# Properties

 Label 2304.3.h.c Level $2304$ Weight $3$ Character orbit 2304.h Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{5} -4 q^{7} +O(q^{10})$$ $$q + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{5} -4 q^{7} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{11} + 8 \zeta_{8}^{2} q^{13} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{17} + 16 \zeta_{8}^{2} q^{19} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{23} -7 q^{25} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{29} -44 q^{31} + ( -12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{35} + 34 \zeta_{8}^{2} q^{37} + ( 33 \zeta_{8} + 33 \zeta_{8}^{3} ) q^{41} -40 \zeta_{8}^{2} q^{43} + ( -60 \zeta_{8} - 60 \zeta_{8}^{3} ) q^{47} -33 q^{49} + ( -27 \zeta_{8} + 27 \zeta_{8}^{3} ) q^{53} + 72 q^{55} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{59} + 50 \zeta_{8}^{2} q^{61} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{65} -8 \zeta_{8}^{2} q^{67} + ( 36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{71} + 16 q^{73} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{77} + 76 q^{79} + ( -84 \zeta_{8} + 84 \zeta_{8}^{3} ) q^{83} + 54 \zeta_{8}^{2} q^{85} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{89} -32 \zeta_{8}^{2} q^{91} + ( 48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{95} + 176 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{7} + O(q^{10})$$ $$4q - 16q^{7} - 28q^{25} - 176q^{31} - 132q^{49} + 288q^{55} + 64q^{73} + 304q^{79} + 704q^{97} + O(q^{100})$$

