# Properties

 Label 8450.2.a.bi Level $8450$ Weight $2$ Character orbit 8450.a Self dual yes Analytic conductor $67.474$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 650) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.30278 −1.30278
1.00000 −2.30278 1.00000 0 −2.30278 0.697224 1.00000 2.30278 0
1.2 1.00000 1.30278 1.00000 0 1.30278 4.30278 1.00000 −1.30278 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.bi 2
5.b even 2 1 8450.2.a.bd 2
13.b even 2 1 8450.2.a.ba 2
13.c even 3 2 650.2.e.e 4
65.d even 2 1 8450.2.a.bl 2
65.n even 6 2 650.2.e.g yes 4
65.q odd 12 4 650.2.o.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.e.e 4 13.c even 3 2
650.2.e.g yes 4 65.n even 6 2
650.2.o.h 8 65.q odd 12 4
8450.2.a.ba 2 13.b even 2 1
8450.2.a.bd 2 5.b even 2 1
8450.2.a.bi 2 1.a even 1 1 trivial
8450.2.a.bl 2 65.d even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} - 3$$ $$T_{7}^{2} - 5 T_{7} + 3$$ $$T_{11}^{2} + 5 T_{11} + 3$$ $$T_{17}^{2} + 5 T_{17} - 23$$ $$T_{31}^{2} + 10 T_{31} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$-3 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$3 - 5 T + T^{2}$$
$11$ $$3 + 5 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-23 + 5 T + T^{2}$$
$19$ $$3 + 8 T + T^{2}$$
$23$ $$-9 + 4 T + T^{2}$$
$29$ $$-1 - 3 T + T^{2}$$
$31$ $$12 + 10 T + T^{2}$$
$37$ $$27 + 11 T + T^{2}$$
$41$ $$( -1 + T )^{2}$$
$43$ $$-81 + T + T^{2}$$
$47$ $$69 - 17 T + T^{2}$$
$53$ $$-27 + 10 T + T^{2}$$
$59$ $$-43 + 6 T + T^{2}$$
$61$ $$-43 + 6 T + T^{2}$$
$67$ $$-9 - 4 T + T^{2}$$
$71$ $$36 + 14 T + T^{2}$$
$73$ $$( 8 + T )^{2}$$
$79$ $$53 - 15 T + T^{2}$$
$83$ $$-81 - 12 T + T^{2}$$
$89$ $$1 - 11 T + T^{2}$$
$97$ $$179 + 27 T + T^{2}$$