Properties

 Label 825.4.a.t 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: 4.4.1540841.1 Defining polynomial: $$x^{4} - 27 x^{2} - 18 x + 92$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} -3 q^{3} + ( 7 - \beta_{1} + \beta_{3} ) q^{4} + ( 3 - 3 \beta_{1} ) q^{6} + ( -8 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + ( -13 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} -3 q^{3} + ( 7 - \beta_{1} + \beta_{3} ) q^{4} + ( 3 - 3 \beta_{1} ) q^{6} + ( -8 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + ( -13 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{8} + 9 q^{9} -11 q^{11} + ( -21 + 3 \beta_{1} - 3 \beta_{3} ) q^{12} + ( 2 - 8 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{13} + ( -16 - 4 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{14} + ( 19 - 13 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} ) q^{16} + ( -18 + 6 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{17} + ( -9 + 9 \beta_{1} ) q^{18} + ( 36 + 10 \beta_{1} + 4 \beta_{3} ) q^{19} + ( 24 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{21} + ( 11 - 11 \beta_{1} ) q^{22} + ( 24 + 8 \beta_{1} + 16 \beta_{3} ) q^{23} + ( 39 - 15 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{24} + ( -94 + 22 \beta_{1} + 20 \beta_{2} - 28 \beta_{3} ) q^{26} -27 q^{27} + ( 8 - 28 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} ) q^{28} + ( 22 + 10 \beta_{1} + 2 \beta_{2} + 18 \beta_{3} ) q^{29} + ( 128 + 28 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} ) q^{31} + ( -65 - 7 \beta_{1} + 10 \beta_{2} - 26 \beta_{3} ) q^{32} + 33 q^{33} + ( 74 + 2 \beta_{1} - 12 \beta_{2} + 26 \beta_{3} ) q^{34} + ( 63 - 9 \beta_{1} + 9 \beta_{3} ) q^{36} + ( 10 - 52 \beta_{1} + 2 \beta_{2} + 18 \beta_{3} ) q^{37} + ( 104 + 60 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} ) q^{38} + ( -6 + 24 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} ) q^{39} + ( 82 + 30 \beta_{1} - 26 \beta_{2} + 30 \beta_{3} ) q^{41} + ( 48 + 12 \beta_{1} - 12 \beta_{2} + 18 \beta_{3} ) q^{42} + ( -140 + 2 \beta_{1} + 19 \beta_{2} - 7 \beta_{3} ) q^{43} + ( -77 + 11 \beta_{1} - 11 \beta_{3} ) q^{44} + ( 88 + 120 \beta_{1} + 32 \beta_{2} - 8 \beta_{3} ) q^{46} + ( 64 - 32 \beta_{1} - 26 \beta_{2} - 10 \beta_{3} ) q^{47} + ( -57 + 39 \beta_{1} + 24 \beta_{2} - 15 \beta_{3} ) q^{48} + ( -127 + 14 \beta_{2} - 6 \beta_{3} ) q^{49} + ( 54 - 18 \beta_{1} - 21 \beta_{2} - 3 \beta_{3} ) q^{51} + ( 306 - 158 \beta_{1} - 56 \beta_{2} + 70 \beta_{3} ) q^{52} + ( 42 + 68 \beta_{1} - 24 \beta_{2} - 8 \beta_{3} ) q^{53} + ( 27 - 27 \beta_{1} ) q^{54} + ( -224 + 52 \beta_{1} + 4 \beta_{2} - 22 \beta_{3} ) q^{56} + ( -108 - 30 \beta_{1} - 12 \beta_{3} ) q^{57} + ( 110 + 134 \beta_{1} + 32 \beta_{2} - 2 \beta_{3} ) q^{58} + ( -244 - 84 \beta_{1} - 48 \beta_{2} + 28 \beta_{3} ) q^{59} + ( 6 + 52 \beta_{1} - 