# Properties

 Label 525.4.a.j 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + 3 q^{3} + (3 \beta - 3) q^{4} + ( - 3 \beta - 3) q^{6} + 7 q^{7} + (5 \beta - 1) q^{8} + 9 q^{9}+O(q^{10})$$ q + (-b - 1) * q^2 + 3 * q^3 + (3*b - 3) * q^4 + (-3*b - 3) * q^6 + 7 * q^7 + (5*b - 1) * q^8 + 9 * q^9 $$q + ( - \beta - 1) q^{2} + 3 q^{3} + (3 \beta - 3) q^{4} + ( - 3 \beta - 3) q^{6} + 7 q^{7} + (5 \beta - 1) q^{8} + 9 q^{9} + (5 \beta - 18) q^{11} + (9 \beta - 9) q^{12} + ( - 17 \beta - 11) q^{13} + ( - 7 \beta - 7) q^{14} + ( - 33 \beta + 5) q^{16} + ( - 27 \beta + 53) q^{17} + ( - 9 \beta - 9) q^{18} + (56 \beta - 56) q^{19} + 21 q^{21} + (8 \beta - 2) q^{22} + ( - 24 \beta - 115) q^{23} + (15 \beta - 3) q^{24} + (45 \beta + 79) q^{26} + 27 q^{27} + (21 \beta - 21) q^{28} + (84 \beta - 73) q^{29} + ( - 13 \beta - 61) q^{31} + (21 \beta + 135) q^{32} + (15 \beta - 54) q^{33} + (\beta + 55) q^{34} + (27 \beta - 27) q^{36} + (19 \beta - 66) q^{37} + ( - 56 \beta - 168) q^{38} + ( - 51 \beta - 33) q^{39} + (45 \beta + 95) q^{41} + ( - 21 \beta - 21) q^{42} + ( - 58 \beta - 373) q^{43} + ( - 54 \beta + 114) q^{44} + (163 \beta + 211) q^{46} + (88 \beta - 120) q^{47} + ( - 99 \beta + 15) q^{48} + 49 q^{49} + ( - 81 \beta + 159) q^{51} + ( - 33 \beta - 171) q^{52} + (89 \beta - 119) q^{53} + ( - 27 \beta - 27) q^{54} + (35 \beta - 7) q^{56} + (168 \beta - 168) q^{57} + ( - 95 \beta - 263) q^{58} + (209 \beta - 325) q^{59} + ( - 273 \beta + 25) q^{61} + (87 \beta + 113) q^{62} + 63 q^{63} + (87 \beta - 259) q^{64} + (24 \beta - 6) q^{66} + (123 \beta - 640) q^{67} + (159 \beta - 483) q^{68} + ( - 72 \beta - 345) q^{69} + (235 \beta + 192) q^{71} + (45 \beta - 9) q^{72} + ( - 88 \beta - 90) q^{73} + (28 \beta - 10) q^{74} + ( - 168 \beta + 840) q^{76} + (35 \beta - 126) q^{77} + (135 \beta + 237) q^{78} + ( - 103 \beta - 162) q^{79} + 81 q^{81} + ( - 185 \beta - 275) q^{82} + (431 \beta - 821) q^{83} + (63 \beta - 63) q^{84} + (489 \beta + 605) q^{86} + (252 \beta - 219) q^{87} + ( - 70 \beta + 118) q^{88} + ( - 522 \beta + 494) q^{89} + ( - 119 \beta - 77) q^{91} + ( - 345 \beta + 57) q^{92} + ( - 39 \beta - 183) q^{93} + ( - 56 \beta - 232) q^{94} + (63 \beta + 405) q^{96} + ( - 704 \beta + 266) q^{97} + ( - 49 \beta - 49) q^{98} + (45 \beta - 162) q^{99}+O(q^{100})$$ q + (-b - 1) * q^2 + 3 * q^3 + (3*b - 3) * q^4 + (-3*b - 3) * q^6 + 7 * q^7 + (5*b - 1) * q^8 + 9 * q^9 + (5*b - 18) * q^11 + (9*b - 9) * q^12 + (-17*b - 11) * q^13 + (-7*b - 7) * q^14 + (-33*b + 5) * q^16 + (-27*b + 53) * q^17 + (-9*b - 9) * q^18 + (56*b - 56) * q^19 + 21 * q^21 + (8*b - 2) * q^22 + (-24*b - 115) * q^23 + (15*b - 3) * q^24 + (45*b + 79) * q^26 + 27 * q^27 + (21*b - 21) * q^28 + (84*b - 73) * q^29 + (-13*b - 61) * q^31 + (21*b + 135) * q^32 + (15*b - 54) * q^33 + (b + 55) * q^34 + (27*b - 27) * q^36 + (19*b - 66) * q^37 + (-56*b - 168) * q^38 + (-51*b - 33) * q^39 + (45*b + 95) * q^41 + (-21*b - 21) * q^42 + (-58*b - 373) * q^43 + (-54*b + 114) * q^44 + (163*b + 211) * q^46 + (88*b - 120) * q^47 + (-99*b + 15) * q^48 + 49 * q^49 + (-81*b + 159) * q^51 + (-33*b - 171) * q^52 + (89*b - 119) * q^53 + (-27*b - 27) * q^54 + (35*b - 7) * q^56 + (168*b - 168) * q^57 + (-95*b - 263) * q^58 + (209*b - 325) * q^59 + (-273*b + 25) * q^61 + (87*b + 113) * q^62 + 63 * q^63 + (87*b - 259) * q^64 + (24*b - 6) * q^66 + (123*b - 640) * q^67 + (159*b - 483) * q^68 + (-72*b - 345) * q^69 + (235*b + 192) * q^71 + (45*b - 9) * q^72 + (-88*b - 90) * q^73 + (28*b - 10) * q^74 + (-168*b + 840) * q^76 + (35*b - 126) * q^77 + (135*b + 237) * q^78 + (-103*b - 162) * q^79 + 81 * q^81 + (-185*b - 275) * q^82 + (431*b - 821) * q^83 + (63*b - 63) * q^84 + (489*b + 605) * q^86 + (252*b - 219) * q^87 + (-70*b + 118) * q^88 + (-522*b + 494) * q^89 + (-119*b - 77) * q^91 + (-345*b + 57) * q^92 + (-39*b - 183) * q^93 + (-56*b - 232) * q^94 + (63*b + 405) * q^96 + (-704*b + 266) * q^97 + (-49*b - 49) * q^98 + (45*b - 162) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 6 * q^3 - 3 * q^4 - 9 * q^6 + 14 * q^7 + 3 * q^8 + 18 * q^9 $$2 q - 3 q^{2} + 6 q^{3} - 3 q^{4} - 9 q^{6} + 14 q^{7} + 3 q^{8} + 18 q^{9} - 31 q^{11} - 9 q^{12} - 39 q^{13} - 21 q^{14} - 23 q^{16} + 79 q^{17} - 27 q^{18} - 56 q^{19} + 42 q^{21} + 4 q^{22} - 254 q^{23} + 9 q^{24} + 203 q^{26} + 54 q^{27} - 21 q^{28} - 62 q^{29} - 135 q^{31} + 291 q^{32} - 93 q^{33} + 111 q^{34} - 27 q^{36} - 113 q^{37} - 392 q^{38} - 117 q^{39} + 235 q^{41} - 63 q^{42} - 804 q^{43} + 174 q^{44} + 585 q^{46} - 152 q^{47} - 69 q^{48} + 98 q^{49} + 237 q^{51} - 375 q^{52} - 149 q^{53} - 81 q^{54} + 21 q^{56} - 168 q^{57} - 621 q^{58} - 441 q^{59} - 223 q^{61} + 313 q^{62} + 126 q^{63} - 431 q^{64} + 12 q^{66} - 1157 q^{67} - 807 q^{68} - 762 q^{69} + 619 q^{71} + 27 q^{72} - 268 q^{73} + 8 q^{74} + 1512 q^{76} - 217 q^{77} + 609 q^{78} - 427 q^{79} + 162 q^{81} - 735 q^{82} - 1211 q^{83} - 63 q^{84} + 1699 q^{86} - 186 q^{87} + 166 q^{88} + 466 q^{89} - 273 q^{91} - 231 q^{92} - 405 q^{93} - 