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Properties

Label 154.2.m.b
Level $154$
Weight $2$
Character orbit 154.m
Analytic conductor $1.230$
Analytic rank $0$
Dimension $24$
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] = [154,2,Mod(9,154)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("154.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 154.m (of order \(15\), degree \(8\), 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: \(1.22969619113\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 4 q^{6} - 5 q^{7} + 6 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 4 q^{6} - 5 q^{7} + 6 q^{8} - 16 q^{9} - 8 q^{10} + 6 q^{11} + 2 q^{12} + 12 q^{13} + 11 q^{14} + 4 q^{15} + 3 q^{16} - 6 q^{17} - 19 q^{18} - 6 q^{20} + 4 q^{21} + 2 q^{22} - 14 q^{23} - 2 q^{24} - 18 q^{25} - 4 q^{26} - 12 q^{27} - 14 q^{28} + 24 q^{29} + 2 q^{30} - 19 q^{31} + 12 q^{32} + 33 q^{33} - 12 q^{34} + 20 q^{35} - 38 q^{36} - 10 q^{37} - 15 q^{38} + 13 q^{39} + 7 q^{40} - 8 q^{41} + 16 q^{42} - 92 q^{43} - 4 q^{44} + 4 q^{45} + 9 q^{46} - 27 q^{47} + 6 q^{48} + 3 q^{49} + 44 q^{50} + 39 q^{51} + 4 q^{52} + 6 q^{53} - 26 q^{54} - 82 q^{55} + 10 q^{56} - 6 q^{57} + 12 q^{58} - 12 q^{59} - 2 q^{60} - 17 q^{61} + 32 q^{62} - 21 q^{63} - 6 q^{64} - 6 q^{65} + 32 q^{66} + 18 q^{67} - 6 q^{68} + 140 q^{69} - 4 q^{70} + 4 q^{71} + 16 q^{72} - q^{73} + 10 q^{74} - 31 q^{75} + 60 q^{76} + 34 q^{77} - 44 q^{78} - 10 q^{79} - 7 q^{80} + 44 q^{81} - 14 q^{82} + 20 q^{83} - 13 q^{84} + 72 q^{85} - q^{86} + 60 q^{87} - q^{88} + 48 q^{89} + 8 q^{90} + 45 q^{91} - 32 q^{92} - 7 q^{93} + 7 q^{94} - 16 q^{95} + 3 q^{96} + 44 q^{97} - 134 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −0.669131 − 0.743145i −3.06161 − 1.36311i −0.104528 + 0.994522i 0.248798 + 0.0528836i 1.03562 + 3.18732i −2.64537 − 0.0451581i 0.809017 − 0.587785i 5.50796 + 6.11721i −0.127178 − 0.220279i
9.2 −0.669131 − 0.743145i −0.735907 − 0.327647i −0.104528 + 0.994522i 2.12235 + 0.451119i 0.248929 + 0.766124i 2.01873 + 1.71018i 0.809017 − 0.587785i −1.57319 − 1.74720i −1.08488 − 1.87907i
9.3 −0.669131 − 0.743145i 2.31936 + 1.03265i −0.104528 + 0.994522i 1.77235 + 0.376725i −0.784551 − 2.41460i −2.18977 + 1.48489i 0.809017 − 0.587785i 2.30570 + 2.56074i −0.905975 − 1.56919i
25.1 −0.913545 − 0.406737i −1.43491 − 0.304999i 0.669131 + 0.743145i −0.00806762 + 0.0767583i 1.18680 + 0.862261i −1.78535 − 1.95256i −0.309017 − 0.951057i −0.774697 − 0.344917i 0.0385906 − 0.0668408i
25.2 −0.913545 − 0.406737i 0.137294 + 0.0291827i 0.669131 + 0.743145i 0.368040 − 3.50166i −0.113554 − 0.0825021i 1.20864 + 2.35355i −0.309017 − 0.951057i −2.72264 − 1.21220i −1.76048 + 3.04923i
25.3 −0.913545 − 0.406737i 0.693087 + 0.147320i 0.669131 + 0.743145i −0.384648 + 3.65968i −0.573246 − 0.416487i −1.85490 + 1.88662i −0.309017 − 0.951057i −2.28197 − 1.01600i 1.83992 − 3.18683i
37.1 −0.913545 + 0.406737i −1.43491 + 0.304999i 0.669131 − 0.743145i −0.00806762 − 0.0767583i 1.18680 − 0.862261i −1.78535 + 1.95256i −0.309017 + 0.951057i −0.774697 + 0.344917i 0.0385906 + 0.0668408i
