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Properties

Label 494.2.h.d
Level $494$
Weight $2$
Character orbit 494.h
Analytic conductor $3.945$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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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] = [494,2,Mod(315,494)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(494, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("494.315"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 494 = 2 \cdot 13 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 494.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,22,0,22,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.94460985985\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 22 q^{2} + 22 q^{4} - 2 q^{5} - 5 q^{7} + 22 q^{8} + 10 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{13} - 5 q^{14} - 3 q^{15} + 22 q^{16} + q^{17} + 10 q^{18} + 12 q^{19} - 2 q^{20} + q^{21} - 2 q^{22}+ \cdots - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
315.1 1.00000 −2.68823 1.00000 −1.17933 + 2.04265i −2.68823 −0.535524 + 0.927555i 1.00000 4.22658 −1.17933 + 2.04265i
315.2 1.00000 −2.31319 1.00000 0.965984 − 1.67313i −2.31319 −2.34460 + 4.06096i 1.00000 2.35083 0.965984 − 1.67313i
315.3 1.00000 −2.03309 1.00000 −0.408503 + 0.707549i −2.03309 1.37607 − 2.38343i 1.00000 1.13347 −0.408503 + 0.707549i
315.4 1.00000 −1.38760 1.00000 1.40461 − 2.43285i −1.38760 1.16976 − 2.02609i 1.00000 −1.07457 1.40461 − 2.43285i
315.5 1.00000 −0.950705 1.00000 −0.586201 + 1.01533i −0.950705 −1.23430 + 2.13788i 1.00000 −2.09616 −0.586201 + 1.01533i
315.6 1.00000 0.403363 1.00000 −2.03730 + 3.52871i 0.403363 −0.716626 + 1.24123i 1.00000 −2.83730 −2.03730 + 3.52871i
315.7 1.00000 0.831715 1.00000 0.104192 − 0.180466i 0.831715 1.81134 − 3.13733i 1.00000 −2.30825 0.104192 − 0.180466i
315.8 1.00000 1.40196 1.00000 2.07009 − 3.58550i 1.40196 −1.49255 + 2.58517i 1.00000 −1.03450 2.07009 − 3.58550i
315.9 1.00000 1.68589 1.00000 0.731502 − 1.26700i 1.68589 0.461554 − 0.799435i 1.00000 −0.157773 0.731502 − 1.26700i
315.10 1.00000 2.37163 1.00000 −0.834227 + 1.44492i 2.37163 1.15324 − 1.99746i 1.00000 2.62465 −0.834227 + 1.44492i
315.11 1.00000 2.67825 1.00000 −1.23081 + 2.13183i 2.67825 −2.14836 + 3.72108i 1.00000 4.17302 −1.23081 + 2.13183i
425.1 1.00000 −2.68823 1.00000 −1.17933 − 2.04265i −2.68823 −0.535524 − 0.927555i 1.00000 4.22658 −1.17933 − 2.04265i
425.2 1.00000 −2.31319 1.00000 0.965984 + 1.67313i −2.31319 −2.34460 − 4.06096i 1.00000 2.35083 0.965984 + 1.67313i
425.3 1.00000 −2.03309 1.00000 −0.408503 − 0.707549i −2.03309 1.37607 + 2.38343i 1.00000 1.13347 −0.408503 − 0.707549i
425.4 1.00000 −1.38760 1.00000 1.40461 + 2.43285i −1.38760 1.16976 + 2.02609i 1.00000 −1.07457 1.40461 + 2.43285i
425.5 1.00000 −0.950705 1.00000 −0.586201 − 1.01533i −0.950705 −1.23430 − 2.13788i 1.00000 −2.09616 −0.586201 − 1.01533i
425.6 1.00000 0.403363 1.00000 −2.03730 − 3.52871i 0.403363 −0.716626 − 1.24123i 1.00000 −2.83730 −2.03730 − 3.52871i
425.7 1.00000 0.831715 1.00000 0.104192 + 0.180466i 0.831715 1.81134 + 3.13733i 1.00000 −2.30825 0.104192 + 0.180466i
425.8 1.00000 1.40196 1.00000 2.07009 + 3.58550i 1.40196 −1.49255 − 2.58517i 1.00000 −1.03450 2.07009 + 3.58550i
425.9 1.00000 1.68589 1.00000 0.731502 + 1.26700i 1.68589 0.461554 + 0.799435i 1.00000 −0.157773 0.731502 + 1.26700i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 315.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
247.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 494.2.h.d yes 22
13.c even 3 1 494.2.e.d ✓ 22
19.c even 3 1 494.2.e.d ✓ 22
247.h even 3 1 inner 494.2.h.d yes 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.e.d ✓ 22 13.c even 3 1
494.2.e.d ✓ 22 19.c even 3 1
494.2.h.d yes 22 1.a even 1 1 trivial
494.2.h.d yes 22 247.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(494, [\chi])\):

\( T_{3}^{11} - 19T_{3}^{9} + T_{3}^{8} + 131T_{3}^{7} - 16T_{3}^{6} - 400T_{3}^{5} + 83T_{3}^{4} + 525T_{3}^{3} - 151T_{3}^{2} - 228T_{3} + 84 \) Copy content Toggle raw display
\( T_{5}^{22} + 2 T_{5}^{21} + 34 T_{5}^{20} + 66 T_{5}^{19} + 760 T_{5}^{18} + 1429 T_{5}^{17} + \cdots + 67081 \) Copy content Toggle raw display