# Properties

 Label 525.4.d.i 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + 4 \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + ( -5 - 7 \beta_{3} ) q^{4} + ( 9 + 3 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} + ( -25 \beta_{1} - 44 \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + 4 \beta_{2} ) q^{2} -3 \beta_{2} q^{3} + ( -5 - 7 \beta_{3} ) q^{4} + ( 9 + 3 \beta_{3} ) q^{6} + 7 \beta_{2} q^{7} + ( -25 \beta_{1} - 44 \beta_{2} ) q^{8} -9 q^{9} + ( -8 - 10 \beta_{3} ) q^{11} + ( 21 \beta_{1} + 36 \beta_{2} ) q^{12} + ( -22 \beta_{1} - 4 \beta_{2} ) q^{13} + ( -21 - 7 \beta_{3} ) q^{14} + ( 117 + 63 \beta_{3} ) q^{16} + ( -28 \beta_{1} - 22 \beta_{2} ) q^{17} + ( -9 \beta_{1} - 36 \beta_{2} ) q^{18} + ( -100 + 26 \beta_{3} ) q^{19} + 21 q^{21} + ( -48 \beta_{1} - 112 \beta_{2} ) q^{22} + ( 56 \beta_{1} + 120 \beta_{2} ) q^{23} + ( -57 - 75 \beta_{3} ) q^{24} + ( 34 + 70 \beta_{3} ) q^{26} + 27 \beta_{2} q^{27} + ( -49 \beta_{1} - 84 \beta_{2} ) q^{28} + ( -26 + 84 \beta_{3} ) q^{29} + ( 156 + 18 \beta_{3} ) q^{31} + ( 169 \beta_{1} + 620 \beta_{2} ) q^{32} + ( 30 \beta_{1} + 54 \beta_{2} ) q^{33} + ( 94 + 106 \beta_{3} ) q^{34} + ( 45 + 63 \beta_{3} ) q^{36} + ( -24 \beta_{1} + 54 \beta_{2} ) q^{37} + ( 4 \beta_{1} - 192 \beta_{2} ) q^{38} + ( 54 - 66 \beta_{3} ) q^{39} + ( 30 + 140 \beta_{3} ) q^{41} + ( 21 \beta_{1} + 84 \beta_{2} ) q^{42} + ( -68 \beta_{1} + 148 \beta_{2} ) q^{43} + ( 320 + 176 \beta_{3} ) q^{44} + ( -416 - 288 \beta_{3} ) q^{46} + ( -108 \beta_{1} - 200 \beta_{2} ) q^{47} + ( -189 \beta_{1} - 540 \beta_{2} ) q^{48} -49 q^{49} + ( 18 - 84 \beta_{3} ) q^{51} + ( 138 \beta_{1} + 664 \beta_{2} ) q^{52} + ( 214 \beta_{1} + 124 \beta_{2} ) q^{53} + ( -81 - 27 \beta_{3} ) q^{54} + ( 133 + 175 \beta_{3} ) q^{56} + ( -78 \beta_{1} + 222 \beta_{2} ) q^{57} + ( 310 \beta_{1} + 568 \beta_{2} ) q^{58} + ( -164 - 36 \beta_{3} ) q^{59} + ( 522 - 252 \beta_{3} ) q^{61} + ( 228 \beta_{1} + 768 \beta_{2} ) q^{62} -63 \beta_{2} q^{63} + ( -1093 - 623 \beta_{3} ) q^{64} + ( -192 - 144 \beta_{3} ) q^{66} + ( -28 \beta_{1} + 380 \beta_{2} ) q^{67} + ( 294 \beta_{1} + 1048 \beta_{2} ) q^{68} + ( 192 + 168 \beta_{3} ) q^{69} + ( 392 - 330 \beta_{3} ) q^{71} + ( 225 \beta_{1} + 396 \beta_{2} ) q^{72} + ( -178 \beta_{1} + 300 \beta_{2} ) q^{73} + ( -138 + 18 \beta_{3} ) q^{74} + ( -228 + 388 \beta_{3} ) q^{76} + ( -70 \beta_{1} - 126 \beta_{2} ) q^{77} + ( -210 \beta_{1} - 312 \beta_{2} ) q^{78} + ( -160 - 88 \beta_{3} ) q^{79} + 81 q^{81} + ( 590 \beta_{1} + 1240 \beta_{2} ) q^{82} + ( -264 \beta_{1} + 436 \beta_{2} ) q^{83} + ( -105 - 147 \beta_{3} ) q^{84} + ( -376 + 56 \beta_{3} ) q^{86} + ( -252 \beta_{1} - 174 \beta_{2} ) q^{87} + ( 640 \beta_{1} + 1792 \beta_{2} ) q^{88} + ( -382 + 728 \beta_{3} ) q^{89} + ( -126 + 154 \beta_{3} ) q^{91} + ( -1120 \beta_{1} - 3008 \beta_{2} ) q^{92} + ( -54 \beta_{1} - 522 \beta_{2} ) q^{93} + ( 708 + 524 \beta_{3} ) q^{94} + ( 1353 + 507 \beta_{3} ) q^{96} + ( -146 \beta_{1} + 176 \beta_{2} ) q^{97} + ( -49 \beta_{1} - 196 \beta_{2} ) q^{98} + ( 72 + 90 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 34q^{4} + 42q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 34q^{4} + 42q^{6} - 36q^{9} - 52q^{11} - 98q^{14} + 594q^{16} - 348q^{19} + 84q^{21} - 378q^{24} + 276q^{26} + 64q^{29} + 660q^{31} + 588q^{34} + 306q^{36} + 84q^{39} + 400q^{41} + 1632q^{44} - 2240q^{46} - 196q^{49} - 96q^{51} - 378q^{54} + 882q^{56} - 728q^{59} + 1584q^{61} - 5618q^{64} - 1056q^{66} + 1104q^{69} + 908q^{71} - 516q^{74} - 136q^{76} - 816q^{79} + 324q^{81} - 714q^{84} - 1392q^{86} - 72q^{89} - 196q^{91} + 3880q^{94} + 6426q^{96} + 468q^{99} + O(q^{100})$$

