Properties

Label 630.2.ca.b
Level $630$
Weight $2$
Character orbit 630.ca
Analytic conductor $5.031$
Analytic rank $0$
Dimension $72$
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] = [630,2,Mod(113,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.ca (of order \(12\), degree \(4\), 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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 4 q^{3} + 12 q^{11} + 4 q^{12} + 36 q^{14} - 20 q^{15} + 36 q^{16} + 24 q^{17} + 12 q^{20} + 4 q^{21} - 8 q^{24} + 12 q^{25} + 32 q^{27} + 32 q^{30} + 44 q^{33} - 12 q^{34} - 4 q^{36} + 24 q^{37} - 36 q^{38} + 40 q^{39} - 48 q^{41} + 4 q^{42} + 48 q^{43} - 68 q^{45} - 48 q^{47} - 8 q^{48} + 16 q^{51} + 4 q^{54} + 24 q^{55} - 84 q^{57} - 48 q^{58} + 4 q^{60} - 8 q^{63} + 24 q^{65} + 8 q^{66} + 12 q^{67} + 12 q^{68} - 64 q^{69} + 16 q^{72} + 84 q^{75} + 24 q^{78} - 72 q^{79} + 48 q^{82} - 60 q^{83} + 24 q^{85} + 72 q^{86} + 4 q^{87} - 48 q^{90} - 24 q^{91} - 64 q^{93} + 132 q^{94} - 60 q^{95} + 36 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 −0.258819 + 0.965926i −1.59099 + 0.684658i −0.866025 0.500000i 2.19432 0.430082i −0.249551 1.71398i −0.965926 0.258819i 0.707107 0.707107i 2.06249 2.17857i −0.152503 + 2.23086i
113.2 −0.258819 + 0.965926i −1.58094 0.707560i −0.866025 0.500000i −1.12367 + 1.93322i 1.09263 1.34394i −0.965926 0.258819i 0.707107 0.707107i 1.99872 + 2.23721i −1.57652 1.58574i
113.3 −0.258819 + 0.965926i −1.38247 + 1.04345i −0.866025 0.500000i −2.13456 + 0.666072i −0.650081 1.60543i −0.965926 0.258819i 0.707107 0.707107i 0.822445 2.88506i −0.0909116 2.23422i
113.4 −0.258819 + 0.965926i −0.970129 1.43487i −0.866025 0.500000i 2.21913 + 0.274732i 1.63707 0.565701i −0.965926 0.258819i 0.707107 0.707107i −1.11770 + 2.78402i −0.839723 + 2.07241i
113.5 −0.258819 + 0.965926i −0.680914 1.59259i −0.866025 0.500000i −1.99168 1.01646i 1.71456 0.245518i −0.965926 0.258819i 0.707107 0.707107i −2.07271 + 2.16884i 1.49731 1.66074i
113.6 −0.258819 + 0.965926i 0.0184135 + 1.73195i −0.866025 0.500000i 1.05084 + 1.97376i −1.67770 0.430476i −0.965926 0.258819i 0.707107 0.707107i −2.99932 + 0.0637827i −2.17848 + 0.504186i
113.7 −0.258819 + 0.965926i 0.995387 + 1.41746i −0.866025 0.500000i 0.0237855 2.23594i −1.62679 + 0.594603i −0.965926 0.258819i 0.707107 0.707107i −1.01841 + 2.82185i 2.15360 + 0.601679i
113.8 −0.258819 + 0.965926i 1.50085 0.864557i −0.866025 0.500000i −1.17878 1.90013i 0.446650 + 1.67347i −0.965926 0.258819i 0.707107 0.707107i 1.50508 2.59513i 2.14047 0.646821i
113.9 −0.258819 + 0.965926i 1.61766 0.619021i −0.866025 0.500000i 0.0745996 + 2.23482i 0.179248 + 1.72275i −0.965926 0.258819i 0.707107 0.707107i 2.23363 2.00273i −2.17798 0.506357i
