Properties

Label 414.3.l.b
Level $414$
Weight $3$
Character orbit 414.l
Analytic conductor $11.281$
Analytic rank $0$
Dimension $80$
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] = [414,3,Mod(19,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 15]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("414.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 414.l (of order \(22\), degree \(10\), 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: \(11.2806829445\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: no (minimal twist has level 138)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 16 q^{4} - 16 q^{13} - 32 q^{16} - 220 q^{17} + 132 q^{19} - 88 q^{20} + 104 q^{23} - 336 q^{25} + 208 q^{26} - 264 q^{28} + 164 q^{29} - 268 q^{31} - 552 q^{35} + 352 q^{37} - 192 q^{41} + 88 q^{43} + 80 q^{46} + 64 q^{47} - 40 q^{49} - 160 q^{50} - 32 q^{52} + 352 q^{53} + 196 q^{55} + 312 q^{58} + 696 q^{59} + 616 q^{61} - 96 q^{62} - 64 q^{64} + 44 q^{67} - 32 q^{70} + 32 q^{71} - 284 q^{73} + 224 q^{77} - 440 q^{79} - 616 q^{82} - 352 q^{83} - 532 q^{85} - 88 q^{89} + 32 q^{92} + 16 q^{94} - 372 q^{95} - 264 q^{97} - 1184 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.587486 1.28641i 0 −1.30972 + 1.51150i −5.01874 + 7.80931i 0 5.11630 0.735612i 2.71386 + 0.796860i 0 12.9944 + 1.86832i
19.2 −0.587486 1.28641i 0 −1.30972 + 1.51150i −0.891807 + 1.38768i 0 9.81972 1.41186i 2.71386 + 0.796860i 0 2.30905 + 0.331992i
19.3 −0.587486 1.28641i 0 −1.30972 + 1.51150i 0.858668 1.33611i 0 −8.36490 + 1.20269i 2.71386 + 0.796860i 0 −2.22325 0.319655i
19.4 −0.587486 1.28641i 0 −1.30972 + 1.51150i 4.44262 6.91285i 0 −10.3360 + 1.48610i 2.71386 + 0.796860i 0 −11.5028 1.65385i
19.5 0.587486 + 1.28641i 0 −1.30972 + 1.51150i −3.94925 + 6.14515i 0 −2.61314 + 0.375713i −2.71386 0.796860i 0 −10.2253 1.47018i
19.6 0.587486 + 1.28641i 0 −1.30972 + 1.51150i −0.791659 + 1.23185i 0 8.65930 1.24502i −2.71386 0.796860i 0 −2.04975 0.294710i
19.7 0.587486 + 1.28641i 0 −1.30972 + 1.51150i 0.634924 0.987962i 0 1.01096 0.145354i −2.71386 0.796860i 0 1.64394 + 0.236362i
19.8 0.587486 + 1.28641i 0 −1.30972 + 1.51150i 3.49672 5.44100i 0 9.55110 1.37324i −2.71386 0.796860i 0 9.05366 + 1.30172i
37.1 −1.35693 + 0.398430i 0 1.68251 1.08128i −8.41654 + 1.21012i 0 −8.45540 3.86145i −1.85223 + 2.13758i 0 10.9385 4.99544i
37.2 −1.35693 + 0.398430i 0 1.68251 1.08128i −3.43203 + 0.493451i 0 4.31074 + 1.96865i −1.85223 + 2.13758i 0 4.46041 2.03700i
37.3 −1.35693 + 0.398430i 0 1.68251 1.08128i 1.17981 0.169632i 0 7.87069 + 3.59442i −1.85223 + 2.13758i 0 −1.53334 + 0.700252i
37.4 −1.35693 + 0.398430i 0 1.68251 1.08128i 7.67653 1.10372i 0 −11.6311 5.31175i −1.85223 + 2.13758i 0 −9.97674 + 4.55623i
37.5 1.35693 0.398430i 0 1.68251 1.08128i −5.59508 + 0.804450i 0 0.513410 + 0.234466i 1.85223 2.13758i 0 −7.27160 + 3.32083i
37.6 1.35693 0.398430i 0 1.68251 1.08128i −4.69745 + 0.675391i 0 −5.04392 2.30348i 1.85223 2.13758i 0 −6.10500 + 2.78806i
37.7 1.35693 0.398430i 0 1.68251 1.08128i 3.60161 0.517833i 0 −10.7332 4.90168i 1.85223 2.13758i 0 4.68080 2.13765i
37.8 1.35693 0.398430i 0 1.68251 1.08128i 3.69869 0.531791i 0 1.55981 + 0.712339i 1.85223 2.13758i 0 4.80697 2.19527i
109.1 −0.587486 + 1.28641i 0 −1.30972 1.51150i −5.01874 7.80931i 0 5.11630 + 0.735612i 2.71386 0.796860i 0 12.9944 1.86832i
109.2 −0.587486 + 1.28641i 0 −1.30972 1.51150i −0.891807 1.38768i 0 9.81972 + 1.41186i 2.71386 0.796860i 0 2.30905 0.331992i
109.3 −0.587486 + 1.28641i 0 −1.30972 1.51150i 0.858668 + 1.33611i 0 −8.36490 1.20269i 2.71386 0.796860i 0 −2.22325 + 0.319655i
109.4 −0.587486 + 1.28641i 0 −1.30972 1.51150i 4.44262 + 6.91285i 0 −10.3360 1.48610i 2.71386 0.796860i 0 −11.5028 + 1.65385i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.3.l.b 80
3.b odd 2 1 138.3.h.a 80
23.d odd 22 1 inner 414.3.l.b 80
69.g even 22 1 138.3.h.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.3.h.a 80 3.b odd 2 1
138.3.h.a 80 69.g even 22 1
414.3.l.b 80 1.a even 1 1 trivial
414.3.l.b 80 23.d odd 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{80} + 68 T_{5}^{78} - 484 T_{5}^{77} + 3194 T_{5}^{76} + 17160 T_{5}^{75} - 202264 T_{5}^{74} + 2039928 T_{5}^{73} - 24535009 T_{5}^{72} + 335882228 T_{5}^{71} - 956558316 T_{5}^{70} + \cdots + 11\!\cdots\!09 \) acting on \(S_{3}^{\mathrm{new}}(414, [\chi])\). Copy content Toggle raw display