Properties

Label 384.3.e.d
Level $384$
Weight $3$
Character orbit 384.e
Analytic conductor $10.463$
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] = [384,3,Mod(257,384)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.257");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.e (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: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 99x^{4} + 170x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3 \)
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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{2} + 1) q^{7} + (\beta_{7} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{2} + 1) q^{7} + (\beta_{7} + \beta_{2}) q^{9} + ( - \beta_{7} + \beta_{5} + \beta_{4}) q^{11} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_1 + 1) q^{13} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{15} + (\beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 3 \beta_1 + 1) q^{17} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + \beta_1 - 3) q^{19} + ( - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{21} + ( - 2 \beta_{7} + 2 \beta_{5} + 4 \beta_{3}) q^{23} + (\beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{2} - 5 \beta_1 - 2) q^{25} + ( - \beta_{7} + \beta_{5} - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + \beta_1 - 6) q^{27} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{3} + 6 \beta_1 - 2) q^{29} + ( - 2 \beta_{7} - 2 \beta_{5} - \beta_{2} + 4 \beta_1 - 9) q^{31} + (2 \beta_{7} - 3 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{33} + ( - \beta_{7} + \beta_{6} + \beta_{5} + 5 \beta_{4} + 8 \beta_{3} + 3 \beta_1 - 1) q^{35} + ( - 3 \beta_{7} - \beta_{6} - 3 \beta_{5} - 4 \beta_{2} + 9 \beta_1 - 1) q^{37} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + \beta_1 - 13) q^{39} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} + 6 \beta_1 - 2) q^{41} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{2} + \beta_1 + 17) q^{43} + ( - 2 \beta_{7} + 4 \beta_{6} + 6 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 10) q^{45} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 6 \beta_1 - 2) q^{47} + (3 \beta_{7} - 5 \beta_{6} + 3 \beta_{5} + 2 \beta_{2} + 9 \beta_1 + 2) q^{49} + ( - 3 \beta_{7} + \beta_{6} - \beta_{5} - 5 \beta_{4} - 4 \beta_{3} - \beta_1 + 23) q^{51} + ( - 4 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + \beta_{3} - 12 \beta_1 + 4) q^{53} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{2} + 4 \beta_1 + 22) q^{55} + ( - \beta_{7} + 4 \beta_{6} + 6 \beta_{4} - 6 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{57} + (\beta_{6} + 8 \beta_{4} + 3 \beta_1 - 1) q^{59} + (\beta_{7} + 3 \beta_{6} + \beta_{5} + 4 \beta_{2} - 11 \beta_1 - 13) q^{61} + ( - 4 \beta_{6} + 4 \beta_{5} + 8 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + 31) q^{63} + ( - 4 \beta_{6} - 6 \beta_{4} + 2 \beta_{3} - 12 \beta_1 + 4) q^{65} + (4 \beta_{7} - 3 \beta_{6} + 4 \beta_{5} + 8 \beta_{2} + \beta_1 - 37) q^{67} + ( - 2 \beta_{6} - 6 \beta_{5} + 6 \beta_{4} - 6 \beta_{2} + 2 \beta_1 + 8) q^{69} + (4 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 10 \beta_{4} + 4 \beta_{3} + 6 \beta_1 - 2) q^{71} + (4 \beta_{6} + 4 \beta_{2} - 12 \beta_1 - 2) q^{73} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} - 9 \beta_{4} - 10 \beta_{2} - 43) q^{75} + (4 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 12 \beta_{4} - 2 \beta_{3} + 