# Properties

 Label 280.2.bv.e Level $280$ Weight $2$ Character orbit 280.bv Analytic conductor $2.236$ Analytic rank $0$ Dimension $160$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.bv (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$160q - 2q^{2} + 12q^{7} + 4q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$160q - 2q^{2} + 12q^{7} + 4q^{8} - 6q^{10} + 6q^{12} - 8q^{15} + 4q^{16} - 12q^{17} - 28q^{18} - 24q^{22} - 16q^{23} + 20q^{25} - 12q^{26} - 46q^{28} + 32q^{30} + 48q^{31} + 18q^{32} - 12q^{33} - 32q^{36} - 48q^{38} + 54q^{40} + 6q^{42} - 64q^{46} - 132q^{47} - 12q^{50} - 20q^{56} - 88q^{57} + 6q^{58} + 34q^{60} - 32q^{63} - 28q^{65} - 180q^{66} + 60q^{68} - 108q^{70} - 160q^{71} + 52q^{72} + 84q^{73} + 48q^{78} - 48q^{80} + 16q^{81} - 90q^{82} - 84q^{86} - 12q^{87} + 44q^{88} + 36q^{92} - 20q^{95} - 48q^{96} - 94q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
117.1 −1.41101 0.0951125i −2.31786 0.621070i 1.98191 + 0.268410i 1.20230 + 1.88533i 3.21146 + 1.09679i −0.485979 2.60074i −2.77096 0.567234i 2.38869 + 1.37911i −1.51714 2.77458i
117.2 −1.40415 + 0.168429i −2.01733 0.540542i 1.94326 0.472997i −1.27480 1.83708i 2.92367 + 0.419225i 2.57779 0.595830i −2.64896 + 0.991460i 1.17936 + 0.680902i 2.09943 + 2.36482i
117.3 −1.39835 + 0.211218i 1.14290 + 0.306239i 1.91077 0.590715i −2.15054 + 0.612514i −1.66286 0.186828i −0.757877 + 2.53488i −2.54716 + 1.22962i −1.38564 0.800000i 2.87784 1.31074i
117.4 −1.31662 + 0.516255i −1.14290 0.306239i 1.46696 1.35942i 2.15054 0.612514i 1.66286 0.186828i −0.757877 + 2.53488i −1.22962 + 2.54716i −1.38564 0.800000i −2.51523 + 1.91668i
117.5 −1.30024 + 0.556211i 2.01733 + 0.540542i 1.38126 1.44642i 1.27480 + 1.83708i −2.92367 + 0.419225i 2.57779 0.595830i −0.991460 + 2.64896i 1.17936 + 0.680902i −2.67936 1.67960i
117.6 −1.29882 0.559518i 1.21391 + 0.325266i 1.37388 + 1.45343i 2.11050 0.738764i −1.39466 1.10167i 2.63915 0.186831i −0.971208 2.65646i −1.23029 0.710311i −3.15452 0.221340i
117.7 −1.24982 0.661775i 3.03007 + 0.811904i 1.12411 + 1.65420i 0.203394 + 2.22680i −3.24974 3.01996i −2.64044 0.167503i −0.310224 2.81136i 5.92404 + 3.42025i 1.21943 2.91770i
117.8 −1.17442 + 0.787876i 2.31786 + 0.621070i 0.758504 1.85059i −1.20230 1.88533i −3.21146 + 1.09679i −0.485979 2.60074i 0.567234 + 2.77096i 2.38869 + 1.37911i 2.89741 + 1.26690i
117.9 −1.16877 0.796228i −2.48518 0.665902i 0.732043 + 1.86121i −2.21132 + 0.331769i 2.37439 + 2.75706i −2.18569 + 1.49090i 0.626360 2.75820i 3.13462 + 1.80978i 2.84868 + 1.37295i
117.10 −0.910407 1.08220i −0.0340321 0.00911887i −0.342320 + 1.97049i −1.86153 + 1.23883i 0.0211146 + 0.0451315i 1.05182 2.42769i 2.44411 1.42349i −2.59700 1.49938i 3.03542 + 0.886711i
117.11 −0.880343 1.10680i 1.90798 + 0.511242i −0.449991 + 1.94872i −1.79531 1.33299i −1.11384 2.56181i 1.57290 + 2.12743i 2.55298 1.21749i 0.780947 + 0.450880i 0.105136 + 3.16053i
117.12 −0.868191 1.11635i −2.84558 0.762471i −0.492489 + 1.93842i 1.93059 1.12820i 1.61932 + 3.83864i 2.58960 + 0.542205i 2.59153 1.13312i 4.91789 + 2.83934i −2.93558 1.17573i
117.13 −0.845055 + 1.13397i −1.21391 0.325266i −0.571766 1.91653i −2.11050 + 0.738764i 1.39466 1.10167i 2.63915 0.186831i 2.65646 + 0.971208i −1.23029 0.710311i 0.945756 3.01754i
