Properties

Label 2736.2.s.ba
Level $2736$
Weight $2$
Character orbit 2736.s
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(577,2736)]
 
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([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.s (of order \(3\), degree \(2\), not 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3}) q^{5} + ( - \beta_{6} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3}) q^{5} + ( - \beta_{6} + 1) q^{7} + ( - \beta_1 + 1) q^{11} + (\beta_{7} + \beta_{5} + \beta_{2} + \beta_1) q^{13} + 2 \beta_{4} q^{17} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{19} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{23} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1) q^{25} + ( - 2 \beta_{7} - 2 \beta_{2}) q^{29} + (\beta_{6} + \beta_{2} - \beta_1) q^{31} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{35} + (2 \beta_{6} + 2 \beta_{2} - 2 \beta_1 + 1) q^{37} + ( - 2 \beta_{7} + 2 \beta_{4}) q^{41} + ( - \beta_{7} + \beta_{5} - 3 \beta_{4}) q^{43} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 2) q^{47} + ( - 2 \beta_{6} + \beta_{2} + \beta_1 - 1) q^{49} + (2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + 3 \beta_1 - 5) q^{53} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} - 3 \beta_{3}) q^{55} + (2 \beta_{7} - \beta_{5} - 3 \beta_{3}) q^{59} + (\beta_{7} - 3 \beta_{5} - 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{61} + ( - 2 \beta_{6} - 2 \beta_{2} - \beta_1 - 5) q^{65} + (2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{67} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{71} + (4 \beta_{5} + \beta_{3}) q^{73} + (\beta_1 + 1) q^{77} + ( - \beta_{7} - 3 \beta_{5} + \beta_{4} - 6 \beta_{3}) q^{79} + ( - 2 \beta_{6} + 2 \beta_{2} + 2 \beta_1 + 2) q^{83} + ( - 4 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} - 4 \beta_1) q^{85} + ( - 2 \beta_{7} - 4 \beta_{6} + \beta_{5} - 4 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + \cdots + 5) q^{89}+ \cdots + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} - 7 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} + 4 q^{7} + 8 q^{11} - 4 q^{17} - 8 q^{19} + 8 q^{23} - 4 q^{25} + 4 q^{31} + 16 q^{37} - 4 q^{41} + 6 q^{43} + 4 q^{47} - 16 q^{49} - 16 q^{53} + 16 q^{55} + 12 q^{59} - 8 q^{61} - 48 q^{65} - 2 q^{67} - 4 q^{71} - 4 q^{73} + 8 q^{77} + 22 q^{79} + 8 q^{83} - 8 q^{85} + 12 q^{89} + 2 q^{91} + 32 q^{95} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 8x^{6} + 21x^{4} - 4x^{3} + 28x^{2} + 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -16\nu^{7} - 199\nu^{6} - 84\nu^{5} - 152\nu^{4} - 4652\nu^{3} - 272\nu^{2} - 132\nu + 3206 ) / 4243 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 152\nu^{7} - 231\nu^{6} + 798\nu^{5} + 1444\nu^{4} + 1764\nu^{3} + 2584\nu^{2} + 1254\nu + 11973 ) / 4243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 754\nu^{7} - 1760\nu^{6} + 6080\nu^{5} - 1323\nu^{4} + 13440\nu^{3} - 12640\nu^{2} + 16828\nu - 5760 ) / 12729 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -815\nu^{7} + 3388\nu^{6} - 11704\nu^{5} + 15594\nu^{4} - 25872\nu^{3} + 24332\nu^{2} - 56579\nu + 11088 ) / 12729 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1052\nu^{7} - 2827\nu^{6} + 9766\nu^{5} - 6978\nu^{4} + 21588\nu^{3} - 20303\nu^{2} + 4436\nu - 9252 ) / 12729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -556\nu^{7} + 510\nu^{6} - 2919\nu^{5} - 5282\nu^{4} - 8909\nu^{3} - 9452\nu^{2} - 4587\nu - 13760 ) / 4243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 602\nu^{7} - 1529\nu^{6} + 5282\nu^{5} - 2767\nu^{4} + 11676\nu^{3} - 10981\nu^{2} + 15574\nu - 5004 ) / 4243 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 3\beta_{3} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{6} + 7\beta_{2} - 3\beta _1 - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{7} + \beta_{5} - 2\beta_{4} + 18\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -53\beta_{7} - 16\beta_{6} + 13\beta_{5} - 16\beta_{4} + 89\beta_{3} - 53\beta_{2} + 13\beta _1 + 89 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{6} - 72\beta_{2} + 11\beta _1 + 130 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 407\beta_{7} - 79\beta_{5} + 122\beta_{4} - 699\beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
0.643668 + 1.11487i
−0.276205 0.478401i
1.39083 + 2.40898i
−0.758290 1.31340i
0.643668 1.11487i
−0.276205 + 0.478401i
1.39083 2.40898i
−0.758290 + 1.31340i
0 0 0 −1.95872 3.39260i 0 −2.04306 0 0 0
577.2 0 0 0 −0.795012 1.37700i 0 3.87834 0 0 0
577.3 0 0 0 −0.412855 0.715087i 0 0.703158 0 0 0
577.4 0 0 0 1.16659 + 2.02059i 0 −0.538445 0 0 0
1873.1 0 0 0 −1.95872 + 3.39260i 0 −2.04306 0 0 0
1873.2 0 0 0 −0.795012 + 1.37700i 0 3.87834 0 0 0
1873.3 0 0 0 −0.412855 + 0.715087i 0 0.703158 0 0 0
1873.4 0 0 0 1.16659 2.02059i 0 −0.538445 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.ba 8
3.b odd 2 1 2736.2.s.bc 8
4.b odd 2 1 1368.2.s.l 8
12.b even 2 1 1368.2.s.m yes 8
19.c even 3 1 inner 2736.2.s.ba 8
57.h odd 6 1 2736.2.s.bc 8
76.g odd 6 1 1368.2.s.l 8
228.m even 6 1 1368.2.s.m yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.s.l 8 4.b odd 2 1
1368.2.s.l 8 76.g odd 6 1
1368.2.s.m yes 8 12.b even 2 1
1368.2.s.m yes 8 228.m even 6 1
2736.2.s.ba 8 1.a even 1 1 trivial
2736.2.s.ba 8 19.c even 3 1 inner
2736.2.s.bc 8 3.b odd 2 1
2736.2.s.bc 8 57.h odd 6 1

