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: $$x^{2} - 2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + 2 \beta ) q^{2} + 3 q^{3} + ( 1 + 4 \beta ) q^{4} + ( 3 + 6 \beta ) q^{6} + 7 q^{7} + ( 9 - 10 \beta ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( 1 + 2 \beta ) q^{2} + 3 q^{3} + ( 1 + 4 \beta ) q^{4} + ( 3 + 6 \beta ) q^{6} + 7 q^{7} + ( 9 - 10 \beta ) q^{8} + 9 q^{9} + ( -8 + 40 \beta ) q^{11} + ( 3 + 12 \beta ) q^{12} + ( 38 + 4 \beta ) q^{13} + ( 7 + 14 \beta ) q^{14} + ( -39 - 24 \beta ) q^{16} + ( 62 + 4 \beta ) q^{17} + ( 9 + 18 \beta ) q^{18} + ( -48 - 32 \beta ) q^{19} + 21 q^{21} + ( 152 + 24 \beta ) q^{22} + ( 8 + 68 \beta ) q^{23} + ( 27 - 30 \beta ) q^{24} + ( 54 + 80 \beta ) q^{26} + 27 q^{27} + ( 7 + 28 \beta ) q^{28} + ( 94 - 108 \beta ) q^{29} + ( -60 + 36 \beta ) q^{31} + ( -207 - 22 \beta ) q^{32} + ( -24 + 120 \beta ) q^{33} + ( 78 + 128 \beta ) q^{34} + ( 9 + 36 \beta ) q^{36} + ( 66 + 132 \beta ) q^{37} + ( -176 - 128 \beta ) q^{38} + ( 114 + 12 \beta ) q^{39} + ( 50 + 160 \beta ) q^{41} + ( 21 + 42 \beta ) q^{42} + ( 268 - 124 \beta ) q^{43} + ( 312 + 8 \beta ) q^{44} + ( 280 + 84 \beta ) q^{46} + ( 464 + 84 \beta ) q^{47} + ( -117 - 72 \beta ) q^{48} + 49 q^{49} + ( 186 + 12 \beta ) q^{51} + ( 70 + 156 \beta ) q^{52} + ( -442 - 128 \beta ) q^{53} + ( 27 + 54 \beta ) q^{54} + ( 63 - 70 \beta ) q^{56} + ( -144 - 96 \beta ) q^{57} + ( -338 + 80 \beta ) q^{58} + ( 52 - 408 \beta ) q^{59} + ( -234 - 84 \beta ) q^{61} + ( 84 - 84 \beta ) q^{62} + 63 q^{63} + ( 17 - 244 \beta ) q^{64} + ( 456 + 72 \beta ) q^{66} + ( 844 - 76 \beta ) q^{67} + ( 94 + 252 \beta ) q^{68} + ( 24 + 204 \beta ) q^{69} + ( -68 - 300 \beta ) q^{71} + ( 81 - 90 \beta ) q^{72} + ( -254 - 644 \beta ) q^{73} + ( 594 + 264 \beta ) q^{74} + ( -304 - 224 \beta ) q^{76} + ( -56 + 280 \beta ) q^{77} + ( 162 + 240 \beta ) q^{78} + ( -216 - 464 \beta ) q^{79} + 81 q^{81} + ( 690 + 260 \beta ) q^{82} + ( 292 + 168 \beta ) q^{83} + ( 21 + 84 \beta ) q^{84} + ( -228 + 412 \beta ) q^{86} + ( 282 - 324 \beta ) q^{87} + ( -872 + 440 \beta ) q^{88} + ( -702 + 224 \beta ) q^{89} + ( 266 + 28 \beta ) q^{91} + ( 552 + 100 \beta ) q^{92} + ( -180 + 108 \beta ) q^{93} + ( 800 + 1012 \beta ) q^{94} + ( -621 - 66 \beta ) q^{96} + ( 594 - 92 \beta ) q^{97} + ( 49 + 98 \beta ) q^{98} + ( -72 + 360 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 6q^{3} + 2q^{4} + 6q^{6} + 14q^{7} + 18q^{8} + 18q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 6q^{3} + 2q^{4} + 6q^{6} + 14q^{7} + 18q^{8} + 18q^{9} - 16q^{11} + 6q^{12} + 76q^{13} + 14q^{14} - 78q^{16} + 124q^{17} + 18q^{18} - 96q^{19} + 42q^{21} + 