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Properties

Label 945.2.cx.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 6 q^{3} - 12 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 6 q^{3} - 12 q^{6} + 6 q^{9} - 6 q^{11} - 42 q^{12} + 3 q^{13} + 42 q^{14} - 9 q^{23} - 24 q^{27} - 36 q^{29} + 12 q^{31} + 60 q^{33} + 60 q^{34} + 42 q^{36} + 42 q^{39} - 54 q^{41} + 63 q^{42} + 45 q^{43} + 30 q^{45} - 63 q^{47} + 18 q^{49} + 72 q^{51} + 102 q^{52} - 123 q^{54} - 27 q^{56} - 12 q^{57} + 48 q^{61} - 99 q^{62} - 66 q^{63} + 144 q^{64} + 3 q^{65} - 18 q^{66} + 54 q^{67} - 117 q^{68} + 198 q^{69} - 21 q^{70} - 36 q^{71} - 237 q^{72} + 6 q^{74} - 108 q^{76} + 27 q^{77} - 48 q^{78} + 9 q^{79} + 144 q^{80} + 6 q^{81} - 18 q^{83} + 69 q^{84} - 18 q^{85} - 3 q^{86} - 30 q^{90} + 18 q^{91} - 96 q^{92} + 12 q^{93} - 174 q^{94} - 60 q^{96} - 45 q^{97} - 234 q^{99}+O(q^{100}) \) 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 −2.73072 − 0.481500i 1.56290 − 0.746563i 5.34563 + 1.94565i 0.939693 + 0.342020i −4.62731 + 1.28612i −1.34078 − 2.28086i −8.85789 − 5.11411i 1.88529 − 2.33360i −2.40136 − 1.38643i
236.2 −2.68821 − 0.474005i −1.72884 − 0.105494i 5.12243 + 1.86441i 0.939693 + 0.342020i 4.59747 + 1.10307i 0.675644 + 2.55803i −8.15849 − 4.71030i 2.97774 + 0.364765i −2.36398 − 1.36484i
236.3 −2.51918 − 0.444199i −1.28377 + 1.16272i 4.26957 + 1.55399i 0.939693 + 0.342020i 3.75054 − 2.35885i −1.38376 − 2.25504i −5.63486 − 3.25329i 0.296155 − 2.98535i −2.21533 − 1.27902i
236.4 −2.32422 − 0.409823i 1.72565 + 0.148740i 3.35467 + 1.22100i 0.939693 + 0.342020i −3.94984 − 1.05292i 1.23983 + 2.33727i −3.20883 − 1.85262i 2.95575 + 0.513348i −2.04389 − 1.18004i
236.5 −2.29821 − 0.405237i −0.275671 − 1.70997i 3.23818 + 1.17860i 0.939693 + 0.342020i −0.0593932 + 4.04159i 2.26525 + 1.36699i −2.92238 − 1.68724i −2.84801 + 0.942780i −2.02101 − 1.16683i
236.6 −2.25960 − 0.398429i 0.781935 − 1.54550i 3.06767 + 1.11654i 0.939693 + 0.342020i −2.38263 + 3.18068i −2.35724 + 1.20142i −2.51274 − 1.45073i −1.77716 − 2.41696i −1.98706 − 1.14723i
236.7 −2.19115 − 0.386359i 1.14111 + 1.30303i 2.77249 + 1.00910i 0.939693 + 0.342020i −1.99690 − 3.29601i −2.20858 + 1.45677i −1.83134 − 1.05733i −0.395758 + 2.97378i −1.92687 − 1.11248i
236.8 −2.17434 − 0.383394i −1.26675 − 1.18124i 2.70136 + 0.983216i 0.939693 + 0.342020i 2.30147 + 3.05408i −2.05721 − 1.66370i −1.67256 − 0.965652i 0.209334 + 2.99269i −1.91208 − 1.10394i
236.9 −2.16306 − 0.381405i −0.962400 + 1.44006i 2.65396 + 0.965962i 0.939693 + 0.342020i 2.63097 − 2.74788i 2.40058 − 1.11230i −1.56792 − 0.905238i −1.14757 − 2.77184i −1.90216 − 1.09821i
