Properties

Label 3465.2.a.bi
Level $3465$
Weight $2$
Character orbit 3465.a
Self dual yes
Analytic conductor $27.668$
Analytic rank $0$
Dimension $3$
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] = [3465,2,Mod(1,3465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3465, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3465.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3465.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: \(27.6681643004\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + q^{5} + q^{7} + ( - 3 \beta_1 + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + q^{5} + q^{7} + ( - 3 \beta_1 + 4) q^{8} + (\beta_{2} + 1) q^{10} - q^{11} + (\beta_1 - 3) q^{13} + (\beta_{2} + 1) q^{14} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + ( - \beta_1 + 5) q^{17} + (4 \beta_{2} + 2 \beta_1) q^{19} + (\beta_{2} - \beta_1 + 2) q^{20} + ( - \beta_{2} - 1) q^{22} + ( - \beta_{2} + 3 \beta_1 + 1) q^{23} + q^{25} + ( - 3 \beta_{2} + 2 \beta_1 - 4) q^{26} + (\beta_{2} - \beta_1 + 2) q^{28} + (2 \beta_{2} - 2 \beta_1) q^{29} + ( - 3 \beta_{2} - 4 \beta_1) q^{31} + (3 \beta_{2} - 4 \beta_1 + 5) q^{32} + (5 \beta_{2} - 2 \beta_1 + 6) q^{34} + q^{35} + ( - \beta_{2} + 5 \beta_1 + 1) q^{37} + 10 q^{38} + ( - 3 \beta_1 + 4) q^{40} + (\beta_{2} - 2 \beta_1 + 4) q^{41} + ( - 3 \beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{2} + \beta_1 - 2) q^{44} + (\beta_{2} + 7 \beta_1 - 5) q^{46} + ( - \beta_1 + 9) q^{47} + q^{49} + (\beta_{2} + 1) q^{50} + ( - 4 \beta_{2} + 5 \beta_1 - 9) q^{52} + ( - \beta_{2} + 5 \beta_1 - 3) q^{53} - q^{55} + ( - 3 \beta_1 + 4) q^{56} + ( - 6 \beta_1 + 8) q^{58} + (3 \beta_{2} + 2 \beta_1 + 2) q^{59} + ( - \beta_{2} - 2 \beta_1 - 2) q^{61} + ( - 5 \beta_1 - 5) q^{62} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + (\beta_1 - 3) q^{65} + (5 \beta_{2} + 3 \beta_1 - 3) q^{67} + (6 \beta_{2} - 7 \beta_1 + 13) q^{68} + (\beta_{2} + 1) q^{70} + (2 \beta_{2} + 2 \beta_1 + 6) q^{71} + (4 \beta_{2} + 3 \beta_1 - 3) q^{73} + (\beta_{2} + 11 \beta_1 - 7) q^{74} + (2 \beta_{2} - 4 \beta_1 + 10) q^{76} - q^{77} + ( - 5 \beta_{2} - 5 \beta_1 - 1) q^{79} + (2 \beta_{2} - 4 \beta_1 + 3) q^{80} + (4 \beta_{2} - 5 \beta_1 + 9) q^{82} + ( - 6 \beta_{2} + 2 \beta_1 - 4) q^{83} + ( - \beta_1 + 5) q^{85} + ( - \beta_{2} + 5 \beta_1 - 11) q^{86} + (3 \beta_1 - 4) q^{88} + ( - 2 \beta_{2} - 2 \beta_1 + 6) q^{89} + (\beta_1 - 3) q^{91} + ( - 3 \beta_{2} + 7 \beta_1 - 11) q^{92} + (9 \beta_{2} - 2 \beta_1 + 10) q^{94} + (4 \beta_{2} + 2 \beta_1) q^{95} - 4 \beta_{2} q^{97} + (\beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} + 3 q^{7} + 9 q^{8} + 3 q^{10} - 3 q^{11} - 8 q^{13} + 3 q^{14} + 5 q^{16} + 14 q^{17} + 2 q^{19} + 5 q^{20} - 3 q^{22} + 6 q^{23} + 3 q^{25} - 10 q^{26} + 5 q^{28} - 2 q^{29} - 4 q^{31} + 11 q^{32} + 16 q^{34} + 3 q^{35} + 8 q^{37} + 30 q^{38} + 9 q^{40} + 10 q^{41} - 2 q^{43} - 5 q^{44} - 8 q^{46} + 26 q^{47} + 3 q^{49} + 3 q^{50} - 22 q^{52} - 4 q^{53} - 3 q^{55} + 9 q^{56} + 18 q^{58} + 8 q^{59} - 8 q^{61} - 20 q^{62} + 33 q^{64} - 8 q^{65} - 6 q^{67} + 32 q^{68} + 3 q^{70} + 20 q^{71} - 6 q^{73} - 10 q^{74} + 26 q^{76} - 3 q^{77} - 8 q^{79} + 5 q^{80} + 22 q^{82} - 10 q^{83} + 14 q^{85} - 28 q^{86} - 9 q^{88} + 16 q^{89} - 8 q^{91} - 26 q^{92} + 28 q^{94} + 2 q^{95} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
