Properties

Label 1638.2.j.n
Level $1638$
Weight $2$
Character orbit 1638.j
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.j (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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + \beta_1 q^{5} + (2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + \beta_1 q^{5} + (2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} - q^{8} + (\beta_{3} + \beta_1) q^{10} + (\beta_{3} + \beta_{2} + \beta_1) q^{11} + q^{13} + (\beta_{3} + \beta_{2} - \beta_1) q^{14} + ( - \beta_{2} - 1) q^{16} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{17} + ( - 3 \beta_{2} - 3) q^{19} + \beta_{3} q^{20} + (\beta_{3} - 1) q^{22} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{23} - 3 \beta_{2} q^{25} + (\beta_{2} + 1) q^{26} + ( - \beta_{3} - 2 \beta_1 - 1) q^{28} - 5 q^{29} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{31} - \beta_{2} q^{32} + (3 \beta_{3} + 1) q^{34} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{35} + ( - 2 \beta_{2} - 5 \beta_1 - 2) q^{37} - 3 \beta_{2} q^{38} - \beta_1 q^{40} + (\beta_{3} - 6) q^{41} + (2 \beta_{3} + 2) q^{43} + ( - \beta_{2} - \beta_1 - 1) q^{44} + (5 \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{46} + (9 \beta_{2} + 9) q^{47} + (2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{49} + 3 q^{50} + \beta_{2} q^{52} + (8 \beta_{3} - \beta_{2} + 8 \beta_1) q^{53} + (\beta_{3} - 2) q^{55} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{56} + ( - 5 \beta_{2} - 5) q^{58} + ( - \beta_{3} - 9 \beta_{2} - \beta_1) q^{59} + (\beta_{2} - 3 \beta_1 + 1) q^{61} + ( - 2 \beta_{3} - 4) q^{62} + q^{64} + \beta_1 q^{65} + (4 \beta_{3} - 9 \beta_{2} + 4 \beta_1) q^{67} + (\beta_{2} - 3 \beta_1 + 1) q^{68} + (\beta_{3} - 4 \beta_{2} - 2) q^{70} + q^{71} + (5 \beta_{3} + 6 \beta_{2} + 5 \beta_1) q^{73} + ( - 5 \beta_{3} - 2 \beta_{2} - 5 \beta_1) q^{74} + 3 q^{76} + ( - 4 \beta_{2} - 2 \beta_1 - 3) q^{77} + ( - 2 \beta_{2} + 8 \beta_1 - 2) q^{79} + ( - \beta_{3} - \beta_1) q^{80} + ( - 6 \beta_{2} - \beta_1 - 6) q^{82} + (4 \beta_{3} + 2) q^{83} + ( - \beta_{3} - 6) q^{85} + (2 \beta_{2} - 2 \beta_1 + 2) q^{86} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{88} + ( - 2 \beta_{2} + 3 \beta_1 - 2) q^{89} + (2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{91} + (5 \beta_{3} + 2) q^{92} + 9 \beta_{2} q^{94} + ( - 3 \beta_{3} - 3 \beta_1) q^{95} + (7 \beta_{3} - 2) q^{97} + ( - 2 \beta_{3} - 4 \beta_1 + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{7} - 4 q^{8} - 2 q^{11} + 4 q^{13} - 2 q^{14} - 2 q^{16} + 2 q^{17} - 6 q^{19} - 4 q^{22} - 4 q^{23} + 6 q^{25} + 2 q^{26} - 4 q^{28} - 20 q^{29} - 8 q^{31} + 2 q^{32} + 4 q^{34} - 12 q^{35} - 4 q^{37} + 6 q^{38} - 24 q^{41} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 18 q^{47} + 10 q^{49} + 12 q^{50} - 2 q^{52} + 2 q^{53} - 8 q^{55} - 2 q^{56} - 10 q^{58} + 18 q^{59} + 2 q^{61} - 16 q^{62} + 4 q^{64} + 18 q^{67} + 2 q^{68} + 4 q^{71} - 12 q^{73} + 4 q^{74} + 12 q^{76} - 4 q^{77} - 4 q^{79} - 12 q^{82} + 8 q^{83} - 24 q^{85} + 4 q^{86} + 2 q^{88} - 4 q^{89} + 2 q^{91} + 8 q^{92} - 18 q^{94} - 8 q^{97} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.707107 1.22474i 0 2.62132 0.358719i −1.00000 0 0.707107 1.22474i
235.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.707107 + 1.22474i 0 −1.62132 + 2.09077i −1.00000 0 −0.707107 + 1.22474i
1171.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.707107 + 1.22474i 0 2.62132 + 0.358719i −1.00000 0 0.707107 + 1.22474i
1171.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.707107 1.22474i 0 −1.62132 2.09077i −1.00000 0 −0.707107 1.22474i
\(n\): 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 1638.2.j.n 4
3.b odd 2 1 546.2.i.h 4
7.c even 3 1 inner 1638.2.j.n 4
21.g even 6 1 3822.2.a.bp 2
21.h odd 6 1 546.2.i.h 4
21.h odd 6 1 3822.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.h 4 3.b odd 2 1
546.2.i.h 4 21.h odd 6 1
1638.2.j.n 4 1.a even 1 1 trivial
1638.2.j.n 4 7.c even 3 1 inner
3822.2.a.bp 2 21.g even 6 1
3822.2.a.bs 2 21.h odd 6 1

Hecke kernels

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

\( T_{5}^{4} + 2T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} + 5T_{11}^{2} - 2T_{11} + 1 \) Copy content Toggle raw display
\( T_{17}^{4} - 2T_{17}^{3} + 21T_{17}^{2} + 34T_{17} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} - 3 T^{2} - 14 T + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289 \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + 62 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$29$ \( (T + 5)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + 62 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 34)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} + 131 T^{2} + \cdots + 16129 \) Copy content Toggle raw display
$59$ \( T^{4} - 18 T^{3} + 245 T^{2} + \cdots + 6241 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + 275 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$71$ \( (T - 1)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 12 T^{3} + 158 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} + 140 T^{2} + \cdots + 15376 \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196 \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T - 94)^{2} \) Copy content Toggle raw display
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