Properties

Label 5635.2.a.w
Level $5635$
Weight $2$
Character orbit 5635.a
Self dual yes
Analytic conductor $44.996$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2777.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) 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
2.36234
0.825785
−0.679643
−1.50848
−1.36234 −2.51572 −0.144030 −1.00000 3.42727 0 2.92090 3.32886 1.36234
1.2 0.174215 0.596155 −1.96965 −1.00000 0.103859 0 −0.691574 −2.64460 −0.174215
1.3 1.67964 −3.26308 0.821201 −1.00000 −5.48081 0 −1.97996 7.64767 −1.67964
1.4 2.50848 −0.817356 4.29248 −1.00000 −2.05032 0 5.75064 −2.33193 −2.50848
\(n\): e.g. 2-40 or 990-1000
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.w 4
7.b odd 2 1 805.2.a.j 4
21.c even 2 1 7245.2.a.bc 4
35.c odd 2 1 4025.2.a.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.j 4 7.b odd 2 1
4025.2.a.l 4 35.c odd 2 1
5635.2.a.w 4 1.a even 1 1 trivial
7245.2.a.bc 4 21.c even 2 1

Hecke kernels

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

\( T_{2}^{4} - 3T_{2}^{3} - T_{2}^{2} + 6T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{4} + 6T_{3}^{3} + 9T_{3}^{2} - T_{3} - 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 7T_{11}^{3} + 14T_{11}^{2} - 5T_{11} - 4 \) Copy content Toggle raw display
\( T_{17}^{4} + 5T_{17}^{3} - 12T_{17}^{2} - 39T_{17} - 22 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} - T^{2} + 6 T - 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 6 T^{3} + 9 T^{2} - T - 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 7 T^{3} + 14 T^{2} - 5 T - 4 \) Copy content Toggle raw display
$13$ \( T^{4} - 5 T^{3} + 5 T - 2 \) Copy content Toggle raw display
$17$ \( T^{4} + 5 T^{3} - 12 T^{2} - 39 T - 22 \) Copy content Toggle raw display
$19$ \( T^{4} + 8 T^{3} - 3 T^{2} - 125 T - 212 \) Copy content Toggle raw display
$23$ \( (T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} - 36 T^{2} + 48 T - 16 \) Copy content Toggle raw display
$31$ \( T^{4} - 49 T^{2} - 117 T - 32 \) Copy content Toggle raw display
$37$ \( T^{4} - 13 T^{3} + 30 T^{2} + \cdots - 506 \) Copy content Toggle raw display
$41$ \( T^{4} + T^{3} - 43 T^{2} + 119 T - 86 \) Copy content Toggle raw display
$43$ \( T^{4} - 14 T^{3} - 11 T^{2} + \cdots - 428 \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} - 79 T^{2} - 511 T - 736 \) Copy content Toggle raw display
$53$ \( T^{4} + T^{3} - 10 T^{2} - 13 T - 2 \) Copy content Toggle raw display
$59$ \( T^{4} - 7 T^{3} - 74 T^{2} + 487 T + 164 \) Copy content Toggle raw display
$61$ \( T^{4} - 7 T^{3} - 54 T^{2} + 297 T - 274 \) Copy content Toggle raw display
$67$ \( T^{4} - 15 T^{3} - 86 T^{2} + \cdots - 1028 \) Copy content Toggle raw display
$71$ \( T^{4} - 71 T^{2} - 199 T - 8 \) Copy content Toggle raw display
$73$ \( T^{4} + 3 T^{3} - 170 T^{2} + \cdots + 1766 \) Copy content Toggle raw display
$79$ \( T^{4} + 14 T^{3} + 3 T^{2} + \cdots - 2192 \) Copy content Toggle raw display
$83$ \( T^{4} + 3 T^{3} - 152 T^{2} + \cdots - 1244 \) Copy content Toggle raw display
$89$ \( T^{4} - 11 T^{3} - 124 T^{2} + \cdots - 5114 \) Copy content Toggle raw display
$97$ \( T^{4} + 9 T^{3} - 147 T^{2} + \cdots + 3362 \) Copy content Toggle raw display
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