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Properties

Label 1386.2.g.a
Level $1386$
Weight $2$
Character orbit 1386.g
Analytic conductor $11.067$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(881,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.881");
 
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.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: \(11.0672657201\)
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: \( x^{16} - 4x^{14} + 8x^{12} + 80x^{10} + 1189x^{8} - 2028x^{6} + 1800x^{4} + 1080x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + \beta_{7} q^{7} - \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + \beta_{7} q^{7} - \beta_1 q^{8} + \beta_{3} q^{10} - \beta_1 q^{11} + \beta_{11} q^{14} + q^{16} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{15} - \beta_{13} + \cdots - \beta_{10}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 8 q^{7} + 16 q^{16} + 16 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{37} + 48 q^{43} - 16 q^{46} + 8 q^{49} - 16 q^{58} - 16 q^{64} + 16 q^{67} - 8 q^{70} - 16 q^{79} + 112 q^{85} - 16 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 8x^{12} + 80x^{10} + 1189x^{8} - 2028x^{6} + 1800x^{4} + 1080x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 96575 \nu^{14} + 438755 \nu^{12} - 939814 \nu^{10} - 7435564 \nu^{8} - 110537567 \nu^{6} + \cdots - 51358644 ) / 86730966 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3833 \nu^{14} + 11990 \nu^{12} - 16306 \nu^{10} - 332152 \nu^{8} - 4851221 \nu^{6} + \cdots - 7232976 ) / 1867860 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3079634 \nu^{15} + 11597979 \nu^{13} - 22413962 \nu^{11} - 248772494 \nu^{9} + \cdots - 5487054570 \nu ) / 2601928980 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12591283 \nu^{15} - 57579817 \nu^{13} + 129110888 \nu^{11} + 952830506 \nu^{9} + \cdots + 3202119162 \nu ) / 7805786940 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14545077 \nu^{15} - 3765719 \nu^{14} - 55196585 \nu^{13} + 11685650 \nu^{12} + \cdots - 9136826568 ) / 5203857960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11395 \nu^{14} + 45868 \nu^{12} - 96938 \nu^{10} - 893276 \nu^{8} - 13578607 \nu^{6} + \cdots - 6275124 ) / 2256660 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14545077 \nu^{15} + 3765719 \nu^{14} - 55196585 \nu^{13} - 11685650 \nu^{12} + \cdots + 3932968608 ) / 5203857960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18506049 \nu^{15} - 1441114 \nu^{14} - 72148880 \nu^{13} + 4446220 \nu^{12} + \cdots - 13615971048 ) / 5203857960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18506049 \nu^{15} + 1441114 \nu^{14} - 72148880 \nu^{13} - 4446220 \nu^{12} + \cdots + 18819829008 ) / 5203857960 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 60462680 \nu^{15} - 34694355 \nu^{14} - 274567721 \nu^{13} + 143608152 \nu^{12} + \cdots - 18889526916 ) / 15611573880 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60462680 \nu^{15} - 17310855 \nu^{14} + 274567721 \nu^{13} + 64632252 \nu^{12} + \cdots - 9644970996 ) / 15611573880 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 81296765 \nu^{15} + 47163030 \nu^{14} + 357230942 \nu^{13} - 210264972 \nu^{12} + \cdots + 25206770376 ) / 15611573880 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 81296765 \nu^{15} - 47163030 \nu^{14} + 357230942 \nu^{13} + 210264972 \nu^{12} + \cdots - 25206770376 ) / 15611573880 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 23298266 \nu^{15} - 93544963 \nu^{13} + 176425470 \nu^{11} + 1897812810 \nu^{9} + \cdots + 37162486386 \nu ) / 2601928980 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15573353 \nu^{15} + 65944679 \nu^{13} - 141046900 \nu^{11} - 1206672370 \nu^{9} + \cdots - 2860280478 \nu ) / 1115112420 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{13} - \beta_{12} - 3\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{14} + \beta_{11} - \beta_{10} + 2\beta_{9} + 2\beta_{8} - \beta_{7} - \beta_{5} + 5\beta_{4} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{13} + 7\beta_{12} + 2\beta_{11} + 2\beta_{10} + 32\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5 \beta_{14} + \beta_{13} + \beta_{12} - 7 \beta_{11} + 7 \beta_{10} + 13 \beta_{9} + 13 \beta_{8} + \cdots - 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 11 \beta_{13} + 11 \beta_{12} + 45 \beta_{11} + 45 \beta_{10} + 11 \beta_{9} - 11 \beta_{8} + \cdots - 68 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 31 \beta_{15} - 2 \beta_{14} + 87 \beta_{13} + 87 \beta_{12} - 67 \beta_{11} + 67 \beta_{10} + \cdots - 58 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 289\beta_{9} - 289\beta_{8} - 190\beta_{7} + 190\beta_{5} - 56\beta_{2} - 1541 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 205 \beta_{15} + 