Properties

Label 6014.2.a.c
Level $6014$
Weight $2$
Character orbit 6014.a
Self dual yes
Analytic conductor $48.022$
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] = [6014,2,Mod(1,6014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6014, 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("6014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6014 = 2 \cdot 31 \cdot 97 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6014.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: \(48.0220317756\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + 2 \beta q^{3} + q^{4} + (\beta + 2) q^{5} - 2 \beta q^{6} + (\beta - 2) q^{7} - q^{8} + 5 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + 2 \beta q^{3} + q^{4} + (\beta + 2) q^{5} - 2 \beta q^{6} + (\beta - 2) q^{7} - q^{8} + 5 q^{9} + ( - \beta - 2) q^{10} + 4 q^{11} + 2 \beta q^{12} + ( - \beta + 2) q^{14} + (4 \beta + 4) q^{15} + q^{16} - \beta q^{17} - 5 q^{18} - 6 q^{19} + (\beta + 2) q^{20} + ( - 4 \beta + 4) q^{21} - 4 q^{22} + 4 \beta q^{23} - 2 \beta q^{24} + (4 \beta + 1) q^{25} + 4 \beta q^{27} + (\beta - 2) q^{28} + ( - 4 \beta + 4) q^{29} + ( - 4 \beta - 4) q^{30} - q^{31} - q^{32} + 8 \beta q^{33} + \beta q^{34} - 2 q^{35} + 5 q^{36} + ( - 4 \beta - 4) q^{37} + 6 q^{38} + ( - \beta - 2) q^{40} + ( - 2 \beta + 6) q^{41} + (4 \beta - 4) q^{42} + 6 \beta q^{43} + 4 q^{44} + (5 \beta + 10) q^{45} - 4 \beta q^{46} + 4 q^{47} + 2 \beta q^{48} + ( - 4 \beta - 1) q^{49} + ( - 4 \beta - 1) q^{50} - 4 q^{51} + (2 \beta + 8) q^{53} - 4 \beta q^{54} + (4 \beta + 8) q^{55} + ( - \beta + 2) q^{56} - 12 \beta q^{57} + (4 \beta - 4) q^{58} + (2 \beta + 10) q^{59} + (4 \beta + 4) q^{60} + ( - 6 \beta - 4) q^{61} + q^{62} + (5 \beta - 10) q^{63} + q^{64} - 8 \beta q^{66} + (6 \beta + 2) q^{67} - \beta q^{68} + 16 q^{69} + 2 q^{70} + ( - 3 \beta + 2) q^{71} - 5 q^{72} + (4 \beta + 2) q^{73} + (4 \beta + 4) q^{74} + (2 \beta + 16) q^{75} - 6 q^{76} + (4 \beta - 8) q^{77} + (2 \beta - 2) q^{79} + (\beta + 2) q^{80} + q^{81} + (2 \beta - 6) q^{82} + (9 \beta - 4) q^{83} + ( - 4 \beta + 4) q^{84} + ( - 2 \beta - 2) q^{85} - 6 \beta q^{86} + (8 \beta - 16) q^{87} - 4 q^{88} + (6 \beta - 2) q^{89} + ( - 5 \beta - 10) q^{90} + 4 \beta q^{92} - 2 \beta q^{93} - 4 q^{94} + ( - 6 \beta - 12) q^{95} - 2 \beta q^{96} + q^{97} + (4 \beta + 1) q^{98} + 20 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 4 q^{7} - 2 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 4 q^{7} - 2 q^{8} + 10 q^{9} - 4 q^{10} + 8 q^{11} + 4 q^{14} + 8 q^{15} + 2 q^{16} - 10 q^{18} - 12 q^{19} + 4 q^{20} + 8 q^{21} - 8 q^{22} + 2 q^{25} - 4 q^{28} + 8 q^{29} - 8 q^{30} - 2 q^{31} - 2 q^{32} - 4 q^{35} + 10 q^{36} - 8 q^{37} + 12 q^{38} - 4 q^{40} + 12 q^{41} - 8 q^{42} + 8 q^{44} + 20 q^{45} + 8 q^{47} - 2 q^{49} - 2 q^{50} - 8 q^{51} + 16 q^{53} + 16 q^{55} + 4 q^{56} - 8 q^{58} + 20 q^{59} + 8 q^{60} - 8 q^{61} + 2 q^{62} - 20 q^{63} + 2 q^{64} + 4 q^{67} + 32 q^{69} + 4 q^{70} + 4 q^{71} - 10 q^{72} + 4 q^{73} + 8 q^{74} + 32 q^{75} - 12 q^{76} - 16 q^{77} - 4 q^{79} + 4 q^{80} + 2 q^{81} - 12 q^{82} - 8 q^{83} + 8 q^{84} - 4 q^{85} - 32 q^{87} - 8 q^{88} - 4 q^{89} - 20 q^{90} - 8 q^{94} - 24 q^{95} + 2 q^{97} + 2 q^{98} + 40 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−1.00000 −2.82843 1.00000 0.585786 2.82843 −3.41421 −1.00000 5.00000 −0.585786
1.2 −1.00000 2.82843 1.00000 3.41421 −2.82843 −0.585786 −1.00000 5.00000 −3.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(31\) \(1\)
\(97\) \(-1\)

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 6014.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6014.2.a.c 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6014))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 8 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2 \) Copy content Toggle raw display
$19$ \( (T + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 32 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$43$ \( T^{2} - 72 \) Copy content Toggle raw display
$47$ \( (T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 16T + 56 \) Copy content Toggle raw display
$59$ \( T^{2} - 20T + 92 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 146 \) Copy content Toggle raw display
$89$ \( T^{2} + 4T - 68 \) Copy content Toggle raw display
$97$ \( (T - 1)^{2} \) Copy content Toggle raw display
show more
show less