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Properties

Label 407.2.q.d
Level $407$
Weight $2$
Character orbit 407.q
Analytic conductor $3.250$
Analytic rank $0$
Dimension $120$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [407,2,Mod(208,407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(407, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("407.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 407 = 11 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 407.q (of order \(12\), 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: \(3.24991136227\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 12 q^{4} - 8 q^{5} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 12 q^{4} - 8 q^{5} + 32 q^{9} + 4 q^{12} - 24 q^{14} - 20 q^{15} + 52 q^{20} - 28 q^{22} - 12 q^{25} + 72 q^{26} + 64 q^{31} + 4 q^{33} - 36 q^{34} - 64 q^{37} + 80 q^{38} - 40 q^{44} - 84 q^{45} - 16 q^{47} + 120 q^{49} - 16 q^{53} - 24 q^{55} + 56 q^{56} - 12 q^{58} - 48 q^{59} - 76 q^{60} - 168 q^{66} - 36 q^{67} - 24 q^{70} + 56 q^{71} + 40 q^{75} - 120 q^{77} - 336 q^{78} + 68 q^{80} - 44 q^{81} - 36 q^{82} - 76 q^{86} - 20 q^{88} - 4 q^{89} - 116 q^{91} - 48 q^{92} + 120 q^{93} - 84 q^{97} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1 −2.58034 + 0.691399i 0.141465 − 0.0816747i 4.44805 − 2.56808i 0.470400 − 1.75556i −0.308557 + 0.308557i 1.22727 − 0.708563i −5.92402 + 5.92402i −1.48666 + 2.57497i 4.85516i
208.2 −2.44229 + 0.654410i −0.350822 + 0.202547i 3.80448 − 2.19652i −1.04648 + 3.90551i 0.724260 − 0.724260i −3.68110 + 2.12528i −4.27846 + 4.27846i −1.41795 + 2.45596i − 10.2232i
208.3 −2.38020 + 0.637773i −2.03381 + 1.17422i 3.52655 − 2.03605i −0.627123 + 2.34046i 4.09199 − 4.09199i 3.14009 − 1.81293i −3.61049 + 3.61049i 1.25758 − 2.17820i − 5.97071i
208.4 −2.31797 + 0.621098i 2.13323 − 1.23162i 3.25517 − 1.87937i −0.538963 + 2.01144i −4.17980 + 4.17980i 3.77725 − 2.18080i −2.98437 + 2.98437i 1.53377 − 2.65656i − 4.99720i
208.5 −2.28360 + 0.611890i −2.30222 + 1.32919i 3.10839 − 1.79463i 0.955412 − 3.56564i 4.44404 − 4.44404i −0.377114 + 0.217727i −2.65679 + 2.65679i 2.03348 − 3.52208i 8.72713i
208.6 −2.08010 + 0.557362i 0.296043 − 0.170920i 2.28413 − 1.31874i 0.347984 − 1.29869i −0.520535 + 0.520535i −1.33681 + 0.771810i −0.970719 + 0.970719i −1.44157 + 2.49688i 2.89537i
208.7 −1.65183 + 0.442606i 2.43339 − 1.40492i 0.800580 − 0.462215i −0.382261 + 1.42662i −3.39771 + 3.39771i −2.26555 + 1.30802i 1.30060 − 1.30060i 2.44758 − 4.23933i − 2.52572i
208.8 −1.54513 + 0.414017i 2.28791 − 1.32092i 0.483971 − 0.279421i 0.972287 − 3.62862i −2.98823 + 2.98823i 3.32173 − 1.91780i 1.63012 − 1.63012i 1.98967 − 3.44622i 6.00924i
208.9 −1.42578 + 0.382037i −1.90067 + 1.09735i 0.154854 − 0.0894051i 0.175060 − 0.653331i 2.29072 − 2.29072i −4.56514 + 2.63569i 1.90086 − 1.90086i 0.908375 − 1.57335i 0.998388i
