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Properties

Label 471.2.p.b
Level $471$
Weight $2$
Character orbit 471.p
Analytic conductor $3.761$
Analytic rank $0$
Dimension $168$
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] = [471,2,Mod(4,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([0, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.p (of order \(26\), degree \(12\), 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: \(3.76095393520\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(14\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 14 q^{3} + 20 q^{4} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q - 14 q^{3} + 20 q^{4} - 14 q^{9} - 6 q^{10} + 2 q^{11} - 188 q^{12} - 26 q^{13} + 13 q^{15} - 24 q^{16} - 18 q^{17} - q^{19} + 13 q^{23} - 34 q^{25} + 65 q^{26} - 14 q^{27} - 13 q^{29} - 6 q^{30} - 12 q^{31} + 2 q^{33} - 26 q^{34} - 42 q^{35} + 20 q^{36} + 27 q^{37} + 13 q^{38} - 16 q^{40} - 13 q^{41} + 26 q^{43} - 24 q^{44} + 13 q^{45} + 34 q^{46} - 22 q^{47} - 24 q^{48} + 74 q^{49} + 21 q^{51} + 50 q^{52} + 26 q^{53} - 52 q^{55} + 62 q^{56} + 25 q^{57} - 98 q^{58} - 26 q^{59} + 26 q^{61} - 78 q^{62} - 13 q^{63} + 34 q^{64} + 26 q^{66} + 36 q^{67} - 22 q^{68} + 52 q^{70} + q^{71} + 13 q^{73} + 130 q^{74} + 18 q^{75} + 91 q^{76} - 13 q^{77} - 65 q^{78} - 13 q^{79} + 26 q^{80} - 14 q^{81} - 10 q^{82} - 52 q^{84} + 117 q^{85} + 70 q^{86} - 52 q^{88} + 35 q^{89} - 32 q^{90} - 182 q^{91} + 52 q^{92} - 12 q^{93} - 78 q^{94} - 65 q^{95} - 65 q^{96} - 156 q^{97} - 52 q^{98} + 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.52118 + 0.956157i −0.748511 − 0.663123i 3.94509 − 3.49504i 2.71463 − 1.87377i 2.52118 + 0.956157i −0.0849820 + 0.0959248i −4.09831 + 7.80867i 0.120537 + 0.992709i −5.05244 + 7.31972i
4.2 −2.34359 + 0.888808i −0.748511 − 0.663123i 3.20543 − 2.83976i −2.35981 + 1.62886i 2.34359 + 0.888808i 0.846681 − 0.955705i −2.65859 + 5.06552i 0.120537 + 0.992709i 4.08269 − 5.91480i
4.3 −1.78773 + 0.677997i −0.748511 − 0.663123i 1.23928 − 1.09790i 2.32761 − 1.60664i 1.78773 + 0.677997i 2.11335 − 2.38548i 0.305962 − 0.582961i 0.120537 + 0.992709i −3.07185 + 4.45034i
4.4 −1.43942 + 0.545900i −0.748511 − 0.663123i 0.276898 − 0.245311i −0.0269007 + 0.0185682i 1.43942 + 0.545900i −0.305765 + 0.345137i 1.16619 − 2.22198i 0.120537 + 0.992709i 0.0285850 − 0.0414126i
4.5 −0.793063 + 0.300769i −0.748511 − 0.663123i −0.958535 + 0.849188i 2.70194 − 1.86502i 0.793063 + 0.300769i −2.18910 + 2.47098i 1.29311 − 2.46381i 0.120537 + 0.992709i −1.58187 + 2.29174i
4.6 −0.486686 + 0.184576i −0.748511 − 0.663123i −1.29423 + 1.14658i −3.20694 + 2.21359i 0.486686 + 0.184576i −2.63118 + 2.96999i 0.902037 − 1.71869i 0.120537 + 0.992709i 1.15220 − 1.66925i
4.7 −0.228995 + 0.0868465i −0.748511 − 0.663123i −1.45212 + 1.28647i −1.22952 + 0.848677i 0.228995 + 0.0868465i 3.30064 − 3.72565i 0.448436 − 0.854423i 0.120537 + 0.992709i 0.207850 − 0.301123i
