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Properties

Label 841.2.e.m
Level $841$
Weight $2$
Character orbit 841.e
Analytic conductor $6.715$
Analytic rank $0$
Dimension $96$
Inner twists $12$

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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] = [841,2,Mod(63,841)] 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("841.63"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(841, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 841 = 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 841.e (of order \(14\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0,0,12,-2,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.71541880999\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 12 q^{4} - 2 q^{5} + 6 q^{6} + 16 q^{9} - 8 q^{13} + 4 q^{16} + 14 q^{20} + 50 q^{22} + 14 q^{23} - 50 q^{24} + 26 q^{25} - 300 q^{28} - 312 q^{30} + 24 q^{34} - 10 q^{35} + 38 q^{36} - 28 q^{38}+ \cdots - 4 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1 −1.96957 + 1.57068i 1.02419 − 0.233765i 0.967136 − 4.23730i 0.913760 + 1.14582i −1.65005 + 2.06910i −0.918266 − 4.02319i 2.56455 + 5.32534i −1.70858 + 0.822811i −3.59944 − 0.821548i
63.2 −1.87192 + 1.49280i 2.04869 − 0.467599i 0.830566 − 3.63895i 1.51379 + 1.89823i −3.13693 + 3.93359i 0.0841952 + 0.368883i 1.79981 + 3.73735i 1.27556 − 0.614275i −5.66736 − 1.29354i
63.3 −1.50785 + 1.20247i 0.472684 − 0.107887i 0.382638 − 1.67645i −1.83920 − 2.30628i −0.583007 + 0.731068i −0.579757 − 2.54008i −0.234672 − 0.487301i −2.49112 + 1.19966i 5.54647 + 1.26595i
63.4 −1.18774 + 0.947194i −3.03673 + 0.693113i 0.0685165 − 0.300190i −1.16875 − 1.46557i 2.95034 − 3.69961i 0.0763836 + 0.334658i −1.11534 − 2.31602i 6.03839 − 2.90794i 2.77636 + 0.633686i
63.5 −1.09008 + 0.869314i −0.0271356 + 0.00619353i −0.0124633 + 0.0546051i 0.714773 + 0.896297i 0.0241960 − 0.0303408i 0.0245796 + 0.107690i −1.24379 − 2.58275i −2.70221 + 1.30132i −1.55833 − 0.355678i
63.6 −0.814626 + 0.649642i −2.10286 + 0.479963i −0.203462 + 0.891425i −0.911707 − 1.14324i 1.40124 − 1.75710i 0.845778 + 3.70560i −1.31753 − 2.73588i 1.48874 − 0.716938i 1.48540 + 0.339033i
63.7 −0.726021 + 0.578982i 1.60882 − 0.367202i −0.253156 + 1.10915i 0.314685 + 0.394602i −0.955433 + 1.19807i 0.924068 + 4.04861i −1.26420 − 2.62515i −0.249447 + 0.120127i −0.456936 − 0.104293i
63.8 −0.0327942 + 0.0261525i −2.95182 + 0.673734i −0.444650 + 1.94814i 1.08614 + 1.36198i 0.0791828 − 0.0992921i −0.456981 − 2.00217i −0.0727657 − 0.151099i 5.55642 − 2.67583i −0.0712384 − 0.0162597i
63.9 0.0327942 − 0.0261525i 2.95182 − 0.673734i −0.444650 + 1.94814i 1.08614 + 1.36198i 0.0791828 − 0.0992921i −0.456981 − 2.00217i 0.0727657 + 0.151099i 5.55642 − 2.67583i 0.0712384 + 0.0162597i
63.10 0.726021 − 0.578982i −1.60882 + 0.367202i −0.253156 + 1.10915i 0.314685 + 0.394602i −0.955433 + 1.19807i 0.924068 + 4.04861i 1.26420 + 2.62515i −0.249447 + 0.120127i 0.456936 + 0.104293i
