# Properties

 Label 525.2.j.a Level 525 Weight 2 Character orbit 525.j Analytic conductor 4.192 Analytic rank 0 Dimension 16 CM no Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: 16.0.6040479020157644046336.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{10} + \beta_{13} - \beta_{15} ) q^{2} + \beta_{7} q^{3} + ( -\beta_{2} + \beta_{11} - 2 \beta_{14} ) q^{4} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{6} ) q^{6} -\beta_{12} q^{7} + ( -\beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{8} + ( -\beta_{2} + \beta_{4} + \beta_{11} - \beta_{14} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{10} + \beta_{13} - \beta_{15} ) q^{2} + \beta_{7} q^{3} + ( -\beta_{2} + \beta_{11} - 2 \beta_{14} ) q^{4} + ( 1 - \beta_{1} - \beta_{3} - 2 \beta_{6} ) q^{6} -\beta_{12} q^{7} + ( -\beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{8} + ( -\beta_{2} + \beta_{4} + \beta_{11} - \beta_{14} ) q^{9} + ( 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{6} ) q^{11} + ( \beta_{10} - 4 \beta_{12} + \beta_{13} - \beta_{15} ) q^{12} + ( -\beta_{7} + \beta_{9} ) q^{13} + ( -\beta_{2} - \beta_{4} ) q^{14} + ( -3 - \beta_{1} + \beta_{3} ) q^{16} + ( -\beta_{10} + \beta_{13} - 2 \beta_{15} ) q^{17} + ( -\beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{18} -4 \beta_{11} q^{19} -\beta_{1} q^{21} + ( -2 \beta_{10} + 8 \beta_{12} - 2 \beta_{13} ) q^{22} -2 \beta_{8} q^{23} + ( -5 \beta_{2} - 4 \beta_{4} - \beta_{11} - 2 \beta_{14} ) q^{24} + ( 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{6} ) q^{26} + ( -2 \beta_{10} - 4 \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{27} + ( -\beta_{5} - \beta_{7} + \beta_{9} ) q^{28} + ( -\beta_{2} + 2 \beta_{4} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{31} + ( 3 \beta_{10} - 3 \beta_{13} + \beta_{15} ) q^{32} + ( -4 \beta_{5} + 2 \beta_{7} - 5 \beta_{8} + \beta_{9} ) q^{33} + ( -2 \beta_{2} + 4 \beta_{11} - 4 \beta_{14} ) q^{34} + ( -5 - 5 \beta_{1} - \beta_{3} - 2 \beta_{6} ) q^{36} + ( 2 \beta_{10} + 2 \beta_{13} ) q^{37} + ( 4 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} ) q^{38} + ( \beta_{2} - \beta_{4} - 4 \beta_{11} + \beta_{14} ) q^{39} + ( 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{6} ) q^{41} + ( \beta_{10} - \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{42} + 8 \beta_{5} q^{43} + ( 6 \beta_{2} + 12 \beta_{4} ) q^{44} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{46} + ( \beta_{10} - \beta_{13} + 4 \beta_{15} ) q^{47} + ( -4 \beta_{5} - 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{48} -\beta_{11} q^{49} + ( -2 \beta_{1} - 3 \beta_{3} - 3 \beta_{6} ) q^{51} + ( -2 \beta_{10} + 8 \beta_{12} - 2 \beta_{13} ) q^{52} + ( -2 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{53} + ( -5 \beta_{2} - 4 \beta_{4} + 5 \beta_{11} - 2 \beta_{14} ) q^{54} + ( \beta_{1} + \beta_{3} + 3 \beta_{6} ) q^{56} -4 \beta_{13} q^{57} + ( 2 \beta_{7} - 2 \beta_{9} ) q^{58} + ( 2 \beta_{2} - 4 \beta_{4} ) q^{59} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{61} + ( 4 \beta_{10} - 4 \beta_{13} + 8 \beta_{15} ) q^{62} + ( -\beta_{5} + \beta_{8} + \beta_{9} ) q^{63} + ( -\beta_{2} - \beta_{11} - 2 \beta_{14} ) q^{64} + ( 8 + 8 \beta_{1} - 2 \beta_{3} + 2 \beta_{6} ) q^{66} + ( 