Properties

Label 1064.2.q.j
Level $1064$
Weight $2$
Character orbit 1064.q
Analytic conductor $8.496$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1064 = 2^{3} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1064.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.49608277506\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -4 + 4 \zeta_{6} ) q^{11} + 2 q^{13} + 2 q^{15} + ( 7 - 7 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( 2 - 6 \zeta_{6} ) q^{21} + 3 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + 4 q^{27} + 4 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} + 8 \zeta_{6} q^{33} + ( 2 + \zeta_{6} ) q^{35} -10 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{39} -9 q^{43} + ( 1 - \zeta_{6} ) q^{45} + 4 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} -14 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{53} -4 q^{55} + 2 q^{57} + ( 12 - 12 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + ( -2 - \zeta_{6} ) q^{63} + 2 \zeta_{6} q^{65} + ( -16 + 16 \zeta_{6} ) q^{67} + 6 q^{69} + 6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} -8 \zeta_{6} q^{75} + ( -4 + 12 \zeta_{6} ) q^{77} -4 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} -9 q^{83} + 7 q^{85} + ( 8 - 8 \zeta_{6} ) q^{87} + 4 \zeta_{6} q^{89} + ( 6 - 4 \zeta_{6} ) q^{91} + 8 \zeta_{6} q^{93} + ( -1 + \zeta_{6} ) q^{95} -4 q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} + 4 q^{7} - q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + q^{5} + 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} + 7 q^{17} + q^{19} - 2 q^{21} + 3 q^{23} + 4 q^{25} + 8 q^{27} + 8 q^{29} - 4 q^{31} + 8 q^{33} + 5 q^{35} - 10 q^{37} + 4 q^{39} - 18 q^{43} + q^{45} + 4 q^{47} + 2 q^{49} - 14 q^{51} - 6 q^{53} - 8 q^{55} + 4 q^{57} + 12 q^{59} + 2 q^{61} - 5 q^{63} + 2 q^{65} - 16 q^{67} + 12 q^{69} + 12 q^{71} - 2 q^{73} - 8 q^{75} + 4 q^{77} - 4 q^{79} + 11 q^{81} - 18 q^{83} + 14 q^{85} + 8 q^{87} + 4 q^{89} + 8 q^{91} + 8 q^{93} - q^{95} - 8 q^{97} + 8 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1064\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(799\) \(913\) \(1009\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.00000 1.73205i 0 0.500000 + 0.866025i 0 2.00000 1.73205i 0 −0.500000 0.866025i 0
457.1 0 1.00000 + 1.73205i 0 0.500000 0.866025i 0 2.00000 + 1.73205i 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1064.2.q.j 2
7.c even 3 1 inner 1064.2.q.j 2
7.c even 3 1 7448.2.a.c 1
7.d odd 6 1 7448.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.q.j 2 1.a even 1 1 trivial
1064.2.q.j 2 7.c even 3 1 inner
7448.2.a.c 1 7.c even 3 1
7448.2.a.v 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1064, [\chi])\):

\( T_{3}^{2} - 2 T_{3} + 4 \)
\( T_{5}^{2} - T_{5} + 1 \)
\( T_{11}^{2} + 4 T_{11} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 4 - 2 T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 7 - 4 T + T^{2} \)
$11$ \( 16 + 4 T + T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( 49 - 7 T + T^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( 9 - 3 T + T^{2} \)
$29$ \( ( -4 + T )^{2} \)
$31$ \( 16 + 4 T + T^{2} \)
$37$ \( 100 + 10 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 9 + T )^{2} \)
$47$ \( 16 - 4 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 144 - 12 T + T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 256 + 16 T + T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( 4 + 2 T + T^{2} \)
$79$ \( 16 + 4 T + T^{2} \)
$83$ \( ( 9 + T )^{2} \)
$89$ \( 16 - 4 T + T^{2} \)
$97$ \( ( 4 + T )^{2} \)
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