# Properties

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

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,2,Mod(131,280)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 0, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("280.131");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 3 q^{2} + 12 q^{3} + q^{4} + 12 q^{5} + 2 q^{6} - 10 q^{7} - 6 q^{8} + 12 q^{9} - 3 q^{10} + 8 q^{11} + 8 q^{12} - 20 q^{13} - 15 q^{14} - 3 q^{16} + 6 q^{17} - 27 q^{18} + 18 q^{19} + 2 q^{20} + 26 q^{21} - 16 q^{22} - 18 q^{23} + 6 q^{24} - 12 q^{25} - 29 q^{26} + 3 q^{28} + 10 q^{30} + 6 q^{31} - 33 q^{32} + 12 q^{33} + 20 q^{34} - 8 q^{35} - 22 q^{36} + 9 q^{38} + 18 q^{39} - 3 q^{40} - 12 q^{42} + 32 q^{43} - 25 q^{44} - 12 q^{45} + 53 q^{46} + 14 q^{48} + 8 q^{49} - 22 q^{51} - 31 q^{52} - 30 q^{53} + 10 q^{54} + 16 q^{55} + 16 q^{56} - 44 q^{57} + 30 q^{58} - 18 q^{59} + 10 q^{60} - 22 q^{61} - 8 q^{62} + 12 q^{63} + 58 q^{64} - 10 q^{65} - 8 q^{66} - 8 q^{67} - 6 q^{68} + 12 q^{69} + 3 q^{70} - 23 q^{72} + 30 q^{73} - 43 q^{74} - 12 q^{75} - 8 q^{76} + 32 q^{77} - 8 q^{78} - 6 q^{79} + 3 q^{80} - 4 q^{81} - 27 q^{82} - 16 q^{84} - 36 q^{86} - 14 q^{87} + 49 q^{88} - 60 q^{89} - 24 q^{90} + 18 q^{91} - 38 q^{92} + 18 q^{93} + 11 q^{94} + 18 q^{95} + 2 q^{96} - 19 q^{98} - 56 q^{99}+O(q^{100})$$ 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 + 8 * q^12 - 20 * q^13 - 15 * q^14 - 3 * q^16 + 6 * q^17 - 27 * q^18 + 18 * q^19 + 2 * q^20 + 26 * q^21 - 16 * q^22 - 18 * q^23 + 6 * q^24 - 12 * q^25 - 29 * q^26 + 3 * q^28 + 10 * q^30 + 6 * q^31 - 33 * q^32 + 12 * q^33 + 20 * q^34 - 8 * q^35 - 22 * q^36 + 9 * q^38 + 18 * q^39 - 3 * q^40 - 12 * q^42 + 32 * q^43 - 25 * q^44 - 12 * q^45 + 53 * q^46 + 14 * q^48 + 8 * q^49 - 22 * q^51 - 31 * q^52 - 30 * q^53 + 10 * q^54 + 16 * q^55 + 16 * q^56 - 44 * q^57 + 30 * q^58 - 18 * q^59 + 10 * q^60 - 22 * q^61 - 8 * q^62 + 12 * q^63 + 58 * q^64 - 10 * q^65 - 8 * q^66 - 8 * q^67 - 6 * q^68 + 12 * q^69 + 3 * q^70 - 23 * q^72 + 30 * q^73 - 43 * q^74 - 12 * q^75 - 8 * q^76 + 32 * q^77 - 8 * q^78 - 6 * q^79 + 3 * q^80 - 4 * q^81 - 27 * q^82 - 16 * q^84 - 36 * q^86 - 14 * q^87 + 49 * q^88 - 60 * q^89 - 24 * q^90 + 18 * q^91 - 38 * q^92 + 18 * q^93 + 11 * q^94 + 18 * q^95 + 2 * q^96 - 19 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
131.1 −1.40227 + 0.183437i −0.908317 + 0.524417i 1.93270 0.514454i 0.500000 0.866025i 1.17750 0.901991i −2.14799 1.54472i −2.61579 + 1.07593i −0.949974 + 1.64540i −0.542272 + 1.30612i
131.2 −1.37955 0.311173i 2.75363 1.58981i 1.80634 + 0.858559i 0.500000 0.866025i −4.29350 + 1.33638i 1.04250 + 2.43170i −2.22479 1.74651i 3.55500 6.15745i −0.959261 + 1.03914i
131.3 −1.31025 0.532213i −0.502680 + 0.290223i 1.43350 + 1.39466i 0.500000 0.866025i 0.813096 0.112730i 2.63362 0.253028i −1.13598 2.59028i −1.33154 + 2.30630i −1.11603 + 0.868601i
131.4 −0.967093 + 1.03186i 2.26702 1.30886i −0.129462 1.99581i 0.500000 0.866025i −0.841856 + 3.60504i −1.72615 2.00510i 2.18459 + 1.79654i 1.92625 3.33637i 0.410069 + 1.35346i
131.5 −0.477071 1.33132i −1.84104 + 1.06293i −1.54481 + 1.27027i 0.500000 0.866025i 2.29340 + 1.94392i −2.17552 1.50569i 2.42811 + 1.45062i 0.759621 1.31570i −1.39149 0.252502i