2 \beta_{10} + 8 \beta_{12} + 2 \beta_{13} ) q^{67} + ( -6 \beta_{7} + 8 \beta_{8} - 6 \beta_{9} ) q^{68} + ( 4 \beta_{2} + 2 \beta_{4} + 2 \beta_{11} + 2 \beta_{14} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{6} ) q^{71} + ( 5 \beta_{10} - 8 \beta_{12} - 5 \beta_{13} + 7 \beta_{15} ) q^{72} + ( 8 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} ) q^{73} + ( -4 \beta_{2} - 8 \beta_{4} ) q^{74} + ( 4 + 4 \beta_{1} - 4 \beta_{3} ) q^{76} + ( -3 \beta_{10} + 3 \beta_{13} - 2 \beta_{15} ) q^{77} + ( 4 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} ) q^{78} + ( 3 \beta_{2} + 8 \beta_{11} + 6 \beta_{14} ) q^{79} + ( 3 - 5 \beta_{1} - 3 \beta_{6} ) q^{81} + ( 4 \beta_{10} + 4 \beta_{13} ) q^{82} + ( 8 \beta_{7} - 2 \beta_{8} + 8 \beta_{9} ) q^{83} + ( \beta_{2} - \beta_{4} - 4 \beta_{11} + 2 \beta_{14} ) q^{84} + ( -8 \beta_{1} - 8 \beta_{3} - 8 \beta_{6} ) q^{86} + ( -5 \beta_{10} - 4 \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{87} + ( 8 \beta_{5} + 8 \beta_{7} - 8 \beta_{9} ) q^{88} + ( -4 \beta_{2} - 8 \beta_{4} ) q^{89} + ( \beta_{1} - \beta_{3} ) q^{91} + ( -6 \beta_{10} + 6 \beta_{13} - 6 \beta_{15} ) q^{92} + ( -8 \beta_{5} + 2 \beta_{8} + 2 \beta_{9} ) q^{93} + ( 4 \beta_{2} - 6 \beta_{11} + 8 \beta_{14} ) q^{94} + ( -5 + \beta_{1} - \beta_{3} + 4 \beta_{6} ) q^{96} + ( -\beta_{10} - \beta_{13} ) q^{97} + ( \beta_{7} - \beta_{8} + \beta_{9} ) q^{98} + ( 8 \beta_{2} + 7 \beta_{4} + 4 \beta_{11} + 7 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 16q^{6} + O(q^{10})$$ $$16q + 16q^{6} - 56q^{16} - 4q^{21} - 16q^{31} - 96q^{36} + 48q^{46} + 4q^{51} + 16q^{61} + 168q^{66} + 96q^{76} + 28q^{81} + 8q^{91} - 72q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 7 x^{12} - 32 x^{8} - 567 x^{4} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$7 \nu^{12} + 32 \nu^{8} + 2368 \nu^{4} - 6561$$$$)/12960$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{14} - 208 \nu^{10} + 4048 \nu^{6} - 1053 \nu^{2}$$$$)/58320$$ $$\beta_{3}$$ $$=$$ $$($$$$13 \nu^{12} - 64 \nu^{8} - 416 \nu^{4} - 7371$$$$)/4320$$ $$\beta_{4}$$ $$=$$ $$($$$$-25 \nu^{14} + 256 \nu^{10} - 496 \nu^{6} + 34911 \nu^{2}$$$$)/58320$$ $$\beta_{5}$$ $$=$$ $$($$$$37 \nu^{13} - 16 \nu^{9} - 1184 \nu^{5} - 20979 \nu$$$$)/19440$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{12} + 32 \nu^{8} - 224 \nu^{4} - 5265$$$$)/1296$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{13} - 1079 \nu$$$$)/480$$ $$\beta_{8}$$ $$=$$ $$($$$$11 \nu^{13} + 4 \nu^{9} + 296 \nu^{5} - 6885 \nu$$$$)/4860$$ $$\beta_{9}$$ $$=$$ $$($$$$-7 \nu^{13} - 32 \nu^{9} + 224 \nu^{5} + 6561 \nu$$$$)/2592$$ $$\beta_{10}$$ $$=$$ $$($$$$31 \nu^{15} + 512 \nu^{11} - 992 \nu^{7} - 17577 \nu^{3}$$$$)/116640$$ $$\beta_{11}$$ $$=$$ $$($$$$-7 \nu^{14} - 32 \nu^{10} + 62 \nu^{6} + 6561 \nu^{2}$$$$)/7290$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{15} + 359 \nu^{3}$$$$)/2160$$ $$\beta_{13}$$ $$=$$ $$($$$$49 \nu^{15} + 224 \nu^{11} + 1024 \nu^{7} - 45927 \nu^{3}$$$$)/69984$$ $$\beta_{14}$$ $$=$$ $$($$$$253 \nu^{14} + 416 \nu^{10} - 8096 \nu^{6} - 143451 \nu^{2}$$$$)/116640$$ $$\beta_{15}$$ $$=$$ $$($$$$-43 \nu^{15} - 104 \nu^{11} + 2024 \nu^{7} + 28593 \nu^{3}$$$$)/43740$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{8} - 