Properties

Label 670.2.a.h
Level $670$
Weight $2$
Character orbit 670.a
Self dual yes
Analytic conductor $5.350$
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] = [670,2,Mod(1,670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(670, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("670.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 670.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: \(5.34997693543\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
2.60168
−0.339877
−2.26180
1.00000 −2.60168 1.00000 −1.00000 −2.60168 −1.60168 1.00000 3.76873 −1.00000
1.2 1.00000 0.339877 1.00000 −1.00000 0.339877 1.33988 1.00000 −2.88448 −1.00000
1.3 1.00000 2.26180 1.00000 −1.00000 2.26180 3.26180 1.00000 2.11575 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(67\) \(1\)

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 670.2.a.h 3
3.b odd 2 1 6030.2.a.bo 3
4.b odd 2 1 5360.2.a.z 3
5.b even 2 1 3350.2.a.k 3
5.c odd 4 2 3350.2.c.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.a.h 3 1.a even 1 1 trivial
3350.2.a.k 3 5.b even 2 1
3350.2.c.j 6 5.c odd 4 2
5360.2.a.z 3 4.b odd 2 1
6030.2.a.bo 3 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(670))\):

\( T_{3}^{3} - 6T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} - 3T_{7}^{2} - 3T_{7} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 6T + 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} - 3 T + 7 \) Copy content Toggle raw display
$11$ \( (T - 3)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 12T - 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 42T - 98 \) Copy content Toggle raw display
$19$ \( T^{3} - 24T - 16 \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} - 12 T + 24 \) Copy content Toggle raw display
$29$ \( T^{3} - 6T^{2} + 18 \) Copy content Toggle raw display
$31$ \( T^{3} - 48T + 114 \) Copy content Toggle raw display
$37$ \( T^{3} - 3 T^{2} - 45 T - 67 \) Copy content Toggle raw display
$41$ \( T^{3} - 12 T^{2} + 6 T + 202 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} - 54 T + 122 \) Copy content Toggle raw display
$47$ \( T^{3} - 108T + 54 \) Copy content Toggle raw display
$53$ \( T^{3} - 12T - 2 \) Copy content Toggle raw display
$59$ \( T^{3} + 12 T^{2} + 6 T - 202 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} - 45 T + 67 \) Copy content Toggle raw display
$67$ \( (T + 1)^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 9 T^{2} + 15 T + 7 \) Copy content Toggle raw display
$73$ \( T^{3} + 24 T^{2} + 186 T + 462 \) Copy content Toggle raw display
$79$ \( T^{3} - 48T + 16 \) Copy content Toggle raw display
$83$ \( T^{3} + 9 T^{2} - 57 T - 29 \) Copy content Toggle raw display
$89$ \( T^{3} + 21 T^{2} + 99 T - 89 \) Copy content Toggle raw display
$97$ \( T^{3} + 21 T^{2} + 123 T + 159 \) Copy content Toggle raw display
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