117.14 −0.751490 + 1.19802i −3.03007 0.811904i −0.870527 1.80061i −0.203394 2.22680i 3.24974 3.01996i −2.64044 0.167503i 2.81136 + 0.310224i 5.92404 + 3.42025i 2.82061 + 1.42975i
117.15 −0.614070 + 1.27394i 2.48518 + 0.665902i −1.24584 1.56457i 2.21132 0.331769i −2.37439 + 2.75706i −2.18569 + 1.49090i 2.75820 0.626360i 3.13462 + 1.80978i −0.935251 + 3.02081i
117.16 −0.464402 1.33579i −0.0539369 0.0144524i −1.56866 + 1.24069i 1.91563 + 1.15341i 0.00574315 + 0.0787600i −1.56007 2.13686i 2.38578 + 1.51922i −2.59538 1.49844i 0.651085 3.09453i
117.17 −0.279851 1.38625i 2.92962 + 0.784989i −1.84337 + 0.775886i 1.27091 1.83978i 0.268333 4.28086i 0.0664755 2.64492i 1.59144 + 2.33823i 5.36839 + 3.09944i −2.90605 1.24694i
117.18 −0.247335 + 1.39242i 0.0340321 + 0.00911887i −1.87765 0.688786i 1.86153 1.23883i −0.0211146 + 0.0451315i 1.05182 2.42769i 1.42349 2.44411i −2.59700 1.49938i 1.26455 + 2.89843i
117.19 −0.240080 1.39369i −0.931735 0.249658i −1.88472 + 0.669192i −0.0962936 2.23399i −0.124254 + 1.35848i −1.59310 + 2.11235i 1.38513 + 2.46605i −1.79228 1.03477i −3.09037 + 0.670540i
117.20 −0.209002 + 1.39868i −1.90798 0.511242i −1.91264 0.584656i 1.79531 + 1.33299i 1.11384 2.56181i 1.57290 + 2.12743i 1.21749 2.55298i 0.780947 + 0.450880i −2.23966 + 2.23247i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 213.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
35.k even 12 1 inner
40.i odd 4 1 inner
56.j odd 6 1 inner
280.bv even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.bv.e 160
5.c odd 4 1 inner 280.2.bv.e 160
7.d odd 6 1 inner 280.2.bv.e 160
8.b even 2 1 inner 280.2.bv.e 160
35.k even 12 1 inner 280.2.bv.e 160
40.i odd 4 1 inner 280.2.bv.e 160
56.j odd 6 1 inner 280.2.bv.e 160
280.bv even 12 1 inner 280.2.bv.e 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bv.e 160 1.a even 1 1 trivial
280.2.bv.e 160 5.c odd 4 1 inner
280.2.bv.e 160 7.d odd 6 1 inner
280.2.bv.e 160 8.b even 2 1 inner
280.2.bv.e 160 35.k even 12 1 inner
280.2.bv.e 160 40.i odd 4 1 inner
280.2.bv.e 160 56.j odd 6 1 inner
280.2.bv.e 160 280.bv even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$70\!\cdots\!12$$$$T_{3}^{136} -$$$$59\!\cdots\!04$$$$T_{3}^{132} +$$$$41\!\cdots\!62$$$$T_{3}^{128} -$$$$24\!\cdots\!24$$$$T_{3}^{124} +$$$$12\!\cdots\!40$$$$T_{3}^{120} -$$$$54\!\cdots\!28$$$$T_{3}^{116} +$$$$20\!\cdots\!28$$$$T_{3}^{112} -$$$$63\!\cdots\!56$$$$T_{3}^{108} +$$$$17\!\cdots\!32$$$$T_{3}^{104} -$$$$39\!\cdots\!00$$$$T_{3}^{100} +$$$$77\!\cdots\!71$$$$T_{3}^{96} -$$$$12\!\cdots\!52$$$$T_{3}^{92} +$$$$18\!\cdots\!24$$$$T_{3}^{88} -$$$$21\!\cdots\!96$$$$T_{3}^{84} +$$$$20\!\cdots\!36$$$$T_{3}^{80} -$$$$16\!\cdots\!64$$$$T_{3}^{76} +$$$$10\!\cdots\!76$$$$T_{3}^{72} -$$$$56\!\cdots\!48$$$$T_{3}^{68} +$$$$22\!\cdots\!98$$$$T_{3}^{64} -$$$$71\!\cdots\!68$$$$T_{3}^{60} +$$$$17\!\cdots\!28$$$$T_{3}^{56} -$$$$31\!\cdots\!00$$$$T_{3}^{52} +$$$$45\!\cdots\!44$$$$T_{3}^{48} -$$$$47\!\cdots\!60$$$$T_{3}^{44} +$$$$36\!\cdots\!32$$$$T_{3}^{40} -$$$$18\!\cdots\!00$$$$T_{3}^{36} +$$$$62\!\cdots\!77$$$$T_{3}^{32} -$$$$63\!\cdots\!00$$$$T_{3}^{28} +$$$$48\!\cdots\!28$$$$T_{3}^{24} -$$$$81\!\cdots\!56$$$$T_{3}^{20} +$$$$11\!\cdots\!36$$$$T_{3}^{16} -$$$$12\!\cdots\!00$$$$T_{3}^{12} +$$$$12\!\cdots\!00$$$$T_{3}^{8} -$$$$19\!\cdots\!00$$$$T_{3}^{4} + 25600000000$$">$$T_{3}^{160} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.