Properties

Label 910.2.a.m
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: \(\Q(\sqrt{2}) \)
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: \( 2 \)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta q^{3} + q^{4} + q^{5} - \beta q^{6} + q^{7} - q^{8} + 5 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta q^{3} + q^{4} + q^{5} - \beta q^{6} + q^{7} - q^{8} + 5 q^{9} - q^{10} + 4 q^{11} + \beta q^{12} + q^{13} - q^{14} + \beta q^{15} + q^{16} + (\beta - 2) q^{17} - 5 q^{18} + ( - \beta - 4) q^{19} + q^{20} + \beta q^{21} - 4 q^{22} + ( - \beta - 4) q^{23} - \beta q^{24} + q^{25} - q^{26} + 2 \beta q^{27} + q^{28} + 6 q^{29} - \beta q^{30} + (2 \beta - 4) q^{31} - q^{32} + 4 \beta q^{33} + ( - \beta + 2) q^{34} + q^{35} + 5 q^{36} + ( - 2 \beta + 2) q^{37} + (\beta + 4) q^{38} + \beta q^{39} - q^{40} + ( - 3 \beta - 2) q^{41} - \beta q^{42} + 4 \beta q^{43} + 4 q^{44} + 5 q^{45} + (\beta + 4) q^{46} - 4 \beta q^{47} + \beta q^{48} + q^{49} - q^{50} + ( - 2 \beta + 8) q^{51} + q^{52} + (\beta + 6) q^{53} - 2 \beta q^{54} + 4 q^{55} - q^{56} + ( - 4 \beta - 8) q^{57} - 6 q^{58} + (\beta + 4) q^{59} + \beta q^{60} + ( - 2 \beta - 6) q^{61} + ( - 2 \beta + 4) q^{62} + 5 q^{63} + q^{64} + q^{65} - 4 \beta q^{66} + ( - 2 \beta - 4) q^{67} + (\beta - 2) q^{68} + ( - 4 \beta - 8) q^{69} - q^{70} + (\beta + 8) q^{71} - 5 q^{72} + ( - 2 \beta + 6) q^{73} + (2 \beta - 2) q^{74} + \beta q^{75} + ( - \beta - 4) q^{76} + 4 q^{77} - \beta q^{78} - 4 \beta q^{79} + q^{80} + q^{81} + (3 \beta + 2) q^{82} + (2 \beta - 8) q^{83} + \beta q^{84} + (\beta - 2) q^{85} - 4 \beta q^{86} + 6 \beta q^{87} - 4 q^{88} + (\beta + 14) q^{89} - 5 q^{90} + q^{91} + ( - \beta - 4) q^{92} + ( - 4 \beta + 16) q^{93} + 4 \beta q^{94} + ( - \beta - 4) q^{95} - \beta q^{96} + ( - 4 \beta + 6) q^{97} - q^{98} + 20 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{7} - 2 q^{8} + 10 q^{9} - 2 q^{10} + 8 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} - 10 q^{18} - 8 q^{19} + 2 q^{20} - 8 q^{22} - 8 q^{23} + 2 q^{25} - 2 q^{26} + 2 q^{28} + 12 q^{29} - 8 q^{31} - 2 q^{32} + 4 q^{34} + 2 q^{35} + 10 q^{36} + 4 q^{37} + 8 q^{38} - 2 q^{40} - 4 q^{41} + 8 q^{44} + 10 q^{45} + 8 q^{46} + 2 q^{49} - 2 q^{50} + 16 q^{51} + 2 q^{52} + 12 q^{53} + 8 q^{55} - 2 q^{56} - 16 q^{57} - 12 q^{58} + 8 q^{59} - 12 q^{61} + 8 q^{62} + 10 q^{63} + 2 q^{64} + 2 q^{65} - 8 q^{67} - 4 q^{68} - 16 q^{69} - 2 q^{70} + 16 q^{71} - 10 q^{72} + 12 q^{73} - 4 q^{74} - 8 q^{76} + 8 q^{77} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 16 q^{83} - 4 q^{85} - 8 q^{88} + 28 q^{89} - 10 q^{90} + 2 q^{91} - 8 q^{92} + 32 q^{93} - 8 q^{95} + 12 q^{97} - 2 q^{98} + 40 q^{99}+O(q^{100}) \) 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 −2.82843 1.00000 1.00000 2.82843 1.00000 −1.00000 5.00000 −1.00000
1.2 −1.00000 2.82843 1.00000 1.00000 −2.82843 1.00000 −1.00000 5.00000 −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.m 2
3.b odd 2 1 8190.2.a.ck 2
4.b odd 2 1 7280.2.a.bc 2
5.b even 2 1 4550.2.a.bn 2
7.b odd 2 1 6370.2.a.bd 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
910.2.a.m 2 1.a even 1 1 trivial
4550.2.a.bn 2 5.b even 2 1
6370.2.a.bd 2 7.b odd 2 1
7280.2.a.bc 2 4.b odd 2 1
8190.2.a.ck 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(910))\):

\( T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 4 \) Copy content Toggle raw display
\( T_{23}^{2} + 8T_{23} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$43$ \( T^{2} - 128 \) Copy content Toggle raw display
$47$ \( T^{2} - 128 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 128 \) Copy content Toggle raw display
$83$ \( T^{2} + 16T + 32 \) Copy content Toggle raw display
$89$ \( T^{2} - 28T + 188 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T - 92 \) Copy content Toggle raw display
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