Properties

Label 722.2.c.g
Level $722$
Weight $2$
Character orbit 722.c
Analytic conductor $5.765$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.76519902594\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( -2 + 2 \zeta_{6} ) q^{5} -3 \zeta_{6} q^{6} -3 q^{7} - q^{8} -6 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( -2 + 2 \zeta_{6} ) q^{5} -3 \zeta_{6} q^{6} -3 q^{7} - q^{8} -6 \zeta_{6} q^{9} + 2 \zeta_{6} q^{10} -2 q^{11} -3 q^{12} -3 \zeta_{6} q^{13} + ( -3 + 3 \zeta_{6} ) q^{14} + 6 \zeta_{6} q^{15} + ( -1 + \zeta_{6} ) q^{16} + ( 1 - \zeta_{6} ) q^{17} -6 q^{18} + 2 q^{20} + ( -9 + 9 \zeta_{6} ) q^{21} + ( -2 + 2 \zeta_{6} ) q^{22} -5 \zeta_{6} q^{23} + ( -3 + 3 \zeta_{6} ) q^{24} + \zeta_{6} q^{25} -3 q^{26} -9 q^{27} + 3 \zeta_{6} q^{28} -3 \zeta_{6} q^{29} + 6 q^{30} + 6 q^{31} + \zeta_{6} q^{32} + ( -6 + 6 \zeta_{6} ) q^{33} -\zeta_{6} q^{34} + ( 6 - 6 \zeta_{6} ) q^{35} + ( -6 + 6 \zeta_{6} ) q^{36} -6 q^{37} -9 q^{39} + ( 2 - 2 \zeta_{6} ) q^{40} + ( 12 - 12 \zeta_{6} ) q^{41} + 9 \zeta_{6} q^{42} + ( 10 - 10 \zeta_{6} ) q^{43} + 2 \zeta_{6} q^{44} + 12 q^{45} -5 q^{46} + 8 \zeta_{6} q^{47} + 3 \zeta_{6} q^{48} + 2 q^{49} + q^{50} -3 \zeta_{6} q^{51} + ( -3 + 3 \zeta_{6} ) q^{52} -3 \zeta_{6} q^{53} + ( -9 + 9 \zeta_{6} ) q^{54} + ( 4 - 4 \zeta_{6} ) q^{55} + 3 q^{56} -3 q^{58} + ( 3 - 3 \zeta_{6} ) q^{59} + ( 6 - 6 \zeta_{6} ) q^{60} + ( 6 - 6 \zeta_{6} ) q^{62} + 18 \zeta_{6} q^{63} + q^{64} + 6 q^{65} + 6 \zeta_{6} q^{66} + 15 \zeta_{6} q^{67} - q^{68} -15 q^{69} -6 \zeta_{6} q^{70} + 6 \zeta_{6} q^{72} + ( 11 - 11 \zeta_{6} ) q^{73} + ( -6 + 6 \zeta_{6} ) q^{74} + 3 q^{75} + 6 q^{77} + ( -9 + 9 \zeta_{6} ) q^{78} + ( -12 + 12 \zeta_{6} ) q^{79} -2 \zeta_{6} q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} -12 \zeta_{6} q^{82} + 2 q^{83} + 9 q^{84} + 2 \zeta_{6} q^{85} -10 \zeta_{6} q^{86} -9 q^{87} + 2 q^{88} + 6 \zeta_{6} q^{89} + ( 12 - 12 \zeta_{6} ) q^{90} + 9 \zeta_{6} q^{91} + ( -5 + 5 \zeta_{6} ) q^{92} + ( 18 - 18 \zeta_{6} ) q^{93} + 8 q^{94} + 3 q^{96} + ( 12 - 12 \zeta_{6} ) q^{97} + ( 2 - 2 \zeta_{6} ) q^{98} + 12 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} - q^{4} - 2 q^{5} - 3 q^{6} - 6 q^{7} - 2 q^{8} - 6 q^{9} + O(q^{10}) \) \( 2 q + q^{2} + 3 q^{3} - q^{4} - 2 q^{5} - 3 q^{6} - 6 q^{7} - 2 q^{8} - 6 q^{9} + 2 q^{10} - 4 q^{11} - 6 q^{12} - 3 q^{13} - 3 q^{14} + 6 q^{15} - q^{16} + q^{17} - 12 q^{18} + 4 q^{20} - 9 q^{21} - 2 q^{22} - 5 q^{23} - 3 q^{24} + q^{25} - 6 q^{26} - 18 q^{27} + 3 q^{28} - 3 q^{29} + 12 q^{30} + 12 q^{31} + q^{32} - 6 q^{33} - q^{34} + 6 q^{35} - 6 q^{36} - 12 q^{37} - 18 q^{39} + 2 q^{40} + 12 q^{41} + 9 q^{42} + 10 q^{43} + 2 q^{44} + 24 q^{45} - 10 q^{46} + 8 q^{47} + 3 q^{48} + 4 q^{49} + 2 q^{50} - 3 q^{51} - 3 q^{52} - 3 q^{53} - 9 q^{54} + 4 q^{55} + 6 q^{56} - 6 q^{58} + 3 q^{59} + 6 q^{60} + 6 q^{62} + 18 q^{63} + 2 q^{64} + 12 q^{65} + 6 q^{66} + 15 q^{67} - 2 q^{68} - 30 q^{69} - 6 q^{70} + 6 q^{72} + 11 q^{73} - 6 q^{74} + 6 q^{75} + 12 q^{77} - 9 q^{78} - 12 q^{79} - 2 q^{80} - 9 q^{81} - 12 q^{82} + 4 q^{83} + 18 q^{84} + 2 q^{85} - 10 q^{86} - 18 q^{87} + 4 q^{88} + 6 q^{89} + 12 q^{90} + 9 q^{91} - 5 q^{92} + 18 q^{93} + 16 q^{94} + 6 q^{96} + 12 q^{97} + 2 q^{98} + 12 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/722\mathbb{Z}\right)^\times\).

