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Properties

Label 2736.2.bm.s
Level $2736$
Weight $2$
Character orbit 2736.bm
Analytic conductor $21.847$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 30x^{5} - 5x^{4} + 114x^{3} + 300x^{2} + 116x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 912)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{5} + (2 \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{5} + (2 \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{7} + (\beta_{6} - \beta_{5} + 1) q^{11} + (\beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{13} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \cdots - 1) q^{17}+ \cdots + ( - 5 \beta_{6} - \beta_{4} - \beta_1 + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} - 4 q^{17} + 6 q^{23} - 12 q^{25} + 12 q^{29} + 28 q^{31} + 18 q^{35} + 12 q^{41} + 18 q^{43} + 12 q^{47} - 24 q^{49} + 6 q^{53} - 12 q^{55} + 10 q^{59} - 4 q^{61} - 6 q^{67} - 8 q^{71} - 8 q^{73} - 28 q^{77} - 14 q^{79} - 8 q^{85} - 54 q^{89} - 26 q^{91} + 38 q^{95} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 30x^{5} - 5x^{4} + 114x^{3} + 300x^{2} + 116x + 19 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 44\nu^{7} - 2059\nu^{6} + 6647\nu^{5} - 15007\nu^{4} + 50938\nu^{3} + 80360\nu^{2} - 278981\nu + 154035 ) / 307713 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 88 \nu^{7} + 4118 \nu^{6} - 13294 \nu^{5} + 30014 \nu^{4} - 101876 \nu^{3} + 146993 \nu^{2} + \cdots - 308070 ) / 307713 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3500 \nu^{7} - 15922 \nu^{6} + 21212 \nu^{5} - 114745 \nu^{4} + 251431 \nu^{3} + 373886 \nu^{2} + \cdots - 1214655 ) / 307713 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3772 \nu^{7} + 8669 \nu^{6} - 10351 \nu^{5} + 111605 \nu^{4} - 2846 \nu^{3} - 315175 \nu^{2} + \cdots - 420924 ) / 307713 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5106 \nu^{7} - 10484 \nu^{6} - 7253 \nu^{5} - 142319 \nu^{4} - 28670 \nu^{3} + 830669 \nu^{2} + \cdots + 414680 ) / 307713 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -1002\nu^{7} + 2264\nu^{6} - 178\nu^{5} + 29375\nu^{4} + 254\nu^{3} - 124940\nu^{2} - 273328\nu - 78329 ) / 43959 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32081 \nu^{7} + 65584 \nu^{6} - 5933 \nu^{5} + 958120 \nu^{4} + 99677 \nu^{3} - 3586874 \nu^{2} + \cdots - 2690259 ) / 307713 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} - 2\beta_{5} + \beta_{4} + 4\beta_{2} + 7\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 35\beta_{6} + 8\beta_{5} + 5\beta_{4} + 9\beta_{3} + 5\beta_{2} - \beta _1 + 47 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 15\beta_{6} - 12\beta_{5} - 2\beta_{4} + 4\beta_{3} + 14\beta_{2} - 2\beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -133\beta_{6} - 187\beta_{5} - 7\beta_{4} + 146\beta_{2} + 191\beta _1 + 56 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -198\beta_{7} + 992\beta_{6} + 47\beta_{5} + 38\beta_{4} + 150\beta_{3} + 335\beta_{2} + 311\beta _1 + 797 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -273\beta_{7} + 1618\beta_{6} - 119\beta_{5} - 485\beta_{4} + 693\beta_{3} + 1192\beta_{2} - 653\beta _1 + 4249 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(-1\) \(1\)

