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Properties

Label 714.2.bj.a
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
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 16 q^{15} + 96 q^{16} + 8 q^{18} - 16 q^{21} + 16 q^{23} - 24 q^{33} + 16 q^{37} + 8 q^{39} - 12 q^{42} - 16 q^{43} - 8 q^{44} + 16 q^{46} - 64 q^{49} - 24 q^{51} + 8 q^{53} - 72 q^{54} + 96 q^{57} + 96 q^{59} - 16 q^{60} - 60 q^{63} - 16 q^{65} - 16 q^{70} - 16 q^{71} + 96 q^{73} + 36 q^{75} + 24 q^{77} - 8 q^{78} + 16 q^{79} + 16 q^{84} - 128 q^{85} + 36 q^{87} + 8 q^{88} - 144 q^{89} - 16 q^{91} + 20 q^{93} + 48 q^{94} - 16 q^{95} - 96 q^{99}+O(q^{100}) \) 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 −1.73178 − 0.0305309i −0.866025 − 0.500000i −0.529055 − 4.01858i −0.477709 + 1.66487i 0.401510 − 2.61511i −0.707107 + 0.707107i 2.99814 + 0.105746i −4.01858 − 0.529055i
59.2 0.258819 − 0.965926i −1.72826 − 0.114472i −0.866025 − 0.500000i 0.197533 + 1.50042i −0.557879 + 1.63975i −2.64338 − 0.111968i −0.707107 + 0.707107i 2.97379 + 0.395675i 1.50042 + 0.197533i
59.3 0.258819 − 0.965926i −1.60092 − 0.661110i −0.866025 − 0.500000i 0.138495 + 1.05197i −1.05293 + 1.37526i 1.85398 − 1.88752i −0.707107 + 0.707107i 2.12587 + 2.11676i 1.05197 + 0.138495i
59.4 0.258819 − 0.965926i −1.59901 + 0.665710i −0.866025 − 0.500000i 0.544358 + 4.13481i 0.229172 + 1.71682i 0.761460 + 2.53381i −0.707107 + 0.707107i 2.11366 − 2.12895i 4.13481 + 0.544358i
59.5 0.258819 − 0.965926i −1.45865 + 0.933989i −0.866025 − 0.500000i −0.0514662 − 0.390924i 0.524637 + 1.65068i 2.64243 + 0.132623i −0.707107 + 0.707107i 1.25533 − 2.72473i −0.390924 − 0.0514662i
59.6 0.258819 − 0.965926i −1.40621 − 1.01122i −0.866025 − 0.500000i −0.311358 − 2.36500i −1.34072 + 1.09657i 0.964260 + 2.46378i −0.707107 + 0.707107i 0.954867 + 2.84398i −2.36500 − 0.311358i
59.7 0.258819 − 0.965926i −1.18951 − 1.25899i −0.866025 − 0.500000i 0.355095 + 2.69721i −1.52396 + 0.823126i −2.11111 − 1.59475i −0.707107 + 0.707107i −0.170135 + 2.99517i 2.69721 + 0.355095i
59.8 0.258819 − 0.965926i −1.05408 + 1.37438i −0.866025 − 0.500000i −0.217209 − 1.64987i 1.05473 + 1.37388i −2.33944 − 1.23573i −0.707107 + 0.707107i −0.777838 − 2.89741i −1.64987 − 0.217209i
59.9 0.258819 − 0.965926i −0.871879 − 1.49661i −0.866025 − 0.500000i 0.175552 + 1.33345i −1.67127 + 0.454820i 0.899493 + 2.48815i −0.707107 + 0.707107i −1.47966 + 2.60972i 1.33345 + 0.175552i
59.10 0.258819 − 0.965926i −0.602718 + 1.62380i −0.866025 − 0.500000i 0.106633 + 0.809960i 1.41248 + 1.00245i 0.277443 − 2.63116i −0.707107 + 0.707107i −2.27346 − 1.95739i 0.809960 + 0.106633i