208.10 −1.33187 + 0.356872i −0.623971 + 0.360250i −0.0855418 + 0.0493876i −0.245162 + 0.914957i 0.702482 − 0.702482i 0.668207 − 0.385790i 2.04629 − 2.04629i −1.24044 + 2.14851i − 1.30609i
208.11 −1.27522 + 0.341693i 1.33038 − 0.768096i −0.222626 + 0.128533i 0.788958 − 2.94443i −1.43407 + 1.43407i −2.10102 + 1.21302i 2.10703 − 2.10703i −0.320056 + 0.554353i 4.02437i
208.12 −1.06314 + 0.284868i 0.0933780 − 0.0539118i −0.682929 + 0.394289i −0.161765 + 0.603715i −0.0839163 + 0.0839163i 3.36076 − 1.94033i 2.17028 − 2.17028i −1.49419 + 2.58801i − 0.687916i
208.13 −0.665855 + 0.178415i −2.63880 + 1.52351i −1.32052 + 0.762403i 0.250156 − 0.933597i 1.48524 − 1.48524i 2.09475 − 1.20941i 1.71813 − 1.71813i 3.14217 − 5.44240i 0.666272i
208.14 −0.340081 + 0.0911246i 1.02764 − 0.593307i −1.62470 + 0.938020i −0.168567 + 0.629101i −0.295416 + 0.295416i 2.92418 − 1.68828i 0.964967 − 0.964967i −0.795974 + 1.37867i − 0.229306i
208.15 −0.162904 + 0.0436499i −0.759156 + 0.438299i −1.70742 + 0.985779i 0.808140 − 3.01602i 0.104538 − 0.104538i −1.35242 + 0.780817i 0.473623 − 0.473623i −1.11579 + 1.93260i 0.526596i
208.16 0.162904 − 0.0436499i −0.759156 + 0.438299i −1.70742 + 0.985779i 0.808140 − 3.01602i −0.104538 + 0.104538i 1.35242 − 0.780817i −0.473623 + 0.473623i −1.11579 + 1.93260i − 0.526596i
208.17 0.340081 − 0.0911246i 1.02764 − 0.593307i −1.62470 + 0.938020i −0.168567 + 0.629101i 0.295416 − 0.295416i −2.92418 + 1.68828i −0.964967 + 0.964967i −0.795974 + 1.37867i 0.229306i
208.18 0.665855 − 0.178415i −2.63880 + 1.52351i −1.32052 + 0.762403i 0.250156 − 0.933597i −1.48524 + 1.48524i −2.09475 + 1.20941i −1.71813 + 1.71813i 3.14217 − 5.44240i − 0.666272i
208.19 1.06314 − 0.284868i 0.0933780 − 0.0539118i −0.682929 + 0.394289i −0.161765 + 0.603715i 0.0839163 − 0.0839163i −3.36076 + 1.94033i −2.17028 + 2.17028i −1.49419 + 2.58801i 0.687916i
208.20 1.27522 − 0.341693i 1.33038 − 0.768096i −0.222626 + 0.128533i 0.788958 − 2.94443i 1.43407 − 1.43407i 2.10102 − 1.21302i −2.10703 + 2.10703i −0.320056 + 0.554353i − 4.02437i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 208.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
37.g odd 12 1 inner
407.q even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 407.2.q.d ✓ 120
11.b odd 2 1 inner 407.2.q.d ✓ 120
37.g odd 12 1 inner 407.2.q.d ✓ 120
407.q even 12 1 inner 407.2.q.d ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
407.2.q.d ✓ 120 1.a even 1 1 trivial
407.2.q.d ✓ 120 11.b odd 2 1 inner
407.2.q.d ✓ 120 37.g odd 12 1 inner
407.2.q.d ✓ 120 407.q even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} - 6 T_{2}^{118} - 204 T_{2}^{116} + 1296 T_{2}^{114} + 24114 T_{2}^{112} + \cdots + 271467011777536 \) acting on \(S_{2}^{\mathrm{new}}(407, [\chi])\). Copy content Toggle raw display