Properties

Label 722.2.e.s
Level $722$
Weight $2$
Character orbit 722.e
Analytic conductor $5.765$
Analytic rank $0$
Dimension $24$
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: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{7} + 12 q^{8} - 6 q^{11} + 6 q^{12} - 24 q^{18} - 12 q^{20} - 54 q^{26} - 12 q^{27} - 12 q^{30} + 78 q^{31} - 24 q^{37} - 36 q^{39} + 66 q^{45} - 30 q^{46} + 36 q^{49} + 18 q^{50} + 12 q^{56} - 12 q^{58} - 12 q^{64} - 12 q^{65} - 18 q^{68} - 60 q^{69} - 48 q^{75} + 24 q^{77} + 36 q^{83} + 12 q^{84} + 78 q^{87} + 6 q^{88} + 72 q^{94} + 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 0.939693 + 0.342020i −0.485104 2.75116i 0.766044 + 0.642788i −1.79605 + 1.50706i 0.485104 2.75116i 0.642040 1.11205i 0.500000 + 0.866025i −4.51450 + 1.64314i −2.20318 + 0.801892i
99.2 0.939693 + 0.342020i −0.222978 1.26457i 0.766044 + 0.642788i 2.83108 2.37556i 0.222978 1.26457i 0.221232 0.383185i 0.500000 + 0.866025i 1.26966 0.462117i 3.47284 1.26401i
99.3 0.939693 + 0.342020i −0.0768330 0.435741i 0.766044 + 0.642788i −0.682922 + 0.573040i 0.0768330 0.435741i −1.26007 + 2.18251i 0.500000 + 0.866025i 2.63511 0.959102i −0.837728 + 0.304908i
99.4 0.939693 + 0.342020i 0.437619 + 2.48186i 0.766044 + 0.642788i −1.88420 + 1.58103i −0.437619 + 2.48186i 1.39680 2.41933i 0.500000 + 0.866025i −3.14904 + 1.14616i −2.31131 + 0.841250i
245.1 −0.173648 + 0.984808i −2.14003 1.79569i −0.939693 0.342020i 2.20318 0.801892i 2.14003 1.79569i 0.642040 + 1.11205i 0.500000 0.866025i 0.834245 + 4.73124i 0.407131 + 2.30896i
245.2 −0.173648 + 0.984808i −0.983662 0.825390i −0.939693 0.342020i −3.47284 + 1.26401i 0.983662 0.825390i 0.221232 + 0.383185i 0.500000 0.866025i −0.234623 1.33061i −0.641755 3.63957i
245.3 −0.173648 + 0.984808i −0.338947 0.284410i −0.939693 0.342020i 0.837728 0.304908i 0.338947 0.284410i −1.26007 2.18251i 0.500000 0.866025i −0.486949 2.76162i 0.154806 + 0.877948i
245.4 −0.173648 + 0.984808i 1.93054 + 1.61992i −0.939693 0.342020i 2.31131 0.841250i −1.93054 + 1.61992i 1.39680 + 2.41933i 0.500000 0.866025i 0.581920 + 3.30023i 0.427114 + 2.42228i
389.1 −0.173648 0.984808i −2.14003 + 1.79569i −0.939693 + 0.342020i 2.20318 + 0.801892i 2.14003 + 1.79569i 0.642040 1.11205i 0.500000 + 0.866025i 0.834245 4.73124i 0.407131 2.30896i
389.2 −0.173648 0.984808i −0.983662 + 0.825390i −0.939693 + 0.342020i −3.47284 1.26401i 0.983662 + 0.825390i 0.221232 0.383185i 0.500000 + 0.866025i −0.234623 + 1.33061i −0.641755 + 3.63957i
389.3 −0.173648 0.984808i −0.338947 + 0.284410i −0.939693 + 0.342020i 0.837728 + 0.304908i 0.338947 + 0.284410i −1.26007 + 2.18251i 0.500000 + 0.866025i −0.486949 + 2.76162i 0.154806 0.877948i
389.4 −0.173648 0.984808i 1.93054 1.61992i −0.939693 + 0.342020i 2.31131 + 0.841250i −1.93054 1.61992i 1.39680 2.41933i 0.500000 + 0.866025i 0.581920 3.30023i 0.427114 2.42228i
415.1 −0.766044 0.642788i −2.36816 + 0.861941i 0.173648 + 0.984808i −0.427114 + 2.42228i 2.36816 + 0.861941i 1.39680 + 2.41933i 0.500000 0.866025i 2.56712 2.15407i 1.88420 1.58103i
415.2 −0.766044 0.642788i 0.415780 0.151331i 0.173648 + 0.984808i −0.154806 + 0.877948i −0.415780 0.151331i −1.26007 2.18251i 0.500000 0.866025i −2.14816 + 1.80252i 0.682922 0.573040i
415.3 −0.766044 0.642788i 1.20664 0.439181i 0.173648 + 0.984808i 0.641755 3.63957i −1.20664 0.439181i 0.221232 + 0.383185i 0.500000 0.866025i −1.03503 + 0.868497i −2.83108 + 2.37556i
415.4 −0.766044 0.642788i 2.62513 0.955469i 0.173648 + 0.984808i −0.407131 + 2.30896i −2.62513 0.955469i 0.642040 + 1.11205i 0.500000 0.866025i 3.68025 3.08810i 1.79605 1.50706i
423.1 0.939693 0.342020i −0.485104 + 2.75116i 0.766044 0.642788i −1.79605 1.50706i 0.485104 + 2.75116i 0.642040 + 1.11205i 0.500000 0.866025i −4.51450 1.64314i −2.20318 0.801892i
423.2 0.939693 0.342020i −0.222978 + 1.26457i 0.766044 0.642788i 2.83108 + 2.37556i 0.222978 + 1.26457i 0.221232 + 0.383185i 0.500000 0.866025i 1.26966 + 0.462117i 3.47284 + 1.26401i
423.3 0.939693 0.342020i −0.0768330 + 0.435741i 0.766044 0.642788i −0.682922 0.573040i 0.0768330 + 0.435741i −1.26007 2.18251i 0.500000 0.866025i 2.63511 + 0.959102i −0.837728 0.304908i
423.4 0.939693 0.342020i 0.437619 2.48186i 0.766044 0.642788i −1.88420 1.58103i −0.437619 2.48186i 1.39680 + 2.41933i 0.500000 0.866025i −3.14904 1.14616i −2.31131 0.841250i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 595.4
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.s 24
19.b odd 2 1 722.2.e.r 24
19.c even 3 2 inner 722.2.e.s 24
19.d odd 6 2 722.2.e.r 24
19.e even 9 1 722.2.a.m 4
19.e even 9 2 722.2.c.n 8
19.e even 9 3 inner 722.2.e.s 24
19.f odd 18 1 722.2.a.n yes 4
19.f odd 18 2 722.2.c.m 8
19.f odd 18 3 722.2.e.r 24
57.j even 18 1 6498.2.a.bx 4
57.l odd 18 1 6498.2.a.ca 4
76.k even 18 1 5776.2.a.bt 4
76.l odd 18 1 5776.2.a.bv 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.m 4 19.e even 9 1
722.2.a.n yes 4 19.f odd 18 1
722.2.c.m 8 19.f odd 18 2
722.2.c.n 8 19.e even 9 2
722.2.e.r 24 19.b odd 2 1
722.2.e.r 24 19.d odd 6 2
722.2.e.r 24 19.f odd 18 3
722.2.e.s 24 1.a even 1 1 trivial
722.2.e.s 24 19.c even 3 2 inner
722.2.e.s 24 19.e even 9 3 inner
5776.2.a.bt 4 76.k even 18 1
5776.2.a.bv 4 76.l odd 18 1
6498.2.a.bx 4 57.j even 18 1
6498.2.a.ca 4 57.l odd 18 1

