Properties

Label 1710.2.l.q
Level $1710$
Weight $2$
Character orbit 1710.l
Analytic conductor $13.654$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.29654208.1
Defining polynomial: \(x^{6} + 14 x^{4} + 49 x^{2} + 12\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( 1 - \beta_{1} ) q^{5} -\beta_{3} q^{7} - q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( 1 - \beta_{1} ) q^{5} -\beta_{3} q^{7} - q^{8} -\beta_{1} q^{10} + ( -1 - \beta_{2} ) q^{11} + ( -\beta_{2} - \beta_{4} ) q^{13} + \beta_{5} q^{14} + ( -1 + \beta_{1} ) q^{16} + ( -1 + \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{2} - \beta_{4} - \beta_{5} ) q^{19} - q^{20} + ( -1 + \beta_{1} + \beta_{4} ) q^{22} + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{23} -\beta_{1} q^{25} -\beta_{2} q^{26} + ( \beta_{3} + \beta_{5} ) q^{28} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{29} + \beta_{2} q^{31} + \beta_{1} q^{32} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{34} + \beta_{5} q^{35} - q^{37} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{38} + ( -1 + \beta_{1} ) q^{40} + ( 2 - 2 \beta_{1} + \beta_{4} + 3 \beta_{5} ) q^{41} + ( -4 + 4 \beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{43} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{44} + ( -1 - \beta_{2} ) q^{46} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{47} + ( 6 + 2 \beta_{2} + 2 \beta_{3} ) q^{49} - q^{50} + \beta_{4} q^{52} + ( 6 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{53} + ( -1 + \beta_{1} + \beta_{4} ) q^{55} + \beta_{3} q^{56} + ( 3 - 2 \beta_{2} + \beta_{3} ) q^{58} + ( 6 - 6 \beta_{1} ) q^{59} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{61} -\beta_{4} q^{62} + q^{64} -\beta_{2} q^{65} + ( -5 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{67} + ( 1 + 2 \beta_{2} + \beta_{3} ) q^{68} + ( \beta_{3} + \beta_{5} ) q^{70} + ( -1 + \beta_{1} + 2 \beta_{4} + \beta_{5} ) q^{71} + ( 7 - 7 \beta_{1} + \beta_{4} + \beta_{5} ) q^{73} + ( -1 + \beta_{1} ) q^{74} + ( -\beta_{3} + \beta_{4} ) q^{76} + ( -4 - 3 \beta_{2} + \beta_{3} ) q^{77} + ( 2 - 2 \beta_{1} - \beta_{4} ) q^{79} + \beta_{1} q^{80} + ( -2 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{82} + ( -2 + 4 \beta_{2} ) q^{83} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{85} + ( 4 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{86} + ( 1 + \beta_{2} ) q^{88} + ( -6 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{89} + ( -4 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} ) q^{91} + ( -1 + \beta_{1} + \beta_{4} ) q^{92} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{94} + ( -\beta_{2} - \beta_{3} - \beta_{5} ) q^{95} + ( -13 + 13 \beta_{1} + \beta_{5} ) q^{97} + ( 6 - 6 \beta_{1} - 2 \beta_{4} - 2 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{2} - 3q^{4} + 3q^{5} + 2q^{7} - 6q^{8} + O(q^{10}) \) \( 6q + 3q^{2} - 3q^{4} + 3q^{5} + 2q^{7} - 6q^{8} - 3q^{10} - 8q^{11} - q^{13} + q^{14} - 3q^{16} - 4q^{17} - 2q^{19} - 6q^{20} - 4q^{22} - 4q^{23} - 3q^{25} - 2q^{26} - q^{28} + 6q^{29} + 2q^{31} + 3q^{32} + 4q^{34} + q^{35} - 6q^{37} - q^{38} - 3q^{40} + 8q^{41} - 11q^{43} + 4q^{44} - 8q^{46} + 36q^{49} - 6q^{50} - q^{52} + 18q^{53} - 4q^{55} - 2q^{56} + 12q^{58} + 18q^{59} + 3q^{61} + q^{62} + 6q^{64} - 2q^{65} - 15q^{67} + 8q^{68} - q^{70} - 4q^{71} + 21q^{73} - 3q^{74} + q^{76} - 32q^{77} + 7q^{79} + 3q^{80} - 8q^{82} - 4q^{83} + 4q^{85} + 11q^{86} + 8q^{88} - 18q^{89} - 15q^{91} - 4q^{92} - q^{95} - 38q^{97} + 18q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 14 x^{4} + 49 x^{2} + 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 7 \nu + 2 \)\()/4\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 5 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} - 9 \nu^{2} - 10 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 12 \nu^{3} - 2 \nu^{2} + 35 \nu - 10 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 12 \nu^{3} + 9 \nu^{2} - 29 \nu + 10 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 5\)
