# Properties

 Label 416.1.t.a Level $416$ Weight $1$ Character orbit 416.t Analytic conductor $0.208$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.207611045255$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.35152.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.143982592.1

## $q$-expansion

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

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

## Character values

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

 $$n$$ $$261$$ $$287$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 − 1.00000i 1.00000i
0 0 0 1.00000 1.00000i 0 0 0 −1.00000 0
385.1 0 0 0 1.00000 + 1.00000i 0 0 0 −1.00000 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.d odd 4 1 inner
52.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.1.t.a 2
3.b odd 2 1 3744.1.bd.b 2
4.b odd 2 1 CM 416.1.t.a 2
8.b even 2 1 832.1.t.a 2
8.d odd 2 1 832.1.t.a 2
12.b even 2 1 3744.1.bd.b 2
13.d odd 4 1 inner 416.1.t.a 2
16.e even 4 1 3328.1.j.a 2
16.e even 4 1 3328.1.j.b 2
16.f odd 4 1 3328.1.j.a 2
16.f odd 4 1 3328.1.j.b 2
39.f even 4 1 3744.1.bd.b 2
52.f even 4 1 inner 416.1.t.a 2
104.j odd 4 1 832.1.t.a 2
104.m even 4 1 832.1.t.a 2
156.l odd 4 1 3744.1.bd.b 2
208.l even 4 1 3328.1.j.a 2
208.m odd 4 1 3328.1.j.a 2
208.r odd 4 1 3328.1.j.b 2
208.s even 4 1 3328.1.j.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.1.t.a 2 1.a even 1 1 trivial
416.1.t.a 2 4.b odd 2 1 CM
416.1.t.a 2 13.d odd 4 1 inner
416.1.t.a 2 52.f even 4 1 inner
832.1.t.a 2 8.b even 2 1
832.1.t.a 2 8.d odd 2 1
832.1.t.a 2 104.j odd 4 1
832.1.t.a 2 104.m even 4 1
3328.1.j.a 2 16.e even 4 1
3328.1.j.a 2 16.f odd 4 1
3328.1.j.a 2 208.l even 4 1
3328.1.j.a 2 208.m odd 4 1
3328.1.j.b 2 16.e even 4 1
3328.1.j.b 2 16.f odd 4 1
3328.1.j.b 2 208.r odd 4 1
3328.1.j.b 2 208.s even 4 1
3744.1.bd.b 2 3.b odd 2 1
3744.1.bd.b 2 12.b even 2 1
3744.1.bd.b 2 39.f even 4 1
3744.1.bd.b 2 156.l odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

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