Properties

Label 2736.3.o.e
Level $2736$
Weight $3$
Character orbit 2736.o
Analytic conductor $74.551$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4 q^{5} + 10 q^{7} +O(q^{10})\) \( q -4 q^{5} + 10 q^{7} + 10 q^{11} + ( 14 - 28 \zeta_{6} ) q^{13} -10 q^{17} -19 q^{19} -20 q^{23} -9 q^{25} + ( -20 + 40 \zeta_{6} ) q^{29} + ( 10 - 20 \zeta_{6} ) q^{31} -40 q^{35} + ( 6 - 12 \zeta_{6} ) q^{37} + ( 20 - 40 \zeta_{6} ) q^{41} + 10 q^{43} -80 q^{47} + 51 q^{49} + ( 24 - 48 \zeta_{6} ) q^{53} -40 q^{55} + ( -20 + 40 \zeta_{6} ) q^{59} -10 q^{61} + ( -56 + 112 \zeta_{6} ) q^{65} + ( -44 + 88 \zeta_{6} ) q^{67} + ( -60 + 120 \zeta_{6} ) q^{71} -10 q^{73} + 100 q^{77} + ( -10 + 20 \zeta_{6} ) q^{79} + 70 q^{83} + 40 q^{85} + ( -60 + 120 \zeta_{6} ) q^{89} + ( 140 - 280 \zeta_{6} ) q^{91} + 76 q^{95} + ( -44 + 88 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{5} + 20q^{7} + O(q^{10}) \) \( 2q - 8q^{5} + 20q^{7} + 20q^{11} - 20q^{17} - 38q^{19} - 40q^{23} - 18q^{25} - 80q^{35} + 20q^{43} - 160q^{47} + 102q^{49} - 80q^{55} - 20q^{61} - 20q^{73} + 200q^{77} + 140q^{83} + 80q^{85} + 152q^{95} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −4.00000 0 10.0000 0 0 0
721.2 0 0 0 −4.00000 0 10.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.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.o.e 2
3.b odd 2 1 912.3.o.a 2
4.b odd 2 1 171.3.c.c 2
12.b even 2 1 57.3.c.a 2
19.b odd 2 1 inner 2736.3.o.e 2
57.d even 2 1 912.3.o.a 2
76.d even 2 1 171.3.c.c 2
228.b odd 2 1 57.3.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.c.a 2 12.b even 2 1
57.3.c.a 2 228.b odd 2 1
171.3.c.c 2 4.b odd 2 1
171.3.c.c 2 76.d even 2 1
912.3.o.a 2 3.b odd 2 1
912.3.o.a 2 57.d even 2 1
2736.3.o.e 2 1.a even 1 1 trivial
2736.3.o.e 2 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5} + 4 \)
\( T_{7} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 4 + T )^{2} \)
$7$ \( ( -10 + T )^{2} \)
$11$ \( ( -10 + T )^{2} \)
$13$ \( 588 + T^{2} \)
$17$ \( ( 10 + T )^{2} \)
$19$ \( ( 19 + T )^{2} \)
$23$ \( ( 20 + T )^{2} \)
$29$ \( 1200 + T^{2} \)
$31$ \( 300 + T^{2} \)
$37$ \( 108 + T^{2} \)
$41$ \( 1200 + T^{2} \)
$43$ \( ( -10 + T )^{2} \)
$47$ \( ( 80 + T )^{2} \)
$53$ \( 1728 + T^{2} \)
$59$ \( 1200 + T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( 5808 + T^{2} \)
$71$ \( 10800 + T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( 300 + T^{2} \)
$83$ \( ( -70 + T )^{2} \)
$89$ \( 10800 + T^{2} \)
$97$ \( 5808 + T^{2} \)
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