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Properties

Label 714.2.bl.b
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
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 48 q^{10} + 16 q^{11} - 16 q^{14} - 32 q^{15} + 96 q^{18} + 32 q^{22} + 16 q^{25} - 32 q^{28} + 32 q^{29} + 80 q^{37} + 64 q^{43} - 16 q^{46} - 48 q^{47} + 64 q^{49} - 16 q^{53} + 16 q^{58} + 48 q^{61} + 64 q^{65} - 16 q^{70} - 96 q^{71} - 144 q^{73} + 80 q^{74} - 48 q^{77} - 64 q^{78} + 64 q^{79} + 48 q^{80} - 64 q^{85} - 32 q^{86} + 144 q^{87} + 16 q^{88} - 144 q^{89} + 128 q^{91} - 64 q^{92} + 144 q^{94} + 48 q^{95} + 80 q^{98} - 32 q^{99}+O(q^{100}) \) 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 −1.73452 + 3.51726i −0.980785 + 0.195090i −0.346709 + 2.62294i −0.382683 + 0.923880i −0.130526 − 0.991445i −3.51726 + 1.73452i
31.2 0.793353 + 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i −0.652532 + 1.32320i −0.980785 + 0.195090i 2.40416 + 1.10454i −0.382683 + 0.923880i −0.130526 − 0.991445i −1.32320 + 0.652532i
31.3 0.793353 + 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i −0.134690 + 0.273124i −0.980785 + 0.195090i −2.04674 − 1.67656i −0.382683 + 0.923880i −0.130526 − 0.991445i −0.273124 + 0.134690i
31.4 0.793353 + 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i 0.511074 − 1.03635i −0.980785 + 0.195090i −1.68570 − 2.03922i −0.382683 + 0.923880i −0.130526 − 0.991445i 1.03635 − 0.511074i
31.5 0.793353 + 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i 1.13476 − 2.30107i −0.980785 + 0.195090i −1.52489 + 2.16211i −0.382683 + 0.923880i −0.130526 − 0.991445i 2.30107 − 1.13476i
31.6 0.793353 + 0.608761i −0.659346 + 0.751840i 0.258819 + 0.965926i 1.38860 − 2.81580i −0.980785 + 0.195090i 2.01950 + 1.70927i −0.382683 + 0.923880i −0.130526 − 0.991445i 2.81580 − 1.38860i
31.7 0.793353 + 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i −1.75902 + 3.56694i 0.980785 − 0.195090i −2.58646 + 0.556975i −0.382683 + 0.923880i −0.130526 − 0.991445i −3.56694 + 1.75902i
31.8 0.793353 + 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i −0.519815 + 1.05408i 0.980785 − 0.195090i 0.635588 + 2.56827i −0.382683 + 0.923880i −0.130526 − 0.991445i −1.05408 + 0.519815i
31.9 0.793353 + 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i −0.399424 + 0.809951i 0.980785 − 0.195090i 2.56450 + 0.650635i −0.382683 + 0.923880i −0.130526 − 0.991445i −0.809951 + 0.399424i
31.10 0.793353 + 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i 0.262727 − 0.532759i 0.980785 − 0.195090i 0.186270 − 2.63919i −0.382683 + 0.923880i −0.130526 − 0.991445i 0.532759 − 0.262727i
31.11 0.793353 + 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i 1.69071 − 3.42843i 0.980785 − 0.195090i −2.54061 − 0.738435i −0.382683 + 0.923880i −0.130526 − 0.991445i 3.42843 − 1.69071i
31.12 0.793353 + 0.608761i 0.659346 − 0.751840i 0.258819 + 0.965926i 1.91454 − 3.88230i 0.980785 − 0.195090i 2.61346 + 0.412098i −0.382683 + 0.923880i −0.130526 − 0.991445i 3.88230 − 1.91454i
61.1 0.608761 + 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i −2.08960 − 0.709325i −0.555570 − 0.831470i 0.0330769 + 2.64554i −0.923880 + 0.382683i 0.991445 + 0.130526i −0.709325 − 2.08960i
61.2 0.608761 + 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i −1.68932 − 0.573447i −0.555570 − 0.831470i 2.64226 − 0.135964i −0.923880 + 0.382683i 0.991445 + 0.130526i −0.573447 − 1.68932i
61.3 0.608761 + 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i −0.708564 − 0.240525i −0.555570 − 0.831470i −2.34517 − 1.22482i −0.923880 + 0.382683i 0.991445 + 0.130526i −0.240525 − 0.708564i
61.4 0.608761 + 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i 0.958150 + 0.325248i −0.555570 − 0.831470i −2.64559 + 0.0294292i −0.923880 + 0.382683i 0.991445 + 0.130526i 0.325248 + 0.958150i
61.5 0.608761 + 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i 2.08197 + 0.706733i −0.555570 − 0.831470i 0.403772 − 2.61476i −0.923880 + 0.382683i 0.991445 + 0.130526i 0.706733 + 2.08197i
61.6 0.608761 + 0.793353i −0.997859 − 0.0654031i −0.258819 + 0.965926i 2.55712 + 0.868026i −0.555570 − 0.831470i 0.231892 + 2.63557i −0.923880 + 0.382683i 0.991445 + 0.130526i 0.868026 + 2.55712i
61.7 0.608761 + 0.793353i 0.997859 + 0.0654031i −0.258819 + 0.965926i −3.63788 − 1.23489i 0.555570 + 0.831470i 1.41963 − 2.23263i −0.923880 + 0.382683i 0.991445 + 0.130526i −1.23489 − 3.63788i
61.8 0.608761 + 0.793353i 0.997859 + 0.0654031i −0.258819 + 0.965926i −2.76442 − 0.938395i 0.555570 + 0.831470i −2.64560 + 0.0286968i −0.923880 + 0.382683i 0.991445 + 0.130526i −0.938395 − 2.76442i
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.b ✓ 192
7.d odd 6 1 inner 714.2.bl.b ✓ 192
17.e odd 16 1 inner 714.2.bl.b ✓ 192
119.s even 48 1 inner 714.2.bl.b ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.bl.b ✓ 192 1.a even 1 1 trivial
714.2.bl.b ✓ 192 7.d odd 6 1 inner
714.2.bl.b ✓ 192 17.e odd 16 1 inner
714.2.bl.b ✓ 192 119.s even 48 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{192} - 8 T_{5}^{190} - 156 T_{5}^{188} - 48 T_{5}^{187} - 5760 T_{5}^{186} + \cdots + 84\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\). Copy content Toggle raw display