14 \beta_{2} - 46 \beta_{3} ) q^{61} + ( 304 + 168 \beta_{1} + 40 \beta_{2} - 12 \beta_{3} ) q^{62} + ( -72 - 18 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{63} + ( -225 - 97 \beta_{1} - 8 \beta_{2} + 9 \beta_{3} ) q^{64} + ( -33 + 33 \beta_{1} ) q^{66} + ( 140 - 32 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{67} + ( 146 + 158 \beta_{1} + 20 \beta_{2} - 68 \beta_{3} ) q^{68} + ( -72 - 24 \beta_{1} - 48 \beta_{3} ) q^{69} + ( -240 + 116 \beta_{1} + 72 \beta_{2} - 20 \beta_{3} ) q^{71} + ( -117 + 45 \beta_{1} + 18 \beta_{2} - 18 \beta_{3} ) q^{72} + ( -250 + 124 \beta_{1} - 47 \beta_{2} - 37 \beta_{3} ) q^{73} + ( -746 + 122 \beta_{1} + 32 \beta_{2} - 64 \beta_{3} ) q^{74} + ( 416 + 76 \beta_{1} - 4 \beta_{2} + 46 \beta_{3} ) q^{76} + ( 88 + 22 \beta_{1} + 11 \beta_{2} - 11 \beta_{3} ) q^{77} + ( 282 - 66 \beta_{1} - 60 \beta_{2} + 84 \beta_{3} ) q^{78} + ( 328 - 70 \beta_{1} - 24 \beta_{2} + 80 \beta_{3} ) q^{79} + 81 q^{81} + ( 442 + 210 \beta_{1} + 112 \beta_{2} - 78 \beta_{3} ) q^{82} + ( 20 + 180 \beta_{1} + 35 \beta_{2} - 27 \beta_{3} ) q^{83} + ( -24 + 84 \beta_{1} + 36 \beta_{2} - 18 \beta_{3} ) q^{84} + ( 92 - 144 \beta_{1} - 52 \beta_{2} + 66 \beta_{3} ) q^{86} + ( -66 - 30 \beta_{1} - 6 \beta_{2} - 54 \beta_{3} ) q^{87} + ( 143 - 55 \beta_{1} - 22 \beta_{2} + 22 \beta_{3} ) q^{88} + ( -254 + 332 \beta_{1} - 12 \beta_{2} + 24 \beta_{3} ) q^{89} + ( 696 - 96 \beta_{1} + 34 \beta_{2} + 2 \beta_{3} ) q^{91} + ( 1272 + 40 \beta_{1} - 80 \beta_{2} + 96 \beta_{3} ) q^{92} + ( -384 - 84 \beta_{1} + 30 \beta_{2} - 30 \beta_{3} ) q^{93} + ( -408 - 48 \beta_{1} + 32 \beta_{2} - 100 \beta_{3} ) q^{94} + ( 195 + 21 \beta_{1} - 30 \beta_{2} + 78 \beta_{3} ) q^{96} + ( 398 + 84 \beta_{1} - 24 \beta_{2} - 28 \beta_{3} ) q^{97} + ( 71 - 135 \beta_{1} - 40 \beta_{2} + 48 \beta_{3} ) q^{98} -99 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 12q^{3} + 26q^{4} + 12q^{6} - 34q^{7} - 48q^{8} + 36q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 12q^{3} + 26q^{4} + 12q^{6} - 34q^{7} - 48q^{8} + 36q^{9} - 44q^{11} - 78q^{12} - 2q^{13} - 52q^{14} + 66q^{16} - 74q^{17} - 36q^{18} + 136q^{19} + 102q^{21} + 44q^{22} + 64q^{23} + 144q^{24} - 320q^{26} - 108q^{27} + 20q^{28} + 52q^{29} + 492q^{31} - 208q^{32} + 132q^{33} + 244q^{34} + 234q^{36} + 4q^{37} + 404q^{38} + 6q^{39} + 268q^{41} + 156q^{42} - 546q^{43} - 286q^{44} + 368q^{46} + 276q^{47} - 198q^{48} - 496q^{49} + 222q^{51} + 1084q^{52} + 184q^{53} + 108q^{54} - 852q^{56} - 408q^{57} + 444q^{58} - 1032q^{59} + 116q^{61} + 1240q^{62} - 306q^{63} - 918q^{64} - 132q^{66} + 552q^{67} + 720q^{68} - 192q^{69} - 920q^{71} - 432q^{72} - 926q^{73} - 2856q^{74} + 1572q^{76} + 374q^{77} + 960q^{78} + 1152q^{79} + 324q^{81} + 1924q^{82} + 134q^{83} - 60q^{84} + 236q^{86} - 156q^{87} + 528q^{88} - 1064q^{89} + 2780q^{91} + 4896q^{92} - 1476q^{93} - 1432q^{94} + 624q^{96} + 1648q^{97} + 188q^{98} - 396q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 27 x^{2} - 18 x + 92$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 20 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{1} + 14$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 21 \beta_{1} + 14$$

