# Properties

 Label 8450.2.a.bu 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: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 130) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_{2} q^{3} + q^{4} -\beta_{2} q^{6} + ( 2 - \beta_{1} - \beta_{2} ) q^{7} - q^{8} + ( -\beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q - q^{2} + \beta_{2} q^{3} + q^{4} -\beta_{2} q^{6} + ( 2 - \beta_{1} - \beta_{2} ) q^{7} - q^{8} + ( -\beta_{1} - \beta_{2} ) q^{9} + ( 2 - 3 \beta_{1} ) q^{11} + \beta_{2} q^{12} + ( -2 + \beta_{1} + \beta_{2} ) q^{14} + q^{16} + ( -1 - \beta_{1} ) q^{17} + ( \beta_{1} + \beta_{2} ) q^{18} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{19} + ( -2 + 3 \beta_{2} ) q^{21} + ( -2 + 3 \beta_{1} ) q^{22} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{23} -\beta_{2} q^{24} + ( -2 - 2 \beta_{2} ) q^{27} + ( 2 - \beta_{1} - \beta_{2} ) q^{28} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 1 + 3 \beta_{1} + 4 \beta_{2} ) q^{31} - q^{32} + ( 3 - 3 \beta_{1} + 2 \beta_{2} ) q^{33} + ( 1 + \beta_{1} ) q^{34} + ( -\beta_{1} - \beta_{2} ) q^{36} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{37} + ( 5 - 2 \beta_{1} - \beta_{2} ) q^{38} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{41} + ( 2 - 3 \beta_{2} ) q^{42} -2 q^{43} + ( 2 - 3 \beta_{1} ) q^{44} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{46} + ( 2 + \beta_{1} + \beta_{2} ) q^{47} + \beta_{2} q^{48} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{49} + ( 1 - \beta_{1} - \beta_{2} ) q^{51} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 2 + 2 \beta_{2} ) q^{54} + ( -2 + \beta_{1} + \beta_{2} ) q^{56} + ( 1 + \beta_{1} - 6 \beta_{2} ) q^{57} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{58} + ( 4 - 4 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 4 \beta_{1} - 3 \beta_{2} ) q^{61} + ( -1 - 3 \beta_{1} - 4 \beta_{2} ) q^{62} + ( 3 - 2 \beta_{2} ) q^{63} + q^{64} + ( -3 + 3 \beta_{1} - 2 \beta_{2} ) q^{66} + ( 4 - 2 \beta_{2} ) q^{67} + ( -1 - \beta_{1} ) q^{68} + ( 6 - 4 \beta_{1} ) q^{69} + ( -6 - 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( \beta_{1} + \beta_{2} ) q^{72} + ( 3 - 3 \beta_{1} - 6 \beta_{2} ) q^{73} + ( -1 - 2 \beta_{1} - 3 \beta_{2} ) q^{74} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{76} + ( 7 - 2 \beta_{1} + \beta_{2} ) q^{77} + ( -3 - 5 \beta_{1} - 4 \beta_{2} ) q^{79} + ( -6 + 5 \beta_{1} + 3 \beta_{2} ) q^{81} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{82} + ( 2 - 2 \beta_{1} + 5 \beta_{2} ) q^{83} + ( -2 + 3 \beta_{2} ) q^{84} + 2 q^{86} + ( 8 - 4 \beta_{1} - 6 \beta_{2} ) q^{87} + ( -2 + 3 \beta_{1} ) q^{88} + ( 2 + \beta_{1} - \beta_{2} ) q^{89} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{92} + ( 9 - \beta_{1} - 3 \beta_{2} ) q^{93} + ( -2 - \beta_{1} - \beta_{2} ) q^{94} -\beta_{2} q^{96} + ( 9 - \beta_{1} + 2 \beta_{2} ) q^{97} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{98} + ( 3 + 4 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} + 5 q^{7} - 3 q^{8} - q^{9} + O(q^{10})$$ $$3 q - 3 q^{2} + 3 q^{4} + 5 q^{7} - 3 q^{8} - q^{9} + 3 q^{11} - 5 q^{14} + 3 q^{16} - 4 q^{17} + q^{18} - 13 q^{19} - 6 q^{21} - 3 q^{22} - 6 q^{27} + 5 q^{28} - 14 q^{29} + 6 q^{31} - 3 q^{32} + 6 q^{33} + 4 q^{34} - q^{36} + 5 q^{37} + 13 q^{38} + 2 q^{41} + 6 q^{42} - 6 q^{43} + 3 q^{44} + 7 q^{47} - 2 q^{49} + 2 q^{51} - 9 q^{53} + 6 q^{54} - 5 q^{56} + 4 q^{57} + 14 q^{58} + 8 q^{59} + 4 q^{61} - 6 q^{62} + 9 q^{63} + 3 q^{64} - 6 q^{66} + 12 q^{67} - 4 q^{68} + 14 q^{69} - 20 q^{71} + q^{72} + 6 q^{73} - 5 q^{74} - 13 q^{76} + 19 q^{77} - 14 q^{79} - 13 q^{81} - 2 q^{82} + 4 q^{83} - 6 q^{84} + 6 q^{86} + 20 q^{87} - 3 q^{88} + 7 q^{89} + 26 q^{93} - 7 q^{94} + 26 q^{97} + 2 q^{98} + 13 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

