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Properties

Label 322.3.h.a
Level $322$
Weight $3$
Character orbit 322.h
Analytic conductor $8.774$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,3,Mod(47,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 322.h (of order \(6\), degree \(2\), 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: \(8.77386451240\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 12 q^{3} - 56 q^{4} - 4 q^{7} + 80 q^{9} - 48 q^{10} - 24 q^{11} - 24 q^{12} + 8 q^{14} + 104 q^{15} - 112 q^{16} + 60 q^{17} + 60 q^{19} - 60 q^{21} - 16 q^{22} + 108 q^{25} - 8 q^{28} - 216 q^{29}+ \cdots - 256 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 −0.707107 + 1.22474i −4.33276 + 2.50152i −1.00000 − 1.73205i 1.21736 + 0.702845i − 7.07537i 6.79744 + 1.67177i 2.82843 8.01522 − 13.8828i −1.72161 + 0.993973i
47.2 −0.707107 + 1.22474i −3.83561 + 2.21449i −1.00000 − 1.73205i −5.49758 − 3.17403i − 6.26352i −0.781511 − 6.95624i 2.82843 5.30792 − 9.19359i 7.77475 − 4.48875i
47.3 −0.707107 + 1.22474i −3.06700 + 1.77073i −1.00000 − 1.73205i 4.80747 + 2.77559i − 5.00838i 6.67083 − 2.12131i 2.82843 1.77098 − 3.06742i −6.79879 + 3.92528i
47.4 −0.707107 + 1.22474i −2.55289 + 1.47391i −1.00000 − 1.73205i 1.43645 + 0.829334i − 4.16885i −2.32604 + 6.60224i 2.82843 −0.155180 + 0.268780i −2.03145 + 1.17286i
47.5 −0.707107 + 1.22474i −1.98245 + 1.14457i −1.00000 − 1.73205i −5.36395 − 3.09688i − 3.23733i −6.39638 + 2.84365i 2.82843 −1.87993 + 3.25613i 7.58578 − 4.37965i
47.6 −0.707107 + 1.22474i −0.543801 + 0.313964i −1.00000 − 1.73205i 7.19802 + 4.15578i − 0.888023i −5.94482 − 3.69583i 2.82843 −4.30285 + 7.45276i −10.1795 + 5.87716i
47.7 −0.707107 + 1.22474i −0.174775 + 0.100906i −1.00000 − 1.73205i 0.489698 + 0.282727i − 0.285406i −4.73142 − 5.15885i 2.82843 −4.47964 + 7.75896i −0.692537 + 0.399837i
47.8 −0.707107 + 1.22474i −0.142548 + 0.0823001i −1.00000 − 1.73205i −1.73588 − 1.00221i − 0.232780i 5.04685 − 4.85071i 2.82843 −4.48645 + 7.77077i 2.45491 − 1.41734i
47.9 −0.707107 + 1.22474i 1.39285 − 0.804161i −1.00000 − 1.73205i 7.16669 + 4.13769i 2.27451i 1.79332 + 6.76639i 2.82843 −3.20665 + 5.55408i −10.1352 + 5.85157i
47.10 −0.707107 + 1.22474i 2.89183 − 1.66960i −1.00000 − 1.73205i 1.32786 + 0.766641i 4.72234i −1.25907 + 6.88584i 2.82843 1.07512 − 1.86217i −1.87788 + 1.08419i
47.11 −0.707107 + 1.22474i 3.13212 − 1.80833i −1.00000 − 1.73205i −6.13932 − 3.54454i 5.11474i −6.88042 − 1.28836i 2.82843 2.04013 − 3.53361i 8.68231 − 5.01273i
47.12 −0.707107 + 1.22474i 3.51603 − 2.02998i −1.00000 − 1.73205i 6.22999 + 3.59689i 5.74166i 6.30679 − 3.03716i 2.82843 3.74166 − 6.48074i −8.81054 + 5.08677i
47.13 −0.707107 + 1.22474i 3.84786 − 2.22156i −1.00000 − 1.73205i 1.14087 + 0.658680i 6.28352i −3.44155 − 6.09555i 2.82843 5.37066 − 9.30226i −1.61343 + 0.931514i
47.14 −0.707107 + 1.22474i 4.85114 − 2.80080i −1.00000 − 1.73205i −3.79239 − 2.18954i 7.92187i 5.56018 + 4.25258i 2.82843 11.1890 − 19.3799i 5.36325 − 3.09647i
47.15 0.707107 − 1.22474i −4.90423 + 2.83146i −1.00000 − 1.73205i −5.92198 − 3.41905i 8.00857i −6.82230 + 1.56724i −2.82843 11.5343 − 19.9780i −8.37494 + 4.83527i
47.16 0.707107 − 1.22474i −3.78100 + 2.18296i −1.00000 − 1.73205i −0.607733 − 0.350875i 6.17434i 6.99651 + 0.221121i −2.82843 5.03062 − 8.71328i −0.859464 + 0.496212i
47.17 0.707107 − 1.22474i −2.69391 + 1.55533i −1.00000 − 1.73205i −3.08901 − 1.78344i 4.39913i 5.18183 + 4.70623i −2.82843 0.338087 − 0.585585i −4.36852 + 2.52217i
47.18 0.707107 − 1.22474i −2.50244 + 1.44479i −1.00000 − 1.73205i 4.59492 + 2.65288i 4.08647i −6.49312 + 2.61521i −2.82843 −0.325186 + 0.563239i 6.49820 − 3.75174i
47.19 0.707107 − 1.22474i −1.19086 + 0.687541i −1.00000 − 1.73205i −8.55945 − 4.94180i 1.94466i 1.07028 − 6.91770i −2.82843 −3.55458 + 6.15671i −12.1049 + 6.98877i
47.20 0.707107 − 1.22474i −0.935211 + 0.539944i −1.00000 − 1.73205i 6.24644 + 3.60638i 1.52719i 6.87255 − 1.32966i −2.82843 −3.91692 + 6.78430i 8.83380 − 5.10020i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.3.h.a ✓ 56
7.d odd 6 1 inner 322.3.h.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.3.h.a ✓ 56 1.a even 1 1 trivial
322.3.h.a ✓ 56 7.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(322, [\chi])\).