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Properties

Label 945.2.bt.c
Level $945$
Weight $2$
Character orbit 945.bt
Analytic conductor $7.546$
Analytic rank $0$
Dimension $120$
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(106,945)]
 
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([10, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bt (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: \(120\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 9 q^{8} - 3 q^{10} - 24 q^{12} + 3 q^{13} + 6 q^{15} - 12 q^{16} + 60 q^{18} - 42 q^{19} + 9 q^{22} + 6 q^{23} - 60 q^{24} - 18 q^{26} - 42 q^{27} + 138 q^{28} - 18 q^{29} + 3 q^{30} - 21 q^{32} + 12 q^{33} + 39 q^{34} + 60 q^{35} + 6 q^{36} - 54 q^{37} - 117 q^{38} - 33 q^{39} - 9 q^{40} + 9 q^{41} + 3 q^{42} + 9 q^{43} + 24 q^{44} + 12 q^{45} - 24 q^{46} - 3 q^{47} + 180 q^{48} + 6 q^{51} + 39 q^{52} - 36 q^{53} - 48 q^{54} + 24 q^{55} + 9 q^{56} - 42 q^{57} - 27 q^{58} + 12 q^{59} + 15 q^{60} + 24 q^{61} - 27 q^{62} + 3 q^{63} - 123 q^{64} + 6 q^{65} - 75 q^{66} - 42 q^{67} - 111 q^{68} + 6 q^{69} + 30 q^{71} + 147 q^{72} - 66 q^{73} - 27 q^{74} - 24 q^{76} - 9 q^{77} - 15 q^{78} - 42 q^{79} - 174 q^{80} - 48 q^{81} + 132 q^{82} + 12 q^{83} - 15 q^{85} + 33 q^{86} + 42 q^{87} + 33 q^{88} + 57 q^{89} - 39 q^{90} - 42 q^{91} - 123 q^{92} - 72 q^{93} + 57 q^{94} - 21 q^{95} - 168 q^{96} + 93 q^{97} + 3 q^{98} + 51 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1 −0.477133 + 2.70596i 1.23733 − 1.21203i −5.21516 − 1.89816i −0.766044 + 0.642788i 2.68933 + 3.92646i −0.939693 + 0.342020i 4.87697 − 8.44715i 0.0619682 − 2.99936i −1.37385 − 2.37958i
106.2 −0.439211 + 2.49089i −1.00935 + 1.40756i −4.13223 − 1.50401i −0.766044 + 0.642788i −3.06275 − 3.13238i −0.939693 + 0.342020i 3.03192 − 5.25144i −0.962437 − 2.84143i −1.26466 − 2.19045i
106.3 −0.438546 + 2.48712i −1.06064 − 1.36932i −4.11405 − 1.49739i −0.766044 + 0.642788i 3.87081 − 2.03742i −0.939693 + 0.342020i 3.00291 − 5.20119i −0.750097 + 2.90471i −1.26274 − 2.18714i
106.4 −0.356337 + 2.02089i 0.837316 + 1.51621i −2.07764 − 0.756197i −0.766044 + 0.642788i −3.36247 + 1.15184i −0.939693 + 0.342020i 0.216466 − 0.374929i −1.59780 + 2.53910i −1.02603 − 1.77714i
106.5 −0.307803 + 1.74564i −1.72387 − 0.168129i −1.07312 − 0.390583i −0.766044 + 0.642788i 0.824105 − 2.95750i −0.939693 + 0.342020i −0.760440 + 1.31712i 2.94346 + 0.579667i −0.886283 − 1.53509i
106.6 −0.266393 + 1.51079i 0.740829 − 1.56562i −0.332132 − 0.120886i −0.766044 + 0.642788i 2.16797 + 1.53631i −0.939693 + 0.342020i −1.26298 + 2.18755i −1.90234 − 2.31972i −0.767047 − 1.32857i
106.7 −0.215184 + 1.22037i 1.68861 − 0.385473i 0.436388 + 0.158832i −0.766044 + 0.642788i 0.107057 + 2.14368i −0.939693 + 0.342020i −1.52693 + 2.64473i 2.70282 − 1.30183i −0.619598 − 1.07317i
