Properties

Label 1638.2.j.i
Level $1638$
Weight $2$
Character orbit 1638.j
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.j (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.

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

Character values

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

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

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
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.50000 0.866025i −1.00000 0 0
1171.1 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 −2.50000 + 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
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.i 2
3.b odd 2 1 546.2.i.d 2
7.c even 3 1 inner 1638.2.j.i 2
21.g even 6 1 3822.2.a.bf 1
21.h odd 6 1 546.2.i.d 2
21.h odd 6 1 3822.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.d 2 3.b odd 2 1
546.2.i.d 2 21.h odd 6 1
1638.2.j.i 2 1.a even 1 1 trivial
1638.2.j.i 2 7.c even 3 1 inner
3822.2.a.u 1 21.h odd 6 1
3822.2.a.bf 1 21.g even 6 1

Hecke kernels

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

\( T_{5} \)
\( T_{11}^{2} + 5 T_{11} + 25 \)
\( T_{17}^{2} - 7 T_{17} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( 25 + 5 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 49 - 7 T + T^{2} \)
$19$ \( 49 + 7 T + T^{2} \)
$23$ \( 4 - 2 T + T^{2} \)
$29$ \( ( -9 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 16 + 4 T + T^{2} \)
$41$ \( ( 4 + T )^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 9 + 3 T + T^{2} \)
$53$ \( 1 - T + T^{2} \)
$59$ \( 49 - 7 T + T^{2} \)
$61$ \( 169 + 13 T + T^{2} \)
$67$ \( 9 + 3 T + T^{2} \)
$71$ \( ( 9 + T )^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 196 + 14 T + T^{2} \)
$83$ \( ( 16 + T )^{2} \)
$89$ \( 144 + 12 T + T^{2} \)
$97$ \( ( -6 + T )^{2} \)
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