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Properties

Label 147.2.m.b
Level $147$
Weight $2$
Character orbit 147.m
Analytic conductor $1.174$
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] = [147,2,Mod(4,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.m (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: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(5\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + q^{2} + 5 q^{3} + 5 q^{4} - 2 q^{5} - 2 q^{6} + 5 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + q^{2} + 5 q^{3} + 5 q^{4} - 2 q^{5} - 2 q^{6} + 5 q^{7} + 6 q^{8} + 5 q^{9} - 34 q^{10} - 11 q^{11} + 5 q^{12} - 2 q^{13} + 40 q^{14} - 3 q^{15} - 31 q^{16} - 9 q^{17} - 6 q^{18} - 29 q^{19} - 43 q^{20} - 11 q^{21} + 9 q^{22} - 4 q^{23} - 24 q^{24} + 55 q^{25} + 36 q^{26} - 10 q^{27} - 57 q^{28} + 4 q^{29} - 20 q^{30} - 39 q^{31} - 92 q^{32} - 18 q^{33} - 36 q^{34} - 33 q^{35} - 10 q^{36} - 24 q^{37} + 118 q^{38} - 6 q^{39} + 35 q^{41} + 38 q^{42} + 2 q^{43} + 40 q^{44} + 12 q^{45} - 40 q^{46} - 5 q^{47} + 76 q^{48} + 129 q^{49} - 176 q^{50} + 54 q^{51} - 6 q^{52} + 26 q^{53} + q^{54} + 2 q^{55} + 63 q^{56} - 12 q^{57} + 11 q^{58} - 41 q^{59} + 32 q^{60} + 6 q^{61} + 36 q^{62} - q^{63} + 74 q^{64} - 51 q^{65} - 15 q^{66} - 55 q^{67} - 22 q^{68} + 8 q^{69} - 68 q^{70} - 66 q^{71} - 24 q^{72} + 24 q^{73} + 28 q^{74} + 41 q^{75} + 3 q^{76} - 34 q^{77} - 30 q^{78} - 51 q^{79} - 5 q^{80} + 5 q^{81} - 41 q^{82} + 30 q^{83} - 30 q^{84} + 68 q^{85} + 110 q^{86} + 19 q^{87} + 129 q^{88} + 75 q^{89} - 16 q^{90} + 5 q^{91} - 4 q^{93} + 38 q^{94} + 36 q^{95} - 57 q^{96} - 168 q^{97} + 227 q^{98} + 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.86275 − 1.27000i −0.733052 − 0.680173i 1.12626 + 2.86965i 3.53817 − 1.09138i 0.501672 + 2.19797i 2.24157 + 1.40547i 0.543186 − 2.37985i 0.0747301 + 0.997204i −7.97679 − 2.46051i
4.2 −0.788109 − 0.537324i −0.733052 − 0.680173i −0.398283 − 1.01481i −2.91381 + 0.898791i 0.212252 + 0.929936i −0.446646 + 2.60778i −0.655894 + 2.87366i 0.0747301 + 0.997204i 2.77934 + 0.857313i
4.3 0.190250 + 0.129710i −0.733052 − 0.680173i −0.711312 − 1.81239i 1.13743 − 0.350851i −0.0512377 − 0.224487i −2.33995 − 1.23477i 0.202234 − 0.886046i 0.0747301 + 0.997204i 0.261905 + 0.0807869i
4.4 1.04710 + 0.713898i −0.733052 − 0.680173i −0.143921 − 0.366706i 1.23854 − 0.382039i −0.282002 − 1.23553i 2.59083 − 0.536271i 0.675095 − 2.95778i 0.0747301 + 0.997204i 1.56961 + 0.484160i
4.5 2.23975 + 1.52704i −0.733052 − 0.680173i 1.95397 + 4.97862i −2.15929 + 0.666054i −0.603206 − 2.64282i 1.59254 − 2.11277i −2.01973 + 8.84903i 0.0747301 + 0.997204i −5.85337 − 1.80553i
16.1 −0.940102 − 2.39534i 0.0747301 + 0.997204i −3.38776 + 3.14338i −3.23329 + 2.20442i 2.31839 − 1.11648i 2.49214 + 0.888404i 6.07754 + 2.92679i −0.988831 + 0.149042i 8.31997 + 5.67246i
16.2 −0.322730 − 0.822302i 0.0747301 + 0.997204i 0.894077 − 0.829582i 0.910959 − 0.621082i 0.795886 − 0.383278i 1.47122 + 2.19898i −2.56248 − 1.23403i −0.988831 + 0.149042i −0.804711 − 0.548643i
