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Properties

Label 945.2.cx.b
Level $945$
Weight $2$
Character orbit 945.cx
Analytic conductor $7.546$
Analytic rank $0$
Dimension $288$
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(236,945)]
 
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([7, 0, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.236");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.cx (of order \(18\), 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: \(288\)
Relative dimension: \(48\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 6 q^{3} - 6 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q + 6 q^{3} - 6 q^{6} + 6 q^{9} - 6 q^{11} + 42 q^{12} - 3 q^{13} + 24 q^{14} - 24 q^{21} - 9 q^{23} + 24 q^{27} + 18 q^{29} - 12 q^{31} - 60 q^{33} - 60 q^{34} - 84 q^{36} - 66 q^{39} + 54 q^{41} - 33 q^{42} - 45 q^{43} + 24 q^{45} - 45 q^{47} - 54 q^{49} - 108 q^{51} - 102 q^{52} - 3 q^{54} + 27 q^{56} - 12 q^{57} + 6 q^{61} - 99 q^{62} + 162 q^{63} + 144 q^{64} + 3 q^{65} - 126 q^{66} - 54 q^{67} + 27 q^{68} + 18 q^{69} + 3 q^{70} - 36 q^{71} + 33 q^{72} + 6 q^{74} + 108 q^{76} + 27 q^{77} - 48 q^{78} + 9 q^{79} - 144 q^{80} + 6 q^{81} - 162 q^{83} - 45 q^{84} - 18 q^{85} - 3 q^{86} + 30 q^{90} + 18 q^{91} - 96 q^{92} + 12 q^{93} + 174 q^{94} + 60 q^{96} + 45 q^{97} + 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
236.1 −2.67973 − 0.472509i −0.527131 − 1.64989i 5.07831 + 1.84835i −0.939693 − 0.342020i 0.632981 + 4.67033i −1.63157 + 2.08278i −8.02212 − 4.63157i −2.44427 + 1.73941i 2.35652 + 1.36054i
236.2 −2.66311 − 0.469577i 1.31694 + 1.12502i 4.99224 + 1.81703i −0.939693 − 0.342020i −2.97887 − 3.61445i −2.56691 − 0.641058i −7.75785 − 4.47899i 0.468667 + 2.96317i 2.34190 + 1.35209i
236.3 −2.59540 − 0.457638i 1.16465 − 1.28202i 4.64726 + 1.69146i −0.939693 − 0.342020i −3.60944 + 2.79436i 2.34283 + 1.22928i −6.72270 − 3.88135i −0.287160 − 2.98622i 2.28235 + 1.31772i
236.4 −2.53042 − 0.446181i −1.55319 − 0.766550i 4.32456 + 1.57401i −0.939693 − 0.342020i 3.58820 + 2.63270i 0.905669 − 2.48591i −5.79022 − 3.34298i 1.82480 + 2.38120i 2.22521 + 1.28473i
236.5 −2.37320 − 0.418460i −0.251058 + 1.71376i 3.57761 + 1.30214i −0.939693 − 0.342020i 1.31295 − 3.96204i 0.256523 + 2.63329i −3.77158 − 2.17752i −2.87394 − 0.860507i 2.08696 + 1.20491i
236.6 −2.21668 − 0.390860i 1.72831 + 0.113838i 2.88151 + 1.04878i −0.939693 − 0.342020i −3.78660 − 0.927867i 2.41856 − 1.07264i −2.07882 − 1.20021i 2.97408 + 0.393492i 1.94931 + 1.12544i
236.7 −2.11581 − 0.373074i −1.42267 + 0.987934i 2.45807 + 0.894664i −0.939693 − 0.342020i 3.37866 − 1.55952i 2.41773 − 1.07451i −1.14580 − 0.661530i 1.04797 − 2.81101i 1.86061 + 1.07422i
236.8 −2.06562 − 0.364225i −0.0353827 + 1.73169i 2.25474 + 0.820660i −0.939693 − 0.342020i 0.703811 − 3.56413i −0.971083 − 2.46110i −0.725590 − 0.418920i −2.99750 − 0.122544i 1.81648 + 1.04874i
