Properties

Label 910.2.a.n
Level $910$
Weight $2$
Character orbit 910.a
Self dual yes
Analytic conductor $7.266$
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] = [910,2,Mod(1,910)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(910, 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("910.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 910 = 2 \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 910.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: \(7.26638658394\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta + 1) q^{3} + q^{4} - q^{5} + ( - \beta - 1) q^{6} - q^{7} - q^{8} + (2 \beta + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta + 1) q^{3} + q^{4} - q^{5} + ( - \beta - 1) q^{6} - q^{7} - q^{8} + (2 \beta + 3) q^{9} + q^{10} + (\beta - 1) q^{11} + (\beta + 1) q^{12} + q^{13} + q^{14} + ( - \beta - 1) q^{15} + q^{16} + 4 q^{17} + ( - 2 \beta - 3) q^{18} + (\beta - 3) q^{19} - q^{20} + ( - \beta - 1) q^{21} + ( - \beta + 1) q^{22} + ( - \beta + 3) q^{23} + ( - \beta - 1) q^{24} + q^{25} - q^{26} + (2 \beta + 10) q^{27} - q^{28} + 2 q^{29} + (\beta + 1) q^{30} + ( - \beta - 7) q^{31} - q^{32} + 4 q^{33} - 4 q^{34} + q^{35} + (2 \beta + 3) q^{36} + (4 \beta + 2) q^{37} + ( - \beta + 3) q^{38} + (\beta + 1) q^{39} + q^{40} + 8 q^{41} + (\beta + 1) q^{42} + ( - \beta - 3) q^{43} + (\beta - 1) q^{44} + ( - 2 \beta - 3) q^{45} + (\beta - 3) q^{46} + 8 q^{47} + (\beta + 1) q^{48} + q^{49} - q^{50} + (4 \beta + 4) q^{51} + q^{52} - 4 \beta q^{53} + ( - 2 \beta - 10) q^{54} + ( - \beta + 1) q^{55} + q^{56} + ( - 2 \beta + 2) q^{57} - 2 q^{58} + (3 \beta - 1) q^{59} + ( - \beta - 1) q^{60} + 2 q^{61} + (\beta + 7) q^{62} + ( - 2 \beta - 3) q^{63} + q^{64} - q^{65} - 4 q^{66} + ( - 6 \beta + 2) q^{67} + 4 q^{68} + (2 \beta - 2) q^{69} - q^{70} + ( - 5 \beta - 1) q^{71} + ( - 2 \beta - 3) q^{72} + ( - 4 \beta - 6) q^{73} + ( - 4 \beta - 2) q^{74} + (\beta + 1) q^{75} + (\beta - 3) q^{76} + ( - \beta + 1) q^{77} + ( - \beta - 1) q^{78} - 8 q^{79} - q^{80} + (6 \beta + 11) q^{81} - 8 q^{82} + 4 q^{83} + ( - \beta - 1) q^{84} - 4 q^{85} + (\beta + 3) q^{86} + (2 \beta + 2) q^{87} + ( - \beta + 1) q^{88} + ( - 4 \beta + 4) q^{89} + (2 \beta + 3) q^{90} - q^{91} + ( - \beta + 3) q^{92} + ( - 8 \beta - 12) q^{93} - 8 q^{94} + ( - \beta + 3) q^{95} + ( - \beta - 1) q^{96} + 6 q^{97} - q^{98} + (\beta + 7) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} + 8 q^{17} - 6 q^{18} - 6 q^{19} - 2 q^{20} - 2 q^{21} + 2 q^{22} + 6 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{26} + 20 q^{27} - 2 q^{28} + 4 q^{29} + 2 q^{30} - 14 q^{31} - 2 q^{32} + 8 q^{33} - 8 q^{34} + 2 q^{35} + 6 q^{36} + 4 q^{37} + 6 q^{38} + 2 q^{39} + 2 q^{40} + 16 q^{41} + 2 q^{42} - 6 q^{43} - 2 q^{44} - 6 q^{45} - 6 q^{46} + 16 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} + 8 q^{51} + 2 q^{52} - 20 q^{54} + 2 q^{55} + 2 q^{56} + 4 q^{57} - 4 q^{58} - 2 q^{59} - 2 q^{60} + 4 q^{61} + 14 q^{62} - 6 q^{63} + 2 q^{64} - 2 q^{65} - 8 q^{66} + 4 q^{67} + 8 q^{68} - 4 q^{69} - 2 q^{70} - 2 q^{71} - 6 q^{72} - 12 q^{73} - 4 q^{74} + 2 q^{75} - 6 q^{76} + 2 q^{77} - 2 q^{78} - 16 q^{79} - 2 q^{80} + 22 q^{81} - 16 q^{82} + 8 q^{83} - 2 q^{84} - 8 q^{85} + 6 q^{86} + 4 q^{87} + 2 q^{88} + 8 q^{89} + 6 q^{90} - 2 q^{91} + 6 q^{92} - 24 q^{93} - 16 q^{94} + 6 q^{95} - 2 q^{96} + 12 q^{97} - 2 q^{98} + 14 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
−1.00000 −1.23607 1.00000 −1.00000 1.23607 −1.00000 −1.00000 −1.47214 1.00000
1.2 −1.00000 3.23607 1.00000 −1.00000 −3.23607 −1.00000 −1.00000 7.47214 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)
\(13\) \( -1 \)

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 910.2.a.n 2
3.b odd 2 1 8190.2.a.cp 2
4.b odd 2 1 7280.2.a.y 2
5.b even 2 1 4550.2.a.bl 2
7.b odd 2 1 6370.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
910.2.a.n 2 1.a even 1 1 trivial
4550.2.a.bl 2 5.b even 2 1
6370.2.a.bc 2 7.b odd 2 1
7280.2.a.y 2 4.b odd 2 1
8190.2.a.cp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(910))\):

\( T_{3}^{2} - 2T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 4 \) Copy content Toggle raw display
\( T_{17} - 4 \) Copy content Toggle raw display
\( T_{23}^{2} - 6T_{23} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$41$ \( (T - 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 80 \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 176 \) Copy content Toggle raw display
$71$ \( T^{2} + 2T - 124 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$97$ \( (T - 6)^{2} \) Copy content Toggle raw display
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