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Properties

Label 272.2.l.b
Level $272$
Weight $2$
Character orbit 272.l
Analytic conductor $2.172$
Analytic rank $0$
Dimension $30$
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: \(30\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 8 q^{6} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} + 8 q^{6} + 2 q^{8} - 6 q^{10} - 10 q^{11} - 2 q^{12} - 6 q^{13} - 22 q^{14} - 28 q^{15} + 2 q^{16} - 30 q^{17} + 8 q^{18} + 10 q^{19} - 10 q^{20} - 4 q^{21} + 28 q^{22} + 18 q^{24} + 6 q^{26} + 4 q^{27} + 8 q^{28} - 18 q^{29} - 56 q^{30} + 48 q^{31} - 38 q^{32} - 12 q^{33} - 2 q^{34} + 16 q^{35} + 20 q^{36} + 6 q^{37} + 42 q^{38} + 10 q^{40} + 12 q^{42} + 2 q^{43} + 22 q^{44} + 42 q^{45} - 14 q^{46} - 32 q^{47} - 62 q^{48} - 38 q^{49} - 28 q^{50} + 2 q^{51} - 10 q^{52} - 6 q^{53} + 40 q^{54} + 96 q^{56} + 62 q^{58} - 14 q^{59} - 28 q^{60} + 6 q^{61} - 74 q^{62} + 84 q^{63} - 38 q^{64} - 12 q^{65} + 26 q^{67} + 2 q^{68} + 12 q^{69} + 48 q^{70} + 44 q^{72} + 42 q^{74} - 26 q^{75} - 38 q^{76} - 28 q^{77} - 56 q^{78} - 72 q^{79} - 38 q^{80} - 10 q^{81} - 76 q^{82} - 34 q^{83} + 104 q^{84} + 2 q^{85} + 34 q^{86} - 2 q^{88} + 70 q^{90} + 16 q^{91} - 40 q^{92} - 24 q^{93} - 44 q^{94} + 76 q^{95} - 126 q^{96} - 28 q^{97} - 30 q^{98} + 54 q^{99}+O(q^{100}) \) 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 −1.41421 − 0.00186943i −0.274984 + 0.274984i 1.99999 + 0.00528753i −2.33984 − 2.33984i 0.389399 − 0.388371i − 0.445993i −2.82840 − 0.0112165i 2.84877i 3.30466 + 3.31341i
69.2 −1.36107 − 0.384035i −0.873530 + 0.873530i 1.70503 + 1.04540i 2.72140 + 2.72140i 1.52440 − 0.853471i 0.811757i −1.91920 − 2.07766i 1.47389i −2.65891 − 4.74914i
69.3 −1.13198 + 0.847715i 0.907888 − 0.907888i 0.562758 − 1.91919i 0.218362 + 0.218362i −0.258081 + 1.79734i − 4.34872i 0.989900 + 2.64955i 1.35148i −0.432289 − 0.0620725i
69.4 −0.812807 + 1.15730i −1.22998 + 1.22998i −0.678689 − 1.88132i −2.77156 − 2.77156i −0.423720 − 2.42319i 0.669690i 2.72890 + 0.743706i − 0.0256965i 5.46027 − 0.954783i
69.5 −0.738062 + 1.20634i 2.06766 − 2.06766i −0.910529 − 1.78071i 2.14842 + 2.14842i 0.968245 + 4.02036i 3.87521i 2.82018 + 0.215865i − 5.55040i −4.17739 + 1.00606i
69.6 −0.405142 − 1.35494i −2.04684 + 2.04684i −1.67172 + 1.09789i −0.112242 − 0.112242i 3.60261 + 1.94409i 1.01296i 2.16485 + 1.82028i − 5.37914i −0.106607 + 0.197555i
69.7 −0.205383 − 1.39922i 1.44510 − 1.44510i −1.91564 + 0.574752i −1.55397 − 1.55397i −2.31881 − 1.72521i − 2.87379i 1.19764 + 2.56235i − 1.17662i −1.85518 + 2.49350i
69.8 0.0871765 + 1.41152i 0.0244593 − 0.0244593i −1.98480 + 0.246104i −1.10948 − 1.10948i 0.0366572 + 0.0323926i 5.11349i −0.520409 − 2.78014i 2.99880i 1.46934 − 1.66279i
