Properties

Label 4650.2.a.ci
Level $4650$
Weight $2$
Character orbit 4650.a
Self dual yes
Analytic conductor $37.130$
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] = [4650,2,Mod(1,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, 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("4650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.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: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1708.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 5 \) 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
3.47090
−0.260711
−2.21018
−1.00000 −1.00000 1.00000 0 1.00000 −2.00000 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 −2.00000 −1.00000 1.00000 0
1.3 −1.00000 −1.00000 1.00000 0 1.00000 −2.00000 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(5\) \( -1 \)
\(31\) \( +1 \)

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 4650.2.a.ci 3
5.b even 2 1 4650.2.a.cp 3
5.c odd 4 2 930.2.d.i 6
15.e even 4 2 2790.2.d.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.i 6 5.c odd 4 2
2790.2.d.j 6 15.e even 4 2
4650.2.a.ci 3 1.a even 1 1 trivial
4650.2.a.cp 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4650))\):

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{3} - 8T_{11}^{2} + 13T_{11} + 8 \) Copy content Toggle raw display
\( T_{13}^{3} + 10T_{13}^{2} + 25T_{13} + 14 \) Copy content Toggle raw display
\( T_{19}^{3} + 2T_{19}^{2} - 35T_{19} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T + 2)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 8 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( T^{3} + 10 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{3} + 2 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( (T + 1)^{3} \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$41$ \( (T - 2)^{3} \) Copy content Toggle raw display
$43$ \( T^{3} - 76T + 16 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$53$ \( T^{3} + 10 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$59$ \( T^{3} + 2 T^{2} + \cdots - 280 \) Copy content Toggle raw display
$61$ \( T^{3} - 14 T^{2} + \cdots + 1372 \) Copy content Toggle raw display
$67$ \( T^{3} - 16 T^{2} + \cdots + 652 \) Copy content Toggle raw display
$71$ \( T^{3} + 8 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$73$ \( T^{3} - 2 T^{2} + \cdots + 392 \) Copy content Toggle raw display
$79$ \( T^{3} + 6 T^{2} + \cdots - 28 \) Copy content Toggle raw display
$83$ \( T^{3} + 2 T^{2} + \cdots + 244 \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} + \cdots - 320 \) Copy content Toggle raw display
$97$ \( T^{3} - 20 T^{2} + \cdots - 236 \) Copy content Toggle raw display
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