↑

Properties

Label 945.2.de.b
Level $945$
Weight $2$
Character orbit 945.de
Analytic conductor $7.546$
Analytic rank $0$
Dimension $288$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(101,945)]
 
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, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.de (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.

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 3 q^{3} + 6 q^{6} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q + 3 q^{3} + 6 q^{6} - 3 q^{9} + 3 q^{11} + 21 q^{12} + 3 q^{13} - 3 q^{14} + 36 q^{21} - 9 q^{23} - 24 q^{27} + 18 q^{29} - 24 q^{31} - 30 q^{33} + 60 q^{34} - 84 q^{36} + 90 q^{38} + 24 q^{39} - 54 q^{41} + 63 q^{42} - 45 q^{43} - 6 q^{45} - 36 q^{47} + 18 q^{49} + 9 q^{51} - 69 q^{52} - 87 q^{54} + 27 q^{56} - 12 q^{57} - 45 q^{59} - 42 q^{61} + 99 q^{62} + 60 q^{63} + 144 q^{64} + 3 q^{65} - 108 q^{66} + 54 q^{67} - 9 q^{68} - 18 q^{69} + 24 q^{70} - 36 q^{71} + 114 q^{72} - 66 q^{74} - 108 q^{76} + 27 q^{77} - 48 q^{78} + 9 q^{79} + 144 q^{80} - 3 q^{81} + 162 q^{83} - 129 q^{84} - 18 q^{85} + 78 q^{86} + 45 q^{87} + 126 q^{88} - 30 q^{90} - 9 q^{91} - 96 q^{92} + 12 q^{93} + 24 q^{94} - 60 q^{96} - 45 q^{97} - 36 q^{98} + 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −1.79925 + 2.14426i 1.70097 − 0.326657i −1.01326 − 5.74649i −0.766044 + 0.642788i −2.36002 + 4.23505i 2.14879 − 1.54360i 9.29682 + 5.36752i 2.78659 − 1.11127i − 2.79913i
101.2 −1.66179 + 1.98044i −1.25287 + 1.19595i −0.813312 − 4.61252i −0.766044 + 0.642788i −0.286509 − 4.46866i −1.44466 + 2.21652i 6.00853 + 3.46903i 0.139384 − 2.99676i − 2.58528i
101.3 −1.65499 + 1.97234i 0.987463 + 1.42300i −0.803839 − 4.55880i −0.766044 + 0.642788i −4.44087 − 0.407430i −2.63428 + 0.246080i 5.86232 + 3.38461i −1.04983 + 2.81031i − 2.57471i
101.4 −1.64649 + 1.96221i −1.71209 + 0.262166i −0.792042 − 4.49189i −0.766044 + 0.642788i 2.30452 − 3.79114i 0.653841 − 2.56369i 5.68150 + 3.28022i 2.86254 − 0.897705i − 2.56148i
101.5 −1.61635 + 1.92629i −0.354272 − 1.69543i −0.750710 − 4.25749i −0.766044 + 0.642788i 3.83852 + 2.05798i 0.628342 + 2.57006i 5.05915 + 2.92090i −2.74898 + 1.20129i − 2.51459i
101.6 −1.43782 + 1.71353i 1.56857 + 0.734557i −0.521551 − 2.95787i −0.766044 + 0.642788i −3.51401 + 1.63163i 0.966325 + 2.46297i 1.94394 + 1.12233i 1.92085 + 2.30441i − 2.23685i
101.7 −1.32371 + 1.57753i −1.69715 − 0.345963i −0.389111 − 2.20676i −0.766044 + 0.642788i 2.79229 − 2.21935i −2.44651 + 1.00727i 0.429453 + 0.247945i 2.76062 + 1.17430i − 2.05932i
