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Properties

Label 280.2.ba.b
Level $280$
Weight $2$
Character orbit 280.ba
Analytic conductor $2.236$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [280,2,Mod(19,280)]
 
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, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("280.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.ba (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: \(80\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 6 q^{4} - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 6 q^{4} - 52 q^{9} - 12 q^{10} - 6 q^{14} + 10 q^{16} - 48 q^{19} + 12 q^{24} - 22 q^{25} + 54 q^{26} - 20 q^{30} - 22 q^{35} + 28 q^{36} + 12 q^{40} + 38 q^{44} + 14 q^{46} + 16 q^{49} - 40 q^{50} + 20 q^{51} + 60 q^{54} - 104 q^{56} - 60 q^{59} - 42 q^{60} - 60 q^{64} - 12 q^{65} + 84 q^{66} - 22 q^{70} + 34 q^{74} - 6 q^{75} - 96 q^{80} - 100 q^{84} - 8 q^{86} - 36 q^{89} + 72 q^{91} - 66 q^{94} + 96 q^{96} + 104 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41178 + 0.0828556i 0.653375 + 1.13168i 1.98627 − 0.233948i 1.67181 + 1.48495i −1.01619 − 1.54355i 1.03631 + 2.43435i −2.78480 + 0.494858i 0.646203 − 1.11926i −2.48327 − 1.95791i
19.2 −1.39533 + 0.230335i −0.929162 − 1.60936i 1.89389 − 0.642786i 2.18875 − 0.457553i 1.66718 + 2.03156i 1.10610 − 2.40345i −2.49455 + 1.33313i −0.226682 + 0.392626i −2.94864 + 1.14258i
19.3 −1.38731 − 0.274535i 1.60233 + 2.77531i 1.84926 + 0.761731i −2.12896 − 0.683763i −1.46101 − 4.29012i −2.23600 + 1.41432i −2.35638 − 1.56444i −3.63491 + 6.29586i 2.76581 + 1.53307i
19.4 −1.36769 + 0.359747i −1.46528 − 2.53794i 1.74116 − 0.984045i −1.36405 + 1.77183i 2.91706 + 2.94399i 1.45426 + 2.21023i −2.02737 + 1.97225i −2.79408 + 4.83948i 1.22819 − 2.91403i
19.5 −1.32026 − 0.506870i 0.360276 + 0.624015i 1.48617 + 1.33840i −1.56432 + 1.59778i −0.159362 − 1.00647i 1.35637 − 2.27162i −1.28373 − 2.52033i 1.24040 − 2.14844i 2.87518 − 1.31657i
19.6 −1.30291 + 0.549923i 0.833565 + 1.44378i 1.39517 − 1.43300i −0.660510 − 2.13629i −1.88003 − 1.42272i 2.56833 − 0.635367i −1.02974 + 2.63432i 0.110340 − 0.191114i 2.03538 + 2.42017i
19.7 −1.24532 − 0.670206i −0.818911 − 1.41839i 1.10165 + 1.66924i 0.951297 + 2.02362i 0.0691894 + 2.31520i −2.64158 − 0.148518i −0.253167 − 2.81707i 0.158771 − 0.274999i 0.171572 − 3.15762i
19.8 −1.21328 + 0.726608i 0.0769760 + 0.133326i 0.944081 − 1.76315i −1.85193 + 1.25314i −0.190269 − 0.105830i −2.33880 − 1.23694i 0.135691 + 2.82517i 1.48815 − 2.57755i 1.33636 − 2.86603i
19.9 −1.20308 − 0.743376i −0.818911 − 1.41839i 0.894783 + 1.78868i −0.951297 − 2.02362i −0.0691894 + 2.31520i 2.64158 + 0.148518i 0.253167 − 2.81707i 0.158771 − 0.274999i −0.359828 + 3.14174i
19.10 −1.09909 − 0.889943i 0.360276 + 0.624015i 0.416004 + 1.95626i 1.56432 − 1.59778i 0.159362 − 1.00647i −1.35637 + 2.27162i 1.28373 − 2.52033i 1.24040 − 2.14844i −3.14126 + 0.363945i
19.11 −1.02807 + 0.971124i 1.39824 + 2.42182i 0.113837 − 1.99676i 2.07888 − 0.823569i −3.78937 − 1.13193i −2.32367 − 1.26513i 1.82207 + 2.16335i −2.41015 + 4.17451i −1.33744 + 2.86553i