236.9 −2.00793 − 0.354052i 1.70846 − 0.284912i 2.02705 + 0.737786i −0.939693 − 0.342020i −3.53134 − 0.0327999i −2.49684 − 0.875094i −0.277471 − 0.160198i 2.83765 − 0.973519i 1.76575 + 1.01945i
236.10 −1.77761 − 0.313441i −0.300698 − 1.70575i 1.18228 + 0.430316i −0.939693 − 0.342020i −0.000128226 3.12642i −2.41447 − 1.08181i 1.15965 + 0.669526i −2.81916 + 1.02583i 1.56321 + 0.902519i
236.11 −1.73322 − 0.305614i 1.31230 − 1.13043i 1.03128 + 0.375354i −0.939693 − 0.342020i −2.61998 + 1.55823i −1.53372 + 2.15585i 1.37562 + 0.794216i 0.444253 − 2.96692i 1.52417 + 0.879980i
236.12 −1.69184 − 0.298317i −1.73201 + 0.0111722i 0.893937 + 0.325366i −0.939693 − 0.342020i 2.93362 + 0.497787i 2.27305 + 1.35397i 1.56022 + 0.900794i 2.99975 − 0.0387009i 1.48778 + 0.858969i
236.13 −1.38086 − 0.243484i 1.32508 + 1.11543i −0.0318822 − 0.0116042i −0.939693 − 0.342020i −1.55816 − 1.86289i −1.33013 + 2.28709i 2.46982 + 1.42595i 0.511652 + 2.95605i 1.21431 + 0.701083i
236.14 −1.35615 − 0.239125i −0.149510 − 1.72559i −0.0974333 − 0.0354628i −0.939693 − 0.342020i −0.209873 + 2.37590i 1.67719 − 2.04623i 2.50880 + 1.44846i −2.95529 + 0.515986i 1.19258 + 0.688534i
236.15 −1.20285 − 0.212094i −1.51658 + 0.836655i −0.477530 − 0.173807i −0.939693 − 0.342020i 2.00166 − 0.684709i −1.29735 − 2.30584i 2.65306 + 1.53175i 1.60002 − 2.53770i 1.05777 + 0.610701i
236.16 −1.17173 − 0.206607i 0.300166 − 1.70584i −0.549124 − 0.199865i −0.939693 − 0.342020i −0.704153 + 1.93677i 1.86667 + 1.87498i 2.66293 + 1.53744i −2.81980 − 1.02407i 1.03040 + 0.594902i
236.17 −1.10429 − 0.194716i −1.59837 − 0.667233i −0.697848 − 0.253996i −0.939693 − 0.342020i 1.63514 + 1.04805i −1.87342 − 1.86823i 2.66336 + 1.53769i 2.10960 + 2.13298i 0.971095 + 0.560662i
236.18 −0.794303 − 0.140057i 1.03134 + 1.39152i −1.26808 − 0.461545i −0.939693 − 0.342020i −0.624307 − 1.24974i 0.601324 − 2.57651i 2.33960 + 1.35077i −0.872662 + 2.87027i 0.698498 + 0.403278i
236.19 −0.769757 − 0.135729i −1.26293 + 1.18533i −1.30528 − 0.475084i −0.939693 − 0.342020i 1.13303 − 0.740998i 1.07261 + 2.41858i 2.29409 + 1.32450i 0.189998 − 2.99398i 0.676913 + 0.390816i
236.20 −0.687844 − 0.121285i 0.395281 + 1.68634i −1.42097 − 0.517189i −0.939693 − 0.342020i −0.0673627 − 1.20788i −2.37930 + 1.15712i 2.12443 + 1.22654i −2.68751 + 1.33316i 0.604880 + 0.349227i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 236.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.bd even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.cx.b ✓ 288
7.d odd 6 1 945.2.de.b yes 288
27.f odd 18 1 945.2.de.b yes 288
189.bd even 18 1 inner 945.2.cx.b ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.cx.b ✓ 288 1.a even 1 1 trivial
945.2.cx.b ✓ 288 189.bd even 18 1 inner
945.2.de.b yes 288 7.d odd 6 1
945.2.de.b yes 288 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} - 2088 T_{2}^{282} + 39 T_{2}^{281} + 324 T_{2}^{280} - 2916 T_{2}^{278} + \cdots + 15\!\cdots\!69 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display