Properties

Label 3024.1.o.d
Level 3024
Weight 1
Character orbit 3024.o
Self dual yes
Analytic conductor 1.509
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
CM discriminant -84
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3024.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.16003008.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{5} + q^{7} +O(q^{10})\) \( q -\beta q^{5} + q^{7} + \beta q^{11} - q^{19} -\beta q^{23} + 2 q^{25} + q^{31} -\beta q^{35} + q^{37} + \beta q^{41} + q^{49} -3 q^{55} + \beta q^{71} + \beta q^{77} + \beta q^{89} + \beta q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{7} + O(q^{10}) \) \( 2q + 2q^{7} - 2q^{19} + 4q^{25} + 2q^{31} + 2q^{37} + 2q^{49} - 6q^{55} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3023.1
1.73205
−1.73205
0 0 0 −1.73205 0 1.00000 0 0 0
3023.2 0 0 0 1.73205 0 1.00000 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
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
3.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.1.o.d yes 2
3.b odd 2 1 inner 3024.1.o.d yes 2
4.b odd 2 1 3024.1.o.a 2
7.b odd 2 1 3024.1.o.a 2
12.b even 2 1 3024.1.o.a 2
21.c even 2 1 3024.1.o.a 2
28.d even 2 1 inner 3024.1.o.d yes 2
84.h odd 2 1 CM 3024.1.o.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.1.o.a 2 4.b odd 2 1
3024.1.o.a 2 7.b odd 2 1
3024.1.o.a 2 12.b even 2 1
3024.1.o.a 2 21.c even 2 1
3024.1.o.d yes 2 1.a even 1 1 trivial
3024.1.o.d yes 2 3.b odd 2 1 inner
3024.1.o.d yes 2 28.d even 2 1 inner
3024.1.o.d yes 2 84.h odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{2} - 3 \)
\( T_{19} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( 1 - T^{2} + T^{4} \)
$13$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$17$ \( ( 1 + T^{2} )^{2} \)
$19$ \( ( 1 + T + T^{2} )^{2} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$31$ \( ( 1 - T + T^{2} )^{2} \)
$37$ \( ( 1 - T + T^{2} )^{2} \)
$41$ \( 1 - T^{2} + T^{4} \)
$43$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$59$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$61$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$67$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$71$ \( 1 - T^{2} + T^{4} \)
$73$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$79$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$83$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$89$ \( 1 - T^{2} + T^{4} \)
$97$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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