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Properties

Label 378.3.bb.a
Level $378$
Weight $3$
Character orbit 378.bb
Analytic conductor $10.300$
Analytic rank $0$
Dimension $216$
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] = [378,3,Mod(29,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.bb (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: \(10.2997539928\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 36 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 36 q^{5} + 12 q^{9} + 24 q^{12} + 60 q^{15} + 96 q^{18} + 144 q^{20} - 72 q^{22} + 72 q^{23} - 36 q^{25} - 108 q^{27} - 216 q^{29} + 180 q^{31} - 144 q^{33} + 144 q^{34} - 240 q^{36} - 360 q^{38} + 48 q^{39} - 108 q^{41} - 180 q^{43} + 636 q^{45} + 324 q^{47} + 48 q^{48} - 144 q^{50} + 504 q^{51} + 360 q^{54} - 492 q^{57} - 324 q^{59} + 864 q^{64} - 576 q^{65} + 1188 q^{67} + 144 q^{68} + 1056 q^{69} + 648 q^{71} + 192 q^{72} + 576 q^{74} + 600 q^{75} + 144 q^{76} + 288 q^{78} + 72 q^{79} - 564 q^{81} - 1188 q^{83} - 168 q^{84} - 360 q^{85} - 792 q^{86} - 120 q^{87} - 288 q^{88} - 1296 q^{89} - 1440 q^{90} - 288 q^{92} - 588 q^{93} - 1224 q^{95} - 192 q^{96} - 1404 q^{97} - 468 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.39273 − 0.245576i −2.97786 − 0.363834i 1.87939 + 0.684040i −3.65567 − 4.35665i 4.05800 + 1.23801i 2.48619 − 0.904900i −2.44949 − 1.41421i 8.73525 + 2.16689i 4.02146 + 6.96538i
29.2 −1.39273 − 0.245576i −2.97427 + 0.392064i 1.87939 + 0.684040i 0.130545 + 0.155578i 4.23863 + 0.184369i −2.48619 + 0.904900i −2.44949 − 1.41421i 8.69257 − 2.33221i −0.143608 − 0.248736i
29.3 −1.39273 − 0.245576i −2.05152 + 2.18890i 1.87939 + 0.684040i 3.50703 + 4.17952i 3.39475 − 2.54474i 2.48619 − 0.904900i −2.44949 − 1.41421i −0.582544 − 8.98113i −3.85795 − 6.68217i
29.4 −1.39273 − 0.245576i −1.82570 − 2.38051i 1.87939 + 0.684040i 1.66525 + 1.98456i 1.95811 + 3.76375i −2.48619 + 0.904900i −2.44949 − 1.41421i −2.33364 + 8.69219i −1.83187 − 3.17290i
29.5 −1.39273 − 0.245576i −1.64950 − 2.50582i 1.87939 + 0.684040i 2.54141 + 3.02874i 1.68193 + 3.89501i 2.48619 − 0.904900i −2.44949 − 1.41421i −3.55832 + 8.26670i −2.79571 − 4.84232i
29.6 −1.39273 − 0.245576i −1.55380 − 2.56626i 1.87939 + 0.684040i −4.10565 − 4.89292i 1.53382 + 3.95568i 2.48619 − 0.904900i −2.44949 − 1.41421i −4.17139 + 7.97493i 4.51647 + 7.82276i
29.7 −1.39273 − 0.245576i −1.47207 + 2.61401i 1.87939 + 0.684040i 1.48112 + 1.76513i 2.69212 − 3.27910i −2.48619 + 0.904900i −2.44949 − 1.41421i −4.66604 − 7.69597i −1.62933 − 2.82208i
