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Properties

Label 819.2.i.b
Level $819$
Weight $2$
Character orbit 819.i
Analytic conductor $6.540$
Analytic rank $0$
Dimension $84$
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] = [819,2,Mod(211,819)]
 
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([2, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.211");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.i (of order \(3\), 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: \(84\)
Relative dimension: \(42\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 4 q^{2} - q^{3} - 42 q^{4} + 8 q^{5} - 6 q^{6} - 84 q^{7} - 18 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 4 q^{2} - q^{3} - 42 q^{4} + 8 q^{5} - 6 q^{6} - 84 q^{7} - 18 q^{8} - q^{9} + 10 q^{11} + 4 q^{12} - 3 q^{13} - 4 q^{14} - 12 q^{15} - 42 q^{16} + q^{17} + 14 q^{18} - 40 q^{20} + q^{21} + 28 q^{23} + 36 q^{24} - 36 q^{25} - 5 q^{26} - 40 q^{27} + 42 q^{28} - 13 q^{29} - 10 q^{30} + 12 q^{32} - 13 q^{33} - 8 q^{35} + 18 q^{36} - 28 q^{38} - 37 q^{39} - 48 q^{41} + 6 q^{42} - 96 q^{44} + 8 q^{45} + 6 q^{46} + 13 q^{47} + 19 q^{48} + 84 q^{49} - 124 q^{50} - 21 q^{51} - 3 q^{52} + 16 q^{53} + 65 q^{54} - 6 q^{55} + 18 q^{56} - 35 q^{57} + 18 q^{58} + 40 q^{59} - 26 q^{60} + 12 q^{61} - 9 q^{62} + q^{63} + 102 q^{64} - 10 q^{65} + 46 q^{66} - 34 q^{68} + 6 q^{69} + 20 q^{71} + 34 q^{72} - 24 q^{73} + 54 q^{74} + 15 q^{75} + 36 q^{76} - 10 q^{77} + 16 q^{78} - 39 q^{79} + 60 q^{80} - 33 q^{81} + 15 q^{82} + 55 q^{83} - 4 q^{84} + 9 q^{85} - 14 q^{86} + 16 q^{87} - 12 q^{88} + 51 q^{89} - 33 q^{90} + 3 q^{91} + 6 q^{92} + 15 q^{93} - 54 q^{94} - 6 q^{95} - 41 q^{96} + 42 q^{97} + 4 q^{98} - 39 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1 −1.37492 + 2.38143i 1.41349 + 1.00102i −2.78079 − 4.81648i −0.0286558 + 0.0496333i −4.32729 + 1.98981i −1.00000 9.79378 0.995919 + 2.82987i −0.0787986 − 0.136483i
211.2 −1.26766 + 2.19565i −1.56521 − 0.741699i −2.21392 − 3.83462i −1.19499 + 2.06978i 3.61266 − 2.49643i −1.00000 6.15534 1.89977 + 2.32183i −3.02968 − 5.24756i
211.3 −1.24515 + 2.15667i 0.673445 − 1.59577i −2.10082 − 3.63873i 1.29826 − 2.24865i 2.60300 + 3.43938i −1.00000 5.48278 −2.09294 − 2.14932i 3.23307 + 5.59983i
211.4 −1.21450 + 2.10357i 1.62863 − 0.589546i −1.95000 − 3.37749i 0.302593 − 0.524106i −0.737813 + 4.14193i −1.00000 4.61507 2.30487 − 1.92031i 0.734994 + 1.27305i
211.5 −1.20394 + 2.08529i −1.73032 − 0.0773757i −1.89895 − 3.28907i 1.70243 − 2.94870i 2.24456 − 3.51506i −1.00000 4.32912 2.98803 + 0.267770i 4.09926 + 7.10012i
211.6 −1.08170 + 1.87357i 0.0286952 + 1.73181i −1.34017 − 2.32124i −0.630431 + 1.09194i −3.27571 − 1.81955i −1.00000 1.47184 −2.99835 + 0.0993896i −1.36388 − 2.36231i
211.7 −1.04516 + 1.81027i −1.30062 + 1.14385i −1.18472 − 2.05200i 0.658571 − 1.14068i −0.711315 − 3.54998i −1.00000 0.772250 0.383230 − 2.97542i 1.37663 + 2.38439i
