Properties

Label 320.4.n.c
Level $320$
Weight $4$
Character orbit 320.n
Analytic conductor $18.881$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,4,Mod(63,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.63");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 320.n (of order \(4\), degree \(2\), not 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: \(18.8806112018\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (11 i + 2) q^{5} - 27 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (11 i + 2) q^{5} - 27 i q^{9} + ( - 37 i + 37) q^{13} + (99 i + 99) q^{17} + (44 i - 117) q^{25} + 284 i q^{29} + (91 i + 91) q^{37} + 472 q^{41} + ( - 54 i + 297) q^{45} + 343 i q^{49} + ( - 27 i + 27) q^{53} + 468 q^{61} + (333 i + 481) q^{65} + ( - 253 i + 253) q^{73} - 729 q^{81} + (1287 i - 891) q^{85} + 176 i q^{89} + ( - 611 i - 611) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 74 q^{13} + 198 q^{17} - 234 q^{25} + 182 q^{37} + 944 q^{41} + 594 q^{45} + 54 q^{53} + 936 q^{61} + 962 q^{65} + 506 q^{73} - 1458 q^{81} - 1782 q^{85} - 1222 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
1.00000i
1.00000i
0 0 0 2.00000 11.0000i 0 0 0 27.0000i 0
127.1 0 0 0 2.00000 + 11.0000i 0 0 0 27.0000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.n.c 2
4.b odd 2 1 CM 320.4.n.c 2
5.c odd 4 1 inner 320.4.n.c 2
8.b even 2 1 20.4.e.a 2
8.d odd 2 1 20.4.e.a 2
20.e even 4 1 inner 320.4.n.c 2
24.f even 2 1 180.4.k.a 2
24.h odd 2 1 180.4.k.a 2
40.e odd 2 1 100.4.e.a 2
40.f even 2 1 100.4.e.a 2
40.i odd 4 1 20.4.e.a 2
40.i odd 4 1 100.4.e.a 2
40.k even 4 1 20.4.e.a 2
40.k even 4 1 100.4.e.a 2
120.q odd 4 1 180.4.k.a 2
120.w even 4 1 180.4.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.e.a 2 8.b even 2 1
20.4.e.a 2 8.d odd 2 1
20.4.e.a 2 40.i odd 4 1
20.4.e.a 2 40.k even 4 1
100.4.e.a 2 40.e odd 2 1
100.4.e.a 2 40.f even 2 1
100.4.e.a 2 40.i odd 4 1
100.4.e.a 2 40.k even 4 1
180.4.k.a 2 24.f even 2 1
180.4.k.a 2 24.h odd 2 1
180.4.k.a 2 120.q odd 4 1
180.4.k.a 2 120.w even 4 1
320.4.n.c 2 1.a even 1 1 trivial
320.4.n.c 2 4.b odd 2 1 CM
320.4.n.c 2 5.c odd 4 1 inner
320.4.n.c 2 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(320, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{13}^{2} - 74T_{13} + 2738 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 74T + 2738 \) Copy content Toggle raw display
$17$ \( T^{2} - 198T + 19602 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 80656 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 182T + 16562 \) Copy content Toggle raw display
$41$ \( (T - 472)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 54T + 1458 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 468)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 506T + 128018 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 30976 \) Copy content Toggle raw display
$97$ \( T^{2} + 1222 T + 746642 \) Copy content Toggle raw display
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