Properties

Label 627.2.a.j
Level $627$
Weight $2$
Character orbit 627.a
Self dual yes
Analytic conductor $5.007$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [627,2,Mod(1,627)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(627, 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("627.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 627 = 3 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 627.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: \(5.00662020673\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.1920025.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 9x^{3} + 8x^{2} + 15x - 4 \) 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 9x^{3} + 8x^{2} + 15x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 6\nu - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + \nu^{3} - 6\nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - \beta_{3} + 7\beta_{2} - 2\beta _1 + 24 \) 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
−2.64918
−1.18000
0.243834
2.19978
2.38557
−2.64918 −1.00000 5.01817 0.850015 2.64918 2.32086 −7.99567 1.00000 −2.25185
1.2 −1.18000 −1.00000 −0.607596 3.33867 1.18000 4.82938 3.07697 1.00000 −3.93964
1.3 0.243834 −1.00000 −1.94055 1.31403 −0.243834 −3.38905 −0.960838 1.00000 0.320406
1.4 2.19978 −1.00000 2.83905 3.77215 −2.19978 0.285221 1.84574 1.00000 8.29791
1.5 2.38557 −1.00000 3.69092 −2.27486 −2.38557 2.95359 4.03380 1.00000 −5.42683
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(11\) \( -1 \)
\(19\) \( +1 \)

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 627.2.a.j 5
3.b odd 2 1 1881.2.a.m 5
11.b odd 2 1 6897.2.a.s 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.2.a.j 5 1.a even 1 1 trivial
1881.2.a.m 5 3.b odd 2 1
6897.2.a.s 5 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(627))\):

\( T_{2}^{5} - T_{2}^{4} - 9T_{2}^{3} + 8T_{2}^{2} + 15T_{2} - 4 \) Copy content Toggle raw display
\( T_{5}^{5} - 7T_{5}^{4} + 8T_{5}^{3} + 31T_{5}^{2} - 66T_{5} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - T^{4} - 9 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 7 T^{4} + \cdots + 32 \) Copy content Toggle raw display
$7$ \( T^{5} - 7 T^{4} + \cdots + 32 \) Copy content Toggle raw display
$11$ \( (T - 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 10 T^{4} + \cdots - 214 \) Copy content Toggle raw display
$17$ \( T^{5} - 17 T^{4} + \cdots + 1282 \) Copy content Toggle raw display
$19$ \( (T + 1)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} + 7 T^{4} + \cdots - 464 \) Copy content Toggle raw display
$29$ \( T^{5} - 2 T^{4} + \cdots + 32 \) Copy content Toggle raw display
$31$ \( T^{5} - 4 T^{4} + \cdots - 12704 \) Copy content Toggle raw display
$37$ \( T^{5} - 11 T^{4} + \cdots + 44 \) Copy content Toggle raw display
$41$ \( T^{5} - T^{4} + \cdots - 3476 \) Copy content Toggle raw display
$43$ \( T^{5} - 6 T^{4} + \cdots - 80 \) Copy content Toggle raw display
$47$ \( T^{5} + 2 T^{4} + \cdots + 928 \) Copy content Toggle raw display
$53$ \( T^{5} - 5 T^{4} + \cdots - 2614 \) Copy content Toggle raw display
$59$ \( T^{5} + 18 T^{4} + \cdots + 188 \) Copy content Toggle raw display
$61$ \( T^{5} - 19 T^{4} + \cdots - 5636 \) Copy content Toggle raw display
$67$ \( T^{5} + T^{4} + \cdots + 89704 \) Copy content Toggle raw display
$71$ \( T^{5} + 29 T^{4} + \cdots - 193840 \) Copy content Toggle raw display
$73$ \( T^{5} - 14 T^{4} + \cdots - 55792 \) Copy content Toggle raw display
$79$ \( T^{5} - 3 T^{4} + \cdots - 34168 \) Copy content Toggle raw display
$83$ \( T^{5} - 4 T^{4} + \cdots - 15748 \) Copy content Toggle raw display
$89$ \( T^{5} - T^{4} + \cdots + 62446 \) Copy content Toggle raw display
$97$ \( T^{5} - 4 T^{4} + \cdots - 4 \) Copy content Toggle raw display
show more
show less