# Properties

 Label 280.2.bj.f Level $280$ Weight $2$ Character orbit 280.bj Analytic conductor $2.236$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 3 q^{2} + 12 q^{3} + q^{4} - 12 q^{5} - 2 q^{6} + 10 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10})$$ 24 * q + 3 * q^2 + 12 * q^3 + q^4 - 12 * q^5 - 2 * q^6 + 10 * q^7 - 6 * q^8 + 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 3 q^{2} + 12 q^{3} + q^{4} - 12 q^{5} - 2 q^{6} + 10 q^{7} - 6 q^{8} + 12 q^{9} - 3 q^{10} + 8 q^{11} + 10 q^{12} + 20 q^{13} + 5 q^{14} - 3 q^{16} + 6 q^{17} + 3 q^{18} + 18 q^{19} - 2 q^{20} - 26 q^{21} - 16 q^{22} + 18 q^{23} - 18 q^{24} - 12 q^{25} - 37 q^{26} - 41 q^{28} - 8 q^{30} - 6 q^{31} + 3 q^{32} + 12 q^{33} - 20 q^{34} - 8 q^{35} - 22 q^{36} + 15 q^{38} - 18 q^{39} + 3 q^{40} - 12 q^{42} + 32 q^{43} + 35 q^{44} + 12 q^{45} - 49 q^{46} - 14 q^{48} + 8 q^{49} - 22 q^{51} - 41 q^{52} + 30 q^{53} + 104 q^{54} - 16 q^{55} + 44 q^{56} - 44 q^{57} - 54 q^{58} - 18 q^{59} - 8 q^{60} + 22 q^{61} + 8 q^{62} - 12 q^{63} + 58 q^{64} - 10 q^{65} + 8 q^{66} - 8 q^{67} + 18 q^{68} - 12 q^{69} - q^{70} - 17 q^{72} + 30 q^{73} + 53 q^{74} - 12 q^{75} + 8 q^{76} - 32 q^{77} - 8 q^{78} + 6 q^{79} - 3 q^{80} - 4 q^{81} - 57 q^{82} + 42 q^{86} + 14 q^{87} - 17 q^{88} - 60 q^{89} + 24 q^{90} + 18 q^{91} - 38 q^{92} - 18 q^{93} + 19 q^{94} - 18 q^{95} - 74 q^{96} + 3 q^{98} - 56 q^{99}+O(q^{100})$$ 24 * q + 3 * q^2 + 12 * q^3 + q^4 - 12 * q^5 - 2 * q^6 + 10 * q^7 - 6 * q^8 + 12 * q^9 - 3 * q^10 + 8 * q^11 + 10 * q^12 + 20 * q^13 + 5 * q^14 - 3 * q^16 + 6 * q^17 + 3 * q^18 + 18 * q^19 - 2 * q^20 - 26 * q^21 - 16 * q^22 + 18 * q^23 - 18 * q^24 - 12 * q^25 - 37 * q^26 - 41 * q^28 - 8 * q^30 - 6 * q^31 + 3 * q^32 + 12 * q^33 - 20 * q^34 - 8 * q^35 - 22 * q^36 + 15 * q^38 - 18 * q^39 + 3 * q^40 - 12 * q^42 + 32 * q^43 + 35 * q^44 + 12 * q^45 - 49 * q^46 - 14 * q^48 + 8 * q^49 - 22 * q^51 - 41 * q^52 + 30 * q^53 + 104 * q^54 - 16 * q^55 + 44 * q^56 - 44 * q^57 - 54 * q^58 - 18 * q^59 - 8 * q^60 + 22 * q^61 + 8 * q^62 - 12 * q^63 + 58 * q^64 - 10 * q^65 + 8 * q^66 - 8 * q^67 + 18 * q^68 - 12 * q^69 - q^70 - 17 * q^72 + 30 * q^73 + 53 * q^74 - 12 * q^75 + 8 * q^76 - 32 * q^77 - 8 * q^78 + 6 * q^79 - 3 * q^80 - 4 * q^81 - 57 * q^82 + 42 * q^86 + 14 * q^87 - 17 * q^88 - 60 * q^89 + 24 * q^90 + 18 * q^91 - 38 * q^92 - 18 * q^93 + 19 * q^94 - 18 * q^95 - 74 * q^96 + 3 * q^98 - 56 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
131.1 −1.41058 + 0.101290i 0.725648 0.418953i 1.97948 0.285754i −0.500000 + 0.866025i −0.981150 + 0.664468i −2.36913 + 1.17781i −2.76328 + 0.603581i −1.14896 + 1.99005i 0.617571 1.27224i