37.2 −0.913545 + 0.406737i 0.137294 − 0.0291827i 0.669131 − 0.743145i 0.368040 + 3.50166i −0.113554 + 0.0825021i 1.20864 − 2.35355i −0.309017 + 0.951057i −2.72264 + 1.21220i −1.76048 − 3.04923i
37.3 −0.913545 + 0.406737i 0.693087 − 0.147320i 0.669131 − 0.743145i −0.384648 − 3.65968i −0.573246 + 0.416487i −1.85490 − 1.88662i −0.309017 + 0.951057i −2.28197 + 1.01600i 1.83992 + 3.18683i
53.1 0.978148 − 0.207912i −0.265383 − 2.52495i 0.913545 − 0.406737i −1.21243 + 1.34654i −0.784551 − 2.41460i 0.735534 − 2.54145i 0.809017 − 0.587785i −3.37052 + 0.716425i −0.905975 + 1.56919i
53.2 0.978148 − 0.207912i 0.0842029 + 0.801137i 0.913545 − 0.406737i −1.45185 + 1.61245i 0.248929 + 0.766124i 2.25030 + 1.39145i 0.809017 − 0.587785i 2.29971 − 0.488819i −1.08488 + 1.87907i
53.3 0.978148 − 0.207912i 0.350311 + 3.33299i 0.913545 − 0.406737i −0.170197 + 0.189023i 1.03562 + 3.18732i −0.860411 − 2.50194i 0.809017 − 0.587785i −8.05163 + 1.71143i −0.127178 + 0.220279i
81.1 0.104528 − 0.994522i −0.474126 − 0.526571i −0.978148 − 0.207912i 3.36170 + 1.49673i −0.573246 + 0.416487i 0.391723 − 2.61659i −0.309017 + 0.951057i 0.261105 − 2.48424i 1.83992 − 3.18683i
81.2 0.104528 − 0.994522i −0.0939198 − 0.104309i −0.978148 − 0.207912i −3.21655 − 1.43210i −0.113554 + 0.0825021i −2.36119 − 1.19364i −0.309017 + 0.951057i 0.311526 − 2.96397i −1.76048 + 3.04923i
81.3 0.104528 − 0.994522i 0.981592 + 1.09017i −0.978148 − 0.207912i 0.0705085 + 0.0313924i 1.18680 − 0.862261i 2.59207 + 0.530253i −0.309017 + 0.951057i 0.0886414 − 0.843366i 0.0385906 − 0.0668408i
93.1 0.978148 + 0.207912i −0.265383 + 2.52495i 0.913545 + 0.406737i −1.21243 − 1.34654i −0.784551 + 2.41460i 0.735534 + 2.54145i 0.809017 + 0.587785i −3.37052 − 0.716425i −0.905975 − 1.56919i
93.2 0.978148 + 0.207912i 0.0842029 − 0.801137i 0.913545 + 0.406737i −1.45185 − 1.61245i 0.248929 − 0.766124i 2.25030 − 1.39145i 0.809017 + 0.587785i 2.29971 + 0.488819i −1.08488 − 1.87907i
93.3 0.978148 + 0.207912i 0.350311 − 3.33299i 0.913545 + 0.406737i −0.170197 − 0.189023i 1.03562 − 3.18732i −0.860411 + 2.50194i 0.809017 + 0.587785i −8.05163 − 1.71143i −0.127178 − 0.220279i
135.1 0.104528 + 0.994522i −0.474126 + 0.526571i −0.978148 + 0.207912i 3.36170 − 1.49673i −0.573246 − 0.416487i 0.391723 + 2.61659i −0.309017 − 0.951057i 0.261105 + 2.48424i 1.83992 + 3.18683i
135.2 0.104528 + 0.994522i −0.0939198 + 0.104309i −0.978148 + 0.207912i −3.21655 + 1.43210i −0.113554 − 0.0825021i −2.36119 + 1.19364i −0.309017 − 0.951057i 0.311526 + 2.96397i −1.76048 − 3.04923i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.m.b ✓ 24
7.c even 3 1 inner 154.2.m.b ✓ 24
11.c even 5 1 inner 154.2.m.b ✓ 24
77.m even 15 1 inner 154.2.m.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.m.b ✓ 24 1.a even 1 1 trivial
154.2.m.b ✓ 24 7.c even 3 1 inner
154.2.m.b ✓ 24 11.c even 5 1 inner
154.2.m.b ✓ 24 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 3 T_{3}^{23} + 8 T_{3}^{22} + 27 T_{3}^{21} - 6 T_{3}^{20} - 194 T_{3}^{19} + 295 T_{3}^{18} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(154, [\chi])\). Copy content Toggle raw display