4.8 0.0562090 − 0.0213172i −0.748511 − 0.663123i −1.49432 + 1.32385i −0.487215 + 0.336300i −0.0562090 − 0.0213172i 0.643615 − 0.726492i −0.111647 + 0.212726i 0.120537 + 0.992709i −0.0202169 + 0.0292892i
4.9 0.888525 − 0.336973i −0.748511 − 0.663123i −0.821095 + 0.727427i 1.01411 − 0.699989i −0.888525 − 0.336973i −2.59236 + 2.92617i −1.36767 + 2.60588i 0.120537 + 0.992709i 0.665184 − 0.963686i
4.10 1.10958 − 0.420806i −0.748511 − 0.663123i −0.442943 + 0.392413i 3.33116 − 2.29933i −1.10958 − 0.420806i 1.72376 − 1.94572i −1.42931 + 2.72333i 0.120537 + 0.992709i 2.72860 − 3.95306i
4.11 1.31555 − 0.498923i −0.748511 − 0.663123i −0.0152704 + 0.0135284i −0.678869 + 0.468589i −1.31555 − 0.498923i −1.42848 + 1.61242i −1.32105 + 2.51706i 0.120537 + 0.992709i −0.659297 + 0.955156i
4.12 1.66479 − 0.631373i −0.748511 − 0.663123i 0.875888 − 0.775969i −3.52651 + 2.43417i −1.66479 − 0.631373i 0.357392 − 0.403412i −0.686630 + 1.30826i 0.120537 + 0.992709i −4.33404 + 6.27894i
4.13 2.07485 − 0.786888i −0.748511 − 0.663123i 2.18880 − 1.93911i −0.506296 + 0.349471i −2.07485 − 0.786888i 1.53981 − 1.73809i 0.953082 − 1.81595i 0.120537 + 0.992709i −0.775495 + 1.12350i
4.14 2.49116 − 0.944772i −0.748511 − 0.663123i 3.81626 − 3.38091i 0.929121 − 0.641326i −2.49116 − 0.944772i 0.0804896 − 0.0908540i 3.83640 − 7.30964i 0.120537 + 0.992709i 1.70868 − 2.47545i
49.1 −1.79122 + 2.02187i 0.120537 − 0.992709i −0.638413 − 5.25780i −3.15747 + 1.19747i 1.79122 + 2.02187i 4.31984 + 0.524524i 7.32805 + 5.05819i −0.970942 − 0.239316i 3.23459 − 8.52892i
49.2 −1.74450 + 1.96913i 0.120537 − 0.992709i −0.593130 − 4.88486i 0.729868 − 0.276802i 1.74450 + 1.96913i −4.57752 − 0.555811i 6.32355 + 4.36483i −0.970942 − 0.239316i −0.728192 + 1.92009i
49.3 −1.39516 + 1.57481i 0.120537 − 0.992709i −0.292485 − 2.40883i 2.72029 − 1.03167i 1.39516 + 1.57481i 2.05397 + 0.249397i 0.738522 + 0.509765i −0.970942 − 0.239316i −2.17056 + 5.72330i
49.4 −1.18778 + 1.34073i 0.120537 − 0.992709i −0.145654 − 1.19957i −2.95965 + 1.12245i 1.18778 + 1.34073i −1.90901 − 0.231796i −1.16694 − 0.805483i −0.970942 − 0.239316i 2.01052 − 5.30131i
49.5 −0.579212 + 0.653795i 0.120537 − 0.992709i 0.149112 + 1.22805i 1.11248 − 0.421910i 0.579212 + 0.653795i −3.40187 − 0.413061i −2.32695 − 1.60618i −0.970942 − 0.239316i −0.368522 + 0.971712i
49.6 −0.485963 + 0.548539i 0.120537 − 0.992709i 0.176338 + 1.45228i −1.26225 + 0.478708i 0.485963 + 0.548539i 4.90995 + 0.596176i −2.08856 − 1.44163i −0.970942 − 0.239316i 0.350817 − 0.925028i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
157.h even 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.p.b ✓ 168
157.h even 26 1 inner 471.2.p.b ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.p.b ✓ 168 1.a even 1 1 trivial
471.2.p.b ✓ 168 157.h even 26 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{168} - 24 T_{2}^{166} + 352 T_{2}^{164} - 4099 T_{2}^{162} + 40399 T_{2}^{160} + \cdots + 9120059001 \) acting on \(S_{2}^{\mathrm{new}}(471, [\chi])\). Copy content Toggle raw display