Properties

Label 1064.2.q.k
Level $1064$
Weight $2$
Character orbit 1064.q
Analytic conductor $8.496$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{3} + ( 1 + \beta_{2} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{7} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{3} + ( 1 + \beta_{2} ) q^{5} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{7} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{9} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{11} + ( -2 - \beta_{3} ) q^{13} + ( -2 - \beta_{3} ) q^{15} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 1 + \beta_{2} ) q^{19} + ( -4 + 4 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{21} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{23} -4 \beta_{2} q^{25} + ( 8 + 8 \beta_{3} ) q^{27} + ( -8 - \beta_{3} ) q^{29} + ( \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{31} + ( 4 - 5 \beta_{1} + 4 \beta_{2} ) q^{33} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{35} + ( 8 + \beta_{1} + 8 \beta_{2} ) q^{37} + ( 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{39} -4 \beta_{3} q^{41} + ( 1 - 3 \beta_{3} ) q^{43} + ( 4 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{45} + ( -3 - 3 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 12 - 8 \beta_{1} + 12 \beta_{2} ) q^{51} + ( \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{53} + ( -1 + 3 \beta_{3} ) q^{55} + ( -2 - \beta_{3} ) q^{57} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 5 + 5 \beta_{2} ) q^{61} + ( 16 + \beta_{1} + 5 \beta_{2} + 7 \beta_{3} ) q^{63} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 8 + 5 \beta_{3} ) q^{69} + ( -12 + 3 \beta_{3} ) q^{71} -7 \beta_{2} q^{73} + ( 8 - 4 \beta_{1} + 8 \beta_{2} ) q^{75} + ( 5 + 2 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{77} + ( 6 + 7 \beta_{1} + 6 \beta_{2} ) q^{79} + ( -12 \beta_{1} + 23 \beta_{2} - 12 \beta_{3} ) q^{81} + ( 5 + 7 \beta_{3} ) q^{83} + ( 4 + 2 \beta_{3} ) q^{85} + ( 10 \beta_{1} - 18 \beta_{2} + 10 \beta_{3} ) q^{87} + ( 2 + 11 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{91} + ( -10 + 4 \beta_{1} - 10 \beta_{2} ) q^{93} + \beta_{2} q^{95} -4 \beta_{3} q^{97} + ( -21 - 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} - 6 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} + 2 q^{5} + 2 q^{7} - 6 q^{9} - 2 q^{11} - 8 q^{13} - 8 q^{15} + 8 q^{17} + 2 q^{19} - 8 q^{21} - 6 q^{23} + 8 q^{25} + 32 q^{27} - 32 q^{29} - 12 q^{31} + 8 q^{33} - 2 q^{35} + 16 q^{37} + 12 q^{39} + 4 q^{43} + 6 q^{45} - 6 q^{47} + 10 q^{49} + 24 q^{51} + 8 q^{53} - 4 q^{55} - 8 q^{57} - 4 q^{59} + 10 q^{61} + 54 q^{63} - 4 q^{65} - 12 q^{67} + 32 q^{69} - 48 q^{71} + 14 q^{73} + 16 q^{75} - 4 q^{77} + 12 q^{79} - 46 q^{81} + 20 q^{83} + 16 q^{85} + 36 q^{87} + 4 q^{89} + 8 q^{91} - 20 q^{93} - 2 q^{95} - 84 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(533\) \(799\) \(913\) \(1009\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2}\) \(1\)

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
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 −1.70711 + 2.95680i 0 0.500000 + 0.866025i 0 −1.62132 + 2.09077i 0 −4.32843 7.49706i 0
305.2 0 −0.292893 + 0.507306i 0 0.500000 + 0.866025i 0 2.62132 0.358719i 0 1.32843 + 2.30090i 0
457.1 0 −1.70711 2.95680i 0 0.500000 0.866025i 0 −1.62132 2.09077i 0 −4.32843 + 7.49706i 0
457.2 0 −0.292893 0.507306i 0 0.500000 0.866025i 0 2.62132 + 0.358719i 0 1.32843 2.30090i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
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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.k 4
7.c even 3 1 inner 1064.2.q.k 4
7.c even 3 1 7448.2.a.bd 2
7.d odd 6 1 7448.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.q.k 4 1.a even 1 1 trivial
1064.2.q.k 4 7.c even 3 1 inner
7448.2.a.x 2 7.d odd 6 1
7448.2.a.bd 2 7.c even 3 1

Hecke kernels

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

\( T_{3}^{4} + 4 T_{3}^{3} + 14 T_{3}^{2} + 8 T_{3} + 4 \)
\( T_{5}^{2} - T_{5} + 1 \)
\( T_{11}^{4} + 2 T_{11}^{3} + 21 T_{11}^{2} - 34 T_{11} + 289 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 4 + 8 T + 14 T^{2} + 4 T^{3} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 49 - 14 T - 3 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( 289 - 34 T + 21 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( ( 2 + 4 T + T^{2} )^{2} \)
$17$ \( 64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( ( 1 - T + T^{2} )^{2} \)
$23$ \( 49 + 42 T + 29 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( ( 62 + 16 T + T^{2} )^{2} \)
$31$ \( 1156 + 408 T + 110 T^{2} + 12 T^{3} + T^{4} \)
$37$ \( 3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4} \)
$41$ \( ( -32 + T^{2} )^{2} \)
$43$ \( ( -17 - 2 T + T^{2} )^{2} \)
$47$ \( 81 - 54 T + 45 T^{2} + 6 T^{3} + T^{4} \)
$53$ \( 196 - 112 T + 50 T^{2} - 8 T^{3} + T^{4} \)
$59$ \( 16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4} \)
$61$ \( ( 25 - 5 T + T^{2} )^{2} \)
$67$ \( 784 + 336 T + 116 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( ( 126 + 24 T + T^{2} )^{2} \)
$73$ \( ( 49 - 7 T + T^{2} )^{2} \)
$79$ \( 3844 + 744 T + 206 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( ( -73 - 10 T + T^{2} )^{2} \)
$89$ \( 56644 + 952 T + 254 T^{2} - 4 T^{3} + T^{4} \)
$97$ \( ( -32 + T^{2} )^{2} \)
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