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Properties

Label 840.2.cq.b
Level $840$
Weight $2$
Character orbit 840.cq
Analytic conductor $6.707$
Analytic rank $0$
Dimension $128$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [840,2,Mod(11,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("840.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.cq (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,0,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(64\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 64 q^{5} + 2 q^{12} + 6 q^{14} + 8 q^{16} - 4 q^{18} - 8 q^{21} + 4 q^{22} + 20 q^{24} - 64 q^{25} + 16 q^{26} - 6 q^{28} + 10 q^{32} + 24 q^{34} + 20 q^{36} + 16 q^{38} + 18 q^{42} - 6 q^{44} + 8 q^{46}+ \cdots - 14 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.41266 + 0.0662262i 1.12178 + 1.31970i 1.99123 − 0.187110i 0.500000 + 0.866025i −1.67209 − 1.79000i −1.22654 + 2.34427i −2.80054 + 0.396195i −0.483227 + 2.96083i −0.763685 − 1.19029i
11.2 −1.41035 − 0.104459i 0.394184 − 1.68660i 1.97818 + 0.294646i 0.500000 + 0.866025i −0.732118 + 2.33752i 2.27220 + 1.35540i −2.75914 − 0.622192i −2.68924 − 1.32966i −0.614712 − 1.27363i
11.3 −1.40194 − 0.185884i −0.299749 + 1.70592i 1.93089 + 0.521198i 0.500000 + 0.866025i 0.737333 − 2.33588i −1.35327 − 2.27347i −2.61012 − 1.08961i −2.82030 − 1.02269i −0.539992 − 1.30706i
11.4 −1.37837 + 0.316382i 1.37790 + 1.04947i 1.79981 − 0.872181i 0.500000 + 0.866025i −2.23129 − 1.01062i 0.185707 − 2.63923i −2.20486 + 1.77161i 0.797221 + 2.89213i −0.963179 − 1.03551i
11.5 −1.37432 − 0.333542i −1.72912 − 0.100648i 1.77750 + 0.916786i 0.500000 + 0.866025i 2.34280 + 0.715058i −1.86938 + 1.87228i −2.13706 − 1.85283i 2.97974 + 0.348066i −0.398303 − 1.35697i
11.6 −1.35317 + 0.410999i −1.60922 + 0.640639i 1.66216 − 1.11231i 0.500000 + 0.866025i 1.91425 − 1.52828i 1.82587 − 1.91473i −1.79203 + 2.18829i 2.17916 − 2.06186i −1.03252 − 0.966383i
11.7 −1.34993 − 0.421525i −0.417719 + 1.68093i 1.64463 + 1.13806i 0.500000 + 0.866025i 1.27244 − 2.09306i 2.02409 + 1.70384i −1.74042 − 2.22956i −2.65102 − 1.40431i −0.309914 − 1.37984i
11.8 −1.31225 − 0.527267i −0.712418 − 1.57875i 1.44398 + 1.38381i 0.500000 + 0.866025i 0.102443 + 2.44735i −0.730618 − 2.54287i −1.16522 − 2.57726i −1.98492 + 2.24946i −0.199496 − 1.40007i
11.9 −1.28706 + 0.586068i 0.965902 − 1.43772i 1.31305 − 1.50861i 0.500000 + 0.866025i −0.400574 + 2.41651i −1.26568 − 2.32337i −0.805825 + 2.71121i −1.13406 − 2.77739i −1.15108 − 0.821593i
11.10 −1.27986 + 0.601628i −0.404828 − 1.68408i 1.27609 − 1.54000i 0.500000 + 0.866025i 1.53131 + 1.91183i −2.46244 + 0.967661i −0.706706 + 2.73872i −2.67223 + 1.36352i −1.16096 − 0.807578i
11.11 −1.23762 − 0.684321i 1.61725 − 0.620085i 1.06341 + 1.69386i 0.500000 + 0.866025i −2.42588 − 0.339287i 2.61911 − 0.374487i −0.156956 − 2.82407i 2.23099 − 2.00566i −0.0261712 − 1.41397i
