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Properties

Label 275.2.i.a
Level $275$
Weight $2$
Character orbit 275.i
Analytic conductor $2.196$
Analytic rank $0$
Dimension $44$
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] = [275,2,Mod(56,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.56");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.i (of order \(5\), degree \(4\), 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - q^{2} - 9 q^{4} - 6 q^{5} + 4 q^{6} + 18 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - q^{2} - 9 q^{4} - 6 q^{5} + 4 q^{6} + 18 q^{8} - 5 q^{9} - q^{10} + 11 q^{11} - 2 q^{13} - 6 q^{14} + 20 q^{15} - 5 q^{16} - 26 q^{17} - 122 q^{18} + 10 q^{19} + 26 q^{20} + 6 q^{21} + q^{22} + 6 q^{23} - 62 q^{24} + 4 q^{25} - 6 q^{26} + 15 q^{27} + 8 q^{29} - 55 q^{30} + 10 q^{31} + 30 q^{32} + 19 q^{34} - 60 q^{35} + 41 q^{36} + 12 q^{37} + 54 q^{38} + 20 q^{39} + 3 q^{40} - 80 q^{42} - 4 q^{43} + 9 q^{44} + 3 q^{45} - 6 q^{46} + 37 q^{47} + 65 q^{48} - 4 q^{49} - 31 q^{50} - 54 q^{51} + 2 q^{52} + 42 q^{53} + 55 q^{54} + q^{55} + 10 q^{56} + 4 q^{58} - 11 q^{59} + 5 q^{60} + 10 q^{61} - 2 q^{62} + 20 q^{63} + 63 q^{65} - 4 q^{66} - 44 q^{67} - 124 q^{68} + 53 q^{69} + 20 q^{70} + 3 q^{71} + 136 q^{72} - 4 q^{73} - 40 q^{74} - 75 q^{75} - 140 q^{76} + 65 q^{78} + 15 q^{79} + 66 q^{80} + 19 q^{81} + 38 q^{82} + 14 q^{83} + 76 q^{84} + 24 q^{85} - 43 q^{86} - 75 q^{87} + 27 q^{88} + 32 q^{89} + 168 q^{90} + 10 q^{91} - 46 q^{92} - 100 q^{93} - 37 q^{94} + 54 q^{95} + 128 q^{96} + 5 q^{97} - 73 q^{98} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1 −0.805707 − 2.47971i −2.65558 + 1.92939i −3.88177 + 2.82027i −2.13873 + 0.652556i 6.92395 + 5.03055i −0.789864 5.90230 + 4.28827i 2.40250 − 7.39413i 3.34134 + 4.77767i
56.2 −0.612340 − 1.88459i 2.09851 − 1.52466i −1.55868 + 1.13244i −2.12551 − 0.694423i −4.15835 − 3.02122i 4.96258 −0.117621 − 0.0854565i 1.15211 − 3.54584i −0.00716883 + 4.43093i
56.3 −0.603017 − 1.85590i 1.46752 − 1.06622i −1.46269 + 1.06270i 2.21905 + 0.275334i −2.86373 − 2.08062i −0.768734 −0.303144 − 0.220247i 0.0897486 − 0.276218i −0.827136 − 4.28436i
56.4 −0.279439 − 0.860024i −0.834018 + 0.605950i 0.956479 − 0.694923i −1.84287 − 1.26642i 0.754188 + 0.547950i −2.69405 −2.32809 − 1.69145i −0.598640 + 1.84242i −0.574182 + 1.93880i
56.5 −0.0280210 − 0.0862398i −2.29785 + 1.66949i 1.61138 − 1.17074i 1.56268 − 1.59938i 0.208365 + 0.151386i −1.41588 −0.292837 − 0.212758i 1.56589 − 4.81931i −0.181719 − 0.0899489i
56.6 0.0824770 + 0.253838i 1.17921 − 0.856743i 1.56040 − 1.13370i −1.17525 + 1.90231i 0.314731 + 0.228666i 2.34233 0.848329 + 0.616347i −0.270534 + 0.832618i −0.579811 − 0.141425i
56.7 0.254741 + 0.784011i 1.52879 − 1.11073i 1.06825 − 0.776131i 1.39336 − 1.74887i 1.26027 + 0.915643i −3.68533 2.21446 + 1.60890i 0.176431 − 0.543000i 1.72608 + 0.646899i
