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Properties

Label 2736.3.h.d
Level $2736$
Weight $3$
Character orbit 2736.h
Analytic conductor $74.551$
Analytic rank $0$
Dimension $12$
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] = [2736,3,Mod(305,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.305");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.h (of order \(2\), degree \(1\), 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: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 36x^{10} + 486x^{8} + 2996x^{6} + 8001x^{4} + 6480x^{2} + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 171)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5} + ( - \beta_{7} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{5} + ( - \beta_{7} + 1) q^{7} + (\beta_{8} - \beta_{5} + \beta_{2}) q^{11} + ( - \beta_{10} + \beta_{6} + \beta_{3} + 1) q^{13} + (\beta_{8} - \beta_{5} - \beta_{2} + \beta_1) q^{17} - \beta_{6} q^{19} + (\beta_{11} - \beta_{8} + \cdots + 3 \beta_1) q^{23}+ \cdots + ( - 4 \beta_{10} + 2 \beta_{9} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 16 q^{7} + 16 q^{13} - 140 q^{25} - 56 q^{31} - 64 q^{37} - 156 q^{43} + 320 q^{49} + 276 q^{55} + 176 q^{61} - 168 q^{67} - 300 q^{73} + 472 q^{79} + 36 q^{85} - 376 q^{91} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 36x^{10} + 486x^{8} + 2996x^{6} + 8001x^{4} + 6480x^{2} + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 9\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 16\nu^{5} + 49\nu^{3} - 66\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 19\nu^{5} + 97\nu^{3} + 99\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 18\nu^{4} + 81\nu^{2} + 40 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} + 25\nu^{6} + 199\nu^{4} + 519\nu^{2} + 180 ) / 12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} + 27\nu^{7} + 243\nu^{5} + 773\nu^{3} + 396\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{8} + 25\nu^{6} + 211\nu^{4} + 651\nu^{2} + 324 ) / 12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} + 29\nu^{8} + 293\nu^{6} + 1195\nu^{4} + 1698\nu^{2} + 648 ) / 24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} + 33\nu^{9} + 399\nu^{7} + 2123\nu^{5} + 4548\nu^{3} + 2160\nu ) / 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{2} - 9\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{7} - 11\beta_{3} + 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{5} - 8\beta_{4} - 32\beta_{2} + 89\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -18\beta_{9} + 18\beta_{7} + 4\beta_{6} + 117\beta_{3} - 526 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -64\beta_{5} + 152\beta_{4} + 414\beta_{2} - 917\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 251\beta_{9} - 239\beta_{7} - 100\beta_{6} - 1255\beta_{3} + 5338 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 48\beta_{8} + 756\beta_{5} - 2160\beta_{4} - 4948\beta_{2} + 9693\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 24\beta_{10} - 3200\beta_{9} + 2852\beta_{7} + 1728\beta_{6} + 13561\beta_{3} - 55674 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 48\beta_{11} - 1584\beta_{8} - 7904\beta_{5} + 27616\beta_{4} + 56938\beta_{2} - 104161\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
− 2.68958i
0.932029i
− 3.23546i
3.35539i
0.545873i
2.42336i
− 2.42336i
− 0.545873i
− 3.35539i
3.23546i
− 0.932029i
2.68958i
0 0 0 − 8.57565i 0 −8.48939 0 0 0
305.2 0 0 0 − 7.54533i 0 12.3747 0 0 0
305.3 0 0 0 − 7.34768i 0 10.6609 0 0 0
305.4 0 0 0 − 4.66206i 0 5.08698 0 0 0
305.5 0 0 0 − 3.60307i 0 −2.53045 0 0 0
305.6 0 0 0 − 0.906051i 0 −9.10278 0 0 0
305.7 0 0 0 0.906051i 0 −9.10278 0 0 0
305.8 0 0 0 3.60307i 0 −2.53045 0 0 0
305.9 0 0 0 4.66206i 0 5.08698 0 0 0
305.10 0 0 0 7.34768i 0 10.6609 0 0 0
305.11 0 0 0 7.54533i 0 12.3747 0 0 0
305.12 0 0 0 8.57565i 0 −8.48939 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.h.d 12
3.b odd 2 1 inner 2736.3.h.d 12
4.b odd 2 1 171.3.b.a ✓ 12
12.b even 2 1 171.3.b.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.b.a ✓ 12 4.b odd 2 1
171.3.b.a ✓ 12 12.b even 2 1
2736.3.h.d 12 1.a even 1 1 trivial
2736.3.h.d 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 220T_{5}^{10} + 18097T_{5}^{8} + 682704T_{5}^{6} + 11564856T_{5}^{4} + 72824832T_{5}^{2} + 52359696 \) acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 220 T^{10} + \cdots + 52359696 \) Copy content Toggle raw display
$7$ \( (T^{6} - 8 T^{5} + \cdots - 131232)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 500 T^{10} + \cdots + 30338064 \) Copy content Toggle raw display
$13$ \( (T^{6} - 8 T^{5} + \cdots - 36384)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 884 T^{10} + \cdots + 2624400 \) Copy content Toggle raw display
$19$ \( (T^{2} - 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{6} + 28 T^{5} + \cdots - 177263136)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 32 T^{5} + \cdots - 4913280)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{6} + 78 T^{5} + \cdots + 2711856592)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 44\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{6} - 88 T^{5} + \cdots - 46699091360)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 84 T^{5} + \cdots - 11774750720)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 86\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{6} + 150 T^{5} + \cdots + 34536530576)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 236 T^{5} + \cdots + 49132153152)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 4402242637760)^{2} \) Copy content Toggle raw display
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