12 \beta_1 - 4) q^{77} + (2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 3 \beta_{2} - 16 \beta_1 - 41) q^{79} + ( - 3 \beta_{7} - \beta_{6} - 5 \beta_{5} + 8 \beta_{4} + 4 \beta_{3} - 3 \beta_1 + 10) q^{81} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + 5 \beta_{4} - 8 \beta_{3} - 6 \beta_1 + 2) q^{83} + (8 \beta_{7} - 8 \beta_{6} + 8 \beta_{5} + 16 \beta_{2} + 8 \beta_1 + 16) q^{85} + (7 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 48) q^{87} + ( - \beta_{7} + 5 \beta_{6} + \beta_{5} - 8 \beta_{4} + 16 \beta_{3} + 15 \beta_1 - 5) q^{89} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 10 \beta_{2} + 2 \beta_1 + 40) q^{91} + (\beta_{7} + 3 \beta_{6} + \beta_{5} + 8 \beta_{4} - 5 \beta_{3} + 4 \beta_{2} - 11 \beta_1 + 27) q^{93} + (2 \beta_{7} - 8 \beta_{6} - 2 \beta_{5} - 16 \beta_{4} - 4 \beta_{3} - 24 \beta_1 + 8) q^{95} + ( - 5 \beta_{7} - \beta_{6} - 5 \beta_{5} - 2 \beta_{2} + 13 \beta_1 + 5) q^{97} + (4 \beta_{7} - \beta_{6} + 4 \beta_{5} - 10 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + \cdots + 55) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 8 q^{7} + 16 q^{15} - 24 q^{19} + 16 q^{21} - 40 q^{25} - 44 q^{27} - 56 q^{31} + 8 q^{33} + 32 q^{37} - 104 q^{39} + 136 q^{43} - 80 q^{45} + 72 q^{49} + 176 q^{51} + 192 q^{55} - 40 q^{57} - 160 q^{61} + 264 q^{63} - 280 q^{67} + 80 q^{69} - 80 q^{73} - 348 q^{75} - 408 q^{79} + 72 q^{81} + 192 q^{85} - 368 q^{87} + 336 q^{91} + 160 q^{93} + 96 q^{97} + 432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 99x^{4} + 170x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 16\nu^{4} + 70\nu^{2} + 6\nu + 63 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} + 11\nu^{2} + 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{7} + 33\nu^{5} + 159\nu^{3} + 193\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} + 36\nu^{5} + 180\nu^{3} + 178\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 16\nu^{5} - 3\nu^{4} - 70\nu^{3} - 27\nu^{2} - 57\nu - 27 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 16\nu^{4} - 70\nu^{2} + 6\nu - 61 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 16\nu^{5} - 3\nu^{4} + 70\nu^{3} - 27\nu^{2} + 63\nu - 27 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + 3\beta _1 - 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - \beta_{6} + 3\beta_{5} + 6\beta_{2} - 3\beta _1 - 53 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} - 13\beta_{6} + 3\beta_{5} - 3\beta_{4} + 12\beta_{3} - 39\beta _1 + 13 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -33\beta_{7} + 11\beta_{6} - 33\beta_{5} - 54\beta_{2} + 33\beta _1 + 367 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21\beta_{7} + 101\beta_{6} - 21\beta_{5} + 57\beta_{4} - 120\beta_{3} + 303\beta _1 - 101 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 159\beta_{7} - 59\beta_{6} + 159\beta_{5} + 222\beta_{2} - 141\beta _1 - 1453 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -54\beta_{7} - 413\beta_{6} + 54\beta_{5} - 351\beta_{4} + 540\beta_{3} - 1239\beta _1 + 413 ) / 6 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
1.32750i
1.32750i
2.98985i
2.98985i
2.55118i
2.55118i
0.888828i
0.888828i
0 −2.69031 1.32750i 0 0.640013i 0 2.72077 0 5.47550 + 7.14275i 0
257.2 0 −2.69031 + 1.32750i 0 0.640013i 0 2.72077 0 5.47550 7.14275i 0
257.3 0 0.246559 2.98985i 0 6.63641i 0 0.578158 0 −8.87842 1.47435i 0
257.4 0 0.246559 + 2.98985i 0 6.63641i 0 0.578158 0 −8.87842 + 1.47435i 0
