Properties

Label 722.2.k.b
Level $722$
Weight $2$
Character orbit 722.k
Analytic conductor $5.765$
Analytic rank $0$
Dimension $1728$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [722,2,Mod(5,722)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(342))
 
chi = DirichletCharacter(H, H._module([232]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("722.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.k (of order \(171\), degree \(108\), 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: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(1728\)
Relative dimension: \(16\) over \(\Q(\zeta_{171})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{171}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1728 q + 3 q^{3} + 3 q^{6} + 3 q^{7} + 48 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1728 q + 3 q^{3} + 3 q^{6} + 3 q^{7} + 48 q^{8} + 3 q^{9} + 3 q^{11} - 12 q^{13} - 12 q^{14} + 6 q^{15} - 45 q^{17} - 84 q^{18} - 18 q^{19} - 6 q^{20} - 24 q^{21} - 57 q^{22} - 45 q^{23} + 3 q^{24} - 6 q^{26} - 240 q^{27} + 6 q^{28} + 18 q^{29} - 6 q^{31} - 117 q^{33} + 12 q^{34} + 12 q^{35} + 3 q^{36} + 30 q^{37} - 105 q^{38} + 30 q^{39} - 3 q^{41} + 12 q^{42} - 108 q^{43} - 21 q^{45} - 3 q^{46} + 426 q^{47} - 6 q^{48} + 45 q^{49} + 39 q^{50} - 21 q^{51} + 6 q^{52} - 252 q^{53} - 9 q^{54} - 18 q^{55} - 6 q^{56} + 24 q^{57} - 12 q^{58} + 3 q^{59} + 6 q^{60} - 6 q^{61} - 18 q^{62} - 12 q^{63} + 48 q^{64} - 24 q^{65} - 3 q^{66} + 9 q^{67} - 114 q^{68} - 237 q^{69} - 216 q^{70} + 18 q^{71} - 6 q^{72} + 30 q^{73} - 39 q^{74} - 204 q^{75} + 6 q^{76} - 33 q^{77} + 18 q^{78} - 6 q^{79} + 489 q^{81} - 3 q^{82} - 237 q^{83} - 15 q^{84} + 24 q^{85} - 12 q^{86} - 39 q^{87} + 3 q^{88} - 12 q^{90} - 45 q^{91} - 6 q^{92} + 564 q^{93} + 36 q^{94} - 228 q^{95} - 3 q^{97} - 21 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.531476 + 0.847073i −1.74688 2.28744i −0.435066 0.900399i −0.524156 1.32529i 2.86606 0.264017i −0.661727 + 0.643738i 0.993931 + 0.110008i −1.39102 + 5.09761i 1.40119 + 0.260362i
5.2 −0.531476 + 0.847073i −1.68664 2.20855i −0.435066 0.900399i 0.858271 + 2.17007i 2.76721 0.254912i −0.102628 + 0.0998380i 0.993931 + 0.110008i −1.24319 + 4.55586i −2.29436 0.426325i
5.3 −0.531476 + 0.847073i −1.35956 1.78026i −0.435066 0.900399i 0.443582 + 1.12157i 2.23058 0.205478i 2.09133 2.03448i 0.993931 + 0.110008i −0.531167 + 1.94654i −1.18580 0.220339i
5.4 −0.531476 + 0.847073i −1.22182 1.59990i −0.435066 0.900399i −0.247493 0.625768i 2.00460 0.184662i −3.31650 + 3.22635i 0.993931 + 0.110008i −0.277084 + 1.01541i 0.661608 + 0.122936i
5.5 −0.531476 + 0.847073i −0.922253 1.20764i −0.435066 0.900399i −0.944641 2.38845i 1.51311 0.139386i 3.33037 3.23984i 0.993931 + 0.110008i 0.181925 0.666690i 2.52525 + 0.469227i
5.6 −0.531476 + 0.847073i −0.499659 0.654274i −0.435066 0.900399i −0.989872 2.50282i 0.819774 0.0755165i 0.742564 0.722377i 0.993931 + 0.110008i 0.611343 2.24035i 2.64616 + 0.491695i
5.7 −0.531476 + 0.847073i −0.370330 0.484925i −0.435066 0.900399i 1.26587 + 3.20067i 0.607588 0.0559702i 0.0459679 0.0447183i 0.993931 + 0.110008i 0.691750 2.53502i −3.38398 0.628792i
