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Properties

Label 407.2.l.b
Level $407$
Weight $2$
Character orbit 407.l
Analytic conductor $3.250$
Analytic rank $0$
Dimension $108$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [407,2,Mod(12,407)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(407, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("407.12");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 407 = 11 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 407.l (of order \(9\), degree \(6\), 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: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 3 q^{3} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 3 q^{3} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 9 q^{9} - 18 q^{10} - 54 q^{11} - 27 q^{12} + 33 q^{13} - 36 q^{15} - 12 q^{17} + 3 q^{18} + 27 q^{20} + 6 q^{21} - 36 q^{24} - 18 q^{25} - 27 q^{26} + 3 q^{27} + 66 q^{28} + 3 q^{29} - 39 q^{30} + 36 q^{31} - 51 q^{32} - 3 q^{33} - 3 q^{34} - 45 q^{35} + 198 q^{36} - 6 q^{37} + 72 q^{38} - 45 q^{39} - 9 q^{40} - 30 q^{41} - 24 q^{42} + 72 q^{45} - 105 q^{46} - 45 q^{47} - 42 q^{48} - 27 q^{49} + 129 q^{50} + 6 q^{51} - 87 q^{52} - 33 q^{53} - 15 q^{54} + 12 q^{55} - 96 q^{56} - 18 q^{57} - 66 q^{58} + 27 q^{59} - 15 q^{60} + 36 q^{61} - 24 q^{62} - 18 q^{63} - 93 q^{64} + 72 q^{65} - 18 q^{67} + 180 q^{68} - 30 q^{69} - 72 q^{70} + 12 q^{71} + 42 q^{72} + 72 q^{73} - 39 q^{74} + 126 q^{75} + 27 q^{76} - 6 q^{77} + 198 q^{78} - 15 q^{79} - 120 q^{80} - 36 q^{81} + 96 q^{82} - 63 q^{83} - 81 q^{84} - 48 q^{85} - 39 q^{86} + 60 q^{87} - 9 q^{88} - 9 q^{89} + 3 q^{90} - 3 q^{91} + 249 q^{92} - 117 q^{93} + 3 q^{94} + 84 q^{95} + 144 q^{96} - 12 q^{97} - 84 q^{98} + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −2.01822 + 1.69349i 0.272602 + 0.228740i 0.858012 − 4.86603i 0.543793 − 0.197924i −0.937538 −1.71611 + 0.624614i 3.87430 + 6.71048i −0.498955 − 2.82971i −0.762310 + 1.32036i
12.2 −1.86788 + 1.56733i −1.35177 − 1.13427i 0.685128 − 3.88556i −1.59269 + 0.579691i 4.30271 −1.39838 + 0.508970i 2.37189 + 4.10824i 0.0197673 + 0.112106i 2.06637 − 3.57906i
12.3 −1.59924 + 1.34192i 1.74066 + 1.46058i 0.409519 − 2.32250i 4.10869 − 1.49544i −4.74372 0.568487 − 0.206912i 0.374034 + 0.647846i 0.375635 + 2.13033i −4.56402 + 7.90512i
12.4 −1.50758 + 1.26501i −2.39647 − 2.01088i 0.325249 − 1.84458i 2.40075 − 0.873801i 6.15664 1.96274 − 0.714378i −0.124933 − 0.216390i 1.17850 + 6.68360i −2.51395 + 4.35429i
12.5 −1.44838 + 1.21534i 2.49937 + 2.09722i 0.273471 − 1.55093i −3.16875 + 1.15333i −6.16886 0.103682 − 0.0377373i −0.401916 − 0.696138i 1.32757 + 7.52900i 3.18788 − 5.52157i
12.6 −1.02126 + 0.856937i 0.290189 + 0.243497i −0.0386697 + 0.219307i −2.28216 + 0.830638i −0.505019 −0.906693 + 0.330009i −1.48160 − 2.56620i −0.496026 − 2.81310i 1.61887 − 2.80396i
