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Properties

Label 630.2.cg.a
Level $630$
Weight $2$
Character orbit 630.cg
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(157,630)]
 
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, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.157");
 
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.cg (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
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 16 q^{11} - 12 q^{15} + 96 q^{16} + 36 q^{17} + 16 q^{18} + 12 q^{21} - 16 q^{23} - 36 q^{27} + 12 q^{30} - 60 q^{33} + 48 q^{35} - 16 q^{36} + 12 q^{41} + 4 q^{42} - 36 q^{45} + 12 q^{46} - 16 q^{50} + 24 q^{51} - 40 q^{53} + 8 q^{56} - 32 q^{57} + 24 q^{58} + 8 q^{60} + 36 q^{61} - 108 q^{63} - 8 q^{65} + 16 q^{71} - 8 q^{72} + 80 q^{77} - 16 q^{78} + 4 q^{81} - 168 q^{83} - 16 q^{86} + 36 q^{90} + 8 q^{92} - 60 q^{93} + 64 q^{95} - 12 q^{96} - 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 −0.965926 + 0.258819i −1.71321 + 0.254780i 0.866025 − 0.500000i −1.99551 − 1.00893i 1.58889 − 0.689510i −0.221143 + 2.63649i −0.707107 + 0.707107i 2.87017 − 0.872982i 2.18864 + 0.458077i
157.2 −0.965926 + 0.258819i −1.70832 − 0.285724i 0.866025 − 0.500000i 1.73208 + 1.41418i 1.72406 − 0.166158i 2.64401 + 0.0960094i −0.707107 + 0.707107i 2.83672 + 0.976216i −2.03908 − 0.917694i
157.3 −0.965926 + 0.258819i −1.64267 + 0.549224i 0.866025 − 0.500000i 1.99630 − 1.00736i 1.44454 − 0.955663i −2.31389 + 1.28293i −0.707107 + 0.707107i 2.39671 − 1.80438i −1.66756 + 1.48971i
157.4 −0.965926 + 0.258819i −1.43992 − 0.962609i 0.866025 − 0.500000i −1.93341 + 1.12335i 1.64000 + 0.557129i 1.38573 − 2.25383i −0.707107 + 0.707107i 1.14677 + 2.77217i 1.57679 − 1.58548i
157.5 −0.965926 + 0.258819i −1.26782 + 1.18010i 0.866025 − 0.500000i −1.44041 + 1.71033i 0.919188 − 1.46802i 2.43402 − 1.03708i −0.707107 + 0.707107i 0.214733 − 2.99231i 0.948666 − 2.02485i
157.6 −0.965926 + 0.258819i −1.20057 − 1.24845i 0.866025 − 0.500000i −0.928450 − 2.03420i 1.48278 + 0.895181i 1.70605 + 2.02223i −0.707107 + 0.707107i −0.117263 + 2.99771i 1.42330 + 1.72459i
157.7 −0.965926 + 0.258819i −1.16870 − 1.27833i 0.866025 − 0.500000i 2.19640 − 0.419297i 1.45974 + 0.932291i −1.80421 − 1.93515i −0.707107 + 0.707107i −0.268265 + 2.98798i −2.01304 + 0.973481i
157.8 −0.965926 + 0.258819i −1.01214 + 1.40555i 0.866025 − 0.500000i 0.201115 + 2.22701i 0.613868 − 1.61962i −2.51031 − 0.835662i −0.707107 + 0.707107i −0.951147 − 2.84523i −0.770654 − 2.09907i
157.9 −0.965926 + 0.258819i −0.798392 + 1.53707i 0.866025 − 0.500000i −1.46535 − 1.68900i 0.373365 − 1.69133i −1.21582 − 2.34985i −0.707107 + 0.707107i −1.72514 − 2.45436i 1.85257 + 1.25219i
157.10 −0.965926 + 0.258819i −0.697697 − 1.58531i 0.866025 − 0.500000i −2.05976 + 0.870292i 1.08423 + 1.35072i −2.59232 + 0.529026i −0.707107 + 0.707107i −2.02644 + 2.21214i 1.76432 − 1.37374i
157.11 −0.965926 + 0.258819i −0.231316 + 1.71654i 0.866025 − 0.500000i 1.97412 + 1.05016i −0.220838 − 1.71791i 1.21830 + 2.34856i −0.707107 + 0.707107i −2.89299 − 0.794125i −2.17866 − 0.503437i
157.12 −0.965926 + 0.258819i −0.0309423 − 1.73177i 0.866025 − 0.500000i 2.05295 − 0.886221i 0.478104 + 1.66476i 2.61918 − 0.374056i −0.707107 + 0.707107i −2.99809 + 0.107170i −1.75363 + 1.38737i
157.13 −0.965926 + 0.258819i 0.100845 + 1.72911i 0.866025 − 0.500000i 1.99067 − 1.01845i −0.544936 − 1.64409i 1.32746 − 2.28864i −0.707107 + 0.707107i −2.97966 + 0.348745i −1.65925 + 1.49897i
157.14 −0.965926 + 0.258819i 0.419695 − 1.68043i 0.866025 − 0.500000i 0.445332 + 2.19127i 0.0295336 + 1.73180i 0.601210 − 2.57654i −0.707107 + 0.707107i −2.64771 − 1.41054i −0.997301 − 2.00135i
157.15 −0.965926 + 0.258819i 0.512590 − 1.65446i 0.866025 − 0.500000i −0.725830 − 2.11499i −0.0669173 + 1.73076i −2.64367 − 0.104885i −0.707107 + 0.707107i −2.47450 − 1.69612i 1.24850 + 1.85506i
157.16 −0.965926 + 0.258819i 0.579077 + 1.63238i 0.866025 − 0.500000i −0.445639 − 2.19121i −0.981837 − 1.42688i −2.40360 + 1.10575i −0.707107 + 0.707107i −2.32934 + 1.89055i 0.997581 + 2.00121i
157.17 −0.965926 + 0.258819i 0.810488 + 1.53072i 0.866025 − 0.500000i −0.924726 + 2.03590i −1.17905 − 1.26879i 0.515728 + 2.59500i −0.707107 + 0.707107i −1.68622 + 2.48126i 0.366288 − 2.20586i
157.18 −0.965926 + 0.258819i 0.973845 + 1.43235i 0.866025 − 0.500000i −2.22463 + 0.225873i −1.31138 − 1.13149i 2.44785 − 1.00400i −0.707107 + 0.707107i −1.10325 + 2.78977i 2.09037 − 0.793954i
157.19 −0.965926 + 0.258819i 1.29834 + 1.14644i 0.866025 − 0.500000i 1.62901 + 1.53177i −1.55082 − 0.771341i −1.13641 − 2.38926i −0.707107 + 0.707107i 0.371355 + 2.97693i −1.96995 − 1.05796i
157.20 −0.965926 + 0.258819i 1.38183 − 1.04429i 0.866025 − 0.500000i −2.17615 − 0.514181i −1.06446 + 1.36635i 2.32202 + 1.26816i −0.707107 + 0.707107i 0.818901 − 2.88607i 2.23508 − 0.0665681i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.48
Significant digits:
Format:

Inner twists

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

Hecke kernels

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