# Properties

 Label 525.4.d.m Level $525$ Weight $4$ Character orbit 525.d Analytic conductor $30.976$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + 3 \beta_{3} q^{4} + ( -6 + 3 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} + ( 5 \beta_{1} + \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + 3 \beta_{3} q^{4} + ( -6 + 3 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} + ( 5 \beta_{1} + \beta_{2} ) q^{8} -9 q^{9} + ( -13 - 5 \beta_{3} ) q^{11} + ( -9 \beta_{1} - 9 \beta_{2} ) q^{12} + ( -17 \beta_{1} + 11 \beta_{2} ) q^{13} + ( 14 - 7 \beta_{3} ) q^{14} + ( -28 + 33 \beta_{3} ) q^{16} + ( 27 \beta_{1} + 53 \beta_{2} ) q^{17} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{18} + 56 \beta_{3} q^{19} + 21 q^{21} + ( -8 \beta_{1} - 2 \beta_{2} ) q^{22} + ( -24 \beta_{1} + 115 \beta_{2} ) q^{23} + ( -12 + 15 \beta_{3} ) q^{24} + ( 124 - 45 \beta_{3} ) q^{26} + 27 \beta_{2} q^{27} + ( 21 \beta_{1} + 21 \beta_{2} ) q^{28} + ( -11 + 84 \beta_{3} ) q^{29} + ( -74 + 13 \beta_{3} ) q^{31} + ( -21 \beta_{1} + 135 \beta_{2} ) q^{32} + ( 15 \beta_{1} + 54 \beta_{2} ) q^{33} + ( -56 + \beta_{3} ) q^{34} -27 \beta_{3} q^{36} + ( -19 \beta_{1} - 66 \beta_{2} ) q^{37} + ( -56 \beta_{1} + 168 \beta_{2} ) q^{38} + ( 84 - 51 \beta_{3} ) q^{39} + ( 140 - 45 \beta_{3} ) q^{41} + ( 21 \beta_{1} - 21 \beta_{2} ) q^{42} + ( -58 \beta_{1} + 373 \beta_{2} ) q^{43} + ( -60 - 54 \beta_{3} ) q^{44} + ( 374 - 163 \beta_{3} ) q^{46} + ( -88 \beta_{1} - 120 \beta_{2} ) q^{47} + ( -99 \beta_{1} - 15 \beta_{2} ) q^{48} -49 q^{49} + ( 78 + 81 \beta_{3} ) q^{51} + ( 33 \beta_{1} - 171 \beta_{2} ) q^{52} + ( 89 \beta_{1} + 119 \beta_{2} ) q^{53} + ( 54 - 27 \beta_{3} ) q^{54} + ( 28 - 35 \beta_{3} ) q^{56} + ( -168 \beta_{1} - 168 \beta_{2} ) q^{57} + ( -95 \beta_{1} + 263 \beta_{2} ) q^{58} + ( 116 + 209 \beta_{3} ) q^{59} + ( -248 + 273 \beta_{3} ) q^{61} + ( -87 \beta_{1} + 113 \beta_{2} ) q^{62} -63 \beta_{2} q^{63} + ( 172 + 87 \beta_{3} ) q^{64} + ( 18 - 24 \beta_{3} ) q^{66} + ( -123 \beta_{1} - 640 \beta_{2} ) q^{67} + ( 159 \beta_{1} + 483 \beta_{2} ) q^{68} + ( 417 - 72 \beta_{3} ) q^{69} + ( 427 - 235 \beta_{3} ) q^{71} + ( -45 \beta_{1} - 9 \beta_{2} ) q^{72} + ( -88 \beta_{1} + 90 \beta_{2} ) q^{73} + ( -18 + 28 \beta_{3} ) q^{74} + ( 672 + 168 \beta_{3} ) q^{76} + ( -35 \beta_{1} - 126 \beta_{2} ) q^{77} + ( 135 \beta_{1} - 237 \beta_{2} ) q^{78} + ( 265 - 103 \beta_{3} ) q^{79} + 81 q^{81} + ( 185 \beta_{1} - 275 \beta_{2} ) q^{82} + ( 431 \beta_{1} + 821 \beta_{2} ) q^{83} + 63 \beta_{3} q^{84} + ( 