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Properties

Label 945.2.bs.b
Level $945$
Weight $2$
Character orbit 945.bs
Analytic conductor $7.546$
Analytic rank $0$
Dimension $300$
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(16,945)]
 
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([4, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bs (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: \(300\)
Relative dimension: \(50\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 300 q + 6 q^{3} + 6 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 300 q + 6 q^{3} + 6 q^{6} + 6 q^{9} + 6 q^{11} - 42 q^{12} - 3 q^{13} + 18 q^{14} - 24 q^{17} - 24 q^{21} - 42 q^{22} + 9 q^{23} + 60 q^{24} + 24 q^{27} + 18 q^{29} + 12 q^{31} + 42 q^{33} + 60 q^{34} + 60 q^{36} + 96 q^{37} - 60 q^{38} + 6 q^{39} - 60 q^{41} + 33 q^{42} + 15 q^{43} + 12 q^{45} + 120 q^{46} + 9 q^{47} - 66 q^{48} + 18 q^{49} + 12 q^{51} - 102 q^{52} + 45 q^{54} - 87 q^{56} - 12 q^{57} - 30 q^{59} + 30 q^{61} - 81 q^{62} + 66 q^{63} - 234 q^{64} - 3 q^{65} - 54 q^{66} + 18 q^{67} + 87 q^{68} - 90 q^{69} - 27 q^{70} + 12 q^{71} - 57 q^{72} + 36 q^{73} - 78 q^{74} + 108 q^{76} - 45 q^{77} - 6 q^{78} - 9 q^{79} - 186 q^{80} + 6 q^{81} + 102 q^{83} - 129 q^{84} + 18 q^{85} + 3 q^{86} - 30 q^{87} - 84 q^{88} - 30 q^{90} + 18 q^{91} - 24 q^{92} - 12 q^{93} - 192 q^{94} + 48 q^{96} - 45 q^{97} - 24 q^{98} - 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.61977 + 0.953517i 1.31188 + 1.13091i 4.42189 − 3.71041i 0.766044 − 0.642788i −4.51517 − 1.71182i 2.43995 − 1.02306i −5.25849 + 9.10797i 0.442081 + 2.96725i −1.39395 + 2.41439i
16.2 −2.49110 + 0.906685i −0.835921 − 1.51698i 3.85139 − 3.23170i 0.766044 − 0.642788i 3.45778 + 3.02103i 0.452182 − 2.60682i −4.01309 + 6.95087i −1.60247 + 2.53615i −1.32548 + 2.29581i
16.3 −2.38273 + 0.867244i −1.57300 − 0.725034i 3.39322 − 2.84725i 0.766044 − 0.642788i 4.37682 + 0.363388i −2.00316 + 1.72840i −3.08022 + 5.33510i 1.94865 + 2.28095i −1.26783 + 2.19594i
16.4 −2.37912 + 0.865928i −1.14606 + 1.29867i 3.37828 − 2.83471i 0.766044 − 0.642788i 1.60206 − 4.08210i −0.604920 + 2.57567i −3.05086 + 5.28425i −0.373093 − 2.97671i −1.26590 + 2.19261i
16.5 −2.33408 + 0.849537i 0.928348 − 1.46225i 3.19414 − 2.68020i 0.766044 − 0.642788i −0.924608 + 4.20167i −2.32745 − 1.25816i −2.69458 + 4.66715i −1.27634 − 2.71495i −1.24194 + 2.15110i
16.6 −2.31250 + 0.841682i −1.66667 + 0.471388i 3.10715 − 2.60721i 0.766044 − 0.642788i 3.45742 − 2.49289i 2.60806 − 0.445008i −2.52994 + 4.38199i 2.55559 − 1.57130i −1.23046 + 2.13121i
16.7 −2.25490 + 0.820718i 1.45156 − 0.944966i 2.87892 − 2.41570i 0.766044 − 0.642788i −2.49759 + 3.32213i 2.46196 + 0.968899i −2.10946 + 3.65369i 1.21408 − 2.74336i −1.19981 + 2.07813i
16.8 −1.96329 + 0.714578i 0.868733 + 1.49843i 1.81178 − 1.52027i 0.766044 − 0.642788i −2.77632 − 2.32108i −2.63354 − 0.253866i −0.381416 + 0.660631i −1.49061 + 2.60348i −1.04464 + 1.80937i
