41} -21 \beta_{1} q^{42} + ( -115 + 6 \beta_{1} + 28 \beta_{2} ) q^{43} + ( -244 + 85 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{44} + ( 136 + 75 \beta_{1} + 18 \beta_{2} + 6 \beta_{3} ) q^{46} + ( -14 + 30 \beta_{1} + 36 \beta_{2} - 4 \beta_{3} ) q^{47} + ( -54 - 9 \beta_{1} ) q^{48} + 49 q^{49} + ( 72 - 21 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{51} + ( -52 + 63 \beta_{1} - 3 \beta_{2} - 15 \beta_{3} ) q^{52} + ( -222 - 17 \beta_{1} + 3 \beta_{2} + 15 \beta_{3} ) q^{53} + 27 \beta_{1} q^{54} + ( 14 + 28 \beta_{1} + 7 \beta_{3} ) q^{56} + ( -54 - 18 \beta_{1} - 12 \beta_{2} ) q^{57} + ( -352 + 63 \beta_{1} - 14 \beta_{2} ) q^{58} + ( -218 + 3 \beta_{1} - 27 \beta_{2} + 17 \beta_{3} ) q^{59} + ( 168 + 87 \beta_{1} + 3 \beta_{2} + 15 \beta_{3} ) q^{61} + ( 120 - 105 \beta_{1} + 45 \beta_{2} - 3 \beta_{3} ) q^{62} -63 q^{63} + ( -322 + 12 \beta_{1} - 46 \beta_{2} + 3 \beta_{3} ) q^{64} + ( 168 - 54 \beta_{1} + 27 \beta_{2} - 9 \beta_{3} ) q^{66} + ( 119 + 105 \beta_{1} - 17 \beta_{2} + 27 \beta_{3} ) q^{67} + ( -436 - 27 \beta_{1} - 39 \beta_{2} + 19 \beta_{3} ) q^{68} + ( 75 + 30 \beta_{1} + 18 \beta_{2} + 6 \beta_{3} ) q^{69} + ( -167 - 45 \beta_{1} - 21 \beta_{2} - 13 \beta_{3} ) q^{71} + ( -18 - 36 \beta_{1} - 9 \beta_{3} ) q^{72} + ( -304 - 162 \beta_{1} + 68 \beta_{2} - 12 \beta_{3} ) q^{73} + ( -840 + 30 \beta_{1} - 57 \beta_{2} - 9 \beta_{3} ) q^{74} + ( 476 + 84 \beta_{1} + 34 \beta_{2} + 6 \beta_{3} ) q^{76} + ( -35 - 35 \beta_{1} + 21 \beta_{2} - 7 \beta_{3} ) q^{77} + ( -96 - 21 \beta_{1} + 27 \beta_{2} - 3 \beta_{3} ) q^{78} + ( 87 - 153 \beta_{1} + 37 \beta_{2} - 3 \beta_{3} ) q^{79} + 81 q^{81} + ( 120 + 27 \beta_{1} + 19 \beta_{2} + 15 \beta_{3} ) q^{82} + ( -370 + 51 \beta_{1} - 33 \beta_{2} - 29 \beta_{3} ) q^{83} + ( 84 + 21 \beta_{2} ) q^{84} + ( -128 - 109 \beta_{1} - 6 \beta_{2} - 28 \beta_{3} ) q^{86} + ( 177 - 90 \beta_{1} + 12 \beta_{3} ) q^{87} + ( -562 + 147 \beta_{1} - 17 \beta_{2} - 18 \beta_{3} ) q^{88} + ( -46 - 140 \beta_{1} + 78 \beta_{2} - 14 \beta_{3} ) q^{89} + ( 14 + 21 \beta_{1} + 7 \beta_{2} - 21 \beta_{3} ) q^{91} + ( -748 - 206 \beta_{1} - 51 \beta_{2} - 2 \beta_{3} ) q^{92} + ( -270 + 27 \beta_{1} - 9 \beta_{2} + 27 \beta_{3} ) q^{93} + ( -424 - 270 \beta_{1} - 14 \beta_{2} - 36 \beta_{3} ) q^{94} + ( 60 - 42 \beta_{1} + 9 \beta_{2} - 24 \beta_{3} ) q^{96} + ( 102 + 168 \beta_{1} - 28 \beta_{2} - 36 \beta_{3} ) q^{97} -49 \beta_{1} q^{98} + ( 45 + 45 \beta_{1} - 27 \beta_{2} + 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{3} + 16q^{4} - 28q^{7} - 9q^{8} + 36q^{9} + O(q^{10})$$ $$4q - 12q^{3} + 16q^{4} - 28q^{7} - 9q^{8} + 36q^{9} + 21q^{11} - 48q^{12} - 5q^{13} + 72q^{16} - 99q^{17} + 72q^{19} + 84q^{21} - 221q^{22} - 102q^{23} + 27q^{24} + 129q^{26} - 