# Properties

 Label 525.4.d.j 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: $$\Q(i, \sqrt{41})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} + 3 \beta_{2} q^{3} + ( -6 + 3 \beta_{3} ) q^{4} + ( 6 - 3 \beta_{3} ) q^{6} -7 \beta_{2} q^{7} + ( -\beta_{1} + 25 \beta_{2} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} ) q^{2} + 3 \beta_{2} q^{3} + ( -6 + 3 \beta_{3} ) q^{4} + ( 6 - 3 \beta_{3} ) q^{6} -7 \beta_{2} q^{7} + ( -\beta_{1} + 25 \beta_{2} ) q^{8} -9 q^{9} + ( 30 + 2 \beta_{3} ) q^{11} + ( 9 \beta_{1} - 9 \beta_{2} ) q^{12} + ( 10 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -14 + 7 \beta_{3} ) q^{14} + ( 14 - 3 \beta_{3} ) q^{16} + ( -12 \beta_{1} - 26 \beta_{2} ) q^{17} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{18} + ( 62 - 2 \beta_{3} ) q^{19} + 21 q^{21} + ( 28 \beta_{1} - 12 \beta_{2} ) q^{22} + ( 48 \beta_{1} + 32 \beta_{2} ) q^{23} + ( -78 + 3 \beta_{3} ) q^{24} + ( -116 + 18 \beta_{3} ) q^{26} -27 \beta_{2} q^{27} + ( -21 \beta_{1} + 21 \beta_{2} ) q^{28} + ( -182 + 12 \beta_{3} ) q^{29} + ( 14 + 38 \beta_{3} ) q^{31} + ( 9 \beta_{1} + 159 \beta_{2} ) q^{32} + ( 6 \beta_{1} + 96 \beta_{2} ) q^{33} + ( 92 + 2 \beta_{3} ) q^{34} + ( 54 - 27 \beta_{3} ) q^{36} + ( 80 \beta_{1} + 134 \beta_{2} ) q^{37} + ( 64 \beta_{1} - 80 \beta_{2} ) q^{38} + ( 24 - 30 \beta_{3} ) q^{39} + ( -46 + 108 \beta_{3} ) q^{41} + ( 21 \beta_{1} - 21 \beta_{2} ) q^{42} + ( -100 \beta_{1} - 248 \beta_{2} ) q^{43} + ( -120 + 84 \beta_{3} ) q^{44} + ( -512 + 64 \beta_{3} ) q^{46} + ( 140 \beta_{1} + 164 \beta_{2} ) q^{47} + ( -9 \beta_{1} + 33 \beta_{2} ) q^{48} -49 q^{49} + ( 42 + 36 \beta_{3} ) q^{51} + ( -54 \beta_{1} + 294 \beta_{2} ) q^{52} + ( -58 \beta_{1} + 462 \beta_{2} ) q^{53} + ( -54 + 27 \beta_{3} ) q^{54} + ( 182 - 7 \beta_{3} ) q^{56} + ( -6 \beta_{1} + 180 \beta_{2} ) q^{57} + ( -194 \beta_{1} + 290 \beta_{2} ) q^{58} + ( -296 + 76 \beta_{3} ) q^{59} + ( -482 + 84 \beta_{3} ) q^{61} + ( -24 \beta_{1} + 328 \beta_{2} ) q^{62} + 63 \beta_{2} q^{63} + ( 322 - 165 \beta_{3} ) q^{64} + ( 120 - 84 \beta_{3} ) q^{66} + ( -228 \beta_{1} + 64 \beta_{2} ) q^{67} + ( -6 \beta_{1} - 282 \beta_{2} ) q^{68} + ( 48 - 144 \beta_{3} ) q^{69} + ( 198 - 86 \beta_{3} ) q^{71} + ( 9 \beta_{1} - 225 \beta_{2} ) q^{72} + ( 38 \beta_{1} + 182 \beta_{2} ) q^{73} + ( -692 + 26 \beta_{3} ) q^{74} + ( -432 + 192 \beta_{3} ) q^{76} + ( -14 \beta_{1} - 224 \beta_{2} ) q^{77} + ( 54 \beta_{1} - 294 \beta_{2} ) q^{78} + ( -912 - 8 \beta_{3} ) q^{79} + 81 q^{81} + ( -154 \beta_{1} + 1018 \beta_{2} ) q^{82} + ( 224 \beta_{1} - 228 \beta_{2} ) q^{83} + ( -126 + 63 \beta_{3} ) q^{84} + ( 704 + 48 \beta_{3} ) q^{86} + ( 36 \beta_{1} - 510 \beta_{2} ) q^{87} + ( 20 \beta_{1} + 780 \beta_{2} ) q^{88} + ( -566 + 336 \beta_{3} ) q^{89} + ( -56 + 70 \beta_{3} ) q^{91} + ( -192 \beta_{1} + 1344 \beta_{2} ) q^{92} + ( 114 \beta_{1} + 156 \beta_{2} ) q^{93} + ( -1352 + 116 \beta_{3} ) q^{94} + ( -450 - 27 \beta_{3} ) q^{96} + ( 278 \beta_{1} + 474 \beta_{2} ) q^{97} + ( -49 \beta_{1} + 49 \beta_{2} ) q^{98} + ( -270 - 18 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 18q^{4} + 18q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 18q^{4} + 18q^{6} - 36q^{9} + 124q^{11} - 42q^{14} + 50q^{16} + 244q^{19} + 84q^{21} - 306q^{24} - 428q^{26} - 704q^{29} + 132q^{31} + 372q^{34} + 162q^{36} + 36q^{39} + 32q^{41} - 312q^{44} - 1920q^{46} - 196q^{49} + 240q^{51} - 162q^{54} + 714q^{56} - 1032q^{59} - 1760q^{61} + 958q^{64} + 312q^{66} - 96q^{69} + 620q^{71} - 2716q^{74} - 1344q^{76} - 3664q^{79} + 324q^{81} - 378q^{84} + 2912q^{86} - 1592q^{89} - 84q^{91} - 5176q^{94} - 1854q^{96} - 1116q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 11 \nu$$$$)/10$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 11$$ $$\nu^{3}$$ $$=$$ $$10 \beta_{2} - 11 \beta_{1}$$