16.9 −1.91145 + 0.695711i 1.72946 − 0.0946986i 1.63754 − 1.37406i 0.766044 − 0.642788i −3.23989 + 1.38422i −0.588576 − 2.57945i −0.140005 + 0.242495i 2.98206 − 0.327555i −1.01706 + 1.76160i
16.10 −1.81933 + 0.662182i 0.336045 − 1.69914i 1.33939 − 1.12388i 0.766044 − 0.642788i 0.513763 + 3.31382i 1.85064 + 1.89080i 0.243517 − 0.421783i −2.77415 − 1.14197i −0.968045 + 1.67670i
16.11 −1.78829 + 0.650884i −1.61685 + 0.621114i 1.24224 − 1.04236i 0.766044 − 0.642788i 2.48713 − 2.16312i −1.78020 − 1.95727i 0.360029 − 0.623589i 2.22843 − 2.00850i −0.951529 + 1.64810i
16.12 −1.61998 + 0.589625i 0.222860 + 1.71765i 0.744596 − 0.624791i 0.766044 − 0.642788i −1.37380 − 2.65116i 0.370923 − 2.61962i 0.886109 − 1.53479i −2.90067 + 0.765592i −0.861975 + 1.49298i
16.13 −1.49115 + 0.542733i −1.63971 − 0.558000i 0.396872 − 0.333015i 0.766044 − 0.642788i 2.74789 − 0.0578625i 2.52346 − 0.795064i 1.17579 − 2.03653i 2.37727 + 1.82991i −0.793423 + 1.37425i
16.14 −1.38604 + 0.504478i 1.71811 − 0.219281i 0.134524 − 0.112879i 0.766044 − 0.642788i −2.27075 + 1.17068i −1.10303 + 2.40485i 1.34548 − 2.33045i 2.90383 − 0.753500i −0.737497 + 1.27738i
16.15 −1.36717 + 0.497608i 1.57341 + 0.724141i 0.0894451 − 0.0750534i 0.766044 − 0.642788i −2.51145 − 0.207080i 0.818785 + 2.51587i 1.36997 − 2.37286i 1.95124 + 2.27874i −0.727455 + 1.25999i
16.16 −1.09209 + 0.397487i −1.66067 − 0.492127i −0.497431 + 0.417394i 0.766044 − 0.642788i 2.00921 − 0.122647i −1.11075 + 2.40130i 1.53950 − 2.66650i 2.51562 + 1.63452i −0.581087 + 1.00647i
16.17 −1.02934 + 0.374649i −0.409577 − 1.68293i −0.612910 + 0.514293i 0.766044 − 0.642788i 1.05210 + 1.57886i −2.63321 − 0.257295i 1.53361 − 2.65630i −2.66449 + 1.37858i −0.547700 + 0.948645i
16.18 −0.959912 + 0.349379i −0.564141 − 1.63760i −0.732724 + 0.614829i 0.766044 − 0.642788i 1.11367 + 1.37486i 2.58082 − 0.582558i 1.51006 − 2.61550i −2.36349 + 1.84768i −0.510758 + 0.884659i
16.19 −0.865566 + 0.315040i −0.889606 + 1.48614i −0.882134 + 0.740198i 0.766044 − 0.642788i 0.301821 − 1.56661i −2.54596 + 0.719795i 1.45147 − 2.51402i −1.41720 − 2.64415i −0.460558 + 0.797710i
16.20 −0.702919 + 0.255842i 0.0922098 + 1.72959i −1.10345 + 0.925903i 0.766044 − 0.642788i −0.507319 − 1.19217i 1.84148 + 1.89973i 1.28678 − 2.22877i −2.98299 + 0.318971i −0.374016 + 0.647814i
See next 80 embeddings (of 300 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.u 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.bs.b ✓ 300
7.c even 3 1 945.2.bu.b yes 300
27.e even 9 1 945.2.bu.b yes 300
189.u even 9 1 inner 945.2.bs.b ✓ 300
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.bs.b ✓ 300 1.a even 1 1 trivial
945.2.bs.b ✓ 300 189.u even 9 1 inner
945.2.bu.b yes 300 7.c even 3 1
945.2.bu.b yes 300 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{300} + 2321 T_{2}^{294} + 15 T_{2}^{293} - 108 T_{2}^{292} - 456 T_{2}^{291} + \cdots + 53\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display