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Properties

Label 322.3.j.a
Level $322$
Weight $3$
Character orbit 322.j
Analytic conductor $8.774$
Analytic rank $0$
Dimension $240$
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] = [322,3,Mod(15,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 17]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.15");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 322.j (of order \(22\), degree \(10\), 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: \(240\)
Relative dimension: \(24\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 8 q^{3} - 48 q^{4} - 112 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 8 q^{3} - 48 q^{4} - 112 q^{9} - 16 q^{12} - 8 q^{13} + 308 q^{15} - 96 q^{16} + 220 q^{17} + 144 q^{18} + 132 q^{19} - 20 q^{23} - 156 q^{25} - 144 q^{26} - 32 q^{27} - 140 q^{29} - 352 q^{30} - 188 q^{31} - 484 q^{33} + 56 q^{35} - 224 q^{36} + 352 q^{37} + 376 q^{39} + 136 q^{41} + 88 q^{43} + 64 q^{46} - 272 q^{47} - 32 q^{48} + 168 q^{49} + 128 q^{50} - 264 q^{51} - 16 q^{52} - 352 q^{53} - 48 q^{54} - 64 q^{55} + 924 q^{57} - 64 q^{58} + 1472 q^{59} + 352 q^{60} + 792 q^{62} - 192 q^{64} + 352 q^{66} + 88 q^{67} - 196 q^{69} + 112 q^{70} - 796 q^{71} - 64 q^{72} - 288 q^{73} - 1056 q^{74} + 1088 q^{75} - 1272 q^{78} + 528 q^{79} - 528 q^{80} - 1376 q^{81} - 64 q^{82} - 1628 q^{83} - 1336 q^{85} + 32 q^{87} + 88 q^{89} + 48 q^{92} - 680 q^{93} + 256 q^{94} + 240 q^{95} + 1980 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −0.201264 + 1.39982i −2.39034 − 5.23411i −1.91899 − 0.563465i −4.63367 + 4.01509i 7.80790 − 2.29261i −1.43040 + 2.22575i 1.17497 − 2.57283i −15.7885 + 18.2209i −4.68782 − 7.29439i
15.2 −0.201264 + 1.39982i −1.55437 − 3.40360i −1.91899 − 0.563465i −2.57535 + 2.23155i 5.07726 − 1.49082i 1.43040 − 2.22575i 1.17497 − 2.57283i −3.27466 + 3.77916i −2.60544 − 4.05415i
15.3 −0.201264 + 1.39982i −1.40484 − 3.07618i −1.91899 − 0.563465i 3.31708 − 2.87427i 4.58883 − 1.34740i 1.43040 − 2.22575i 1.17497 − 2.57283i −1.59553 + 1.84134i 3.35585 + 5.22180i
15.4 −0.201264 + 1.39982i −0.989563 − 2.16684i −1.91899 − 0.563465i −0.242716 + 0.210315i 3.23235 − 0.949103i −1.43040 + 2.22575i 1.17497 − 2.57283i 2.17778 − 2.51329i −0.245553 − 0.382088i
15.5 −0.201264 + 1.39982i −0.429611 − 0.940716i −1.91899 − 0.563465i −5.46755 + 4.73766i 1.40330 − 0.412045i −1.43040 + 2.22575i 1.17497 − 2.57283i 5.19336 − 5.99346i −5.53145 − 8.60710i
15.6 −0.201264 + 1.39982i 0.156968 + 0.343713i −1.91899 − 0.563465i −1.96268 + 1.70068i −0.512728 + 0.150551i 1.43040 − 2.22575i 1.17497 − 2.57283i 5.80025 − 6.69384i −1.98562 − 3.08969i
15.7 −0.201264 + 1.39982i 0.393781 + 0.862259i −1.91899 − 0.563465i 1.48495 − 1.28672i −1.28626 + 0.377680i −1.43040 + 2.22575i 1.17497 − 2.57283i 5.30532 − 6.12266i 1.50230 + 2.33763i