Hecke kernels

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

\( T_{5}^{8} + 4T_{5}^{7} + 20T_{5}^{6} + 24T_{5}^{5} + 108T_{5}^{4} + 176T_{5}^{3} + 352T_{5}^{2} + 240T_{5} + 144 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} - 8T_{7}^{2} + 2T_{7} + 3 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} - 4T_{11}^{2} + 12T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{8} + 30T_{13}^{6} - 16T_{13}^{5} + 719T_{13}^{4} - 240T_{13}^{3} + 5494T_{13}^{2} + 1448T_{13} + 32761 \) 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
$5$ \( T^{8} + 4 T^{7} + 20 T^{6} + 24 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} - 8 T^{2} + 2 T + 3)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} - 4 T^{2} + 12 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 30 T^{6} - 16 T^{5} + \cdots + 32761 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + 48 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + 28 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} + 124 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$29$ \( T^{8} + 64 T^{6} + 448 T^{5} + \cdots + 36864 \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} - 24 T^{2} + 18 T + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} - 78 T^{2} + 320 T + 197)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} + 128 T^{6} + \cdots + 409600 \) Copy content Toggle raw display
$43$ \( T^{8} - 6 T^{7} + 112 T^{6} + \cdots + 2455489 \) Copy content Toggle raw display
$47$ \( T^{8} - 4 T^{7} + 96 T^{6} + \cdots + 409600 \) Copy content Toggle raw display
$53$ \( T^{8} + 16 T^{7} + 300 T^{6} + \cdots + 26998416 \) Copy content Toggle raw display
$59$ \( T^{8} - 12 T^{7} + 156 T^{6} + \cdots + 15376 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} + 134 T^{6} + \cdots + 299209 \) Copy content Toggle raw display
$67$ \( T^{8} + 2 T^{7} + 128 T^{6} + \cdots + 2002225 \) Copy content Toggle raw display
$71$ \( T^{8} + 4 T^{7} + 176 T^{6} + \cdots + 11505664 \) Copy content Toggle raw display
$73$ \( T^{8} + 4 T^{7} + 170 T^{6} + \cdots + 25281 \) Copy content Toggle raw display
$79$ \( T^{8} - 22 T^{7} + 428 T^{6} + \cdots + 139129 \) Copy content Toggle raw display
$83$ \( (T^{4} - 4 T^{3} - 160 T^{2} - 64 T + 3392)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 12 T^{7} + 268 T^{6} + \cdots + 39388176 \) Copy content Toggle raw display
$97$ \( T^{8} - 24 T^{7} + 432 T^{6} + \cdots + 25600 \) Copy content Toggle raw display
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