211.8 −0.904784 + 1.56713i 1.28980 − 1.15603i −0.637267 − 1.10378i −0.505643 + 0.875800i 0.644664 + 3.06725i −1.00000 −1.31278 0.327177 − 2.98211i −0.914996 − 1.58482i
211.9 −0.856462 + 1.48344i 1.02367 + 1.39717i −0.467056 − 0.808964i 1.33080 − 2.30502i −2.94936 + 0.321927i −1.00000 −1.82579 −0.904187 + 2.86050i 2.27957 + 3.94833i
211.10 −0.825549 + 1.42989i −1.55708 + 0.758626i −0.363064 − 0.628845i −1.58467 + 2.74474i 0.200689 − 2.85274i −1.00000 −2.10329 1.84897 − 2.36248i −2.61645 − 4.53183i
211.11 −0.660257 + 1.14360i −0.447639 − 1.67321i 0.128120 + 0.221911i 1.08773 − 1.88401i 2.20903 + 0.592827i −1.00000 −2.97940 −2.59924 + 1.49798i 1.43637 + 2.48786i
211.12 −0.594217 + 1.02921i 1.41445 − 0.999670i 0.293813 + 0.508899i −1.81394 + 3.14184i 0.188385 + 2.04979i −1.00000 −3.07522 1.00132 − 2.82796i −2.15575 − 3.73387i
211.13 −0.544167 + 0.942525i −0.139507 − 1.72642i 0.407765 + 0.706269i 0.0159251 − 0.0275830i 1.70311 + 0.807973i −1.00000 −3.06424 −2.96108 + 0.481697i 0.0173318 + 0.0300196i
211.14 −0.536046 + 0.928458i −1.67759 − 0.430901i 0.425310 + 0.736659i −0.517577 + 0.896469i 1.29934 − 1.32659i −1.00000 −3.05613 2.62865 + 1.44576i −0.554889 − 0.961096i
211.15 −0.511214 + 0.885449i −0.431806 + 1.67736i 0.477320 + 0.826743i 1.98245 − 3.43371i −1.26447 − 1.23983i −1.00000 −3.02091 −2.62709 − 1.44859i 2.02692 + 3.51072i
211.16 −0.486262 + 0.842230i 0.650917 + 1.60509i 0.527099 + 0.912963i −1.38506 + 2.39899i −1.66837 − 0.232271i −1.00000 −2.97028 −2.15261 + 2.08956i −1.34700 − 2.33307i
211.17 −0.311459 + 0.539463i −1.69351 − 0.363332i 0.805987 + 1.39601i 0.790671 − 1.36948i 0.723464 − 0.800424i −1.00000 −2.24996 2.73598 + 1.23062i 0.492523 + 0.853075i
211.18 −0.284550 + 0.492855i 1.51071 + 0.847212i 0.838063 + 1.45157i −1.14305 + 1.97982i −0.847424 + 0.503485i −1.00000 −2.09208 1.56446 + 2.55978i −0.650508 − 1.12671i
211.19 −0.0848538 + 0.146971i −1.28656 + 1.15964i 0.985600 + 1.70711i 0.659518 − 1.14232i −0.0612635 − 0.287487i −1.00000 −0.673942 0.310481 − 2.98389i 0.111925 + 0.193860i
211.20 −0.0496196 + 0.0859438i 1.60415 + 0.653227i 0.995076 + 1.72352i 0.934995 − 1.61946i −0.135738 + 0.105454i −1.00000 −0.395980 2.14659 + 2.09575i 0.0927882 + 0.160714i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.f even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.i.b ✓ 84
9.c even 3 1 819.2.t.a yes 84
13.c even 3 1 819.2.t.a yes 84
117.f even 3 1 inner 819.2.i.b ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.i.b ✓ 84 1.a even 1 1 trivial
819.2.i.b ✓ 84 117.f even 3 1 inner
819.2.t.a yes 84 9.c even 3 1
819.2.t.a yes 84 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{84} - 4 T_{2}^{83} + 71 T_{2}^{82} - 246 T_{2}^{81} + 2590 T_{2}^{80} - 8098 T_{2}^{79} + \cdots + 12117361 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display