Properties

Label 999.2.c.e
Level $999$
Weight $2$
Character orbit 999.c
Analytic conductor $7.977$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 999 = 3^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 999.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.97705516193\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 q^{4} + q^{7} +O(q^{10})\) \( q + 2 q^{4} + q^{7} + ( -3 + 6 \zeta_{6} ) q^{13} + 4 q^{16} + ( -3 + 6 \zeta_{6} ) q^{19} + 5 q^{25} + 2 q^{28} + ( 6 - 12 \zeta_{6} ) q^{31} + ( 7 - 3 \zeta_{6} ) q^{37} + ( -6 + 12 \zeta_{6} ) q^{43} -6 q^{49} + ( -6 + 12 \zeta_{6} ) q^{52} + ( 9 - 18 \zeta_{6} ) q^{61} + 8 q^{64} -5 q^{67} + 7 q^{73} + ( -6 + 12 \zeta_{6} ) q^{76} + ( -3 + 6 \zeta_{6} ) q^{79} + ( -3 + 6 \zeta_{6} ) q^{91} + ( 3 - 6 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{4} + 2q^{7} + O(q^{10}) \) \( 2q + 4q^{4} + 2q^{7} + 8q^{16} + 10q^{25} + 4q^{28} + 11q^{37} - 12q^{49} + 16q^{64} - 10q^{67} + 14q^{73} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/999\mathbb{Z}\right)^\times\).

\(n\) \(298\) \(704\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 2.00000 0 0 1.00000 0 0 0
406.2 0 0 2.00000 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
37.b even 2 1 inner
111.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 999.2.c.e 2
3.b odd 2 1 CM 999.2.c.e 2
37.b even 2 1 inner 999.2.c.e 2
111.d odd 2 1 inner 999.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
999.2.c.e 2 1.a even 1 1 trivial
999.2.c.e 2 3.b odd 2 1 CM
999.2.c.e 2 37.b even 2 1 inner
999.2.c.e 2 111.d 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 \(S_{2}^{\mathrm{new}}(999, [\chi])\):

\( T_{2} \)
\( T_{5} \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( 27 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 27 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 108 + T^{2} \)
$37$ \( 37 - 11 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 108 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 243 + T^{2} \)
$67$ \( ( 5 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -7 + T )^{2} \)
$79$ \( 27 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 27 + T^{2} \)
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