69.9 0.321310 + 1.37723i −2.20735 + 2.20735i −1.79352 + 0.885036i 2.24231 + 2.24231i −3.74927 − 2.33078i 3.15246i −1.79517 − 2.18572i − 6.74479i −2.36770 + 3.80866i
69.10 0.526800 − 1.31243i −1.66106 + 1.66106i −1.44496 − 1.38278i −0.573635 − 0.573635i 1.30499 + 3.05508i 1.90621i −2.57601 + 1.16797i − 2.51826i −1.05505 + 0.450667i
69.11 1.09253 + 0.897981i 2.12262 − 2.12262i 0.387258 + 1.96215i −0.495683 − 0.495683i 4.22510 − 0.412960i − 0.121447i −1.33888 + 2.49146i − 6.01102i −0.0964362 − 0.986665i
69.12 1.15607 − 0.814551i 0.561979 − 0.561979i 0.673012 − 1.88336i 1.41199 + 1.41199i 0.191928 − 1.10745i − 1.77930i −0.756045 − 2.72551i 2.36836i 2.78251 + 0.482227i
69.13 1.25559 − 0.650757i 1.55623 − 1.55623i 1.15303 − 1.63417i −2.63792 − 2.63792i 0.941264 − 2.96671i 4.77816i 0.384292 − 2.80220i − 1.84368i −5.02881 − 1.59551i
69.14 1.27512 + 0.611614i 0.148531 − 0.148531i 1.25186 + 1.55976i −0.0348255 − 0.0348255i 0.280239 − 0.0985512i 1.59619i 0.642295 + 2.75453i 2.95588i −0.0231069 − 0.0657065i
69.15 1.35405 + 0.408098i −1.54071 + 1.54071i 1.66691 + 1.10517i 1.88667 + 1.88667i −2.71496 + 1.45744i − 3.34688i 1.80607 + 2.17672i − 1.74756i 1.78470 + 3.32460i
205.1 −1.41421 + 0.00186943i −0.274984 − 0.274984i 1.99999 − 0.00528753i −2.33984 + 2.33984i 0.389399 + 0.388371i 0.445993i −2.82840 + 0.0112165i − 2.84877i 3.30466 − 3.31341i
205.2 −1.36107 + 0.384035i −0.873530 − 0.873530i 1.70503 − 1.04540i 2.72140 − 2.72140i 1.52440 + 0.853471i − 0.811757i −1.91920 + 2.07766i − 1.47389i −2.65891 + 4.74914i
205.3 −1.13198 − 0.847715i 0.907888 + 0.907888i 0.562758 + 1.91919i 0.218362 − 0.218362i −0.258081 − 1.79734i 4.34872i 0.989900 − 2.64955i − 1.35148i −0.432289 + 0.0620725i
205.4 −0.812807 − 1.15730i −1.22998 − 1.22998i −0.678689 + 1.88132i −2.77156 + 2.77156i −0.423720 + 2.42319i − 0.669690i 2.72890 − 0.743706i 0.0256965i 5.46027 + 0.954783i
205.5 −0.738062 − 1.20634i 2.06766 + 2.06766i −0.910529 + 1.78071i 2.14842 − 2.14842i 0.968245 − 4.02036i − 3.87521i 2.82018 − 0.215865i 5.55040i −4.17739 − 1.00606i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.15
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.b ✓ 30
4.b odd 2 1 1088.2.l.b 30
16.e even 4 1 inner 272.2.l.b ✓ 30
16.f odd 4 1 1088.2.l.b 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
272.2.l.b ✓ 30 1.a even 1 1 trivial
272.2.l.b ✓ 30 16.e even 4 1 inner
1088.2.l.b 30 4.b odd 2 1
1088.2.l.b 30 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} + 2 T_{3}^{29} + 2 T_{3}^{28} - 4 T_{3}^{27} + 204 T_{3}^{26} + 392 T_{3}^{25} + 384 T_{3}^{24} + \cdots + 128 \) acting on \(S_{2}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display