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Properties

Label 630.2.bq.a
Level $630$
Weight $2$
Character orbit 630.bq
Analytic conductor $5.031$
Analytic rank $0$
Dimension $96$
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(79,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
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("630.79");
 
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.bq (of order \(6\), degree \(2\), 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: \(96\)
Relative dimension: \(48\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 48 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 48 q^{4} - 4 q^{6} + 2 q^{9} - 8 q^{11} - 2 q^{14} - 2 q^{15} - 48 q^{16} + 14 q^{21} - 2 q^{24} + 24 q^{26} + 10 q^{29} - 2 q^{30} - 34 q^{35} - 8 q^{36} + 8 q^{39} + 30 q^{41} - 4 q^{44} + 10 q^{45} - 6 q^{46} + 12 q^{49} - 12 q^{50} - 8 q^{51} + 4 q^{54} + 12 q^{55} - 4 q^{56} - 24 q^{59} - 10 q^{60} - 6 q^{61} - 96 q^{64} + 18 q^{65} - 16 q^{66} + 36 q^{69} + 6 q^{70} - 32 q^{71} - 14 q^{75} + 70 q^{81} + 22 q^{84} - 8 q^{86} - 66 q^{89} - 22 q^{90} - 12 q^{94} + 30 q^{95} + 2 q^{96} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1 −0.866025 + 0.500000i −1.73154 + 0.0419304i 0.500000 − 0.866025i 1.97698 − 1.04476i 1.47860 − 0.902084i 2.18670 + 1.48941i 1.00000i 2.99648 − 0.145208i −1.18974 + 1.89328i
79.2 −0.866025 + 0.500000i −1.72799 + 0.118521i 0.500000 − 0.866025i −0.557856 − 2.16536i 1.43722 − 0.966638i 0.346870 − 2.62291i 1.00000i 2.97191 − 0.409608i 1.56580 + 1.59633i
79.3 −0.866025 + 0.500000i −1.62633 + 0.595853i 0.500000 − 0.866025i −0.409739 + 2.19821i 1.11052 − 1.32919i −1.55043 − 2.14387i 1.00000i 2.28992 − 1.93811i −0.744259 − 2.10857i
79.4 −0.866025 + 0.500000i −1.58075 − 0.707966i 0.500000 − 0.866025i 0.730833 + 2.11326i 1.72296 − 0.177260i −0.600499 + 2.57670i 1.00000i 1.99757 + 2.23824i −1.68955 − 1.46472i
79.5 −0.866025 + 0.500000i −1.41090 + 1.00467i 0.500000 − 0.866025i −2.01759 + 0.964026i 0.719540 − 1.57552i 2.27785 + 1.34588i 1.00000i 0.981277 − 2.83498i 1.26527 − 1.84366i
79.6 −0.866025 + 0.500000i −1.30741 − 1.13608i 0.500000 − 0.866025i −2.15250 − 0.605607i 1.70029 + 0.330172i −1.94150 + 1.79738i 1.00000i 0.418632 + 2.97065i 2.16692 − 0.551777i
79.7 −0.866025 + 0.500000i −1.00790 − 1.40859i 0.500000 − 0.866025i 1.55413 − 1.60770i 1.57716 + 0.715928i −2.57509 − 0.607380i 1.00000i −0.968275 + 2.83944i −0.542063 + 2.16937i
79.8 −0.866025 + 0.500000i −1.00775 + 1.40870i 0.500000 − 0.866025i 1.91489 − 1.15464i 0.168388 − 1.72385i −2.59099 + 0.535498i 1.00000i −0.968876 − 2.83924i −1.08102 + 1.95739i
79.9 −0.866025 + 0.500000i −0.728087 + 1.57159i 0.500000 − 0.866025i 1.89610 + 1.18525i −0.155252 − 1.72508i 0.873692 − 2.49733i 1.00000i −1.93978 − 2.28851i −2.23469 − 0.0784052i
79.10 −0.866025 + 0.500000i −0.386161 − 1.68845i 0.500000 − 0.866025i −2.23165 − 0.140481i 1.17865 + 1.26916i 0.387288 − 2.61725i 1.00000i −2.70176 + 1.30403i 2.00291 − 0.994165i
