Properties

Label 1710.2.l.c
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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 1 - \zeta_{6} ) q^{5} -5 q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( 1 - \zeta_{6} ) q^{5} -5 q^{7} + q^{8} + \zeta_{6} q^{10} - q^{11} -6 \zeta_{6} q^{13} + ( 5 - 5 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 4 - 4 \zeta_{6} ) q^{17} + ( -3 + 5 \zeta_{6} ) q^{19} - q^{20} + ( 1 - \zeta_{6} ) q^{22} + 7 \zeta_{6} q^{23} -\zeta_{6} q^{25} + 6 q^{26} + 5 \zeta_{6} q^{28} + 6 \zeta_{6} q^{29} -\zeta_{6} q^{32} + 4 \zeta_{6} q^{34} + ( -5 + 5 \zeta_{6} ) q^{35} + 7 q^{37} + ( -2 - 3 \zeta_{6} ) q^{38} + ( 1 - \zeta_{6} ) q^{40} + ( -5 + 5 \zeta_{6} ) q^{41} + ( -6 + 6 \zeta_{6} ) q^{43} + \zeta_{6} q^{44} -7 q^{46} -8 \zeta_{6} q^{47} + 18 q^{49} + q^{50} + ( -6 + 6 \zeta_{6} ) q^{52} + 11 \zeta_{6} q^{53} + ( -1 + \zeta_{6} ) q^{55} -5 q^{56} -6 q^{58} + ( -8 + 8 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} + q^{64} -6 q^{65} -12 \zeta_{6} q^{67} -4 q^{68} -5 \zeta_{6} q^{70} + ( 2 - 2 \zeta_{6} ) q^{71} + ( 2 - 2 \zeta_{6} ) q^{73} + ( -7 + 7 \zeta_{6} ) q^{74} + ( 5 - 2 \zeta_{6} ) q^{76} + 5 q^{77} + ( -10 + 10 \zeta_{6} ) q^{79} + \zeta_{6} q^{80} -5 \zeta_{6} q^{82} + 10 q^{83} -4 \zeta_{6} q^{85} -6 \zeta_{6} q^{86} - q^{88} + 13 \zeta_{6} q^{89} + 30 \zeta_{6} q^{91} + ( 7 - 7 \zeta_{6} ) q^{92} + 8 q^{94} + ( 2 + 3 \zeta_{6} ) q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} + ( -18 + 18 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} + q^{5} - 10q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} + q^{5} - 10q^{7} + 2q^{8} + q^{10} - 2q^{11} - 6q^{13} + 5q^{14} - q^{16} + 4q^{17} - q^{19} - 2q^{20} + q^{22} + 7q^{23} - q^{25} + 12q^{26} + 5q^{28} + 6q^{29} - q^{32} + 4q^{34} - 5q^{35} + 14q^{37} - 7q^{38} + q^{40} - 5q^{41} - 6q^{43} + q^{44} - 14q^{46} - 8q^{47} + 36q^{49} + 2q^{50} - 6q^{52} + 11q^{53} - q^{55} - 10q^{56} - 12q^{58} - 8q^{59} + 4q^{61} + 2q^{64} - 12q^{65} - 12q^{67} - 8q^{68} - 5q^{70} + 2q^{71} + 2q^{73} - 7q^{74} + 8q^{76} + 10q^{77} - 10q^{79} + q^{80} - 5q^{82} + 20q^{83} - 4q^{85} - 6q^{86} - 2q^{88} + 13q^{89} + 30q^{91} + 7q^{92} + 16q^{94} + 7q^{95} - 2q^{97} - 18q^{98} + O(q^{100}) \)

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 −5.00000 1.00000 0 0.500000 0.866025i
1531.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −5.00000 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.c 2
3.b odd 2 1 570.2.i.e 2
19.c even 3 1 inner 1710.2.l.c 2
57.h odd 6 1 570.2.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.e 2 3.b odd 2 1
570.2.i.e 2 57.h odd 6 1
1710.2.l.c 2 1.a even 1 1 trivial
1710.2.l.c 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 \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7} + 5 \)
\( T_{11} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( ( 5 + T )^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 36 + 6 T + T^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( 19 + T + T^{2} \)
$23$ \( 49 - 7 T + T^{2} \)
$29$ \( 36 - 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -7 + T )^{2} \)
$41$ \( 25 + 5 T + T^{2} \)
$43$ \( 36 + 6 T + T^{2} \)
$47$ \( 64 + 8 T + T^{2} \)
$53$ \( 121 - 11 T + T^{2} \)
$59$ \( 64 + 8 T + T^{2} \)
$61$ \( 16 - 4 T + T^{2} \)
$67$ \( 144 + 12 T + T^{2} \)
$71$ \( 4 - 2 T + T^{2} \)
$73$ \( 4 - 2 T + T^{2} \)
$79$ \( 100 + 10 T + T^{2} \)
$83$ \( ( -10 + T )^{2} \)
$89$ \( 169 - 13 T + T^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
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