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Properties

Label 441.2.bb.e
Level $441$
Weight $2$
Character orbit 441.bb
Analytic conductor $3.521$
Analytic rank $0$
Dimension $60$
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] = [441,2,Mod(37,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 32]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.bb (of order \(21\), degree \(12\), 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(5\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 147)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - q^{2} + 5 q^{4} + 2 q^{5} + 5 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - q^{2} + 5 q^{4} + 2 q^{5} + 5 q^{7} - 6 q^{8} - 34 q^{10} + 11 q^{11} - 2 q^{13} - 40 q^{14} - 31 q^{16} + 9 q^{17} - 29 q^{19} + 43 q^{20} + 9 q^{22} + 4 q^{23} + 55 q^{25} - 36 q^{26} - 57 q^{28} - 4 q^{29} - 39 q^{31} + 92 q^{32} - 36 q^{34} + 33 q^{35} - 24 q^{37} - 118 q^{38} - 35 q^{41} + 2 q^{43} - 40 q^{44} - 40 q^{46} + 5 q^{47} + 129 q^{49} + 176 q^{50} - 6 q^{52} - 26 q^{53} + 2 q^{55} - 63 q^{56} + 11 q^{58} + 41 q^{59} + 6 q^{61} - 36 q^{62} + 74 q^{64} + 51 q^{65} - 55 q^{67} + 22 q^{68} - 68 q^{70} + 66 q^{71} + 24 q^{73} - 28 q^{74} + 3 q^{76} + 34 q^{77} - 51 q^{79} + 5 q^{80} - 41 q^{82} - 30 q^{83} + 68 q^{85} - 110 q^{86} + 129 q^{88} - 75 q^{89} + 5 q^{91} + 38 q^{94} - 36 q^{95} - 168 q^{97} - 227 q^{98}+O(q^{100}) \) 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 −2.23975 + 1.52704i 0 1.95397 − 4.97862i 2.15929 + 0.666054i 0 1.59254 + 2.11277i 2.01973 + 8.84903i 0 −5.85337 + 1.80553i
37.2 −1.04710 + 0.713898i 0 −0.143921 + 0.366706i −1.23854 − 0.382039i 0 2.59083 + 0.536271i −0.675095 − 2.95778i 0 1.56961 − 0.484160i
37.3 −0.190250 + 0.129710i 0 −0.711312 + 1.81239i −1.13743 − 0.350851i 0 −2.33995 + 1.23477i −0.202234 − 0.886046i 0 0.261905 − 0.0807869i
37.4 0.788109 − 0.537324i 0 −0.398283 + 1.01481i 2.91381 + 0.898791i 0 −0.446646 − 2.60778i 0.655894 + 2.87366i 0 2.77934 − 0.857313i
37.5 1.86275 − 1.27000i 0 1.12626 − 2.86965i −3.53817 − 1.09138i 0 2.24157 − 1.40547i −0.543186 − 2.37985i 0 −7.97679 + 2.46051i
46.1 −0.820550 + 2.09073i 0 −2.23173 − 2.07075i 0.297708 + 0.202974i 0 2.33428 + 1.24544i 2.11349 − 1.01780i 0 −0.668648 + 0.455876i
46.2 −0.489645 + 1.24760i 0 0.149360 + 0.138586i 2.98135 + 2.03265i 0 −2.47781 − 0.927615i −2.66107 + 1.28150i 0 −3.99573 + 2.72425i
46.3 −0.317978 + 0.810194i 0 0.910799 + 0.845098i −2.75602 − 1.87902i 0 −2.42550 + 1.05686i −2.54264 + 1.22447i 0 2.39873 − 1.63543i
46.4 0.322730 − 0.822302i 0 0.894077 + 0.829582i −0.910959 − 0.621082i 0 1.47122 − 2.19898i 2.56248 − 1.23403i 0 −0.804711 + 0.548643i
46.5 0.940102 − 2.39534i 0 −3.38776 − 3.14338i 3.23329 + 2.20442i 0 2.49214 − 0.888404i −6.07754 + 2.92679i 0 8.31997 − 5.67246i
100.1 −2.27757 − 0.702538i 0 3.04130 + 2.07352i 3.42100 − 0.515632i 0 2.20464 − 1.46273i −2.49792 − 3.13230i 0 −8.15382 − 1.22899i
100.2 −1.64310 − 0.506829i 0 0.790427 + 0.538904i −1.86453 + 0.281032i 0 −2.42176 + 1.06541i 1.11855 + 1.40262i 0 3.20604 + 0.483233i
100.3 −0.380106 − 0.117247i 0 −1.52174 − 1.03751i −0.996754 + 0.150237i 0 2.64575 − 0.00303574i 0.952800 + 1.19477i 0 0.396487 + 0.0597608i
100.4 1.48554 + 0.458227i 0 0.344369 + 0.234787i −3.62814 + 0.546855i 0 −2.49763 − 0.872846i −1.53457 − 1.92429i 0 −5.64032 − 0.850142i
100.5 1.85967 + 0.573632i 0 1.47684 + 1.00689i 3.25272 − 0.490269i 0 −0.730570 + 2.54289i −0.257935 − 0.323440i 0 6.33022 + 0.954128i
109.1 −1.60276 + 1.48714i 0 0.207780 − 2.77264i −0.252155 + 0.642481i 0 −0.451208 − 2.60699i 1.06386 + 1.33404i 0 −0.551316 − 1.40473i
109.2 −0.502390 + 0.466150i 0 −0.114360 + 1.52603i 1.18268 − 3.01342i 0 −2.63978 + 0.177638i −1.50851 − 1.89161i 0 0.810538 + 2.06522i
109.3 −0.227517 + 0.211105i 0 −0.142261 + 1.89835i −1.53122 + 3.90148i 0 2.06200 + 1.65775i −0.755410 − 0.947254i 0 −0.475244 − 1.21090i
109.4 1.03880 − 0.963867i 0 0.000608887 − 0.00812503i 0.107830 − 0.274746i 0 −1.39076 + 2.25073i 1.75989 + 2.20683i 0 −0.152805 − 0.389340i
109.5 2.02691 − 1.88070i 0 0.421883 − 5.62963i −0.259960 + 0.662367i 0 2.63989 + 0.175969i −6.28460 − 7.88063i 0 0.718799 + 1.83147i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.bb.e 60
3.b odd 2 1 147.2.m.b ✓ 60
49.g even 21 1 inner 441.2.bb.e 60
147.n odd 42 1 147.2.m.b ✓ 60
147.n odd 42 1 7203.2.a.n 30
147.o even 42 1 7203.2.a.m 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.m.b ✓ 60 3.b odd 2 1
147.2.m.b ✓ 60 147.n odd 42 1
441.2.bb.e 60 1.a even 1 1 trivial
441.2.bb.e 60 49.g even 21 1 inner
7203.2.a.m 30 147.o even 42 1
7203.2.a.n 30 147.n odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + T_{2}^{59} - 7 T_{2}^{58} - 8 T_{2}^{57} + 14 T_{2}^{56} - 31 T_{2}^{55} + 115 T_{2}^{54} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\). Copy content Toggle raw display