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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(722, [\chi])\):

\( T_{3}^{2} - 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

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