Properties

Label 950.2.a.f
Level $950$
Weight $2$
Character orbit 950.a
Self dual yes
Analytic conductor $7.586$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta + 1) q^{3} + q^{4} + ( - \beta - 1) q^{6} + ( - \beta + 3) q^{7} - q^{8} + 2 \beta q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta + 1) q^{3} + q^{4} + ( - \beta - 1) q^{6} + ( - \beta + 3) q^{7} - q^{8} + 2 \beta q^{9} + \beta q^{11} + (\beta + 1) q^{12} + ( - 2 \beta + 3) q^{13} + (\beta - 3) q^{14} + q^{16} + q^{17} - 2 \beta q^{18} - q^{19} + (2 \beta + 1) q^{21} - \beta q^{22} + (3 \beta + 5) q^{23} + ( - \beta - 1) q^{24} + (2 \beta - 3) q^{26} + ( - \beta + 1) q^{27} + ( - \beta + 3) q^{28} + ( - 2 \beta - 3) q^{29} + ( - 3 \beta + 2) q^{31} - q^{32} + (\beta + 2) q^{33} - q^{34} + 2 \beta q^{36} + 6 \beta q^{37} + q^{38} + (\beta - 1) q^{39} + 3 \beta q^{41} + ( - 2 \beta - 1) q^{42} + (3 \beta + 6) q^{43} + \beta q^{44} + ( - 3 \beta - 5) q^{46} + (\beta + 1) q^{48} + ( - 6 \beta + 4) q^{49} + (\beta + 1) q^{51} + ( - 2 \beta + 3) q^{52} + ( - 6 \beta - 3) q^{53} + (\beta - 1) q^{54} + (\beta - 3) q^{56} + ( - \beta - 1) q^{57} + (2 \beta + 3) q^{58} + ( - 7 \beta - 3) q^{59} + ( - 3 \beta + 10) q^{61} + (3 \beta - 2) q^{62} + (6 \beta - 4) q^{63} + q^{64} + ( - \beta - 2) q^{66} + (3 \beta + 9) q^{67} + q^{68} + (8 \beta + 11) q^{69} + (\beta - 12) q^{71} - 2 \beta q^{72} + ( - 6 \beta + 3) q^{73} - 6 \beta q^{74} - q^{76} + (3 \beta - 2) q^{77} + ( - \beta + 1) q^{78} + (6 \beta + 2) q^{79} + ( - 6 \beta - 1) q^{81} - 3 \beta q^{82} + ( - 6 \beta + 6) q^{83} + (2 \beta + 1) q^{84} + ( - 3 \beta - 6) q^{86} + ( - 5 \beta - 7) q^{87} - \beta q^{88} - 5 \beta q^{89} + ( - 9 \beta + 13) q^{91} + (3 \beta + 5) q^{92} + ( - \beta - 4) q^{93} + ( - \beta - 1) q^{96} + ( - 4 \beta - 6) q^{97} + (6 \beta - 4) q^{98} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{12} + 6 q^{13} - 6 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{21} + 10 q^{23} - 2 q^{24} - 6 q^{26} + 2 q^{27} + 6 q^{28} - 6 q^{29} + 4 q^{31} - 2 q^{32} + 4 q^{33} - 2 q^{34} + 2 q^{38} - 2 q^{39} - 2 q^{42} + 12 q^{43} - 10 q^{46} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 6 q^{52} - 6 q^{53} - 2 q^{54} - 6 q^{56} - 2 q^{57} + 6 q^{58} - 6 q^{59} + 20 q^{61} - 4 q^{62} - 8 q^{63} + 2 q^{64} - 4 q^{66} + 18 q^{67} + 2 q^{68} + 22 q^{69} - 24 q^{71} + 6 q^{73} - 2 q^{76} - 4 q^{77} + 2 q^{78} + 4 q^{79} - 2 q^{81} + 12 q^{83} + 2 q^{84} - 12 q^{86} - 14 q^{87} + 26 q^{91} + 10 q^{92} - 8 q^{93} - 2 q^{96} - 12 q^{97} - 8 q^{98} + 8 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−1.00000 −0.414214 1.00000 0 0.414214 4.41421 −1.00000 −2.82843 0
1.2 −1.00000 2.41421 1.00000 0 −2.41421 1.58579 −1.00000 2.82843 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.f 2
3.b odd 2 1 8550.2.a.cb 2
4.b odd 2 1 7600.2.a.v 2
5.b even 2 1 950.2.a.g 2
5.c odd 4 2 190.2.b.a 4
15.d odd 2 1 8550.2.a.bn 2
15.e even 4 2 1710.2.d.c 4
20.d odd 2 1 7600.2.a.bg 2
20.e even 4 2 1520.2.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.a 4 5.c odd 4 2
950.2.a.f 2 1.a even 1 1 trivial
950.2.a.g 2 5.b even 2 1
1520.2.d.e 4 20.e even 4 2
1710.2.d.c 4 15.e even 4 2
7600.2.a.v 2 4.b odd 2 1
7600.2.a.bg 2 20.d odd 2 1
8550.2.a.bn 2 15.d odd 2 1
8550.2.a.cb 2 3.b odd 2 1

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 6T_{7} + 7 \) Copy content Toggle raw display
\( T_{11}^{2} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 2 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 1 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 10T + 7 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$37$ \( T^{2} - 72 \) Copy content Toggle raw display
$41$ \( T^{2} - 18 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 63 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T - 89 \) Copy content Toggle raw display
$61$ \( T^{2} - 20T + 82 \) Copy content Toggle raw display
$67$ \( T^{2} - 18T + 63 \) Copy content Toggle raw display
$71$ \( T^{2} + 24T + 142 \) Copy content Toggle raw display
$73$ \( T^{2} - 6T - 63 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$83$ \( T^{2} - 12T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} - 50 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
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