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Properties

Label 209.2.k.c
Level $209$
Weight $2$
Character orbit 209.k
Analytic conductor $1.669$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [209,2,Mod(18,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.18");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.k (of order \(10\), 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: \(1.66887340224\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 32 q^{4} - 10 q^{5} - 20 q^{6} + 20 q^{7} + 32 q^{9} - 28 q^{11} + 20 q^{16} + 10 q^{17} - 10 q^{19} - 52 q^{20} - 20 q^{23} + 40 q^{24} + 12 q^{25} + 16 q^{26} + 10 q^{28} + 20 q^{30} + 60 q^{35}+ \cdots - 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1 −2.04958 − 1.48911i 2.98595 + 0.970193i 1.36531 + 4.20198i −2.80019 + 2.03445i −4.67522 − 6.43489i 0.130591 − 0.0424315i 1.89316 − 5.82655i 5.54755 + 4.03053i 8.76873
18.2 −1.65927 − 1.20553i −1.03335 − 0.335757i 0.681836 + 2.09848i −2.27071 + 1.64977i 1.30985 + 1.80285i 2.26837 − 0.737037i 0.130856 − 0.402733i −1.47196 − 1.06944i 5.75655
18.3 −1.50855 − 1.09603i −1.74193 − 0.565986i 0.456422 + 1.40472i 0.620262 − 0.450647i 2.00745 + 2.76302i −2.46234 + 0.800063i −0.301354 + 0.927471i 0.286918 + 0.208458i −1.42962
18.4 −1.47842 − 1.07413i 1.96879 + 0.639697i 0.413920 + 1.27391i 3.04034 − 2.20894i −2.22357 − 3.06047i 0.571977 − 0.185846i −0.373003 + 1.14799i 1.03986 + 0.755500i −6.86759
18.5 −0.899298 − 0.653378i 1.75604 + 0.570573i −0.236200 − 0.726950i −0.523027 + 0.380001i −1.20641 − 1.66048i 3.53379 − 1.14820i −0.949561 + 2.92245i 0.331086 + 0.240548i 0.718642
18.6 −0.465575 − 0.338260i −1.32976 − 0.432066i −0.515694 − 1.58714i 0.698529 − 0.507511i 0.472954 + 0.650965i 1.64346 − 0.533991i −0.652440 + 2.00800i −0.845462 − 0.614264i −0.496888
18.7 −0.300425 − 0.218271i −3.03656 − 0.986639i −0.575421 − 1.77096i −0.574232 + 0.417204i 0.696903 + 0.959205i −2.06780 + 0.671870i −0.443184 + 1.36398i 5.82021 + 4.22863i 0.263577
18.8 0.300425 + 0.218271i 3.03656 + 0.986639i −0.575421 − 1.77096i −0.574232 + 0.417204i 0.696903 + 0.959205i −2.06780 + 0.671870i 0.443184 − 1.36398i 5.82021 + 4.22863i −0.263577
18.9 0.465575 + 0.338260i 1.32976 + 0.432066i −0.515694 − 1.58714i 0.698529 − 0.507511i 0.472954 + 0.650965i 1.64346 − 0.533991i 0.652440 − 2.00800i −0.845462 − 0.614264i 0.496888
18.10 0.899298 + 0.653378i −1.75604 − 0.570573i −0.236200 − 0.726950i −0.523027 + 0.380001i −1.20641 − 1.66048i 3.53379 − 1.14820i 0.949561 − 2.92245i 0.331086 + 0.240548i −0.718642
18.11 1.47842 + 1.07413i −1.96879 − 0.639697i 0.413920 + 1.27391i 3.04034 − 2.20894i −2.22357 − 3.06047i 0.571977 − 0.185846i 0.373003 − 1.14799i 1.03986 + 0.755500i 6.86759
18.12 1.50855 + 1.09603i 1.74193 + 0.565986i 0.456422 + 1.40472i 0.620262 − 0.450647i 2.00745 + 2.76302i −2.46234 + 0.800063i 0.301354 − 0.927471i 0.286918 + 0.208458i 1.42962
18.13 1.65927 + 1.20553i 1.03335 + 0.335757i 0.681836 + 2.09848i −2.27071 + 1.64977i 1.30985 + 1.80285i 2.26837 − 0.737037i −0.130856 + 0.402733i −1.47196 − 1.06944i −5.75655
18.14 2.04958 + 1.48911i −2.98595 − 0.970193i 1.36531 + 4.20198i −2.80019 + 2.03445i −4.67522 − 6.43489i 0.130591 − 0.0424315i −1.89316 + 5.82655i 5.54755 + 4.03053i −8.76873
94.1 −0.805271 − 2.47837i −0.729604 + 1.00421i −3.87582 + 2.81595i 0.147707 − 0.454595i 3.07634 + 0.999563i 2.60156 + 3.58074i 5.88358 + 4.27467i 0.450928 + 1.38781i −1.24560
94.2 −0.787101 − 2.42245i 1.24680 − 1.71608i −3.63069 + 2.63785i 0.405883 − 1.24918i −5.13846 − 1.66959i −1.04090 − 1.43268i 5.12646 + 3.72459i −0.463350 − 1.42604i −3.34554
94.3 −0.713349 − 2.19546i −0.368052 + 0.506580i −2.69316 + 1.95669i −1.07334 + 3.30339i 1.37473 + 0.446676i −1.86145 − 2.56207i 2.48186 + 1.80318i 0.805890 + 2.48027i 8.01814
94.4 −0.548706 − 1.68874i −1.65185 + 2.27357i −0.932742 + 0.677676i −0.00109312 + 0.00336429i 4.74586 + 1.54202i −0.981625 − 1.35109i −1.21684 − 0.884085i −1.51348 − 4.65802i 0.00628122
94.5 −0.497006 − 1.52963i 1.67150 − 2.30062i −0.474709 + 0.344896i −0.600188 + 1.84719i −4.34983 − 1.41335i 0.354186 + 0.487495i −1.83886 − 1.33601i −1.57189 − 4.83779i 3.12380
94.6 −0.366018 − 1.12649i 0.922652 − 1.26992i 0.483026 − 0.350939i 1.00302 − 3.08698i −1.76826 − 0.574543i 1.99434 + 2.74498i −2.48862 − 1.80809i 0.165637 + 0.509779i −3.84457
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
19.b odd 2 1 inner
209.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.k.c ✓ 56
11.d odd 10 1 inner 209.2.k.c ✓ 56
19.b odd 2 1 inner 209.2.k.c ✓ 56
209.k even 10 1 inner 209.2.k.c ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.k.c ✓ 56 1.a even 1 1 trivial
209.2.k.c ✓ 56 11.d odd 10 1 inner
209.2.k.c ✓ 56 19.b odd 2 1 inner
209.2.k.c ✓ 56 209.k even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} + 30 T_{2}^{54} + 476 T_{2}^{52} + 5344 T_{2}^{50} + 49318 T_{2}^{48} + 390884 T_{2}^{46} + \cdots + 24413481 \) acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\). Copy content Toggle raw display