Properties

Label 1305.2.c.g
Level $1305$
Weight $2$
Character orbit 1305.c
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(784,1305)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.784");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.c (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: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 435)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{4} + (2 \beta_1 + 1) q^{5} + \beta_{2} q^{7} + 3 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{4} + (2 \beta_1 + 1) q^{5} + \beta_{2} q^{7} + 3 \beta_1 q^{8} + (\beta_1 - 2) q^{10} + (\beta_{3} + 2) q^{11} - 2 \beta_{2} q^{13} - \beta_{3} q^{14} - q^{16} + (2 \beta_{2} + 2 \beta_1) q^{17} + (\beta_{3} + 2) q^{19} + (2 \beta_1 + 1) q^{20} + (\beta_{2} + 2 \beta_1) q^{22} - 3 \beta_{2} q^{23} + (4 \beta_1 - 3) q^{25} + 2 \beta_{3} q^{26} + \beta_{2} q^{28} - q^{29} + ( - \beta_{3} + 2) q^{31} + 5 \beta_1 q^{32} + ( - 2 \beta_{3} - 2) q^{34} + ( - 2 \beta_{3} + \beta_{2}) q^{35} - 6 \beta_1 q^{37} + (\beta_{2} + 2 \beta_1) q^{38} + (3 \beta_1 - 6) q^{40} + ( - 2 \beta_{3} - 2) q^{41} + ( - 2 \beta_{2} + 4 \beta_1) q^{43} + (\beta_{3} + 2) q^{44} + 3 \beta_{3} q^{46} - 2 \beta_{2} q^{47} - q^{49} + ( - 3 \beta_1 - 4) q^{50} - 2 \beta_{2} q^{52} - 4 \beta_1 q^{53} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{55} - 3 \beta_{3} q^{56} - \beta_1 q^{58} - 4 q^{59} + ( - 2 \beta_{3} + 6) q^{61} + ( - \beta_{2} + 2 \beta_1) q^{62} - 7 q^{64} + (4 \beta_{3} - 2 \beta_{2}) q^{65} + (\beta_{2} - 4 \beta_1) q^{67} + (2 \beta_{2} + 2 \beta_1) q^{68} + ( - \beta_{3} - 2 \beta_{2}) q^{70} + ( - 2 \beta_{3} - 4) q^{71} + (2 \beta_{2} + 2 \beta_1) q^{73} + 6 q^{74} + (\beta_{3} + 2) q^{76} + (2 \beta_{2} + 8 \beta_1) q^{77} + ( - \beta_{3} + 6) q^{79} + ( - 2 \beta_1 - 1) q^{80} + ( - 2 \beta_{2} - 2 \beta_1) q^{82} + (3 \beta_{2} - 4 \beta_1) q^{83} + ( - 4 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{85}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 4 q^{5} - 8 q^{10} + 8 q^{11} - 4 q^{16} + 8 q^{19} + 4 q^{20} - 12 q^{25} - 4 q^{29} + 8 q^{31} - 8 q^{34} - 24 q^{40} - 8 q^{41} + 8 q^{44} - 4 q^{49} - 16 q^{50} + 8 q^{55} - 16 q^{59} + 24 q^{61} - 28 q^{64} - 16 q^{71} + 24 q^{74} + 8 q^{76} + 24 q^{79} - 4 q^{80} - 16 q^{85} - 16 q^{86} - 24 q^{89} + 64 q^{91} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
784.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.00000i 0 1.00000 1.00000 2.00000i 0 2.82843i 3.00000i 0 −2.00000 1.00000i
784.2 1.00000i 0 1.00000 1.00000 2.00000i 0 2.82843i 3.00000i 0 −2.00000 1.00000i
784.3 1.00000i 0 1.00000 1.00000 + 2.00000i 0 2.82843i 3.00000i 0 −2.00000 + 1.00000i
784.4 1.00000i 0 1.00000 1.00000 + 2.00000i 0 2.82843i 3.00000i 0 −2.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.c.g 4
3.b odd 2 1 435.2.c.c 4
5.b even 2 1 inner 1305.2.c.g 4
5.c odd 4 1 6525.2.a.n 2
5.c odd 4 1 6525.2.a.bd 2
15.d odd 2 1 435.2.c.c 4
15.e even 4 1 2175.2.a.k 2
15.e even 4 1 2175.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.c 4 3.b odd 2 1
435.2.c.c 4 15.d odd 2 1
1305.2.c.g 4 1.a even 1 1 trivial
1305.2.c.g 4 5.b even 2 1 inner
2175.2.a.k 2 15.e even 4 1
2175.2.a.s 2 15.e even 4 1
6525.2.a.n 2 5.c odd 4 1
6525.2.a.bd 2 5.c odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$47$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T + 4)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$79$ \( (T^{2} - 12 T + 28)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 176T^{2} + 3136 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
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