Properties

Label 722.2.e.d
Level $722$
Weight $2$
Character orbit 722.e
Analytic conductor $5.765$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $6$

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Newspace parameters

Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{18} + \zeta_{18}^{4} ) q^{2} -\zeta_{18} q^{3} -\zeta_{18}^{5} q^{4} -4 \zeta_{18}^{4} q^{5} + ( \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{6} + ( -3 + 3 \zeta_{18}^{3} ) q^{7} + \zeta_{18}^{3} q^{8} -2 \zeta_{18}^{2} q^{9} +O(q^{10})\) \( q + ( -\zeta_{18} + \zeta_{18}^{4} ) q^{2} -\zeta_{18} q^{3} -\zeta_{18}^{5} q^{4} -4 \zeta_{18}^{4} q^{5} + ( \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{6} + ( -3 + 3 \zeta_{18}^{3} ) q^{7} + \zeta_{18}^{3} q^{8} -2 \zeta_{18}^{2} q^{9} + 4 \zeta_{18}^{2} q^{10} -2 \zeta_{18}^{3} q^{11} + ( -1 + \zeta_{18}^{3} ) q^{12} + ( -\zeta_{18}^{2} + \zeta_{18}^{5} ) q^{13} -3 \zeta_{18}^{4} q^{14} + 4 \zeta_{18}^{5} q^{15} -\zeta_{18} q^{16} + ( 3 \zeta_{18} - 3 \zeta_{18}^{4} ) q^{17} + 2 q^{18} -4 q^{20} + ( 3 \zeta_{18} - 3 \zeta_{18}^{4} ) q^{21} + 2 \zeta_{18} q^{22} + \zeta_{18}^{5} q^{23} -\zeta_{18}^{4} q^{24} + ( -11 \zeta_{18}^{2} + 11 \zeta_{18}^{5} ) q^{25} + ( 1 - \zeta_{18}^{3} ) q^{26} + 5 \zeta_{18}^{3} q^{27} + 3 \zeta_{18}^{2} q^{28} + 5 \zeta_{18}^{2} q^{29} -4 \zeta_{18}^{3} q^{30} + ( -8 + 8 \zeta_{18}^{3} ) q^{31} + ( \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{32} + 2 \zeta_{18}^{4} q^{33} + 3 \zeta_{18}^{5} q^{34} + 12 \zeta_{18} q^{35} + ( -2 \zeta_{18} + 2 \zeta_{18}^{4} ) q^{36} + 2 q^{37} + q^{39} + ( 4 \zeta_{18} - 4 \zeta_{18}^{4} ) q^{40} -8 \zeta_{18} q^{41} + 3 \zeta_{18}^{5} q^{42} + 4 \zeta_{18}^{4} q^{43} + ( -2 \zeta_{18}^{2} + 2 \zeta_{18}^{5} ) q^{44} + ( -8 + 8 \zeta_{18}^{3} ) q^{45} -\zeta_{18}^{3} q^{46} + 8 \zeta_{18}^{2} q^{47} + \zeta_{18}^{2} q^{48} -2 \zeta_{18}^{3} q^{49} + ( 11 - 11 \zeta_{18}^{3} ) q^{50} + ( -3 \zeta_{18}^{2} + 3 \zeta_{18}^{5} ) q^{51} + \zeta_{18}^{4} q^{52} -\zeta_{18}^{5} q^{53} -5 \zeta_{18} q^{54} + ( -8 \zeta_{18} + 8 \zeta_{18}^{4} ) q^{55} -3 q^{56} -5 q^{58} + ( -15 \zeta_{18} + 15 \zeta_{18}^{4} ) q^{59} + 4 \zeta_{18} q^{60} -2 \zeta_{18}^{5} q^{61} -8 \zeta_{18}^{4} q^{62} + ( 6 \zeta_{18}^{2} - 6 \zeta_{18}^{5} ) q^{63} + ( -1 + \zeta_{18}^{3} ) q^{64} + 4 \zeta_{18}^{3} q^{65} -2 \zeta_{18}^{2} q^{66} -3 \zeta_{18}^{2} q^{67} -3 \zeta_{18}^{3} q^{68} + ( 1 - \zeta_{18}^{3} ) q^{69} + ( -12 \zeta_{18}^{2} + 12 \zeta_{18}^{5} ) q^{70} -2 \zeta_{18}^{4} q^{71} -2 \zeta_{18}^{5} q^{72} -9 \zeta_{18} q^{73} + ( -2 \zeta_{18} + 2 \zeta_{18}^{4} ) q^{74} + 11 q^{75} + 6 q^{77} + ( -\zeta_{18} + \zeta_{18}^{4} ) q^{78} -10 \zeta_{18} q^{79} + 4 \zeta_{18}^{5} q^{80} + \zeta_{18}^{4} q^{81} + ( 8 \zeta_{18}^{2} - 8 \zeta_{18}^{5} ) q^{82} + ( 6 - 6 \zeta_{18}^{3} ) q^{83} -3 \zeta_{18}^{3} q^{84} -12 \zeta_{18}^{2} q^{85} -4 \zeta_{18}^{2} q^{86} -5 \zeta_{18}^{3} q^{87} + ( 2 - 2 \zeta_{18}^{3} ) q^{88} -8 \zeta_{18}^{4} q^{90} -3 \zeta_{18}^{5} q^{91} + \zeta_{18} q^{92} + ( 8 \zeta_{18} - 8 \zeta_{18}^{4} ) q^{93} -8 q^{94} - q^{96} + ( 2 \zeta_{18} - 2 \zeta_{18}^{4} ) q^{97} + 2 \zeta_{18} q^{98} + 4 \zeta_{18}^{5} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 