↑

Properties

Label 616.2.cf.a
Level $616$
Weight $2$
Character orbit 616.cf
Analytic conductor $4.919$
Analytic rank $0$
Dimension $736$
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [616,2,Mod(51,616)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(616, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 15, 10, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("616.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 616.cf (of order \(30\), 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: \(4.91878476451\)
Analytic rank: \(0\)
Dimension: \(736\)
Relative dimension: \(92\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 736 q - 5 q^{2} - 6 q^{3} - 5 q^{4} - 20 q^{6} - 20 q^{8} + 78 q^{9} - 4 q^{11} - 20 q^{12} - 2 q^{14} - 17 q^{16} - 10 q^{17} - 30 q^{18} - 10 q^{19} + 12 q^{20} - 52 q^{22} - 35 q^{24} - 82 q^{25} + 9 q^{26}+ \cdots + 80 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −1.41169 − 0.0844492i −2.99930 + 0.637521i 1.98574 + 0.238432i 0.695182 − 0.0730666i 4.28792 − 0.646693i −1.86517 + 1.87647i −2.78311 − 0.504286i 5.84872 − 2.60402i −0.987552 + 0.0444397i
51.2 −1.40999 − 0.109229i 1.97246 − 0.419258i 1.97614 + 0.308025i −1.48697 + 0.156287i −2.82694 + 0.375699i −2.02544 + 1.70223i −2.75269 − 0.650164i 0.974165 − 0.433726i 2.11368 − 0.0579419i
51.3 −1.40799 + 0.132557i 1.67650 − 0.356351i 1.96486 − 0.373278i −3.32660 + 0.349640i −2.31326 + 0.723971i −0.234433 − 2.63534i −2.71701 + 0.786027i −0.0569647 + 0.0253623i 4.63746 − 0.933254i
51.4 −1.40659 − 0.146647i 0.680000 − 0.144539i 1.95699 + 0.412543i 1.47432 − 0.154957i −0.977678 + 0.103587i 2.64484 − 0.0693628i −2.69218 − 0.867265i −2.29913 + 1.02364i −2.09649 + 0.00175716i
51.5 −1.38602 − 0.280983i 2.48521 − 0.528248i 1.84210 + 0.778894i 4.09963 − 0.430888i −3.59298 + 0.0338608i −1.05533 + 2.42617i −2.33433 − 1.59716i 3.15660 − 1.40541i −5.80323 − 0.554704i
51.6 −1.37688 + 0.322818i −0.588994 + 0.125195i 1.79158 − 0.888960i 3.61881 − 0.380353i 0.770558 − 0.362515i −1.79507 − 1.94364i −2.17981 + 1.80234i −2.40940 + 1.07273i −4.85987 + 1.69192i
51.7 −1.37415 − 0.334228i −2.36635 + 0.502983i 1.77658 + 0.918559i −2.99460 + 0.314746i 3.41983 + 0.0997248i 2.26166 − 1.37292i −2.13429 − 1.85602i 2.60598 − 1.16026i 4.22024 + 0.568372i
51.8 −1.36302 − 0.377057i −0.0163400 + 0.00347317i 1.71566 + 1.02787i 0.739653 − 0.0777407i 0.0235813 + 0.00142710i 1.14899 − 2.38324i −1.95091 − 2.04791i −2.74038 + 1.22010i −1.03748 − 0.172929i
51.9 −1.35707 − 0.397932i −1.83873 + 0.390834i 1.68330 + 1.08005i −0.578730 + 0.0608270i 2.65082 + 0.201298i −2.11729 − 1.58654i −1.85458 − 2.13554i 0.487535 − 0.217065i 0.809584 + 0.147748i
51.10 −1.34805 + 0.427497i −0.900599 + 0.191428i 1.63449 − 1.15258i −1.83379 + 0.192739i 1.13222 − 0.643058i −1.09808 + 2.40712i −1.71066 + 2.25247i −1.96620 + 0.875410i 2.38965 − 1.04376i
