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Properties

Label 1089.2.e.p
Level $1089$
Weight $2$
Character orbit 1089.e
Analytic conductor $8.696$
Analytic rank $0$
Dimension $36$
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] = [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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} + 9 q^{3} - 12 q^{4} + q^{5} - q^{6} - q^{7} - 12 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} + 9 q^{3} - 12 q^{4} + q^{5} - q^{6} - q^{7} - 12 q^{8} - q^{9} - 4 q^{10} - 8 q^{12} - 3 q^{13} - 5 q^{15} + 8 q^{16} - 40 q^{17} + 17 q^{18} - 6 q^{19} + 5 q^{20} - 8 q^{21} + 10 q^{23} - 57 q^{24} - 7 q^{25} + 4 q^{26} - 9 q^{27} + 38 q^{28} + 21 q^{29} + 12 q^{30} - 6 q^{31} + 9 q^{32} + 4 q^{34} - 76 q^{35} - 65 q^{36} + 14 q^{37} - 13 q^{38} + 42 q^{39} + 20 q^{41} + 18 q^{42} - 4 q^{43} + 20 q^{45} + 16 q^{46} - 7 q^{47} + 10 q^{48} - 7 q^{49} + 25 q^{50} - 4 q^{51} + 19 q^{52} + 62 q^{53} + 17 q^{54} + 57 q^{56} - 18 q^{57} - 12 q^{58} + 12 q^{59} + 11 q^{60} + 16 q^{61} - 38 q^{62} + 5 q^{63} - 32 q^{64} + 42 q^{65} + 5 q^{67} + 51 q^{68} + 31 q^{69} - 8 q^{70} + 26 q^{71} + 51 q^{72} + 9 q^{74} - 16 q^{75} + 8 q^{76} - 32 q^{78} + 2 q^{79} + 92 q^{80} + 47 q^{81} - 68 q^{82} + 36 q^{83} + 60 q^{84} - 25 q^{85} - 26 q^{86} - 60 q^{87} - 28 q^{89} + 163 q^{90} - 30 q^{91} + 15 q^{92} - 39 q^{93} - 4 q^{94} + 64 q^{95} - 21 q^{96} + 16 q^{97} - 164 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1 −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
364.2 −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
364.3 −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
364.4 −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
364.5 −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
364.6 −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
364.7 −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
364.8 −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
364.9 −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
364.10 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
364.11 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
364.12 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
364.13 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
364.14 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
364.15 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
364.16 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
364.17 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
364.18 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
727.1 −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
727.2 −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
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.p 36
9.c even 3 1 inner 1089.2.e.p 36
9.c even 3 1 9801.2.a.cm 18
9.d odd 6 1 9801.2.a.cp 18
11.b odd 2 1 1089.2.e.o 36
11.c even 5 2 99.2.m.b ✓ 72
33.h odd 10 2 297.2.n.b 72
99.g even 6 1 9801.2.a.cn 18
99.h odd 6 1 1089.2.e.o 36
99.h odd 6 1 9801.2.a.co 18
99.m even 15 2 99.2.m.b ✓ 72
99.m even 15 2 891.2.f.f 36
99.n odd 30 2 297.2.n.b 72
99.n odd 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.c even 5 2
99.2.m.b ✓ 72 99.m even 15 2
297.2.n.b 72 33.h odd 10 2
297.2.n.b 72 99.n odd 30 2
891.2.f.e 36 99.n odd 30 2
891.2.f.f 36 99.m even 15 2
1089.2.e.o 36 11.b odd 2 1
1089.2.e.o 36 99.h odd 6 1
1089.2.e.p 36 1.a even 1 1 trivial
1089.2.e.p 36 9.c even 3 1 inner
9801.2.a.cm 18 9.c even 3 1
9801.2.a.cn 18 99.g even 6 1
9801.2.a.co 18 99.h odd 6 1
9801.2.a.cp 18 9.d odd 6 1

Hecke kernels

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

\( T_{2}^{36} - 2 T_{2}^{35} + 26 T_{2}^{34} - 40 T_{2}^{33} + 370 T_{2}^{32} - 489 T_{2}^{31} + \cdots + 3025 \) 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