# Properties

 Label 525.4.d.f Level $525$ Weight $4$ Character orbit 525.d Analytic conductor $30.976$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q -3 i q^{3} + 8 q^{4} -7 i q^{7} -9 q^{9} +O(q^{10})$$ $$q -3 i q^{3} + 8 q^{4} -7 i q^{7} -9 q^{9} + 42 q^{11} -24 i q^{12} + 20 i q^{13} + 64 q^{16} -66 i q^{17} -38 q^{19} -21 q^{21} + 12 i q^{23} + 27 i q^{27} -56 i q^{28} + 258 q^{29} + 146 q^{31} -126 i q^{33} -72 q^{36} -434 i q^{37} + 60 q^{39} -282 q^{41} + 20 i q^{43} + 336 q^{44} + 72 i q^{47} -192 i q^{48} -49 q^{49} -198 q^{51} + 160 i q^{52} + 336 i q^{53} + 114 i q^{57} + 360 q^{59} -682 q^{61} + 63 i q^{63} + 512 q^{64} -812 i q^{67} -528 i q^{68} + 36 q^{69} + 810 q^{71} -124 i q^{73} -304 q^{76} -294 i q^{77} -1136 q^{79} + 81 q^{81} + 156 i q^{83} -168 q^{84} -774 i q^{87} + 1038 q^{89} + 140 q^{91} + 96 i q^{92} -438 i q^{93} -1208 i q^{97} -378 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 16q^{4} - 18q^{9} + O(q^{10})$$ $$2q + 16q^{4} - 18q^{9} + 84q^{11} + 128q^{16} - 76q^{19} - 42q^{21} + 516q^{29} + 292q^{31} - 144q^{36} + 120q^{39} - 564q^{41} + 672q^{44} - 98q^{49} - 396q^{51} + 720q^{59} - 1364q^{61} + 1024q^{64} + 72q^{69} + 1620q^{71} - 608q^{76} - 2272q^{79} + 162q^{81} - 336q^{84} + 2076q^{89} + 280q^{91} - 756q^{99} + O(q^{100})$$

## 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
0 3.00000i 8.00000 0 0 7.00000i 0 −9.00000 0
274.2 0 3.00000i 8.00000 0 0 7.00000i 0 −9.00000 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 525.4.d.f 2
5.b even 2 1 inner 525.4.d.f 2
5.c odd 4 1 105.4.a.a 1
5.c odd 4 1 525.4.a.e 1
15.e even 4 1 315.4.a.d 1
15.e even 4 1 1575.4.a.f 1
20.e even 4 1 1680.4.a.s 1
35.f even 4 1 735.4.a.c 1
105.k odd 4 1 2205.4.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.a 1 5.c odd 4 1
315.4.a.d 1 15.e even 4 1
525.4.a.e 1 5.c odd 4 1
525.4.d.f 2 1.a even 1 1 trivial
525.4.d.f 2 5.b even 2 1 inner
735.4.a.c 1 35.f even 4 1
1575.4.a.f 1 15.e even 4 1
1680.4.a.s 1 20.e even 4 1
2205.4.a.o 1 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}$$ $$T_{11} - 42$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 8 T^{2} )^{2}$$
$3$ $$1 + 9 T^{2}$$
$5$ 1
$7$ $$1 + 49 T^{2}$$
$11$ $$( 1 - 42 T + 1331 T^{2} )^{2}$$
$13$ $$1 - 3994 T^{2} + 4826809 T^{4}$$
$17$ $$1 - 5470 T^{2} + 24137569 T^{4}$$
$19$ $$( 1 + 38 T + 6859 T^{2} )^{2}$$
$23$ $$1 - 24190 T^{2} + 148035889 T^{4}$$
$29$ $$( 1 - 258 T + 24389 T^{2} )^{2}$$
$31$ $$( 1 - 146 T + 29791 T^{2} )^{2}$$
$37$ $$1 + 87050 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 + 282 T + 68921 T^{2} )^{2}$$
$43$ $$1 - 158614 T^{2} + 6321363049 T^{4}$$
$47$ $$1 - 202462 T^{2} + 10779215329 T^{4}$$
$53$ $$1 - 184858 T^{2} + 22164361129 T^{4}$$
$59$ $$( 1 - 360 T + 205379 T^{2} )^{2}$$
$61$ $$( 1 + 682 T + 226981 T^{2} )^{2}$$
$67$ $$1 + 57818 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 - 810 T + 357911 T^{2} )^{2}$$
$73$ $$1 - 762658 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 + 1136 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 1119238 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 - 1038 T + 704969 T^{2} )^{2}$$
$97$ $$1 - 366082 T^{2} + 832972004929 T^{4}$$