# Properties

 Label 2450.2.a.bi Level $2450$ Weight $2$ Character orbit 2450.a Self dual yes Analytic conductor $19.563$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.5633484952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 490) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{8} -3 q^{9} +O(q^{10})$$ $$q - q^{2} + q^{4} - q^{8} -3 q^{9} -4 q^{11} + 3 \beta q^{13} + q^{16} -3 \beta q^{17} + 3 q^{18} + 4 \beta q^{19} + 4 q^{22} -3 \beta q^{26} -4 q^{29} -4 \beta q^{31} - q^{32} + 3 \beta q^{34} -3 q^{36} + 6 q^{37} -4 \beta q^{38} -\beta q^{41} + 12 q^{43} -4 q^{44} + 3 \beta q^{52} + 12 q^{53} + 4 q^{58} -8 \beta q^{59} + 5 \beta q^{61} + 4 \beta q^{62} + q^{64} + 12 q^{67} -3 \beta q^{68} + 8 q^{71} + 3 q^{72} -3 \beta q^{73} -6 q^{74} + 4 \beta q^{76} + 9 q^{81} + \beta q^{82} + 12 \beta q^{83} -12 q^{86} + 4 q^{88} -3 \beta q^{89} + 3 \beta q^{97} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} - 6q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} - 6q^{9} - 8q^{11} + 2q^{16} + 6q^{18} + 8q^{22} - 8q^{29} - 2q^{32} - 6q^{36} + 12q^{37} + 24q^{43} - 8q^{44} + 24q^{53} + 8q^{58} + 2q^{64} + 24q^{67} + 16q^{71} + 6q^{72} - 12q^{74} + 18q^{81} - 24q^{86} + 8q^{88} + 24q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
−1.00000 0 1.00000 0 0 0 −1.00000 −3.00000 0
1.2 −1.00000 0 1.00000 0 0 0 −1.00000 −3.00000 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
$$2$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.2.a.bi 2
5.b even 2 1 2450.2.a.bo 2
5.c odd 4 2 490.2.c.g 4
7.b odd 2 1 inner 2450.2.a.bi 2
35.c odd 2 1 2450.2.a.bo 2
35.f even 4 2 490.2.c.g 4
35.k even 12 4 490.2.i.d 8
35.l odd 12 4 490.2.i.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.c.g 4 5.c odd 4 2
490.2.c.g 4 35.f even 4 2
490.2.i.d 8 35.k even 12 4
490.2.i.d 8 35.l odd 12 4
2450.2.a.bi 2 1.a even 1 1 trivial
2450.2.a.bi 2 7.b odd 2 1 inner
2450.2.a.bo 2 5.b even 2 1
2450.2.a.bo 2 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2450))$$:

 $$T_{3}$$ $$T_{11} + 4$$ $$T_{13}^{2} - 18$$ $$T_{17}^{2} - 18$$ $$T_{19}^{2} - 32$$ $$T_{23}$$ $$T_{37} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$-18 + T^{2}$$
$17$ $$-18 + T^{2}$$
$19$ $$-32 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$-32 + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$-2 + T^{2}$$
$43$ $$( -12 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$-128 + T^{2}$$
$61$ $$-50 + T^{2}$$
$67$ $$( -12 + T )^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$-18 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$-288 + T^{2}$$
$89$ $$-18 + T^{2}$$
$97$ $$-18 + T^{2}$$