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Properties

Label 294.3.k.a
Level $294$
Weight $3$
Character orbit 294.k
Analytic conductor $8.011$
Analytic rank $0$
Dimension $120$
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] = [294,3,Mod(13,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 294.k (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: \(8.01091977219\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 40 q^{4} - 8 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 40 q^{4} - 8 q^{7} + 60 q^{9} - 116 q^{11} + 24 q^{14} - 60 q^{15} - 80 q^{16} + 196 q^{17} + 56 q^{20} - 12 q^{21} + 128 q^{22} - 32 q^{23} + 60 q^{25} - 224 q^{26} - 16 q^{28} - 80 q^{29} + 304 q^{35} + 120 q^{36} + 20 q^{37} + 280 q^{38} + 240 q^{39} + 112 q^{40} + 392 q^{41} + 120 q^{42} + 48 q^{43} + 160 q^{44} + 84 q^{45} + 96 q^{46} - 532 q^{47} - 36 q^{49} - 128 q^{50} - 120 q^{51} - 216 q^{53} + 532 q^{55} - 176 q^{56} + 72 q^{57} - 160 q^{58} - 1008 q^{59} - 120 q^{60} - 364 q^{61} - 392 q^{62} + 24 q^{63} - 160 q^{64} - 280 q^{65} - 88 q^{67} - 296 q^{70} + 184 q^{71} - 252 q^{73} + 288 q^{74} + 396 q^{77} - 240 q^{78} - 200 q^{79} - 180 q^{81} + 364 q^{83} - 24 q^{84} + 208 q^{85} - 664 q^{86} + 252 q^{87} - 304 q^{88} + 1036 q^{89} + 280 q^{91} - 64 q^{92} + 168 q^{93} + 112 q^{94} + 472 q^{95} + 192 q^{98} - 72 q^{99}+O(q^{100}) \) 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 −1.35417 + 1.07992i −1.80194 + 0.867767i −4.74601 + 3.78481i 1.91509 + 1.52723i −2.00414 − 6.70697i 1.76350 + 2.21135i 0.667563 − 2.92478i 6.71187 + 5.35254i
13.2 −0.314692 − 1.37876i −1.35417 + 1.07992i −1.80194 + 0.867767i −2.01788 + 1.60921i 1.91509 + 1.52723i 6.60251 − 2.32526i 1.76350 + 2.21135i 0.667563 − 2.92478i 2.85371 + 2.27576i
13.3 −0.314692 − 1.37876i −1.35417 + 1.07992i −1.80194 + 0.867767i −0.446656 + 0.356196i 1.91509 + 1.52723i −1.92431 + 6.73031i 1.76350 + 2.21135i 0.667563 − 2.92478i 0.631666 + 0.503737i
13.4 −0.314692 − 1.37876i −1.35417 + 1.07992i −1.80194 + 0.867767i 2.26537 − 1.80657i 1.91509 + 1.52723i −6.97849 − 0.548346i 1.76350 + 2.21135i 0.667563 − 2.92478i −3.20372 − 2.55488i
13.5 −0.314692 − 1.37876i −1.35417 + 1.07992i −1.80194 + 0.867767i 7.02530 − 5.60249i 1.91509 + 1.52723i 3.37345 + 6.13350i 1.76350 + 2.21135i 0.667563 − 2.92478i −9.93527 − 7.92312i
13.6 −0.314692 − 1.37876i 1.35417 − 1.07992i −1.80194 + 0.867767i −5.30458 + 4.23026i −1.91509 − 1.52723i 6.99053 + 0.364055i 1.76350 + 2.21135i 0.667563 − 2.92478i 7.50181 + 5.98249i
13.7 −0.314692 − 1.37876i 1.35417 − 1.07992i −1.80194 + 0.867767i −2.16106 + 1.72339i −1.91509 − 1.52723i −0.743019 + 6.96045i 1.76350 + 2.21135i 0.667563 − 2.92478i 3.05620 + 2.43724i
