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Properties

Label 112.4.m.a
Level $112$
Weight $4$
Character orbit 112.m
Analytic conductor $6.608$
Analytic rank $0$
Dimension $34$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,4,Mod(29,112)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.29");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 112.m (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: \(6.60821392064\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 20 q^{4} + 40 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q - 20 q^{4} + 40 q^{6} + 60 q^{10} - 78 q^{11} - 172 q^{12} - 28 q^{14} + 120 q^{15} - 160 q^{16} + 340 q^{17} + 180 q^{18} + 24 q^{19} + 336 q^{20} + 20 q^{22} + 128 q^{24} - 256 q^{26} - 132 q^{27} - 84 q^{28} - 66 q^{29} - 228 q^{30} + 140 q^{32} - 792 q^{33} - 192 q^{34} + 1876 q^{36} - 746 q^{37} + 624 q^{38} - 4 q^{40} - 84 q^{42} + 334 q^{43} - 164 q^{44} + 1340 q^{45} + 480 q^{46} - 2252 q^{48} - 1666 q^{49} - 2008 q^{50} + 744 q^{51} - 288 q^{52} + 1014 q^{53} - 1072 q^{54} + 28 q^{56} + 648 q^{58} - 1020 q^{59} - 2028 q^{60} - 2136 q^{61} + 1236 q^{62} + 1638 q^{63} - 1760 q^{64} - 2112 q^{65} + 408 q^{66} + 230 q^{67} + 3468 q^{68} - 1732 q^{69} - 756 q^{70} + 3312 q^{72} + 1884 q^{74} + 4636 q^{75} - 3616 q^{76} + 546 q^{77} + 5108 q^{78} + 4412 q^{79} - 1660 q^{80} - 162 q^{81} - 4344 q^{82} + 1340 q^{83} + 624 q^{85} - 4496 q^{86} + 2888 q^{88} + 5824 q^{90} - 6708 q^{92} + 1376 q^{93} + 6452 q^{94} - 3040 q^{95} + 200 q^{96} - 3492 q^{97} - 6166 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −2.79139 − 0.456203i −2.44465 − 2.44465i 7.58376 + 2.54688i −3.18766 + 3.18766i 5.70872 + 7.93923i − 7.00000i −20.0074 − 10.5691i − 15.0474i 10.3522 − 7.44380i
29.2 −2.56375 + 1.19465i −4.61003 − 4.61003i 5.14561 − 6.12558i 8.40946 − 8.40946i 17.3263 + 6.31157i − 7.00000i −5.87410 + 21.8517i 15.5047i −11.5133 + 31.6061i
29.3 −2.40478 − 1.48897i 4.44908 + 4.44908i 3.56594 + 7.16129i 8.91772 − 8.91772i −4.07452 − 17.3236i − 7.00000i 2.08762 − 22.5309i 12.5887i −34.7234 + 8.16696i
29.4 −2.27164 + 1.68513i 1.30949 + 1.30949i 2.32067 − 7.65601i −9.34882 + 9.34882i −5.18135 − 0.768023i − 7.00000i 7.62967 + 21.3023i − 23.5705i 5.48314 − 36.9911i
29.5 −1.82459 − 2.16122i 0.260272 + 0.260272i −1.34173 + 7.88668i −13.6585 + 13.6585i 0.0876146 − 1.03740i − 7.00000i 19.4930 − 11.4902i − 26.8645i 54.4403 + 4.59783i
29.6 −1.22568 + 2.54906i 2.34659 + 2.34659i −4.99540 − 6.24868i 8.07684 − 8.07684i −8.85776 + 3.10542i − 7.00000i 22.0510 − 5.07467i − 15.9871i 10.6887 + 30.4880i
29.7 −0.944292 − 2.66614i −0.369181 − 0.369181i −6.21663 + 5.03523i 8.66479 − 8.66479i −0.635673 + 1.33290i − 7.00000i 19.2950 + 11.8197i − 26.7274i −31.2837 − 14.9195i
