Properties

Label 931.2.a.e
Level $931$
Weight $2$
Character orbit 931.a
Self dual yes
Analytic conductor $7.434$
Analytic rank $1$
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] = [931,2,Mod(1,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, 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("931.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.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: \(7.43407242818\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
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.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + \beta q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + ( - \beta + 2) q^{6} + (\beta - 3) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + \beta q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + ( - \beta + 2) q^{6} + (\beta - 3) q^{8} - q^{9} + ( - \beta + 1) q^{10} + (\beta + 1) q^{11} + (\beta - 4) q^{12} + ( - 3 \beta - 2) q^{13} - \beta q^{15} + 3 q^{16} + 4 q^{17} + ( - \beta + 1) q^{18} + q^{19} + (2 \beta - 1) q^{20} + q^{22} + ( - \beta - 7) q^{23} + ( - 3 \beta + 2) q^{24} - 4 q^{25} + (\beta - 4) q^{26} - 4 \beta q^{27} + ( - \beta - 8) q^{29} + (\beta - 2) q^{30} + ( - 3 \beta + 2) q^{31} + (\beta + 3) q^{32} + (\beta + 2) q^{33} + (4 \beta - 4) q^{34} + (2 \beta - 1) q^{36} + ( - 5 \beta + 2) q^{37} + (\beta - 1) q^{38} + ( - 2 \beta - 6) q^{39} + ( - \beta + 3) q^{40} + (2 \beta + 6) q^{41} + ( - 5 \beta + 1) q^{43} + ( - \beta - 3) q^{44} + q^{45} + ( - 6 \beta + 5) q^{46} + ( - \beta + 3) q^{47} + 3 \beta q^{48} + ( - 4 \beta + 4) q^{50} + 4 \beta q^{51} + (\beta + 10) q^{52} + 3 \beta q^{53} + (4 \beta - 8) q^{54} + ( - \beta - 1) q^{55} + \beta q^{57} + ( - 7 \beta + 6) q^{58} + (2 \beta - 4) q^{59} + ( - \beta + 4) q^{60} + (2 \beta - 9) q^{61} + (5 \beta - 8) q^{62} + (2 \beta - 7) q^{64} + (3 \beta + 2) q^{65} + \beta q^{66} + (2 \beta + 4) q^{67} + ( - 8 \beta + 4) q^{68} + ( - 7 \beta - 2) q^{69} + (3 \beta - 10) q^{71} + ( - \beta + 3) q^{72} + (10 \beta - 1) q^{73} + (7 \beta - 12) q^{74} - 4 \beta q^{75} + ( - 2 \beta + 1) q^{76} + ( - 4 \beta + 2) q^{78} - \beta q^{79} - 3 q^{80} - 5 q^{81} + (4 \beta - 2) q^{82} + (3 \beta - 5) q^{83} - 4 q^{85} + (6 \beta - 11) q^{86} + ( - 8 \beta - 2) q^{87} + ( - 2 \beta - 1) q^{88} + ( - \beta + 10) q^{89} + (\beta - 1) q^{90} + (13 \beta - 3) q^{92} + (2 \beta - 6) q^{93} + (4 \beta - 5) q^{94} - q^{95} + (3 \beta + 2) q^{96} + (2 \beta + 6) q^{97} + ( - \beta - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{6} - 6 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{6} - 6 q^{8} - 2 q^{9} + 2 q^{10} + 2 q^{11} - 8 q^{12} - 4 q^{13} + 6 q^{16} + 8 q^{17} + 2 q^{18} + 2 q^{19} - 2 q^{20} + 2 q^{22} - 14 q^{23} + 4 q^{24} - 8 q^{25} - 8 q^{26} - 16 q^{29} - 4 q^{30} + 4 q^{31} + 6 q^{32} + 4 q^{33} - 8 q^{34} - 2 q^{36} + 4 q^{37} - 2 q^{38} - 12 q^{39} + 6 q^{40} + 12 q^{41} + 2 q^{43} - 6 q^{44} + 2 q^{45} + 10 q^{46} + 6 q^{47} + 8 q^{50} + 20 q^{52} - 16 q^{54} - 2 q^{55} + 12 q^{58} - 8 q^{59} + 8 q^{60} - 18 q^{61} - 16 q^{62} - 14 q^{64} + 4 q^{65} + 8 q^{67} + 8 q^{68} - 4 q^{69} - 20 q^{71} + 6 q^{72} - 2 q^{73} - 24 q^{74} + 2 q^{76} + 4 q^{78} - 6 q^{80} - 10 q^{81} - 4 q^{82} - 10 q^{83} - 8 q^{85} - 22 q^{86} - 4 q^{87} - 2 q^{88} + 20 q^{89} - 2 q^{90} - 6 q^{92} - 12 q^{93} - 10 q^{94} - 2 q^{95} + 4 q^{96} + 12 q^{97} - 2 q^{99}+O(q^{100}) \) 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
−2.41421 −1.41421 3.82843 −1.00000 3.41421 0 −4.41421 −1.00000 2.41421
1.2 0.414214 1.41421 −1.82843 −1.00000 0.585786 0 −1.58579 −1.00000 −0.414214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(19\) \(-1\)

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 931.2.a.e 2
3.b odd 2 1 8379.2.a.bl 2
7.b odd 2 1 931.2.a.f 2
7.c even 3 2 931.2.f.i 4
7.d odd 6 2 133.2.f.c 4
21.c even 2 1 8379.2.a.bi 2
21.g even 6 2 1197.2.j.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.f.c 4 7.d odd 6 2
931.2.a.e 2 1.a even 1 1 trivial
931.2.a.f 2 7.b odd 2 1
931.2.f.i 4 7.c even 3 2
1197.2.j.e 4 21.g even 6 2
8379.2.a.bi 2 21.c even 2 1
8379.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(931))\):

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 14T + 47 \) Copy content Toggle raw display
$29$ \( T^{2} + 16T + 62 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 46 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T - 49 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 7 \) Copy content Toggle raw display
$53$ \( T^{2} - 18 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} + 18T + 73 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$71$ \( T^{2} + 20T + 82 \) Copy content Toggle raw display
$73$ \( T^{2} + 2T - 199 \) Copy content Toggle raw display
$79$ \( T^{2} - 2 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 7 \) Copy content Toggle raw display
$89$ \( T^{2} - 20T + 98 \) Copy content Toggle raw display
$97$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
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