131.6 −0.447296 + 1.34161i −0.219454 + 0.126702i −1.59985 1.20020i 0.500000 0.866025i −0.0718241 0.351096i −0.978876 + 2.45801i 2.32581 1.60954i −1.46789 + 2.54247i 0.938224 + 1.05818i
131.7 −0.394992 1.35793i 1.75472 1.01309i −1.68796 + 1.07275i 0.500000 0.866025i −2.06881 1.98263i 1.63843 2.07739i 2.12345 + 1.86841i 0.552704 0.957311i −1.37350 0.336893i
131.8 0.617571 + 1.27224i 0.725648 0.418953i −1.23721 + 1.57140i 0.500000 0.866025i 0.981150 + 0.664468i 2.36913 1.17781i −2.76328 0.603581i −1.14896 + 1.99005i 1.41058 + 0.101290i
131.9 0.743926 1.20274i 1.94732 1.12428i −0.893147 1.78949i 0.500000 0.866025i 0.0964439 3.17849i −2.47009 + 0.947962i −2.81672 0.257032i 1.02803 1.78060i −0.669637 1.24563i
131.10 1.03665 0.961958i −2.66758 + 1.54013i 0.149275 1.99442i 0.500000 0.866025i −1.28380 + 4.16266i −2.53597 + 0.754231i −1.76380 2.21111i 3.24397 5.61873i −0.314756 1.37874i
131.11 1.06817 + 0.926830i 1.90624 1.10057i 0.281971 + 1.98002i 0.500000 0.866025i 3.05623 + 0.591168i 0.584379 + 2.58041i −1.53395 + 2.37634i 0.922503 1.59782i 1.33674 0.461647i
131.12 1.41221 + 0.0753205i 0.784482 0.452921i 1.98865 + 0.212736i 0.500000 0.866025i 1.14196 0.580530i −1.23347 2.34063i 2.79237 + 0.450214i −1.08973 + 1.88746i 0.771333 1.18535i
171.1 −1.40227 0.183437i −0.908317 0.524417i 1.93270 + 0.514454i 0.500000 + 0.866025i 1.17750 + 0.901991i −2.14799 + 1.54472i −2.61579 1.07593i −0.949974 1.64540i −0.542272 1.30612i
171.2 −1.37955 + 0.311173i 2.75363 + 1.58981i 1.80634 0.858559i 0.500000 + 0.866025i −4.29350 1.33638i 1.04250 2.43170i −2.22479 + 1.74651i 3.55500 + 6.15745i −0.959261 1.03914i
171.3 −1.31025 + 0.532213i −0.502680 0.290223i 1.43350 1.39466i 0.500000 + 0.866025i 0.813096 + 0.112730i 2.63362 + 0.253028i −1.13598 + 2.59028i −1.33154 2.30630i −1.11603 0.868601i
171.4 −0.967093 1.03186i 2.26702 + 1.30886i −0.129462 + 1.99581i 0.500000 + 0.866025i −0.841856 3.60504i −1.72615 + 2.00510i 2.18459 1.79654i 1.92625 + 3.33637i 0.410069 1.35346i
171.5 −0.477071 + 1.33132i −1.84104 1.06293i −1.54481 1.27027i 0.500000 + 0.866025i 2.29340 1.94392i −2.17552 + 1.50569i 2.42811 1.45062i 0.759621 + 1.31570i −1.39149 + 0.252502i
171.6 −0.447296 1.34161i −0.219454 0.126702i −1.59985 + 1.20020i 0.500000 + 0.866025i −0.0718241 + 0.351096i −0.978876 2.45801i 2.32581 + 1.60954i −1.46789 2.54247i 0.938224 1.05818i
171.7 −0.394992 + 1.35793i 1.75472 + 1.01309i −1.68796 1.07275i 0.500000 + 0.866025i −2.06881 + 1.98263i 1.63843 + 2.07739i 2.12345 1.86841i 0.552704 + 0.957311i −1.37350 + 0.336893i
171.8 0.617571 1.27224i 0.725648 + 0.418953i −1.23721 1.57140i 0.500000 + 0.866025i 0.981150 0.664468i 2.36913 + 1.17781i −2.76328 + 0.603581i −1.14896 1.99005i 1.41058 0.101290i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.12 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.e 24
4.b odd 2 1 1120.2.bz.f 24
7.d odd 6 1 280.2.bj.f yes 24
8.b even 2 1 1120.2.bz.e 24
8.d odd 2 1 280.2.bj.f yes 24
28.f even 6 1 1120.2.bz.e 24
56.j odd 6 1 1120.2.bz.f 24
56.m even 6 1 inner 280.2.bj.e 24

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