# Properties

 Label 525.2.d.c Level 525 Weight 2 Character orbit 525.d 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.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ 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.

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

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
2.23607i 1.00000i −3.00000 0 −2.23607 1.00000i 2.23607i −1.00000 0
274.2 2.23607i 1.00000i −3.00000 0 2.23607 1.00000i 2.23607i −1.00000 0
274.3 2.23607i 1.00000i −3.00000 0 2.23607 1.00000i 2.23607i −1.00000 0
274.4 2.23607i 1.00000i −3.00000 0 −2.23607 1.00000i 2.23607i −1.00000 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 525.2.d.c 4
3.b odd 2 1 1575.2.d.d 4
5.b even 2 1 inner 525.2.d.c 4
5.c odd 4 1 105.2.a.b 2
5.c odd 4 1 525.2.a.g 2
15.d odd 2 1 1575.2.d.d 4
15.e even 4 1 315.2.a.d 2
15.e even 4 1 1575.2.a.r 2
20.e even 4 1 1680.2.a.v 2
20.e even 4 1 8400.2.a.cx 2
35.f even 4 1 735.2.a.k 2
35.f even 4 1 3675.2.a.y 2
35.k even 12 2 735.2.i.i 4
35.l odd 12 2 735.2.i.k 4
40.i odd 4 1 6720.2.a.cx 2
40.k even 4 1 6720.2.a.cs 2
60.l odd 4 1 5040.2.a.bw 2
105.k odd 4 1 2205.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.b 2 5.c odd 4 1
315.2.a.d 2 15.e even 4 1
525.2.a.g 2 5.c odd 4 1
525.2.d.c 4 1.a even 1 1 trivial
525.2.d.c 4 5.b even 2 1 inner
735.2.a.k 2 35.f even 4 1
735.2.i.i 4 35.k even 12 2
735.2.i.k 4 35.l odd 12 2
1575.2.a.r 2 15.e even 4 1
1575.2.d.d 4 3.b odd 2 1
1575.2.d.d 4 15.d odd 2 1
1680.2.a.v 2 20.e even 4 1
2205.2.a.w 2 105.k odd 4 1
3675.2.a.y 2 35.f even 4 1
5040.2.a.bw 2 60.l odd 4 1
6720.2.a.cs 2 40.k even 4 1
6720.2.a.cx 2 40.i odd 4 1
8400.2.a.cx 2 20.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 $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{2} + 5$$ $$T_{11}^{2} - 4 T_{11} - 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} + 4 T^{4} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ 
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 6 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{4}$$
$31$ $$( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 20 T^{2} + 1558 T^{4} + 27380 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 6 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 + 4 T^{2} - 698 T^{4} + 8836 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 44 T^{2} + 982 T^{4} - 123596 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 + 38 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 2 T + 61 T^{2} )^{4}$$
$67$ $$( 1 - 118 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$1 - 124 T^{2} + 9382 T^{4} - 660796 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$1 - 44 T^{2} - 6218 T^{4} - 303116 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{4}$$
$97$ $$1 - 316 T^{2} + 42502 T^{4} - 2973244 T^{6} + 88529281 T^{8}$$