Properties

Label 670.2.a.g
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: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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.

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

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) 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

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
−1.48119
0.311108
2.17009
−1.00000 −0.481194 1.00000 1.00000 0.481194 −1.48119 −1.00000 −2.76845 −1.00000
1.2 −1.00000 1.31111 1.00000 1.00000 −1.31111 0.311108 −1.00000 −1.28100 −1.00000
1.3 −1.00000 3.17009 1.00000 1.00000 −3.17009 2.17009 −1.00000 7.04945 −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.g 3
3.b odd 2 1 6030.2.a.br 3
4.b odd 2 1 5360.2.a.x 3
5.b even 2 1 3350.2.a.l 3
5.c odd 4 2 3350.2.c.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.a.g 3 1.a even 1 1 trivial
3350.2.a.l 3 5.b even 2 1
3350.2.c.h 6 5.c odd 4 2
5360.2.a.x 3 4.b odd 2 1
6030.2.a.br 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(670))\):

\( T_{3}^{3} - 4T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} - T_{7}^{2} - 3T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 4 T^{2} + 2 T + 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - T^{2} - 3T + 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} - 16 T - 10 \) Copy content Toggle raw display
$17$ \( T^{3} - 8 T^{2} + 6 T + 2 \) Copy content Toggle raw display
$19$ \( T^{3} + 8 T^{2} - 40 T - 304 \) Copy content Toggle raw display
$23$ \( T^{3} - 10 T^{2} + 20 T + 8 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 20 T - 50 \) Copy content Toggle raw display
$31$ \( T^{3} - 4T + 2 \) Copy content Toggle raw display
$37$ \( T^{3} + T^{2} - 65 T - 151 \) Copy content Toggle raw display
$41$ \( T^{3} - 12 T^{2} - 10 T + 338 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} - 126 T - 806 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} - 16 T + 130 \) Copy content Toggle raw display
$53$ \( T^{3} - 12 T^{2} - 72 T + 918 \) Copy content Toggle raw display
$59$ \( T^{3} + 4 T^{2} - 154 T + 10 \) Copy content Toggle raw display
$61$ \( T^{3} - 3 T^{2} - 97 T + 349 \) Copy content Toggle raw display
$67$ \( (T + 1)^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 13 T^{2} - 37 T - 13 \) Copy content Toggle raw display
$73$ \( T^{3} - 16 T^{2} + 26 T + 338 \) Copy content Toggle raw display
$79$ \( T^{3} + 12 T^{2} - 96 T - 80 \) Copy content Toggle raw display
$83$ \( T^{3} - 31 T^{2} + 287 T - 685 \) Copy content Toggle raw display
$89$ \( T^{3} - 11 T^{2} + 3 T + 167 \) Copy content Toggle raw display
$97$ \( T^{3} - 9 T^{2} - 205 T + 2029 \) Copy content Toggle raw display
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