↑

Properties

Label 312.2.bo.a
Level $312$
Weight $2$
Character orbit 312.bo
Analytic conductor $2.491$
Analytic rank $0$
Dimension $208$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [312,2,Mod(149,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 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("312.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.bo (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: \(2.49133254306\)
Analytic rank: \(0\)
Dimension: \(208\)
Relative dimension: \(52\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 12 q^{4} + 6 q^{6} - 16 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q - 12 q^{4} + 6 q^{6} - 16 q^{7} - 4 q^{9} - 12 q^{10} + 4 q^{15} - 12 q^{16} - 20 q^{18} + 4 q^{22} + 34 q^{24} - 36 q^{28} - 54 q^{30} - 16 q^{31} + 4 q^{33} + 12 q^{34} + 12 q^{36} - 20 q^{39} + 8 q^{40} - 28 q^{42} - 56 q^{46} - 34 q^{48} - 72 q^{49} - 64 q^{52} - 54 q^{54} - 40 q^{55} - 20 q^{57} + 28 q^{58} - 12 q^{60} + 20 q^{63} - 8 q^{66} - 136 q^{70} + 60 q^{72} - 32 q^{73} + 64 q^{76} - 68 q^{78} - 32 q^{79} - 4 q^{81} - 72 q^{82} + 8 q^{84} - 4 q^{87} + 60 q^{88} - 72 q^{94} + 128 q^{96} - 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −1.40956 − 0.114580i −1.62230 + 0.606751i 1.97374 + 0.323017i −2.87411 + 2.87411i 2.35626 − 0.669371i −0.883084 + 3.29572i −2.74511 − 0.681465i 2.26371 − 1.96866i 4.38057 − 3.72193i
149.2 −1.39442 + 0.235779i −1.09050 + 1.34566i 1.88882 − 0.657551i 2.03684 − 2.03684i 1.20334 − 2.13354i 0.560344 − 2.09123i −2.47877 + 1.36225i −0.621621 − 2.93489i −2.35997 + 3.32046i
149.3 −1.38813 − 0.270358i 1.71893 − 0.212806i 1.85381 + 0.750584i 0.426563 − 0.426563i −2.44363 − 0.169323i 0.338276 − 1.26246i −2.37041 − 1.54310i 2.90943 − 0.731595i −0.707450 + 0.476801i
149.4 −1.37935 + 0.312097i −0.809071 − 1.53147i 1.80519 − 0.860981i −0.593362 + 0.593362i 1.59396 + 1.85992i −0.416671 + 1.55504i −2.22127 + 1.75099i −1.69081 + 2.47814i 0.633265 − 1.00364i
149.5 −1.35060 + 0.419389i 0.809071 + 1.53147i 1.64823 − 1.13285i 0.593362 − 0.593362i −1.73501 − 1.72909i −0.416671 + 1.55504i −1.75099 + 2.22127i −1.69081 + 2.47814i −0.552544 + 1.05024i
149.6 −1.32549 + 0.493019i 1.09050 − 1.34566i 1.51386 − 1.30699i −2.03684 + 2.03684i −0.782012 + 2.32131i 0.560344 − 2.09123i −1.36225 + 2.47877i −0.621621 − 2.93489i 1.69562 − 3.70402i
149.7 −1.32339 − 0.498630i −1.65353 − 0.515585i 1.50274 + 1.31977i 2.22918 − 2.22918i 1.93119 + 1.50682i −0.526194 + 1.96378i −1.33063 − 2.49588i 2.46834 + 1.70507i −4.06161 + 1.83854i
149.8 −1.25682 − 0.648392i −0.547676 + 1.64318i 1.15918 + 1.62982i −0.262561 + 0.262561i 1.75375 − 1.71007i 0.743546 − 2.77495i −0.400109 − 2.79998i −2.40010 − 1.79986i 0.500234 − 0.159749i
149.9 −1.24100 − 0.678179i −0.927073 − 1.46306i 1.08015 + 1.68324i −1.35807 + 1.35807i 0.158280 + 2.44437i 1.20082 − 4.48154i −0.198923 − 2.82142i −1.28107 + 2.71272i 2.60637 − 0.764344i