59.11 0.258819 − 0.965926i −0.175135 + 1.72317i −0.866025 − 0.500000i −0.515450 − 3.91523i 1.61913 + 0.615158i 2.01502 + 1.71456i −0.707107 + 0.707107i −2.93866 − 0.603576i −3.91523 − 0.515450i
59.12 0.258819 − 0.965926i 0.0296224 − 1.73180i −0.866025 − 0.500000i −0.0719386 − 0.546428i −1.66512 − 0.476835i 0.808048 − 2.51934i −0.707107 + 0.707107i −2.99825 − 0.102600i −0.546428 − 0.0719386i
59.13 0.258819 − 0.965926i 0.368990 + 1.69229i −0.866025 − 0.500000i 0.158595 + 1.20465i 1.73013 + 0.0815800i −2.03502 + 1.69077i −0.707107 + 0.707107i −2.72769 + 1.24888i 1.20465 + 0.158595i
59.14 0.258819 − 0.965926i 0.421176 − 1.68006i −0.866025 − 0.500000i 0.335130 + 2.54557i −1.51381 − 0.841657i −1.93361 + 1.80586i −0.707107 + 0.707107i −2.64522 − 1.41521i 2.54557 + 0.335130i
59.15 0.258819 − 0.965926i 0.448690 + 1.67292i −0.866025 − 0.500000i 0.383140 + 2.91023i 1.73205 0.000416401i 2.61878 − 0.376791i −0.707107 + 0.707107i −2.59735 + 1.50125i 2.91023 + 0.383140i
59.16 0.258819 − 0.965926i 0.622504 − 1.61632i −0.866025 − 0.500000i −0.290096 − 2.20350i −1.40013 − 1.01963i 1.85562 + 1.88592i −0.707107 + 0.707107i −2.22498 − 2.01233i −2.20350 − 0.290096i
59.17 0.258819 − 0.965926i 0.924679 − 1.46457i −0.866025 − 0.500000i −0.494322 − 3.75475i −1.17534 − 1.27223i −2.58718 + 0.553618i −0.707107 + 0.707107i −1.28994 − 2.70852i −3.75475 − 0.494322i
59.18 0.258819 − 0.965926i 1.16464 + 1.28203i −0.866025 − 0.500000i −0.197975 − 1.50377i 1.53978 − 0.793143i −0.829648 + 2.51231i −0.707107 + 0.707107i −0.287218 + 2.98622i −1.50377 − 0.197975i
59.19 0.258819 − 0.965926i 1.36619 − 1.06468i −0.866025 − 0.500000i 0.370197 + 2.81193i −0.674804 − 1.59519i 2.35410 − 1.20756i −0.707107 + 0.707107i 0.732926 − 2.90909i 2.81193 + 0.370197i
59.20 0.258819 − 0.965926i 1.44599 + 0.953475i −0.866025 − 0.500000i −0.0556739 − 0.422885i 1.29524 − 1.14994i −2.16737 − 1.51739i −0.707107 + 0.707107i 1.18177 + 2.75743i −0.422885 − 0.0556739i
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.a ✓ 192
3.b odd 2 1 714.2.bj.b yes 192
7.d odd 6 1 inner 714.2.bj.a ✓ 192
17.d even 8 1 714.2.bj.b yes 192
21.g even 6 1 714.2.bj.b yes 192
51.g odd 8 1 inner 714.2.bj.a ✓ 192
119.r odd 24 1 714.2.bj.b yes 192
357.bj even 24 1 inner 714.2.bj.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.bj.a ✓ 192 1.a even 1 1 trivial
714.2.bj.a ✓ 192 7.d odd 6 1 inner
714.2.bj.a ✓ 192 51.g odd 8 1 inner
714.2.bj.a ✓ 192 357.bj even 24 1 inner
714.2.bj.b yes 192 3.b odd 2 1
714.2.bj.b yes 192 17.d even 8 1
714.2.bj.b yes 192 21.g even 6 1
714.2.bj.b yes 192 119.r odd 24 1

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