# Properties

 Label 525.4.a.x Level $525$ Weight $4$ Character orbit 525.a Self dual yes Analytic conductor $30.976$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

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## 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: $$5$$ Coefficient field: 5.5.78066700.1 Defining polynomial: $$x^{5} - x^{4} - 18 x^{3} + 7 x^{2} + 30 x - 10$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}\cdot 5$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} -3 q^{3} + ( 5 - \beta_{1} - \beta_{3} ) q^{4} + 3 \beta_{1} q^{6} + 7 q^{7} + ( 6 - 5 \beta_{1} - \beta_{2} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} -3 q^{3} + ( 5 - \beta_{1} - \beta_{3} ) q^{4} + 3 \beta_{1} q^{6} + 7 q^{7} + ( 6 - 5 \beta_{1} - \beta_{2} ) q^{8} + 9 q^{9} + ( 15 + 8 \beta_{1} + \beta_{4} ) q^{11} + ( -15 + 3 \beta_{1} + 3 \beta_{3} ) q^{12} + ( -1 - 6 \beta_{1} - \beta_{2} + \beta_{4} ) q^{13} -7 \beta_{1} q^{14} + ( 29 - 5 \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} ) q^{16} + ( 18 - 6 \beta_{1} + \beta_{2} - 8 \beta_{3} - 2 \beta_{4} ) q^{17} -9 \beta_{1} q^{18} + ( 33 - 2 \beta_{1} + \beta_{2} - 6 \beta_{3} + \beta_{4} ) q^{19} -21 q^{21} + ( -102 - 10 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} ) q^{22} + ( -27 - 12 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{23} + ( -18 + 15 \beta_{1} + 3 \beta_{2} ) q^{24} + ( 84 - 10 \beta_{1} + \beta_{2} - 12 \beta_{3} - 2 \beta_{4} ) q^{26} -27 q^{27} + ( 35 - 7 \beta_{1} - 7 \beta_{3} ) q^{28} + ( 72 - 8 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{29} + ( 65 - 26 \beta_{1} - \beta_{2} + 8 \beta_{3} - \beta_{4} ) q^{31} + ( -18 - 25 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} - 2 \beta_{4} ) q^{32} + ( -45 - 24 \beta_{1} - 3 \beta_{4} ) q^{33} + ( 14 - 72 \beta_{1} - 11 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} ) q^{34} + ( 45 - 9 \beta_{1} - 9 \beta_{3} ) q^{36} + ( -74 + 24 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} + 8 \beta_{4} ) q^{37} + ( -18 - 78 \beta_{1} - 3 \beta_{2} + 14 \beta_{3} + 2 \beta_{4} ) q^{38} + ( 3 + 18 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} ) q^{39} + ( 171 + 32 \beta_{1} + 3 \beta_{2} + 20 \beta_{3} + 7 \beta_{4} ) q^{41} + 21 \beta_{1} q^{42} + ( -122 - 16 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} ) q^{43} + ( 72 + 102 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{44} + ( 132 + 4 \beta_{1} + 3 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} ) q^{46} + ( 138 - 36 \beta_{1} - 6 \beta_{2} - 10 \beta_{3} + 8 \beta_{4} ) q^{47} + ( -87 + 15 \beta_{1} + 3 \beta_{2} + 15 \beta_{3} + 6 \beta_{4} ) q^{48} + 49 q^{49} + ( -54 + 18 \beta_{1} - 3 \beta_{2} + 24 \beta_{3} + 6 \beta_{4} ) q^{51} + ( 46 - 122 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} ) q^{52} + ( 15 - 14 \beta_{1} + 11 \beta_{2} + 14 \beta_{3} - \beta_{4} ) q^{53} + 27 \beta_{1} q^{54} + ( 42 - 35 \beta_{1} - 7 \beta_{2} ) q^{56} + ( -99 + 6 \beta_{1} - 3 \beta_{2} + 18 \beta_{3} - 3 \beta_{4} ) q^{57} + ( 70 - 126 \beta_{1} - 8 \beta_{2} - 18 \beta_{3} - 4 \beta_{4} ) q^{58} + ( 144 - 28 \beta_{1} - 12 \beta_{2} + 28 \beta_{3} - 8 \beta_{4} ) q^{59} + ( 118 - 36 \beta_{1} + 16 \beta_{2} + 18 \beta_{3} - 10 \beta_{4} ) q^{61} + ( 396 - 34 \beta_{1} + 5 \beta_{2} - 44 \beta_{3} - 2 \beta_{4} ) q^{62} + 63 q^{63} + ( 13 - 49 \beta_{1} - 10 \beta_{2} + 7 \beta_{3} + 12 \beta_{4} ) q^{64} + ( 306 + 30 \beta_{1} - 6 \beta_{2} - 30 \beta_{3} ) q^{66} + ( 52 + 64 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} ) q^{67} + ( 882 - 24 \beta_{1} - 9 \beta_{2} - 98 \beta_{3} - 6 \beta_{4} ) q^{68} + ( 81 + 36 \beta_{1} - 9 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} ) q^{69} + ( 291 - 16 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - 17 \beta_{4} ) q^{71} + ( 54 - 45 \beta_{1} - 9 \beta_{2} ) q^{72} + ( -87 + 102 \beta_{1} + 13 \beta_{2} + 24 \beta_{3} - \beta_{4} ) q^{73} + ( -234 + 148 \beta_{1} + 28 \beta_{2} + 46 \beta_{3} + 4 \beta_{4} ) q^{74} + ( 864 + 42 \beta_{1} + 7 \beta_{2} - 64 \beta_{3} - 14 \beta_{4} ) q^{76} + ( 105 + 56 \beta_{1} + 7 \beta_{4} ) q^{77} + ( -252 + 30 \beta_{1} - 3 \beta_{2} + 36 \beta_{3} + 6 \beta_{4} ) q^{78} + ( -182 - 96 \beta_{1} + 14 \beta_{2} + 36 \beta_{3} - 2 \beta_{4} ) q^{79} + 81 q^{81} + ( -274 - 14 \beta_{1} + 37 \beta_{2} + 50 \beta_{3} + 6 \beta_{4} ) q^{82} + ( 312 + 48 \beta_{1} - 4 \beta_{2} - 20 \beta_{3} + 4 \beta_{4} ) q^{83} + ( -105 + 21 \beta_{1} + 21 \beta_{3} ) q^{84} + ( 198 + 148 \beta_{1} - 12 \beta_{2} - 18 \beta_{3} + 4 \beta_{4} ) q^{86} + ( -216 + 24 \beta_{1} + 6 \beta_{2} + 18 \beta_{3} ) q^{87} + ( -594 + 118 \beta_{1} - 16 \beta_{2} + 114 \beta_{3} + 24 \beta_{4} ) q^{88} + ( 81 + 72 \beta_{1} - 15 \beta_{2} - 8 \beta_{3} + 25 \beta_{4} ) q^{89} + ( -7 - 42 \beta_{1} - 7 \beta_{2} + 7 \beta_{4} ) q^{91} + ( 276 + 68 \beta_{1} + 7 \beta_{2} + 40 \beta_{3} - 2 \beta_{4} ) q^{92} + ( -195 + 78 \beta_{1} + 3 \beta_{2} - 24 \beta_{3} + 3 \beta_{4} ) q^{93} + ( 438 - 280 \beta_{1} - 58 \beta_{3} - 12 \beta_{4} ) q^{94} + ( 54 + 75 \beta_{1} + 6 \beta_{2} + 36 \beta_{3} + 6 \beta_{4} ) q^{96} + ( 