Properties

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

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( 1 - \beta_{1} ) q^{5} + ( 1 - \beta_{3} ) q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( 1 - \beta_{1} ) q^{5} + ( 1 - \beta_{3} ) q^{7} - q^{8} -\beta_{1} q^{10} + 3 q^{11} -2 \beta_{2} q^{13} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{14} + ( -1 + \beta_{1} ) q^{16} + ( -4 + 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{19} - q^{20} + ( 3 - 3 \beta_{1} ) q^{22} + ( -\beta_{1} - 2 \beta_{2} ) q^{23} -\beta_{1} q^{25} -2 \beta_{3} q^{26} + ( -\beta_{1} + \beta_{2} ) q^{28} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{29} + ( 4 - 2 \beta_{3} ) q^{31} + \beta_{1} q^{32} + ( 4 \beta_{1} - \beta_{2} ) q^{34} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{35} + ( -5 - 2 \beta_{3} ) q^{37} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{38} + ( -1 + \beta_{1} ) q^{40} + ( -3 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{41} + ( 2 - 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{43} -3 \beta_{1} q^{44} + ( -1 - 2 \beta_{3} ) q^{46} + ( -6 \beta_{1} + 2 \beta_{2} ) q^{47} -2 \beta_{3} q^{49} - q^{50} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{52} + ( -5 \beta_{1} - \beta_{2} ) q^{53} + ( 3 - 3 \beta_{1} ) q^{55} + ( -1 + \beta_{3} ) q^{56} + ( 2 + 3 \beta_{3} ) q^{58} + ( -6 + 6 \beta_{1} ) q^{59} + ( -10 \beta_{1} - \beta_{2} ) q^{61} + ( 4 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{62} + q^{64} -2 \beta_{3} q^{65} + 3 \beta_{2} q^{67} + ( 4 - \beta_{3} ) q^{68} + ( -\beta_{1} + \beta_{2} ) q^{70} + ( 4 - 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{71} + ( -8 + 8 \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} + ( -5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{74} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{76} + ( 3 - 3 \beta_{3} ) q^{77} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{79} + \beta_{1} q^{80} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{82} + ( -2 - 4 \beta_{3} ) q^{83} + ( 4 \beta_{1} - \beta_{2} ) q^{85} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{86} -3 q^{88} + ( -3 \beta_{1} - \beta_{2} ) q^{89} + ( 12 \beta_{1} - 2 \beta_{2} ) q^{91} + ( -1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{92} + ( -6 + 2 \beta_{3} ) q^{94} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{95} + ( 16 - 16 \beta_{1} - \beta_{2} + \beta_{3} ) q^{97} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{4} + 2q^{5} + 4q^{7} - 4q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{4} + 2q^{5} + 4q^{7} - 4q^{8} - 2q^{10} + 12q^{11} + 2q^{14} - 2q^{16} - 8q^{17} - 2q^{19} - 4q^{20} + 6q^{22} - 2q^{23} - 2q^{25} - 2q^{28} + 4q^{29} + 16q^{31} + 2q^{32} + 8q^{34} + 2q^{35} - 20q^{37} + 2q^{38} - 2q^{40} - 6q^{41} + 4q^{43} - 6q^{44} - 4q^{46} - 12q^{47} - 4q^{50} - 10q^{53} + 6q^{55} - 4q^{56} + 8q^{58} - 12q^{59} - 20q^{61} + 8q^{62} + 4q^{64} + 16q^{68} - 2q^{70} + 8q^{71} - 16q^{73} - 10q^{74} + 4q^{76} + 12q^{77} - 4q^{79} + 2q^{80} + 6q^{82} - 8q^{83} + 8q^{85} - 4q^{86} - 12q^{88} - 6q^{89} + 24q^{91} - 2q^{92} - 24q^{94} + 2q^{95} + 32q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + 4 \beta_{2}\)\()/3\)

Character values

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

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

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
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −1.44949 −1.00000 0 −0.500000 + 0.866025i
1261.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 3.44949 −1.00000 0 −0.500000 + 0.866025i
1531.1 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −1.44949 −1.00000 0 −0.500000 0.866025i
1531.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 3.44949 −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.n 4
3.b odd 2 1 570.2.i.f 4
19.c even 3 1 inner 1710.2.l.n 4
57.h odd 6 1 570.2.i.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.f 4 3.b odd 2 1
570.2.i.f 4 57.h odd 6 1
1710.2.l.n 4 1.a even 1 1 trivial
1710.2.l.n 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 \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7}^{2} - 2 T_{7} - 5 \)
\( T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( ( -5 - 2 T + T^{2} )^{2} \)
$11$ \( ( -3 + T )^{4} \)
$13$ \( 576 + 24 T^{2} + T^{4} \)
$17$ \( 100 + 80 T + 54 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( 361 + 38 T - 15 T^{2} + 2 T^{3} + T^{4} \)
$23$ \( 529 - 46 T + 27 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( 2500 + 200 T + 66 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( ( -8 - 8 T + T^{2} )^{2} \)
$37$ \( ( 1 + 10 T + T^{2} )^{2} \)
$41$ \( 2025 - 270 T + 81 T^{2} + 6 T^{3} + T^{4} \)
$43$ \( 8464 + 368 T + 108 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( 144 + 144 T + 132 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( 361 + 190 T + 81 T^{2} + 10 T^{3} + T^{4} \)
$59$ \( ( 36 + 6 T + T^{2} )^{2} \)
$61$ \( 8836 + 1880 T + 306 T^{2} + 20 T^{3} + T^{4} \)
$67$ \( 2916 + 54 T^{2} + T^{4} \)
$71$ \( 1444 + 304 T + 102 T^{2} - 8 T^{3} + T^{4} \)
$73$ \( 3364 + 928 T + 198 T^{2} + 16 T^{3} + T^{4} \)
$79$ \( 400 - 80 T + 36 T^{2} + 4 T^{3} + T^{4} \)
$83$ \( ( -92 + 4 T + T^{2} )^{2} \)
$89$ \( 9 + 18 T + 33 T^{2} + 6 T^{3} + T^{4} \)
$97$ \( 62500 - 8000 T + 774 T^{2} - 32 T^{3} + T^{4} \)
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