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Properties

Label 810.2.w.a
Level $810$
Weight $2$
Character orbit 810.w
Analytic conductor $6.468$
Analytic rank $0$
Dimension $1944$
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] = [810,2,Mod(23,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(108))
 
chi = DirichletCharacter(H, H._module([22, 81]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.w (of order \(108\), degree \(36\), 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: \(6.46788256372\)
Analytic rank: \(0\)
Dimension: \(1944\)
Relative dimension: \(54\) over \(\Q(\zeta_{108})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{108}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1944 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1944 q + 36 q^{20} + 108 q^{30} + 108 q^{35} + 36 q^{41} + 216 q^{51} + 108 q^{67} - 144 q^{72} - 576 q^{77} - 576 q^{87} - 144 q^{92} + 36 q^{93} + 108 q^{97}+O(q^{100}) \) 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 −0.989442 + 0.144932i −1.73193 + 0.0200796i 0.957990 − 0.286803i 2.19579 − 0.422505i 1.71074 − 0.270880i −1.03819 − 0.0302082i −0.906308 + 0.422618i 2.99919 − 0.0695530i −2.11137 + 0.736284i
23.2 −0.989442 + 0.144932i −1.72613 − 0.143043i 0.957990 − 0.286803i −0.303554 − 2.21537i 1.72864 − 0.108639i −2.33951 − 0.0680727i −0.906308 + 0.422618i 2.95908 + 0.493822i 0.621426 + 2.14798i
23.3 −0.989442 + 0.144932i −1.72037 − 0.200785i 0.957990 − 0.286803i −1.85060 + 1.25510i 1.73131 − 0.0506719i −2.85262 − 0.0830029i −0.906308 + 0.422618i 2.91937 + 0.690850i 1.64916 − 1.51006i
23.4 −0.989442 + 0.144932i −1.69878 + 0.337832i 0.957990 − 0.286803i −0.579902 + 2.15956i 1.63189 − 0.580473i 2.86135 + 0.0832567i −0.906308 + 0.422618i 2.77174 − 1.14781i 0.260789 − 2.22081i
23.5 −0.989442 + 0.144932i −1.59988 − 0.663620i 0.957990 − 0.286803i 1.84421 + 1.26447i 1.67917 + 0.424740i 2.81768 + 0.0819861i −0.906308 + 0.422618i 2.11922 + 2.12342i −2.00800 − 0.983839i
23.6 −0.989442 + 0.144932i −1.26007 − 1.18837i 0.957990 − 0.286803i 0.248557 − 2.22221i 1.41899 + 0.993204i 3.95411 + 0.115053i −0.906308 + 0.422618i 0.175532 + 2.99486i 0.0761368 + 2.23477i
23.7 −0.989442 + 0.144932i −1.21160 + 1.23775i 0.957990 − 0.286803i −2.06476 − 0.858353i 1.01941 − 1.40029i 0.637598 + 0.0185522i −0.906308 + 0.422618i −0.0640749 − 2.99932i 2.16736 + 0.550041i
23.8 −0.989442 + 0.144932i −1.00497 + 1.41068i 0.957990 − 0.286803i −2.23559 − 0.0464100i 0.789908 − 1.54144i 3.04427 + 0.0885793i −0.906308 + 0.422618i −0.980063 − 2.83540i 2.21871 − 0.278088i
23.9 −0.989442 + 0.144932i −0.865062 − 1.50056i 0.957990 − 0.286803i 2.22919 − 0.175246i 1.07341 + 1.35934i −2.04645 − 0.0595457i −0.906308 + 0.422618i −1.50333 + 2.59615i −2.18025 + 0.496477i
23.10 −0.989442 + 0.144932i −0.835230 + 1.51736i 0.957990 − 0.286803i 0.465389 + 2.18710i 0.606497 − 1.62239i −1.39952 − 0.0407218i −0.906308 + 0.422618i −1.60478 − 2.53469i −0.777456 − 2.09656i
23.11 −0.989442 + 0.144932i −0.714289 + 1.57791i 0.957990 − 0.286803i 1.79217 − 1.33721i 0.478058 − 1.66477i −3.01476 − 0.0877205i −0.906308 + 0.422618i −1.97958 − 2.25416i −1.57945 + 1.58283i
23.12 −0.989442 + 0.144932i −0.686142 − 1.59035i 0.957990 − 0.286803i −2.10154 − 0.763888i 0.909390 + 1.47411i −2.89740 − 0.0843058i −0.906308 + 0.422618i −2.05842 + 2.18241i 2.19006 + 0.451242i
23.13 −0.989442 + 0.144932i −0.107384 − 1.72872i 0.957990 − 0.286803i 0.876175 + 2.05726i 0.356797 + 1.69490i −3.77684 − 0.109895i −0.906308 + 0.422618i −2.97694 + 0.371275i −1.16509 − 1.90855i
23.14 −0.989442 + 0.144932i 0.000411202 1.73205i 0.957990 − 0.286803i 2.15694 + 0.589579i −0.251436 − 1.71370i 4.85974 + 0.141404i −0.906308 + 0.422618i −3.00000 + 0.00142444i −2.21962 − 0.270744i
23.15 −0.989442 + 0.144932i 0.128776 − 1.72726i 0.957990 − 0.286803i −1.90840 + 1.16533i 0.122918 + 1.72768i 3.82371 + 0.111259i −0.906308 + 0.422618i −2.96683 − 0.444858i 1.71936 − 1.42962i
23.16 −0.989442 + 0.144932i 0.148656 − 1.72566i 0.957990 − 0.286803i 0.753503 − 2.10529i 0.103016 + 1.72898i −0.00438391 0.000127559i −0.906308 + 0.422618i −2.95580 − 0.513061i −0.440424 + 2.19227i
23.17 −0.989442 + 0.144932i 0.212730 + 1.71894i 0.957990 − 0.286803i −0.709703 − 2.12045i −0.459612 − 1.66996i 0.0329090 0.000957555i −0.906308 + 0.422618i −2.90949 + 0.731338i 1.00953 + 1.99521i
23.18 −0.989442 + 0.144932i 0.691018 + 1.58824i 0.957990 − 0.286803i 0.152632 + 2.23085i −0.913908 − 1.47132i 0.335576 + 0.00976425i −0.906308 + 0.422618i −2.04499 + 2.19500i −0.474342 − 2.18518i
23.19 −0.989442 + 0.144932i 0.934576 − 1.45828i 0.957990 − 0.286803i 1.98946 + 1.02081i −0.713358 + 1.57833i 3.55193 + 0.103350i −0.906308 + 0.422618i −1.25313 − 2.72574i −2.11640 − 0.721693i
23.20 −0.989442 + 0.144932i 1.19124 − 1.25736i 0.957990 − 0.286803i −0.817927 + 2.08110i −0.996433 + 1.41673i −2.78353 − 0.0809924i −0.906308 + 0.422618i −0.161888 − 2.99563i 0.507673 − 2.17767i
See next 80 embeddings (of 1944 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
81.h odd 54 1 inner
405.x even 108 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.w.a ✓ 1944
5.c odd 4 1 inner 810.2.w.a ✓ 1944
81.h odd 54 1 inner 810.2.w.a ✓ 1944
405.x even 108 1 inner 810.2.w.a ✓ 1944
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.2.w.a ✓ 1944 1.a even 1 1 trivial
810.2.w.a ✓ 1944 5.c odd 4 1 inner
810.2.w.a ✓ 1944 81.h odd 54 1 inner
810.2.w.a ✓ 1944 405.x even 108 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(810, [\chi])\).