2 \beta_{7} + \beta_{5}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{14} + 3 \beta_{11} + 3 \beta_{4} + 2 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} - 2 \beta_{13} - 4 \beta_{12} + 2 \beta_{10}$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{6} + 10 \beta_{1} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{9} + 15 \beta_{8} - 15 \beta_{5}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-5 \beta_{11} + 8 \beta_{4} + 16 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$($$$$35 \beta_{15} + 26 \beta_{13} - 35 \beta_{12}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$39 \beta_{6} - 70 \beta_{3} + 39$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$-74 \beta_{9} + 37 \beta_{8} - 74 \beta_{7} - 68 \beta_{5}$$ $$\nu^{10}$$ $$=$$ $$($$$$-62 \beta_{14} - 253 \beta_{11} + 253 \beta_{4}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$93 \beta_{15} + 93 \beta_{12} + 506 \beta_{10}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$80 \beta_{6} + 160 \beta_{3} + 160 \beta_{1} + 679$$ $$\nu^{13}$$ $$=$$ $$($$$$1079 \beta_{8} - 1198 \beta_{7} + 1079 \beta_{5}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$2158 \beta_{14} + 1797 \beta_{11} + 1797 \beta_{4} + 2158 \beta_{2}$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$359 \beta_{15} - 718 \beta_{13} - 3596 \beta_{12} + 718 \beta_{10}$$

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-\beta_{11}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
218.1
 1.73122 − 0.0537601i 0.0537601 − 1.73122i 1.47240 + 0.912166i −0.912166 − 1.47240i 0.912166 + 1.47240i −1.47240 − 0.912166i −0.0537601 + 1.73122i −1.73122 + 0.0537601i 1.73122 + 0.0537601i 0.0537601 + 1.73122i 1.47240 − 0.912166i −0.912166 + 1.47240i 0.912166 − 1.47240i −1.47240 + 0.912166i −0.0537601 − 1.73122i −1.73122 − 0.0537601i
−1.78498 1.78498i −1.47240 0.912166i 4.37228i 0 1.00000 + 4.25639i 0.707107 0.707107i 4.23447 4.23447i 1.33591 + 2.68614i 0
218.2 −1.78498 1.78498i 0.912166 + 1.47240i 4.37228i 0 1.00000 4.25639i −0.707107 + 0.707107i 4.23447 4.23447i −1.33591 + 2.68614i 0
218.3 −0.560232 0.560232i −1.73122 + 0.0537601i 1.37228i 0 1.00000 + 0.939764i −0.707107 + 0.707107i −1.88926 + 1.88926i 2.99422 0.186141i 0
218.4 −0.560232 0.560232i −0.0537601 + 1.73122i 1.37228i 0 1.00000 0.939764i 0.707107 0.707107i −1.88926 + 1.88926i −2.99422 0.186141i 0
218.5 0.560232 + 0.560232i 0.0537601 1.73122i 1.37228i 0 1.00000 0.939764i −0.707107 + 0.707107i 1.88926 1.88926i −2.99422 0.186141i 0
218.6 0.560232 + 0.560232i 1.73122 0.0537601i 1.37228i 0 1.00000 + 0.939764i 0.707107 0.707107i 1.88926 1.88926i 2.99422 0.186141i 0
218.7 1.78498 + 1.78498i −0.912166 1.47240i 4.37228i 0 1.00000 4.25639i 0.707107 0.707107i −4.23447 + 4.23447i −1.33591 + 2.68614i 0
218.8 1.78498 + 1.78498i 1.47240 + 0.912166i 4.37228i 0 1.00000 + 4.25639i −0.707107 + 0.707107i −4.23447 + 4.23447i 1.33591 + 2.68614i 0
407.1 −1.78498 + 1.78498i −1.47240 + 0.912166i 4.37228i 0 1.00000 4.25639i 0.707107 + 0.707107i 4.23447 + 4.23447i 1.33591 2.68614i 0
407.2 −1.78498 + 1.78498i 0.912166 1.47240i 4.37228i 0 1.00000 + 4.25639i −0.707107 0.707107i 4.23447 + 4.23447i −1.33591 2.68614i 0