16.3 0.317978 + 0.810194i 0.0747301 + 0.997204i 0.910799 − 0.845098i 2.75602 − 1.87902i −0.784166 + 0.377634i −2.42550 − 1.05686i 2.54264 + 1.22447i −0.988831 + 0.149042i 2.39873 + 1.63543i
16.4 0.489645 + 1.24760i 0.0747301 + 0.997204i 0.149360 − 0.138586i −2.98135 + 2.03265i −1.20752 + 0.581509i −2.47781 + 0.927615i 2.66107 + 1.28150i −0.988831 + 0.149042i −3.99573 − 2.72425i
16.5 0.820550 + 2.09073i 0.0747301 + 0.997204i −2.23173 + 2.07075i −0.297708 + 0.202974i −2.02356 + 0.974496i 2.33428 − 1.24544i −2.11349 − 1.01780i −0.988831 + 0.149042i −0.668648 − 0.455876i
25.1 −1.85967 + 0.573632i 0.365341 + 0.930874i 1.47684 − 1.00689i −3.25272 − 0.490269i −1.21339 − 1.52155i −0.730570 − 2.54289i 0.257935 − 0.323440i −0.733052 + 0.680173i 6.33022 − 0.954128i
25.2 −1.48554 + 0.458227i 0.365341 + 0.930874i 0.344369 − 0.234787i 3.62814 + 0.546855i −0.969279 − 1.21544i −2.49763 + 0.872846i 1.53457 − 1.92429i −0.733052 + 0.680173i −5.64032 + 0.850142i
25.3 0.380106 − 0.117247i 0.365341 + 0.930874i −1.52174 + 1.03751i 0.996754 + 0.150237i 0.248010 + 0.310995i 2.64575 + 0.00303574i −0.952800 + 1.19477i −0.733052 + 0.680173i 0.396487 − 0.0597608i
25.4 1.64310 − 0.506829i 0.365341 + 0.930874i 0.790427 − 0.538904i 1.86453 + 0.281032i 1.07209 + 1.34435i −2.42176 − 1.06541i −1.11855 + 1.40262i −0.733052 + 0.680173i 3.20604 − 0.483233i
25.5 2.27757 − 0.702538i 0.365341 + 0.930874i 3.04130 − 2.07352i −3.42100 − 0.515632i 1.48607 + 1.86347i 2.20464 + 1.46273i 2.49792 − 3.13230i −0.733052 + 0.680173i −8.15382 + 1.22899i
37.1 −1.86275 + 1.27000i −0.733052 + 0.680173i 1.12626 − 2.86965i 3.53817 + 1.09138i 0.501672 − 2.19797i 2.24157 − 1.40547i 0.543186 + 2.37985i 0.0747301 − 0.997204i −7.97679 + 2.46051i
37.2 −0.788109 + 0.537324i −0.733052 + 0.680173i −0.398283 + 1.01481i −2.91381 − 0.898791i 0.212252 − 0.929936i −0.446646 − 2.60778i −0.655894 − 2.87366i 0.0747301 − 0.997204i 2.77934 − 0.857313i
37.3 0.190250 − 0.129710i −0.733052 + 0.680173i −0.711312 + 1.81239i 1.13743 + 0.350851i −0.0512377 + 0.224487i −2.33995 + 1.23477i 0.202234 + 0.886046i 0.0747301 − 0.997204i 0.261905 − 0.0807869i
37.4 1.04710 − 0.713898i −0.733052 + 0.680173i −0.143921 + 0.366706i 1.23854 + 0.382039i −0.282002 + 1.23553i 2.59083 + 0.536271i 0.675095 + 2.95778i 0.0747301 − 0.997204i 1.56961 − 0.484160i
37.5 2.23975 − 1.52704i −0.733052 + 0.680173i 1.95397 − 4.97862i −2.15929 − 0.666054i −0.603206 + 2.64282i 1.59254 + 2.11277i −2.01973 − 8.84903i 0.0747301 − 0.997204i −5.85337 + 1.80553i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
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 147.2.m.b ✓ 60
3.b odd 2 1 441.2.bb.e 60
49.g even 21 1 inner 147.2.m.b ✓ 60
49.g even 21 1 7203.2.a.n 30
49.h odd 42 1 7203.2.a.m 30
147.n odd 42 1 441.2.bb.e 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.m.b ✓ 60 1.a even 1 1 trivial
147.2.m.b ✓ 60 49.g even 21 1 inner
441.2.bb.e 60 3.b odd 2 1
441.2.bb.e 60 147.n odd 42 1
7203.2.a.m 30 49.h odd 42 1
7203.2.a.n 30 49.g even 21 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 \(S_{2}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display