Properties

Label 3072.2.d.i
Level $3072$
Weight $2$
Character orbit 3072.d
Analytic conductor $24.530$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( \beta_{4} + \beta_{6} ) q^{5} + ( 1 - \beta_{1} ) q^{7} - q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + ( \beta_{4} + \beta_{6} ) q^{5} + ( 1 - \beta_{1} ) q^{7} - q^{9} + ( \beta_{5} - \beta_{6} - \beta_{7} ) q^{11} + ( -2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{13} + ( -1 - \beta_{3} ) q^{15} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} + ( \beta_{4} + \beta_{7} ) q^{21} + 2 \beta_{2} q^{23} + ( -1 + 2 \beta_{2} - 2 \beta_{3} ) q^{25} -\beta_{4} q^{27} + ( -3 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{29} + ( -3 - \beta_{1} ) q^{31} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{33} + ( 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{35} + ( 4 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( 2 + \beta_{1} - \beta_{3} ) q^{39} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{41} + ( \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{43} + ( -\beta_{4} - \beta_{6} ) q^{45} + 2 \beta_{2} q^{47} + ( 1 - 2 \beta_{1} + 4 \beta_{2} ) q^{49} + ( -\beta_{5} + \beta_{6} - \beta_{7} ) q^{51} + ( 5 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{53} + ( 4 - 4 \beta_{2} ) q^{55} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{57} + 4 \beta_{5} q^{59} + ( -4 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{61} + ( -1 + \beta_{1} ) q^{63} + ( -2 + \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{65} + ( -4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{67} -2 \beta_{5} q^{69} + ( -2 + 2 \beta_{1} ) q^{71} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{73} + ( -\beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{75} + ( -6 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{77} + ( -3 - \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{79} + q^{81} + ( 5 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{83} + ( 6 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{85} + ( 3 + 2 \beta_{2} - \beta_{3} ) q^{87} + ( 2 - 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{91} + ( -3 \beta_{4} + \beta_{7} ) q^{93} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{95} + ( 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -\beta_{5} + \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} - 8q^{9} + O(q^{10}) \) \( 8q + 8q^{7} - 8q^{9} - 8q^{15} - 8q^{25} - 24q^{31} + 16q^{39} + 8q^{49} + 32q^{55} - 8q^{63} - 16q^{65} - 16q^{71} + 16q^{73} - 24q^{79} + 8q^{81} + 24q^{87} + 16q^{89} + 48q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{4} + 2 \nu^{3} - 7 \nu^{2} + 6 \nu - 5 \)
\(\beta_{2}\)\(=\)\( \nu^{6} - 3 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} + 19 \nu^{2} - 12 \nu + 4 \)
\(\beta_{3}\)\(=\)\( -\nu^{6} + 3 \nu^{5} - 11 \nu^{4} + 17 \nu^{3} - 24 \nu^{2} + 16 \nu - 5 \)
\(\beta_{4}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31 \)
\(\beta_{5}\)\(=\)\( 10 \nu^{7} - 35 \nu^{6} + 123 \nu^{5} - 220 \nu^{4} + 325 \nu^{3} - 285 \nu^{2} + 166 \nu - 42 \)
\(\beta_{6}\)\(=\)\( 10 \nu^{7} - 35 \nu^{6} + 123 \nu^{5} - 220 \nu^{4} + 325 \nu^{3} - 285 \nu^{2} + 168 \nu - 43 \)
\(\beta_{7}\)\(=\)\( -28 \nu^{7} + 98 \nu^{6} - 342 \nu^{5} + 610 \nu^{4} - 890 \nu^{3} + 774 \nu^{2} - 440 \nu + 109 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - \beta_{5} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_{1} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} - 5 \beta_{6} + 7 \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 3 \beta_{1} - 10\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{7} - 6 \beta_{6} + 8 \beta_{5} + 6 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} + 11 \beta_{6} - 13 \beta_{5} - 20 \beta_{4} - 25 \beta_{3} - 25 \beta_{2} + 15 \beta_{1} + 52\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(10 \beta_{7} + 32 \beta_{6} - 40 \beta_{5} - 45 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} - \beta_{1} - 6\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-3 \beta_{7} + 11 \beta_{6} - 19 \beta_{5} + 133 \beta_{3} + 147 \beta_{2} - 63 \beta_{1} - 236\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1537.1
