Properties

Label 384.4.d.b
Level $384$
Weight $4$
Character orbit 384.d
Analytic conductor $22.657$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,4,Mod(193,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.193");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), 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: \(22.6567334422\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 8 i q^{5} + 12 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} + 8 i q^{5} + 12 q^{7} - 9 q^{9} + 12 i q^{11} + 20 i q^{13} - 24 q^{15} + 62 q^{17} + 108 i q^{19} + 36 i q^{21} - 72 q^{23} + 61 q^{25} - 27 i q^{27} - 128 i q^{29} - 204 q^{31} - 36 q^{33} + 96 i q^{35} + 228 i q^{37} - 60 q^{39} - 22 q^{41} + 204 i q^{43} - 72 i q^{45} - 600 q^{47} - 199 q^{49} + 186 i q^{51} - 256 i q^{53} - 96 q^{55} - 324 q^{57} + 828 i q^{59} - 84 i q^{61} - 108 q^{63} - 160 q^{65} + 348 i q^{67} - 216 i q^{69} + 456 q^{71} + 822 q^{73} + 183 i q^{75} + 144 i q^{77} - 1356 q^{79} + 81 q^{81} + 108 i q^{83} + 496 i q^{85} + 384 q^{87} - 938 q^{89} + 240 i q^{91} - 612 i q^{93} - 864 q^{95} + 1278 q^{97} - 108 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 24 q^{7} - 18 q^{9} - 48 q^{15} + 124 q^{17} - 144 q^{23} + 122 q^{25} - 408 q^{31} - 72 q^{33} - 120 q^{39} - 44 q^{41} - 1200 q^{47} - 398 q^{49} - 192 q^{55} - 648 q^{57} - 216 q^{63} - 320 q^{65} + 912 q^{71} + 1644 q^{73} - 2712 q^{79} + 162 q^{81} + 768 q^{87} - 1876 q^{89} - 1728 q^{95} + 2556 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
1.00000i
1.00000i
0 3.00000i 0 8.00000i 0 12.0000 0 −9.00000 0
193.2 0 3.00000i 0 8.00000i 0 12.0000 0 −9.00000 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 384.4.d.b yes 2
3.b odd 2 1 1152.4.d.g 2
4.b odd 2 1 384.4.d.a 2
8.b even 2 1 inner 384.4.d.b yes 2
8.d odd 2 1 384.4.d.a 2
12.b even 2 1 1152.4.d.b 2
16.e even 4 1 768.4.a.b 1
16.e even 4 1 768.4.a.c 1
16.f odd 4 1 768.4.a.a 1
16.f odd 4 1 768.4.a.d 1
24.f even 2 1 1152.4.d.b 2
24.h odd 2 1 1152.4.d.g 2
48.i odd 4 1 2304.4.a.e 1
48.i odd 4 1 2304.4.a.k 1
48.k even 4 1 2304.4.a.f 1
48.k even 4 1 2304.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.a 2 4.b odd 2 1
384.4.d.a 2 8.d odd 2 1
384.4.d.b yes 2 1.a even 1 1 trivial
384.4.d.b yes 2 8.b even 2 1 inner
768.4.a.a 1 16.f odd 4 1
768.4.a.b 1 16.e even 4 1
768.4.a.c 1 16.e even 4 1
768.4.a.d 1 16.f odd 4 1
1152.4.d.b 2 12.b even 2 1
1152.4.d.b 2 24.f even 2 1
1152.4.d.g 2 3.b odd 2 1
1152.4.d.g 2 24.h odd 2 1
2304.4.a.e 1 48.i odd 4 1
2304.4.a.f 1 48.k even 4 1
2304.4.a.k 1 48.i odd 4 1
2304.4.a.l 1 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{7} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 64 \) Copy content Toggle raw display
$7$ \( (T - 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 144 \) Copy content Toggle raw display
$13$ \( T^{2} + 400 \) Copy content Toggle raw display
$17$ \( (T - 62)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 11664 \) Copy content Toggle raw display
$23$ \( (T + 72)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16384 \) Copy content Toggle raw display
$31$ \( (T + 204)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 51984 \) Copy content Toggle raw display
$41$ \( (T + 22)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 41616 \) Copy content Toggle raw display
$47$ \( (T + 600)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 65536 \) Copy content Toggle raw display
$59$ \( T^{2} + 685584 \) Copy content Toggle raw display
$61$ \( T^{2} + 7056 \) Copy content Toggle raw display
$67$ \( T^{2} + 121104 \) Copy content Toggle raw display
$71$ \( (T - 456)^{2} \) Copy content Toggle raw display
$73$ \( (T - 822)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1356)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 11664 \) Copy content Toggle raw display
$89$ \( (T + 938)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1278)^{2} \) Copy content Toggle raw display
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