Properties

Label 722.2.e.f
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: \(\Q(\zeta_{18})\)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{18} + \zeta_{18}^{4} ) q^{2} -\zeta_{18} q^{3} -\zeta_{18}^{5} q^{4} + ( \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{6} + ( 1 - \zeta_{18}^{3} ) q^{7} + \zeta_{18}^{3} q^{8} -2 \zeta_{18}^{2} q^{9} +O(q^{10})\) \( q + ( -\zeta_{18} + \zeta_{18}^{4} ) q^{2} -\zeta_{18} q^{3} -\zeta_{18}^{5} q^{4} + ( \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{6} + ( 1 - \zeta_{18}^{3} ) q^{7} + \zeta_{18}^{3} q^{8} -2 \zeta_{18}^{2} q^{9} + 6 \zeta_{18}^{3} q^{11} + ( -1 + \zeta_{18}^{3} ) q^{12} + ( -5 \zeta_{18}^{2} + 5 \zeta_{18}^{5} ) q^{13} + \zeta_{18}^{4} q^{14} -\zeta_{18} q^{16} + ( 3 \zeta_{18} - 3 \zeta_{18}^{4} ) q^{17} + 2 q^{18} + ( -\zeta_{18} + \zeta_{18}^{4} ) q^{21} -6 \zeta_{18} q^{22} -3 \zeta_{18}^{5} q^{23} -\zeta_{18}^{4} q^{24} + ( 5 \zeta_{18}^{2} - 5 \zeta_{18}^{5} ) q^{25} + ( 5 - 5 \zeta_{18}^{3} ) q^{26} + 5 \zeta_{18}^{3} q^{27} -\zeta_{18}^{2} q^{28} + 9 \zeta_{18}^{2} q^{29} + ( 4 - 4 \zeta_{18}^{3} ) q^{31} + ( \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{32} -6 \zeta_{18}^{4} q^{33} + 3 \zeta_{18}^{5} q^{34} + ( -2 \zeta_{18} + 2 \zeta_{18}^{4} ) q^{36} + 2 q^{37} + 5 q^{39} -\zeta_{18}^{5} q^{42} + 8 \zeta_{18}^{4} q^{43} + ( 6 \zeta_{18}^{2} - 6 \zeta_{18}^{5} ) q^{44} + 3 \zeta_{18}^{3} q^{46} + \zeta_{18}^{2} q^{48} + 6 \zeta_{18}^{3} q^{49} + ( -5 + 5 \zeta_{18}^{3} ) q^{50} + ( -3 \zeta_{18}^{2} + 3 \zeta_{18}^{5} ) q^{51} + 5 \zeta_{18}^{4} q^{52} + 3 \zeta_{18}^{5} q^{53} -5 \zeta_{18} q^{54} + q^{56} -9 q^{58} + ( 9 \zeta_{18} - 9 \zeta_{18}^{4} ) q^{59} + 10 \zeta_{18}^{5} q^{61} + 4 \zeta_{18}^{4} q^{62} + ( -2 \zeta_{18}^{2} + 2 \zeta_{18}^{5} ) q^{63} + ( -1 + \zeta_{18}^{3} ) q^{64} + 6 \zeta_{18}^{2} q^{66} + 5 \zeta_{18}^{2} q^{67} -3 \zeta_{18}^{3} q^{68} + ( -3 + 3 \zeta_{18}^{3} ) q^{69} -6 \zeta_{18}^{4} q^{71} -2 \zeta_{18}^{5} q^{72} + 7 \zeta_{18} q^{73} + ( -2 \zeta_{18} + 2 \zeta_{18}^{4} ) q^{74} -5 q^{75} + 6 q^{77} + ( -5 \zeta_{18} + 5 \zeta_{18}^{4} ) q^{78} + 10 \zeta_{18} q^{79} + \zeta_{18}^{4} q^{81} + ( 6 - 6 \zeta_{18}^{3} ) q^{83} + \zeta_{18}^{3} q^{84} -8 \zeta_{18}^{2} q^{86} -9 \zeta_{18}^{3} q^{87} + ( -6 + 6 \zeta_{18}^{3} ) q^{88} + ( 12 \zeta_{18}^{2} - 12 \zeta_{18}^{5} ) q^{89} + 5 \zeta_{18}^{5} q^{91} -3 \zeta_{18} q^{92} + ( -4 \zeta_{18} + 4 \zeta_{18}^{4} ) q^{93} - q^{96} + ( -10 \zeta_{18} + 10 \zeta_{18}^{4} ) q^{97} -6 \zeta_{18} q^{98} -12 \zeta_{18}^{5} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{7} + 3q^{8} + O(q^{10}) \) \( 6q + 3q^{7} + 3q^{8} + 18q^{11} - 3q^{12} + 12q^{18} + 15q^{26} + 15q^{27} + 12q^{31} + 12q^{37} + 30q^{39} + 9q^{46} + 18q^{49} - 15q^{50} + 6q^{56} - 54q^{58} - 3q^{64} - 9q^{68} - 9q^{69} - 30q^{75} + 36q^{77} + 18q^{83} + 3q^{84} - 27q^{87} - 18q^{88} - 6q^{96} + O(q^{100}) \)

