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Properties

Label 147.2.i.b
Level $147$
Weight $2$
Character orbit 147.i
Analytic conductor $1.174$
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] = [147,2,Mod(22,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.i (of order \(7\), degree \(6\), 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: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - q^{2} - 6 q^{3} - 9 q^{4} - 4 q^{5} - q^{6} - 6 q^{7} - 15 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - q^{2} - 6 q^{3} - 9 q^{4} - 4 q^{5} - q^{6} - 6 q^{7} - 15 q^{8} - 6 q^{9} + 10 q^{10} - 7 q^{11} - 9 q^{12} - 12 q^{13} - q^{14} + 3 q^{15} - 3 q^{16} - 3 q^{17} + 6 q^{18} + 6 q^{19} + 25 q^{20} + q^{21} - 21 q^{22} - 20 q^{23} + 6 q^{24} - 2 q^{25} + 6 q^{26} - 6 q^{27} - q^{28} - 22 q^{29} - 4 q^{30} + 16 q^{31} + 26 q^{32} + 6 q^{34} - 9 q^{35} - 9 q^{36} + 32 q^{37} + 17 q^{38} + 23 q^{39} - 21 q^{40} - 5 q^{41} - 8 q^{42} - 34 q^{43} + 2 q^{44} + 3 q^{45} - 32 q^{46} - 7 q^{47} + 74 q^{48} + 20 q^{49} + 236 q^{50} + 18 q^{51} + 20 q^{52} - 32 q^{53} - q^{54} - 17 q^{55} - 39 q^{56} - 8 q^{57} - 53 q^{58} - q^{59} - 17 q^{60} + 14 q^{61} - 60 q^{62} + 22 q^{63} - 21 q^{64} - 39 q^{65} - 21 q^{66} - 22 q^{67} - 110 q^{68} - 20 q^{69} - 40 q^{70} + 36 q^{71} + 6 q^{72} - 11 q^{73} - 46 q^{74} - 9 q^{75} - 101 q^{76} - 17 q^{77} + 6 q^{78} - 14 q^{79} - 112 q^{80} - 6 q^{81} + 2 q^{82} + 12 q^{83} - 22 q^{84} - 44 q^{85} + 184 q^{86} - q^{87} + 204 q^{88} + 12 q^{89} + 10 q^{90} - 16 q^{91} - 105 q^{92} - 19 q^{93} - 5 q^{94} + 18 q^{95} - 9 q^{96} + 172 q^{97} + q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −2.37817 + 1.14527i −0.222521 + 0.974928i 3.09710 − 3.88364i −0.398449 + 1.74572i −0.587361 − 2.57340i 0.566955 + 2.58429i −1.74291 + 7.63618i −0.900969 − 0.433884i −1.05174 − 4.60796i
22.2 −1.60360 + 0.772253i −0.222521 + 0.974928i 0.728177 − 0.913105i 0.987738 − 4.32756i −0.396056 − 1.73524i −2.63181 − 0.271221i 0.329556 − 1.44388i −0.900969 − 0.433884i 1.75804 + 7.70246i
22.3 −1.11145 + 0.535248i −0.222521 + 0.974928i −0.298141 + 0.373858i −0.661767 + 2.89939i −0.274506 − 1.20269i −0.553581 − 2.58719i 0.680276 − 2.98049i −0.900969 − 0.433884i −0.816369 − 3.57675i
22.4 0.402966 − 0.194058i −0.222521 + 0.974928i −1.12226 + 1.40727i −0.162101 + 0.710213i 0.0995244 + 0.436045i −1.42812 + 2.22721i −0.378189 + 1.65695i −0.900969 − 0.433884i 0.0725012 + 0.317649i
22.5 1.54657 − 0.744788i −0.222521 + 0.974928i 0.590184 − 0.740067i 0.580458 − 2.54315i 0.381971 + 1.67352i 2.64218 − 0.137475i −0.402375 + 1.76292i −0.900969 − 0.433884i −0.996391 − 4.36547i
22.6 2.24272 − 1.08004i −0.222521 + 0.974928i 2.61635 − 3.28080i −0.222387 + 0.974342i 0.553907 + 2.42683i −2.64454 − 0.0800852i 1.21655 − 5.33004i −0.900969 − 0.433884i 0.553574 + 2.42537i
43.1 −1.57410 + 1.97386i −0.900969 − 0.433884i −0.973283 − 4.26423i −0.136210 − 0.0655954i 2.27464 − 1.09541i 0.357057 − 2.62155i 5.39974 + 2.60038i 0.623490 + 0.781831i 0.343884 − 0.165606i
