# Properties

 Label 280.2.br.a Level $280$ Weight $2$ Character orbit 280.br Analytic conductor $2.236$ Analytic rank $0$ Dimension $176$ 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.br (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$176$$ Relative dimension: $$44$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$176q - 2q^{2} - 4q^{3} - 16q^{6} - 20q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$176q - 2q^{2} - 4q^{3} - 16q^{6} - 20q^{8} - 2q^{10} - 8q^{11} - 14q^{12} + 4q^{16} - 4q^{17} - 16q^{18} + 8q^{20} - 4q^{25} - 20q^{26} - 40q^{27} - 34q^{28} - 12q^{30} + 18q^{32} + 20q^{33} - 32q^{35} - 48q^{36} - 16q^{38} - 22q^{40} - 32q^{41} + 30q^{42} - 16q^{43} + 20q^{46} + 36q^{48} + 68q^{50} - 8q^{51} - 32q^{52} - 52q^{56} - 40q^{57} - 14q^{58} - 50q^{60} + 56q^{62} - 4q^{65} - 60q^{66} - 28q^{67} - 40q^{68} + 64q^{70} - 48q^{72} - 4q^{73} - 4q^{75} - 144q^{76} + 184q^{78} - 40q^{80} + 32q^{81} + 74q^{82} - 16q^{83} - 56q^{86} - 64q^{88} - 56q^{90} + 16q^{91} + 44q^{92} - 104q^{96} - 48q^{97} + 70q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1 −1.41382 0.0333641i 0.388267 + 1.44903i 1.99777 + 0.0943417i 2.17362 0.524773i −0.500594 2.06162i 2.60799 0.445401i −2.82134 0.200036i 0.649136 0.374779i −3.09061 + 0.669413i
67.2 −1.41151 + 0.0874515i −0.147176 0.549269i 1.98470 0.246877i −1.31051 + 1.81179i 0.255774 + 0.762426i 0.374780 2.61907i −2.77983 + 0.522034i 2.31804 1.33832i 1.69135 2.67196i
67.3 −1.38856 + 0.268130i 0.262906 + 0.981180i 1.85621 0.744631i −1.42407 1.72396i −0.628146 1.29194i 0.0598247 + 2.64507i −2.37781 + 1.53167i 1.70448 0.984083i 2.43965 + 2.01199i
67.4 −1.38301 0.295435i −0.482920 1.80228i 1.82544 + 0.817180i 0.915641 2.04000i 0.135426 + 2.63525i −2.62546 0.327068i −2.28317 1.66947i −0.416933 + 0.240716i −1.86903 + 2.55083i
67.5 −1.32896 + 0.483609i 0.835437 + 3.11790i 1.53224 1.28539i 0.804876 + 2.08619i −2.61810 3.73952i −2.64531 + 0.0485395i −1.41466 + 2.44923i −6.42524 + 3.70961i −2.07854 2.38320i
67.6 −1.32319 0.499155i −0.833838 3.11193i 1.50169 + 1.32096i −2.20028 + 0.398453i −0.450005 + 4.53390i 1.42374 + 2.23002i −1.32766 2.49746i −6.39072 + 3.68968i 3.11029 + 0.571051i
67.7 −1.24593 + 0.669082i −0.569871 2.12679i 1.10466 1.66725i 2.01915 + 0.960747i 2.13301 + 2.26853i 0.895661 + 2.48954i −0.260793 + 2.81638i −1.60039 + 0.923986i −3.15853 + 0.153958i
67.8 −1.18837 0.766666i 0.205610 + 0.767348i 0.824448 + 1.82217i 1.68075 + 1.47481i 0.343958 1.06953i −2.28076 + 1.34095i 0.417242 2.79748i 2.05153 1.18445i −0.866669 3.04120i
67.9 −1.17517 0.786756i 0.673346 + 2.51296i 0.762030 + 1.84914i −1.71938 + 1.42959i 1.18579 3.48291i 2.61222 + 0.419863i 0.559308 2.77258i −3.26351 + 1.88419i 3.14530 0.327278i
67.10 −1.16325 + 0.804274i −0.540340 2.01658i 0.706287 1.87114i 0.161306 2.23024i 2.25043 + 1.91120i 1.67330 2.04940i 0.683321 + 2.74464i −1.17654 + 0.679275i 1.60609 + 2.72406i