63.11 0.814626 − 0.649642i 2.10286 − 0.479963i −0.203462 + 0.891425i −0.911707 − 1.14324i 1.40124 − 1.75710i 0.845778 + 3.70560i 1.31753 + 2.73588i 1.48874 − 0.716938i −1.48540 − 0.339033i
63.12 1.09008 − 0.869314i 0.0271356 − 0.00619353i −0.0124633 + 0.0546051i 0.714773 + 0.896297i 0.0241960 − 0.0303408i 0.0245796 + 0.107690i 1.24379 + 2.58275i −2.70221 + 1.30132i 1.55833 + 0.355678i
63.13 1.18774 − 0.947194i 3.03673 − 0.693113i 0.0685165 − 0.300190i −1.16875 − 1.46557i 2.95034 − 3.69961i 0.0763836 + 0.334658i 1.11534 + 2.31602i 6.03839 − 2.90794i −2.77636 − 0.633686i
63.14 1.50785 − 1.20247i −0.472684 + 0.107887i 0.382638 − 1.67645i −1.83920 − 2.30628i −0.583007 + 0.731068i −0.579757 − 2.54008i 0.234672 + 0.487301i −2.49112 + 1.19966i −5.54647 − 1.26595i
63.15 1.87192 − 1.49280i −2.04869 + 0.467599i 0.830566 − 3.63895i 1.51379 + 1.89823i −3.13693 + 3.93359i 0.0841952 + 0.368883i −1.79981 − 3.73735i 1.27556 − 0.614275i 5.66736 + 1.29354i
63.16 1.96957 − 1.57068i −1.02419 + 0.233765i 0.967136 − 4.23730i 0.913760 + 1.14582i −1.65005 + 2.06910i −0.918266 − 4.02319i −2.56455 − 5.32534i −1.70858 + 0.822811i 3.59944 + 0.821548i
196.1 −2.45602 + 0.560570i −0.455808 + 0.946496i 3.91586 − 1.88578i −0.326117 − 1.42881i 0.588897 − 2.58012i −3.71798 − 1.79049i −4.62116 + 3.68525i 1.18238 + 1.48265i 1.60190 + 3.32638i
196.2 −2.33424 + 0.532775i −0.911751 + 1.89327i 3.36289 − 1.61948i −0.540264 − 2.36705i 1.11956 − 4.90511i 0.340899 + 0.164169i −3.24315 + 2.58633i −0.882711 − 1.10689i 2.52221 + 5.23743i
196.3 −1.88026 + 0.429157i −0.210364 + 0.436826i 1.54927 − 0.746089i 0.656401 + 2.87588i 0.208073 − 0.911627i −2.34739 − 1.13044i 0.422864 − 0.337223i 1.72391 + 2.16171i −2.46841 − 5.12571i
196.4 −1.48109 + 0.338050i 1.35147 − 2.80636i 0.277417 − 0.133597i 0.417123 + 1.82754i −1.05296 + 4.61334i 0.309271 + 0.148937i 2.00977 − 1.60274i −4.17870 − 5.23992i −1.23560 − 2.56574i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner
29.d even 7 5 inner
29.e even 14 5 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 841.2.e.m 96
29.b even 2 1 inner 841.2.e.m 96
29.c odd 4 1 841.2.d.p 48
29.c odd 4 1 841.2.d.q 48
29.d even 7 1 841.2.b.f 16
29.d even 7 5 inner 841.2.e.m 96
29.e even 14 1 841.2.b.f 16
29.e even 14 5 inner 841.2.e.m 96
29.f odd 28 1 841.2.a.i ✓ 8
29.f odd 28 1 841.2.a.j yes 8
29.f odd 28 5 841.2.d.p 48
29.f odd 28 5 841.2.d.q 48
87.k even 28 1 7569.2.a.bd 8
87.k even 28 1 7569.2.a.bi 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
841.2.a.i ✓ 8 29.f odd 28 1
841.2.a.j yes 8 29.f odd 28 1
841.2.b.f 16 29.d even 7 1
841.2.b.f 16 29.e even 14 1
841.2.d.p 48 29.c odd 4 1
841.2.d.p 48 29.f odd 28 5
841.2.d.q 48 29.c odd 4 1
841.2.d.q 48 29.f odd 28 5
841.2.e.m 96 1.a even 1 1 trivial
841.2.e.m 96 29.b even 2 1 inner
841.2.e.m 96 29.d even 7 5 inner
841.2.e.m 96 29.e even 14 5 inner
7569.2.a.bd 8 87.k even 28 1
7569.2.a.bi 8 87.k even 28 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 22 T_{2}^{94} + 291 T_{2}^{92} - 3025 T_{2}^{90} + 27345 T_{2}^{88} - 226083 T_{2}^{86} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(841, [\chi])\). Copy content Toggle raw display