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Properties

Label 371.2.j.a
Level $371$
Weight $2$
Character orbit 371.j
Analytic conductor $2.962$
Analytic rank $0$
Dimension $68$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [371,2,Mod(158,371)] 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("371.158"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(371, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 371.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.96244991499\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 32 q^{4} - 24 q^{6} + 28 q^{9} - 10 q^{10} - 2 q^{11} - 24 q^{15} - 20 q^{16} + 8 q^{17} - 44 q^{24} + 40 q^{25} - 12 q^{28} - 32 q^{29} + 56 q^{36} - 10 q^{37} + 26 q^{38} + 38 q^{40} - 34 q^{42}+ \cdots + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
158.1 −2.29615 + 1.32568i 1.65891 + 0.957770i 2.51486 − 4.35586i 3.12458 − 1.80398i −5.07879 −1.93332 − 1.80618i 8.03286i 0.334647 + 0.579626i −4.78299 + 8.28438i
158.2 −2.24165 + 1.29422i 0.585673 + 0.338139i 2.35000 − 4.07032i −0.969845 + 0.559940i −1.75050 −0.269365 + 2.63200i 6.98877i −1.27132 − 2.20200i 1.44937 − 2.51038i
158.3 −2.11529 + 1.22126i −1.71796 − 0.991865i 1.98297 − 3.43461i 2.97534 − 1.71781i 4.84532 2.61520 + 0.400920i 4.80189i 0.467593 + 0.809895i −4.19581 + 7.26736i
158.4 −2.05333 + 1.18549i 2.86409 + 1.65358i 1.81078 − 3.13637i −2.83086 + 1.63440i −7.84123 1.81640 − 1.92372i 3.84471i 3.96867 + 6.87393i 3.87514 − 6.71193i
158.5 −1.93373 + 1.11644i −1.09304 − 0.631067i 1.49289 − 2.58575i −0.161677 + 0.0933442i 2.81820 −0.108627 − 2.64352i 2.20111i −0.703509 − 1.21851i 0.208427 − 0.361006i
158.6 −1.88372 + 1.08756i −1.67328 − 0.966070i 1.36559 − 2.36527i −1.27017 + 0.733332i 4.20265 −2.24358 + 1.40227i 1.59042i 0.366584 + 0.634943i 1.59509 − 2.76278i
158.7 −1.59618 + 0.921557i 1.23756 + 0.714506i 0.698536 − 1.20990i −1.21143 + 0.699420i −2.63383 −1.95296 − 1.78492i − 1.11127i −0.478963 − 0.829587i 1.28911 − 2.23281i
158.8 −1.49056 + 0.860575i 2.33924 + 1.35056i 0.481178 − 0.833425i 2.31992 − 1.33941i −4.64904 0.938979 + 2.47352i − 1.78594i 2.14804 + 3.72052i −2.30532 + 3.99293i
158.9 −1.20195 + 0.693947i 0.0854188 + 0.0493166i −0.0368746 + 0.0638687i −3.60059 + 2.07880i −0.136892 2.26905 + 1.36067i − 2.87814i −1.49514 − 2.58965i 2.88516 − 4.99724i
158.10 −1.18587 + 0.684662i 0.557483 + 0.321863i −0.0624753 + 0.108210i −0.156700 + 0.0904708i −0.881470 2.50374 − 0.855160i − 2.90975i −1.29281 − 2.23921i 0.123884 − 0.214573i
158.11 −1.08438 + 0.626070i −0.855062 − 0.493670i −0.216073 + 0.374250i 2.11748 − 1.22253i 1.23629 −0.140835 + 2.64200i − 3.04539i −1.01258 − 1.75384i −1.53077 + 2.65138i
158.12 −1.07408 + 0.620121i −2.79305 − 1.61257i −0.230900 + 0.399931i 2.11222 − 1.21949i 3.99996 −2.17989 − 1.49936i − 3.05323i 3.70077 + 6.40992i −1.51246 + 2.61966i
158.13 −0.773620 + 0.446650i 1.71574 + 0.990585i −0.601008 + 1.04098i −0.890459 + 0.514107i −1.76978 −2.64441 − 0.0841604i − 2.86036i 0.462518 + 0.801104i 0.459251 − 0.795447i
158.14 −0.506391 + 0.292365i −1.65813 − 0.957324i −0.829045 + 1.43595i −3.13323 + 1.80897i 1.11955 −2.41619 + 1.07797i − 2.13900i 0.332940 + 0.576669i 1.05776 − 1.83210i
158.15 −0.454170 + 0.262215i −0.139865 − 0.0807510i −0.862486 + 1.49387i 2.86782 − 1.65574i 0.0846966 −0.284975 − 2.63036i − 1.95349i −1.48696 − 2.57549i −0.868319 + 1.50397i
158.16 −0.427749 + 0.246961i −1.74693 − 1.00859i −0.878021 + 1.52078i 0.320681 − 0.185145i 0.996329 2.22175 + 1.43661i − 1.85519i 0.534507 + 0.925794i −0.0914474 + 0.158391i
158.17 −0.173544 + 0.100196i 2.42642 + 1.40089i −0.979922 + 1.69727i 1.34894 − 0.778811i −0.561454 1.80905 − 1.93063i − 0.793518i 2.42501 + 4.20024i −0.156067 + 0.270316i
158.18 0.173544 − 0.100196i −2.42642 − 1.40089i −0.979922 + 1.69727i −1.34894 + 0.778811i −0.561454 1.80905 − 1.93063i 0.793518i 2.42501 + 4.20024i −0.156067 + 0.270316i
158.19 0.427749 − 0.246961i 1.74693 + 1.00859i −0.878021 + 1.52078i −0.320681 + 0.185145i 0.996329 2.22175 + 1.43661i 1.85519i 0.534507 + 0.925794i −0.0914474 + 0.158391i
158.20 0.454170 − 0.262215i 0.139865 + 0.0807510i −0.862486 + 1.49387i −2.86782 + 1.65574i 0.0846966 −0.284975 − 2.63036i 1.95349i −1.48696 − 2.57549i −0.868319 + 1.50397i
See all 68 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 158.34
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
53.b even 2 1 inner
371.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 371.2.j.a ✓ 68
7.c even 3 1 inner 371.2.j.a ✓ 68
53.b even 2 1 inner 371.2.j.a ✓ 68
371.j even 6 1 inner 371.2.j.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
371.2.j.a ✓ 68 1.a even 1 1 trivial
371.2.j.a ✓ 68 7.c even 3 1 inner
371.2.j.a ✓ 68 53.b even 2 1 inner
371.2.j.a ✓ 68 371.j even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(371, [\chi])\).