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Properties

Label 112.3.k.a
Level $112$
Weight $3$
Character orbit 112.k
Analytic conductor $3.052$
Analytic rank $0$
Dimension $48$
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] = [112,3,Mod(43,112)]
 
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([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.43");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 112.k (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: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{4} + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{4} + 12 q^{6} - 40 q^{10} + 16 q^{11} - 108 q^{12} - 14 q^{14} + 66 q^{16} + 30 q^{18} - 64 q^{19} + 84 q^{20} + 94 q^{22} - 64 q^{23} + 40 q^{24} - 196 q^{26} - 96 q^{27} + 16 q^{29} - 72 q^{30} + 160 q^{32} - 28 q^{34} + 64 q^{36} - 48 q^{37} - 224 q^{38} + 384 q^{39} + 180 q^{40} + 176 q^{43} - 114 q^{44} - 256 q^{46} + 52 q^{48} + 336 q^{49} + 6 q^{50} - 192 q^{51} - 48 q^{52} - 80 q^{53} - 288 q^{54} - 512 q^{55} - 98 q^{56} - 50 q^{58} - 288 q^{59} + 512 q^{60} - 64 q^{61} + 156 q^{62} + 126 q^{64} - 32 q^{65} - 116 q^{66} + 80 q^{67} - 32 q^{68} + 192 q^{69} + 168 q^{70} + 26 q^{72} + 330 q^{74} + 608 q^{75} + 672 q^{76} + 112 q^{77} - 352 q^{78} - 980 q^{80} - 432 q^{81} - 76 q^{82} - 160 q^{83} + 320 q^{85} - 542 q^{86} - 896 q^{87} - 214 q^{88} + 1144 q^{90} - 12 q^{92} + 96 q^{93} + 660 q^{94} + 184 q^{96} + 496 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.99968 − 0.0358652i 2.45353 + 2.45353i 3.99743 + 0.143438i 3.05079 + 3.05079i −4.81828 − 4.99427i 2.64575 −7.98842 − 0.430198i 3.03965i −5.99119 − 6.21002i
43.2 −1.94663 + 0.458959i −3.68547 − 3.68547i 3.57871 − 1.78684i −6.05295 − 6.05295i 8.86573 + 5.48276i 2.64575 −6.14634 + 5.12080i 18.1655i 14.5609 + 9.00479i
43.3 −1.91895 + 0.563601i −0.868082 − 0.868082i 3.36471 − 2.16304i 1.32483 + 1.32483i 2.15505 + 1.17655i −2.64575 −5.23761 + 6.04711i − 7.49287i −3.28895 − 1.79560i
43.4 −1.85609 − 0.744930i −0.942254 − 0.942254i 2.89016 + 2.76532i −1.50963 − 1.50963i 1.04700 + 2.45082i −2.64575 −3.30444 − 7.28565i − 7.22432i 1.67744 + 3.92657i
43.5 −1.74640 − 0.974721i −3.20388 − 3.20388i 2.09984 + 3.40451i 6.09982 + 6.09982i 2.47238 + 8.71816i 2.64575 −0.348720 − 7.99240i 11.5297i −4.70712 − 16.5984i
43.6 −1.54056 − 1.27541i 1.08720 + 1.08720i 0.746640 + 3.92970i −4.05229 − 4.05229i −0.288265 − 3.06152i 2.64575 3.86175 − 7.00620i − 6.63600i 1.07444 + 11.4111i
43.7 −1.29247 − 1.52628i 3.85711 + 3.85711i −0.659032 + 3.94534i 0.652426 + 0.652426i 0.901804 − 10.8722i −2.64575 6.87345 − 4.09337i 20.7545i 0.152539 − 1.83902i
