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Properties

Label 714.2.bl.a
Level $714$
Weight $2$
Character orbit 714.bl
Analytic conductor $5.701$
Analytic rank $0$
Dimension $192$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [714,2,Mod(31,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([0, 8, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("714.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.bl (of order \(48\), degree \(16\), 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: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(12\) over \(\Q(\zeta_{48})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 192 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 48 q^{10} + 16 q^{11} - 16 q^{14} + 32 q^{15} - 96 q^{18} + 32 q^{22} + 16 q^{25} + 32 q^{28} - 32 q^{29} + 64 q^{35} - 48 q^{37} - 64 q^{43} - 16 q^{46} + 48 q^{47} + 64 q^{49} + 16 q^{53} + 16 q^{58} + 48 q^{61} - 64 q^{65} + 16 q^{70} - 32 q^{71} + 240 q^{73} + 48 q^{74} + 16 q^{77} - 64 q^{78} - 64 q^{79} + 48 q^{80} - 64 q^{85} + 32 q^{86} + 144 q^{87} + 16 q^{88} + 144 q^{89} - 64 q^{91} + 48 q^{94} + 80 q^{95} - 80 q^{98} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −0.793353 − 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i −1.16153 + 2.35536i 0.980785 − 0.195090i 1.07493 − 2.41754i 0.382683 − 0.923880i −0.130526 − 0.991445i 2.35536 − 1.16153i
31.2 −0.793353 − 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i −0.360559 + 0.731140i 0.980785 − 0.195090i 2.34297 − 1.22902i 0.382683 − 0.923880i −0.130526 − 0.991445i 0.731140 − 0.360559i
31.3 −0.793353 − 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i −0.0355436 + 0.0720753i 0.980785 − 0.195090i −2.59947 − 0.492676i 0.382683 − 0.923880i −0.130526 − 0.991445i 0.0720753 − 0.0355436i
31.4 −0.793353 − 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i −0.00979683 + 0.0198660i 0.980785 − 0.195090i 0.596978 + 2.57752i 0.382683 − 0.923880i −0.130526 − 0.991445i 0.0198660 − 0.00979683i
31.5 −0.793353 − 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i 1.19451 − 2.42223i 0.980785 − 0.195090i −2.62745 − 0.310617i 0.382683 − 0.923880i −0.130526 − 0.991445i −2.42223 + 1.19451i
31.6 −0.793353 − 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i 1.56264 − 3.16873i 0.980785 − 0.195090i 2.34138 − 1.23204i 0.382683 − 0.923880i −0.130526 − 0.991445i −3.16873 + 1.56264i
31.7 −0.793353 − 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i −1.58464 + 3.21333i −0.980785 + 0.195090i −0.808165 − 2.51930i 0.382683 − 0.923880i −0.130526 − 0.991445i 3.21333 − 1.58464i
31.8 −0.793353 − 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i −0.928428 + 1.88267i −0.980785 + 0.195090i 2.18840 + 1.48691i 0.382683 − 0.923880i −0.130526 − 0.991445i 1.88267 − 0.928428i
31.9 −0.793353 − 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i 0.0475892 − 0.0965013i −0.980785 + 0.195090i −2.05854 − 1.66205i 0.382683 − 0.923880i −0.130526 − 0.991445i −0.0965013 + 0.0475892i
31.10 −0.793353 − 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i 0.121926 − 0.247241i −0.980785 + 0.195090i −1.88236 + 1.85922i 0.382683 − 0.923880i −0.130526 − 0.991445i −0.247241 + 0.121926i
31.11 −0.793353 − 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i 1.35865 − 2.75506i −0.980785 + 0.195090i −0.276034 + 2.63131i 0.382683 − 0.923880i −0.130526 − 0.991445i −2.75506 + 1.35865i
31.12 −0.793353 − 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i 1.49760 − 3.03683i −0.980785 + 0.195090i 1.91292 − 1.82777i 0.382683 − 0.923880i −0.130526 − 0.991445i −3.03683 + 1.49760i
61.1 −0.608761 − 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i −3.98166 − 1.35159i 0.555570 + 0.831470i 1.67414 + 2.04872i 0.923880 − 0.382683i 0.991445 + 0.130526i 1.35159 + 3.98166i
61.2 −0.608761 − 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i −2.03737 − 0.691594i 0.555570 + 0.831470i −2.38560 − 1.14407i 0.923880 − 0.382683i 0.991445 + 0.130526i 0.691594 + 2.03737i
61.3 −0.608761 − 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i −1.46767 − 0.498208i 0.555570 + 0.831470i −2.46893 + 0.950993i 0.923880 − 0.382683i 0.991445 + 0.130526i 0.498208 + 1.46767i
61.4 −0.608761 − 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i −0.619157 − 0.210176i 0.555570 + 0.831470i 0.818611 − 2.51592i 0.923880 − 0.382683i 0.991445 + 0.130526i 0.210176 + 0.619157i
61.5 −0.608761 − 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i 2.71295 + 0.920921i 0.555570 + 0.831470i −0.695280 + 2.55276i 0.923880 − 0.382683i 0.991445 + 0.130526i −0.920921 − 2.71295i
61.6 −0.608761 − 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i 3.00327 + 1.01947i 0.555570 + 0.831470i 2.54392 − 0.726955i 0.923880 − 0.382683i 0.991445 + 0.130526i −1.01947 − 3.00327i
61.7 −0.608761 − 0.793353i 0.997859 + 0.0654031i −0.258819 + 0.965926i −3.70938 − 1.25917i −0.555570 − 0.831470i −1.85824 − 1.88333i 0.923880 − 0.382683i 0.991445 + 0.130526i 1.25917 + 3.70938i
61.8 −0.608761 − 0.793353i 0.997859 + 0.0654031i −0.258819 + 0.965926i −1.25442 − 0.425817i −0.555570 − 0.831470i −1.07390 + 2.41800i 0.923880 − 0.382683i 0.991445 + 0.130526i 0.425817 + 1.25442i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
17.e odd 16 1 inner
119.s even 48 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.bl.a ✓ 192
7.d odd 6 1 inner 714.2.bl.a ✓ 192
17.e odd 16 1 inner 714.2.bl.a ✓ 192
119.s even 48 1 inner 714.2.bl.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.bl.a ✓ 192 1.a even 1 1 trivial
714.2.bl.a ✓ 192 7.d odd 6 1 inner
714.2.bl.a ✓ 192 17.e odd 16 1 inner
714.2.bl.a ✓ 192 119.s even 48 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{192} - 8 T_{5}^{190} + 4 T_{5}^{188} + 48 T_{5}^{187} + 7680 T_{5}^{186} - 21360 T_{5}^{185} + \cdots + 62\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\). Copy content Toggle raw display