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Properties

Label 353.2.d.a
Level $353$
Weight $2$
Character orbit 353.d
Analytic conductor $2.819$
Analytic rank $0$
Dimension $112$
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] = [353,2,Mod(70,353)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(353, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("353.70");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 353.d (of order \(8\), degree \(4\), 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: \(2.81871919135\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 112 q^{4} - 4 q^{5} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 112 q^{4} - 4 q^{5} + 8 q^{6} - 4 q^{7} + 4 q^{9} - 4 q^{10} - 16 q^{12} + 16 q^{13} + 24 q^{14} + 96 q^{16} - 12 q^{18} - 4 q^{19} - 24 q^{22} - 24 q^{23} - 20 q^{24} - 44 q^{25} + 8 q^{26} + 24 q^{27} - 44 q^{28} + 44 q^{30} + 4 q^{31} - 36 q^{33} + 24 q^{35} + 20 q^{36} - 4 q^{37} + 32 q^{38} + 8 q^{39} - 32 q^{40} - 32 q^{41} + 56 q^{42} - 48 q^{43} - 104 q^{45} + 76 q^{46} - 44 q^{47} + 4 q^{48} + 8 q^{49} + 20 q^{50} - 32 q^{51} - 72 q^{52} + 20 q^{53} + 64 q^{54} + 16 q^{55} - 24 q^{56} - 20 q^{57} - 64 q^{58} + 60 q^{59} - 176 q^{60} - 12 q^{62} + 28 q^{63} - 104 q^{64} + 4 q^{65} + 64 q^{66} + 8 q^{67} - 48 q^{69} + 8 q^{71} + 56 q^{72} + 4 q^{74} + 92 q^{75} - 20 q^{77} + 72 q^{78} + 24 q^{79} - 36 q^{80} - 72 q^{82} - 4 q^{85} - 44 q^{86} + 24 q^{87} + 344 q^{88} + 16 q^{89} - 44 q^{90} + 44 q^{92} + 76 q^{93} + 148 q^{94} - 20 q^{95} + 4 q^{96} + 72 q^{97} - 64 q^{98} + 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1 − 2.66968i 1.01032 − 2.43914i −5.12717 0.986227 + 0.408509i −6.51170 − 2.69724i −1.64994 + 0.683429i 8.34855i −2.80731 − 2.80731i 1.09059 − 2.63291i
70.2 − 2.51526i −1.09809 + 2.65103i −4.32654 −0.682238 − 0.282592i 6.66804 + 2.76199i 2.41790 − 1.00153i 5.85184i −3.70085 − 3.70085i −0.710793 + 1.71601i
70.3 − 2.43542i 0.0205309 − 0.0495660i −3.93127 −3.05819 − 1.26675i −0.120714 − 0.0500013i −2.14446 + 0.888264i 4.70346i 2.11929 + 2.11929i −3.08506 + 7.44799i
70.4 − 2.31363i 0.188513 − 0.455110i −3.35288 1.56952 + 0.650117i −1.05296 − 0.436149i 4.83424 − 2.00241i 3.13006i 1.94973 + 1.94973i 1.50413 − 3.63129i
70.5 − 1.94881i 0.950949 − 2.29579i −1.79787 −1.10376 − 0.457193i −4.47407 − 1.85322i 0.111116 − 0.0460256i − 0.393911i −2.24504 − 2.24504i −0.890983 + 2.15102i
70.6 − 1.92834i −0.879044 + 2.12220i −1.71850 −0.196825 − 0.0815274i 4.09232 + 1.69510i −3.32203 + 1.37603i − 0.542835i −1.60969 − 1.60969i −0.157213 + 0.379545i
70.7 − 1.87940i 0.216299 − 0.522191i −1.53215 3.31047 + 1.37124i −0.981406 − 0.406512i −2.63832 + 1.09283i − 0.879287i 1.89542 + 1.89542i 2.57711 − 6.22170i
70.8 − 1.61906i −0.332369 + 0.802411i −0.621342 1.62005 + 0.671048i 1.29915 + 0.538125i 1.26627 − 0.524507i − 2.23212i 1.58793 + 1.58793i 1.08646 − 2.62296i
70.9 − 1.19786i −0.452259 + 1.09185i 0.565136 −2.52884 − 1.04748i 1.30788 + 0.541742i 0.465424 − 0.192785i − 3.07267i 1.13372 + 1.13372i −1.25473 + 3.02919i
70.10 − 1.04483i −1.13780 + 2.74690i 0.908322 2.49473 + 1.03335i 2.87006 + 1.18882i 1.89543 − 0.785113i − 3.03871i −4.12956 − 4.12956i 1.07968 − 2.60658i
70.11 − 0.999206i 0.722463 − 1.74418i 1.00159 −2.60544 − 1.07921i −1.74279 − 0.721889i 3.34997 − 1.38760i − 2.99920i −0.398887 − 0.398887i −1.07835 + 2.60337i
70.12 − 0.644316i 1.16882 − 2.82178i 1.58486 2.63385 + 1.09098i −1.81812 − 0.753090i −1.66519 + 0.689743i − 2.30978i −4.47500 − 4.47500i 0.702933 − 1.69703i
70.13 − 0.542172i 0.276574 − 0.667710i 1.70605 0.993606 + 0.411565i −0.362014 − 0.149951i −0.280668 + 0.116256i − 2.00932i 1.75198 + 1.75198i 0.223139 − 0.538706i
70.14 − 0.154958i −0.404731 + 0.977107i 1.97599 −2.51182 − 1.04043i 0.151410 + 0.0627162i −3.04156 + 1.25985i − 0.616110i 1.33039 + 1.33039i −0.161222 + 0.389225i
70.15 0.0800420i 0.991093 − 2.39271i 1.99359 −3.06810 − 1.27085i 0.191517 + 0.0793290i −4.68639 + 1.94117i 0.319655i −2.62148 − 2.62148i 0.101721 − 0.245577i
70.16 0.291441i −0.651084 + 1.57186i 1.91506 2.34278 + 0.970412i −0.458104 − 0.189753i −2.74727 + 1.13796i 1.14101i 0.0744977 + 0.0744977i −0.282818 + 0.682784i
70.17 0.594777i −0.632440 + 1.52685i 1.64624 −1.23835 − 0.512939i −0.908133 − 0.376161i 4.37225 − 1.81104i 2.16870i 0.190044 + 0.190044i 0.305085 − 0.736539i
70.18 0.852572i 0.329423 − 0.795298i 1.27312 0.104782 + 0.0434020i 0.678049 + 0.280857i −0.206301 + 0.0854525i 2.79057i 1.59734 + 1.59734i −0.0370033 + 0.0893339i
70.19 0.995774i 0.900672 − 2.17442i 1.00843 −0.216087 − 0.0895061i 2.16523 + 0.896866i 1.50773 − 0.624522i 2.99572i −1.79555 − 1.79555i 0.0891279 − 0.215174i
70.20 1.49454i −1.15502 + 2.78846i −0.233663 1.19836 + 0.496375i −4.16747 − 1.72622i −1.00476 + 0.416185i 2.63987i −4.32010 − 4.32010i −0.741855 + 1.79100i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
353.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 353.2.d.a ✓ 112
353.d even 8 1 inner 353.2.d.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
353.2.d.a ✓ 112 1.a even 1 1 trivial
353.2.d.a ✓ 112 353.d even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(353, [\chi])\).