113.10 0.258819 0.965926i −1.70384 0.311322i −0.866025 0.500000i 1.82858 + 1.28698i −0.741701 + 1.56521i 0.965926 + 0.258819i −0.707107 + 0.707107i 2.80616 + 1.06089i 1.71640 1.43317i
113.11 0.258819 0.965926i −1.32530 + 1.11516i −0.866025 0.500000i 0.902382 2.04590i 0.734145 + 1.56877i 0.965926 + 0.258819i −0.707107 + 0.707107i 0.512852 2.95584i −1.74263 1.40115i
113.12 0.258819 0.965926i −0.955974 + 1.44434i −0.866025 0.500000i −2.22543 0.217885i 1.14770 + 1.29722i 0.965926 + 0.258819i −0.707107 + 0.707107i −1.17223 2.76150i −0.786443 + 2.09320i
113.13 0.258819 0.965926i −0.718418 1.57603i −0.866025 0.500000i 0.920277 + 2.03791i −1.70827 + 0.286032i 0.965926 + 0.258819i −0.707107 + 0.707107i −1.96775 + 2.26450i 2.20666 0.361468i
113.14 0.258819 0.965926i −0.221516 + 1.71783i −0.866025 0.500000i −0.743840 + 2.10872i 1.60196 + 0.658574i 0.965926 + 0.258819i −0.707107 + 0.707107i −2.90186 0.761052i 1.84435 + 1.26427i
113.15 0.258819 0.965926i −0.137151 1.72661i −0.866025 0.500000i −2.16332 0.565716i −1.70328 0.314402i 0.965926 + 0.258819i −0.707107 + 0.707107i −2.96238 + 0.473615i −1.10635 + 1.94319i
113.16 0.258819 0.965926i 1.15104 + 1.29426i −0.866025 0.500000i 2.23016 + 0.162443i 1.54807 0.776846i 0.965926 + 0.258819i −0.707107 + 0.707107i −0.350194 + 2.97949i 0.734115 2.11213i
113.17 0.258819 0.965926i 1.52110 0.828409i −0.866025 0.500000i 0.418681 2.19652i −0.406492 1.68368i 0.965926 + 0.258819i −0.707107 + 0.707107i 1.62748 2.52018i −2.01331 0.972916i
113.18 0.258819 0.965926i 1.73114 0.0560710i −0.866025 0.500000i −2.03351 + 0.929968i 0.393892 1.68667i 0.965926 + 0.258819i −0.707107 + 0.707107i 2.99371 0.194134i 0.371969 + 2.20491i
407.1 −0.258819 0.965926i −1.59099 0.684658i −0.866025 + 0.500000i 2.19432 + 0.430082i −0.249551 + 1.71398i −0.965926 + 0.258819i 0.707107 + 0.707107i 2.06249 + 2.17857i −0.152503 2.23086i
407.2 −0.258819 0.965926i −1.58094 + 0.707560i −0.866025 + 0.500000i −1.12367 1.93322i 1.09263 + 1.34394i −0.965926 + 0.258819i 0.707107 + 0.707107i 1.99872 2.23721i −1.57652 + 1.58574i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.ca.b yes 72
5.c odd 4 1 630.2.ca.a 72
9.d odd 6 1 630.2.ca.a 72
45.l even 12 1 inner 630.2.ca.b yes 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.ca.a 72 5.c odd 4 1
630.2.ca.a 72 9.d odd 6 1
630.2.ca.b yes 72 1.a even 1 1 trivial
630.2.ca.b yes 72 45.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{72} + 144 T_{13}^{70} - 24 T_{13}^{69} + 7071 T_{13}^{68} - 4800 T_{13}^{67} + 23184 T_{13}^{66} - 246276 T_{13}^{65} - 8607399 T_{13}^{64} + 2669072 T_{13}^{63} - 143310792 T_{13}^{62} + \cdots + 19\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display