Properties

 Label 2352.2.h.f Level $2352$ Weight $2$ Character orbit 2352.h Analytic conductor $18.781$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.h (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$18.7808145554$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{5} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{5} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{11} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{13} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{15} + 2 \zeta_{8}^{2} q^{17} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{23} + q^{25} + ( 5 + \zeta_{8} + \zeta_{8}^{3} ) q^{27} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{29} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{31} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{33} + 6 q^{37} + ( \zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{39} + 10 \zeta_{8}^{2} q^{41} + 12 \zeta_{8}^{2} q^{43} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{45} + 6 q^{47} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{51} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{53} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{55} + 12 q^{59} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{61} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{65} -12 \zeta_{8}^{2} q^{67} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{69} + ( 9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{71} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{73} + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{75} -12 \zeta_{8}^{2} q^{79} + ( -7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} + 6 q^{83} + 4 q^{85} + ( 12 + 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{87} + 2 \zeta_{8}^{2} q^{89} + ( -12 - 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{93} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{97} + ( 3 \zeta_{8} + 12 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 4q^{9} + 4q^{25} + 20q^{27} + 24q^{37} + 24q^{47} + 48q^{59} - 4q^{75} - 28q^{81} + 24q^{83} + 16q^{85} + 48q^{87} - 48q^{93} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1471$$ $$1765$$ $$2257$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2255.1
 −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i
0 −1.00000 1.41421i 0 2.00000i 0 0 0 −1.00000 + 2.82843i 0
2255.2 0 −1.00000 1.41421i 0 2.00000i 0 0 0 −1.00000 + 2.82843i 0
2255.3 0 −1.00000 + 1.41421i 0 2.00000i 0 0 0 −1.00000 2.82843i 0
2255.4 0 −1.00000 + 1.41421i 0 2.00000i 0 0 0 −1.00000 2.82843i 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
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.h.f 4
3.b odd 2 1 2352.2.h.k yes 4
4.b odd 2 1 2352.2.h.k yes 4
7.b odd 2 1 2352.2.h.k yes 4
12.b even 2 1 inner 2352.2.h.f 4
21.c even 2 1 inner 2352.2.h.f 4
28.d even 2 1 inner 2352.2.h.f 4
84.h odd 2 1 2352.2.h.k yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.h.f 4 1.a even 1 1 trivial
2352.2.h.f 4 12.b even 2 1 inner
2352.2.h.f 4 21.c even 2 1 inner
2352.2.h.f 4 28.d even 2 1 inner
2352.2.h.k yes 4 3.b odd 2 1
2352.2.h.k yes 4 4.b odd 2 1
2352.2.h.k yes 4 7.b odd 2 1
2352.2.h.k yes 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2352, [\chi])$$:

 $$T_{5}^{2} + 4$$ $$T_{11}^{2} - 18$$ $$T_{13}^{2} - 2$$ $$T_{47} - 6$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + 2 T + T^{2} )^{2}$$
$5$ $$( 4 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( -18 + T^{2} )^{2}$$
$13$ $$( -2 + T^{2} )^{2}$$
$17$ $$( 4 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( -18 + T^{2} )^{2}$$
$29$ $$( 72 + T^{2} )^{2}$$
$31$ $$( 72 + T^{2} )^{2}$$
$37$ $$( -6 + T )^{4}$$
$41$ $$( 100 + T^{2} )^{2}$$
$43$ $$( 144 + T^{2} )^{2}$$
$47$ $$( -6 + T )^{4}$$
$53$ $$( 32 + T^{2} )^{2}$$
$59$ $$( -12 + T )^{4}$$
$61$ $$( -2 + T^{2} )^{2}$$
$67$ $$( 144 + T^{2} )^{2}$$
$71$ $$( -162 + T^{2} )^{2}$$
$73$ $$( -98 + T^{2} )^{2}$$
$79$ $$( 144 + T^{2} )^{2}$$
$83$ $$( -6 + T )^{4}$$
$89$ $$( 4 + T^{2} )^{2}$$
$97$ $$( -50 + T^{2} )^{2}$$