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Properties

Label 417.2.j.a
Level $417$
Weight $2$
Character orbit 417.j
Analytic conductor $3.330$
Analytic rank $0$
Dimension $968$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [417,2,Mod(8,417)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(417, base_ring=CyclotomicField(46))
 
chi = DirichletCharacter(H, H._module([23, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("417.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 417 = 3 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 417.j (of order \(46\), degree \(22\), 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: \(3.32976176429\)
Analytic rank: \(0\)
Dimension: \(968\)
Relative dimension: \(44\) over \(\Q(\zeta_{46})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{46}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 968 q - 23 q^{3} - 86 q^{4} - 21 q^{6} - 34 q^{7} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 968 q - 23 q^{3} - 86 q^{4} - 21 q^{6} - 34 q^{7} - 15 q^{9} - 46 q^{10} - 23 q^{12} - 54 q^{13} - 23 q^{15} - 62 q^{16} - 23 q^{18} - 46 q^{19} + 46 q^{21} - 46 q^{22} - 94 q^{24} - 2 q^{25} + 46 q^{27} + 6 q^{28} - 65 q^{30} - 146 q^{31} - 23 q^{33} + 102 q^{34} + 15 q^{36} - 62 q^{37} - 23 q^{39} - 138 q^{40} - 22 q^{42} - 47 q^{45} + 74 q^{46} - 23 q^{48} - 62 q^{49} - 30 q^{51} - 14 q^{52} - 293 q^{54} - 102 q^{55} - 29 q^{57} - 46 q^{58} - 23 q^{60} - 46 q^{61} - 45 q^{63} + 130 q^{64} + 36 q^{66} - 34 q^{67} - 17 q^{69} - 46 q^{70} - 299 q^{72} - 46 q^{73} + 115 q^{75} - 46 q^{76} + 41 q^{78} - 26 q^{79} + 41 q^{81} - 46 q^{82} - 253 q^{84} + 92 q^{85} - 23 q^{87} - 322 q^{88} + 115 q^{90} - 86 q^{91} - 23 q^{93} + 368 q^{94} - 92 q^{96} - 9 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −0.184153 + 2.69222i −0.852710 − 1.50761i −5.23276 − 0.719226i −0.207406 − 0.477498i 4.21584 − 2.01805i 4.18301 + 1.17202i 1.80188 − 8.67115i −1.54577 + 2.57111i 1.32372 − 0.470451i
8.2 −0.175059 + 2.55928i −1.01405 + 1.40418i −4.53787 − 0.623716i −1.28913 − 2.96787i −3.41615 − 2.84104i 0.588134 + 0.164788i 1.34683 − 6.48128i −0.943415 − 2.84780i 7.82127 − 2.77968i
8.3 −0.170468 + 2.49215i 1.64869 − 0.530859i −4.20037 − 0.577328i 1.56618 + 3.60572i 1.04193 + 4.19928i 2.30567 + 0.646017i 1.13836 − 5.47809i 2.43638 − 1.75045i −9.25296 + 3.28850i
8.4 −0.166131 + 2.42875i 0.0789474 − 1.73025i −3.88985 − 0.534648i 0.573734 + 1.32087i 4.18923 + 0.479192i −4.63840 − 1.29962i 0.954155 − 4.59165i −2.98753 − 0.273198i −3.30337 + 1.17402i
8.5 −0.157298 + 2.29962i 1.27281 − 1.17472i −3.28214 − 0.451120i −1.26521 − 2.91281i 2.50119 + 3.11176i −0.318802 − 0.0893240i 0.615751 − 2.96316i 0.240085 − 2.99038i 6.89737 − 2.45133i
8.6 −0.156117 + 2.28235i 0.193666 + 1.72119i −3.20338 − 0.440295i 0.682619 + 1.57155i −3.95859 + 0.173308i −0.380241 − 0.106538i 0.574125 − 2.76284i −2.92499 + 0.666673i −3.69339 + 1.31263i
8.7 −0.151488 + 2.21467i −1.64751 − 0.534515i −2.90046 − 0.398659i −0.750994 − 1.72896i 1.43335 − 3.56773i −2.36125 − 0.661592i 0.419001 − 2.01634i 2.42859 + 1.76124i 3.94285 − 1.40129i
