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Properties

Label 945.2.do.a
Level $945$
Weight $2$
Character orbit 945.do
Analytic conductor $7.546$
Analytic rank $0$
Dimension $1680$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(2,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([2, 9, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.do (of order \(36\), degree \(12\), 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: \(1680\)
Relative dimension: \(140\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1680 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 36 q^{6} - 12 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1680 q - 6 q^{2} - 6 q^{3} - 6 q^{5} - 36 q^{6} - 12 q^{7} - 36 q^{8} + 6 q^{10} + 12 q^{11} - 6 q^{12} - 24 q^{13} - 24 q^{15} - 12 q^{16} - 18 q^{17} - 6 q^{18} - 24 q^{20} - 24 q^{21} - 24 q^{22} + 42 q^{23} - 6 q^{25} - 24 q^{27} - 24 q^{28} + 30 q^{30} - 12 q^{31} + 90 q^{32} - 6 q^{33} - 18 q^{35} - 84 q^{36} - 12 q^{37} - 30 q^{38} - 54 q^{40} - 72 q^{41} - 108 q^{42} - 24 q^{43} - 6 q^{45} - 24 q^{46} - 6 q^{47} - 96 q^{48} - 72 q^{50} - 12 q^{51} - 30 q^{52} - 90 q^{53} - 48 q^{55} - 72 q^{56} - 24 q^{57} - 6 q^{58} + 186 q^{60} + 24 q^{61} - 36 q^{62} - 54 q^{63} + 138 q^{65} - 12 q^{66} - 6 q^{67} + 114 q^{68} - 12 q^{70} - 72 q^{71} + 18 q^{72} - 12 q^{73} - 150 q^{75} - 156 q^{77} - 84 q^{78} + 12 q^{81} - 12 q^{82} - 24 q^{83} - 24 q^{85} - 120 q^{86} - 102 q^{87} - 6 q^{88} + 30 q^{90} - 12 q^{91} + 108 q^{92} + 138 q^{93} - 66 q^{95} - 444 q^{96} - 24 q^{97} - 360 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.76676 + 0.242060i 0.815510 + 1.52805i 5.62674 − 0.992146i 0.110577 + 2.23333i −2.62620 − 4.03035i 0.601683 + 2.57643i −9.96228 + 2.66939i −1.66989 + 2.49228i −0.846539 − 6.15232i
2.2 −2.71415 + 0.237457i −0.472407 − 1.66638i 5.34058 − 0.941689i −1.98419 + 1.03102i 1.67787 + 4.41063i 2.64434 − 0.0864946i −9.00815 + 2.41373i −2.55366 + 1.57442i 5.14056 − 3.26948i
2.3 −2.68840 + 0.235205i 1.70823 − 0.286299i 5.20258 − 0.917355i 1.52258 − 1.63761i −4.52506 + 1.17147i 2.16379 − 1.52250i −8.55743 + 2.29296i 2.83607 − 0.978126i −3.70813 + 4.76067i
2.4 −2.66896 + 0.233504i 1.58537 − 0.697579i 5.09921 − 0.899129i −2.08358 + 0.811596i −4.06839 + 2.23200i −2.43266 − 1.04028i −8.22391 + 2.20359i 2.02677 − 2.21184i 5.37149 − 2.65264i
2.5 −2.66225 + 0.232916i −0.733987 + 1.56884i 5.06368 − 0.892864i 2.23005 − 0.163979i 1.58864 − 4.34760i −2.62108 − 0.360472i −8.11010 + 2.17309i −1.92253 − 2.30302i −5.89874 + 0.955966i
2.6 −2.64390 + 0.231312i −1.73199 + 0.0144212i 4.96711 − 0.875836i 1.81377 + 1.30776i 4.57588 − 0.438758i 1.34802 − 2.27658i −7.80284 + 2.09076i 2.99958 − 0.0499546i −5.09794 − 3.03803i
2.7 −2.63358 + 0.230409i 0.589797 + 1.62854i 4.91306 − 0.866305i −1.62802 − 1.53282i −1.92851 − 4.15300i 0.271151 − 2.63182i −7.63222 + 2.04505i −2.30428 + 1.92102i 4.64071 + 3.66170i
2.8 −2.62626 + 0.229768i −1.69352 + 0.363310i 4.87482 − 0.859563i −0.553660 − 2.16644i 4.36414 − 1.34326i 2.48666 + 0.903602i −7.51212 + 2.01287i 2.73601 − 1.23055i 1.95183 + 5.56242i