108q^{27} - 112q^{28} - 240q^{29} + 351q^{31} - 72q^{32} - 63q^{33} - 285q^{34} + 144q^{36} - 399q^{37} - 324q^{38} + 15q^{39} + 381q^{41} - 460q^{43} - 975q^{44} + 550q^{46} - 60q^{47} - 216q^{48} + 196q^{49} + 297q^{51} - 223q^{52} - 873q^{53} + 63q^{56} - 216q^{57} - 1408q^{58} - 855q^{59} + 687q^{61} + 477q^{62} - 252q^{63} - 1285q^{64} + 663q^{66} + 503q^{67} - 1725q^{68} + 306q^{69} - 681q^{71} - 81q^{72} - 1228q^{73} - 3369q^{74} + 1910q^{76} - 147q^{77} - 387q^{78} + 345q^{79} + 324q^{81} + 495q^{82} - 1509q^{83} + 336q^{84} - 540q^{86} + 720q^{87} - 2266q^{88} - 198q^{89} + 35q^{91} - 2994q^{92} - 1053q^{93} - 1732q^{94} + 216q^{96} + 372q^{97} + 189q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 24 x^{2} - 3 x + 46$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 12$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 20 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 12$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 20 \beta_{1} + 2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 4.75345 1.37627 −1.52801 −4.60171
−4.75345 −3.00000 14.5953 0 14.2604 −7.00000 −31.3504 9.00000 0
1.2 −1.37627 −3.00000 −6.10588 0 4.12881 −7.00000 19.4135 9.00000 0
1.3 1.52801 −3.00000 −5.66519 0 −4.58402 −7.00000 −20.8805 9.00000 0
1.4 4.60171 −3.00000 13.1758 0 −13.8051 −7.00000 23.8174 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.t 4 1.a even 1 1 trivial
525.4.a.u yes 4 5.b even 2 1
525.4.d.n 8 5.c odd 4 2
1575.4.a.bj 4 3.b odd 2 1
1575.4.a.bk 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2}^{4} - 24 T_{2}^{2} + 3 T_{2} + 46$$ $$T_{11}^{4} - 21 T_{11}^{3} - 2643 T_{11}^{2} + 61617 T_{11} - 304010$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T^{2} + 3 T^{3} + 46 T^{4} + 24 T^{5} + 512 T^{6} + 4096 T^{8}$$
$3$ $$( 1 + 3 T )^{4}$$
$5$ 1
$7$ $$( 1 + 7 T )^{4}$$
$11$ $$1 - 21 T + 2681 T^{2} - 22236 T^{3} + 3289690 T^{4} - 29596116 T^{5} + 4749555041 T^{6} - 49516901511 T^{7} + 3138428376721 T^{8}$$
$13$ $$1 + 5 T + 1848 T^{2} - 69653 T^{3} - 988138 T^{4} - 153027641 T^{5} + 8919943032 T^{6} + 53022496865 T^{7} + 23298085122481 T^{8}$$
$17$ $$1 + 99 T + 13298 T^{2} + 508917 T^{3} + 56706418 T^{4} + 2500309221 T^{5} + 320981392562 T^{6} + 11740199773203 T^{7} + 582622237229761 T^{8}$$
$19$ $$1 - 72 T + 25164 T^{2} - 1424592 T^{3} + 252125750 T^{4} - 9771276528 T^{5} + 1183862549484 T^{6} - 23233514240088 T^{7} + 2213314919066161 T^{8}$$
$23$ $$1 + 102 T + 38528 T^{2} + 3477336 T^{3} + 641077657 T^{4} + 42308747112 T^{5} + 5703526731392 T^{6} + 183717571469226 T^{7} + 21914624432020321 T^{8}$$
$29$ $$1 + 240 T + 86430 T^{2} + 14441664 T^{3} + 3156579983 T^{4} + 352217743296 T^{5} + 51410579634030 T^{6} + 3481715034208560 T^{7} + 353814783205469041 T^{8}$$
