Properties

Label 714.2.a.l
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: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} - q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + (\beta + 1) q^{10} + ( - \beta + 1) q^{11} + q^{12} + ( - \beta + 3) q^{13} - q^{14} + (\beta + 1) q^{15} + q^{16} + q^{17} + q^{18} + ( - 2 \beta - 2) q^{19} + (\beta + 1) q^{20} - q^{21} + ( - \beta + 1) q^{22} + (2 \beta - 2) q^{23} + q^{24} + 3 \beta q^{25} + ( - \beta + 3) q^{26} + q^{27} - q^{28} + ( - 4 \beta + 2) q^{29} + (\beta + 1) q^{30} + ( - 4 \beta + 4) q^{31} + q^{32} + ( - \beta + 1) q^{33} + q^{34} + ( - \beta - 1) q^{35} + q^{36} + (\beta + 1) q^{37} + ( - 2 \beta - 2) q^{38} + ( - \beta + 3) q^{39} + (\beta + 1) q^{40} + 2 \beta q^{41} - q^{42} + (\beta - 9) q^{43} + ( - \beta + 1) q^{44} + (\beta + 1) q^{45} + (2 \beta - 2) q^{46} + (4 \beta - 4) q^{47} + q^{48} + q^{49} + 3 \beta q^{50} + q^{51} + ( - \beta + 3) q^{52} + (3 \beta - 9) q^{53} + q^{54} + ( - \beta - 3) q^{55} - q^{56} + ( - 2 \beta - 2) q^{57} + ( - 4 \beta + 2) q^{58} - 4 q^{59} + (\beta + 1) q^{60} + ( - 4 \beta + 2) q^{61} + ( - 4 \beta + 4) q^{62} - q^{63} + q^{64} + (\beta - 1) q^{65} + ( - \beta + 1) q^{66} + (3 \beta + 5) q^{67} + q^{68} + (2 \beta - 2) q^{69} + ( - \beta - 1) q^{70} - 8 q^{71} + q^{72} + (5 \beta + 1) q^{73} + (\beta + 1) q^{74} + 3 \beta q^{75} + ( - 2 \beta - 2) q^{76} + (\beta - 1) q^{77} + ( - \beta + 3) q^{78} + (3 \beta + 1) q^{79} + (\beta + 1) q^{80} + q^{81} + 2 \beta q^{82} + ( - 7 \beta + 7) q^{83} - q^{84} + (\beta + 1) q^{85} + (\beta - 9) q^{86} + ( - 4 \beta + 2) q^{87} + ( - \beta + 1) q^{88} + (3 \beta - 5) q^{89} + (\beta + 1) q^{90} + (\beta - 3) q^{91} + (2 \beta - 2) q^{92} + ( - 4 \beta + 4) q^{93} + (4 \beta - 4) q^{94} + ( - 6 \beta - 10) q^{95} + q^{96} + (\beta + 13) q^{97} + q^{98} + ( - \beta + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{10} + q^{11} + 2 q^{12} + 5 q^{13} - 2 q^{14} + 3 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 6 q^{19} + 3 q^{20} - 2 q^{21} + q^{22} - 2 q^{23} + 2 q^{24} + 3 q^{25} + 5 q^{26} + 2 q^{27} - 2 q^{28} + 3 q^{30} + 4 q^{31} + 2 q^{32} + q^{33} + 2 q^{34} - 3 q^{35} + 2 q^{36} + 3 q^{37} - 6 q^{38} + 5 q^{39} + 3 q^{40} + 2 q^{41} - 2 q^{42} - 17 q^{43} + q^{44} + 3 q^{45} - 2 q^{46} - 4 q^{47} + 2 q^{48} + 2 q^{49} + 3 q^{50} + 2 q^{51} + 5 q^{52} - 15 q^{53} + 2 q^{54} - 7 q^{55} - 2 q^{56} - 6 q^{57} - 8 q^{59} + 3 q^{60} + 4 q^{62} - 2 q^{63} + 2 q^{64} - q^{65} + q^{66} + 13 q^{67} + 2 q^{68} - 2 q^{69} - 3 q^{70} - 16 q^{71} + 2 q^{72} + 7 q^{73} + 3 q^{74} + 3 q^{75} - 6 q^{76} - q^{77} + 5 q^{78} + 5 q^{79} + 3 q^{80} + 2 q^{81} + 2 q^{82} + 7 q^{83} - 2 q^{84} + 3 q^{85} - 17 q^{86} + q^{88} - 7 q^{89} + 3 q^{90} - 5 q^{91} - 2 q^{92} + 4 q^{93} - 4 q^{94} - 26 q^{95} + 2 q^{96} + 27 q^{97} + 2 q^{98} + q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
1.00000 1.00000 1.00000 −0.561553 1.00000 −1.00000 1.00000 1.00000 −0.561553
1.2 1.00000 1.00000 1.00000 3.56155 1.00000 −1.00000 1.00000 1.00000 3.56155
\(n\): 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.l 2
3.b odd 2 1 2142.2.a.w 2
4.b odd 2 1 5712.2.a.bl 2
7.b odd 2 1 4998.2.a.bx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.a.l 2 1.a even 1 1 trivial
2142.2.a.w 2 3.b odd 2 1
4998.2.a.bx 2 7.b odd 2 1
5712.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(714))\):

\( T_{5}^{2} - 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 4 \) 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
$5$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 68 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} + 17T + 68 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 15T + 18 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 68 \) Copy content Toggle raw display
$67$ \( T^{2} - 13T + 4 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 7T - 94 \) Copy content Toggle raw display
$79$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 7T - 196 \) Copy content Toggle raw display
$89$ \( T^{2} + 7T - 26 \) Copy content Toggle raw display
$97$ \( T^{2} - 27T + 178 \) Copy content Toggle raw display
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