Properties

Label 6422.2.a.r
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
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] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, 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("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + q^{4} + (\beta_{2} - 2 \beta_1) q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{7} + q^{8} + ( - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + q^{4} + (\beta_{2} - 2 \beta_1) q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{7} + q^{8} + ( - \beta_1 + 3) q^{9} + (\beta_{2} - 2 \beta_1) q^{10} - 2 \beta_{2} q^{11} + (\beta_{2} + \beta_1 - 1) q^{12} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{14} + ( - 5 \beta_{2} + 3 \beta_1 - 4) q^{15} + q^{16} + (4 \beta_{2} - 3 \beta_1 + 4) q^{17} + ( - \beta_1 + 3) q^{18} - q^{19} + (\beta_{2} - 2 \beta_1) q^{20} + (4 \beta_{2} - 6 \beta_1 + 4) q^{21} - 2 \beta_{2} q^{22} + (4 \beta_1 - 4) q^{23} + (\beta_{2} + \beta_1 - 1) q^{24} + ( - \beta_{2} + \beta_1) q^{25} + ( - 2 \beta_{2} + \beta_1 - 3) q^{27} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{28} + (4 \beta_{2} - 4 \beta_1 + 2) q^{29} + ( - 5 \beta_{2} + 3 \beta_1 - 4) q^{30} + ( - 5 \beta_{2} + \beta_1 - 3) q^{31} + q^{32} + (2 \beta_{2} - 2 \beta_1 - 4) q^{33} + (4 \beta_{2} - 3 \beta_1 + 4) q^{34} + (2 \beta_{2} + 2 \beta_1 - 4) q^{35} + ( - \beta_1 + 3) q^{36} + (2 \beta_{2} + 2 \beta_1 - 6) q^{37} - q^{38} + (\beta_{2} - 2 \beta_1) q^{40} + ( - 6 \beta_{2} + 2 \beta_1 - 2) q^{41} + (4 \beta_{2} - 6 \beta_1 + 4) q^{42} + (4 \beta_{2} - 2 \beta_1 + 4) q^{43} - 2 \beta_{2} q^{44} + (4 \beta_{2} - 6 \beta_1 + 3) q^{45} + (4 \beta_1 - 4) q^{46} + (\beta_{2} + \beta_1 - 1) q^{48} + ( - 4 \beta_1 + 1) q^{49} + ( - \beta_{2} + \beta_1) q^{50} + ( - 6 \beta_{2} + 11 \beta_1 - 5) q^{51} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{53} + ( - 2 \beta_{2} + \beta_1 - 3) q^{54} + (6 \beta_{2} - 2 \beta_1 + 2) q^{55} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{56} + ( - \beta_{2} - \beta_1 + 1) q^{57} + (4 \beta_{2} - 4 \beta_1 + 2) q^{58} + ( - 7 \beta_{2} + 6 \beta_1 - 8) q^{59} + ( - 5 \beta_{2} + 3 \beta_1 - 4) q^{60} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{61} + ( - 5 \beta_{2} + \beta_1 - 3) q^{62} + ( - 6 \beta_{2} + 8 \beta_1 - 8) q^{63} + q^{64} + (2 \beta_{2} - 2 \beta_1 - 4) q^{66} + ( - 7 \beta_{2} + \beta_1 - 5) q^{67} + (4 \beta_{2} - 3 \beta_1 + 4) q^{68} + (4 \beta_{2} - 8 \beta_1 + 16) q^{69} + (2 \beta_{2} + 2 \beta_1 - 4) q^{70} + (2 \beta_{2} - 7 \beta_1 + 8) q^{71} + ( - \beta_1 + 3) q^{72} + (8 \beta_{2} + \beta_1) q^{73} + (2 \beta_{2} + 2 \beta_1 - 6) q^{74} + (3 \beta_{2} - 2 \beta_1 + 1) q^{75} - q^{76} + ( - 4 \beta_{2} + 4 \beta_1) q^{77} + (2 \beta_{2} - 5 \beta_1 + 8) q^{79} + (\beta_{2} - 2 \beta_1) q^{80} + (\beta_{2} - 3 \beta_1 - 7) q^{81} + ( - 6 \beta_{2} + 2 \beta_1 - 2) q^{82} + ( - 6 \beta_{2} + 2 \beta_1 + 4) q^{83} + (4 \beta_{2} - 6 \beta_1 + 4) q^{84} + ( - 5 \beta_{2} - 4 \beta_1 + 5) q^{85} + (4 \beta_{2} - 2 \beta_1 + 4) q^{86} + ( - 10 \beta_{2} + 10 \beta_1 - 6) q^{87} - 2 \beta_{2} q^{88} + (8 \beta_{2} - 4 \beta_1 + 4) q^{89} + (4 \beta_{2} - 6 \beta_1 + 3) q^{90} + (4 \beta_1 - 4) q^{92} + (4 \beta_{2} - 9 \beta_1 - 4) q^{93} + ( - \beta_{2} + 2 \beta_1) q^{95} + (\beta_{2} + \beta_1 - 1) q^{96} + ( - 2 \beta_1 - 6) q^{97} + ( - 4 \beta_1 + 1) q^{98} + ( - 4 