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

## 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
 0.311108 2.17009 −1.48119
−1.00000 −2.21432 1.00000 0 2.21432 3.90321 −1.00000 1.90321 0
1.2 −1.00000 0.539189 1.00000 0 −0.539189 −0.709275 −1.00000 −2.70928 0
1.3 −1.00000 1.67513 1.00000 0 −1.67513 1.80606 −1.00000 −0.193937 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.bu 3
5.b even 2 1 8450.2.a.cb 3
5.c odd 4 2 1690.2.b.c 6
13.b even 2 1 8450.2.a.ca 3
13.c even 3 2 650.2.e.k 6
65.d even 2 1 8450.2.a.bt 3
65.f even 4 2 1690.2.c.c 6
65.h odd 4 2 1690.2.b.b 6
65.k even 4 2 1690.2.c.b 6
65.n even 6 2 650.2.e.j 6
65.q odd 12 4 130.2.n.a 12
195.bl even 12 4 1170.2.bp.h 12
260.bj even 12 4 1040.2.dh.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.n.a 12 65.q odd 12 4
650.2.e.j 6 65.n even 6 2
650.2.e.k 6 13.c even 3 2
1040.2.dh.b 12 260.bj even 12 4
1170.2.bp.h 12 195.bl even 12 4
1690.2.b.b 6 65.h odd 4 2
1690.2.b.c 6 5.c odd 4 2
1690.2.c.b 6 65.k even 4 2
1690.2.c.c 6 65.f even 4 2
8450.2.a.bt 3 65.d even 2 1
8450.2.a.bu 3 1.a even 1 1 trivial
8450.2.a.ca 3 13.b even 2 1
8450.2.a.cb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8450))$$:

 $$T_{3}^{3} - 4 T_{3} + 2$$ $$T_{7}^{3} - 5 T_{7}^{2} + 3 T_{7} + 5$$ $$T_{11}^{3} - 3 T_{11}^{2} - 27 T_{11} + 31$$ $$T_{17}^{3} + 4 T_{17}^{2} + 2 T_{17} - 2$$ $$T_{31}^{3} - 6 T_{31}^{2} - 58 T_{31} + 218$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$2 - 4 T + T^{3}$$
$5$ $$T^{3}$$
$7$ $$5 + 3 T - 5 T^{2} + T^{3}$$
$11$ $$31 - 27 T - 3 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$-2 + 2 T + 4 T^{2} + T^{3}$$
$19$ $$5 + 43 T + 13 T^{2} + T^{3}$$
$23$ $$-76 - 40 T + T^{3}$$
$29$ $$-152 + 28 T + 14 T^{2} + T^{3}$$
$31$ $$218 - 58 T - 6 T^{2} + T^{3}$$
$37$ $$107 - 29 T - 5 T^{2} + T^{3}$$
$41$ $$20 - 44 T - 2 T^{2} + T^{3}$$
$43$ $$( 2 + T )^{3}$$
$47$ $$-1 + 11 T - 7 T^{2} + T^{3}$$
$53$ $$-13 - 7 T + 9 T^{2} + T^{3}$$
$59$ $$272 - 32 T - 8 T^{2} + T^{3}$$
$61$ $$610 - 108 T - 4 T^{2} + T^{3}$$
$67$ $$-16 + 32 T - 12 T^{2} + T^{3}$$
$71$ $$-464 + 40 T + 20 T^{2} + T^{3}$$
$73$ $$-270 - 126 T - 6 T^{2} + T^{3}$$
$79$ $$-158 - 42 T + 14 T^{2} + T^{3}$$
$83$ $$46 - 128 T - 4 T^{2} + T^{3}$$
$89$ $$19 + 7 T - 7 T^{2} + T^{3}$$
$97$ $$-466 + 202 T - 26 T^{2} + T^{3}$$