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
 −4.17080 −2.63835 1.60719 5.20196
−5.17080 −3.00000 18.7372 0 15.5124 11.1745 −55.5199 9.00000 0
1.2 −3.63835 −3.00000 5.23763 0 10.9151 −20.8444 10.0505 9.00000 0
1.3 0.607192 −3.00000 −7.63132 0 −1.82158 −8.95080 −9.49121 9.00000 0
1.4 4.20196 −3.00000 9.65650 0 −12.6059 −15.3793 6.96057 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.t 4
3.b odd 2 1 2475.4.a.be 4
5.b even 2 1 165.4.a.h 4
5.c odd 4 2 825.4.c.p 8
15.d odd 2 1 495.4.a.m 4
55.d odd 2 1 1815.4.a.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.h 4 5.b even 2 1
495.4.a.m 4 15.d odd 2 1
825.4.a.t 4 1.a even 1 1 trivial
825.4.c.p 8 5.c odd 4 2
1815.4.a.t 4 55.d odd 2 1
2475.4.a.be 4 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(825))$$:

 $$T_{2}^{4} + 4 T_{2}^{3} - 21 T_{2}^{2} - 68 T_{2} + 48$$ $$T_{7}^{4} + 34 T_{7}^{3} + 140 T_{7}^{2} - 4336 T_{7} - 32064$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$48 - 68 T - 21 T^{2} + 4 T^{3} + T^{4}$$
$3$ $$( 3 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$-32064 - 4336 T + 140 T^{2} + 34 T^{3} + T^{4}$$
$11$ $$( 11 + T )^{4}$$
$13$ $$68144 - 144984 T - 6584 T^{2} + 2 T^{3} + T^{4}$$
$17$ $$23770800 - 251400 T - 10384 T^{2} + 74 T^{3} + T^{4}$$
$19$ $$576 + 23472 T + 1780 T^{2} - 136 T^{3} + T^{4}$$
$23$ $$247529472 + 1599488 T - 39488 T^{2} - 64 T^{3} + T^{4}$$
$29$ $$315474624 + 2737584 T - 57364 T^{2} - 52 T^{3} + T^{4}$$
$31$ $$-903269376 + 8037504 T + 40496 T^{2} - 492 T^{3} + T^{4}$$
$37$ $$1260009136 - 1497776 T - 131408 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$-6228069696 + 73712016 T - 188244 T^{2} - 268 T^{3} + T^{4}$$
$43$ $$-1200551616 - 12862032 T + 42908 T^{2} + 546 T^{3} + T^{4}$$
$47$ $$5628791808 + 14965632 T - 189776 T^{2} - 276 T^{3} + T^{4}$$
$53$ $$13030473936 + 7410656 T - 318392 T^{2} - 184 T^{3} + T^{4}$$
$59$ $$-2612441088 - 266215936 T - 143792 T^{2} + 1032 T^{3} + T^{4}$$
$61$ $$-8546777488 - 133895280 T - 551360 T^{2} - 116 T^{3} + T^{4}$$
$67$ $$-909580544 + 14141568 T + 35584 T^{2} - 552 T^{3} + T^{4}$$
$71$ $$291456592896 - 487321600 T - 886640 T^{2} + 920 T^{3} + T^{4}$$
$73$ $$-133750796272 - 1247813640 T - 1199936 T^{2} + 926 T^{3} + T^{4}$$
$79$ $$48148922944 + 252278064 T - 419564 T^{2} - 1152 T^{3} + T^{4}$$
$83$ $$-17388663552 + 261254304 T - 977028 T^{2} - 134 T^{3} + T^{4}$$
$89$ $$479041129296 - 2802564256 T - 2716632 T^{2} + 1064 T^{3} + T^{4}$$
$97$ $$1607443600 + 68378560 T + 429784 T^{2} - 1648 T^{3} + T^{4}$$