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Properties

Label 112.3.l.b
Level $112$
Weight $3$
Character orbit 112.l
Analytic conductor $3.052$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,3,Mod(13,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 112.l (of order \(4\), 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: \(3.05177896084\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{2} - 8 q^{4} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 4 q^{2} - 8 q^{4} - 16 q^{8} + 40 q^{14} - 8 q^{15} + 48 q^{16} + 196 q^{18} - 20 q^{21} - 120 q^{22} - 96 q^{29} - 40 q^{30} - 184 q^{32} - 100 q^{35} + 160 q^{36} - 128 q^{37} - 144 q^{42} - 72 q^{43} - 448 q^{44} - 168 q^{46} + 192 q^{49} - 364 q^{50} - 128 q^{51} + 88 q^{53} + 56 q^{56} + 408 q^{58} + 504 q^{60} + 444 q^{63} + 256 q^{64} - 8 q^{65} + 440 q^{67} - 112 q^{70} + 592 q^{72} - 408 q^{74} + 12 q^{77} + 664 q^{78} - 8 q^{79} + 64 q^{81} - 576 q^{84} + 96 q^{85} + 256 q^{86} + 448 q^{88} - 388 q^{91} - 1192 q^{92} + 32 q^{93} - 776 q^{95} + 540 q^{98} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.97215 + 0.332609i −3.31524 + 3.31524i 3.77874 − 1.31191i −4.66000 − 4.66000i 5.43547 − 7.64083i −2.13987 + 6.66491i −7.01589 + 3.84412i − 12.9816i 10.7402 + 7.64027i
13.2 −1.97215 + 0.332609i 3.31524 − 3.31524i 3.77874 − 1.31191i 4.66000 + 4.66000i −5.43547 + 7.64083i 2.13987 + 6.66491i −7.01589 + 3.84412i − 12.9816i −10.7402 − 7.64027i
13.3 −1.91394 + 0.580376i −2.42317 + 2.42317i 3.32633 − 2.22161i 4.90814 + 4.90814i 3.23145 − 6.04416i 5.19813 − 4.68822i −5.07702 + 6.18255i − 2.74353i −12.2424 − 6.54531i
13.4 −1.91394 + 0.580376i 2.42317 − 2.42317i 3.32633 − 2.22161i −4.90814 − 4.90814i −3.23145 + 6.04416i −5.19813 − 4.68822i −5.07702 + 6.18255i − 2.74353i 12.2424 + 6.54531i
13.5 −1.88371 − 0.672048i −0.746365 + 0.746365i 3.09670 + 2.53188i 0.822232 + 0.822232i 1.90753 − 0.904340i −6.88091 − 1.28574i −4.13173 − 6.85046i 7.88588i −0.996265 − 2.10142i
13.6 −1.88371 − 0.672048i 0.746365 − 0.746365i 3.09670 + 2.53188i −0.822232 − 0.822232i −1.90753 + 0.904340i 6.88091 − 1.28574i −4.13173 − 6.85046i 7.88588i 0.996265 + 2.10142i
13.7 −1.10346 − 1.66805i −3.53466 + 3.53466i −1.56475 + 3.68124i −0.208662 − 0.208662i 9.79632 + 1.99561i −1.63496 − 6.80639i 7.86712 − 1.45204i − 15.9876i −0.117807 + 0.578308i
13.8 −1.10346 − 1.66805i 3.53466 − 3.53466i −1.56475 + 3.68124i 0.208662 + 0.208662i −9.79632 − 1.99561i 1.63496 − 6.80639i 7.86712 − 1.45204i − 15.9876i 0.117807 − 0.578308i
13.9 −1.08539 + 1.67986i −2.28266 + 2.28266i −1.64387 − 3.64660i −2.90674 − 2.90674i −1.35698 − 6.31211i 2.26451 − 6.62359i 7.91002 + 1.19649i − 1.42104i 8.03786 − 1.72799i
