Properties

Label 800.4.c.d
Level $800$
Weight $4$
Character orbit 800.c
Analytic conductor $47.202$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,4,Mod(449,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("800.449");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), 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: \(47.2015280046\)
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: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + 5 i q^{3} + 10 i q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 i q^{3} + 10 i q^{7} + 2 q^{9} + 15 q^{11} + 8 i q^{13} + 21 i q^{17} + 105 q^{19} - 50 q^{21} - 10 i q^{23} + 145 i q^{27} + 20 q^{29} + 230 q^{31} + 75 i q^{33} + 54 i q^{37} - 40 q^{39} - 195 q^{41} - 300 i q^{43} + 480 i q^{47} + 243 q^{49} - 105 q^{51} + 322 i q^{53} + 525 i q^{57} + 560 q^{59} - 730 q^{61} + 20 i q^{63} - 255 i q^{67} + 50 q^{69} + 40 q^{71} + 317 i q^{73} + 150 i q^{77} - 830 q^{79} - 671 q^{81} + 75 i q^{83} + 100 i q^{87} + 705 q^{89} - 80 q^{91} + 1150 i q^{93} + 1434 i q^{97} + 30 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} + 30 q^{11} + 210 q^{19} - 100 q^{21} + 40 q^{29} + 460 q^{31} - 80 q^{39} - 390 q^{41} + 486 q^{49} - 210 q^{51} + 1120 q^{59} - 1460 q^{61} + 100 q^{69} + 80 q^{71} - 1660 q^{79} - 1342 q^{81} + 1410 q^{89} - 160 q^{91} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 5.00000i 0 0 0 10.0000i 0 2.00000 0
449.2 0 5.00000i 0 0 0 10.0000i 0 2.00000 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
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 800.4.c.d 2
4.b odd 2 1 800.4.c.c 2
5.b even 2 1 inner 800.4.c.d 2
5.c odd 4 1 800.4.a.c yes 1
5.c odd 4 1 800.4.a.j yes 1
20.d odd 2 1 800.4.c.c 2
20.e even 4 1 800.4.a.b 1
20.e even 4 1 800.4.a.i yes 1
40.i odd 4 1 1600.4.a.k 1
40.i odd 4 1 1600.4.a.bp 1
40.k even 4 1 1600.4.a.l 1
40.k even 4 1 1600.4.a.bq 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.4.a.b 1 20.e even 4 1
800.4.a.c yes 1 5.c odd 4 1
800.4.a.i yes 1 20.e even 4 1
800.4.a.j yes 1 5.c odd 4 1
800.4.c.c 2 4.b odd 2 1
800.4.c.c 2 20.d odd 2 1
800.4.c.d 2 1.a even 1 1 trivial
800.4.c.d 2 5.b even 2 1 inner
1600.4.a.k 1 40.i odd 4 1
1600.4.a.l 1 40.k even 4 1
1600.4.a.bp 1 40.i odd 4 1
1600.4.a.bq 1 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{2} + 25 \) Copy content Toggle raw display
\( T_{11} - 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 25 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 100 \) Copy content Toggle raw display
$11$ \( (T - 15)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{2} + 441 \) Copy content Toggle raw display
$19$ \( (T - 105)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 100 \) Copy content Toggle raw display
$29$ \( (T - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T - 230)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2916 \) Copy content Toggle raw display
$41$ \( (T + 195)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 90000 \) Copy content Toggle raw display
$47$ \( T^{2} + 230400 \) Copy content Toggle raw display
$53$ \( T^{2} + 103684 \) Copy content Toggle raw display
$59$ \( (T - 560)^{2} \) Copy content Toggle raw display
$61$ \( (T + 730)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 65025 \) Copy content Toggle raw display
$71$ \( (T - 40)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100489 \) Copy content Toggle raw display
$79$ \( (T + 830)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 5625 \) Copy content Toggle raw display
$89$ \( (T - 705)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2056356 \) Copy content Toggle raw display
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