83 + 162 \beta_{1} - \beta_{2} - 20 \beta_{3} - 7 \beta_{4} ) q^{97} -49 \beta_{1} q^{98} + ( 135 + 72 \beta_{1} + 9 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + q^{2} - 15q^{3} + 27q^{4} - 3q^{6} + 35q^{7} + 33q^{8} + 45q^{9} + O(q^{10})$$ $$5q + q^{2} - 15q^{3} + 27q^{4} - 3q^{6} + 35q^{7} + 33q^{8} + 45q^{9} + 66q^{11} - 81q^{12} - 2q^{13} + 7q^{14} + 155q^{16} + 108q^{17} + 9q^{18} + 174q^{19} - 105q^{21} - 506q^{22} - 116q^{23} - 99q^{24} + 446q^{26} - 135q^{27} + 189q^{28} + 370q^{29} + 342q^{31} - 55q^{32} - 198q^{33} + 112q^{34} + 243q^{36} - 408q^{37} - 34q^{38} + 6q^{39} + 802q^{41} - 21q^{42} - 584q^{43} + 290q^{44} + 640q^{46} + 716q^{47} - 465q^{48} + 245q^{49} - 324q^{51} + 338q^{52} + 98q^{53} - 27q^{54} + 231q^{56} - 522q^{57} + 482q^{58} + 704q^{59} + 650q^{61} + 2070q^{62} + 315q^{63} + 75q^{64} + 1518q^{66} + 180q^{67} + 4520q^{68} + 348q^{69} + 1470q^{71} + 297q^{72} - 534q^{73} - 1312q^{74} + 4370q^{76} + 462q^{77} - 1338q^{78} - 820q^{79} + 405q^{81} - 1338q^{82} + 1520q^{83} - 567q^{84} + 832q^{86} - 1110q^{87} - 3258q^{88} + 286q^{89} - 14q^{91} + 1288q^{92} - 1026q^{93} + 2540q^{94} + 165q^{96} + 278q^{97} + 49q^{98} + 594q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 18 x^{3} + 7 x^{2} + 30 x - 10$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{4} + 3 \nu^{3} + 27 \nu^{2} - 20 \nu + 10$$$$)/15$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{4} + 24 \nu^{3} - 84 \nu^{2} - 320 \nu + 70$$$$)/15$$ $$\beta_{3}$$ $$=$$ $$($$$$8 \nu^{4} - 12 \nu^{3} - 138 \nu^{2} + 140 \nu + 155$$$$)/15$$ $$\beta_{4}$$ $$=$$ $$($$$$-32 \nu^{4} + 18 \nu^{3} + 582 \nu^{2} + 100 \nu - 785$$$$)/15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + 4 \beta_{3} + \beta_{2} + 2 \beta_{1} + 5$$$$)/20$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} - \beta_{3} + \beta_{2} - 18 \beta_{1} + 70$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{4} + 23 \beta_{3} + 12 \beta_{2} + 4 \beta_{1} + 70$$$$)/10$$ $$\nu^{4}$$ $$=$$ $$($$$$38 \beta_{4} + 2 \beta_{3} + 53 \beta_{2} - 644 \beta_{1} + 2150$$$$)/20$$

## 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
 −1.37042 1.35311 0.329739 4.40248 −3.71490
−4.88936 −3.00000 15.9059 0 14.6681 7.00000 −38.6546 9.00000 0
1.2 −2.20666 −3.00000 −3.13065 0 6.61998 7.00000 24.5616 9.00000 0
1.3 −0.428319 −3.00000 −7.81654 0 1.28496 7.00000 6.77452 9.00000 0
1.4 3.33774 −3.00000 3.14050 0 −10.0132 7.00000 −16.2197 9.00000 0
1.5 5.18660 −3.00000 18.9008 0 −15.5598 7.00000 56.5383 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.x 5
3.b odd 2 1 1575.4.a.bo 5
