# Properties

 Label 825.4.a.s Level $825$ Weight $4$ Character orbit 825.a Self dual yes Analytic conductor $48.677$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.23612.1 Defining polynomial: $$x^{3} - x^{2} - 20 x + 26$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + ( 3 + 3 \beta_{1} ) q^{6} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{7} + ( 15 + 3 \beta_{1} + 4 \beta_{2} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 7 + \beta_{1} + \beta_{2} ) q^{4} + ( 3 + 3 \beta_{1} ) q^{6} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{7} + ( 15 + 3 \beta_{1} + 4 \beta_{2} ) q^{8} + 9 q^{9} + 11 q^{11} + ( 21 + 3 \beta_{1} + 3 \beta_{2} ) q^{12} + ( 4 - 12 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -22 + 10 \beta_{1} + 4 \beta_{2} ) q^{14} + ( 9 + 23 \beta_{1} + 7 \beta_{2} ) q^{16} + ( 76 - 10 \beta_{1} - 6 \beta_{2} ) q^{17} + ( 9 + 9 \beta_{1} ) q^{18} + ( 48 + 2 \beta_{1} + 4 \beta_{2} ) q^{19} + ( 6 - 6 \beta_{1} + 6 \beta_{2} ) q^{21} + ( 11 + 11 \beta_{1} ) q^{22} + ( 60 + 20 \beta_{1} + 8 \beta_{2} ) q^{23} + ( 45 + 9 \beta_{1} + 12 \beta_{2} ) q^{24} + ( -168 - 4 \beta_{1} - 18 \beta_{2} ) q^{26} + 27 q^{27} + ( 110 + 10 \beta_{1} + 6 \beta_{2} ) q^{28} + ( 34 - 34 \beta_{1} + 20 \beta_{2} ) q^{29} + ( -24 + 4 \beta_{1} - 16 \beta_{2} ) q^{31} + ( 225 + 13 \beta_{1} + 12 \beta_{2} ) q^{32} + 33 q^{33} + ( -76 + 52 \beta_{1} - 28 \beta_{2} ) q^{34} + ( 63 + 9 \beta_{1} + 9 \beta_{2} ) q^{36} + ( 122 + 24 \beta_{1} + 4 \beta_{2} ) q^{37} + ( 84 + 64 \beta_{1} + 14 \beta_{2} ) q^{38} + ( 12 - 36 \beta_{1} - 6 \beta_{2} ) q^{39} + ( -66 + 2 \beta_{1} - 20 \beta_{2} ) q^{41} + ( -66 + 30 \beta_{1} + 12 \beta_{2} ) q^{42} + ( 150 + 74 \beta_{1} - 14 \beta_{2} ) q^{43} + ( 77 + 11 \beta_{1} + 11 \beta_{2} ) q^{44} + ( 356 + 92 \beta_{1} + 44 \beta_{2} ) q^{46} + ( 36 - 48 \beta_{1} - 28 \beta_{2} ) q^{47} + ( 27 + 69 \beta_{1} + 21 \beta_{2} ) q^{48} + ( -51 - 4 \beta_{1} - 20 \beta_{2} ) q^{49} + ( 228 - 30 \beta_{1} - 18 \beta_{2} ) q^{51} + ( -292 - 144 \beta_{1} - 42 \beta_{2} ) q^{52} + ( 42 + 32 \beta_{1} - 52 \beta_{2} ) q^{53} + ( 27 + 27 \beta_{1} ) q^{54} + ( 438 + 54 \beta_{1} - 4 \beta_{2} ) q^{56} + ( 144 + 6 \beta_{1} + 12 \beta_{2} ) q^{57} + ( -402 + 114 \beta_{1} + 26 \beta_{2} ) q^{58} + ( -308 - 120 \beta_{1} - 4 \beta_{2} ) q^{59} + ( 218 - 12 \beta_{1} - 44 \beta_{2} ) q^{61} + ( -88 \beta_{1} - 44 \beta_{2} ) q^{62} + ( 18 - 18 \beta_{1} + 18 \beta_{2} ) q^{63} + ( 359 + 89 \beta_{1} - 7 \beta_{2} ) q^{64} + ( 33 + 33 \beta_{1} ) q^{66} + ( 128 - 148 \beta_{1} + 28 \beta_{2} ) q^{67} + ( -12 - 108 \beta_{1} + 16 \beta_{2} ) q^{68} + ( 180 + 60 \beta_{1} + 24 \beta_{2} ) q^{69} + ( -216 + 104 \beta_{1} - 16 \beta_{2} ) q^{71} + ( 135 + 27 \beta_{1} + 36 \beta_{2} ) q^{72} + ( -356 + 168 \beta_{1} - 14 \beta_{2} ) q^{73} + ( 466 + 138 \beta_{1} + 36 \beta_{2} ) q^{74} + ( 624 + 124 \beta_{1} + 74 \beta_{2} ) q^{76} + ( 22 - 22 \beta_{1} + 22 \beta_{2} ) q^{77} + ( -504 - 12 \beta_{1} - 54 \beta_{2} ) q^{78} + ( -524 - 14 \beta_{1} - 52 \beta_{2} ) q^{79} + 81 q^{81} + ( -78 - 146 \beta_{1} - 58 \beta_{2} ) q^{82} + ( 582 - 164 \beta_{1} - 70 \beta_{2} ) q^{83} + ( 330 + 30 \beta_{1} + 18 \beta_{2} ) q^{84} + ( 1158 + 94 \beta_{1} + 32 \beta_{2} ) q^{86} + ( 102 - 102 \beta_{1} + 60 \beta_{2} ) q^{87} + ( 165 + 33 \beta_{1} + 44 \beta_{2} ) q^{88} + ( -730 + 68 \beta_{1} ) q^{89} + ( 56 - 176 \beta_{1} + 4 \beta_{2} ) q^{91} + ( 1252 + 372 \beta_{1} + 160 \beta_{2} ) q^{92} + ( -72 + 12 \beta_{1} - 48 \beta_{2} ) q^{93} + ( -692 - 76 \beta_{1} - 132 \beta_{2} ) q^{94} + ( 675 + 39 \beta_{1} + 36 \beta_{2} ) q^{96} + ( -286 + 240 \beta_{1} - 64 \beta_{2} ) q^{97} + ( -147 - 131 \beta_{1} - 64 \beta_{2} ) q^{98} + 99 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 4 q^{2} + 9 q^{3} + 22 q^{4} + 12 q^{6} + 4 q^{7} + 48 q^{8} + 27 q^{9} + O(q^{10})$$ $$3 q + 4 q^{2} + 9 q^{3} + 22 q^{4} + 12 q^{6} + 4 q^{7} + 48 q^{8} + 27 q^{9} + 33 q^{11} + 66 q^{12} - 56 q^{14} + 50 q^{16} + 218 q^{17} + 36 q^{18} + 146 q^{19} + 12 q^{21} + 44 q^{22} + 200 q^{23} + 144 q^{24} - 508 q^{26} + 81 q^{27} + 340 q^{28} + 68 q^{29} - 68 q^{31} + 688 q^{32} + 99 q^{33} - 176 q^{34} + 198 q^{36} + 390 q^{37} + 316 q^{38} - 196 q^{41} - 168 q^{42} + 524 q^{43} + 242 q^{44} + 1160 q^{46} + 60 q^{47} + 150 q^{48} - 157 q^{49} + 654 q^{51} - 1020 q^{52} + 158 q^{53} + 108 q^{54} + 1368 q^{56} + 438 q^{57} - 1092 q^{58} - 1044 q^{59} + 642 q^{61} - 88 q^{62} + 36 q^{63} + 1166 q^{64} + 132 q^{66} + 236 q^{67} - 144 q^{68} + 600 q^{69} - 544 q^{71} + 432 q^{72} - 900 q^{73} + 1536 q^{74} + 1996 q^{76} + 44 q^{77} - 1524 q^{78} - 1586 q^{79} + 243 q^{81} - 380 q^{82} + 1582 q^{83} + 1020 q^{84} + 3568 q^{86} + 204 q^{87} + 528 q^{88} - 2122 q^{89} - 8 q^{91} + 4128 q^{92} - 204 q^{93} - 2152 q^{94} + 2064 q^{96} - 618 q^{97} - 572 q^{98} + 297 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 20 x + 26$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 14$$

