Properties

Label 209.2.a.c
Level $209$
Weight $2$
Character orbit 209.a
Self dual yes
Analytic conductor $1.669$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [209,2,Mod(1,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("209.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.a (trivial)

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: yes
Analytic conductor: \(1.66887340224\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{3} q^{3} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{4} - \beta_{3}) q^{6} + ( - \beta_{2} - \beta_1 + 2) q^{7} + (\beta_{2} + 2 \beta_1) q^{8} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{3} q^{3} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{4} - \beta_{3}) q^{6} + ( - \beta_{2} - \beta_1 + 2) q^{7} + (\beta_{2} + 2 \beta_1) q^{8} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{9} + (\beta_{4} - \beta_{2} - \beta_1 + 3) q^{10} + q^{11} + (\beta_{4} - \beta_{2} + \beta_1 + 1) q^{12} + (\beta_{4} - \beta_1 + 1) q^{13} + ( - \beta_{4} - 2 \beta_{3} - 3 \beta_1 - 1) q^{14} + ( - 2 \beta_{4} - 3 \beta_{3} + \cdots + 2) q^{15}+ \cdots + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{8} + 4 q^{9} + 12 q^{10} + 5 q^{11} + 6 q^{12} + 4 q^{13} - 14 q^{14} + 3 q^{15} + 8 q^{16} - 4 q^{17} - 20 q^{18} - 5 q^{19} - 8 q^{20} + 10 q^{21} + 2 q^{22} + 3 q^{23} - 14 q^{24} + 6 q^{25} - 6 q^{26} - 11 q^{27} - 10 q^{28} + 10 q^{29} + 6 q^{30} + 11 q^{31} + 14 q^{32} + q^{33} - 4 q^{34} - 8 q^{35} - 26 q^{36} + q^{37} - 2 q^{38} + 2 q^{39} - 16 q^{40} + 2 q^{41} - 16 q^{42} + 20 q^{43} + 6 q^{44} - 28 q^{45} - 4 q^{46} - 20 q^{47} + 4 q^{48} + 3 q^{49} - 32 q^{50} + 24 q^{51} + 6 q^{52} - 14 q^{53} + 16 q^{54} - 5 q^{55} - 38 q^{56} - q^{57} - 6 q^{58} + 3 q^{59} - 40 q^{60} - 10 q^{61} - 6 q^{62} + 24 q^{63} - 2 q^{66} + 9 q^{67} + 24 q^{68} - 5 q^{69} + 50 q^{70} + 23 q^{71} - 12 q^{72} + 8 q^{74} - 18 q^{75} - 6 q^{76} + 6 q^{77} - 22 q^{78} + 44 q^{79} - 18 q^{80} + q^{81} - 30 q^{82} - 14 q^{83} + 14 q^{84} - 12 q^{85} + 52 q^{86} + 28 q^{87} + 6 q^{88} - 27 q^{89} + 26 q^{90} + 24 q^{91} + 58 q^{92} - 27 q^{93} - 8 q^{94} + 5 q^{95} + 50 q^{96} + 15 q^{97} - 10 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 7\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 3\beta_{3} + 8\beta_{2} + 10\beta _1 + 6 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.245526
1.71250
−1.15351
−1.51908
2.71457
−2.18524 2.15766 2.77529 −3.43077 −4.71500 3.93972 −1.69419 1.65548 7.49706
1.2 −0.779856 −2.98063 −1.39182 −3.49235 2.32446 1.06736 2.64513 5.88418 2.72353
1.3 0.484093 2.26452 −1.76565 0.637602 1.09624 2.66942 −1.82293 2.12805 0.308658
1.4 1.82669 −0.563416 1.33679 2.34577 −1.02918 1.69239 −1.21147 −2.68256 4.28499
1.5 2.65432 0.121872 5.04540 −1.06025 0.323487 −3.36889 8.08346 −2.98515 −2.81425
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.a.c 5
3.b odd 2 1 1881.2.a.k 5
4.b odd 2 1 3344.2.a.t 5
5.b even 2 1 5225.2.a.h 5
11.b odd 2 1 2299.2.a.n 5
19.b odd 2 1 3971.2.a.h 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.c 5 1.a even 1 1 trivial
1881.2.a.k 5 3.b odd 2 1
2299.2.a.n 5 11.b odd 2 1
3344.2.a.t 5 4.b odd 2 1
3971.2.a.h 5 19.b odd 2 1
5225.2.a.h 5 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 2T_{2}^{4} - 6T_{2}^{3} + 10T_{2}^{2} + 5T_{2} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(209))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 2 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( T^{5} - T^{4} - 9 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{5} + 5 T^{4} + \cdots + 19 \) Copy content Toggle raw display
$7$ \( T^{5} - 6 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T - 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{5} + 4 T^{4} + \cdots - 64 \) Copy content Toggle raw display
$19$ \( (T + 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} - 3 T^{4} + \cdots - 784 \) Copy content Toggle raw display
$29$ \( T^{5} - 10 T^{4} + \cdots + 490 \) Copy content Toggle raw display
$31$ \( T^{5} - 11 T^{4} + \cdots - 757 \) Copy content Toggle raw display
$37$ \( T^{5} - T^{4} + \cdots - 3088 \) Copy content Toggle raw display
$41$ \( T^{5} - 2 T^{4} + \cdots - 4112 \) Copy content Toggle raw display
$43$ \( T^{5} - 20 T^{4} + \cdots + 11266 \) Copy content Toggle raw display
$47$ \( T^{5} + 20 T^{4} + \cdots + 13184 \) Copy content Toggle raw display
$53$ \( T^{5} + 14 T^{4} + \cdots + 30304 \) Copy content Toggle raw display
$59$ \( T^{5} - 3 T^{4} + \cdots - 2000 \) Copy content Toggle raw display
$61$ \( T^{5} + 10 T^{4} + \cdots - 736 \) Copy content Toggle raw display
$67$ \( T^{5} - 9 T^{4} + \cdots + 17689 \) Copy content Toggle raw display
$71$ \( T^{5} - 23 T^{4} + \cdots + 19081 \) Copy content Toggle raw display
$73$ \( T^{5} - 340 T^{3} + \cdots + 155392 \) Copy content Toggle raw display
$79$ \( T^{5} - 44 T^{4} + \cdots - 36800 \) Copy content Toggle raw display
$83$ \( T^{5} + 14 T^{4} + \cdots - 3908 \) Copy content Toggle raw display
$89$ \( T^{5} + 27 T^{4} + \cdots + 320 \) Copy content Toggle raw display
$97$ \( T^{5} - 15 T^{4} + \cdots - 37456 \) Copy content Toggle raw display
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