# Properties

 Label 49.6.a.b Level $49$ Weight $6$ Character orbit 49.a Self dual yes Analytic conductor $7.859$ Analytic rank $0$ Dimension $1$ CM discriminant -7 Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.85880717084$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 11 q^{2} + 89 q^{4} + 627 q^{8} - 243 q^{9}+O(q^{10})$$ q + 11 * q^2 + 89 * q^4 + 627 * q^8 - 243 * q^9 $$q + 11 q^{2} + 89 q^{4} + 627 q^{8} - 243 q^{9} - 76 q^{11} + 4049 q^{16} - 2673 q^{18} - 836 q^{22} - 4952 q^{23} - 3125 q^{25} + 7282 q^{29} + 24475 q^{32} - 21627 q^{36} - 8886 q^{37} + 11748 q^{43} - 6764 q^{44} - 54472 q^{46} - 34375 q^{50} + 24550 q^{53} + 80102 q^{58} + 139657 q^{64} + 69364 q^{67} - 2224 q^{71} - 152361 q^{72} - 97746 q^{74} + 80168 q^{79} + 59049 q^{81} + 129228 q^{86} - 47652 q^{88} - 440728 q^{92} + 18468 q^{99}+O(q^{100})$$ q + 11 * q^2 + 89 * q^4 + 627 * q^8 - 243 * q^9 - 76 * q^11 + 4049 * q^16 - 2673 * q^18 - 836 * q^22 - 4952 * q^23 - 3125 * q^25 + 7282 * q^29 + 24475 * q^32 - 21627 * q^36 - 8886 * q^37 + 11748 * q^43 - 6764 * q^44 - 54472 * q^46 - 34375 * q^50 + 24550 * q^53 + 80102 * q^58 + 139657 * q^64 + 69364 * q^67 - 2224 * q^71 - 152361 * q^72 - 97746 * q^74 + 80168 * q^79 + 59049 * q^81 + 129228 * q^86 - 47652 * q^88 - 440728 * q^92 + 18468 * q^99

## 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
 0
11.0000 0 89.0000 0 0 0 627.000 −243.000 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
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.6.a.b 1
3.b odd 2 1 441.6.a.a 1
4.b odd 2 1 784.6.a.g 1
7.b odd 2 1 CM 49.6.a.b 1
7.c even 3 2 49.6.c.a 2
7.d odd 6 2 49.6.c.a 2
21.c even 2 1 441.6.a.a 1
28.d even 2 1 784.6.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.b 1 1.a even 1 1 trivial
49.6.a.b 1 7.b odd 2 1 CM
49.6.c.a 2 7.c even 3 2
49.6.c.a 2 7.d odd 6 2
441.6.a.a 1 3.b odd 2 1
441.6.a.a 1 21.c even 2 1
784.6.a.g 1 4.b odd 2 1
784.6.a.g 1 28.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 $$S_{6}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2} - 11$$ T2 - 11 $$T_{3}$$ T3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 11$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 76$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 4952$$
$29$ $$T - 7282$$
$31$ $$T$$
$37$ $$T + 8886$$
$41$ $$T$$
$43$ $$T - 11748$$
$47$ $$T$$
$53$ $$T - 24550$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T - 69364$$
$71$ $$T + 2224$$
$73$ $$T$$
$79$ $$T - 80168$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$
show more
show less