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Properties

Label 833.2.z.a
Level $833$
Weight $2$
Character orbit 833.z
Analytic conductor $6.652$
Analytic rank $0$
Dimension $984$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [833,2,Mod(16,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([20, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("833.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.z (of order \(42\), 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: \(6.65153848837\)
Analytic rank: \(0\)
Dimension: \(984\)
Relative dimension: \(82\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 984 q - 26 q^{2} + 54 q^{4} - 28 q^{8} - 104 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 984 q - 26 q^{2} + 54 q^{4} - 28 q^{8} - 104 q^{9} - 12 q^{13} - 20 q^{15} + 46 q^{16} - 19 q^{17} - 32 q^{18} - 60 q^{19} - 40 q^{21} - 98 q^{25} - 40 q^{26} - 2 q^{30} - 26 q^{32} - 30 q^{33} - 66 q^{34} - 2 q^{35} + 140 q^{36} + 58 q^{38} - 208 q^{42} + 8 q^{43} - 170 q^{47} - 144 q^{49} - 44 q^{50} + 46 q^{51} - 26 q^{52} - 16 q^{53} + 100 q^{55} - 70 q^{59} + 28 q^{60} - 260 q^{64} - 40 q^{66} - 26 q^{67} + 25 q^{68} + 144 q^{69} + 6 q^{70} + 26 q^{72} - 104 q^{76} - 58 q^{77} - 18 q^{81} - 124 q^{83} + 122 q^{84} + 34 q^{85} - 196 q^{86} - 2 q^{87} - 86 q^{89} + 350 q^{93} + 90 q^{94} - 258 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −0.984633 − 2.50880i −1.97895 + 0.148302i −3.85849 + 3.58015i 2.07477 + 3.04313i 2.32060 + 4.81877i −0.975942 − 2.45917i 7.92469 + 3.81633i 0.927749 − 0.139836i 5.59172 − 8.20155i
16.2 −0.984633 − 2.50880i 1.97895 − 0.148302i −3.85849 + 3.58015i −2.07477 − 3.04313i −2.32060 − 4.81877i 0.975942 + 2.45917i 7.92469 + 3.81633i 0.927749 − 0.139836i −5.59172 + 8.20155i
16.3 −0.975811 − 2.48633i −2.87120 + 0.215167i −3.76350 + 3.49202i 0.000134072 0 0.000196647i 3.33673 + 6.92879i 0.0484030 + 2.64531i 7.54187 + 3.63197i 5.23103 − 0.788451i 0.000358101 0 0.000525237i
16.4 −0.975811 − 2.48633i 2.87120 − 0.215167i −3.76350 + 3.49202i −0.000134072 0 0.000196647i −3.33673 − 6.92879i −0.0484030 − 2.64531i 7.54187 + 3.63197i 5.23103 − 0.788451i −0.000358101 0 0.000525237i
16.5 −0.951570 − 2.42456i −1.19808 + 0.0897840i −3.50691 + 3.25393i −0.822697 − 1.20667i 1.35775 + 2.81939i 2.59020 − 0.539323i 6.53309 + 3.14617i −1.53915 + 0.231989i −2.14280 + 3.14291i
16.6 −0.951570 − 2.42456i 1.19808 − 0.0897840i −3.50691 + 3.25393i 0.822697 + 1.20667i −1.35775 − 2.81939i −2.59020 + 0.539323i 6.53309 + 3.14617i −1.53915 + 0.231989i 2.14280 − 3.14291i
16.7 −0.913949 − 2.32871i −1.01028 + 0.0757103i −3.12146 + 2.89629i −1.79088 − 2.62673i 1.09966 + 2.28346i −2.64567 + 0.0205131i 5.08968 + 2.45106i −1.95155 + 0.294149i −4.48011 + 6.57112i
16.8 −0.913949 − 2.32871i 1.01028 − 0.0757103i −3.12146 + 2.89629i 1.79088 + 2.62673i −1.09966 − 2.28346i 2.64567 − 0.0205131i 5.08968 + 2.45106i −1.95155 + 0.294149i 4.48011 − 6.57112i
