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Properties

Label 819.2.bb.c
Level $819$
Weight $2$
Character orbit 819.bb
Analytic conductor $6.540$
Analytic rank $0$
Dimension $212$
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] = [819,2,Mod(146,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.146");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.bb (of order \(6\), degree \(2\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(212\)
Relative dimension: \(106\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 212 q - 208 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 212 q - 208 q^{4} + 8 q^{9} - 3 q^{14} - 18 q^{15} + 216 q^{16} - 16 q^{18} - 26 q^{21} + 52 q^{22} + 24 q^{23} - 88 q^{25} - 5 q^{28} - 10 q^{30} - 21 q^{35} - 54 q^{36} + 12 q^{37} + 60 q^{39} - 24 q^{42} + 12 q^{43} + 22 q^{46} - 10 q^{49} - 12 q^{50} + 38 q^{51} + 33 q^{56} - 2 q^{57} + 28 q^{58} + 28 q^{60} + 4 q^{63} - 180 q^{64} - 12 q^{65} - 38 q^{67} + 36 q^{70} - 78 q^{71} + 86 q^{72} + 66 q^{74} - 60 q^{77} + 34 q^{78} - 14 q^{79} + 28 q^{81} + 108 q^{84} - 48 q^{85} - 138 q^{86} - 4 q^{88} - 3 q^{91} - 114 q^{92} - 80 q^{93} + 99 q^{98} - 2 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1 − 2.80368i −0.269893 + 1.71089i −5.86060 −0.715742 − 1.23970i 4.79679 + 0.756693i 2.40401 + 1.10486i 10.8239i −2.85432 − 0.923518i −3.47572 + 2.00671i
146.2 − 2.80368i 0.269893 − 1.71089i −5.86060 0.715742 + 1.23970i −4.79679 − 0.756693i −0.245172 + 2.63437i 10.8239i −2.85432 − 0.923518i 3.47572 − 2.00671i
146.3 − 2.75985i −1.66798 − 0.466740i −5.61677 1.87265 + 3.24352i −1.28813 + 4.60337i 0.365740 − 2.62035i 9.98173i 2.56431 + 1.55702i 8.95163 − 5.16823i
146.4 − 2.75985i 1.66798 + 0.466740i −5.61677 −1.87265 − 3.24352i 1.28813 − 4.60337i −2.45216 − 0.993435i 9.98173i 2.56431 + 1.55702i −8.95163 + 5.16823i
146.5 − 2.52358i −1.73195 − 0.0185617i −4.36843 −1.43443 − 2.48450i −0.0468419 + 4.37071i 1.77216 − 1.96455i 5.97692i 2.99931 + 0.0642959i −6.26982 + 3.61989i
146.6 − 2.52358i 1.73195 + 0.0185617i −4.36843 1.43443 + 2.48450i 0.0468419 − 4.37071i −2.58743 + 0.552456i 5.97692i 2.99931 + 0.0642959i 6.26982 − 3.61989i
146.7 − 2.48495i −1.66649 + 0.472045i −4.17499 0.148529 + 0.257259i 1.17301 + 4.14114i −1.36240 + 2.26801i 5.40476i 2.55435 − 1.57331i 0.639278 − 0.369087i
146.8 − 2.48495i 1.66649 − 0.472045i −4.17499 −0.148529 − 0.257259i −1.17301 − 4.14114i 2.64535 − 0.0458637i 5.40476i 2.55435 − 1.57331i −0.639278 + 0.369087i
146.9 − 2.38201i −1.21203 + 1.23733i −3.67398 −0.794329 − 1.37582i 2.94734 + 2.88706i −2.42305 − 1.06245i 3.98744i −0.0619762 − 2.99936i −3.27721 + 1.89210i
146.10 − 2.38201i 1.21203 − 1.23733i −3.67398 0.794329 + 1.37582i −2.94734 − 2.88706i 0.291415 − 2.62965i 3.98744i −0.0619762 − 2.99936i 3.27721 − 1.89210i
