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Properties

Label 287.2.l.a
Level $287$
Weight $2$
Character orbit 287.l
Analytic conductor $2.292$
Analytic rank $0$
Dimension $104$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [287,2,Mod(27,287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(287, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("287.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 287 = 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 287.l (of order \(8\), degree \(4\), 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: \(104\)
Relative dimension: \(26\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 8 q^{2} - 4 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 8 q^{2} - 4 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{11} - 8 q^{14} - 32 q^{15} - 96 q^{16} - 16 q^{18} - 24 q^{21} + 24 q^{22} + 12 q^{28} - 16 q^{29} + 48 q^{30} - 72 q^{32} - 44 q^{35} + 56 q^{36} + 32 q^{37} - 8 q^{39} - 8 q^{42} - 32 q^{43} + 72 q^{44} - 88 q^{46} - 12 q^{49} + 80 q^{50} + 32 q^{51} - 8 q^{53} + 104 q^{56} - 32 q^{57} - 96 q^{58} + 112 q^{60} - 132 q^{63} + 80 q^{65} + 56 q^{67} + 148 q^{70} - 104 q^{71} + 32 q^{74} - 16 q^{77} - 168 q^{78} - 124 q^{84} + 8 q^{85} - 64 q^{88} + 8 q^{91} + 104 q^{92} - 64 q^{93} - 8 q^{95} - 8 q^{98} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −1.82241 − 1.82241i −1.24228 − 2.99913i 4.64235i −1.81451 − 1.81451i −3.20170 + 7.72960i −0.372244 − 2.61943i 4.81545 − 4.81545i −5.33022 + 5.33022i 6.61356i
27.2 −1.82241 − 1.82241i 1.24228 + 2.99913i 4.64235i 1.81451 + 1.81451i 3.20170 − 7.72960i 1.58900 + 2.11544i 4.81545 − 4.81545i −5.33022 + 5.33022i − 6.61356i
27.3 −1.72064 − 1.72064i −0.369209 − 0.891351i 3.92122i 2.00511 + 2.00511i −0.898418 + 2.16897i −2.56283 + 0.657199i 3.30574 − 3.30574i 1.46313 − 1.46313i − 6.90017i
27.4 −1.72064 − 1.72064i 0.369209 + 0.891351i 3.92122i −2.00511 − 2.00511i 0.898418 − 2.16897i −2.27690 + 1.34748i 3.30574 − 3.30574i 1.46313 − 1.46313i 6.90017i
27.5 −1.55293 − 1.55293i −0.699693 − 1.68921i 2.82318i 1.29416 + 1.29416i −1.53665 + 3.70980i 2.12474 + 1.57654i 1.27834 − 1.27834i −0.242535 + 0.242535i − 4.01948i
27.6 −1.55293 − 1.55293i 0.699693 + 1.68921i 2.82318i −1.29416 − 1.29416i 1.53665 − 3.70980i 0.387642 − 2.61720i 1.27834 − 1.27834i −0.242535 + 0.242535i 4.01948i
27.7 −1.17122 − 1.17122i −0.481548 − 1.16256i 0.743529i 0.511687 + 0.511687i −0.797616 + 1.92562i 1.67326 − 2.04944i −1.47161 + 1.47161i 1.00167 − 1.00167i − 1.19860i
27.8 −1.17122 − 1.17122i 0.481548 + 1.16256i 0.743529i −0.511687 − 0.511687i 0.797616 − 1.92562i 2.63235 + 0.265998i −1.47161 + 1.47161i 1.00167 − 1.00167i 1.19860i
27.9 −0.717741 − 0.717741i −0.472410 − 1.14050i − 0.969695i −1.70830 − 1.70830i −0.479514 + 1.15765i −2.54594 − 0.719863i −2.13147 + 2.13147i 1.04376 − 1.04376i 2.45224i
27.10 −0.717741 − 0.717741i 0.472410 + 1.14050i − 0.969695i 1.70830 + 1.70830i 0.479514 − 1.15765i −1.29123 + 2.30927i −2.13147 + 2.13147i 1.04376 − 1.04376i − 2.45224i
27.11 −0.300045 − 0.300045i −0.219160 − 0.529099i − 1.81995i 1.73401 + 1.73401i −0.0929957 + 0.224511i −1.42530 − 2.22902i −1.14615 + 1.14615i 1.88941 − 1.88941i − 1.04056i
27.12 −0.300045 − 0.300045i 0.219160 + 0.529099i − 1.81995i −1.73401 − 1.73401i 0.0929957 − 0.224511i 0.568314 + 2.58399i −1.14615 + 1.14615i 1.88941 − 1.88941i 1.04056i
27.13 −0.0425840 − 0.0425840i −0.956654 − 2.30957i − 1.99637i −1.57284 − 1.57284i −0.0576124 + 0.139089i 2.61468 − 0.404284i −0.170181 + 0.170181i −2.29759 + 2.29759i 0.133955i
27.14 −0.0425840 − 0.0425840i 0.956654 + 2.30957i − 1.99637i 1.57284 + 1.57284i 0.0576124 − 0.139089i 2.13473 − 1.56299i −0.170181 + 0.170181i −2.29759 + 2.29759i − 0.133955i
27.15 0.279520 + 0.279520i −0.739386 − 1.78504i − 1.84374i 2.85060 + 2.85060i 0.292280 − 0.705627i 1.56158 + 2.13576i 1.07440 − 1.07440i −0.518341 + 0.518341i 1.59360i
27.16 0.279520 + 0.279520i 0.739386 + 1.78504i − 1.84374i −2.85060 − 2.85060i −0.292280 + 0.705627i −0.406011 − 2.61441i 1.07440 − 1.07440i −0.518341 + 0.518341i − 1.59360i
27.17 0.652493 + 0.652493i −1.09762 − 2.64989i − 1.14851i 0.202079 + 0.202079i 1.01284 − 2.44522i −2.64411 − 0.0932944i 2.05438 − 2.05438i −3.69581 + 3.69581i 0.263710i
27.18 0.652493 + 0.652493i 1.09762 + 2.64989i − 1.14851i −0.202079 − 0.202079i −1.01284 + 2.44522i −1.80370 + 1.93563i 2.05438 − 2.05438i −3.69581 + 3.69581i − 0.263710i
27.19 0.814711 + 0.814711i −0.0360204 − 0.0869609i − 0.672492i 0.788699 + 0.788699i 0.0415018 − 0.100194i −0.714546 − 2.54743i 2.17731 − 2.17731i 2.11506 − 2.11506i 1.28512i
27.20 0.814711 + 0.814711i 0.0360204 + 0.0869609i − 0.672492i −0.788699 − 0.788699i −0.0415018 + 0.100194i 1.29605 + 2.30657i 2.17731 − 2.17731i 2.11506 − 2.11506i − 1.28512i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
41.e odd 8 1 inner
287.l even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.2.l.a ✓ 104
7.b odd 2 1 inner 287.2.l.a ✓ 104
41.e odd 8 1 inner 287.2.l.a ✓ 104
287.l even 8 1 inner 287.2.l.a ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.l.a ✓ 104 1.a even 1 1 trivial
287.2.l.a ✓ 104 7.b odd 2 1 inner
287.2.l.a ✓ 104 41.e odd 8 1 inner
287.2.l.a ✓ 104 287.l even 8 1 inner

Hecke kernels

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