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Properties

Label 800.4.c.i
Level $800$
Weight $4$
Character orbit 800.c
Analytic conductor $47.202$
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] = [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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 160)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 2 \beta_1) q^{3} + ( - 5 \beta_{2} - 2 \beta_1) q^{7} + (4 \beta_{3} - 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 2 \beta_1) q^{3} + ( - 5 \beta_{2} - 2 \beta_1) q^{7} + (4 \beta_{3} - 13) q^{9} + (3 \beta_{3} - 32) q^{11} + (8 \beta_{2} - 3 \beta_1) q^{13} + (24 \beta_{2} + \beta_1) q^{17} + (4 \beta_{3} + 104) q^{19} + (8 \beta_{3} - 104) q^{21} + (11 \beta_{2} + 30 \beta_1) q^{23} + (18 \beta_{2} - 68 \beta_1) q^{27} + ( - 8 \beta_{3} + 146) q^{29} + ( - 3 \beta_{3} - 88) q^{31} + (56 \beta_{2} - 136 \beta_1) q^{33} + ( - 16 \beta_{2} - 89 \beta_1) q^{37} + ( - 19 \beta_{3} + 216) q^{39} + (20 \beta_{3} + 50) q^{41} + ( - 37 \beta_{2} - 94 \beta_1) q^{43} + ( - 13 \beta_{2} + 70 \beta_1) q^{47} + ( - 20 \beta_{3} - 273) q^{49} + ( - 47 \beta_{3} + 568) q^{51} + (24 \beta_{2} - 79 \beta_1) q^{53} + ( - 72 \beta_{2} + 112 \beta_1) q^{57} + (10 \beta_{3} + 360) q^{59} + (16 \beta_{3} - 634) q^{61} + (33 \beta_{2} - 454 \beta_1) q^{63} + (47 \beta_{2} + 186 \beta_1) q^{67} + (8 \beta_{3} + 24) q^{69} + (109 \beta_{3} + 24) q^{71} + ( - 120 \beta_{2} + 235 \beta_1) q^{73} + (136 \beta_{2} - 296 \beta_1) q^{77} + (86 \beta_{3} + 16) q^{79} + (4 \beta_{3} + 625) q^{81} + (47 \beta_{2} - 398 \beta_1) q^{83} + ( - 210 \beta_{2} + 484 \beta_1) q^{87} + (24 \beta_{3} + 390) q^{89} + (\beta_{3} + 936) q^{91} + (64 \beta_{2} - 104 \beta_1) q^{93} + (40 \beta_{2} + 305 \beta_1) q^{97} + ( - 167 \beta_{3} + 1568) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 52 q^{9} - 128 q^{11} + 416 q^{19} - 416 q^{21} + 584 q^{29} - 352 q^{31} + 864 q^{39} + 200 q^{41} - 1092 q^{49} + 2272 q^{51} + 1440 q^{59} - 2536 q^{61} + 96 q^{69} + 96 q^{71} + 64 q^{79} + 2500 q^{81} + 1560 q^{89} + 3744 q^{91} + 6272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 6\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} + 12\nu ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 6\beta_{2} ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−1.22474 + 1.22474i
1.22474 + 1.22474i
1.22474 − 1.22474i
−1.22474 − 1.22474i
0 − 8.89898i 0 0 0 − 20.4949i 0 −52.1918 0
449.2 0 − 0.898979i 0 0 0 − 28.4949i 0 26.1918 0
449.3 0 0.898979i 0 0 0 28.4949i 0 26.1918 0
449.4 0 8.89898i 0 0 0 20.4949i 0 −52.1918 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.i 4
4.b odd 2 1 800.4.c.k 4
5.b even 2 1 inner 800.4.c.i 4
5.c odd 4 1 160.4.a.c ✓ 2
5.c odd 4 1 800.4.a.s 2
15.e even 4 1 1440.4.a.t 2
20.d odd 2 1 800.4.c.k 4
20.e even 4 1 160.4.a.g yes 2
20.e even 4 1 800.4.a.m 2
40.i odd 4 1 320.4.a.s 2
40.i odd 4 1 1600.4.a.cd 2
40.k even 4 1 320.4.a.o 2
40.k even 4 1 1600.4.a.cn 2
60.l odd 4 1 1440.4.a.x 2
80.i odd 4 1 1280.4.d.x 4
80.j even 4 1 1280.4.d.q 4
80.s even 4 1 1280.4.d.q 4
80.t odd 4 1 1280.4.d.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.c ✓ 2 5.c odd 4 1
160.4.a.g yes 2 20.e even 4 1
320.4.a.o 2 40.k even 4 1
320.4.a.s 2 40.i odd 4 1
800.4.a.m 2 20.e even 4 1
800.4.a.s 2 5.c odd 4 1
800.4.c.i 4 1.a even 1 1 trivial
800.4.c.i 4 5.b even 2 1 inner
800.4.c.k 4 4.b odd 2 1
800.4.c.k 4 20.d odd 2 1
1280.4.d.q 4 80.j even 4 1
1280.4.d.q 4 80.s even 4 1
1280.4.d.x 4 80.i odd 4 1
1280.4.d.x 4 80.t odd 4 1
1440.4.a.t 2 15.e even 4 1
1440.4.a.x 2 60.l odd 4 1
1600.4.a.cd 2 40.i odd 4 1
1600.4.a.cn 2 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 \(S_{4}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{4} + 80T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} + 64T_{11} + 160 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1232 T^{2} + 341056 \) Copy content Toggle raw display
$11$ \( (T^{2} + 64 T + 160)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 3144 T^{2} + 2250000 \) Copy content Toggle raw display
$17$ \( T^{4} + 27656 T^{2} + 190992400 \) Copy content Toggle raw display
$19$ \( (T^{2} - 208 T + 9280)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 13008 T^{2} + 484416 \) Copy content Toggle raw display
$29$ \( (T^{2} - 292 T + 15172)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 176 T + 6880)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 75656 T^{2} + 652291600 \) Copy content Toggle raw display
$41$ \( (T^{2} - 100 T - 35900)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 136400 T^{2} + 6190144 \) Copy content Toggle raw display
$47$ \( T^{4} + 47312 T^{2} + 241615936 \) Copy content Toggle raw display
$53$ \( T^{4} + 77576 T^{2} + 124099600 \) Copy content Toggle raw display
$59$ \( (T^{2} - 720 T + 120000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 1268 T + 377380)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 7287695424 \) Copy content Toggle raw display
$71$ \( (T^{2} - 48 T - 1140000)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 15550090000 \) Copy content Toggle raw display
$79$ \( (T^{2} - 32 T - 709760)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 337096360000 \) Copy content Toggle raw display
$89$ \( (T^{2} - 780 T + 96804)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 111355690000 \) Copy content Toggle raw display
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