520 q^{94} + 873 q^{96} - 172 q^{97} - 147 q^{98} - 279 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 6 * q^3 - 3 * q^4 - 9 * q^6 + 14 * q^7 + 3 * q^8 + 18 * q^9 - 31 * q^11 - 9 * q^12 - 39 * q^13 - 21 * q^14 - 23 * q^16 + 79 * q^17 - 27 * q^18 - 56 * q^19 + 42 * q^21 + 4 * q^22 - 254 * q^23 + 9 * q^24 + 203 * q^26 + 54 * q^27 - 21 * q^28 - 62 * q^29 - 135 * q^31 + 291 * q^32 - 93 * q^33 + 111 * q^34 - 27 * q^36 - 113 * q^37 - 392 * q^38 - 117 * q^39 + 235 * q^41 - 63 * q^42 - 804 * q^43 + 174 * q^44 + 585 * q^46 - 152 * q^47 - 69 * q^48 + 98 * q^49 + 237 * q^51 - 375 * q^52 - 149 * q^53 - 81 * q^54 + 21 * q^56 - 168 * q^57 - 621 * q^58 - 441 * q^59 - 223 * q^61 + 313 * q^62 + 126 * q^63 - 431 * q^64 + 12 * q^66 - 1157 * q^67 - 807 * q^68 - 762 * q^69 + 619 * q^71 + 27 * q^72 - 268 * q^73 + 8 * q^74 + 1512 * q^76 - 217 * q^77 + 609 * q^78 - 427 * q^79 + 162 * q^81 - 735 * q^82 - 1211 * q^83 - 63 * q^84 + 1699 * q^86 - 186 * q^87 + 166 * q^88 + 466 * q^89 - 273 * q^91 - 231 * q^92 - 405 * q^93 - 520 * q^94 + 873 * q^96 - 172 * q^97 - 147 * q^98 - 279 * 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
 2.56155 −1.56155
−3.56155 3.00000 4.68466 0 −10.6847 7.00000 11.8078 9.00000 0
1.2 0.561553 3.00000 −7.68466 0 1.68466 7.00000 −8.80776 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.j 2
3.b odd 2 1 1575.4.a.x 2
5.b even 2 1 525.4.a.m yes 2
5.c odd 4 2 525.4.d.m 4
15.d odd 2 1 1575.4.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.j 2 1.a even 1 1 trivial
525.4.a.m yes 2 5.b even 2 1
525.4.d.m 4 5.c odd 4 2
1575.4.a.o 2 15.d odd 2 1
1575.4.a.x 2 3.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 3T_{2} - 2$$ T2^2 + 3*T2 - 2 $$T_{11}^{2} + 31T_{11} + 134$$ T11^2 + 31*T11 + 134

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T - 2$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 7)^{2}$$
$11$ $$T^{2} + 31T + 134$$
$13$ $$T^{2} + 39T - 848$$
$17$ $$T^{2} - 79T - 1538$$
$19$ $$T^{2} + 56T - 12544$$
$23$ $$T^{2} + 254T + 13681$$
$29$ $$T^{2} + 62T - 29027$$
$31$ $$T^{2} + 135T + 3838$$
$37$ $$T^{2} + 113T + 1658$$
$41$ $$T^{2} - 235T + 5200$$
$43$ $$T^{2} + 804T + 147307$$
$47$ $$T^{2} + 152T - 27136$$
$53$ $$T^{2} + 149T - 28114$$
$59$ $$T^{2} + 441T - 137024$$
$61$ $$T^{2} + 223T - 304316$$
$67$ $$T^{2} + 1157 T + 270364$$
$71$ $$T^{2} - 619T - 138916$$
$73$ $$T^{2} + 268T - 14956$$
$79$ $$T^{2} + 427T + 494$$
$83$ $$T^{2} + 1211 T - 422854$$
$89$ $$T^{2} - 466 T - 1103768$$
$97$ $$T^{2} + 172 T - 2098972$$