Properties

Label 1710.2.d.g
Level $1710$
Weight $2$
Character orbit 1710.d
Analytic conductor $13.654$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1369.1
0.258819 0.965926i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 0.965926i
1.00000i 0 −1.00000 −1.41421 1.73205i 0 1.03528i 1.00000i 0 −1.73205 + 1.41421i
1369.2 1.00000i 0 −1.00000 −1.41421 + 1.73205i 0 3.86370i 1.00000i 0 1.73205 + 1.41421i
1369.3 1.00000i 0 −1.00000 1.41421 1.73205i 0 1.03528i 1.00000i 0 −1.73205 1.41421i
1369.4 1.00000i 0 −1.00000 1.41421 + 1.73205i 0 3.86370i 1.00000i 0 1.73205 1.41421i
1369.5 1.00000i 0 −1.00000 −1.41421 1.73205i 0 3.86370i 1.00000i 0 1.73205 1.41421i
1369.6 1.00000i 0 −1.00000 −1.41421 + 1.73205i 0 1.03528i 1.00000i 0 −1.73205 1.41421i
1369.7 1.00000i 0 −1.00000 1.41421 1.73205i 0 3.86370i 1.00000i 0 1.73205 + 1.41421i
1369.8 1.00000i 0 −1.00000 1.41421 + 1.73205i 0 1.03528i 1.00000i 0 −1.73205 + 1.41421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1369.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.d.g 8
3.b odd 2 1 inner 1710.2.d.g 8
5.b even 2 1 inner 1710.2.d.g 8
5.c odd 4 1 8550.2.a.cu 4
5.c odd 4 1 8550.2.a.cv 4
15.d odd 2 1 inner 1710.2.d.g 8
15.e even 4 1 8550.2.a.cu 4
15.e even 4 1 8550.2.a.cv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.d.g 8 1.a even 1 1 trivial
1710.2.d.g 8 3.b odd 2 1 inner
1710.2.d.g 8 5.b even 2 1 inner
1710.2.d.g 8 15.d odd 2 1 inner
8550.2.a.cu 4 5.c odd 4 1
8550.2.a.cu 4 15.e even 4 1
8550.2.a.cv 4 5.c odd 4 1
8550.2.a.cv 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7}^{4} + 16 T_{7}^{2} + 16 \)
\( T_{11}^{4} - 16 T_{11}^{2} + 16 \)
\( T_{13}^{4} + 16 T_{13}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( T^{8} \)
$5$ \( ( 25 + 2 T^{2} + T^{4} )^{2} \)
$7$ \( ( 16 + 16 T^{2} + T^{4} )^{2} \)
$11$ \( ( 16 - 16 T^{2} + T^{4} )^{2} \)
$13$ \( ( 16 + 16 T^{2} + T^{4} )^{2} \)
$17$ \( ( 16 + 56 T^{2} + T^{4} )^{2} \)
$19$ \( ( -1 + T )^{8} \)
$23$ \( ( 64 + 32 T^{2} + T^{4} )^{2} \)
$29$ \( ( 1936 - 112 T^{2} + T^{4} )^{2} \)
$31$ \( ( -32 + 8 T + T^{2} )^{4} \)
$37$ \( ( 144 + 48 T^{2} + T^{4} )^{2} \)
$41$ \( ( 144 - 48 T^{2} + T^{4} )^{2} \)
$43$ \( ( 144 + 48 T^{2} + T^{4} )^{2} \)
$47$ \( ( 576 + 96 T^{2} + T^{4} )^{2} \)
$53$ \( ( 36 + T^{2} )^{4} \)
$59$ \( ( 7744 - 208 T^{2} + T^{4} )^{2} \)
$61$ \( ( 2 + T )^{8} \)
$67$ \( ( 2304 + 192 T^{2} + T^{4} )^{2} \)
$71$ \( ( 4096 - 256 T^{2} + T^{4} )^{2} \)
$73$ \( ( 2304 + 192 T^{2} + T^{4} )^{2} \)
$79$ \( ( -4 + T )^{8} \)
$83$ \( ( 5184 + 288 T^{2} + T^{4} )^{2} \)
$89$ \( ( 16 - 16 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1936 + 304 T^{2} + T^{4} )^{2} \)
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