# Properties

 Label 525.4.a.l 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + 3 q^{3} + (2 \beta + 1) q^{4} + (3 \beta + 3) q^{6} + 7 q^{7} + ( - 5 \beta + 9) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b + 1) * q^2 + 3 * q^3 + (2*b + 1) * q^4 + (3*b + 3) * q^6 + 7 * q^7 + (-5*b + 9) * q^8 + 9 * q^9 $$q + (\beta + 1) q^{2} + 3 q^{3} + (2 \beta + 1) q^{4} + (3 \beta + 3) q^{6} + 7 q^{7} + ( - 5 \beta + 9) q^{8} + 9 q^{9} + (20 \beta - 8) q^{11} + (6 \beta + 3) q^{12} + (2 \beta + 38) q^{13} + (7 \beta + 7) q^{14} + ( - 12 \beta - 39) q^{16} + (2 \beta + 62) q^{17} + (9 \beta + 9) q^{18} + ( - 16 \beta - 48) q^{19} + 21 q^{21} + (12 \beta + 152) q^{22} + (34 \beta + 8) q^{23} + ( - 15 \beta + 27) q^{24} + (40 \beta + 54) q^{26} + 27 q^{27} + (14 \beta + 7) q^{28} + ( - 54 \beta + 94) q^{29} + (18 \beta - 60) q^{31} + ( - 11 \beta - 207) q^{32} + (60 \beta - 24) q^{33} + (64 \beta + 78) q^{34} + (18 \beta + 9) q^{36} + (66 \beta + 66) q^{37} + ( - 64 \beta - 176) q^{38} + (6 \beta + 114) q^{39} + (80 \beta + 50) q^{41} + (21 \beta + 21) q^{42} + ( - 62 \beta + 268) q^{43} + (4 \beta + 312) q^{44} + (42 \beta + 280) q^{46} + (42 \beta + 464) q^{47} + ( - 36 \beta - 117) q^{48} + 49 q^{49} + (6 \beta + 186) q^{51} + (78 \beta + 70) q^{52} + ( - 64 \beta - 442) q^{53} + (27 \beta + 27) q^{54} + ( - 35 \beta + 63) q^{56} + ( - 48 \beta - 144) q^{57} + (40 \beta - 338) q^{58} + ( - 204 \beta + 52) q^{59} + ( - 42 \beta - 234) q^{61} + ( - 42 \beta + 84) q^{62} + 63 q^{63} + ( - 122 \beta + 17) q^{64} + (36 \beta + 456) q^{66} + ( - 38 \beta + 844) q^{67} + (126 \beta + 94) q^{68} + (102 \beta + 24) q^{69} + ( - 150 \beta - 68) q^{71} + ( - 45 \beta + 81) q^{72} + ( - 322 \beta - 254) q^{73} + (132 \beta + 594) q^{74} + ( - 112 \beta - 304) q^{76} + (140 \beta - 56) q^{77} + (120 \beta + 162) q^{78} + ( - 232 \beta - 216) q^{79} + 81 q^{81} + (130 \beta + 690) q^{82} + (84 \beta + 292) q^{83} + (42 \beta + 21) q^{84} + (206 \beta - 228) q^{86} + ( - 162 \beta + 282) q^{87} + (220 \beta - 872) q^{88} + (112 \beta - 702) q^{89} + (14 \beta + 266) q^{91} + (50 \beta + 552) q^{92} + (54 \beta - 180) q^{93} + (506 \beta + 800) q^{94} + ( - 33 \beta - 621) q^{96} + ( - 46 \beta + 594) q^{97} + (49 \beta + 49) q^{98} + (180 \beta - 72) q^{99}+O(q^{100})$$ q + (b + 1) * q^2 + 3 * q^3 + (2*b + 1) * q^4 + (3*b + 3) * q^6 + 7 * q^7 + (-5*b + 9) * q^8 + 9 * q^9 + (20*b - 8) * q^11 + (6*b + 3) * q^12 + (2*b + 38) * q^13 + (7*b + 7) * q^14 + (-12*b - 39) * q^16 + (2*b + 62) * q^17 + (9*b + 9) * q^18 + (-16*b - 48) * q^19 + 21 * q^21 + (12*b + 152) * q^22 + (34*b + 8) * q^23 + (-15*b + 27) * q^24 + (40*b + 54) * q^26 + 27 * q^27 + (14*b + 7) * q^28 + (-54*b + 94) * q^29 + (18*b - 60) * q^31 + (-11*b - 207) * q^32 + (60*b - 24) * q^33 + (64*b + 78) * q^34 + (18*b + 9) * q^36 + (66*b + 66) * q^37 + (-64*b - 176) * q^38 + (6*b + 114) * q^39 + (80*b + 50) * q^41 + (21*b + 21) * q^42 + (-62*b + 268) * q^43 + (4*b + 312) * q^44 + (42*b + 280) * q^46 + (42*b + 464) * q^47 + (-36*b - 117) * q^48 + 49 * q^49 + (6*b + 186) * q^51 + (78*b + 