29.8 −1.39273 − 0.245576i 0.195285 − 2.99364i 1.87939 + 0.684040i −6.02798 − 7.18387i −1.00714 + 4.12137i −2.48619 + 0.904900i −2.44949 − 1.41421i −8.92373 − 1.16923i 6.63116 + 11.4855i
29.9 −1.39273 − 0.245576i 0.299176 + 2.98504i 1.87939 + 0.684040i 2.57053 + 3.06344i 0.316383 − 4.23083i 2.48619 − 0.904900i −2.44949 − 1.41421i −8.82099 + 1.78611i −2.82775 − 4.89780i
29.10 −1.39273 − 0.245576i 0.654838 + 2.92766i 1.87939 + 0.684040i −2.61405 − 3.11530i −0.193049 − 4.23825i −2.48619 + 0.904900i −2.44949 − 1.41421i −8.14238 + 3.83428i 2.87562 + 4.98072i
29.11 −1.39273 − 0.245576i 0.946892 − 2.84665i 1.87939 + 0.684040i 2.00924 + 2.39452i −2.01783 + 3.73207i 2.48619 − 0.904900i −2.44949 − 1.41421i −7.20679 − 5.39093i −2.21029 − 3.82833i
29.12 −1.39273 − 0.245576i 0.999018 − 2.82877i 1.87939 + 0.684040i −1.93099 − 2.30127i −2.08604 + 3.69438i −2.48619 + 0.904900i −2.44949 − 1.41421i −7.00392 − 5.65199i 2.12421 + 3.67924i
29.13 −1.39273 − 0.245576i 1.35004 + 2.67907i 1.87939 + 0.684040i −2.85923 − 3.40750i −1.22232 − 4.06275i 2.48619 − 0.904900i −2.44949 − 1.41421i −5.35479 + 7.23368i 3.14533 + 5.44787i
29.14 −1.39273 − 0.245576i 1.90265 − 2.31947i 1.87939 + 0.684040i 6.30522 + 7.51427i −3.21947 + 2.76315i −2.48619 + 0.904900i −2.44949 − 1.41421i −1.75988 − 8.82626i −6.93614 − 12.0137i
29.15 −1.39273 − 0.245576i 2.46426 + 1.71097i 1.87939 + 0.684040i 3.47068 + 4.13619i −3.01187 − 2.98808i −2.48619 + 0.904900i −2.44949 − 1.41421i 3.14516 + 8.43255i −3.81796 − 6.61291i
29.16 −1.39273 − 0.245576i 2.83022 + 0.994906i 1.87939 + 0.684040i −3.33285 − 3.97194i −3.69741 − 2.08067i 2.48619 − 0.904900i −2.44949 − 1.41421i 7.02032 + 5.63161i 3.66635 + 6.35030i
29.17 −1.39273 − 0.245576i 2.88030 − 0.838969i 1.87939 + 0.684040i −1.16071 − 1.38328i −4.21751 + 0.461124i −2.48619 + 0.904900i −2.44949 − 1.41421i 7.59226 − 4.83297i 1.27685 + 2.21157i
29.18 −1.39273 − 0.245576i 2.98491 + 0.300568i 1.87939 + 0.684040i 4.64426 + 5.53481i −4.08335 − 1.15163i 2.48619 − 0.904900i −2.44949 − 1.41421i 8.81932 + 1.79433i −5.10898 − 8.84901i
29.19 1.39273 + 0.245576i −2.91431 + 0.711886i 1.87939 + 0.684040i 5.47307 + 6.52255i −4.23367 + 0.275780i 2.48619 − 0.904900i 2.44949 + 1.41421i 7.98644 − 4.14932i 6.02072 + 10.4282i
29.20 1.39273 + 0.245576i −2.83187 − 0.990197i 1.87939 + 0.684040i −0.611728 − 0.729029i −3.70086 − 2.07452i 2.48619 − 0.904900i 2.44949 + 1.41421i 7.03902 + 5.60823i −0.672939 − 1.16556i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.3.bb.a ✓ 216
27.f odd 18 1 inner 378.3.bb.a ✓ 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.3.bb.a ✓ 216 1.a even 1 1 trivial
378.3.bb.a ✓ 216 27.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(378, [\chi])\).