29.8 −0.769500 + 2.72174i −5.25462 − 5.25462i −6.81574 − 4.18876i −8.21141 + 8.21141i 18.3451 − 10.2583i − 7.00000i 16.6454 − 15.3274i 28.2221i −16.0306 − 28.6680i
29.9 0.351132 − 2.80655i 4.19498 + 4.19498i −7.75341 − 1.97094i −12.4679 + 12.4679i 13.2464 − 10.3004i − 7.00000i −8.25400 + 21.0683i 8.19565i 30.6138 + 39.3696i
29.10 0.523412 − 2.77958i −5.84565 − 5.84565i −7.45208 − 2.90972i −4.14767 + 4.14767i −19.3081 + 13.1887i − 7.00000i −11.9883 + 19.1906i 41.3432i 9.35782 + 13.6997i
29.11 0.708933 + 2.73814i 5.83752 + 5.83752i −6.99483 + 3.88232i −5.63760 + 5.63760i −11.8455 + 20.1223i − 7.00000i −15.5892 − 16.4005i 41.1532i −19.4332 − 11.4399i
29.12 1.44063 + 2.43405i −3.02072 − 3.02072i −3.84915 + 7.01313i −5.61058 + 5.61058i 3.00082 − 11.7043i − 7.00000i −22.6155 + 0.734340i − 8.75046i −21.7392 − 5.57362i
29.13 1.70047 − 2.26018i 6.81099 + 6.81099i −2.21682 − 7.68672i 12.3048 − 12.3048i 26.9759 − 3.81218i − 7.00000i −21.1430 − 8.06063i 65.7791i −6.88711 − 48.7348i
29.14 2.24016 − 1.72676i 0.259297 + 0.259297i 2.03663 − 7.73642i −0.0107678 + 0.0107678i 1.02861 + 0.133124i − 7.00000i −8.79651 − 20.8476i − 26.8655i −0.00552826 + 0.0427151i
29.15 2.44137 + 1.42819i −2.06034 − 2.06034i 3.92055 + 6.97347i 12.0043 − 12.0043i −2.08749 − 7.97261i − 7.00000i −0.387943 + 22.6241i − 18.5100i 46.4513 − 12.1625i
29.16 2.63214 − 1.03531i −6.34135 − 6.34135i 5.85627 − 5.45014i 5.02217 − 5.02217i −23.2565 − 10.1260i − 7.00000i 9.77193 − 20.4086i 53.4254i 8.01953 − 18.4185i
29.17 2.75739 + 0.629938i 4.47833 + 4.47833i 7.20636 + 3.47396i −1.11910 + 1.11910i 9.52742 + 15.1696i − 7.00000i 17.6823 + 14.1186i 13.1109i −3.79075 + 2.38082i
85.1 −2.79139 + 0.456203i −2.44465 + 2.44465i 7.58376 − 2.54688i −3.18766 − 3.18766i 5.70872 − 7.93923i 7.00000i −20.0074 + 10.5691i 15.0474i 10.3522 + 7.44380i
85.2 −2.56375 − 1.19465i −4.61003 + 4.61003i 5.14561 + 6.12558i 8.40946 + 8.40946i 17.3263 − 6.31157i 7.00000i −5.87410 − 21.8517i − 15.5047i −11.5133 − 31.6061i
85.3 −2.40478 + 1.48897i 4.44908 − 4.44908i 3.56594 − 7.16129i 8.91772 + 8.91772i −4.07452 + 17.3236i 7.00000i 2.08762 + 22.5309i − 12.5887i −34.7234 − 8.16696i
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.4.m.a ✓ 34
4.b odd 2 1 448.4.m.a 34
16.e even 4 1 inner 112.4.m.a ✓ 34
16.f odd 4 1 448.4.m.a 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.m.a ✓ 34 1.a even 1 1 trivial
112.4.m.a ✓ 34 16.e even 4 1 inner
448.4.m.a 34 4.b odd 2 1
448.4.m.a 34 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{34} + 44 T_{3}^{31} + 17172 T_{3}^{30} + 4496 T_{3}^{29} + 968 T_{3}^{28} + 295272 T_{3}^{27} + \cdots + 15\!\cdots\!28 \) acting on \(S_{4}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display