79.11 −0.866025 + 0.500000i −0.0735223 − 1.73049i 0.500000 − 0.866025i −0.928700 + 2.03409i 0.928917 + 1.46189i 2.46103 + 0.971242i 1.00000i −2.98919 + 0.254459i −0.212767 − 2.22592i
79.12 −0.866025 + 0.500000i 0.0366265 + 1.73166i 0.500000 − 0.866025i −1.78400 + 1.34809i −0.897551 − 1.48135i −2.32863 + 1.25598i 1.00000i −2.99732 + 0.126850i 0.870943 − 2.05948i
79.13 −0.866025 + 0.500000i 0.217830 + 1.71830i 0.500000 − 0.866025i 1.41556 + 1.73095i −1.04780 − 1.37918i 1.74017 + 1.99294i 1.00000i −2.90510 + 0.748595i −2.09139 − 0.791267i
79.14 −0.866025 + 0.500000i 0.260217 + 1.71239i 0.500000 − 0.866025i −0.750434 − 2.10638i −1.08155 − 1.35287i 2.48341 − 0.912521i 1.00000i −2.86457 + 0.891186i 1.70309 + 1.44896i
79.15 −0.866025 + 0.500000i 0.340160 − 1.69832i 0.500000 − 0.866025i 1.70959 + 1.44129i 0.554573 + 1.64087i −2.11221 − 1.59329i 1.00000i −2.76858 − 1.15540i −2.20119 − 0.393399i
79.16 −0.866025 + 0.500000i 0.835730 + 1.51709i 0.500000 − 0.866025i −0.688958 − 2.12728i −1.48231 − 0.895972i −2.47408 − 0.937525i 1.00000i −1.60311 + 2.53575i 1.66030 + 1.49780i
79.17 −0.866025 + 0.500000i 1.00450 − 1.41102i 0.500000 − 0.866025i 0.354381 − 2.20781i −0.164412 + 1.72423i 1.56393 − 2.13404i 1.00000i −0.981960 − 2.83474i 0.797001 + 2.08921i
79.18 −0.866025 + 0.500000i 1.14415 − 1.30036i 0.500000 − 0.866025i −1.49821 − 1.65993i −0.340682 + 1.69822i 0.276001 + 2.63132i 1.00000i −0.381857 − 2.97560i 2.12745 + 0.688440i
79.19 −0.866025 + 0.500000i 1.42620 − 0.982831i 0.500000 − 0.866025i 2.17928 − 0.500735i −0.743709 + 1.56426i −1.03354 + 2.43553i 1.00000i 1.06809 − 2.80343i −1.63695 + 1.52329i
79.20 −0.866025 + 0.500000i 1.60031 − 0.662588i 0.500000 − 0.866025i 0.412019 + 2.19778i −1.05461 + 1.37397i 2.44216 − 1.01778i 1.00000i 2.12195 − 2.12069i −1.45571 − 1.69732i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
63.g even 3 1 inner
315.bo even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bq.a yes 96
3.b odd 2 1 1890.2.bq.a 96
5.b even 2 1 inner 630.2.bq.a yes 96
7.c even 3 1 630.2.ba.a ✓ 96
9.c even 3 1 630.2.ba.a ✓ 96
9.d odd 6 1 1890.2.ba.a 96
15.d odd 2 1 1890.2.bq.a 96
21.h odd 6 1 1890.2.ba.a 96
35.j even 6 1 630.2.ba.a ✓ 96
45.h odd 6 1 1890.2.ba.a 96
45.j even 6 1 630.2.ba.a ✓ 96
63.g even 3 1 inner 630.2.bq.a yes 96
63.n odd 6 1 1890.2.bq.a 96
105.o odd 6 1 1890.2.ba.a 96
315.v odd 6 1 1890.2.bq.a 96
315.bo even 6 1 inner 630.2.bq.a yes 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.ba.a ✓ 96 7.c even 3 1
630.2.ba.a ✓ 96 9.c even 3 1
630.2.ba.a ✓ 96 35.j even 6 1
630.2.ba.a ✓ 96 45.j even 6 1
630.2.bq.a yes 96 1.a even 1 1 trivial
630.2.bq.a yes 96 5.b even 2 1 inner
630.2.bq.a yes 96 63.g even 3 1 inner
630.2.bq.a yes 96 315.bo even 6 1 inner
1890.2.ba.a 96 9.d odd 6 1
1890.2.ba.a 96 21.h odd 6 1
1890.2.ba.a 96 45.h odd 6 1
1890.2.ba.a 96 105.o odd 6 1
1890.2.bq.a 96 3.b odd 2 1
1890.2.bq.a 96 15.d odd 2 1
1890.2.bq.a 96 63.n odd 6 1
1890.2.bq.a 96 315.v odd 6 1

Hecke kernels

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