Properties

Label 103.2.a.a
Level $103$
Weight $2$
Character orbit 103.a
Self dual yes
Analytic conductor $0.822$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [103,2,Mod(1,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 103.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: \(0.822459140819\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} - q^{3} + 3 \beta q^{4} + (\beta - 2) q^{5} + (\beta + 1) q^{6} - q^{7} + ( - 4 \beta - 1) q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} - q^{3} + 3 \beta q^{4} + (\beta - 2) q^{5} + (\beta + 1) q^{6} - q^{7} + ( - 4 \beta - 1) q^{8} - 2 q^{9} + q^{10} + ( - \beta - 1) q^{11} - 3 \beta q^{12} - 3 \beta q^{13} + (\beta + 1) q^{14} + ( - \beta + 2) q^{15} + (3 \beta + 5) q^{16} + ( - \beta - 4) q^{17} + (2 \beta + 2) q^{18} + (3 \beta + 1) q^{19} + ( - 3 \beta + 3) q^{20} + q^{21} + (3 \beta + 2) q^{22} + (4 \beta - 2) q^{23} + (4 \beta + 1) q^{24} - 3 \beta q^{25} + (6 \beta + 3) q^{26} + 5 q^{27} - 3 \beta q^{28} + ( - 2 \beta - 2) q^{29} - q^{30} + ( - 6 \beta + 3) q^{31} + ( - 3 \beta - 6) q^{32} + (\beta + 1) q^{33} + (6 \beta + 5) q^{34} + ( - \beta + 2) q^{35} - 6 \beta q^{36} + (6 \beta - 3) q^{37} + ( - 7 \beta - 4) q^{38} + 3 \beta q^{39} + (3 \beta - 2) q^{40} + (8 \beta - 4) q^{41} + ( - \beta - 1) q^{42} + ( - 6 \beta + 1) q^{43} + ( - 6 \beta - 3) q^{44} + ( - 2 \beta + 4) q^{45} + ( - 6 \beta - 2) q^{46} + (5 \beta - 4) q^{47} + ( - 3 \beta - 5) q^{48} - 6 q^{49} + (6 \beta + 3) q^{50} + (\beta + 4) q^{51} + ( - 9 \beta - 9) q^{52} + (5 \beta - 7) q^{53} + ( - 5 \beta - 5) q^{54} + q^{55} + (4 \beta + 1) q^{56} + ( - 3 \beta - 1) q^{57} + (6 \beta + 4) q^{58} + ( - \beta + 8) q^{59} + (3 \beta - 3) q^{60} + ( - 3 \beta + 9) q^{61} + (9 \beta + 3) q^{62} + 2 q^{63} + (6 \beta - 1) q^{64} + (3 \beta - 3) q^{65} + ( - 3 \beta - 2) q^{66} + (12 \beta - 5) q^{67} + ( - 15 \beta - 3) q^{68} + ( - 4 \beta + 2) q^{69} - q^{70} + ( - 5 \beta + 4) q^{71} + (8 \beta + 2) q^{72} + ( - 3 \beta - 6) q^{73} + ( - 9 \beta - 3) q^{74} + 3 \beta q^{75} + (12 \beta + 9) q^{76} + (\beta + 1) q^{77} + ( - 6 \beta - 3) q^{78} + ( - 9 \beta + 8) q^{79} + (2 \beta - 7) q^{80} + q^{81} + ( - 12 \beta - 4) q^{82} + ( - 7 \beta + 5) q^{83} + 3 \beta q^{84} + ( - 3 \beta + 7) q^{85} + (11 \beta + 5) q^{86} + (2 \beta + 2) q^{87} + (9 \beta + 5) q^{88} + (6 \beta - 12) q^{89} - 2 q^{90} + 3 \beta q^{91} + (6 \beta + 12) q^{92} + (6 \beta - 3) q^{93} + ( - 6 \beta - 1) q^{94} + ( - 2 \beta + 1) q^{95} + (3 \beta + 6) q^{96} + ( - 6 \beta + 8) q^{97} + (6 \beta + 6) q^{98} + (2 \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + 2 q^{10} - 3 q^{11} - 3 q^{12} - 3 q^{13} + 3 q^{14} + 3 q^{15} + 13 q^{16} - 9 q^{17} + 6 q^{18} + 5 q^{19} + 3 q^{20} + 2 q^{21} + 7 q^{22} + 6 q^{24} - 3 q^{25} + 12 q^{26} + 10 q^{27} - 3 q^{28} - 6 q^{29} - 2 q^{30} - 15 q^{32} + 3 q^{33} + 16 q^{34} + 3 q^{35} - 6 q^{36} - 15 q^{38} + 3 q^{39} - q^{40} - 3 q^{42} - 4 q^{43} - 12 q^{44} + 6 q^{45} - 10 q^{46} - 3 q^{47} - 13 q^{48} - 12 q^{49} + 12 q^{50} + 9 q^{51} - 27 q^{52} - 9 q^{53} - 15 q^{54} + 2 q^{55} + 6 q^{56} - 5 q^{57} + 14 q^{58} + 15 q^{59} - 3 q^{60} + 15 q^{61} + 15 q^{62} + 4 q^{63} + 4 q^{64} - 3 q^{65} - 7 q^{66} + 2 q^{67} - 21 q^{68} - 2 q^{70} + 3 q^{71} + 12 q^{72} - 15 q^{73} - 15 q^{74} + 3 q^{75} + 30 q^{76} + 3 q^{77} - 12 q^{78} + 7 q^{79} - 12 q^{80} + 2 q^{81} - 20 q^{82} + 3 q^{83} + 3 q^{84} + 11 q^{85} + 21 q^{86} + 6 q^{87} + 19 q^{88} - 18 q^{89} - 4 q^{90} + 3 q^{91} + 30 q^{92} - 8 q^{94} + 15 q^{96} + 10 q^{97} + 18 q^{98} + 6 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
−2.61803 −1.00000 4.85410 −0.381966 2.61803 −1.00000 −7.47214 −2.00000 1.00000
1.2 −0.381966 −1.00000 −1.85410 −2.61803 0.381966 −1.00000 1.47214 −2.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(103\) \( +1 \)

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 103.2.a.a 2
3.b odd 2 1 927.2.a.b 2
4.b odd 2 1 1648.2.a.f 2
5.b even 2 1 2575.2.a.g 2
7.b odd 2 1 5047.2.a.a 2
8.b even 2 1 6592.2.a.t 2
8.d odd 2 1 6592.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.2.a.a 2 1.a even 1 1 trivial
927.2.a.b 2 3.b odd 2 1
1648.2.a.f 2 4.b odd 2 1
2575.2.a.g 2 5.b even 2 1
5047.2.a.a 2 7.b odd 2 1
6592.2.a.h 2 8.d odd 2 1
6592.2.a.t 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(103))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$17$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T - 5 \) Copy content Toggle raw display
$23$ \( T^{2} - 20 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$31$ \( T^{2} - 45 \) Copy content Toggle raw display
$37$ \( T^{2} - 45 \) Copy content Toggle raw display
$41$ \( T^{2} - 80 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$47$ \( T^{2} + 3T - 29 \) Copy content Toggle raw display
$53$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$59$ \( T^{2} - 15T + 55 \) Copy content Toggle raw display
$61$ \( T^{2} - 15T + 45 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T - 179 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$73$ \( T^{2} + 15T + 45 \) Copy content Toggle raw display
$79$ \( T^{2} - 7T - 89 \) Copy content Toggle raw display
$83$ \( T^{2} - 3T - 59 \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
show more
show less