# Properties

 Label 8450.2.a.bo Level $8450$ Weight $2$ Character orbit 8450.a Self dual yes Analytic conductor $67.474$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1690) 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.

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

## 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
 1.80194 0.445042 −1.24698
−1.00000 −1.80194 1.00000 0 1.80194 −1.55496 −1.00000 0.246980 0
1.2 −1.00000 −0.445042 1.00000 0 0.445042 −3.24698 −1.00000 −2.80194 0
1.3 −1.00000 1.24698 1.00000 0 −1.24698 −0.198062 −1.00000 −1.44504 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.bo 3
5.b even 2 1 1690.2.a.s yes 3
13.b even 2 1 8450.2.a.bz 3
65.d even 2 1 1690.2.a.q 3
65.g odd 4 2 1690.2.d.j 6
65.l even 6 2 1690.2.e.q 6
65.n even 6 2 1690.2.e.o 6
65.s odd 12 4 1690.2.l.l 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1690.2.a.q 3 65.d even 2 1
1690.2.a.s yes 3 5.b even 2 1
1690.2.d.j 6 65.g odd 4 2
1690.2.e.o 6 65.n even 6 2
1690.2.e.q 6 65.l even 6 2
1690.2.l.l 12 65.s odd 12 4
8450.2.a.bo 3 1.a even 1 1 trivial
8450.2.a.bz 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8450))$$:

 $$T_{3}^{3} + T_{3}^{2} - 2 T_{3} - 1$$ $$T_{7}^{3} + 5 T_{7}^{2} + 6 T_{7} + 1$$ $$T_{11}^{3} - 4 T_{11}^{2} - 4 T_{11} + 8$$ $$T_{17}^{3} + 10 T_{17}^{2} + 24 T_{17} + 8$$ $$T_{31}^{3} - 8 T_{31}^{2} - 44 T_{31} + 344$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$-1 - 2 T + T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$1 + 6 T + 5 T^{2} + T^{3}$$
$11$ $$8 - 4 T - 4 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$8 + 24 T + 10 T^{2} + T^{3}$$
$19$ $$8 + 12 T - 8 T^{2} + T^{3}$$
$23$ $$139 - 46 T - 3 T^{2} + T^{3}$$
$29$ $$-1 - 4 T - 3 T^{2} + T^{3}$$
$31$ $$344 - 44 T - 8 T^{2} + T^{3}$$
$37$ $$-64 - 16 T + 8 T^{2} + T^{3}$$
$41$ $$827 - 74 T - 11 T^{2} + T^{3}$$
$43$ $$43 - 30 T - T^{2} + T^{3}$$
$47$ $$-71 + 26 T + 15 T^{2} + T^{3}$$
$53$ $$-568 - 116 T + 4 T^{2} + T^{3}$$
$59$ $$-344 + 164 T - 24 T^{2} + T^{3}$$
$61$ $$-533 - 16 T + 15 T^{2} + T^{3}$$
$67$ $$169 - 78 T + 5 T^{2} + T^{3}$$
$71$ $$-1112 + 332 T - 32 T^{2} + T^{3}$$
$73$ $$104 + 160 T - 26 T^{2} + T^{3}$$
$79$ $$344 - 120 T - 2 T^{2} + T^{3}$$
$83$ $$-287 - 154 T + 7 T^{2} + T^{3}$$
$89$ $$-2197 - 130 T + 17 T^{2} + T^{3}$$
$97$ $$1112 - 184 T - 6 T^{2} + T^{3}$$