↑

Properties

Label 1386.2.bk.b
Level $1386$
Weight $2$
Character orbit 1386.bk
Analytic conductor $11.067$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(703,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.bk (of order \(6\), degree \(2\), 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: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + \cdots + 13417 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + (\beta_{13} + 1) q^{4} + ( - \beta_{13} - \beta_{9} - 1) q^{5} + ( - \beta_{15} + \beta_{13} + \beta_{11} + \cdots + 1) q^{7}+ \cdots + (\beta_{12} + \beta_{11}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{2} + (\beta_{13} + 1) q^{4} + ( - \beta_{13} - \beta_{9} - 1) q^{5} + ( - \beta_{15} + \beta_{13} + \beta_{11} + \cdots + 1) q^{7}+ \cdots + (2 \beta_{15} - \beta_{14} + \beta_{12} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 12 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 12 q^{5} + 6 q^{7} - 2 q^{10} + 4 q^{11} - 8 q^{14} - 8 q^{16} + 10 q^{19} + 2 q^{22} + 4 q^{23} + 10 q^{25} - 12 q^{26} + 6 q^{31} - 8 q^{35} + 14 q^{37} + 12 q^{38} + 2 q^{40} + 32 q^{41} - 4 q^{44} - 18 q^{46} + 24 q^{47} - 6 q^{49} + 14 q^{55} - 4 q^{56} - 28 q^{61} - 8 q^{62} - 16 q^{64} + 72 q^{65} - 16 q^{67} - 30 q^{70} + 56 q^{71} + 44 q^{73} + 24 q^{74} + 20 q^{76} + 52 q^{77} + 30 q^{79} + 12 q^{80} - 12 q^{82} + 8 q^{83} + 12 q^{86} - 2 q^{88} + 36 q^{89} - 8 q^{91} + 8 q^{92} - 14 q^{94} + 72 q^{95} - 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + \cdots + 13417 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3325577 \nu^{14} + 23279039 \nu^{13} - 218220364 \nu^{12} + 1006694677 \nu^{11} + \cdots - 13304183911 ) / 3261539424 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14496748 \nu^{15} + 165765669 \nu^{14} - 1387066295 \nu^{13} + 16433893584 \nu^{12} + \cdots + 8734585660679 ) / 316369324128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14496748 \nu^{15} - 383216889 \nu^{14} + 2455811611 \nu^{13} - 20082727968 \nu^{12} + \cdots + 1933997835005 ) / 316369324128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28661011 \nu^{15} + 376248067 \nu^{14} - 3019246022 \nu^{13} + 19609879751 \nu^{12} + \cdots + 1766457326490 ) / 316369324128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30920939 \nu^{15} - 47258327 \nu^{14} - 211956648 \nu^{13} - 7396232221 \nu^{12} + \cdots - 2047012171496 ) / 158184662064 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30920939 \nu^{15} + 511072412 \nu^{14} - 4120271821 \nu^{13} + 28521203647 \nu^{12} + \cdots + 1740262265133 ) / 158184662064 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7915220 \nu^{15} - 40953259 \nu^{14} + 414028833 \nu^{13} - 1460644352 \nu^{12} + \cdots - 132462757465 ) / 28760847648 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7979 \nu^{15} + 68912 \nu^{14} - 628546 \nu^{13} + 3357412 \nu^{12} - 16625487 \nu^{11} + \cdots + 285645792 ) / 18627492 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7979 \nu^{15} + 68912 \nu^{14} - 628546 \nu^{13} + 3357412 \nu^{12} - 16625487 \nu^{11} + \cdots + 304273284 ) / 18627492 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6671 \nu^{15} + 41254 \nu^{14} - 402799 \nu^{13} + 1707247 \nu^{12} - 8612564 \nu^{11} + \cdots + 82787651 ) / 7787936 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24636003 \nu^{15} - 180367241 \nu^{14} + 1683650240 \nu^{13} - 8042787727 \nu^{12} + \cdots - 463004009716 ) / 28760847648 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24636003 \nu^{15} - 189172804 \nu^{14} + 1745289181 \nu^{13} - 8640627827 \nu^{12} + \cdots - 503813289929 ) / 28760847648 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15958 \nu^{15} - 119685 \nu^{14} + 1130119 \nu^{13} - 5530551 \nu^{12} + 27795985 \nu^{11} + \cdots - 518381770 ) / 18627492 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 731076296 \nu^{15} + 5564679969 \nu^{14} - 51759660791 \nu^{13} + 255254450652 \nu^{12} + \cdots + 18336335470607 ) / 316369324128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 791321153 \nu^{15} - 5699211694 \nu^{14} + 53761281981 \nu^{13} - 254911441289 \nu^{12} + \cdots - 16525376998129 ) / 316369324128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{9} + \beta_{8} + 1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{13} - 2\beta_{11} - \beta_{10} - 2\beta_{9} + \beta_{7} - 2\beta_{6} - 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{15} + \beta_{14} - 5 \beta_{13} - \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + 9 \beta_{9} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 8 \beta_{15} - 10 \beta_{14} + 2 \beta_{13} + 5 \beta_{12} + 17 \beta_{11} + 10 \beta_{10} + \cdots + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 28 \beta_{15} - 22 \beta_{14} + 64 \beta_{13} + 24 \beta_{12} + 23 \beta_{11} + 58 \beta_{10} + \cdots + 162 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 52 \beta_{15} + 75 \beta_{14} + 61 \beta_{13} - 74 \beta_{12} - 170 \beta_{11} - 65 \beta_{10} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 325 \beta_{15} + 298 \beta_{14} - 657 \beta_{13} - 450 \beta_{12} - 323 \beta_{11} - 852 \beta_{10} + \cdots - 1701 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 228 \beta_{15} - 448 \beta_{14} - 1368 \beta_{13} + 585 \beta_{12} + 1711 \beta_{11} - 150 \beta_{10} + \cdots - 1786 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3506 \beta_{15} - 3490 \beta_{14} + 5934 \beta_{13} + 6780 \beta_{12} + 4769 \beta_{11} + \cdots + 16429 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 775 \beta_{15} + 1126 \beta_{14} + 20300 \beta_{13} - 160 \beta_{12} - 15824 \beta_{11} + \cdots + 36293 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 35787 \beta_{15} + 37693 \beta_{14} - 47572 \beta_{13} - 87239 \beta_{12} - 66603 \beta_{11} + \cdots - 143099 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 39984 \beta_{15} + 26340 \beta_{14} - 258878 \beta_{13} - 94424 \beta_{12} + 124230 \beta_{11} + \cdots - 542199 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 342678 \beta_{15} - 375858 \beta_{14} + 321008 \beta_{13} + 984284 \beta_{12} + 862898 \beta_{11} + \cdots + 1048389 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 729089 \beta_{15} - 672801 \beta_{14} + 3058731 \beta_{13} + 2237884 \beta_{12} - 659178 \beta_{11} + \cdots + 7051750 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3012048 \beta_{15} + 3370541 \beta_{14} - 1368219 \beta_{13} - 9668935 \beta_{12} - 10414656 \beta_{11} + \cdots - 4830300 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(1\) \(-\beta_{13}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
0.500000 + 2.40229i
0.500000 + 1.35798i
0.500000 − 1.56688i
0.500000 − 3.19339i
0.500000 + 3.43554i
0.500000 + 0.921602i
0.500000 − 0.0286340i
0.500000 − 3.32851i
0.500000 − 2.40229i
0.500000 − 1.35798i
0.500000 + 1.56688i
0.500000 + 3.19339i
0.500000 − 3.43554i
0.500000 − 0.921602i
0.500000 + 0.0286340i
0.500000 + 3.32851i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i −2.83045 + 1.63416i 0 2.54243 + 0.732142i 1.00000i 0 1.63416 − 2.83045i
703.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.92604 + 1.11200i 0 −2.45660 − 0.982398i 1.00000i 0 1.11200 − 1.92604i
703.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.606961 − 0.350429i 0 1.82993 − 1.91085i 1.00000i 0 −0.350429 + 0.606961i
703.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.01555 − 1.16368i 0 1.31629 + 2.29508i 1.00000i 0 −1.16368 + 2.01555i
703.5 0.866025 − 0.500000i 0 0.500000 − 0.866025i −3.72526 + 2.15078i 0 1.43173 + 2.22489i − 1.00000i 0 −2.15078 + 3.72526i
703.6 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.54813 + 0.893814i 0 −0.165362 − 2.64058i − 1.00000i 0 −0.893814 + 1.54813i
