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Properties

Label 384.4.d.f
Level $384$
Weight $4$
Character orbit 384.d
Analytic conductor $22.657$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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: \(8\)
Coefficient field: 8.0.1534132224.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{6} q^{5} - \beta_{2} q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{6} q^{5} - \beta_{2} q^{7} - 9 q^{9} + ( - \beta_{3} - 4 \beta_1) q^{11} + ( - \beta_{7} - \beta_{6}) q^{13} + (\beta_{4} + \beta_{2}) q^{15} + ( - \beta_{5} + 30) q^{17} + ( - 3 \beta_{3} + 4 \beta_1) q^{19} + (\beta_{7} - 2 \beta_{6}) q^{21} + ( - 4 \beta_{4} + 2 \beta_{2}) q^{23} + ( - 2 \beta_{5} - 43) q^{25} - 9 \beta_1 q^{27} + (2 \beta_{7} + 5 \beta_{6}) q^{29} + (4 \beta_{4} + 5 \beta_{2}) q^{31} + ( - \beta_{5} + 36) q^{33} + (3 \beta_{3} + 40 \beta_1) q^{35} + ( - \beta_{7} + 17 \beta_{6}) q^{37} + (3 \beta_{4} - 6 \beta_{2}) q^{39} + ( - 5 \beta_{5} - 102) q^{41} + ( - 3 \beta_{3} - 44 \beta_1) q^{43} + 9 \beta_{6} q^{45} + ( - 8 \beta_{4} - 14 \beta_{2}) q^{47} + ( - 8 \beta_{5} + 209) q^{49} + (9 \beta_{3} + 30 \beta_1) q^{51} + ( - 2 \beta_{7} - 13 \beta_{6}) q^{53} + (4 \beta_{4} - 4 \beta_{2}) q^{55} + ( - 3 \beta_{5} - 36) q^{57} + (16 \beta_{3} - 156 \beta_1) q^{59} + (5 \beta_{7} - 49 \beta_{6}) q^{61} + 9 \beta_{2} q^{63} + ( - 9 \beta_{5} - 144) q^{65} + ( - 12 \beta_{3} + 196 \beta_1) q^{67} + ( - 6 \beta_{7} - 24 \beta_{6}) q^{69} + (12 \beta_{4} + 30 \beta_{2}) q^{71} + ( - 8 \beta_{5} - 242) q^{73} + (18 \beta_{3} - 43 \beta_1) q^{75} + ( - 12 \beta_{7} + 16 \beta_{6}) q^{77} + ( - 20 \beta_{4} - 11 \beta_{2}) q^{79} + 81 q^{81} + (17 \beta_{3} + 284 \beta_1) q^{83} + (8 \beta_{7} + 26 \beta_{6}) q^{85} + ( - 9 \beta_{4} + 9 \beta_{2}) q^{87} + ( - 6 \beta_{5} + 918) q^{89} + ( - 63 \beta_{3} - 432 \beta_1) q^{91} + ( - \beta_{7} + 38 \beta_{6}) q^{93} + (28 \beta_{4} + 4 \beta_{2}) q^{95} + (2 \beta_{5} - 370) q^{97} + (9 \beta_{3} + 36 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 72 q^{9} + 240 q^{17} - 344 q^{25} + 288 q^{33} - 816 q^{41} + 1672 q^{49} - 288 q^{57} - 1152 q^{65} - 1936 q^{73} + 648 q^{81} + 7344 q^{89} - 2960 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 107x^{4} + 210x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{5} - 33\nu^{3} - 87\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} + 6\nu^{4} - 2\nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{5} + 104\nu^{3} + 328\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 8\nu^{6} + 96\nu^{4} + 272\nu^{2} - 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 24\nu^{6} + 288\nu^{4} + 864\nu^{2} + 72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{7} + 31\nu^{5} + 155\nu^{3} + 253\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 10\nu^{7} + 131\nu^{5} + 487\nu^{3} + 377\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + 10\beta_{6} - 3\beta_{3} + 16\beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 3\beta_{4} - 216 ) / 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} - 15\beta_{6} + 6\beta_{3} - 16\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{5} + 19\beta_{4} - 8\beta_{2} + 1320 ) / 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -74\beta_{7} + 370\beta_{6} - 177\beta_{3} + 176\beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 19\beta_{5} - 60\beta_{4} + 48\beta_{2} - 4104 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 470\beta_{7} - 2302\beta_{6} + 1263\beta_{3} + 208\beta_1 ) / 96 \) Copy content Toggle raw display