70) * q^52 + (-64*b - 442) * q^53 + (27*b + 27) * q^54 + (-35*b + 63) * q^56 + (-48*b - 144) * q^57 + (40*b - 338) * q^58 + (-204*b + 52) * q^59 + (-42*b - 234) * q^61 + (-42*b + 84) * q^62 + 63 * q^63 + (-122*b + 17) * q^64 + (36*b + 456) * q^66 + (-38*b + 844) * q^67 + (126*b + 94) * q^68 + (102*b + 24) * q^69 + (-150*b - 68) * q^71 + (-45*b + 81) * q^72 + (-322*b - 254) * q^73 + (132*b + 594) * q^74 + (-112*b - 304) * q^76 + (140*b - 56) * q^77 + (120*b + 162) * q^78 + (-232*b - 216) * q^79 + 81 * q^81 + (130*b + 690) * q^82 + (84*b + 292) * q^83 + (42*b + 21) * q^84 + (206*b - 228) * q^86 + (-162*b + 282) * q^87 + (220*b - 872) * q^88 + (112*b - 702) * q^89 + (14*b + 266) * q^91 + (50*b + 552) * q^92 + (54*b - 180) * q^93 + (506*b + 800) * q^94 + (-33*b - 621) * q^96 + (-46*b + 594) * q^97 + (49*b + 49) * q^98 + (180*b - 72) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 6 q^{3} + 2 q^{4} + 6 q^{6} + 14 q^{7} + 18 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 6 * q^3 + 2 * q^4 + 6 * q^6 + 14 * q^7 + 18 * q^8 + 18 * q^9 $$2 q + 2 q^{2} + 6 q^{3} + 2 q^{4} + 6 q^{6} + 14 q^{7} + 18 q^{8} + 18 q^{9} - 16 q^{11} + 6 q^{12} + 76 q^{13} + 14 q^{14} - 78 q^{16} + 124 q^{17} + 18 q^{18} - 96 q^{19} + 42 q^{21} + 304 q^{22} + 16 q^{23} + 54 q^{24} + 108 q^{26} + 54 q^{27} + 14 q^{28} + 188 q^{29} - 120 q^{31} - 414 q^{32} - 48 q^{33} + 156 q^{34} + 18 q^{36} + 132 q^{37} - 352 q^{38} + 228 q^{39} + 100 q^{41} + 42 q^{42} + 536 q^{43} + 624 q^{44} + 560 q^{46} + 928 q^{47} - 234 q^{48} + 98 q^{49} + 372 q^{51} + 140 q^{52} - 884 q^{53} + 54 q^{54} + 126 q^{56} - 288 q^{57} - 676 q^{58} + 104 q^{59} - 468 q^{61} + 168 q^{62} + 126 q^{63} + 34 q^{64} + 912 q^{66} + 1688 q^{67} + 188 q^{68} + 48 q^{69} - 136 q^{71} + 162 q^{72} - 508 q^{73} + 1188 q^{74} - 608 q^{76} - 112 q^{77} + 324 q^{78} - 432 q^{79} + 162 q^{81} + 1380 q^{82} + 584 q^{83} + 42 q^{84} - 456 q^{86} + 564 q^{87} - 1744 q^{88} - 1404 q^{89} + 532 q^{91} + 1104 q^{92} - 360 q^{93} + 1600 q^{94} - 1242 q^{96} + 1188 q^{97} + 98 q^{98} - 144 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 6 * q^3 + 2 * q^4 + 6 * q^6 + 14 * q^7 + 18 * q^8 + 18 * q^9 - 16 * q^11 + 6 * q^12 + 76 * q^13 + 14 * q^14 - 78 * q^16 + 124 * q^17 + 18 * q^18 - 96 * q^19 + 42 * q^21 + 304 * q^22 + 16 * q^23 + 54 * q^24 + 108 * q^26 + 54 * q^27 + 14 * q^28 + 188 * q^29 - 120 * q^31 - 414 * q^32 - 48 * q^33 + 156 * q^34 + 18 * q^36 + 132 * q^37 - 352 * q^38 + 228 * q^39 + 100 * q^41 + 42 * q^42 + 536 * q^43 + 624 * q^44 + 560 * q^46 + 928 * q^47 - 234 * q^48 + 98 * q^49 + 372 * q^51 + 140 * q^52 - 884 * q^53 + 54 * q^54 + 126 * q^56 - 288 * q^57 - 676 * q^58 + 104 * q^59 - 468 * q^61 + 168 * q^62 + 126 * q^63 + 34 * q^64 + 912 * q^66 + 1688 * q^67 + 188 * q^68 + 48 * q^69 - 136 * q^71 + 162 * q^72 - 508 * q^73 + 1188 * q^74 - 608 * q^76 - 112 * q^77 + 324 * q^78 - 432 * q^79 + 162 * q^81 + 1380 * q^82 + 584 * q^83 + 42 * q^84 - 456 * q^86 + 564 * q^87 - 1744 * q^88 - 1404 * q^89 + 532 * q^91 + 1104 * q^92 - 360 * q^93 + 1600 * q^94 - 1242 * q^96 + 1188 * q^97 + 98 * q^98 - 144 * q^99

