Properties

Label 688.2.i.d
Level $688$
Weight $2$
Character orbit 688.i
Analytic conductor $5.494$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.49370765906\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + 3 \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} + 3 \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} + 5 \zeta_{6} q^{13} -\zeta_{6} q^{15} -3 \zeta_{6} q^{17} + ( 1 - \zeta_{6} ) q^{19} + 3 q^{21} + ( -7 + 7 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + 5 q^{27} -3 \zeta_{6} q^{29} + ( 5 - 5 \zeta_{6} ) q^{31} + 3 q^{35} + ( 9 - 9 \zeta_{6} ) q^{37} + 5 q^{39} -10 q^{41} + ( 7 - 6 \zeta_{6} ) q^{43} + 2 q^{45} + 8 q^{47} + ( -2 + 2 \zeta_{6} ) q^{49} -3 q^{51} + ( 5 - 5 \zeta_{6} ) q^{53} -\zeta_{6} q^{57} -12 q^{59} + 13 \zeta_{6} q^{61} + ( -6 + 6 \zeta_{6} ) q^{63} + 5 q^{65} + ( -3 + 3 \zeta_{6} ) q^{67} + 7 \zeta_{6} q^{69} -\zeta_{6} q^{71} -11 \zeta_{6} q^{73} + 4 q^{75} -5 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + ( 9 - 9 \zeta_{6} ) q^{83} -3 q^{85} -3 q^{87} + ( 1 - \zeta_{6} ) q^{89} + ( -15 + 15 \zeta_{6} ) q^{91} -5 \zeta_{6} q^{93} -\zeta_{6} q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5} + 3 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + q^{3} + q^{5} + 3 q^{7} + 2 q^{9} + 5 q^{13} - q^{15} - 3 q^{17} + q^{19} + 6 q^{21} - 7 q^{23} + 4 q^{25} + 10 q^{27} - 3 q^{29} + 5 q^{31} + 6 q^{35} + 9 q^{37} + 10 q^{39} - 20 q^{41} + 8 q^{43} + 4 q^{45} + 16 q^{47} - 2 q^{49} - 6 q^{51} + 5 q^{53} - q^{57} - 24 q^{59} + 13 q^{61} - 6 q^{63} + 10 q^{65} - 3 q^{67} + 7 q^{69} - q^{71} - 11 q^{73} + 8 q^{75} - 5 q^{79} - q^{81} + 9 q^{83} - 6 q^{85} - 6 q^{87} + q^{89} - 15 q^{91} - 5 q^{93} - q^{95} - 4 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 0.500000 0.866025i 0 1.50000 + 2.59808i 0 1.00000 + 1.73205i 0
337.1 0 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.50000 2.59808i 0 1.00000 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.i.d 2
4.b odd 2 1 43.2.c.a 2
12.b even 2 1 387.2.h.a 2
43.c even 3 1 inner 688.2.i.d 2
172.f even 6 1 1849.2.a.a 1
172.g odd 6 1 43.2.c.a 2
172.g odd 6 1 1849.2.a.c 1
516.p even 6 1 387.2.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.c.a 2 4.b odd 2 1
43.2.c.a 2 172.g odd 6 1
387.2.h.a 2 12.b even 2 1
387.2.h.a 2 516.p even 6 1
688.2.i.d 2 1.a even 1 1 trivial
688.2.i.d 2 43.c even 3 1 inner
1849.2.a.a 1 172.f even 6 1
1849.2.a.c 1 172.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{5}^{2} - T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 9 - 3 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 25 - 5 T + T^{2} \)
$17$ \( 9 + 3 T + T^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( 49 + 7 T + T^{2} \)
$29$ \( 9 + 3 T + T^{2} \)
$31$ \( 25 - 5 T + T^{2} \)
$37$ \( 81 - 9 T + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( 43 - 8 T + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 25 - 5 T + T^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( 169 - 13 T + T^{2} \)
$67$ \( 9 + 3 T + T^{2} \)
$71$ \( 1 + T + T^{2} \)
$73$ \( 121 + 11 T + T^{2} \)
$79$ \( 25 + 5 T + T^{2} \)
$83$ \( 81 - 9 T + T^{2} \)
$89$ \( 1 - T + T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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