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Properties

Label 117.3.y.a
Level $117$
Weight $3$
Character orbit 117.y
Analytic conductor $3.188$
Analytic rank $0$
Dimension $104$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,3,Mod(31,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.y (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: \(3.18801909302\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{2} - 8 q^{3} - 2 q^{5} + 10 q^{6} - 60 q^{8} - 8 q^{9} - 14 q^{11} - 2 q^{13} - 4 q^{14} - 38 q^{15} + 156 q^{16} - 86 q^{18} - 36 q^{19} - 58 q^{20} + 106 q^{21} - 4 q^{22} - 48 q^{24} - 128 q^{26}+ \cdots - 200 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −3.62659 − 0.971741i −2.94231 − 0.585489i 8.74375 + 5.04820i −1.52926 − 5.70728i 10.1016 + 4.98249i 2.26965 − 8.47046i −16.1850 − 16.1850i 8.31441 + 3.44538i 22.1840i
31.2 −3.62452 − 0.971189i 1.24318 − 2.73029i 8.72987 + 5.04019i 0.305966 + 1.14188i −7.15758 + 8.68865i −3.11346 + 11.6196i −16.1333 − 16.1333i −5.90899 − 6.78851i − 4.43593i
31.3 −3.32897 − 0.891995i −0.797857 + 2.89196i 6.82228 + 3.93885i 2.07873 + 7.75792i 5.23566 − 8.91556i 1.39866 − 5.21987i −9.44985 − 9.44985i −7.72685 − 4.61474i − 27.6801i
31.4 −3.18343 − 0.852998i 2.65511 + 1.39656i 5.94252 + 3.43092i −1.23046 − 4.59214i −7.26110 − 6.71066i 0.411953 − 1.53743i −6.66931 − 6.66931i 5.09923 + 7.41605i 15.6684i
31.5 −2.62588 − 0.703603i −1.51101 + 2.59169i 2.93610 + 1.69516i −0.871516 − 3.25254i 5.79125 − 5.74232i −3.16613 + 11.8161i 1.17198 + 1.17198i −4.43370 − 7.83213i 9.15399i
31.6 −2.48463 − 0.665755i −1.77959 − 2.41518i 2.26606 + 1.30831i 1.74973 + 6.53009i 2.81370 + 7.18559i 0.485382 − 1.81147i 2.51620 + 2.51620i −2.66615 + 8.59602i − 17.3897i
31.7 −2.32589 − 0.623221i 2.30915 − 1.91516i 1.55728 + 0.899094i 0.326628 + 1.21899i −6.56440 + 3.01534i 2.50428 − 9.34611i 3.74897 + 3.74897i 1.66434 − 8.84477i − 3.03881i
31.8 −1.66139 − 0.445168i −0.924780 − 2.85391i −0.902057 − 0.520803i −1.70724 − 6.37150i 0.265953 + 5.15314i −0.446010 + 1.66453i 6.13171 + 6.13171i −7.28956 + 5.27847i 11.3456i
31.9 −1.60696 − 0.430585i 1.68465 + 2.48233i −1.06717 − 0.616132i 0.359970 + 1.34342i −1.63832 − 4.71440i −0.198185 + 0.739637i 6.15513 + 6.15513i −3.32390 + 8.36371i − 2.31383i
31.10 −1.33961 − 0.358947i −2.93992 + 0.597373i −1.79839 − 1.03830i −0.613360 − 2.28909i 4.15277 + 0.255030i 0.0472052 − 0.176172i 5.95909 + 5.95909i 8.28629 − 3.51246i 3.28665i
31.11 −1.06825 − 0.286237i 2.91836 − 0.695104i −2.40487 − 1.38845i 1.81354 + 6.76822i −3.31650 − 0.0927971i −3.04303 + 11.3567i 5.29964 + 5.29964i 8.03366 − 4.05713i − 7.74925i
31.12 −0.514097 − 0.137752i −2.55437 + 1.57328i −3.21878 − 1.85836i 1.42653 + 5.32387i 1.52991 − 0.456949i 2.51662 − 9.39214i 2.90415 + 2.90415i 4.04958 − 8.03747i − 2.93349i
31.13 0.149774 + 0.0401318i 0.0310137 + 2.99984i −3.44328 − 1.98798i −1.98994 − 7.42657i −0.115744 + 0.450542i 1.37301 − 5.12415i −0.874500 − 0.874500i −8.99808 + 0.186072i − 1.19217i
31.14 0.181674 + 0.0486794i 2.99854 + 0.0936789i −3.43347 − 1.98231i −1.04393 − 3.89601i 0.540196 + 0.162986i 1.72376 − 6.43316i −1.05925 − 1.05925i 8.98245 + 0.561799i − 0.758622i
31.15 0.531767 + 0.142486i −2.42608 − 1.76470i −3.20163 − 1.84846i 1.12819 + 4.21046i −1.03866 − 1.28409i −2.13692 + 7.97511i −2.99626 − 2.99626i 2.77170 + 8.56257i 2.39973i
31.16 0.583000 + 0.156214i 1.09433 − 2.79329i −3.14862 − 1.81785i −0.592695 − 2.21197i 1.07434 − 1.45754i −0.518405 + 1.93471i −3.25881 − 3.25881i −6.60490 − 6.11353i − 1.38217i
31.17 1.07169 + 0.287157i −0.645473 + 2.92974i −2.39805 − 1.38451i 0.951873 + 3.55244i −1.53304 + 2.95441i −2.20661 + 8.23517i −5.31050 − 5.31050i −8.16673 − 3.78213i 4.08044i
31.18 1.55517 + 0.416705i 2.60419 + 1.48937i −1.21920 − 0.703908i 2.23994 + 8.35955i 3.42932 + 3.40139i 2.18521 − 8.15533i −6.15658 − 6.15658i 4.56358 + 7.75718i 13.9339i
31.19 1.88597 + 0.505345i −2.06210 − 2.17894i −0.162580 − 0.0938654i −0.0385330 − 0.143807i −2.78794 − 5.15148i 3.35702 − 12.5286i −5.78170 − 5.78170i −0.495516 + 8.98635i − 0.290689i
31.20 1.91711 + 0.513688i −2.98190 + 0.329042i −0.0526611 − 0.0304039i −2.49739 − 9.32037i −5.88566 − 0.900957i −1.04281 + 3.89184i −5.69903 − 5.69903i 8.78346 − 1.96234i − 19.1511i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
13.d odd 4 1 inner
117.y odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.3.y.a ✓ 104
3.b odd 2 1 351.3.bb.a 104
9.c even 3 1 inner 117.3.y.a ✓ 104
9.d odd 6 1 351.3.bb.a 104
13.d odd 4 1 inner 117.3.y.a ✓ 104
39.f even 4 1 351.3.bb.a 104
117.y odd 12 1 inner 117.3.y.a ✓ 104
117.z even 12 1 351.3.bb.a 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.3.y.a ✓ 104 1.a even 1 1 trivial
117.3.y.a ✓ 104 9.c even 3 1 inner
117.3.y.a ✓ 104 13.d odd 4 1 inner
117.3.y.a ✓ 104 117.y odd 12 1 inner
351.3.bb.a 104 3.b odd 2 1
351.3.bb.a 104 9.d odd 6 1
351.3.bb.a 104 39.f even 4 1
351.3.bb.a 104 117.z even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(117, [\chi])\).