## 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
 −1.41421 1.41421
−1.82843 3.00000 −4.65685 0 −5.48528 7.00000 23.1421 9.00000 0
1.2 3.82843 3.00000 6.65685 0 11.4853 7.00000 −5.14214 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.l 2
3.b odd 2 1 1575.4.a.q 2
5.b even 2 1 105.4.a.e 2
5.c odd 4 2 525.4.d.l 4
15.d odd 2 1 315.4.a.k 2
20.d odd 2 1 1680.4.a.bo 2
35.c odd 2 1 735.4.a.o 2
105.g even 2 1 2205.4.a.bb 2

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

 $$T_{2}^{2} - 2T_{2} - 7$$ T2^2 - 2*T2 - 7 $$T_{11}^{2} + 16T_{11} - 3136$$ T11^2 + 16*T11 - 3136

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 7$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 7)^{2}$$
$11$ $$T^{2} + 16T - 3136$$
$13$ $$T^{2} - 76T + 1412$$
$17$ $$T^{2} - 124T + 3812$$
$19$ $$T^{2} + 96T + 256$$
$23$ $$T^{2} - 16T - 9184$$
$29$ $$T^{2} - 188T - 14492$$
$31$ $$T^{2} + 120T + 1008$$
$37$ $$T^{2} - 132T - 30492$$
$41$ $$T^{2} - 100T - 48700$$
$43$ $$T^{2} - 536T + 41072$$
$47$ $$T^{2} - 928T + 201184$$
$53$ $$T^{2} + 884T + 162596$$
$59$ $$T^{2} - 104T - 330224$$
$61$ $$T^{2} + 468T + 40644$$
$67$ $$T^{2} - 1688 T + 700784$$
$71$ $$T^{2} + 136T - 175376$$
$73$ $$T^{2} + 508T - 764956$$
$79$ $$T^{2} + 432T - 383936$$
$83$ $$T^{2} - 584T + 28816$$
$89$ $$T^{2} + 1404 T + 392452$$
$97$ $$T^{2} - 1188 T + 335908$$