## 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 −4.24264 0 −4.00000 0 0 0
2177.2 0 0 0 −4.24264 0 −4.00000 0 0 0
2177.3 0 0 0 4.24264 0 −4.00000 0 0 0
2177.4 0 0 0 4.24264 0 −4.00000 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
3.b odd 2 1 inner
8.b 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.c 4
3.b odd 2 1 inner 2304.3.h.c 4
4.b odd 2 1 2304.3.h.f 4
8.b even 2 1 inner 2304.3.h.c 4
8.d odd 2 1 2304.3.h.f 4
12.b even 2 1 2304.3.h.f 4
16.e even 4 1 144.3.e.b 2
16.e even 4 1 576.3.e.f 2
16.f odd 4 1 18.3.b.a 2
16.f odd 4 1 576.3.e.c 2
24.f even 2 1 2304.3.h.f 4
24.h odd 2 1 inner 2304.3.h.c 4
48.i odd 4 1 144.3.e.b 2
48.i odd 4 1 576.3.e.f 2
48.k even 4 1 18.3.b.a 2
48.k even 4 1 576.3.e.c 2
80.i odd 4 1 3600.3.c.b 4
80.j even 4 1 450.3.b.b 4
80.k odd 4 1 450.3.d.f 2
80.q even 4 1 3600.3.l.d 2
80.s even 4 1 450.3.b.b 4
80.t odd 4 1 3600.3.c.b 4
112.j even 4 1 882.3.b.a 2
112.u odd 12 2 882.3.s.b 4
112.v even 12 2 882.3.s.d 4
144.u even 12 2 162.3.d.b 4
144.v odd 12 2 162.3.d.b 4
144.w odd 12 2 1296.3.q.f 4
144.x even 12 2 1296.3.q.f 4
176.i even 4 1 2178.3.c.d 2
208.l even 4 1 3042.3.d.a 4
208.o odd 4 1 3042.3.c.e 2
208.s even 4 1 3042.3.d.a 4
240.t even 4 1 450.3.d.f 2
240.z odd 4 1 450.3.b.b 4
240.bb even 4 1 3600.3.c.b 4
240.bd odd 4 1 450.3.b.b 4
240.bf even 4 1 3600.3.c.b 4
240.bm odd 4 1 3600.3.l.d 2
336.v odd 4 1 882.3.b.a 2
336.br odd 12 2 882.3.s.d 4
336.bu even 12 2 882.3.s.b 4
528.s odd 4 1 2178.3.c.d 2
624.s odd 4 1 3042.3.d.a 4
624.v even 4 1 3042.3.c.e 2
624.bo odd 4 1 3042.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 16.f odd 4 1
18.3.b.a 2 48.k even 4 1
144.3.e.b 2 16.e even 4 1
144.3.e.b 2 48.i odd 4 1
162.3.d.b 4 144.u even 12 2
162.3.d.b 4 144.v odd 12 2
450.3.b.b 4 80.j even 4 1
450.3.b.b 4 80.s even 4 1
450.3.b.b 4 240.z odd 4 1
450.3.b.b 4 240.bd odd 4 1
450.3.d.f 2 80.k odd 4 1
450.3.d.f 2 240.t even 4 1
576.3.e.c 2 16.f odd 4 1
576.3.e.c 2 48.k even 4 1
576.3.e.f 2 16.e even 4 1
576.3.e.f 2 48.i odd 4 1
882.3.b.a 2 112.j even 4 1
882.3.b.a 2 336.v odd 4 1
882.3.s.b 4 112.u odd 12 2
882.3.s.b 4 336.bu even 12 2
882.3.s.d 4 112.v even 12 2
882.3.s.d 4 336.br odd 12 2
1296.3.q.f 4 144.w odd 12 2
1296.3.q.f 4 144.x even 12 2
2178.3.c.d 2 176.i even 4 1
2178.3.c.d 2 528.s odd 4 1
2304.3.h.c 4 1.a even 1 1 trivial
2304.3.h.c 4 3.b odd 2 1 inner
2304.3.h.c 4 8.b even 2 1 inner
2304.3.h.c 4 24.h odd 2 1 inner
2304.3.h.f 4 4.b odd 2 1
2304.3.h.f 4 8.d odd 2 1
2304.3.h.f 4 12.b even 2 1
2304.3.h.f 4 24.f even 2 1
3042.3.c.e 2 208.o odd 4 1
3042.3.c.e 2 624.v even 4 1
3042.3.d.a 4 208.l even 4 1
3042.3.d.a 4 208.s even 4 1
3042.3.d.a 4 624.s odd 4 1
3042.3.d.a 4 624.bo odd 4 1
3600.3.c.b 4 80.i odd 4 1
3600.3.c.b 4 80.t odd 4 1
3600.3.c.b 4 240.bb even 4 1
3600.3.c.b 4 240.bf even 4 1
3600.3.l.d 2 80.q even 4 1
3600.3.l.d 2 240.bm odd 4 1

## Hecke kernels

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

 $$T_{5}^{2} - 18$$ $$T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -18 + T^{2} )^{2}$$
$7$ $$( 4 + T )^{4}$$
$11$ $$( -288 + T^{2} )^{2}$$
$13$ $$( 64 + T^{2} )^{2}$$
$17$ $$( 162 + T^{2} )^{2}$$
$19$ $$( 256 + T^{2} )^{2}$$
$23$ $$( 288 + T^{2} )^{2}$$
$29$ $$( -18 + T^{2} )^{2}$$
$31$ $$( 44 + T )^{4}$$
$37$ $$( 1156 + T^{2} )^{2}$$
$41$ $$( 2178 + T^{2} )^{2}$$
$43$ $$( 1600 + T^{2} )^{2}$$
$47$ $$( 7200 + T^{2} )^{2}$$
$53$ $$( -1458 + T^{2} )^{2}$$
$59$ $$( -1152 + T^{2} )^{2}$$
$61$ $$( 2500 + T^{2} )^{2}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( 2592 + T^{2} )^{2}$$
$73$ $$( -16 + T )^{4}$$
$79$ $$( -76 + T )^{4}$$
$83$ $$( -14112 + T^{2} )^{2}$$
$89$ $$( 162 + T^{2} )^{2}$$
$97$ $$( -176 + T )^{4}$$