# Properties

 Label 722.2.e.r Level $722$ Weight $2$ Character orbit 722.e Analytic conductor $5.765$ Analytic rank $0$ Dimension $24$ CM no Inner twists $6$

# Related objects

## 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.

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

## 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 −0.437619 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
99.2 −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
99.3 −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
99.4 −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
245.1 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
245.2 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
245.3 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
245.4 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
389.1 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
389.2 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
389.3 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
389.4 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
415.1 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
415.2 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
415.3 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
415.4 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
423.1 −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
423.2 −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
423.3 −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
423.4 −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
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 595.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.r 24
19.b odd 2 1 722.2.e.s 24
19.c even 3 2 inner 722.2.e.r 24
19.d odd 6 2 722.2.e.s 24
19.e even 9 1 722.2.a.n yes 4
19.e even 9 2 722.2.c.m 8
19.e even 9 3 inner 722.2.e.r 24
19.f odd 18 1 722.2.a.m 4
19.f odd 18 2 722.2.c.n 8
19.f odd 18 3 722.2.e.s 24
57.j even 18 1 6498.2.a.ca 4
57.l odd 18 1 6498.2.a.bx 4
76.k even 18 1 5776.2.a.bv 4
76.l odd 18 1 5776.2.a.bt 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.m 4 19.f odd 18 1
722.2.a.n yes 4 19.e even 9 1
722.2.c.m 8 19.e even 9 2
722.2.c.n 8 19.f odd 18 2
722.2.e.r 24 1.a even 1 1 trivial
722.2.e.r 24 19.c even 3 2 inner
722.2.e.r 24 19.e even 9 3 inner
722.2.e.s 24 19.b odd 2 1
722.2.e.s 24 19.d odd 6 2
722.2.e.s 24 19.f odd 18 3
5776.2.a.bt 4 76.l odd 18 1
5776.2.a.bv 4 76.k even 18 1
6498.2.a.bx 4 57.l odd 18 1
6498.2.a.ca 4 57.j even 18 1

## Hecke kernels

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

 $$T_{3}^{24} + 8 T_{3}^{21} + 400 T_{3}^{18} - 1152 T_{3}^{15} + 119104 T_{3}^{12} + 259072 T_{3}^{9} + 568320 T_{3}^{6} + 49152 T_{3}^{3} + 4096$$ T3^24 + 8*T3^21 + 400*T3^18 - 1152*T3^15 + 119104*T3^12 + 259072*T3^9 + 568320*T3^6 + 49152*T3^3 + 4096 $$T_{5}^{24} + 22 T_{5}^{21} + 1710 T_{5}^{18} - 48048 T_{5}^{15} + 1278099 T_{5}^{12} - 12617792 T_{5}^{9} + 102640310 T_{5}^{6} - 72280142 T_{5}^{3} + 47045881$$ T5^24 + 22*T5^21 + 1710*T5^18 - 48048*T5^15 + 1278099*T5^12 - 12617792*T5^9 + 102640310*T5^6 - 72280142*T5^3 + 47045881 $$T_{7}^{8} - 2T_{7}^{7} + 10T_{7}^{6} - 12T_{7}^{5} + 64T_{7}^{4} - 88T_{7}^{3} + 120T_{7}^{2} - 48T_{7} + 16$$ T7^8 - 2*T7^7 + 10*T7^6 - 12*T7^5 + 64*T7^4 - 88*T7^3 + 120*T7^2 - 48*T7 + 16 $$T_{13}^{24} + 642 T_{13}^{21} + 281870 T_{13}^{18} + 67527952 T_{13}^{15} + 11801523939 T_{13}^{12} + 1049930053408 T_{13}^{9} + 64940441655990 T_{13}^{6} + 1829557198438 T_{13}^{3} + \cdots + 51520374361$$ T13^24 + 642*T13^21 + 281870*T13^18 + 67527952*T13^15 + 11801523939*T13^12 + 1049930053408*T13^9 + 64940441655990*T13^6 + 1829557198438*T13^3 + 51520374361