# Properties

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

# Learn more about

## 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: yes 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{2} -3 i q^{3} - q^{4} + 9 q^{6} -7 i q^{7} + 21 i q^{8} -9 q^{9} +O(q^{10})$$ $$q + 3 i q^{2} -3 i q^{3} - q^{4} + 9 q^{6} -7 i q^{7} + 21 i q^{8} -9 q^{9} -6 q^{11} + 3 i q^{12} + 41 i q^{13} + 21 q^{14} -71 q^{16} -27 i q^{17} -27 i q^{18} + 4 q^{19} -21 q^{21} -18 i q^{22} + 75 i q^{23} + 63 q^{24} -123 q^{26} + 27 i q^{27} + 7 i q^{28} + 123 q^{29} -205 q^{31} -45 i q^{32} + 18 i q^{33} + 81 q^{34} + 9 q^{36} + 262 i q^{37} + 12 i q^{38} + 123 q^{39} + 57 q^{41} -63 i q^{42} + 407 i q^{43} + 6 q^{44} -225 q^{46} + 60 i q^{47} + 213 i q^{48} -49 q^{49} -81 q^{51} -41 i q^{52} + 327 i q^{53} -81 q^{54} + 147 q^{56} -12 i q^{57} + 369 i q^{58} -33 q^{59} -427 q^{61} -615 i q^{62} + 63 i q^{63} -433 q^{64} -54 q^{66} + 628 i q^{67} + 27 i q^{68} + 225 q^{69} + 300 q^{71} -189 i q^{72} + 98 i q^{73} -786 q^{74} -4 q^{76} + 42 i q^{77} + 369 i q^{78} -686 q^{79} + 81 q^{81} + 171 i q^{82} + 1401 i q^{83} + 21 q^{84} -1221 q^{86} -369 i q^{87} -126 i q^{88} -714 q^{89} + 287 q^{91} -75 i q^{92} + 615 i q^{93} -180 q^{94} -135 q^{96} -494 i q^{97} -147 i q^{98} + 54 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 18q^{6} - 18q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 18q^{6} - 18q^{9} - 12q^{11} + 42q^{14} - 142q^{16} + 8q^{19} - 42q^{21} + 126q^{24} - 246q^{26} + 246q^{29} - 410q^{31} + 162q^{34} + 18q^{36} + 246q^{39} + 114q^{41} + 12q^{44} - 450q^{46} - 98q^{49} - 162q^{51} - 162q^{54} + 294q^{56} - 66q^{59} - 854q^{61} - 866q^{64} - 108q^{66} + 450q^{69} + 600q^{71} - 1572q^{74} - 8q^{76} - 1372q^{79} + 162q^{81} + 42q^{84} - 2442q^{86} - 1428q^{89} + 574q^{91} - 360q^{94} - 270q^{96} + 108q^{99} + O(q^{100})$$

## 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
3.00000i 3.00000i −1.00000 0 9.00000 7.00000i 21.0000i −9.00000 0
274.2 3.00000i 3.00000i −1.00000 0 9.00000 7.00000i 21.0000i −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.d 2
5.b even 2 1 inner 525.4.d.d 2
5.c odd 4 1 525.4.a.c 1
5.c odd 4 1 525.4.a.h yes 1
15.e even 4 1 1575.4.a.a 1
15.e even 4 1 1575.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.c 1 5.c odd 4 1
525.4.a.h yes 1 5.c odd 4 1
525.4.d.d 2 1.a even 1 1 trivial
525.4.d.d 2 5.b even 2 1 inner
1575.4.a.a 1 15.e even 4 1
1575.4.a.j 1 15.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 $$S_{4}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{2} + 9$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 7 T^{2} + 64 T^{4}$$
$3$ $$1 + 9 T^{2}$$
$5$ 1
$7$ $$1 + 49 T^{2}$$
$11$ $$( 1 + 6 T + 1331 T^{2} )^{2}$$
$13$ $$1 - 2713 T^{2} + 4826809 T^{4}$$
$17$ $$1 - 9097 T^{2} + 24137569 T^{4}$$
$19$ $$( 1 - 4 T + 6859 T^{2} )^{2}$$
$23$ $$1 - 18709 T^{2} + 148035889 T^{4}$$
$29$ $$( 1 - 123 T + 24389 T^{2} )^{2}$$
$31$ $$( 1 + 205 T + 29791 T^{2} )^{2}$$
$37$ $$1 - 32662 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 - 57 T + 68921 T^{2} )^{2}$$
$43$ $$1 + 6635 T^{2} + 6321363049 T^{4}$$
$47$ $$1 - 204046 T^{2} + 10779215329 T^{4}$$
$53$ $$1 - 190825 T^{2} + 22164361129 T^{4}$$
$59$ $$( 1 + 33 T + 205379 T^{2} )^{2}$$
$61$ $$( 1 + 427 T + 226981 T^{2} )^{2}$$
$67$ $$1 - 207142 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 - 300 T + 357911 T^{2} )^{2}$$
$73$ $$1 - 768430 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 + 686 T + 493039 T^{2} )^{2}$$
$83$ $$1 + 819227 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 + 714 T + 704969 T^{2} )^{2}$$
$97$ $$1 - 1581310 T^{2} + 832972004929 T^{4}$$
show more
show less