Newform orbit 768.4.c.b

• Label: 768.4.c.b
• Level: $768$
• Weight: $4$
• Character orbit: 768.c
• Analytic conductor: $45.313$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (3*b - 3) * q^3 + 4*b * q^5 - 12*b * q^7 + (-18*b - 9) * q^9 - 30 * q^11 - 72 * q^13 + (-12*b - 24) * q^15 - 36*b * q^17 + 18*b * q^19 + (36*b + 72) * q^21 + 144 * q^23 + 93 * q^25 + (27*b + 135) * q^27 - 4*b * q^29 + 156*b * q^31 + (-90*b + 90) * q^33 + 96 * q^35 - 72 * q^37 + (-216*b + 216) * q^39 - 216*b * q^41 + 162*b * q^43 + (-36*b + 144) * q^45 - 576 * q^47 + 55 * q^49 + (108*b + 216) * q^51 - 364*b * q^53 - 120*b * q^55 + (-54*b - 108) * q^57 + 414 * q^59 + 504 * q^61 + (108*b - 432) * q^63 - 288*b * q^65 + 558*b * q^67 + (432*b - 432) * q^69 + 720 * q^71 + 178 * q^73 + (279*b - 279) * q^75 + 360*b * q^77 + 684*b * q^79 + (324*b - 567) * q^81 + 438 * q^83 + 288 * q^85 + (12*b + 24) * q^87 + 612*b * q^89 + 864*b * q^91 + (-468*b - 936) * q^93 - 144 * q^95 + 650 * q^97 + (540*b + 270) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 6 * q^3 - 18 * q^9 - 60 * q^11 - 144 * q^13 - 48 * q^15 + 144 * q^21 + 288 * q^23 + 186 * q^25 + 270 * q^27 + 180 * q^33 + 192 * q^35 - 144 * q^37 + 432 * q^39 + 288 * q^45 - 1152 * q^47 + 110 * q^49 + 432 * q^51 - 216 * q^57 + 828 * q^59 + 1008 * q^61 - 864 * q^63 - 864 * q^69 + 1440 * q^71 + 356 * q^73 - 558 * q^75 - 1134 * q^81 + 876 * q^83 + 576 * q^85 + 48 * q^87 - 1872 * q^93 - 288 * q^95 + 1300 * q^97 + 540 * q^99

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

	−	1.41421i
		1.41421i

0

−3.00000

−

4.24264i

0

−

5.65685i

0

16.9706i

0

−9.00000

+

25.4558i

0

767.2

0

−3.00000

+

4.24264i

0

5.65685i

0

−

16.9706i

0

−9.00000

−

25.4558i

0

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.b		2
3.b	odd	2	1		768.4.c.h		2
4.b	odd	2	1		768.4.c.h		2
8.b	even	2	1		768.4.c.i		2
8.d	odd	2	1		768.4.c.c		2
12.b	even	2	1	inner	768.4.c.b		2
16.e	even	4	2		384.4.f.c	✓	4
16.f	odd	4	2		384.4.f.d	yes	4
24.f	even	2	1		768.4.c.i		2
24.h	odd	2	1		768.4.c.c		2
48.i	odd	4	2		384.4.f.d	yes	4
48.k	even	4	2		384.4.f.c	✓	4

    

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

Hecke kernels

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

\( T_{5}^{2} + 32 \)

\( T_{7}^{2} + 288 \)

\( T_{11} + 30 \)

\( T_{13} + 72 \)

\( T_{23} - 144 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 6T + 27 \)
$5$	\( T^{2} + 32 \)
$7$	\( T^{2} + 288 \)
$11$	\( (T + 30)^{2} \)