Properties

Label 6615.2.a.p
Level $6615$
Weight $2$
Character orbit 6615.a
Self dual yes
Analytic conductor $52.821$
Analytic rank $1$
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] = [6615,2,Mod(1,6615)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6615, 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("6615.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6615 = 3^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6615.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: \(52.8210409371\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 1) q^{4} - q^{5} - 3 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta + 1) q^{4} - q^{5} - 3 q^{8} + \beta q^{10} + 2 \beta q^{11} + ( - 2 \beta - 2) q^{13} + (\beta - 2) q^{16} + ( - 2 \beta + 3) q^{17} + ( - 2 \beta + 1) q^{19} + ( - \beta - 1) q^{20} + ( - 2 \beta - 6) q^{22} - 3 q^{23} + q^{25} + (4 \beta + 6) q^{26} + ( - 2 \beta + 6) q^{29} + (2 \beta + 1) q^{31} + (\beta + 3) q^{32} + ( - \beta + 6) q^{34} + 2 q^{37} + (\beta + 6) q^{38} + 3 q^{40} + 2 \beta q^{41} + (2 \beta - 4) q^{43} + (4 \beta + 6) q^{44} + 3 \beta q^{46} + 4 \beta q^{47} - \beta q^{50} + ( - 6 \beta - 8) q^{52} + (2 \beta - 3) q^{53} - 2 \beta q^{55} + ( - 4 \beta + 6) q^{58} + ( - 2 \beta + 6) q^{59} + (4 \beta - 5) q^{61} + ( - 3 \beta - 6) q^{62} + ( - 6 \beta + 1) q^{64} + (2 \beta + 2) q^{65} + (4 \beta - 10) q^{67} + ( - \beta - 3) q^{68} + ( - 2 \beta + 12) q^{71} + ( - 2 \beta - 8) q^{73} - 2 \beta q^{74} + ( - 3 \beta - 5) q^{76} + ( - 2 \beta - 7) q^{79} + ( - \beta + 2) q^{80} + ( - 2 \beta - 6) q^{82} - 3 q^{83} + (2 \beta - 3) q^{85} + (2 \beta - 6) q^{86} - 6 \beta q^{88} + 6 \beta q^{89} + ( - 3 \beta - 3) q^{92} + ( - 4 \beta - 12) q^{94} + (2 \beta - 1) q^{95} - 8 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{4} - 2 q^{5} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 3 q^{4} - 2 q^{5} - 6 q^{8} + q^{10} + 2 q^{11} - 6 q^{13} - 3 q^{16} + 4 q^{17} - 3 q^{20} - 14 q^{22} - 6 q^{23} + 2 q^{25} + 16 q^{26} + 10 q^{29} + 4 q^{31} + 7 q^{32} + 11 q^{34} + 4 q^{37} + 13 q^{38} + 6 q^{40} + 2 q^{41} - 6 q^{43} + 16 q^{44} + 3 q^{46} + 4 q^{47} - q^{50} - 22 q^{52} - 4 q^{53} - 2 q^{55} + 8 q^{58} + 10 q^{59} - 6 q^{61} - 15 q^{62} - 4 q^{64} + 6 q^{65} - 16 q^{67} - 7 q^{68} + 22 q^{71} - 18 q^{73} - 2 q^{74} - 13 q^{76} - 16 q^{79} + 3 q^{80} - 14 q^{82} - 6 q^{83} - 4 q^{85} - 10 q^{86} - 6 q^{88} + 6 q^{89} - 9 q^{92} - 28 q^{94} - 16 q^{97}+O(q^{100}) \) 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
2.30278
−1.30278
−2.30278 0 3.30278 −1.00000 0 0 −3.00000 0 2.30278
1.2 1.30278 0 −0.302776 −1.00000 0 0 −3.00000 0 −1.30278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 6615.2.a.p 2
3.b odd 2 1 6615.2.a.v 2
7.b odd 2 1 135.2.a.c 2
21.c even 2 1 135.2.a.d yes 2
28.d even 2 1 2160.2.a.ba 2
35.c odd 2 1 675.2.a.p 2
35.f even 4 2 675.2.b.i 4
56.e even 2 1 8640.2.a.ck 2
56.h odd 2 1 8640.2.a.cr 2
63.l odd 6 2 405.2.e.k 4
63.o even 6 2 405.2.e.j 4
84.h odd 2 1 2160.2.a.y 2
105.g even 2 1 675.2.a.k 2
105.k odd 4 2 675.2.b.h 4
168.e odd 2 1 8640.2.a.cy 2
168.i even 2 1 8640.2.a.df 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.c 2 7.b odd 2 1
135.2.a.d yes 2 21.c even 2 1
405.2.e.j 4 63.o even 6 2
405.2.e.k 4 63.l odd 6 2
675.2.a.k 2 105.g even 2 1
675.2.a.p 2 35.c odd 2 1
675.2.b.h 4 105.k odd 4 2
675.2.b.i 4 35.f even 4 2
2160.2.a.y 2 84.h odd 2 1
2160.2.a.ba 2 28.d even 2 1
6615.2.a.p 2 1.a even 1 1 trivial
6615.2.a.v 2 3.b odd 2 1
8640.2.a.ck 2 56.e even 2 1
8640.2.a.cr 2 56.h odd 2 1
8640.2.a.cy 2 168.e odd 2 1
8640.2.a.df 2 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6615))\):

\( T_{2}^{2} + T_{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 13 \) Copy content Toggle raw display
$23$ \( (T + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 12 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$59$ \( T^{2} - 10T + 12 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T - 43 \) Copy content Toggle raw display
$67$ \( T^{2} + 16T + 12 \) Copy content Toggle raw display
$71$ \( T^{2} - 22T + 108 \) Copy content Toggle raw display
$73$ \( T^{2} + 18T + 68 \) Copy content Toggle raw display
$79$ \( T^{2} + 16T + 51 \) Copy content Toggle raw display
$83$ \( (T + 3)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 108 \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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