Properties

Label 1305.2.a.t
Level $1305$
Weight $2$
Character orbit 1305.a
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
Dimension $7$
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] = [1305,2,Mod(1,1305)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + ( - \beta_{5} + 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + ( - \beta_{5} + 1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + \beta_1 q^{10} + (\beta_{6} - \beta_{2} - 1) q^{11} + ( - \beta_{4} + 1) q^{13} + ( - \beta_{6} - \beta_{4} + 2 \beta_1 - 1) q^{14} + (\beta_{4} + \beta_{2} + 3) q^{16} + (\beta_{4} + 1) q^{17} + ( - \beta_{6} + \beta_{3} + \beta_1 + 2) q^{19} + (\beta_{2} + 2) q^{20} + (2 \beta_{6} + \beta_{5} - \beta_{3} + \cdots + 1) q^{22}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{5} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 13 q^{4} + 7 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 13 q^{4} + 7 q^{5} + 10 q^{7} + q^{10} - 3 q^{11} + 6 q^{13} - 9 q^{14} + 21 q^{16} + 8 q^{17} + 10 q^{19} + 13 q^{20} + 9 q^{22} + 11 q^{23} + 7 q^{25} + 3 q^{26} + 25 q^{28} - 7 q^{29} + 18 q^{31} + q^{32} - q^{34} + 10 q^{35} + 13 q^{37} + 12 q^{38} - 13 q^{41} + 9 q^{43} - 37 q^{44} - 8 q^{46} + 2 q^{47} + 21 q^{49} + q^{50} - q^{52} + 5 q^{53} - 3 q^{55} - 30 q^{56} - q^{58} - 8 q^{59} + 14 q^{61} - 8 q^{62} + 8 q^{64} + 6 q^{65} + 14 q^{67} + 27 q^{68} - 9 q^{70} - 8 q^{71} + 3 q^{73} - 34 q^{74} + 4 q^{76} - 28 q^{77} + 4 q^{79} + 21 q^{80} - 20 q^{82} + 17 q^{83} + 8 q^{85} + 4 q^{86} + 26 q^{88} - 20 q^{89} + 12 q^{91} + 60 q^{92} - 21 q^{94} + 10 q^{95} + 13 q^{97} - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + \nu^{4} - 8\nu^{3} - 7\nu^{2} + 11\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + \nu^{5} - 10\nu^{4} - 7\nu^{3} + 25\nu^{2} + 6\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - \beta_{4} + 8\beta_{3} + 37\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} - 2\beta_{5} + 11\beta_{4} - \beta_{3} + 45\beta_{2} - \beta _1 + 141 \) 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
−2.66072
−2.16940
−0.897436
0.431560
1.28209
2.43890
2.57501
−2.66072 0 5.07943 1.00000 0 4.68271 −8.19351 0 −2.66072
1.2 −2.16940 0 2.70628 1.00000 0 −0.484966 −1.53221 0 −2.16940
1.3 −0.897436 0 −1.19461 1.00000 0 3.83036 2.86696 0 −0.897436
1.4 0.431560 0 −1.81376 1.00000 0 −2.42505 −1.64586 0 0.431560
1.5 1.28209 0 −0.356256 1.00000 0 3.04834 −3.02092 0 1.28209
1.6 2.43890 0 3.94822 1.00000 0 3.59688 4.75150 0 2.43890
1.7 2.57501 0 4.63069 1.00000 0 −2.24826 6.77405 0 2.57501
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(29\) \(1\)

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 1305.2.a.t yes 7
3.b odd 2 1 1305.2.a.s 7
5.b even 2 1 6525.2.a.bv 7
15.d odd 2 1 6525.2.a.bw 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.a.s 7 3.b odd 2 1
1305.2.a.t yes 7 1.a even 1 1 trivial
6525.2.a.bv 7 5.b even 2 1
6525.2.a.bw 7 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1305))\):

\( T_{2}^{7} - T_{2}^{6} - 13T_{2}^{5} + 12T_{2}^{4} + 47T_{2}^{3} - 37T_{2}^{2} - 35T_{2} + 18 \) Copy content Toggle raw display
\( T_{7}^{7} - 10T_{7}^{6} + 15T_{7}^{5} + 114T_{7}^{4} - 280T_{7}^{3} - 400T_{7}^{2} + 956T_{7} + 520 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} + \cdots + 18 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( (T - 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 10 T^{6} + \cdots + 520 \) Copy content Toggle raw display
$11$ \( T^{7} + 3 T^{6} + \cdots - 12 \) Copy content Toggle raw display
$13$ \( T^{7} - 6 T^{6} + \cdots + 1408 \) Copy content Toggle raw display
$17$ \( T^{7} - 8 T^{6} + \cdots + 384 \) Copy content Toggle raw display
$19$ \( T^{7} - 10 T^{6} + \cdots + 6656 \) Copy content Toggle raw display
$23$ \( T^{7} - 11 T^{6} + \cdots - 4752 \) Copy content Toggle raw display
$29$ \( (T + 1)^{7} \) Copy content Toggle raw display
$31$ \( T^{7} - 18 T^{6} + \cdots - 1408 \) Copy content Toggle raw display
$37$ \( T^{7} - 13 T^{6} + \cdots - 3328 \) Copy content Toggle raw display
$41$ \( T^{7} + 13 T^{6} + \cdots - 2496 \) Copy content Toggle raw display
$43$ \( T^{7} - 9 T^{6} + \cdots - 171008 \) Copy content Toggle raw display
$47$ \( T^{7} - 2 T^{6} + \cdots - 23040 \) Copy content Toggle raw display
$53$ \( T^{7} - 5 T^{6} + \cdots + 734208 \) Copy content Toggle raw display
$59$ \( T^{7} + 8 T^{6} + \cdots + 49152 \) Copy content Toggle raw display
$61$ \( T^{7} - 14 T^{6} + \cdots + 2422656 \) Copy content Toggle raw display
$67$ \( T^{7} - 14 T^{6} + \cdots + 366536 \) Copy content Toggle raw display
$71$ \( T^{7} + 8 T^{6} + \cdots + 245760 \) Copy content Toggle raw display
$73$ \( T^{7} - 3 T^{6} + \cdots + 2022656 \) Copy content Toggle raw display
$79$ \( T^{7} - 4 T^{6} + \cdots + 27712 \) Copy content Toggle raw display
$83$ \( T^{7} - 17 T^{6} + \cdots - 52800 \) Copy content Toggle raw display
$89$ \( T^{7} + 20 T^{6} + \cdots - 26400 \) Copy content Toggle raw display
$97$ \( T^{7} - 13 T^{6} + \cdots + 249856 \) Copy content Toggle raw display
show more
show less