Properties

 Label 8450.2.a.br Level $8450$ Weight $2$ Character orbit 8450.a Self dual yes Analytic conductor $67.474$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$8450 = 2 \cdot 5^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8450.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$67.4735897080$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2808.1 Defining polynomial: $$x^{3} - 9 x - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 650) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} -\beta_{1} q^{3} + q^{4} + \beta_{1} q^{6} + ( -1 - \beta_{1} ) q^{7} - q^{8} + ( 3 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q - q^{2} -\beta_{1} q^{3} + q^{4} + \beta_{1} q^{6} + ( -1 - \beta_{1} ) q^{7} - q^{8} + ( 3 + \beta_{2} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} ) q^{11} -\beta_{1} q^{12} + ( 1 + \beta_{1} ) q^{14} + q^{16} -3 q^{17} + ( -3 - \beta_{2} ) q^{18} + ( 2 - \beta_{1} ) q^{19} + ( 6 + \beta_{1} + \beta_{2} ) q^{21} + ( -1 + \beta_{1} - \beta_{2} ) q^{22} + ( \beta_{1} + \beta_{2} ) q^{23} + \beta_{1} q^{24} + ( -2 - 3 \beta_{1} ) q^{27} + ( -1 - \beta_{1} ) q^{28} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{29} + ( 5 + \beta_{2} ) q^{31} - q^{32} + ( 4 - 4 \beta_{1} + \beta_{2} ) q^{33} + 3 q^{34} + ( 3 + \beta_{2} ) q^{36} + ( 4 - \beta_{1} + \beta_{2} ) q^{37} + ( -2 + \beta_{1} ) q^{38} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{41} + ( -6 - \beta_{1} - \beta_{2} ) q^{42} -4 q^{43} + ( 1 - \beta_{1} + \beta_{2} ) q^{44} + ( -\beta_{1} - \beta_{2} ) q^{46} + ( 1 + 2 \beta_{1} + \beta_{2} ) q^{47} -\beta_{1} q^{48} + ( 2 \beta_{1} + \beta_{2} ) q^{49} + 3 \beta_{1} q^{51} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{53} + ( 2 + 3 \beta_{1} ) q^{54} + ( 1 + \beta_{1} ) q^{56} + ( 6 - 2 \beta_{1} + \beta_{2} ) q^{57} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{58} + ( 5 + 4 \beta_{1} - \beta_{2} ) q^{59} + ( -5 - 3 \beta_{1} ) q^{61} + ( -5 - \beta_{2} ) q^{62} + ( -5 - 6 \beta_{1} - \beta_{2} ) q^{63} + q^{64} + ( -4 + 4 \beta_{1} - \beta_{2} ) q^{66} + ( 3 - \beta_{1} - \beta_{2} ) q^{67} -3 q^{68} + ( -8 - 3 \beta_{1} - \beta_{2} ) q^{69} + ( -3 - \beta_{2} ) q^{72} + ( 2 + 4 \beta_{1} - \beta_{2} ) q^{73} + ( -4 + \beta_{1} - \beta_{2} ) q^{74} + ( 2 - \beta_{1} ) q^{76} + ( 3 - 3 \beta_{1} ) q^{77} + ( -6 + \beta_{1} - \beta_{2} ) q^{79} + ( 9 + 2 \beta_{1} ) q^{81} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{82} + ( -5 - \beta_{1} + \beta_{2} ) q^{83} + ( 6 + \beta_{1} + \beta_{2} ) q^{84} + 4 q^{86} + ( 10 + 2 \beta_{2} ) q^{87} + ( -1 + \beta_{1} - \beta_{2} ) q^{88} + ( -8 + 2 \beta_{1} + \beta_{2} ) q^{89} + ( \beta_{1} + \beta_{2} ) q^{92} + ( -2 - 8 \beta_{1} ) q^{93} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{94} + \beta_{1} q^{96} + ( -10 + 2 \beta_{1} ) q^{97} + ( -2 \beta_{1} - \beta_{2} ) q^{98} + ( 19 - 4 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{9} + O(q^{10})$$ $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{9} + 3 q^{11} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 9 q^{18} + 6 q^{19} + 18 q^{21} - 3 q^{22} - 6 q^{27} - 3 q^{28} - 9 q^{29} + 15 q^{31} - 3 q^{32} + 12 q^{33} + 9 q^{34} + 9 q^{36} + 12 q^{37} - 6 q^{38} + 6 q^{41} - 18 q^{42} - 12 q^{43} + 3 q^{44} + 3 q^{47} - 9 q^{53} + 6 q^{54} + 3 q^{56} + 18 q^{57} + 9 q^{58} + 15 q^{59} - 15 q^{61} - 15 q^{62} - 15 q^{63} + 3 q^{64} - 12 q^{66} + 9 q^{67} - 9 q^{68} - 24 q^{69} - 9 q^{72} + 6 q^{73} - 12 q^{74} + 6 q^{76} + 9 q^{77} - 18 q^{79} + 27 q^{81} - 6 q^{82} - 15 q^{83} + 18 q^{84} + 12 q^{86} + 30 q^{87} - 3 q^{88} - 24 q^{89} - 6 q^{93} - 3 q^{94} - 30 q^{97} + 57 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 9 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 6$$

