# Properties

 Label 8450.2.a.x Level $8450$ Weight $2$ Character orbit 8450.a Self dual yes Analytic conductor $67.474$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 650) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
1.00000 2.00000 1.00000 0 2.00000 5.00000 1.00000 1.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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.x 1
5.b even 2 1 8450.2.a.a 1
13.b even 2 1 650.2.a.f 1
39.d odd 2 1 5850.2.a.bc 1
52.b odd 2 1 5200.2.a.i 1
65.d even 2 1 650.2.a.h yes 1
65.h odd 4 2 650.2.b.b 2
195.e odd 2 1 5850.2.a.bb 1
195.s even 4 2 5850.2.e.ba 2
260.g odd 2 1 5200.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.a.f 1 13.b even 2 1
650.2.a.h yes 1 65.d even 2 1
650.2.b.b 2 65.h odd 4 2
5200.2.a.i 1 52.b odd 2 1
5200.2.a.bc 1 260.g odd 2 1
5850.2.a.bb 1 195.e odd 2 1
5850.2.a.bc 1 39.d odd 2 1
5850.2.e.ba 2 195.s even 4 2
8450.2.a.a 1 5.b even 2 1
8450.2.a.x 1 1.a even 1 1 trivial

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-2 + T$$
$5$ $$T$$
$7$ $$-5 + T$$
$11$ $$-3 + T$$
$13$ $$T$$
$17$ $$3 + T$$
$19$ $$-4 + T$$
$23$ $$6 + T$$
$29$ $$-9 + T$$
$31$ $$5 + T$$
$37$ $$-2 + T$$
$41$ $$T$$
$43$ $$2 + T$$
$47$ $$9 + T$$
$53$ $$-9 + T$$
$59$ $$-9 + T$$
$61$ $$1 + T$$
$67$ $$-5 + T$$
$71$ $$T$$
$73$ $$-14 + T$$
$79$ $$16 + T$$
$83$ $$15 + T$$
$89$ $$-6 + T$$
$97$ $$-8 + T$$