Properties

Label 5070.2.a.bt
Level $5070$
Weight $2$
Character orbit 5070.a
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
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] = [5070,2,Mod(1,5070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5070, 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("5070.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.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: \(40.4841538248\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
0.445042
1.80194
−1.24698
1.00000 −1.00000 1.00000 1.00000 −1.00000 −1.69202 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 −1.35690 1.00000 1.00000 1.00000
1.3 1.00000 −1.00000 1.00000 1.00000 −1.00000 3.04892 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(13\) \(1\)

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 5070.2.a.bt yes 3
13.b even 2 1 5070.2.a.bk 3
13.d odd 4 2 5070.2.b.s 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5070.2.a.bk 3 13.b even 2 1
5070.2.a.bt yes 3 1.a even 1 1 trivial
5070.2.b.s 6 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7}^{3} - 7T_{7} - 7 \) Copy content Toggle raw display
\( T_{11}^{3} + 14T_{11}^{2} + 63T_{11} + 91 \) Copy content Toggle raw display
\( T_{17}^{3} - 3T_{17}^{2} - 4T_{17} + 13 \) Copy content Toggle raw display
\( T_{31}^{3} + 14T_{31}^{2} + 35T_{31} - 7 \) 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
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 7T - 7 \) Copy content Toggle raw display
$11$ \( T^{3} + 14 T^{2} + 63 T + 91 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 3 T^{2} - 4 T + 13 \) Copy content Toggle raw display
$19$ \( T^{3} + 7 T^{2} - 28 T - 203 \) Copy content Toggle raw display
$23$ \( T^{3} + 5 T^{2} - 78 T - 419 \) Copy content Toggle raw display
$29$ \( T^{3} - 12 T^{2} - 15 T + 377 \) Copy content Toggle raw display
$31$ \( T^{3} + 14 T^{2} + 35 T - 7 \) Copy content Toggle raw display
$37$ \( T^{3} - 4 T^{2} - 25 T + 71 \) Copy content Toggle raw display
$41$ \( T^{3} + 13 T^{2} - 2 T - 139 \) Copy content Toggle raw display
$43$ \( T^{3} + 5 T^{2} - 8 T - 41 \) Copy content Toggle raw display
$47$ \( T^{3} + 21 T^{2} + 119 T + 203 \) Copy content Toggle raw display
$53$ \( T^{3} - 3 T^{2} - 18 T + 13 \) Copy content Toggle raw display
$59$ \( T^{3} + 7 T^{2} - 70 T + 91 \) Copy content Toggle raw display
$61$ \( T^{3} + 17 T^{2} - 4 T - 601 \) Copy content Toggle raw display
$67$ \( T^{3} + 5 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$71$ \( T^{3} + 13 T^{2} - 30 T - 13 \) Copy content Toggle raw display
$73$ \( T^{3} - 13 T^{2} - 58 T + 503 \) Copy content Toggle raw display
$79$ \( T^{3} + 7 T^{2} - 56 T - 91 \) Copy content Toggle raw display
$83$ \( T^{3} + 12 T^{2} - 43 T - 587 \) Copy content Toggle raw display
$89$ \( T^{3} + 3 T^{2} - 88 T - 293 \) Copy content Toggle raw display
$97$ \( T^{3} - 21T - 7 \) Copy content Toggle raw display
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