# Properties

 Label 525.4.a.r Level $525$ Weight $4$ Character orbit 525.a Self dual yes Analytic conductor $30.976$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.2292.1 Defining polynomial: $$x^{3} - x^{2} - 13 x + 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -3 q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} -3 \beta_{1} q^{6} -7 q^{7} + ( 8 - 2 \beta_{1} + \beta_{2} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -3 q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} -3 \beta_{1} q^{6} -7 q^{7} + ( 8 - 2 \beta_{1} + \beta_{2} ) q^{8} + 9 q^{9} + ( -26 + 8 \beta_{1} - 4 \beta_{2} ) q^{11} + ( -3 - 3 \beta_{1} - 3 \beta_{2} ) q^{12} + ( 36 - 5 \beta_{1} + \beta_{2} ) q^{13} -7 \beta_{1} q^{14} + ( -27 + 2 \beta_{1} - 10 \beta_{2} ) q^{16} + ( 50 - 3 \beta_{1} - 5 \beta_{2} ) q^{17} + 9 \beta_{1} q^{18} + ( -44 - 7 \beta_{1} - \beta_{2} ) q^{19} + 21 q^{21} + ( 76 - 34 \beta_{1} + 8 \beta_{2} ) q^{22} + ( -52 - 23 \beta_{1} + \beta_{2} ) q^{23} + ( -24 + 6 \beta_{1} - 3 \beta_{2} ) q^{24} + ( -46 + 35 \beta_{1} - 5 \beta_{2} ) q^{26} -27 q^{27} + ( -7 - 7 \beta_{1} - 7 \beta_{2} ) q^{28} + ( 66 - 34 \beta_{1} - 6 \beta_{2} ) q^{29} + ( -104 - 47 \beta_{1} + 7 \beta_{2} ) q^{31} + ( -36 - 49 \beta_{1} - 6 \beta_{2} ) q^{32} + ( 78 - 24 \beta_{1} + 12 \beta_{2} ) q^{33} + ( -22 + 27 \beta_{1} - 3 \beta_{2} ) q^{34} + ( 9 + 9 \beta_{1} + 9 \beta_{2} ) q^{36} + ( 84 + 32 \beta_{1} + 32 \beta_{2} ) q^{37} + ( -62 - 55 \beta_{1} - 7 \beta_{2} ) q^{38} + ( -108 + 15 \beta_{1} - 3 \beta_{2} ) q^{39} + ( -34 - 61 \beta_{1} + 43 \beta_{2} ) q^{41} + 21 \beta_{1} q^{42} + ( -4 - 108 \beta_{1} - 12 \beta_{2} ) q^{43} + ( -106 + 10 \beta_{1} - 2 \beta_{2} ) q^{44} + ( -208 - 71 \beta_{1} - 23 \beta_{2} ) q^{46} + ( -52 + 82 \beta_{1} - 50 \beta_{2} ) q^{47} + ( 81 - 6 \beta_{1} + 30 \beta_{2} ) q^{48} + 49 q^{49} + ( -150 + 9 \beta_{1} + 15 \beta_{2} ) q^{51} + ( 32 + 9 \beta_{1} + 27 \beta_{2} ) q^{52} + ( 144 - 105 \beta_{1} - 27 \beta_{2} ) q^{53} -27 \beta_{1} q^{54} + ( -56 + 14 \beta_{1} - 7 \beta_{2} ) q^{56} + ( 132 + 21 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -300 + 8 \beta_{1} - 34 \beta_{2} ) q^{58} + ( -332 + 112 \beta_{1} - 4 \beta_{2} ) q^{59} + ( -306 + 106 \beta_{1} + 58 \beta_{2} ) q^{61} + ( -430 - 123 \beta_{1} - 47 \beta_{2} ) q^{62} -63 q^{63} + ( -219 - 125 \beta_{1} + 31 \beta_{2} ) q^{64} + ( -228 + 102 \beta_{1} - 24 \beta_{2} ) q^{66} + ( 20 - 90 \beta_{1} + 66 \beta_{2} ) q^{67} + ( -154 + 17 \beta_{1} + 67 \beta_{2} ) q^{68} + ( 156 + 69 \beta_{1} - 3 \beta_{2} ) q^{69} + ( -374 + 136 \beta_{1} + 32 \beta_{2} ) q^{71} + ( 72 - 18 \beta_{1} + 9 \beta_{2} ) q^{72} + ( 452 + 133 \beta_{1} - 65 \beta_{2} ) q^{73} + ( 256 + 244 \beta_{1} + 