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Properties

Label 272.2.l.c
Level $272$
Weight $2$
Character orbit 272.l
Analytic conductor $2.172$
Analytic rank $0$
Dimension $32$
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] = [272,2,Mod(69,272)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(272, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("272.69");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 272.l (of order \(4\), degree \(2\), 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: \(2.17193093498\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{4} - 16 q^{6} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{4} - 16 q^{6} - 18 q^{8} - 6 q^{10} - 4 q^{11} + 2 q^{12} + 14 q^{14} + 24 q^{15} + 26 q^{16} + 32 q^{17} + 10 q^{18} - 14 q^{20} - 8 q^{22} - 50 q^{24} - 6 q^{26} + 12 q^{27} - 8 q^{29} + 36 q^{30} - 24 q^{31} + 10 q^{32} + 12 q^{35} + 16 q^{36} - 8 q^{37} - 34 q^{38} - 38 q^{40} - 12 q^{42} - 12 q^{43} + 38 q^{44} - 40 q^{45} + 26 q^{46} + 32 q^{47} + 30 q^{48} - 32 q^{49} - 2 q^{50} - 54 q^{52} + 8 q^{53} - 52 q^{54} - 36 q^{56} - 26 q^{58} - 12 q^{59} + 28 q^{60} + 6 q^{62} - 40 q^{63} + 10 q^{64} + 28 q^{66} + 8 q^{67} - 2 q^{68} - 32 q^{70} - 32 q^{72} - 14 q^{74} + 8 q^{75} + 30 q^{76} + 56 q^{77} + 108 q^{78} + 48 q^{79} + 90 q^{80} - 64 q^{81} + 32 q^{82} + 4 q^{83} - 8 q^{84} - 54 q^{86} - 66 q^{88} - 66 q^{90} - 28 q^{91} + 24 q^{92} - 24 q^{93} + 12 q^{94} - 40 q^{95} + 70 q^{96} + 12 q^{98} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1 −1.40483 + 0.162642i −2.37297 + 2.37297i 1.94710 − 0.456969i −0.208645 − 0.208645i 2.94767 − 3.71956i − 3.10686i −2.66102 + 0.958643i − 8.26197i 0.327046 + 0.259177i
69.2 −1.38688 − 0.276678i 2.03099 − 2.03099i 1.84690 + 0.767442i 1.79233 + 1.79233i −3.37869 + 2.25482i − 4.22080i −2.34910 − 1.57535i − 5.24987i −1.98986 − 2.98166i
69.3 −1.37038 + 0.349388i 0.724061 − 0.724061i 1.75586 − 0.957585i 0.100057 + 0.100057i −0.739257 + 1.24521i 2.62804i −2.07161 + 1.92573i 1.95147i −0.172075 − 0.102157i
69.4 −1.20363 − 0.742481i 1.79938 − 1.79938i 0.897443 + 1.78734i −2.01194 − 2.01194i −3.50178 + 0.829777i 1.70564i 0.246881 − 2.81763i − 3.47550i 0.927799 + 3.91545i
69.5 −0.974409 + 1.02495i −1.15555 + 1.15555i −0.101055 − 1.99745i 1.52867 + 1.52867i −0.0584053 − 2.31035i 1.16151i 2.14576 + 1.84275i 0.329430i −3.05636 + 0.0772643i
69.6 −0.241180 − 1.39350i 0.142166 − 0.142166i −1.88366 + 0.672166i 1.73453 + 1.73453i −0.232396 − 0.163821i 3.44232i 1.39096 + 2.46277i 2.95958i 1.99873 − 2.83540i
69.7 −0.193744 + 1.40088i −1.42790 + 1.42790i −1.92493 − 0.542825i −0.427384 − 0.427384i −1.72366 − 2.27696i − 3.74015i 1.13338 − 2.59142i − 1.07777i 0.681517 − 0.515910i
