Properties

Label 945.2.bu.b
Level $945$
Weight $2$
Character orbit 945.bu
Analytic conductor $7.546$
Analytic rank $0$
Dimension $300$
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] = [945,2,Mod(121,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bu (of order \(9\), degree \(6\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(300\)
Relative dimension: \(50\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 300 q - 3 q^{3} + 6 q^{6} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 300 q - 3 q^{3} + 6 q^{6} - 3 q^{9} - 3 q^{11} + 21 q^{12} - 3 q^{13} - 27 q^{14} + 48 q^{17} + 12 q^{21} - 42 q^{22} + 9 q^{23} - 30 q^{24} + 24 q^{27} + 18 q^{29} - 24 q^{31} - 48 q^{33} + 60 q^{34} + 60 q^{36} - 48 q^{37} + 30 q^{38} - 12 q^{39} - 60 q^{41} + 33 q^{42} + 15 q^{43} - 24 q^{45} - 60 q^{46} + 36 q^{47} - 66 q^{48} - 36 q^{49} - 51 q^{51} + 69 q^{52} + 9 q^{54} + 21 q^{56} - 12 q^{57} + 15 q^{59} - 60 q^{61} - 81 q^{62} - 60 q^{63} - 234 q^{64} - 3 q^{65} + 36 q^{66} + 18 q^{67} - 21 q^{68} - 90 q^{69} + 18 q^{70} + 12 q^{71} + 24 q^{72} - 18 q^{73} + 102 q^{74} + 108 q^{76} - 45 q^{77} - 6 q^{78} - 9 q^{79} - 186 q^{80} - 3 q^{81} + 102 q^{83} + 69 q^{84} + 18 q^{85} - 78 q^{86} + 15 q^{87} + 42 q^{88} - 30 q^{90} - 9 q^{91} - 24 q^{92} - 12 q^{93} + 6 q^{94} - 114 q^{96} - 45 q^{97} + 12 q^{98} - 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 −2.58756 0.941793i 1.52967 0.812463i 4.27638 + 3.58831i −0.939693 + 0.342020i −4.72329 + 0.661657i 0.0897486 + 2.64423i −4.93231 8.54301i 1.67981 2.48561i 2.75362
121.2 −2.53332 0.922054i 1.21130 + 1.23804i 4.03545 + 3.38615i −0.939693 + 0.342020i −1.92707 4.25325i 2.18214 1.49608i −4.40499 7.62967i −0.0655076 + 2.99928i 2.69591
121.3 −2.46003 0.895377i −0.0168643 1.73197i 3.71795 + 3.11973i −0.939693 + 0.342020i −1.50928 + 4.27579i 0.727092 2.54388i −3.73503 6.46926i −2.99943 + 0.0584170i 2.61791
121.4 −2.42951 0.884268i −0.581447 + 1.63154i 3.58848 + 3.01110i −0.939693 + 0.342020i 2.85535 3.44968i −2.48598 0.905496i −3.47020 6.01057i −2.32384 1.89731i 2.58543
121.5 −2.42501 0.882631i −1.65869 + 0.498745i 3.56954 + 2.99520i −0.939693 + 0.342020i 4.46254 + 0.254549i 1.74013 + 1.99297i −3.43186 5.94416i 2.50251 1.65453i 2.58064
121.6 −2.09407 0.762179i −1.17216 1.27516i 2.27212 + 1.90653i −0.939693 + 0.342020i 1.48270 + 3.56367i 2.53229 0.766504i −1.07639 1.86437i −0.252059 + 2.98939i 2.22846
121.7 −2.03980 0.742428i −1.45939 0.932834i 2.07751 + 1.74324i −0.939693 + 0.342020i 2.28431 + 2.98629i −2.64053 + 0.166113i −0.772773 1.33848i 1.25964 + 2.72274i 2.17071
121.8 −1.98350 0.721936i 1.73038 + 0.0761166i 1.88100 + 1.57835i −0.939693 + 0.342020i −3.37726 1.40020i −2.60857 + 0.441982i −0.480705 0.832606i 2.98841 + 0.263421i 2.11080
