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Properties

Label 714.2.bj.b
Level $714$
Weight $2$
Character orbit 714.bj
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(59,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([12, 4, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("714.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.bj (of order \(24\), degree \(8\), 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: \(24\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-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
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 32 q^{15} + 96 q^{16} + 8 q^{18} + 16 q^{21} - 16 q^{23} - 24 q^{33} + 16 q^{37} + 8 q^{39} - 36 q^{42} - 16 q^{43} + 8 q^{44} + 72 q^{45} + 16 q^{46} - 64 q^{49} - 24 q^{51} - 8 q^{53} - 96 q^{57} - 96 q^{59} + 8 q^{60} - 36 q^{63} + 16 q^{65} - 16 q^{70} + 16 q^{71} + 96 q^{73} + 36 q^{75} - 24 q^{77} + 72 q^{78} + 16 q^{79} + 16 q^{84} - 128 q^{85} - 108 q^{87} + 8 q^{88} + 144 q^{89} - 16 q^{91} - 44 q^{93} + 48 q^{94} + 16 q^{95} - 224 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −0.258819 + 0.965926i −1.72447 + 0.161911i −0.866025 − 0.500000i −0.106633 − 0.809960i 0.289931 − 1.70761i 0.277443 − 2.63116i 0.707107 − 0.707107i 2.94757 − 0.558419i 0.809960 + 0.106633i
59.2 −0.258819 + 0.965926i −1.70979 − 0.276823i −0.866025 − 0.500000i 0.515450 + 3.91523i 0.709916 − 1.57988i 2.01502 + 1.71456i 0.707107 − 0.707107i 2.84674 + 0.946616i −3.91523 − 0.515450i
59.3 −0.258819 + 0.965926i −1.60036 + 0.662446i −0.866025 − 0.500000i 0.217209 + 1.64987i −0.225669 − 1.71729i −2.33944 − 1.23573i 0.707107 − 0.707107i 2.12233 − 2.12031i −1.64987 − 0.217209i
59.4 −0.258819 + 0.965926i −1.53913 − 0.794414i −0.866025 − 0.500000i −0.158595 − 1.20465i 1.16570 − 1.28107i −2.03502 + 1.69077i 0.707107 − 0.707107i 1.73781 + 2.44541i 1.20465 + 0.158595i
59.5 −0.258819 + 0.965926i −1.49979 − 0.866386i −0.866025 − 0.500000i −0.383140 − 2.91023i 1.22504 − 1.22445i 2.61878 − 0.376791i 0.707107 − 0.707107i 1.49875 + 2.59880i 2.91023 + 0.383140i
59.6 −0.258819 + 0.965926i −1.27969 + 1.16721i −0.866025 − 0.500000i 0.0514662 + 0.390924i −0.796235 − 1.53818i 2.64243 + 0.132623i 0.707107 − 0.707107i 0.275218 − 2.98735i −0.390924 − 0.0514662i
59.7 −0.258819 + 0.965926i −1.05688 + 1.37223i −0.866025 − 0.500000i −0.544358 − 4.13481i −1.05193 − 1.37603i 0.761460 + 2.53381i 0.707107 − 0.707107i −0.766008 − 2.90056i 4.13481 + 0.544358i
59.8 −0.258819 + 0.965926i −0.936917 − 1.45677i −0.866025 − 0.500000i 0.197975 + 1.50377i 1.64963 − 0.527952i −0.829648 + 2.51231i 0.707107 − 0.707107i −1.24437 + 2.72975i −1.50377 − 0.197975i
59.9 −0.258819 + 0.965926i −0.546737 − 1.64350i −0.866025 − 0.500000i 0.0556739 + 0.422885i 1.72900 − 0.102739i −2.16737 − 1.51739i 0.707107 − 0.707107i −2.40216 + 1.79712i −0.422885 − 0.0556739i
59.10 −0.258819 + 0.965926i −0.420553 − 1.68022i −0.866025 − 0.500000i −0.486395 − 3.69453i 1.73181 + 0.0286494i −1.55974 − 2.13710i 0.707107 − 0.707107i −2.64627 + 1.41324i 3.69453 + 0.486395i