56.8 0.356257 + 1.09645i −1.81672 + 1.31993i 0.542759 − 0.394337i 1.31962 + 1.80516i −2.09445 − 1.52171i 0.858963 2.49112 + 1.80990i 0.631227 − 1.94272i −1.50913 + 2.09000i
56.9 0.520804 + 1.60287i −0.371336 + 0.269791i −0.679923 + 0.493993i −0.281625 − 2.21826i −0.625834 − 0.454695i 4.10024 1.58105 + 1.14870i −0.861948 + 2.65280i 3.40892 − 1.60669i
56.10 0.637729 + 1.96273i 1.68996 − 1.22783i −1.82756 + 1.32780i 0.646723 + 2.14050i 3.48763 + 2.53391i −1.60088 −0.432416 − 0.314168i 0.421352 − 1.29679i −3.78879 + 2.63440i
56.11 0.785533 + 2.41762i 0.0115222 − 0.00837137i −3.60980 + 2.62268i −2.19549 − 0.424071i 0.0292899 + 0.0212803i −1.30938 −5.06316 − 3.67860i −0.926988 + 2.85298i −0.699385 − 5.64098i
111.1 −2.17723 + 1.58185i 0.123865 − 0.381217i 1.62005 − 4.98601i −1.71157 + 1.43893i 0.333346 + 1.02593i −3.22798 2.69664 + 8.29940i 2.29707 + 1.66892i 1.45032 − 5.84034i
111.2 −1.92758 + 1.40047i −0.601854 + 1.85232i 1.13621 − 3.49690i −0.970345 − 2.01455i −1.43399 − 4.41336i 3.79591 1.23462 + 3.79978i −0.641798 − 0.466294i 4.69173 + 2.52427i
111.3 −1.45510 + 1.05719i −0.100383 + 0.308946i 0.381627 − 1.17453i 1.24221 + 1.85927i −0.180549 − 0.555672i 2.47200 −0.425202 − 1.30864i 2.34168 + 1.70133i −3.77316 − 1.39217i
111.4 −1.20213 + 0.873402i 0.586463 − 1.80495i 0.0642629 − 0.197781i 1.62445 − 1.53661i 0.871437 + 2.68201i −0.147804 −0.822860 − 2.53250i −0.486847 − 0.353715i −0.610735 + 3.26601i
111.5 −0.197944 + 0.143815i −0.402147 + 1.23768i −0.599535 + 1.84518i 1.81773 + 1.30225i −0.0983942 − 0.302826i −4.61247 −0.297905 − 0.916858i 1.05692 + 0.767896i −0.547092 + 0.00364272i
111.6 0.212312 − 0.154254i 0.527031 − 1.62203i −0.596752 + 1.83661i −0.425792 + 2.19515i −0.138310 − 0.425674i 1.81767 0.318799 + 0.981163i 0.0738195 + 0.0536330i 0.248210 + 0.531739i
111.7 0.641500 − 0.466077i −1.00865 + 3.10430i −0.423740 + 1.30414i −1.93291 − 1.12421i 0.799794 + 2.46151i 0.454208 0.826061 + 2.54235i −6.19223 − 4.49892i −1.76393 + 0.179703i
111.8 0.706495 − 0.513299i −0.196025 + 0.603302i −0.382374 + 1.17683i 0.683987 − 2.12889i 0.171184 + 0.526849i 0.831214 0.873632 + 2.68876i 2.10150 + 1.52683i −0.609522 − 1.85514i
111.9 1.12055 − 0.814127i 0.964216 − 2.96755i −0.0252045 + 0.0775715i 2.16576 − 0.556295i −1.33551 − 4.11029i −3.94703 0.890934 + 2.74201i −5.44960 − 3.95937i 1.97395 − 2.38657i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.i.a ✓ 44
25.d even 5 1 inner 275.2.i.a ✓ 44
25.d even 5 1 6875.2.a.f 22
25.e even 10 1 6875.2.a.e 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.i.a ✓ 44 1.a even 1 1 trivial
275.2.i.a ✓ 44 25.d even 5 1 inner
6875.2.a.e 22 25.e even 10 1
6875.2.a.f 22 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{44} + T_{2}^{43} + 16 T_{2}^{42} + 11 T_{2}^{41} + 160 T_{2}^{40} + 34 T_{2}^{39} + 1292 T_{2}^{38} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display