12.7 −0.411950 + 0.345667i 1.74349 + 1.46296i −0.297079 + 1.68482i 1.10972 − 0.403906i −1.22393 −4.44001 + 1.61603i −0.997767 − 1.72818i 0.378555 + 2.14689i −0.317533 + 0.549983i
12.8 −0.252578 + 0.211938i −1.56515 − 1.31331i −0.328419 + 1.86255i 1.51467 − 0.551297i 0.673662 −2.17450 + 0.791452i −0.641511 − 1.11113i 0.203947 + 1.15664i −0.265732 + 0.460262i
12.9 −0.142323 + 0.119423i −0.338736 − 0.284234i −0.341302 + 1.93562i 3.26427 − 1.18810i 0.0821541 1.70616 − 0.620990i −0.368373 − 0.638040i −0.486991 − 2.76186i −0.322694 + 0.558923i
12.10 0.140273 − 0.117703i 1.45492 + 1.22082i −0.341474 + 1.93659i 0.471761 − 0.171707i 0.347779 4.79032 − 1.74353i 0.363156 + 0.629005i 0.105437 + 0.597965i 0.0459648 − 0.0796133i
12.11 0.188572 − 0.158231i −1.10952 − 0.930997i −0.336774 + 1.90994i −3.79960 + 1.38294i −0.356537 3.46185 − 1.26001i 0.484869 + 0.839818i −0.156667 − 0.888505i −0.497676 + 0.862000i
12.12 0.828915 − 0.695542i −2.55397 − 2.14304i −0.143975 + 0.816526i −1.36351 + 0.496275i −3.60760 0.142005 − 0.0516855i 1.53066 + 2.65117i 1.40922 + 7.99208i −0.785049 + 1.35974i
12.13 1.15178 − 0.966455i 1.44586 + 1.21322i 0.0452566 − 0.256663i 0.316311 − 0.115128i 2.83783 0.273437 − 0.0995230i 1.30761 + 2.26485i 0.0976628 + 0.553874i 0.253053 − 0.438301i
12.14 1.27441 − 1.06936i −0.251482 − 0.211018i 0.133299 − 0.755976i −3.31364 + 1.20607i −0.546143 −4.51385 + 1.64291i 1.02509 + 1.77551i −0.502230 − 2.84829i −2.93322 + 5.08048i
12.15 1.67568 − 1.40606i −0.785440 − 0.659062i 0.483594 − 2.74260i 0.366475 − 0.133386i −2.24283 1.87283 − 0.681655i −0.858466 − 1.48691i −0.338392 − 1.91912i 0.426545 − 0.738798i
12.16 1.81360 − 1.52179i 2.38971 + 2.00521i 0.625998 − 3.55021i 0.512516 − 0.186541i 7.38547 −4.06919 + 1.48107i −1.89988 − 3.29068i 1.16892 + 6.62930i 0.645623 − 1.11825i
12.17 1.87572 − 1.57392i −2.14663 − 1.80124i 0.693817 − 3.93483i 3.08595 − 1.12319i −6.86148 −1.72168 + 0.626640i −2.44311 − 4.23159i 0.842626 + 4.77877i 4.02056 − 6.96381i
12.18 2.08651 − 1.75079i 1.52081 + 1.27611i 0.940962 − 5.33646i −3.17456 + 1.15545i 5.40740 3.73983 − 1.36119i −4.65594 − 8.06433i 0.163463 + 0.927044i −4.60081 + 7.96884i
34.1 −2.01822 − 1.69349i 0.272602 − 0.228740i 0.858012 + 4.86603i 0.543793 + 0.197924i −0.937538 −1.71611 − 0.624614i 3.87430 − 6.71048i −0.498955 + 2.82971i −0.762310 − 1.32036i
34.2 −1.86788 − 1.56733i −1.35177 + 1.13427i 0.685128 + 3.88556i −1.59269 − 0.579691i 4.30271 −1.39838 − 0.508970i 2.37189 − 4.10824i 0.0197673 − 0.112106i 2.06637 + 3.57906i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 407.2.l.b ✓ 108
37.f even 9 1 inner 407.2.l.b ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
407.2.l.b ✓ 108 1.a even 1 1 trivial
407.2.l.b ✓ 108 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{108} + 7 T_{2}^{105} + 39 T_{2}^{103} + 893 T_{2}^{102} - 504 T_{2}^{101} + 462 T_{2}^{100} + \cdots + 1672482816 \) acting on \(S_{2}^{\mathrm{new}}(407, [\chi])\). Copy content Toggle raw display