# Properties

 Label 9450.2.a.eb Level 9450 Weight 2 Character orbit 9450.a Self dual yes Analytic conductor 75.459 Analytic rank 1 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} - q^{7} - q^{8} + \beta q^{11} + ( 2 - 2 \beta ) q^{13} + q^{14} + q^{16} -4 q^{17} + ( 4 - \beta ) q^{19} -\beta q^{22} + ( 1 + 2 \beta ) q^{23} + ( -2 + 2 \beta ) q^{26} - q^{28} + ( 2 + 2 \beta ) q^{29} + ( -2 - \beta ) q^{31} - q^{32} + 4 q^{34} + ( 2 + 3 \beta ) q^{37} + ( -4 + \beta ) q^{38} + ( -3 - 4 \beta ) q^{41} + ( -8 - 2 \beta ) q^{43} + \beta q^{44} + ( -1 - 2 \beta ) q^{46} + ( -8 + 2 \beta ) q^{47} + q^{49} + ( 2 - 2 \beta ) q^{52} -8 \beta q^{53} + q^{56} + ( -2 - 2 \beta ) q^{58} + ( 2 + 4 \beta ) q^{59} + 8 \beta q^{61} + ( 2 + \beta ) q^{62} + q^{64} + ( -2 + 6 \beta ) q^{67} -4 q^{68} + ( -6 - \beta ) q^{71} + ( 10 - 4 \beta ) q^{73} + ( -2 - 3 \beta ) q^{74} + ( 4 - \beta ) q^{76} -\beta q^{77} + ( 2 - 4 \beta ) q^{79} + ( 3 + 4 \beta ) q^{82} + ( -6 + 2 \beta ) q^{83} + ( 8 + 2 \beta ) q^{86} -\beta q^{88} - q^{89} + ( -2 + 2 \beta ) q^{91} + ( 1 + 2 \beta ) q^{92} + ( 8 - 2 \beta ) q^{94} + 4 \beta q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{7} - 2q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{7} - 2q^{8} + 4q^{13} + 2q^{14} + 2q^{16} - 8q^{17} + 8q^{19} + 2q^{23} - 4q^{26} - 2q^{28} + 4q^{29} - 4q^{31} - 2q^{32} + 8q^{34} + 4q^{37} - 8q^{38} - 6q^{41} - 16q^{43} - 2q^{46} - 16q^{47} + 2q^{49} + 4q^{52} + 2q^{56} - 4q^{58} + 4q^{59} + 4q^{62} + 2q^{64} - 4q^{67} - 8q^{68} - 12q^{71} + 20q^{73} - 4q^{74} + 8q^{76} + 4q^{79} + 6q^{82} - 12q^{83} + 16q^{86} - 2q^{89} - 4q^{91} + 2q^{92} + 16q^{94} - 2q^{98} + O(q^{100})$$

## 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
 −1.73205 1.73205
−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.eb 2
3.b odd 2 1 9450.2.a.er 2
5.b even 2 1 9450.2.a.eu 2
5.c odd 4 2 1890.2.g.n 4
15.d odd 2 1 9450.2.a.ei 2
15.e even 4 2 1890.2.g.q yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.g.n 4 5.c odd 4 2
1890.2.g.q yes 4 15.e even 4 2
9450.2.a.eb 2 1.a even 1 1 trivial
9450.2.a.ei 2 15.d odd 2 1
9450.2.a.er 2 3.b odd 2 1
9450.2.a.eu 2 5.b even 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9450))$$:

 $$T_{11}^{2} - 3$$ $$T_{13}^{2} - 4 T_{13} - 8$$ $$T_{17} + 4$$ $$T_{19}^{2} - 8 T_{19} + 13$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ 
$5$ 
$7$ $$( 1 + T )^{2}$$
$11$ $$1 + 19 T^{2} + 121 T^{4}$$
$13$ $$1 - 4 T + 18 T^{2} - 52 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 4 T + 17 T^{2} )^{2}$$
$19$ $$1 - 8 T + 51 T^{2} - 152 T^{3} + 361 T^{4}$$
$23$ $$1 - 2 T + 35 T^{2} - 46 T^{3} + 529 T^{4}$$
$29$ $$1 - 4 T + 50 T^{2} - 116 T^{3} + 841 T^{4}$$
$31$ $$1 + 4 T + 63 T^{2} + 124 T^{3} + 961 T^{4}$$
$37$ $$1 - 4 T + 51 T^{2} - 148 T^{3} + 1369 T^{4}$$
$41$ $$1 + 6 T + 43 T^{2} + 246 T^{3} + 1681 T^{4}$$
$43$ $$1 + 16 T + 138 T^{2} + 688 T^{3} + 1849 T^{4}$$
$47$ $$1 + 16 T + 146 T^{2} + 752 T^{3} + 2209 T^{4}$$
$53$ $$1 - 86 T^{2} + 2809 T^{4}$$
$59$ $$1 - 4 T + 74 T^{2} - 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 70 T^{2} + 3721 T^{4}$$
$67$ $$1 + 4 T + 30 T^{2} + 268 T^{3} + 4489 T^{4}$$
$71$ $$1 + 12 T + 175 T^{2} + 852 T^{3} + 5041 T^{4}$$
$73$ $$1 - 20 T + 198 T^{2} - 1460 T^{3} + 5329 T^{4}$$
$79$ $$1 - 4 T + 114 T^{2} - 316 T^{3} + 6241 T^{4}$$
$83$ $$1 + 12 T + 190 T^{2} + 996 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + T + 89 T^{2} )^{2}$$
$97$ $$1 + 146 T^{2} + 9409 T^{4}$$