Properties

Label 1638.2.r.o
Level $1638$
Weight $2$
Character orbit 1638.r
Analytic conductor $13.079$
Analytic rank $0$
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: \(0\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 q^{5} + \zeta_{6} q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -3 q^{5} + \zeta_{6} q^{7} - q^{8} + ( -3 + 3 \zeta_{6} ) q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} + ( 3 + \zeta_{6} ) q^{13} + q^{14} + ( -1 + \zeta_{6} ) q^{16} -5 \zeta_{6} q^{17} -4 \zeta_{6} q^{19} + 3 \zeta_{6} q^{20} -4 \zeta_{6} q^{22} + ( -4 + 4 \zeta_{6} ) q^{23} + 4 q^{25} + ( 4 - 3 \zeta_{6} ) q^{26} + ( 1 - \zeta_{6} ) q^{28} + ( -9 + 9 \zeta_{6} ) q^{29} + \zeta_{6} q^{32} -5 q^{34} -3 \zeta_{6} q^{35} + ( -7 + 7 \zeta_{6} ) q^{37} -4 q^{38} + 3 q^{40} + ( 7 - 7 \zeta_{6} ) q^{41} -8 \zeta_{6} q^{43} -4 q^{44} + 4 \zeta_{6} q^{46} -12 q^{47} + ( -1 + \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{50} + ( 1 - 4 \zeta_{6} ) q^{52} -7 q^{53} + ( -12 + 12 \zeta_{6} ) q^{55} -\zeta_{6} q^{56} + 9 \zeta_{6} q^{58} -8 \zeta_{6} q^{59} -7 \zeta_{6} q^{61} + q^{64} + ( -9 - 3 \zeta_{6} ) q^{65} + ( -12 + 12 \zeta_{6} ) q^{67} + ( -5 + 5 \zeta_{6} ) q^{68} -3 q^{70} + 12 \zeta_{6} q^{71} - q^{73} + 7 \zeta_{6} q^{74} + ( -4 + 4 \zeta_{6} ) q^{76} + 4 q^{77} -16 q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} -7 \zeta_{6} q^{82} + 8 q^{83} + 15 \zeta_{6} q^{85} -8 q^{86} + ( -4 + 4 \zeta_{6} ) q^{88} + ( -6 + 6 \zeta_{6} ) q^{89} + ( -1 + 4 \zeta_{6} ) q^{91} + 4 q^{92} + ( -12 + 12 \zeta_{6} ) q^{94} + 12 \zeta_{6} q^{95} -18 \zeta_{6} q^{97} + \zeta_{6} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 6q^{5} + q^{7} - 2q^{8} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 6q^{5} + q^{7} - 2q^{8} - 3q^{10} + 4q^{11} + 7q^{13} + 2q^{14} - q^{16} - 5q^{17} - 4q^{19} + 3q^{20} - 4q^{22} - 4q^{23} + 8q^{25} + 5q^{26} + q^{28} - 9q^{29} + q^{32} - 10q^{34} - 3q^{35} - 7q^{37} - 8q^{38} + 6q^{40} + 7q^{41} - 8q^{43} - 8q^{44} + 4q^{46} - 24q^{47} - q^{49} + 4q^{50} - 2q^{52} - 14q^{53} - 12q^{55} - q^{56} + 9q^{58} - 8q^{59} - 7q^{61} + 2q^{64} - 21q^{65} - 12q^{67} - 5q^{68} - 6q^{70} + 12q^{71} - 2q^{73} + 7q^{74} - 4q^{76} + 8q^{77} - 32q^{79} + 3q^{80} - 7q^{82} + 16q^{83} + 15q^{85} - 16q^{86} - 4q^{88} - 6q^{89} + 2q^{91} + 8q^{92} - 12q^{94} + 12q^{95} - 18q^{97} + q^{98} + O(q^{100}) \)

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 −3.00000 0 0.500000 0.866025i −1.00000 0 −1.50000 2.59808i
1387.1 0.500000 0.866025i 0 −0.500000 0.866025i −3.00000 0 0.500000 + 0.866025i −1.00000 0 −1.50000 + 2.59808i
\(n\): 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.o 2
3.b odd 2 1 546.2.l.b 2
13.c even 3 1 inner 1638.2.r.o 2
39.h odd 6 1 7098.2.a.a 1
39.i odd 6 1 546.2.l.b 2
39.i odd 6 1 7098.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.l.b 2 3.b odd 2 1
546.2.l.b 2 39.i odd 6 1
1638.2.r.o 2 1.a even 1 1 trivial
1638.2.r.o 2 13.c even 3 1 inner
7098.2.a.a 1 39.h odd 6 1
7098.2.a.v 1 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 \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 3 + T )^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( 16 - 4 T + T^{2} \)
$13$ \( 13 - 7 T + T^{2} \)
$17$ \( 25 + 5 T + T^{2} \)
$19$ \( 16 + 4 T + T^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( 81 + 9 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 49 + 7 T + T^{2} \)
$41$ \( 49 - 7 T + T^{2} \)
$43$ \( 64 + 8 T + T^{2} \)
$47$ \( ( 12 + T )^{2} \)
$53$ \( ( 7 + T )^{2} \)
$59$ \( 64 + 8 T + T^{2} \)
$61$ \( 49 + 7 T + T^{2} \)
$67$ \( 144 + 12 T + T^{2} \)
$71$ \( 144 - 12 T + T^{2} \)
$73$ \( ( 1 + T )^{2} \)
$79$ \( ( 16 + T )^{2} \)
$83$ \( ( -8 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( 324 + 18 T + T^{2} \)
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