Properties

Label 1710.2.l.a
Level $1710$
Weight $2$
Character orbit 1710.l
Analytic conductor $13.654$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.6544187456\)
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 570)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1261.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 1.00000 0 −0.500000 + 0.866025i
1531.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 1.00000 0 −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.l.a 2
3.b odd 2 1 570.2.i.d 2
19.c even 3 1 inner 1710.2.l.a 2
57.h odd 6 1 570.2.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.d 2 3.b odd 2 1
570.2.i.d 2 57.h odd 6 1
1710.2.l.a 2 1.a even 1 1 trivial
1710.2.l.a 2 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7} + 1 \)
\( T_{11} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 6 + T )^{2} \)
$13$ \( 25 + 5 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 19 - 8 T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 36 - 6 T + T^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( ( -11 + T )^{2} \)
$41$ \( 36 - 6 T + T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( 144 - 12 T + T^{2} \)
$53$ \( 144 + 12 T + T^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 49 - 7 T + T^{2} \)
$67$ \( 1 - T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 1 - T + T^{2} \)
$79$ \( 49 - 7 T + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 144 + 12 T + T^{2} \)
$97$ \( 196 + 14 T + T^{2} \)
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