Hecke kernels

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

\( T_{3}^{24} - 8 T_{3}^{21} + 400 T_{3}^{18} + 1152 T_{3}^{15} + 119104 T_{3}^{12} - 259072 T_{3}^{9} + 568320 T_{3}^{6} - 49152 T_{3}^{3} + 4096 \) Copy content Toggle raw display
\( T_{5}^{24} + 22 T_{5}^{21} + 1710 T_{5}^{18} - 48048 T_{5}^{15} + 1278099 T_{5}^{12} - 12617792 T_{5}^{9} + 102640310 T_{5}^{6} - 72280142 T_{5}^{3} + 47045881 \) Copy content Toggle raw display
\( T_{7}^{8} - 2T_{7}^{7} + 10T_{7}^{6} - 12T_{7}^{5} + 64T_{7}^{4} - 88T_{7}^{3} + 120T_{7}^{2} - 48T_{7} + 16 \) Copy content Toggle raw display
\( T_{13}^{24} - 642 T_{13}^{21} + 281870 T_{13}^{18} - 67527952 T_{13}^{15} + 11801523939 T_{13}^{12} - 1049930053408 T_{13}^{9} + 64940441655990 T_{13}^{6} - 1829557198438 T_{13}^{3} + \cdots + 51520374361 \) Copy content Toggle raw display