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Properties

Label 816.2.bl.d
Level $816$
Weight $2$
Character orbit 816.bl
Analytic conductor $6.516$
Analytic rank $0$
Dimension $72$
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] = [816,2,Mod(13,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.bl (of order \(4\), 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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 2 q^{2} - 8 q^{5} - 2 q^{6} - 4 q^{8} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 2 q^{2} - 8 q^{5} - 2 q^{6} - 4 q^{8} - 72 q^{9} - 18 q^{10} + 8 q^{14} + 4 q^{16} - 4 q^{17} - 2 q^{18} - 8 q^{20} + 6 q^{22} - 16 q^{23} + 8 q^{24} + 72 q^{25} + 26 q^{26} + 4 q^{28} + 2 q^{30} + 12 q^{32} + 6 q^{34} + 32 q^{37} + 14 q^{38} - 8 q^{40} - 24 q^{41} + 8 q^{42} - 4 q^{44} + 8 q^{45} - 2 q^{46} + 16 q^{47} + 16 q^{48} + 8 q^{50} - 4 q^{51} - 32 q^{52} + 2 q^{54} - 76 q^{56} - 28 q^{58} - 32 q^{59} - 8 q^{60} + 48 q^{61} - 20 q^{62} - 12 q^{64} - 8 q^{65} - 22 q^{66} - 48 q^{68} - 16 q^{69} + 48 q^{70} + 4 q^{72} + 8 q^{73} - 24 q^{74} + 44 q^{76} + 6 q^{78} - 8 q^{79} - 92 q^{80} + 72 q^{81} + 14 q^{82} + 24 q^{84} + 8 q^{85} - 34 q^{86} + 24 q^{87} + 52 q^{88} + 18 q^{90} + 32 q^{92} - 20 q^{94} - 12 q^{96} - 24 q^{97} - 2 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.41091 − 0.0965758i − 1.00000i 1.98135 + 0.272520i 1.39253 −0.0965758 + 1.41091i −2.15553 − 2.15553i −2.76919 − 0.575852i −1.00000 −1.96473 − 0.134485i
13.2 −1.39453 + 0.235132i − 1.00000i 1.88943 − 0.655798i 2.25027 0.235132 + 1.39453i 1.34832 + 1.34832i −2.48066 + 1.35879i −1.00000 −3.13806 + 0.529111i
13.3 −1.39097 − 0.255364i − 1.00000i 1.86958 + 0.710405i 1.20145 −0.255364 + 1.39097i 2.01365 + 2.01365i −2.41911 − 1.46557i −1.00000 −1.67118 − 0.306806i
13.4 −1.37665 − 0.323789i − 1.00000i 1.79032 + 0.891487i −2.95467 −0.323789 + 1.37665i −0.222688 − 0.222688i −2.17599 − 1.80695i −1.00000 4.06755 + 0.956690i
13.5 −1.27265 + 0.616744i − 1.00000i 1.23925 − 1.56979i −2.96351 0.616744 + 1.27265i 3.17509 + 3.17509i −0.608969 + 2.76209i −1.00000 3.77149 − 1.82773i
13.6 −1.23158 + 0.695138i − 1.00000i 1.03357 − 1.71223i −1.59923 0.695138 + 1.23158i −2.23229 − 2.23229i −0.0826772 + 2.82722i −1.00000 1.96957 − 1.11169i
13.7 −1.13691 − 0.841087i − 1.00000i 0.585145 + 1.91249i 0.471982 −0.841087 + 1.13691i −2.18831 − 2.18831i 0.943309 − 2.66649i −1.00000 −0.536602 − 0.396978i
13.8 −1.10822 − 0.878544i − 1.00000i 0.456322 + 1.94725i −3.30647 −0.878544 + 1.10822i 0.861755 + 0.861755i 1.20503 − 2.55889i −1.00000 3.66431 + 2.90488i
