Properties

Label 768.6.d.c
Level 768
Weight 6
Character orbit 768.d
Analytic conductor 123.175
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 768.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(123.174773616\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -9 i q^{3} + 66 i q^{5} -176 q^{7} -81 q^{9} +O(q^{10})\) \( q -9 i q^{3} + 66 i q^{5} -176 q^{7} -81 q^{9} + 60 i q^{11} -658 i q^{13} + 594 q^{15} -414 q^{17} + 956 i q^{19} + 1584 i q^{21} -600 q^{23} -1231 q^{25} + 729 i q^{27} + 5574 i q^{29} -3592 q^{31} + 540 q^{33} -11616 i q^{35} + 8458 i q^{37} -5922 q^{39} -19194 q^{41} -13316 i q^{43} -5346 i q^{45} -19680 q^{47} + 14169 q^{49} + 3726 i q^{51} + 31266 i q^{53} -3960 q^{55} + 8604 q^{57} -26340 i q^{59} -31090 i q^{61} + 14256 q^{63} + 43428 q^{65} -16804 i q^{67} + 5400 i q^{69} -6120 q^{71} + 25558 q^{73} + 11079 i q^{75} -10560 i q^{77} + 74408 q^{79} + 6561 q^{81} -6468 i q^{83} -27324 i q^{85} + 50166 q^{87} + 32742 q^{89} + 115808 i q^{91} + 32328 i q^{93} -63096 q^{95} + 166082 q^{97} -4860 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 352q^{7} - 162q^{9} + O(q^{10}) \) \( 2q - 352q^{7} - 162q^{9} + 1188q^{15} - 828q^{17} - 1200q^{23} - 2462q^{25} - 7184q^{31} + 1080q^{33} - 11844q^{39} - 38388q^{41} - 39360q^{47} + 28338q^{49} - 7920q^{55} + 17208q^{57} + 28512q^{63} + 86856q^{65} - 12240q^{71} + 51116q^{73} + 148816q^{79} + 13122q^{81} + 100332q^{87} + 65484q^{89} - 126192q^{95} + 332164q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
385.1
1.00000i
1.00000i
0 9.00000i 0 66.0000i 0 −176.000 0 −81.0000 0
385.2 0 9.00000i 0 66.0000i 0 −176.000 0 −81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.6.d.c 2
4.b odd 2 1 768.6.d.p 2
8.b even 2 1 inner 768.6.d.c 2
8.d odd 2 1 768.6.d.p 2
16.e even 4 1 6.6.a.a 1
16.e even 4 1 192.6.a.o 1
16.f odd 4 1 48.6.a.c 1
16.f odd 4 1 192.6.a.g 1
48.i odd 4 1 18.6.a.b 1
48.i odd 4 1 576.6.a.j 1
48.k even 4 1 144.6.a.j 1
48.k even 4 1 576.6.a.i 1
80.i odd 4 1 150.6.c.b 2
80.q even 4 1 150.6.a.d 1
80.t odd 4 1 150.6.c.b 2
112.l odd 4 1 294.6.a.m 1
112.w even 12 2 294.6.e.g 2
112.x odd 12 2 294.6.e.a 2
144.w odd 12 2 162.6.c.h 2
144.x even 12 2 162.6.c.e 2
176.l odd 4 1 726.6.a.a 1
208.p even 4 1 1014.6.a.c 1
240.bb even 4 1 450.6.c.j 2
240.bf even 4 1 450.6.c.j 2
240.bm odd 4 1 450.6.a.m 1
336.y even 4 1 882.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 16.e even 4 1
18.6.a.b 1 48.i odd 4 1
48.6.a.c 1 16.f odd 4 1
144.6.a.j 1 48.k even 4 1
150.6.a.d 1 80.q even 4 1
150.6.c.b 2 80.i odd 4 1
150.6.c.b 2 80.t odd 4 1
162.6.c.e 2 144.x even 12 2
162.6.c.h 2 144.w odd 12 2
192.6.a.g 1 16.f odd 4 1
192.6.a.o 1 16.e even 4 1
294.6.a.m 1 112.l odd 4 1
294.6.e.a 2 112.x odd 12 2
294.6.e.g 2 112.w even 12 2
450.6.a.m 1 240.bm odd 4 1
450.6.c.j 2 240.bb even 4 1
450.6.c.j 2 240.bf even 4 1
576.6.a.i 1 48.k even 4 1
576.6.a.j 1 48.i odd 4 1
726.6.a.a 1 176.l odd 4 1
768.6.d.c 2 1.a even 1 1 trivial
768.6.d.c 2 8.b even 2 1 inner
768.6.d.p 2 4.b odd 2 1
768.6.d.p 2 8.d odd 2 1
882.6.a.a 1 336.y even 4 1
1014.6.a.c 1 208.p even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(768, [\chi])\):

\( T_{5}^{2} + 4356 \)
\( T_{7} + 176 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 81 T^{2} \)
$5$ \( 1 - 1894 T^{2} + 9765625 T^{4} \)
$7$ \( ( 1 + 176 T + 16807 T^{2} )^{2} \)
$11$ \( 1 - 318502 T^{2} + 25937424601 T^{4} \)
$13$ \( 1 - 309622 T^{2} + 137858491849 T^{4} \)
$17$ \( ( 1 + 414 T + 1419857 T^{2} )^{2} \)
$19$ \( 1 - 4038262 T^{2} + 6131066257801 T^{4} \)
$23$ \( ( 1 + 600 T + 6436343 T^{2} )^{2} \)
$29$ \( 1 - 9952822 T^{2} + 420707233300201 T^{4} \)
$31$ \( ( 1 + 3592 T + 28629151 T^{2} )^{2} \)
$37$ \( 1 - 67150150 T^{2} + 4808584372417849 T^{4} \)
$41$ \( ( 1 + 19194 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 - 116701030 T^{2} + 21611482313284249 T^{4} \)
$47$ \( ( 1 + 19680 T + 229345007 T^{2} )^{2} \)
$53$ \( 1 + 141171770 T^{2} + 174887470365513049 T^{4} \)
$59$ \( 1 - 736052998 T^{2} + 511116753300641401 T^{4} \)
$61$ \( 1 - 722604502 T^{2} + 713342911662882601 T^{4} \)
$67$ \( 1 - 2417875798 T^{2} + 1822837804551761449 T^{4} \)
$71$ \( ( 1 + 6120 T + 1804229351 T^{2} )^{2} \)
$73$ \( ( 1 - 25558 T + 2073071593 T^{2} )^{2} \)
$79$ \( ( 1 - 74408 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 - 7836246262 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( ( 1 - 32742 T + 5584059449 T^{2} )^{2} \)
$97$ \( ( 1 - 166082 T + 8587340257 T^{2} )^{2} \)
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