0.311108
2.17009
−1.48119
−1.21432 0 −0.525428 1.00000 0 1.00000 3.06668 0 −1.21432
1.2 1.53919 0 0.369102 1.00000 0 1.00000 −2.51026 0 1.53919
1.3 2.67513 0 5.15633 1.00000 0 1.00000 8.44358 0 2.67513
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 3465.2.a.bi 3
3.b odd 2 1 385.2.a.e 3
12.b even 2 1 6160.2.a.bo 3
15.d odd 2 1 1925.2.a.w 3
15.e even 4 2 1925.2.b.m 6
21.c even 2 1 2695.2.a.h 3
33.d even 2 1 4235.2.a.p 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.e 3 3.b odd 2 1
1925.2.a.w 3 15.d odd 2 1
1925.2.b.m 6 15.e even 4 2
2695.2.a.h 3 21.c even 2 1
3465.2.a.bi 3 1.a even 1 1 trivial
4235.2.a.p 3 33.d even 2 1
6160.2.a.bo 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3465))\):

\( T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5 \) Copy content Toggle raw display
\( T_{13}^{3} + 8T_{13}^{2} + 18T_{13} + 10 \) Copy content Toggle raw display
\( T_{17}^{3} - 14T_{17}^{2} + 62T_{17} - 86 \) Copy content Toggle raw display
\( T_{19}^{3} - 2T_{19}^{2} - 60T_{19} + 200 \) Copy content Toggle raw display
\( T_{23}^{3} - 6T_{23}^{2} - 28T_{23} + 148 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3T^{2} - T + 5 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( (T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + 18 T + 10 \) Copy content Toggle raw display
$17$ \( T^{3} - 14 T^{2} + 62 T - 86 \) Copy content Toggle raw display
$19$ \( T^{3} - 2 T^{2} - 60 T + 200 \) Copy content Toggle raw display
$23$ \( T^{3} - 6 T^{2} - 28 T + 148 \) Copy content Toggle raw display
$29$ \( T^{3} + 2 T^{2} - 36 T - 104 \) Copy content Toggle raw display
$31$ \( T^{3} + 4 T^{2} - 60 T + 50 \) Copy content Toggle raw display
$37$ \( T^{3} - 8 T^{2} - 76 T + 436 \) Copy content Toggle raw display
$41$ \( T^{3} - 10 T^{2} + 12 T - 2 \) Copy content Toggle raw display
$43$ \( T^{3} + 2 T^{2} - 44 T - 20 \) Copy content Toggle raw display
$47$ \( T^{3} - 26 T^{2} + 222 T - 622 \) Copy content Toggle raw display
$53$ \( T^{3} + 4 T^{2} - 92 T + 68 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 16 T + 130 \) Copy content Toggle raw display
$61$ \( T^{3} + 8 T^{2} + 8 T + 2 \) Copy content Toggle raw display
$67$ \( T^{3} + 6 T^{2} - 88 T + 76 \) Copy content Toggle raw display
$71$ \( T^{3} - 20 T^{2} + 112 T - 160 \) Copy content Toggle raw display
$73$ \( T^{3} + 6 T^{2} - 58 T - 46 \) Copy content Toggle raw display
$79$ \( T^{3} + 8 T^{2} - 112 T - 244 \) Copy content Toggle raw display
$83$ \( T^{3} + 10 T^{2} - 148 T - 488 \) Copy content Toggle raw display
$89$ \( T^{3} - 16 T^{2} + 64 T - 32 \) Copy content Toggle raw display
$97$ \( T^{3} - 64T - 128 \) Copy content Toggle raw display
show more
show less