16.9 −0.814125 − 2.07436i −3.17637 + 0.238036i −2.17405 + 2.01722i −2.10681 − 3.09012i 3.07974 + 6.39514i 0.895786 − 2.48949i 1.93896 + 0.933753i 7.06620 − 1.06506i −4.69480 + 6.88601i
16.10 −0.814125 − 2.07436i 3.17637 − 0.238036i −2.17405 + 2.01722i 2.10681 + 3.09012i −3.07974 − 6.39514i −0.895786 + 2.48949i 1.93896 + 0.933753i 7.06620 − 1.06506i 4.69480 − 6.88601i
16.11 −0.787522 − 2.00657i −1.27991 + 0.0959163i −1.94004 + 1.80010i 0.370339 + 0.543187i 1.20042 + 2.49271i 2.59992 + 0.490345i 1.25563 + 0.604681i −1.33751 + 0.201598i 0.798295 − 1.17088i
16.12 −0.787522 − 2.00657i 1.27991 − 0.0959163i −1.94004 + 1.80010i −0.370339 − 0.543187i −1.20042 − 2.49271i −2.59992 − 0.490345i 1.25563 + 0.604681i −1.33751 + 0.201598i −0.798295 + 1.17088i
16.13 −0.787363 − 2.00617i −0.748728 + 0.0561094i −1.93867 + 1.79882i −0.115073 − 0.168781i 0.702085 + 1.45790i 0.563729 + 2.58500i 1.25173 + 0.602802i −2.40905 + 0.363106i −0.247999 + 0.363748i
16.14 −0.787363 − 2.00617i 0.748728 − 0.0561094i −1.93867 + 1.79882i 0.115073 + 0.168781i −0.702085 − 1.45790i −0.563729 − 2.58500i 1.25173 + 0.602802i −2.40905 + 0.363106i 0.247999 − 0.363748i
16.15 −0.738430 − 1.88149i −2.04404 + 0.153179i −1.52862 + 1.41835i 0.941996 + 1.38165i 1.79758 + 3.73272i −1.66954 − 2.05247i 0.155303 + 0.0747898i 1.18813 − 0.179082i 1.90397 − 2.79261i
16.16 −0.738430 − 1.88149i 2.04404 − 0.153179i −1.52862 + 1.41835i −0.941996 − 1.38165i −1.79758 − 3.73272i 1.66954 + 2.05247i 0.155303 + 0.0747898i 1.18813 − 0.179082i −1.90397 + 2.79261i
16.17 −0.665446 − 1.69553i −0.686439 + 0.0514415i −0.965898 + 0.896222i 2.13309 + 3.12867i 0.544009 + 1.12965i −1.31506 + 2.29578i −1.11980 − 0.539265i −2.49794 + 0.376504i 3.88530 − 5.69868i
16.18 −0.665446 − 1.69553i 0.686439 − 0.0514415i −0.965898 + 0.896222i −2.13309 − 3.12867i −0.544009 − 1.12965i 1.31506 − 2.29578i −1.11980 − 0.539265i −2.49794 + 0.376504i −3.88530 + 5.69868i
16.19 −0.643253 − 1.63898i −2.71438 + 0.203415i −0.806383 + 0.748214i 0.718869 + 1.05439i 2.07942 + 4.31797i −2.27471 + 1.35119i −1.42764 − 0.687515i 4.35998 − 0.657161i 1.26571 − 1.85645i
16.20 −0.643253 − 1.63898i 2.71438 − 0.203415i −0.806383 + 0.748214i −0.718869 − 1.05439i −2.07942 − 4.31797i 2.27471 − 1.35119i −1.42764 − 0.687515i 4.35998 − 0.657161i −1.26571 + 1.85645i
See next 80 embeddings (of 984 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
49.g even 21 1 inner
833.z even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.z.a ✓ 984
17.b even 2 1 inner 833.2.z.a ✓ 984
49.g even 21 1 inner 833.2.z.a ✓ 984
833.z even 42 1 inner 833.2.z.a ✓ 984
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
833.2.z.a ✓ 984 1.a even 1 1 trivial
833.2.z.a ✓ 984 17.b even 2 1 inner
833.2.z.a ✓ 984 49.g even 21 1 inner
833.2.z.a ✓ 984 833.z even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(833, [\chi])\).