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Properties

Label 588.2.e.f
Level $588$
Weight $2$
Character orbit 588.e
Analytic conductor $4.695$
Analytic rank $0$
Dimension $24$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,2,Mod(491,588)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) 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("588.491"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{16} - 12 q^{18} - 24 q^{25} - 48 q^{30} + 12 q^{36} + 72 q^{46} - 24 q^{57} + 72 q^{58} + 72 q^{60} - 48 q^{64} + 108 q^{72} - 24 q^{78} - 24 q^{81} - 48 q^{85} - 48 q^{88} + 48 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1 −1.29871 − 0.559784i −1.60392 − 0.653796i 1.37328 + 1.45399i 3.97283i 1.71704 + 1.74694i 0 −0.969574 − 2.65705i 2.14510 + 2.09727i 2.22392 − 5.15954i
491.2 −1.29871 − 0.559784i 1.60392 + 0.653796i 1.37328 + 1.45399i − 3.97283i −1.71704 − 1.74694i 0 −0.969574 − 2.65705i 2.14510 + 2.09727i −2.22392 + 5.15954i
491.3 −1.29871 + 0.559784i −1.60392 + 0.653796i 1.37328 − 1.45399i − 3.97283i 1.71704 − 1.74694i 0 −0.969574 + 2.65705i 2.14510 − 2.09727i 2.22392 + 5.15954i
491.4 −1.29871 + 0.559784i 1.60392 − 0.653796i 1.37328 − 1.45399i 3.97283i −1.71704 + 1.74694i 0 −0.969574 + 2.65705i 2.14510 − 2.09727i −2.22392 − 5.15954i
491.5 −1.05533 − 0.941421i −0.406768 − 1.68361i 0.227452 + 1.98702i 1.25365i −1.15571 + 2.15971i 0 1.63059 − 2.31110i −2.66908 + 1.36968i 1.18022 − 1.32302i
491.6 −1.05533 − 0.941421i 0.406768 + 1.68361i 0.227452 + 1.98702i − 1.25365i 1.15571 − 2.15971i 0 1.63059 − 2.31110i −2.66908 + 1.36968i −1.18022 + 1.32302i
491.7 −1.05533 + 0.941421i −0.406768 + 1.68361i 0.227452 − 1.98702i − 1.25365i −1.15571 − 2.15971i 0 1.63059 + 2.31110i −2.66908 − 1.36968i 1.18022 + 1.32302i
491.8 −1.05533 + 0.941421i 0.406768 − 1.68361i 0.227452 − 1.98702i 1.25365i 1.15571 + 2.15971i 0 1.63059 + 2.31110i −2.66908 − 1.36968i −1.18022 − 1.32302i
491.9 −0.446802 − 1.34178i −1.32740 + 1.11266i −1.60074 + 1.19902i − 0.803124i 2.08603 + 1.28394i 0 2.32403 + 1.61211i 0.523976 − 2.95389i −1.07761 + 0.358837i
491.10 −0.446802 − 1.34178i 1.32740 − 1.11266i −1.60074 + 1.19902i 0.803124i −2.08603 − 1.28394i 0 2.32403 + 1.61211i 0.523976 − 2.95389i 1.07761 − 0.358837i
491.11 −0.446802 + 1.34178i −1.32740 − 1.11266i −1.60074 − 1.19902i 0.803124i 2.08603 − 1.28394i 0 2.32403 − 1.61211i 0.523976 + 2.95389i −1.07761 − 0.358837i
491.12 −0.446802 + 1.34178i 1.32740 + 1.11266i −1.60074 − 1.19902i − 0.803124i −2.08603 + 1.28394i 0 2.32403 − 1.61211i 0.523976 + 2.95389i 1.07761 + 0.358837i
491.13 0.446802 − 1.34178i −1.32740 + 1.11266i −1.60074 − 1.19902i 0.803124i 0.899858 + 2.27821i 0 −2.32403 + 1.61211i 0.523976 − 2.95389i 1.07761 + 0.358837i
491.14 0.446802 − 1.34178i 1.32740 − 1.11266i −1.60074 − 1.19902i − 0.803124i −0.899858 − 2.27821i 0 −2.32403 + 1.61211i 0.523976 − 2.95389i −1.07761 − 0.358837i