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
−0.213988 + 0.172868i
−0.654220 + 2.95767i
−1.27736 − 1.04884i
3.14556 − 0.349646i
−0.213988 − 0.172868i
−0.654220 − 2.95767i
−1.27736 + 1.04884i
3.14556 + 0.349646i
0 0 0 −2.00488 + 3.47255i 0 − 1.92982i 0 0 0
559.2 0 0 0 0.353597 − 0.612447i 0 0.0924751i 0 0 0
559.3 0 0 0 0.912850 − 1.58110i 0 − 4.99333i 0 0 0
559.4 0 0 0 1.73843 − 3.01105i 0 3.36658i 0 0 0
1855.1 0 0 0 −2.00488 − 3.47255i 0 1.92982i 0 0 0
1855.2 0 0 0 0.353597 + 0.612447i 0 − 0.0924751i 0 0 0
1855.3 0 0 0 0.912850 + 1.58110i 0 4.99333i 0 0 0
1855.4 0 0 0 1.73843 + 3.01105i 0 − 3.36658i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.4
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.s 8
3.b odd 2 1 912.2.bb.g ✓ 8
4.b odd 2 1 2736.2.bm.r 8
12.b even 2 1 912.2.bb.h yes 8
19.d odd 6 1 2736.2.bm.r 8
57.f even 6 1 912.2.bb.h yes 8
76.f even 6 1 inner 2736.2.bm.s 8
228.n odd 6 1 912.2.bb.g ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bb.g ✓ 8 3.b odd 2 1
912.2.bb.g ✓ 8 228.n odd 6 1
912.2.bb.h yes 8 12.b even 2 1
912.2.bb.h yes 8 57.f even 6 1
2736.2.bm.r 8 4.b odd 2 1
2736.2.bm.r 8 19.d odd 6 1
2736.2.bm.s 8 1.a even 1 1 trivial
2736.2.bm.s 8 76.f even 6 1 inner

Hecke kernels

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

\( T_{5}^{8} - 2T_{5}^{7} + 18T_{5}^{6} - 44T_{5}^{5} + 286T_{5}^{4} - 576T_{5}^{3} + 1044T_{5}^{2} - 648T_{5} + 324 \) Copy content Toggle raw display
\( T_{7}^{8} + 40T_{7}^{6} + 418T_{7}^{4} + 1056T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{8} + 40T_{11}^{6} + 520T_{11}^{4} + 2376T_{11}^{2} + 2916 \) Copy content Toggle raw display
\( T_{23}^{8} - 6T_{23}^{7} - 26T_{23}^{6} + 228T_{23}^{5} + 1378T_{23}^{4} + 912T_{23}^{3} - 4140T_{23}^{2} - 2736T_{23} + 12996 \) Copy content Toggle raw display
\( T_{31}^{4} - 14T_{31}^{3} + 10T_{31}^{2} + 282T_{31} - 45 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 324 \) Copy content Toggle raw display
$7$ \( T^{8} + 40 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{8} + 40 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$13$ \( T^{8} - 26 T^{6} + \cdots + 225 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + \cdots + 129600 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 12996 \) Copy content Toggle raw display
$29$ \( T^{8} - 12 T^{7} + \cdots + 746496 \) Copy content Toggle raw display
$31$ \( (T^{4} - 14 T^{3} + \cdots - 45)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 148 T^{6} + \cdots + 81225 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + \cdots + 129600 \) Copy content Toggle raw display
$43$ \( T^{8} - 18 T^{7} + \cdots + 1946025 \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( T^{8} - 6 T^{7} + \cdots + 36 \) Copy content Toggle raw display
$59$ \( T^{8} - 10 T^{7} + \cdots + 14152644 \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{7} + \cdots + 1590121 \) Copy content Toggle raw display
$67$ \( T^{8} + 6 T^{7} + \cdots + 693889 \) Copy content Toggle raw display
$71$ \( T^{8} + 8 T^{7} + \cdots + 627264 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} + \cdots + 393129 \) Copy content Toggle raw display
$79$ \( T^{8} + 14 T^{7} + \cdots + 5331481 \) Copy content Toggle raw display
$83$ \( T^{8} + 504 T^{6} + \cdots + 13483584 \) Copy content Toggle raw display
$89$ \( T^{8} + 54 T^{7} + \cdots + 61496964 \) Copy content Toggle raw display
$97$ \( T^{8} - 60 T^{7} + \cdots + 10890000 \) Copy content Toggle raw display
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