# Properties

 Label 525.4.d.k Level $525$ Weight $4$ Character orbit 525.d Analytic conductor $30.976$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -2 \beta_{1} + \beta_{2} ) q^{2} -3 \beta_{1} q^{3} + ( -1 + 4 \beta_{3} ) q^{4} + ( -6 + 3 \beta_{3} ) q^{6} -7 \beta_{1} q^{7} + ( 6 \beta_{1} - \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + ( -2 \beta_{1} + \beta_{2} ) q^{2} -3 \beta_{1} q^{3} + ( -1 + 4 \beta_{3} ) q^{4} + ( -6 + 3 \beta_{3} ) q^{6} -7 \beta_{1} q^{7} + ( 6 \beta_{1} - \beta_{2} ) q^{8} -9 q^{9} + ( -46 - 2 \beta_{3} ) q^{11} + ( 3 \beta_{1} - 12 \beta_{2} ) q^{12} + ( -4 \beta_{1} + 38 \beta_{2} ) q^{13} + ( -14 + 7 \beta_{3} ) q^{14} + ( 9 + 24 \beta_{3} ) q^{16} + ( -22 \beta_{1} + 44 \beta_{2} ) q^{17} + ( 18 \beta_{1} - 9 \beta_{2} ) q^{18} + ( 54 - 26 \beta_{3} ) q^{19} -21 q^{21} + ( 82 \beta_{1} - 42 \beta_{2} ) q^{22} + ( 160 \beta_{1} + 20 \beta_{2} ) q^{23} + ( 18 - 3 \beta_{3} ) q^{24} + ( -198 + 80 \beta_{3} ) q^{26} + 27 \beta_{1} q^{27} + ( 7 \beta_{1} - 28 \beta_{2} ) q^{28} + ( 118 - 12 \beta_{3} ) q^{29} + ( -30 + 102 \beta_{3} ) q^{31} + ( 150 \beta_{1} - 47 \beta_{2} ) q^{32} + ( 138 \beta_{1} + 6 \beta_{2} ) q^{33} + ( -264 + 110 \beta_{3} ) q^{34} + ( 9 - 36 \beta_{3} ) q^{36} + ( 102 \beta_{1} + 24 \beta_{2} ) q^{37} + ( -238 \beta_{1} + 106 \beta_{2} ) q^{38} + ( -12 + 114 \beta_{3} ) q^{39} + ( 22 - 80 \beta_{3} ) q^{41} + ( 42 \beta_{1} - 21 \beta_{2} ) q^{42} + ( -68 \beta_{1} - 128 \beta_{2} ) q^{43} + ( 6 - 182 \beta_{3} ) q^{44} + ( 220 - 120 \beta_{3} ) q^{46} + ( 200 \beta_{1} - 168 \beta_{2} ) q^{47} + ( -27 \beta_{1} - 72 \beta_{2} ) q^{48} -49 q^{49} + ( -66 + 132 \beta_{3} ) q^{51} + ( 764 \beta_{1} - 54 \beta_{2} ) q^{52} + ( -8 \beta_{1} - 86 \beta_{2} ) q^{53} + ( 54 - 27 \beta_{3} ) q^{54} + ( 42 - 7 \beta_{3} ) q^{56} + ( -162 \beta_{1} + 78 \beta_{2} ) q^{57} + ( -296 \beta_{1} + 142 \beta_{2} ) q^{58} + ( 232 + 36 \beta_{3} ) q^{59} + ( -342 + 84 \beta_{3} ) q^{61} + ( 570 \beta_{1} - 234 \beta_{2} ) q^{62} + 63 \beta_{1} q^{63} + ( 607 - 52 \beta_{3} ) q^{64} + ( 246 - 126 \beta_{3} ) q^{66} + ( 368 \beta_{1} + 164 \beta_{2} ) q^{67} + ( 902 \beta_{1} - 132 \beta_{2} ) q^{68} + ( 480 + 60 \beta_{3} ) q^{69} + ( -370 - 138 \beta_{3} ) q^{71} + ( -54 \beta_{1} + 9 \beta_{2} ) q^{72} + ( -212 \beta_{1} + 122 \beta_{2} ) q^{73} + ( 84 - 54 \beta_{3} ) q^{74} + ( -574 + 242 \beta_{3} ) q^{76} + ( 322 \beta_{1} + 14 \beta_{2} ) q^{77} + ( 594 \beta_{1} - 240 \beta_{2} ) q^{78} + ( 204 + 484 \beta_{3} ) q^{79} + 81 q^{81} + ( -444 \beta_{1} + 182 \beta_{2} ) q^{82} + ( -304 \beta_{1} + 84 \beta_{2} ) q^{83} + ( 21 - 84 \beta_{3} ) q^{84} + ( 504 - 188 \beta_{3} ) q^{86} + ( -354 \beta_{1} + 36 \beta_{2} ) q^{87} + ( -266 \beta_{1} + 34 \beta_{2} ) q^{88} + ( 666 + 112 \beta_{3} ) q^{89} + ( -28 + 266 \beta_{3} ) q^{91} + ( 240 \beta_{1} + 620 \beta_{2} ) q^{92} + ( 90 \beta_{1} - 306 \beta_{2} ) q^{93} + ( 1240 - 536 \beta_{3} ) q^{94} + ( 450 - 141 \beta_{3} ) q^{96} + ( -1224 \beta_{1} - 86 \beta_{2} ) q^{97} + ( 98 \beta_{1} - 49 \beta_{2} ) q^{98} + ( 414 + 18 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 24q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 24q^{6} - 36q^{9} - 184q^{11} - 56q^{14} + 36q^{16} + 216q^{19} - 84q^{21} + 72q^{24} - 792q^{26} + 472q^{29} - 120q^{31} - 1056q^{34} + 36q^{36} - 48q^{39} + 88q^{41} + 24q^{44} + 880q^{46} - 196q^{49} - 264q^{51} + 216q^{54} + 168q^{56} + 928q^{59} - 1368q^{61} + 2428q^{64} + 984q^{66} + 1920q^{69} - 1480q^{71} + 336q^{74} - 2296q^{76} + 816q^{79} + 324q^{81} + 84q^{84} + 2016q^{86} + 2664q^{89} - 112q^{91} + 4960q^{94} + 1800q^{96} + 1656q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2 \beta_{1}$$