43.8 −1.05702 + 1.69785i 2.92752 + 2.92752i −1.76542 − 3.58933i 4.98965 + 4.98965i −8.06494 + 1.87606i −2.64575 7.96024 + 0.796563i 8.14073i −13.7459 + 3.19755i
43.9 −1.03460 + 1.71161i −3.53870 − 3.53870i −1.85919 − 3.54167i 3.55906 + 3.55906i 9.71800 − 2.39571i −2.64575 7.98547 + 0.482011i 16.0447i −9.77393 + 2.40950i
43.10 −0.787197 + 1.83856i −0.990832 − 0.990832i −2.76064 − 2.89463i −0.763051 − 0.763051i 2.60169 − 1.04173i 2.64575 7.49513 − 2.79697i − 7.03650i 2.00359 − 0.802247i
43.11 −0.385023 − 1.96259i −2.39230 − 2.39230i −3.70352 + 1.51128i −2.43386 − 2.43386i −3.77401 + 5.61619i 2.64575 4.39197 + 6.68660i 2.44621i −3.83958 + 5.71376i
43.12 0.147360 + 1.99456i −0.767547 − 0.767547i −3.95657 + 0.587838i −4.42732 − 4.42732i 1.41782 − 1.64403i −2.64575 −1.75552 − 7.80501i − 7.82174i 8.17815 − 9.48297i
43.13 0.347519 − 1.96958i 1.49790 + 1.49790i −3.75846 − 1.36893i −6.69821 − 6.69821i 3.47078 − 2.42968i −2.64575 −4.00235 + 6.92684i − 4.51259i −15.5204 + 10.8649i
43.14 0.451604 + 1.94835i 0.788779 + 0.788779i −3.59211 + 1.75976i 5.09254 + 5.09254i −1.18060 + 1.89303i 2.64575 −5.05083 − 6.20396i − 7.75565i −7.62222 + 12.2218i
43.15 0.540712 + 1.92552i 4.17892 + 4.17892i −3.41526 + 2.08231i −3.72669 − 3.72669i −5.78700 + 10.3062i 2.64575 −5.85620 − 5.45023i 25.9267i 5.16075 − 9.19088i
43.16 0.734503 − 1.86024i 2.79252 + 2.79252i −2.92101 − 2.73271i 3.47789 + 3.47789i 7.24589 − 3.14366i 2.64575 −7.22900 + 3.42660i 6.59638i 9.02424 − 3.91520i
43.17 0.826031 − 1.82145i −3.51111 − 3.51111i −2.63535 − 3.00914i 1.37873 + 1.37873i −9.29559 + 3.49502i −2.64575 −7.65788 + 2.31450i 15.6558i 3.65015 − 1.37241i
43.18 1.44611 − 1.38159i −0.955204 − 0.955204i 0.182440 − 3.99584i −1.52170 − 1.52170i −2.70102 − 0.0616290i 2.64575 −5.25677 − 6.03046i − 7.17517i −4.30288 − 0.0981785i
43.19 1.52422 + 1.29490i 1.50939 + 1.50939i 0.646466 + 3.94741i 0.797814 + 0.797814i 0.346126 + 4.25515i −2.64575 −4.12615 + 6.85382i − 4.44347i 0.182951 + 2.24913i
43.20 1.77943 − 0.913031i 3.11584 + 3.11584i 2.33275 − 3.24935i −1.25115 − 1.25115i 8.38928 + 2.69956i −2.64575 1.18421 − 7.91187i 10.4169i −3.36867 − 1.08399i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f 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.k.a ✓ 48
4.b odd 2 1 448.3.k.a 48
8.b even 2 1 896.3.k.a 48
8.d odd 2 1 896.3.k.b 48
16.e even 4 1 448.3.k.a 48
16.e even 4 1 896.3.k.b 48
16.f odd 4 1 inner 112.3.k.a ✓ 48
16.f odd 4 1 896.3.k.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.3.k.a ✓ 48 1.a even 1 1 trivial
112.3.k.a ✓ 48 16.f odd 4 1 inner
448.3.k.a 48 4.b odd 2 1
448.3.k.a 48 16.e even 4 1
896.3.k.a 48 8.b even 2 1
896.3.k.a 48 16.f odd 4 1
896.3.k.b 48 8.d odd 2 1
896.3.k.b 48 16.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(112, [\chi])\).