Properties

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

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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: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + (\beta_1 - 1) q^{5} - \beta_{3} q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} + (\beta_1 - 1) q^{5} - \beta_{3} q^{7} + 3 q^{8} + ( - \beta_1 + 1) q^{10} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{11} + (\beta_{2} + 2 \beta_1 - 1) q^{13} + \beta_{3} q^{14} - q^{16} + 2 q^{17} - \beta_{3} q^{19} + ( - \beta_1 + 1) q^{20} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{22} + \beta_{3} q^{23} + (\beta_{3} + \beta_{2} - \beta_1 - 2) q^{25} + ( - \beta_{2} - 2 \beta_1 + 1) q^{26} + \beta_{3} q^{28} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{29} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{31} - 5 q^{32} - 2 q^{34} + (\beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{35} + ( - 2 \beta_{2} + 4) q^{37} + \beta_{3} q^{38} + (3 \beta_1 - 3) q^{40} + (2 \beta_{2} + 4 \beta_1 - 2) q^{41} + (\beta_{2} + 7) q^{43} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{44} - \beta_{3} q^{46} + ( - 3 \beta_{2} + 7) q^{47} - 5 q^{49} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{50} + ( - \beta_{2} - 2 \beta_1 + 1) q^{52} + (\beta_{2} + 2 \beta_1 - 1) q^{53} + (2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{55} - 3 \beta_{3} q^{56} + (\beta_{3} - 2 \beta_{2} + 1) q^{58} + ( - 2 \beta_{2} - 2) q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{61}+ \cdots + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{4} - 3 q^{5} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{4} - 3 q^{5} + 12 q^{8} + 3 q^{10} - 4 q^{16} + 8 q^{17} + 3 q^{20} - 7 q^{25} - 20 q^{32} - 8 q^{34} + 6 q^{35} + 12 q^{37} - 9 q^{40} + 30 q^{43} + 22 q^{47} - 20 q^{49} + 7 q^{50} - 21 q^{55} - 12 q^{59} + 28 q^{64} - 27 q^{65} - 8 q^{68} - 6 q^{70} - 24 q^{71} - 12 q^{73} - 12 q^{74} - 36 q^{77} + 3 q^{80} - 6 q^{85} - 30 q^{86} + 12 q^{91} - 22 q^{94} + 6 q^{95} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 25\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - 4 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 6 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 4\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{3} - 4\beta_{2} - 4\beta _1 - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{2} - 4 \) 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
1.28078 + 2.21837i
1.28078 2.21837i
−0.780776 + 1.35234i
−0.780776 1.35234i
−1.00000 0 −1.00000 −1.78078 1.35234i 0 3.46410i 3.00000 0 1.78078 + 1.35234i
289.2 −1.00000 0 −1.00000 −1.78078 + 1.35234i 0 3.46410i 3.00000 0 1.78078 1.35234i
289.3 −1.00000 0 −1.00000 0.280776 2.21837i 0 3.46410i 3.00000 0 −0.280776 + 2.21837i
289.4 −1.00000 0 −1.00000 0.280776 + 2.21837i 0 3.46410i 3.00000 0 −0.280776 2.21837i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.e 4
3.b odd 2 1 145.2.d.c yes 4
5.b even 2 1 1305.2.f.i 4
12.b even 2 1 2320.2.j.c 4
15.d odd 2 1 145.2.d.a 4
15.e even 4 2 725.2.c.f 8
29.b even 2 1 1305.2.f.i 4
60.h even 2 1 2320.2.j.a 4
87.d odd 2 1 145.2.d.a 4
145.d even 2 1 inner 1305.2.f.e 4
348.b even 2 1 2320.2.j.a 4
435.b odd 2 1 145.2.d.c yes 4
435.p even 4 2 725.2.c.f 8
1740.k even 2 1 2320.2.j.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.d.a 4 15.d odd 2 1
145.2.d.a 4 87.d odd 2 1
145.2.d.c yes 4 3.b odd 2 1
145.2.d.c yes 4 435.b odd 2 1
725.2.c.f 8 15.e even 4 2
725.2.c.f 8 435.p even 4 2
1305.2.f.e 4 1.a even 1 1 trivial
1305.2.f.e 4 145.d even 2 1 inner
1305.2.f.i 4 5.b even 2 1
1305.2.f.i 4 29.b even 2 1
2320.2.j.a 4 60.h even 2 1
2320.2.j.a 4 348.b even 2 1
2320.2.j.c 4 12.b even 2 1
2320.2.j.c 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 \(S_{2}^{\mathrm{new}}(1305, [\chi])\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 39T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{4} + 27T^{2} + 144 \) Copy content Toggle raw display
$17$ \( (T - 2)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 10T^{2} + 841 \) Copy content Toggle raw display
$31$ \( T^{4} + 63T^{2} + 36 \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 108T^{2} + 2304 \) Copy content Toggle raw display
$43$ \( (T^{2} - 15 T + 52)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 11 T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 27T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 156T^{2} + 576 \) Copy content Toggle raw display
$67$ \( T^{4} + 156T^{2} + 576 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T - 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 279 T^{2} + 12996 \) Copy content Toggle raw display
$83$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 108T^{2} + 2304 \) Copy content Toggle raw display
$97$ \( (T^{2} - 68)^{2} \) Copy content Toggle raw display
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