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Properties

Label 392.2.u.a
Level $392$
Weight $2$
Character orbit 392.u
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(27,392)]
 
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([7, 7, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.u (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.

\(\operatorname{Tr}(f)(q) = \) \( 324 q - 5 q^{2} - 14 q^{3} - 5 q^{4} - 7 q^{6} - 11 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 324 q - 5 q^{2} - 14 q^{3} - 5 q^{4} - 7 q^{6} - 11 q^{8} + 40 q^{9} - 7 q^{10} - 10 q^{11} + 28 q^{12} - 45 q^{14} + 3 q^{16} - 14 q^{17} - 24 q^{18} - 7 q^{20} + 11 q^{22} - 49 q^{24} - 56 q^{25} - 7 q^{26} - 14 q^{27} + 20 q^{28} - 28 q^{30} - 25 q^{32} - 14 q^{33} + 21 q^{34} + 10 q^{35} + 17 q^{36} + 7 q^{38} - 63 q^{40} - 14 q^{41} + 17 q^{42} - 10 q^{43} - 39 q^{44} + 21 q^{46} - 20 q^{49} + 104 q^{50} - 62 q^{51} - 7 q^{52} - 28 q^{54} - 62 q^{56} + 4 q^{57} - 11 q^{58} - 14 q^{59} - 59 q^{60} - 21 q^{62} - 35 q^{64} + 10 q^{65} - 35 q^{66} - 48 q^{67} + 39 q^{70} - 92 q^{72} - 14 q^{73} - 59 q^{74} - 14 q^{75} + 126 q^{76} - 27 q^{78} - 100 q^{81} - 112 q^{82} - 14 q^{83} + 22 q^{84} - 87 q^{86} + 35 q^{88} - 14 q^{89} - 126 q^{90} + 130 q^{91} + 66 q^{92} - 154 q^{94} + 238 q^{96} - 57 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −1.41007 + 0.108234i 0.363292 + 0.0829191i 1.97657 − 0.305235i −0.719538 + 3.15250i −0.521241 − 0.0776006i −0.538315 − 2.59041i −2.75406 + 0.644335i −2.57780 − 1.24140i 0.673386 − 4.52311i
27.2 −1.40955 + 0.114767i 3.19528 + 0.729303i 1.97366 − 0.323541i 0.672455 − 2.94622i −4.58761 − 0.661273i 2.64563 + 0.0253243i −2.74483 + 0.682558i 6.97505 + 3.35901i −0.609728 + 4.23001i
27.3 −1.40350 − 0.173750i −2.88239 − 0.657886i 1.93962 + 0.487715i 0.182778 − 0.800805i 3.93112 + 1.42416i −2.53639 + 0.752811i −2.63752 − 1.02152i 5.17244 + 2.49092i −0.395669 + 1.09217i
27.4 −1.38947 + 0.263385i 1.79940 + 0.410702i 1.86126 − 0.731930i 0.219181 − 0.960295i −2.60839 − 0.0967234i −2.48567 − 0.906341i −2.39338 + 1.50722i 0.366269 + 0.176386i −0.0516188 + 1.39203i
27.5 −1.38826 + 0.269678i −1.88034 − 0.429176i 1.85455 − 0.748769i 0.0299159 − 0.131070i 2.72615 + 0.0887210i 2.57060 + 0.626107i −2.37267 + 1.53962i 0.648587 + 0.312343i −0.00618435 + 0.190028i
27.6 −1.35559 − 0.402969i 0.439740 + 0.100368i 1.67523 + 1.09252i −0.130698 + 0.572627i −0.555660 − 0.313258i 2.49833 − 0.870836i −1.83067 − 2.15607i −2.51961 − 1.21338i 0.407924 − 0.723579i
27.7 −1.30005 + 0.556652i −1.12183 − 0.256050i 1.38028 − 1.44735i −0.901885 + 3.95141i 1.60097 − 0.291589i −0.620980 + 2.57184i −0.988762 + 2.64997i −1.50997 − 0.727164i −1.02706 − 5.63909i
27.8 −1.28062 − 0.600020i 2.16604 + 0.494385i 1.27995 + 1.53679i −0.539135 + 2.36210i −2.47723 − 1.93279i −0.349806 + 2.62252i −0.717021 − 2.73603i 1.74442 + 0.840068i 2.10773 − 2.70145i
