Properties

 Label 936.1.o.c Level $936$ Weight $1$ Character orbit 936.o Analytic conductor $0.467$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -39 Inner twists $8$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$936 = 2^{3} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 936.o (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.467124851824$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.32448.1

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{5} - \zeta_{8}^{3} q^{8} +O(q^{10})$$ q - z * q^2 + z^2 * q^4 + (-z^3 + z) * q^5 - z^3 * q^8 $$q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{5} - \zeta_{8}^{3} q^{8} + ( - \zeta_{8}^{2} - 1) q^{10} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} + \zeta_{8}^{2} q^{13} - q^{16} + (\zeta_{8}^{3} + \zeta_{8}) q^{20} + (\zeta_{8}^{2} - 1) q^{22} + q^{25} - \zeta_{8}^{3} q^{26} + \zeta_{8} q^{32} + ( - \zeta_{8}^{2} + 1) q^{40} + (\zeta_{8}^{3} + \zeta_{8}) q^{41} + q^{43} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{44} + (\zeta_{8}^{3} - \zeta_{8}) q^{47} - q^{49} - \zeta_{8} q^{50} - q^{52} - \zeta_{8}^{2} q^{55} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{59} - \zeta_{8}^{2} q^{61} - \zeta_{8}^{2} q^{64} + (\zeta_{8}^{3} + \zeta_{8}) q^{65} + (\zeta_{8}^{3} - \zeta_{8}) q^{71} + \zeta_{8}^{2} q^{79} + (\zeta_{8}^{3} - \zeta_{8}) q^{80} + ( - \zeta_{8}^{2} + 1) q^{82} + (\zeta_{8}^{3} + \zeta_{8}) q^{83} - 2 \zeta_{8} q^{86} + ( - \zeta_{8}^{2} - 1) q^{88} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{89} + (\zeta_{8}^{2} + 1) q^{94} + \zeta_{8} q^{98} +O(q^{100})$$ q - z * q^2 + z^2 * q^4 + (-z^3 + z) * q^5 - z^3 * q^8 + (-z^2 - 1) * q^10 + (-z^3 - z) * q^11 + z^2 * q^13 - q^16 + (z^3 + z) * q^20 + (z^2 - 1) * q^22 + q^25 - z^3 * q^26 + z * q^32 + (-z^2 + 1) * q^40 + (z^3 + z) * q^41 + q^43 + (-z^3 + z) * q^44 + (z^3 - z) * q^47 - q^49 - z * q^50 - q^52 - z^2 * q^55 + (-z^3 - z) * q^59 - z^2 * q^61 - z^2 * q^64 + (z^3 + z) * q^65 + (z^3 - z) * q^71 + z^2 * q^79 + (z^3 - z) * q^80 + (-z^2 + 1) * q^82 + (z^3 + z) * q^83 - 2*z * q^86 + (-z^2 - 1) * q^88 + (-z^3 - z) * q^89 + (z^2 + 1) * q^94 + z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{10} - 4 q^{16} - 4 q^{22} + 4 q^{25} + 4 q^{40} + 8 q^{43} - 4 q^{49} - 4 q^{52} + 4 q^{82} - 4 q^{88} + 4 q^{94}+O(q^{100})$$ 4 * q - 4 * q^10 - 4 * q^16 - 4 * q^22 + 4 * q^25 + 4 * q^40 + 8 * q^43 - 4 * q^49 - 4 * q^52 + 4 * q^82 - 4 * q^88 + 4 * q^94

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$209$$ $$469$$ $$703$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
−0.707107 0.707107i 0 1.00000i 1.41421 0 0 0.707107 0.707107i 0 −1.00000 1.00000i
883.2 −0.707107 + 0.707107i 0 1.00000i 1.41421 0 0 0.707107 + 0.707107i 0 −1.00000 + 1.00000i
883.3 0.707107 0.707107i 0 1.00000i −1.41421 0 0 −0.707107 0.707107i 0 −1.00000 + 1.00000i
883.4 0.707107 + 0.707107i 0 1.00000i −1.41421 0 0 −0.707107 + 0.707107i 0 −1.00000 1.00000i
 $$n$$: 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
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
8.d odd 2 1 inner
13.b even 2 1 inner
24.f even 2 1 inner
104.h odd 2 1 inner
312.h even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.1.o.c 4
3.b odd 2 1 inner 936.1.o.c 4
4.b odd 2 1 3744.1.o.c 4
8.b even 2 1 3744.1.o.c 4
8.d odd 2 1 inner 936.1.o.c 4
12.b even 2 1 3744.1.o.c 4
13.b even 2 1 inner 936.1.o.c 4
24.f even 2 1 inner 936.1.o.c 4
24.h odd 2 1 3744.1.o.c 4
39.d odd 2 1 CM 936.1.o.c 4
52.b odd 2 1 3744.1.o.c 4
104.e even 2 1 3744.1.o.c 4
104.h odd 2 1 inner 936.1.o.c 4
156.h even 2 1 3744.1.o.c 4
312.b odd 2 1 3744.1.o.c 4
312.h even 2 1 inner 936.1.o.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.1.o.c 4 1.a even 1 1 trivial
936.1.o.c 4 3.b odd 2 1 inner
936.1.o.c 4 8.d odd 2 1 inner
936.1.o.c 4 13.b even 2 1 inner
936.1.o.c 4 24.f even 2 1 inner
936.1.o.c 4 39.d odd 2 1 CM
936.1.o.c 4 104.h odd 2 1 inner
936.1.o.c 4 312.h even 2 1 inner
3744.1.o.c 4 4.b odd 2 1
3744.1.o.c 4 8.b even 2 1
3744.1.o.c 4 12.b even 2 1
3744.1.o.c 4 24.h odd 2 1
3744.1.o.c 4 52.b odd 2 1
3744.1.o.c 4 104.e even 2 1
3744.1.o.c 4 156.h even 2 1
3744.1.o.c 4 312.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2$$ acting on $$S_{1}^{\mathrm{new}}(936, [\chi])$$.

Hecke characteristic polynomials

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