# Properties

 Label 784.2.x.m Level $784$ Weight $2$ Character orbit 784.x Analytic conductor $6.260$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.x (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 2 q^{2} + 4 q^{3} + 6 q^{4} + 4 q^{5} - 8 q^{6} - 8 q^{8}+O(q^{10})$$ 24 * q - 2 * q^2 + 4 * q^3 + 6 * q^4 + 4 * q^5 - 8 * q^6 - 8 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 2 q^{2} + 4 q^{3} + 6 q^{4} + 4 q^{5} - 8 q^{6} - 8 q^{8} - 4 q^{10} + 8 q^{12} - 48 q^{15} - 10 q^{16} - 8 q^{17} + 40 q^{20} + 28 q^{22} - 8 q^{24} - 20 q^{26} - 8 q^{27} - 8 q^{29} + 28 q^{30} - 8 q^{31} - 12 q^{32} - 16 q^{34} - 32 q^{36} + 20 q^{37} + 16 q^{38} - 8 q^{40} + 32 q^{43} - 14 q^{44} + 40 q^{45} + 28 q^{46} + 16 q^{47} - 32 q^{48} + 88 q^{50} + 16 q^{51} - 16 q^{52} - 4 q^{53} + 64 q^{54} - 14 q^{58} - 16 q^{59} - 60 q^{60} - 20 q^{61} - 16 q^{62} - 36 q^{64} - 32 q^{65} + 12 q^{66} - 24 q^{67} - 28 q^{68} + 8 q^{69} - 6 q^{72} + 38 q^{74} - 40 q^{75} - 96 q^{76} - 152 q^{78} - 24 q^{79} + 24 q^{80} + 44 q^{81} - 16 q^{82} + 40 q^{83} - 16 q^{85} - 38 q^{86} + 14 q^{88} + 80 q^{90} + 64 q^{92} + 48 q^{93} - 24 q^{94} - 16 q^{96} - 96 q^{97} + 64 q^{99}+O(q^{100})$$ 24 * q - 2 * q^2 + 4 * q^3 + 6 * q^4 + 4 * q^5 - 8 * q^6 - 8 * q^8 - 4 * q^10 + 8 * q^12 - 48 * q^15 - 10 * q^16 - 8 * q^17 + 40 * q^20 + 28 * q^22 - 8 * q^24 - 20 * q^26 - 8 * q^27 - 8 * q^29 + 28 * q^30 - 8 * q^31 - 12 * q^32 - 16 * q^34 - 32 * q^36 + 20 * q^37 + 16 * q^38 - 8 * q^40 + 32 * q^43 - 14 * q^44 + 40 * q^45 + 28 * q^46 + 16 * q^47 - 32 * q^48 + 88 * q^50 + 16 * q^51 - 16 * q^52 - 4 * q^53 + 64 * q^54 - 14 * q^58 - 16 * q^59 - 60 * q^60 - 20 * q^61 - 16 * q^62 - 36 * q^64 - 32 * q^65 + 12 * q^66 - 24 * q^67 - 28 * q^68 + 8 * q^69 - 6 * q^72 + 38 * q^74 - 40 * q^75 - 96 * q^76 - 152 * q^78 - 24 * q^79 + 24 * q^80 + 44 * q^81 - 16 * q^82 + 40 * q^83 - 16 * q^85 - 38 * q^86 + 14 * q^88 + 80 * q^90 + 64 * q^92 + 48 * q^93 - 24 * q^94 - 16 * q^96 - 96 * q^97 + 64 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1 −1.40268 0.180222i −0.152579 0.569433i 1.93504 + 0.505589i 0.414228 1.54592i 0.111396 + 0.826233i 0 −2.62313 1.05792i 2.29710 1.32623i −0.859639 + 2.09378i
165.2 −1.29835 + 0.560616i 0.752181 + 2.80718i 1.37142 1.45575i −0.998394 + 3.72606i −2.55034 3.22301i 0 −0.964462 + 2.65891i −4.71639 + 2.72301i −0.792625 5.39744i
165.3 −0.329210 1.37536i −0.231031 0.862219i −1.78324 + 0.905566i 0.857449 3.20004i −1.10981 + 0.601602i 0 1.83254 + 2.15448i 1.90803 1.10160i −4.68350 0.125816i
