Properties

Label 2535.2.a.ba
Level $2535$
Weight $2$
Character orbit 2535.a
Self dual yes
Analytic conductor $20.242$
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] = [2535,2,Mod(1,2535)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2535, 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("2535.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2535.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: \(20.2420769124\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
2.60168
−0.339877
−2.26180
−2.60168 −1.00000 4.76873 −1.00000 2.60168 −3.60168 −7.20336 1.00000 2.60168
1.2 0.339877 −1.00000 −1.88448 −1.00000 −0.339877 −0.660123 −1.32025 1.00000 −0.339877
1.3 2.26180 −1.00000 3.11575 −1.00000 −2.26180 1.26180 2.52360 1.00000 −2.26180
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(13\) \( +1 \)

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 2535.2.a.ba 3
3.b odd 2 1 7605.2.a.bw 3
13.b even 2 1 2535.2.a.bb 3
13.e even 6 2 195.2.i.d 6
39.d odd 2 1 7605.2.a.bv 3
39.h odd 6 2 585.2.j.f 6
65.l even 6 2 975.2.i.l 6
65.r odd 12 4 975.2.bb.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.d 6 13.e even 6 2
585.2.j.f 6 39.h odd 6 2
975.2.i.l 6 65.l even 6 2
975.2.bb.k 12 65.r odd 12 4
2535.2.a.ba 3 1.a even 1 1 trivial
2535.2.a.bb 3 13.b even 2 1
7605.2.a.bv 3 39.d odd 2 1
7605.2.a.bw 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2535))\):

\( T_{2}^{3} - 6T_{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 3T_{7} - 3 \) Copy content Toggle raw display
\( T_{11}^{3} - 24T_{11} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 6T + 2 \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$11$ \( T^{3} - 24T - 16 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 42T - 98 \) Copy content Toggle raw display
$19$ \( T^{3} + 12 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{3} - 48T + 96 \) Copy content Toggle raw display
$29$ \( T^{3} - 6T^{2} + 14 \) Copy content Toggle raw display
$31$ \( T^{3} + 3 T^{2} + \cdots - 363 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$41$ \( T^{3} - 24T - 26 \) Copy content Toggle raw display
$43$ \( T^{3} - 9 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$47$ \( T^{3} - 12 T^{2} + \cdots + 206 \) Copy content Toggle raw display
$53$ \( T^{3} - 24T + 16 \) Copy content Toggle raw display
$59$ \( T^{3} - 6 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$61$ \( T^{3} + 3 T^{2} + \cdots - 67 \) Copy content Toggle raw display
$67$ \( T^{3} + 9 T^{2} + \cdots - 351 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} + \cdots + 682 \) Copy content Toggle raw display
$73$ \( T^{3} + 21 T^{2} + \cdots - 89 \) Copy content Toggle raw display
$79$ \( T^{3} + 3 T^{2} + \cdots - 363 \) Copy content Toggle raw display
$83$ \( T^{3} - 18 T^{2} + \cdots + 168 \) Copy content Toggle raw display
$89$ \( T^{3} + 30 T^{2} + \cdots + 882 \) Copy content Toggle raw display
$97$ \( T^{3} + 15 T^{2} + \cdots - 589 \) Copy content Toggle raw display
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