Properties

Label 100.6.g.a
Level $100$
Weight $6$
Character orbit 100.g
Analytic conductor $16.038$
Analytic rank $0$
Dimension $52$
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] = [100,6,Mod(21,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.21");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 100.g (of order \(5\), 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: \(16.0383819813\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(13\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 2 q^{3} + 116 q^{5} - 42 q^{7} - 1153 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q + 2 q^{3} + 116 q^{5} - 42 q^{7} - 1153 q^{9} - 5 q^{11} + 1458 q^{13} + 2418 q^{15} - 2329 q^{17} + 1912 q^{19} - 4818 q^{21} + 3594 q^{23} + 934 q^{25} + 3206 q^{27} - 1458 q^{29} + 5532 q^{31} + 2435 q^{33} - 5603 q^{35} + 22043 q^{37} - 9938 q^{39} + 4567 q^{41} - 35390 q^{43} - 59359 q^{45} - 5859 q^{47} + 165974 q^{49} + 68014 q^{51} + 20151 q^{53} - 97855 q^{55} - 241368 q^{57} - 116271 q^{59} + 39134 q^{61} + 262808 q^{63} + 190502 q^{65} + 40883 q^{67} - 51844 q^{69} - 109999 q^{71} - 187802 q^{73} - 164833 q^{75} + 102220 q^{77} + 122216 q^{79} - 264922 q^{81} + 125394 q^{83} + 83764 q^{85} + 205117 q^{87} - 107222 q^{89} + 58608 q^{91} - 490158 q^{93} - 82634 q^{95} + 129683 q^{97} + 302280 q^{99}+O(q^{100}) \) 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 −23.4577 17.0430i 0 52.3337 19.6516i 0 137.238 0 184.708 + 568.472i 0
21.2 0 −23.1010 16.7838i 0 −35.4147 + 43.2528i 0 −103.774 0 176.867 + 544.339i 0
21.3 0 −12.3717 8.98853i 0 −17.8429 52.9776i 0 −136.040 0 −2.82702 8.70068i 0
21.4 0 −11.3077 8.21554i 0 −48.4390 27.9045i 0 250.128 0 −14.7217 45.3088i 0
21.5 0 −9.09673 6.60916i 0 38.0613 + 40.9431i 0 −35.2774 0 −36.0216 110.863i 0
21.6 0 −6.56136 4.76711i 0 −37.0820 + 41.8321i 0 −4.12841 0 −54.7650 168.549i 0
21.7 0 1.62583 + 1.18123i 0 43.6006 34.9856i 0 115.151 0 −73.8431 227.266i 0
21.8 0 5.48004 + 3.98148i 0 53.5629 16.0003i 0 −205.169 0 −60.9125 187.469i 0
21.9 0 10.7159 + 7.78556i 0 −54.0256 + 14.3608i 0 −52.1911 0 −20.8755 64.2481i 0
21.10 0 11.4456 + 8.31568i 0 9.13427 + 55.1504i 0 206.581 0 −13.2410 40.7517i 0
21.11 0 14.0415 + 10.2018i 0 −37.5181 41.4415i 0 −48.5199 0 17.9974 + 55.3901i 0
21.12 0 20.8083 + 15.1181i 0 18.7145 52.6761i 0 120.829 0 129.338 + 398.061i 0
21.13 0 22.2789 + 16.1866i 0 31.6166 + 46.1020i 0 −146.877 0 159.254 + 490.133i 0
41.1 0 −8.70966 26.8056i 0 43.6109 + 34.9726i 0 −69.2087 0 −446.089 + 324.103i 0
41.2 0 −7.57963 23.3277i 0 −36.9556 41.9438i 0 −216.676 0 −290.140 + 210.799i 0
41.3 0 −5.66638 17.4393i 0 −28.2721 + 48.2254i 0 144.604 0 −75.4310 + 54.8038i 0
41.4 0 −5.45584 16.7914i 0 38.9708 40.0784i 0 174.835 0 −55.5923 + 40.3902i 0
41.5 0 −2.99812 9.22728i 0 −51.6337 21.4233i 0 70.0668 0 120.437 87.5028i 0
41.6 0 −1.45295 4.47171i 0 45.8662 31.9577i 0 −98.5151 0 178.706 129.838i 0
41.7 0 0.438384 + 1.34921i 0 37.4300 + 41.5210i 0 −49.5774 0 194.963 141.649i 0
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.6.g.a 52
25.d even 5 1 inner 100.6.g.a 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.6.g.a 52 1.a even 1 1 trivial
100.6.g.a 52 25.d even 5 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(100, [\chi])\).