131.2 −1.33674 0.461647i 1.90624 1.10057i 1.57376 + 1.23421i −0.500000 + 0.866025i −3.05623 + 0.591168i −0.584379 2.58041i −1.53395 2.37634i 0.922503 1.59782i 1.06817 0.926830i
131.3 −0.938224 + 1.05818i −0.219454 + 0.126702i −0.239473 1.98561i −0.500000 + 0.866025i 0.0718241 0.351096i 0.978876 2.45801i 2.32581 + 1.60954i −1.46789 + 2.54247i −0.447296 1.34161i
131.4 −0.771333 1.18535i 0.784482 0.452921i −0.810092 + 1.82859i −0.500000 + 0.866025i −1.14196 0.580530i 1.23347 + 2.34063i 2.79237 0.450214i −1.08973 + 1.88746i 1.41221 0.0753205i
131.5 −0.410069 + 1.35346i 2.26702 1.30886i −1.66369 1.11002i −0.500000 + 0.866025i 0.841856 + 3.60504i 1.72615 + 2.00510i 2.18459 1.79654i 1.92625 3.33637i −0.967093 1.03186i
131.6 0.314756 1.37874i −2.66758 + 1.54013i −1.80186 0.867935i −0.500000 + 0.866025i 1.28380 + 4.16266i 2.53597 0.754231i −1.76380 + 2.21111i 3.24397 5.61873i 1.03665 + 0.961958i
131.7 0.542272 + 1.30612i −0.908317 + 0.524417i −1.41188 + 1.41654i −0.500000 + 0.866025i −1.17750 0.901991i 2.14799 + 1.54472i −2.61579 1.07593i −0.949974 + 1.64540i −1.40227 0.183437i
131.8 0.669637 1.24563i 1.94732 1.12428i −1.10317 1.66823i −0.500000 + 0.866025i −0.0964439 3.17849i 2.47009 0.947962i −2.81672 + 0.257032i 1.02803 1.78060i 0.743926 + 1.20274i
131.9 0.959261 + 1.03914i 2.75363 1.58981i −0.159637 + 1.99362i −0.500000 + 0.866025i 4.29350 + 1.33638i −1.04250 2.43170i −2.22479 + 1.74651i 3.55500 6.15745i −1.37955 + 0.311173i
131.10 1.11603 + 0.868601i −0.502680 + 0.290223i 0.491065 + 1.93878i −0.500000 + 0.866025i −0.813096 0.112730i −2.63362 + 0.253028i −1.13598 + 2.59028i −1.33154 + 2.30630i −1.31025 + 0.532213i
131.11 1.37350 0.336893i 1.75472 1.01309i 1.77301 0.925446i −0.500000 + 0.866025i 2.06881 1.98263i −1.63843 + 2.07739i 2.12345 1.86841i 0.552704 0.957311i −0.394992 + 1.35793i
131.12 1.39149 0.252502i −1.84104 + 1.06293i 1.87249 0.702708i −0.500000 + 0.866025i −2.29340 + 1.94392i 2.17552 + 1.50569i 2.42811 1.45062i 0.759621 1.31570i −0.477071 + 1.33132i
171.1 −1.41058 0.101290i 0.725648 + 0.418953i 1.97948 + 0.285754i −0.500000 0.866025i −0.981150 0.664468i −2.36913 1.17781i −2.76328 0.603581i −1.14896 1.99005i 0.617571 + 1.27224i
171.2 −1.33674 + 0.461647i 1.90624 + 1.10057i 1.57376 1.23421i −0.500000 0.866025i −3.05623 0.591168i −0.584379 + 2.58041i −1.53395 + 2.37634i 0.922503 + 1.59782i 1.06817 + 0.926830i
171.3 −0.938224 1.05818i −0.219454 0.126702i −0.239473 + 1.98561i −0.500000 0.866025i 0.0718241 + 0.351096i 0.978876 + 2.45801i 2.32581 1.60954i −1.46789 2.54247i −0.447296 + 1.34161i
