# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
524.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
−1.73205 1.73205 1.00000 0 −3.00000 1.73205 2.00000i 1.73205 3.00000 0
524.2 −1.73205 1.73205 1.00000 0 −3.00000 1.73205 + 2.00000i 1.73205 3.00000 0
524.3 1.73205 −1.73205 1.00000 0 −3.00000 −1.73205 2.00000i −1.73205 3.00000 0
524.4 1.73205 −1.73205 1.00000 0 −3.00000 −1.73205 + 2.00000i −1.73205 3.00000 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
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.g.b 4
3.b odd 2 1 525.2.g.c 4
5.b even 2 1 inner 525.2.g.b 4
5.c odd 4 1 105.2.b.a 2
5.c odd 4 1 525.2.b.a 2
7.b odd 2 1 525.2.g.c 4
15.d odd 2 1 525.2.g.c 4
15.e even 4 1 105.2.b.b yes 2
15.e even 4 1 525.2.b.b 2
20.e even 4 1 1680.2.f.b 2
21.c even 2 1 inner 525.2.g.b 4
35.c odd 2 1 525.2.g.c 4
35.f even 4 1 105.2.b.b yes 2
35.f even 4 1 525.2.b.b 2
35.k even 12 1 735.2.s.a 2
35.k even 12 1 735.2.s.f 2
35.l odd 12 1 735.2.s.b 2
35.l odd 12 1 735.2.s.d 2
60.l odd 4 1 1680.2.f.c 2
105.g even 2 1 inner 525.2.g.b 4
105.k odd 4 1 105.2.b.a 2
105.k odd 4 1 525.2.b.a 2
105.w odd 12 1 735.2.s.b 2
105.w odd 12 1 735.2.s.d 2
105.x even 12 1 735.2.s.a 2
105.x even 12 1 735.2.s.f 2
140.j odd 4 1 1680.2.f.c 2
420.w even 4 1 1680.2.f.b 2

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

## Hecke kernels

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

 $$T_{2}^{2} - 3$$ $$T_{41} - 6$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} + 4 T^{4} )^{2}$$
$3$ $$( 1 - 3 T^{2} )^{2}$$
$5$ 
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 10 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 13 T^{2} )^{4}$$
$17$ $$( 1 + 2 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 34 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 10 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 50 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )^{2}( 1 + 12 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 6 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 22 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 50 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 53 T^{2} )^{4}$$
$59$ $$( 1 - 12 T + 59 T^{2} )^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 70 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 130 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 98 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 83 T^{2} )^{4}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{4}$$
$97$ $$( 1 + 146 T^{2} + 9409 T^{4} )^{2}$$