\(n\) \(363\)
\(\chi(n)\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
429.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 1.50000 2.59808i −0.500000 0.866025i −1.00000 + 1.73205i −1.50000 2.59808i −3.00000 −1.00000 −3.00000 5.19615i 1.00000 + 1.73205i
653.1 0.500000 + 0.866025i 1.50000 + 2.59808i −0.500000 + 0.866025i −1.00000 1.73205i −1.50000 + 2.59808i −3.00000 −1.00000 −3.00000 + 5.19615i 1.00000 1.73205i
\(n\): 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 722.2.c.g 2
19.b odd 2 1 722.2.c.a 2
19.c even 3 1 722.2.a.a 1
19.c even 3 1 inner 722.2.c.g 2
19.d odd 6 1 722.2.a.f yes 1
19.d odd 6 1 722.2.c.a 2
19.e even 9 6 722.2.e.h 6
19.f odd 18 6 722.2.e.g 6
57.f even 6 1 6498.2.a.a 1
57.h odd 6 1 6498.2.a.m 1
76.f even 6 1 5776.2.a.a 1
76.g odd 6 1 5776.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.a 1 19.c even 3 1
722.2.a.f yes 1 19.d odd 6 1
722.2.c.a 2 19.b odd 2 1
722.2.c.a 2 19.d odd 6 1
722.2.c.g 2 1.a even 1 1 trivial
722.2.c.g 2 19.c even 3 1 inner
722.2.e.g 6 19.f odd 18 6
722.2.e.h 6 19.e even 9 6
5776.2.a.a 1 76.f even 6 1
5776.2.a.q 1 76.g odd 6 1
6498.2.a.a 1 57.f even 6 1
6498.2.a.m 1 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\):

\( T_{3}^{2} - 3 T_{3} + 9 \)
\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{7} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 9 - 3 T + T^{2} \)
$5$ \( 4 + 2 T + T^{2} \)
$7$ \( ( 3 + T )^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 9 + 3 T + T^{2} \)
$17$ \( 1 - T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( 25 + 5 T + T^{2} \)
$29$ \( 9 + 3 T + T^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( 144 - 12 T + T^{2} \)
$43$ \( 100 - 10 T + T^{2} \)
$47$ \( 64 - 8 T + T^{2} \)
$53$ \( 9 + 3 T + T^{2} \)
$59$ \( 9 - 3 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 225 - 15 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 121 - 11 T + T^{2} \)
$79$ \( 144 + 12 T + T^{2} \)
$83$ \( ( -2 + T )^{2} \)
$89$ \( 36 - 6 T + T^{2} \)
$97$ \( 144 - 12 T + T^{2} \)
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