Properties

Label 1305.4.a.a
Level $1305$
Weight $4$
Character orbit 1305.a
Self dual yes
Analytic conductor $76.997$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(76.9974925575\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{2} + 17 q^{4} - 5 q^{5} + 16 q^{7} - 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} + 17 q^{4} - 5 q^{5} + 16 q^{7} - 45 q^{8} + 25 q^{10} + 44 q^{11} + 78 q^{13} - 80 q^{14} + 89 q^{16} - 18 q^{17} - 28 q^{19} - 85 q^{20} - 220 q^{22} - 184 q^{23} + 25 q^{25} - 390 q^{26} + 272 q^{28} - 29 q^{29} - 224 q^{31} - 85 q^{32} + 90 q^{34} - 80 q^{35} + 254 q^{37} + 140 q^{38} + 225 q^{40} + 78 q^{41} - 260 q^{43} + 748 q^{44} + 920 q^{46} - 312 q^{47} - 87 q^{49} - 125 q^{50} + 1326 q^{52} - 574 q^{53} - 220 q^{55} - 720 q^{56} + 145 q^{58} - 180 q^{59} - 610 q^{61} + 1120 q^{62} - 287 q^{64} - 390 q^{65} - 340 q^{67} - 306 q^{68} + 400 q^{70} - 296 q^{71} + 394 q^{73} - 1270 q^{74} - 476 q^{76} + 704 q^{77} - 960 q^{79} - 445 q^{80} - 390 q^{82} + 908 q^{83} + 90 q^{85} + 1300 q^{86} - 1980 q^{88} + 990 q^{89} + 1248 q^{91} - 3128 q^{92} + 1560 q^{94} + 140 q^{95} + 1234 q^{97} + 435 q^{98}+O(q^{100}) \) 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
0
−5.00000 0 17.0000 −5.00000 0 16.0000 −45.0000 0 25.0000
\(n\): e.g. 2-40 or 990-1000
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 1305.4.a.a 1
3.b odd 2 1 435.4.a.c 1
15.d odd 2 1 2175.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.4.a.c 1 3.b odd 2 1
1305.4.a.a 1 1.a even 1 1 trivial
2175.4.a.a 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1305))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 16 \) Copy content Toggle raw display
$11$ \( T - 44 \) Copy content Toggle raw display
$13$ \( T - 78 \) Copy content Toggle raw display
$17$ \( T + 18 \) Copy content Toggle raw display
$19$ \( T + 28 \) Copy content Toggle raw display
$23$ \( T + 184 \) Copy content Toggle raw display
$29$ \( T + 29 \) Copy content Toggle raw display
$31$ \( T + 224 \) Copy content Toggle raw display
$37$ \( T - 254 \) Copy content Toggle raw display
$41$ \( T - 78 \) Copy content Toggle raw display
$43$ \( T + 260 \) Copy content Toggle raw display
$47$ \( T + 312 \) Copy content Toggle raw display
$53$ \( T + 574 \) Copy content Toggle raw display
$59$ \( T + 180 \) Copy content Toggle raw display
$61$ \( T + 610 \) Copy content Toggle raw display
$67$ \( T + 340 \) Copy content Toggle raw display
$71$ \( T + 296 \) Copy content Toggle raw display
$73$ \( T - 394 \) Copy content Toggle raw display
$79$ \( T + 960 \) Copy content Toggle raw display
$83$ \( T - 908 \) Copy content Toggle raw display
$89$ \( T - 990 \) Copy content Toggle raw display
$97$ \( T - 1234 \) Copy content Toggle raw display
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