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Properties

Label 392.2.x.a
Level $392$
Weight $2$
Character orbit 392.x
Analytic conductor $3.130$
Analytic rank $0$
Dimension $324$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,2,Mod(29,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 7, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.x (of order \(14\), degree \(6\), 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: \(3.13013575923\)
Analytic rank: \(0\)
Dimension: \(324\)
Relative dimension: \(54\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 324 q - 5 q^{2} - 5 q^{4} - 9 q^{6} - 12 q^{7} + q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 324 q - 5 q^{2} - 5 q^{4} - 9 q^{6} - 12 q^{7} + q^{8} + 40 q^{9} - 3 q^{10} + 14 q^{12} + 27 q^{14} - 22 q^{15} - 13 q^{16} - 10 q^{17} - 4 q^{18} - 11 q^{20} + 3 q^{22} - 10 q^{23} + 29 q^{24} + 36 q^{25} - 27 q^{26} - 44 q^{28} - 40 q^{30} - 96 q^{31} - 5 q^{32} - 22 q^{33} - 33 q^{34} - 23 q^{36} - 11 q^{38} - 22 q^{39} - 35 q^{40} - 10 q^{41} + 3 q^{42} - 11 q^{44} - 11 q^{46} - 14 q^{47} - 4 q^{48} - 20 q^{49} + 96 q^{50} - 3 q^{52} + 58 q^{54} - 46 q^{55} - 84 q^{56} + 36 q^{57} + 21 q^{58} - 103 q^{60} + 11 q^{62} - 24 q^{63} + 13 q^{64} - 30 q^{65} + 25 q^{66} - 40 q^{68} + 51 q^{70} - 90 q^{71} + 58 q^{72} + 6 q^{73} - 15 q^{74} - 98 q^{76} + 81 q^{78} - 24 q^{79} - 92 q^{80} - 20 q^{81} - 80 q^{82} - 70 q^{84} + 5 q^{86} - 46 q^{87} - 107 q^{88} + 6 q^{89} + 162 q^{90} + 60 q^{92} - 28 q^{94} + 64 q^{95} - 194 q^{96} - 24 q^{97} + 71 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.41276 + 0.0640294i 2.51819 − 0.574761i 1.99180 − 0.180917i 0.830662 − 0.189593i −3.52081 + 0.973240i 2.46362 − 0.964667i −2.80236 + 0.383127i 3.30805 − 1.59307i −1.16139 + 0.321037i
29.2 −1.41090 + 0.0967767i −3.22192 + 0.735383i 1.98127 − 0.273084i −0.174140 + 0.0397462i 4.47464 − 1.34936i 0.240470 + 2.63480i −2.76894 + 0.577034i 7.13710 − 3.43704i 0.241847 − 0.0729305i
29.3 −1.40625 + 0.149862i 0.657936 − 0.150170i 1.95508 − 0.421487i 1.54419 − 0.352452i −0.902718 + 0.309776i −0.766317 + 2.53234i −2.68617 + 0.885709i −2.29258 + 1.10405i −2.11871 + 0.727052i
29.4 −1.39568 − 0.228187i −1.07585 + 0.245556i 1.89586 + 0.636954i 1.58392 − 0.361520i 1.55758 − 0.0972227i −2.46729 − 0.955226i −2.50068 − 1.32160i −1.60575 + 0.773289i −2.29315 + 0.143136i
29.5 −1.37838 − 0.316326i −1.49102 + 0.340315i 1.79988 + 0.872036i −2.46433 + 0.562468i 2.16284 + 0.00256300i 1.26691 − 2.32270i −2.20507 − 1.77135i −0.595591 + 0.286822i 3.57471 + 0.00423609i
29.6 −1.36233 + 0.379566i 1.89053 − 0.431500i 1.71186 − 1.03418i −2.58553 + 0.590131i −2.41173 + 1.30542i −2.33357 − 1.24678i −1.93957 + 2.05866i 0.684992 − 0.329875i 3.29834 − 1.78533i
29.7 −1.35979 − 0.388538i 1.66139 − 0.379202i 1.69808 + 1.05666i −3.54703 + 0.809586i −2.40648 − 0.129877i 0.860431 + 2.50193i −1.89848 − 2.09661i −0.0864767 + 0.0416450i 5.13778 + 0.277284i
