# Properties

 Label 525.4.d.o 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: $$x^{8} + 40 x^{6} + 488 x^{4} + 1945 x^{2} + 1936$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{4} ) q^{2} -3 \beta_{4} q^{3} + ( -5 + \beta_{2} - 3 \beta_{3} ) q^{4} + ( 6 + 3 \beta_{3} ) q^{6} -7 \beta_{4} q^{7} + ( -5 \beta_{1} - 23 \beta_{4} + \beta_{5} - 5 \beta_{7} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{4} ) q^{2} -3 \beta_{4} q^{3} + ( -5 + \beta_{2} - 3 \beta_{3} ) q^{4} + ( 6 + 3 \beta_{3} ) q^{6} -7 \beta_{4} q^{7} + ( -5 \beta_{1} - 23 \beta_{4} + \beta_{5} - 5 \beta_{7} ) q^{8} -9 q^{9} + ( 11 - 4 \beta_{2} - 3 \beta_{3} - \beta_{6} ) q^{11} + ( 9 \beta_{1} + 9 \beta_{4} + 3 \beta_{7} ) q^{12} + ( 3 \beta_{1} - 11 \beta_{4} - 3 \beta_{5} + 4 \beta_{7} ) q^{13} + ( 14 + 7 \beta_{3} ) q^{14} + ( 60 - 12 \beta_{2} + 27 \beta_{3} + 6 \beta_{6} ) q^{16} + ( 17 \beta_{1} + 17 \beta_{4} + 3 \beta_{5} ) q^{17} + ( -9 \beta_{1} - 9 \beta_{4} ) q^{18} + ( -10 - 20 \beta_{2} - 6 \beta_{3} ) q^{19} -21 q^{21} + ( 30 \beta_{1} - 26 \beta_{4} - 3 \beta_{5} ) q^{22} + ( -39 \beta_{4} + 8 \beta_{5} - 4 \beta_{7} ) q^{23} + ( -69 + 15 \beta_{2} - 15 \beta_{3} - 3 \beta_{6} ) q^{24} + ( -8 + 26 \beta_{2} - 5 \beta_{3} - 7 \beta_{6} ) q^{26} + 27 \beta_{4} q^{27} + ( 21 \beta_{1} + 21 \beta_{4} + 7 \beta_{7} ) q^{28} + ( -105 - 4 \beta_{2} - 16 \beta_{3} + 2 \beta_{6} ) q^{29} + ( -12 + 12 \beta_{2} + 3 \beta_{3} - 15 \beta_{6} ) q^{31} + ( 95 \beta_{1} + 143 \beta_{4} - 10 \beta_{5} + 41 \beta_{7} ) q^{32} + ( 9 \beta_{1} - 54 \beta_{4} - 3 \beta_{5} - 12 \beta_{7} ) q^{33} + ( -224 + 2 \beta_{2} - 57 \beta_{3} + 3 \beta_{6} ) q^{34} + ( 45 - 9 \beta_{2} + 27 \beta_{3} ) q^{36} + ( -51 \beta_{1} + 42 \beta_{4} + 9 \beta_{5} - 12 \beta_{7} ) q^{37} + ( 84 \beta_{1} - 84 \beta_{4} - 20 \beta_{5} + 34 \beta_{7} ) q^{38} + ( -36 - 12 \beta_{2} + 9 \beta_{3} + 9 \beta_{6} ) q^{39} + ( -118 + 12 \beta_{2} - 5 \beta_{3} + 21 \beta_{6} ) q^{41} + ( -21 \beta_{1} - 21 \beta_{4} ) q^{42} + ( 114 \beta_{1} - 43 \beta_{4} - 4 \beta_{7} ) q^{43} + ( -187 + 13 \beta_{2} - 52 \beta_{3} - 11 \beta_{6} ) q^{44} + ( 70 - 48 \beta_{2} + 39 \beta_{3} + 12 \beta_{6} ) q^{46} + ( 34 \beta_{1} + 172 \beta_{4} - 4 \beta_{5} + 8 \beta_{7} ) q^{47} + ( -81 \beta_{1} - 135 \beta_{4} + 18 \beta_{5} - 36 \beta_{7} ) q^{48} -49 q^{49} + ( 102 + 51 \beta_{3} - 9 \beta_{6} ) q^{51} + ( -105 \beta_{1} - 157 \beta_{4} + 9 \beta_{5} - 60 \beta_{7} ) q^{52} + ( 151 \beta_{1} - 251 \beta_{4} + 3 \beta_{5} + 12 \beta_{7} ) q^{53} + ( -54 - 27 \beta_{3} ) q^{54} + ( -161 + 35 \beta_{2} - 35 \beta_{3} - 7 \beta_{6} ) q^{56} + ( 18 \beta_{1} - 48 \beta_{4} - 60 \beta_{7} ) q^{57} + ( -105 \beta_{1} - 241 \beta_{4} - 6 \beta_{5} + 2 \beta_{7} ) q^{58} + ( -84 + 20 \beta_{2} - 73 \beta_{3} - \beta_{6} ) q^{59} + ( -366 - 12 \beta_{2} - 33 \beta_{3} + 21 \beta_{6} ) q^{61} + ( -39 \beta_{1} - 63 \beta_{4} + 27 \beta_{5} - 96 \beta_{7} ) q^{62} + 63 \beta_{4} q^{63} + ( -841 + 131 \beta_{2} - 261 \beta_{3} - 3 \beta_{6} ) q^{64} + ( 12 + 90 \beta_{3} + 9 \beta_{6} ) q^{66} + ( -63 \beta_{1} + 424 \beta_{4} - 21 \beta_{5} + 52 \beta_{7} ) q^{67} + ( -161 \beta_{1} - 581 \beta_{4} + 23 \beta_{5} - 46 \beta_{7} ) q^{68} + ( -105 + 12 \beta_{2} - 24 \beta_{6} ) q^{69} + ( -537 - 80 \beta_{2} - 137 \beta_{3} - 11 \beta_{6} ) q^{71} + ( 45 \beta_{1} + 207 \beta_{4} - 9 \beta_{5} + 45 \beta_{7} ) q^{72} + ( -18 \beta_{1} - 114 \beta_{4} + 36 \beta_{5} - 8 \beta_{7} ) q^{73} + ( 468 - 120 \beta_{2} + 90 \beta_{3} + 21 \beta_{6} ) q^{74} + ( -816 + 92 \beta_{2} - 228 \beta_{3} - 54 \beta_{6} ) q^{76} + ( 21 \beta_{1} - 126 \beta_{4} - 7 \beta_{5} - 28 \beta_{7} ) q^{77} + ( 15 \beta_{1} + 87 \beta_{4} - 21 \beta_{5} + 78 \beta_{7} ) q^{78} + ( -445 - 104 \beta_{2} + 33 \beta_{3} + 9 \beta_{6} ) q^{79} + 81 q^{81} + ( -225 \beta_{1} - 25 \beta_{4} - 9 \beta_{5} + 76 \beta_{7} ) q^{82} + ( 79 \beta_{1} - 71 \beta_{4} - 37 \beta_{5} + 116 \beta_{7} ) q^{83} + ( 105 - 21 \beta_{2} + 63 \beta_{3} ) q^{84} + ( -1168 + 106 \beta_{2} - 169 \beta_{3} + 4 \beta_{6} ) q^{86} + ( 48 \beta_{1} + 255 \beta_{4} + 6 \beta_{5} - 12 \beta_{7} ) q^{87} + ( -42 \beta_{1} - 916 \beta_{4} - 133 \beta_{7} ) q^{88} + ( 434 + 28 \beta_{2} + 356 \beta_{3} + 10 \beta_{6} ) q^{89} + ( -84 - 28 \beta_{2} + 21 \beta_{3} + 21 \beta_{6} ) q^{91} + ( 325 \beta_{1} + 133 \beta_{4} + 4 \beta_{5} + 163 \beta_{7} ) q^{92} + ( -9 \beta_{1} + 81 \beta_{4} - 45 \beta_{5} + 36 \beta_{7} ) q^{93} + ( -714 + 70 \beta_{2} - 264 \beta_{3} - 12 \beta_{6} ) q^{94} + ( 591 - 123 \beta_{2} + 285 \beta_{3} + 30 \beta_{6} ) q^{96} + ( -204 \beta_{1} + 754 \beta_{4} - 24 \beta_{5} + 8 \beta_{7} ) q^{97} + ( -49 \beta_{1} - 49 \beta_{4} ) q^{98} + ( -99 + 36 \beta_{2} + 27 \beta_{3} + 9 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 32q^{4} + 36q^{6} - 72q^{9} + O(q^{10})$$ $$8q - 32q^{4} + 36q^{6} - 72q^{9} + 114q^{11} + 84q^{14} + 432q^{16} + 24q^{19} - 168q^{21} - 558q^{24} - 162q^{26} - 756q^{29} - 186q^{31} - 1566q^{34} + 288q^{36} - 258q^{39} - 930q^{41} - 1362q^{44} + 620q^{46} - 392q^{49} + 594q^{51} - 324q^{54} - 1302q^{56} - 462q^{59} - 2706q^{61} - 6214q^{64} - 246q^{66} - 936q^{69} - 3450q^{71} + 3906q^{74} - 6092q^{76} - 3258q^{79} + 