Properties

Label 717.2.a.e
Level $717$
Weight $2$
Character orbit 717.a
Self dual yes
Analytic conductor $5.725$
Analytic rank $0$
Dimension $7$
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] = [717,2,Mod(1,717)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(717, 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("717.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 717 = 3 \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 717.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: \(5.72527382493\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 22x^{3} - 5x^{2} - 7x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + q^{3} + (\beta_{6} + 1) q^{4} + \beta_{6} q^{5} - \beta_{2} q^{6} + (\beta_{4} + \beta_1 + 1) q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + q^{3} + (\beta_{6} + 1) q^{4} + \beta_{6} q^{5} - \beta_{2} q^{6} + (\beta_{4} + \beta_1 + 1) q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{8} + q^{9} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1) q^{10} + (\beta_{2} + \beta_1 + 2) q^{11} + (\beta_{6} + 1) q^{12} + (\beta_{3} - 1) q^{13} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 1) q^{14} + \beta_{6} q^{15} + (\beta_{6} - \beta_{5} - \beta_{4} - 1) q^{16} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{17} - \beta_{2} q^{18} + ( - \beta_{6} - \beta_{5} - \beta_1 - 1) q^{19} + (2 \beta_{6} - \beta_{5} - \beta_{4} + 4) q^{20} + (\beta_{4} + \beta_1 + 1) q^{21} + ( - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 - 3) q^{22} + (\beta_{6} + \beta_{3} + 3) q^{23} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{24} + (\beta_{6} - \beta_{5} - \beta_{4} - 1) q^{25} + (\beta_{6} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{26} + q^{27} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_1 + 1) q^{28} + ( - 3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 2) q^{29} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1) q^{30} + (\beta_{5} - \beta_{4} - \beta_1 - 1) q^{31} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{32} + (\beta_{2} + \beta_1 + 2) q^{33} + ( - 3 \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_1 - 4) q^{34} + ( - \beta_{6} - \beta_{5} + \beta_{4}) q^{35} + (\beta_{6} + 1) q^{36} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + 2) q^{37} + ( - \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{38} + (\beta_{3} - 1) q^{39} + ( - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 1) q^{40} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 + 4) q^{41} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 1) q^{42} + (2 \beta_{5} - 3 \beta_{4} - \beta_{3} + 1) q^{43} + (3 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{44} + \beta_{6} q^{45} + (\beta_{5} - \beta_{4} - 4 \beta_{2} + \beta_1 + 1) q^{46} + ( - 3 \beta_{6} - 2 \beta_{3} - 2 \beta_1 + 2) q^{47} + (\beta_{6} - \beta_{5} - \beta_{4} - 1) q^{48} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{49} + ( - 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{50} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{51} + ( - 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_1 - 3) q^{52} + (2 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{53} - \beta_{2} q^{54} + (3 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{55} + ( - \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{56} + ( - \beta_{6} - \beta_{5} - \beta_1 - 1) q^{57} + (4 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{58} + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 - 1) q^{59} + (2 \beta_{6} - \beta_{5} - \beta_{4} + 4) q^{60} + ( - 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{61} + (2 \beta_{6} - 2 \beta_{4} + \beta_1 - 1) q^{62} + (\beta_{4} + \beta_1 + 1) q^{63} + (\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{64} + ( - 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 - 2) q^{65} + ( - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 - 