Properties

Label 201.2.a.d
Level $201$
Weight $2$
Character orbit 201.a
Self dual yes
Analytic conductor $1.605$
Analytic rank $0$
Dimension $3$
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] = [201,2,Mod(1,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, 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("201.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 201.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: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + \beta_1 q^{5} + ( - \beta_{2} - 1) q^{6} + (\beta_{2} + \beta_1) q^{7} + ( - 3 \beta_1 + 4) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} - q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + \beta_1 q^{5} + ( - \beta_{2} - 1) q^{6} + (\beta_{2} + \beta_1) q^{7} + ( - 3 \beta_1 + 4) q^{8} + q^{9} + (2 \beta_1 - 1) q^{10} + ( - \beta_{2} + \beta_1 + 3) q^{11} + ( - \beta_{2} + \beta_1 - 2) q^{12} + ( - \beta_{2} + \beta_1 - 3) q^{13} + (\beta_1 + 2) q^{14} - \beta_1 q^{15} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + ( - \beta_{2} - 3 \beta_1 + 1) q^{17} + (\beta_{2} + 1) q^{18} + ( - 3 \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_{2} + 2 \beta_1 - 3) q^{20} + ( - \beta_{2} - \beta_1) q^{21} + (3 \beta_{2} + 3 \beta_1 - 1) q^{22} + ( - 2 \beta_{2} - 3 \beta_1 + 2) q^{23} + (3 \beta_1 - 4) q^{24} + (\beta_{2} + \beta_1 - 3) q^{25} + ( - 3 \beta_{2} + 3 \beta_1 - 7) q^{26} - q^{27} + q^{28} + 4 \beta_1 q^{29} + ( - 2 \beta_1 + 1) q^{30} + ( - 2 \beta_{2} - 4 \beta_1 + 5) q^{31} + (3 \beta_{2} - 4 \beta_1 + 5) q^{32} + (\beta_{2} - \beta_1 - 3) q^{33} + (\beta_{2} - 5 \beta_1 + 1) q^{34} + (\beta_{2} + 2 \beta_1 + 1) q^{35} + (\beta_{2} - \beta_1 + 2) q^{36} + (\beta_{2} - 3 \beta_1 - 2) q^{37} + ( - \beta_{2} + 5 \beta_1 - 11) q^{38} + (\beta_{2} - \beta_1 + 3) q^{39} + ( - 3 \beta_{2} + \beta_1 - 6) q^{40} + (3 \beta_{2} - 2 \beta_1 + 1) q^{41} + ( - \beta_1 - 2) q^{42} - q^{43} + (\beta_{2} + \beta_1 - 1) q^{44} + \beta_1 q^{45} + (2 \beta_{2} - 4 \beta_1 - 1) q^{46} + (3 \beta_{2} + 3 \beta_1 + 5) q^{47} + ( - 2 \beta_{2} + 4 \beta_1 - 3) q^{48} + (2 \beta_1 - 4) q^{49} + ( - 3 \beta_{2} + \beta_1 - 1) q^{50} + (\beta_{2} + 3 \beta_1 - 1) q^{51} + ( - 5 \beta_{2} + 7 \beta_1 - 13) q^{52} + ( - 5 \beta_{2} - 2 \beta_1 + 3) q^{53} + ( - \beta_{2} - 1) q^{54} + (\beta_{2} + 3 \beta_1 + 3) q^{55} + (\beta_{2} - 2 \beta_1 - 3) q^{56} + (3 \beta_{2} - \beta_1 + 1) q^{57} + (8 \beta_1 - 4) q^{58} + (5 \beta_{2} + 5) q^{59} + (\beta_{2} - 2 \beta_1 + 3) q^{60} + ( - \beta_{2} - 5 \beta_1 + 1) q^{61} + (5 \beta_{2} - 6 \beta_1 + 3) q^{62} + (\beta_{2} + \beta_1) q^{63} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + (\beta_{2} - 3 \beta_1 + 3) q^{65} + ( - 3 \beta_{2} - 3 \beta_1 + 1) q^{66} + q^{67} + (3 \beta_{2} - 5 \beta_1 + 7) q^{68} + (2 \beta_{2} + 3 \beta_1 - 2) q^{69} + (\beta_{2} + 3 \beta_1 + 2) q^{70} + ( - \beta_{2} + 3 \beta_1 + 5) q^{71} + ( - 3 \beta_1 + 4) q^{72} + ( - 2 \beta_{2} + 2 \beta_1 - 7) q^{73} + ( - 2 \beta_{2} - 7 \beta_1 + 4) q^{74} + ( - \beta_{2} - \beta_1 + 3) q^{75} + ( - 5 \beta_{2} + 9 \beta_1 - 17) q^{76} + (5 \beta_{2} + 5 \beta_1 - 1) q^{77} + (3 \beta_{2} - 3 \beta_1 + 7) q^{78} + ( - 2 \beta_1 + 10) q^{79} + ( - 4 \beta_{2} + \beta_1 - 10) q^{80} + q^{81} + (\beta_{2} - 7 \beta_1 + 12) q^{82} + (3 \beta_{2} + 2 \beta_1 - 3) q^{83} - q^{84} + ( - 3 \beta_{2} - 3 \beta_1 - 5) q^{85} + ( - \beta_{2} - 1) q^{86} - 4 \beta_1 q^{87} + ( - 7 \beta_{2} - 5 \beta_1 + 3) q^{88} + ( - 5 \beta_{2} + 3 \beta_1 - 3) q^{89} + (2 \beta_1 - 1) q^{90} + ( - \beta_{2} - \beta_1 - 1) q^{91} + (3 \beta_{2} - 4 \beta_1 + 5) q^{92} + (2 \beta_{2} + 4 \beta_1 - 5) q^{93} + (5 \beta_{2} + 3 \beta_1 + 11) q^{94} + (\beta_{2} - 3 \beta_1 + 5) q^{95} + ( - 3 \beta_{2} + 4 \beta_1 - 5) q^{96} + (7 \beta_{2} + 7 \beta_1 - 5) q^{97} + ( - 4 \beta_{2} + 4 \beta_1 - 6) q^{98} + ( - \beta_{2} + \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} + q^{5} - 3 q^{6} + q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 5 q^{4} + q^{5} - 3 q^{6} + q^{7} + 9 q^{8} + 3 q^{9} - q^{10} + 10 q^{11} - 5 q^{12} - 8 q^{13} + 7 q^{14} - q^{15} + 5 q^{16} + 3 q^{18} - 2 q^{19} - 7 q^{20} - q^{21} + 3 q^{23} - 9 q^{24} - 8 q^{25} - 18 q^{26} - 3 q^{27} + 3 q^{28} + 4 q^{29} + q^{30} + 11 q^{31} + 11 q^{32} - 10 q^{33} - 2 q^{34} + 5 q^{35} + 5 q^{36} - 9 q^{37} - 28 q^{38} + 8 q^{39} - 17 q^{40} + q^{41} - 7 q^{42} - 3 q^{43} - 2 q^{44} + q^{45} - 7 q^{46} + 18 q^{47} - 5 q^{48} - 10 q^{49} - 2 q^{50} - 32 q^{52} + 7 q^{53} - 3 q^{54} + 12 q^{55} - 11 q^{56} + 2 q^{57} - 4 q^{58} + 15 q^{59} + 7 q^{60} - 2 q^{61} + 3 q^{62} + q^{63} + 33 q^{64} + 6 q^{65} + 3 q^{67} + 16 q^{68} - 3 q^{69} + 9 q^{70} + 18 q^{71} + 9 q^{72} - 19 q^{73} + 5 q^{74} + 8 q^{75} - 42 q^{76} + 2 q^{77} + 18 q^{78} + 28 q^{79} - 29 q^{80} + 3 q^{81} + 29 q^{82} - 7 q^{83} - 3 q^{84} - 18 q^{85} - 3 q^{86} - 4 q^{87} + 4 q^{88} - 6 q^{89} - q^{90} - 4 q^{91} + 11 q^{92} - 11 q^{93} + 36 q^{94} + 12 q^{95} - 11 q^{96} - 8 q^{97} - 14 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) 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

