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Properties

Label 384.6.c.a
Level $384$
Weight $6$
Character orbit 384.c
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
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] = [384,6,Mod(383,384)]
 
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, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.383");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.c (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: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + \cdots + 2870280625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{88}\cdot 3^{14}\cdot 41^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_1 q^{5} + \beta_{9} q^{7} + ( - \beta_{11} + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_1 q^{5} + \beta_{9} q^{7} + ( - \beta_{11} + \beta_{6}) q^{9} + ( - \beta_{4} + \beta_{3} - 47) q^{11} + (\beta_{7} + 3 \beta_{3}) q^{13} + ( - \beta_{13} - \beta_{7} + \beta_{6} + \cdots - 42) q^{15}+ \cdots + ( - 6 \beta_{19} + 44 \beta_{18} + \cdots - 18919) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{3} - 948 q^{11} - 852 q^{15} - 1640 q^{21} - 328 q^{23} - 12500 q^{25} - 2030 q^{27} + 2836 q^{33} + 7184 q^{35} - 15056 q^{37} + 12980 q^{39} - 11800 q^{45} - 36640 q^{47} - 33388 q^{49} - 1936 q^{51} + 15404 q^{57} - 62908 q^{59} - 73264 q^{61} - 23608 q^{63} + 84024 q^{69} - 34888 q^{71} + 52568 q^{73} - 115698 q^{75} + 55444 q^{81} + 225172 q^{83} + 30112 q^{85} + 225700 q^{87} + 148016 q^{93} - 418616 q^{95} + 7600 q^{97} - 378260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 306 x^{18} + 37827 x^{16} + 2442168 x^{14} + 88368509 x^{12} + 1774000974 x^{10} + \cdots + 2870280625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 87\!\cdots\!54 \nu^{19} + \cdots + 43\!\cdots\!50 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\!\cdots\!64 \nu^{19} + \cdots + 61\!\cdots\!00 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 45\!\cdots\!26 \nu^{19} + \cdots + 13\!\cdots\!25 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 45\!\cdots\!26 \nu^{19} + \cdots + 15\!\cdots\!75 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\!\cdots\!26 \nu^{19} + \cdots - 11\!\cdots\!25 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 46\!\cdots\!64 \nu^{19} + \cdots - 31\!\cdots\!00 \nu ) / 50\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!78 \nu^{19} + \cdots + 32\!\cdots\!25 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22\!\cdots\!74 \nu^{19} + \cdots + 31\!\cdots\!75 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 61\!\cdots\!54 \nu^{19} + \cdots - 73\!\cdots\!25 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 45\!\cdots\!26 \nu^{19} + \cdots - 13\!\cdots\!25 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 29\!\cdots\!96 \nu^{19} + \cdots + 76\!\cdots\!75 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!26 \nu^{19} + \cdots + 42\!\cdots\!75 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!64 \nu^{19} + \cdots - 13\!\cdots\!50 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!62 \nu^{19} + \cdots - 74\!\cdots\!75 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 44\!\cdots\!06 \nu^{19} + \cdots + 34\!\cdots\!25 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 53\!\cdots\!98 \nu^{19} + \cdots - 18\!\cdots\!25 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 35\!\cdots\!18 \nu^{19} + \cdots - 86\!\cdots\!50 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 15\!\cdots\!18 \nu^{19} + \cdots - 22\!\cdots\!75 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 51\!\cdots\!88 \nu^{19} + \cdots - 13\!\cdots\!50 ) / 47\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 158 \beta_{19} - 32 \beta_{18} - 190 \beta_{17} + 222 \beta_{16} + 148 \beta_{15} + 18 \beta_{14} + \cdots + 290 ) / 94464 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 22 \beta_{19} - 536 \beta_{18} - 22 \beta_{17} - 558 \beta_{16} + 487 \beta_{15} - 66 \beta_{14} + \cdots - 2886677 ) / 94464 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 12588 \beta_{19} - 60 \beta_{18} + 7608 \beta_{17} - 7548 \beta_{16} - 7574 \beta_{15} + \cdots - 52090 ) / 94464 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11516 \beta_{19} + 67004 \beta_{18} - 11516 \beta_{17} + 55488 \beta_{16} - 93969 \beta_{15} + \cdots + 169463777 ) / 94464 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1157992 \beta_{19} + 142616 \beta_{18} - 333464 \beta_{17} + 190848 \beta_{16} + 555190 \beta_{15} + \cdots + 4975110 ) / 94464 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9063 \beta_{19} - 40098 \beta_{18} + 9063 \beta_{17} - 31035 \beta_{16} + 72948 \beta_{15} + \cdots - 72004531 ) / 576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 109740394 \beta_{19} - 18581960 \beta_{18} + 15180842 \beta_{17} + 3401118 \beta_{16} + \cdots - 455565084 ) / 94464 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 150155950 \beta_{19} + 619978528 \beta_{18} - 150155950 \beta_{17} + 469822578 \beta_{16} + \cdots + 916127584893 ) / 