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
 − 1.56155i 2.56155i − 2.56155i 1.56155i
5.56155i 3.00000i −22.9309 0 16.6847 7.00000i 83.0388i −9.00000 0
274.2 1.43845i 3.00000i 5.93087 0 4.31534 7.00000i 20.0388i −9.00000 0
274.3 1.43845i 3.00000i 5.93087 0 4.31534 7.00000i 20.0388i −9.00000 0
274.4 5.56155i 3.00000i −22.9309 0 16.6847 7.00000i 83.0388i −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.i 4
5.b even 2 1 inner 525.4.d.i 4
5.c odd 4 1 105.4.a.c 2
5.c odd 4 1 525.4.a.p 2
15.e even 4 1 315.4.a.m 2
15.e even 4 1 1575.4.a.m 2
20.e even 4 1 1680.4.a.bk 2
35.f even 4 1 735.4.a.k 2
105.k odd 4 1 2205.4.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.c 2 5.c odd 4 1
315.4.a.m 2 15.e even 4 1
525.4.a.p 2 5.c odd 4 1
525.4.d.i 4 1.a even 1 1 trivial
525.4.d.i 4 5.b even 2 1 inner
735.4.a.k 2 35.f even 4 1
1575.4.a.m 2 15.e even 4 1
1680.4.a.bk 2 20.e even 4 1
2205.4.a.bh 2 105.k odd 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 33 T_{2}^{2} + 64$$ $$T_{11}^{2} + 26 T_{11} - 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2} - 80 T^{4} + 64 T^{6} + 4096 T^{8}$$
$3$ $$( 1 + 9 T^{2} )^{2}$$
$5$ 1
$7$ $$( 1 + 49 T^{2} )^{2}$$
$11$ $$( 1 + 26 T + 2406 T^{2} + 34606 T^{3} + 1771561 T^{4} )^{2}$$
$13$ $$1 - 4576 T^{2} + 14485390 T^{4} - 22087477984 T^{6} + 23298085122481 T^{8}$$
$17$ $$1 - 12860 T^{2} + 88767046 T^{4} - 310409137340 T^{6} + 582622237229761 T^{8}$$
$19$ $$( 1 + 174 T + 18414 T^{2} + 1193466 T^{3} + 47045881 T^{4} )^{2}$$
$23$ $$1 - 5084 T^{2} - 148699226 T^{4} - 752614459676 T^{6} + 21914624432020321 T^{8}$$
$29$ $$( 1 - 32 T + 19046 T^{2} - 780448 T^{3} + 594823321 T^{4} )^{2}$$
$31$ $$( 1 - 330 T + 85430 T^{2} - 9831030 T^{3} + 887503681 T^{4} )^{2}$$
$37$ $$1 - 189004 T^{2} + 14019426870 T^{4} - 484932554206636 T^{6} + 6582952005840035281 T^{8}$$
$41$ $$( 1 - 200 T + 64542 T^{2} - 13784200 T^{3} + 4750104241 T^{4} )^{2}$$
$43$ $$1 - 212476 T^{2} + 21325427350 T^{4} - 1343137935199324 T^{6} + 39959630797262576401 T^{8}$$
$47$ $$1 - 273516 T^{2} + 36034474214 T^{4} - 2948287859926764 T^{6} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 - 205664 T^{2} + 54678146734 T^{4} - 4558411167234656 T^{6} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$( 1 + 364 T + 438374 T^{2} + 74757956 T^{3} + 42180533641 T^{4} )^{2}$$
$61$ $$( 1 - 792 T + 340886 T^{2} - 179768952 T^{3} + 51520374361 T^{4} )^{2}$$
$67$ $$1 - 885916 T^{2} + 375059568694 T^{4} - 80138528097631804 T^{6} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$( 1 - 454 T + 304526 T^{2} - 162491594 T^{3} + 128100283921 T^{4} )^{2}$$
$73$ $$1 - 984112 T^{2} + 463281832126 T^{4} - 148929828101720368 T^{6} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$( 1 + 408 T + 994782 T^{2} + 201159912 T^{3} + 243087455521 T^{4} )^{2}$$
$83$ $$1 - 1049484 T^{2} + 546979674134 T^{4} - 343118690804791596 T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$( 1 + 36 T - 842170 T^{2} + 25378884 T^{3} + 496981290961 T^{4} )^{2}$$
$97$ $$1 - 3345504 T^{2} + 4441575836990 T^{4} - 2786711174377989216 T^{6} +$$$$69\!\cdots\!41$$$$T^{8}$$