5.8 −0.531476 + 0.847073i −0.0835538 0.109409i −0.435066 0.900399i 0.639670 + 1.61736i 0.137084 0.0126280i −2.97508 + 2.89420i 0.993931 + 0.110008i 0.784769 2.87590i −1.70999 0.317740i
5.9 −0.531476 + 0.847073i 0.279342 + 0.365781i −0.435066 0.900399i 0.735163 + 1.85880i −0.458307 + 0.0422186i 2.96468 2.88408i 0.993931 + 0.110008i 0.733994 2.68982i −1.96527 0.365174i
5.10 −0.531476 + 0.847073i 0.438670 + 0.574412i −0.435066 0.900399i 0.193124 + 0.488301i −0.719711 + 0.0662989i 0.269022 0.261709i 0.993931 + 0.110008i 0.652240 2.39023i −0.516267 0.0959298i
5.11 −0.531476 + 0.847073i 0.768509 + 1.00632i −0.435066 0.900399i −1.07532 2.71886i −1.26087 + 0.116150i −2.64556 + 2.57364i 0.993931 + 0.110008i 0.367690 1.34745i 2.87457 + 0.534137i
5.12 −0.531476 + 0.847073i 0.899901 + 1.17837i −0.435066 0.900399i −0.586197 1.48216i −1.47644 + 0.136008i 0.0870356 0.0846696i 0.993931 + 0.110008i 0.211031 0.773354i 1.56704 + 0.291179i
5.13 −0.531476 + 0.847073i 1.47525 + 1.93175i −0.435066 0.900399i 1.05726 + 2.67321i −2.42040 + 0.222964i −1.80140 + 1.75243i 0.993931 + 0.110008i −0.765547 + 2.80546i −2.82631 0.525169i
5.14 −0.531476 + 0.847073i 1.55637 + 2.03798i −0.435066 0.900399i 0.345356 + 0.873206i −2.55349 + 0.235224i 3.65514 3.55578i 0.993931 + 0.110008i −0.941297 + 3.44952i −0.923217 0.171547i
5.15 −0.531476 + 0.847073i 1.58354 + 2.07356i −0.435066 0.900399i −1.42441 3.60152i −2.59807 + 0.239331i 1.09260 1.06290i 0.993931 + 0.110008i −1.00227 + 3.67296i 3.80780 + 0.707542i
5.16 −0.531476 + 0.847073i 1.96711 + 2.57582i −0.435066 0.900399i −0.113996 0.288231i −3.22738 + 0.297302i 0.284348 0.276618i 0.993931 + 0.110008i −1.97554 + 7.23965i 0.304739 + 0.0566248i
9.1 0.832107 0.554615i −2.95221 + 0.108524i 0.384804 0.922998i 0.183409 0.672127i −2.39637 + 1.72765i −2.37256 + 0.531704i −0.191711 0.981451i 5.71188 0.420509i −0.220156 0.661003i
9.2 0.832107 0.554615i −2.84570 + 0.104609i 0.384804 0.922998i 1.02364 3.75128i −2.30991 + 1.66531i 2.65546 0.595103i −0.191711 0.981451i 5.09516 0.375107i −1.22874 3.68919i
9.3 0.832107 0.554615i −2.36433 + 0.0869136i 0.384804 0.922998i −1.09705 + 4.02028i −1.91917 + 1.38362i 0.826170 0.185149i −0.191711 0.981451i 2.59061 0.190721i 1.31685 + 3.95374i
9.4 0.832107 0.554615i −1.40272 + 0.0515645i 0.384804 0.922998i −0.278167 + 1.01938i −1.13862 + 0.820879i −0.689106 + 0.154432i −0.191711 0.981451i −1.02693 + 0.0756027i 0.333900 + 1.00251i
See next 80 embeddings (of 1728 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
361.k even 171 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.k.b 1728
361.k even 171 1 inner 722.2.k.b 1728
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.k.b 1728 1.a even 1 1 trivial
722.2.k.b 1728 361.k even 171 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{1728} - 3 T_{3}^{1727} + 3 T_{3}^{1726} + 80 T_{3}^{1725} - 351 T_{3}^{1724} + 561 T_{3}^{1723} + 1127 T_{3}^{1722} - 4224 T_{3}^{1721} + 18387 T_{3}^{1720} - 146332 T_{3}^{1719} + 901479 T_{3}^{1718} - 1695279 T_{3}^{1717} + \cdots + 51\!\cdots\!89 \) acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\). Copy content Toggle raw display