Properties

 Label 441.6.a.o Level $441$ Weight $6$ Character orbit 441.a Self dual yes Analytic conductor $70.729$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 441.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$70.7292645375$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 25 q^{4} - 57 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 25 * q^4 - 57*b * q^8 $$q + \beta q^{2} - 25 q^{4} - 57 \beta q^{8} - 302 \beta q^{11} + 401 q^{16} - 2114 q^{22} - 418 \beta q^{23} - 3125 q^{25} + 2036 \beta q^{29} + 2225 \beta q^{32} + 8886 q^{37} - 11748 q^{43} + 7550 \beta q^{44} - 2926 q^{46} - 3125 \beta q^{50} - 12364 \beta q^{53} + 14252 q^{58} + 2743 q^{64} + 69364 q^{67} + 32098 \beta q^{71} + 8886 \beta q^{74} + 80168 q^{79} - 11748 \beta q^{86} + 120498 q^{88} + 10450 \beta q^{92} +O(q^{100})$$ q + b * q^2 - 25 * q^4 - 57*b * q^8 - 302*b * q^11 + 401 * q^16 - 2114 * q^22 - 418*b * q^23 - 3125 * q^25 + 2036*b * q^29 + 2225*b * q^32 + 8886 * q^37 - 11748 * q^43 + 7550*b * q^44 - 2926 * q^46 - 3125*b * q^50 - 12364*b * q^53 + 14252 * q^58 + 2743 * q^64 + 69364 * q^67 + 32098*b * q^71 + 8886*b * q^74 + 80168 * q^79 - 11748*b * q^86 + 120498 * q^88 + 10450*b * q^92 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 50 q^{4}+O(q^{10})$$ 2 * q - 50 * q^4 $$2 q - 50 q^{4} + 802 q^{16} - 4228 q^{22} - 6250 q^{25} + 17772 q^{37} - 23496 q^{43} - 5852 q^{46} + 28504 q^{58} + 5486 q^{64} + 138728 q^{67} + 160336 q^{79} + 240996 q^{88}+O(q^{100})$$ 2 * q - 50 * q^4 + 802 * q^16 - 4228 * q^22 - 6250 * q^25 + 17772 * q^37 - 23496 * q^43 - 5852 * q^46 + 28504 * q^58 + 5486 * q^64 + 138728 * q^67 + 160336 * q^79 + 240996 * q^88

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
 −2.64575 2.64575
−2.64575 0 −25.0000 0 0 0 150.808 0 0
1.2 2.64575 0 −25.0000 0 0 0 −150.808 0 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
$$3$$ $$1$$
$$7$$ $$-1$$

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.o 2
3.b odd 2 1 inner 441.6.a.o 2
7.b odd 2 1 CM 441.6.a.o 2
21.c even 2 1 inner 441.6.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.6.a.o 2 1.a even 1 1 trivial
441.6.a.o 2 3.b odd 2 1 inner
441.6.a.o 2 7.b odd 2 1 CM
441.6.a.o 2 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(441))$$:

 $$T_{2}^{2} - 7$$ T2^2 - 7 $$T_{5}$$ T5 $$T_{13}$$ T13

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 7$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 638428$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 1223068$$
$29$ $$T^{2} - 29017072$$
$31$ $$T^{2}$$
$37$ $$(T - 8886)^{2}$$
$41$ $$T^{2}$$
$43$ $$(T + 11748)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 1070079472$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$(T - 69364)^{2}$$
$71$ $$T^{2} - 7211971228$$
$73$ $$T^{2}$$
$79$ $$(T - 80168)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$