# Properties

 Label 525.4.d.c 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 21) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{2} -3 i q^{3} - q^{4} + 9 q^{6} -7 i q^{7} + 21 i q^{8} -9 q^{9} +O(q^{10})$$ $$q + 3 i q^{2} -3 i q^{3} - q^{4} + 9 q^{6} -7 i q^{7} + 21 i q^{8} -9 q^{9} -36 q^{11} + 3 i q^{12} -34 i q^{13} + 21 q^{14} -71 q^{16} -42 i q^{17} -27 i q^{18} + 124 q^{19} -21 q^{21} -108 i q^{22} + 63 q^{24} + 102 q^{26} + 27 i q^{27} + 7 i q^{28} -102 q^{29} -160 q^{31} -45 i q^{32} + 108 i q^{33} + 126 q^{34} + 9 q^{36} -398 i q^{37} + 372 i q^{38} -102 q^{39} -318 q^{41} -63 i q^{42} -268 i q^{43} + 36 q^{44} -240 i q^{47} + 213 i q^{48} -49 q^{49} -126 q^{51} + 34 i q^{52} -498 i q^{53} -81 q^{54} + 147 q^{56} -372 i q^{57} -306 i q^{58} + 132 q^{59} + 398 q^{61} -480 i q^{62} + 63 i q^{63} -433 q^{64} -324 q^{66} -92 i q^{67} + 42 i q^{68} -720 q^{71} -189 i q^{72} -502 i q^{73} + 1194 q^{74} -124 q^{76} + 252 i q^{77} -306 i q^{78} + 1024 q^{79} + 81 q^{81} -954 i q^{82} -204 i q^{83} + 21 q^{84} + 804 q^{86} + 306 i q^{87} -756 i q^{88} -354 q^{89} -238 q^{91} + 480 i q^{93} + 720 q^{94} -135 q^{96} + 286 i q^{97} -147 i q^{98} + 324 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 18q^{6} - 18q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 18q^{6} - 18q^{9} - 72q^{11} + 42q^{14} - 142q^{16} + 248q^{19} - 42q^{21} + 126q^{24} + 204q^{26} - 204q^{29} - 320q^{31} + 252q^{34} + 18q^{36} - 204q^{39} - 636q^{41} + 72q^{44} - 98q^{49} - 252q^{51} - 162q^{54} + 294q^{56} + 264q^{59} + 796q^{61} - 866q^{64} - 648q^{66} - 1440q^{71} + 2388q^{74} - 248q^{76} + 2048q^{79} + 162q^{81} + 42q^{84} + 1608q^{86} - 708q^{89} - 476q^{91} + 1440q^{94} - 270q^{96} + 648q^{99} + O(q^{100})$$

## 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.c 2
5.b even 2 1 inner 525.4.d.c 2
5.c odd 4 1 21.4.a.a 1
5.c odd 4 1 525.4.a.g 1
15.e even 4 1 63.4.a.c 1
15.e even 4 1 1575.4.a.b 1
20.e even 4 1 336.4.a.f 1
35.f even 4 1 147.4.a.c 1
35.k even 12 2 147.4.e.g 2
35.l odd 12 2 147.4.e.i 2
40.i odd 4 1 1344.4.a.ba 1
40.k even 4 1 1344.4.a.n 1
60.l odd 4 1 1008.4.a.v 1
105.k odd 4 1 441.4.a.j 1
105.w odd 12 2 441.4.e.d 2
105.x even 12 2 441.4.e.b 2
140.j odd 4 1 2352.4.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 5.c odd 4 1
63.4.a.c 1 15.e even 4 1
147.4.a.c 1 35.f even 4 1
147.4.e.g 2 35.k even 12 2
147.4.e.i 2 35.l odd 12 2
336.4.a.f 1 20.e even 4 1
441.4.a.j 1 105.k odd 4 1
441.4.e.b 2 105.x even 12 2
441.4.e.d 2 105.w odd 12 2
525.4.a.g 1 5.c odd 4 1
525.4.d.c 2 1.a even 1 1 trivial
525.4.d.c 2 5.b even 2 1 inner
1008.4.a.v 1 60.l odd 4 1
1344.4.a.n 1 40.k even 4 1
1344.4.a.ba 1 40.i odd 4 1
1575.4.a.b 1 15.e even 4 1
2352.4.a.r 1 140.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 $$S_{4}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{2} + 9$$ $$T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 7 T^{2} + 64 T^{4}$$
$3$ $$1 + 9 T^{2}$$
$5$ 1
$7$ $$1 + 49 T^{2}$$
$11$ $$( 1 + 36 T + 1331 T^{2} )^{2}$$
$13$ $$1 - 3238 T^{2} + 4826809 T^{4}$$
$17$ $$1 - 8062 T^{2} + 24137569 T^{4}$$
$19$ $$( 1 - 124 T + 6859 T^{2} )^{2}$$
$23$ $$( 1 - 12167 T^{2} )^{2}$$
$29$ $$( 1 + 102 T + 24389 T^{2} )^{2}$$
$31$ $$( 1 + 160 T + 29791 T^{2} )^{2}$$
$37$ $$1 + 57098 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 + 318 T + 68921 T^{2} )^{2}$$
$43$ $$1 - 87190 T^{2} + 6321363049 T^{4}$$
$47$ $$1 - 150046 T^{2} + 10779215329 T^{4}$$
$53$ $$1 - 49750 T^{2} + 22164361129 T^{4}$$
$59$ $$( 1 - 132 T + 205379 T^{2} )^{2}$$
$61$ $$( 1 - 398 T + 226981 T^{2} )^{2}$$
$67$ $$1 - 593062 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 + 720 T + 357911 T^{2} )^{2}$$
$73$ $$1 - 526030 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 - 1024 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 1101958 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 + 354 T + 704969 T^{2} )^{2}$$
$97$ $$1 - 1743550 T^{2} + 832972004929 T^{4}$$