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
 3.10548 −0.223462 −2.88202
−1.00000 −3.10548 1.00000 0 3.10548 −4.10548 −1.00000 6.64402 0
1.2 −1.00000 0.223462 1.00000 0 −0.223462 −0.776538 −1.00000 −2.95006 0
1.3 −1.00000 2.88202 1.00000 0 −2.88202 1.88202 −1.00000 5.30604 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
$$2$$ $$1$$
$$5$$ $$1$$
$$13$$ $$-1$$

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 8450.2.a.br 3
5.b even 2 1 8450.2.a.ce 3
13.b even 2 1 8450.2.a.cd 3
13.d odd 4 2 650.2.d.c 6
65.d even 2 1 8450.2.a.bq 3
65.f even 4 2 650.2.c.f 6
65.g odd 4 2 650.2.d.d yes 6
65.k even 4 2 650.2.c.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.c.e 6 65.k even 4 2
650.2.c.f 6 65.f even 4 2
650.2.d.c 6 13.d odd 4 2
650.2.d.d yes 6 65.g odd 4 2
8450.2.a.bq 3 65.d even 2 1
8450.2.a.br 3 1.a even 1 1 trivial
8450.2.a.cd 3 13.b even 2 1
8450.2.a.ce 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8450))$$:

 $$T_{3}^{3} - 9 T_{3} + 2$$ $$T_{7}^{3} + 3 T_{7}^{2} - 6 T_{7} - 6$$ $$T_{11}^{3} - 3 T_{11}^{2} - 27 T_{11} + 45$$ $$T_{17} + 3$$ $$T_{31}^{3} - 15 T_{31}^{2} + 48 T_{31} + 60$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$2 - 9 T + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-6 - 6 T + 3 T^{2} + T^{3}$$
$11$ $$45 - 27 T - 3 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$( 3 + T )^{3}$$
$19$ $$12 + 3 T - 6 T^{2} + T^{3}$$
$23$ $$-24 - 42 T + T^{3}$$
$29$ $$-240 - 24 T + 9 T^{2} + T^{3}$$
$31$ $$60 + 48 T - 15 T^{2} + T^{3}$$
$37$ $$72 + 18 T - 12 T^{2} + T^{3}$$
$41$ $$360 - 63 T - 6 T^{2} + T^{3}$$
$43$ $$( 4 + T )^{3}$$
$47$ $$-144 - 72 T - 3 T^{2} + T^{3}$$
$53$ $$-12 - 24 T + 9 T^{2} + T^{3}$$
$59$ $$1224 - 72 T - 15 T^{2} + T^{3}$$
$61$ $$-226 - 6 T + 15 T^{2} + T^{3}$$
$67$ $$123 - 15 T - 9 T^{2} + T^{3}$$
$71$ $$T^{3}$$
$73$ $$900 - 135 T - 6 T^{2} + T^{3}$$
$79$ $$20 + 78 T + 18 T^{2} + T^{3}$$
$83$ $$-9 + 45 T + 15 T^{2} + T^{3}$$
$89$ $$-306 + 117 T + 24 T^{2} + T^{3}$$
$97$ $$624 + 264 T + 30 T^{2} + T^{3}$$