51.11 −1.34434 + 0.439028i −0.915055 + 0.194501i 1.61451 − 1.18041i 0.860030 − 0.0903928i 1.14476 − 0.663211i 1.99733 + 1.73513i −1.65222 + 2.29569i −1.94114 + 0.864252i −1.11649 + 0.499096i
51.12 −1.32992 + 0.480938i 1.19186 − 0.253337i 1.53740 − 1.27922i 1.40132 − 0.147284i −1.46324 + 0.910130i −2.47696 − 0.929875i −1.42939 + 2.44066i −1.38429 + 0.616326i −1.79281 + 0.869824i
51.13 −1.29648 + 0.564935i −3.09335 + 0.657512i 1.36170 − 1.46485i 2.66505 − 0.280108i 3.63900 − 2.59999i 2.55037 − 0.703989i −0.937862 + 2.66841i 6.39587 − 2.84762i −3.29693 + 1.86873i
51.14 −1.29182 − 0.575508i −0.233347 + 0.0495995i 1.33758 + 1.48690i −3.64337 + 0.382933i 0.329987 + 0.0702199i −0.115064 + 2.64325i −0.872184 − 2.69059i −2.68865 + 1.19706i 4.92694 + 1.60211i
51.15 −1.26003 − 0.642133i 3.23791 − 0.688239i 1.17533 + 1.61821i −1.15744 + 0.121652i −4.52179 − 1.21197i 2.59490 + 0.516239i −0.441839 − 2.79370i 7.26976 − 3.23670i 1.53652 + 0.589946i
51.16 −1.24599 + 0.668957i 3.19991 − 0.680162i 1.10499 − 1.66703i 1.44966 − 0.152366i −3.53207 + 2.98808i 0.628560 − 2.57000i −0.261644 + 2.81630i 7.03618 − 3.13271i −1.70434 + 1.15961i
51.17 −1.19993 − 0.748444i 2.53079 − 0.537936i 0.879663 + 1.79616i −1.22485 + 0.128737i −3.43938 − 1.24867i −1.61622 − 2.09472i 0.288793 − 2.81365i 3.37488 − 1.50259i 1.56609 + 0.762258i
51.18 −1.19739 + 0.752505i −1.95444 + 0.415428i 0.867474 − 1.80208i −3.14392 + 0.330439i 2.02761 − 1.96815i −2.19655 − 1.47484i 0.317371 + 2.81057i 0.906602 − 0.403645i 3.51583 − 2.76148i
51.19 −1.14655 − 0.827899i −2.23736 + 0.475565i 0.629168 + 1.89846i 3.57136 − 0.375365i 2.95897 + 1.30704i 2.47613 − 0.932082i 0.850357 − 2.69757i 2.03896 − 0.907805i −4.40552 − 2.52635i
51.20 −1.13554 + 0.842936i 0.664577 − 0.141260i 0.578919 − 1.91438i −2.98592 + 0.313833i −0.635583 + 0.720603i 2.14808 − 1.54459i 0.956312 + 2.66185i −2.31893 + 1.03245i 3.12610 − 2.87331i
See next 80 embeddings (of 736 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.d odd 2 1 inner
11.d odd 10 1 inner
56.k odd 6 1 inner
77.o odd 30 1 inner
88.k even 10 1 inner
616.cf even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 616.2.cf.a ✓ 736
7.c even 3 1 inner 616.2.cf.a ✓ 736
8.d odd 2 1 inner 616.2.cf.a ✓ 736
11.d odd 10 1 inner 616.2.cf.a ✓ 736
56.k odd 6 1 inner 616.2.cf.a ✓ 736
77.o odd 30 1 inner 616.2.cf.a ✓ 736
88.k even 10 1 inner 616.2.cf.a ✓ 736
616.cf even 30 1 inner 616.2.cf.a ✓ 736
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.cf.a ✓ 736 1.a even 1 1 trivial
616.2.cf.a ✓ 736 7.c even 3 1 inner
616.2.cf.a ✓ 736 8.d odd 2 1 inner
616.2.cf.a ✓ 736 11.d odd 10 1 inner
616.2.cf.a ✓ 736 56.k odd 6 1 inner
616.2.cf.a ✓ 736 77.o odd 30 1 inner
616.2.cf.a ✓ 736 88.k even 10 1 inner
616.2.cf.a ✓ 736 616.cf even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(616, [\chi])\).