Properties

Label 392.2.q.b
Level $392$
Weight $2$
Character orbit 392.q
Analytic conductor $3.130$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [392,2,Mod(57,392)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("392.57"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 0, 12])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 392.q (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [42,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.13013575923\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 5 q^{3} + 4 q^{5} - q^{7} - 2 q^{9} + 2 q^{11} + 14 q^{13} - q^{15} + 2 q^{17} - 26 q^{19} - 9 q^{21} - 10 q^{23} - 17 q^{25} + 11 q^{27} + 21 q^{29} - 32 q^{31} + 14 q^{33} + 25 q^{35} + 6 q^{37}+ \cdots + 94 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1 0 −2.04391 + 0.984293i 0 −1.11822 + 0.538506i 0 1.05076 + 2.42815i 0 1.33825 − 1.67811i 0
57.2 0 −2.01889 + 0.972248i 0 2.40901 − 1.16012i 0 −2.28083 − 1.34082i 0 1.26019 − 1.58023i 0
57.3 0 −0.928435 + 0.447111i 0 −0.587115 + 0.282740i 0 2.17796 − 1.50216i 0 −1.20839 + 1.51527i 0
57.4 0 0.636649 − 0.306594i 0 −1.28687 + 0.619724i 0 −1.64956 + 2.06857i 0 −1.55915 + 1.95511i 0
57.5 0 0.805030 − 0.387682i 0 3.56435 − 1.71650i 0 2.63621 + 0.224522i 0 −1.37269 + 1.72130i 0
57.6 0 2.27666 − 1.09638i 0 2.01217 − 0.969012i 0 −2.56866 − 0.634011i 0 2.11065 − 2.64667i 0
57.7 0 2.89639 − 1.39483i 0 −2.06790 + 0.995846i 0 2.61451 + 0.405358i 0 4.57304 − 5.73441i 0
113.1 0 −1.67812 + 2.10430i 0 −2.08811 + 2.61840i 0 −1.71849 − 2.01166i 0 −0.944420 − 4.13777i 0
113.2 0 −1.44889 + 1.81685i 0 0.799185 − 1.00215i 0 −0.480566 + 2.60174i 0 −0.534104 − 2.34006i 0
113.3 0 −0.162528 + 0.203804i 0 −0.622878 + 0.781064i 0 2.35862 − 1.19872i 0 0.652442 + 2.85854i 0
113.4 0 0.0714866 − 0.0896413i 0 1.38008 − 1.73057i 0 −2.33937 − 1.23585i 0 0.664638 + 2.91197i 0
113.5 0 0.921166 − 1.15511i 0 −2.23802 + 2.80639i 0 −1.64258 + 2.07411i 0 0.181840 + 0.796694i 0
113.6 0 1.22867 − 1.54070i 0 1.69839 − 2.12971i 0 2.43170 + 1.04250i 0 −0.196573 − 0.861244i 0
113.7 0 1.84570 − 2.31444i 0 0.101846 − 0.127711i 0 −1.88075 − 1.86086i 0 −1.28244 − 5.61873i 0
169.1 0 −0.567507 + 2.48641i 0 0.909831 − 3.98623i 0 1.44045 + 2.21925i 0 −3.15726 − 1.52046i 0
169.2 0 −0.414931 + 1.81793i 0 −0.300295 + 1.31568i 0 −2.62249 + 0.350041i 0 −0.429806 − 0.206984i 0
169.3 0 −0.384316 + 1.68380i 0 −0.419945 + 1.83990i 0 2.57784 + 0.595623i 0 0.0154284 + 0.00742991i 0
169.4 0 0.220159 − 0.964580i 0 −0.150203 + 0.658081i 0 0.828814 + 2.51258i 0 1.82096 + 0.876929i 0
169.5 0 0.284478 − 1.24638i 0 0.437465 − 1.91666i 0 1.37017 − 2.26332i 0 1.23038 + 0.592517i 0
169.6 0 0.301716 − 1.32190i 0 −0.856276 + 3.75159i 0 −0.648507 − 2.56504i 0 1.04651 + 0.503972i 0
See all 42 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 57.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.2.q.b ✓ 42
4.b odd 2 1 784.2.u.f 42
49.e even 7 1 inner 392.2.q.b ✓ 42
196.k odd 14 1 784.2.u.f 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.q.b ✓ 42 1.a even 1 1 trivial
392.2.q.b ✓ 42 49.e even 7 1 inner
784.2.u.f 42 4.b odd 2 1
784.2.u.f 42 196.k odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{42} - 5 T_{3}^{41} + 24 T_{3}^{40} - 78 T_{3}^{39} + 291 T_{3}^{38} - 818 T_{3}^{37} + \cdots + 3265249 \) acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\). Copy content Toggle raw display