Properties

Label 1305.2.d.b
Level $1305$
Weight $2$
Character orbit 1305.d
Analytic conductor $10.420$
Analytic rank $0$
Dimension $6$
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(811,1305)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.811");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.d (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: \(6\)
Coefficient field: 6.0.16516096.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 13x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} - \beta_{2} q^{7} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} - \beta_{2} q^{7} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{8} - \beta_1 q^{10} + (\beta_{5} - \beta_1) q^{11} + (\beta_{3} - 1) q^{13} + ( - \beta_{5} - \beta_{4} + 3 \beta_1) q^{14} + (\beta_{3} - 2 \beta_{2} + 4) q^{16} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{17} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{19} + ( - \beta_{2} + 1) q^{20} + ( - \beta_{2} + 2) q^{22} + (\beta_{3} - \beta_{2} + 1) q^{23} + q^{25} + ( - \beta_{4} - 2 \beta_1) q^{26} + ( - \beta_{3} + 3 \beta_{2} - 9) q^{28} + (\beta_{5} - \beta_{4} - \beta_{2} + \cdots + 1) q^{29}+ \cdots + ( - 2 \beta_{5} - 3 \beta_{4} + 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 6 q^{5} - 4 q^{13} + 26 q^{16} + 6 q^{20} + 12 q^{22} + 8 q^{23} + 6 q^{25} - 56 q^{28} + 6 q^{29} + 16 q^{34} - 20 q^{38} + 14 q^{49} + 20 q^{52} - 28 q^{53} + 4 q^{58} + 16 q^{59} - 28 q^{62} - 46 q^{64} + 4 q^{65} - 16 q^{67} + 56 q^{71} - 40 q^{74} - 26 q^{80} - 56 q^{82} + 24 q^{83} - 4 q^{86} - 44 q^{88} - 16 q^{91} - 48 q^{92} + 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 9x^{4} + 13x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} + 8\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} - 8\nu^{3} - 7\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 9\nu^{3} + 13\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} - 8\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} - 9\beta_{4} + 41\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
811.1
2.68667i
1.30397i
0.285442i
0.285442i
1.30397i
2.68667i
2.68667i 0 −5.21819 −1.00000 0 4.21819 8.64620i 0 2.68667i
811.2 1.30397i 0 0.299664 −1.00000 0 −1.29966 2.99869i 0 1.30397i
811.3 0.285442i 0 1.91852 −1.00000 0 −2.91852 1.11851i 0 0.285442i
811.4 0.285442i 0 1.91852 −1.00000 0 −2.91852 1.11851i 0 0.285442i
811.5 1.30397i 0 0.299664 −1.00000 0 −1.29966 2.99869i 0 1.30397i
811.6 2.68667i 0 −5.21819 −1.00000 0 4.21819 8.64620i 0 2.68667i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 811.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b 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.d.b 6
3.b odd 2 1 145.2.c.b 6
12.b even 2 1 2320.2.g.i 6
15.d odd 2 1 725.2.c.e 6
15.e even 4 2 725.2.d.c 12
29.b even 2 1 inner 1305.2.d.b 6
87.d odd 2 1 145.2.c.b 6
87.f even 4 2 4205.2.a.m 6
348.b even 2 1 2320.2.g.i 6
435.b odd 2 1 725.2.c.e 6
435.p even 4 2 725.2.d.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.c.b 6 3.b odd 2 1
145.2.c.b 6 87.d odd 2 1
725.2.c.e 6 15.d odd 2 1
725.2.c.e 6 435.b odd 2 1
725.2.d.c 12 15.e even 4 2
725.2.d.c 12 435.p even 4 2
1305.2.d.b 6 1.a even 1 1 trivial
1305.2.d.b 6 29.b even 2 1 inner
2320.2.g.i 6 12.b even 2 1
2320.2.g.i 6 348.b even 2 1
4205.2.a.m 6 87.f even 4 2

Hecke kernels

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

\( T_{2}^{6} + 9T_{2}^{4} + 13T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{23}^{3} - 4T_{23}^{2} - 26T_{23} + 96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 9 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} - 14 T - 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 16 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{3} + 2 T^{2} - 24 T - 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 52 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{6} + 56 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$23$ \( (T^{3} - 4 T^{2} - 26 T + 96)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} + \cdots + 24389 \) Copy content Toggle raw display
$31$ \( T^{6} + 152 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$37$ \( T^{6} + 100 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$41$ \( T^{6} + 184 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( T^{6} + 176 T^{4} + \cdots + 121104 \) Copy content Toggle raw display
$47$ \( T^{6} + 16 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( (T^{3} + 14 T^{2} + \cdots - 576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 8 T^{2} - 4 T + 48)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 104 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$67$ \( (T^{3} + 8 T^{2} - 10 T - 8)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 28 T^{2} + \cdots - 576)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 252 T^{4} + \cdots + 419904 \) Copy content Toggle raw display
$79$ \( T^{6} + 176 T^{4} + \cdots + 121104 \) Copy content Toggle raw display
$83$ \( (T^{3} - 12 T^{2} + \cdots - 24)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 160 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$97$ \( T^{6} + 28 T^{4} + \cdots + 576 \) Copy content Toggle raw display
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