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 0 −0.173648 + 0.984808i 0.500000 0.866025i 0.500000 + 0.866025i 1.87939 0.684040i 0
245.1 −0.173648 + 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i 0 −0.766044 + 0.642788i 0.500000 + 0.866025i 0.500000 0.866025i −0.347296 1.96962i 0
389.1 −0.173648 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i 0 −0.766044 0.642788i 0.500000 0.866025i 0.500000 + 0.866025i −0.347296 + 1.96962i 0
415.1 −0.766044 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i 0 0.939693 + 0.342020i 0.500000 + 0.866025i 0.500000 0.866025i −1.53209 + 1.28558i 0
423.1 0.939693 0.342020i 0.173648 0.984808i 0.766044 0.642788i 0 −0.173648 0.984808i 0.500000 + 0.866025i 0.500000 0.866025i 1.87939 + 0.684040i 0
595.1 −0.766044 + 0.642788i −0.939693 0.342020i 0.173648 0.984808i 0 0.939693 0.342020i 0.500000 0.866025i 0.500000 + 0.866025i −1.53209 1.28558i 0
\(n\): 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.f 6
19.b odd 2 1 722.2.e.e 6
19.c even 3 2 inner 722.2.e.f 6
19.d odd 6 2 722.2.e.e 6
19.e even 9 1 38.2.a.a 1
19.e even 9 2 722.2.c.e 2
19.e even 9 3 inner 722.2.e.f 6
19.f odd 18 1 722.2.a.e 1
19.f odd 18 2 722.2.c.c 2
19.f odd 18 3 722.2.e.e 6
57.j even 18 1 6498.2.a.f 1
57.l odd 18 1 342.2.a.e 1
76.k even 18 1 5776.2.a.m 1
76.l odd 18 1 304.2.a.c 1
95.p even 18 1 950.2.a.d 1
95.q odd 36 2 950.2.b.b 2
133.y odd 18 1 1862.2.a.b 1
152.t even 18 1 1216.2.a.e 1
152.u odd 18 1 1216.2.a.m 1
209.q odd 18 1 4598.2.a.p 1
228.v even 18 1 2736.2.a.n 1
247.bn even 18 1 6422.2.a.h 1
285.bd odd 18 1 8550.2.a.m 1
380.ba odd 18 1 7600.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 19.e even 9 1
304.2.a.c 1 76.l odd 18 1
342.2.a.e 1 57.l odd 18 1
722.2.a.e 1 19.f odd 18 1
722.2.c.c 2 19.f odd 18 2
722.2.c.e 2 19.e even 9 2
722.2.e.e 6 19.b odd 2 1
722.2.e.e 6 19.d odd 6 2
722.2.e.e 6 19.f odd 18 3
722.2.e.f 6 1.a even 1 1 trivial
722.2.e.f 6 19.c even 3 2 inner
722.2.e.f 6 19.e even 9 3 inner
950.2.a.d 1 95.p even 18 1
950.2.b.b 2 95.q odd 36 2
1216.2.a.e 1 152.t even 18 1
1216.2.a.m 1 152.u odd 18 1
1862.2.a.b 1 133.y odd 18 1
2736.2.a.n 1 228.v even 18 1
4598.2.a.p 1 209.q odd 18 1
5776.2.a.m 1 76.k even 18 1
6422.2.a.h 1 247.bn even 18 1
6498.2.a.f 1 57.j even 18 1
7600.2.a.n 1 380.ba odd 18 1
8550.2.a.m 1 285.bd odd 18 1

