# Properties

 Label 525.4.a.s 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: 4.4.26729725.1 Defining polynomial: $$x^{4} - 2 x^{3} - 18 x^{2} + 19 x + 44$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{2} -3 q^{3} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( 3 + 3 \beta_{1} ) q^{6} + 7 q^{7} + ( -23 - 5 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{2} -3 q^{3} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( 3 + 3 \beta_{1} ) q^{6} + 7 q^{7} + ( -23 - 5 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{8} + 9 q^{9} + ( 18 - 3 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{11} + ( -9 - 9 \beta_{1} - 3 \beta_{2} ) q^{12} + ( -11 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{13} + ( -7 - 7 \beta_{1} ) q^{14} + ( 45 + 27 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{16} + ( -17 - 17 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{17} + ( -9 - 9 \beta_{1} ) q^{18} + ( -16 + 6 \beta_{1} - 20 \beta_{2} ) q^{19} -21 q^{21} + ( 26 - 30 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{22} + ( -39 + 4 \beta_{2} - 8 \beta_{3} ) q^{23} + ( 69 + 15 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} ) q^{24} + ( -29 - 5 \beta_{1} - 19 \beta_{2} - 7 \beta_{3} ) q^{26} -27 q^{27} + ( 21 + 21 \beta_{1} + 7 \beta_{2} ) q^{28} + ( 85 + 16 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -27 + 3 \beta_{1} + 3 \beta_{2} - 15 \beta_{3} ) q^{31} + ( -143 - 95 \beta_{1} - 31 \beta_{2} - 10 \beta_{3} ) q^{32} + ( -54 + 9 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} ) q^{33} + ( 169 + 57 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{34} + ( 27 + 27 \beta_{1} + 9 \beta_{2} ) q^{36} + ( -42 + 51 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} ) q^{37} + ( -84 + 84 \beta_{1} + 14 \beta_{2} + 20 \beta_{3} ) q^{38} + ( 33 - 9 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} ) q^{39} + ( -125 - 5 \beta_{1} - 33 \beta_{2} + 21 \beta_{3} ) q^{41} + ( 21 + 21 \beta_{1} ) q^{42} + ( -43 + 114 \beta_{1} - 4 \beta_{2} ) q^{43} + ( 148 + 52 \beta_{1} + 2 \beta_{2} + 11 \beta_{3} ) q^{44} + ( 79 + 39 \beta_{1} + 36 \beta_{2} + 12 \beta_{3} ) q^{46} + ( -172 - 34 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{47} + ( -135 - 81 \beta_{1} - 18 \beta_{2} - 18 \beta_{3} ) q^{48} + 49 q^{49} + ( 51 + 51 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} ) q^{51} + ( 157 + 105 \beta_{1} + 51 \beta_{2} + 9 \beta_{3} ) q^{52} + ( -251 + 151 \beta_{1} + 15 \beta_{2} - 3 \beta_{3} ) q^{53} + ( 27 + 27 \beta_{1} ) q^{54} + ( -161 - 35 \beta_{1} - 28 \beta_{2} - 7 \beta_{3} ) q^{56} + ( 48 - 18 \beta_{1} + 60 \beta_{2} ) q^{57} + ( -241 - 105 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{58} + ( 31 + 73 \beta_{1} + 19 \beta_{2} + \beta_{3} ) q^{59} + ( -321 - 33 \beta_{1} - 9 \beta_{2} + 21 \beta_{3} ) q^{61} + ( 63 + 39 \beta_{1} + 69 \beta_{2} + 27 \beta_{3} ) q^{62} + 63 q^{63} + ( 711 + 261 \beta_{1} + 128 \beta_{2} + 3 \beta_{3} ) q^{64} + ( -78 + 90 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{66} + ( -424 + 63 \beta_{1} - 31 \beta_{2} - 21 \beta_{3} ) q^{67} + ( -581 - 161 \beta_{1} - 23 \beta_{2} - 23 \beta_{3} ) q^{68} + ( 117 - 12 \beta_{2} + 24 \beta_{3} ) q^{69} + ( -320 - 137 \beta_{1} + 91 \beta_{2} - 11 \beta_{3} ) q^{71} + ( -207 - 45 \beta_{1} - 36 \beta_{2} - 9 \beta_{3} ) q^{72} + ( -114 - 18 \beta_{1} + 28 \beta_{2} - 36 \beta_{3} ) q^{73} + ( -498 - 90 \beta_{1} - 99 \beta_{2} - 21 \beta_{3} ) q^{74} + ( -680 - 228 \beta_{1} - 38 \beta_{2} - 54 \beta_{3} ) q^{76} + ( 126 - 21 \beta_{1} + 35 \beta_{2} - 7 \beta_{3} ) q^{77} + ( 87 + 15 \beta_{1} + 57 \beta_{2} + 21 \beta_{3} ) q^{78} + ( 374 - 33 \beta_{1} - 95 \beta_{2} - 9 \beta_{3} ) q^{79} + 81 q^{81} + ( 25 + 225 \beta_{1} - 67 \beta_{2} - 9 \beta_{3} ) q^{82} + ( -71 + 79 \beta_{1} + 79 \beta_{2} + 37 \beta_{3} ) q^{83} + ( -63 - 63 \beta_{1} - 21 \beta_{2} ) q^{84} + ( -1105 - 169 \beta_{1} - 110 \beta_{2} + 4 \beta_{3} ) q^{86} + ( -255 - 48 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{87} + ( -916 - 42 \beta_{1} - 133 \beta_{2} ) q^{88} + ( -50 - 356 \beta_{1} + 38 \beta_{2} - 10 \beta_{3} ) q^{89} + ( -77 + 21 \beta_{1} + 7 \beta_{2} + 21 \beta_{3} ) q^{91} + ( -133 - 325 \beta_{1} - 167 \beta_{2} + 4 \beta_{3} ) q^{92} + ( 81 - 9 \beta_{1} - 9 \beta_{2} + 45 \beta_{3} ) q^{93} + ( 520 + 264 \beta_{1} + 58 \beta_{2} + 12 \beta_{3} ) q^{94} + ( 429 + 285 \beta_{1} + 93 \beta_{2} + 30 \beta_{3} ) q^{96} + ( -754 + 204 \beta_{1} + 16 \beta_{2} - 24 \beta_{3} ) q^{97} + ( -49 - 49 \beta_{1} ) q^{98} + ( 162 - 27 \beta_{1} + 45 \beta_{2} - 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{2} - 12q^{3} + 16q^{4} + 18q^{6} + 28q^{7} - 93q^{8} + 36q^{9} + O(q^{10})$$ $$4q - 6q^{2} - 12q^{3} + 16q^{4} + 18q^{6} + 28q^{7} - 93q^{8} + 36q^{9} + 57q^{11} - 48q^{12} - 43q^{13} - 42q^{14} + 216q^{16} - 99q^{17} - 54q^{18} - 12q^{19} - 84q^{21} + 41q^{22} - 156q^{23} + 279q^{24} - 81q^{26} - 108q^{27} + 112q^{28} + 378q^{29} - 93q^{31} - 690q^{32} - 171q^{33} + 783q^{34} + 144q^{36} - 81q^{37} - 216q^{38} + 129q^{39} - 465q^{41} + 126q^{42} + 64q^{43} + 681q^{44} + 310q^{46} - 744q^{47} - 648q^{48} + 196q^{49} + 297q^{51} + 727q^{52} - 729q^{53} + 162q^{54} - 651q^{56} + 36q^{57} - 1172q^{58} + 231q^{59} - 1353q^{61} + 165q^{62} + 252q^{63} + 3107q^{64} - 123q^{66} - 1487q^{67} - 2577q^{68} + 468q^{69} - 1725q^{71} - 837q^{72} - 512q^{73} - 1953q^{74} - 3046q^{76} + 399q^{77} + 243q^{78} + 1629q^{79} + 324q^{81} + 693q^{82} - 321q^{83} - 336q^{84} - 4542q^{86} - 1134q^{87} - 3482q^{88} - 978q^{89} - 301q^{91} - 852q^{92} + 279q^{93} + 2480q^{94} + 2070q^{96} - 2616q^{97} - 294q^{98} + 513q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 18 x^{2} + 19 x + 44$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 10$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 14 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 10$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 15 \beta_{1} + 8$$

## 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
 4.56826 2.21734 −1.21734 −3.56826
−5.56826 −3.00000 23.0055 0 16.7048 7.00000 −83.5546 9.00000 0
1.2 −3.21734 −3.00000 2.35129 0 9.65203 7.00000 18.1738 9.00000 0
1.3 0.217342 −3.00000 −7.95276 0 −0.652027 7.00000 −3.46721 9.00000 0
1.4 2.56826 −3.00000 −1.40404 0 −7.70478 7.00000 −24.1520 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.s 4
3.b odd 2 1 1575.4.a.bm 4
5.b even 2 1 525.4.a.v yes 4
5.c odd 4 2 525.4.d.o 8
15.d odd 2 1 1575.4.a.bf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.s 4 1.a even 1 1 trivial
525.4.a.v yes 4 5.b even 2 1
525.4.d.o 8 5.c odd 4 2
1575.4.a.bf 4 15.d odd 2 1
1575.4.a.bm 4 3.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 6 T_{2}^{3} - 6 T_{2}^{2} - 45 T_{2} + 10$$ $$T_{11}^{4} - 57 T_{11}^{3} - 1323 T_{11}^{2} + 44829 T_{11} + 99334$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T + 26 T^{2} + 99 T^{3} + 298 T^{4} + 792 T^{5} + 1664 T^{6} + 3072 T^{7} + 4096 T^{8}$$
$3$ $$( 1 + 3 T )^{4}$$
$5$ 1
$7$ $$( 1 - 7 T )^{4}$$
$11$ $$1 - 57 T + 4001 T^{2} - 182772 T^{3} + 7206874 T^{4} - 243269532 T^{5} + 7088015561 T^{6} - 134403018387 T^{7} + 3138428376721 T^{8}$$
