Properties

Label 950.2.a.n
Level $950$
Weight $2$
Character orbit 950.a
Self dual yes
Analytic conductor $7.586$
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] = [950,2,Mod(1,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, 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("950.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
3.12489
−0.363328
−1.76156
1.00000 −2.12489 1.00000 0 −2.12489 4.12489 1.00000 1.51514 0
1.2 1.00000 1.36333 1.00000 0 1.36333 0.636672 1.00000 −1.14134 0
1.3 1.00000 2.76156 1.00000 0 2.76156 −0.761557 1.00000 4.62620 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( -1 \)
\(19\) \( -1 \)

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 950.2.a.n 3
3.b odd 2 1 8550.2.a.ck 3
4.b odd 2 1 7600.2.a.bi 3
5.b even 2 1 950.2.a.i 3
5.c odd 4 2 190.2.b.b 6
15.d odd 2 1 8550.2.a.cl 3
15.e even 4 2 1710.2.d.d 6
20.d odd 2 1 7600.2.a.cd 3
20.e even 4 2 1520.2.d.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 5.c odd 4 2
950.2.a.i 3 5.b even 2 1
950.2.a.n 3 1.a even 1 1 trivial
1520.2.d.j 6 20.e even 4 2
1710.2.d.d 6 15.e even 4 2
7600.2.a.bi 3 4.b odd 2 1
7600.2.a.cd 3 20.d odd 2 1
8550.2.a.ck 3 3.b odd 2 1
8550.2.a.cl 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(950))\):

\( T_{3}^{3} - 2T_{3}^{2} - 5T_{3} + 8 \) Copy content Toggle raw display
\( T_{7}^{3} - 4T_{7}^{2} - T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{3} - 10T_{11} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 2 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 4T^{2} - T + 2 \) Copy content Toggle raw display
$11$ \( T^{3} - 10T - 8 \) Copy content Toggle raw display
$13$ \( T^{3} - 8 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 49T + 122 \) Copy content Toggle raw display
$29$ \( T^{3} + 8 T^{2} + \cdots - 410 \) Copy content Toggle raw display
$31$ \( T^{3} - 4 T^{2} + \cdots + 232 \) Copy content Toggle raw display
$37$ \( T^{3} - 14 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} + \cdots - 100 \) Copy content Toggle raw display
$43$ \( T^{3} - 18 T^{2} + \cdots - 148 \) Copy content Toggle raw display
$47$ \( T^{3} + 14 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$53$ \( T^{3} - 16 T^{2} + \cdots - 106 \) Copy content Toggle raw display
$59$ \( T^{3} - 2 T^{2} + \cdots + 80 \) Copy content Toggle raw display
$61$ \( T^{3} + 30 T^{2} + \cdots + 892 \) Copy content Toggle raw display
$67$ \( T^{3} - 2 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( T^{3} + 8 T^{2} + \cdots - 1016 \) Copy content Toggle raw display
$73$ \( T^{3} - 10 T^{2} + \cdots - 164 \) Copy content Toggle raw display
$79$ \( T^{3} - 228T - 880 \) Copy content Toggle raw display
$83$ \( T^{3} + 6 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$89$ \( T^{3} + 14 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$97$ \( T^{3} - 10 T^{2} + \cdots + 488 \) Copy content Toggle raw display
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