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Properties

Label 1386.2.e.b
Level $1386$
Weight $2$
Character orbit 1386.e
Analytic conductor $11.067$
Analytic rank $0$
Dimension $8$
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(307,1386)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.307");
 
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.e (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: \(8\)
Coefficient field: 8.0.12745506816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 23x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 154)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{7} + \beta_{3} - \beta_1) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{7} + \beta_{3} - \beta_1) q^{7} + \beta_1 q^{8} + \beta_{3} q^{10} + ( - \beta_{5} - \beta_1 - 1) q^{11} - \beta_{7} q^{13} + (\beta_{6} + \beta_{2} - 1) q^{14} + q^{16} + ( - \beta_{7} + \beta_{3}) q^{17} + ( - 2 \beta_{7} - \beta_{3}) q^{19} + \beta_{2} q^{20} - \beta_{4} q^{22} - 4 q^{23} + (\beta_{5} - \beta_{4} + 1) q^{25} - \beta_{6} q^{26} + ( - \beta_{7} - \beta_{3} + \beta_1) q^{28} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{29} + ( - \beta_{6} - 3 \beta_{2}) q^{31} - \beta_1 q^{32} + ( - \beta_{6} + \beta_{2}) q^{34} + (\beta_{5} + \beta_{4} + \cdots + 4 \beta_1) q^{35}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{3} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 4 q^{11} - 8 q^{14} + 8 q^{16} - 4 q^{22} - 32 q^{23} - 8 q^{37} + 4 q^{44} + 40 q^{49} + 56 q^{53} + 8 q^{56} - 24 q^{58} - 8 q^{64} - 24 q^{67} + 24 q^{70} - 16 q^{71} - 4 q^{77} - 16 q^{86} + 4 q^{88} - 24 q^{91} + 32 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 23x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 24\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 24\nu^{3} + 5\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 24\nu^{3} - 5\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} - \nu^{4} - 67\nu^{2} - 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} + \nu^{4} - 67\nu^{2} + 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} - \nu^{5} - 115\nu^{3} - 24\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + \nu^{5} - 115\nu^{3} + 24\nu ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} + 6\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} + 5\beta_{3} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{5} - 5\beta_{4} - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 5\beta_{6} + 24\beta_{3} - 24\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{5} - 12\beta_{4} - 67\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -24\beta_{7} - 24\beta_{6} - 115\beta_{3} - 115\beta_{2} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.54779 + 1.54779i
0.323042 + 0.323042i
−0.323042 − 0.323042i
−1.54779 − 1.54779i
−1.54779 + 1.54779i
−0.323042 + 0.323042i
0.323042 − 0.323042i
1.54779 − 1.54779i
− 1.00000i 0 −1.00000 − 3.09557i 0 −2.44949 − 1.00000i 1.00000i 0 −3.09557
307.2 − 1.00000i 0 −1.00000 − 0.646084i 0 2.44949 − 1.00000i 1.00000i 0 −0.646084
307.3 − 1.00000i 0 −1.00000 0.646084i 0 −2.44949 − 1.00000i 1.00000i 0 0.646084
307.4 − 1.00000i 0 −1.00000 3.09557i 0 2.44949 − 1.00000i 1.00000i 0 3.09557
307.5 1.00000i 0 −1.00000 − 3.09557i 0 2.44949 + 1.00000i − 1.00000i 0 3.09557
307.6 1.00000i 0 −1.00000 − 0.646084i 0 −2.44949 + 1.00000i − 1.00000i 0 0.646084
307.7 1.00000i 0 −1.00000 0.646084i 0 2.44949 + 1.00000i − 1.00000i 0 −0.646084
307.8 1.00000i 0 −1.00000 3.09557i 0 −2.44949 + 1.00000i − 1.00000i 0 −3.09557
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b 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.e.b 8
3.b odd 2 1 154.2.c.a ✓ 8
7.b odd 2 1 inner 1386.2.e.b 8
11.b odd 2 1 inner 1386.2.e.b 8
12.b even 2 1 1232.2.e.e 8
21.c even 2 1 154.2.c.a ✓ 8
21.g even 6 2 1078.2.i.b 16
21.h odd 6 2 1078.2.i.b 16
33.d even 2 1 154.2.c.a ✓ 8
77.b even 2 1 inner 1386.2.e.b 8
84.h odd 2 1 1232.2.e.e 8
132.d odd 2 1 1232.2.e.e 8
231.h odd 2 1 154.2.c.a ✓ 8
231.k odd 6 2 1078.2.i.b 16
231.l even 6 2 1078.2.i.b 16
924.n even 2 1 1232.2.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.c.a ✓ 8 3.b odd 2 1
154.2.c.a ✓ 8 21.c even 2 1
154.2.c.a ✓ 8 33.d even 2 1
154.2.c.a ✓ 8 231.h odd 2 1
1078.2.i.b 16 21.g even 6 2
1078.2.i.b 16 21.h odd 6 2
1078.2.i.b 16 231.k odd 6 2
1078.2.i.b 16 231.l even 6 2
1232.2.e.e 8 12.b even 2 1
1232.2.e.e 8 84.h odd 2 1
1232.2.e.e 8 132.d odd 2 1
1232.2.e.e 8 924.n even 2 1
1386.2.e.b 8 1.a even 1 1 trivial
1386.2.e.b 8 7.b odd 2 1 inner
1386.2.e.b 8 11.b odd 2 1 inner
1386.2.e.b 8 77.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{4} + 10T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{4} - 10T_{13}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 10 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 10 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 14)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 34 T^{2} + 100)^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 60 T^{2} + 144)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 76 T^{2} + 100)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 20)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 124 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 176 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 124 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 14 T + 28)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 10 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 90 T^{2} + 324)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T - 12)^{4} \) Copy content Toggle raw display
$71$ \( (T + 2)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - 300 T^{2} + 10404)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 250 T^{2} + 13924)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 96)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 328 T^{2} + 18496)^{2} \) Copy content Toggle raw display
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