Properties

Label 714.2.a.k
Level $714$
Weight $2$
Character orbit 714.a
Self dual yes
Analytic conductor $5.701$
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] = [714,2,Mod(1,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, 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("714.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.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: \(5.70131870432\)
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 \( q - q^{2} + q^{3} + q^{4} + \beta q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - \beta q^{10} + \beta q^{11} + q^{12} + ( - \beta + 2) q^{13} + q^{14} + \beta q^{15} + q^{16} - q^{17} - q^{18} + 6 q^{19} + \beta q^{20} - q^{21} - \beta q^{22} + ( - 2 \beta + 4) q^{23} - q^{24} + (\beta + 5) q^{25} + (\beta - 2) q^{26} + q^{27} - q^{28} - 2 \beta q^{29} - \beta q^{30} - q^{32} + \beta q^{33} + q^{34} - \beta q^{35} + q^{36} + (\beta - 2) q^{37} - 6 q^{38} + ( - \beta + 2) q^{39} - \beta q^{40} + ( - 2 \beta + 6) q^{41} + q^{42} + ( - \beta + 6) q^{43} + \beta q^{44} + \beta q^{45} + (2 \beta - 4) q^{46} - 8 q^{47} + q^{48} + q^{49} + ( - \beta - 5) q^{50} - q^{51} + ( - \beta + 2) q^{52} + ( - \beta - 4) q^{53} - q^{54} + (\beta + 10) q^{55} + q^{56} + 6 q^{57} + 2 \beta q^{58} + (2 \beta - 2) q^{59} + \beta q^{60} - 2 q^{61} - q^{63} + q^{64} + (\beta - 10) q^{65} - \beta q^{66} + (\beta + 6) q^{67} - q^{68} + ( - 2 \beta + 4) q^{69} + \beta q^{70} + (4 \beta - 4) q^{71} - q^{72} + ( - \beta - 8) q^{73} + ( - \beta + 2) q^{74} + (\beta + 5) q^{75} + 6 q^{76} - \beta q^{77} + (\beta - 2) q^{78} + (\beta + 10) q^{79} + \beta q^{80} + q^{81} + (2 \beta - 6) q^{82} + ( - \beta + 12) q^{83} - q^{84} - \beta q^{85} + (\beta - 6) q^{86} - 2 \beta q^{87} - \beta q^{88} + ( - \beta + 12) q^{89} - \beta q^{90} + (\beta - 2) q^{91} + ( - 2 \beta + 4) q^{92} + 8 q^{94} + 6 \beta q^{95} - q^{96} + ( - \beta + 4) q^{97} - q^{98} + \beta q^{99} +O(q^{100}) \) 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 \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - q^{10} + q^{11} + 2 q^{12} + 3 q^{13} + 2 q^{14} + q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 12 q^{19} + q^{20} - 2 q^{21} - q^{22} + 6 q^{23} - 2 q^{24} + 11 q^{25} - 3 q^{26} + 2 q^{27} - 2 q^{28} - 2 q^{29} - q^{30} - 2 q^{32} + q^{33} + 2 q^{34} - q^{35} + 2 q^{36} - 3 q^{37} - 12 q^{38} + 3 q^{39} - q^{40} + 10 q^{41} + 2 q^{42} + 11 q^{43} + q^{44} + q^{45} - 6 q^{46} - 16 q^{47} + 2 q^{48} + 2 q^{49} - 11 q^{50} - 2 q^{51} + 3 q^{52} - 9 q^{53} - 2 q^{54} + 21 q^{55} + 2 q^{56} + 12 q^{57} + 2 q^{58} - 2 q^{59} + q^{60} - 4 q^{61} - 2 q^{63} + 2 q^{64} - 19 q^{65} - q^{66} + 13 q^{67} - 2 q^{68} + 6 q^{69} + q^{70} - 4 q^{71} - 2 q^{72} - 17 q^{73} + 3 q^{74} + 11 q^{75} + 12 q^{76} - q^{77} - 3 q^{78} + 21 q^{79} + q^{80} + 2 q^{81} - 10 q^{82} + 23 q^{83} - 2 q^{84} - q^{85} - 11 q^{86} - 2 q^{87} - q^{88} + 23 q^{89} - q^{90} - 3 q^{91} + 6 q^{92} + 16 q^{94} + 6 q^{95} - 2 q^{96} + 7 q^{97} - 2 q^{98} + q^{99}+O(q^{100}) \) 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
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)
\(17\) \(1\)

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 714.2.a.k 2
3.b odd 2 1 2142.2.a.y 2
4.b odd 2 1 5712.2.a.bi 2
7.b odd 2 1 4998.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.a.k 2 1.a even 1 1 trivial
2142.2.a.y 2 3.b odd 2 1
4998.2.a.bs 2 7.b odd 2 1
5712.2.a.bi 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(714))\):

\( T_{5}^{2} - T_{5} - 10 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 10 \) 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
$11$ \( T^{2} - T - 10 \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 8 \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6T - 32 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 40 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3T - 8 \) Copy content Toggle raw display
$41$ \( T^{2} - 10T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 11T + 20 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 9T + 10 \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 40 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 13T + 32 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 160 \) Copy content Toggle raw display
$73$ \( T^{2} + 17T + 62 \) Copy content Toggle raw display
$79$ \( T^{2} - 21T + 100 \) Copy content Toggle raw display
$83$ \( T^{2} - 23T + 122 \) Copy content Toggle raw display
$89$ \( T^{2} - 23T + 122 \) Copy content Toggle raw display
$97$ \( T^{2} - 7T + 2 \) Copy content Toggle raw display
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