9q^{7} + 3q^{8} + O(q^{10}) \) \( 6q - 9q^{7} + 3q^{8} - 6q^{11} - 3q^{12} + 12q^{18} - 24q^{20} + 3q^{26} + 15q^{27} - 12q^{30} - 24q^{31} + 12q^{37} + 6q^{39} - 24q^{45} - 3q^{46} - 6q^{49} + 33q^{50} - 18q^{56} - 30q^{58} - 3q^{64} + 12q^{65} - 9q^{68} + 3q^{69} + 66q^{75} + 36q^{77} + 18q^{83} - 9q^{84} - 15q^{87} + 6q^{88} - 48q^{94} - 6q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(363\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.173648 0.984808i
−0.766044 0.642788i
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
0.939693 + 0.342020i
0.939693 + 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i −3.06418 + 2.57115i −0.173648 + 0.984808i −1.50000 + 2.59808i 0.500000 + 0.866025i 1.87939 0.684040i −3.75877 + 1.36808i
245.1 −0.173648 + 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 3.75877 1.36808i −0.766044 + 0.642788i −1.50000 2.59808i 0.500000 0.866025i −0.347296 1.96962i 0.694593 + 3.93923i
389.1 −0.173648 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 3.75877 + 1.36808i −0.766044 0.642788i −1.50000 + 2.59808i 0.500000 + 0.866025i −0.347296 + 1.96962i 0.694593 3.93923i
415.1 −0.766044 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i −0.694593 + 3.93923i 0.939693 + 0.342020i −1.50000 2.59808i 0.500000 0.866025i −1.53209 + 1.28558i 3.06418 2.57115i
423.1 0.939693 0.342020i 0.173648 0.984808i 0.766044 0.642788i −3.06418 2.57115i −0.173648 0.984808i −1.50000 2.59808i 0.500000 0.866025i 1.87939 + 0.684040i −3.75877 1.36808i
595.1 −0.766044 + 0.642788i −0.939693 0.342020i 0.173648 0.984808i −0.694593 3.93923i 0.939693 0.342020i −1.50000 + 2.59808i 0.500000 + 0.866025i −1.53209 1.28558i 3.06418 + 2.57115i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 595.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 2 inner
19.e even 9 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.e.d 6
19.b odd 2 1 722.2.e.c 6
19.c even 3 2 inner 722.2.e.d 6
19.d odd 6 2 722.2.e.c 6
19.e even 9 1 722.2.a.b 1
19.e even 9 2 722.2.c.f 2
19.e even 9 3 inner 722.2.e.d 6
19.f odd 18 1 38.2.a.b 1
19.f odd 18 2 722.2.c.d 2
19.f odd 18 3 722.2.e.c 6
57.j even 18 1 342.2.a.d 1
57.l odd 18 1 6498.2.a.y 1
76.k even 18 1 304.2.a.d 1
76.l odd 18 1 5776.2.a.d 1
95.o odd 18 1 950.2.a.b 1
95.r even 36 2 950.2.b.c 2
133.ba even 18 1 1862.2.a.f 1
152.s odd 18 1 1216.2.a.n 1
152.v even 18 1 1216.2.a.g 1
209.p even 18 1 4598.2.a.a 1
228.u odd 18 1 2736.2.a.w 1
247.bl odd 18 1 6422.2.a.b 1
285.bf even 18 1 8550.2.a.u 1
380.bb even 18 1 7600.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 19.f odd 18 1
304.2.a.d 1 76.k even 18 1
342.2.a.d 1 57.j even 18 1
722.2.a.b 1 19.e even 9 1
722.2.c.d 2 19.f odd 18 2
722.2.c.f 2 19.e even 9 2
722.2.e.c 6 19.b odd 2 1
722.2.e.c 6 19.d odd 6 2
722.2.e.c 6 19.f odd 18 3
722.2.e.d 6 1.a even 1 1 trivial
722.2.e.d 6 19.c even 3 2 inner
722.2.e.d 6 19.e even 9 3 inner
950.2.a.b 1 95.o odd 18 1
950.2.b.c 2 95.r even 36 2
1216.2.a.g 1 152.v even 18 1
1216.2.a.n 1 152.s odd 18 1
1862.2.a.f 1 133.ba even 18 1
2736.2.a.w 1 228.u odd 18 1
4598.2.a.a 1 209.p even 18 1
5776.2.a.d 1 76.l odd 18 1
6422.2.a.b 1 247.bl odd 18 1
6498.2.a.y 1 57.l odd 18 1
7600.2.a.h 1 380.bb even 18 1
8550.2.a.u 1 285.bf even 18 1

