# Properties

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

# Related objects

## 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: $$4$$ Coefficient field: 4.4.91035289.2 Defining polynomial: $$x^{4} - 2 x^{3} - 285 x^{2} + 286 x + 19616$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} -3 q^{3} + ( -4 - \beta_{3} ) q^{4} + 3 \beta_{3} q^{6} + ( 3 + \beta_{2} ) q^{7} + ( 4 + 11 \beta_{3} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} -3 q^{3} + ( -4 - \beta_{3} ) q^{4} + 3 \beta_{3} q^{6} + ( 3 + \beta_{2} ) q^{7} + ( 4 + 11 \beta_{3} ) q^{8} + 9 q^{9} -11 q^{11} + ( 12 + 3 \beta_{3} ) q^{12} + ( -2 + \beta_{1} + \beta_{2} + 8 \beta_{3} ) q^{13} + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{14} + ( -12 + 15 \beta_{3} ) q^{16} + ( -31 + \beta_{2} - 20 \beta_{3} ) q^{17} -9 \beta_{3} q^{18} + ( 29 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{19} + ( -9 - 3 \beta_{2} ) q^{21} + 11 \beta_{3} q^{22} + ( 15 - 2 \beta_{1} + 2 \beta_{2} + 24 \beta_{3} ) q^{23} + ( -12 - 33 \beta_{3} ) q^{24} + ( -34 - \beta_{1} - 3 \beta_{2} + 12 \beta_{3} ) q^{26} -27 q^{27} + ( -14 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{28} + ( 72 + 3 \beta_{1} - 3 \beta_{2} + 20 \beta_{3} ) q^{29} + ( -42 + 3 \beta_{1} + 3 \beta_{2} - 18 \beta_{3} ) q^{31} + ( -92 - 61 \beta_{3} ) q^{32} + 33 q^{33} + ( 78 - \beta_{1} + \beta_{2} + 11 \beta_{3} ) q^{34} + ( -36 - 9 \beta_{3} ) q^{36} + ( -143 - \beta_{1} + 3 \beta_{2} - 78 \beta_{3} ) q^{37} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 29 \beta_{3} ) q^{38} + ( 6 - 3 \beta_{1} - 3 \beta_{2} - 24 \beta_{3} ) q^{39} + ( -27 - 4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 6 + 3 \beta_{1} - 3 \beta_{2} + 9 \beta_{3} ) q^{42} + ( -58 + 3 \beta_{1} - \beta_{2} - 108 \beta_{3} ) q^{43} + ( 44 + 11 \beta_{3} ) q^{44} + ( -100 - 2 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} ) q^{46} + ( -241 + \beta_{1} - 3 \beta_{2} - 148 \beta_{3} ) q^{47} + ( 36 - 45 \beta_{3} ) q^{48} + ( 212 + \beta_{1} + 5 \beta_{2} - 80 \beta_{3} ) q^{49} + ( 93 - 3 \beta_{2} + 60 \beta_{3} ) q^{51} + ( -26 - 5 \beta_{1} - 7 \beta_{2} - 20 \beta_{3} ) q^{52} + ( 18 + 8 \beta_{1} + 8 \beta_{2} - 26 \beta_{3} ) q^{53} + 27 \beta_{3} q^{54} + ( 34 + 11 \beta_{1} - 7 \beta_{2} + 33 \beta_{3} ) q^{56} + ( -87 - 3 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{57} + ( -74 + 3 \beta_{1} - 15 \beta_{2} - 46 \beta_{3} ) q^{58} + ( 199 - 5 \beta_{1} - 15 \beta_{2} - 70 \beta_{3} ) q^{59} + ( -74 - 4 \beta_{1} + 18 \beta_{2} + 18 \beta_{3} ) q^{61} + ( 66 - 3 \beta_{1} - 9 \beta_{2} + 30 \beta_{3} ) q^{62} + ( 27 + 9 \beta_{2} ) q^{63} + ( 340 - 89 \beta_{3} ) q^{64} -33 \beta_{3} q^{66} + ( -106 + 14 \beta_{1} + 6 \beta_{2} - 82 \beta_{3} ) q^{67} + ( 202 - \beta_{1} - 3 \beta_{2} + 91 \beta_{3} ) q^{68} + ( -45 + 6 \beta_{1} - 6 \beta_{2} - 72 \beta_{3} ) q^{69} + ( 425 - 4 \beta_{1} - 2 \beta_{2} + 88 \beta_{3} ) q^{71} + ( 36 + 99 \beta_{3} ) q^{72} + ( 94 - 20 \beta_{1} - 2 \beta_{2} - 94 \beta_{3} ) q^{73} + ( 306 - 3 \beta_{1} + 7 \beta_{2} + 63 \beta_{3} ) q^{74} + ( -112 - 6 \beta_{1} - 10 \beta_{2} - 21 \beta_{3} ) q^{76} + ( -33 - 11 \beta_{2} ) q^{77} + ( 102 + 3 \beta_{1} + 9 \beta_{2} - 36 \beta_{3} ) q^{78} + ( 545 - 2 \beta_{1} - 11 \beta_{2} - 188 \beta_{3} ) q^{79} + 81 q^{81} + ( 10 - 3 \beta_{1} + 19 \beta_{2} + 15 \beta_{3} ) q^{82} + ( 370 + 7 \beta_{1} - 5 \beta_{2} + 262 \beta_{3} ) q^{83} + ( 42 + 3 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{84} + ( 434 + \beta_{1} - 13 \beta_{2} - 44 \beta_{3} ) q^{86} + ( -216 - 9 \beta_{1} + 9 \beta_{2} - 60 \beta_{3} ) q^{87} + ( -44 - 121 \beta_{3} ) q^{88} + ( 156 + 5 \beta_{1} + 15 \beta_{2} - 222 \beta_{3} ) q^{89} + ( 780 + 12 \beta_{1} - 6 \beta_{2} + 488 \beta_{3} ) q^{91} + ( -160 + 6 \beta_{1} + 2 \beta_{2} - 91 \beta_{3} ) q^{92} + ( 126 - 9 \beta_{1} - 9 \beta_{2} + 54 \beta_{3} ) q^{93} + ( 598 + 3 \beta_{1} - 7 \beta_{2} + 95 \beta_{3} ) q^{94} + ( 276 + 183 \beta_{3} ) q^{96} + ( 41 - 10 \beta_{1} + 32 \beta_{2} - 20 \beta_{3} ) q^{97} + ( 310 - 5 \beta_{1} + \beta_{2} - 290 \beta_{3} ) q^{98} -99 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 12 q^{3} - 14 q^{4} - 6 q^{6} + 11 q^{7} - 6 q^{8} + 36 q^{9} + O(q^{10})$$ $$4 q + 2 q^{2} - 12 q^{3} - 14 q^{4} - 6 q^{6} + 11 q^{7} - 6 q^{8} + 36 q^{9} - 44 q^{11} + 42 q^{12} - 25 q^{13} - 3 q^{14} - 78 q^{16} - 85 q^{17} + 18 q^{18} + 118 q^{19} - 33 q^{21} - 22 q^{22} + 10 q^{23} + 18 q^{24} - 157 q^{26} - 108 q^{27} - 47 q^{28} + 251 q^{29} - 135 q^{31} - 246 q^{32} + 132 q^{33} + 289 q^{34} - 126 q^{36} - 419 q^{37} + 76 q^{38} + 75 q^{39} - 103 q^{41} + 9 q^{42} - 15 q^{43} + 154 q^{44} - 420 q^{46} - 665 q^{47} + 234 q^{48} + 1003 q^{49} + 255 q^{51} - 57 q^{52} + 116 q^{53} - 54 q^{54} + 77 q^{56} - 354 q^{57} - 189 q^{58} + 951 q^{59} - 350 q^{61} + 213 q^{62} + 99 q^{63} + 1538 q^{64} + 66 q^{66} - 266 q^{67} + 629 q^{68} - 30 q^{69} + 1526 q^{71} - 54 q^{72} + 566 q^{73} + 1091 q^{74} - 396 q^{76} - 121 q^{77} + 471 q^{78} + 2567 q^{79} + 324 q^{81} - 9 q^{82} + 961 q^{83} + 141 q^{84} + 1837 q^{86} - 753 q^{87} + 66 q^{88} + 1053 q^{89} + 2150 q^{91} - 460 q^{92} + 405 q^{93} + 2209 q^{94} + 738 q^{96} + 172 q^{97} + 1819 q^{98} - 396 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 285 x^{2} + 286 x + 19616$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$4 \nu - 