29.8 −1.27422 + 0.613491i −0.572572 + 0.130686i 1.24726 − 1.56344i 2.54476 − 0.580825i 0.649407 − 0.517790i 2.64313 + 0.117655i −0.630118 + 2.75735i −2.39215 + 1.15200i −2.88625 + 2.30129i
29.9 −1.21441 + 0.724704i −2.60623 + 0.594854i 0.949607 − 1.76018i 1.50359 − 0.343185i 2.73395 − 2.61114i −0.359787 − 2.62117i 0.122395 + 2.82578i 3.73566 − 1.79900i −1.57728 + 1.50643i
29.10 −1.15610 + 0.814518i −1.32789 + 0.303083i 0.673120 − 1.88332i −3.64556 + 0.832076i 1.28831 − 1.43199i 1.72642 + 2.00487i 0.755810 + 2.72557i −1.03147 + 0.496729i 3.53688 − 3.93134i
29.11 −1.15159 − 0.820880i −1.66139 + 0.379202i 0.652312 + 1.89063i 3.54703 − 0.809586i 2.22452 + 0.927119i 0.860431 + 2.50193i 0.800788 − 2.71270i −0.0864767 + 0.0416450i −4.74929 − 1.97937i
29.12 −1.10672 − 0.880437i 1.49102 − 0.340315i 0.449662 + 1.94880i 2.46433 − 0.562468i −1.94976 − 0.936112i 1.26691 − 2.32270i 1.21814 − 2.55267i −0.595591 + 0.286822i −3.22254 − 1.54719i
29.13 −1.09080 + 0.900083i 2.90876 − 0.663907i 0.379700 − 1.96363i 3.50354 − 0.799660i −2.57532 + 3.34232i −2.44296 − 1.01585i 1.35325 + 2.48369i 5.31723 − 2.56064i −3.10191 + 4.02575i
29.14 −1.04860 − 0.948916i 1.07585 − 0.245556i 0.199115 + 1.99006i −1.58392 + 0.361520i −1.36115 − 0.763403i −2.46729 − 0.955226i 1.67961 − 2.27572i −1.60575 + 0.773289i 2.00395 + 1.12392i
29.15 −0.971811 + 1.02742i 0.358224 − 0.0817623i −0.111165 − 1.99691i −0.00845386 + 0.00192954i −0.264123 + 0.447503i −1.86200 + 1.87962i 2.15969 + 1.82641i −2.58127 + 1.24307i 0.00623312 − 0.0105608i
29.16 −0.838129 + 1.13910i 2.81013 − 0.641393i −0.595081 − 1.90942i −1.97018 + 0.449680i −1.62464 + 3.73857i 2.34414 + 1.22679i 2.67377 + 0.922484i 4.78252 − 2.30314i 1.13903 − 2.62111i
29.17 −0.830783 − 1.14446i −2.51819 + 0.574761i −0.619598 + 1.90160i −0.830662 + 0.189593i 2.74987 + 2.40448i 2.46362 − 0.964667i 2.69107 − 0.870713i 3.30805 − 1.59307i 0.907082 + 0.793152i
29.18 −0.804018 − 1.16342i 3.22192 − 0.735383i −0.707111 + 1.87083i 0.174140 − 0.0397462i −3.44605 − 3.15720i 0.240470 + 2.63480i 2.74509 − 0.681508i 7.13710 − 3.43704i −0.186253 − 0.170642i
29.19 −0.759616 − 1.19289i −0.657936 + 0.150170i −0.845966 + 1.81228i −1.54419 + 0.352452i 0.678915 + 0.670773i −0.766317 + 2.53234i 2.80445 − 0.367490i −2.29258 + 1.10405i 1.59343 + 1.57432i
29.20 −0.731810 + 1.21015i −2.33979 + 0.534043i −0.928910 − 1.77119i −2.40522 + 0.548975i 1.06601 − 3.22231i −2.28369 − 1.33595i 2.82319 + 0.172059i 2.48653 − 1.19745i 1.09582 − 3.31241i
See next 80 embeddings (of 324 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
49.e even 7 1 inner
392.x even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.2.x.a ✓ 324
8.b even 2 1 inner 392.2.x.a ✓ 324
49.e even 7 1 inner 392.2.x.a ✓ 324
392.x even 14 1 inner 392.2.x.a ✓ 324
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.x.a ✓ 324 1.a even 1 1 trivial
392.2.x.a ✓ 324 8.b even 2 1 inner
392.2.x.a ✓ 324 49.e even 7 1 inner
392.2.x.a ✓ 324 392.x even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(392, [\chi])\).