13.9 −1.03074 − 0.968288i − 1.00000i 0.124836 + 1.99610i 4.19403 −0.968288 + 1.03074i 0.848621 + 0.848621i 1.80413 − 2.17833i −1.00000 −4.32295 − 4.06103i
13.10 −1.02419 + 0.975212i − 1.00000i 0.0979225 − 1.99760i 3.55662 0.975212 + 1.02419i −2.62071 − 2.62071i 1.84779 + 2.14141i −1.00000 −3.64265 + 3.46846i
13.11 −0.871316 + 1.11392i − 1.00000i −0.481618 − 1.94115i 1.78157 1.11392 + 0.871316i 1.41668 + 1.41668i 2.58191 + 1.15487i −1.00000 −1.55231 + 1.98452i
13.12 −0.746512 + 1.20113i − 1.00000i −0.885440 − 1.79332i −1.52429 1.20113 + 0.746512i 0.675370 + 0.675370i 2.81501 + 0.275203i −1.00000 1.13790 − 1.83087i
13.13 −0.686566 − 1.23638i − 1.00000i −1.05725 + 1.69771i 2.01124 −1.23638 + 0.686566i −1.39556 − 1.39556i 2.82488 + 0.141575i −1.00000 −1.38085 − 2.48665i
13.14 −0.274441 − 1.38733i − 1.00000i −1.84936 + 0.761481i −2.36619 −1.38733 + 0.274441i 1.31188 + 1.31188i 1.56397 + 2.35669i −1.00000 0.649380 + 3.28268i
13.15 −0.213189 + 1.39805i − 1.00000i −1.90910 − 0.596098i 2.25156 1.39805 + 0.213189i −1.99225 − 1.99225i 1.24038 − 2.54194i −1.00000 −0.480008 + 3.14780i
13.16 −0.135293 + 1.40773i − 1.00000i −1.96339 − 0.380911i −2.06081 1.40773 + 0.135293i −2.95533 − 2.95533i 0.801852 − 2.71239i −1.00000 0.278813 − 2.90106i
13.17 −0.126615 − 1.40853i − 1.00000i −1.96794 + 0.356683i 0.580846 −1.40853 + 0.126615i 0.405584 + 0.405584i 0.751570 + 2.72675i −1.00000 −0.0735437 − 0.818142i
13.18 −0.109244 + 1.40999i − 1.00000i −1.97613 − 0.308066i −1.08323 1.40999 + 0.109244i 1.05243 + 1.05243i 0.650250 − 2.75267i −1.00000 0.118337 − 1.52734i
13.19 −0.0253084 − 1.41399i − 1.00000i −1.99872 + 0.0715715i −4.20239 −1.41399 + 0.0253084i −3.33388 − 3.33388i 0.151786 + 2.82435i −1.00000 0.106356 + 5.94213i
13.20 0.166614 − 1.40436i − 1.00000i −1.94448 − 0.467975i 3.35217 −1.40436 − 0.166614i 2.43776 + 2.43776i −0.981186 + 2.65279i −1.00000 0.558520 − 4.70767i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
272.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.bl.d yes 72
16.e even 4 1 816.2.s.d ✓ 72
17.c even 4 1 816.2.s.d ✓ 72
272.j even 4 1 inner 816.2.bl.d yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.s.d ✓ 72 16.e even 4 1
816.2.s.d ✓ 72 17.c even 4 1
816.2.bl.d yes 72 1.a even 1 1 trivial
816.2.bl.d yes 72 272.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(816, [\chi])\):

\( T_{5}^{36} + 4 T_{5}^{35} - 100 T_{5}^{34} - 400 T_{5}^{33} + 4488 T_{5}^{32} + 17984 T_{5}^{31} + \cdots - 582025216 \) Copy content Toggle raw display
\( T_{7}^{72} + 8 T_{7}^{69} + 2368 T_{7}^{68} + 576 T_{7}^{67} + 32 T_{7}^{66} + 5344 T_{7}^{65} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display