# Properties

 Label 280.2.bl.b Level $280$ Weight $2$ Character orbit 280.bl Analytic conductor $2.236$ Analytic rank $0$ Dimension $60$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$30$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$60q + 4q^{2} - 2q^{4} + 8q^{7} + 4q^{8} + 38q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$60q + 4q^{2} - 2q^{4} + 8q^{7} + 4q^{8} + 38q^{9} + 2q^{10} - 10q^{12} + 20q^{14} - 22q^{16} - 12q^{17} + 6q^{18} - 16q^{20} - 28q^{22} - 2q^{23} - 32q^{24} + 30q^{25} - 4q^{26} - 22q^{28} - 6q^{32} - 16q^{34} + 20q^{36} - 42q^{38} + 4q^{40} - 36q^{41} - 62q^{42} + 14q^{44} + 28q^{46} - 30q^{47} + 124q^{48} + 12q^{49} + 8q^{50} + 8q^{52} - 40q^{54} - 44q^{55} + 8q^{56} - 32q^{57} - 14q^{58} + 14q^{60} - 16q^{62} - 74q^{63} + 4q^{64} + 2q^{65} + 60q^{66} + 28q^{68} + 10q^{70} - 8q^{71} - 72q^{72} - 52q^{74} - 12q^{76} - 40q^{78} + 32q^{79} - 22q^{81} - 16q^{82} - 8q^{84} - 44q^{86} + 48q^{87} - 16q^{88} + 4q^{89} - 48q^{90} + 84q^{92} - 56q^{94} + 2q^{95} - 16q^{96} - 96q^{97} - 80q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
221.1 −1.40842 + 0.127929i 1.92077 + 1.10896i 1.96727 0.360354i −0.866025 + 0.500000i −2.84711 1.31615i 2.33530 1.24353i −2.72463 + 0.759198i 0.959573 + 1.66203i 1.15576 0.814997i
221.2 −1.40765 0.136130i −2.20766 1.27459i 1.96294 + 0.383247i −0.866025 + 0.500000i 2.93409 + 2.09470i −0.879470 2.49530i −2.71095 0.806691i 1.74917 + 3.02964i 1.28712 0.585931i
221.3 −1.33035 + 0.479764i 2.73909 + 1.58141i 1.53965 1.27651i 0.866025 0.500000i −4.40265 0.789713i −2.49644 + 0.876244i −1.43585 + 2.43687i 3.50174 + 6.06518i −0.912233 + 1.08066i
221.4 −1.21704 + 0.720280i −1.50534 0.869106i 0.962395 1.75322i −0.866025 + 0.500000i 2.45806 0.0265219i −0.830882 + 2.51190i 0.0915345 + 2.82695i 0.0106917 + 0.0185186i 0.693852 1.23230i
221.5 −1.11578 + 0.868928i 0.501254 + 0.289399i 0.489927 1.93906i 0.866025 0.500000i −0.810756 + 0.112648i 1.67292 2.04972i 1.13826 + 2.58928i −1.33250 2.30795i −0.531829 + 1.31040i
221.6 −1.10700 0.880081i −2.81779 1.62685i 0.450915 + 1.94851i 0.866025 0.500000i 1.68754 + 4.28082i −0.452341 + 2.60680i 1.21568 2.55385i 3.79330 + 6.57019i −1.39873 0.208671i
221.7 −1.02369 0.975735i −0.992927 0.573266i 0.0958836 + 1.99770i −0.866025 + 0.500000i 0.457093 + 1.55568i 2.64418 + 0.0910290i 1.85107 2.13858i −0.842731 1.45965i 1.37441 + 0.333166i
221.8 −0.900839 1.09018i 0.214333 + 0.123745i −0.376980 + 1.96415i −0.866025 + 0.500000i −0.0581749 0.345135i −2.36404 + 1.18800i 2.48087 1.35841i −1.46937 2.54503i 1.32524 + 0.493703i
221.9 −0.867668 + 1.11676i −1.96523 1.13463i −0.494304 1.93795i 0.866025 0.500000i 2.97227 1.21021i −2.64045 0.167479i 2.59312 + 1.12948i 1.07475 + 1.86152i −0.193043 + 1.40098i
221.10 −0.785061 + 1.17630i 1.69968 + 0.981313i −0.767360 1.84693i −0.866025 + 0.500000i −2.48867 + 1.22895i 1.68530 + 2.03955i 2.77497 + 0.547309i 0.425950 + 0.737767i 0.0917329 1.41124i
221.11 −0.493703 1.32524i −0.214333 0.123745i −1.51251 + 1.30855i 0.866025 0.500000i −0.0581749 + 0.345135i −2.36404 + 1.18800i 2.48087 + 1.35841i −1.46937 2.54503i −1.09018 0.900839i