2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 150 \nu$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{2} - \nu - 150$$$$)/14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$56 \beta_{3} + \beta_{1} + 602$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$56 \beta_{3} + 56 \beta_{2} + 151 \beta_{1} + 902$$$$)/4$$

## 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
 −12.6191 13.6191 11.1952 −10.1952
−1.56155 −3.00000 −5.56155 0 4.68466 −16.7054 21.1771 9.00000 0
1.2 −1.56155 −3.00000 −5.56155 0 4.68466 24.2670 21.1771 9.00000 0
1.3 2.56155 −3.00000 −1.43845 0 −7.68466 −25.6772 −24.1771 9.00000 0
1.4 2.56155 −3.00000 −1.43845 0 −7.68466 29.1157 −24.1771 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.v yes 4
3.b odd 2 1 2475.4.a.bb 4
5.b even 2 1 825.4.a.u 4
5.c odd 4 2 825.4.c.q 8
15.d odd 2 1 2475.4.a.bd 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.u 4 5.b even 2 1
825.4.a.v yes 4 1.a even 1 1 trivial
825.4.c.q 8 5.c odd 4 2
2475.4.a.bb 4 3.b odd 2 1
2475.4.a.bd 4 15.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 4$$ $$T_{7}^{4} - 11 T_{7}^{3} - 1127 T_{7}^{2} + 7047 T_{7} + 303074$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -4 - T + T^{2} )^{2}$$
$3$ $$( 3 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$303074 + 7047 T - 1127 T^{2} - 11 T^{3} + T^{4}$$
$11$ $$( 11 + T )^{4}$$
$13$ $$-1706944 - 254480 T - 5940 T^{2} + 25 T^{3} + T^{4}$$
$17$ $$-1313888 - 175321 T - 1693 T^{2} + 85 T^{3} + T^{4}$$
$19$ $$-6797601 + 456966 T - 3664 T^{2} - 118 T^{3} + T^{4}$$
$23$ $$20292331 + 1754090 T - 29068 T^{2} - 10 T^{3} + T^{4}$$
$29$ $$-42323912 + 8773676 T - 32818 T^{2} - 251 T^{3} + T^{4}$$
$31$ $$92275200 - 265680 T - 45522 T^{2} + 135 T^{3} + T^{4}$$
$37$ $$-881580586 - 14013283 T + 393 T^{2} + 419 T^{3} + T^{4}$$
$41$ $$1343813418 - 4808235 T - 82279 T^{2} + 103 T^{3} + T^{4}$$
$43$ $$795036672 - 3533856 T - 142988 T^{2} + 15 T^{3} + T^{4}$$
$47$ $$1019998564 - 59584093 T - 39841 T^{2} + 665 T^{3} + T^{4}$$
$53$ $$3274784192 + 50311744 T - 354924 T^{2} - 116 T^{3} + T^{4}$$
$59$ $$-37958294854 + 234590839 T - 75259 T^{2} - 951 T^{3} + T^{4}$$
$61$ $$10189762592 - 33725752 T - 426732 T^{2} + 350 T^{3} + T^{4}$$
$67$ $$122607629152 + 49237592 T - 950812 T^{2} + 266 T^{3} + T^{4}$$
$71$ $$6876641459 - 132382682 T + 732444 T^{2} - 1526 T^{3} + T^{4}$$
$73$ $$625767485216 + 743014072 T - 1787100 T^{2} - 566 T^{3} + T^{4}$$
$79$ $$-51952998816 - 399027621 T + 1997131 T^{2} - 2567 T^{3} + T^{4}$$
$83$ $$-55004349600 + 473542320 T - 486734 T^{2} - 961 T^{3} + T^{4}$$
$89$ $$-134213353376 + 576920604 T - 338640 T^{2} - 1053 T^{3} + T^{4}$$
$97$ $$27014010329 + 119728676 T - 1707858 T^{2} - 172 T^{3} + T^{4}$$