## 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
 − 3.70156i − 2.70156i 2.70156i 3.70156i
4.70156i 3.00000i −14.1047 0 14.1047 7.00000i 28.7016i −9.00000 0
274.2 1.70156i 3.00000i 5.10469 0 −5.10469 7.00000i 22.2984i −9.00000 0
274.3 1.70156i 3.00000i 5.10469 0 −5.10469 7.00000i 22.2984i −9.00000 0
274.4 4.70156i 3.00000i −14.1047 0 14.1047 7.00000i 28.7016i −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.j 4
5.b even 2 1 inner 525.4.d.j 4
5.c odd 4 1 105.4.a.g 2
5.c odd 4 1 525.4.a.i 2
15.e even 4 1 315.4.a.g 2
15.e even 4 1 1575.4.a.y 2
20.e even 4 1 1680.4.a.y 2
35.f even 4 1 735.4.a.q 2
105.k odd 4 1 2205.4.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 5.c odd 4 1
315.4.a.g 2 15.e even 4 1
525.4.a.i 2 5.c odd 4 1
525.4.d.j 4 1.a even 1 1 trivial
525.4.d.j 4 5.b even 2 1 inner
735.4.a.q 2 35.f even 4 1
1575.4.a.y 2 15.e even 4 1
1680.4.a.y 2 20.e even 4 1
2205.4.a.v 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 $$S_{4}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{4} + 25 T_{2}^{2} + 64$$ $$T_{11}^{2} - 62 T_{11} + 920$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 7 T^{2} + 48 T^{4} - 448 T^{6} + 4096 T^{8}$$
$3$ $$( 1 + 9 T^{2} )^{2}$$
$5$ 1
$7$ $$( 1 + 49 T^{2} )^{2}$$
$11$ $$( 1 - 62 T + 3582 T^{2} - 82522 T^{3} + 1771561 T^{4} )^{2}$$
$13$ $$1 - 6720 T^{2} + 20906318 T^{4} - 32436156480 T^{6} + 23298085122481 T^{8}$$
$17$ $$1 - 15900 T^{2} + 109116038 T^{4} - 383787347100 T^{6} + 582622237229761 T^{8}$$
$19$ $$( 1 - 122 T + 17398 T^{2} - 836798 T^{3} + 47045881 T^{4} )^{2}$$
$23$ $$1 - 1308 T^{2} + 290453798 T^{4} - 193630942812 T^{6} + 21914624432020321 T^{8}$$
$29$ $$( 1 + 352 T + 78278 T^{2} + 8584928 T^{3} + 594823321 T^{4} )^{2}$$
$31$ $$( 1 - 66 T + 45870 T^{2} - 1966206 T^{3} + 887503681 T^{4} )^{2}$$
$37$ $$1 - 53740 T^{2} + 3534883318 T^{4} - 137882137219660 T^{6} + 6582952005840035281 T^{8}$$
$41$ $$( 1 - 16 T + 18350 T^{2} - 1102736 T^{3} + 4750104241 T^{4} )^{2}$$
$43$ $$1 - 34620 T^{2} - 3131277802 T^{4} - 218845588756380 T^{6} + 39959630797262576401 T^{8}$$
$47$ $$1 + 4180 T^{2} + 14462189158 T^{4} + 45057120075220 T^{6} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 - 44384 T^{2} + 11570351278 T^{4} - 983743004349536 T^{6} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$( 1 + 516 T + 418118 T^{2} + 105975564 T^{3} + 42180533641 T^{4} )^{2}$$
$61$ $$( 1 + 880 T + 575238 T^{2} + 199743280 T^{3} + 51520374361 T^{4} )^{2}$$
$67$ $$1 - 74012 T^{2} + 114756705078 T^{4} - 6695005781092028 T^{6} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$( 1 - 310 T + 664038 T^{2} - 110952410 T^{3} + 128100283921 T^{4} )^{2}$$
$73$ $$1 - 1473328 T^{2} + 843769310398 T^{4} - 222964952949919792 T^{6} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$( 1 + 1832 T + 1824478 T^{2} + 903247448 T^{3} + 243087455521 T^{4} )^{2}$$
$83$ $$1 - 1027340 T^{2} + 679923446038 T^{4} - 335878923176908460 T^{6} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$( 1 + 796 T + 411158 T^{2} + 561155324 T^{3} + 496981290961 T^{4} )^{2}$$
$97$ $$1 - 1841920 T^{2} + 2158510258558 T^{4} - 1534267795318823680 T^{6} +$$$$69\!\cdots\!41$$$$T^{8}$$