Properties

Label 5070.2.a.bi
Level $5070$
Weight $2$
Character orbit 5070.a
Self dual yes
Analytic conductor $40.484$
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] = [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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + (\beta + 1) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + (\beta + 1) q^{7} + q^{8} + q^{9} + q^{10} + ( - 2 \beta + 1) q^{11} + q^{12} + (\beta + 1) q^{14} + q^{15} + q^{16} + 2 \beta q^{17} + q^{18} + ( - \beta - 1) q^{19} + q^{20} + (\beta + 1) q^{21} + ( - 2 \beta + 1) q^{22} - 3 \beta q^{23} + q^{24} + q^{25} + q^{27} + (\beta + 1) q^{28} + (\beta + 4) q^{29} + q^{30} + (3 \beta - 2) q^{31} + q^{32} + ( - 2 \beta + 1) q^{33} + 2 \beta q^{34} + (\beta + 1) q^{35} + q^{36} + (2 \beta - 1) q^{37} + ( - \beta - 1) q^{38} + q^{40} + ( - 4 \beta + 6) q^{41} + (\beta + 1) q^{42} + (\beta + 2) q^{43} + ( - 2 \beta + 1) q^{44} + q^{45} - 3 \beta q^{46} + 7 q^{47} + q^{48} + (3 \beta - 2) q^{49} + q^{50} + 2 \beta q^{51} + ( - \beta + 7) q^{53} + q^{54} + ( - 2 \beta + 1) q^{55} + (\beta + 1) q^{56} + ( - \beta - 1) q^{57} + (\beta + 4) q^{58} + (\beta + 8) q^{59} + q^{60} + 6 q^{61} + (3 \beta - 2) q^{62} + (\beta + 1) q^{63} + q^{64} + ( - 2 \beta + 1) q^{66} + (4 \beta + 4) q^{67} + 2 \beta q^{68} - 3 \beta q^{69} + (\beta + 1) q^{70} + (2 \beta - 10) q^{71} + q^{72} - 6 \beta q^{73} + (2 \beta - 1) q^{74} + q^{75} + ( - \beta - 1) q^{76} + ( - 3 \beta - 7) q^{77} + ( - \beta + 10) q^{79} + q^{80} + q^{81} + ( - 4 \beta + 6) q^{82} + (2 \beta - 4) q^{83} + (\beta + 1) q^{84} + 2 \beta q^{85} + (\beta + 2) q^{86} + (\beta + 4) q^{87} + ( - 2 \beta + 1) q^{88} + (5 \beta - 11) q^{89} + q^{90} - 3 \beta q^{92} + (3 \beta - 2) q^{93} + 7 q^{94} + ( - \beta - 1) q^{95} + q^{96} + (2 \beta - 4) q^{97} + (3 \beta - 2) q^{98} + ( - 2 \beta + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 3 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 3 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 3 q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 3 q^{19} + 2 q^{20} + 3 q^{21} - 3 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27} + 3 q^{28} + 9 q^{29} + 2 q^{30} - q^{31} + 2 q^{32} + 2 q^{34} + 3 q^{35} + 2 q^{36} - 3 q^{38} + 2 q^{40} + 8 q^{41} + 3 q^{42} + 5 q^{43} + 2 q^{45} - 3 q^{46} + 14 q^{47} + 2 q^{48} - q^{49} + 2 q^{50} + 2 q^{51} + 13 q^{53} + 2 q^{54} + 3 q^{56} - 3 q^{57} + 9 q^{58} + 17 q^{59} + 2 q^{60} + 12 q^{61} - q^{62} + 3 q^{63} + 2 q^{64} + 12 q^{67} + 2 q^{68} - 3 q^{69} + 3 q^{70} - 18 q^{71} + 2 q^{72} - 6 q^{73} + 2 q^{75} - 3 q^{76} - 17 q^{77} + 19 q^{79} + 2 q^{80} + 2 q^{81} + 8 q^{82} - 6 q^{83} + 3 q^{84} + 2 q^{85} + 5 q^{86} + 9 q^{87} - 17 q^{89} + 2 q^{90} - 3 q^{92} - q^{93} + 14 q^{94} - 3 q^{95} + 2 q^{96} - 6 q^{97} - q^{98}+O(q^{100}) \) 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
−1.56155
2.56155
1.00000 1.00000 1.00000 1.00000 1.00000 −0.561553 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 3.56155 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.bi 2
13.b even 2 1 5070.2.a.bb 2
13.c even 3 2 390.2.i.g 4
13.d odd 4 2 5070.2.b.r 4
39.i odd 6 2 1170.2.i.o 4
65.n even 6 2 1950.2.i.bi 4
65.q odd 12 4 1950.2.z.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.g 4 13.c even 3 2
1170.2.i.o 4 39.i odd 6 2
1950.2.i.bi 4 65.n even 6 2
1950.2.z.n 8 65.q odd 12 4
5070.2.a.bb 2 13.b even 2 1
5070.2.a.bi 2 1.a even 1 1 trivial
5070.2.b.r 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5070))\):

\( T_{7}^{2} - 3T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 17 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 16 \) Copy content Toggle raw display
\( T_{31}^{2} + T_{31} - 38 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 17 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 16 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$37$ \( T^{2} - 17 \) Copy content Toggle raw display
$41$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$43$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$47$ \( (T - 7)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 13T + 38 \) Copy content Toggle raw display
$59$ \( T^{2} - 17T + 68 \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 12T - 32 \) Copy content Toggle raw display
$71$ \( T^{2} + 18T + 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 6T - 144 \) Copy content Toggle raw display
$79$ \( T^{2} - 19T + 86 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$89$ \( T^{2} + 17T - 34 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
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