Properties

Label 1305.2.d.a
Level $1305$
Weight $2$
Character orbit 1305.d
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(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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + (\beta_{2} - 1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + q^{5} + (\beta_{2} - 1) q^{7} + \beta_{3} q^{8} + \beta_1 q^{10} + ( - 3 \beta_{3} + \beta_1) q^{11} + ( - 2 \beta_{2} + 2) q^{13} + (\beta_{3} - 3 \beta_1) q^{14} + (2 \beta_{2} - 1) q^{16} + (3 \beta_{3} + \beta_1) q^{17} + ( - 3 \beta_{3} + 3 \beta_1) q^{19} + \beta_{2} q^{20} + (\beta_{2} + 1) q^{22} + (3 \beta_{2} + 3) q^{23} + q^{25} + ( - 2 \beta_{3} + 6 \beta_1) q^{26} + ( - \beta_{2} + 3) q^{28} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{29} + ( - 3 \beta_{3} + 3 \beta_1) q^{31} + (4 \beta_{3} - 5 \beta_1) q^{32} + (\beta_{2} - 5) q^{34} + (\beta_{2} - 1) q^{35} + (3 \beta_{3} - 3 \beta_1) q^{37} + (3 \beta_{2} - 3) q^{38} + \beta_{3} q^{40} + (4 \beta_{3} + 2 \beta_1) q^{41} + ( - 3 \beta_{3} + 3 \beta_1) q^{43} + ( - 5 \beta_{3} + \beta_1) q^{44} + (3 \beta_{3} - 3 \beta_1) q^{46} + ( - 3 \beta_{3} - 5 \beta_1) q^{47} + ( - 2 \beta_{2} - 3) q^{49} + \beta_1 q^{50} + (2 \beta_{2} - 6) q^{52} + ( - 3 \beta_{3} + \beta_1) q^{55} + (\beta_{3} - \beta_1) q^{56} + (3 \beta_{3} + \beta_{2} - 6 \beta_1 - 1) q^{58} - 6 q^{59} + 6 \beta_1 q^{61} + (3 \beta_{2} - 3) q^{62} + ( - \beta_{2} + 4) q^{64} + ( - 2 \beta_{2} + 2) q^{65} + (7 \beta_{2} - 1) q^{67} + (7 \beta_{3} - 5 \beta_1) q^{68} + (\beta_{3} - 3 \beta_1) q^{70} - 6 q^{71} + ( - 3 \beta_{3} + 9 \beta_1) q^{73} + ( - 3 \beta_{2} + 3) q^{74} + ( - 3 \beta_{3} - 3 \beta_1) q^{76} - 2 \beta_{3} q^{77} + (9 \beta_{3} - 3 \beta_1) q^{79} + (2 \beta_{2} - 1) q^{80} + (2 \beta_{2} - 8) q^{82} + ( - 3 \beta_{2} + 3) q^{83} + (3 \beta_{3} + \beta_1) q^{85} + (3 \beta_{2} - 3) q^{86} + (3 \beta_{2} + 5) q^{88} + 4 \beta_1 q^{89} + (4 \beta_{2} - 8) q^{91} + (3 \beta_{2} + 9) q^{92} + ( - 5 \beta_{2} + 13) q^{94} + ( - 3 \beta_{3} + 3 \beta_1) q^{95} + ( - 3 \beta_{3} - 3 \beta_1) q^{97} + ( - 2 \beta_{3} + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 4 q^{7} + 8 q^{13} - 4 q^{16} + 4 q^{22} + 12 q^{23} + 4 q^{25} + 12 q^{28} - 20 q^{34} - 4 q^{35} - 12 q^{38} - 12 q^{49} - 24 q^{52} - 4 q^{58} - 24 q^{59} - 12 q^{62} + 16 q^{64} + 8 q^{65} - 4 q^{67} - 24 q^{71} + 12 q^{74} - 4 q^{80} - 32 q^{82} + 12 q^{83} - 12 q^{86} + 20 q^{88} - 32 q^{91} + 36 q^{92} + 52 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) 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
1.93185i
0.517638i
0.517638i
1.93185i
1.93185i 0 −1.73205 1.00000 0 −2.73205 0.517638i 0 1.93185i
811.2 0.517638i 0 1.73205 1.00000 0 0.732051 1.93185i 0 0.517638i
811.3 0.517638i 0 1.73205 1.00000 0 0.732051 1.93185i 0 0.517638i
811.4 1.93185i 0 −1.73205 1.00000 0 −2.73205 0.517638i 0 1.93185i
\(n\): e.g. 2-40 or 990-1000
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.a 4
3.b odd 2 1 145.2.c.a 4
12.b even 2 1 2320.2.g.e 4
15.d odd 2 1 725.2.c.d 4
15.e even 4 2 725.2.d.b 8
29.b even 2 1 inner 1305.2.d.a 4
87.d odd 2 1 145.2.c.a 4
87.f even 4 2 4205.2.a.g 4
348.b even 2 1 2320.2.g.e 4
435.b odd 2 1 725.2.c.d 4
435.p even 4 2 725.2.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.c.a 4 3.b odd 2 1
145.2.c.a 4 87.d odd 2 1
725.2.c.d 4 15.d odd 2 1
725.2.c.d 4 435.b odd 2 1
725.2.d.b 8 15.e even 4 2
725.2.d.b 8 435.p even 4 2
1305.2.d.a 4 1.a even 1 1 trivial
1305.2.d.a 4 29.b even 2 1 inner
2320.2.g.e 4 12.b even 2 1
2320.2.g.e 4 348.b even 2 1
4205.2.a.g 4 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 \(S_{2}^{\mathrm{new}}(1305, [\chi])\):

\( T_{2}^{4} + 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{23}^{2} - 6T_{23} - 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 28T^{2} + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 52T^{2} + 484 \) Copy content Toggle raw display
$19$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 50T^{2} + 841 \) Copy content Toggle raw display
$31$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 112T^{2} + 2704 \) Copy content Toggle raw display
$43$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 196T^{2} + 8836 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T + 6)^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 144T^{2} + 1296 \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T - 146)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 252T^{2} + 324 \) Copy content Toggle raw display
$79$ \( T^{4} + 252T^{2} + 324 \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 64T^{2} + 256 \) Copy content Toggle raw display
$97$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
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