Properties

 Label 1792.2.f.f Level $1792$ Weight $2$ Character orbit 1792.f Analytic conductor $14.309$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1792 = 2^{8} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1792.f (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} -\beta_{2} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} -\beta_{2} q^{5} + ( -\beta_{1} + \beta_{3} ) q^{7} + 3 q^{9} -2 \beta_{1} q^{11} -\beta_{2} q^{13} -6 \beta_{1} q^{15} + 2 \beta_{2} q^{17} -\beta_{3} q^{19} + ( 6 - \beta_{2} ) q^{21} -4 \beta_{1} q^{23} - q^{25} + 4 q^{29} + 2 \beta_{3} q^{31} -2 \beta_{2} q^{33} + ( -6 \beta_{1} - \beta_{3} ) q^{35} -8 q^{37} -6 \beta_{1} q^{39} + 6 \beta_{1} q^{43} -3 \beta_{2} q^{45} -2 \beta_{3} q^{47} + ( 5 - 2 \beta_{2} ) q^{49} + 12 \beta_{1} q^{51} + 4 q^{53} -2 \beta_{3} q^{55} -6 q^{57} + \beta_{3} q^{59} -3 \beta_{2} q^{61} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{63} -6 q^{65} -2 \beta_{1} q^{67} -4 \beta_{2} q^{69} + 10 \beta_{1} q^{71} + 6 \beta_{2} q^{73} -\beta_{3} q^{75} + ( -2 - 2 \beta_{2} ) q^{77} + 6 \beta_{1} q^{79} -9 q^{81} + \beta_{3} q^{83} + 12 q^{85} + 4 \beta_{3} q^{87} -6 \beta_{2} q^{89} + ( -6 \beta_{1} - \beta_{3} ) q^{91} + 12 q^{93} + 6 \beta_{1} q^{95} + 2 \beta_{2} q^{97} -6 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} + 24q^{21} - 4q^{25} + 16q^{29} - 32q^{37} + 20q^{49} + 16q^{53} - 24q^{57} - 24q^{65} - 8q^{77} - 36q^{81} + 48q^{85} + 48q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 3 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 3 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 3 \beta_{2}$$$$)/2$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1791.1
 −1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i 1.22474 − 1.22474i
0 −2.44949 0 2.44949i 0 −2.44949 + 1.00000i 0 3.00000 0
1791.2 0 −2.44949 0 2.44949i 0 −2.44949 1.00000i 0 3.00000 0
1791.3 0 2.44949 0 2.44949i 0 2.44949 1.00000i 0 3.00000 0
1791.4 0 2.44949 0 2.44949i 0 2.44949 + 1.00000i 0 3.00000 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
7.b odd 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 1792.2.f.f 4
4.b odd 2 1 inner 1792.2.f.f 4
7.b odd 2 1 inner 1792.2.f.f 4
8.b even 2 1 1792.2.f.e 4
8.d odd 2 1 1792.2.f.e 4
16.e even 4 1 56.2.e.b 4
16.e even 4 1 224.2.e.b 4
16.f odd 4 1 56.2.e.b 4
16.f odd 4 1 224.2.e.b 4
28.d even 2 1 inner 1792.2.f.f 4
48.i odd 4 1 504.2.p.f 4
48.i odd 4 1 2016.2.p.e 4
48.k even 4 1 504.2.p.f 4
48.k even 4 1 2016.2.p.e 4
56.e even 2 1 1792.2.f.e 4
56.h odd 2 1 1792.2.f.e 4
112.j even 4 1 56.2.e.b 4
112.j even 4 1 224.2.e.b 4
112.l odd 4 1 56.2.e.b 4
112.l odd 4 1 224.2.e.b 4
112.u odd 12 2 392.2.m.f 8
112.u odd 12 2 1568.2.q.e 8
112.v even 12 2 392.2.m.f 8
112.v even 12 2 1568.2.q.e 8
112.w even 12 2 392.2.m.f 8
112.w even 12 2 1568.2.q.e 8
112.x odd 12 2 392.2.m.f 8
112.x odd 12 2 1568.2.q.e 8
336.v odd 4 1 504.2.p.f 4
336.v odd 4 1 2016.2.p.e 4
336.y even 4 1 504.2.p.f 4
336.y even 4 1 2016.2.p.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.e.b 4 16.e even 4 1
56.2.e.b 4 16.f odd 4 1
56.2.e.b 4 112.j even 4 1
56.2.e.b 4 112.l odd 4 1
224.2.e.b 4 16.e even 4 1
224.2.e.b 4 16.f odd 4 1
224.2.e.b 4 112.j even 4 1
224.2.e.b 4 112.l odd 4 1
392.2.m.f 8 112.u odd 12 2
392.2.m.f 8 112.v even 12 2
392.2.m.f 8 112.w even 12 2
392.2.m.f 8 112.x odd 12 2
504.2.p.f 4 48.i odd 4 1
504.2.p.f 4 48.k even 4 1
504.2.p.f 4 336.v odd 4 1
504.2.p.f 4 336.y even 4 1
1568.2.q.e 8 112.u odd 12 2
1568.2.q.e 8 112.v even 12 2
1568.2.q.e 8 112.w even 12 2
1568.2.q.e 8 112.x odd 12 2
1792.2.f.e 4 8.b even 2 1
1792.2.f.e 4 8.d odd 2 1
1792.2.f.e 4 56.e even 2 1
1792.2.f.e 4 56.h odd 2 1
1792.2.f.f 4 1.a even 1 1 trivial
1792.2.f.f 4 4.b odd 2 1 inner
1792.2.f.f 4 7.b odd 2 1 inner
1792.2.f.f 4 28.d even 2 1 inner
2016.2.p.e 4 48.i odd 4 1
2016.2.p.e 4 48.k even 4 1
2016.2.p.e 4 336.v odd 4 1
2016.2.p.e 4 336.y even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1792, [\chi])$$:

 $$T_{3}^{2} - 6$$ $$T_{5}^{2} + 6$$ $$T_{29} - 4$$ $$T_{31}^{2} - 24$$ $$T_{37} + 8$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -6 + T^{2} )^{2}$$
$5$ $$( 6 + T^{2} )^{2}$$
$7$ $$49 - 10 T^{2} + T^{4}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$( 6 + T^{2} )^{2}$$
$17$ $$( 24 + T^{2} )^{2}$$
$19$ $$( -6 + T^{2} )^{2}$$
$23$ $$( 16 + T^{2} )^{2}$$
$29$ $$( -4 + T )^{4}$$
$31$ $$( -24 + T^{2} )^{2}$$
$37$ $$( 8 + T )^{4}$$
$41$ $$T^{4}$$
$43$ $$( 36 + T^{2} )^{2}$$
$47$ $$( -24 + T^{2} )^{2}$$
$53$ $$( -4 + T )^{4}$$
$59$ $$( -6 + T^{2} )^{2}$$
$61$ $$( 54 + T^{2} )^{2}$$
$67$ $$( 4 + T^{2} )^{2}$$
$71$ $$( 100 + T^{2} )^{2}$$
$73$ $$( 216 + T^{2} )^{2}$$
$79$ $$( 36 + T^{2} )^{2}$$
$83$ $$( -6 + T^{2} )^{2}$$
$89$ $$( 216 + T^{2} )^{2}$$
$97$ $$( 24 + T^{2} )^{2}$$