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Properties

Label 384.2.j.b
Level $384$
Weight $2$
Character orbit 384.j
Analytic conductor $3.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,2,Mod(97,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.j (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.06625543762\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{6} - \beta_{2}) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4}) q^{7} - \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{6} - \beta_{2}) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4}) q^{7} - \beta_{4} q^{9} + (\beta_{7} + \beta_{4} - \beta_1 + 1) q^{11} + ( - \beta_{7} + 2 \beta_{3} - \beta_1) q^{13} + ( - \beta_1 - 1) q^{15} + (\beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{17} + (\beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_1 + 1) q^{19} + (\beta_{7} + \beta_{6} + \beta_{2} - \beta_1) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{23} + ( - 2 \beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{25} - \beta_{2} q^{27} + ( - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 2) q^{29} + ( - \beta_{6} + \beta_{5} + \beta_1 + 3) q^{31} + (\beta_{6} - \beta_{5} - \beta_{3} + \beta_{2}) q^{33} + ( - \beta_{7} + 3 \beta_{4} - \beta_1 - 3) q^{35} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 4 \beta_{2} + \beta_1 + 2) q^{37} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4}) q^{39} + (2 \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{3} + 3 \beta_{2}) q^{41} + ( - \beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_1 + 1) q^{43} + ( - \beta_{5} + \beta_{3}) q^{45} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{47} + ( - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 1) q^{49} + ( - \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_1 - 1) q^{51} + ( - \beta_{6} - 2 \beta_{4} + 5 \beta_{2} - 2) q^{53} + ( - 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2}) q^{55} + (2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2}) q^{57} + ( - 4 \beta_{4} - 4) q^{59} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + \beta_1 - 2) q^{61} + (\beta_{6} - \beta_{5} - \beta_1 + 1) q^{63} + ( - \beta_{6} + \beta_{5} - 5 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 2) q^{65} + (2 \beta_{7} - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_1 + 2) q^{67} + ( - 2 \beta_{4} - 2) q^{69} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4}) q^{71} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2}) q^{73} + ( - 2 \beta_{6} - 2 \beta_{4} + \beta_{2} - 2) q^{75} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 2 \beta_1 - 2) q^{77} + (\beta_{6} - \beta_{5} + 4 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 3) q^{79} - q^{81} + ( - \beta_{7} - 4 \beta_{5} - 5 \beta_{4} - \beta_1 + 5) q^{83} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 6 \beta_{2} + 2 \beta_1 + 2) q^{85} + ( - \beta_{7} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{87} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} - 6 \beta_{2}) q^{89} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + 4 \beta_{2} + \beta_1 + 1) q^{91} + (\beta_{7} + \beta_{5} - 3 \beta_{3} + \beta_1) q^{93} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 6) q^{95} + ( - 2 \beta_{6} + 2 \beta_{5} - 6 \beta_{3} + 6 \beta_{2}) q^{97} + ( - \beta_{7} - \beta_{4} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{11} - 8 q^{15} + 8 q^{19} + 16 q^{29} + 24 q^{31} - 24 q^{35} + 16 q^{37} + 8 q^{43} - 8 q^{49} - 8 q^{51} - 16 q^{53} - 32 q^{59} - 16 q^{61} + 8 q^{63} - 16 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{75} - 16 q^{77} - 24 q^{79} - 8 q^{81} + 40 q^{83} + 16 q^{85} + 8 q^{91} - 48 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( -\nu^{6} + 3\nu^{5} - 11\nu^{4} + 17\nu^{3} - 24\nu^{2} + 16\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 5\nu^{7} - 18\nu^{6} + 63\nu^{5} - 115\nu^{4} + 170\nu^{3} - 152\nu^{2} + 89\nu - 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 8\nu^{7} - 28\nu^{6} + 98\nu^{5} - 175\nu^{4} + 256\nu^{3} - 223\nu^{2} + 126\nu - 31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 9\nu^{7} - 31\nu^{6} + 108\nu^{5} - 190\nu^{4} + 275\nu^{3} - 236\nu^{2} + 131\nu - 33 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 9\nu^{7} - 32\nu^{6} + 111\nu^{5} - 200\nu^{4} + 290\nu^{3} - 253\nu^{2} + 141\nu - 33 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 10\nu^{7} - 35\nu^{6} + 123\nu^{5} - 