27.9 −1.24578 + 0.669356i −1.12183 − 0.256050i 1.10393 − 1.66774i 0.901885 − 3.95141i 1.56894 − 0.431921i 0.620980 − 2.57184i −0.258936 + 2.81655i −1.50997 − 0.727164i 1.52135 + 5.52627i
27.10 −1.20616 − 0.738362i 0.779271 + 0.177863i 0.909642 + 1.78117i 0.811161 − 3.55393i −0.808597 − 0.789916i −2.34361 − 1.22781i 0.217973 − 2.82002i −2.12728 − 1.02444i −3.60248 + 3.68767i
27.11 −1.18475 − 0.772249i −1.77562 − 0.405273i 0.807264 + 1.82984i −0.298047 + 1.30583i 1.79069 + 1.85136i −0.422692 − 2.61177i 0.456688 − 2.79131i 0.285659 + 0.137566i 1.36154 − 1.31691i
27.12 −1.07641 + 0.917246i −1.88034 − 0.429176i 0.317320 − 1.97467i −0.0299159 + 0.131070i 2.41768 − 1.26277i −2.57060 − 0.626107i 1.46969 + 2.41661i 0.648587 + 0.312343i −0.0880218 − 0.168526i
27.13 −1.07224 + 0.922114i 1.79940 + 0.410702i 0.299411 − 1.97746i −0.219181 + 0.960295i −2.30811 + 1.21888i 2.48567 + 0.906341i 1.50240 + 2.39641i 0.366269 + 0.176386i −0.650487 − 1.23178i
27.14 −0.968568 + 1.03047i 3.19528 + 0.729303i −0.123751 − 1.99617i −0.672455 + 2.94622i −3.84638 + 2.58628i −2.64563 − 0.0253243i 2.17686 + 1.80590i 6.97505 + 3.35901i −2.38468 − 3.54656i
27.15 −0.963783 + 1.03495i 0.363292 + 0.0829191i −0.142246 − 1.99494i 0.719538 − 3.15250i −0.435952 + 0.296074i 0.538315 + 2.59041i 2.20175 + 1.77547i −2.57780 − 1.24140i 2.56920 + 3.78301i
27.16 −0.932941 − 1.06284i −2.99639 − 0.683905i −0.259243 + 1.98313i 0.912691 − 3.99876i 2.06857 + 3.82271i 2.62811 − 0.305057i 2.34960 − 1.57461i 5.80769 + 2.79684i −5.10151 + 2.76056i
27.17 −0.925507 − 1.06932i −1.08753 − 0.248221i −0.286875 + 1.97932i −0.221810 + 0.971813i 0.741087 + 1.39264i −2.19023 + 1.48421i 2.38202 − 1.52511i −1.58181 − 0.761757i 1.24446 − 0.662235i
27.18 −0.775177 − 1.18284i 2.71851 + 0.620482i −0.798201 + 1.83381i −0.210841 + 0.923753i −1.37340 − 3.69654i 0.540344 − 2.58999i 2.78785 − 0.477391i 4.30240 + 2.07192i 1.25609 − 0.466682i
27.19 −0.739225 + 1.20563i −2.88239 − 0.657886i −0.907094 − 1.78246i −0.182778 + 0.800805i 2.92390 − 2.98877i 2.53639 − 0.752811i 2.81954 + 0.224022i 5.17244 + 2.49092i −0.830361 − 0.812338i
27.20 −0.704926 − 1.22600i −0.505554 − 0.115389i −1.00616 + 1.72848i 0.0593783 − 0.260153i 0.214910 + 0.701150i 1.92851 + 1.81131i 2.82839 + 0.0151011i −2.46064 − 1.18498i −0.360805 + 0.110591i
See next 80 embeddings (of 324 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
49.f odd 14 1 inner
392.u 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.u.a ✓ 324
8.d odd 2 1 inner 392.2.u.a ✓ 324
49.f odd 14 1 inner 392.2.u.a ✓ 324
392.u even 14 1 inner 392.2.u.a ✓ 324
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.u.a ✓ 324 1.a even 1 1 trivial
392.2.u.a ✓ 324 8.d odd 2 1 inner
392.2.u.a ✓ 324 49.f odd 14 1 inner
392.2.u.a ✓ 324 392.u even 14 1 inner

Hecke kernels

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