59.11 −0.258819 + 0.965926i −0.418727 + 1.68067i −0.866025 − 0.500000i 0.529055 + 4.01858i −1.51503 − 0.839450i 0.401510 − 2.61511i 0.707107 − 0.707107i −2.64933 − 1.40749i −4.01858 − 0.529055i
59.12 −0.258819 + 0.965926i −0.336736 + 1.69900i −0.866025 − 0.500000i −0.197533 − 1.50042i −1.55396 − 0.764996i −2.64338 − 0.111968i 0.707107 − 0.707107i −2.77322 − 1.14423i 1.50042 + 0.197533i
59.13 −0.258819 + 0.965926i 0.224236 + 1.71747i −0.866025 − 0.500000i −0.138495 − 1.05197i −1.71699 − 0.227920i 1.85398 − 1.88752i 0.707107 − 0.707107i −2.89944 + 0.770239i 1.05197 + 0.138495i
59.14 −0.258819 + 0.965926i 0.342079 − 1.69793i −0.866025 − 0.500000i 0.347164 + 2.63697i 1.55154 + 0.769880i 2.47085 − 0.946003i 0.707107 − 0.707107i −2.76596 − 1.16165i −2.63697 − 0.347164i
59.15 −0.258819 + 0.965926i 0.612809 + 1.62002i −0.866025 − 0.500000i 0.311358 + 2.36500i −1.72343 + 0.172636i 0.964260 + 2.46378i 0.707107 − 0.707107i −2.24893 + 1.98553i −2.36500 − 0.311358i
59.16 −0.258819 + 0.965926i 0.762057 − 1.55540i −0.866025 − 0.500000i 0.271033 + 2.05870i 1.30517 + 1.13866i −2.09182 − 1.61996i 0.707107 − 0.707107i −1.83854 − 2.37061i −2.05870 − 0.271033i
59.17 −0.258819 + 0.965926i 0.829792 − 1.52034i −0.866025 − 0.500000i −0.101619 − 0.771873i 1.25377 + 1.19501i 0.375334 + 2.61899i 0.707107 − 0.707107i −1.62289 − 2.52314i 0.771873 + 0.101619i
59.18 −0.258819 + 0.965926i 0.908228 + 1.47483i −0.866025 − 0.500000i −0.355095 − 2.69721i −1.65964 + 0.495567i −2.11111 − 1.59475i 0.707107 − 0.707107i −1.35025 + 2.67896i 2.69721 + 0.355095i
59.19 −0.258819 + 0.965926i 1.21995 + 1.22952i −0.866025 − 0.500000i −0.175552 − 1.33345i −1.50337 + 0.860159i 0.899493 + 2.48815i 0.707107 − 0.707107i −0.0234389 + 2.99991i 1.33345 + 0.175552i
59.20 −0.258819 + 0.965926i 1.38199 − 1.04408i −0.866025 − 0.500000i −0.370197 − 2.81193i 0.650813 + 1.60513i 2.35410 − 1.20756i 0.707107 − 0.707107i 0.819814 − 2.88581i 2.81193 + 0.370197i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
51.g odd 8 1 inner
357.bj even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.bj.b yes 192
3.b odd 2 1 714.2.bj.a ✓ 192
7.d odd 6 1 inner 714.2.bj.b yes 192
17.d even 8 1 714.2.bj.a ✓ 192
21.g even 6 1 714.2.bj.a ✓ 192
51.g odd 8 1 inner 714.2.bj.b yes 192
119.r odd 24 1 714.2.bj.a ✓ 192
357.bj even 24 1 inner 714.2.bj.b yes 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.bj.a ✓ 192 3.b odd 2 1
714.2.bj.a ✓ 192 17.d even 8 1
714.2.bj.a ✓ 192 21.g even 6 1
714.2.bj.a ✓ 192 119.r odd 24 1
714.2.bj.b yes 192 1.a even 1 1 trivial
714.2.bj.b yes 192 7.d odd 6 1 inner
714.2.bj.b yes 192 51.g odd 8 1 inner
714.2.bj.b yes 192 357.bj even 24 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{192} + 24 T_{5}^{187} + 856 T_{5}^{186} + 11904 T_{5}^{185} - 149626 T_{5}^{184} + \cdots + 41\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\). Copy content Toggle raw display