165.4 0.114575 + 1.40956i −0.509227 1.90046i −1.97375 + 0.323001i −0.792364 + 2.95714i 2.62048 0.935533i 0 −0.681431 2.74511i −0.754366 + 0.435533i −4.25907 0.778075i
165.5 1.03822 0.960264i 0.219607 + 0.819585i 0.155788 1.99392i −0.356864 + 1.33183i 1.01502 + 0.640026i 0 −1.75295 2.21972i 1.97458 1.14003i 0.908409 + 1.72541i
165.6 1.37745 0.320359i −0.811002 3.02670i 1.79474 0.882557i 0.143894 0.537019i −2.08675 3.90932i 0 2.18943 1.79064i −5.90511 + 3.40932i 0.0261678 0.785815i
373.1 −1.35072 0.418990i −0.819585 0.219607i 1.64889 + 1.13188i 1.33183 0.356864i 1.01502 + 0.640026i 0 −1.75295 2.21972i −1.97458 1.14003i −1.94846 0.0760019i
373.2 −1.02649 + 0.972785i 0.862219 + 0.231031i 0.107378 1.99712i −3.20004 + 0.857449i −1.10981 + 0.601602i 0 1.83254 + 2.15448i −1.90803 1.10160i 2.45071 3.99312i
373.3 −0.966164 1.03273i 3.02670 + 0.811002i −0.133053 + 1.99557i −0.537019 + 0.143894i −2.08675 3.90932i 0 2.18943 1.79064i 5.90511 + 3.40932i 0.667452 + 0.415570i
373.4 0.545265 + 1.30487i 0.569433 + 0.152579i −1.40537 + 1.42300i −1.54592 + 0.414228i 0.111396 + 0.826233i 0 −2.62313 1.05792i −2.29710 1.32623i −1.38345 1.79136i
373.5 1.13468 + 0.844095i −2.80718 0.752181i 0.575008 + 1.91556i 3.72606 0.998394i −2.55034 3.22301i 0 −0.964462 + 2.65891i 4.71639 + 2.72301i 5.07063 + 2.01229i
373.6 1.16343 0.804007i 1.90046 + 0.509227i 0.707146 1.87081i 2.95714 0.792364i 2.62048 0.935533i 0 −0.681431 2.74511i 0.754366 + 0.435533i 2.80337 3.29942i
557.1 −1.35072 + 0.418990i −0.819585 + 0.219607i 1.64889 1.13188i 1.33183 + 0.356864i 1.01502 0.640026i 0 −1.75295 + 2.21972i −1.97458 + 1.14003i −1.94846 + 0.0760019i
557.2 −1.02649 0.972785i 0.862219 0.231031i 0.107378 + 1.99712i −3.20004 0.857449i −1.10981 0.601602i 0 1.83254 2.15448i −1.90803 + 1.10160i 2.45071 + 3.99312i
557.3 −0.966164 + 1.03273i 3.02670 0.811002i −0.133053 1.99557i −0.537019 0.143894i −2.08675 + 3.90932i 0 2.18943 + 1.79064i 5.90511 3.40932i 0.667452 0.415570i
557.4 0.545265 1.30487i 0.569433 0.152579i −1.40537 1.42300i −1.54592 0.414228i 0.111396 0.826233i 0 −2.62313 + 1.05792i −2.29710 + 1.32623i −1.38345 + 1.79136i
557.5 1.13468 0.844095i −2.80718 + 0.752181i 0.575008 1.91556i 3.72606 + 0.998394i −2.55034 + 3.22301i 0 −0.964462 2.65891i 4.71639 2.72301i 5.07063 2.01229i
557.6 1.16343 + 0.804007i 1.90046 0.509227i 0.707146 + 1.87081i 2.95714 + 0.792364i 2.62048 + 0.935533i 0 −0.681431 + 2.74511i 0.754366 0.435533i 2.80337 + 3.29942i
765.1 −1.40268 + 0.180222i −0.152579 + 0.569433i 1.93504 0.505589i 0.414228 + 1.54592i 0.111396 0.826233i 0 −2.62313 + 1.05792i 2.29710 + 1.32623i −0.859639 2.09378i
