↑

Properties

Label 98.3.f.a
Level $98$
Weight $3$
Character orbit 98.f
Analytic conductor $2.670$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [98,3,Mod(13,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 98.f (of order \(14\), degree \(6\), 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: \(2.67030659073\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 16 q^{4} + 28 q^{6} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 16 q^{4} + 28 q^{6} - 28 q^{9} + 58 q^{11} + 72 q^{15} - 32 q^{16} - 98 q^{17} + 16 q^{18} - 28 q^{20} - 104 q^{22} + 106 q^{23} + 44 q^{25} + 112 q^{26} - 210 q^{27} + 100 q^{29} + 8 q^{30} - 140 q^{35} - 56 q^{36} - 94 q^{37} - 140 q^{38} - 316 q^{39} - 112 q^{40} - 196 q^{41} + 112 q^{42} + 44 q^{43} - 80 q^{44} + 56 q^{45} - 108 q^{46} + 14 q^{47} + 56 q^{49} - 32 q^{50} + 132 q^{51} + 140 q^{52} - 438 q^{53} + 252 q^{54} + 224 q^{55} + 112 q^{56} + 466 q^{57} + 40 q^{58} + 504 q^{59} + 144 q^{60} + 420 q^{61} + 196 q^{62} - 126 q^{63} - 64 q^{64} + 56 q^{65} - 64 q^{67} - 56 q^{69} + 224 q^{70} - 638 q^{71} + 32 q^{72} + 126 q^{73} - 288 q^{74} + 924 q^{75} - 210 q^{77} + 368 q^{78} + 48 q^{79} - 572 q^{81} + 700 q^{83} - 420 q^{84} - 460 q^{85} + 404 q^{86} - 812 q^{87} + 184 q^{88} + 532 q^{89} - 672 q^{90} - 28 q^{91} - 40 q^{92} - 1462 q^{93} - 280 q^{94} - 92 q^{95} + 504 q^{98} - 176 q^{99}+O(q^{100}) \) 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 −0.314692 − 1.37876i −3.24594 + 2.58855i −1.80194 + 0.867767i 4.28385 − 3.41626i 4.59046 + 3.66077i 1.92912 − 6.72893i 1.76350 + 2.21135i 1.83285 − 8.03025i −6.05828 − 4.83132i
13.2 −0.314692 − 1.37876i −1.68004 + 1.33979i −1.80194 + 0.867767i −1.60064 + 1.27647i 2.37594 + 1.89475i 5.75428 + 3.98602i 1.76350 + 2.21135i −0.975178 + 4.27253i 2.26365 + 1.80520i
13.3 −0.314692 − 1.37876i −0.0363883 + 0.0290187i −1.80194 + 0.867767i −6.36830 + 5.07855i 0.0514608 + 0.0410386i −6.98972 − 0.379245i 1.76350 + 2.21135i −2.00221 + 8.77224i 9.00614 + 7.18216i
13.4 −0.314692 − 1.37876i 3.23347 − 2.57861i −1.80194 + 0.867767i 0.0517229 − 0.0412476i −4.57282 − 3.64670i 2.77213 − 6.42770i 1.76350 + 2.21135i 1.80343 − 7.90135i −0.0731472 − 0.0583329i
13.5 0.314692 + 1.37876i −2.64179 + 2.10676i −1.80194 + 0.867767i −1.71101 + 1.36448i −3.73605 − 2.97940i −4.32818 − 5.50153i −1.76350 − 2.21135i 0.537935 − 2.35685i −2.41973 − 1.92967i
13.6 0.314692 + 1.37876i −0.0697289 + 0.0556069i −1.80194 + 0.867767i −3.21239 + 2.56179i −0.0986116 − 0.0786401i 0.217869 + 6.99661i −1.76350 − 2.21135i −2.00092 + 8.76660i −4.54300 − 3.62292i
13.7 0.314692 + 1.37876i 0.157246 − 0.125399i −1.80194 + 0.867767i 6.91995 − 5.51848i 0.222379 + 0.177341i 5.38677 − 4.47021i −1.76350 − 2.21135i −1.99369 + 8.73491i 9.78629 + 7.80431i
