# Properties

 Label 8450.2.a.bf Level $8450$ Weight $2$ Character orbit 8450.a Self dual yes Analytic conductor $67.474$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
−1.00000 −0.732051 1.00000 0 0.732051 3.00000 −1.00000 −2.46410 0
1.2 −1.00000 2.73205 1.00000 0 −2.73205 3.00000 −1.00000 4.46410 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.bf 2
5.b even 2 1 1690.2.a.m 2
13.b even 2 1 8450.2.a.bm 2
13.f odd 12 2 650.2.m.a 4
65.d even 2 1 1690.2.a.j 2
65.g odd 4 2 1690.2.d.f 4
65.l even 6 2 1690.2.e.n 4
65.n even 6 2 1690.2.e.l 4
65.o even 12 2 650.2.n.b 4
65.s odd 12 2 130.2.l.a 4
65.s odd 12 2 1690.2.l.g 4
65.t even 12 2 650.2.n.a 4
195.bh even 12 2 1170.2.bs.c 4
260.bc even 12 2 1040.2.da.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.a 4 65.s odd 12 2
650.2.m.a 4 13.f odd 12 2
650.2.n.a 4 65.t even 12 2
650.2.n.b 4 65.o even 12 2
1040.2.da.a 4 260.bc even 12 2
1170.2.bs.c 4 195.bh even 12 2
1690.2.a.j 2 65.d even 2 1
1690.2.a.m 2 5.b even 2 1
1690.2.d.f 4 65.g odd 4 2
1690.2.e.l 4 65.n even 6 2
1690.2.e.n 4 65.l even 6 2
1690.2.l.g 4 65.s odd 12 2
8450.2.a.bf 2 1.a even 1 1 trivial
8450.2.a.bm 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8450))$$:

 $$T_{3}^{2} - 2T_{3} - 2$$ T3^2 - 2*T3 - 2 $$T_{7} - 3$$ T7 - 3 $$T_{11} - 3$$ T11 - 3 $$T_{17}^{2} - 6T_{17} - 18$$ T17^2 - 6*T17 - 18 $$T_{31}^{2} - 6T_{31} + 6$$ T31^2 - 6*T31 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - 2T - 2$$
$5$ $$T^{2}$$
$7$ $$(T - 3)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 6T - 18$$
$19$ $$T^{2} - 6T - 3$$
$23$ $$T^{2} - 12T + 24$$
$29$ $$T^{2} + 12T + 24$$
$31$ $$T^{2} - 6T + 6$$
$37$ $$T^{2} - 12T + 9$$
$41$ $$T^{2} - 108$$
$43$ $$(T - 2)^{2}$$
$47$ $$(T - 3)^{2}$$
$53$ $$T^{2} - 6T - 3$$
$59$ $$T^{2} - 108$$
$61$ $$T^{2} - 2T - 26$$
$67$ $$T^{2}$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} - 6T - 66$$
$79$ $$T^{2} - 2T - 26$$
$83$ $$T^{2} - 6T - 18$$
$89$ $$T^{2} - 24T + 117$$
$97$ $$T^{2} - 6T - 138$$