$13$ $$1 + 43 T + 4464 T^{2} + 221093 T^{3} + 13201958 T^{4} + 485741321 T^{5} + 21546875376 T^{6} + 455993473039 T^{7} + 23298085122481 T^{8}$$
$17$ $$1 + 99 T + 13790 T^{2} + 926229 T^{3} + 90687946 T^{4} + 4550563077 T^{5} + 332857076510 T^{6} + 11740199773203 T^{7} + 582622237229761 T^{8}$$
$19$ $$1 + 12 T - 10212 T^{2} + 154020 T^{3} + 95206646 T^{4} + 1056423180 T^{5} - 480432536772 T^{6} + 3872252373348 T^{7} + 2213314919066161 T^{8}$$
$23$ $$1 + 156 T + 28082 T^{2} + 3613896 T^{3} + 479553835 T^{4} + 43970272632 T^{5} + 4157143834898 T^{6} + 280979815188228 T^{7} + 21914624432020321 T^{8}$$
$29$ $$1 - 378 T + 143760 T^{2} - 29729988 T^{3} + 5852097929 T^{4} - 725084677332 T^{5} + 85511800626960 T^{6} - 5483701178878482 T^{7} + 353814783205469041 T^{8}$$
$31$ $$1 + 93 T + 16330 T^{2} - 936891 T^{3} + 385099458 T^{4} - 27910919781 T^{5} + 14492935110730 T^{6} + 2458884860942403 T^{7} + 787662783788549761 T^{8}$$
$37$ $$1 + 81 T + 107185 T^{2} + 469746 T^{3} + 5937897606 T^{4} + 23794044138 T^{5} + 275007385148665 T^{6} + 10526900923401237 T^{7} + 6582952005840035281 T^{8}$$
$41$ $$1 + 465 T + 105716 T^{2} + 566235 T^{3} - 2050285754 T^{4} + 39025482435 T^{5} + 502162019941556 T^{6} + 152232599493191865 T^{7} + 22563490300366186081 T^{8}$$
$43$ $$1 - 64 T + 64662 T^{2} + 2492176 T^{3} + 5477627255 T^{4} + 198145437232 T^{5} + 408751977474438 T^{6} - 32165927163957952 T^{7} + 39959630797262576401 T^{8}$$
$47$ $$1 + 744 T + 588276 T^{2} + 247094976 T^{3} + 101020317878 T^{4} + 25654141693248 T^{5} + 6341153676882804 T^{6} + 832633071988458648 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 + 729 T + 333446 T^{2} + 137056347 T^{3} + 63585804370 T^{4} + 20404537772319 T^{5} + 7390617561020534 T^{6} + 2405527658423754957 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$1 - 231 T + 701280 T^{2} - 148812447 T^{3} + 204019205798 T^{4} - 30562951552413 T^{5} + 29580364631760480 T^{6} - 2001152034109290909 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$
$61$ $$1 + 1353 T + 1374808 T^{2} + 931878711 T^{3} + 511309944510 T^{4} + 211518761701491 T^{5} + 70830622834497688 T^{6} + 15822179663604592773 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$
$67$ $$1 + 1487 T + 1603371 T^{2} + 1080212116 T^{3} + 674138302124 T^{4} + 324887836644508 T^{5} + 145038346676691699 T^{6} + 40456116647290586189 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$1 + 1725 T + 1434819 T^{2} + 706812600 T^{3} + 347044577276 T^{4} + 252976004478600 T^{5} + 183800721275245299 T^{6} + 79088663739324578475 T^{7} +$$$$16\!\cdots\!41$$$$T^{8}$$
$73$ $$1 + 512 T + 1030980 T^{2} + 484092568 T^{3} + 566560915526 T^{4} + 188320238525656 T^{5} + 156022560619433220 T^{6} + 30142252394633171456 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$1 - 1629 T + 1989837 T^{2} - 1837769472 T^{3} + 1504809477470 T^{4} - 906092022705408 T^{5} + 483704413231540077 T^{6} -$$$$19\!\cdots\!51$$$$T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$
$83$ $$1 + 321 T + 661434 T^{2} + 144132717 T^{3} + 79870144034 T^{4} + 82413213855279 T^{5} + 216249478918951146 T^{6} + 60007821940880469363 T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$1 + 978 T + 526228 T^{2} - 159100050 T^{3} - 360762420714 T^{4} - 112160603148450 T^{5} + 261525470779825108 T^{6} +$$$$34\!\cdots\!02$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$
$97$ $$1 + 2616 T + 5183100 T^{2} + 6945540936 T^{3} + 7789652954246 T^{4} + 6339007682681928 T^{5} + 4317377198747499900 T^{6} +$$$$19\!\cdots\!72$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$