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Properties

Label 270.2.r.a
Level $270$
Weight $2$
Character orbit 270.r
Analytic conductor $2.156$
Analytic rank $0$
Dimension $216$
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] = [270,2,Mod(23,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([22, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.r (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: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q + 12 q^{6} + 24 q^{11} - 24 q^{20} + 24 q^{23} + 36 q^{25} + 12 q^{27} - 72 q^{30} - 12 q^{33} - 108 q^{35} - 12 q^{36} - 36 q^{38} - 72 q^{41} - 48 q^{42} - 60 q^{45} - 48 q^{47} - 12 q^{48} - 48 q^{50} - 192 q^{51} - 12 q^{56} - 36 q^{57} + 36 q^{61} + 84 q^{63} + 24 q^{65} - 72 q^{67} + 36 q^{68} - 48 q^{72} - 36 q^{75} - 240 q^{77} + 24 q^{78} - 24 q^{81} - 60 q^{83} + 72 q^{86} - 252 q^{87} - 48 q^{92} - 96 q^{93} - 60 q^{95} - 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.422618 − 0.906308i −1.64369 − 0.546145i −0.642788 + 0.766044i 0.0834598 + 2.23451i 0.199679 + 1.72050i 0.151473 − 1.73135i 0.965926 + 0.258819i 2.40345 + 1.79539i 1.98988 − 1.01998i
23.2 −0.422618 − 0.906308i −1.11741 + 1.32341i −0.642788 + 0.766044i 0.315168 − 2.21375i 1.67165 + 0.453419i −0.161670 + 1.84790i 0.965926 + 0.258819i −0.502804 − 2.95756i −2.13953 + 0.649930i
23.3 −0.422618 − 0.906308i −0.485244 + 1.66269i −0.642788 + 0.766044i −1.90917 + 1.16407i 1.71198 − 0.262903i 0.141869 − 1.62157i 0.965926 + 0.258819i −2.52908 − 1.61362i 1.86186 + 1.23834i
23.4 −0.422618 − 0.906308i −0.269916 − 1.71089i −0.642788 + 0.766044i −1.54034 + 1.62092i −1.43652 + 0.967680i −0.313747 + 3.58615i 0.965926 + 0.258819i −2.85429 + 0.923593i 2.12002 + 0.710989i
23.5 −0.422618 − 0.906308i 0.108126 − 1.72867i −0.642788 + 0.766044i 1.61864 − 1.54272i −1.61241 + 0.632573i 0.128886 − 1.47317i 0.965926 + 0.258819i −2.97662 − 0.373831i −2.08225 − 0.815008i
23.6 −0.422618 − 0.906308i 0.670666 + 1.59694i −0.642788 + 0.766044i 2.12921 + 0.682969i 1.16388 − 1.28272i −0.276354 + 3.15874i 0.965926 + 0.258819i −2.10041 + 2.14202i −0.280865 − 2.21836i
23.7 −0.422618 − 0.906308i 1.48032 + 0.899255i −0.642788 + 0.766044i −0.228325 − 2.22438i 0.189392 − 1.72167i 0.448089 − 5.12168i 0.965926 + 0.258819i 1.38268 + 2.66237i −1.91948 + 1.14700i
23.8 −0.422618 − 0.906308i 1.53677 − 0.798965i −0.642788 + 0.766044i 1.54295 + 1.61843i −1.37357 − 1.05513i 4.48025e−5 0 0.000512095i 0.965926 + 0.258819i 1.72331 − 2.45565i 0.814713 − 2.08236i
23.9 −0.422618 − 0.906308i 1.57567 + 0.719210i −0.642788 + 0.766044i −2.23484 − 0.0739999i −0.0140815 − 1.73199i −0.290253 + 3.31761i 0.965926 + 0.258819i 1.96547 + 2.26648i 0.877419 + 2.05673i