146.11 − 2.37909i −0.320668 − 1.70211i −3.66005 −1.59188 − 2.75722i −4.04946 + 0.762898i 2.31037 − 1.28927i 3.94941i −2.79434 + 1.09162i −6.55967 + 3.78723i
146.12 − 2.37909i 0.320668 + 1.70211i −3.66005 1.59188 + 2.75722i 4.04946 − 0.762898i −2.27172 + 1.35620i 3.94941i −2.79434 + 1.09162i 6.55967 − 3.78723i
146.13 − 2.36481i −0.615663 − 1.61894i −3.59234 −0.452072 − 0.783012i −3.82849 + 1.45593i −1.71183 − 2.01733i 3.76560i −2.24192 + 1.99344i −1.85168 + 1.06907i
146.14 − 2.36481i 0.615663 + 1.61894i −3.59234 0.452072 + 0.783012i 3.82849 − 1.45593i −0.891142 − 2.49116i 3.76560i −2.24192 + 1.99344i 1.85168 − 1.06907i
146.15 − 2.22340i −0.988282 − 1.42243i −2.94353 0.710789 + 1.23112i −3.16263 + 2.19735i 2.23791 + 1.41130i 2.09784i −1.04660 + 2.81152i 2.73728 − 1.58037i
146.16 − 2.22340i 0.988282 + 1.42243i −2.94353 −0.710789 − 1.23112i 3.16263 − 2.19735i 0.103265 + 2.64374i 2.09784i −1.04660 + 2.81152i −2.73728 + 1.58037i
146.17 − 2.18798i −0.761689 + 1.55558i −2.78724 1.66870 + 2.89027i 3.40357 + 1.66656i 2.62915 + 0.295925i 1.72245i −1.83966 − 2.36974i 6.32384 − 3.65107i
146.18 − 2.18798i 0.761689 − 1.55558i −2.78724 −1.66870 − 2.89027i −3.40357 − 1.66656i −1.05830 + 2.42487i 1.72245i −1.83966 − 2.36974i −6.32384 + 3.65107i
146.19 − 1.98985i −1.47885 − 0.901662i −1.95949 0.522157 + 0.904402i −1.79417 + 2.94269i 2.06814 + 1.65009i − 0.0806136i 1.37401 + 2.66685i 1.79962 − 1.03901i
146.20 − 1.98985i 1.47885 + 0.901662i −1.95949 −0.522157 − 0.904402i 1.79417 − 2.94269i 0.394951 + 2.61611i − 0.0806136i 1.37401 + 2.66685i −1.79962 + 1.03901i
See next 80 embeddings (of 212 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.106
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
117.k odd 6 1 inner
819.bb even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.bb.c ✓ 212
7.b odd 2 1 inner 819.2.bb.c ✓ 212
9.d odd 6 1 819.2.de.c yes 212
13.c even 3 1 819.2.de.c yes 212
63.o even 6 1 819.2.de.c yes 212
91.n odd 6 1 819.2.de.c yes 212
117.k odd 6 1 inner 819.2.bb.c ✓ 212
819.bb even 6 1 inner 819.2.bb.c ✓ 212
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.bb.c ✓ 212 1.a even 1 1 trivial
819.2.bb.c ✓ 212 7.b odd 2 1 inner
819.2.bb.c ✓ 212 117.k odd 6 1 inner
819.2.bb.c ✓ 212 819.bb even 6 1 inner
819.2.de.c yes 212 9.d odd 6 1
819.2.de.c yes 212 13.c even 3 1
819.2.de.c yes 212 63.o even 6 1
819.2.de.c yes 212 91.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{106} + 158 T_{2}^{104} + 12087 T_{2}^{102} + 596501 T_{2}^{100} + 21348277 T_{2}^{98} + \cdots + 157323 \) Copy content Toggle raw display
\( T_{5}^{212} + 309 T_{5}^{210} + 49440 T_{5}^{208} + 5412803 T_{5}^{206} + 453445698 T_{5}^{204} + \cdots + 28\!\cdots\!01 \) Copy content Toggle raw display