Properties

 Label 416.1.bl.a Level $416$ Weight $1$ Character orbit 416.bl Analytic conductor $0.208$ Analytic rank $0$ Dimension $4$ Projective image $D_{12}$ CM discriminant -4 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$416 = 2^{5} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 416.bl (of order $$12$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.207611045255$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{12}$$ Projective field: Galois closure of 12.0.469804094334435328.7

$q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{5} + \zeta_{12}^{4}) q^{5} + \zeta_{12}^{2} q^{9}+O(q^{10})$$ q + (z^5 + z^4) * q^5 + z^2 * q^9 $$q + (\zeta_{12}^{5} + \zeta_{12}^{4}) q^{5} + \zeta_{12}^{2} q^{9} + \zeta_{12} q^{13} + \zeta_{12}^{5} q^{17} + ( - \zeta_{12}^{4} - \zeta_{12}^{3} - \zeta_{12}^{2}) q^{25} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{29} + (\zeta_{12}^{3} - \zeta_{12}^{2}) q^{37} + ( - \zeta_{12}^{5} - 1) q^{41} + ( - \zeta_{12} - 1) q^{45} + \zeta_{12} q^{49} + q^{53} - \zeta_{12}^{2} q^{61} + (\zeta_{12}^{5} - 1) q^{65} + ( - \zeta_{12}^{5} + \zeta_{12}^{4}) q^{73} + \zeta_{12}^{4} q^{81} + ( - \zeta_{12}^{4} - \zeta_{12}^{3}) q^{85} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{89} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{97} +O(q^{100})$$ q + (z^5 + z^4) * q^5 + z^2 * q^9 + z * q^13 + z^5 * q^17 + (-z^4 - z^3 - z^2) * q^25 + (-z^5 - z^3) * q^29 + (z^3 - z^2) * q^37 + (-z^5 - 1) * q^41 + (-z - 1) * q^45 + z * q^49 + q^53 - z^2 * q^61 + (z^5 - 1) * q^65 + (-z^5 + z^4) * q^73 + z^4 * q^81 + (-z^4 - z^3) * q^85 + (-z^4 + z) * q^89 + (z^5 - z^2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^5 + 2 * q^9 $$4 q - 2 q^{5} + 2 q^{9} - 2 q^{37} - 4 q^{41} - 4 q^{45} + 4 q^{53} - 2 q^{61} - 4 q^{65} - 2 q^{73} - 2 q^{81} + 2 q^{85} + 2 q^{89} - 2 q^{97}+O(q^{100})$$ 4 * q - 2 * q^5 + 2 * q^9 - 2 * q^37 - 4 * q^41 - 4 * q^45 + 4 * q^53 - 2 * q^61 - 4 * q^65 - 2 * q^73 - 2 * q^81 + 2 * q^85 + 2 * q^89 - 2 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$287$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{12}^{5}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0 0 0.366025 + 0.366025i 0 0 0 0.500000 + 0.866025i 0
97.1 0 0 0 −1.36603 + 1.36603i 0 0 0 0.500000 + 0.866025i 0
193.1 0 0 0 −1.36603 1.36603i 0 0 0 0.500000 0.866025i 0
353.1 0 0 0 0.366025 0.366025i 0 0 0 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
13.f odd 12 1 inner
52.l even 12 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.1.bl.a 4
3.b odd 2 1 3744.1.gs.c 4
4.b odd 2 1 CM 416.1.bl.a 4
8.b even 2 1 832.1.bl.a 4
8.d odd 2 1 832.1.bl.a 4
12.b even 2 1 3744.1.gs.c 4
13.f odd 12 1 inner 416.1.bl.a 4
16.e even 4 1 3328.1.bv.a 4
16.e even 4 1 3328.1.bv.b 4
16.f odd 4 1 3328.1.bv.a 4
16.f odd 4 1 3328.1.bv.b 4
39.k even 12 1 3744.1.gs.c 4
52.l even 12 1 inner 416.1.bl.a 4
104.u even 12 1 832.1.bl.a 4
104.x odd 12 1 832.1.bl.a 4
156.v odd 12 1 3744.1.gs.c 4
208.be odd 12 1 3328.1.bv.b 4
208.bf even 12 1 3328.1.bv.b 4
208.bk even 12 1 3328.1.bv.a 4
208.bl odd 12 1 3328.1.bv.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.bl.a 4 1.a even 1 1 trivial
416.1.bl.a 4 4.b odd 2 1 CM
416.1.bl.a 4 13.f odd 12 1 inner
416.1.bl.a 4 52.l even 12 1 inner
832.1.bl.a 4 8.b even 2 1
832.1.bl.a 4 8.d odd 2 1
832.1.bl.a 4 104.u even 12 1
832.1.bl.a 4 104.x odd 12 1
3328.1.bv.a 4 16.e even 4 1
3328.1.bv.a 4 16.f odd 4 1
3328.1.bv.a 4 208.bk even 12 1
3328.1.bv.a 4 208.bl odd 12 1
3328.1.bv.b 4 16.e even 4 1
3328.1.bv.b 4 16.f odd 4 1
3328.1.bv.b 4 208.be odd 12 1
3328.1.bv.b 4 208.bf even 12 1
3744.1.gs.c 4 3.b odd 2 1
3744.1.gs.c 4 12.b even 2 1
3744.1.gs.c 4 39.k even 12 1
3744.1.gs.c 4 156.v odd 12 1

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(416, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - T^{2} + 1$$
$17$ $$T^{4} - T^{2} + 1$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4} + 3T^{2} + 9$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 2 T^{3} + 5 T^{2} + 4 T + 1$$
$41$ $$T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$(T - 1)^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + T + 1)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 2 T^{3} + 2 T^{2} - 2 T + 1$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$97$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$