23.10 0.422618 + 0.906308i −1.71880 + 0.213803i −0.642788 + 0.766044i 1.33372 − 1.79477i −0.920170 − 1.46741i 0.318607 − 3.64169i −0.965926 − 0.258819i 2.90858 − 0.734972i 2.19027 + 0.450258i
23.11 0.422618 + 0.906308i −1.09878 + 1.33891i −0.642788 + 0.766044i −1.53755 − 1.62356i −1.67783 − 0.429980i −0.298488 + 3.41173i −0.965926 − 0.258819i −0.585380 − 2.94233i 0.821648 − 2.07964i
23.12 0.422618 + 0.906308i −0.931860 − 1.46001i −0.642788 + 0.766044i 1.60755 + 1.55428i 0.929400 − 1.46158i −0.119477 + 1.36563i −0.965926 − 0.258819i −1.26327 + 2.72105i −0.729270 + 2.11380i
23.13 0.422618 + 0.906308i −0.572880 − 1.63457i −0.642788 + 0.766044i −2.22872 + 0.181179i 1.23931 − 1.21000i 0.347359 − 3.97033i −0.965926 − 0.258819i −2.34362 + 1.87282i −1.10610 − 1.94333i
23.14 0.422618 + 0.906308i −0.384296 + 1.68888i −0.642788 + 0.766044i 2.13029 + 0.679602i −1.69306 + 0.365461i −0.165615 + 1.89299i −0.965926 − 0.258819i −2.70463 − 1.29806i 0.284371 + 2.21791i
23.15 0.422618 + 0.906308i 0.931314 + 1.46036i −0.642788 + 0.766044i −1.83318 + 1.28042i −0.929946 + 1.46123i 0.0540957 − 0.618317i −0.965926 − 0.258819i −1.26531 + 2.72011i −1.93519 − 1.12029i
23.16 0.422618 + 0.906308i 1.27806 − 1.16900i −0.642788 + 0.766044i −0.954076 − 2.02231i 1.59960 + 0.664279i 0.108110 − 1.23570i −0.965926 − 0.258819i 0.266890 − 2.98810i 1.42963 − 1.71935i
23.17 0.422618 + 0.906308i 1.48818 − 0.886189i −0.642788 + 0.766044i −0.781119 + 2.09520i 1.43209 + 0.974226i −0.399506 + 4.56637i −0.965926 − 0.258819i 1.42934 − 2.63761i −2.22901 + 0.177535i
23.18 0.422618 + 0.906308i 1.71720 + 0.226328i −0.642788 + 0.766044i 2.03983 + 0.916011i 0.520597 + 1.65196i 0.326578 − 3.73281i −0.965926 − 0.258819i 2.89755 + 0.777302i 0.0318831 + 2.23584i
47.1 −0.422618 + 0.906308i −1.64369 + 0.546145i −0.642788 − 0.766044i 0.0834598 − 2.23451i 0.199679 − 1.72050i 0.151473 + 1.73135i 0.965926 − 0.258819i 2.40345 − 1.79539i 1.98988 + 1.01998i
47.2 −0.422618 + 0.906308i −1.11741 − 1.32341i −0.642788 − 0.766044i 0.315168 + 2.21375i 1.67165 − 0.453419i −0.161670 − 1.84790i 0.965926 − 0.258819i −0.502804 + 2.95756i −2.13953 − 0.649930i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
27.f odd 18 1 inner
135.q even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.2.r.a ✓ 216
3.b odd 2 1 810.2.s.a 216
5.c odd 4 1 inner 270.2.r.a ✓ 216
15.e even 4 1 810.2.s.a 216
27.e even 9 1 810.2.s.a 216
27.f odd 18 1 inner 270.2.r.a ✓ 216
135.q even 36 1 inner 270.2.r.a ✓ 216
135.r odd 36 1 810.2.s.a 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.r.a ✓ 216 1.a even 1 1 trivial
270.2.r.a ✓ 216 5.c odd 4 1 inner
270.2.r.a ✓ 216 27.f odd 18 1 inner
270.2.r.a ✓ 216 135.q even 36 1 inner
810.2.s.a 216 3.b odd 2 1
810.2.s.a 216 15.e even 4 1
810.2.s.a 216 27.e even 9 1
810.2.s.a 216 135.r odd 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(270, [\chi])\).