Properties

Label 627.2.a.g
Level $627$
Weight $2$
Character orbit 627.a
Self dual yes
Analytic conductor $5.007$
Analytic rank $0$
Dimension $3$
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] = [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: \(3\)
Coefficient field: 3.3.169.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x - 1 \) 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) 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
2.65109
−0.273891
−1.37720
−1.65109 1.00000 0.726109 2.27389 −1.65109 1.37720 2.10331 1.00000 −3.75441
1.2 1.27389 1.00000 −0.377203 3.37720 1.27389 −2.65109 −3.02830 1.00000 4.30219
1.3 2.37720 1.00000 3.65109 −0.651093 2.37720 0.273891 3.92498 1.00000 −1.54778
\(n\): e.g. 2-40 or 990-1000
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.g 3
3.b odd 2 1 1881.2.a.f 3
11.b odd 2 1 6897.2.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
627.2.a.g 3 1.a even 1 1 trivial
1881.2.a.f 3 3.b odd 2 1
6897.2.a.m 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(627))\):

\( T_{2}^{3} - 2T_{2}^{2} - 3T_{2} + 5 \) Copy content Toggle raw display
\( T_{5}^{3} - 5T_{5}^{2} + 4T_{5} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2 T^{2} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 5 T^{2} + \cdots + 5 \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 4T + 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 9 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$17$ \( T^{3} - 6T^{2} - T + 5 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 7T^{2} - T - 47 \) Copy content Toggle raw display
$29$ \( T^{3} - 4 T^{2} + \cdots - 25 \) Copy content Toggle raw display
$31$ \( T^{3} + 14 T^{2} + \cdots + 79 \) Copy content Toggle raw display
$37$ \( T^{3} + 5 T^{2} + \cdots - 5 \) Copy content Toggle raw display
$41$ \( T^{3} - 3 T^{2} + \cdots + 155 \) Copy content Toggle raw display
$43$ \( T^{3} + 16 T^{2} + \cdots + 131 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} + \cdots + 655 \) Copy content Toggle raw display
$53$ \( T^{3} + 2 T^{2} + \cdots - 5 \) Copy content Toggle raw display
$59$ \( T^{3} + 3 T^{2} + \cdots - 155 \) Copy content Toggle raw display
$61$ \( T^{3} - 9 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$67$ \( T^{3} + 5 T^{2} + \cdots - 395 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} + \cdots + 53 \) Copy content Toggle raw display
$73$ \( T^{3} + 4 T^{2} + \cdots - 79 \) Copy content Toggle raw display
$79$ \( T^{3} - 16 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$83$ \( T^{3} - 9 T^{2} + \cdots - 79 \) Copy content Toggle raw display
$89$ \( T^{3} + 4 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$97$ \( T^{3} - 2 T^{2} + \cdots - 775 \) Copy content Toggle raw display
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