# Properties

 Label 525.4.d.e 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} + 3 i q^{3} + 4 q^{4} -6 q^{6} + 7 i q^{7} + 24 i q^{8} -9 q^{9} +O(q^{10})$$ $$q + 2 i q^{2} + 3 i q^{3} + 4 q^{4} -6 q^{6} + 7 i q^{7} + 24 i q^{8} -9 q^{9} -21 q^{11} + 12 i q^{12} + 24 i q^{13} -14 q^{14} -16 q^{16} + 22 i q^{17} -18 i q^{18} -16 q^{19} -21 q^{21} -42 i q^{22} -25 i q^{23} -72 q^{24} -48 q^{26} -27 i q^{27} + 28 i q^{28} -167 q^{29} + 10 q^{31} + 160 i q^{32} -63 i q^{33} -44 q^{34} -36 q^{36} + 133 i q^{37} -32 i q^{38} -72 q^{39} -168 q^{41} -42 i q^{42} -97 i q^{43} -84 q^{44} + 50 q^{46} + 400 i q^{47} -48 i q^{48} -49 q^{49} -66 q^{51} + 96 i q^{52} -182 i q^{53} + 54 q^{54} -168 q^{56} -48 i q^{57} -334 i q^{58} -488 q^{59} + 28 q^{61} + 20 i q^{62} -63 i q^{63} -448 q^{64} + 126 q^{66} + 967 i q^{67} + 88 i q^{68} + 75 q^{69} -285 q^{71} -216 i q^{72} -838 i q^{73} -266 q^{74} -64 q^{76} -147 i q^{77} -144 i q^{78} + 469 q^{79} + 81 q^{81} -336 i q^{82} -406 i q^{83} -84 q^{84} + 194 q^{86} -501 i q^{87} -504 i q^{88} -324 q^{89} -168 q^{91} -100 i q^{92} + 30 i q^{93} -800 q^{94} -480 q^{96} + 114 i q^{97} -98 i q^{98} + 189 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{4} - 12q^{6} - 18q^{9} + O(q^{10})$$ $$2q + 8q^{4} - 12q^{6} - 18q^{9} - 42q^{11} - 28q^{14} - 32q^{16} - 32q^{19} - 42q^{21} - 144q^{24} - 96q^{26} - 334q^{29} + 20q^{31} - 88q^{34} - 72q^{36} - 144q^{39} - 336q^{41} - 168q^{44} + 100q^{46} - 98q^{49} - 132q^{51} + 108q^{54} - 336q^{56} - 976q^{59} + 56q^{61} - 896q^{64} + 252q^{66} + 150q^{69} - 570q^{71} - 532q^{74} - 128q^{76} + 938q^{79} + 162q^{81} - 168q^{84} + 388q^{86} - 648q^{89} - 336q^{91} - 1600q^{94} - 960q^{96} + 378q^{99} + O(q^{100})$$

## 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
2.00000i 3.00000i 4.00000 0 −6.00000 7.00000i 24.0000i −9.00000 0
274.2 2.00000i 3.00000i 4.00000 0 −6.00000 7.00000i 24.0000i −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.e 2
5.b even 2 1 inner 525.4.d.e 2
5.c odd 4 1 525.4.a.d 1
5.c odd 4 1 525.4.a.f yes 1
15.e even 4 1 1575.4.a.d 1
15.e even 4 1 1575.4.a.h 1

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

 $$T_{2}^{2} + 4$$ $$T_{11} + 21$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$49 + T^{2}$$
$11$ $$( 21 + T )^{2}$$
$13$ $$576 + T^{2}$$
$17$ $$484 + T^{2}$$
$19$ $$( 16 + T )^{2}$$
$23$ $$625 + T^{2}$$
$29$ $$( 167 + T )^{2}$$
$31$ $$( -10 + T )^{2}$$
$37$ $$17689 + T^{2}$$
$41$ $$( 168 + T )^{2}$$
$43$ $$9409 + T^{2}$$
$47$ $$160000 + T^{2}$$
$53$ $$33124 + T^{2}$$
$59$ $$( 488 + T )^{2}$$
$61$ $$( -28 + T )^{2}$$
$67$ $$935089 + T^{2}$$
$71$ $$( 285 + T )^{2}$$
$73$ $$702244 + T^{2}$$
$79$ $$( -469 + T )^{2}$$
$83$ $$164836 + T^{2}$$
$89$ $$( 324 + T )^{2}$$
$97$ $$12996 + T^{2}$$