Properties

Label 966.2.a.n
Level $966$
Weight $2$
Character orbit 966.a
Self dual yes
Analytic conductor $7.714$
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] = [966,2,Mod(1,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("966.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.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: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + \beta q^{5} - q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + \beta q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - \beta q^{10} - 4 q^{11} + q^{12} + (\beta + 2) q^{13} - q^{14} + \beta q^{15} + q^{16} + 4 q^{17} - q^{18} + ( - 2 \beta + 2) q^{19} + \beta q^{20} + q^{21} + 4 q^{22} + q^{23} - q^{24} + (\beta + 5) q^{25} + ( - \beta - 2) q^{26} + q^{27} + q^{28} + ( - \beta + 4) q^{29} - \beta q^{30} - 2 q^{31} - q^{32} - 4 q^{33} - 4 q^{34} + \beta q^{35} + q^{36} + ( - \beta + 8) q^{37} + (2 \beta - 2) q^{38} + (\beta + 2) q^{39} - \beta q^{40} + (\beta - 4) q^{41} - q^{42} + ( - \beta + 2) q^{43} - 4 q^{44} + \beta q^{45} - q^{46} - 3 \beta q^{47} + q^{48} + q^{49} + ( - \beta - 5) q^{50} + 4 q^{51} + (\beta + 2) q^{52} + ( - 2 \beta - 2) q^{53} - q^{54} - 4 \beta q^{55} - q^{56} + ( - 2 \beta + 2) q^{57} + (\beta - 4) q^{58} - 2 \beta q^{59} + \beta q^{60} - 2 q^{61} + 2 q^{62} + q^{63} + q^{64} + (3 \beta + 10) q^{65} + 4 q^{66} + 4 \beta q^{67} + 4 q^{68} + q^{69} - \beta q^{70} - 2 \beta q^{71} - q^{72} + (2 \beta - 6) q^{73} + (\beta - 8) q^{74} + (\beta + 5) q^{75} + ( - 2 \beta + 2) q^{76} - 4 q^{77} + ( - \beta - 2) q^{78} + 8 q^{79} + \beta q^{80} + q^{81} + ( - \beta + 4) q^{82} + (2 \beta + 6) q^{83} + q^{84} + 4 \beta q^{85} + (\beta - 2) q^{86} + ( - \beta + 4) q^{87} + 4 q^{88} + ( - 2 \beta - 4) q^{89} - \beta q^{90} + (\beta + 2) q^{91} + q^{92} - 2 q^{93} + 3 \beta q^{94} - 20 q^{95} - q^{96} + ( - \beta + 10) q^{97} - q^{98} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} - q^{10} - 8 q^{11} + 2 q^{12} + 5 q^{13} - 2 q^{14} + q^{15} + 2 q^{16} + 8 q^{17} - 2 q^{18} + 2 q^{19} + q^{20} + 2 q^{21} + 8 q^{22} + 2 q^{23} - 2 q^{24} + 11 q^{25} - 5 q^{26} + 2 q^{27} + 2 q^{28} + 7 q^{29} - q^{30} - 4 q^{31} - 2 q^{32} - 8 q^{33} - 8 q^{34} + q^{35} + 2 q^{36} + 15 q^{37} - 2 q^{38} + 5 q^{39} - q^{40} - 7 q^{41} - 2 q^{42} + 3 q^{43} - 8 q^{44} + q^{45} - 2 q^{46} - 3 q^{47} + 2 q^{48} + 2 q^{49} - 11 q^{50} + 8 q^{51} + 5 q^{52} - 6 q^{53} - 2 q^{54} - 4 q^{55} - 2 q^{56} + 2 q^{57} - 7 q^{58} - 2 q^{59} + q^{60} - 4 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} + 23 q^{65} + 8 q^{66} + 4 q^{67} + 8 q^{68} + 2 q^{69} - q^{70} - 2 q^{71} - 2 q^{72} - 10 q^{73} - 15 q^{74} + 11 q^{75} + 2 q^{76} - 8 q^{77} - 5 q^{78} + 16 q^{79} + q^{80} + 2 q^{81} + 7 q^{82} + 14 q^{83} + 2 q^{84} + 4 q^{85} - 3 q^{86} + 7 q^{87} + 8 q^{88} - 10 q^{89} - q^{90} + 5 q^{91} + 2 q^{92} - 4 q^{93} + 3 q^{94} - 40 q^{95} - 2 q^{96} + 19 q^{97} - 2 q^{98} - 8 q^{99}+O(q^{100}) \) 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
−2.70156
3.70156
−1.00000 1.00000 1.00000 −2.70156 −1.00000 1.00000 −1.00000 1.00000 2.70156
1.2 −1.00000 1.00000 1.00000 3.70156 −1.00000 1.00000 −1.00000 1.00000 −3.70156
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)
\(23\) \( -1 \)

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 966.2.a.n 2
3.b odd 2 1 2898.2.a.bb 2
4.b odd 2 1 7728.2.a.bc 2
7.b odd 2 1 6762.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.n 2 1.a even 1 1 trivial
2898.2.a.bb 2 3.b odd 2 1
6762.2.a.bo 2 7.b odd 2 1
7728.2.a.bc 2 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(966))\):

\( T_{5}^{2} - T_{5} - 10 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 5T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 10 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T - 4 \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 40 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 7T + 2 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 15T + 46 \) Copy content Toggle raw display
$41$ \( T^{2} + 7T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} - 3T - 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 3T - 90 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 32 \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 40 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 160 \) Copy content Toggle raw display
$71$ \( T^{2} + 2T - 40 \) Copy content Toggle raw display
$73$ \( T^{2} + 10T - 16 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 10T - 16 \) Copy content Toggle raw display
$97$ \( T^{2} - 19T + 80 \) Copy content Toggle raw display
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