Properties

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

Related objects

Downloads

Learn more

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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{5} - 5 q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + ( - \zeta_{6} + 1) q^{5} - 5 q^{7} + q^{8} + \zeta_{6} q^{10} + 2 q^{11} + \zeta_{6} q^{13} + ( - 5 \zeta_{6} + 5) q^{14} + (\zeta_{6} - 1) q^{16} + (6 \zeta_{6} - 6) q^{17} + (2 \zeta_{6} + 3) q^{19} - q^{20} + (2 \zeta_{6} - 2) q^{22} - 4 \zeta_{6} q^{23} - \zeta_{6} q^{25} - q^{26} + 5 \zeta_{6} q^{28} - 6 \zeta_{6} q^{29} + 7 q^{31} - \zeta_{6} q^{32} - 6 \zeta_{6} q^{34} + (5 \zeta_{6} - 5) q^{35} + q^{37} + (3 \zeta_{6} - 5) q^{38} + ( - \zeta_{6} + 1) q^{40} + ( - 9 \zeta_{6} + 9) q^{43} - 2 \zeta_{6} q^{44} + 4 q^{46} + 18 q^{49} + q^{50} + ( - \zeta_{6} + 1) q^{52} - 8 \zeta_{6} q^{53} + ( - 2 \zeta_{6} + 2) q^{55} - 5 q^{56} + 6 q^{58} + ( - 12 \zeta_{6} + 12) q^{59} + 3 \zeta_{6} q^{61} + (7 \zeta_{6} - 7) q^{62} + q^{64} + q^{65} + 9 \zeta_{6} q^{67} + 6 q^{68} - 5 \zeta_{6} q^{70} + ( - 6 \zeta_{6} + 6) q^{71} + ( - 7 \zeta_{6} + 7) q^{73} + (\zeta_{6} - 1) q^{74} + ( - 5 \zeta_{6} + 2) q^{76} - 10 q^{77} + ( - 5 \zeta_{6} + 5) q^{79} + \zeta_{6} q^{80} + 14 q^{83} + 6 \zeta_{6} q^{85} + 9 \zeta_{6} q^{86} + 2 q^{88} - 8 \zeta_{6} q^{89} - 5 \zeta_{6} q^{91} + (4 \zeta_{6} - 4) q^{92} + ( - 3 \zeta_{6} + 5) q^{95} + ( - 14 \zeta_{6} + 14) q^{97} + (18 \zeta_{6} - 18) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + q^{5} - 10 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + q^{5} - 10 q^{7} + 2 q^{8} + q^{10} + 4 q^{11} + q^{13} + 5 q^{14} - q^{16} - 6 q^{17} + 8 q^{19} - 2 q^{20} - 2 q^{22} - 4 q^{23} - q^{25} - 2 q^{26} + 5 q^{28} - 6 q^{29} + 14 q^{31} - q^{32} - 6 q^{34} - 5 q^{35} + 2 q^{37} - 7 q^{38} + q^{40} + 9 q^{43} - 2 q^{44} + 8 q^{46} + 36 q^{49} + 2 q^{50} + q^{52} - 8 q^{53} + 2 q^{55} - 10 q^{56} + 12 q^{58} + 12 q^{59} + 3 q^{61} - 7 q^{62} + 2 q^{64} + 2 q^{65} + 9 q^{67} + 12 q^{68} - 5 q^{70} + 6 q^{71} + 7 q^{73} - q^{74} - q^{76} - 20 q^{77} + 5 q^{79} + q^{80} + 28 q^{83} + 6 q^{85} + 9 q^{86} + 4 q^{88} - 8 q^{89} - 5 q^{91} - 4 q^{92} + 7 q^{95} + 14 q^{97} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.d 2
3.b odd 2 1 1710.2.l.e yes 2
19.c even 3 1 inner 1710.2.l.d 2
57.h odd 6 1 1710.2.l.e yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.l.d 2 1.a even 1 1 trivial
1710.2.l.d 2 19.c even 3 1 inner
1710.2.l.e yes 2 3.b odd 2 1
1710.2.l.e yes 2 57.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 \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7} + 5 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( (T + 5)^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( (T - 7)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$67$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$83$ \( (T - 14)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
show more
show less