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Properties

Label 1089.2.e.o
Level $1089$
Weight $2$
Character orbit 1089.e
Analytic conductor $8.696$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,2,Mod(364,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1089.e (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: \(8.69570878012\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} + 9 q^{3} - 12 q^{4} + q^{5} + q^{6} + q^{7} + 12 q^{8} - q^{9} + 4 q^{10} - 8 q^{12} + 3 q^{13} - 5 q^{15} + 8 q^{16} + 40 q^{17} - 17 q^{18} + 6 q^{19} + 5 q^{20} + 8 q^{21} + 10 q^{23}+ \cdots + 164 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1 −1.30084 − 2.25313i 1.22607 + 1.22342i −2.38439 + 4.12988i −1.02604 + 1.77715i 1.16159 − 4.35397i 1.59481 + 2.76230i 7.20347 0.00649615 + 2.99999i 5.33887
364.2 −1.22272 − 2.11781i 1.57627 − 0.717885i −1.99007 + 3.44691i 1.50144 − 2.60057i −3.44768 − 2.46048i 1.94598 + 3.37054i 4.84231 1.96928 − 2.26317i −7.34333
364.3 −1.00432 − 1.73953i 0.413284 + 1.68202i −1.01731 + 1.76203i −0.00272589 + 0.00472139i 2.51086 − 2.40820i −1.91390 − 3.31498i 0.0695356 −2.65839 + 1.39030i 0.0109507
364.4 −0.947338 − 1.64084i 0.157872 − 1.72484i −0.794900 + 1.37681i −1.75651 + 3.04236i −2.97974 + 1.37497i −1.71600 − 2.97220i −0.777197 −2.95015 − 0.544607i 6.65603
364.5 −0.741755 − 1.28476i −1.22126 + 1.22822i −0.100401 + 0.173899i 0.0361636 − 0.0626372i 2.48384 + 0.657978i 0.975110 + 1.68894i −2.66913 −0.0170643 − 2.99995i −0.107298
364.6 −0.728693 − 1.26213i 1.66454 − 0.478847i −0.0619858 + 0.107362i 1.50302 − 2.60330i −1.81731 − 1.75194i −0.825091 − 1.42910i −2.73410 2.54141 − 1.59412i −4.38094
364.7 −0.584430 − 1.01226i −1.73005 − 0.0832210i 0.316882 − 0.548856i 1.58346 − 2.74264i 0.926852 + 1.79990i 0.124437 + 0.215531i −3.07850 2.98615 + 0.287953i −3.70170
364.8 −0.332160 − 0.575318i 1.61109 − 0.635904i 0.779339 − 1.34985i −0.558312 + 0.967024i −0.900989 − 0.715670i −1.95227 − 3.38143i −2.36410 2.19125 − 2.04900i 0.741796
364.9 −0.245247 − 0.424780i −0.361332 − 1.69394i 0.879708 − 1.52370i −0.854089 + 1.47932i −0.630937 + 0.568921i 1.70526 + 2.95360i −1.84397 −2.73888 + 1.22415i 0.837851
364.10 0.142574 + 0.246946i 0.0364826 + 1.73167i 0.959345 − 1.66163i −1.35437 + 2.34583i −0.422426 + 0.255900i −2.03667 − 3.52761i 1.11741 −2.99734 + 0.126351i −0.772391
364.11 0.288906 + 0.500400i −1.33220 + 1.10691i 0.833067 − 1.44291i −1.50221 + 2.60191i −0.938776 − 0.346842i −0.582160 − 1.00833i 2.11834 0.549521 − 2.94924i −1.73599
364.12 0.338834 + 0.586878i 0.461570 − 1.66942i 0.770383 − 1.33434i −0.145397 + 0.251836i 1.13614 − 0.294770i 1.67774 + 2.90594i 2.39947 −2.57391 − 1.54111i −0.197063