648q^{81} + 672q^{84} - 9084q^{86} + 1956q^{89} - 602q^{91} - 4960q^{94} + 4140q^{96} - 1026q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 40 x^{6} + 488 x^{4} + 1945 x^{2} + 1936$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{6} - 61 \nu^{4} - 323 \nu^{2} + 575$$$$)/147$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{6} - 61 \nu^{4} - 470 \nu^{2} - 895$$$$)/147$$ $$\beta_{4}$$ $$=$$ $$($$$$-17 \nu^{7} - 592 \nu^{5} - 5612 \nu^{3} - 12385 \nu$$$$)/6468$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 62 \nu^{5} + 1159 \nu^{3} + 5960 \nu$$$$)/231$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 41 \nu^{4} + 466 \nu^{2} + 1088$$$$)/21$$ $$\beta_{7}$$ $$=$$ $$($$$$41 \nu^{7} + 1618 \nu^{5} + 17720 \nu^{3} + 42235 \nu$$$$)/3234$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} - 10$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{7} + \beta_{5} - 8 \beta_{4} - 15 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{6} + 29 \beta_{3} - 22 \beta_{2} + 159$$ $$\nu^{5}$$ $$=$$ $$61 \beta_{7} - 22 \beta_{5} + 258 \beta_{4} + 265 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-61 \beta_{6} - 723 \beta_{3} + 436 \beta_{2} - 2947$$ $$\nu^{7}$$ $$=$$ $$-1464 \beta_{7} + 436 \beta_{5} - 6724 \beta_{4} - 5005 \beta_{1}$$

## 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
 − 4.56826i − 2.21734i − 3.56826i − 1.21734i 1.21734i 3.56826i 2.21734i 4.56826i
5.56826i 3.00000i −23.0055 0 16.7048 7.00000i 83.5546i −9.00000 0
274.2 3.21734i 3.00000i −2.35129 0 9.65203 7.00000i 18.1738i −9.00000 0
274.3 2.56826i 3.00000i 1.40404 0 −7.70478 7.00000i 24.1520i −9.00000 0
274.4 0.217342i 3.00000i 7.95276 0 −0.652027 7.00000i 3.46721i −9.00000 0
274.5 0.217342i 3.00000i 7.95276 0 −0.652027 7.00000i 3.46721i −9.00000 0
274.6 2.56826i 3.00000i 1.40404 0 −7.70478 7.00000i 24.1520i −9.00000 0
274.7 3.21734i 3.00000i −2.35129 0 9.65203 7.00000i 18.1738i −9.00000 0
274.8 5.56826i 3.00000i −23.0055 0 16.7048 7.00000i 83.5546i −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.o 8
5.b even 2 1 inner 525.4.d.o 8
5.c odd 4 1 525.4.a.s 4
5.c odd 4 1 525.4.a.v yes 4
15.e even 4 1 1575.4.a.bf 4
15.e even 4 1 1575.4.a.bm 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.s 4 5.c odd 4 1
525.4.a.v yes 4 5.c odd 4 1
525.4.d.o 8 1.a even 1 1 trivial
525.4.d.o 8 5.b even 2 1 inner
1575.4.a.bf 4 15.e even 4 1
1575.4.a.bm 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 $$S_{4}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{8} + 48 T_{2}^{6} + 596 T_{2}^{4} + 2145 T_{2}^{2} + 100$$ $$T_{11}^{4} - 57 T_{11}^{3} - 1323 T_{11}^{2} + 44829 T_{11} + 99334$$