69.8 −0.0578728 + 1.41303i 1.71777 − 1.71777i −1.99330 − 0.163552i −1.50377 − 1.50377i 2.32785 + 2.52667i − 1.59335i 0.346462 − 2.80713i − 2.90147i 2.21190 − 2.03785i
69.9 0.120143 − 1.40910i −0.775950 + 0.775950i −1.97113 − 0.338586i 2.17252 + 2.17252i 1.00017 + 1.18662i − 4.50381i −0.713919 + 2.73684i 1.79580i 3.32232 − 2.80029i
69.10 0.551708 − 1.30216i 1.96031 − 1.96031i −1.39124 − 1.43682i 1.05453 + 1.05453i −1.47112 − 3.63416i 1.60770i −2.63853 + 1.01891i − 4.68564i 1.95497 − 0.791377i
69.11 0.591533 + 1.28456i 0.915574 − 0.915574i −1.30018 + 1.51972i 2.93727 + 2.93727i 1.71770 + 0.634516i − 1.91420i −2.72126 − 0.771193i 1.32345i −2.03560 + 5.51058i
69.12 0.790050 − 1.17295i −0.442194 + 0.442194i −0.751643 − 1.85338i −2.08286 − 2.08286i 0.169318 + 0.868028i − 0.974550i −2.76777 − 0.582622i 2.60893i −4.08866 + 0.797536i
69.13 0.808880 + 1.16005i −0.0958959 + 0.0958959i −0.691426 + 1.87668i −0.447567 − 0.447567i −0.188812 − 0.0336756i 0.608862i −2.73632 + 0.715922i 2.98161i 0.157171 − 0.881227i
69.14 1.19599 + 0.754723i −2.19829 + 2.19829i 0.860787 + 1.80528i −2.96739 − 2.96739i −4.28823 + 0.970033i 2.30405i −0.332996 + 2.80876i − 6.66494i −1.30941 − 5.78853i
69.15 1.36516 − 0.369229i −1.27073 + 1.27073i 1.72734 − 1.00812i 0.524628 + 0.524628i −1.26556 + 2.20394i 2.96126i 1.98588 − 2.01403i − 0.229507i 0.909911 + 0.522495i
69.16 1.40946 + 0.115879i 0.449216 − 0.449216i 1.97314 + 0.326653i −2.19499 − 2.19499i 0.685206 − 0.581096i − 4.36565i 2.74321 + 0.689050i 2.59641i −2.83939 − 3.34810i
205.1 −1.40483 − 0.162642i −2.37297 − 2.37297i 1.94710 + 0.456969i −0.208645 + 0.208645i 2.94767 + 3.71956i 3.10686i −2.66102 − 0.958643i 8.26197i 0.327046 − 0.259177i
205.2 −1.38688 + 0.276678i 2.03099 + 2.03099i 1.84690 − 0.767442i 1.79233 − 1.79233i −3.37869 − 2.25482i 4.22080i −2.34910 + 1.57535i 5.24987i −1.98986 + 2.98166i
205.3 −1.37038 − 0.349388i 0.724061 + 0.724061i 1.75586 + 0.957585i 0.100057 − 0.100057i −0.739257 − 1.24521i − 2.62804i −2.07161 − 1.92573i − 1.95147i −0.172075 + 0.102157i
205.4 −1.20363 + 0.742481i 1.79938 + 1.79938i 0.897443 − 1.78734i −2.01194 + 2.01194i −3.50178 − 0.829777i − 1.70564i 0.246881 + 2.81763i 3.47550i 0.927799 − 3.91545i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.2.l.c ✓ 32
4.b odd 2 1 1088.2.l.c 32
16.e even 4 1 inner 272.2.l.c ✓ 32
16.f odd 4 1 1088.2.l.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
272.2.l.c ✓ 32 1.a even 1 1 trivial
272.2.l.c ✓ 32 16.e even 4 1 inner
1088.2.l.c 32 4.b odd 2 1
1088.2.l.c 32 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 4 T_{3}^{29} + 232 T_{3}^{28} - 64 T_{3}^{27} + 8 T_{3}^{26} - 168 T_{3}^{25} + 18080 T_{3}^{24} + \cdots + 2304 \) acting on \(S_{2}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display