# Properties

 Label 8450.2.a.bj 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: $$\Q(\sqrt{10})$$ Defining polynomial: $$x^{2} - 10$$ 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 $$\beta = \sqrt{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - q^{7} + q^{8} + 7 q^{9} +O(q^{10})$$ $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - q^{7} + q^{8} + 7 q^{9} + ( 1 + \beta ) q^{11} + \beta q^{12} - q^{14} + q^{16} + ( 2 - \beta ) q^{17} + 7 q^{18} + ( 5 + \beta ) q^{19} -\beta q^{21} + ( 1 + \beta ) q^{22} -6 q^{23} + \beta q^{24} + 4 \beta q^{27} - q^{28} + ( 4 - 2 \beta ) q^{29} + ( -4 + \beta ) q^{31} + q^{32} + ( 10 + \beta ) q^{33} + ( 2 - \beta ) q^{34} + 7 q^{36} + ( 3 + \beta ) q^{37} + ( 5 + \beta ) q^{38} + ( 4 - 2 \beta ) q^{41} -\beta q^{42} -2 q^{43} + ( 1 + \beta ) q^{44} -6 q^{46} -3 q^{47} + \beta q^{48} -6 q^{49} + ( -10 + 2 \beta ) q^{51} + ( -1 - \beta ) q^{53} + 4 \beta q^{54} - q^{56} + ( 10 + 5 \beta ) q^{57} + ( 4 - 2 \beta ) q^{58} + ( 4 - 2 \beta ) q^{59} + ( 2 + 3 \beta ) q^{61} + ( -4 + \beta ) q^{62} -7 q^{63} + q^{64} + ( 10 + \beta ) q^{66} + ( 4 + 2 \beta ) q^{67} + ( 2 - \beta ) q^{68} -6 \beta q^{69} + ( -2 - 2 \beta ) q^{71} + 7 q^{72} + ( 6 + \beta ) q^{73} + ( 3 + \beta ) q^{74} + ( 5 + \beta ) q^{76} + ( -1 - \beta ) q^{77} + ( -4 + 3 \beta ) q^{79} + 19 q^{81} + ( 4 - 2 \beta ) q^{82} -3 \beta q^{83} -\beta q^{84} -2 q^{86} + ( -20 + 4 \beta ) q^{87} + ( 1 + \beta ) q^{88} + ( 1 - 2 \beta ) q^{89} -6 q^{92} + ( 10 - 4 \beta ) q^{93} -3 q^{94} + \beta q^{96} + ( 2 + 3 \beta ) q^{97} -6 q^{98} + ( 7 + 7 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 14 q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 14 q^{9} + 2 q^{11} - 2 q^{14} + 2 q^{16} + 4 q^{17} + 14 q^{18} + 10 q^{19} + 2 q^{22} - 12 q^{23} - 2 q^{28} + 8 q^{29} - 8 q^{31} + 2 q^{32} + 20 q^{33} + 4 q^{34} + 14 q^{36} + 6 q^{37} + 10 q^{38} + 8 q^{41} - 4 q^{43} + 2 q^{44} - 12 q^{46} - 6 q^{47} - 12 q^{49} - 20 q^{51} - 2 q^{53} - 2 q^{56} + 20 q^{57} + 8 q^{58} + 8 q^{59} + 4 q^{61} - 8 q^{62} - 14 q^{63} + 2 q^{64} + 20 q^{66} + 8 q^{67} + 4 q^{68} - 4 q^{71} + 14 q^{72} + 12 q^{73} + 6 q^{74} + 10 q^{76} - 2 q^{77} - 8 q^{79} + 38 q^{81} + 8 q^{82} - 4 q^{86} - 40 q^{87} + 2 q^{88} + 2 q^{89} - 12 q^{92} + 20 q^{93} - 6 q^{94} + 4 q^{97} - 12 q^{98} + 14 q^{99} + O(q^{100})$$

## 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
 −3.16228 3.16228
1.00000 −3.16228 1.00000 0 −3.16228 −1.00000 1.00000 7.00000 0
1.2 1.00000 3.16228 1.00000 0 3.16228 −1.00000 1.00000 7.00000 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.bj 2
5.b even 2 1 1690.2.a.k 2
13.b even 2 1 8450.2.a.bc 2
13.e even 6 2 650.2.e.h 4
65.d even 2 1 1690.2.a.n 2
65.g odd 4 2 1690.2.d.g 4
65.l even 6 2 130.2.e.c 4
65.n even 6 2 1690.2.e.m 4
65.r odd 12 4 650.2.o.g 8
65.s odd 12 4 1690.2.l.k 8
195.y odd 6 2 1170.2.i.q 4
260.w odd 6 2 1040.2.q.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.c 4 65.l even 6 2
650.2.e.h 4 13.e even 6 2
650.2.o.g 8 65.r odd 12 4
1040.2.q.m 4 260.w odd 6 2
1170.2.i.q 4 195.y odd 6 2
1690.2.a.k 2 5.b even 2 1
1690.2.a.n 2 65.d even 2 1
1690.2.d.g 4 65.g odd 4 2
1690.2.e.m 4 65.n even 6 2
1690.2.l.k 8 65.s odd 12 4
8450.2.a.bc 2 13.b even 2 1
8450.2.a.bj 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8450))$$:

 $$T_{3}^{2} - 10$$ $$T_{7} + 1$$ $$T_{11}^{2} - 2 T_{11} - 9$$ $$T_{17}^{2} - 4 T_{17} - 6$$ $$T_{31}^{2} + 8 T_{31} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-10 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-9 - 2 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-6 - 4 T + T^{2}$$
$19$ $$15 - 10 T + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$-24 - 8 T + T^{2}$$
$31$ $$6 + 8 T + T^{2}$$
$37$ $$-1 - 6 T + T^{2}$$
$41$ $$-24 - 8 T + T^{2}$$
$43$ $$( 2 + T )^{2}$$
$47$ $$( 3 + T )^{2}$$
$53$ $$-9 + 2 T + T^{2}$$
$59$ $$-24 - 8 T + T^{2}$$
$61$ $$-86 - 4 T + T^{2}$$
$67$ $$-24 - 8 T + T^{2}$$
$71$ $$-36 + 4 T + T^{2}$$
$73$ $$26 - 12 T + T^{2}$$
$79$ $$-74 + 8 T + T^{2}$$
$83$ $$-90 + T^{2}$$
$89$ $$-39 - 2 T + T^{2}$$
$97$ $$-86 - 4 T + T^{2}$$