Properties

 Label 2304.3.h.e Level $2304$ Weight $3$ Character orbit 2304.h Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $8$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2304.h (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{5} +O(q^{10})$$ $$q + ( -7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{5} + 24 \zeta_{8}^{2} q^{13} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{17} + 73 q^{25} + ( 41 \zeta_{8} - 41 \zeta_{8}^{3} ) q^{29} + 70 \zeta_{8}^{2} q^{37} + ( -31 \zeta_{8} - 31 \zeta_{8}^{3} ) q^{41} -49 q^{49} + ( -17 \zeta_{8} + 17 \zeta_{8}^{3} ) q^{53} + 22 \zeta_{8}^{2} q^{61} + ( -168 \zeta_{8} - 168 \zeta_{8}^{3} ) q^{65} -96 q^{73} + 98 \zeta_{8}^{2} q^{85} + ( 41 \zeta_{8} + 41 \zeta_{8}^{3} ) q^{89} + 144 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 292q^{25} - 196q^{49} - 384q^{73} + 576q^{97} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2177.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 −9.89949 0 0 0 0 0
2177.2 0 0 0 −9.89949 0 0 0 0 0
2177.3 0 0 0 9.89949 0 0 0 0 0
2177.4 0 0 0 9.89949 0 0 0 0 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 $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.h.e 4
3.b odd 2 1 inner 2304.3.h.e 4
4.b odd 2 1 CM 2304.3.h.e 4
8.b even 2 1 inner 2304.3.h.e 4
8.d odd 2 1 inner 2304.3.h.e 4
12.b even 2 1 inner 2304.3.h.e 4
16.e even 4 1 288.3.e.b 2
16.e even 4 1 576.3.e.e 2
16.f odd 4 1 288.3.e.b 2
16.f odd 4 1 576.3.e.e 2
24.f even 2 1 inner 2304.3.h.e 4
24.h odd 2 1 inner 2304.3.h.e 4
48.i odd 4 1 288.3.e.b 2
48.i odd 4 1 576.3.e.e 2
48.k even 4 1 288.3.e.b 2
48.k even 4 1 576.3.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.e.b 2 16.e even 4 1
288.3.e.b 2 16.f odd 4 1
288.3.e.b 2 48.i odd 4 1
288.3.e.b 2 48.k even 4 1
576.3.e.e 2 16.e even 4 1
576.3.e.e 2 16.f odd 4 1
576.3.e.e 2 48.i odd 4 1
576.3.e.e 2 48.k even 4 1
2304.3.h.e 4 1.a even 1 1 trivial
2304.3.h.e 4 3.b odd 2 1 inner
2304.3.h.e 4 4.b odd 2 1 CM
2304.3.h.e 4 8.b even 2 1 inner
2304.3.h.e 4 8.d odd 2 1 inner
2304.3.h.e 4 12.b even 2 1 inner
2304.3.h.e 4 24.f even 2 1 inner
2304.3.h.e 4 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}^{2} - 98$$ $$T_{7}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -98 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 576 + T^{2} )^{2}$$
$17$ $$( 98 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -3362 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 4900 + T^{2} )^{2}$$
$41$ $$( 1922 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -578 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 484 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 96 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 3362 + T^{2} )^{2}$$
$97$ $$( -144 + T )^{4}$$