364.13 0.503287 + 0.871719i 0.903894 + 1.47749i 0.493403 − 0.854600i 1.99556 − 3.45642i −0.833038 + 1.53154i 1.37814 + 2.38701i 3.00644 −1.36595 + 2.67099i 4.01737
364.14 0.719313 + 1.24589i 0.553057 − 1.64138i −0.0348220 + 0.0603135i 1.74427 − 3.02116i 2.44279 − 0.491619i −0.168921 − 0.292580i 2.77706 −2.38826 − 1.81555i 5.01870
364.15 0.773068 + 1.33899i 1.56806 + 0.735660i −0.195269 + 0.338216i −0.296016 + 0.512715i 0.227172 + 2.66833i 0.360975 + 0.625227i 2.48845 1.91761 + 2.30711i −0.915362
364.16 1.06943 + 1.85231i 1.72628 + 0.141233i −1.28737 + 2.22979i −0.837993 + 1.45145i 1.58454 + 3.34865i −0.647817 − 1.12205i −1.22928 2.96011 + 0.487615i −3.58471
364.17 1.10759 + 1.91839i −1.72523 − 0.153513i −1.45349 + 2.51752i 0.311236 − 0.539077i −1.61634 − 3.47971i 1.28962 + 2.23368i −2.00911 2.95287 + 0.529693i 1.37888
364.18 1.16450 + 2.01697i −1.02841 − 1.39369i −1.71213 + 2.96549i 0.158510 − 0.274547i 1.61346 − 3.69723i −0.709246 − 1.22845i −3.31708 −0.884754 + 2.86657i 0.738340
727.1 −1.30084 + 2.25313i 1.22607 − 1.22342i −2.38439 − 4.12988i −1.02604 − 1.77715i 1.16159 + 4.35397i 1.59481 − 2.76230i 7.20347 0.00649615 − 2.99999i 5.33887
727.2 −1.22272 + 2.11781i 1.57627 + 0.717885i −1.99007 − 3.44691i 1.50144 + 2.60057i −3.44768 + 2.46048i 1.94598 − 3.37054i 4.84231 1.96928 + 2.26317i −7.34333
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.2.e.o 36
9.c even 3 1 inner 1089.2.e.o 36
9.c even 3 1 9801.2.a.co 18
9.d odd 6 1 9801.2.a.cn 18
11.b odd 2 1 1089.2.e.p 36
11.d odd 10 2 99.2.m.b ✓ 72
33.f even 10 2 297.2.n.b 72
99.g even 6 1 9801.2.a.cp 18
99.h odd 6 1 1089.2.e.p 36
99.h odd 6 1 9801.2.a.cm 18
99.o odd 30 2 99.2.m.b ✓ 72
99.o odd 30 2 891.2.f.f 36
99.p even 30 2 297.2.n.b 72
99.p even 30 2 891.2.f.e 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.m.b ✓ 72 11.d odd 10 2
99.2.m.b ✓ 72 99.o odd 30 2
297.2.n.b 72 33.f even 10 2
297.2.n.b 72 99.p even 30 2
891.2.f.e 36 99.p even 30 2
891.2.f.f 36 99.o odd 30 2
1089.2.e.o 36 1.a even 1 1 trivial
1089.2.e.o 36 9.c even 3 1 inner
1089.2.e.p 36 11.b odd 2 1
1089.2.e.p 36 99.h odd 6 1
9801.2.a.cm 18 99.h odd 6 1
9801.2.a.cn 18 9.d odd 6 1
9801.2.a.co 18 9.c even 3 1
9801.2.a.cp 18 99.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1089, [\chi])\):

\( T_{2}^{36} + 2 T_{2}^{35} + 26 T_{2}^{34} + 40 T_{2}^{33} + 370 T_{2}^{32} + 489 T_{2}^{31} + \cdots + 3025 \) Copy content Toggle raw display
\( T_{5}^{36} - T_{5}^{35} + 49 T_{5}^{34} + 6 T_{5}^{33} + 1393 T_{5}^{32} + 978 T_{5}^{31} + 27003 T_{5}^{30} + \cdots + 1 \) Copy content Toggle raw display