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Properties

Label 378.3.x.a
Level $378$
Weight $3$
Character orbit 378.x
Analytic conductor $10.300$
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] = [378,3,Mod(103,378)]
 
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([14, 15]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.103");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.x (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: \(288\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 36 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 36 q^{6} - 12 q^{9} + 36 q^{11} + 36 q^{14} - 24 q^{15} + 186 q^{21} + 126 q^{23} + 36 q^{29} - 72 q^{30} + 162 q^{35} - 48 q^{36} + 246 q^{39} - 48 q^{42} + 324 q^{47} - 72 q^{49} + 288 q^{50} - 198 q^{51} + 270 q^{53} + 432 q^{54} - 144 q^{56} + 354 q^{57} + 270 q^{59} + 72 q^{60} + 138 q^{63} - 1152 q^{64} - 360 q^{65} + 72 q^{69} + 180 q^{70} + 216 q^{71} - 96 q^{72} + 216 q^{74} - 1170 q^{75} + 234 q^{77} - 96 q^{78} - 36 q^{79} + 324 q^{81} - 24 q^{84} - 360 q^{85} + 144 q^{86} - 990 q^{87} - 198 q^{91} - 180 q^{92} + 168 q^{93} - 72 q^{95} + 216 q^{98} - 216 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −0.245576 − 1.39273i −2.88152 − 0.834786i −1.87939 + 0.684040i −2.42933 − 0.428357i −0.455001 + 4.21817i 6.98943 − 0.384530i 1.41421 + 2.44949i 7.60626 + 4.81090i 3.48860i
103.2 −0.245576 − 1.39273i −2.88111 + 0.836175i −1.87939 + 0.684040i −2.78557 − 0.491170i 1.87210 + 3.80726i −2.31660 − 6.60555i 1.41421 + 2.44949i 7.60162 − 4.81823i 4.00016i
103.3 −0.245576 − 1.39273i −2.87341 + 0.862258i −1.87939 + 0.684040i 1.69625 + 0.299094i 1.90653 + 3.79014i −3.79765 + 5.88029i 1.41421 + 2.44949i 7.51302 − 4.95525i − 2.43586i
103.4 −0.245576 − 1.39273i −2.63824 + 1.42818i −1.87939 + 0.684040i 9.06832 + 1.59899i 2.63696 + 3.32362i 6.91230 − 1.10459i 1.41421 + 2.44949i 4.92058 − 7.53578i − 13.0224i
103.5 −0.245576 − 1.39273i −2.62193 − 1.45790i −1.87939 + 0.684040i −6.72969 − 1.18663i −1.38658 + 4.00966i −6.13450 − 3.37164i 1.41421 + 2.44949i 4.74904 + 7.64504i 9.66404i
103.6 −0.245576 − 1.39273i −2.51315 − 1.63832i −1.87939 + 0.684040i 7.97184 + 1.40565i −1.66457 + 3.90246i −6.84004 − 1.48792i 1.41421 + 2.44949i 3.63180 + 8.23469i − 11.4478i
103.7 −0.245576 − 1.39273i −1.41523 − 2.64521i −1.87939 + 0.684040i −7.73280 − 1.36350i −3.33651 + 2.62063i −0.535573 + 6.97948i 1.41421 + 2.44949i −4.99426 + 7.48715i 11.1045i
103.8 −0.245576 − 1.39273i −1.33034 − 2.68890i −1.87939 + 0.684040i 4.28833 + 0.756148i −3.41821 + 2.51313i 1.57679 − 6.82010i 1.41421 + 2.44949i −5.46040 + 7.15430i − 6.15817i
103.9 −0.245576 − 1.39273i −1.25381 + 2.72543i −1.87939 + 0.684040i −7.95206 − 1.40216i 4.10369 + 1.07692i −6.38212 + 2.87551i 1.41421 + 2.44949i −5.85592 − 6.83434i 11.4194i
103.10 −0.245576 − 1.39273i −0.821670 + 2.88528i −1.87939 + 0.684040i 3.00275 + 0.529466i 4.22020 + 0.435807i 3.81210 + 5.87094i 1.41421 + 2.44949i −7.64972 − 4.74150i − 4.31204i
103.11 −0.245576 − 1.39273i −0.633702 + 2.93231i −1.87939 + 0.684040i −0.496740 − 0.0875886i 4.23953 + 0.162473i 1.31770 − 6.87486i 1.41421 + 2.44949i −8.19684 − 3.71642i 0.713333i
103.12 −0.245576 − 1.39273i −0.585121 − 2.94239i −1.87939 + 0.684040i 0.493487 + 0.0870150i −3.95425 + 1.53749i 6.41302 + 2.80592i 1.41421 + 2.44949i −8.31527 + 3.44330i − 0.708662i
103.13 −0.245576 − 1.39273i 0.411232 − 2.97168i −1.87939 + 0.684040i 6.35944 + 1.12134i −4.23973 + 0.157039i −0.735425 + 6.96126i 1.41421 + 2.44949i −8.66178 − 2.44410i − 9.13234i
103.14 −0.245576 − 1.39273i 0.976562 + 2.83660i −1.87939 + 0.684040i −4.53175 − 0.799069i 3.71080 − 2.05669i 4.16912 + 5.62303i 1.41421 + 2.44949i −7.09265 + 5.54024i 6.50772i
103.15 −0.245576 − 1.39273i 1.29146 − 2.70779i −1.87939 + 0.684040i 1.51147 + 0.266513i −4.08837 − 1.13369i −6.83845 + 1.49521i 1.41421 + 2.44949i −5.66424 − 6.99402i − 2.17052i
103.16 −0.245576 − 1.39273i 1.40646 − 2.64988i −1.87939 + 0.684040i −5.01328 − 0.883977i −4.03596 − 1.30807i −6.54341 − 2.48672i 1.41421 + 2.44949i −5.04375 − 7.45390i 7.19922i
103.17 −0.245576 − 1.39273i 1.53514 + 2.57747i −1.87939 + 0.684040i 3.89578 + 0.686931i 3.21272 − 2.77099i −6.90502 − 1.14924i 1.41421 + 2.44949i −4.28670 + 7.91354i − 5.59446i
103.18 −0.245576 − 1.39273i 1.89729 + 2.32385i −1.87939 + 0.684040i 2.83526 + 0.499933i 2.77056 − 3.21309i 5.74585 − 3.99815i 1.41421 + 2.44949i −1.80056 + 8.81805i − 4.07152i
103.19 −0.245576 − 1.39273i 2.40164 − 1.79781i −1.87939 + 0.684040i −3.74775 − 0.660829i −3.09364 − 2.90334i 6.98532 − 0.453118i 1.41421 + 2.44949i 2.53579 − 8.63538i 5.38188i
103.20 −0.245576 − 1.39273i 2.62145 + 1.45877i −1.87939 + 0.684040i −9.32067 − 1.64349i 1.38791 − 4.00920i 2.37932 − 6.58322i 1.41421 + 2.44949i 4.74397 + 7.64819i 13.3848i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.x 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.x.a ✓ 288
7.d odd 6 1 378.3.be.a yes 288
27.e even 9 1 378.3.be.a yes 288
189.x odd 18 1 inner 378.3.x.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.3.x.a ✓ 288 1.a even 1 1 trivial
378.3.x.a ✓ 288 189.x odd 18 1 inner
378.3.be.a yes 288 7.d odd 6 1
378.3.be.a yes 288 27.e even 9 1

Hecke kernels

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