13.10 −1.08539 + 1.67986i 2.28266 − 2.28266i −1.64387 − 3.64660i 2.90674 + 2.90674i 1.35698 + 6.31211i −2.26451 − 6.62359i 7.91002 + 1.19649i − 1.42104i −8.03786 + 1.72799i
13.11 −0.981392 − 1.74266i −0.752012 + 0.752012i −2.07374 + 3.42047i −6.01979 − 6.01979i 2.04852 + 0.572483i 4.46479 + 5.39126i 7.99587 + 0.257000i 7.86896i −4.58268 + 16.3982i
13.12 −0.981392 − 1.74266i 0.752012 − 0.752012i −2.07374 + 3.42047i 6.01979 + 6.01979i −2.04852 − 0.572483i −4.46479 + 5.39126i 7.99587 + 0.257000i 7.86896i 4.58268 − 16.3982i
13.13 −0.213649 + 1.98856i −3.60463 + 3.60463i −3.90871 − 0.849706i 6.37616 + 6.37616i −6.39788 − 7.93813i −5.25829 + 4.62065i 2.52478 − 7.59115i − 16.9867i −14.0416 + 11.3171i
13.14 −0.213649 + 1.98856i 3.60463 − 3.60463i −3.90871 − 0.849706i −6.37616 − 6.37616i 6.39788 + 7.93813i 5.25829 + 4.62065i 2.52478 − 7.59115i − 16.9867i 14.0416 − 11.3171i
13.15 0.159836 − 1.99360i −1.79667 + 1.79667i −3.94890 − 0.637301i 3.85269 + 3.85269i 3.29468 + 3.86902i 6.95068 + 0.829499i −1.90170 + 7.77068i 2.54394i 8.29653 − 7.06493i
13.16 0.159836 − 1.99360i 1.79667 − 1.79667i −3.94890 − 0.637301i −3.85269 − 3.85269i −3.29468 − 3.86902i −6.95068 + 0.829499i −1.90170 + 7.77068i 2.54394i −8.29653 + 7.06493i
13.17 0.392844 + 1.96104i −1.00109 + 1.00109i −3.69135 + 1.54076i −4.03888 − 4.03888i −2.35644 − 1.56990i −6.50171 − 2.59380i −4.47162 − 6.63360i 6.99566i 6.33376 − 9.50706i
13.18 0.392844 + 1.96104i 1.00109 − 1.00109i −3.69135 + 1.54076i 4.03888 + 4.03888i 2.35644 + 1.56990i 6.50171 − 2.59380i −4.47162 − 6.63360i 6.99566i −6.33376 + 9.50706i
13.19 0.870222 − 1.80075i −2.99374 + 2.99374i −2.48543 − 3.13411i −0.657693 − 0.657693i 2.78577 + 7.99621i −6.11030 + 3.41529i −7.80663 + 1.74828i − 8.92498i −1.75668 + 0.612005i
13.20 0.870222 − 1.80075i 2.99374 − 2.99374i −2.48543 − 3.13411i 0.657693 + 0.657693i −2.78577 − 7.99621i 6.11030 + 3.41529i −7.80663 + 1.74828i − 8.92498i 1.75668 − 0.612005i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
16.e even 4 1 inner
112.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.3.l.b ✓ 56
4.b odd 2 1 448.3.l.b 56
7.b odd 2 1 inner 112.3.l.b ✓ 56
16.e even 4 1 inner 112.3.l.b ✓ 56
16.f odd 4 1 448.3.l.b 56
28.d even 2 1 448.3.l.b 56
112.j even 4 1 448.3.l.b 56
112.l odd 4 1 inner 112.3.l.b ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.3.l.b ✓ 56 1.a even 1 1 trivial
112.3.l.b ✓ 56 7.b odd 2 1 inner
112.3.l.b ✓ 56 16.e even 4 1 inner
112.3.l.b ✓ 56 112.l odd 4 1 inner
448.3.l.b 56 4.b odd 2 1
448.3.l.b 56 16.f odd 4 1
448.3.l.b 56 28.d even 2 1
448.3.l.b 56 112.j even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} + 3548 T_{3}^{52} + 5173600 T_{3}^{48} + 4043100096 T_{3}^{44} + 1853536759904 T_{3}^{40} + \cdots + 59\!\cdots\!36 \) acting on \(S_{3}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display