32 \beta_{2} ) q^{74} + ( -136 - 89 \beta_{1} - 47 \beta_{2} ) q^{76} + ( 182 - 56 \beta_{1} + 28 \beta_{2} ) q^{77} + ( 138 - 105 \beta_{1} + 15 \beta_{2} ) q^{78} + ( -464 - 198 \beta_{1} + 30 \beta_{2} ) q^{79} + 81 q^{81} + ( -592 + 77 \beta_{1} - 61 \beta_{2} ) q^{82} + ( -356 + 252 \beta_{1} + 56 \beta_{2} ) q^{83} + ( 21 + 21 \beta_{1} + 21 \beta_{2} ) q^{84} + ( -960 - 160 \beta_{1} - 108 \beta_{2} ) q^{86} + ( -198 + 102 \beta_{1} + 18 \beta_{2} ) q^{87} + ( -516 + 168 \beta_{1} - 54 \beta_{2} ) q^{88} + ( -650 - 31 \beta_{1} - 43 \beta_{2} ) q^{89} + ( -252 + 35 \beta_{1} - 7 \beta_{2} ) q^{91} + ( -200 - 187 \beta_{1} - 79 \beta_{2} ) q^{92} + ( 312 + 141 \beta_{1} - 21 \beta_{2} ) q^{93} + ( 788 - 170 \beta_{1} + 82 \beta_{2} ) q^{94} + ( 108 + 147 \beta_{1} + 18 \beta_{2} ) q^{96} + ( 556 + 183 \beta_{1} - 27 \beta_{2} ) q^{97} + 49 \beta_{1} q^{98} + ( -234 + 72 \beta_{1} - 36 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{2} - 9q^{3} + 3q^{4} - 3q^{6} - 21q^{7} + 21q^{8} + 27q^{9} + O(q^{10})$$ $$3q + q^{2} - 9q^{3} + 3q^{4} - 3q^{6} - 21q^{7} + 21q^{8} + 27q^{9} - 66q^{11} - 9q^{12} + 102q^{13} - 7q^{14} - 69q^{16} + 152q^{17} + 9q^{18} - 138q^{19} + 63q^{21} + 186q^{22} - 180q^{23} - 63q^{24} - 98q^{26} - 81q^{27} - 21q^{28} + 170q^{29} - 366q^{31} - 151q^{32} + 198q^{33} - 36q^{34} + 27q^{36} + 252q^{37} - 234q^{38} - 306q^{39} - 206q^{41} + 21q^{42} - 108q^{43} - 306q^{44} - 672q^{46} - 24q^{47} + 207q^{48} + 147q^{49} - 456q^{51} + 78q^{52} + 354q^{53} - 27q^{54} - 147q^{56} + 414q^{57} - 858q^{58} - 880q^{59} - 870q^{61} - 1366q^{62} - 189q^{63} - 813q^{64} - 558q^{66} - 96q^{67} - 512q^{68} + 540q^{69} - 1018q^{71} + 189q^{72} + 1554q^{73} + 980q^{74} - 450q^{76} + 462q^{77} + 294q^{78} - 1620q^{79} + 243q^{81} - 1638q^{82} - 872q^{83} + 63q^{84} - 2932q^{86} - 510q^{87} - 1326q^{88} - 1938q^{89} - 714q^{91} - 708q^{92} + 1098q^{93} + 2112q^{94} + 453q^{96} + 1878q^{97} + 49q^{98} - 594q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 13 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 9$$

## 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
 −3.18296 0.0765073 4.10645
−3.18296 −3.00000 2.13122 0 9.54887 −7.00000 18.6801 9.00000 0
1.2 0.0765073 −3.00000 −7.99415 0 −0.229522 −7.00000 −1.22367 9.00000 0
1.3 4.10645 −3.00000 8.86293 0 −12.3193 −7.00000 3.54358 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.r 3
3.b odd 2 1 1575.4.a.bc 3
5.b even 2 1 525.4.a.q 3
5.c odd 4 2 105.4.d.a 6
15.d odd 2 1 1575.4.a.bd 3
15.e even 4 2 315.4.d.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.d.a 6 5.c odd 4 2
315.4.d.a 6 15.e even 4 2
525.4.a.q 3 5.b even 2 1
525.4.a.r 3 1.a even 1 1 trivial
1575.4.a.bc 3 3.b odd 2 1
