# Properties

 Label 525.2.n.a Level 525 Weight 2 Character orbit 525.n Analytic conductor 4.192 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{3} q^{3} + ( 1 - \zeta_{10} ) q^{4} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{5} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{6} - q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} -\zeta_{10} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{3} q^{3} + ( 1 - \zeta_{10} ) q^{4} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{5} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{6} - q^{7} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{8} -\zeta_{10} q^{9} + ( 3 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{10} + ( 2 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{11} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{12} + ( 4 - 3 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{13} + ( -1 + \zeta_{10}^{3} ) q^{14} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{15} + ( 3 + 3 \zeta_{10}^{2} ) q^{16} + ( 4 \zeta_{10} - 5 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{17} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{18} + ( -\zeta_{10} - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{19} + ( -1 + \zeta_{10} - 2 \zeta_{10}^{3} ) q^{20} -\zeta_{10}^{3} q^{21} + ( 2 - 2 \zeta_{10} ) q^{22} + ( 1 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{23} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{24} + 5 q^{25} + ( 5 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{26} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{27} + ( -1 + \zeta_{10} ) q^{28} + ( -6 + 6 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{29} + ( \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{30} + ( -3 \zeta_{10} + \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{31} + ( 4 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{32} + ( -2 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{33} + ( -1 + 4 \zeta_{10} - \zeta_{10}^{2} ) q^{34} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{35} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{36} + ( 3 - 6 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{37} + ( -3 - \zeta_{10} - 3 \zeta_{10}^{2} ) q^{38} + ( -1 - 3 \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{39} + 5 \zeta_{10}^{2} q^{40} + ( -5 + 3 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{41} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{42} + q^{43} + ( -4 \zeta_{10} + 6 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{44} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{45} + ( 1 - \zeta_{10} ) q^{46} -12 \zeta_{10}^{3} q^{47} + ( -3 + 3 \zeta_{10}^{3} ) q^{48} + q^{49} + ( 5 - 5 \zeta_{10}^{3} ) q^{50} + ( 1 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{51} + ( 4 - 7 \zeta_{10} + 7 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{52} + ( 5 - 5 \zeta_{10} - \zeta_{10}^{3} ) q^{53} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{54} + ( -2 + 8 \zeta_{10} - 8 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{55} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{56} + ( 3 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{57} + ( 3 \zeta_{10} + 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{58} + ( -7 + \zeta_{10} - 7 \zeta_{10}^{2} ) q^{59} + ( -1 + 3 \zeta_{10} - \zeta_{10}^{2} ) q^{60} + ( -4 + 10 \zeta_{10} - 10 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{61} + ( -2 - 3 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{62} + \zeta_{10} q^{63} + ( -3 + \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{64} + ( -2 + 11 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{65} + ( 2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{66} + ( 2 \zeta_{10} - 9 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{67} + ( 4 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{68} + ( -\zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{69} + ( -3 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{70} + ( 4 - 4 \zeta_{10} + \zeta_{10}^{3} ) q^{71} + ( 2 - 2 \zeta_{10} - \zeta_{10}^{3} ) q^{72} + ( -11 + 6 \zeta_{10} - 6 \zeta_{10}^{2} + 11 \zeta_{10}^{3} ) q^{73} + ( -3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{74} + 5 \zeta_{10}^{3} q^{75} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{76} + ( -2 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{77} + ( -1 + \zeta_{10} + 5 \zeta_{10}^{3} ) q^{78} + ( -9 + 9 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{79} + ( 3 + 6 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{80} + \zeta_{10}^{2} q^{81} + ( -7 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{82} + ( -8 \zeta_{10} - 8 \zeta_{10}^{3} ) q^{83} + ( -1 + \zeta_{10} - \zeta_{10}^{2} ) q^{84} + ( -6 \zeta_{10} + 13 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{85} + ( 1 - \zeta_{10}^{3} ) q^{86} + ( -6 + 3 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{87} + ( 6 - 8 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{88} + ( -2 - 2 \zeta_{10} + 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{89} + ( -3 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{90} + ( -4 + 3 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{91} + ( -2 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{92} + ( 2 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{93} + ( -12 \zeta_{10} - 12 \zeta_{10}^{3} ) q^{94} + ( -5 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{95} + ( 1 - \zeta_{10} + 4 \zeta_{10}^{3} ) q^{96} + ( 4 - 4 \zeta_{10} - 9 \zeta_{10}^{3} ) q^{97} + ( 1 - \zeta_{10}^{3} ) q^{98} + ( -2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 3q^{2} + q^{3} + 3q^{4} + 2q^{6} - 4q^{7} + 5q^{8} - q^{9} + O(q^{10})$$ $$4q + 3q^{2} + q^{3} + 3q^{4} + 2q^{6} - 4q^{7} + 5q^{8} - q^{9} + 5q^{10} - 2q^{11} + 2q^{12} + 9q^{13} - 3q^{14} - 5q^{15} + 9q^{16} + 13q^{17} - 2q^{18} - 5q^{20} - q^{21} + 6q^{22} - q^{23} + 20q^{25} + 18q^{26} + q^{27} - 3q^{28} - 15q^{29} - 7q^{31} + 18q^{32} - 8q^{33} + q^{34} - 2q^{36} + 3q^{37} - 10q^{38} - 9q^{39} - 5q^{40} - 12q^{41} - 2q^{42} + 4q^{43} - 14q^{44} - 5q^{45} + 3q^{46} - 12q^{47} - 9q^{48} + 4q^{49} + 15q^{50} + 12q^{51} - 2q^{52} + 14q^{53} + 2q^{54} + 10q^{55} - 5q^{56} + 10q^{57} - 20q^{59} + 8q^{61} - 9q^{62} + q^{63} - 7q^{64} + 5q^{65} + 4q^{66} + 13q^{67} + 26q^{68} - 4q^{69} - 5q^{70} + 13q^{71} + 5q^{72} - 21q^{73} + 6q^{74} + 5q^{75} + 2q^{77} + 2q^{78} - 25q^{79} + 15q^{80} - q^{81} - 24q^{82} - 16q^{83} - 2q^{84} - 25q^{85} + 3q^{86} - 15q^{87} + 10q^{88} - 10q^{89} - 10q^{90} - 9q^{91} - 7q^{92} + 2q^{93} - 24q^{94} - 10q^{95} + 7q^{96} + 3q^{97} + 3q^{98} - 12q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
106.1
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
0.190983 + 0.587785i 0.809017 0.587785i 1.30902 0.951057i −2.23607 0.500000 + 0.363271i −1.00000 1.80902 + 1.31433i 0.309017 0.951057i −0.427051 1.31433i
211.1 1.30902 0.951057i −0.309017 + 0.951057i 0.190983 0.587785i 2.23607 0.500000 + 1.53884i −1.00000 0.690983 + 2.12663i −0.809017 0.587785i 2.92705 2.12663i
316.1 1.30902 + 0.951057i −0.309017 0.951057i 0.190983 + 0.587785i 2.23607 0.500000 1.53884i −1.00000 0.690983 2.12663i −0.809017 + 0.587785i 2.92705 + 2.12663i
421.1 0.190983 0.587785i 0.809017 + 0.587785i 1.30902 + 0.951057i −2.23607 0.500000 0.363271i −1.00000 1.80902 1.31433i 0.309017 + 0.951057i −0.427051 + 1.31433i
 $$n$$: 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
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.n.a 4
25.d even 5 1 inner 525.2.n.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.n.a 4 1.a even 1 1 trivial
525.2.n.a 4 25.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 3 T_{2}^{3} + 4 T_{2}^{2} - 2 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + 2 T^{2} + T^{4} + 8 T^{6} - 24 T^{7} + 16 T^{8}$$
$3$ $$1 - T + T^{2} - T^{3} + T^{4}$$
$5$ $$( 1 - 5 T^{2} )^{2}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 374 T^{5} + 1573 T^{6} + 2662 T^{7} + 14641 T^{8}$$
$13$ $$1 - 9 T + 48 T^{2} - 235 T^{3} + 1011 T^{4} - 3055 T^{5} + 8112 T^{6} - 19773 T^{7} + 28561 T^{8}$$
$17$ $$1 - 13 T + 52 T^{2} + 25 T^{3} - 729 T^{4} + 425 T^{5} + 15028 T^{6} - 63869 T^{7} + 83521 T^{8}$$
$19$ $$1 - 9 T^{2} + 70 T^{3} + 291 T^{4} + 1330 T^{5} - 3249 T^{6} + 130321 T^{8}$$
$23$ $$1 + T - 17 T^{2} + 65 T^{3} + 576 T^{4} + 1495 T^{5} - 8993 T^{6} + 12167 T^{7} + 279841 T^{8}$$
$29$ $$1 + 15 T + 106 T^{2} + 675 T^{3} + 4171 T^{4} + 19575 T^{5} + 89146 T^{6} + 365835 T^{7} + 707281 T^{8}$$
$31$ $$1 + 7 T + 3 T^{2} + 119 T^{3} + 1640 T^{4} + 3689 T^{5} + 2883 T^{6} + 208537 T^{7} + 923521 T^{8}$$
$37$ $$1 - 3 T + 17 T^{2} - 225 T^{3} + 2116 T^{4} - 8325 T^{5} + 23273 T^{6} - 151959 T^{7} + 1874161 T^{8}$$
$41$ $$1 + 12 T + 53 T^{2} + 444 T^{3} + 4405 T^{4} + 18204 T^{5} + 89093 T^{6} + 827052 T^{7} + 2825761 T^{8}$$
$43$ $$( 1 - T + 43 T^{2} )^{4}$$
$47$ $$1 + 12 T + 97 T^{2} + 600 T^{3} + 2641 T^{4} + 28200 T^{5} + 214273 T^{6} + 1245876 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 14 T + 43 T^{2} - 160 T^{3} + 2961 T^{4} - 8480 T^{5} + 120787 T^{6} - 2084278 T^{7} + 7890481 T^{8}$$
$59$ $$1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 31270 T^{5} + 456011 T^{6} + 4107580 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 8 T + 123 T^{2} - 766 T^{3} + 10325 T^{4} - 46726 T^{5} + 457683 T^{6} - 1815848 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 13 T + 12 T^{2} + 895 T^{3} - 9919 T^{4} + 59965 T^{5} + 53868 T^{6} - 3909919 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 13 T - 2 T^{2} + 349 T^{3} + 405 T^{4} + 24779 T^{5} - 10082 T^{6} - 4652843 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 21 T + 93 T^{2} - 1405 T^{3} - 22044 T^{4} - 102565 T^{5} + 495597 T^{6} + 8169357 T^{7} + 28398241 T^{8}$$
$79$ $$1 + 25 T + 231 T^{2} + 1505 T^{3} + 12896 T^{4} + 118895 T^{5} + 1441671 T^{6} + 12325975 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 16 T + 173 T^{2} + 2200 T^{3} + 26921 T^{4} + 182600 T^{5} + 1191797 T^{6} + 9148592 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 10 T - 29 T^{2} + 200 T^{3} + 10101 T^{4} + 17800 T^{5} - 229709 T^{6} + 7049690 T^{7} + 62742241 T^{8}$$
$97$ $$1 - 3 T + 12 T^{2} - 865 T^{3} + 11511 T^{4} - 83905 T^{5} + 112908 T^{6} - 2738019 T^{7} + 88529281 T^{8}$$