Properties

Label 630.2.bw.a
Level $630$
Weight $2$
Character orbit 630.bw
Analytic conductor $5.031$
Analytic rank $0$
Dimension $192$
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] = [630,2,Mod(103,630)]
 
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([4, 9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.103");
 
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.bw (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: \(192\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 8 q^{11} - 12 q^{15} - 192 q^{16} + 36 q^{17} - 8 q^{18} + 8 q^{23} + 36 q^{27} + 12 q^{30} + 48 q^{35} - 16 q^{36} - 12 q^{41} + 4 q^{42} + 48 q^{45} + 12 q^{46} - 16 q^{50} - 24 q^{51} - 40 q^{53} - 4 q^{56} - 32 q^{57} - 12 q^{58} - 16 q^{60} - 12 q^{63} + 16 q^{65} - 36 q^{68} + 16 q^{71} - 8 q^{72} + 96 q^{75} - 64 q^{77} - 16 q^{78} + 16 q^{81} + 168 q^{83} + 8 q^{86} - 36 q^{87} - 36 q^{90} + 8 q^{92} - 48 q^{93} - 128 q^{95} - 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −0.707107 + 0.707107i −1.71556 0.238440i 1.00000i 0.973575 + 2.01300i 1.38169 1.04448i 0.404878 + 2.61459i 0.707107 + 0.707107i 2.88629 + 0.818117i −2.11182 0.734981i
103.2 −0.707107 + 0.707107i −1.67295 0.448583i 1.00000i −0.239265 + 2.22323i 1.50015 0.865760i −0.403547 2.61479i 0.707107 + 0.707107i 2.59755 + 1.50092i −1.40288 1.74125i
103.3 −0.707107 + 0.707107i −1.53699 + 0.798544i 1.00000i 2.11145 0.736070i 0.522158 1.65147i 2.64560 0.0283592i 0.707107 + 0.707107i 1.72466 2.45470i −0.972537 + 2.01350i
103.4 −0.707107 + 0.707107i −1.38967 + 1.03383i 1.00000i 2.18560 + 0.472406i 0.251617 1.71368i −2.55035 0.704070i 0.707107 + 0.707107i 0.862380 2.87338i −1.87949 + 1.21141i
103.5 −0.707107 + 0.707107i −1.37690 + 1.05079i 1.00000i −2.23501 0.0688901i 0.230594 1.71663i −2.60633 + 0.455003i 0.707107 + 0.707107i 0.791690 2.89365i 1.62910 1.53168i
103.6 −0.707107 + 0.707107i −1.34565 1.09051i 1.00000i 1.81952 1.29974i 1.72263 0.180414i 1.89634 1.84497i 0.707107 + 0.707107i 0.621572 + 2.93490i −0.367541 + 2.20565i
103.7 −0.707107 + 0.707107i −1.12396 1.31785i 1.00000i −1.44027 1.71045i 1.72662 + 0.137102i −2.58180 0.578175i 0.707107 + 0.707107i −0.473446 + 2.96241i 2.22789 + 0.191049i
103.8 −0.707107 + 0.707107i −0.949637 1.44851i 1.00000i −2.17592 0.515133i 1.69575 + 0.352758i 2.58119 0.580908i 0.707107 + 0.707107i −1.19638 + 2.75112i 1.90286 1.17435i
103.9 −0.707107 + 0.707107i −0.721761 + 1.57450i 1.00000i −0.997387 + 2.00130i −0.602980 1.62370i 0.960462 2.46526i 0.707107 + 0.707107i −1.95812 2.27283i −0.709876 2.12040i
103.10 −0.707107 + 0.707107i −0.688185 1.58947i 1.00000i 1.94881 1.09641i 1.61054 + 0.637302i −0.414054 + 2.61315i 0.707107 + 0.707107i −2.05280 + 2.18769i −0.602738 + 2.15330i
103.11 −0.707107 + 0.707107i −0.374933 + 1.69098i 1.00000i −0.935649 2.03090i −0.930588 1.46082i 2.14864 + 1.54381i 0.707107 + 0.707107i −2.71885 1.26801i 2.09767 + 0.774460i
103.12 −0.707107 + 0.707107i 0.0979337 1.72928i 1.00000i −1.65263 + 1.50626i 1.15354 + 1.29204i 0.935970 + 2.47466i 0.707107 + 0.707107i −2.98082 0.338709i 0.103501 2.23367i
103.13 −0.707107 + 0.707107i 0.272070 + 1.71055i 1.00000i −1.78688 + 1.34427i −1.40192 1.01716i −0.883234 + 2.49397i 0.707107 + 0.707107i −2.85196 + 0.930777i 0.312975 2.21406i
103.14 −0.707107 + 0.707107i 0.379200 + 1.69003i 1.00000i 0.792977 2.09074i −1.46317 0.926898i −1.57559 2.12544i 0.707107 + 0.707107i −2.71241 + 1.28172i 0.917656 + 2.03909i
103.15 −0.707107 + 0.707107i 0.757106 1.55782i 1.00000i 1.59411 + 1.56806i 0.566187 + 1.63690i 2.62505 + 0.330352i 0.707107 + 0.707107i −1.85358 2.35886i −2.23599 + 0.0184185i
103.16 −0.707107 + 0.707107i 0.894028 1.48348i 1.00000i −1.22625 1.86984i 0.416805 + 1.68115i 1.29629 2.30643i 0.707107 + 0.707107i −1.40143 2.65255i 2.18927 + 0.455093i
103.17 −0.707107 + 0.707107i 0.929118 1.46176i 1.00000i 1.34455 1.78667i 0.376634 + 1.69061i −2.51243 0.829274i 0.707107 + 0.707107i −1.27348 2.71629i 0.312627 + 2.21411i
103.18 −0.707107 + 0.707107i 1.20754 + 1.24171i 1.00000i 2.21622 0.297244i −1.73188 0.0241584i −0.456895 + 2.60600i 0.707107 + 0.707107i −0.0836789 + 2.99883i −1.35692 + 1.77729i
103.19 −0.707107 + 0.707107i 1.45946 0.932722i 1.00000i 0.207084 + 2.22646i −0.372462 + 1.69153i −2.32758 + 1.25792i 0.707107 + 0.707107i 1.26006 2.72254i −1.72077 1.42791i
103.20 −0.707107 + 0.707107i 1.47488 + 0.908144i 1.00000i −2.17109 0.535116i −1.68505 + 0.400744i −1.11920 2.39737i 0.707107 + 0.707107i 1.35055 + 2.67881i 1.91358 1.15681i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
63.t odd 6 1 inner
315.bs 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.bw.a 192
5.c odd 4 1 inner 630.2.bw.a 192
7.d odd 6 1 630.2.cg.a yes 192
9.c even 3 1 630.2.cg.a yes 192
35.k even 12 1 630.2.cg.a yes 192
45.k odd 12 1 630.2.cg.a yes 192
63.t odd 6 1 inner 630.2.bw.a 192
315.bs even 12 1 inner 630.2.bw.a 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bw.a 192 1.a even 1 1 trivial
630.2.bw.a 192 5.c odd 4 1 inner
630.2.bw.a 192 63.t odd 6 1 inner
630.2.bw.a 192 315.bs even 12 1 inner
630.2.cg.a yes 192 7.d odd 6 1
630.2.cg.a yes 192 9.c even 3 1
630.2.cg.a yes 192 35.k even 12 1
630.2.cg.a yes 192 45.k odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(630, [\chi])\).