# Properties

 Label 784.2.x.p Level $784$ Weight $2$ Character orbit 784.x Analytic conductor $6.260$ Analytic rank $0$ Dimension $96$ CM no Inner twists $8$

# 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: $$96$$ Relative dimension: $$24$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96 q + 4 q^{4}+O(q^{10})$$ 96 * q + 4 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$96 q + 4 q^{4} + 8 q^{11} + 64 q^{15} - 36 q^{16} - 20 q^{18} - 56 q^{22} - 32 q^{29} - 96 q^{30} - 40 q^{32} + 80 q^{36} + 16 q^{37} + 16 q^{43} - 4 q^{44} - 64 q^{46} - 56 q^{50} - 16 q^{53} + 20 q^{58} - 8 q^{60} + 88 q^{64} - 40 q^{67} + 196 q^{72} + 28 q^{74} + 112 q^{78} - 80 q^{79} + 48 q^{81} + 108 q^{86} + 100 q^{88} + 128 q^{95} - 80 q^{99}+O(q^{100})$$ 96 * q + 4 * q^4 + 8 * q^11 + 64 * q^15 - 36 * q^16 - 20 * q^18 - 56 * q^22 - 32 * q^29 - 96 * q^30 - 40 * q^32 + 80 * q^36 + 16 * q^37 + 16 * q^43 - 4 * q^44 - 64 * q^46 - 56 * q^50 - 16 * q^53 + 20 * q^58 - 8 * q^60 + 88 * q^64 - 40 * q^67 + 196 * q^72 + 28 * q^74 + 112 * q^78 - 80 * q^79 + 48 * q^81 + 108 * q^86 + 100 * q^88 + 128 * q^95 - 80 * 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 −1.40346 0.174081i −0.548135 2.04567i 1.93939 + 0.488631i 0.207992 0.776238i 0.413172 + 2.96643i 0 −2.63679 1.02339i −1.28622 + 0.742602i −0.427037 + 1.05321i
165.2 −1.40346 0.174081i 0.548135 + 2.04567i 1.93939 + 0.488631i −0.207992 + 0.776238i −0.413172 2.96643i 0 −2.63679 1.02339i −1.28622 + 0.742602i 0.427037 1.05321i
165.3 −1.28409 + 0.592547i −0.349212 1.30328i 1.29778 1.52177i −0.743228 + 2.77376i 1.22067 + 1.46660i 0 −0.764745 + 2.72308i 1.02150 0.589763i −0.689213 4.00216i
165.4 −1.28409 + 0.592547i 0.349212 + 1.30328i 1.29778 1.52177i 0.743228 2.77376i −1.22067 1.46660i 0 −0.764745 + 2.72308i 1.02150 0.589763i 0.689213 + 4.00216i
165.5 −0.947436 + 1.04994i −0.0278645 0.103992i −0.204729 1.98949i 1.01195 3.77665i 0.135585 + 0.0692696i 0 2.28281 + 1.66997i 2.58804 1.49420i 3.00648 + 4.64061i
165.6 −0.947436 + 1.04994i 0.0278645 + 0.103992i −0.204729 1.98949i −1.01195 + 3.77665i −0.135585 0.0692696i 0 2.28281 + 1.66997i 2.58804 1.49420i −3.00648 4.64061i
165.7 −0.843680 1.13499i −0.164681 0.614599i −0.576410 + 1.91514i −0.389432 + 1.45338i −0.558626 + 0.705436i 0 2.65997 0.961542i 2.24746 1.29757i 1.97813 0.784185i
165.8 −0.843680 1.13499i 0.164681 + 0.614599i −0.576410 + 1.91514i 0.389432 1.45338i 0.558626 0.705436i 0 2.65997 0.961542i 2.24746 1.29757i −1.97813 + 0.784185i
165.9 −0.646444 1.25782i −0.729365 2.72203i −1.16422 + 1.62622i −0.691444 + 2.58050i −2.95233 + 2.67705i 0 2.79809 + 0.413120i −4.27938 + 2.47070i 3.69279 0.798439i
165.10 −0.646444 1.25782i 0.729365 + 2.72203i −1.16422 + 1.62622i 0.691444 2.58050i 2.95233 2.67705i 0 2.79809 + 0.413120i −4.27938 + 2.47070i −3.69279 + 0.798439i
165.11 0.0679220 + 1.41258i −0.856173 3.19528i −1.99077 + 0.191891i 0.238507 0.890122i 4.45544 1.42644i 0 −0.406279 2.79910i −6.87872 + 3.97143i 1.27357 + 0.276452i
165.12 0.0679220 + 1.41258i 0.856173 + 3.19528i −1.99077 + 0.191891i −0.238507 + 0.890122i −4.45544 + 1.42644i 0 −0.406279 2.79910i −6.87872 + 3.97143i −1.27357 0.276452i
165.13 0.464209 1.33586i −0.0888704 0.331669i −1.56902 1.24023i 0.613128 2.28822i −0.484316 0.0352455i 0 −2.38512 + 1.52026i 2.49597 1.44105i −2.77212 1.88126i
165.14 0.464209 1.33586i 0.0888704 + 0.331669i −1.56902 1.24023i −0.613128 + 2.28822i 0.484316 + 0.0352455i 0 −2.38512 + 1.52026i 2.49597 1.44105i 2.77212 + 1.88126i
165.15 0.581133 + 1.28930i −0.554865 2.07078i −1.32457 + 1.49850i −0.213735 + 0.797669i 2.34740 1.91879i 0 −2.70177 0.836931i −1.38220 + 0.798013i −1.15264 + 0.187984i
165.16 0.581133 + 1.28930i 0.554865 + 2.07078i −1.32457 + 1.49850i 0.213735 0.797669i −2.34740 + 1.91879i 0 −2.70177 0.836931i −1.38220 + 0.798013i 1.15264 0.187984i
165.17 0.977831 1.02169i −0.757585 2.82735i −0.0876948 1.99808i −0.967109 + 3.60930i −3.62946 1.99065i 0 −2.12716 1.86418i −4.82188 + 2.78391i 2.74191 + 4.51737i
165.18 0.977831 1.02169i 0.757585 + 2.82735i −0.0876948 1.99808i 0.967109 3.60930i 3.62946 + 1.99065i 0 −2.12716 1.86418i −4.82188 + 2.78391i −2.74191 4.51737i
165.19 1.22699 + 0.703197i −0.269295 1.00502i 1.01103 + 1.72564i 0.290260 1.08326i 0.376306 1.42253i 0 0.0270652 + 2.82830i 1.66052 0.958704i 1.11790 1.12505i
165.20 1.22699 + 0.703197i 0.269295 + 1.00502i 1.01103 + 1.72564i −0.290260 + 1.08326i −0.376306 + 1.42253i 0 0.0270652 + 2.82830i 1.66052 0.958704i −1.11790 + 1.12505i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 765.24 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
16.e even 4 1 inner
112.l odd 4 1 inner
112.w even 12 1 inner
112.x odd 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.p 96
7.b odd 2 1 inner 784.2.x.p 96
7.c even 3 1 784.2.m.l 48
7.c even 3 1 inner 784.2.x.p 96
7.d odd 6 1 784.2.m.l 48
7.d odd 6 1 inner 784.2.x.p 96
16.e even 4 1 inner 784.2.x.p 96
112.l odd 4 1 inner 784.2.x.p 96
112.w even 12 1 784.2.m.l 48
112.w even 12 1 inner 784.2.x.p 96
112.x odd 12 1 784.2.m.l 48
112.x odd 12 1 inner 784.2.x.p 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.m.l 48 7.c even 3 1
784.2.m.l 48 7.d odd 6 1
784.2.m.l 48 112.w even 12 1
784.2.m.l 48 112.x odd 12 1
784.2.x.p 96 1.a even 1 1 trivial
784.2.x.p 96 7.b odd 2 1 inner
784.2.x.p 96 7.c even 3 1 inner
784.2.x.p 96 7.d odd 6 1 inner
784.2.x.p 96 16.e even 4 1 inner
784.2.x.p 96 112.l odd 4 1 inner
784.2.x.p 96 112.w even 12 1 inner
784.2.x.p 96 112.x odd 12 1 inner

