# Properties

 Label 768.4.c.n Level $768$ Weight $4$ Character orbit 768.c Analytic conductor $45.313$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$45.3134668844$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-6}, \sqrt{-14})$$ Defining polynomial: $$x^{4} + 10 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 384) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( -2 \beta_{1} - \beta_{2} ) q^{5} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 15 + 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( -2 \beta_{1} - \beta_{2} ) q^{5} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{7} + ( 15 + 3 \beta_{2} ) q^{9} + ( -32 + \beta_{1} + 2 \beta_{3} ) q^{11} + ( 28 - 4 \beta_{1} - 8 \beta_{3} ) q^{13} + ( -24 + 9 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{15} + ( -16 \beta_{1} + 2 \beta_{2} ) q^{17} + ( -7 \beta_{1} - 16 \beta_{2} ) q^{19} + ( 36 + 18 \beta_{1} - 9 \beta_{2} + 8 \beta_{3} ) q^{21} + ( 80 + 4 \beta_{1} + 8 \beta_{3} ) q^{23} + ( -27 + 16 \beta_{1} + 32 \beta_{3} ) q^{25} + ( -27 \beta_{1} + 3 \beta_{3} ) q^{27} + ( -14 \beta_{1} - 23 \beta_{2} ) q^{29} + ( -7 \beta_{1} - 6 \beta_{2} ) q^{31} + ( 42 + 3 \beta_{2} - 32 \beta_{3} ) q^{33} + ( 32 + 4 \beta_{1} + 8 \beta_{3} ) q^{35} + ( 124 + 28 \beta_{1} + 56 \beta_{3} ) q^{37} + ( -168 - 12 \beta_{2} + 28 \beta_{3} ) q^{39} + ( 16 \beta_{1} - 24 \beta_{2} ) q^{41} + ( 49 \beta_{1} - 16 \beta_{2} ) q^{43} + ( 168 - 54 \beta_{1} - 15 \beta_{2} - 48 \beta_{3} ) q^{45} + ( 32 \beta_{1} + 64 \beta_{3} ) q^{47} + ( -97 - 48 \beta_{1} - 96 \beta_{3} ) q^{49} + ( -192 - 18 \beta_{1} + 48 \beta_{2} - 8 \beta_{3} ) q^{51} + ( 70 \beta_{1} - 45 \beta_{2} ) q^{53} + ( 78 \beta_{1} + 44 \beta_{2} ) q^{55} + ( -84 + 144 \beta_{1} + 21 \beta_{2} + 64 \beta_{3} ) q^{57} + ( -5 \beta_{1} - 10 \beta_{3} ) q^{59} + ( 28 - 68 \beta_{1} - 136 \beta_{3} ) q^{61} + ( 336 + 81 \beta_{1} - 30 \beta_{2} + 72 \beta_{3} ) q^{63} + ( -112 \beta_{1} - 76 \beta_{2} ) q^{65} + ( 91 \beta_{1} + 32 \beta_{2} ) q^{67} + ( 168 + 12 \beta_{2} + 80 \beta_{3} ) q^{69} + ( 80 + 100 \beta_{1} + 200 \beta_{3} ) q^{71} + ( 154 + 48 \beta_{1} + 96 \beta_{3} ) q^{73} + ( 672 + 48 \beta_{2} - 27 \beta_{3} ) q^{75} + ( -68 \beta_{1} + 46 \beta_{2} ) q^{77} + ( 189 \beta_{1} + 2 \beta_{2} ) q^{79} + ( -279 + 90 \beta_{2} ) q^{81} + ( 672 - 37 \beta_{1} - 74 \beta_{3} ) q^{83} + ( -656 + 48 \beta_{1} + 96 \beta_{3} ) q^{85} + ( -168 + 207 \beta_{1} + 42 \beta_{2} + 92 \beta_{3} ) q^{87} + ( -128 \beta_{1} + 26 \beta_{2} ) q^{89} + ( -28 \beta_{1} + 16 \beta_{2} ) q^{91} + ( -84 + 54 \beta_{1} + 21 \beta_{2} + 24 \beta_{3} ) q^{93} + ( -1232 + 156 \beta_{1} + 312 \beta_{3} ) q^{95} + ( 714 - 32 \beta_{1} - 64 \beta_{3} ) q^{97} + ( -480 - 27 \beta_{1} - 96 \beta_{2} + 