15.8 −0.201264 + 1.39982i 0.551983 + 1.20867i −1.91899 − 0.563465i 3.97234 − 3.44205i −1.80302 + 0.529414i 1.43040 − 2.22575i 1.17497 − 2.57283i 4.73754 − 5.46741i 4.01876 + 6.25332i
15.9 −0.201264 + 1.39982i 1.12331 + 2.45971i −1.91899 − 0.563465i −5.83730 + 5.05805i −3.66923 + 1.07738i 1.43040 − 2.22575i 1.17497 − 2.57283i 1.10540 − 1.27570i −5.90551 − 9.18916i
15.10 −0.201264 + 1.39982i 1.74692 + 3.82522i −1.91899 − 0.563465i 1.96464 − 1.70237i −5.70621 + 1.67549i −1.43040 + 2.22575i 1.17497 − 2.57283i −5.68684 + 6.56296i 1.98760 + 3.09276i
15.11 −0.201264 + 1.39982i 1.82589 + 3.99814i −1.91899 − 0.563465i 6.64281 − 5.75602i −5.96415 + 1.75123i 1.43040 − 2.22575i 1.17497 − 2.57283i −6.75749 + 7.79856i 6.72044 + 10.4572i
15.12 −0.201264 + 1.39982i 2.36775 + 5.18465i −1.91899 − 0.563465i −5.63924 + 4.88643i −7.73412 + 2.27094i −1.43040 + 2.22575i 1.17497 − 2.57283i −15.3806 + 17.7502i −5.70514 − 8.87738i
15.13 0.201264 − 1.39982i −2.22483 − 4.87171i −1.91899 − 0.563465i 3.64048 − 3.15450i −7.26729 + 2.13387i −1.43040 + 2.22575i −1.17497 + 2.57283i −12.8899 + 14.8757i −3.68303 − 5.73090i
15.14 0.201264 − 1.39982i −1.80401 − 3.95022i −1.91899 − 0.563465i −1.53881 + 1.33339i −5.89268 + 1.73025i 1.43040 − 2.22575i −1.17497 + 2.57283i −6.45607 + 7.45070i 1.55680 + 2.42242i
15.15 0.201264 − 1.39982i −1.79572 − 3.93208i −1.91899 − 0.563465i −5.39698 + 4.67651i −5.86562 + 1.72230i −1.43040 + 2.22575i −1.17497 + 2.57283i −6.34291 + 7.32011i 5.46005 + 8.49600i
15.16 0.201264 − 1.39982i −0.881617 − 1.93047i −1.91899 − 0.563465i 5.29356 − 4.58689i −2.87975 + 0.845571i 1.43040 − 2.22575i −1.17497 + 2.57283i 2.94427 − 3.39787i −5.35542 − 8.33319i
15.17 0.201264 − 1.39982i −0.141297 − 0.309398i −1.91899 − 0.563465i −2.21531 + 1.91957i −0.461539 + 0.135520i 1.43040 − 2.22575i −1.17497 + 2.57283i 5.81798 − 6.71431i 2.24120 + 3.48737i
15.18 0.201264 − 1.39982i 0.0276705 + 0.0605900i −1.91899 − 0.563465i −7.31210 + 6.33597i 0.0903841 − 0.0265392i 1.43040 − 2.22575i −1.17497 + 2.57283i 5.89084 − 6.79839i 7.39755 + 11.5108i
15.19 0.201264 − 1.39982i 0.120703 + 0.264303i −1.91899 − 0.563465i −0.332863 + 0.288428i 0.394270 − 0.115768i −1.43040 + 2.22575i −1.17497 + 2.57283i 5.83846 − 6.73794i 0.336753 + 0.523998i
15.20 0.201264 − 1.39982i 0.465580 + 1.01948i −1.91899 − 0.563465i −2.47539 + 2.14494i 1.52079 − 0.446544i −1.43040 + 2.22575i −1.17497 + 2.57283i 5.07118 − 5.85245i 2.50432 + 3.89680i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.3.j.a ✓ 240
23.d odd 22 1 inner 322.3.j.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.3.j.a ✓ 240 1.a even 1 1 trivial
322.3.j.a ✓ 240 23.d odd 22 1 inner

Hecke kernels

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