## Hecke kernels

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

 $$T_{3}^{96} - 336 T_{3}^{92} + 69072 T_{3}^{88} - 8966400 T_{3}^{84} + 849400992 T_{3}^{80} - 58586222592 T_{3}^{76} + 3083985715200 T_{3}^{72} - 122067528105984 T_{3}^{68} + \cdots + 4294967296$$ T3^96 - 336*T3^92 + 69072*T3^88 - 8966400*T3^84 + 849400992*T3^80 - 58586222592*T3^76 + 3083985715200*T3^72 - 122067528105984*T3^68 + 3747155880329984*T3^64 - 88448475147563008*T3^60 + 1627396910840127488*T3^56 - 22758390154016522240*T3^52 + 241458515147210006528*T3^48 - 1823413996320809680896*T3^44 + 9831047739401653125120*T3^40 - 31949050074833315954688*T3^36 + 72004952424752735125504*T3^32 - 78460929427749666291712*T3^28 + 59502764111697222303744*T3^24 - 10114973896195002335232*T3^20 + 1373137317150878334976*T3^16 - 19096564361003532288*T3^12 + 240532834278703104*T3^8 - 32311038967808*T3^4 + 4294967296 $$T_{5}^{96} - 832 T_{5}^{92} + 426416 T_{5}^{88} - 139298048 T_{5}^{84} + 33435938208 T_{5}^{80} - 5731077684224 T_{5}^{76} + 736377886682112 T_{5}^{72} + \cdots + 60\!\cdots\!36$$ T5^96 - 832*T5^92 + 426416*T5^88 - 139298048*T5^84 + 33435938208*T5^80 - 5731077684224*T5^76 + 736377886682112*T5^72 - 66735808233768960*T5^68 + 4515884835488973568*T5^64 - 216645259463118340096*T5^60 + 7643664485211123138560*T5^56 - 177299015397965436387328*T5^52 + 2798689268917314753372160*T5^48 - 19349397154483816932835328*T5^44 + 92511098480941416888598528*T5^40 - 230617296993716713008136192*T5^36 + 402066795035085285202395136*T5^32 - 451216726118344511929188352*T5^28 + 370730570978095952534962176*T5^24 - 214307052860740945829691392*T5^20 + 92560209871836212382138368*T5^16 - 27836113651082923379523584*T5^12 + 6009447160366829444005888*T5^8 - 751198847848347384938496*T5^4 + 60336661037197280935936