19.12 −0.931410 − 1.06418i 1.60233 + 2.77531i −0.264952 + 1.98237i 2.12896 + 0.683763i 1.46101 − 4.29012i 2.23600 − 1.41432i 2.35638 − 1.56444i −3.63491 + 6.29586i −1.25529 − 2.90246i
19.13 −0.881474 + 1.10590i −1.13535 − 1.96648i −0.446008 − 1.94964i 1.70095 − 1.45147i 3.17551 + 0.477826i −0.551936 + 2.58754i 2.54924 + 1.22531i −1.07804 + 1.86722i 0.105836 + 3.16051i
19.14 −0.634137 − 1.26407i 0.653375 + 1.13168i −1.19574 + 1.60319i −1.67181 − 1.48495i 1.01619 − 1.54355i −1.03631 − 2.43435i 2.78480 + 0.494858i 0.646203 − 1.11926i −0.816920 + 3.05494i
19.15 −0.516996 + 1.31633i 1.13535 + 1.96648i −1.46543 − 1.36107i −0.406539 + 2.19880i −3.17551 + 0.477826i −0.551936 + 2.58754i 2.54924 − 1.22531i −1.07804 + 1.86722i −2.68416 − 1.67191i
19.16 −0.498189 − 1.32356i −0.929162 − 1.60936i −1.50361 + 1.31877i −2.18875 + 0.457553i −1.66718 + 2.03156i −1.10610 + 2.40345i 2.49455 + 1.33313i −0.226682 + 0.392626i 1.69601 + 2.66900i
19.17 −0.372297 − 1.36433i −1.46528 − 2.53794i −1.72279 + 1.01587i 1.36405 − 1.77183i −2.91706 + 2.94399i −1.45426 − 2.21023i 2.02737 + 1.97225i −2.79408 + 4.83948i −2.92519 − 1.20136i
19.18 −0.326985 + 1.37589i −1.39824 − 2.42182i −1.78616 − 0.899793i 0.326208 + 2.21215i 3.78937 − 1.13193i −2.32367 − 1.26513i 1.82207 − 2.16335i −2.41015 + 4.17451i −3.15034 − 0.274512i
19.19 −0.175210 − 1.40332i 0.833565 + 1.44378i −1.93860 + 0.491749i 0.660510 + 2.13629i 1.88003 − 1.42272i −2.56833 + 0.635367i 1.02974 + 2.63432i 0.110340 − 0.191114i 2.88216 − 1.30120i
19.20 −0.0226228 + 1.41403i −0.0769760 − 0.133326i −1.99898 − 0.0639789i 0.159286 − 2.23039i 0.190269 − 0.105830i −2.33880 − 1.23694i 0.135691 − 2.82517i 1.48815 − 2.57755i 3.15024 + 0.275693i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
8.d odd 2 1 inner
35.i odd 6 1 inner
40.e odd 2 1 inner
56.m even 6 1 inner
280.ba 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.ba.b ✓ 80
4.b odd 2 1 1120.2.bq.b 80
5.b even 2 1 inner 280.2.ba.b ✓ 80
7.d odd 6 1 inner 280.2.ba.b ✓ 80
8.b even 2 1 1120.2.bq.b 80
8.d odd 2 1 inner 280.2.ba.b ✓ 80
20.d odd 2 1 1120.2.bq.b 80
28.f even 6 1 1120.2.bq.b 80
35.i odd 6 1 inner 280.2.ba.b ✓ 80
40.e odd 2 1 inner 280.2.ba.b ✓ 80
40.f even 2 1 1120.2.bq.b 80
56.j odd 6 1 1120.2.bq.b 80
56.m even 6 1 inner 280.2.ba.b ✓ 80
140.s even 6 1 1120.2.bq.b 80
280.ba even 6 1 inner 280.2.ba.b ✓ 80
280.bk odd 6 1 1120.2.bq.b 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.ba.b ✓ 80 1.a even 1 1 trivial
280.2.ba.b ✓ 80 5.b even 2 1 inner
280.2.ba.b ✓ 80 7.d odd 6 1 inner
280.2.ba.b ✓ 80 8.d odd 2 1 inner
280.2.ba.b ✓ 80 35.i odd 6 1 inner
280.2.ba.b ✓ 80 40.e odd 2 1 inner
280.2.ba.b ✓ 80 56.m even 6 1 inner
280.2.ba.b ✓ 80 280.ba even 6 1 inner
1120.2.bq.b 80 4.b odd 2 1
1120.2.bq.b 80 8.b even 2 1
1120.2.bq.b 80 20.d odd 2 1
1120.2.bq.b 80 28.f even 6 1
1120.2.bq.b 80 40.f even 2 1
1120.2.bq.b 80 56.j odd 6 1
1120.2.bq.b 80 140.s even 6 1
1120.2.bq.b 80 280.bk odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 43 T_{3}^{38} + 1073 T_{3}^{36} + 18024 T_{3}^{34} + 226738 T_{3}^{32} + 2204218 T_{3}^{30} + \cdots + 3701776 \) acting on \(S_{2}^{\mathrm{new}}(280, [\chi])\). Copy content Toggle raw display