703.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.725202 + 0.418696i 0 −2.44037 + 1.02205i − 1.00000i 0 −0.418696 + 0.725202i
703.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i 2.13257 − 1.23124i 0 0.941950 − 2.47239i − 1.00000i 0 1.23124 − 2.13257i
901.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −2.83045 − 1.63416i 0 2.54243 − 0.732142i − 1.00000i 0 1.63416 + 2.83045i
901.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.92604 − 1.11200i 0 −2.45660 + 0.982398i − 1.00000i 0 1.11200 + 1.92604i
901.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.606961 + 0.350429i 0 1.82993 + 1.91085i − 1.00000i 0 −0.350429 − 0.606961i
901.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.01555 + 1.16368i 0 1.31629 − 2.29508i − 1.00000i 0 −1.16368 − 2.01555i
901.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −3.72526 − 2.15078i 0 1.43173 − 2.22489i 1.00000i 0 −2.15078 − 3.72526i
901.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.54813 − 0.893814i 0 −0.165362 + 2.64058i 1.00000i 0 −0.893814 − 1.54813i
901.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.725202 − 0.418696i 0 −2.44037 − 1.02205i 1.00000i 0 −0.418696 − 0.725202i
901.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.13257 + 1.23124i 0 0.941950 + 2.47239i 1.00000i 0 1.23124 + 2.13257i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.bk.b 16
3.b odd 2 1 462.2.p.b yes 16
7.d odd 6 1 1386.2.bk.a 16
11.b odd 2 1 1386.2.bk.a 16
21.g even 6 1 462.2.p.a ✓ 16
21.g even 6 1 3234.2.e.a 16
21.h odd 6 1 3234.2.e.b 16
33.d even 2 1 462.2.p.a ✓ 16
77.i even 6 1 inner 1386.2.bk.b 16
231.k odd 6 1 462.2.p.b yes 16
231.k odd 6 1 3234.2.e.b 16
231.l even 6 1 3234.2.e.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.p.a ✓ 16 21.g even 6 1
462.2.p.a ✓ 16 33.d even 2 1
462.2.p.b yes 16 3.b odd 2 1
462.2.p.b yes 16 231.k odd 6 1
1386.2.bk.a 16 7.d odd 6 1
1386.2.bk.a 16 11.b odd 2 1
1386.2.bk.b 16 1.a even 1 1 trivial
1386.2.bk.b 16 77.i even 6 1 inner
3234.2.e.a 16 21.g even 6 1
3234.2.e.a 16 231.l even 6 1
3234.2.e.b 16 21.h odd 6 1
3234.2.e.b 16 231.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{16} + 12 T_{5}^{15} + 47 T_{5}^{14} - 12 T_{5}^{13} - 484 T_{5}^{12} - 312 T_{5}^{11} + \cdots + 35344 \) Copy content Toggle raw display
\( T_{13}^{8} - 64T_{13}^{6} + 8T_{13}^{5} + 836T_{13}^{4} - 1168T_{13}^{3} - 592T_{13}^{2} + 1216T_{13} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 12 T^{15} + \cdots + 35344 \) Copy content Toggle raw display
$7$ \( T^{16} - 6 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} - 64 T^{6} + \cdots - 128)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 53 T^{14} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{16} - 10 T^{15} + \cdots + 2768896 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 24039882304 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 12810617856 \) Copy content Toggle raw display
$31$ \( T^{16} - 6 T^{15} + \cdots + 262144 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 125622042624 \) Copy content Toggle raw display
$41$ \( (T^{8} - 16 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 23084548096 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1344737098384 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 26488213504 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 109280491776 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 3489501184 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2403352576 \) Copy content Toggle raw display
$71$ \( (T^{8} - 28 T^{7} + \cdots + 4193232)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 119793516544 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 1130708969104 \) Copy content Toggle raw display
$83$ \( (T^{8} - 4 T^{7} + \cdots + 5604)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 96546188427264 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 68597371921 \) Copy content Toggle raw display
show more
show less