171.4 −0.771333 + 1.18535i 0.784482 + 0.452921i −0.810092 1.82859i −0.500000 0.866025i −1.14196 + 0.580530i 1.23347 2.34063i 2.79237 + 0.450214i −1.08973 1.88746i 1.41221 + 0.0753205i
171.5 −0.410069 1.35346i 2.26702 + 1.30886i −1.66369 + 1.11002i −0.500000 0.866025i 0.841856 3.60504i 1.72615 2.00510i 2.18459 + 1.79654i 1.92625 + 3.33637i −0.967093 + 1.03186i
171.6 0.314756 + 1.37874i −2.66758 1.54013i −1.80186 + 0.867935i −0.500000 0.866025i 1.28380 4.16266i 2.53597 + 0.754231i −1.76380 2.21111i 3.24397 + 5.61873i 1.03665 0.961958i
171.7 0.542272 1.30612i −0.908317 0.524417i −1.41188 1.41654i −0.500000 0.866025i −1.17750 + 0.901991i 2.14799 1.54472i −2.61579 + 1.07593i −0.949974 1.64540i −1.40227 + 0.183437i
171.8 0.669637 + 1.24563i 1.94732 + 1.12428i −1.10317 + 1.66823i −0.500000 0.866025i −0.0964439 + 3.17849i 2.47009 + 0.947962i −2.81672 0.257032i 1.02803 + 1.78060i 0.743926 1.20274i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 171.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.m even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.bj.f yes 24
4.b odd 2 1 1120.2.bz.e 24
7.d odd 6 1 280.2.bj.e 24
8.b even 2 1 1120.2.bz.f 24
8.d odd 2 1 280.2.bj.e 24
28.f even 6 1 1120.2.bz.f 24
56.j odd 6 1 1120.2.bz.e 24
56.m even 6 1 inner 280.2.bj.f yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bj.e 24 7.d odd 6 1
280.2.bj.e 24 8.d odd 2 1
280.2.bj.f yes 24 1.a even 1 1 trivial
280.2.bj.f yes 24 56.m even 6 1 inner
1120.2.bz.e 24 4.b odd 2 1
1120.2.bz.e 24 56.j odd 6 1
1120.2.bz.f 24 8.b even 2 1
1120.2.bz.f 24 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$:

 $$T_{3}^{24} - 12 T_{3}^{23} + 48 T_{3}^{22} - 494 T_{3}^{20} + 492 T_{3}^{19} + 5528 T_{3}^{18} - 18120 T_{3}^{17} + 2835 T_{3}^{16} + 73188 T_{3}^{15} - 67428 T_{3}^{14} - 269256 T_{3}^{13} + 637314 T_{3}^{12} - 189744 T_{3}^{11} + \cdots + 4096$$ T3^24 - 12*T3^23 + 48*T3^22 - 494*T3^20 + 492*T3^19 + 5528*T3^18 - 18120*T3^17 + 2835*T3^16 + 73188*T3^15 - 67428*T3^14 - 269256*T3^13 + 637314*T3^12 - 189744*T3^11 - 764820*T3^10 + 536664*T3^9 + 776561*T3^8 - 947640*T3^7 - 99516*T3^6 + 446496*T3^5 + 11984*T3^4 - 151872*T3^3 + 8704*T3^2 + 21504*T3 + 4096 $$T_{13}^{12} - 10 T_{13}^{11} - 34 T_{13}^{10} + 512 T_{13}^{9} + 221 T_{13}^{8} - 9034 T_{13}^{7} - 424 T_{13}^{6} + 69028 T_{13}^{5} + 22412 T_{13}^{4} - 196672 T_{13}^{3} - 135600 T_{13}^{2} + 108288 T_{13} + 81088$$ T13^12 - 10*T13^11 - 34*T13^10 + 512*T13^9 + 221*T13^8 - 9034*T13^7 - 424*T13^6 + 69028*T13^5 + 22412*T13^4 - 196672*T13^3 - 135600*T13^2 + 108288*T13 + 81088