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Properties

Label 498.2.f.b
Level $498$
Weight $2$
Character orbit 498.f
Analytic conductor $3.977$
Analytic rank $0$
Dimension $560$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [498,2,Mod(5,498)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(498, base_ring=CyclotomicField(82))
 
chi = DirichletCharacter(H, H._module([41, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("498.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 498 = 2 \cdot 3 \cdot 83 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 498.f (of order \(82\), degree \(40\), 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: \(3.97655002066\)
Analytic rank: \(0\)
Dimension: \(560\)
Relative dimension: \(14\) over \(\Q(\zeta_{82})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{82}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 560 q + 14 q^{2} + q^{3} - 14 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} + 14 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 560 q + 14 q^{2} + q^{3} - 14 q^{4} - 2 q^{5} - q^{6} + 2 q^{7} + 14 q^{8} - 7 q^{9} + 2 q^{10} + q^{12} - 2 q^{14} - 2 q^{15} - 14 q^{16} + 7 q^{18} - 2 q^{20} + 11 q^{21} - q^{24} - 16 q^{25} + 10 q^{27} + 2 q^{28} + 2 q^{30} - 2 q^{31} + 14 q^{32} + 9 q^{33} - 4 q^{35} - 7 q^{36} + 119 q^{39} + 2 q^{40} - 11 q^{42} + 2 q^{45} - 12 q^{47} - 40 q^{48} - 12 q^{49} + 16 q^{50} + 9 q^{51} - 2 q^{53} - 10 q^{54} - 2 q^{56} - 6 q^{57} - 2 q^{60} + 8 q^{61} + 2 q^{62} + 7 q^{63} - 14 q^{64} - 9 q^{66} - 117 q^{69} + 4 q^{70} + 4 q^{71} + 7 q^{72} - 569 q^{75} + 287 q^{77} - 201 q^{78} - 2 q^{80} - 159 q^{81} - 30 q^{83} - 112 q^{84} - 161 q^{87} - 14 q^{89} - 207 q^{90} + 54 q^{93} + 12 q^{94} - 492 q^{95} - q^{96} + 12 q^{98} - 108 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0.859570 − 0.511019i −1.66650 − 0.472006i 0.477720 − 0.878512i −0.170470 − 0.474639i −1.67367 + 0.445888i −1.87302 + 4.17811i −0.0383027 − 0.999266i 2.55442 + 1.57319i −0.389080 − 0.320872i
5.2 0.859570 − 0.511019i −1.59654 − 0.671621i 0.477720 − 0.878512i 0.154538 + 0.430279i −1.71554 + 0.238554i 1.70440 − 3.80198i −0.0383027 − 0.999266i 2.09785 + 2.14453i 0.352717 + 0.290883i
5.3 0.859570 − 0.511019i −1.27096 + 1.17671i 0.477720 − 0.878512i 0.810358 + 2.25628i −0.491161 + 1.66095i −0.0872382 + 0.194601i −0.0383027 − 0.999266i 0.230700 − 2.99112i 1.84956 + 1.52532i
5.4 0.859570 − 0.511019i −1.24557 + 1.20356i 0.477720 − 0.878512i −0.206927 − 0.576146i −0.455609 + 1.67105i 0.846494 − 1.88826i −0.0383027 − 0.999266i 0.102877 − 2.99824i −0.472289 − 0.389494i
5.5 0.859570 − 0.511019i −0.843297 + 1.51289i 0.477720 − 0.878512i −1.34702 − 3.75051i 0.0482451 + 1.73138i −1.95298 + 4.35649i −0.0383027 − 0.999266i −1.57770 − 2.55164i −3.07444 − 2.53547i
5.6 0.859570 − 0.511019i −0.839649 − 1.51492i 0.477720 − 0.878512i −0.104174 − 0.290052i −1.49589 − 0.873105i −0.341738 + 0.762310i −0.0383027 − 0.999266i −1.58998 + 2.54401i −0.237767 − 0.196085i
5.7 0.859570 − 0.511019i −0.538793 − 1.64612i 0.477720 − 0.878512i −1.39249 − 3.87712i −1.30433 − 1.13962i 0.354238 − 0.790194i −0.0383027 − 0.999266i −2.41940 + 1.77383i −3.17823 − 2.62107i
