Properties

Label 1710.2.l.k
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: \(\Q(\sqrt{-3}, \sqrt{19})\)
Defining polynomial: \(x^{4} + 19 x^{2} + 361\)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{2} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + \beta_{3} q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{2} + \beta_{2} q^{4} + ( -1 - \beta_{2} ) q^{5} + \beta_{3} q^{7} + q^{8} + \beta_{2} q^{10} + 3 q^{11} -4 \beta_{2} q^{13} + \beta_{1} q^{14} + ( -1 - \beta_{2} ) q^{16} + ( -1 + \beta_{1} - \beta_{2} ) q^{17} -\beta_{1} q^{19} + q^{20} + ( -3 - 3 \beta_{2} ) q^{22} + 3 \beta_{2} q^{23} + \beta_{2} q^{25} -4 q^{26} + ( -\beta_{1} - \beta_{3} ) q^{28} + ( -\beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{29} + ( -2 + 2 \beta_{3} ) q^{31} + \beta_{2} q^{32} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + \beta_{1} q^{35} -7 q^{37} + ( \beta_{1} + \beta_{3} ) q^{38} + ( -1 - \beta_{2} ) q^{40} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{41} + ( 10 + 10 \beta_{2} ) q^{43} + 3 \beta_{2} q^{44} + 3 q^{46} + 6 \beta_{2} q^{47} + 12 q^{49} + q^{50} + ( 4 + 4 \beta_{2} ) q^{52} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{53} + ( -3 - 3 \beta_{2} ) q^{55} + \beta_{3} q^{56} + ( -5 + \beta_{3} ) q^{58} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{59} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{61} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{62} + q^{64} -4 q^{65} + ( \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{67} + ( 1 + \beta_{3} ) q^{68} + ( -\beta_{1} - \beta_{3} ) q^{70} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{71} + ( -9 + \beta_{1} - 9 \beta_{2} ) q^{73} + ( 7 + 7 \beta_{2} ) q^{74} -\beta_{3} q^{76} + 3 \beta_{3} q^{77} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{79} + \beta_{2} q^{80} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{82} + 6 q^{83} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{85} -10 \beta_{2} q^{86} + 3 q^{88} + ( -\beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{89} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{91} + ( -3 - 3 \beta_{2} ) q^{92} + 6 q^{94} + ( \beta_{1} + \beta_{3} ) q^{95} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{97} + ( -12 - 12 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} - 2q^{5} + 4q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} - 2q^{5} + 4q^{8} - 2q^{10} + 12q^{11} + 8q^{13} - 2q^{16} - 2q^{17} + 4q^{20} - 6q^{22} - 6q^{23} - 2q^{25} - 16q^{26} + 10q^{29} - 8q^{31} - 2q^{32} - 2q^{34} - 28q^{37} - 2q^{40} - 4q^{41} + 20q^{43} - 6q^{44} + 12q^{46} - 12q^{47} + 48q^{49} + 4q^{50} + 8q^{52} - 6q^{55} - 20q^{58} + 8q^{59} - 2q^{61} + 4q^{62} + 4q^{64} - 16q^{65} - 14q^{67} + 4q^{68} + 6q^{71} - 18q^{73} + 14q^{74} - 8q^{79} - 2q^{80} - 4q^{82} + 24q^{83} - 2q^{85} + 20q^{86} + 12q^{88} - 8q^{89} - 6q^{92} + 24q^{94} + 2q^{97} - 24q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 19 x^{2} + 361\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/19\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/19\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(19 \beta_{2}\)
\(\nu^{3}\)\(=\)\(19 \beta_{3}\)

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
2.17945 + 3.77492i
−2.17945 3.77492i
2.17945 3.77492i
−2.17945 + 3.77492i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −4.35890 1.00000 0 −0.500000 + 0.866025i
1261.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 4.35890 1.00000 0 −0.500000 + 0.866025i
1531.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −4.35890 1.00000 0 −0.500000 0.866025i
1531.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 4.35890 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.k 4
3.b odd 2 1 570.2.i.h 4
19.c even 3 1 inner 1710.2.l.k 4
57.h odd 6 1 570.2.i.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.h 4 3.b odd 2 1
570.2.i.h 4 57.h odd 6 1
1710.2.l.k 4 1.a even 1 1 trivial
1710.2.l.k 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 \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7}^{2} - 19 \)
\( T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( ( -19 + T^{2} )^{2} \)
$11$ \( ( -3 + T )^{4} \)
$13$ \( ( 16 - 4 T + T^{2} )^{2} \)
$17$ \( 324 - 36 T + 22 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( 361 + 19 T^{2} + T^{4} \)
$23$ \( ( 9 + 3 T + T^{2} )^{2} \)
$29$ \( 36 - 60 T + 94 T^{2} - 10 T^{3} + T^{4} \)
$31$ \( ( -72 + 4 T + T^{2} )^{2} \)
$37$ \( ( 7 + T )^{4} \)
$41$ \( 225 - 60 T + 31 T^{2} + 4 T^{3} + T^{4} \)
$43$ \( ( 100 - 10 T + T^{2} )^{2} \)
$47$ \( ( 36 + 6 T + T^{2} )^{2} \)
$53$ \( 29241 + 171 T^{2} + T^{4} \)
$59$ \( 3600 + 480 T + 124 T^{2} - 8 T^{3} + T^{4} \)
$61$ \( 324 - 36 T + 22 T^{2} + 2 T^{3} + T^{4} \)
$67$ \( 900 + 420 T + 166 T^{2} + 14 T^{3} + T^{4} \)
$71$ \( 26244 + 972 T + 198 T^{2} - 6 T^{3} + T^{4} \)
$73$ \( 3844 + 1116 T + 262 T^{2} + 18 T^{3} + T^{4} \)
$79$ \( 3600 - 480 T + 124 T^{2} + 8 T^{3} + T^{4} \)
$83$ \( ( -6 + T )^{4} \)
$89$ \( 9 - 24 T + 67 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( 28900 + 340 T + 174 T^{2} - 2 T^{3} + T^{4} \)
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