\(\nu^{3}\)\(=\)\((\)\(-14 \beta_{5} - 14 \beta_{4} - 7 \beta_{3} - 7 \beta_{2} + 12 \beta_{1} - 6\)\()/3\)
\(\nu^{4}\)\(=\)\(-2 \beta_{3} - 9 \beta_{2} + 35\)
\(\nu^{5}\)\(=\)\((\)\(98 \beta_{5} + 110 \beta_{4} + 49 \beta_{3} + 55 \beta_{2} - 144 \beta_{1} + 72\)\()/3\)

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
2.35084i
2.86514i
0.514306i
2.35084i
2.86514i
0.514306i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 −4.59821 −1.00000 0 −0.500000 + 0.866025i
1261.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 1.75353 −1.00000 0 −0.500000 + 0.866025i
1261.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 + 0.866025i 0 3.84469 −1.00000 0 −0.500000 + 0.866025i
1531.1 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 −4.59821 −1.00000 0 −0.500000 0.866025i
1531.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 1.75353 −1.00000 0 −0.500000 0.866025i
1531.3 0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 0.866025i 0 3.84469 −1.00000 0 −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1531.3
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.q 6
3.b odd 2 1 570.2.i.j 6
19.c even 3 1 inner 1710.2.l.q 6
57.h odd 6 1 570.2.i.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.i.j 6 3.b odd 2 1
570.2.i.j 6 57.h odd 6 1
1710.2.l.q 6 1.a even 1 1 trivial
1710.2.l.q 6 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 \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7}^{3} - T_{7}^{2} - 19 T_{7} + 31 \)
\( T_{11}^{3} + 4 T_{11}^{2} - 11 T_{11} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{3} \)
$3$ \( T^{6} \)
$5$ \( ( 1 - T + T^{2} )^{3} \)
$7$ \( ( 31 - 19 T - T^{2} + T^{3} )^{2} \)
$11$ \( ( -6 - 11 T + 4 T^{2} + T^{3} )^{2} \)
$13$ \( 64 - 128 T + 248 T^{2} - 32 T^{3} + 17 T^{4} + T^{5} + T^{6} \)
$17$ \( 46656 + 10800 T + 3364 T^{2} + 232 T^{3} + 66 T^{4} + 4 T^{5} + T^{6} \)
$19$ \( 6859 + 722 T + 152 T^{2} + 140 T^{3} + 8 T^{4} + 2 T^{5} + T^{6} \)
$23$ \( 36 + 66 T + 145 T^{2} - 32 T^{3} + 27 T^{4} + 4 T^{5} + T^{6} \)
$29$ \( 467856 - 69768 T + 14508 T^{2} - 756 T^{3} + 138 T^{4} - 6 T^{5} + T^{6} \)
$31$ \( ( -8 - 16 T - T^{2} + T^{3} )^{2} \)
$37$ \( ( 1 + T )^{6} \)
$41$ \( 1077444 - 129750 T + 23929 T^{2} - 1076 T^{3} + 189 T^{4} - 8 T^{5} + T^{6} \)
$43$ \( 1296 + 864 T + 972 T^{2} - 192 T^{3} + 145 T^{4} + 11 T^{5} + T^{6} \)
$47$ \( 20736 - 12096 T + 7056 T^{2} - 288 T^{3} + 84 T^{4} + T^{6} \)
$53$ \( 11664 - 9396 T + 5625 T^{2} - 1350 T^{3} + 237 T^{4} - 18 T^{5} + T^{6} \)
$59$ \( ( 36 - 6 T + T^{2} )^{3} \)
$61$ \( 4 - 36 T + 330 T^{2} + 50 T^{3} + 27 T^{4} - 3 T^{5} + T^{6} \)
$67$ \( 4 + 108 T + 2886 T^{2} + 806 T^{3} + 171 T^{4} + 15 T^{5} + T^{6} \)
$71$ \( 46656 + 10800 T + 3364 T^{2} + 232 T^{3} + 66 T^{4} + 4 T^{5} + T^{6} \)
$73$ \( 45796 - 26964 T + 11382 T^{2} - 2218 T^{3} + 315 T^{4} - 21 T^{5} + T^{6} \)
$79$ \( 144 + 84 T^{2} - 24 T^{3} + 49 T^{4} - 7 T^{5} + T^{6} \)
$83$ \( ( -1032 - 260 T + 2 T^{2} + T^{3} )^{2} \)
$89$ \( 11664 + 9396 T + 5625 T^{2} + 1350 T^{3} + 237 T^{4} + 18 T^{5} + T^{6} \)
$97$ \( 3283344 + 837144 T + 144588 T^{2} + 13932 T^{3} + 982 T^{4} + 38 T^{5} + T^{6} \)
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