# Properties

 Label 825.4.c.k Level $825$ Weight $4$ Character orbit 825.c Analytic conductor $48.677$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.36142572544.1 Defining polynomial: $$x^{6} + 53 x^{4} + 632 x^{2} + 484$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + 3 \beta_{3} q^{3} + ( -10 + \beta_{4} ) q^{4} + ( -3 + 3 \beta_{2} ) q^{6} + ( -2 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} ) q^{7} + ( -9 \beta_{1} + 3 \beta_{3} - 2 \beta_{5} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + 3 \beta_{3} q^{3} + ( -10 + \beta_{4} ) q^{4} + ( -3 + 3 \beta_{2} ) q^{6} + ( -2 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} ) q^{7} + ( -9 \beta_{1} + 3 \beta_{3} - 2 \beta_{5} ) q^{8} -9 q^{9} + 11 q^{11} + ( -30 \beta_{3} - 3 \beta_{5} ) q^{12} + ( 38 \beta_{3} - 2 \beta_{5} ) q^{13} + ( 28 - 16 \beta_{2} - 6 \beta_{4} ) q^{14} + ( 60 - 2 \beta_{2} - 5 \beta_{4} ) q^{16} + ( -2 \beta_{1} + 34 \beta_{3} + 2 \beta_{5} ) q^{17} + ( -9 \beta_{1} - 9 \beta_{3} ) q^{18} + ( 24 - 14 \beta_{2} - 8 \beta_{4} ) q^{19} + ( 12 - 6 \beta_{2} - 6 \beta_{4} ) q^{21} + ( 11 \beta_{1} + 11 \beta_{3} ) q^{22} + ( 24 \beta_{1} + 48 \beta_{3} + 4 \beta_{5} ) q^{23} + ( -9 - 27 \beta_{2} - 6 \beta_{4} ) q^{24} + ( -48 + 24 \beta_{2} - 4 \beta_{4} ) q^{26} -27 \beta_{3} q^{27} + ( 38 \beta_{1} + 238 \beta_{3} + 12 \beta_{5} ) q^{28} + ( 62 + 34 \beta_{2} ) q^{29} + ( 96 - 40 \beta_{2} + 8 \beta_{4} ) q^{31} + ( 21 \beta_{1} + 93 \beta_{3} - 4 \beta_{5} ) q^{32} + 33 \beta_{3} q^{33} + ( 10 + 50 \beta_{2} + 2 \beta_{4} ) q^{34} + ( 90 - 9 \beta_{4} ) q^{36} + ( -20 \beta_{1} - 286 \beta_{3} ) q^{37} + ( 66 \beta_{1} + 222 \beta_{3} + 30 \beta_{5} ) q^{38} + ( -114 - 6 \beta_{4} ) q^{39} + ( 30 + 66 \beta_{2} ) q^{41} + ( 48 \beta_{1} + 84 \beta_{3} + 18 \beta_{5} ) q^{42} + ( 22 \beta_{1} + 48 \beta_{3} - 14 \beta_{5} ) q^{43} + ( -110 + 11 \beta_{4} ) q^{44} + ( -436 + 52 \beta_{2} + 32 \beta_{4} ) q^{46} + ( -168 \beta_{3} + 4 \beta_{5} ) q^{47} + ( 6 \beta_{1} + 180 \beta_{3} + 15 \beta_{5} ) q^{48} + ( -117 + 72 \beta_{2} ) q^{49} + ( -102 - 6 \beta_{2} + 6 \beta_{4} ) q^{51} + ( 4 \beta_{1} - 172 \beta_{3} - 32 \beta_{5} ) q^{52} + ( 96 \beta_{1} + 126 \beta_{3} + 16 \beta_{5} ) q^{53} + ( 27 - 27 \beta_{2} ) q^{54} + ( -600 + 156 \beta_{2} + 14 \beta_{4} ) q^{56} + ( 42 \beta_{1} + 72 \beta_{3} + 24 \beta_{5} ) q^{57} + ( 96 \beta_{1} - 516 \beta_{3} - 34 \beta_{5} ) q^{58} + ( -172 - 32 \beta_{2} + 8 \beta_{4} ) q^{59} + ( 134 + 12 \beta_{2} - 68 \beta_{4} ) q^{61} + ( 816 \beta_{3} + 24 \beta_{5} ) q^{62} + ( 18 \beta_{1} + 36 \beta_{3} + 18 \beta_{5} ) q^{63} + ( 10 + 28 \beta_{2} - 27 \beta_{4} ) q^{64} + ( -33 + 33 \beta_{2} ) q^{66} + ( 100 \beta_{1} + 176 \beta_{3} + 16 \beta_{5} ) q^{67} + ( 30 \beta_{1} - 558 \beta_{3} - 38 \beta_{5} ) q^{68} + ( -144 + 72 \beta_{2} + 12 \beta_{4} ) q^{69} + ( -324 + 60 \beta_{2} + 28 \beta_{4} ) q^{71} + ( 81 \beta_{1} - 27 \beta_{3} + 18 \beta_{5} ) q^{72} + ( -36 \beta_{1} + 194 \beta_{3} - 38 \beta_{5} ) q^{73} + ( 626 - 266 \beta_{2} - 20 \beta_{4} ) q^{74} + ( -1002 + 254 \beta_{2} + 62 \beta_{4} ) q^{76} + ( -22 \beta_{1} - 44 \beta_{3} - 22 \beta_{5} ) q^{77} + ( -72 \beta_{1} - 144 \beta_{3} + 12 \beta_{5} ) q^{78} + ( 212 - 94 \beta_{2} + 8 \beta_{4} ) q^{79} + 81 q^{81} + ( 96 \beta_{1} - 1092 \beta_{3} - 66 \beta_{5} ) q^{82} + ( 180 \beta_{1} + 60 \beta_{3} + 54 \beta_{5} ) q^{83} + ( -714 + 114 \beta_{2} + 36 \beta_{4} ) q^{84} + ( -492 - 72 \beta_{2} - 6 \beta_{4} ) q^{86} + ( -102 \beta_{1} + 186 \beta_{3} ) q^{87} + ( -99 \beta_{1} + 33 \beta_{3} - 22 \beta_{5} ) q^{88} + ( -254 - 28 \beta_{2} + 120 \beta_{4} ) q^{89} + ( -244 - 40 \beta_{2} - 92 \beta_{4} ) q^{91} + ( -416 \beta_{1} - 776 \beta_{3} - 84 \beta_{5} ) q^{92} + ( 120 \beta_{1} + 288 \beta_{3} - 24 \beta_{5} ) q^{93} + ( 188 - 140 \beta_{2} + 8 \beta_{4} ) q^{94} + ( -279 + 63 \beta_{2} - 12 \beta_{4} ) q^{96} + ( 100 \beta_{1} - 658 \beta_{3} + 44 \beta_{5} ) q^{97} + ( -45 \beta_{1} - 1341 \beta_{3} - 72 \beta_{5} ) q^{98} -99 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 60q^{4} - 12q^{6} - 54q^{9} + O(q^{10})$$ $$6q - 60q^{4} - 12q^{6} - 54q^{9} + 66q^{11} + 136q^{14} + 356q^{16} + 116q^{19} + 60q^{21} - 108q^{24} - 240q^{26} + 440q^{29} + 496q^{31} + 160q^{34} + 540q^{36} - 684q^{39} + 312q^{41} - 660q^{44} - 2512q^{46} - 558q^{49} - 624q^{51} + 108q^{54} - 3288q^{56} - 1096q^{59} + 828q^{61} + 116q^{64} - 132q^{66} - 720q^{69} - 1824q^{71} + 3224q^{74} - 5504q^{76} + 1084q^{79} + 486q^{81} - 4056q^{84} - 3096q^{86} - 1580q^{89} - 1544q^{91} + 848q^{94} - 1548q^{96} - 594q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 53 x^{4} + 632 x^{2} + 484$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} + 27 \nu^{2} - 22$$$$)/48$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 75 \nu^{3} + 1226 \nu$$$$)/1056$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 51 \nu^{2} + 386$$$$)/24$$ $$\beta_{5}$$ $$=$$ $$($$$$-13 \nu^{5} - 623 \nu^{3} - 6082 \nu$$$$)/352$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - 2 \beta_{2} - 17$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + 39 \beta_{3} - 28 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-27 \beta_{4} + 102 \beta_{2} + 481$$ $$\nu^{5}$$ $$=$$ $$-75 \beta_{5} - 1869 \beta_{3} + 874 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 4.06484i − 5.97123i − 0.906392i 0.906392i 5.97123i 4.06484i