$31$ $$1 - 351 T + 102010 T^{2} - 21726711 T^{3} + 4440563682 T^{4} - 647260447401 T^{5} + 90534250498810 T^{6} - 9280307378395521 T^{7} + 787662783788549761 T^{8}$$
$37$ $$1 + 399 T + 123385 T^{2} + 25924734 T^{3} + 6891314910 T^{4} + 1313165551302 T^{5} + 316572152974465 T^{6} + 51854734178235723 T^{7} + 6582952005840035281 T^{8}$$
$41$ $$1 - 381 T + 273692 T^{2} - 71526783 T^{3} + 28134293782 T^{4} - 4929697411143 T^{5} + 1300065529927772 T^{6} - 124732517004099141 T^{7} + 22563490300366186081 T^{8}$$
$43$ $$1 + 460 T + 241338 T^{2} + 78643232 T^{3} + 29430041627 T^{4} + 6252687446624 T^{5} + 1525585115519562 T^{6} + 231192601490947780 T^{7} + 39959630797262576401 T^{8}$$
$47$ $$1 + 60 T + 130548 T^{2} + 11848116 T^{3} + 16872903254 T^{4} + 1230106947468 T^{5} + 1407205002770292 T^{6} + 67147828386166020 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 + 873 T + 701906 T^{2} + 368908875 T^{3} + 161980949146 T^{4} + 54922046583375 T^{5} + 15557298062611874 T^{6} + 2880693615643262109 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$1 + 855 T + 767616 T^{2} + 344685519 T^{3} + 195821792246 T^{4} + 70791167206701 T^{5} + 32378452511369856 T^{6} + 7406861424949972845 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$
$61$ $$1 - 687 T + 715504 T^{2} - 385624593 T^{3} + 235308406830 T^{4} - 87529455743733 T^{5} + 36863033936792944 T^{6} - 8033878365777054867 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$
$67$ $$1 - 503 T + 452451 T^{2} - 193160692 T^{3} + 211333484900 T^{4} - 58095589207996 T^{5} + 40927985470746219 T^{6} - 13684886801336358341 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$1 + 681 T + 1310019 T^{2} + 668258544 T^{3} + 678493576820 T^{4} + 239177083741584 T^{5} + 167813805841904499 T^{6} + 31222828989263790111 T^{7} +$$$$16\!\cdots\!41$$$$T^{8}$$
$73$ $$1 + 1228 T + 618972 T^{2} - 228294532 T^{3} - 319091414218 T^{4} - 88810453955044 T^{5} + 93671648714554908 T^{6} + 72294308477752997164 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$1 - 345 T + 1235121 T^{2} - 721148448 T^{3} + 716751093026 T^{4} - 355554309653472 T^{5} + 300242421150553041 T^{6} - 41348800614003320055 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$
$83$ $$1 + 1509 T + 2187990 T^{2} + 1588008513 T^{3} + 1474125264218 T^{4} + 908002623622731 T^{5} + 715342267527638310 T^{6} +$$$$28\!\cdots\!27$$$$T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$1 + 198 T + 1184884 T^{2} - 492558966 T^{3} + 650980125750 T^{4} - 347238801702054 T^{5} + 588865179959033524 T^{6} + 69370567934082071382 T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$
$97$ $$1 - 372 T + 1910724 T^{2} - 1429297356 T^{3} + 2012229372278 T^{4} - 1304481105792588 T^{5} + 1591579601145958596 T^{6} -$$$$28\!\cdots\!24$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$