221.12 −0.333166 1.37441i 0.992927 + 0.573266i −1.77800 + 0.915813i 0.866025 0.500000i 0.457093 1.55568i 2.64418 + 0.0910290i 1.85107 + 2.13858i −0.842731 1.45965i −0.975735 1.02369i
221.13 −0.278951 + 1.38643i −0.433972 0.250554i −1.84437 0.773493i −0.866025 + 0.500000i 0.468432 0.531779i 0.366713 2.62021i 1.58688 2.34133i −1.37445 2.38061i −0.451636 1.34016i
221.14 −0.208671 1.39873i 2.81779 + 1.62685i −1.91291 + 0.583749i −0.866025 + 0.500000i 1.68754 4.28082i −0.452341 + 2.60680i 1.21568 + 2.55385i 3.79330 + 6.57019i 0.880081 + 1.10700i
221.15 −0.0340493 + 1.41380i −1.12126 0.647359i −1.99768 0.0962781i 0.866025 0.500000i 0.953416 1.56320i 2.45429 + 0.988163i 0.204138 2.82105i −0.661853 1.14636i 0.677414 + 1.24141i
221.16 0.149219 + 1.40632i 2.56394 + 1.48029i −1.95547 + 0.419699i 0.866025 0.500000i −1.69917 + 3.82660i 2.64452 + 0.0805794i −0.882024 2.68738i 2.88251 + 4.99266i 0.832387 + 1.14330i
221.17 0.456776 + 1.33842i −0.970453 0.560291i −1.58271 + 1.22271i −0.866025 + 0.500000i 0.306623 1.55480i −1.15341 + 2.38110i −2.35944 1.55982i −0.872147 1.51060i −1.06479 0.930714i
221.18 0.585931 1.28712i 2.20766 + 1.27459i −1.31337 1.50833i 0.866025 0.500000i 2.93409 2.09470i −0.879470 2.49530i −2.71095 + 0.806691i 1.74917 + 3.02964i −0.136130 1.40765i
221.19 0.683186 + 1.23825i −2.17556 1.25606i −1.06651 + 1.69191i 0.866025 0.500000i 0.0690023 3.55200i −0.986215 2.45507i −2.82363 0.164724i 1.65537 + 2.86718i 1.21078 + 0.730761i
221.20 0.730761 + 1.21078i 2.17556 + 1.25606i −0.931976 + 1.76958i −0.866025 + 0.500000i 0.0690023 + 3.55200i −0.986215 2.45507i −2.82363 + 0.164724i 1.65537 + 2.86718i −1.23825 0.683186i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 261.30 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
8.b even 2 1 inner
56.p 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.bl.b 60
4.b odd 2 1 1120.2.cb.b 60
7.c even 3 1 inner 280.2.bl.b 60
8.b even 2 1 inner 280.2.bl.b 60
8.d odd 2 1 1120.2.cb.b 60
28.g odd 6 1 1120.2.cb.b 60
56.k odd 6 1 1120.2.cb.b 60
56.p even 6 1 inner 280.2.bl.b 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bl.b 60 1.a even 1 1 trivial
280.2.bl.b 60 7.c even 3 1 inner
280.2.bl.b 60 8.b even 2 1 inner
280.2.bl.b 60 56.p even 6 1 inner
1120.2.cb.b 60 4.b odd 2 1
1120.2.cb.b 60 8.d odd 2 1
1120.2.cb.b 60 28.g odd 6 1
1120.2.cb.b 60 56.k odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$43\!\cdots\!72$$$$T_{3}^{38} +$$$$20\!\cdots\!69$$$$T_{3}^{36} -$$$$86\!\cdots\!12$$$$T_{3}^{34} +$$$$30\!\cdots\!86$$$$T_{3}^{32} -$$$$94\!\cdots\!84$$$$T_{3}^{30} +$$$$24\!\cdots\!70$$$$T_{3}^{28} -$$$$55\!\cdots\!16$$$$T_{3}^{26} +$$$$10\!\cdots\!82$$$$T_{3}^{24} -$$$$16\!\cdots\!36$$$$T_{3}^{22} +$$$$21\!\cdots\!01$$$$T_{3}^{20} -$$$$22\!\cdots\!96$$$$T_{3}^{18} +$$$$19\!\cdots\!42$$$$T_{3}^{16} -$$$$12\!\cdots\!36$$$$T_{3}^{14} +$$$$65\!\cdots\!89$$$$T_{3}^{12} -$$$$23\!\cdots\!76$$$$T_{3}^{10} +$$$$62\!\cdots\!68$$$$T_{3}^{8} -$$$$11\!\cdots\!96$$$$T_{3}^{6} +$$$$13\!\cdots\!76$$$$T_{3}^{4} - 692636160000 T_{3}^{2} + 25600000000$$">$$T_{3}^{60} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.