149.10 −1.16343 + 0.804012i 1.62230 − 0.606751i 0.707131 − 1.87082i 2.87411 − 2.87411i −1.39959 + 2.01026i −0.883084 + 3.29572i 0.681465 + 2.74511i 2.26371 − 1.96866i −1.03300 + 5.65465i
149.11 −1.06698 + 0.928202i −1.71893 + 0.212806i 0.276882 − 1.98074i −0.426563 + 0.426563i 1.63653 − 1.82257i 0.338276 − 1.26246i 1.54310 + 2.37041i 2.90943 − 0.731595i 0.0591966 − 0.851070i
149.12 −1.02509 − 0.974265i −0.0830849 + 1.73006i 0.101615 + 1.99742i 0.638351 − 0.638351i 1.77070 − 1.69252i −1.27601 + 4.76213i 1.84185 − 2.14653i −2.98619 − 0.287483i −1.27629 + 0.0324435i
149.13 −0.992587 − 1.00736i 1.46646 − 0.921677i −0.0295415 + 1.99978i −2.52058 + 2.52058i −2.38405 − 0.562409i −0.567074 + 2.11635i 2.04382 − 1.95520i 1.30102 − 2.70321i 5.04102 + 0.0372319i
149.14 −0.909207 − 1.08321i 0.489521 − 1.66144i −0.346687 + 1.96972i 1.52459 − 1.52459i −2.24476 + 0.980334i 0.00327369 − 0.0122176i 2.44883 − 1.41535i −2.52074 − 1.62662i −3.03762 − 0.265283i
149.15 −0.896777 + 1.09352i 1.65353 + 0.515585i −0.391584 − 1.96129i −2.22918 + 2.22918i −2.04665 + 1.34581i −0.526194 + 1.96378i 2.49588 + 1.33063i 2.46834 + 1.70507i −0.438582 − 4.43673i
149.16 −0.837035 − 1.13990i 1.46386 + 0.925798i −0.598744 + 1.90827i 2.34301 − 2.34301i −0.169989 − 2.44358i 0.863251 − 3.22170i 2.67641 − 0.914783i 1.28580 + 2.71048i −4.63199 − 0.709617i
149.17 −0.764239 + 1.18993i 0.547676 − 1.64318i −0.831877 − 1.81879i 0.262561 − 0.262561i 1.53672 + 1.90748i 0.743546 − 2.77495i 2.79998 + 0.400109i −2.40010 − 1.79986i 0.111771 + 0.513089i
149.18 −0.735645 + 1.20782i 0.927073 + 1.46306i −0.917652 − 1.77705i 1.35807 − 1.35807i −2.44910 + 0.0434451i 1.20082 − 4.48154i 2.82142 + 0.198923i −1.28107 + 2.71272i 0.641243 + 2.63935i
149.19 −0.605430 − 1.27807i −1.62596 + 0.596879i −1.26691 + 1.54756i −1.50349 + 1.50349i 1.74725 + 1.71671i 0.436719 − 1.62986i 2.74491 + 0.682257i 2.28747 − 1.94100i 2.83182 + 1.01130i
149.20 −0.497838 − 1.32369i −1.31229 − 1.13043i −1.50431 + 1.31797i −0.582075 + 0.582075i −0.843035 + 2.29985i −0.611177 + 2.28094i 2.49349 + 1.33511i 0.444236 + 2.96693i 1.06027 + 0.480708i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
13.f odd 12 1 inner
24.h odd 2 1 inner
39.k even 12 1 inner
104.x odd 12 1 inner
312.bo even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.2.bo.a ✓ 208
3.b odd 2 1 inner 312.2.bo.a ✓ 208
8.b even 2 1 inner 312.2.bo.a ✓ 208
13.f odd 12 1 inner 312.2.bo.a ✓ 208
24.h odd 2 1 inner 312.2.bo.a ✓ 208
39.k even 12 1 inner 312.2.bo.a ✓ 208
104.x odd 12 1 inner 312.2.bo.a ✓ 208
312.bo even 12 1 inner 312.2.bo.a ✓ 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bo.a ✓ 208 1.a even 1 1 trivial
312.2.bo.a ✓ 208 3.b odd 2 1 inner
312.2.bo.a ✓ 208 8.b even 2 1 inner
312.2.bo.a ✓ 208 13.f odd 12 1 inner
312.2.bo.a ✓ 208 24.h odd 2 1 inner
312.2.bo.a ✓ 208 39.k even 12 1 inner
312.2.bo.a ✓ 208 104.x odd 12 1 inner
312.2.bo.a ✓ 208 312.bo even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(312, [\chi])\).