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
0.311108
2.17009
−1.48119
−1.21432 −1.00000 −0.525428 0.311108 1.21432 −1.90321 3.06668 1.00000 −0.377784
1.2 1.53919 −1.00000 0.369102 2.17009 −1.53919 2.70928 −2.51026 1.00000 3.34017
1.3 2.67513 −1.00000 5.15633 −1.48119 −2.67513 0.193937 8.44358 1.00000 −3.96239
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(67\) \(-1\)

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 201.2.a.d 3
3.b odd 2 1 603.2.a.i 3
4.b odd 2 1 3216.2.a.u 3
5.b even 2 1 5025.2.a.m 3
7.b odd 2 1 9849.2.a.ba 3
12.b even 2 1 9648.2.a.bn 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.a.d 3 1.a even 1 1 trivial
603.2.a.i 3 3.b odd 2 1
3216.2.a.u 3 4.b odd 2 1
5025.2.a.m 3 5.b even 2 1
9648.2.a.bn 3 12.b even 2 1
9849.2.a.ba 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(201))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3T^{2} - T + 5 \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - T^{2} - 3T + 1 \) Copy content Toggle raw display
$7$ \( T^{3} - T^{2} - 5T + 1 \) Copy content Toggle raw display
$11$ \( T^{3} - 10 T^{2} + 24 T + 4 \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 28T + 52 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 44 T - 20 \) Copy content Toggle raw display
$23$ \( T^{3} - 3 T^{2} - 31 T + 95 \) Copy content Toggle raw display
$29$ \( T^{3} - 4 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$31$ \( T^{3} - 11 T^{2} - 13 T + 295 \) Copy content Toggle raw display
$37$ \( T^{3} + 9 T^{2} - 13 T - 169 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} - 61 T - 97 \) Copy content Toggle raw display
$43$ \( (T + 1)^{3} \) Copy content Toggle raw display
$47$ \( T^{3} - 18 T^{2} + 60 T + 52 \) Copy content Toggle raw display
$53$ \( T^{3} - 7 T^{2} - 77 T - 131 \) Copy content Toggle raw display
$59$ \( T^{3} - 15 T^{2} - 25 T + 625 \) Copy content Toggle raw display
$61$ \( T^{3} + 2 T^{2} - 76 T + 116 \) Copy content Toggle raw display
$67$ \( (T - 1)^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 18 T^{2} + 68 T + 100 \) Copy content Toggle raw display
$73$ \( T^{3} + 19 T^{2} + 83 T + 97 \) Copy content Toggle raw display
$79$ \( T^{3} - 28 T^{2} + 248 T - 688 \) Copy content Toggle raw display
$83$ \( T^{3} + 7 T^{2} - 21 T - 25 \) Copy content Toggle raw display
$89$ \( T^{3} + 6 T^{2} - 148 T + 116 \) Copy content Toggle raw display
$97$ \( T^{3} + 8 T^{2} - 240 T - 932 \) Copy content Toggle raw display
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