94464 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 10412254104 \beta_{19} + 1923147732 \beta_{18} - 677615484 \beta_{17} - 1245532248 \beta_{16} + \cdots + 41783882666 ) / 94464 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 14308089724 \beta_{19} - 57863278012 \beta_{18} + 14308089724 \beta_{17} - 43555188288 \beta_{16} + \cdots - 76164955966193 ) / 94464 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 982668597896 \beta_{19} - 185692209688 \beta_{18} + 26830889248 \beta_{17} + 158861320440 \beta_{16} + \cdots - 3851636553484 ) / 94464 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 4085546226 \beta_{19} + 16416458988 \beta_{18} - 4085546226 \beta_{17} + 12330912762 \beta_{16} + \cdots + 20156931540638 ) / 288 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 92227714270454 \beta_{19} + 17490873542560 \beta_{18} - 633412128262 \beta_{17} + \cdots + 356237369188494 ) / 94464 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 124793470842386 \beta_{19} - 500555902335512 \beta_{18} + 124793470842386 \beta_{17} + \cdots - 58\!\cdots\!29 ) / 94464 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 86\!\cdots\!96 \beta_{19} + \cdots - 33\!\cdots\!78 ) / 94464 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 11\!\cdots\!60 \beta_{19} + \cdots + 53\!\cdots\!49 ) / 94464 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 80\!\cdots\!16 \beta_{19} + \cdots + 30\!\cdots\!14 ) / 94464 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 65\!\cdots\!59 \beta_{19} + \cdots - 29\!\cdots\!47 ) / 576 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 74\!\cdots\!58 \beta_{19} + \cdots - 28\!\cdots\!48 ) / 94464 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
7.62048i
− 7.62048i
− 0.671758i
0.671758i
0.406991i
− 0.406991i
− 9.62962i
9.62962i
5.94193i
− 5.94193i
− 6.14819i
6.14819i
− 7.32004i
7.32004i
4.52328i
− 4.52328i
2.58939i
− 2.58939i
− 0.852576i
0.852576i
0 −15.4378 − 2.16198i 0 81.6092i 0 91.7503i 0 233.652 + 66.7525i 0
383.2 0 −15.4378 + 2.16198i 0 − 81.6092i 0 − 91.7503i 0 233.652 − 66.7525i 0
383.3 0 −15.1145 − 3.81453i 0 40.5648i 0 81.4905i 0 213.899 + 115.310i 0
383.4 0 −15.1145 + 3.81453i 0 − 40.5648i 0 − 81.4905i 0 213.899 − 115.310i 0
383.5 0 −10.1204 − 11.8566i 0 − 19.3214i 0 − 82.7816i 0 −38.1561 + 239.986i 0
383.6 0 −10.1204 + 11.8566i 0 19.3214i 0 82.7816i 0 −38.1561 − 239.986i 0
383.7 0 −5.16061 − 14.7095i 0 48.6693i 0 − 167.754i 0 −189.736 + 151.820i 0
383.8 0 −5.16061 + 14.7095i 0 − 48.6693i 0 167.754i 0 −189.736 − 151.820i 0
383.9 0 −2.25525 − 15.4245i 0 − 86.6842i 0 110.468i 0 −232.828 + 69.5720i 0
383.10 0 −2.25525 + 15.4245i 0 86.6842i 0 − 110.468i 0 −232.828 − 69.5720i 0
383.11 0 −1.67629 − 15.4981i 0 6.76651i 0 132.568i 0 −237.380 + 51.9584i 0
383.12 0 −1.67629 + 15.4981i 0 − 6.76651i 0 − 132.568i 0 −237.380 − 51.9584i 0
383.13 0 10.3565 − 11.6509i 0 68.3281i 0 − 220.652i 0 −28.4850 − 241.325i 0
383.14 0 10.3565 + 11.6509i 0 − 68.3281i 0 220.652i 0 −28.4850 + 241.325i 0
383.15 0 10.5848 − 11.4438i 0 − 101.915i 0 − 13.9460i 0 −18.9231 − 242.262i 0
383.16 0 10.5848 + 11.4438i 0 101.915i 0 13.9460i 0 −18.9231 + 242.262i 0
383.17 0 12.3458 − 9.51744i 0 26.7752i 0 70.6872i 0 61.8366 − 235.001i 0
383.18 0 12.3458 + 9.51744i 0 − 26.7752i 0 − 70.6872i 0 61.8366 + 235.001i 0
383.19 0 15.4777 − 1.85457i 0 55.8580i 0 225.953i 0 236.121 − 57.4091i 0
383.20 0 15.4777 + 1.85457i 0 − 55.8580i 0 − 225.953i 0 236.121 + 57.4091i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 383.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.6.c.a ✓ 20
3.b odd 2 1 384.6.c.d yes 20
4.b odd 2 1 384.6.c.d yes 20
8.b even 2 1 384.6.c.c yes 20
8.d odd 2 1 384.6.c.b yes 20
12.b even 2 1 inner 384.6.c.a ✓ 20
24.f even 2 1 384.6.c.c yes 20
24.h odd 2 1 384.6.c.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.6.c.a ✓ 20 1.a even 1 1 trivial
384.6.c.a ✓ 20 12.b even 2 1 inner
384.6.c.b yes 20 8.d odd 2 1
384.6.c.b yes 20 24.h odd 2 1
384.6.c.c yes 20 8.b even 2 1
384.6.c.c yes 20 24.f even 2 1
384.6.c.d yes 20 3.b odd 2 1
384.6.c.d yes 20 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(384, [\chi])\):

\( T_{11}^{10} + 474 T_{11}^{9} - 742804 T_{11}^{8} - 368139504 T_{11}^{7} + 129628916560 T_{11}^{6} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
\( T_{13}^{10} - 1985572 T_{13}^{8} - 389121024 T_{13}^{7} + 1179051645856 T_{13}^{6} + \cdots + 30\!\cdots\!80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 36\!\cdots\!16 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 30\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 45\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 90\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 30\!\cdots\!20)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 30\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 17\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 16\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 91\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
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