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 8 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 2 q^{14} - 4 q^{15} + 3 q^{16} + 5 q^{17} + 8 q^{18} - 3 q^{19} - 3 q^{20} + 2 q^{21} + 2 q^{22} - 8 q^{23} - 3 q^{24} + 2 q^{25} - 6 q^{27} - 2 q^{28} - 2 q^{29} - 4 q^{30} - 3 q^{31} + 3 q^{32} - 16 q^{33} + 5 q^{34} - 12 q^{35} + 8 q^{36} - 18 q^{37} - 3 q^{38} - 3 q^{40} + 2 q^{41} + 2 q^{42} + 6 q^{43} + 2 q^{44} - q^{45} - 8 q^{46} - 3 q^{48} - q^{49} + 2 q^{50} + 2 q^{51} - 8 q^{53} - 6 q^{54} - 2 q^{55} - 2 q^{56} + 3 q^{57} - 2 q^{58} - 11 q^{59} - 4 q^{60} + 13 q^{61} - 3 q^{62} - 10 q^{63} + 3 q^{64} - 16 q^{66} - 7 q^{67} + 5 q^{68} + 36 q^{69} - 12 q^{70} + 15 q^{71} + 8 q^{72} - 7 q^{73} - 18 q^{74} - 2 q^{75} - 3 q^{76} + 8 q^{77} + 17 q^{79} - 3 q^{80} - 25 q^{81} + 2 q^{82} + 20 q^{83} + 2 q^{84} + 16 q^{85} + 6 q^{86} + 2 q^{87} + 2 q^{88} - q^{90} - 8 q^{92} - 25 q^{93} + 3 q^{95} - 3 q^{96} - 20 q^{97} - q^{98} + 10 q^{99}+O(q^{100}) \) 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
−1.24698
0.445042
1.80194
1.00000 −2.69202 1.00000 2.04892 −2.69202 −3.60388 1.00000 4.24698 2.04892
1.2 1.00000 −2.35690 1.00000 −2.69202 −2.35690 2.49396 1.00000 2.55496 −2.69202
1.3 1.00000 2.04892 1.00000 −2.35690 2.04892 −0.890084 1.00000 1.19806 −2.35690
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(-1\)
\(19\) \(1\)

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 6422.2.a.r yes 3
13.b even 2 1 6422.2.a.l 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6422.2.a.l 3 13.b even 2 1
6422.2.a.r yes 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3}^{3} + 3T_{3}^{2} - 4T_{3} - 13 \) Copy content Toggle raw display
\( T_{5}^{3} + 3T_{5}^{2} - 4T_{5} - 13 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 8T_{7} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 3 T^{2} - 4 T - 13 \) Copy content Toggle raw display
$5$ \( T^{3} + 3 T^{2} - 4 T - 13 \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 8 T - 8 \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 8 T + 8 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 5 T^{2} - 22 T + 97 \) Copy content Toggle raw display
$19$ \( (T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 8 T^{2} - 16 T - 64 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 36 T - 8 \) Copy content Toggle raw display
$31$ \( T^{3} + 3 T^{2} - 46 T - 97 \) Copy content Toggle raw display
$37$ \( T^{3} + 18 T^{2} + 80 T - 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} - 64 T - 104 \) Copy content Toggle raw display
$43$ \( T^{3} - 6 T^{2} - 16 T + 104 \) Copy content Toggle raw display
$47$ \( T^{3} \) Copy content Toggle raw display
$53$ \( T^{3} + 8 T^{2} - 44 T - 8 \) Copy content Toggle raw display
$59$ \( T^{3} + 11 T^{2} - 60 T - 533 \) Copy content Toggle raw display
$61$ \( T^{3} - 13 T^{2} + 12 T + 223 \) Copy content Toggle raw display
$67$ \( T^{3} + 7 T^{2} - 84 T - 301 \) Copy content Toggle raw display
$71$ \( T^{3} - 15 T^{2} - 16 T + 43 \) Copy content Toggle raw display
$73$ \( T^{3} + 7 T^{2} - 154 T - 791 \) Copy content Toggle raw display
$79$ \( T^{3} - 17 T^{2} + 52 T - 43 \) Copy content Toggle raw display
$83$ \( T^{3} - 20 T^{2} + 68 T - 8 \) Copy content Toggle raw display
$89$ \( T^{3} - 112T + 448 \) Copy content Toggle raw display
$97$ \( T^{3} + 20 T^{2} + 124 T + 232 \) Copy content Toggle raw display
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