407.3 −0.560232 + 0.560232i −1.73122 0.0537601i 1.37228i 0 1.00000 0.939764i −0.707107 0.707107i −1.88926 1.88926i 2.99422 + 0.186141i 0
407.4 −0.560232 + 0.560232i −0.0537601 1.73122i 1.37228i 0 1.00000 + 0.939764i 0.707107 + 0.707107i −1.88926 1.88926i −2.99422 + 0.186141i 0
407.5 0.560232 0.560232i 0.0537601 + 1.73122i 1.37228i 0 1.00000 + 0.939764i −0.707107 0.707107i 1.88926 + 1.88926i −2.99422 + 0.186141i 0
407.6 0.560232 0.560232i 1.73122 + 0.0537601i 1.37228i 0 1.00000 0.939764i 0.707107 + 0.707107i 1.88926 + 1.88926i 2.99422 + 0.186141i 0
407.7 1.78498 1.78498i −0.912166 + 1.47240i 4.37228i 0 1.00000 + 4.25639i 0.707107 + 0.707107i −4.23447 4.23447i −1.33591 2.68614i 0
407.8 1.78498 1.78498i 1.47240 0.912166i 4.37228i 0 1.00000 4.25639i −0.707107 0.707107i −4.23447 4.23447i 1.33591 2.68614i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 407.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.j.a 16
3.b odd 2 1 inner 525.2.j.a 16
5.b even 2 1 inner 525.2.j.a 16
5.c odd 4 2 inner 525.2.j.a 16
15.d odd 2 1 inner 525.2.j.a 16
15.e even 4 2 inner 525.2.j.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.j.a 16 1.a even 1 1 trivial
525.2.j.a 16 3.b odd 2 1 inner
525.2.j.a 16 5.b even 2 1 inner
525.2.j.a 16 5.c odd 4 2 inner
525.2.j.a 16 15.d odd 2 1 inner
525.2.j.a 16 15.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 41 T_{2}^{4} + 16$$ acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} + 24 T^{8} + 16 T^{12} + 256 T^{16} )^{2}$$
$3$ $$1 - 7 T^{4} - 32 T^{8} - 567 T^{12} + 6561 T^{16}$$
$5$ 
$7$ $$( 1 + T^{4} )^{4}$$
$11$ $$( 1 + 7 T^{2} + 180 T^{4} + 847 T^{6} + 14641 T^{8} )^{4}$$
$13$ $$( 1 - 47 T^{4} + 47568 T^{8} - 1342367 T^{12} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 + 193 T^{4} - 1920 T^{8} + 16119553 T^{12} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 - 22 T^{2} + 361 T^{4} )^{8}$$
$23$ $$( 1 + 98 T^{4} + 279841 T^{8} )^{4}$$
$29$ $$( 1 + 73 T^{2} + 2808 T^{4} + 61393 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 + 2 T + 30 T^{2} + 62 T^{3} + 961 T^{4} )^{8}$$
$37$ $$( 1 - 2012 T^{4} + 3915558 T^{8} - 3770811932 T^{12} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 8 T^{2} + 78 T^{4} + 13448 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 - 3214 T^{4} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 - 2087 T^{4} + 6045360 T^{8} - 10183894247 T^{12} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 - 3836 T^{4} + 14179686 T^{8} - 30267885116 T^{12} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 + 64 T^{2} + 4686 T^{4} + 222784 T^{6} + 12117361 T^{8} )^{4}$$
$61$ $$( 1 - 2 T + 90 T^{2} - 122 T^{3} + 3721 T^{4} )^{8}$$
$67$ $$( 1 + 388 T^{4} - 29618010 T^{8} + 7818634948 T^{12} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 208 T^{2} + 19710 T^{4} - 1048528 T^{6} + 25411681 T^{8} )^{4}$$
$73$ $$( 1 - 188 T^{4} - 49166394 T^{8} - 5338869308 T^{12} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 83 T^{2} + 1656 T^{4} - 518003 T^{6} + 38950081 T^{8} )^{4}$$
$83$ $$( 1 - 9692 T^{4} + 102045030 T^{8} - 459966047132 T^{12} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 + 52 T^{2} - 2490 T^{4} + 411892 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 + 31201 T^{4} + 419298624 T^{8} + 2762202096481 T^{12} + 7837433594376961 T^{16} )^{2}$$