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Properties

Label 384.5.g.a
Level $384$
Weight $5$
Character orbit 384.g
Analytic conductor $39.694$
Analytic rank $0$
Dimension $16$
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] = [384,5,Mod(127,384)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.127");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 384.g (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: \(39.6940658242\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 104 x^{14} - 588 x^{13} + 3846 x^{12} - 15796 x^{11} + 66992 x^{10} - 203224 x^{9} + \cdots + 2196513 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{86}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + (\beta_1 - 6) q^{5} + \beta_{11} q^{7} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{3} + (\beta_1 - 6) q^{5} + \beta_{11} q^{7} - 27 q^{9} - \beta_{15} q^{11} + ( - \beta_{6} - 30) q^{13} + ( - \beta_{10} - 6 \beta_{8}) q^{15} + (\beta_{7} + \beta_{6} - \beta_{3} + \cdots - 30) q^{17}+ \cdots + 27 \beta_{15} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 96 q^{5} - 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 96 q^{5} - 432 q^{9} - 480 q^{13} - 480 q^{17} + 2672 q^{25} + 3360 q^{29} - 1120 q^{37} + 1440 q^{41} + 2592 q^{45} - 2480 q^{49} - 3552 q^{53} - 7488 q^{57} - 18272 q^{61} - 1344 q^{65} - 8480 q^{73} + 17280 q^{77} + 11664 q^{81} + 29888 q^{85} + 18720 q^{89} - 28800 q^{93} + 13088 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 104 x^{14} - 588 x^{13} + 3846 x^{12} - 15796 x^{11} + 66992 x^{10} - 203224 x^{9} + \cdots + 2196513 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 187552476416 \nu^{14} - 1312867334912 \nu^{13} + 18046258978138 \nu^{12} + \cdots + 22\!\cdots\!72 ) / 103424768719305 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2490484857664 \nu^{14} - 17433394003648 \nu^{13} + 233692901170892 \nu^{12} + \cdots + 16\!\cdots\!28 ) / 517123843596525 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27558205376 \nu^{14} - 192907437632 \nu^{13} + 2700762176578 \nu^{12} + \cdots + 48\!\cdots\!52 ) / 5331173645325 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 251702906608 \nu^{14} + 1761920346256 \nu^{13} - 22765377324824 \nu^{12} + \cdots + 51\!\cdots\!84 ) / 47011258508775 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4465986482512 \nu^{14} - 31261905377584 \nu^{13} + 423396726326036 \nu^{12} + \cdots + 39\!\cdots\!24 ) / 517123843596525 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5802356658272 \nu^{14} + 40616496607904 \nu^{13} - 531654652160416 \nu^{12} + \cdots - 22\!\cdots\!44 ) / 517123843596525 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18956498563472 \nu^{14} + 132695489944304 \nu^{13} + \cdots - 19\!\cdots\!44 ) / 517123843596525 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2374573312 \nu^{15} + 17809299840 \nu^{14} - 234591101928 \nu^{13} + \cdots + 13\!\cdots\!37 ) / 5691981010375 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 57\!\cdots\!68 \nu^{15} + \cdots + 27\!\cdots\!93 ) / 30\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 26\!\cdots\!88 \nu^{15} + \cdots + 18\!\cdots\!88 ) / 10\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 80\!\cdots\!48 \nu^{15} + \cdots + 41\!\cdots\!48 ) / 30\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12\!\cdots\!68 \nu^{15} + \cdots - 16\!\cdots\!68 ) / 30\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 62\!\cdots\!72 \nu^{15} + \cdots - 29\!\cdots\!57 ) / 61\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 62\!\cdots\!48 \nu^{15} + \cdots - 30\!\cdots\!48 ) / 61\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 47\!\cdots\!04 \nu^{15} + \cdots + 27\!\cdots\!04 ) / 36\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 24 \beta_{15} - 33 \beta_{14} + 48 \beta_{13} - 18 \beta_{12} - 93 \beta_{11} - 16 \beta_{10} + \cdots + 1536 ) / 3072 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 24 \beta_{15} - 33 \beta_{14} + 48 \beta_{13} - 18 \beta_{12} - 93 \beta_{11} - 16 \beta_{10} + \cdots - 27648 ) / 3072 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 376 \beta_{15} + 457 \beta_{14} - 368 \beta_{13} + 30 \beta_{12} + 1257 \beta_{11} + 464 \beta_{10} + \cdots - 28160 ) / 2048 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 288 \beta_{15} + 351 \beta_{14} - 288 \beta_{13} + 27 \beta_{12} + 966 \beta_{11} + 352 \beta_{10} + \cdots + 121728 ) / 768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 34344 \beta_{15} - 33369 \beta_{14} + 15480 \beta_{13} + 2790 \beta_{12} - 107565 \beta_{11} + \cdots + 2575872 ) / 6144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9068 \beta_{15} - 8930 \beta_{14} + 4354 \beta_{13} + 651 \beta_{12} - 28509 \beta_{11} + \cdots - 1958400 ) / 512 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1020240 \beta_{15} + 864399 \beta_{14} - 288036 \beta_{13} - 112698 \beta_{12} + 3067779 \beta_{11} + \cdots - 91367424 ) / 6144 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 287136 \beta_{15} + 247767 \beta_{14} - 87588 \beta_{13} - 30420 \beta_{12} + 867861 \beta_{11} + \cdots + 38655936 ) / 384 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 9479896 \beta_{15} - 7525783 \beta_{14} + 2175800 \beta_{13} + 1079266 \beta_{12} - 27972811 \beta_{11} + \cdots + 1114140160 ) / 2048 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 44357016 \beta_{15} - 35845761 \beta_{14} + 10881240 \beta_{13} + 4973235 \beta_{12} + \cdots - 4024214784 ) / 1536 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 734711736 \beta_{15} + 567742335 \beta_{14} - 156666096 \beta_{13} - 83085630 \beta_{12} + \cdots - 120205423104 ) / 6144 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 536517936 \beta_{15} + 420956847 \beta_{14} - 120256860 \beta_{13} - 60461260 \beta_{12} + \cdots + 32667021056 ) / 512 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 17022992544 \beta_{15} - 13022279289 \beta_{14} + 3550677564 \beta_{13} + 1910161182 \beta_{12} + \cdots + 4194204521472 ) / 6144 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 55766736780 \beta_{15} - 43212175509 \beta_{14} + 12074845482 \beta_{13} + 6268600986 \beta_{12} + \cdots - 2080893828096 ) / 1536 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 53257703988 \beta_{15} + 40559649571 \beta_{14} - 11021769170 \beta_{13} - 5945657018 \beta_{12} + \cdots - 23618873322240 ) / 1024 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 + 5.62551i
0.500000 − 2.68460i
0.500000 − 1.33456i
0.500000 − 3.64904i
0.500000 + 1.23482i
0.500000 − 3.21129i
0.500000 + 2.27039i
0.500000 + 1.74877i
0.500000 − 5.62551i
0.500000 + 2.68460i
0.500000 + 1.33456i
0.500000 + 3.64904i
0.500000 − 1.23482i
0.500000 + 3.21129i
0.500000 − 2.27039i
0.500000 − 1.74877i
0 − 5.19615i 0 −42.6459 0 − 78.8689i 0 −27.0000 0
127.2 0 − 5.19615i 0 −38.1647 0 50.0160i 0 −27.0000 0
127.3 0 − 5.19615i 0 −31.6602 0 17.9843i 0 −27.0000 0
127.4 0 − 5.19615i 0 −8.12430 0 42.5107i 0 −27.0000 0
127.5 0 − 5.19615i 0 −2.60435 0 − 77.5593i 0 −27.0000 0
127.6 0 − 5.19615i 0 18.0608 0 15.9380i 0 −27.0000 0
127.7 0 − 5.19615i 0 24.8934 0 52.9149i 0 −27.0000 0
127.8 0 − 5.19615i 0 32.2453 0 − 22.9356i 0 −27.0000 0
127.9 0 5.19615i 0 −42.6459 0 78.8689i 0 −27.0000 0
127.10 0 5.19615i 0 −38.1647 0 − 50.0160i 0 −27.0000 0
127.11 0 5.19615i 0 −31.6602 0 − 17.9843i 0 −27.0000 0
127.12 0 5.19615i 0 −8.12430 0 − 42.5107i 0 −27.0000 0
127.13 0 5.19615i 0 −2.60435 0 77.5593i 0 −27.0000 0
127.14 0 5.19615i 0 18.0608 0 − 15.9380i 0 −27.0000 0
127.15 0 5.19615i 0 24.8934 0 − 52.9149i 0 −27.0000 0
127.16 0 5.19615i 0 32.2453 0 22.9356i 0 −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.5.g.a ✓ 16
3.b odd 2 1 1152.5.g.f 16
4.b odd 2 1 inner 384.5.g.a ✓ 16
8.b even 2 1 384.5.g.b yes 16
8.d odd 2 1 384.5.g.b yes 16
12.b even 2 1 1152.5.g.f 16
16.e even 4 1 768.5.b.h 16
16.e even 4 1 768.5.b.i 16
16.f odd 4 1 768.5.b.h 16
16.f odd 4 1 768.5.b.i 16
24.f even 2 1 1152.5.g.c 16
24.h odd 2 1 1152.5.g.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.g.a ✓ 16 1.a even 1 1 trivial
384.5.g.a ✓ 16 4.b odd 2 1 inner
384.5.g.b yes 16 8.b even 2 1
384.5.g.b yes 16 8.d odd 2 1
768.5.b.h 16 16.e even 4 1
768.5.b.h 16 16.f odd 4 1
768.5.b.i 16 16.e even 4 1
768.5.b.i 16 16.f odd 4 1
1152.5.g.c 16 24.f even 2 1
1152.5.g.c 16 24.h odd 2 1
1152.5.g.f 16 3.b odd 2 1
1152.5.g.f 16 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 48 T_{5}^{7} - 2016 T_{5}^{6} - 96768 T_{5}^{5} + 1356800 T_{5}^{4} + 55670784 T_{5}^{3} + \cdots - 15806087168 \) acting on \(S_{5}^{\mathrm{new}}(384, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 27)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} + 48 T^{7} + \cdots - 15806087168)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 24\!\cdots\!48)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 27\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 28\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 44\!\cdots\!88)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 18\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 78\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 28\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 99\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 61\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 33\!\cdots\!52)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 37\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 16\!\cdots\!24)^{2} \) Copy content Toggle raw display
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