5.06484i 3.00000i −17.6526 0 −15.1945 27.4348i 48.8887i −9.00000 0
199.2 4.97123i 3.00000i −16.7131 0 14.9137 5.48376i 43.3148i −9.00000 0
199.3 1.90639i 3.00000i 4.36567 0 −5.71918 22.9186i 23.5738i −9.00000 0
199.4 1.90639i 3.00000i 4.36567 0 −5.71918 22.9186i 23.5738i −9.00000 0
199.5 4.97123i 3.00000i −16.7131 0 14.9137 5.48376i 43.3148i −9.00000 0
199.6 5.06484i 3.00000i −17.6526 0 −15.1945 27.4348i 48.8887i −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 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 825.4.c.k 6
5.b even 2 1 inner 825.4.c.k 6
5.c odd 4 1 165.4.a.e 3
5.c odd 4 1 825.4.a.r 3
15.e even 4 1 495.4.a.k 3
15.e even 4 1 2475.4.a.t 3
55.e even 4 1 1815.4.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.e 3 5.c odd 4 1
495.4.a.k 3 15.e even 4 1
825.4.a.r 3 5.c odd 4 1
825.4.c.k 6 1.a even 1 1 trivial
825.4.c.k 6 5.b even 2 1 inner
1815.4.a.r 3 55.e even 4 1
2475.4.a.t 3 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 $$S_{4}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{6} + 54 T_{2}^{4} + 817 T_{2}^{2} + 2304$$ $$T_{7}^{6} + 1308 T_{7}^{4} + 433776 T_{7}^{2} + 11888704$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2304 + 817 T^{2} + 54 T^{4} + T^{6}$$
$3$ $$( 9 + T^{2} )^{3}$$
$5$ $$T^{6}$$
$7$ $$11888704 + 433776 T^{2} + 1308 T^{4} + T^{6}$$
$11$ $$( -11 + T )^{6}$$
$13$ $$1385030656 + 5293696 T^{2} + 5572 T^{4} + T^{6}$$
$17$ $$71368704 + 6038080 T^{2} + 5232 T^{4} + T^{6}$$
$19$ $$( -65520 - 11496 T - 58 T^{2} + T^{3} )^{2}$$
$23$ $$21970354176 + 150835456 T^{2} + 35872 T^{4} + T^{6}$$
$29$ $$( 629760 - 14308 T - 220 T^{2} + T^{3} )^{2}$$
$31$ $$( 9589248 - 38592 T - 248 T^{2} + T^{3} )^{2}$$
$37$ $$346232953016896 + 18787813168 T^{2} + 255148 T^{4} + T^{6}$$
$41$ $$( -3013632 - 106596 T - 156 T^{2} + T^{3} )^{2}$$
$43$ $$2089181160000 + 2372281200 T^{2} + 104764 T^{4} + T^{6}$$
$47$ $$19116342706176 + 2348323072 T^{2} + 89632 T^{4} + T^{6}$$
$53$ $$13353645381696 + 47253191728 T^{2} + 523660 T^{4} + T^{6}$$
$59$ $$( 1206720 + 57584 T + 548 T^{2} + T^{3} )^{2}$$
$61$ $$( 342344792 - 681332 T - 414 T^{2} + T^{3} )^{2}$$
$67$ $$66187206344704 + 49710422016 T^{2} + 596688 T^{4} + T^{6}$$
$71$ $$( -2867712 + 97888 T + 912 T^{2} + T^{3} )^{2}$$
$73$ $$700057090622464 + 42811915648 T^{2} + 583012 T^{4} + T^{6}$$
$79$ $$( 88503440 - 161224 T - 542 T^{2} + T^{3} )^{2}$$
$83$ $$188640853279490304 + 1191022995600 T^{2} + 2182680 T^{4} + T^{6}$$
$89$ $$( -1941629400 - 2118532 T + 790 T^{2} + T^{3} )^{2}$$
$97$ $$9618771531512896 + 1342315546672 T^{2} + 2367052 T^{4} + T^{6}$$