Properties

Label 1710.2.l.l
Level $1710$
Weight $2$
Character orbit 1710.l
Analytic conductor $13.654$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{73})\)
Defining polynomial: \(x^{4} - x^{3} + 19 x^{2} + 18 x + 324\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{2} -\beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{5} - q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{2} -\beta_{2} q^{4} + ( 1 - \beta_{2} ) q^{5} - q^{7} + q^{8} + \beta_{2} q^{10} + ( 1 - \beta_{3} ) q^{11} + ( -\beta_{1} + \beta_{2} ) q^{13} + ( 1 - \beta_{2} ) q^{14} + ( -1 + \beta_{2} ) q^{16} + ( 1 - \beta_{1} ) q^{19} - q^{20} + ( -1 + \beta_{1} + \beta_{3} ) q^{22} + \beta_{1} q^{23} -\beta_{2} q^{25} -\beta_{3} q^{26} + \beta_{2} q^{28} -6 \beta_{2} q^{29} + ( -2 + \beta_{3} ) q^{31} -\beta_{2} q^{32} + ( -1 + \beta_{2} ) q^{35} - q^{37} + ( \beta_{2} - \beta_{3} ) q^{38} + ( 1 - \beta_{2} ) q^{40} + ( 1 - \beta_{1} - \beta_{3} ) q^{41} + ( -4 - \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{43} -\beta_{1} q^{44} + ( -1 + \beta_{3} ) q^{46} -6 q^{49} + q^{50} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( -\beta_{1} + 6 \beta_{2} ) q^{53} + ( 1 - \beta_{1} - \beta_{3} ) q^{55} - q^{56} + 6 q^{58} + ( -4 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{59} + ( \beta_{1} + \beta_{2} ) q^{61} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{62} + q^{64} + \beta_{3} q^{65} + ( -\beta_{1} + 7 \beta_{2} ) q^{67} -\beta_{2} q^{70} + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{71} + ( 8 - \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{73} + ( 1 - \beta_{2} ) q^{74} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( -1 + \beta_{3} ) q^{77} + ( 4 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{79} + \beta_{2} q^{80} + \beta_{1} q^{82} -6 q^{83} + ( \beta_{1} - 5 \beta_{2} ) q^{86} + ( 1 - \beta_{3} ) q^{88} + ( -3 \beta_{1} + 6 \beta_{2} ) q^{89} + ( \beta_{1} - \beta_{2} ) q^{91} + ( 1 - \beta_{1} - \beta_{3} ) q^{92} + ( -\beta_{2} + \beta_{3} ) q^{95} + ( 10 - 10 \beta_{2} ) q^{97} + ( 6 - 6 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} + 2q^{5} - 4q^{7} + 4q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} + 2q^{5} - 4q^{7} + 4q^{8} + 2q^{10} + 2q^{11} + q^{13} + 2q^{14} - 2q^{16} + 3q^{19} - 4q^{20} - q^{22} + q^{23} - 2q^{25} - 2q^{26} + 2q^{28} - 12q^{29} - 6q^{31} - 2q^{32} - 2q^{35} - 4q^{37} + 2q^{40} + q^{41} - 9q^{43} - q^{44} - 2q^{46} - 24q^{49} + 4q^{50} + q^{52} + 11q^{53} + q^{55} - 4q^{56} + 24q^{58} - 10q^{59} + 3q^{61} + 3q^{62} + 4q^{64} + 2q^{65} + 13q^{67} - 2q^{70} + 2q^{71} + 15q^{73} + 2q^{74} - 3q^{76} - 2q^{77} + 5q^{79} + 2q^{80} + q^{82} - 24q^{83} - 9q^{86} + 2q^{88} + 9q^{89} - q^{91} + q^{92} + 20q^{97} + 12q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 19 x^{2} + 18 x + 324\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 19 \nu^{2} - 19 \nu + 324 \)\()/342\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 37 \)\()/19\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 18 \beta_{2} + \beta_{1} - 19\)
\(\nu^{3}\)\(=\)\(19 \beta_{3} - 37\)

Character values

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

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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
−1.88600 + 3.26665i
2.38600 4.13267i
−1.88600 3.26665i
2.38600 + 4.13267i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −1.00000 1.00000 0 0.500000 0.866025i
1261.2 −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
1531.2 −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.l 4
3.b odd 2 1 570.2.i.g 4
19.c even 3 1 inner 1710.2.l.l 4
57.h odd 6 1 570.2.i.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.g 4 3.b odd 2 1
570.2.i.g 4 57.h odd 6 1
1710.2.l.l 4 1.a even 1 1 trivial
1710.2.l.l 4 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 \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7} + 1 \)
\( T_{11}^{2} - T_{11} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( ( -18 - T + T^{2} )^{2} \)
$13$ \( 324 + 18 T + 19 T^{2} - T^{3} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( 361 - 57 T + 22 T^{2} - 3 T^{3} + T^{4} \)
$23$ \( 324 + 18 T + 19 T^{2} - T^{3} + T^{4} \)
$29$ \( ( 36 + 6 T + T^{2} )^{2} \)
$31$ \( ( -16 + 3 T + T^{2} )^{2} \)
$37$ \( ( 1 + T )^{4} \)
$41$ \( 324 + 18 T + 19 T^{2} - T^{3} + T^{4} \)
$43$ \( 4 + 18 T + 79 T^{2} + 9 T^{3} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( 144 - 132 T + 109 T^{2} - 11 T^{3} + T^{4} \)
$59$ \( 2304 - 480 T + 148 T^{2} + 10 T^{3} + T^{4} \)
$61$ \( 256 + 48 T + 25 T^{2} - 3 T^{3} + T^{4} \)
$67$ \( 576 - 312 T + 145 T^{2} - 13 T^{3} + T^{4} \)
$71$ \( 5184 + 144 T + 76 T^{2} - 2 T^{3} + T^{4} \)
$73$ \( 1444 - 570 T + 187 T^{2} - 15 T^{3} + T^{4} \)
$79$ \( 24964 + 790 T + 183 T^{2} - 5 T^{3} + T^{4} \)
$83$ \( ( 6 + T )^{4} \)
$89$ \( 20736 + 1296 T + 225 T^{2} - 9 T^{3} + T^{4} \)
$97$ \( ( 100 - 10 T + T^{2} )^{2} \)
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