# Properties

 Label 8450.2.a.be Level $8450$ Weight $2$ Character orbit 8450.a Self dual yes Analytic conductor $67.474$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} + ( 1 - \beta ) q^{7} - q^{8} + ( 2 + \beta ) q^{9} +O(q^{10})$$ $$q - q^{2} + \beta q^{3} + q^{4} -\beta q^{6} + ( 1 - \beta ) q^{7} - q^{8} + ( 2 + \beta ) q^{9} + ( 1 + \beta ) q^{11} + \beta q^{12} + ( -1 + \beta ) q^{14} + q^{16} + ( -2 + \beta ) q^{17} + ( -2 - \beta ) q^{18} -5 q^{19} -5 q^{21} + ( -1 - \beta ) q^{22} + ( 1 - 2 \beta ) q^{23} -\beta q^{24} + 5 q^{27} + ( 1 - \beta ) q^{28} + ( -1 - \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} - q^{32} + ( 5 + 2 \beta ) q^{33} + ( 2 - \beta ) q^{34} + ( 2 + \beta ) q^{36} + ( 2 - 3 \beta ) q^{37} + 5 q^{38} + ( -1 - 4 \beta ) q^{41} + 5 q^{42} + ( -3 + 5 \beta ) q^{43} + ( 1 + \beta ) q^{44} + ( -1 + 2 \beta ) q^{46} + ( -2 + \beta ) q^{47} + \beta q^{48} + ( -1 - \beta ) q^{49} + ( 5 - \beta ) q^{51} + ( 1 - 2 \beta ) q^{53} -5 q^{54} + ( -1 + \beta ) q^{56} -5 \beta q^{57} + ( 1 + \beta ) q^{58} + ( -1 + 2 \beta ) q^{59} + ( -3 - 2 \beta ) q^{61} + ( 4 + 2 \beta ) q^{62} + ( -3 - 2 \beta ) q^{63} + q^{64} + ( -5 - 2 \beta ) q^{66} + 11 q^{67} + ( -2 + \beta ) q^{68} + ( -10 - \beta ) q^{69} + ( -2 - 2 \beta ) q^{71} + ( -2 - \beta ) q^{72} + ( 4 - 4 \beta ) q^{73} + ( -2 + 3 \beta ) q^{74} -5 q^{76} + ( -4 - \beta ) q^{77} + ( 1 + 5 \beta ) q^{79} + ( -6 + 2 \beta ) q^{81} + ( 1 + 4 \beta ) q^{82} + ( 5 - 4 \beta ) q^{83} -5 q^{84} + ( 3 - 5 \beta ) q^{86} + ( -5 - 2 \beta ) q^{87} + ( -1 - \beta ) q^{88} + ( -9 + 3 \beta ) q^{89} + ( 1 - 2 \beta ) q^{92} + ( -10 - 6 \beta ) q^{93} + ( 2 - \beta ) q^{94} -\beta q^{96} + ( -12 + \beta ) q^{97} + ( 1 + \beta ) q^{98} + ( 7 + 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + 5 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + 5 q^{9} + 3 q^{11} + q^{12} - q^{14} + 2 q^{16} - 3 q^{17} - 5 q^{18} - 10 q^{19} - 10 q^{21} - 3 q^{22} - q^{24} + 10 q^{27} + q^{28} - 3 q^{29} - 10 q^{31} - 2 q^{32} + 12 q^{33} + 3 q^{34} + 5 q^{36} + q^{37} + 10 q^{38} - 6 q^{41} + 10 q^{42} - q^{43} + 3 q^{44} - 3 q^{47} + q^{48} - 3 q^{49} + 9 q^{51} - 10 q^{54} - q^{56} - 5 q^{57} + 3 q^{58} - 8 q^{61} + 10 q^{62} - 8 q^{63} + 2 q^{64} - 12 q^{66} + 22 q^{67} - 3 q^{68} - 21 q^{69} - 6 q^{71} - 5 q^{72} + 4 q^{73} - q^{74} - 10 q^{76} - 9 q^{77} + 7 q^{79} - 10 q^{81} + 6 q^{82} + 6 q^{83} - 10 q^{84} + q^{86} - 12 q^{87} - 3 q^{88} - 15 q^{89} - 26 q^{93} + 3 q^{94} - q^{96} - 23 q^{97} + 3 q^{98} + 18 q^{99} + O(q^{100})$$

## 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
 −1.79129 2.79129
−1.00000 −1.79129 1.00000 0 1.79129 2.79129 −1.00000 0.208712 0
1.2 −1.00000 2.79129 1.00000 0 −2.79129 −1.79129 −1.00000 4.79129 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.be 2
5.b even 2 1 8450.2.a.bh 2
13.b even 2 1 8450.2.a.bk 2
13.e even 6 2 650.2.e.d 4
65.d even 2 1 8450.2.a.bb 2
65.l even 6 2 650.2.e.i yes 4
65.r odd 12 4 650.2.o.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.e.d 4 13.e even 6 2
650.2.e.i yes 4 65.l even 6 2
650.2.o.f 8 65.r odd 12 4
8450.2.a.bb 2 65.d even 2 1
8450.2.a.be 2 1.a even 1 1 trivial
8450.2.a.bh 2 5.b even 2 1
8450.2.a.bk 2 13.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8450))$$:

 $$T_{3}^{2} - T_{3} - 5$$ $$T_{7}^{2} - T_{7} - 5$$ $$T_{11}^{2} - 3 T_{11} - 3$$ $$T_{17}^{2} + 3 T_{17} - 3$$ $$T_{31}^{2} + 10 T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-5 - T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-5 - T + T^{2}$$
$11$ $$-3 - 3 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-3 + 3 T + T^{2}$$
$19$ $$( 5 + T )^{2}$$
$23$ $$-21 + T^{2}$$
$29$ $$-3 + 3 T + T^{2}$$
$31$ $$4 + 10 T + T^{2}$$
$37$ $$-47 - T + T^{2}$$
$41$ $$-75 + 6 T + T^{2}$$
$43$ $$-131 + T + T^{2}$$
$47$ $$-3 + 3 T + T^{2}$$
$53$ $$-21 + T^{2}$$
$59$ $$-21 + T^{2}$$
$61$ $$-5 + 8 T + T^{2}$$
$67$ $$( -11 + T )^{2}$$
$71$ $$-12 + 6 T + T^{2}$$
$73$ $$-80 - 4 T + T^{2}$$
$79$ $$-119 - 7 T + T^{2}$$
$83$ $$-75 - 6 T + T^{2}$$
$89$ $$9 + 15 T + T^{2}$$
$97$ $$127 + 23 T + T^{2}$$