Properties

Label 2736.2.bm.l
Level $2736$
Weight $2$
Character orbit 2736.bm
Analytic conductor $21.847$
Analytic rank $0$
Dimension $6$
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] = [2736,2,Mod(559,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.bm (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.31726512.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 10x^{4} + 3x^{3} + 84x^{2} - 27x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 304)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1 - 1) q^{5} + (\beta_{5} - \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1 - 1) q^{5} + (\beta_{5} - \beta_{4}) q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 1) q^{11} + (2 \beta_{3} + 4) q^{13} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_1 + 1) q^{17} + ( - \beta_{5} + \beta_{4} + \beta_1 - 3) q^{19} + \beta_{4} q^{23} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{25} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{29} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{31} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{35} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{37} + ( - 3 \beta_{5} - 2 \beta_{2} + \beta_1) q^{41} + (2 \beta_{3} - 2) q^{43} + ( - \beta_{4} - 2 \beta_{3} - 4) q^{47} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{49} + (4 \beta_{3} + 8) q^{53} + (\beta_{5} - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{55} + (\beta_{5} - 2 \beta_{4} + 9 \beta_{3} + 9) q^{59} + (3 \beta_{3} - \beta_{2} + \beta_1) q^{61} + ( - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{65} + ( - 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{67} + (6 \beta_{3} + 6) q^{71} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{73} + ( - \beta_{2} + 7) q^{77} + (6 \beta_{3} - 2 \beta_1 + 6) q^{79} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 2 \beta_1) q^{83} + (4 \beta_{5} - 2 \beta_{4} - 10 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{85} + (3 \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 + 6) q^{89} + (4 \beta_{5} - 2 \beta_{4}) q^{91} + ( - \beta_{5} + \beta_{4} + 8 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{95} + ( - 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} + 18 q^{13} + 2 q^{17} - 17 q^{19} - 5 q^{25} + 12 q^{29} - 4 q^{31} - 6 q^{35} - 3 q^{41} - 18 q^{43} - 18 q^{47} + 2 q^{49} + 36 q^{53} + 12 q^{55} + 27 q^{59} - 10 q^{61} + 11 q^{67} + 18 q^{71} - 5 q^{73} + 40 q^{77} + 16 q^{79} + 26 q^{85} + 24 q^{89} - 6 q^{95} + 21 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 10x^{4} + 3x^{3} + 84x^{2} - 27x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 10\nu^{4} - 100\nu^{3} + 84\nu^{2} - 27\nu + 270 ) / 813 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\nu^{5} - 29\nu^{4} + 290\nu^{3} + 190\nu^{2} + 2436\nu - 783 ) / 813 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -161\nu^{5} - 16\nu^{4} - 1466\nu^{3} - 1923\nu^{2} - 14916\nu - 5310 ) / 2439 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 191\nu^{5} - 284\nu^{4} + 2027\nu^{3} - 597\nu^{2} + 15726\nu - 10107 ) / 2439 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} + \beta_{4} + 6\beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} - \beta_{4} - 10\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{5} - 20\beta_{4} - 57\beta_{3} - 16\beta _1 - 57 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32\beta_{5} - 16\beta_{4} - 66\beta_{3} + 103\beta_{2} - 103\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−1.35887 + 2.35363i
0.162698 0.281802i