220\nu^{4} + 325\nu^{3} - 285\nu^{2} + 168\nu - 43 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - \beta_{5} - 3\beta_{4} + 2\beta_{3} + 5\beta_{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{7} - \beta_{6} + 3\beta_{5} - 6\beta_{4} + 12\beta_{3} + 4\beta_{2} - 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{7} - 10\beta_{6} + 5\beta_{5} + 10\beta_{4} + 6\beta_{3} - 19\beta_{2} - 5\beta _1 + 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 22\beta_{7} - 9\beta_{6} - 11\beta_{5} + 45\beta_{4} - 48\beta_{3} - 32\beta_{2} + 5\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{7} + 33\beta_{6} - 30\beta_{5} - 83\beta_{3} + 64\beta_{2} + 35\beta _1 - 118 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.500000 + 2.10607i
0.500000 − 0.691860i
0.500000 − 1.44392i
0.500000 + 0.0297061i
0.500000 − 2.10607i
0.500000 + 0.691860i
0.500000 + 1.44392i
0.500000 − 0.0297061i
0 −0.707107 + 0.707107i 0 −1.27133 − 1.27133i 0 − 0.158942i 0 − 1.00000i 0
97.2 0 −0.707107 + 0.707107i 0 2.68554 + 2.68554i 0 2.15894i 0 − 1.00000i 0
97.3 0 0.707107 − 0.707107i 0 −1.74912 − 1.74912i 0 − 2.55765i 0 − 1.00000i 0
97.4 0 0.707107 − 0.707107i 0 0.334904 + 0.334904i 0 4.55765i 0 − 1.00000i 0
289.1 0 −0.707107 − 0.707107i 0 −1.27133 + 1.27133i 0 0.158942i 0 1.00000i 0
289.2 0 −0.707107 − 0.707107i 0 2.68554 − 2.68554i 0 − 2.15894i 0 1.00000i 0
289.3 0 0.707107 + 0.707107i 0 −1.74912 + 1.74912i 0 2.55765i 0 1.00000i 0
289.4 0 0.707107 + 0.707107i 0 0.334904 − 0.334904i 0 − 4.55765i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.j.b 8
3.b odd 2 1 1152.2.k.c 8
4.b odd 2 1 384.2.j.a 8
8.b even 2 1 48.2.j.a ✓ 8
8.d odd 2 1 192.2.j.a 8
12.b even 2 1 1152.2.k.f 8
16.e even 4 1 48.2.j.a ✓ 8
16.e even 4 1 inner 384.2.j.b 8
16.f odd 4 1 192.2.j.a 8
16.f odd 4 1 384.2.j.a 8
24.f even 2 1 576.2.k.b 8
24.h odd 2 1 144.2.k.b 8
32.g even 8 1 3072.2.a.i 4
32.g even 8 1 3072.2.a.t 4
32.g even 8 2 3072.2.d.f 8
32.h odd 8 1 3072.2.a.n 4
32.h odd 8 1 3072.2.a.o 4
32.h odd 8 2 3072.2.d.i 8
48.i odd 4 1 144.2.k.b 8
48.i odd 4 1 1152.2.k.c 8
48.k even 4 1 576.2.k.b 8
48.k even 4 1 1152.2.k.f 8
96.o even 8 1 9216.2.a.x 4
96.o even 8 1 9216.2.a.bn 4
96.p odd 8 1 9216.2.a.y 4
96.p odd 8 1 9216.2.a.bo 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a ✓ 8 8.b even 2 1
48.2.j.a ✓ 8 16.e even 4 1
144.2.k.b 8 24.h odd 2 1
144.2.k.b 8 48.i odd 4 1
192.2.j.a 8 8.d odd 2 1
192.2.j.a 8 16.f odd 4 1
384.2.j.a 8 4.b odd 2 1
384.2.j.a 8 16.f odd 4 1
384.2.j.b 8 1.a even 1 1 trivial
384.2.j.b 8 16.e even 4 1 inner
576.2.k.b 8 24.f even 2 1
576.2.k.b 8 48.k even 4 1
1152.2.k.c 8 3.b odd 2 1
1152.2.k.c 8 48.i odd 4 1
1152.2.k.f 8 12.b even 2 1
1152.2.k.f 8 48.k even 4 1
3072.2.a.i 4 32.g even 8 1
3072.2.a.n 4 32.h odd 8 1
3072.2.a.o 4 32.h odd 8 1
3072.2.a.t 4 32.g even 8 1
3072.2.d.f 8 32.g even 8 2
3072.2.d.i 8 32.h odd 8 2
9216.2.a.x 4 96.o even 8 1
9216.2.a.y 4 96.p odd 8 1
9216.2.a.bn 4 96.o even 8 1
9216.2.a.bo 4 96.p odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} - 8T_{11}^{7} + 32T_{11}^{6} - 256T_{11}^{3} + 2048T_{11}^{2} + 2048T_{11} + 1024 \) acting on \(S_{2}^{\mathrm{new}}(384, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 16 T^{5} + 128 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{8} + 32 T^{6} + 264 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{8} + 64 T^{5} + 776 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{4} - 32 T^{2} + 64 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{7} + 32 T^{6} + 32 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 61504 \) Copy content Toggle raw display
$31$ \( (T^{4} - 12 T^{3} + 40 T^{2} - 24 T - 28)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$41$ \( T^{8} + 128 T^{6} + 3872 T^{4} + \cdots + 12544 \) Copy content Toggle raw display
$43$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 12544 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 18496 \) Copy content Toggle raw display
$59$ \( (T^{2} + 8 T + 32)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{8} + 256 T^{6} + 8320 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( (T^{4} + 12 T^{3} - 168 T^{2} + \cdots - 10108)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 40 T^{7} + 800 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{8} + 464 T^{6} + 62304 T^{4} + \cdots + 3625216 \) Copy content Toggle raw display
$97$ \( (T^{4} - 224 T^{2} + 768 T + 512)^{2} \) Copy content Toggle raw display
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