# Properties

 Label 525.4.a.p Level $525$ Weight $4$ Character orbit 525.a Self dual yes Analytic conductor $30.976$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 - \beta ) q^{2} + 3 q^{3} + ( 12 - 7 \beta ) q^{4} + ( 12 - 3 \beta ) q^{6} + 7 q^{7} + ( 44 - 25 \beta ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( 4 - \beta ) q^{2} + 3 q^{3} + ( 12 - 7 \beta ) q^{4} + ( 12 - 3 \beta ) q^{6} + 7 q^{7} + ( 44 - 25 \beta ) q^{8} + 9 q^{9} + ( -18 + 10 \beta ) q^{11} + ( 36 - 21 \beta ) q^{12} + ( 4 - 22 \beta ) q^{13} + ( 28 - 7 \beta ) q^{14} + ( 180 - 63 \beta ) q^{16} + ( -22 + 28 \beta ) q^{17} + ( 36 - 9 \beta ) q^{18} + ( 74 + 26 \beta ) q^{19} + 21 q^{21} + ( -112 + 48 \beta ) q^{22} + ( -120 + 56 \beta ) q^{23} + ( 132 - 75 \beta ) q^{24} + ( 104 - 70 \beta ) q^{26} + 27 q^{27} + ( 84 - 49 \beta ) q^{28} + ( -58 + 84 \beta ) q^{29} + ( 174 - 18 \beta ) q^{31} + ( 620 - 169 \beta ) q^{32} + ( -54 + 30 \beta ) q^{33} + ( -200 + 106 \beta ) q^{34} + ( 108 - 63 \beta ) q^{36} + ( 54 + 24 \beta ) q^{37} + ( 192 + 4 \beta ) q^{38} + ( 12 - 66 \beta ) q^{39} + ( 170 - 140 \beta ) q^{41} + ( 84 - 21 \beta ) q^{42} + ( -148 - 68 \beta ) q^{43} + ( -496 + 176 \beta ) q^{44} + ( -704 + 288 \beta ) q^{46} + ( -200 + 108 \beta ) q^{47} + ( 540 - 189 \beta ) q^{48} + 49 q^{49} + ( -66 + 84 \beta ) q^{51} + ( 664 - 138 \beta ) q^{52} + ( -124 + 214 \beta ) q^{53} + ( 108 - 27 \beta ) q^{54} + ( 308 - 175 \beta ) q^{56} + ( 222 + 78 \beta ) q^{57} + ( -568 + 310 \beta ) q^{58} + ( 200 - 36 \beta ) q^{59} + ( 270 + 252 \beta ) q^{61} + ( 768 - 228 \beta ) q^{62} + 63 q^{63} + ( 1716 - 623 \beta ) q^{64} + ( -336 + 144 \beta ) q^{66} + ( 380 + 28 \beta ) q^{67} + ( -1048 + 294 \beta ) q^{68} + ( -360 + 168 \beta ) q^{69} + ( 62 + 330 \beta ) q^{71} + ( 396 - 225 \beta ) q^{72} + ( -300 - 178 \beta ) q^{73} + ( 120 + 18 \beta ) q^{74} + ( 160 - 388 \beta ) q^{76} + ( -126 + 70 \beta ) q^{77} + ( 312 - 210 \beta ) q^{78} + ( 248 - 88 \beta ) q^{79} + 81 q^{81} + ( 1240 - 590 \beta ) q^{82} + ( -436 - 264 \beta ) q^{83} + ( 252 - 147 \beta ) q^{84} + ( -320 - 56 \beta ) q^{86} + ( -174 + 252 \beta ) q^{87} + ( -1792 + 640 \beta ) q^{88} + ( -346 + 728 \beta ) q^{89} + ( 28 - 154 \beta ) q^{91} + ( -3008 + 1120 \beta ) q^{92} + ( 522 - 54 \beta ) q^{93} + ( -1232 + 524 \beta ) q^{94} + ( 1860 - 507 \beta ) q^{96} + ( 176 + 146 \beta ) q^{97} + ( 196 - 49 \beta ) q^{98} + ( -162 + 90 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 7 q^{2} + 6 q^{3} + 17 q^{4} + 21 q^{6} + 14 q^{7} + 63 q^{8} + 18 