# Properties

 Label 2304.2.k.b Level $2304$ Weight $2$ Character orbit 2304.k Analytic conductor $18.398$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2304.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.3975326257$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 768) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{7}+O(q^{10})$$ q - b2 * q^7 $$q - \beta_{2} q^{7} + (\beta_1 - 1) q^{13} - 6 q^{17} + (3 \beta_{3} - 3 \beta_{2}) q^{19} + 6 \beta_{2} q^{23} - 5 \beta_1 q^{25} + ( - 6 \beta_1 + 6) q^{29} + \beta_{3} q^{31} + ( - 5 \beta_1 - 5) q^{37} - 6 \beta_1 q^{41} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{43} - 6 \beta_{3} q^{47} + 5 q^{49} + ( - 6 \beta_1 - 6) q^{53} + (5 \beta_1 - 5) q^{61} + (2 \beta_{3} - 2 \beta_{2}) q^{67} + 6 \beta_{2} q^{71} - \beta_{3} q^{79} + (6 \beta_{3} - 6 \beta_{2}) q^{83} - 6 \beta_1 q^{89} + (\beta_{3} + \beta_{2}) q^{91} - 12 q^{97}+O(q^{100})$$ q - b2 * q^7 + (b1 - 1) * q^13 - 6 * q^17 + (3*b3 - 3*b2) * q^19 + 6*b2 * q^23 - 5*b1 * q^25 + (-6*b1 + 6) * q^29 + b3 * q^31 + (-5*b1 - 5) * q^37 - 6*b1 * q^41 + (-3*b3 - 3*b2) * q^43 - 6*b3 * q^47 + 5 * q^49 + (-6*b1 - 6) * q^53 + (5*b1 - 5) * q^61 + (2*b3 - 2*b2) * q^67 + 6*b2 * q^71 - b3 * q^79 + (6*b3 - 6*b2) * q^83 - 6*b1 * q^89 + (b3 + b2) * q^91 - 12 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{13} - 24 q^{17} + 24 q^{29} - 20 q^{37} + 20 q^{49} - 24 q^{53} - 20 q^{61} - 48 q^{97}+O(q^{100})$$ 4 * q - 4 * q^13 - 24 * q^17 + 24 * q^29 - 20 * q^37 + 20 * q^49 - 24 * q^53 - 20 * q^61 - 48 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

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

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 0 0 1.41421i 0 0 0
577.2 0 0 0 0 0 1.41421i 0 0 0
1729.1 0 0 0 0 0 1.41421i 0 0 0
1729.2 0 0 0 0 0 1.41421i 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 inner
16.e even 4 1 inner
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.k.b 4
3.b odd 2 1 768.2.j.b 4
4.b odd 2 1 inner 2304.2.k.b 4
8.b even 2 1 2304.2.k.c 4
8.d odd 2 1 2304.2.k.c 4
12.b even 2 1 768.2.j.b 4
16.e even 4 1 inner 2304.2.k.b 4
16.e even 4 1 2304.2.k.c 4
16.f odd 4 1 inner 2304.2.k.b 4
16.f odd 4 1 2304.2.k.c 4
24.f even 2 1 768.2.j.c yes 4
24.h odd 2 1 768.2.j.c yes 4
32.g even 8 1 9216.2.a.n 2
32.g even 8 1 9216.2.a.o 2
32.h odd 8 1 9216.2.a.n 2
32.h odd 8 1 9216.2.a.o 2
48.i odd 4 1 768.2.j.b 4
48.i odd 4 1 768.2.j.c yes 4
48.k even 4 1 768.2.j.b 4
48.k even 4 1 768.2.j.c yes 4
96.o even 8 1 3072.2.a.a 2
96.o even 8 1 3072.2.a.g 2
96.o even 8 2 3072.2.d.a 4
96.p odd 8 1 3072.2.a.a 2
96.p odd 8 1 3072.2.a.g 2
96.p odd 8 2 3072.2.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.b 4 3.b odd 2 1
768.2.j.b 4 12.b even 2 1
768.2.j.b 4 48.i odd 4 1
768.2.j.b 4 48.k even 4 1
768.2.j.c yes 4 24.f even 2 1
768.2.j.c yes 4 24.h odd 2 1
768.2.j.c yes 4 48.i odd 4 1
768.2.j.c yes 4 48.k even 4 1
2304.2.k.b 4 1.a even 1 1 trivial
2304.2.k.b 4 4.b odd 2 1 inner
2304.2.k.b 4 16.e even 4 1 inner
2304.2.k.b 4 16.f odd 4 1 inner
2304.2.k.c 4 8.b even 2 1
2304.2.k.c 4 8.d odd 2 1
2304.2.k.c 4 16.e even 4 1
2304.2.k.c 4 16.f odd 4 1
3072.2.a.a 2 96.o even 8 1
3072.2.a.a 2 96.p odd 8 1
3072.2.a.g 2 96.o even 8 1
3072.2.a.g 2 96.p odd 8 1
3072.2.d.a 4 96.o even 8 2
3072.2.d.a 4 96.p odd 8 2
9216.2.a.n 2 32.g even 8 1
9216.2.a.n 2 32.h odd 8 1
9216.2.a.o 2 32.g even 8 1
9216.2.a.o 2 32.h odd 8 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2304, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 2$$ T7^2 + 2 $$T_{13}^{2} + 2T_{13} + 2$$ T13^2 + 2*T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 2)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 2 T + 2)^{2}$$
$17$ $$(T + 6)^{4}$$
$19$ $$T^{4} + 1296$$
$23$ $$(T^{2} + 72)^{2}$$
$29$ $$(T^{2} - 12 T + 72)^{2}$$
$31$ $$(T^{2} - 2)^{2}$$
$37$ $$(T^{2} + 10 T + 50)^{2}$$
$41$ $$(T^{2} + 36)^{2}$$
$43$ $$T^{4} + 1296$$
$47$ $$(T^{2} - 72)^{2}$$
$53$ $$(T^{2} + 12 T + 72)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 10 T + 50)^{2}$$
$67$ $$T^{4} + 256$$
$71$ $$(T^{2} + 72)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} - 2)^{2}$$
$83$ $$T^{4} + 20736$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$(T + 12)^{4}$$