# Properties

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

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 0 1.00000 1.00000i 3.00000i −1.00000 0
274.2 1.00000i 1.00000i 1.00000 0 1.00000 1.00000i 3.00000i −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.b 2
3.b odd 2 1 1575.2.d.b 2
5.b even 2 1 inner 525.2.d.b 2
5.c odd 4 1 105.2.a.a 1
5.c odd 4 1 525.2.a.a 1
15.d odd 2 1 1575.2.d.b 2
15.e even 4 1 315.2.a.a 1
15.e even 4 1 1575.2.a.h 1
20.e even 4 1 1680.2.a.f 1
20.e even 4 1 8400.2.a.co 1
35.f even 4 1 735.2.a.f 1
35.f even 4 1 3675.2.a.f 1
35.k even 12 2 735.2.i.b 2
35.l odd 12 2 735.2.i.a 2
40.i odd 4 1 6720.2.a.p 1
40.k even 4 1 6720.2.a.bk 1
60.l odd 4 1 5040.2.a.d 1
105.k odd 4 1 2205.2.a.b 1

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

 $$T_{2}^{2} + 1$$ $$T_{11}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T^{2} + 4 T^{4}$$
$3$ $$1 + T^{2}$$
$5$ 
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}$$
$23$ $$1 + 18 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$1 - 30 T^{2} + 2209 T^{4}$$
$53$ $$1 - 6 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 2 T + 61 T^{2} )^{2}$$
$67$ $$1 - 118 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$1 - 142 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{2}$$
$83$ $$1 - 150 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 8 T + 97 T^{2} )( 1 + 8 T + 97 T^{2} )$$
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