Properties

 Label 525.4.d.b 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: 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 3 i q^{3} -8 q^{4} -12 q^{6} -7 i q^{7} -9 q^{9} +O(q^{10})$$ $$q + 4 i q^{2} + 3 i q^{3} -8 q^{4} -12 q^{6} -7 i q^{7} -9 q^{9} + 62 q^{11} -24 i q^{12} + 62 i q^{13} + 28 q^{14} -64 q^{16} + 84 i q^{17} -36 i q^{18} -100 q^{19} + 21 q^{21} + 248 i q^{22} + 42 i q^{23} -248 q^{26} -27 i q^{27} + 56 i q^{28} + 10 q^{29} -48 q^{31} -256 i q^{32} + 186 i q^{33} -336 q^{34} + 72 q^{36} -246 i q^{37} -400 i q^{38} -186 q^{39} -248 q^{41} + 84 i q^{42} -68 i q^{43} -496 q^{44} -168 q^{46} + 324 i q^{47} -192 i q^{48} -49 q^{49} -252 q^{51} -496 i q^{52} -258 i q^{53} + 108 q^{54} -300 i q^{57} + 40 i q^{58} -120 q^{59} + 622 q^{61} -192 i q^{62} + 63 i q^{63} + 512 q^{64} -744 q^{66} + 904 i q^{67} -672 i q^{68} -126 q^{69} -678 q^{71} + 642 i q^{73} + 984 q^{74} + 800 q^{76} -434 i q^{77} -744 i q^{78} -740 q^{79} + 81 q^{81} -992 i q^{82} -468 i q^{83} -168 q^{84} + 272 q^{86} + 30 i q^{87} -200 q^{89} + 434 q^{91} -336 i q^{92} -144 i q^{93} -1296 q^{94} + 768 q^{96} -1266 i q^{97} -196 i q^{98} -558 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 16q^{4} - 24q^{6} - 18q^{9} + O(q^{10})$$ $$2q - 16q^{4} - 24q^{6} - 18q^{9} + 124q^{11} + 56q^{14} - 128q^{16} - 200q^{19} + 42q^{21} - 496q^{26} + 20q^{29} - 96q^{31} - 672q^{34} + 144q^{36} - 372q^{39} - 496q^{41} - 992q^{44} - 336q^{46} - 98q^{49} - 504q^{51} + 216q^{54} - 240q^{59} + 1244q^{61} + 1024q^{64} - 1488q^{66} - 252q^{69} - 1356q^{71} + 1968q^{74} + 1600q^{76} - 1480q^{79} + 162q^{81} - 336q^{84} + 544q^{86} - 400q^{89} + 868q^{91} - 2592q^{94} + 1536q^{96} - 1116q^{99} + O(q^{100})$$

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
4.00000i 3.00000i −8.00000 0 −12.0000 7.00000i 0 −9.00000 0
274.2 4.00000i 3.00000i −8.00000 0 −12.0000 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.b 2
5.b even 2 1 inner 525.4.d.b 2
5.c odd 4 1 21.4.a.b 1
5.c odd 4 1 525.4.a.b 1
15.e even 4 1 63.4.a.a 1
15.e even 4 1 1575.4.a.k 1
20.e even 4 1 336.4.a.h 1
35.f even 4 1 147.4.a.g 1
35.k even 12 2 147.4.e.b 2
35.l odd 12 2 147.4.e.c 2
40.i odd 4 1 1344.4.a.w 1
40.k even 4 1 1344.4.a.i 1
60.l odd 4 1 1008.4.a.m 1
105.k odd 4 1 441.4.a.b 1
105.w odd 12 2 441.4.e.n 2
105.x even 12 2 441.4.e.m 2
140.j odd 4 1 2352.4.a.l 1

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

 $$T_{2}^{2} + 16$$ $$T_{11} - 62$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$49 + T^{2}$$
$11$ $$( -62 + T )^{2}$$
$13$ $$3844 + T^{2}$$
$17$ $$7056 + T^{2}$$
$19$ $$( 100 + T )^{2}$$
$23$ $$1764 + T^{2}$$
$29$ $$( -10 + T )^{2}$$
$31$ $$( 48 + T )^{2}$$
$37$ $$60516 + T^{2}$$
$41$ $$( 248 + T )^{2}$$
$43$ $$4624 + T^{2}$$
$47$ $$104976 + T^{2}$$
$53$ $$66564 + T^{2}$$
$59$ $$( 120 + T )^{2}$$
$61$ $$( -622 + T )^{2}$$
$67$ $$817216 + T^{2}$$
$71$ $$( 678 + T )^{2}$$
$73$ $$412164 + T^{2}$$
$79$ $$( 740 + T )^{2}$$
$83$ $$219024 + T^{2}$$
$89$ $$( 200 + T )^{2}$$
$97$ $$1602756 + T^{2}$$