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Properties

Label 644.4.i.b
Level $644$
Weight $4$
Character orbit 644.i
Analytic conductor $37.997$
Analytic rank $0$
Dimension $44$
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] = [644,4,Mod(93,644)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(644, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("644.93");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 644.i (of order \(3\), 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: \(37.9972300437\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 12 q^{3} + 10 q^{5} - 6 q^{7} - 238 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 12 q^{3} + 10 q^{5} - 6 q^{7} - 238 q^{9} + 28 q^{11} - 152 q^{13} + 208 q^{15} - 52 q^{17} + 38 q^{19} - 10 q^{21} + 506 q^{23} - 516 q^{25} - 876 q^{27} - 100 q^{29} + 230 q^{31} + 424 q^{33} + 98 q^{35} + 18 q^{37} - 350 q^{39} + 784 q^{41} - 336 q^{43} + 1156 q^{45} + 452 q^{47} + 546 q^{49} - 498 q^{51} - 508 q^{53} - 3084 q^{55} - 1916 q^{57} + 508 q^{59} + 1386 q^{61} + 1290 q^{63} + 360 q^{65} - 1896 q^{67} + 552 q^{69} - 3352 q^{71} + 990 q^{73} + 3328 q^{75} + 1328 q^{77} + 524 q^{79} - 4486 q^{81} - 1120 q^{83} - 5296 q^{85} + 3700 q^{87} + 1216 q^{89} + 1438 q^{91} + 366 q^{93} + 90 q^{95} + 716 q^{97} + 5716 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1 0 −4.78776 − 8.29265i 0 1.66948 − 2.89162i 0 −18.1145 + 3.85540i 0 −32.3454 + 56.0238i 0
93.2 0 −4.53140 − 7.84862i 0 7.01387 − 12.1484i 0 18.1867 − 3.49896i 0 −27.5672 + 47.7478i 0
93.3 0 −3.96641 − 6.87002i 0 −0.583194 + 1.01012i 0 −10.1039 − 15.5213i 0 −17.9648 + 31.1159i 0
93.4 0 −3.38884 − 5.86964i 0 −9.31873 + 16.1405i 0 17.4817 + 6.11464i 0 −9.46847 + 16.3999i 0
93.5 0 −2.62528 − 4.54712i 0 1.52021 − 2.63307i 0 4.67102 + 17.9215i 0 −0.284198 + 0.492245i 0
93.6 0 −2.07015 − 3.58560i 0 −7.90404 + 13.6902i 0 −15.5002 − 10.1363i 0 4.92898 − 8.53725i 0
93.7 0 −1.91623 − 3.31900i 0 8.26993 − 14.3239i 0 17.5855 − 5.80951i 0 6.15616 − 10.6628i 0
93.8 0 −1.77144 − 3.06822i 0 −3.18380 + 5.51450i 0 10.8024 − 15.0435i 0 7.22402 − 12.5124i 0
93.9 0 −1.05987 − 1.83574i 0 −6.53650 + 11.3216i 0 −13.3762 + 12.8093i 0 11.2534 − 19.4914i 0
93.10 0 −0.439555 − 0.761331i 0 5.30456 − 9.18776i 0 −16.9718 + 7.41335i 0 13.1136 − 22.7134i 0
93.11 0 0.118737 + 0.205659i 0 3.62281 − 6.27489i 0 1.70982 − 18.4412i 0 13.4718 − 23.3338i 0
93.12 0 1.08350 + 1.87668i 0 −5.65862 + 9.80102i 0 10.5174 − 15.2441i 0 11.1520 − 19.3159i 0
93.13 0 1.11951 + 1.93905i 0 9.00947 − 15.6049i 0 −3.98915 − 18.0855i 0 10.9934 − 19.0411i 0
93.14 0 1.35692 + 2.35025i 0 −4.93765 + 8.55225i 0 7.85503 + 16.7720i 0 9.81756 − 17.0045i 0
93.15 0 1.88039 + 3.25692i 0 0.00989341 − 0.0171359i 0 12.8752 + 13.3127i 0 6.42830 − 11.1341i 0
93.16 0 2.29830 + 3.98077i 0 4.80691 − 8.32582i 0 −17.6263 + 5.68442i 0 2.93562 − 5.08465i 0
93.17 0 2.71868 + 4.70889i 0 −4.49925 + 7.79293i 0 −18.1496 − 3.68694i 0 −1.28241 + 2.22119i 0
93.18 0 3.73245 + 6.46479i 0 5.71334 − 9.89579i 0 −6.28887 + 17.4198i 0 −14.3624 + 24.8763i 0
93.19 0 4.30025 + 7.44826i 0 6.78212 − 11.7470i 0 13.5482 + 12.6272i 0 −23.4844 + 40.6761i 0
93.20 0 4.31937 + 7.48137i 0 −8.44548 + 14.6280i 0 −11.8579 − 14.2264i 0 −23.8140 + 41.2470i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 644.4.i.b ✓ 44
7.c even 3 1 inner 644.4.i.b ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.4.i.b ✓ 44 1.a even 1 1 trivial
644.4.i.b ✓ 44 7.c even 3 1 inner