Properties

Label 1305.2.d.f
Level $1305$
Weight $2$
Character orbit 1305.d
Analytic conductor $10.420$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 13x^{8} + 56x^{6} + 89x^{4} + 41x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + q^{5} + \beta_{7} q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + q^{5} + \beta_{7} q^{7} + \beta_{3} q^{8} + \beta_1 q^{10} + ( - \beta_{9} - \beta_{6}) q^{11} + (\beta_{4} - \beta_{2} + 1) q^{13} - \beta_{6} q^{14} + \beta_{4} q^{16} + ( - \beta_{9} + \beta_{8} - \beta_{3}) q^{17} + ( - \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{19}+ \cdots + (\beta_{9} + \beta_{8} + \cdots - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{4} + 10 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{4} + 10 q^{5} - 4 q^{7} + 4 q^{13} - 2 q^{16} - 6 q^{20} + 10 q^{22} + 10 q^{23} + 10 q^{25} - 2 q^{28} + 2 q^{34} - 4 q^{35} + 32 q^{38} - 10 q^{49} - 34 q^{52} - 14 q^{53} - 18 q^{58} + 32 q^{59} + 4 q^{62} + 36 q^{64} + 4 q^{65} - 8 q^{67} - 8 q^{71} - 32 q^{74} - 2 q^{80} - 16 q^{82} + 18 q^{83} - 56 q^{86} + 24 q^{91} - 26 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 13x^{8} + 56x^{6} + 89x^{4} + 41x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 6\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} + 9\nu^{4} + 20\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{9} - 12\nu^{7} - 46\nu^{5} - 59\nu^{3} - 12\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} + 12\nu^{6} + 46\nu^{4} + 59\nu^{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{9} - 13\nu^{7} - 54\nu^{5} - 73\nu^{3} - 13\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} + 13\nu^{7} + 56\nu^{5} + 87\nu^{3} + 31\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + \beta_{8} - 7\beta_{3} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{5} - 9\beta_{4} + 34\beta_{2} - 71 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8\beta_{9} - 10\beta_{8} + 2\beta_{6} + 42\beta_{3} - 97\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{7} - 12\beta_{5} + 62\beta_{4} - 191\beta_{2} + 373 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 50\beta_{9} + 74\beta_{8} - 26\beta_{6} - 241\beta_{3} + 514\beta_1 \) 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
2.38254i
2.09920i
1.49936i
0.732188i
0.364257i
0.364257i
0.732188i
1.49936i
2.09920i
2.38254i
2.38254i 0 −3.67649 1.00000 0 1.34246 3.99431i 0 2.38254i
811.2 2.09920i 0 −2.40666 1.00000 0 −2.25380 0.853652i 0 2.09920i
811.3 1.49936i 0 −0.248070 1.00000 0 0.522278 2.62677i 0 1.49936i
811.4 0.732188i 0 1.46390 1.00000 0 −4.08783 2.53623i 0 0.732188i
811.5 0.364257i 0 1.86732 1.00000 0 2.47690 1.40870i 0 0.364257i
811.6 0.364257i 0 1.86732 1.00000 0 2.47690 1.40870i 0 0.364257i
811.7 0.732188i 0 1.46390 1.00000 0 −4.08783 2.53623i 0 0.732188i
811.8 1.49936i 0 −0.248070 1.00000 0 0.522278 2.62677i 0 1.49936i
811.9 2.09920i 0 −2.40666 1.00000 0 −2.25380 0.853652i 0 2.09920i
811.10 2.38254i 0 −3.67649 1.00000 0 1.34246 3.99431i 0 2.38254i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 811.10
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.f yes 10
3.b odd 2 1 1305.2.d.e 10
29.b even 2 1 inner 1305.2.d.f yes 10
87.d odd 2 1 1305.2.d.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.d.e 10 3.b odd 2 1
1305.2.d.e 10 87.d odd 2 1
1305.2.d.f yes 10 1.a even 1 1 trivial
1305.2.d.f yes 10 29.b even 2 1 inner

Hecke kernels

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

\( T_{2}^{10} + 13T_{2}^{8} + 56T_{2}^{6} + 89T_{2}^{4} + 41T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{23}^{5} - 5T_{23}^{4} - 86T_{23}^{3} + 270T_{23}^{2} + 1728T_{23} + 648 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 13 T^{8} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} + 2 T^{4} - 13 T^{3} + \cdots - 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 47 T^{8} + \cdots + 6724 \) Copy content Toggle raw display
$13$ \( (T^{5} - 2 T^{4} - 25 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 62 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( T^{10} + 140 T^{8} + \cdots + 186624 \) Copy content Toggle raw display
$23$ \( (T^{5} - 5 T^{4} + \cdots + 648)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} - 43 T^{8} + \cdots + 20511149 \) Copy content Toggle raw display
$31$ \( T^{10} + 212 T^{8} + \cdots + 6718464 \) Copy content Toggle raw display
$37$ \( T^{10} + 229 T^{8} + \cdots + 186624 \) Copy content Toggle raw display
$41$ \( T^{10} + 197 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 215737344 \) Copy content Toggle raw display
$47$ \( T^{10} + 314 T^{8} + \cdots + 63744256 \) Copy content Toggle raw display
$53$ \( (T^{5} + 7 T^{4} + \cdots + 6480)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 16 T^{4} + \cdots - 46656)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 512 T^{8} + \cdots + 74649600 \) Copy content Toggle raw display
$67$ \( (T^{5} + 4 T^{4} + \cdots + 2836)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 4 T^{4} + \cdots + 3456)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 967458816 \) Copy content Toggle raw display
$79$ \( T^{10} + 140 T^{8} + \cdots + 186624 \) Copy content Toggle raw display
$83$ \( (T^{5} - 9 T^{4} + \cdots - 38664)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 132342016 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 191102976 \) Copy content Toggle raw display
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