Properties

Label 2070.2.a.w
Level $2070$
Weight $2$
Character orbit 2070.a
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $2$
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] = [2070,2,Mod(1,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, 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("2070.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.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: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + (\beta + 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - q^{5} + (\beta + 1) q^{7} + q^{8} - q^{10} + ( - \beta + 4) q^{11} + (\beta + 1) q^{13} + (\beta + 1) q^{14} + q^{16} - 3 \beta q^{17} + (3 \beta - 1) q^{19} - q^{20} + ( - \beta + 4) q^{22} + q^{23} + q^{25} + (\beta + 1) q^{26} + (\beta + 1) q^{28} + (2 \beta - 2) q^{29} + ( - 3 \beta - 1) q^{31} + q^{32} - 3 \beta q^{34} + ( - \beta - 1) q^{35} + 8 q^{37} + (3 \beta - 1) q^{38} - q^{40} + ( - 3 \beta + 6) q^{41} - 4 \beta q^{43} + ( - \beta + 4) q^{44} + q^{46} + (2 \beta - 2) q^{47} + (3 \beta - 3) q^{49} + q^{50} + (\beta + 1) q^{52} + (4 \beta + 2) q^{53} + (\beta - 4) q^{55} + (\beta + 1) q^{56} + (2 \beta - 2) q^{58} + ( - 2 \beta + 8) q^{59} + 5 \beta q^{61} + ( - 3 \beta - 1) q^{62} + q^{64} + ( - \beta - 1) q^{65} - 4 q^{67} - 3 \beta q^{68} + ( - \beta - 1) q^{70} + (\beta + 14) q^{71} + ( - 6 \beta + 8) q^{73} + 8 q^{74} + (3 \beta - 1) q^{76} + (2 \beta + 1) q^{77} + ( - 8 \beta + 4) q^{79} - q^{80} + ( - 3 \beta + 6) q^{82} + ( - 4 \beta - 2) q^{83} + 3 \beta q^{85} - 4 \beta q^{86} + ( - \beta + 4) q^{88} + (3 \beta + 4) q^{91} + q^{92} + (2 \beta - 2) q^{94} + ( - 3 \beta + 1) q^{95} + (\beta + 4) q^{97} + (3 \beta - 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8} - 2 q^{10} + 7 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} + q^{19} - 2 q^{20} + 7 q^{22} + 2 q^{23} + 2 q^{25} + 3 q^{26} + 3 q^{28} - 2 q^{29} - 5 q^{31} + 2 q^{32} - 3 q^{34} - 3 q^{35} + 16 q^{37} + q^{38} - 2 q^{40} + 9 q^{41} - 4 q^{43} + 7 q^{44} + 2 q^{46} - 2 q^{47} - 3 q^{49} + 2 q^{50} + 3 q^{52} + 8 q^{53} - 7 q^{55} + 3 q^{56} - 2 q^{58} + 14 q^{59} + 5 q^{61} - 5 q^{62} + 2 q^{64} - 3 q^{65} - 8 q^{67} - 3 q^{68} - 3 q^{70} + 29 q^{71} + 10 q^{73} + 16 q^{74} + q^{76} + 4 q^{77} - 2 q^{80} + 9 q^{82} - 8 q^{83} + 3 q^{85} - 4 q^{86} + 7 q^{88} + 11 q^{91} + 2 q^{92} - 2 q^{94} - q^{95} + 9 q^{97} - 3 q^{98}+O(q^{100}) \) 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
−1.30278
2.30278
1.00000 0 1.00000 −1.00000 0 −0.302776 1.00000 0 −1.00000
1.2 1.00000 0 1.00000 −1.00000 0 3.30278 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( +1 \)
\(23\) \( -1 \)

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 2070.2.a.w 2
3.b odd 2 1 230.2.a.b 2
12.b even 2 1 1840.2.a.j 2
15.d odd 2 1 1150.2.a.m 2
15.e even 4 2 1150.2.b.f 4
24.f even 2 1 7360.2.a.bu 2
24.h odd 2 1 7360.2.a.bc 2
60.h even 2 1 9200.2.a.ca 2
69.c even 2 1 5290.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 3.b odd 2 1
1150.2.a.m 2 15.d odd 2 1
1150.2.b.f 4 15.e even 4 2
1840.2.a.j 2 12.b even 2 1
2070.2.a.w 2 1.a even 1 1 trivial
5290.2.a.j 2 69.c even 2 1
7360.2.a.bc 2 24.h odd 2 1
7360.2.a.bu 2 24.f even 2 1
9200.2.a.ca 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2070))\):

\( T_{7}^{2} - 3T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 7T_{11} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} - 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} - 27 \) Copy content Toggle raw display
\( T_{29}^{2} + 2T_{29} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 27 \) Copy content Toggle raw display
$19$ \( T^{2} - T - 29 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$31$ \( T^{2} + 5T - 23 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 9T - 9 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 48 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 12 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T - 75 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 29T + 207 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T - 92 \) Copy content Toggle raw display
$79$ \( T^{2} - 208 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 9T + 17 \) Copy content Toggle raw display
show more
show less