3) q^{66} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_1 + 2) q^{67} + (4 \beta_{6} - 3 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 4 \beta_1 - 1) q^{68} + (\beta_{6} + \beta_{3} + 3) q^{69} + ( - \beta_{6} - \beta_{5} + 3 \beta_{4} + 2 \beta_{2} + \beta_1 + 1) q^{70} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{71} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{72} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{73} + ( - 2 \beta_{4} - \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 3) q^{74} + (\beta_{6} - \beta_{5} - \beta_{4} - 1) q^{75} + (3 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 4) q^{76} + ( - \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{77} + (\beta_{6} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{78} + (3 \beta_{6} - \beta_{4} + \beta_{3} + 1) q^{79} + (5 \beta_{6} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{80} + q^{81} + (5 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{82} + ( - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 + 5) q^{83} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_1 + 1) q^{84} + (4 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 3) q^{85} + (3 \beta_{6} + \beta_{5} - 6 \beta_{4} + \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 4) q^{86} + ( - 3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 2) q^{87} + ( - 5 \beta_{6} + 3 \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{88} + (\beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 5) q^{89} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1) q^{90} + (2 \beta_{6} - \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_1 - 4) q^{91} + (4 \beta_{6} - 2 \beta_{4} - \beta_1 + 5) q^{92} + (\beta_{5} - \beta_{4} - \beta_1 - 1) q^{93} + (\beta_{6} - 3 \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_1 - 2) q^{94} + (\beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 3) q^{95} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{96} + ( - 3 \beta_{6} + \beta_{5} - 3 \beta_{4} - \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{97} + (3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_1 + 3) q^{98} + (\beta_{2} + \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{3} + 10 q^{4} + 3 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{3} + 10 q^{4} + 3 q^{5} + 3 q^{7} + 7 q^{9} + 11 q^{11} + 10 q^{12} - 5 q^{13} + 6 q^{14} + 3 q^{15} - 4 q^{16} + 11 q^{17} - 8 q^{19} + 34 q^{20} + 3 q^{21} - 19 q^{22} + 26 q^{23} - 4 q^{25} + 6 q^{26} + 7 q^{27} - 2 q^{28} + 6 q^{29} - 2 q^{31} - 5 q^{32} + 11 q^{33} - 40 q^{34} - 5 q^{35} + 10 q^{36} + 12 q^{37} - 9 q^{38} - 5 q^{39} - 5 q^{40} + 26 q^{41} + 6 q^{42} + 10 q^{43} + 3 q^{44} + 3 q^{45} + 6 q^{46} + 7 q^{47} - 4 q^{48} + 2 q^{49} - 5 q^{50} + 11 q^{51} - 22 q^{52} + 2 q^{53} - 8 q^{55} - 9 q^{56} - 8 q^{57} - 6 q^{58} + 6 q^{59} + 34 q^{60} - 8 q^{61} - 2 q^{62} + 3 q^{63} - 18 q^{64} - 17 q^{65} - 19 q^{66} + 24 q^{67} - 9 q^{68} + 26 q^{69} - 3 q^{70} - 25 q^{71} - 16 q^{73} - 9 q^{74} - 4 q^{75} - 32 q^{76} + 24 q^{77} + 6 q^{78} + 19 q^{79} + 18 q^{80} + 7 q^{81} + 39 q^{82} + 37 q^{83} - 2 q^{84} - 20 q^{85} - q^{86} + 6 q^{87} - 21 q^{88} + 29 q^{89} - 19 q^{91} + 52 q^{92} - 2 q^{93} - 22 q^{94} - 24 q^{95} - 5 q^{96} - 12 q^{97} + 23 q^{98} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 22x^{3} - 5x^{2} - 7x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} + \nu^{3} - 7\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - \nu^{5} - 10\nu^{4} + 7\nu^{3} + 21\nu^{2} + 2\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{6} + \nu^{5} + 10\nu^{4} - 7\nu^{3} - 21\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - \nu^{5} - 10\nu^{4} + 7\nu^{3} + 22\nu^{2} + \nu - 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - \nu^{5} - 10\nu^{4} + 8\nu^{3} + 22\nu^{2} - 4\nu - 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{6} - \nu^{5} - 21\nu^{4} + 6\nu^{3} + 51\nu^{2} + 11\nu - 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{4} + \beta_{3} - \beta_{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{5} - 2\beta_{4} + 5\beta_{3} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{5} + 16\beta_{4} + 7\beta_{3} - 7\beta_{2} + 2\beta _1 + 36 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{6} + 14\beta_{5} - 18\beta_{4} + 31\beta_{3} + 31\beta_{2} + 2\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{6} - 20\beta_{5} + 114\beta_{4} + 43\beta_{3} - 53\beta_{2} + 22\beta _1 + 234 ) / 2 \) 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
