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Properties

Label 356.2.m.a
Level $356$
Weight $2$
Character orbit 356.m
Analytic conductor $2.843$
Analytic rank $0$
Dimension $140$
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] = [356,2,Mod(5,356)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(356, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([0, 35]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("356.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 356 = 2^{2} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 356.m (of order \(44\), degree \(20\), 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: \(2.84267431196\)
Analytic rank: \(0\)
Dimension: \(140\)
Relative dimension: \(7\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 140 q - 4 q^{3} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 140 q - 4 q^{3} + 22 q^{9} + 4 q^{11} - 4 q^{13} + 2 q^{15} - 12 q^{19} + 4 q^{23} + 6 q^{25} - 46 q^{27} + 4 q^{29} + 2 q^{31} - 10 q^{33} + 42 q^{35} - 14 q^{37} - 20 q^{39} - 8 q^{41} - 2 q^{43} + 44 q^{45} + 74 q^{51} - 88 q^{53} - 48 q^{57} - 40 q^{59} + 84 q^{61} - 6 q^{63} + 10 q^{65} + 16 q^{67} - 88 q^{69} + 30 q^{73} - 70 q^{75} + 112 q^{77} - 44 q^{79} - 88 q^{81} - 54 q^{83} - 190 q^{85} + 8 q^{87} - 116 q^{89} - 64 q^{91} - 42 q^{93} - 64 q^{95} - 86 q^{97} - 110 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0 −2.27872 + 1.70583i 0 1.21571 − 1.89169i 0 −0.112894 − 0.518965i 0 1.43752 − 4.89576i 0
5.2 0 −1.16980 + 0.875701i 0 −2.03855 + 3.17205i 0 −1.06086 − 4.87668i 0 −0.243619 + 0.829691i 0
5.3 0 −1.14335 + 0.855897i 0 −0.129369 + 0.201301i 0 −0.00796087 − 0.0365955i 0 −0.270520 + 0.921307i 0
5.4 0 0.274104 − 0.205192i 0 −0.567105 + 0.882433i 0 0.538235 + 2.47423i 0 −0.812168 + 2.76599i 0
5.5 0 0.662806 − 0.496171i 0 2.37564 − 3.69657i 0 0.816692 + 3.75427i 0 −0.652071 + 2.22075i 0
5.6 0 1.48088 − 1.10858i 0 0.506457 − 0.788063i 0 −0.709860 − 3.26317i 0 0.118878 − 0.404862i 0
5.7 0 1.77410 − 1.32808i 0 −2.29018 + 3.56359i 0 0.536643 + 2.46691i 0 0.538455 − 1.83381i 0
9.1 0 −3.06848 − 0.219462i 0 −0.730840 + 2.48901i 0 0.516735 − 0.282159i 0 6.39794 + 0.919885i 0
9.2 0 −2.48173 − 0.177497i 0 1.09454 − 3.72766i 0 −1.30449 + 0.712305i 0 3.15801 + 0.454053i 0
9.3 0 −1.74659 − 0.124919i 0 −0.161368 + 0.549567i 0 0.949060 − 0.518226i 0 0.0655101 + 0.00941893i 0
9.4 0 −0.160360 − 0.0114692i 0 −0.307114 + 1.04593i 0 −3.54732 + 1.93698i 0 −2.94388 − 0.423266i 0
9.5 0 0.255156 + 0.0182491i 0 0.450053 − 1.53274i 0 2.24280 − 1.22466i 0 −2.90469 − 0.417632i 0
9.6 0 2.44360 + 0.174770i 0 −0.761558 + 2.59363i 0 −1.52657 + 0.833568i 0 2.97117 + 0.427189i 0
9.7 0 2.66677 + 0.190731i 0 0.221813 − 0.755427i 0 2.66978 − 1.45781i 0 4.10584 + 0.590331i 0
17.1 0 −2.55709 − 0.556261i 0 2.10940 − 1.82780i 0 −0.236752 + 3.31023i 0 3.50040 + 1.59858i 0
17.2 0 −2.41124 − 0.524533i 0 −0.759026 + 0.657699i 0 0.110201 − 1.54082i 0 2.81004 + 1.28330i 0
17.3 0 −0.662423 − 0.144101i 0 −2.09565 + 1.81589i 0 0.101655 − 1.42133i 0 −2.31086 − 1.05533i 0
17.4 0 −0.509318 − 0.110795i 0 2.85245 − 2.47166i 0 0.275564 − 3.85289i 0 −2.48177 − 1.13338i 0
17.5 0 0.916441 + 0.199359i 0 −1.30965 + 1.13482i 0 −0.329427 + 4.60599i 0 −1.92878 − 0.880843i 0
17.6 0 1.28626 + 0.279809i 0 1.85372 − 1.60626i 0 −0.00854372 + 0.119457i 0 −1.15272 − 0.526431i 0
See next 80 embeddings (of 140 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.g even 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 356.2.m.a ✓ 140
89.g even 44 1 inner 356.2.m.a ✓ 140
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
356.2.m.a ✓ 140 1.a even 1 1 trivial
356.2.m.a ✓ 140 89.g even 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(356, [\chi])\).