257.5 0 1.57844 2.55118i 0 1.31534i 0 −10.2329 0 −4.01705 8.05378i 0
257.6 0 1.57844 + 2.55118i 0 1.31534i 0 −10.2329 0 −4.01705 + 8.05378i 0
257.7 0 2.86531 0.888828i 0 8.59176i 0 10.9340 0 7.41997 5.09353i 0
257.8 0 2.86531 + 0.888828i 0 8.59176i 0 10.9340 0 7.41997 + 5.09353i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.e.d yes 8
3.b odd 2 1 inner 384.3.e.d yes 8
4.b odd 2 1 384.3.e.a 8
8.b even 2 1 384.3.e.b yes 8
8.d odd 2 1 384.3.e.c yes 8
12.b even 2 1 384.3.e.a 8
16.e even 4 2 768.3.h.g 16
16.f odd 4 2 768.3.h.h 16
24.f even 2 1 384.3.e.c yes 8
24.h odd 2 1 384.3.e.b yes 8
48.i odd 4 2 768.3.h.g 16
48.k even 4 2 768.3.h.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.e.a 8 4.b odd 2 1
384.3.e.a 8 12.b even 2 1
384.3.e.b yes 8 8.b even 2 1
384.3.e.b yes 8 24.h odd 2 1
384.3.e.c yes 8 8.d odd 2 1
384.3.e.c yes 8 24.f even 2 1
384.3.e.d yes 8 1.a even 1 1 trivial
384.3.e.d yes 8 3.b odd 2 1 inner
768.3.h.g 16 16.e even 4 2
768.3.h.g 16 48.i odd 4 2
768.3.h.h 16 16.f odd 4 2
768.3.h.h 16 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\):

\( T_{7}^{4} - 4T_{7}^{3} - 108T_{7}^{2} + 368T_{7} - 176 \) Copy content Toggle raw display
\( T_{13}^{4} - 360T_{13}^{2} + 256T_{13} + 7824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + 8 T^{6} + 4 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} + 120 T^{6} + 3504 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} - 108 T^{2} + 368 T - 176)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 488 T^{6} + \cdots + 20214016 \) Copy content Toggle raw display
$13$ \( (T^{4} - 360 T^{2} + 256 T + 7824)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 1248 T^{6} + \cdots + 991494144 \) Copy content Toggle raw display
$19$ \( (T^{4} + 12 T^{3} - 780 T^{2} + \cdots - 98352)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 2336 T^{6} + \cdots + 48132849664 \) Copy content Toggle raw display
$29$ \( T^{8} + 3704 T^{6} + \cdots + 78767790336 \) Copy content Toggle raw display
$31$ \( (T^{4} + 28 T^{3} - 828 T^{2} + \cdots + 50256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 16 T^{3} - 3528 T^{2} + \cdots - 488432)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 10976 T^{6} + \cdots + 3916979306496 \) Copy content Toggle raw display
$43$ \( (T^{4} - 68 T^{3} - 588 T^{2} + \cdots - 917424)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 5376 T^{6} + \cdots + 424144797696 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 870911869798656 \) Copy content Toggle raw display
$59$ \( T^{8} + 8840 T^{6} + \cdots + 12745356964096 \) Copy content Toggle raw display
$61$ \( (T^{4} + 80 T^{3} - 2184 T^{2} + \cdots - 4584688)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 140 T^{3} + 900 T^{2} + \cdots - 6784816)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 22688 T^{6} + \cdots + 1706597351424 \) Copy content Toggle raw display
$73$ \( (T^{4} + 40 T^{3} - 5544 T^{2} + \cdots - 899312)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 204 T^{3} + 7524 T^{2} + \cdots - 38762928)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 13416 T^{6} + \cdots + 56638749360384 \) Copy content Toggle raw display
$89$ \( T^{8} + 45632 T^{6} + \cdots + 92\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} - 48 T^{3} - 8136 T^{2} + \cdots + 8416272)^{2} \) Copy content Toggle raw display
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