Properties

Label 1560.4.g.a
Level $1560$
Weight $4$
Character orbit 1560.g
Analytic conductor $92.043$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1560,4,Mod(961,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.961");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1560.g (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: \(92.0429796090\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + 5 i q^{5} + 8 i q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + 5 i q^{5} + 8 i q^{7} + 9 q^{9} - 12 i q^{11} + (26 i - 39) q^{13} + 15 i q^{15} - 36 q^{17} + 124 i q^{19} + 24 i q^{21} - 42 q^{23} - 25 q^{25} + 27 q^{27} + 76 q^{29} - 174 i q^{31} - 36 i q^{33} - 40 q^{35} - 132 i q^{37} + (78 i - 117) q^{39} - 354 i q^{41} - 380 q^{43} + 45 i q^{45} + 96 i q^{47} + 279 q^{49} - 108 q^{51} - 202 q^{53} + 60 q^{55} + 372 i q^{57} - 764 i q^{59} - 882 q^{61} + 72 i q^{63} + ( - 195 i - 130) q^{65} + 502 i q^{67} - 126 q^{69} + 164 i q^{71} + 686 i q^{73} - 75 q^{75} + 96 q^{77} - 280 q^{79} + 81 q^{81} - 664 i q^{83} - 180 i q^{85} + 228 q^{87} + 1350 i q^{89} + ( - 312 i - 208) q^{91} - 522 i q^{93} - 620 q^{95} - 1526 i q^{97} - 108 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 18 q^{9} - 78 q^{13} - 72 q^{17} - 84 q^{23} - 50 q^{25} + 54 q^{27} + 152 q^{29} - 80 q^{35} - 234 q^{39} - 760 q^{43} + 558 q^{49} - 216 q^{51} - 404 q^{53} + 120 q^{55} - 1764 q^{61} - 260 q^{65} - 252 q^{69} - 150 q^{75} + 192 q^{77} - 560 q^{79} + 162 q^{81} + 456 q^{87} - 416 q^{91} - 1240 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
1.00000i
1.00000i
0 3.00000 0 5.00000i 0 8.00000i 0 9.00000 0
961.2 0 3.00000 0 5.00000i 0 8.00000i 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.4.g.a 2
13.b even 2 1 inner 1560.4.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.4.g.a 2 1.a even 1 1 trivial
1560.4.g.a 2 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 64 \) acting on \(S_{4}^{\mathrm{new}}(1560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{2} + 64 \) Copy content Toggle raw display
$11$ \( T^{2} + 144 \) Copy content Toggle raw display
$13$ \( T^{2} + 78T + 2197 \) Copy content Toggle raw display
$17$ \( (T + 36)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 15376 \) Copy content Toggle raw display
$23$ \( (T + 42)^{2} \) Copy content Toggle raw display
$29$ \( (T - 76)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 30276 \) Copy content Toggle raw display
$37$ \( T^{2} + 17424 \) Copy content Toggle raw display
$41$ \( T^{2} + 125316 \) Copy content Toggle raw display
$43$ \( (T + 380)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9216 \) Copy content Toggle raw display
$53$ \( (T + 202)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 583696 \) Copy content Toggle raw display
$61$ \( (T + 882)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 252004 \) Copy content Toggle raw display
$71$ \( T^{2} + 26896 \) Copy content Toggle raw display
$73$ \( T^{2} + 470596 \) Copy content Toggle raw display
$79$ \( (T + 280)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 440896 \) Copy content Toggle raw display
$89$ \( T^{2} + 1822500 \) Copy content Toggle raw display
$97$ \( T^{2} + 2328676 \) Copy content Toggle raw display
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