Properties

Label 1305.2.f.h
Level $1305$
Weight $2$
Character orbit 1305.f
Analytic conductor $10.420$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(289,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 8\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).

\(n\) \(146\) \(262\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
1.61803i
1.61803i
0.618034i
0.618034i
−2.23607 0 3.00000 1.00000 2.00000i 0 2.00000i −2.23607 0 −2.23607 + 4.47214i
289.2 −2.23607 0 3.00000 1.00000 + 2.00000i 0 2.00000i −2.23607 0 −2.23607 4.47214i
289.3 2.23607 0 3.00000 1.00000 2.00000i 0 2.00000i 2.23607 0 2.23607 4.47214i
289.4 2.23607 0 3.00000 1.00000 + 2.00000i 0 2.00000i 2.23607 0 2.23607 + 4.47214i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
29.b even 2 1 inner
145.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.2.f.h 4
3.b odd 2 1 145.2.d.b 4
5.b even 2 1 inner 1305.2.f.h 4
12.b even 2 1 2320.2.j.b 4
15.d odd 2 1 145.2.d.b 4
15.e even 4 1 725.2.c.a 2
15.e even 4 1 725.2.c.b 2
29.b even 2 1 inner 1305.2.f.h 4
60.h even 2 1 2320.2.j.b 4
87.d odd 2 1 145.2.d.b 4
145.d even 2 1 inner 1305.2.f.h 4
348.b even 2 1 2320.2.j.b 4
435.b odd 2 1 145.2.d.b 4
435.p even 4 1 725.2.c.a 2
435.p even 4 1 725.2.c.b 2
1740.k even 2 1 2320.2.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.d.b 4 3.b odd 2 1
145.2.d.b 4 15.d odd 2 1
145.2.d.b 4 87.d odd 2 1
145.2.d.b 4 435.b odd 2 1
725.2.c.a 2 15.e even 4 1
725.2.c.a 2 435.p even 4 1
725.2.c.b 2 15.e even 4 1
725.2.c.b 2 435.p even 4 1
1305.2.f.h 4 1.a even 1 1 trivial
1305.2.f.h 4 5.b even 2 1 inner
1305.2.f.h 4 29.b even 2 1 inner
1305.2.f.h 4 145.d even 2 1 inner
2320.2.j.b 4 12.b even 2 1
2320.2.j.b 4 60.h even 2 1
2320.2.j.b 4 348.b even 2 1
2320.2.j.b 4 1740.k even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1305, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 29)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T + 4)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
show more
show less