11.12 −1.20068 + 0.747246i −1.44104 − 0.960934i 0.883246 − 1.79440i 0.500000 + 0.866025i 2.44828 + 0.0769572i 1.36448 + 2.26676i 0.280367 + 2.81450i 1.15321 + 2.76950i −1.24747 − 0.666193i
11.13 −1.18635 + 0.769790i 1.72178 + 0.188349i 0.814847 − 1.82648i 0.500000 + 0.866025i −2.18762 + 1.10196i 2.62384 + 0.339789i 0.439313 + 2.79410i 2.92905 + 0.648591i −1.25983 − 0.642513i
11.14 −1.12992 − 0.850457i 1.73196 + 0.0178511i 0.553446 + 1.92190i 0.500000 + 0.866025i −1.94180 − 1.49313i −2.53783 − 0.747941i 1.00914 − 2.64228i 2.99936 + 0.0618346i 0.171556 − 1.40377i
11.15 −1.10428 − 0.883495i −1.72228 + 0.183676i 0.438874 + 1.95125i 0.500000 + 0.866025i 2.06416 + 1.31880i 2.64252 + 0.130688i 1.23928 − 2.54247i 2.93253 − 0.632683i 0.212988 − 1.39808i
11.16 −1.06316 − 0.932574i 1.39682 + 1.02416i 0.260612 + 1.98295i 0.500000 + 0.866025i −0.529941 − 2.39148i −0.825730 + 2.51360i 1.57217 − 2.35123i 0.902212 + 2.86112i 0.276053 − 1.38701i
11.17 −0.952183 + 1.04563i 0.593587 + 1.62716i −0.186694 − 1.99127i 0.500000 + 0.866025i −2.26662 − 0.928683i −2.45676 − 0.981994i 2.25990 + 1.70084i −2.29531 + 1.93172i −1.38164 − 0.301799i
11.18 −0.937277 − 1.05901i −1.04183 + 1.38369i −0.243023 + 1.98518i 0.500000 + 0.866025i 2.44183 − 0.193594i −2.64094 + 0.159435i 2.33011 − 1.60330i −0.829200 − 2.88313i 0.448495 − 1.34121i
11.19 −0.867056 + 1.11723i −1.59043 + 0.685964i −0.496426 − 1.93741i 0.500000 + 0.866025i 0.612606 − 2.37165i −2.34127 − 1.23226i 2.59497 + 1.12522i 2.05891 − 2.18195i −1.40108 − 0.192276i
11.20 −0.739970 − 1.20517i 0.999729 − 1.41441i −0.904890 + 1.78358i 0.500000 + 0.866025i −2.44437 − 0.158230i 1.31611 − 2.29518i 2.81912 − 0.229249i −1.00109 − 2.82804i 0.673727 − 1.24342i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.64
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
24.f even 2 1 inner
168.v even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.cq.b yes 128
3.b odd 2 1 840.2.cq.a ✓ 128
7.c even 3 1 inner 840.2.cq.b yes 128
8.d odd 2 1 840.2.cq.a ✓ 128
21.h odd 6 1 840.2.cq.a ✓ 128
24.f even 2 1 inner 840.2.cq.b yes 128
56.k odd 6 1 840.2.cq.a ✓ 128
168.v even 6 1 inner 840.2.cq.b yes 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.cq.a ✓ 128 3.b odd 2 1
840.2.cq.a ✓ 128 8.d odd 2 1
840.2.cq.a ✓ 128 21.h odd 6 1
840.2.cq.a ✓ 128 56.k odd 6 1
840.2.cq.b yes 128 1.a even 1 1 trivial
840.2.cq.b yes 128 7.c even 3 1 inner
840.2.cq.b yes 128 24.f even 2 1 inner
840.2.cq.b yes 128 168.v even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{64} + 388 T_{23}^{62} - 968 T_{23}^{61} + 83901 T_{23}^{60} - 351512 T_{23}^{59} + \cdots + 15\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\). Copy content Toggle raw display