# Properties

 Label 9450.2.a.ed Level 9450 Weight 2 Character orbit 9450.a Self dual yes Analytic conductor 75.459 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 9450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.4586299101$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1890) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} - q^{7} - q^{8} + \beta q^{11} + ( -2 + \beta ) q^{13} + q^{14} + q^{16} + \beta q^{17} + 2 q^{19} -\beta q^{22} -\beta q^{23} + ( 2 - \beta ) q^{26} - q^{28} + 3 q^{29} + ( -1 + 2 \beta ) q^{31} - q^{32} -\beta q^{34} + ( -5 + \beta ) q^{37} -2 q^{38} + ( 3 - \beta ) q^{41} + ( -2 - \beta ) q^{43} + \beta q^{44} + \beta q^{46} + ( 3 + 2 \beta ) q^{47} + q^{49} + ( -2 + \beta ) q^{52} + 6 q^{53} + q^{56} -3 q^{58} + ( -3 + \beta ) q^{59} + ( -1 - \beta ) q^{61} + ( 1 - 2 \beta ) q^{62} + q^{64} + ( -2 + 2 \beta ) q^{67} + \beta q^{68} + ( 3 - \beta ) q^{71} + ( -11 + \beta ) q^{73} + ( 5 - \beta ) q^{74} + 2 q^{76} -\beta q^{77} + ( 2 + \beta ) q^{79} + ( -3 + \beta ) q^{82} + ( 2 + \beta ) q^{86} -\beta q^{88} + ( -6 + 2 \beta ) q^{89} + ( 2 - \beta ) q^{91} -\beta q^{92} + ( -3 - 2 \beta ) q^{94} + ( 10 - 2 \beta ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{7} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{7} - 2q^{8} + q^{11} - 3q^{13} + 2q^{14} + 2q^{16} + q^{17} + 4q^{19} - q^{22} - q^{23} + 3q^{26} - 2q^{28} + 6q^{29} - 2q^{32} - q^{34} - 9q^{37} - 4q^{38} + 5q^{41} - 5q^{43} + q^{44} + q^{46} + 8q^{47} + 2q^{49} - 3q^{52} + 12q^{53} + 2q^{56} - 6q^{58} - 5q^{59} - 3q^{61} + 2q^{64} - 2q^{67} + q^{68} + 5q^{71} - 21q^{73} + 9q^{74} + 4q^{76} - q^{77} + 5q^{79} - 5q^{82} + 5q^{86} - q^{88} - 10q^{89} + 3q^{91} - q^{92} - 8q^{94} + 18q^{97} - 2q^{98} + O(q^{100})$$

## 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
 −3.77200 4.77200
−1.00000 0 1.00000 0 0 −1.00000 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 −1.00000 −1.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 9450.2.a.ed 2
3.b odd 2 1 9450.2.a.eo 2
5.b even 2 1 1890.2.a.bb yes 2
15.d odd 2 1 1890.2.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.a.z 2 15.d odd 2 1
1890.2.a.bb yes 2 5.b even 2 1
9450.2.a.ed 2 1.a even 1 1 trivial
9450.2.a.eo 2 3.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9450))$$:

 $$T_{11}^{2} - T_{11} - 18$$ $$T_{13}^{2} + 3 T_{13} - 16$$ $$T_{17}^{2} - T_{17} - 18$$ $$T_{19} - 2$$