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Properties

Label 840.2.cq.a
Level $840$
Weight $2$
Character orbit 840.cq
Analytic conductor $6.707$
Analytic rank $0$
Dimension $128$
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] = [840,2,Mod(11,840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(6.70743376979\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 64 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 64 q^{5} - 12 q^{12} - 6 q^{14} + 8 q^{16} + 14 q^{18} + 8 q^{21} + 4 q^{22} - 64 q^{25} - 16 q^{26} - 6 q^{28} - 10 q^{32} + 24 q^{34} + 20 q^{36} - 16 q^{38} + 16 q^{42} + 6 q^{44} + 8 q^{46} - 80 q^{48} + 8 q^{49} + 20 q^{51} - 6 q^{52} - 32 q^{54} - 18 q^{56} - 8 q^{58} - 12 q^{60} - 20 q^{62} - 28 q^{63} - 14 q^{66} + 32 q^{67} - 2 q^{68} + 40 q^{69} - 6 q^{70} - 48 q^{71} + 4 q^{72} - 8 q^{73} - 26 q^{74} - 24 q^{76} - 12 q^{78} + 8 q^{80} + 4 q^{81} - 12 q^{82} - 22 q^{84} + 46 q^{86} - 36 q^{87} - 20 q^{88} - 28 q^{90} - 24 q^{91} - 40 q^{92} + 12 q^{94} + 34 q^{96} + 32 q^{97} + 14 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.41420 − 0.00628611i 1.33156 − 1.10768i 1.99992 + 0.0177796i −0.500000 − 0.866025i −1.89005 + 1.55811i 1.13902 − 2.38802i −2.82818 − 0.0377157i 0.546093 − 2.94988i 0.701656 + 1.22788i
11.2 −1.41399 − 0.0249901i −1.06497 − 1.36596i 1.99875 + 0.0706716i −0.500000 − 0.866025i 1.47172 + 1.95807i −0.648913 + 2.56494i −2.82445 − 0.149878i −0.731690 + 2.90940i 0.685354 + 1.23705i
11.3 −1.40779 − 0.134661i 0.125358 − 1.72751i 1.96373 + 0.379149i −0.500000 − 0.866025i −0.409106 + 2.41508i −2.48277 + 0.914246i −2.71346 − 0.798201i −2.96857 − 0.433114i 0.587274 + 1.28651i
11.4 −1.40108 + 0.192276i −1.59043 + 0.685964i 1.92606 − 0.538787i −0.500000 − 0.866025i 2.09642 − 1.26689i 2.34127 + 1.23226i −2.59497 + 1.12522i 2.05891 − 2.18195i 0.867056 + 1.11723i
11.5 −1.39450 − 0.235299i −0.842351 + 1.51342i 1.88927 + 0.656250i −0.500000 − 0.866025i 1.53077 − 1.91226i 0.0685655 − 2.64486i −2.48017 − 1.35968i −1.58089 − 2.54966i 0.493476 + 1.32532i
11.6 −1.38164 + 0.301799i 0.593587 + 1.62716i 1.81783 − 0.833952i −0.500000 − 0.866025i −1.31120 − 2.06900i 2.45676 + 0.981994i −2.25990 + 1.70084i −2.29531 + 1.93172i 0.952183 + 1.04563i
11.7 −1.35661 − 0.399521i 1.55219 + 0.768565i 1.68077 + 1.08399i −0.500000 − 0.866025i −1.79866 − 1.66278i −2.29369 − 1.31870i −1.84706 − 2.14205i 1.81862 + 2.38592i 0.332308 + 1.37462i
11.8 −1.28418 − 0.592356i −1.58003 − 0.709572i 1.29823 + 1.52138i −0.500000 − 0.866025i 1.60873 + 1.84716i 2.01207 − 1.71801i −0.765958 − 2.72274i 1.99302 + 2.24229i 0.129094 + 1.40831i
11.9 −1.25983 + 0.642513i 1.72178 + 0.188349i 1.17435 − 1.61892i −0.500000 − 0.866025i −2.29017 + 0.868978i −2.62384 − 0.339789i −0.439313 + 2.79410i 2.92905 + 0.648591i 1.18635 + 0.769790i
11.10 −1.24747 + 0.666193i −1.44104 − 0.960934i 1.11237 − 1.66211i −0.500000 − 0.866025i 2.43783 + 0.238726i −1.36448 − 2.26676i −0.280367 + 2.81450i 1.15321 + 2.76950i 1.20068 + 0.747246i
11.11 −1.24611 − 0.668746i −1.33144 + 1.10782i 1.10556 + 1.66666i −0.500000 − 0.866025i 2.39997 − 0.490059i −2.03960 + 1.68524i −0.263069 − 2.81617i 0.545482 − 2.94999i 0.0439012 + 1.41353i
11.12 −1.22169 − 0.712380i 1.62293 − 0.605058i 0.985031 + 1.74061i −0.500000 − 0.866025i −2.41374 − 0.416952i 0.768777 + 2.53160i 0.0365752 − 2.82819i 2.26781 − 1.96394i −0.00609602 + 1.41420i
11.13 −1.16096 + 0.807578i −0.404828 − 1.68408i 0.695636 − 1.87512i −0.500000 − 0.866025i 1.83001 + 1.62821i 2.46244 − 0.967661i 0.706706 + 2.73872i −2.67223 + 1.36352i 1.27986 + 0.601628i
11.14 −1.15108 + 0.821593i 0.965902 − 1.43772i 0.649970 − 1.89144i −0.500000 − 0.866025i 0.0693879 + 2.44851i 1.26568 + 2.32337i 0.805825 + 2.71121i −1.13406 − 2.77739i 1.28706 + 0.586068i
11.15 −1.10278 − 0.885363i 0.354983 + 1.69528i 0.432265 + 1.95273i −0.500000 − 0.866025i 1.10947 − 2.18382i 0.582248 + 2.58089i 1.25218 − 2.53615i −2.74797 + 1.20359i −0.215355 + 1.39772i
11.16 −1.03252 + 0.966383i −1.60922 + 0.640639i 0.132207 − 1.99563i −0.500000 − 0.866025i 1.04245 − 2.21660i −1.82587 + 1.91473i 1.79203 + 2.18829i 2.17916 − 2.06186i 1.35317 + 0.410999i
11.17 −0.963179 + 1.03551i 1.37790 + 1.04947i −0.144572 − 1.99477i −0.500000 − 0.866025i −2.41391 + 0.416004i −0.185707 + 2.63923i 2.20486 + 1.77161i 0.797221 + 2.89213i 1.37837 + 0.316382i
11.18 −0.871113 − 1.11407i −0.0504933 − 1.73131i −0.482323 + 1.94097i −0.500000 − 0.866025i −1.88483 + 1.56442i 1.99150 − 1.74181i 2.58254 − 1.15346i −2.99490 + 0.174839i −0.529260 + 1.31144i
11.19 −0.817897 − 1.15371i 0.611810 + 1.62040i −0.662089 + 1.88723i −0.500000 − 0.866025i 1.36907 − 2.03117i −1.51455 − 2.16936i 2.71884 − 0.779702i −2.25138 + 1.98275i −0.590193 + 1.28517i
11.20 −0.766440 − 1.18852i −1.22851 − 1.22097i −0.825141 + 1.82185i −0.500000 − 0.866025i −0.509562 + 2.39590i −2.61918 + 0.374000i 2.79772 − 0.415646i 0.0184728 + 2.99994i −0.646065 + 1.25801i
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:
Format:

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

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