1094 - 489 \beta_{3} ) q^{86} + ( -252 \beta_{1} - 219 \beta_{2} ) q^{87} + ( -70 \beta_{1} - 118 \beta_{2} ) q^{88} + ( 28 - 522 \beta_{3} ) q^{89} + ( -196 + 119 \beta_{3} ) q^{91} + ( 345 \beta_{1} + 57 \beta_{2} ) q^{92} + ( -39 \beta_{1} + 183 \beta_{2} ) q^{93} + ( 288 - 56 \beta_{3} ) q^{94} + ( 468 - 63 \beta_{3} ) q^{96} + ( 704 \beta_{1} + 266 \beta_{2} ) q^{97} + ( -49 \beta_{1} + 49 \beta_{2} ) q^{98} + ( 117 + 45 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4} - 18 q^{6} - 36 q^{9} + O(q^{10})$$ $$4 q + 6 q^{4} - 18 q^{6} - 36 q^{9} - 62 q^{11} + 42 q^{14} - 46 q^{16} + 112 q^{19} + 84 q^{21} - 18 q^{24} + 406 q^{26} + 124 q^{29} - 270 q^{31} - 222 q^{34} - 54 q^{36} + 234 q^{39} + 470 q^{41} - 348 q^{44} + 1170 q^{46} - 196 q^{49} + 474 q^{51} + 162 q^{54} + 42 q^{56} + 882 q^{59} - 446 q^{61} + 862 q^{64} + 24 q^{66} + 1524 q^{69} + 1238 q^{71} - 16 q^{74} + 3024 q^{76} + 854 q^{79} + 324 q^{81} + 126 q^{84} + 3398 q^{86} - 932 q^{89} - 546 q^{91} + 1040 q^{94} + 1746 q^{96} + 558 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 5 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 − 2.56155i − 1.56155i 1.56155i 2.56155i
3.56155i 3.00000i −4.68466 0 −10.6847 7.00000i 11.8078i −9.00000 0
274.2 0.561553i 3.00000i 7.68466 0 1.68466 7.00000i 8.80776i −9.00000 0
274.3 0.561553i 3.00000i 7.68466 0 1.68466 7.00000i 8.80776i −9.00000 0
274.4 3.56155i 3.00000i −4.68466 0 −10.6847 7.00000i 11.8078i −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{4} + 13 T_{2}^{2} + 4$$ $$T_{11}^{2} + 31 T_{11} + 134$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 13 T^{2} + T^{4}$$
$3$ $$( 9 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 49 + T^{2} )^{2}$$
$11$ $$( 134 + 31 T + T^{2} )^{2}$$
$13$ $$719104 + 3217 T^{2} + T^{4}$$
$17$ $$2365444 + 9317 T^{2} + T^{4}$$
$19$ $$( -12544 - 56 T + T^{2} )^{2}$$
$23$ $$187169761 + 37154 T^{2} + T^{4}$$
$29$ $$( -29027 - 62 T + T^{2} )^{2}$$
$31$ $$( 3838 + 135 T + T^{2} )^{2}$$
$37$ $$2748964 + 9453 T^{2} + T^{4}$$
$41$ $$( 5200 - 235 T + T^{2} )^{2}$$
$43$ $$21699352249 + 351802 T^{2} + T^{4}$$
$47$ $$736362496 + 77376 T^{2} + T^{4}$$
$53$ $$790396996 + 78429 T^{2} + T^{4}$$
$59$ $$( -137024 - 441 T + T^{2} )^{2}$$
$61$ $$( -304316 + 223 T + T^{2} )^{2}$$
$67$ $$73096692496 + 797921 T^{2} + T^{4}$$
$71$ $$( -138916 - 619 T + T^{2} )^{2}$$
$73$ $$223681936 + 101736 T^{2} + T^{4}$$
$79$ $$( 494 - 427 T + T^{2} )^{2}$$
$83$ $$178805505316 + 2312229 T^{2} + T^{4}$$
$89$ $$( -1103768 + 466 T + T^{2} )^{2}$$
$97$ $$4405683456784 + 4227528 T^{2} + T^{4}$$