101.8 −1.30168 + 1.55128i 1.43133 − 0.975343i −0.364809 − 2.06893i −0.766044 + 0.642788i −0.350103 + 3.48998i −2.33045 + 1.25261i 0.176866 + 0.102114i 1.09741 − 2.79208i − 2.02506i
101.9 −1.27297 + 1.51707i 0.570838 − 1.63528i −0.333747 − 1.89277i −0.766044 + 0.642788i 1.75418 + 2.94767i 2.12105 − 1.58150i −0.133827 − 0.0772652i −2.34829 − 1.86696i − 1.98040i
101.10 −1.23878 + 1.47632i 1.73041 − 0.0754804i −0.297650 − 1.68806i −0.766044 + 0.642788i −2.03216 + 2.64814i −1.33595 − 2.28369i −0.477169 − 0.275494i 2.98861 − 0.261223i − 1.92720i
101.11 −1.19784 + 1.42753i −0.0314593 + 1.73177i −0.255727 − 1.45030i −0.766044 + 0.642788i −2.43447 − 2.11929i 2.63705 − 0.214365i −0.851026 − 0.491340i −2.99802 − 0.108960i − 1.86351i
101.12 −1.01549 + 1.21022i −1.63743 − 0.564642i −0.0861045 − 0.488323i −0.766044 + 0.642788i 2.34614 − 1.40826i 1.82649 + 1.91415i −2.05793 − 1.18814i 2.36236 + 1.84912i − 1.57983i
101.13 −0.962927 + 1.14757i −0.326259 − 1.70105i −0.0423959 − 0.240439i −0.766044 + 0.642788i 2.26623 + 1.26358i −2.63955 − 0.181112i −2.27795 − 1.31518i −2.78711 + 1.10996i − 1.49805i
101.14 −0.953253 + 1.13604i −0.774124 − 1.54943i −0.0346050 − 0.196254i −0.766044 + 0.642788i 2.49815 + 0.597560i 2.20488 − 1.46237i −2.31269 − 1.33523i −1.80146 + 2.39890i − 1.48300i
101.15 −0.942630 + 1.12338i −1.15546 + 1.29032i −0.0261411 − 0.148254i −0.766044 + 0.642788i −0.360352 − 2.51431i −0.393555 − 2.61632i −2.34882 − 1.35609i −0.329838 − 2.98181i − 1.46647i
101.16 −0.814966 + 0.971239i 1.43791 + 0.965616i 0.0681613 + 0.386562i −0.766044 + 0.642788i −2.10969 + 0.609610i 2.14519 + 1.54859i −2.62700 − 1.51670i 1.13517 + 2.77694i − 1.26786i
101.17 −0.652627 + 0.777771i 0.319084 + 1.70241i 0.168291 + 0.954427i −0.766044 + 0.642788i −1.53232 − 0.862862i −2.64277 + 0.125462i −2.61072 − 1.50730i −2.79637 + 1.08642i − 1.01531i
101.18 −0.487806 + 0.581344i −0.761722 + 1.55556i 0.247290 + 1.40245i −0.766044 + 0.642788i −0.532746 − 1.20164i 0.509410 + 2.59625i −2.25037 − 1.29925i −1.83956 − 2.36981i − 0.758891i
101.19 −0.399745 + 0.476398i −1.69367 + 0.362610i 0.280138 + 1.58874i −0.766044 + 0.642788i 0.504289 − 0.951812i −2.53842 + 0.745945i −1.94601 − 1.12353i 2.73703 − 1.22828i − 0.621893i
101.20 −0.382508 + 0.455855i 0.971199 − 1.43415i 0.285805 + 1.62088i −0.766044 + 0.642788i 0.282271 + 0.991297i −1.00845 − 2.44602i −1.87891 − 1.08479i −1.11354 − 2.78568i − 0.595076i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.48
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} - 2088 T_{2}^{282} + 3 T_{2}^{281} - 324 T_{2}^{280} + 2943 T_{2}^{278} + \cdots + 15\!\cdots\!69 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display