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Properties

Label 2736.3.o.o
Level $2736$
Weight $3$
Character orbit 2736.o
Analytic conductor $74.551$
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] = [2736,3,Mod(721,2736)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (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: \(8\)
Coefficient field: 8.0.5661086492196864.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 54x^{6} + 753x^{4} + 3152x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 342)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + ( - \beta_{7} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + ( - \beta_{7} - 1) q^{7} + ( - \beta_{5} - \beta_1) q^{11} + \beta_{2} q^{13} + (4 \beta_{5} + \beta_1) q^{17} + (\beta_{7} - \beta_{3} - \beta_{2} + 1) q^{19} + ( - 5 \beta_{5} + 6 \beta_1) q^{23} + (\beta_{7} - 4) q^{25} - \beta_{6} q^{29} - 6 \beta_{3} q^{31} + ( - 8 \beta_{5} + 7 \beta_1) q^{35} - 4 \beta_{3} q^{37} + ( - \beta_{6} + 3 \beta_{4}) q^{41} + ( - 3 \beta_{7} + 1) q^{43} + (3 \beta_{5} - 11 \beta_1) q^{47} + (3 \beta_{7} + 30) q^{49} + ( - \beta_{6} - 3 \beta_{4}) q^{53} + ( - \beta_{7} + 27) q^{55} + ( - 2 \beta_{6} - \beta_{4}) q^{59} + ( - 5 \beta_{7} - 17) q^{61} + (2 \beta_{6} - 5 \beta_{4}) q^{65} + (2 \beta_{3} - 3 \beta_{2}) q^{67} + (2 \beta_{6} - \beta_{4}) q^{71} + ( - 3 \beta_{7} + 65) q^{73} + ( - 12 \beta_{5} + \beta_1) q^{77} + ( - 4 \beta_{3} - \beta_{2}) q^{79} + (7 \beta_{5} + 6 \beta_1) q^{83} + (7 \beta_{7} - 45) q^{85} + (\beta_{6} + 13 \beta_{4}) q^{89} + (12 \beta_{3} - 7 \beta_{2}) q^{91} + ( - 2 \beta_{6} + 8 \beta_{5} + \cdots - 7 \beta_1) q^{95}+ \cdots + (4 \beta_{3} - 5 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{7} + 12 q^{19} - 28 q^{25} - 4 q^{43} + 252 q^{49} + 212 q^{55} - 156 q^{61} + 508 q^{73} - 332 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 54x^{6} + 753x^{4} + 3152x^{2} + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 46\nu^{4} + 385\nu^{2} + 216 ) / 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 34\nu^{5} + 143\nu^{3} + 3708\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 43\nu^{5} + 262\nu^{3} - 504\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 46\nu^{5} + 397\nu^{3} + 492\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} - 89\nu^{4} - 665\nu^{2} - 72 ) / 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 52\nu^{5} - 655\nu^{3} - 2244\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{6} - 89\nu^{4} - 656\nu^{2} + 54 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + 2\beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 2\beta_{5} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{6} + 19\beta_{4} - 29\beta_{3} - 10\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -35\beta_{7} + 76\beta_{5} + 24\beta _1 + 370 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -135\beta_{6} - 677\beta_{4} + 955\beta_{3} + 284\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1225\beta_{7} - 2726\beta_{5} - 1068\beta _1 - 11846 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5019\beta_{6} + 23881\beta_{4} - 32909\beta_{3} - 9340\beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
− 0.214926i
0.214926i
− 5.92486i
5.92486i
− 3.09643i
3.09643i
− 3.04335i
3.04335i
0 0 0 −5.50871 0 −10.3459 0 0 0
721.2 0 0 0 −5.50871 0 −10.3459 0 0 0
721.3 0 0 0 −3.55726 0 7.34590 0 0 0
721.4 0 0 0 −3.55726 0 7.34590 0 0 0
721.5 0 0 0 3.55726 0 7.34590 0 0 0
721.6 0 0 0 3.55726 0 7.34590 0 0 0
721.7 0 0 0 5.50871 0 −10.3459 0 0 0
721.8 0 0 0 5.50871 0 −10.3459 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.o 8
3.b odd 2 1 inner 2736.3.o.o 8
4.b odd 2 1 342.3.d.c ✓ 8
12.b even 2 1 342.3.d.c ✓ 8
19.b odd 2 1 inner 2736.3.o.o 8
57.d even 2 1 inner 2736.3.o.o 8
76.d even 2 1 342.3.d.c ✓ 8
228.b odd 2 1 342.3.d.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.3.d.c ✓ 8 4.b odd 2 1
342.3.d.c ✓ 8 12.b even 2 1
342.3.d.c ✓ 8 76.d even 2 1
342.3.d.c ✓ 8 228.b odd 2 1
2736.3.o.o 8 1.a even 1 1 trivial
2736.3.o.o 8 3.b odd 2 1 inner
2736.3.o.o 8 19.b odd 2 1 inner
2736.3.o.o 8 57.d 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 \(S_{3}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{4} - 43T_{5}^{2} + 384 \) Copy content Toggle raw display
\( T_{7}^{2} + 3T_{7} - 76 \) 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
$5$ \( (T^{4} - 43 T^{2} + 384)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 3 T - 76)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 109 T^{2} + 1014)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 552 T^{2} + 31104)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 859 T^{2} + 11616)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2098 T^{2} + 221184)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2826 T^{2} + 1971216)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 4644 T^{2} + 4478976)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2064 T^{2} + 884736)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4338 T^{2} + 419904)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + T - 704)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 4957 T^{2} + 1821606)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 3906 T^{2} + 746496)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 11304 T^{2} + 31539456)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 39 T - 1576)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 6204 T^{2} + 146016)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 11592 T^{2} + 29942784)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 127 T + 3328)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2136 T^{2} + 13824)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 4642 T^{2} + 1247616)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 26226 T^{2} + 106007616)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 18264 T^{2} + 55296)^{2} \) Copy content Toggle raw display
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