Properties

Label 1710.2.l.j
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: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{2} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + \beta_{3} q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{2} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + \beta_{3} q^{7} + q^{8} + \beta_{2} q^{10} + ( 1 - 2 \beta_{3} ) q^{11} + \beta_{1} q^{14} + ( -1 - \beta_{2} ) q^{16} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{17} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{19} + q^{20} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{22} -\beta_{2} q^{23} + \beta_{2} q^{25} + ( -\beta_{1} - \beta_{3} ) q^{28} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( 2 - 2 \beta_{3} ) q^{31} + \beta_{2} q^{32} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{34} + \beta_{1} q^{35} + ( 3 - 2 \beta_{3} ) q^{37} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{38} + ( -1 - \beta_{2} ) q^{40} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{41} + ( -6 - 6 \beta_{2} ) q^{43} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{44} - q^{46} + ( -4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{47} + q^{50} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{53} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{55} + \beta_{3} q^{56} + ( 1 - \beta_{3} ) q^{58} + 2 \beta_{1} q^{59} + ( 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{61} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{62} + q^{64} + ( -\beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{67} + ( 3 - \beta_{3} ) q^{68} + ( -\beta_{1} - \beta_{3} ) q^{70} + ( -11 - \beta_{1} - 11 \beta_{2} ) q^{71} + ( -11 - \beta_{1} - 11 \beta_{2} ) q^{73} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{74} + ( -2 - 4 \beta_{2} + \beta_{3} ) q^{76} + ( -14 + \beta_{3} ) q^{77} -2 \beta_{1} q^{79} + \beta_{2} q^{80} + ( \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{82} + ( -6 + 4 \beta_{3} ) q^{83} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{85} + 6 \beta_{2} q^{86} + ( 1 - 2 \beta_{3} ) q^{88} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{89} + ( 1 + \beta_{2} ) q^{92} + ( -2 + 4 \beta_{3} ) q^{94} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{95} + ( -1 + \beta_{1} - \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} - 2q^{5} + 4q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} - 2q^{5} + 4q^{8} - 2q^{10} + 4q^{11} - 2q^{16} - 6q^{17} - 12q^{19} + 4q^{20} - 2q^{22} + 2q^{23} - 2q^{25} - 2q^{29} + 8q^{31} - 2q^{32} - 6q^{34} + 12q^{37} + 12q^{38} - 2q^{40} - 12q^{41} - 12q^{43} - 2q^{44} - 4q^{46} + 4q^{47} + 4q^{50} - 4q^{53} - 2q^{55} + 4q^{58} + 2q^{61} - 4q^{62} + 4q^{64} + 14q^{67} + 12q^{68} - 22q^{71} - 22q^{73} - 6q^{74} - 56q^{77} - 2q^{80} - 12q^{82} - 24q^{83} - 6q^{85} - 12q^{86} + 4q^{88} + 2q^{92} - 8q^{94} + 12q^{95} - 2q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

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
1.32288 + 2.29129i
−1.32288 2.29129i
1.32288 2.29129i
−1.32288 + 2.29129i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −2.64575 1.00000 0 −0.500000 + 0.866025i
1261.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 2.64575 1.00000 0 −0.500000 + 0.866025i
1531.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −2.64575 1.00000 0 −0.500000 0.866025i
1531.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 2.64575 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.j 4
3.b odd 2 1 570.2.i.i 4
19.c even 3 1 inner 1710.2.l.j 4
57.h odd 6 1 570.2.i.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.i 4 3.b odd 2 1
570.2.i.i 4 57.h odd 6 1
1710.2.l.j 4 1.a even 1 1 trivial
1710.2.l.j 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 \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7}^{2} - 7 \)
\( T_{11}^{2} - 2 T_{11} - 27 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( ( -27 - 2 T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 4 + 12 T + 34 T^{2} + 6 T^{3} + T^{4} \)
$19$ \( 361 + 228 T + 67 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( ( 1 - T + T^{2} )^{2} \)
$29$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
$31$ \( ( -24 - 4 T + T^{2} )^{2} \)
$37$ \( ( -19 - 6 T + T^{2} )^{2} \)
$41$ \( 841 + 348 T + 115 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( ( 36 + 6 T + T^{2} )^{2} \)
$47$ \( 11664 + 432 T + 124 T^{2} - 4 T^{3} + T^{4} \)
$53$ \( 9 - 12 T + 19 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( 784 + 28 T^{2} + T^{4} \)
$61$ \( 3844 + 124 T + 66 T^{2} - 2 T^{3} + T^{4} \)
$67$ \( 1764 - 588 T + 154 T^{2} - 14 T^{3} + T^{4} \)
$71$ \( 12996 + 2508 T + 370 T^{2} + 22 T^{3} + T^{4} \)
$73$ \( 12996 + 2508 T + 370 T^{2} + 22 T^{3} + T^{4} \)
$79$ \( 784 + 28 T^{2} + T^{4} \)
$83$ \( ( -76 + 12 T + T^{2} )^{2} \)
$89$ \( 30625 + 175 T^{2} + T^{4} \)
$97$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
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