## 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
 −4.59056 1.32906 4.26150
−3.59056 3.00000 4.89212 0 −10.7717 16.1465 11.1590 9.00000 0
1.2 2.32906 3.00000 −2.57547 0 6.98719 −22.4672 −24.6309 9.00000 0
1.3 5.26150 3.00000 19.6833 0 15.7845 10.3207 61.4719 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
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 825.4.a.s 3
3.b odd 2 1 2475.4.a.s 3
5.b even 2 1 165.4.a.d 3
5.c odd 4 2 825.4.c.l 6
15.d odd 2 1 495.4.a.l 3
55.d odd 2 1 1815.4.a.s 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.d 3 5.b even 2 1
495.4.a.l 3 15.d odd 2 1
825.4.a.s 3 1.a even 1 1 trivial
825.4.c.l 6 5.c odd 4 2
1815.4.a.s 3 55.d odd 2 1
2475.4.a.s 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(825))$$:

 $$T_{2}^{3} - 4 T_{2}^{2} - 15 T_{2} + 44$$ $$T_{7}^{3} - 4 T_{7}^{2} - 428 T_{7} + 3744$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$44 - 15 T - 4 T^{2} + T^{3}$$
$3$ $$( -3 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$3744 - 428 T - 4 T^{2} + T^{3}$$
$11$ $$( -11 + T )^{3}$$
$13$ $$34144 - 3560 T + T^{3}$$
$17$ $$235104 + 9680 T - 218 T^{2} + T^{3}$$
$19$ $$-30960 + 5376 T - 146 T^{2} + T^{3}$$
$23$ $$-1664 - 2672 T - 200 T^{2} + T^{3}$$
$29$ $$3163056 - 54364 T - 68 T^{2} + T^{3}$$
$31$ $$-1812096 - 23232 T + 68 T^{2} + T^{3}$$
$37$ $$-618952 + 36460 T - 390 T^{2} + T^{3}$$
$41$ $$-4364208 - 26076 T + 196 T^{2} + T^{3}$$
$43$ $$31273920 - 28668 T - 524 T^{2} + T^{3}$$
$47$ $$20966976 - 135920 T - 60 T^{2} + T^{3}$$
$53$ $$-39574952 - 260852 T - 158 T^{2} + T^{3}$$
$59$ $$-84227264 + 64144 T + 1044 T^{2} + T^{3}$$
$61$ $$22757384 - 60548 T - 642 T^{2} + T^{3}$$
$67$ $$-87537664 - 462208 T - 236 T^{2} + T^{3}$$
$71$ $$6553600 - 129728 T + 544 T^{2} + T^{3}$$
$73$ $$5609344 - 299576 T + 900 T^{2} + T^{3}$$
$79$ $$-14694992 + 562208 T + 1586 T^{2} + T^{3}$$
$83$ $$924645384 - 307644 T - 1582 T^{2} + T^{3}$$
$89$ $$293444632 + 1406940 T + 2122 T^{2} + T^{3}$$
$97$ $$223543736 - 1291700 T + 618 T^{2} + T^{3}$$
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