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Properties

Label 297.2.n.b
Level $297$
Weight $2$
Character orbit 297.n
Analytic conductor $2.372$
Analytic rank $0$
Dimension $72$
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] = [297,2,Mod(37,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.n (of order \(15\), degree \(8\), not 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: \(2.37155694003\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + q^{2} + 11 q^{4} + 8 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + q^{2} + 11 q^{4} + 8 q^{5} - 2 q^{7} - 6 q^{8} - 8 q^{10} + 2 q^{11} - 11 q^{13} + 10 q^{14} - 9 q^{16} + 20 q^{17} + 8 q^{19} + 45 q^{20} - 16 q^{22} - 20 q^{23} + 11 q^{25} + 12 q^{26} - 54 q^{28} + 23 q^{29} + 3 q^{31} - 18 q^{32} + 8 q^{34} - 18 q^{35} - 42 q^{37} + q^{38} - 25 q^{40} - 10 q^{41} - 8 q^{43} - 38 q^{44} - 18 q^{46} + 34 q^{47} + q^{49} - 27 q^{52} - 4 q^{53} + 18 q^{55} - 114 q^{56} + q^{58} + 16 q^{59} - 3 q^{61} - 184 q^{62} + 26 q^{64} - 84 q^{65} + 10 q^{67} + 23 q^{68} - 46 q^{70} + 48 q^{71} - 40 q^{73} - 68 q^{74} + 16 q^{76} + 26 q^{77} + 19 q^{79} + 56 q^{80} + 94 q^{82} - 7 q^{83} + 25 q^{85} + 77 q^{86} + 18 q^{88} + 56 q^{89} + 20 q^{91} - 50 q^{92} - 63 q^{94} + 77 q^{95} - 33 q^{97} + 328 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −0.255617 + 2.43204i 0 −3.89317 − 0.827519i −0.313886 − 2.98642i 0 2.60423 − 2.89229i 1.49636 − 4.60531i 0 7.34333
37.2 −0.209960 + 1.99763i 0 −1.99016 − 0.423021i 0.000569867 0.00542192i 0 −2.56130 + 2.84461i 0.0214877 − 0.0661323i 0 −0.0109507
37.3 −0.122179 + 1.16246i 0 0.619915 + 0.131767i −0.331034 − 3.14958i 0 0.166529 − 0.184949i −0.951310 + 2.92783i 0 3.70170
37.4 −0.0694404 + 0.660681i 0 1.52462 + 0.324067i 0.116719 + 1.11051i 0 −2.61265 + 2.90164i −0.730548 + 2.24840i 0 −0.741796
37.5 −0.0512706 + 0.487807i 0 1.72097 + 0.365803i 0.178553 + 1.69882i 0 2.28209 − 2.53451i −0.569818 + 1.75372i 0 −0.837851
37.6 0.0603978 − 0.574647i 0 1.62972 + 0.346409i 0.314048 + 2.98797i 0 −0.779082 + 0.865259i 0.654602 − 2.01466i 0 1.73599
37.7 0.150377 − 1.43074i 0 −0.0681222 − 0.0144798i −0.364652 − 3.46943i 0 −0.226061 + 0.251066i 0.858159 − 2.64114i 0 −5.01870
37.8 0.161615 − 1.53767i 0 −0.382004 − 0.0811974i 0.0618842 + 0.588789i 0 0.483079 − 0.536514i 0.768973 − 2.36665i 0 0.915362
37.9 0.231548 − 2.20304i 0 −2.84345 − 0.604395i −0.0650661 − 0.619063i 0 1.72584 − 1.91674i −0.620850 + 1.91078i 0 −1.37888
64.1 −2.54483 + 0.540921i 0 4.35649 − 1.93963i 2.00724 + 0.426651i 0 −0.333407 + 3.17215i −5.82773 + 4.23409i 0 −5.33887
