Properties

Label 1638.2.r.f
Level $1638$
Weight $2$
Character orbit 1638.r
Analytic conductor $13.079$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -\zeta_{6} q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -\zeta_{6} q^{7} + q^{8} + ( -3 + 3 \zeta_{6} ) q^{11} + ( 4 - 3 \zeta_{6} ) q^{13} + q^{14} + ( -1 + \zeta_{6} ) q^{16} -3 \zeta_{6} q^{17} -5 \zeta_{6} q^{19} -3 \zeta_{6} q^{22} + ( -6 + 6 \zeta_{6} ) q^{23} -5 q^{25} + ( -1 + 4 \zeta_{6} ) q^{26} + ( -1 + \zeta_{6} ) q^{28} + ( -9 + 9 \zeta_{6} ) q^{29} + 8 q^{31} -\zeta_{6} q^{32} + 3 q^{34} + ( -8 + 8 \zeta_{6} ) q^{37} + 5 q^{38} + ( 3 - 3 \zeta_{6} ) q^{41} -8 \zeta_{6} q^{43} + 3 q^{44} -6 \zeta_{6} q^{46} -3 q^{47} + ( -1 + \zeta_{6} ) q^{49} + ( 5 - 5 \zeta_{6} ) q^{50} + ( -3 - \zeta_{6} ) q^{52} + 3 q^{53} -\zeta_{6} q^{56} -9 \zeta_{6} q^{58} -6 \zeta_{6} q^{59} + 7 \zeta_{6} q^{61} + ( -8 + 8 \zeta_{6} ) q^{62} + q^{64} + ( -2 + 2 \zeta_{6} ) q^{67} + ( -3 + 3 \zeta_{6} ) q^{68} -6 \zeta_{6} q^{71} -16 q^{73} -8 \zeta_{6} q^{74} + ( -5 + 5 \zeta_{6} ) q^{76} + 3 q^{77} -13 q^{79} + 3 \zeta_{6} q^{82} -18 q^{83} + 8 q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} + ( -15 + 15 \zeta_{6} ) q^{89} + ( -3 - \zeta_{6} ) q^{91} + 6 q^{92} + ( 3 - 3 \zeta_{6} ) q^{94} + 16 \zeta_{6} q^{97} -\zeta_{6} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - q^{7} + 2q^{8} - 3q^{11} + 5q^{13} + 2q^{14} - q^{16} - 3q^{17} - 5q^{19} - 3q^{22} - 6q^{23} - 10q^{25} + 2q^{26} - q^{28} - 9q^{29} + 16q^{31} - q^{32} + 6q^{34} - 8q^{37} + 10q^{38} + 3q^{41} - 8q^{43} + 6q^{44} - 6q^{46} - 6q^{47} - q^{49} + 5q^{50} - 7q^{52} + 6q^{53} - q^{56} - 9q^{58} - 6q^{59} + 7q^{61} - 8q^{62} + 2q^{64} - 2q^{67} - 3q^{68} - 6q^{71} - 32q^{73} - 8q^{74} - 5q^{76} + 6q^{77} - 26q^{79} + 3q^{82} - 36q^{83} + 16q^{86} - 3q^{88} - 15q^{89} - 7q^{91} + 12q^{92} + 3q^{94} + 16q^{97} - q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 + 0.866025i 1.00000 0 0
1387.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −0.500000 0.866025i 1.00000 0 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
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.f 2
3.b odd 2 1 546.2.l.e 2
13.c even 3 1 inner 1638.2.r.f 2
39.h odd 6 1 7098.2.a.ba 1
39.i odd 6 1 546.2.l.e 2
39.i odd 6 1 7098.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.e 2 3.b odd 2 1
546.2.l.e 2 39.i odd 6 1
1638.2.r.f 2 1.a even 1 1 trivial
1638.2.r.f 2 13.c even 3 1 inner
7098.2.a.m 1 39.i odd 6 1
7098.2.a.ba 1 39.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 \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5} \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{17}^{2} + 3 T_{17} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( 9 + 3 T + T^{2} \)
$13$ \( 13 - 5 T + T^{2} \)
$17$ \( 9 + 3 T + T^{2} \)
$19$ \( 25 + 5 T + T^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( 81 + 9 T + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 64 + 8 T + T^{2} \)
$41$ \( 9 - 3 T + T^{2} \)
$43$ \( 64 + 8 T + T^{2} \)
$47$ \( ( 3 + T )^{2} \)
$53$ \( ( -3 + T )^{2} \)
$59$ \( 36 + 6 T + T^{2} \)
$61$ \( 49 - 7 T + T^{2} \)
$67$ \( 4 + 2 T + T^{2} \)
$71$ \( 36 + 6 T + T^{2} \)
$73$ \( ( 16 + T )^{2} \)
$79$ \( ( 13 + T )^{2} \)
$83$ \( ( 18 + T )^{2} \)
$89$ \( 225 + 15 T + T^{2} \)
$97$ \( 256 - 16 T + T^{2} \)
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