Properties

Label 5635.2.a.ba
Level $5635$
Weight $2$
Character orbit 5635.a
Self dual yes
Analytic conductor $44.996$
Analytic rank $1$
Dimension $6$
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] = [5635,2,Mod(1,5635)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5635, 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("5635.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5635 = 5 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5635.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: \(44.9957015390\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.169449536.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} - 2x^{3} + 13x^{2} + 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 8x^{4} - 2x^{3} + 13x^{2} + 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} + \nu^{4} + 7\nu^{3} - 4\nu^{2} - 9\nu + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 8\nu^{3} - 2\nu^{2} + 13\nu + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 8\beta_{3} + 2\beta_{2} + 27\beta _1 + 11 \) 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
−2.19211
−1.49211
−0.497659
0.165121
1.44884
2.56791
−2.19211 2.19211 2.80536 1.00000 −4.80536 0 −1.76543 1.80536 −2.19211
1.2 −1.49211 1.49211 0.226383 1.00000 −2.22638 0 2.64643 −0.773617 −1.49211
1.3 −0.497659 0.497659 −1.75234 1.00000 −0.247665 0 1.86739 −2.75234 −0.497659
1.4 0.165121 −0.165121 −1.97273 1.00000 −0.0272650 0 −0.655983 −2.97273 0.165121
1.5 1.44884 −1.44884 0.0991459 1.00000 −2.09915 0 −2.75404 −0.900854 1.44884
1.6 2.56791 −2.56791 4.59418 1.00000 −6.59418 0 6.66164 3.59418 2.56791
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)
\(23\) \(1\)

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 5635.2.a.ba yes 6
7.b odd 2 1 5635.2.a.z 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5635.2.a.z 6 7.b odd 2 1
5635.2.a.ba yes 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5635))\):

\( T_{2}^{6} - 8T_{2}^{4} - 2T_{2}^{3} + 13T_{2}^{2} + 4T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{6} - 8T_{3}^{4} + 2T_{3}^{3} + 13T_{3}^{2} - 4T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{6} - 4T_{11}^{5} - 15T_{11}^{4} + 34T_{11}^{3} + 62T_{11}^{2} - 46T_{11} - 27 \) Copy content Toggle raw display
\( T_{17}^{6} - 27T_{17}^{4} + 16T_{17}^{3} + 172T_{17}^{2} - 232T_{17} + 79 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 8 T^{4} - 2 T^{3} + 13 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 8 T^{4} + 2 T^{3} + 13 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 4 T^{5} - 15 T^{4} + 34 T^{3} + \cdots - 27 \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} + 5 T^{4} - 104 T^{3} + \cdots - 49 \) Copy content Toggle raw display
$17$ \( T^{6} - 27 T^{4} + 16 T^{3} + 172 T^{2} + \cdots + 79 \) Copy content Toggle raw display
$19$ \( T^{6} - 2 T^{5} - 46 T^{4} + 66 T^{3} + \cdots - 763 \) Copy content Toggle raw display
$23$ \( (T + 1)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + 8 T^{5} - 84 T^{4} + \cdots - 6976 \) Copy content Toggle raw display
$31$ \( T^{6} + 8 T^{5} - 98 T^{4} + \cdots - 16607 \) Copy content Toggle raw display
$37$ \( T^{6} + 16 T^{5} - 15 T^{4} + \cdots - 20665 \) Copy content Toggle raw display
$41$ \( T^{6} + 22 T^{5} + 123 T^{4} + \cdots - 36739 \) Copy content Toggle raw display
$43$ \( T^{6} - 8 T^{5} - 116 T^{4} + \cdots - 19893 \) Copy content Toggle raw display
$47$ \( T^{6} - 2 T^{5} - 174 T^{4} + \cdots - 41661 \) Copy content Toggle raw display
$53$ \( T^{6} - 4 T^{5} - 53 T^{4} + 20 T^{3} + \cdots - 185 \) Copy content Toggle raw display
$59$ \( T^{6} + 18 T^{5} + 9 T^{4} + \cdots + 2121 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} - 127 T^{4} + 4 T^{3} + \cdots + 5 \) Copy content Toggle raw display
$67$ \( T^{6} - 4 T^{5} - 231 T^{4} + \cdots - 8353 \) Copy content Toggle raw display
$71$ \( T^{6} - 186 T^{4} - 790 T^{3} + \cdots - 3467 \) Copy content Toggle raw display
$73$ \( T^{6} + 12 T^{5} - 221 T^{4} + \cdots + 214587 \) Copy content Toggle raw display
$79$ \( T^{6} + 30 T^{5} + 250 T^{4} + \cdots + 91161 \) Copy content Toggle raw display
$83$ \( T^{6} + 32 T^{5} + 225 T^{4} + \cdots - 24753 \) Copy content Toggle raw display
$89$ \( T^{6} - 28 T^{5} + 177 T^{4} + \cdots - 5747 \) Copy content Toggle raw display
$97$ \( T^{6} + 4 T^{5} - 291 T^{4} + \cdots - 382285 \) Copy content Toggle raw display
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