q^{9} + O(q^{10})$$ $$2 q + 7 q^{2} + 6 q^{3} + 17 q^{4} + 21 q^{6} + 14 q^{7} + 63 q^{8} + 18 q^{9} - 26 q^{11} + 51 q^{12} - 14 q^{13} + 49 q^{14} + 297 q^{16} - 16 q^{17} + 63 q^{18} + 174 q^{19} + 42 q^{21} - 176 q^{22} - 184 q^{23} + 189 q^{24} + 138 q^{26} + 54 q^{27} + 119 q^{28} - 32 q^{29} + 330 q^{31} + 1071 q^{32} - 78 q^{33} - 294 q^{34} + 153 q^{36} + 132 q^{37} + 388 q^{38} - 42 q^{39} + 200 q^{41} + 147 q^{42} - 364 q^{43} - 816 q^{44} - 1120 q^{46} - 292 q^{47} + 891 q^{48} + 98 q^{49} - 48 q^{51} + 1190 q^{52} - 34 q^{53} + 189 q^{54} + 441 q^{56} + 522 q^{57} - 826 q^{58} + 364 q^{59} + 792 q^{61} + 1308 q^{62} + 126 q^{63} + 2809 q^{64} - 528 q^{66} + 788 q^{67} - 1802 q^{68} - 552 q^{69} + 454 q^{71} + 567 q^{72} - 778 q^{73} + 258 q^{74} - 68 q^{76} - 182 q^{77} + 414 q^{78} + 408 q^{79} + 162 q^{81} + 1890 q^{82} - 1136 q^{83} + 357 q^{84} - 696 q^{86} - 96 q^{87} - 2944 q^{88} + 36 q^{89} - 98 q^{91} - 4896 q^{92} + 990 q^{93} - 1940 q^{94} + 3213 q^{96} + 498 q^{97} + 343 q^{98} - 234 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.56155 −1.56155
1.43845 3.00000 −5.93087 0 4.31534 7.00000 −20.0388 9.00000 0
1.2 5.56155 3.00000 22.9309 0 16.6847 7.00000 83.0388 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.4.a.p 2
3.b odd 2 1 1575.4.a.m 2
5.b even 2 1 105.4.a.c 2
5.c odd 4 2 525.4.d.i 4
15.d odd 2 1 315.4.a.m 2
20.d odd 2 1 1680.4.a.bk 2
35.c odd 2 1 735.4.a.k 2
105.g even 2 1 2205.4.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.c 2 5.b even 2 1
315.4.a.m 2 15.d odd 2 1
525.4.a.p 2 1.a even 1 1 trivial
525.4.d.i 4 5.c odd 4 2
735.4.a.k 2 35.c odd 2 1
1575.4.a.m 2 3.b odd 2 1
1680.4.a.bk 2 20.d odd 2 1
2205.4.a.bh 2 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2}^{2} - 7 T_{2} + 8$$ $$T_{11}^{2} + 26 T_{11} - 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$8 - 7 T + T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-256 + 26 T + T^{2}$$
$13$ $$-2008 + 14 T + T^{2}$$
$17$ $$-3268 + 16 T + T^{2}$$
$19$ $$4696 - 174 T + T^{2}$$
$23$ $$-4864 + 184 T + T^{2}$$
$29$ $$-29732 + 32 T + T^{2}$$
$31$ $$25848 - 330 T + T^{2}$$
$37$ $$1908 - 132 T + T^{2}$$
$41$ $$-73300 - 200 T + T^{2}$$
$43$ $$13472 + 364 T + T^{2}$$
$47$ $$-28256 + 292 T + T^{2}$$
$53$ $$-194344 + 34 T + T^{2}$$
$59$ $$27616 - 364 T + T^{2}$$
$61$ $$-113076 - 792 T + T^{2}$$
$67$ $$151904 - 788 T + T^{2}$$
$71$ $$-411296 - 454 T + T^{2}$$
$73$ $$16664 + 778 T + T^{2}$$
$79$ $$8704 - 408 T + T^{2}$$
$83$ $$26416 + 1136 T + T^{2}$$
$89$ $$-2252108 - 36 T + T^{2}$$
$97$ $$-28592 - 498 T + T^{2}$$