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

Character values

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

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

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
0.0690906i
2.63019i
− 2.21597i
− 2.48330i
2.48330i
2.21597i
− 2.63019i
− 0.0690906i
0 − 3.00000i 0 − 17.4288i 0 2.99032 0 −9.00000 0
193.2 0 − 3.00000i 0 − 5.67763i 0 33.0917 0 −9.00000 0
193.3 0 − 3.00000i 0 5.67763i 0 −33.0917 0 −9.00000 0
193.4 0 − 3.00000i 0 17.4288i 0 −2.99032 0 −9.00000 0
193.5 0 3.00000i 0 − 17.4288i 0 −2.99032 0 −9.00000 0
193.6 0 3.00000i 0 − 5.67763i 0 −33.0917 0 −9.00000 0
193.7 0 3.00000i 0 5.67763i 0 33.0917 0 −9.00000 0
193.8 0 3.00000i 0 17.4288i 0 2.99032 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 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.f ✓ 8
3.b odd 2 1 1152.4.d.p 8
4.b odd 2 1 inner 384.4.d.f ✓ 8
8.b even 2 1 inner 384.4.d.f ✓ 8
8.d odd 2 1 inner 384.4.d.f ✓ 8
12.b even 2 1 1152.4.d.p 8
16.e even 4 1 768.4.a.u 4
16.e even 4 1 768.4.a.v 4
16.f odd 4 1 768.4.a.u 4
16.f odd 4 1 768.4.a.v 4
24.f even 2 1 1152.4.d.p 8
24.h odd 2 1 1152.4.d.p 8
48.i odd 4 1 2304.4.a.by 4
48.i odd 4 1 2304.4.a.cb 4
48.k even 4 1 2304.4.a.by 4
48.k even 4 1 2304.4.a.cb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.f ✓ 8 1.a even 1 1 trivial
384.4.d.f ✓ 8 4.b odd 2 1 inner
384.4.d.f ✓ 8 8.b even 2 1 inner
384.4.d.f ✓ 8 8.d odd 2 1 inner
768.4.a.u 4 16.e even 4 1
768.4.a.u 4 16.f odd 4 1
768.4.a.v 4 16.e even 4 1
768.4.a.v 4 16.f odd 4 1
1152.4.d.p 8 3.b odd 2 1
1152.4.d.p 8 12.b even 2 1
1152.4.d.p 8 24.f even 2 1
1152.4.d.p 8 24.h odd 2 1
2304.4.a.by 4 48.i odd 4 1
2304.4.a.by 4 48.k even 4 1
2304.4.a.cb 4 48.i odd 4 1
2304.4.a.cb 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 \(S_{4}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{4} + 336T_{5}^{2} + 9792 \) Copy content Toggle raw display
\( T_{7}^{4} - 1104T_{7}^{2} + 9792 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 336 T^{2} + 9792)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 1104 T^{2} + 9792)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 1312 T^{2} + 135424)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 8640 T^{2} + 12690432)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 60 T - 3708)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 9504 T^{2} + 19927296)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 53568 T^{2} + 621831168)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 41040 T^{2} + 419577408)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 55248 T^{2} + 461095488)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 107136 T^{2} + 2487324672)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 204 T - 104796)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 44064 T^{2} + 164249856)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 302400 T^{2} + 21332616192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 87888 T^{2} + 806557248)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 700192 T^{2} + 7735554304)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 1040256 T^{2} + 270509248512)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 838944 T^{2} + 73992704256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 1104192 T^{2} + 297070322688)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 484 T - 236348)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 1039824 T^{2} + 1904357952)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1747744 T^{2} + 334010020096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 1836 T + 676836)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 740 T + 118468)^{4} \) Copy content Toggle raw display
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