Properties

Label 1064.2.q.l
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: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -\beta_{2} + \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + 2 \beta_{2} q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -\beta_{2} + \beta_{3} ) q^{7} + ( -1 + \beta_{2} ) q^{9} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + 2 q^{13} + 2 q^{15} + ( 1 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{17} + ( 1 - \beta_{2} ) q^{19} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{21} + ( 1 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{23} + 4 \beta_{2} q^{25} + 4 q^{27} -4 q^{29} + 4 \beta_{2} q^{31} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{33} + ( -1 - \beta_{1} + \beta_{3} ) q^{35} + ( -2 + 2 \beta_{2} ) q^{37} + 4 \beta_{2} q^{39} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} - q^{43} + \beta_{2} q^{45} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 4 - 3 \beta_{1} - 4 \beta_{2} ) q^{49} + ( 8 - 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{51} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{53} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + 2 q^{57} + 4 \beta_{2} q^{59} + ( -9 + \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 1 + \beta_{1} - \beta_{3} ) q^{63} + ( 2 - 2 \beta_{2} ) q^{65} + ( -2 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{69} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{71} + ( -1 + 2 \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{73} + ( -8 + 8 \beta_{2} ) q^{75} + ( -4 + 3 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} ) q^{77} + ( 6 + 2 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{79} + 11 \beta_{2} q^{81} + ( 3 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{85} -8 \beta_{2} q^{87} + ( -10 + 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -8 + 8 \beta_{2} ) q^{93} -\beta_{2} q^{95} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{97} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 2q^{5} - 3q^{7} - 2q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 2q^{5} - 3q^{7} - 2q^{9} + 3q^{11} + 8q^{13} + 8q^{15} - 7q^{17} + 2q^{19} + 6q^{21} + 8q^{25} + 16q^{27} - 16q^{29} + 8q^{31} - 6q^{33} - 6q^{35} - 4q^{37} + 8q^{39} + 12q^{41} - 4q^{43} + 2q^{45} - 3q^{47} + 5q^{49} + 14q^{51} - 2q^{53} + 6q^{55} + 8q^{57} + 8q^{59} - 17q^{61} + 6q^{63} + 4q^{65} + 12q^{71} - 15q^{73} - 16q^{75} - 24q^{77} + 14q^{79} + 22q^{81} + 16q^{83} - 14q^{85} - 16q^{87} - 18q^{89} - 6q^{91} - 16q^{93} - 2q^{95} + 4q^{97} - 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 5 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu + 5 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} + 4 \beta_{1} + 5\)

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
−1.63746 + 1.52274i
2.13746 0.656712i
−1.63746 1.52274i
2.13746 + 0.656712i
0 1.00000 1.73205i 0 0.500000 + 0.866025i 0 −2.63746 + 0.209313i 0 −0.500000 0.866025i 0
305.2 0 1.00000 1.73205i 0 0.500000 + 0.866025i 0 1.13746 + 2.38876i 0 −0.500000 0.866025i 0
457.1 0 1.00000 + 1.73205i 0 0.500000 0.866025i 0 −2.63746 0.209313i 0 −0.500000 + 0.866025i 0
457.2 0 1.00000 + 1.73205i 0 0.500000 0.866025i 0 1.13746 2.38876i 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.l 4
7.c even 3 1 inner 1064.2.q.l 4
7.c even 3 1 7448.2.a.w 2
7.d odd 6 1 7448.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.q.l 4 1.a even 1 1 trivial
1064.2.q.l 4 7.c even 3 1 inner
7448.2.a.w 2 7.c even 3 1
7448.2.a.be 2 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 \(S_{2}^{\mathrm{new}}(1064, [\chi])\):

\( T_{3}^{2} - 2 T_{3} + 4 \)
\( T_{5}^{2} - T_{5} + 1 \)
\( T_{11}^{4} - 3 T_{11}^{3} + 21 T_{11}^{2} + 36 T_{11} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 4 - 2 T + T^{2} )^{2} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 49 + 21 T + 2 T^{2} + 3 T^{3} + T^{4} \)
$11$ \( 144 + 36 T + 21 T^{2} - 3 T^{3} + T^{4} \)
$13$ \( ( -2 + T )^{4} \)
$17$ \( 4 - 14 T + 51 T^{2} + 7 T^{3} + T^{4} \)
$19$ \( ( 1 - T + T^{2} )^{2} \)
$23$ \( 3249 + 57 T^{2} + T^{4} \)
$29$ \( ( 4 + T )^{4} \)
$31$ \( ( 16 - 4 T + T^{2} )^{2} \)
$37$ \( ( 4 + 2 T + T^{2} )^{2} \)
$41$ \( ( -48 - 6 T + T^{2} )^{2} \)
$43$ \( ( 1 + T )^{4} \)
$47$ \( 144 - 36 T + 21 T^{2} + 3 T^{3} + T^{4} \)
$53$ \( 3136 - 112 T + 60 T^{2} + 2 T^{3} + T^{4} \)
$59$ \( ( 16 - 4 T + T^{2} )^{2} \)
$61$ \( 3364 + 986 T + 231 T^{2} + 17 T^{3} + T^{4} \)
$67$ \( T^{4} \)
$71$ \( ( -48 - 6 T + T^{2} )^{2} \)
$73$ \( 1764 + 630 T + 183 T^{2} + 15 T^{3} + T^{4} \)
$79$ \( 64 + 112 T + 204 T^{2} - 14 T^{3} + T^{4} \)
$83$ \( ( -41 - 8 T + T^{2} )^{2} \)
$89$ \( 576 + 432 T + 300 T^{2} + 18 T^{3} + T^{4} \)
$97$ \( ( -56 - 2 T + T^{2} )^{2} \)
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