13.8 0.314692 + 1.37876i 4.28318 − 3.41572i −1.80194 + 0.867767i 2.99371 − 2.38740i 6.05733 + 4.83056i −6.43428 + 2.75681i −1.76350 − 2.21135i 4.67577 − 20.4859i 4.23374 + 3.37630i
27.1 −1.27416 + 0.613604i −3.92177 − 0.895119i 1.24698 − 1.56366i −1.66380 − 0.379752i 5.54622 − 1.26589i 5.42772 + 4.42039i −0.629384 + 2.75751i 6.47033 + 3.11595i 2.35297 − 0.537050i
27.2 −1.27416 + 0.613604i −2.02737 − 0.462735i 1.24698 − 1.56366i 1.67952 + 0.383339i 2.86714 − 0.654406i −5.62716 + 4.16354i −0.629384 + 2.75751i −4.21260 − 2.02868i −2.37520 + 0.542123i
27.3 −1.27416 + 0.613604i 0.323025 + 0.0737284i 1.24698 − 1.56366i −6.28380 − 1.43424i −0.456827 + 0.104268i 2.27845 − 6.61881i −0.629384 + 2.75751i −8.00981 − 3.85732i 8.88663 − 2.02832i
27.4 −1.27416 + 0.613604i 2.93774 + 0.670521i 1.24698 − 1.56366i 7.13650 + 1.62886i −4.15460 + 0.948259i −5.15034 − 4.74068i −0.629384 + 2.75751i 0.0720136 + 0.0346799i −10.0925 + 2.30356i
27.5 1.27416 − 0.613604i −3.16750 − 0.722962i 1.24698 − 1.56366i −5.37915 − 1.22776i −4.47953 + 1.02242i −5.22764 − 4.65529i 0.629384 − 2.75751i 1.40169 + 0.675020i −7.60727 + 1.73631i
27.6 1.27416 − 0.613604i −1.26988 − 0.289841i 1.24698 − 1.56366i 4.19788 + 0.958139i −1.79588 + 0.409898i 6.75731 − 1.82724i 0.629384 − 2.75751i −6.58014 − 3.16883i 5.93670 − 1.35501i
27.7 1.27416 − 0.613604i 2.89960 + 0.661815i 1.24698 − 1.56366i 3.15515 + 0.720143i 4.10065 − 0.935948i −6.18915 + 3.27023i 0.629384 − 2.75751i −0.139036 − 0.0669563i 4.46206 − 1.01844i
27.8 1.27416 − 0.613604i 4.22616 + 0.964593i 1.24698 − 1.56366i −5.89122 − 1.34463i 5.97669 − 1.36414i 6.37392 − 2.89365i 0.629384 − 2.75751i 8.82126 + 4.24809i −8.33145 + 1.90160i
41.1 −0.881748 − 1.10568i −1.63522 − 3.39556i −0.445042 + 1.94986i −0.900324 − 1.86954i −2.31255 + 4.80205i −6.99908 + 0.113345i 2.54832 − 1.22721i −3.24450 + 4.06848i −1.27325 + 2.64393i
41.2 −0.881748 − 1.10568i −0.450780 − 0.936055i −0.445042 + 1.94986i 2.86498 + 5.94920i −0.637499 + 1.32378i 2.19881 + 6.64569i 2.54832 − 1.22721i 4.93841 − 6.19257i 4.05170 − 8.41343i
41.3 −0.881748 − 1.10568i 0.340617 + 0.707297i −0.445042 + 1.94986i −1.35475 − 2.81318i 0.481704 − 1.00027i 3.17901 − 6.23650i 2.54832 − 1.22721i 5.22716 − 6.55465i −1.91591 + 3.97843i
41.4 −0.881748 − 1.10568i 2.27785 + 4.73000i −0.445042 + 1.94986i −1.68283 − 3.49444i 3.22136 − 6.68923i 5.06466 + 4.83211i 2.54832 − 1.22721i −11.5729 + 14.5119i −2.37989 + 4.94188i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.3.f.a ✓ 48
49.f odd 14 1 inner 98.3.f.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.3.f.a ✓ 48 1.a even 1 1 trivial
98.3.f.a ✓ 48 49.f odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(98, [\chi])\).