↑

Properties

Label 287.2.e.d
Level $287$
Weight $2$
Character orbit 287.e
Analytic conductor $2.292$
Analytic rank $0$
Dimension $34$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [287,2,Mod(165,287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(287, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("287.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 287 = 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 287.e (of order \(3\), 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: \(34\)
Relative dimension: \(17\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 3 q^{2} - q^{3} - 25 q^{4} + q^{5} + 4 q^{6} - 2 q^{7} + 18 q^{8} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q - 3 q^{2} - q^{3} - 25 q^{4} + q^{5} + 4 q^{6} - 2 q^{7} + 18 q^{8} - 26 q^{9} + 2 q^{10} - 15 q^{11} - 4 q^{12} - 10 q^{13} + 21 q^{14} + 48 q^{15} - 33 q^{16} - 4 q^{17} - 10 q^{18} - 5 q^{19} - 52 q^{20} + 12 q^{21} + 32 q^{22} - 12 q^{23} - 16 q^{24} - 24 q^{25} - 31 q^{26} - 22 q^{27} + 60 q^{28} + 28 q^{29} + 33 q^{30} + 3 q^{31} - 16 q^{32} - 4 q^{33} - 48 q^{34} + 45 q^{35} + 114 q^{36} - 24 q^{37} - 45 q^{39} - 36 q^{40} + 34 q^{41} + 65 q^{42} + 28 q^{43} + 9 q^{44} + 21 q^{45} - 44 q^{46} - 19 q^{47} - 120 q^{48} - 10 q^{49} - 8 q^{50} - 2 q^{51} + 25 q^{52} - 4 q^{53} - 68 q^{54} + 18 q^{55} + 25 q^{56} - 24 q^{57} + q^{58} + 27 q^{59} - 66 q^{60} + q^{61} - 46 q^{62} + 37 q^{63} + 150 q^{64} - 22 q^{65} + 16 q^{66} - 49 q^{67} - 45 q^{68} + 24 q^{69} + 73 q^{70} + 80 q^{71} + 23 q^{72} + 14 q^{73} - 33 q^{74} - 27 q^{75} - 18 q^{76} - 20 q^{77} - 24 q^{78} - 61 q^{79} + 82 q^{80} - 53 q^{81} - 3 q^{82} - 36 q^{83} + 188 q^{84} - 26 q^{85} + 4 q^{86} + 17 q^{87} - 74 q^{88} - 18 q^{89} - 40 q^{90} + 7 q^{91} + 56 q^{92} + 36 q^{93} + 5 q^{94} - 20 q^{95} - 148 q^{96} + 52 q^{97} + 142 q^{98} + 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1 −1.36755 − 2.36867i −1.26123 + 2.18452i −2.74039 + 4.74650i 0.813928 + 1.40976i 6.89919 −1.98316 − 1.75131i 9.52032 −1.68141 − 2.91228i 2.22618 − 3.85585i
165.2 −1.28712 − 2.22936i −0.234126 + 0.405518i −2.31337 + 4.00687i −1.12112 − 1.94184i 1.20539 1.72083 + 2.00966i 6.76185 1.39037 + 2.40819i −2.88604 + 4.99876i
165.3 −1.26475 − 2.19061i 1.34808 − 2.33495i −2.19919 + 3.80911i −0.322251 − 0.558155i −6.81996 −0.460755 − 2.60532i 6.06670 −2.13465 − 3.69733i −0.815135 + 1.41186i
165.4 −1.18475 − 2.05204i 0.823159 − 1.42575i −1.80725 + 3.13025i 2.01488 + 3.48988i −3.90094 0.327045 + 2.62546i 3.82554 0.144819 + 0.250833i 4.77425 − 8.26924i
165.5 −0.878715 − 1.52198i −1.61064 + 2.78971i −0.544281 + 0.942722i −0.960818 − 1.66419i 5.66118 −0.778115 + 2.52874i −1.60179 −3.68832 − 6.38836i −1.68857 + 2.92469i
165.6 −0.604772 − 1.04749i −0.0752022 + 0.130254i 0.268503 − 0.465061i −1.66371 − 2.88164i 0.181921 −1.99521 − 1.73756i −3.06862 1.48869 + 2.57849i −2.01233 + 3.48546i
