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Properties

Label 287.2.f.a
Level $287$
Weight $2$
Character orbit 287.f
Analytic conductor $2.292$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [287,2,Mod(50,287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(287, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("287.50");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 287 = 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 287.f (of order \(4\), 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: \(2.29170653801\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{3} - 36 q^{4} + 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{3} - 36 q^{4} + 8 q^{6} - 32 q^{10} - 8 q^{11} + 16 q^{12} + 16 q^{13} - 8 q^{15} + 28 q^{16} + 20 q^{17} - 12 q^{18} - 20 q^{19} + 4 q^{22} + 16 q^{23} - 12 q^{24} - 40 q^{25} - 20 q^{26} - 20 q^{27} - 12 q^{29} + 4 q^{30} + 32 q^{34} + 4 q^{35} - 16 q^{38} + 64 q^{40} + 16 q^{41} + 32 q^{42} + 8 q^{44} + 72 q^{45} - 24 q^{47} - 40 q^{48} - 64 q^{51} - 96 q^{52} + 8 q^{53} + 52 q^{54} - 8 q^{55} - 88 q^{57} - 36 q^{58} + 48 q^{59} + 52 q^{60} - 8 q^{63} - 84 q^{64} - 44 q^{65} + 56 q^{66} + 40 q^{67} - 60 q^{68} + 28 q^{69} - 8 q^{70} + 20 q^{71} + 80 q^{72} - 20 q^{75} - 4 q^{76} + 12 q^{78} - 12 q^{79} + 16 q^{81} - 52 q^{82} + 40 q^{83} + 8 q^{85} + 80 q^{86} + 96 q^{88} - 8 q^{89} - 20 q^{92} - 64 q^{93} + 52 q^{94} + 68 q^{96} - 60 q^{97} - 4 q^{98} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1 − 2.46506i 0.589206 + 0.589206i −4.07653 − 2.39319i 1.45243 − 1.45243i 0.707107 + 0.707107i 5.11877i − 2.30567i −5.89935
50.2 − 2.24636i −1.79418 − 1.79418i −3.04612 − 1.17005i −4.03037 + 4.03037i −0.707107 − 0.707107i 2.34997i 3.43817i −2.62835
50.3 − 1.99257i 1.04836 + 1.04836i −1.97032 − 1.71001i 2.08892 − 2.08892i −0.707107 − 0.707107i − 0.0591377i − 0.801887i −3.40730
50.4 − 1.72512i 2.12657 + 2.12657i −0.976043 − 0.471226i 3.66859 − 3.66859i 0.707107 + 0.707107i − 1.76645i 6.04458i −0.812921
50.5 − 1.55423i −0.820177 − 0.820177i −0.415620 1.07119i −1.27474 + 1.27474i −0.707107 − 0.707107i − 2.46249i − 1.65462i 1.66487
50.6 − 1.43253i −1.43055 − 1.43055i −0.0521511 − 4.15342i −2.04931 + 2.04931i 0.707107 + 0.707107i − 2.79036i 1.09294i −5.94992
50.7 − 1.13705i 1.99666 + 1.99666i 0.707122 2.87203i 2.27029 − 2.27029i −0.707107 − 0.707107i − 3.07813i 4.97327i 3.26564
50.8 − 1.08678i −0.429589 − 0.429589i 0.818911 2.95294i −0.466868 + 0.466868i 0.707107 + 0.707107i − 3.06353i − 2.63091i 3.20919
50.9 − 0.447748i 1.09022 + 1.09022i 1.79952 − 1.12660i 0.488146 − 0.488146i 0.707107 + 0.707107i − 1.70123i − 0.622822i −0.504431
50.10 − 0.124298i −0.249942 − 0.249942i 1.98455 − 1.52864i −0.0310673 + 0.0310673i −0.707107 − 0.707107i − 0.495271i − 2.87506i −0.190006
50.11 0.701559i 2.17674 + 2.17674i 1.50781 − 3.99632i −1.52711 + 1.52711i −0.707107 − 0.707107i 2.46094i 6.47641i 2.80365
50.12 0.710667i 0.708215 + 0.708215i 1.49495 3.05444i −0.503305 + 0.503305i 0.707107 + 0.707107i 2.48375i − 1.99686i −2.17069
50.13 0.748721i −1.37188 − 1.37188i 1.43942 − 2.22584i 1.02716 − 1.02716i 0.707107 + 0.707107i 2.57516i 0.764137i 1.66653
50.14 0.865152i −1.36546 − 1.36546i 1.25151 1.87589i 1.18133 − 1.18133i −0.707107 − 0.707107i 2.81305i 0.728936i −1.62293
50.15 1.38058i 1.24569 + 1.24569i 0.0939893 1.25269i −1.71978 + 1.71978i −0.707107 − 0.707107i 2.89093i 0.103480i −1.72944
50.16 1.70500i −1.24229 − 1.24229i −0.907032 2.38000i 2.11811 − 2.11811i 0.707107 + 0.707107i 1.86351i 0.0865659i −4.05791
50.17 2.30053i 1.78480 + 1.78480i −3.29242 − 0.414800i −4.10598 + 4.10598i 0.707107 + 0.707107i − 2.97325i 3.37102i 0.954258
50.18 2.36496i −0.345245 − 0.345245i −3.59302 − 3.38746i 0.816490 − 0.816490i −0.707107 − 0.707107i − 3.76741i − 2.76161i 8.01120
50.19 2.69233i −2.23891 − 2.23891i −5.24863 − 1.01652i 6.02789 − 6.02789i 0.707107 + 0.707107i − 8.74637i 7.02546i 2.73681
50.20 2.74225i 0.521769 + 0.521769i −5.51991 4.13490i −1.43082 + 1.43082i −0.707107 − 0.707107i − 9.65245i − 2.45551i −11.3389
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.2.f.a ✓ 40
41.c even 4 1 inner 287.2.f.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.f.a ✓ 40 1.a even 1 1 trivial
287.2.f.a ✓ 40 41.c even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(287, [\chi])\).