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Properties

Label 105.3.t.b
Level $105$
Weight $3$
Character orbit 105.t
Analytic conductor $2.861$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,3,Mod(11,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 105.t (of order \(6\), 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: \(2.86104277578\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{3} + 36 q^{4} - 24 q^{6} - 58 q^{7} - 2 q^{9} + 20 q^{10} - 42 q^{12} - 100 q^{13} + 20 q^{15} - 12 q^{16} - 14 q^{18} + 50 q^{19} - 12 q^{21} + 256 q^{22} - 140 q^{24} + 90 q^{25} + 4 q^{27}+ \cdots + 384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −3.31814 + 1.91573i 1.84164 + 2.36819i 5.34002 − 9.24919i −1.93649 + 1.11803i −10.6476 − 4.32991i −5.86414 + 3.82255i 25.5943i −2.21669 + 8.72274i 4.28370 − 7.41958i
11.2 −2.61597 + 1.51033i −0.469377 − 2.96305i 2.56220 − 4.43785i 1.93649 − 1.11803i 5.70307 + 7.04234i −4.99419 + 4.90490i 3.39641i −8.55937 + 2.78158i −3.37720 + 5.84948i
11.3 −2.60397 + 1.50340i 2.88780 − 0.812762i 2.52044 − 4.36553i 1.93649 − 1.11803i −6.29785 + 6.45794i −0.494553 − 6.98251i 3.12973i 7.67884 − 4.69420i −3.36171 + 5.82265i
11.4 −2.46897 + 1.42546i −2.92531 − 0.665249i 2.06389 − 3.57476i −1.93649 + 1.11803i 8.17081 − 2.52744i −5.93505 − 3.71149i 0.364297i 8.11489 + 3.89212i 3.18743 − 5.52080i
11.5 −2.31825 + 1.33844i 2.46021 − 1.71679i 1.58287 − 2.74161i −1.93649 + 1.11803i −3.40557 + 7.27281i 4.89419 + 5.00469i − 2.23323i 3.10528 − 8.44732i 2.99285 − 5.18377i
11.6 −1.50527 + 0.869067i −1.65926 + 2.49937i −0.489445 + 0.847743i −1.93649 + 1.11803i 0.325501 − 5.20423i 2.93407 + 6.35541i − 8.65398i −3.49374 − 8.29420i 1.94329 − 3.36588i
11.7 −0.987174 + 0.569945i −0.202464 − 2.99316i −1.35032 + 2.33883i −1.93649 + 1.11803i 1.90581 + 2.83938i 2.61061 − 6.49498i − 7.63801i −8.91802 + 1.21201i 1.27444 − 2.20739i
11.8 −0.860118 + 0.496589i 2.51213 + 1.63987i −1.50680 + 2.60985i 1.93649 − 1.11803i −2.97507 − 0.162987i −1.66850 + 6.79824i − 6.96575i 3.62163 + 8.23916i −1.11041 + 1.92328i
11.9 −0.644768 + 0.372257i −2.67627 + 1.35556i −1.72285 + 2.98406i 1.93649 − 1.11803i 1.22096 − 1.87029i −5.98242 − 3.63464i − 5.54343i 5.32489 − 7.25573i −0.832392 + 1.44175i
11.10 0.644768 − 0.372257i 0.164184 + 2.99550i −1.72285 + 2.98406i −1.93649 + 1.11803i 1.22096 + 1.87029i −5.98242 − 3.63464i 5.54343i −8.94609 + 0.983626i −0.832392 + 1.44175i
11.11 0.860118 − 0.496589i −2.67624 − 1.35563i −1.50680 + 2.60985i −1.93649 + 1.11803i −2.97507 + 0.162987i −1.66850 + 6.79824i 6.96575i 5.32451 + 7.25600i −1.11041 + 1.92328i
11.12 0.987174 − 0.569945i 2.69338 − 1.32124i −1.35032 + 2.33883i 1.93649 − 1.11803i 1.90581 − 2.83938i 2.61061 − 6.49498i 7.63801i 5.50864 − 7.11722i 1.27444 − 2.20739i
11.13 1.50527 − 0.869067i −1.33489 + 2.68664i −0.489445 + 0.847743i 1.93649 − 1.11803i 0.325501 + 5.20423i 2.93407 + 6.35541i 8.65398i −5.43612 − 7.17277i 1.94329 − 3.36588i
11.14 2.31825 − 1.33844i 0.256676 − 2.98900i 1.58287 − 2.74161i 1.93649 − 1.11803i −3.40557 − 7.27281i 4.89419 + 5.00469i 2.23323i −8.86824 − 1.53441i 2.99285 − 5.18377i
11.15 2.46897 − 1.42546i 2.03878 + 2.20077i 2.06389 − 3.57476i 1.93649 − 1.11803i 8.17081 + 2.52744i −5.93505 − 3.71149i − 0.364297i −0.686770 + 8.97376i 3.18743 − 5.52080i
11.16 2.60397 − 1.50340i −0.740030 − 2.90729i 2.52044 − 4.36553i −1.93649 + 1.11803i −6.29785 − 6.45794i −0.494553 − 6.98251i − 3.12973i −7.90471 + 4.30297i −3.36171 + 5.82265i
11.17 2.61597 − 1.51033i 2.80077 − 1.07503i 2.56220 − 4.43785i −1.93649 + 1.11803i 5.70307 − 7.04234i −4.99419 + 4.90490i − 3.39641i 6.68860 − 6.02184i −3.37720 + 5.84948i
11.18 3.31814 − 1.91573i −2.97174 − 0.410813i 5.34002 − 9.24919i 1.93649 − 1.11803i −10.6476 + 4.32991i −5.86414 + 3.82255i − 25.5943i 8.66247 + 2.44166i 4.28370 − 7.41958i
86.1 −3.31814 − 1.91573i 1.84164 − 2.36819i 5.34002 + 9.24919i −1.93649 − 1.11803i −10.6476 + 4.32991i −5.86414 − 3.82255i − 25.5943i −2.21669 − 8.72274i 4.28370 + 7.41958i
86.2 −2.61597 − 1.51033i −0.469377 + 2.96305i 2.56220 + 4.43785i 1.93649 + 1.11803i 5.70307 − 7.04234i −4.99419 − 4.90490i − 3.39641i −8.55937 − 2.78158i −3.37720 − 5.84948i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.t.b ✓ 36
3.b odd 2 1 inner 105.3.t.b ✓ 36
7.c even 3 1 inner 105.3.t.b ✓ 36
21.h odd 6 1 inner 105.3.t.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.t.b ✓ 36 1.a even 1 1 trivial
105.3.t.b ✓ 36 3.b odd 2 1 inner
105.3.t.b ✓ 36 7.c even 3 1 inner
105.3.t.b ✓ 36 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 54 T_{2}^{34} + 1713 T_{2}^{32} - 36282 T_{2}^{30} + 573854 T_{2}^{28} - 6959778 T_{2}^{26} + \cdots + 22915904400 \) acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\). Copy content Toggle raw display