Properties

Label 671.2.bj.a
Level $671$
Weight $2$
Character orbit 671.bj
Analytic conductor $5.358$
Analytic rank $0$
Dimension $240$
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] = [671,2,Mod(21,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.bj (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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 12 q^{4} - 12 q^{5} - 240 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 12 q^{4} - 12 q^{5} - 240 q^{9} - 2 q^{11} + 8 q^{12} - 12 q^{14} + 8 q^{15} + 88 q^{16} - 80 q^{20} - 2 q^{22} + 40 q^{23} + 104 q^{25} + 44 q^{26} - 12 q^{31} + 14 q^{33} - 16 q^{34} - 24 q^{36} - 24 q^{37} - 8 q^{38} + 8 q^{42} - 38 q^{44} - 96 q^{45} + 12 q^{47} - 72 q^{48} - 120 q^{49} - 8 q^{53} + 16 q^{55} - 60 q^{56} + 104 q^{58} - 12 q^{59} - 6 q^{66} - 52 q^{67} + 100 q^{69} - 96 q^{70} - 36 q^{71} + 24 q^{75} + 10 q^{77} + 60 q^{78} - 60 q^{80} + 240 q^{81} - 124 q^{82} - 76 q^{86} + 32 q^{89} - 44 q^{91} - 196 q^{92} + 20 q^{93} + 36 q^{97} - 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −0.726621 2.71179i 0.275133i −5.09375 + 2.94088i −2.13341 1.23173i −0.746103 + 0.199918i 0.0793007 + 0.295954i 7.70593 + 7.70593i 2.92430 −1.79000 + 6.68036i
21.2 −0.671334 2.50545i 2.02545i −4.09455 + 2.36399i 1.82518 + 1.05377i −5.07467 + 1.35975i 1.18834 + 4.43495i 5.00344 + 5.00344i −1.10246 1.41486 5.28033i
21.3 −0.669128 2.49722i 3.34229i −4.05632 + 2.34192i 2.14227 + 1.23684i −8.34643 + 2.23642i −1.05755 3.94683i 4.90629 + 4.90629i −8.17091 1.65521 6.17733i
21.4 −0.662120 2.47107i 0.560439i −3.93572 + 2.27229i 1.44440 + 0.833927i −1.38488 + 0.371078i −0.344015 1.28388i 4.60300 + 4.60300i 2.68591 1.10432 4.12138i
21.5 −0.658834 2.45880i 2.40391i −3.87959 + 2.23988i 2.22593 + 1.28514i 5.91074 1.58378i −0.483143 1.80311i 4.46350 + 4.46350i −2.77879 1.69339 6.31981i
21.6 −0.640358 2.38985i 2.85448i −3.56927 + 2.06072i −0.196828 0.113639i 6.82178 1.82789i 1.10056 + 4.10735i 3.71144 + 3.71144i −5.14805 −0.145539 + 0.543159i
21.7 −0.592067 2.20962i 1.97237i −2.79985 + 1.61649i −3.10552 1.79297i −4.35821 + 1.16778i −0.874617 3.26411i 1.99442 + 1.99442i −0.890261 −2.12312 + 7.92360i
21.8 −0.578074 2.15740i 1.37027i −2.58816 + 1.49427i 0.616533 + 0.355955i 2.95622 0.792118i −0.224432 0.837593i 1.56124 + 1.56124i 1.12236 0.411537 1.53588i
21.9 −0.545851 2.03714i 2.79757i −2.11995 + 1.22395i −1.59773 0.922448i −5.69905 + 1.52706i 0.747293 + 2.78894i 0.667953 + 0.667953i −4.82641 −1.00704 + 3.75832i
21.10 −0.529457 1.97596i 0.158033i −1.89204 + 1.09237i −0.468236 0.270336i −0.312268 + 0.0836718i 0.436603 + 1.62942i 0.267227 + 0.267227i 2.97503 −0.286263 + 1.06835i
21.11 −0.521873 1.94766i 0.552867i −1.78897 + 1.03286i −3.49214 2.01619i 1.07680 0.288527i 0.957365 + 3.57294i 0.0937038 + 0.0937038i 2.69434 −2.10439 + 7.85369i
21.12 −0.505082 1.88499i 0.997508i −1.56603 + 0.904149i −1.66368 0.960523i 1.88029 0.503823i −1.28528 4.79673i −0.264530 0.264530i 2.00498 −0.970286 + 3.62116i
21.13 −0.485244 1.81096i 0.156634i −1.31205 + 0.757513i 2.73556 + 1.57937i 0.283658 0.0760059i 0.685855 + 2.55964i −0.642937 0.642937i 2.97547 1.53277 5.72036i
21.14 −0.462016 1.72427i 3.01358i −1.02758 + 0.593276i −3.33285 1.92422i 5.19622 1.39232i −0.0492920 0.183960i −1.02677 1.02677i −6.08169 −1.77804 + 6.63574i
21.15 −0.450896 1.68277i 1.09351i −0.896348 + 0.517507i 2.84449 + 1.64227i −1.84013 + 0.493062i −1.06406 3.97113i −1.18874 1.18874i 1.80422 1.48098 5.52711i
21.16 −0.390456 1.45720i 2.15184i −0.238929 + 0.137945i 3.43790 + 1.98487i 3.13566 0.840198i 1.08358 + 4.04398i −1.83919 1.83919i −1.63041 1.55001 5.78472i
21.17 −0.387783 1.44723i 2.94302i −0.212038 + 0.122420i −0.127261 0.0734740i 4.25922 1.14126i −0.459320 1.71420i −1.85949 1.85949i −5.66139 −0.0569840 + 0.212667i
21.18 −0.375407 1.40104i 2.58780i −0.0899266 + 0.0519191i −0.272209 0.157160i −3.62561 + 0.971479i 0.0181465 + 0.0677236i −1.94476 1.94476i −3.69672 −0.117998 + 0.440375i
21.19 −0.358764 1.33893i 1.90630i 0.0680400 0.0392829i 1.78909 + 1.03293i −2.55239 + 0.683912i −0.270943 1.01117i −2.03733 2.03733i −0.633976 0.741157 2.76603i
21.20 −0.328754 1.22693i 0.178685i 0.334784 0.193287i −1.39664 0.806352i −0.219233 + 0.0587433i 0.860816 + 3.21261i −2.14355 2.14355i 2.96807 −0.530182 + 1.97867i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
61.h odd 12 1 inner
671.bj even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.bj.a 240
11.b odd 2 1 inner 671.2.bj.a 240
61.h odd 12 1 inner 671.2.bj.a 240
671.bj even 12 1 inner 671.2.bj.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.bj.a 240 1.a even 1 1 trivial
671.2.bj.a 240 11.b odd 2 1 inner
671.2.bj.a 240 61.h odd 12 1 inner
671.2.bj.a 240 671.bj even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).