13.8 −0.314692 − 1.37876i 1.35417 − 1.07992i −1.80194 + 0.867767i −1.58454 + 1.26363i −1.91509 − 1.52723i −4.82444 + 5.07196i 1.76350 + 2.21135i 0.667563 − 2.92478i 2.24088 + 1.78704i
13.9 −0.314692 − 1.37876i 1.35417 − 1.07992i −1.80194 + 0.867767i 4.26103 − 3.39806i −1.91509 − 1.52723i −5.97560 − 3.64584i 1.76350 + 2.21135i 0.667563 − 2.92478i −6.02601 − 4.80559i
13.10 −0.314692 − 1.37876i 1.35417 − 1.07992i −1.80194 + 0.867767i 5.66395 − 4.51685i −1.91509 − 1.52723i 6.99881 − 0.129299i 1.76350 + 2.21135i 0.667563 − 2.92478i −8.01003 − 6.38779i
13.11 0.314692 + 1.37876i −1.35417 + 1.07992i −1.80194 + 0.867767i −6.94562 + 5.53895i −1.91509 − 1.52723i 6.95603 + 0.783380i −1.76350 − 2.21135i 0.667563 − 2.92478i −9.82259 − 7.83325i
13.12 0.314692 + 1.37876i −1.35417 + 1.07992i −1.80194 + 0.867767i −5.37229 + 4.28426i −1.91509 − 1.52723i −6.88185 + 1.28070i −1.76350 − 2.21135i 0.667563 − 2.92478i −7.59756 − 6.05886i
13.13 0.314692 + 1.37876i −1.35417 + 1.07992i −1.80194 + 0.867767i 1.53711 − 1.22580i −1.91509 − 1.52723i 4.13123 − 5.65092i −1.76350 − 2.21135i 0.667563 − 2.92478i 2.17380 + 1.73355i
13.14 0.314692 + 1.37876i −1.35417 + 1.07992i −1.80194 + 0.867767i 2.98825 − 2.38305i −1.91509 − 1.52723i 2.78336 + 6.42284i −1.76350 − 2.21135i 0.667563 − 2.92478i 4.22602 + 3.37014i
13.15 0.314692 + 1.37876i −1.35417 + 1.07992i −1.80194 + 0.867767i 5.56085 − 4.43463i −1.91509 − 1.52723i −6.94109 + 0.906248i −1.76350 − 2.21135i 0.667563 − 2.92478i 7.86423 + 6.27152i
13.16 0.314692 + 1.37876i 1.35417 − 1.07992i −1.80194 + 0.867767i −4.84203 + 3.86139i 1.91509 + 1.52723i −0.315497 − 6.99289i −1.76350 − 2.21135i 0.667563 − 2.92478i −6.84766 − 5.46083i
13.17 0.314692 + 1.37876i 1.35417 − 1.07992i −1.80194 + 0.867767i −3.54264 + 2.82516i 1.91509 + 1.52723i 6.96258 − 0.722841i −1.76350 − 2.21135i 0.667563 − 2.92478i −5.01005 − 3.99538i
13.18 0.314692 + 1.37876i 1.35417 − 1.07992i −1.80194 + 0.867767i −3.06560 + 2.44473i 1.91509 + 1.52723i −4.30459 + 5.52001i −1.76350 − 2.21135i 0.667563 − 2.92478i −4.33541 − 3.45737i
13.19 0.314692 + 1.37876i 1.35417 − 1.07992i −1.80194 + 0.867767i 2.50405 − 1.99691i 1.91509 + 1.52723i −4.02933 − 5.72403i −1.76350 − 2.21135i 0.667563 − 2.92478i 3.54126 + 2.82406i
13.20 0.314692 + 1.37876i 1.35417 − 1.07992i −1.80194 + 0.867767i 5.50919 − 4.39343i 1.91509 + 1.52723i 5.11177 + 4.78224i −1.76350 − 2.21135i 0.667563 − 2.92478i 7.79117 + 6.21325i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.20
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 294.3.k.a ✓ 120
49.f odd 14 1 inner 294.3.k.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.3.k.a ✓ 120 1.a even 1 1 trivial
294.3.k.a ✓ 120 49.f odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(294, [\chi])\).