# Properties

 Label 525.2.b.f Level 525 Weight 2 Character orbit 525.b Analytic conductor 4.192 Analytic rank 0 Dimension 4 CM discriminant -35 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{5}, \sqrt{-7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + 2 q^{4} + ( \beta_{1} - \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + 2 q^{4} + ( \beta_{1} - \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{9} + ( -1 + 2 \beta_{2} ) q^{11} + 2 \beta_{3} q^{12} + ( \beta_{1} - \beta_{3} ) q^{13} + 4 q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} + ( 4 - \beta_{2} ) q^{21} + ( -3 \beta_{1} - \beta_{3} ) q^{27} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{28} + ( 1 - 2 \beta_{2} ) q^{29} + ( -6 \beta_{1} - \beta_{3} ) q^{33} + ( -2 + 2 \beta_{2} ) q^{36} + ( 4 - \beta_{2} ) q^{39} + ( -2 + 4 \beta_{2} ) q^{44} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{47} + 4 \beta_{3} q^{48} -7 q^{49} + ( 2 + \beta_{2} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{52} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{63} + 8 q^{64} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{68} + ( 2 - 4 \beta_{2} ) q^{71} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{73} + ( 7 \beta_{1} + 7 \beta_{3} ) q^{77} + q^{79} + ( -8 - \beta_{2} ) q^{81} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{83} + ( 8 - 2 \beta_{2} ) q^{84} + ( 6 \beta_{1} + \beta_{3} ) q^{87} -7 q^{91} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{97} + ( -17 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} - 2q^{9} + O(q^{10})$$ $$4q + 8q^{4} - 2q^{9} + 16q^{16} + 14q^{21} - 4q^{36} + 14q^{39} - 28q^{49} + 10q^{51} + 32q^{64} + 4q^{79} - 34q^{81} + 28q^{84} - 28q^{91} - 70q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 2 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 7 \nu + 8$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} + 7 \nu + 2$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} + 3 \nu + 2$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 3 \beta_{1} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{3} - 2 \beta_{2} + 3 \beta_{1} - 9$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 0.809017 + 2.14046i 0.809017 − 2.14046i −0.309017 − 0.817582i −0.309017 + 0.817582i
0 −1.11803 1.32288i 2.00000 0 0 2.64575i 0 −0.500000 + 2.95804i 0
251.2 0 −1.11803 + 1.32288i 2.00000 0 0 2.64575i 0 −0.500000 2.95804i 0
251.3 0 1.11803 1.32288i 2.00000 0 0 2.64575i 0 −0.500000 2.95804i 0
251.4 0 1.11803 + 1.32288i 2.00000 0 0 2.64575i 0 −0.500000 + 2.95804i 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
105.g 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.b.f 4
3.b odd 2 1 inner 525.2.b.f 4
5.b even 2 1 inner 525.2.b.f 4
5.c odd 4 2 105.2.g.b 4
7.b odd 2 1 inner 525.2.b.f 4
15.d odd 2 1 inner 525.2.b.f 4
15.e even 4 2 105.2.g.b 4
20.e even 4 2 1680.2.k.b 4
21.c even 2 1 inner 525.2.b.f 4
35.c odd 2 1 CM 525.2.b.f 4
35.f even 4 2 105.2.g.b 4
35.k even 12 4 735.2.p.b 8
35.l odd 12 4 735.2.p.b 8
60.l odd 4 2 1680.2.k.b 4
105.g even 2 1 inner 525.2.b.f 4
105.k odd 4 2 105.2.g.b 4
105.w odd 12 4 735.2.p.b 8
105.x even 12 4 735.2.p.b 8
140.j odd 4 2 1680.2.k.b 4
420.w even 4 2 1680.2.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.b 4 5.c odd 4 2
105.2.g.b 4 15.e even 4 2
105.2.g.b 4 35.f even 4 2
105.2.g.b 4 105.k odd 4 2
525.2.b.f 4 1.a even 1 1 trivial
525.2.b.f 4 3.b odd 2 1 inner
525.2.b.f 4 5.b even 2 1 inner
525.2.b.f 4 7.b odd 2 1 inner
525.2.b.f 4 15.d odd 2 1 inner
525.2.b.f 4 21.c even 2 1 inner
525.2.b.f 4 35.c odd 2 1 CM
525.2.b.f 4 105.g even 2 1 inner
735.2.p.b 8 35.k even 12 4
735.2.p.b 8 35.l odd 12 4
735.2.p.b 8 105.w odd 12 4
735.2.p.b 8 105.x even 12 4
1680.2.k.b 4 20.e even 4 2
1680.2.k.b 4 60.l odd 4 2
1680.2.k.b 4 140.j odd 4 2
1680.2.k.b 4 420.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}$$ $$T_{11}^{2} + 35$$ $$T_{17}^{2} - 5$$ $$T_{37}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{4}$$
$3$ $$1 + T^{2} + 9 T^{4}$$
$5$ 
$7$ $$( 1 + 7 T^{2} )^{2}$$
$11$ $$( 1 - 3 T + 11 T^{2} )^{2}( 1 + 3 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 19 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 29 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{4}$$
$23$ $$( 1 - 23 T^{2} )^{4}$$
$29$ $$( 1 - 9 T + 29 T^{2} )^{2}( 1 + 9 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 + 37 T^{2} )^{4}$$
$41$ $$( 1 + 41 T^{2} )^{4}$$
$43$ $$( 1 + 43 T^{2} )^{4}$$
$47$ $$( 1 - 31 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{4}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$( 1 - 61 T^{2} )^{4}$$
$67$ $$( 1 + 67 T^{2} )^{4}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{2}( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 34 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 86 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{4}$$
$97$ $$( 1 + 149 T^{2} + 9409 T^{4} )^{2}$$