# Properties

 Label 525.4.a.o Level $525$ Weight $4$ Character orbit 525.a Self dual yes Analytic conductor $30.976$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + 2 \beta ) q^{2} -3 q^{3} + ( -3 + 8 \beta ) q^{4} + ( -3 - 6 \beta ) q^{6} + 7 q^{7} + ( 5 + 2 \beta ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( 1 + 2 \beta ) q^{2} -3 q^{3} + ( -3 + 8 \beta ) q^{4} + ( -3 - 6 \beta ) q^{6} + 7 q^{7} + ( 5 + 2 \beta ) q^{8} + 9 q^{9} + ( -48 + 4 \beta ) q^{11} + ( 9 - 24 \beta ) q^{12} + ( 34 - 76 \beta ) q^{13} + ( 7 + 14 \beta ) q^{14} + ( 33 - 48 \beta ) q^{16} + ( -22 + 88 \beta ) q^{17} + ( 9 + 18 \beta ) q^{18} + ( -28 - 52 \beta ) q^{19} -21 q^{21} + ( -40 - 84 \beta ) q^{22} + ( 180 - 40 \beta ) q^{23} + ( -15 - 6 \beta ) q^{24} + ( -118 - 160 \beta ) q^{26} -27 q^{27} + ( -21 + 56 \beta ) q^{28} + ( -106 - 24 \beta ) q^{29} + ( 72 - 204 \beta ) q^{31} + ( -103 - 94 \beta ) q^{32} + ( 144 - 12 \beta ) q^{33} + ( 154 + 220 \beta ) q^{34} + ( -27 + 72 \beta ) q^{36} + ( -126 + 48 \beta ) q^{37} + ( -132 - 212 \beta ) q^{38} + ( -102 + 228 \beta ) q^{39} + ( -58 + 160 \beta ) q^{41} + ( -21 - 42 \beta ) q^{42} + ( -196 + 256 \beta ) q^{43} + ( 176 - 364 \beta ) q^{44} + ( 100 + 240 \beta ) q^{46} + ( -32 - 336 \beta ) q^{47} + ( -99 + 144 \beta ) q^{48} + 49 q^{49} + ( 66 - 264 \beta ) q^{51} + ( -710 - 108 \beta ) q^{52} + ( -94 + 172 \beta ) q^{53} + ( -27 - 54 \beta ) q^{54} + ( 35 + 14 \beta ) q^{56} + ( 84 + 156 \beta ) q^{57} + ( -154 - 284 \beta ) q^{58} + ( -268 + 72 \beta ) q^{59} + ( -258 - 168 \beta ) q^{61} + ( -336 - 468 \beta ) q^{62} + 63 q^{63} + ( -555 - 104 \beta ) q^{64} + ( 120 + 252 \beta ) q^{66} + ( -532 + 328 \beta ) q^{67} + ( 770 + 264 \beta ) q^{68} + ( -540 + 120 \beta ) q^{69} + ( -508 + 276 \beta ) q^{71} + ( 45 + 18 \beta ) q^{72} + ( -90 - 244 \beta ) q^{73} + ( -30 - 108 \beta ) q^{74} + ( -332 - 484 \beta ) q^{76} + ( -336 + 28 \beta ) q^{77} + ( 354 + 480 \beta ) q^{78} + ( -688 + 968 \beta ) q^{79} + 81 q^{81} + ( 262 + 364 \beta ) q^{82} + ( -220 - 168 \beta ) q^{83} + ( 63 - 168 \beta ) q^{84} + ( 316 + 376 \beta ) q^{86} + ( 318 + 72 \beta ) q^{87} + ( -232 - 68 \beta ) q^{88} + ( -778 + 224 \beta ) q^{89} + ( 238 - 532 \beta ) q^{91} + ( -860 + 1240 \beta ) q^{92} + ( -216 + 612 \beta ) q^{93} + ( -704 - 1072 \beta ) q^{94} + ( 309 + 282 \beta ) q^{96} + ( 1310 - 172 \beta ) q^{97} + ( 49 + 98 \beta ) q^{98} + ( -432 + 36 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} - 6q^{3} + 2q^{4} - 12q^{6} + 14q^{7} + 12q^{8} + 18q^{9} + O(q^{10})$$ $$2q + 4q^{2} - 6q^{3} + 2q^{4} - 12q^{6} + 14q^{7} + 12q^{8} + 18q^{9} - 92q^{11} - 6q^{12} - 8q^{13} + 28q^{14} + 18q^{16} + 44q^{17} + 36q^{18} - 108q^{19} - 42q^{21} - 164q^{22} + 320q^{23} - 36q^{24} - 396q^{26} - 54q^{27} + 14q^{28} - 236q^{29} - 60q^{31} - 300q^{32} + 276q^{33} + 528q^{34} + 18q^{36} - 204q^{37} - 476q^{38} + 24q^{39} + 44q^{41} - 84q^{42} - 136q^{43} - 12q^{44} + 440q^{46} - 