Hecke kernels

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

\( T_{3}^{6} + T_{3}^{3} + 1 \)
\( T_{5} \)
\( T_{7}^{2} - T_{7} + 1 \)
\( T_{13}^{6} + 125 T_{13}^{3} + 15625 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{3} + T^{6} \)
$3$ \( 1 - 8 T^{3} + 37 T^{6} - 216 T^{9} + 729 T^{12} \)
$5$ \( 1 - 125 T^{6} + 15625 T^{12} \)
$7$ \( ( 1 - 5 T + 7 T^{2} )^{3}( 1 + 4 T + 7 T^{2} )^{3} \)
$11$ \( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{3} \)
$13$ \( ( 1 - 89 T^{3} + 2197 T^{6} )( 1 + 19 T^{3} + 2197 T^{6} ) \)
$17$ \( 1 - 126 T^{3} + 10963 T^{6} - 619038 T^{9} + 24137569 T^{12} \)
$19$ 1
$23$ \( 1 - 180 T^{3} + 20233 T^{6} - 2190060 T^{9} + 148035889 T^{12} \)
$29$ \( 1 - 54 T^{3} - 21473 T^{6} - 1317006 T^{9} + 594823321 T^{12} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )^{3}( 1 + 7 T + 31 T^{2} )^{3} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{6} \)
$41$ \( 1 - 68921 T^{6} + 4750104241 T^{12} \)
$43$ \( ( 1 - 449 T^{3} + 79507 T^{6} )( 1 - 71 T^{3} + 79507 T^{6} ) \)
$47$ \( 1 - 103823 T^{6} + 10779215329 T^{12} \)
$53$ \( 1 + 450 T^{3} + 53623 T^{6} + 66994650 T^{9} + 22164361129 T^{12} \)
$59$ \( 1 - 864 T^{3} + 541117 T^{6} - 177447456 T^{9} + 42180533641 T^{12} \)
$61$ \( 1 + 830 T^{3} + 461919 T^{6} + 188394230 T^{9} + 51520374361 T^{12} \)
$67$ \( ( 1 - 1007 T^{3} + 300763 T^{6} )( 1 + 127 T^{3} + 300763 T^{6} ) \)
$71$ \( 1 + 1062 T^{3} + 769933 T^{6} + 380101482 T^{9} + 128100283921 T^{12} \)
$73$ \( ( 1 + 271 T^{3} + 389017 T^{6} )( 1 + 919 T^{3} + 389017 T^{6} ) \)
$79$ \( 1 + 1370 T^{3} + 1383861 T^{6} + 675463430 T^{9} + 243087455521 T^{12} \)
$83$ \( ( 1 - 6 T - 47 T^{2} - 498 T^{3} + 6889 T^{4} )^{3} \)
$89$ \( 1 + 1476 T^{3} + 1473607 T^{6} + 1040534244 T^{9} + 496981290961 T^{12} \)
$97$ \( 1 + 1910 T^{3} + 2735427 T^{6} + 1743205430 T^{9} + 832972004929 T^{12} \)
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