5.b even 2 1 525.4.a.w 5
5.c odd 4 2 105.4.d.b 10
15.d odd 2 1 1575.4.a.bp 5
15.e even 4 2 315.4.d.b 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.d.b 10 5.c odd 4 2
315.4.d.b 10 15.e even 4 2
525.4.a.w 5 5.b even 2 1
525.4.a.x 5 1.a even 1 1 trivial
1575.4.a.bo 5 3.b odd 2 1
1575.4.a.bp 5 15.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{5} - T_{2}^{4} - 33 T_{2}^{3} + 17 T_{2}^{2} + 200 T_{2} + 80$$ $$T_{11}^{5} - 66 T_{11}^{4} - 2100 T_{11}^{3} + 140456 T_{11}^{2} + 1472448 T_{11} - 55852416$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$80 + 200 T + 17 T^{2} - 33 T^{3} - T^{4} + T^{5}$$
$3$ $$( 3 + T )^{5}$$
$5$ $$T^{5}$$
$7$ $$( -7 + T )^{5}$$
$11$ $$-55852416 + 1472448 T + 140456 T^{2} - 2100 T^{3} - 66 T^{4} + T^{5}$$
$13$ $$-41380960 + 4336080 T + 15504 T^{2} - 4152 T^{3} + 2 T^{4} + T^{5}$$
$17$ $$-232018688 + 37019968 T + 1541792 T^{2} - 16940 T^{3} - 108 T^{4} + T^{5}$$
$19$ $$-784374624 - 40426032 T + 1513744 T^{2} - 3640 T^{3} - 174 T^{4} + T^{5}$$
$23$ $$488160000 - 45825600 T - 2700320 T^{2} - 19676 T^{3} + 116 T^{4} + T^{5}$$
$29$ $$1150048 - 14986160 T - 1029840 T^{2} + 38440 T^{3} - 370 T^{4} + T^{5}$$
$31$ $$52737095200 - 1618226480 T + 13826640 T^{2} - 6904 T^{3} - 342 T^{4} + T^{5}$$
$37$ $$315202167808 + 3074210048 T - 30738752 T^{2} - 89600 T^{3} + 408 T^{4} + T^{5}$$
$41$ $$531402107648 - 12602553024 T + 74825096 T^{2} + 47852 T^{3} - 802 T^{4} + T^{5}$$
$43$ $$-132088069120 - 5499585280 T - 36399168 T^{2} + 15872 T^{3} + 584 T^{4} + T^{5}$$
$47$ $$65728742400 - 13646008320 T + 87933440 T^{2} - 11696 T^{3} - 716 T^{4} + T^{5}$$
$53$ $$462468251232 + 8691280848 T - 12761488 T^{2} - 241080 T^{3} - 98 T^{4} + T^{5}$$
$59$ $$-33724261457920 + 47252515840 T + 380481408 T^{2} - 599408 T^{3} - 704 T^{4} + T^{5}$$
$61$ $$1105143174112 + 47217562320 T + 214899120 T^{2} - 445560 T^{3} - 650 T^{4} + T^{5}$$
$67$ $$1579424171008 - 34615928320 T + 233215360 T^{2} - 479120 T^{3} - 180 T^{4} + T^{5}$$
$71$ $$237519904000 - 40906747200 T + 376607000 T^{2} + 92060 T^{3} - 1470 T^{4} + T^{5}$$
$73$ $$-1941655936032 + 8346269136 T + 99791920 T^{2} - 603064 T^{3} + 534 T^{4} + T^{5}$$
$79$ $$43229481181184 + 35855795200 T - 590031680 T^{2} - 728400 T^{3} + 820 T^{4} + T^{5}$$
$83$ $$128527499264 - 9802956800 T - 59036160 T^{2} + 667840 T^{3} - 1520 T^{4} + T^{5}$$
$89$ $$1125486676224 + 41779572288 T + 206349496 T^{2} - 1347380 T^{3} - 286 T^{4} + T^{5}$$
$97$ $$-19868737339616 + 145383131216 T + 6097680 T^{2} - 1133656 T^{3} - 278 T^{4} + T^{5}$$
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