400q^{47} - 54q^{48} + 98q^{49} - 132q^{51} - 1528q^{52} - 16q^{53} - 108q^{54} + 84q^{56} + 324q^{57} - 592q^{58} - 464q^{59} - 684q^{61} - 1140q^{62} + 126q^{63} - 1214q^{64} + 492q^{66} - 736q^{67} + 1804q^{68} - 960q^{69} - 740q^{71} + 108q^{72} - 424q^{73} - 168q^{74} - 1148q^{76} - 644q^{77} + 1188q^{78} - 408q^{79} + 162q^{81} + 888q^{82} - 608q^{83} - 42q^{84} + 1008q^{86} + 708q^{87} - 532q^{88} - 1332q^{89} - 56q^{91} - 480q^{92} + 180q^{93} - 2480q^{94} + 900q^{96} + 2448q^{97} + 196q^{98} - 828q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
−0.236068 −3.00000 −7.94427 0 0.708204 7.00000 3.76393 9.00000 0
1.2 4.23607 −3.00000 9.94427 0 −12.7082 7.00000 8.23607 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.o 2
3.b odd 2 1 1575.4.a.n 2
5.b even 2 1 105.4.a.d 2
5.c odd 4 2 525.4.d.k 4
15.d odd 2 1 315.4.a.l 2
20.d odd 2 1 1680.4.a.bd 2
35.c odd 2 1 735.4.a.m 2
105.g even 2 1 2205.4.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 5.b even 2 1
315.4.a.l 2 15.d odd 2 1
525.4.a.o 2 1.a even 1 1 trivial
525.4.d.k 4 5.c odd 4 2
735.4.a.m 2 35.c odd 2 1
1575.4.a.n 2 3.b odd 2 1
1680.4.a.bd 2 20.d odd 2 1
2205.4.a.be 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2}^{2} - 4 T_{2} - 1$$ $$T_{11}^{2} + 92 T_{11} + 2096$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 15 T^{2} - 32 T^{3} + 64 T^{4}$$
$3$ $$( 1 + 3 T )^{2}$$
$5$ 1
$7$ $$( 1 - 7 T )^{2}$$
$11$ $$1 + 92 T + 4758 T^{2} + 122452 T^{3} + 1771561 T^{4}$$
$13$ $$1 + 8 T - 2810 T^{2} + 17576 T^{3} + 4826809 T^{4}$$
$17$ $$1 - 44 T + 630 T^{2} - 216172 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 108 T + 13254 T^{2} + 740772 T^{3} + 47045881 T^{4}$$
$23$ $$1 - 320 T + 47934 T^{2} - 3893440 T^{3} + 148035889 T^{4}$$
$29$ $$1 + 236 T + 61982 T^{2} + 5755804 T^{3} + 594823321 T^{4}$$
$31$ $$1 + 60 T + 8462 T^{2} + 1787460 T^{3} + 887503681 T^{4}$$
$37$ $$1 + 204 T + 108830 T^{2} + 10333212 T^{3} + 2565726409 T^{4}$$
$41$ $$1 - 44 T + 106326 T^{2} - 3032524 T^{3} + 4750104241 T^{4}$$
$43$ $$1 + 136 T + 81718 T^{2} + 10812952 T^{3} + 6321363049 T^{4}$$
$47$ $$1 + 400 T + 106526 T^{2} + 41529200 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 16 T + 260838 T^{2} + 2382032 T^{3} + 22164361129 T^{4}$$
$59$ $$1 + 464 T + 458102 T^{2} + 95295856 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 684 T + 535646 T^{2} + 155255004 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 736 T + 602470 T^{2} + 221361568 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 740 T + 757502 T^{2} + 264854140 T^{3} + 128100283921 T^{4}$$
$73$ $$1 + 424 T + 748558 T^{2} + 164943208 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 408 T - 143586 T^{2} + 201159912 T^{3} + 243087455521 T^{4}$$
$83$ $$1 + 608 T + 1200710 T^{2} + 347646496 T^{3} + 326940373369 T^{4}$$
$89$ $$1 + 1332 T + 1790774 T^{2} + 939018708 T^{3} + 496981290961 T^{4}$$
$97$ $$1 - 2448 T + 3286542 T^{2} - 2234223504 T^{3} + 832972004929 T^{4}$$