Properties

Label 2070.2.a.u
Level $2070$
Weight $2$
Character orbit 2070.a
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + \beta q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} + \beta q^{7} - q^{8} + q^{10} + ( - 3 \beta + 1) q^{11} + (5 \beta - 4) q^{13} - \beta q^{14} + q^{16} + (5 \beta - 3) q^{17} - 3 \beta q^{19} - q^{20} + (3 \beta - 1) q^{22} - q^{23} + q^{25} + ( - 5 \beta + 4) q^{26} + \beta q^{28} + ( - 2 \beta + 8) q^{29} + ( - 5 \beta + 6) q^{31} - q^{32} + ( - 5 \beta + 3) q^{34} - \beta q^{35} + ( - 4 \beta + 4) q^{37} + 3 \beta q^{38} + q^{40} + (7 \beta + 1) q^{41} + ( - 3 \beta + 1) q^{44} + q^{46} - 6 \beta q^{47} + (\beta - 6) q^{49} - q^{50} + (5 \beta - 4) q^{52} + (4 \beta + 2) q^{53} + (3 \beta - 1) q^{55} - \beta q^{56} + (2 \beta - 8) q^{58} + (6 \beta + 2) q^{59} + (7 \beta - 5) q^{61} + (5 \beta - 6) q^{62} + q^{64} + ( - 5 \beta + 4) q^{65} + ( - 4 \beta + 12) q^{67} + (5 \beta - 3) q^{68} + \beta q^{70} + ( - 5 \beta + 1) q^{71} + ( - 2 \beta + 2) q^{73} + (4 \beta - 4) q^{74} - 3 \beta q^{76} + ( - 2 \beta - 3) q^{77} + (4 \beta + 4) q^{79} - q^{80} + ( - 7 \beta - 1) q^{82} + ( - 8 \beta + 2) q^{83} + ( - 5 \beta + 3) q^{85} + (3 \beta - 1) q^{88} + ( - 4 \beta + 8) q^{89} + (\beta + 5) q^{91} - q^{92} + 6 \beta q^{94} + 3 \beta q^{95} + (\beta + 13) q^{97} + ( - \beta + 6) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} - 3 q^{13} - q^{14} + 2 q^{16} - q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 2 q^{23} + 2 q^{25} + 3 q^{26} + q^{28} + 14 q^{29} + 7 q^{31} - 2 q^{32} + q^{34} - q^{35} + 4 q^{37} + 3 q^{38} + 2 q^{40} + 9 q^{41} - q^{44} + 2 q^{46} - 6 q^{47} - 11 q^{49} - 2 q^{50} - 3 q^{52} + 8 q^{53} + q^{55} - q^{56} - 14 q^{58} + 10 q^{59} - 3 q^{61} - 7 q^{62} + 2 q^{64} + 3 q^{65} + 20 q^{67} - q^{68} + q^{70} - 3 q^{71} + 2 q^{73} - 4 q^{74} - 3 q^{76} - 8 q^{77} + 12 q^{79} - 2 q^{80} - 9 q^{82} - 4 q^{83} + q^{85} + q^{88} + 12 q^{89} + 11 q^{91} - 2 q^{92} + 6 q^{94} + 3 q^{95} + 27 q^{97} + 11 q^{98}+O(q^{100}) \) 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
−0.618034
1.61803
−1.00000 0 1.00000 −1.00000 0 −0.618034 −1.00000 0 1.00000
1.2 −1.00000 0 1.00000 −1.00000 0 1.61803 −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.u 2
3.b odd 2 1 230.2.a.c 2
12.b even 2 1 1840.2.a.l 2
15.d odd 2 1 1150.2.a.j 2
15.e even 4 2 1150.2.b.i 4
24.f even 2 1 7360.2.a.bn 2
24.h odd 2 1 7360.2.a.bh 2
60.h even 2 1 9200.2.a.bu 2
69.c even 2 1 5290.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.c 2 3.b odd 2 1
1150.2.a.j 2 15.d odd 2 1
1150.2.b.i 4 15.e even 4 2
1840.2.a.l 2 12.b even 2 1
2070.2.a.u 2 1.a even 1 1 trivial
5290.2.a.o 2 69.c even 2 1
7360.2.a.bh 2 24.h odd 2 1
7360.2.a.bn 2 24.f even 2 1
9200.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2070))\):

\( T_{7}^{2} - T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 11 \) Copy content Toggle raw display
\( T_{13}^{2} + 3T_{13} - 29 \) Copy content Toggle raw display
\( T_{17}^{2} + T_{17} - 31 \) Copy content Toggle raw display
\( T_{29}^{2} - 14T_{29} + 44 \) 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
$7$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T - 29 \) Copy content Toggle raw display
$17$ \( T^{2} + T - 31 \) Copy content Toggle raw display
$19$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$31$ \( T^{2} - 7T - 19 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$41$ \( T^{2} - 9T - 41 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$61$ \( T^{2} + 3T - 59 \) Copy content Toggle raw display
$67$ \( T^{2} - 20T + 80 \) Copy content Toggle raw display
$71$ \( T^{2} + 3T - 29 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$97$ \( T^{2} - 27T + 181 \) Copy content Toggle raw display
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