67.11 −1.04840 + 0.949136i −0.0956638 0.357022i 0.198283 1.99015i −1.68428 + 1.47078i 0.439157 + 0.283504i −2.10200 + 1.60674i 1.68104 + 2.27467i 2.47976 1.43169i 0.369825 3.14058i
67.12 −0.986956 1.01288i −0.565335 2.10986i −0.0518375 + 1.99933i 1.71378 + 1.43630i −1.57907 + 2.65495i 1.45748 2.20811i 2.07623 1.92074i −1.53383 + 0.885555i −0.236634 3.15341i
67.13 −0.942490 + 1.05438i 0.438259 + 1.63560i −0.223426 1.98748i −0.615814 2.14960i −2.13760 1.07945i −0.885584 2.49314i 2.30613 + 1.63760i 0.114945 0.0663635i 2.84689 + 1.37667i
67.14 −0.786782 1.17515i 0.552175 + 2.06075i −0.761947 + 1.84917i 1.28687 1.82865i 1.98724 2.27025i −0.126396 2.64273i 2.77254 0.559495i −1.34370 + 0.775786i −3.16142 0.0735182i
67.15 −0.784588 1.17661i −0.0488512 0.182315i −0.768843 + 1.84632i −0.546472 2.16826i −0.176187 + 0.200521i 1.82312 + 1.91735i 2.77563 0.543964i 2.56722 1.48219i −2.12246 + 2.34418i
67.16 −0.624657 + 1.26878i 0.629416 + 2.34901i −1.21961 1.58511i 1.82068 1.29812i −3.37355 0.668737i 0.511566 + 2.59582i 2.77299 0.557266i −2.52362 + 1.45701i 0.509724 + 3.12093i
67.17 −0.431412 1.34680i −0.409884 1.52971i −1.62777 + 1.16206i −2.22655 0.206144i −1.88339 + 1.21197i −2.08935 1.62314i 2.26730 + 1.69096i 0.426076 0.245995i 0.682924 + 3.08766i
67.18 −0.421789 + 1.34985i −0.220018 0.821117i −1.64419 1.13870i 2.20849 + 0.350108i 1.20119 + 0.0493476i −2.38534 1.14463i 2.23058 1.73911i 1.97225 1.13868i −1.40411 + 2.83346i
67.19 −0.309645 + 1.37990i −0.220018 0.821117i −1.80824 0.854556i −2.20849 0.350108i 1.20119 0.0493476i 2.38534 + 1.14463i 1.73911 2.23058i 1.97225 1.13868i 1.16696 2.93908i
67.20 −0.110222 1.40991i 0.0164275 + 0.0613082i −1.97570 + 0.310806i −0.336260 + 2.21064i 0.0846285 0.0299188i 2.52833 + 0.779437i 0.655974 + 2.75131i 2.59459 1.49799i 3.15387 + 0.230436i
See next 80 embeddings (of 176 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 163.44 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.c even 3 1 inner
8.d odd 2 1 inner
35.l odd 12 1 inner
40.k even 4 1 inner
56.k odd 6 1 inner
280.br 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.br.a 176
5.c odd 4 1 inner 280.2.br.a 176
7.c even 3 1 inner 280.2.br.a 176
8.d odd 2 1 inner 280.2.br.a 176
35.l odd 12 1 inner 280.2.br.a 176
40.k even 4 1 inner 280.2.br.a 176
56.k odd 6 1 inner 280.2.br.a 176
280.br even 12 1 inner 280.2.br.a 176

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.br.a 176 1.a even 1 1 trivial
280.2.br.a 176 5.c odd 4 1 inner
280.2.br.a 176 7.c even 3 1 inner
280.2.br.a 176 8.d odd 2 1 inner
280.2.br.a 176 35.l odd 12 1 inner
280.2.br.a 176 40.k even 4 1 inner
280.2.br.a 176 56.k odd 6 1 inner
280.2.br.a 176 280.br even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(280, [\chi])$$.