28 \beta_{14} + 589 \beta_{13} + 589 \beta_{12} - 479 \beta_{11} + 479 \beta_{10} + \cdots + 412 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 753 \beta_{13} - 753 \beta_{12} - 1877 \beta_{11} - 1877 \beta_{10} + 753 \beta_{9} - 753 \beta_{8} + \cdots - 4274 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 274 \beta_{15} + 1383 \beta_{14} + 1027 \beta_{13} + 1027 \beta_{12} - 1877 \beta_{11} + 1877 \beta_{10} + \cdots + 7396 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 12341\beta_{13} - 12341\beta_{12} - 11422\beta_{11} - 11422\beta_{10} + 4656\beta_{6} - 53908\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 2328 \beta_{15} + 9409 \beta_{14} - 8039 \beta_{13} - 8039 \beta_{12} + 12341 \beta_{11} - 12341 \beta_{10} + \cdots + 51224 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 42169 \beta_{13} - 42169 \beta_{12} - 82029 \beta_{11} - 82029 \beta_{10} - 42169 \beta_{9} + \cdots + 234634 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 64331 \beta_{15} + 18406 \beta_{14} - 188529 \beta_{13} - 188529 \beta_{12} + 166367 \beta_{11} + \cdots + 142604 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−0.876932 − 2.11710i
2.43348 − 1.00798i
−0.221383 − 0.534465i
1.12256 − 0.464978i
−1.12256 + 0.464978i
0.221383 + 0.534465i
−2.43348 + 1.00798i
0.876932 + 2.11710i
−0.876932 + 2.11710i
2.43348 + 1.00798i
−0.221383 + 0.534465i
1.12256 + 0.464978i
−1.12256 − 0.464978i
0.221383 − 0.534465i
−2.43348 − 1.00798i
0.876932 − 2.11710i
− 1.00000i 0 −1.00000 −4.23420 0 −2.59178 + 0.531652i 1.00000i 0 4.23420i
881.2 − 1.00000i 0 −1.00000 −2.01596 0 1.78450 + 1.95335i 1.00000i 0 2.01596i
881.3 − 1.00000i 0 −1.00000 −1.06893 0 0.884677 − 2.49346i 1.00000i 0 1.06893i
881.4 − 1.00000i 0 −1.00000 −0.929956 0 −2.07739 − 1.63843i 1.00000i 0 0.929956i
881.5 − 1.00000i 0 −1.00000 0.929956 0 −2.07739 + 1.63843i 1.00000i 0 − 0.929956i
881.6 − 1.00000i 0 −1.00000 1.06893 0 0.884677 + 2.49346i 1.00000i 0 − 1.06893i
881.7 − 1.00000i 0 −1.00000 2.01596 0 1.78450 − 1.95335i 1.00000i 0 − 2.01596i
881.8 − 1.00000i 0 −1.00000 4.23420 0 −2.59178 − 0.531652i 1.00000i 0 − 4.23420i
881.9 1.00000i 0 −1.00000 −4.23420 0 −2.59178 − 0.531652i − 1.00000i 0 − 4.23420i
881.10 1.00000i 0 −1.00000 −2.01596 0 1.78450 − 1.95335i − 1.00000i 0 − 2.01596i
881.11 1.00000i 0 −1.00000 −1.06893 0 0.884677 + 2.49346i − 1.00000i 0 − 1.06893i
881.12 1.00000i 0 −1.00000 −0.929956 0 −2.07739 + 1.63843i − 1.00000i 0 − 0.929956i
881.13 1.00000i 0 −1.00000 0.929956 0 −2.07739 − 1.63843i − 1.00000i 0 0.929956i
881.14 1.00000i 0 −1.00000 1.06893 0 0.884677 − 2.49346i − 1.00000i 0 1.06893i
881.15 1.00000i 0 −1.00000 2.01596 0 1.78450 + 1.95335i − 1.00000i 0 2.01596i
881.16 1.00000i 0 −1.00000 4.23420 0 −2.59178 + 0.531652i − 1.00000i 0 4.23420i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.g.a ✓ 16
3.b odd 2 1 inner 1386.2.g.a ✓ 16
7.b odd 2 1 inner 1386.2.g.a ✓ 16
21.c even 2 1 inner 1386.2.g.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.g.a ✓ 16 1.a even 1 1 trivial
1386.2.g.a ✓ 16 3.b odd 2 1 inner
1386.2.g.a ✓ 16 7.b odd 2 1 inner
1386.2.g.a ✓ 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24T_{5}^{6} + 118T_{5}^{4} - 168T_{5}^{2} + 72 \) acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 24 T^{6} + \cdots + 72)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T^{8} - 84 T^{6} + \cdots + 83232)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 104 T^{6} + \cdots + 72)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 120 T^{6} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 108 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 268 T^{6} + \cdots + 6139008)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots + 544)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 132 T^{6} + \cdots + 23328)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 12 T^{3} + \cdots - 16)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 196 T^{6} + \cdots + 288)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 260 T^{6} + \cdots + 8714304)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 96 T^{6} + \cdots + 18432)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 528 T^{6} + \cdots + 294912)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} - 70 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 312 T^{6} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 196 T^{6} + \cdots + 288)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots - 292)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 348 T^{6} + \cdots + 1492992)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 420 T^{6} + \cdots + 1936512)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 480 T^{6} + \cdots + 18874368)^{2} \) Copy content Toggle raw display
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