1575.4.a.bd 3 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(525))$$:

 $$T_{2}^{3} - T_{2}^{2} - 13 T_{2} + 1$$ $$T_{11}^{3} + 66 T_{11}^{2} - 276 T_{11} - 6120$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 11 T^{2} - 15 T^{3} + 88 T^{4} - 64 T^{5} + 512 T^{6}$$
$3$ $$( 1 + 3 T )^{3}$$
$5$ 1
$7$ $$( 1 + 7 T )^{3}$$
$11$ $$1 + 66 T + 3717 T^{2} + 169572 T^{3} + 4947327 T^{4} + 116923026 T^{5} + 2357947691 T^{6}$$
$13$ $$1 - 102 T + 9675 T^{2} - 476884 T^{3} + 21255975 T^{4} - 492334518 T^{5} + 10604499373 T^{6}$$
$17$ $$1 - 152 T + 20867 T^{2} - 1561824 T^{3} + 102519571 T^{4} - 3668910488 T^{5} + 118587876497 T^{6}$$
$19$ $$1 + 138 T + 26205 T^{2} + 1963716 T^{3} + 179740095 T^{4} + 6492331578 T^{5} + 322687697779 T^{6}$$
$23$ $$1 + 180 T + 40221 T^{2} + 4151304 T^{3} + 489368907 T^{4} + 26646460020 T^{5} + 1801152661463 T^{6}$$
$29$ $$1 - 170 T + 65051 T^{2} - 6611676 T^{3} + 1586528839 T^{4} - 101119964570 T^{5} + 14507145975869 T^{6}$$
$31$ $$1 + 366 T + 102201 T^{2} + 18296380 T^{3} + 3044669991 T^{4} + 324826347246 T^{5} + 26439622160671 T^{6}$$
$37$ $$1 - 252 T + 99399 T^{2} - 17307224 T^{3} + 5034857547 T^{4} - 646563055068 T^{5} + 129961739795077 T^{6}$$
$41$ $$1 + 206 T + 68783 T^{2} + 10173252 T^{3} + 4740593143 T^{4} + 978521473646 T^{5} + 327381934393961 T^{6}$$
$43$ $$1 + 108 T + 76905 T^{2} + 30874696 T^{3} + 6114485835 T^{4} + 682707209292 T^{5} + 502592611936843 T^{6}$$
$47$ $$1 + 24 T + 84141 T^{2} + 25877328 T^{3} + 8735771043 T^{4} + 258701167896 T^{5} + 1119130473102767 T^{6}$$
$53$ $$1 - 354 T + 295827 T^{2} - 51866076 T^{3} + 44041836279 T^{4} - 7846183839666 T^{5} + 3299763591802133 T^{6}$$
$59$ $$1 + 880 T + 706697 T^{2} + 338588064 T^{3} + 145140723163 T^{4} + 37118869604080 T^{5} + 8662995818654939 T^{6}$$
$61$ $$1 + 870 T + 582363 T^{2} + 282478148 T^{3} + 132185336103 T^{4} + 44822725694070 T^{5} + 11694146092834141 T^{6}$$
$67$ $$1 + 96 T + 555537 T^{2} + 22556608 T^{3} + 167084974731 T^{4} + 8684004688224 T^{5} + 27206534396294947 T^{6}$$
$71$ $$1 + 1018 T + 1108049 T^{2} + 595462884 T^{3} + 396582925639 T^{4} + 130406089031578 T^{5} + 45848500718449031 T^{6}$$
$73$ $$1 - 1554 T + 1505463 T^{2} - 1009418700 T^{3} + 585650699871 T^{4} - 235173387653106 T^{5} + 58871586708267913 T^{6}$$
$79$ $$1 + 1620 T + 1787517 T^{2} + 1338821912 T^{3} + 881315594163 T^{4} + 393801677944020 T^{5} + 119851595982618319 T^{6}$$
$83$ $$1 + 872 T + 923489 T^{2} + 308826096 T^{3} + 528039004843 T^{4} + 285092005577768 T^{5} + 186940255267540403 T^{6}$$
$89$ $$1 + 1938 T + 3246255 T^{2} + 2913932892 T^{3} + 2288509141095 T^{4} + 963149741882418 T^{5} + 350356403707485209 T^{6}$$
$97$ $$1 - 1878 T + 3431919 T^{2} - 3287491396 T^{3} + 3132219809487 T^{4} - 1564321425256662 T^{5} + 760231058654565217 T^{6}$$