165.7 −0.587784 − 1.01807i −1.36430 + 2.36304i 0.309020 − 0.535238i 1.41426 + 2.44958i 3.20766 2.04575 − 1.67777i −3.07768 −2.22264 − 3.84973i 1.66256 − 2.87965i
165.8 −0.557713 − 0.965987i 1.37705 − 2.38511i 0.377913 − 0.654564i 0.672309 + 1.16447i −3.07199 −2.64555 + 0.0328867i −3.07392 −2.29251 − 3.97075i 0.749910 − 1.29888i
165.9 −0.190962 − 0.330755i 0.175243 − 0.303530i 0.927067 − 1.60573i 1.39512 + 2.41642i −0.133859 2.22712 + 1.42827i −1.47198 1.43858 + 2.49169i 0.532830 − 0.922889i
165.10 0.145387 + 0.251818i 0.633158 − 1.09666i 0.957725 − 1.65883i −1.61895 − 2.80410i 0.368212 −1.14416 + 2.38556i 1.13851 0.698223 + 1.20936i 0.470748 − 0.815360i
165.11 0.451240 + 0.781570i 1.66259 − 2.87969i 0.592766 − 1.02670i 0.469181 + 0.812645i 3.00091 1.14720 + 2.38410i 2.87488 −4.02843 − 6.97744i −0.423426 + 0.733395i
165.12 0.486022 + 0.841815i −0.753325 + 1.30480i 0.527565 − 0.913769i −1.41815 − 2.45631i −1.46453 2.62766 − 0.308901i 2.96972 0.365002 + 0.632201i 1.37851 − 2.38765i
165.13 0.834451 + 1.44531i 0.106774 − 0.184938i −0.392616 + 0.680030i −0.501668 − 0.868914i 0.356391 2.44503 − 1.01086i 2.02733 1.47720 + 2.55858i 0.837234 − 1.45013i
165.14 0.889478 + 1.54062i −0.603897 + 1.04598i −0.582342 + 1.00865i 1.26642 + 2.19350i −2.14861 −0.862133 − 2.50135i 1.48599 0.770616 + 1.33475i −2.25290 + 3.90213i
165.15 0.992766 + 1.71952i −1.44392 + 2.50094i −0.971167 + 1.68211i −1.95720 − 3.38997i −5.73388 −2.26620 + 1.36540i 0.114499 −2.66979 − 4.62421i 3.88608 − 6.73089i
165.16 1.24077 + 2.14907i −0.704660 + 1.22051i −2.07901 + 3.60095i 0.507237 + 0.878561i −3.49728 −2.53048 − 0.772431i −5.35521 0.506908 + 0.877991i −1.25873 + 2.18018i
165.17 1.38401 + 2.39717i 1.42525 − 2.46860i −2.83094 + 4.90334i 1.51053 + 2.61632i 7.89019 1.12514 − 2.39459i −10.1361 −2.56265 − 4.43864i −4.18117 + 7.24200i
247.1 −1.36755 + 2.36867i −1.26123 − 2.18452i −2.74039 − 4.74650i 0.813928 − 1.40976i 6.89919 −1.98316 + 1.75131i 9.52032 −1.68141 + 2.91228i 2.22618 + 3.85585i
247.2 −1.28712 + 2.22936i −0.234126 − 0.405518i −2.31337 − 4.00687i −1.12112 + 1.94184i 1.20539 1.72083 − 2.00966i 6.76185 1.39037 − 2.40819i −2.88604 − 4.99876i
247.3 −1.26475 + 2.19061i 1.34808 + 2.33495i −2.19919 − 3.80911i −0.322251 + 0.558155i −6.81996 −0.460755 + 2.60532i 6.06670 −2.13465 + 3.69733i −0.815135 − 1.41186i
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.2.e.d ✓ 34
7.c even 3 1 inner 287.2.e.d ✓ 34
7.c even 3 1 2009.2.a.s 17
7.d odd 6 1 2009.2.a.r 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.e.d ✓ 34 1.a even 1 1 trivial
287.2.e.d ✓ 34 7.c even 3 1 inner
2009.2.a.r 17 7.d odd 6 1
2009.2.a.s 17 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{34} + 3 T_{2}^{33} + 34 T_{2}^{32} + 79 T_{2}^{31} + 609 T_{2}^{30} + 1233 T_{2}^{29} + \cdots + 215296 \) acting on \(S_{2}^{\mathrm{new}}(287, [\chi])\). Copy content Toggle raw display