121.9 −1.96173 0.714012i −1.68738 + 0.390828i 1.80649 + 1.51582i −0.939693 + 0.342020i 3.58924 + 0.438111i −0.670414 + 2.55940i −0.373899 0.647612i 2.69451 1.31895i 2.08763
121.10 −1.79482 0.653260i 1.11049 + 1.32921i 1.26253 + 1.05939i −0.939693 + 0.342020i −1.12481 3.11113i 2.30235 + 1.30352i 0.336049 + 0.582054i −0.533607 + 2.95216i 1.91000
121.11 −1.75998 0.640580i −0.825720 + 1.52256i 1.15509 + 0.969237i −0.939693 + 0.342020i 2.42857 2.15073i 1.72037 2.01005i 0.460867 + 0.798245i −1.63637 2.51442i 1.87293
121.12 −1.74920 0.636657i 1.25509 1.19363i 1.12228 + 0.941706i −0.939693 + 0.342020i −2.95534 + 1.28883i −1.04616 2.43014i 0.497908 + 0.862403i 0.150506 2.99622i 1.86146
121.13 −1.70664 0.621166i 0.271277 + 1.71067i 0.994686 + 0.834641i −0.939693 + 0.342020i 0.599641 3.08802i −1.70013 + 2.02720i 0.637048 + 1.10340i −2.85282 + 0.928134i 1.81617
121.14 −1.37934 0.502038i 1.10379 1.33479i 0.118445 + 0.0993874i −0.939693 + 0.342020i −2.19261 + 1.28698i −0.935185 + 2.47496i 1.35438 + 2.34586i −0.563305 2.94664i 1.46786
121.15 −1.35055 0.491562i −0.692406 1.58763i 0.0502756 + 0.0421862i −0.939693 + 0.342020i 0.154713 + 2.48454i 2.22682 + 1.42873i 1.39007 + 2.40767i −2.04115 + 2.19857i 1.43723
121.16 −1.01830 0.370629i −1.72022 + 0.202110i −0.632529 0.530755i −0.939693 + 0.342020i 1.82660 + 0.431755i −0.446552 2.60779i 1.53104 + 2.65183i 2.91830 0.695348i 1.08365
121.17 −1.00833 0.367004i 1.68428 + 0.403993i −0.650043 0.545450i −0.939693 + 0.342020i −1.55005 1.02550i 1.93105 + 1.80860i 1.52832 + 2.64714i 2.67358 + 1.36087i 1.07305
121.18 −0.854220 0.310911i −0.695679 1.58620i −0.899063 0.754403i −0.939693 + 0.342020i 0.101096 + 1.57126i −1.67338 2.04934i 1.44249 + 2.49846i −2.03206 + 2.20697i 0.909042
121.19 −0.688400 0.250557i −1.54572 + 0.781513i −1.12097 0.940608i −0.939693 + 0.342020i 1.25988 0.150703i −2.64451 + 0.0810115i 1.26858 + 2.19725i 1.77848 2.41599i 0.732580
121.20 −0.664313 0.241790i 1.70853 + 0.284468i −1.14924 0.964326i −0.939693 + 0.342020i −1.06622 0.602082i 0.613933 2.57354i 1.23724 + 2.14296i 2.83816 + 0.972043i 0.706947
See next 80 embeddings (of 300 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.w even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.bu.b yes 300
7.c even 3 1 945.2.bs.b 300
27.e even 9 1 945.2.bs.b 300
189.w even 9 1 inner 945.2.bu.b yes 300
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.bs.b 300 7.c even 3 1
945.2.bs.b 300 27.e even 9 1
945.2.bu.b yes 300 1.a even 1 1 trivial
945.2.bu.b yes 300 189.w even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{300} + 2321 T_{2}^{294} - 3 T_{2}^{293} - 108 T_{2}^{292} - 456 T_{2}^{291} - 963 T_{2}^{290} - 702 T_{2}^{289} + 3027341 T_{2}^{288} + 13719 T_{2}^{287} - 108759 T_{2}^{286} - 993437 T_{2}^{285} + \cdots + 53\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display