## 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
 − 1.61803i − 0.618034i 0.618034i 1.61803i
4.23607i 3.00000i −9.94427 0 −12.7082 7.00000i 8.23607i −9.00000 0
274.2 0.236068i 3.00000i 7.94427 0 0.708204 7.00000i 3.76393i −9.00000 0
274.3 0.236068i 3.00000i 7.94427 0 0.708204 7.00000i 3.76393i −9.00000 0
274.4 4.23607i 3.00000i −9.94427 0 −12.7082 7.00000i 8.23607i −9.00000 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
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.k 4
5.b even 2 1 inner 525.4.d.k 4
5.c odd 4 1 105.4.a.d 2
5.c odd 4 1 525.4.a.o 2
15.e even 4 1 315.4.a.l 2
15.e even 4 1 1575.4.a.n 2
20.e even 4 1 1680.4.a.bd 2
35.f even 4 1 735.4.a.m 2
105.k odd 4 1 2205.4.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 5.c odd 4 1
315.4.a.l 2 15.e even 4 1
525.4.a.o 2 5.c odd 4 1
525.4.d.k 4 1.a even 1 1 trivial
525.4.d.k 4 5.b even 2 1 inner
735.4.a.m 2 35.f even 4 1
1575.4.a.n 2 15.e even 4 1
1680.4.a.bd 2 20.e even 4 1
2205.4.a.be 2 105.k odd 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 18 T_{2}^{2} + 1$$ $$T_{11}^{2} + 92 T_{11} + 2096$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 14 T^{2} + 97 T^{4} - 896 T^{6} + 4096 T^{8}$$
$3$ $$( 1 + 9 T^{2} )^{2}$$
$5$ 1
$7$ $$( 1 + 49 T^{2} )^{2}$$
$11$ $$( 1 + 92 T + 4758 T^{2} + 122452 T^{3} + 1771561 T^{4} )^{2}$$
$13$ $$1 + 5684 T^{2} + 17268502 T^{4} + 27435582356 T^{6} + 23298085122481 T^{8}$$
$17$ $$1 + 676 T^{2} + 29648902 T^{4} + 16316996644 T^{6} + 582622237229761 T^{8}$$
$19$ $$( 1 - 108 T + 13254 T^{2} - 740772 T^{3} + 47045881 T^{4} )^{2}$$
$23$ $$1 + 6532 T^{2} + 101938534 T^{4} + 966970426948 T^{6} + 21914624432020321 T^{8}$$
$29$ $$( 1 - 236 T + 61982 T^{2} - 5755804 T^{3} + 594823321 T^{4} )^{2}$$
$31$ $$( 1 + 60 T + 8462 T^{2} + 1787460 T^{3} + 887503681 T^{4} )^{2}$$
$37$ $$1 - 176044 T^{2} + 12759471222 T^{4} - 451680739945996 T^{6} + 6582952005840035281 T^{8}$$
$41$ $$( 1 - 44 T + 106326 T^{2} - 3032524 T^{3} + 4750104241 T^{4} )^{2}$$
$43$ $$1 - 144940 T^{2} + 16379434678 T^{4} - 916218360322060 T^{6} + 39959630797262576401 T^{8}$$
$47$ $$1 - 53052 T^{2} - 317140666 T^{4} - 571858931634108 T^{6} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 - 521420 T^{2} + 112288959478 T^{4} - 11556941179883180 T^{6} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$( 1 - 464 T + 458102 T^{2} - 95295856 T^{3} + 42180533641 T^{4} )^{2}$$
$61$ $$( 1 + 684 T + 535646 T^{2} + 155255004 T^{3} + 51520374361 T^{4} )^{2}$$
$67$ $$1 - 663244 T^{2} + 218042637142 T^{4} - 59995979223296236 T^{6} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$( 1 + 740 T + 757502 T^{2} + 264854140 T^{3} + 128100283921 T^{4} )^{2}$$
$73$ $$1 - 1317340 T^{2} + 723135691558 T^{4} - 199358629659551260 T^{6} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$( 1 - 408 T - 143586 T^{2} - 201159912 T^{3} + 243087455521 T^{4} )^{2}$$
$83$ $$1 - 2031756 T^{2} + 1672847111702 T^{4} - 664263065234705964 T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$( 1 - 1332 T + 1790774 T^{2} - 939018708 T^{3} + 496981290961 T^{4} )^{2}$$
$97$ $$1 - 580380 T^{2} + 1528544052038 T^{4} - 483440292220693020 T^{6} +$$$$69\!\cdots\!41$$$$T^{8}$$