1.69617 2.93786i
−1.35887 2.35363i
0.162698 + 0.281802i
1.69617 + 2.93786i
0 0 0 −1.85887 + 3.21966i 0 2.36936i 0 0 0
559.2 0 0 0 −0.337302 + 0.584224i 0 3.59084i 0 0 0
559.3 0 0 0 1.19617 2.07183i 0 1.22147i 0 0 0
1855.1 0 0 0 −1.85887 3.21966i 0 2.36936i 0 0 0
1855.2 0 0 0 −0.337302 0.584224i 0 3.59084i 0 0 0
1855.3 0 0 0 1.19617 + 2.07183i 0 1.22147i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.bm.l 6
3.b odd 2 1 304.2.n.d 6
4.b odd 2 1 2736.2.bm.m 6
12.b even 2 1 304.2.n.e yes 6
19.d odd 6 1 2736.2.bm.m 6
24.f even 2 1 1216.2.n.d 6
24.h odd 2 1 1216.2.n.e 6
57.f even 6 1 304.2.n.e yes 6
76.f even 6 1 inner 2736.2.bm.l 6
228.n odd 6 1 304.2.n.d 6
456.s odd 6 1 1216.2.n.e 6
456.v even 6 1 1216.2.n.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.d 6 3.b odd 2 1
304.2.n.d 6 228.n odd 6 1
304.2.n.e yes 6 12.b even 2 1
304.2.n.e yes 6 57.f even 6 1
1216.2.n.d 6 24.f even 2 1
1216.2.n.d 6 456.v even 6 1
1216.2.n.e 6 24.h odd 2 1
1216.2.n.e 6 456.s odd 6 1
2736.2.bm.l 6 1.a even 1 1 trivial
2736.2.bm.l 6 76.f even 6 1 inner
2736.2.bm.m 6 4.b odd 2 1
2736.2.bm.m 6 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{6} + 2T_{5}^{5} + 12T_{5}^{4} - 4T_{5}^{3} + 76T_{5}^{2} + 48T_{5} + 36 \) Copy content Toggle raw display
\( T_{7}^{6} + 20T_{7}^{4} + 100T_{7}^{2} + 108 \) Copy content Toggle raw display
\( T_{11}^{6} + 29T_{11}^{4} + 235T_{11}^{2} + 507 \) Copy content Toggle raw display
\( T_{23}^{6} - 10T_{23}^{4} + 100T_{23}^{2} - 180T_{23} + 108 \) Copy content Toggle raw display
\( T_{31}^{3} + 2T_{31}^{2} - 54T_{31} + 66 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + 12 T^{4} - 4 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{6} + 20 T^{4} + 100 T^{2} + \cdots + 108 \) Copy content Toggle raw display
$11$ \( T^{6} + 29 T^{4} + 235 T^{2} + \cdots + 507 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 12)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} - 2 T^{5} + 48 T^{4} + 112 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( T^{6} + 17 T^{5} + 144 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 10 T^{4} + 100 T^{2} + \cdots + 108 \) Copy content Toggle raw display
$29$ \( T^{6} - 12 T^{5} + 36 T^{4} + \cdots + 2700 \) Copy content Toggle raw display
$31$ \( (T^{3} + 2 T^{2} - 54 T + 66)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 72 T^{4} + 864 T^{2} + \cdots + 2700 \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} - 96 T^{4} + \cdots + 54675 \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 12)^{3} \) Copy content Toggle raw display
$47$ \( T^{6} + 18 T^{5} + 134 T^{4} + \cdots + 12 \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 48)^{3} \) Copy content Toggle raw display
$59$ \( T^{6} - 27 T^{5} + 516 T^{4} + \cdots + 263169 \) Copy content Toggle raw display
$61$ \( T^{6} + 10 T^{5} + 76 T^{4} + 228 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$67$ \( T^{6} - 11 T^{5} + 160 T^{4} + \cdots + 245025 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T + 36)^{3} \) Copy content Toggle raw display
$73$ \( T^{6} + 5 T^{5} + 50 T^{4} - 91 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$79$ \( T^{6} - 16 T^{5} + 208 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$83$ \( T^{6} + 113 T^{4} + 3031 T^{2} + \cdots + 3 \) Copy content Toggle raw display
$89$ \( T^{6} - 24 T^{5} + 120 T^{4} + \cdots + 97200 \) Copy content Toggle raw display
$97$ \( T^{6} - 21 T^{5} + 84 T^{4} + \cdots + 151875 \) Copy content Toggle raw display
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