491.15 0.446802 + 1.34178i −1.32740 − 1.11266i −1.60074 + 1.19902i − 0.803124i 0.899858 − 2.27821i 0 −2.32403 − 1.61211i 0.523976 + 2.95389i 1.07761 − 0.358837i
491.16 0.446802 + 1.34178i 1.32740 + 1.11266i −1.60074 + 1.19902i 0.803124i −0.899858 + 2.27821i 0 −2.32403 − 1.61211i 0.523976 + 2.95389i −1.07761 + 0.358837i
491.17 1.05533 − 0.941421i −0.406768 − 1.68361i 0.227452 − 1.98702i − 1.25365i −2.01426 − 1.39383i 0 −1.63059 − 2.31110i −2.66908 + 1.36968i −1.18022 − 1.32302i
491.18 1.05533 − 0.941421i 0.406768 + 1.68361i 0.227452 − 1.98702i 1.25365i 2.01426 + 1.39383i 0 −1.63059 − 2.31110i −2.66908 + 1.36968i 1.18022 + 1.32302i
491.19 1.05533 + 0.941421i −0.406768 + 1.68361i 0.227452 + 1.98702i 1.25365i −2.01426 + 1.39383i 0 −1.63059 + 2.31110i −2.66908 − 1.36968i −1.18022 + 1.32302i
491.20 1.05533 + 0.941421i 0.406768 − 1.68361i 0.227452 + 1.98702i − 1.25365i 2.01426 − 1.39383i 0 −1.63059 + 2.31110i −2.66908 − 1.36968i 1.18022 − 1.32302i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 491.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.e.f ✓ 24
3.b odd 2 1 inner 588.2.e.f ✓ 24
4.b odd 2 1 inner 588.2.e.f ✓ 24
7.b odd 2 1 inner 588.2.e.f ✓ 24
7.c even 3 1 588.2.n.d 24
7.c even 3 1 588.2.n.h 24
7.d odd 6 1 588.2.n.d 24
7.d odd 6 1 588.2.n.h 24
12.b even 2 1 inner 588.2.e.f ✓ 24
21.c even 2 1 inner 588.2.e.f ✓ 24
21.g even 6 1 588.2.n.d 24
21.g even 6 1 588.2.n.h 24
21.h odd 6 1 588.2.n.d 24
21.h odd 6 1 588.2.n.h 24
28.d even 2 1 inner 588.2.e.f ✓ 24
28.f even 6 1 588.2.n.d 24
28.f even 6 1 588.2.n.h 24
28.g odd 6 1 588.2.n.d 24
28.g odd 6 1 588.2.n.h 24
84.h odd 2 1 inner 588.2.e.f ✓ 24
84.j odd 6 1 588.2.n.d 24
84.j odd 6 1 588.2.n.h 24
84.n even 6 1 588.2.n.d 24
84.n even 6 1 588.2.n.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.e.f ✓ 24 1.a even 1 1 trivial
588.2.e.f ✓ 24 3.b odd 2 1 inner
588.2.e.f ✓ 24 4.b odd 2 1 inner
588.2.e.f ✓ 24 7.b odd 2 1 inner
588.2.e.f ✓ 24 12.b even 2 1 inner
588.2.e.f ✓ 24 21.c even 2 1 inner
588.2.e.f ✓ 24 28.d even 2 1 inner
588.2.e.f ✓ 24 84.h odd 2 1 inner
588.2.n.d 24 7.c even 3 1
588.2.n.d 24 7.d odd 6 1
588.2.n.d 24 21.g even 6 1
588.2.n.d 24 21.h odd 6 1
588.2.n.d 24 28.f even 6 1
588.2.n.d 24 28.g odd 6 1
588.2.n.d 24 84.j odd 6 1
588.2.n.d 24 84.n even 6 1
588.2.n.h 24 7.c even 3 1
588.2.n.h 24 7.d odd 6 1
588.2.n.h 24 21.g even 6 1
588.2.n.h 24 21.h odd 6 1
588.2.n.h 24 28.f even 6 1
588.2.n.h 24 28.g odd 6 1
588.2.n.h 24 84.j odd 6 1
588.2.n.h 24 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{6} + 18T_{5}^{4} + 36T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{6} - 42T_{11}^{4} + 480T_{11}^{2} - 1536 \) Copy content Toggle raw display
\( T_{13}^{6} - 48T_{13}^{4} + 576T_{13}^{2} - 392 \) Copy content Toggle raw display