Properties

Label 1638.2.r.z
Level $1638$
Weight $2$
Character orbit 1638.r
Analytic conductor $13.079$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.r (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{3} ) q^{5} + ( 1 - \beta_{2} ) q^{7} - q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( 1 - \beta_{3} ) q^{5} + ( 1 - \beta_{2} ) q^{7} - q^{8} + ( \beta_{1} + \beta_{2} ) q^{10} + \beta_{1} q^{11} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + q^{14} -\beta_{2} q^{16} + ( -5 + 5 \beta_{2} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{20} + ( \beta_{1} + \beta_{3} ) q^{22} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{23} -3 \beta_{3} q^{25} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{26} + \beta_{2} q^{28} -\beta_{2} q^{29} -2 \beta_{3} q^{31} + ( 1 - \beta_{2} ) q^{32} -5 q^{34} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( 5 \beta_{1} - 3 \beta_{2} ) q^{37} + 3 \beta_{3} q^{38} + ( -1 + \beta_{3} ) q^{40} + ( 2 \beta_{1} - \beta_{2} ) q^{41} + ( 8 + 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{43} + \beta_{3} q^{44} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{46} -3 \beta_{3} q^{47} -\beta_{2} q^{49} + 3 \beta_{1} q^{50} + ( 1 - \beta_{1} + \beta_{3} ) q^{52} + ( 9 + 2 \beta_{3} ) q^{53} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{55} + ( -1 + \beta_{2} ) q^{56} + ( 1 - \beta_{2} ) q^{58} + ( 8 - 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{59} + ( 1 - \beta_{2} ) q^{61} + 2 \beta_{1} q^{62} + q^{64} + ( -4 + 3 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{67} -5 \beta_{2} q^{68} + ( 1 - \beta_{3} ) q^{70} + ( -4 + 4 \beta_{2} ) q^{71} + ( -5 - \beta_{3} ) q^{73} + ( 3 + 5 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} ) q^{74} -3 \beta_{1} q^{76} -\beta_{3} q^{77} + ( 4 + \beta_{3} ) q^{79} + ( -\beta_{1} - \beta_{2} ) q^{80} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{82} -4 \beta_{3} q^{83} + ( -5 + 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{85} + ( 8 + 2 \beta_{3} ) q^{86} -\beta_{1} q^{88} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -1 + \beta_{1} - \beta_{3} ) q^{91} + ( 4 - 2 \beta_{3} ) q^{92} + 3 \beta_{1} q^{94} + ( -12 + 6 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} ) q^{95} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{97} + ( 1 - \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{4} + 6q^{5} + 2q^{7} - 4q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{4} + 6q^{5} + 2q^{7} - 4q^{8} + 3q^{10} + q^{11} - 2q^{13} + 4q^{14} - 2q^{16} - 10q^{17} - 3q^{19} - 3q^{20} - q^{22} - 10q^{23} + 6q^{25} - q^{26} + 2q^{28} - 2q^{29} + 4q^{31} + 2q^{32} - 20q^{34} + 3q^{35} - q^{37} - 6q^{38} - 6q^{40} + 14q^{43} - 2q^{44} + 10q^{46} + 6q^{47} - 2q^{49} + 3q^{50} + q^{52} + 32q^{53} + 10q^{55} - 2q^{56} + 2q^{58} + 18q^{59} + 2q^{61} + 2q^{62} + 4q^{64} - 3q^{65} + 6q^{67} - 10q^{68} + 6q^{70} - 8q^{71} - 18q^{73} + q^{74} - 3q^{76} + 2q^{77} + 14q^{79} - 3q^{80} + 8q^{83} - 15q^{85} + 28q^{86} - q^{88} - 7q^{89} - q^{91} + 20q^{92} + 3q^{94} - 30q^{95} - 8q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 5 x^{2} + 4 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 5 \nu^{2} - 5 \nu + 16 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} - 4\)
\(\nu^{3}\)\(=\)\(5 \beta_{3} - 4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
−0.780776 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i
1.28078 2.21837i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.561553 0 0.500000 0.866025i −1.00000 0 −0.280776 0.486319i
757.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 3.56155 0 0.500000 0.866025i −1.00000 0 1.78078 + 3.08440i
1387.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.561553 0 0.500000 + 0.866025i −1.00000 0 −0.280776 + 0.486319i
1387.2 0.500000 0.866025i 0 −0.500000 0.866025i 3.56155 0 0.500000 + 0.866025i −1.00000 0 1.78078 3.08440i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.r.z 4
3.b odd 2 1 546.2.l.i 4
13.c even 3 1 inner 1638.2.r.z 4
39.h odd 6 1 7098.2.a.bq 2
39.i odd 6 1 546.2.l.i 4
39.i odd 6 1 7098.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.i 4 3.b odd 2 1
546.2.l.i 4 39.i odd 6 1
1638.2.r.z 4 1.a even 1 1 trivial
1638.2.r.z 4 13.c even 3 1 inner
7098.2.a.bq 2 39.h odd 6 1
7098.2.a.bw 2 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{2} - 3 T_{5} - 2 \)
\( T_{11}^{4} - T_{11}^{3} + 5 T_{11}^{2} + 4 T_{11} + 16 \)
\( T_{17}^{2} + 5 T_{17} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( -2 - 3 T + T^{2} )^{2} \)
$7$ \( ( 1 - T + T^{2} )^{2} \)
$11$ \( 16 + 4 T + 5 T^{2} - T^{3} + T^{4} \)
$13$ \( ( 13 + T + T^{2} )^{2} \)
$17$ \( ( 25 + 5 T + T^{2} )^{2} \)
$19$ \( 1296 - 108 T + 45 T^{2} + 3 T^{3} + T^{4} \)
$23$ \( 64 + 80 T + 92 T^{2} + 10 T^{3} + T^{4} \)
$29$ \( ( 1 + T + T^{2} )^{2} \)
$31$ \( ( -16 - 2 T + T^{2} )^{2} \)
$37$ \( 11236 - 106 T + 107 T^{2} + T^{3} + T^{4} \)
$41$ \( 289 + 17 T^{2} + T^{4} \)
$43$ \( 1024 - 448 T + 164 T^{2} - 14 T^{3} + T^{4} \)
$47$ \( ( -36 - 3 T + T^{2} )^{2} \)
$53$ \( ( 47 - 16 T + T^{2} )^{2} \)
$59$ \( 4096 - 1152 T + 260 T^{2} - 18 T^{3} + T^{4} \)
$61$ \( ( 1 - T + T^{2} )^{2} \)
$67$ \( 64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4} \)
$71$ \( ( 16 + 4 T + T^{2} )^{2} \)
$73$ \( ( 16 + 9 T + T^{2} )^{2} \)
$79$ \( ( 8 - 7 T + T^{2} )^{2} \)
$83$ \( ( -64 - 4 T + T^{2} )^{2} \)
$89$ \( 676 - 182 T + 75 T^{2} + 7 T^{3} + T^{4} \)
$97$ \( 2704 - 416 T + 116 T^{2} + 8 T^{3} + T^{4} \)
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