Properties

Label 1205.2.b.d
Level $1205$
Weight $2$
Character orbit 1205.b
Analytic conductor $9.622$
Analytic rank $0$
Dimension $66$
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] = [1205,2,Mod(724,1205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1205.724");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1205 = 5 \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1205.b (of order \(2\), degree \(1\), 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: \(9.62197344356\)
Analytic rank: \(0\)
Dimension: \(66\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 78 q^{4} + 16 q^{6} - 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 78 q^{4} + 16 q^{6} - 90 q^{9} + q^{10} + 48 q^{11} - 30 q^{14} - 3 q^{15} + 98 q^{16} - 12 q^{19} - 10 q^{20} + 18 q^{21} - 42 q^{24} + 6 q^{25} + 48 q^{26} - 56 q^{29} - 5 q^{30} + 48 q^{31} - 8 q^{34} + 3 q^{35} + 158 q^{36} - 84 q^{39} - 6 q^{40} + 56 q^{41} - 144 q^{44} - 13 q^{45} + 36 q^{46} - 98 q^{49} + 2 q^{50} + 44 q^{51} - 86 q^{54} + 3 q^{55} + 104 q^{56} - 108 q^{59} + 7 q^{60} + 22 q^{61} - 136 q^{64} + 15 q^{65} + 74 q^{66} - 20 q^{69} - 32 q^{70} + 212 q^{71} - 84 q^{74} - 9 q^{75} + 6 q^{76} - 66 q^{79} - 21 q^{80} + 162 q^{81} + 52 q^{84} - 48 q^{85} + 100 q^{86} - 54 q^{89} - 155 q^{90} + 72 q^{91} + 96 q^{94} - 5 q^{95} + 122 q^{96} - 112 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
724.1 2.81437i 0.650611i −5.92068 2.22726 + 0.198227i −1.83106 2.61686i 11.0342i 2.57671 0.557884 6.26834i
724.2 2.73687i 1.37229i −5.49044 −1.04813 1.97520i 3.75578 0.639768i 9.55288i 1.11681 −5.40587 + 2.86858i
724.3 2.70326i 3.35528i −5.30763 0.474898 + 2.18506i 9.07021 3.43095i 8.94141i −8.25791 5.90678 1.28377i
724.4 2.60935i 2.94245i −4.80870 −1.75698 1.38312i −7.67788 2.28979i 7.32887i −5.65803 −3.60904 + 4.58458i
724.5 2.55975i 2.19931i −4.55230 −2.23588 0.0286488i −5.62967 4.06530i 6.53325i −1.83695 −0.0733336 + 5.72330i
724.6 2.48606i 2.54168i −4.18051 2.19521 0.425495i 6.31876 3.51286i 5.42088i −3.46011 −1.05781 5.45743i
724.7 2.48219i 1.89252i −4.16124 −1.31757 + 1.80665i 4.69758 3.72710i 5.36461i −0.581628 4.48445 + 3.27046i
724.8 2.37391i 0.703313i −3.63543 1.41667 1.73005i −1.66960 4.47866i 3.88235i 2.50535 −4.10697 3.36303i
724.9 2.33352i 1.59052i −3.44531 −0.227123 + 2.22450i −3.71151 3.91727i 3.37267i 0.470249 5.19092 + 0.529996i
724.10 2.31942i 2.66924i −3.37972 0.686453 2.12809i 6.19110 0.0358936i 3.20016i −4.12486 −4.93595 1.59218i
724.11 2.20997i 2.26852i −2.88397 −2.23543 0.0532972i 5.01336 0.372373i 1.95355i −2.14617 −0.117785 + 4.94024i
724.12 2.06213i 2.32228i −2.25239 2.09119 + 0.791781i −4.78886 3.49367i 0.520471i −2.39299 1.63276 4.31232i
724.13 1.90752i 0.479938i −1.63863 −2.22437 0.228456i −0.915491 1.00580i 0.689327i 2.76966 −0.435785 + 4.24302i
724.14 1.85162i 0.305536i −1.42851 2.21384 0.314518i −0.565737 2.51443i 1.05818i 2.90665 −0.582370 4.09920i
724.15 1.85071i 2.57343i −1.42511 1.63741 + 1.52279i −4.76266 3.52351i 1.06395i −3.62254 2.81824 3.03036i
724.16 1.67685i 3.17567i −0.811815 −1.72522 + 1.42254i −5.32511 0.164609i 1.99240i −7.08488 2.38538 + 2.89293i
724.17 1.64329i 3.10101i −0.700400 1.37323 + 1.76472i 5.09585 0.459168i 2.13562i −6.61624 2.89995 2.25661i
724.18 1.54552i 0.0372080i −0.388631 −0.615725 + 2.14962i 0.0575057 5.06287i 2.49040i 2.99862 3.32229 + 0.951615i
724.19 1.47177i 1.07414i −0.166115 1.15452 1.91496i 1.58089 3.64779i 2.69906i 1.84623 −2.81839 1.69919i
724.20 1.38861i 0.0366779i 0.0717484 0.203412 2.22680i 0.0509315 3.69082i 2.87686i 2.99865 −3.09216 0.282461i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 724.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1205.2.b.d 66
5.b even 2 1 inner 1205.2.b.d 66
5.c odd 4 2 6025.2.a.q 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.b.d 66 1.a even 1 1 trivial
1205.2.b.d 66 5.b even 2 1 inner
6025.2.a.q 66 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} + 105 T_{2}^{64} + 5239 T_{2}^{62} + 165314 T_{2}^{60} + 3703934 T_{2}^{58} + 62721100 T_{2}^{56} + 834250271 T_{2}^{54} + 8942833038 T_{2}^{52} + 78658767871 T_{2}^{50} + \cdots + 1841449 \) acting on \(S_{2}^{\mathrm{new}}(1205, [\chi])\). Copy content Toggle raw display