8.8 −0.139167 + 2.03454i 1.45288 + 0.942945i −2.13862 − 0.293947i −1.21188 − 2.79003i −2.12065 + 2.82472i 4.80033 + 1.34499i 0.0658584 − 0.316928i 1.22171 + 2.73997i 5.84509 − 2.07735i
8.9 −0.129625 + 1.89504i −1.66812 + 0.466222i −1.59302 − 0.218955i 0.906673 + 2.08737i −0.667281 − 3.22160i 3.65675 + 1.02457i −0.151494 + 0.729030i 2.56527 − 1.55543i −4.07319 + 1.44761i
8.10 −0.125309 + 1.83195i 1.71860 + 0.215470i −1.35898 − 0.186787i 0.132442 + 0.304911i −0.610086 + 3.12139i −2.11890 − 0.593688i −0.234708 + 1.12948i 2.90715 + 0.740610i −0.575179 + 0.204419i
8.11 −0.112281 + 1.64149i −1.64258 − 0.549489i −0.700497 − 0.0962810i −0.0793764 − 0.182743i 1.08641 − 2.63457i −0.875777 − 0.245381i −0.432805 + 2.08277i 2.39612 + 1.80516i 0.308882 − 0.109777i
8.12 −0.0993084 + 1.45184i 0.441638 − 1.67480i −0.116598 − 0.0160261i 0.442028 + 1.01765i 2.38768 + 0.807509i 2.44118 + 0.683987i −0.557304 + 2.68189i −2.60991 − 1.47931i −1.52136 + 0.540692i
8.13 −0.0850772 + 1.24378i −1.01028 + 1.40689i 0.441609 + 0.0606978i −0.978046 − 2.25169i −1.66391 − 1.37627i −3.48623 − 0.976795i −0.620360 + 2.98533i −0.958664 − 2.84270i 2.88382 − 1.02491i
8.14 −0.0791416 + 1.15701i −0.961844 − 1.44044i 0.648966 + 0.0891983i 1.48064 + 3.40877i 1.74272 − 0.998864i 0.245818 + 0.0688751i −0.626464 + 3.01471i −1.14971 + 2.77095i −4.06116 + 1.44334i
8.15 −0.0749107 + 1.09516i −1.17345 + 1.27398i 0.787619 + 0.108256i 1.54601 + 3.55927i −1.30730 − 1.38054i −3.88686 − 1.08905i −0.624231 + 3.00396i −0.246045 − 2.98989i −4.01376 + 1.42649i
8.16 −0.0612827 + 0.895922i 1.27466 + 1.17271i 1.18245 + 0.162524i 0.577124 + 1.32867i −1.12877 + 1.07013i 0.752562 + 0.210858i −0.583486 + 2.80789i 0.249497 + 2.98961i −1.22575 + 0.435633i
8.17 −0.0610558 + 0.892605i 1.31916 − 1.12242i 1.18836 + 0.163336i −0.507137 − 1.16755i 0.921336 + 1.24601i −0.578352 − 0.162047i −0.582411 + 2.80271i 0.480344 − 2.96130i 1.07312 − 0.381387i
8.18 −0.0550760 + 0.805183i 0.120833 + 1.72783i 1.33609 + 0.183641i −0.873800 − 2.01169i −1.39787 + 0.00213054i 2.08055 + 0.582943i −0.549855 + 2.64605i −2.97080 + 0.417558i 1.66790 − 0.592773i
8.19 −0.0293260 + 0.428731i −1.55428 + 0.764334i 1.79842 + 0.247187i −0.638416 − 1.46978i −0.282113 − 0.688784i 0.782570 + 0.219266i −0.333581 + 1.60528i 1.83159 − 2.37598i 0.648863 − 0.230606i
8.20 −0.0239216 + 0.349721i −1.37787 − 1.04951i 1.85964 + 0.255602i −1.06781 − 2.45835i 0.399997 − 0.456764i 3.60244 + 1.00936i −0.276513 + 1.33065i 0.797050 + 2.89218i 0.885279 − 0.314628i
See next 80 embeddings (of 968 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
139.f odd 46 1 inner
417.j even 46 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 417.2.j.a ✓ 968
3.b odd 2 1 inner 417.2.j.a ✓ 968
139.f odd 46 1 inner 417.2.j.a ✓ 968
417.j even 46 1 inner 417.2.j.a ✓ 968
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
417.2.j.a ✓ 968 1.a even 1 1 trivial
417.2.j.a ✓ 968 3.b odd 2 1 inner
417.2.j.a ✓ 968 139.f odd 46 1 inner
417.2.j.a ✓ 968 417.j even 46 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(417, [\chi])\).