106.8 −0.214144 + 1.21447i −1.56639 + 0.739211i 0.450299 + 0.163895i −0.766044 + 0.642788i −0.562319 − 2.06063i −0.939693 + 0.342020i −1.52868 + 2.64776i 1.90713 − 2.31578i −0.616604 − 1.06799i
106.9 −0.107469 + 0.609489i 0.309875 + 1.70411i 1.51946 + 0.553038i −0.766044 + 0.642788i −1.07194 + 0.00572638i −0.939693 + 0.342020i −1.11926 + 1.93861i −2.80795 + 1.05612i −0.309446 − 0.535976i
106.10 −0.0154371 + 0.0875479i −0.416016 − 1.68135i 1.87196 + 0.681337i −0.766044 + 0.642788i 0.153621 − 0.0104663i −0.939693 + 0.342020i −0.177446 + 0.307345i −2.65386 + 1.39894i −0.0444493 − 0.0769884i
106.11 0.0670559 − 0.380293i −1.50204 − 0.862486i 1.73926 + 0.633038i −0.766044 + 0.642788i −0.428718 + 0.513380i −0.939693 + 0.342020i 0.743527 − 1.28783i 1.51224 + 2.59097i 0.193080 + 0.334424i
106.12 0.113311 − 0.642619i −0.799467 + 1.53651i 1.47927 + 0.538409i −0.766044 + 0.642788i 0.896799 + 0.687856i −0.939693 + 0.342020i 1.16614 − 2.01982i −1.72170 − 2.45677i 0.326266 + 0.565109i
106.13 0.131377 − 0.745076i 1.63578 + 0.569396i 1.34151 + 0.488268i −0.766044 + 0.642788i 0.639148 − 1.14398i −0.939693 + 0.342020i 1.29661 − 2.24580i 2.35158 + 1.86282i 0.378285 + 0.655209i
106.14 0.151677 − 0.860201i 0.910009 − 1.47373i 1.16245 + 0.423095i −0.766044 + 0.642788i −1.12968 − 1.00632i −0.939693 + 0.342020i 1.41373 − 2.44866i −1.34377 − 2.68222i 0.436736 + 0.756448i
106.15 0.271551 − 1.54004i −1.59775 − 0.668715i −0.418601 − 0.152358i −0.766044 + 0.642788i −1.46372 + 2.27902i −0.939693 + 0.342020i 1.21549 − 2.10529i 2.10564 + 2.13689i 0.781899 + 1.35429i
106.16 0.310156 − 1.75898i −1.51522 + 0.839108i −1.11843 − 0.407075i −0.766044 + 0.642788i 1.00602 + 2.92550i −0.939693 + 0.342020i 0.723191 − 1.25260i 1.59180 − 2.54287i 0.893058 + 1.54682i
106.17 0.334712 − 1.89825i 0.249884 + 1.71393i −1.61193 − 0.586695i −0.766044 + 0.642788i 3.33711 + 0.0993313i −0.939693 + 0.342020i 0.274307 − 0.475113i −2.87512 + 0.856569i 0.963766 + 1.66929i
106.18 0.403656 − 2.28925i 1.60248 − 0.657303i −3.19833 − 1.16410i −0.766044 + 0.642788i −0.857876 − 3.93381i −0.939693 + 0.342020i −1.63137 + 2.82561i 2.13591 − 2.10663i 1.16228 + 2.01313i
106.19 0.439061 − 2.49004i −0.122834 − 1.72769i −4.12815 − 1.50252i −0.766044 + 0.642788i −4.35595 − 0.452699i −0.939693 + 0.342020i −3.02540 + 5.24014i −2.96982 + 0.424439i 1.26423 + 2.18971i
106.20 0.441452 − 2.50360i 1.33541 + 1.10303i −4.19375 − 1.52640i −0.766044 + 0.642788i 3.35108 − 2.85639i −0.939693 + 0.342020i −3.13062 + 5.42239i 0.566629 + 2.94600i 1.27111 + 2.20163i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e 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.bt.c ✓ 120
27.e even 9 1 inner 945.2.bt.c ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.bt.c ✓ 120 1.a even 1 1 trivial
945.2.bt.c ✓ 120 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} - 7 T_{2}^{117} + 3 T_{2}^{116} + 33 T_{2}^{115} + 1004 T_{2}^{114} - 174 T_{2}^{113} + \cdots + 139314069504 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display