64.2 −1.85327 + 0.393925i 0 1.45235 − 0.646630i 3.43625 + 0.730397i 0 0.358742 − 3.41320i 0.628765 − 0.456825i 0 −6.65603
64.3 −1.45109 + 0.308439i 0 0.183441 − 0.0816732i −0.0707467 − 0.0150377i 0 −0.203853 + 1.93954i 2.15937 − 1.56887i 0 0.107298
64.4 −1.42554 + 0.303007i 0 0.113254 − 0.0504238i −2.94034 − 0.624989i 0 0.172491 − 1.64114i 2.21193 − 1.60706i 0 4.38094
64.5 0.278917 − 0.0592857i 0 −1.75281 + 0.780402i 2.64954 + 0.563177i 0 0.425779 − 4.05102i −0.904002 + 0.656796i 0 0.772391
64.6 0.662860 − 0.140895i 0 −1.40756 + 0.626686i 0.284440 + 0.0604596i 0 −0.350744 + 3.33711i −1.94121 + 1.41037i 0 0.197063
64.7 0.984579 − 0.209279i 0 −0.901493 + 0.401371i −3.90391 − 0.829803i 0 −0.288110 + 2.74118i −2.43227 + 1.76714i 0 −4.01737
64.8 2.09212 − 0.444695i 0 2.35214 − 1.04724i 1.63936 + 0.348457i 0 0.135431 − 1.28854i 0.994512 − 0.722555i 0 3.58471
64.9 2.27811 − 0.484227i 0 3.12821 − 1.39277i −0.310092 − 0.0659121i 0 0.148273 − 1.41072i 2.68358 − 1.94973i 0 −0.738340
91.1 −2.02366 − 0.900991i 0 1.94515 + 2.16031i 0.568657 − 0.253182i 0 −2.52287 + 0.536252i −0.620850 − 1.91078i 0 −1.37888
91.2 −1.41247 − 0.628870i 0 0.261321 + 0.290226i −0.540848 + 0.240801i 0 −0.706174 + 0.150102i 0.768973 + 2.36665i 0 0.915362
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.2.n.b 72
3.b odd 2 1 99.2.m.b ✓ 72
9.c even 3 1 inner 297.2.n.b 72
9.c even 3 1 891.2.f.e 36
9.d odd 6 1 99.2.m.b ✓ 72
9.d odd 6 1 891.2.f.f 36
11.c even 5 1 inner 297.2.n.b 72
33.f even 10 1 1089.2.e.o 36
33.h odd 10 1 99.2.m.b ✓ 72
33.h odd 10 1 1089.2.e.p 36
99.m even 15 1 inner 297.2.n.b 72
99.m even 15 1 891.2.f.e 36
99.m even 15 1 9801.2.a.cp 18
99.n odd 30 1 99.2.m.b ✓ 72
99.n odd 30 1 891.2.f.f 36
99.n odd 30 1 1089.2.e.p 36
99.n odd 30 1 9801.2.a.cm 18
99.o odd 30 1 9801.2.a.cn 18
99.p even 30 1 1089.2.e.o 36
99.p even 30 1 9801.2.a.co 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.m.b ✓ 72 3.b odd 2 1
99.2.m.b ✓ 72 9.d odd 6 1
99.2.m.b ✓ 72 33.h odd 10 1
99.2.m.b ✓ 72 99.n odd 30 1
297.2.n.b 72 1.a even 1 1 trivial
297.2.n.b 72 9.c even 3 1 inner
297.2.n.b 72 11.c even 5 1 inner
297.2.n.b 72 99.m even 15 1 inner
891.2.f.e 36 9.c even 3 1
891.2.f.e 36 99.m even 15 1
891.2.f.f 36 9.d odd 6 1
891.2.f.f 36 99.n odd 30 1
1089.2.e.o 36 33.f even 10 1
1089.2.e.o 36 99.p even 30 1
1089.2.e.p 36 33.h odd 10 1
1089.2.e.p 36 99.n odd 30 1
9801.2.a.cm 18 99.n odd 30 1
9801.2.a.cn 18 99.o odd 30 1
9801.2.a.co 18 99.p even 30 1
9801.2.a.cp 18 99.m even 15 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - T_{2}^{71} - 14 T_{2}^{70} + 19 T_{2}^{69} + 76 T_{2}^{68} - 112 T_{2}^{67} + \cdots + 9150625 \) acting on \(S_{2}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display