43.2 −0.894323 + 1.12145i −0.900969 − 0.433884i −0.0127847 − 0.0560133i 1.03743 + 0.499602i 1.29233 − 0.622355i 2.02504 + 1.70270i −2.51042 − 1.20895i 0.623490 + 0.781831i −1.48808 + 0.716620i
43.3 −0.385632 + 0.483568i −0.900969 − 0.433884i 0.359916 + 1.57690i −3.63892 − 1.75241i 0.557255 − 0.268360i −2.64500 − 0.0631583i −2.01584 − 0.970778i 0.623490 + 0.781831i 2.25069 − 1.08388i
43.4 0.494936 − 0.620630i −0.900969 − 0.433884i 0.304822 + 1.33551i 1.66830 + 0.803409i −0.715203 + 0.344424i 0.814382 − 2.51730i 2.41013 + 1.16066i 0.623490 + 0.781831i 1.32432 − 0.637759i
43.5 1.34250 − 1.68344i −0.900969 − 0.433884i −0.586625 − 2.57017i −2.99194 − 1.44084i −1.93997 + 0.934238i 2.53827 − 0.746462i −1.23434 − 0.594424i 0.623490 + 0.781831i −6.44225 + 3.10242i
43.6 1.64011 − 2.05663i −0.900969 − 0.433884i −1.09473 − 4.79635i 3.33882 + 1.60789i −2.37003 + 1.14134i −2.39773 + 1.11843i −6.91975 − 3.33238i 0.623490 + 0.781831i 8.78287 − 4.22961i
64.1 −0.569099 + 2.49338i 0.623490 − 0.781831i −4.09115 − 1.97020i −1.87405 + 2.34998i 1.59458 + 1.99954i −2.56325 + 0.655556i 4.05157 − 5.08051i −0.222521 − 0.974928i −4.79289 − 6.01009i
64.2 −0.441918 + 1.93617i 0.623490 − 0.781831i −1.75153 − 0.843490i 2.19876 − 2.75716i 1.23823 + 1.55269i 1.51600 + 2.16835i −0.0692834 + 0.0868787i −0.222521 − 0.974928i 4.36666 + 5.47561i
64.3 −0.132418 + 0.580160i 0.623490 − 0.781831i 1.48289 + 0.714121i 0.595901 − 0.747236i 0.371026 + 0.465252i −0.416022 − 2.61284i −1.35272 + 1.69625i −0.222521 − 0.974928i 0.354609 + 0.444665i
64.4 −0.0515349 + 0.225789i 0.623490 − 0.781831i 1.75361 + 0.844495i −2.14953 + 2.69543i 0.144398 + 0.181069i 1.10205 + 2.40530i −0.569846 + 0.714564i −0.222521 − 0.974928i −0.497824 − 0.624251i
64.5 0.368659 − 1.61520i 0.623490 − 0.781831i −0.671021 − 0.323147i 0.768029 − 0.963077i −1.03296 − 1.29529i −1.85883 + 1.88275i 1.29659 − 1.62588i −0.222521 − 0.974928i −1.27242 − 1.59557i
64.6 0.603790 − 2.64538i 0.623490 − 0.781831i −4.83152 − 2.32674i −0.940078 + 1.17882i −1.69178 − 2.12143i 2.57696 − 0.599410i −5.68875 + 7.13347i −0.222521 − 0.974928i 2.55081 + 3.19862i
85.1 −0.569099 − 2.49338i 0.623490 + 0.781831i −4.09115 + 1.97020i −1.87405 − 2.34998i 1.59458 − 1.99954i −2.56325 − 0.655556i 4.05157 + 5.08051i −0.222521 + 0.974928i −4.79289 + 6.01009i
85.2 −0.441918 − 1.93617i 0.623490 + 0.781831i −1.75153 + 0.843490i 2.19876 + 2.75716i 1.23823 − 1.55269i 1.51600 − 2.16835i −0.0692834 − 0.0868787i −0.222521 + 0.974928i 4.36666 − 5.47561i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.2.i.b ✓ 36
3.b odd 2 1 441.2.u.d 36
49.e even 7 1 inner 147.2.i.b ✓ 36
49.e even 7 1 7203.2.a.h 18
49.f odd 14 1 7203.2.a.g 18
147.l odd 14 1 441.2.u.d 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.i.b ✓ 36 1.a even 1 1 trivial
147.2.i.b ✓ 36 49.e even 7 1 inner
441.2.u.d 36 3.b odd 2 1
441.2.u.d 36 147.l odd 14 1
7203.2.a.g 18 49.f odd 14 1
7203.2.a.h 18 49.e even 7 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + T_{2}^{35} + 11 T_{2}^{34} + 17 T_{2}^{33} + 93 T_{2}^{32} + 142 T_{2}^{31} + 689 T_{2}^{30} + \cdots + 123201 \) acting on \(S_{2}^{\mathrm{new}}(147, [\chi])\). Copy content Toggle raw display