Properties

Label 714.2.i.i
Level $714$
Weight $2$
Character orbit 714.i
Analytic conductor $5.701$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + q^{6} + ( -1 - 2 \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + q^{6} + ( -1 - 2 \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{10} + ( 2 - 2 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} + q^{13} + ( 2 - 3 \zeta_{6} ) q^{14} - q^{15} -\zeta_{6} q^{16} + ( -1 + \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{18} -4 \zeta_{6} q^{19} + q^{20} + ( -3 + \zeta_{6} ) q^{21} + 2 q^{22} -8 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} + ( 4 - 4 \zeta_{6} ) q^{25} + \zeta_{6} q^{26} - q^{27} + ( 3 - \zeta_{6} ) q^{28} + 3 q^{29} -\zeta_{6} q^{30} + ( 3 - 3 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} -2 \zeta_{6} q^{33} - q^{34} + ( -2 + 3 \zeta_{6} ) q^{35} + q^{36} -4 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} + ( 1 - \zeta_{6} ) q^{39} + \zeta_{6} q^{40} + q^{41} + ( -1 - 2 \zeta_{6} ) q^{42} -4 q^{43} + 2 \zeta_{6} q^{44} + ( -1 + \zeta_{6} ) q^{45} + ( 8 - 8 \zeta_{6} ) q^{46} + 9 \zeta_{6} q^{47} - q^{48} + ( -3 + 8 \zeta_{6} ) q^{49} + 4 q^{50} + \zeta_{6} q^{51} + ( -1 + \zeta_{6} ) q^{52} -\zeta_{6} q^{54} -2 q^{55} + ( 1 + 2 \zeta_{6} ) q^{56} -4 q^{57} + 3 \zeta_{6} q^{58} + ( -11 + 11 \zeta_{6} ) q^{59} + ( 1 - \zeta_{6} ) q^{60} + 3 q^{62} + ( -2 + 3 \zeta_{6} ) q^{63} + q^{64} -\zeta_{6} q^{65} + ( 2 - 2 \zeta_{6} ) q^{66} + ( 12 - 12 \zeta_{6} ) q^{67} -\zeta_{6} q^{68} -8 q^{69} + ( -3 + \zeta_{6} ) q^{70} + 4 q^{71} + \zeta_{6} q^{72} + ( -6 + 6 \zeta_{6} ) q^{73} + ( 4 - 4 \zeta_{6} ) q^{74} -4 \zeta_{6} q^{75} + 4 q^{76} + ( -6 + 2 \zeta_{6} ) q^{77} + q^{78} + ( -1 + \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + \zeta_{6} q^{82} + 9 q^{83} + ( 2 - 3 \zeta_{6} ) q^{84} + q^{85} -4 \zeta_{6} q^{86} + ( 3 - 3 \zeta_{6} ) q^{87} + ( -2 + 2 \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} - q^{90} + ( -1 - 2 \zeta_{6} ) q^{91} + 8 q^{92} -3 \zeta_{6} q^{93} + ( -9 + 9 \zeta_{6} ) q^{94} + ( -4 + 4 \zeta_{6} ) q^{95} -\zeta_{6} q^{96} + 8 q^{97} + ( -8 + 5 \zeta_{6} ) q^{98} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{3} - q^{4} - q^{5} + 2q^{6} - 4q^{7} - 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} + q^{3} - q^{4} - q^{5} + 2q^{6} - 4q^{7} - 2q^{8} - q^{9} + q^{10} + 2q^{11} + q^{12} + 2q^{13} + q^{14} - 2q^{15} - q^{16} - q^{17} + q^{18} - 4q^{19} + 2q^{20} - 5q^{21} + 4q^{22} - 8q^{23} - q^{24} + 4q^{25} + q^{26} - 2q^{27} + 5q^{28} + 6q^{29} - q^{30} + 3q^{31} + q^{32} - 2q^{33} - 2q^{34} - q^{35} + 2q^{36} - 4q^{37} + 4q^{38} + q^{39} + q^{40} + 2q^{41} - 4q^{42} - 8q^{43} + 2q^{44} - q^{45} + 8q^{46} + 9q^{47} - 2q^{48} + 2q^{49} + 8q^{50} + q^{51} - q^{52} - q^{54} - 4q^{55} + 4q^{56} - 8q^{57} + 3q^{58} - 11q^{59} + q^{60} + 6q^{62} - q^{63} + 2q^{64} - q^{65} + 2q^{66} + 12q^{67} - q^{68} - 16q^{69} - 5q^{70} + 8q^{71} + q^{72} - 6q^{73} + 4q^{74} - 4q^{75} + 8q^{76} - 10q^{77} + 2q^{78} - q^{80} - q^{81} + q^{82} + 18q^{83} + q^{84} + 2q^{85} - 4q^{86} + 3q^{87} - 2q^{88} + 6q^{89} - 2q^{90} - 4q^{91} + 16q^{92} - 3q^{93} - 9q^{94} - 4q^{95} - q^{96} + 16q^{97} - 11q^{98} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/714\mathbb{Z}\right)^\times\).

\(n\) \(239\) \(409\) \(547\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
205.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −2.00000 + 1.73205i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
613.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −2.00000 1.73205i −1.00000 −0.500000 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.i.i 2
7.c even 3 1 inner 714.2.i.i 2
7.c even 3 1 4998.2.a.e 1
7.d odd 6 1 4998.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.i.i 2 1.a even 1 1 trivial
714.2.i.i 2 7.c even 3 1 inner
4998.2.a.e 1 7.c even 3 1
4998.2.a.m 1 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\):

\( T_{5}^{2} + T_{5} + 1 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( 4 - 2 T + T^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( 1 + T + T^{2} \)
$19$ \( 16 + 4 T + T^{2} \)
$23$ \( 64 + 8 T + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( 9 - 3 T + T^{2} \)
$37$ \( 16 + 4 T + T^{2} \)
$41$ \( ( -1 + T )^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( 81 - 9 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 121 + 11 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 144 - 12 T + T^{2} \)
$71$ \( ( -4 + T )^{2} \)
$73$ \( 36 + 6 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( -9 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( ( -8 + T )^{2} \)
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