0.500000 0.691860i
0.500000 1.44392i
0.500000 + 0.0297061i
0.500000 + 2.10607i
0.500000 2.10607i
0.500000 0.0297061i
0.500000 + 1.44392i
0.500000 + 0.691860i
0 1.00000i 0 3.79793i 0 2.15894 0 −1.00000 0
1537.2 0 1.00000i 0 2.47363i 0 −2.55765 0 −1.00000 0
1537.3 0 1.00000i 0 0.473626i 0 4.55765 0 −1.00000 0
1537.4 0 1.00000i 0 1.79793i 0 −0.158942 0 −1.00000 0
1537.5 0 1.00000i 0 1.79793i 0 −0.158942 0 −1.00000 0
1537.6 0 1.00000i 0 0.473626i 0 4.55765 0 −1.00000 0
1537.7 0 1.00000i 0 2.47363i 0 −2.55765 0 −1.00000 0
1537.8 0 1.00000i 0 3.79793i 0 2.15894 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1537.8
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 3072.2.d.i 8
4.b odd 2 1 3072.2.d.f 8
8.b even 2 1 inner 3072.2.d.i 8
8.d odd 2 1 3072.2.d.f 8
16.e even 4 1 3072.2.a.n 4
16.e even 4 1 3072.2.a.o 4
16.f odd 4 1 3072.2.a.i 4
16.f odd 4 1 3072.2.a.t 4
32.g even 8 2 192.2.j.a 8
32.g even 8 2 384.2.j.a 8
32.h odd 8 2 48.2.j.a 8
32.h odd 8 2 384.2.j.b 8
48.i odd 4 1 9216.2.a.x 4
48.i odd 4 1 9216.2.a.bn 4
48.k even 4 1 9216.2.a.y 4
48.k even 4 1 9216.2.a.bo 4
96.o even 8 2 144.2.k.b 8
96.o even 8 2 1152.2.k.c 8
96.p odd 8 2 576.2.k.b 8
96.p odd 8 2 1152.2.k.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 32.h odd 8 2
144.2.k.b 8 96.o even 8 2
192.2.j.a 8 32.g even 8 2
384.2.j.a 8 32.g even 8 2
384.2.j.b 8 32.h odd 8 2
576.2.k.b 8 96.p odd 8 2
1152.2.k.c 8 96.o even 8 2
1152.2.k.f 8 96.p odd 8 2
3072.2.a.i 4 16.f odd 4 1
3072.2.a.n 4 16.e even 4 1
3072.2.a.o 4 16.e even 4 1
3072.2.a.t 4 16.f odd 4 1
3072.2.d.f 8 4.b odd 2 1
3072.2.d.f 8 8.d odd 2 1
3072.2.d.i 8 1.a even 1 1 trivial
3072.2.d.i 8 8.b even 2 1 inner
9216.2.a.x 4 48.i odd 4 1
9216.2.a.y 4 48.k even 4 1
9216.2.a.bn 4 48.i odd 4 1
9216.2.a.bo 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3072, [\chi])\):

\( T_{5}^{8} + 24 T_{5}^{6} + 160 T_{5}^{4} + 320 T_{5}^{2} + 64 \)
\( T_{7}^{4} - 4 T_{7}^{3} - 8 T_{7}^{2} + 24 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( 64 + 320 T^{2} + 160 T^{4} + 24 T^{6} + T^{8} \)
$7$ \( ( 4 + 24 T - 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$11$ \( 1024 + 2560 T^{2} + 640 T^{4} + 48 T^{6} + T^{8} \)
$13$ \( 16 + 2272 T^{2} + 792 T^{4} + 56 T^{6} + T^{8} \)
$17$ \( ( 16 - 64 T - 32 T^{2} + T^{4} )^{2} \)
$19$ \( 256 + 5120 T^{2} + 1056 T^{4} + 64 T^{6} + T^{8} \)
$23$ \( ( -8 + T^{2} )^{4} \)
$29$ \( 61504 + 20288 T^{2} + 2208 T^{4} + 88 T^{6} + T^{8} \)
$31$ \( ( -28 + 24 T + 40 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$37$ \( 1106704 + 159584 T^{2} + 7768 T^{4} + 152 T^{6} + T^{8} \)
$41$ \( ( -112 - 192 T - 64 T^{2} + T^{4} )^{2} \)
$43$ \( 12544 + 44032 T^{2} + 8992 T^{4} + 192 T^{6} + T^{8} \)
$47$ \( ( -8 + T^{2} )^{4} \)
$53$ \( 18496 + 40256 T^{2} + 4768 T^{4} + 152 T^{6} + T^{8} \)
$59$ \( ( 32 + T^{2} )^{4} \)
$61$ \( 1106704 + 159584 T^{2} + 7768 T^{4} + 152 T^{6} + T^{8} \)
$67$ \( 65536 + 65536 T^{2} + 8704 T^{4} + 256 T^{6} + T^{8} \)
$71$ \( ( 64 - 192 T - 32 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$73$ \( ( 64 - 64 T - 96 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$79$ \( ( -10108 - 2888 T - 168 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$83$ \( 1024 + 39424 T^{2} + 46720 T^{4} + 432 T^{6} + T^{8} \)
$89$ \( ( -1904 + 1632 T - 200 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$97$ \( ( 512 + 768 T - 224 T^{2} + T^{4} )^{2} \)
show more
show less