30 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 60 q^{9} + O(q^{10})$$ $$4 q + 60 q^{9} - 128 q^{11} + 112 q^{13} - 96 q^{15} + 144 q^{21} + 320 q^{23} - 108 q^{25} + 168 q^{33} + 128 q^{35} + 496 q^{37} - 672 q^{39} + 672 q^{45} - 388 q^{49} - 768 q^{51} - 336 q^{57} + 112 q^{61} + 1344 q^{63} + 672 q^{69} + 320 q^{71} + 616 q^{73} + 2688 q^{75} - 1116 q^{81} + 2688 q^{83} - 2624 q^{85} - 672 q^{87} - 336 q^{93} - 4928 q^{95} + 2856 q^{97} - 1920 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 10 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{3} - 8 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + 12 \nu$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} + 8 \nu + 10$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} + \beta_{1} - 10$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{2} - 3 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
767.1
 3.09557i − 3.09557i − 0.646084i 0.646084i
0 −4.58258 2.44949i 0 17.2813i 0 0.269691i 0 15.0000 + 22.4499i 0
767.2 0 −4.58258 + 2.44949i 0 17.2813i 0 0.269691i 0 15.0000 22.4499i 0
767.3 0 4.58258 2.44949i 0 2.31464i 0 29.6636i 0 15.0000 22.4499i 0
767.4 0 4.58258 + 2.44949i 0 2.31464i 0 29.6636i 0 15.0000 + 22.4499i 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.c.n 4
3.b odd 2 1 768.4.c.p 4
4.b odd 2 1 768.4.c.p 4
8.b even 2 1 768.4.c.o 4
8.d odd 2 1 768.4.c.m 4
12.b even 2 1 inner 768.4.c.n 4
16.e even 4 2 384.4.f.g 8
16.f odd 4 2 384.4.f.h yes 8
24.f even 2 1 768.4.c.o 4
24.h odd 2 1 768.4.c.m 4
48.i odd 4 2 384.4.f.h yes 8
48.k even 4 2 384.4.f.g 8

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}^{4} + 304 T_{5}^{2} + 1600$$ $$T_{7}^{4} + 880 T_{7}^{2} + 64$$ $$T_{11}^{2} + 64 T_{11} + 940$$ $$T_{13}^{2} - 56 T_{13} - 560$$ $$T_{23}^{2} - 160 T_{23} + 5056$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$729 - 30 T^{2} + T^{4}$$
$5$ $$1600 + 304 T^{2} + T^{4}$$
$7$ $$64 + 880 T^{2} + T^{4}$$
$11$ $$( 940 + 64 T + T^{2} )^{2}$$
$13$ $$( -560 - 56 T + T^{2} )^{2}$$
$17$ $$35046400 + 12736 T^{2} + T^{4}$$
$19$ $$173185600 + 31024 T^{2} + T^{4}$$
$23$ $$( 5056 - 160 T + T^{2} )^{2}$$
$29$ $$621006400 + 68656 T^{2} + T^{4}$$
$31$ $$705600 + 6384 T^{2} + T^{4}$$
$37$ $$( -50480 - 248 T + T^{2} )^{2}$$
$41$ $$681836544 + 76800 T^{2} + T^{4}$$
$43$ $$1873850944 + 143920 T^{2} + T^{4}$$
$47$ $$( -86016 + T^{2} )^{2}$$
$53$ $$17640000 + 462000 T^{2} + T^{4}$$
$59$ $$( -2100 + T^{2} )^{2}$$
$61$ $$( -387632 - 56 T + T^{2} )^{2}$$
$67$ $$19993960000 + 512176 T^{2} + T^{4}$$
$71$ $$( -833600 - 160 T + T^{2} )^{2}$$
$73$ $$( -169820 - 308 T + T^{2} )^{2}$$
$79$ $$734586126400 + 1715056 T^{2} + T^{4}$$
$83$ $$( 336588 - 1344 T + T^{2} )^{2}$$
$89$ $$126280729600 + 862144 T^{2} + T^{4}$$
$97$ $$( 423780 - 1428 T + T^{2} )^{2}$$