Hecke kernels

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

\( T_{3}^{6} + T_{3}^{3} + 1 \)
\( T_{5}^{6} - 64 T_{5}^{3} + 4096 \)
\( T_{7}^{2} + 3 T_{7} + 9 \)
\( T_{13}^{6} + T_{13}^{3} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{3} + T^{6} \)
$3$ \( 1 + T^{3} + T^{6} \)
$5$ \( 4096 - 64 T^{3} + T^{6} \)
$7$ \( ( 9 + 3 T + T^{2} )^{3} \)
$11$ \( ( 4 + 2 T + T^{2} )^{3} \)
$13$ \( 1 + T^{3} + T^{6} \)
$17$ \( 729 + 27 T^{3} + T^{6} \)
$19$ \( T^{6} \)
$23$ \( 1 - T^{3} + T^{6} \)
$29$ \( 15625 + 125 T^{3} + T^{6} \)
$31$ \( ( 64 + 8 T + T^{2} )^{3} \)
$37$ \( ( -2 + T )^{6} \)
$41$ \( 262144 + 512 T^{3} + T^{6} \)
$43$ \( 4096 + 64 T^{3} + T^{6} \)
$47$ \( 262144 + 512 T^{3} + T^{6} \)
$53$ \( 1 + T^{3} + T^{6} \)
$59$ \( 11390625 - 3375 T^{3} + T^{6} \)
$61$ \( 64 + 8 T^{3} + T^{6} \)
$67$ \( 729 - 27 T^{3} + T^{6} \)
$71$ \( 64 - 8 T^{3} + T^{6} \)
$73$ \( 531441 + 729 T^{3} + T^{6} \)
$79$ \( 1000000 + 1000 T^{3} + T^{6} \)
$83$ \( ( 36 - 6 T + T^{2} )^{3} \)
$89$ \( T^{6} \)
$97$ \( 64 + 8 T^{3} + T^{6} \)
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