2.9 −2.57748 + 0.225500i −1.30469 − 1.13921i 4.62293 − 0.815147i 0.914018 − 2.04073i 3.61969 + 2.64207i −2.21541 + 1.44636i −6.73335 + 1.80420i 0.404415 + 2.97262i −1.89568 + 5.46604i
2.10 −2.55828 + 0.223820i −1.37638 − 1.05146i 4.52506 − 0.797890i 0.0260964 + 2.23592i 3.75651 + 2.38186i −2.46399 + 0.963729i −6.43668 + 1.72470i 0.788868 + 2.89442i −0.567205 − 5.71425i
2.11 −2.52822 + 0.221190i 0.256485 − 1.71296i 4.37335 − 0.771139i 2.23577 − 0.0367868i −0.269560 + 4.38746i 1.26262 + 2.32504i −5.98341 + 1.60325i −2.86843 − 0.878694i −5.64437 + 0.587535i
2.12 −2.47378 + 0.216428i −1.24785 + 1.20120i 4.10313 − 0.723492i −2.21044 − 0.337590i 2.82693 − 3.24157i −2.18593 + 1.49054i −5.19642 + 1.39238i 0.114248 − 2.99782i 5.54120 + 0.356724i
2.13 −2.34206 + 0.204903i 1.72467 − 0.159690i 3.47363 − 0.612494i 1.93897 + 1.11373i −4.00656 + 0.727395i −1.72790 + 2.00358i −3.46814 + 0.929286i 2.94900 − 0.550827i −4.76939 − 2.21111i
2.14 −2.33659 + 0.204425i 0.652910 − 1.60428i 3.44826 − 0.608021i 0.757838 − 2.10373i −1.19763 + 3.88201i −1.10331 − 2.40472i −3.40168 + 0.911478i −2.14742 − 2.09490i −1.34070 + 5.07048i
2.15 −2.30986 + 0.202087i −0.622516 − 1.61631i 3.32501 − 0.586289i −1.48727 − 1.66974i 1.76456 + 3.60766i −0.389684 − 2.61690i −3.08248 + 0.825947i −2.22495 + 2.01236i 3.77281 + 3.55632i
2.16 −2.25310 + 0.197121i 1.56424 + 0.743734i 3.06801 − 0.540972i 0.548650 + 2.16771i −3.67101 − 1.36736i 0.0213089 − 2.64567i −2.43661 + 0.652887i 1.89372 + 2.32676i −1.66347 − 4.77593i
2.17 −2.24797 + 0.196672i 1.65068 + 0.524636i 3.04505 − 0.536925i −2.19277 + 0.437900i −3.81386 − 0.854721i 1.49648 + 2.18187i −2.38026 + 0.637787i 2.44951 + 1.73202i 4.84315 − 1.41564i
2.18 −2.21921 + 0.194156i 1.46015 − 0.931648i 2.91759 − 0.514450i −1.44831 − 1.70364i −3.05949 + 2.35102i 1.89717 + 1.84411i −2.07131 + 0.555005i 1.26406 − 2.72069i 3.54488 + 3.49954i
2.19 −2.19767 + 0.192271i 0.295033 + 1.70674i 2.82316 − 0.497800i 0.351401 − 2.20828i −0.976541 − 3.69412i −0.166853 + 2.64048i −1.84688 + 0.494869i −2.82591 + 1.00709i −0.347674 + 4.92064i
2.20 −2.13789 + 0.187041i −0.329821 + 1.70036i 2.56596 − 0.452448i −1.66435 + 1.49329i 0.387083 − 3.69686i 2.61310 − 0.414362i −1.25525 + 0.336344i −2.78244 − 1.12163i 3.27890 − 3.50379i
See next 80 embeddings (of 1680 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
189.bc odd 18 1 inner
945.do even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.do.a yes 1680
5.c odd 4 1 inner 945.2.do.a yes 1680
7.c even 3 1 945.2.dh.a ✓ 1680
27.f odd 18 1 945.2.dh.a ✓ 1680
35.l odd 12 1 945.2.dh.a ✓ 1680
135.q even 36 1 945.2.dh.a ✓ 1680
189.bc odd 18 1 inner 945.2.do.a yes 1680
945.do even 36 1 inner 945.2.do.a yes 1680
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.dh.a ✓ 1680 7.c even 3 1
945.2.dh.a ✓ 1680 27.f odd 18 1
945.2.dh.a ✓ 1680 35.l odd 12 1
945.2.dh.a ✓ 1680 135.q even 36 1
945.2.do.a yes 1680 1.a even 1 1 trivial
945.2.do.a yes 1680 5.c odd 4 1 inner
945.2.do.a yes 1680 189.bc odd 18 1 inner
945.2.do.a yes 1680 945.do even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(945, [\chi])\).