5.8 0.859570 − 0.511019i 0.676077 − 1.59465i 0.477720 − 0.878512i 1.01029 + 2.81294i −0.233763 − 1.71620i 0.839187 − 1.87196i −0.0383027 − 0.999266i −2.08584 − 2.15622i 2.30588 + 1.90164i
5.9 0.859570 − 0.511019i 0.761721 + 1.55556i 0.477720 − 0.878512i 0.287863 + 0.801496i 1.44968 + 0.947862i −0.810050 + 1.80697i −0.0383027 − 0.999266i −1.83956 + 2.36981i 0.657017 + 0.541838i
5.10 0.859570 − 0.511019i 1.24353 − 1.20567i 0.477720 − 0.878512i −0.439021 − 1.22237i 0.452781 − 1.67182i 0.453257 − 1.01107i −0.0383027 − 0.999266i 0.0927311 − 2.99857i −1.00202 − 0.826360i
5.11 0.859570 − 0.511019i 1.29747 + 1.14742i 0.477720 − 0.878512i 1.26013 + 3.50858i 1.70162 + 0.323251i 2.15171 − 4.79980i −0.0383027 − 0.999266i 0.366872 + 2.97748i 2.87612 + 2.37192i
5.12 0.859570 − 0.511019i 1.48237 + 0.895873i 0.477720 − 0.878512i −1.13740 − 3.16686i 1.73201 + 0.0125475i 0.320195 − 0.714254i −0.0383027 − 0.999266i 1.39482 + 2.65602i −2.59600 − 2.14091i
5.13 0.859570 − 0.511019i 1.69496 − 0.356507i 0.477720 − 0.878512i −0.369367 − 1.02843i 1.27476 − 1.17260i 0.174605 − 0.389488i −0.0383027 − 0.999266i 2.74581 − 1.20853i −0.843043 − 0.695252i
5.14 0.859570 − 0.511019i 1.72048 − 0.199887i 0.477720 − 0.878512i 0.985451 + 2.74379i 1.37672 − 1.05101i −1.99586 + 4.45213i −0.0383027 − 0.999266i 2.92009 − 0.687803i 2.24919 + 1.85489i
35.1 0.973695 − 0.227854i −1.73042 − 0.0751194i 0.896166 − 0.443720i −4.17771 − 0.320741i −1.70202 + 0.321139i 0.860019 + 3.12956i 0.771489 − 0.636242i 2.98871 + 0.259976i −4.14089 + 0.639601i
35.2 0.973695 − 0.227854i −1.64015 + 0.556696i 0.896166 − 0.443720i 0.658027 + 0.0505197i −1.47016 + 0.915766i −0.478443 − 1.74103i 0.771489 − 0.636242i 2.38018 − 1.82613i 0.652229 − 0.100743i
35.3 0.973695 − 0.227854i −1.51668 − 0.836465i 0.896166 − 0.443720i 2.92675 + 0.224700i −1.66738 − 0.468881i 1.19858 + 4.36158i 0.771489 − 0.636242i 1.60065 + 2.53730i 2.90096 − 0.448081i
35.4 0.973695 − 0.227854i −1.13007 − 1.31261i 0.896166 − 0.443720i 0.467578 + 0.0358981i −1.39942 − 1.02060i −0.241893 − 0.880237i 0.771489 − 0.636242i −0.445903 + 2.96668i 0.463458 − 0.0715854i
35.5 0.973695 − 0.227854i −1.06073 − 1.36925i 0.896166 − 0.443720i −2.53248 − 0.194430i −1.34482 − 1.09154i −0.942345 − 3.42914i 0.771489 − 0.636242i −0.749686 + 2.90482i −2.51016 + 0.387718i
35.6 0.973695 − 0.227854i −0.983871 + 1.42548i 0.896166 − 0.443720i 4.21220 + 0.323390i −0.633190 + 1.61216i −0.0735604 − 0.267682i 0.771489 − 0.636242i −1.06400 − 2.80498i 4.17509 − 0.644882i
See next 80 embeddings (of 560 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
249.f even 82 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 498.2.f.b yes 560
3.b odd 2 1 498.2.f.a ✓ 560
83.d odd 82 1 498.2.f.a ✓ 560
249.f even 82 1 inner 498.2.f.b yes 560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
498.2.f.a ✓ 560 3.b odd 2 1
498.2.f.a ✓ 560 83.d odd 82 1
498.2.f.b yes 560 1.a even 1 1 trivial
498.2.f.b yes 560 249.f even 82 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{560} + 2 T_{5}^{559} + 45 T_{5}^{558} + 84 T_{5}^{557} + 1241 T_{5}^{556} + 2370 T_{5}^{555} + \cdots + 45\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(498, [\chi])\). Copy content Toggle raw display