236.10 −1.78560 − 0.314850i 0.473342 + 1.66612i 1.20987 + 0.440355i 0.939693 + 0.342020i −0.320624 − 3.12406i −0.782470 − 2.52740i 1.11877 + 0.645923i −2.55189 + 1.57729i −1.57023 − 0.906575i
236.11 −1.76111 − 0.310532i −1.13100 + 1.31181i 1.12571 + 0.409725i 0.939693 + 0.342020i 2.39918 − 1.95903i 0.198720 + 2.63828i 1.24212 + 0.717141i −0.441670 − 2.96731i −1.54870 − 0.894141i
236.12 −1.68254 − 0.296677i 1.27794 − 1.16914i 0.863525 + 0.314298i 0.939693 + 0.342020i −2.49703 + 1.58798i 1.93664 − 1.80262i 1.59953 + 0.923487i 0.266241 − 2.98816i −1.47960 − 0.854246i
236.13 −1.41196 − 0.248967i −1.73185 + 0.0262502i 0.0522696 + 0.0190246i 0.939693 + 0.342020i 2.45185 + 0.394110i −2.23339 + 1.41844i 2.41425 + 1.39387i 2.99862 − 0.0909230i −1.24166 − 0.716872i
236.14 −1.40572 − 0.247866i −1.04572 − 1.38075i 0.0352256 + 0.0128211i 0.939693 + 0.342020i 1.12774 + 2.20015i 2.61177 − 0.422668i 2.42600 + 1.40065i −0.812953 + 2.88775i −1.23617 − 0.713703i
236.15 −1.21161 − 0.213639i −1.73064 + 0.0698820i −0.457035 − 0.166347i 0.939693 + 0.342020i 2.11179 + 0.285063i 1.97939 − 1.75557i 2.64915 + 1.52949i 2.99023 − 0.241881i −1.06547 − 0.615149i
236.16 −0.942667 − 0.166218i 1.65861 − 0.499000i −1.01839 − 0.370665i 0.939693 + 0.342020i −1.64646 + 0.194700i 0.525573 + 2.59302i 2.55633 + 1.47590i 2.50200 − 1.65530i −0.828967 − 0.478605i
236.17 −0.940491 − 0.165834i 1.41925 + 0.992842i −1.02236 − 0.372110i 0.939693 + 0.342020i −1.17014 − 1.16912i 1.56986 − 2.12968i 2.55392 + 1.47451i 1.02853 + 2.81818i −0.827054 − 0.477500i
236.18 −0.938834 − 0.165542i 1.73052 + 0.0729159i −1.02538 − 0.373208i 0.939693 + 0.342020i −1.61260 − 0.354928i −2.58547 − 0.561548i 2.55207 + 1.47344i 2.98937 + 0.252364i −0.825597 − 0.476658i
236.19 −0.714447 − 0.125976i −0.397583 − 1.68580i −1.38482 − 0.504033i 0.939693 + 0.342020i 0.0716811 + 1.25450i −0.713860 + 2.54763i 2.18243 + 1.26003i −2.68386 + 1.34049i −0.628274 − 0.362734i
236.20 −0.683240 − 0.120474i 0.731306 − 1.57009i −1.42708 − 0.519416i 0.939693 + 0.342020i −0.688812 + 0.984646i −1.63510 − 2.08001i 2.11413 + 1.22059i −1.93038 − 2.29644i −0.600831 − 0.346890i
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.a ✓ 288
7.d odd 6 1 945.2.de.a yes 288
27.f odd 18 1 945.2.de.a yes 288
189.bd even 18 1 inner 945.2.cx.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.cx.a ✓ 288 1.a even 1 1 trivial
945.2.cx.a ✓ 288 189.bd even 18 1 inner
945.2.de.a yes 288 7.d odd 6 1
945.2.de.a yes 288 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} - 2088 T_{2}^{282} - 15 T_{2}^{281} - 324 T_{2}^{280} + 2916 T_{2}^{278} + \cdots + 14\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display