304q^{22} + 16q^{23} + 54q^{24} + 108q^{26} + 54q^{27} + 14q^{28} + 188q^{29} - 120q^{31} - 414q^{32} - 48q^{33} + 156q^{34} + 18q^{36} + 132q^{37} - 352q^{38} + 228q^{39} + 100q^{41} + 42q^{42} + 536q^{43} + 624q^{44} + 560q^{46} + 928q^{47} - 234q^{48} + 98q^{49} + 372q^{51} + 140q^{52} - 884q^{53} + 54q^{54} + 126q^{56} - 288q^{57} - 676q^{58} + 104q^{59} - 468q^{61} + 168q^{62} + 126q^{63} + 34q^{64} + 912q^{66} + 1688q^{67} + 188q^{68} + 48q^{69} - 136q^{71} + 162q^{72} - 508q^{73} + 1188q^{74} - 608q^{76} - 112q^{77} + 324q^{78} - 432q^{79} + 162q^{81} + 1380q^{82} + 584q^{83} + 42q^{84} - 456q^{86} + 564q^{87} - 1744q^{88} - 1404q^{89} + 532q^{91} + 1104q^{92} - 360q^{93} + 1600q^{94} - 1242q^{96} + 1188q^{97} + 98q^{98} - 144q^{99} + O(q^{100})$$

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
 −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 $$S_{4}^{\mathrm{new}}(\Gamma_0(525))$$:

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 9 T^{2} - 16 T^{3} + 64 T^{4}$$
$3$ $$( 1 - 3 T )^{2}$$
$5$ 1
$7$ $$( 1 - 7 T )^{2}$$
$11$ $$1 + 16 T - 474 T^{2} + 21296 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 76 T + 5806 T^{2} - 166972 T^{3} + 4826809 T^{4}$$
$17$ $$1 - 124 T + 13638 T^{2} - 609212 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 96 T + 13974 T^{2} + 658464 T^{3} + 47045881 T^{4}$$
$23$ $$1 - 16 T + 15150 T^{2} - 194672 T^{3} + 148035889 T^{4}$$
$29$ $$1 - 188 T + 34286 T^{2} - 4585132 T^{3} + 594823321 T^{4}$$
$31$ $$1 + 120 T + 60590 T^{2} + 3574920 T^{3} + 887503681 T^{4}$$
$37$ $$1 - 132 T + 70814 T^{2} - 6686196 T^{3} + 2565726409 T^{4}$$
$41$ $$1 - 100 T + 89142 T^{2} - 6892100 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 536 T + 200086 T^{2} - 42615752 T^{3} + 6321363049 T^{4}$$
$47$ $$1 - 928 T + 408830 T^{2} - 96347744 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 884 T + 460350 T^{2} + 131607268 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 104 T + 80534 T^{2} - 21359416 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 468 T + 494606 T^{2} + 106227108 T^{3} + 51520374361 T^{4}$$
$67$ $$1 - 1688 T + 1302310 T^{2} - 507687944 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 136 T + 540446 T^{2} + 48675896 T^{3} + 128100283921 T^{4}$$
$73$ $$1 + 508 T + 13078 T^{2} + 197620636 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 432 T + 602142 T^{2} + 212992848 T^{3} + 243087455521 T^{4}$$
$83$ $$1 - 584 T + 1172390 T^{2} - 333923608 T^{3} + 326940373369 T^{4}$$
$89$ $$1 + 1404 T + 1802390 T^{2} + 989776476 T^{3} + 496981290961 T^{4}$$
$97$ $$1 - 1188 T + 2161254 T^{2} - 1084255524 T^{3} + 832972004929 T^{4}$$