0.456608
2.29537
−0.727328
0.326576
2.51519
−2.67022
−1.19620
−2.51244 1.00000 4.31233 3.31233 −2.51244 −0.339486 −5.80958 1.00000 −8.32202
1.2 −1.83393 1.00000 1.36331 0.363312 −1.83393 −2.69764 1.16764 1.00000 −0.666290
1.3 −1.51425 1.00000 0.292958 −0.707042 −1.51425 3.59927 2.58489 1.00000 1.07064
1.4 −0.0204073 1.00000 −1.99958 −2.99958 −0.0204073 −1.53276 0.0816207 1.00000 0.0612134
1.5 1.42661 1.00000 0.0352187 −0.964781 1.42661 2.45740 −2.80298 1.00000 −1.37637
1.6 2.03375 1.00000 2.13614 1.13614 2.03375 4.00620 0.276881 1.00000 2.31063
1.7 2.42067 1.00000 3.85962 2.85962 2.42067 −2.49298 4.50152 1.00000 6.92219
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(239\) \(1\)

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 717.2.a.e 7
3.b odd 2 1 2151.2.a.f 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
717.2.a.e 7 1.a even 1 1 trivial
2151.2.a.f 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(717))\):

\( T_{2}^{7} - 12T_{2}^{5} + 44T_{2}^{3} + T_{2}^{2} - 49T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{7} - 3T_{5}^{6} - 11T_{5}^{5} + 31T_{5}^{4} + 19T_{5}^{3} - 38T_{5}^{2} - 12T_{5} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 12 T^{5} + 44 T^{3} + T^{2} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 3 T^{6} - 11 T^{5} + 31 T^{4} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( T^{7} - 3 T^{6} - 21 T^{5} + 43 T^{4} + \cdots - 124 \) Copy content Toggle raw display
$11$ \( T^{7} - 11 T^{6} + 22 T^{5} + \cdots + 592 \) Copy content Toggle raw display
$13$ \( T^{7} + 5 T^{6} - 25 T^{5} - 169 T^{4} + \cdots + 152 \) Copy content Toggle raw display
$17$ \( T^{7} - 11 T^{6} - 21 T^{5} + \cdots - 6376 \) Copy content Toggle raw display
$19$ \( T^{7} + 8 T^{6} - 30 T^{5} - 341 T^{4} + \cdots - 620 \) Copy content Toggle raw display
$23$ \( T^{7} - 26 T^{6} + 254 T^{5} + \cdots + 512 \) Copy content Toggle raw display
$29$ \( T^{7} - 6 T^{6} - 139 T^{5} + \cdots + 73880 \) Copy content Toggle raw display
$31$ \( T^{7} + 2 T^{6} - 40 T^{5} - 91 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$37$ \( T^{7} - 12 T^{6} - 52 T^{5} + \cdots + 11608 \) Copy content Toggle raw display
$41$ \( T^{7} - 26 T^{6} + 175 T^{5} + \cdots + 9208 \) Copy content Toggle raw display
$43$ \( T^{7} - 10 T^{6} - 158 T^{5} + \cdots + 2564 \) Copy content Toggle raw display
$47$ \( T^{7} - 7 T^{6} - 173 T^{5} + \cdots + 464896 \) Copy content Toggle raw display
$53$ \( T^{7} - 2 T^{6} - 96 T^{5} + 89 T^{4} + \cdots + 928 \) Copy content Toggle raw display
$59$ \( T^{7} - 6 T^{6} - 171 T^{5} + \cdots - 24320 \) Copy content Toggle raw display
$61$ \( T^{7} + 8 T^{6} - 156 T^{5} + \cdots - 47728 \) Copy content Toggle raw display
$67$ \( T^{7} - 24 T^{6} - 56 T^{5} + \cdots - 3733504 \) Copy content Toggle raw display
$71$ \( T^{7} + 25 T^{6} + 179 T^{5} + \cdots - 37696 \) Copy content Toggle raw display
$73$ \( T^{7} + 16 T^{6} - 26 T^{5} + \cdots + 3496 \) Copy content Toggle raw display
$79$ \( T^{7} - 19 T^{6} + 12 T^{5} + \cdots + 115900 \) Copy content Toggle raw display
$83$ \( T^{7} - 37 T^{6} + 496 T^{5} + \cdots - 69584 \) Copy content Toggle raw display
$89$ \( T^{7} - 29 T^{6} + 184 T^{5} + \cdots - 3800 \) Copy content Toggle raw display
$97$ \( T^{7} + 12 T^{6} - 231 T^{5} + \cdots - 335144 \) Copy content Toggle raw display
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