765.2 −1.29835 0.560616i 0.752181 2.80718i 1.37142 + 1.45575i −0.998394 3.72606i −2.55034 + 3.22301i 0 −0.964462 2.65891i −4.71639 2.72301i −0.792625 + 5.39744i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 765.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.m 24
7.b odd 2 1 784.2.x.l 24
7.c even 3 1 784.2.m.h 12
7.c even 3 1 inner 784.2.x.m 24
7.d odd 6 1 112.2.m.d 12
7.d odd 6 1 784.2.x.l 24
16.e even 4 1 inner 784.2.x.m 24
28.f even 6 1 448.2.m.d 12
56.j odd 6 1 896.2.m.g 12
56.m even 6 1 896.2.m.h 12
112.l odd 4 1 784.2.x.l 24
112.v even 12 1 448.2.m.d 12
112.v even 12 1 896.2.m.h 12
112.w even 12 1 784.2.m.h 12
112.w even 12 1 inner 784.2.x.m 24
112.x odd 12 1 112.2.m.d 12
112.x odd 12 1 784.2.x.l 24
112.x odd 12 1 896.2.m.g 12
224.bc odd 24 2 7168.2.a.bj 12
224.be even 24 2 7168.2.a.bi 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.d 12 7.d odd 6 1
112.2.m.d 12 112.x odd 12 1
448.2.m.d 12 28.f even 6 1
448.2.m.d 12 112.v even 12 1
784.2.m.h 12 7.c even 3 1
784.2.m.h 12 112.w even 12 1
784.2.x.l 24 7.b odd 2 1
784.2.x.l 24 7.d odd 6 1
784.2.x.l 24 112.l odd 4 1
784.2.x.l 24 112.x odd 12 1
784.2.x.m 24 1.a even 1 1 trivial
784.2.x.m 24 7.c even 3 1 inner
784.2.x.m 24 16.e even 4 1 inner
784.2.x.m 24 112.w even 12 1 inner
896.2.m.g 12 56.j odd 6 1
896.2.m.g 12 112.x odd 12 1
896.2.m.h 12 56.m even 6 1
896.2.m.h 12 112.v even 12 1
7168.2.a.bi 12 224.be even 24 2
7168.2.a.bj 12 224.bc odd 24 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$:

 $$T_{3}^{24} - 4 T_{3}^{23} + 8 T_{3}^{22} - 24 T_{3}^{21} - 28 T_{3}^{20} + 288 T_{3}^{19} - 640 T_{3}^{18} + 1720 T_{3}^{17} + 3052 T_{3}^{16} - 20848 T_{3}^{15} + 41248 T_{3}^{14} - 101520 T_{3}^{13} + 182160 T_{3}^{12} - 122368 T_{3}^{11} + \cdots + 4096$$ T3^24 - 4*T3^23 + 8*T3^22 - 24*T3^21 - 28*T3^20 + 288*T3^19 - 640*T3^18 + 1720*T3^17 + 3052*T3^16 - 20848*T3^15 + 41248*T3^14 - 101520*T3^13 + 182160*T3^12 - 122368*T3^11 + 51872*T3^10 + 288*T3^9 - 80496*T3^8 + 68544*T3^7 - 28032*T3^6 - 2560*T3^5 + 44800*T3^4 - 36864*T3^3 + 18432*T3^2 - 12288*T3 + 4096 $$T_{5}^{24} - 4 T_{5}^{23} + 8 T_{5}^{22} - 56 T_{5}^{21} - 28 T_{5}^{20} + 712 T_{5}^{19} - 1056 T_{5}^{18} + 7160 T_{5}^{17} + 5772 T_{5}^{16} - 84656 T_{5}^{15} + 115392 T_{5}^{14} - 809680 T_{5}^{13} + 1242704 T_{5}^{12} + \cdots + 5308416$$ T5^24 - 4*T5^23 + 8*T5^22 - 56*T5^21 - 28*T5^20 + 712*T5^19 - 1056*T5^18 + 7160*T5^17 + 5772*T5^16 - 84656*T5^15 + 115392*T5^14 - 809680*T5^13 + 1242704*T5^12 + 2631680*T5^11 - 442336*T5^10 + 6569248*T5^9 - 2129648*T5^8 - 14186496*T5^7 - 460800*T5^6 - 28035072*T5^5 + 31177728*T5^4 + 35389440*T5^3 + 10616832*T5^2 + 10616832*T5 + 5308416