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Properties

Label 684.3.s.a
Level $684$
Weight $3$
Character orbit 684.s
Analytic conductor $18.638$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,3,Mod(445,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.445");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.s (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: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - q^{7} + 4 q^{9} - 6 q^{11} + 33 q^{15} - 21 q^{17} - 20 q^{19} - 48 q^{23} - 200 q^{25} - 63 q^{27} - 27 q^{29} - 24 q^{31} + 27 q^{33} - 54 q^{35} - 81 q^{39} - 18 q^{41} - 152 q^{43} + 188 q^{45} - 12 q^{47} - 267 q^{49} - 126 q^{51} - 36 q^{53} + 126 q^{57} - 135 q^{59} - 7 q^{61} - 190 q^{63} - 288 q^{65} + 48 q^{69} - 81 q^{71} + 55 q^{73} + 165 q^{75} + 30 q^{77} + 28 q^{81} - 93 q^{83} + 306 q^{87} + 216 q^{89} + 96 q^{91} + 24 q^{93} + 288 q^{95} - 241 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
445.1 0 −2.99896 + 0.0790095i 0 −2.76024 − 4.78088i 0 0.838417 + 1.45218i 0 8.98751 − 0.473893i 0
445.2 0 −2.99241 − 0.213197i 0 2.77840 + 4.81232i 0 0.506731 + 0.877684i 0 8.90909 + 1.27595i 0
445.3 0 −2.96123 − 0.480768i 0 −0.920304 − 1.59401i 0 3.31490 + 5.74157i 0 8.53772 + 2.84733i 0
445.4 0 −2.93454 + 0.623289i 0 3.64833 + 6.31909i 0 −4.80976 − 8.33075i 0 8.22302 − 3.65813i 0
445.5 0 −2.72361 + 1.25776i 0 0.340706 + 0.590120i 0 −2.79581 − 4.84248i 0 5.83609 − 6.85128i 0
445.6 0 −2.67140 + 1.36514i 0 −4.45215 − 7.71135i 0 −2.28143 − 3.95155i 0 5.27278 − 7.29368i 0
445.7 0 −2.67131 − 1.36533i 0 3.94439 + 6.83189i 0 6.38121 + 11.0526i 0 5.27176 + 7.29442i 0
445.8 0 −2.56169 − 1.56133i 0 −1.88587 − 3.26643i 0 −0.501515 − 0.868649i 0 4.12453 + 7.99927i 0
445.9 0 −2.34179 + 1.87511i 0 1.96284 + 3.39973i 0 2.56510 + 4.44288i 0 1.96795 − 8.78221i 0
445.10 0 −2.21812 − 2.01989i 0 −2.05295 − 3.55582i 0 −6.48935 − 11.2399i 0 0.840121 + 8.96070i 0
445.11 0 −2.18497 − 2.05570i 0 3.46044 + 5.99365i 0 −3.78976 − 6.56406i 0 0.548164 + 8.98329i 0
445.12 0 −1.99772 + 2.23810i 0 0.267948 + 0.464100i 0 4.62209 + 8.00569i 0 −1.01820 − 8.94222i 0
445.13 0 −1.91383 − 2.31025i 0 −2.98110 − 5.16342i 0 6.83689 + 11.8418i 0 −1.67447 + 8.84286i 0
445.14 0 −1.40415 + 2.65111i 0 −1.52868 − 2.64775i 0 −5.58622 − 9.67561i 0 −5.05674 − 7.44509i 0
445.15 0 −1.26845 − 2.71865i 0 1.21776 + 2.10923i 0 −1.46858 − 2.54365i 0 −5.78207 + 6.89693i 0
445.16 0 −0.719501 + 2.91244i 0 −1.98064 − 3.43056i 0 2.79341 + 4.83834i 0 −7.96464 − 4.19101i 0
445.17 0 −0.522544 − 2.95414i 0 −3.49763 − 6.05807i 0 0.133800 + 0.231748i 0 −8.45389 + 3.08734i 0
445.18 0 −0.455700 + 2.96519i 0 1.91242 + 3.31241i 0 −3.97487 − 6.88467i 0 −8.58467 − 2.70247i 0
445.19 0 −0.283015 − 2.98662i 0 1.65331 + 2.86362i 0 0.469266 + 0.812792i 0 −8.83980 + 1.69052i 0
445.20 0 −0.224370 − 2.99160i 0 1.33945 + 2.31999i 0 3.94899 + 6.83985i 0 −8.89932 + 1.34245i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 445.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.s.a ✓ 80
3.b odd 2 1 2052.3.s.a 80
9.c even 3 1 684.3.bl.a yes 80
9.d odd 6 1 2052.3.bl.a 80
19.d odd 6 1 684.3.bl.a yes 80
57.f even 6 1 2052.3.bl.a 80
171.i odd 6 1 inner 684.3.s.a ✓ 80
171.t even 6 1 2052.3.s.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.s.a ✓ 80 1.a even 1 1 trivial
684.3.s.a ✓ 80 171.i odd 6 1 inner
684.3.bl.a yes 80 9.c even 3 1
684.3.bl.a yes 80 19.d odd 6 1
2052.3.s.a 80 3.b odd 2 1
2052.3.s.a 80 171.t even 6 1
2052.3.bl.a 80 9.d odd 6 1
2052.3.bl.a 80 57.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(684, [\chi])\).