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Properties

Label 800.4.c.o
Level $800$
Weight $4$
Character orbit 800.c
Analytic conductor $47.202$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: 8.0.1135425807366400.12
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 52x^{6} + 664x^{4} + 337x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{7} + \beta_{3}) q^{7} + (\beta_{4} - 24) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{7} + \beta_{3}) q^{7} + (\beta_{4} - 24) q^{9} + (3 \beta_{5} - 4 \beta_1) q^{11} + (\beta_{6} + 18 \beta_{2}) q^{13} + (\beta_{6} + 15 \beta_{2}) q^{17} + (2 \beta_{5} - 5 \beta_1) q^{19} + ( - \beta_{4} + 56) q^{21} + ( - 6 \beta_{7} - 16 \beta_{3}) q^{23} + ( - \beta_{7} + 48 \beta_{3}) q^{27} + ( - 3 \beta_{4} + 18) q^{29} + ( - 10 \beta_{5} - 16 \beta_1) q^{31} + ( - 4 \beta_{6} - 189 \beta_{2}) q^{33} + ( - 3 \beta_{6} - 232 \beta_{2}) q^{37} + (\beta_{5} - 69 \beta_1) q^{39} + (4 \beta_{4} + 181) q^{41} + (11 \beta_{7} + 13 \beta_{3}) q^{43} + ( - 11 \beta_{7} + 31 \beta_{3}) q^{47} + ( - 4 \beta_{4} + 27) q^{49} + (\beta_{5} - 66 \beta_1) q^{51} - 222 \beta_{2} q^{53} + ( - 5 \beta_{6} - 245 \beta_{2}) q^{57} + ( - 25 \beta_{5} + 53 \beta_1) q^{59} + (\beta_{4} + 444) q^{61} + (28 \beta_{7} - 80 \beta_{3}) q^{63} + (25 \beta_{7} + 46 \beta_{3}) q^{67} + (16 \beta_{4} - 846) q^{69} + (29 \beta_{5} - \beta_1) q^{71} + (17 \beta_{6} - 265 \beta_{2}) q^{73} + (19 \beta_{6} - 556 \beta_{2}) q^{77} + (9 \beta_{5} - 107 \beta_1) q^{79} + ( - 21 \beta_{4} + 1795) q^{81} + ( - 37 \beta_{7} + 114 \beta_{3}) q^{83} + (3 \beta_{7} - 171 \beta_{3}) q^{87} + (5 \beta_{4} - 225) q^{89} + ( - 34 \beta_{5} + 74 \beta_1) q^{91} + ( - 16 \beta_{6} - 866 \beta_{2}) q^{93} + (10 \beta_{6} + 126 \beta_{2}) q^{97} + (77 \beta_{5} + 285 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 192 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 192 q^{9} + 448 q^{21} + 144 q^{29} + 1448 q^{41} + 216 q^{49} + 3552 q^{61} - 6768 q^{69} + 14360 q^{81} - 1800 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 52x^{6} + 664x^{4} + 337x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{6} + 158\nu^{4} + 1280\nu^{2} + 323 ) / 599 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\nu^{7} + 1208\nu^{5} + 15746\nu^{3} + 11591\nu ) / 3594 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -23\nu^{7} - 1208\nu^{5} - 15746\nu^{3} - 4403\nu ) / 3594 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -8\nu^{6} - 316\nu^{4} - 164\nu^{2} + 30502 ) / 599 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 196\nu^{6} + 10138\nu^{4} + 127412\nu^{2} + 32599 ) / 599 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 52\nu^{5} - 670\nu^{3} - 493\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2467\nu^{7} - 127696\nu^{5} - 1607674\nu^{3} - 449143\nu ) / 3594 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 2\beta _1 - 52 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 3\beta_{6} - 105\beta_{3} - 154\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} - 27\beta_{4} - 103\beta _1 + 1376 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{7} + 65\beta_{6} + 1428\beta_{3} + 3312\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -79\beta_{5} + 1493\beta_{4} + 8055\beta _1 - 76070 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 393\beta_{7} - 2387\beta_{6} - 40067\beta_{3} - 121620\beta_{2} ) / 4 \) 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
4.54854i
5.54854i
0.610728i
− 0.389272i
− 0.610728i
0.389272i
− 4.54854i
− 5.54854i
0 − 10.0971i 0 0 0 10.5923i 0 −74.9510 0
449.2 0 − 10.0971i 0 0 0 10.5923i 0 −74.9510 0
449.3 0 − 0.221457i 0 0 0 22.7992i 0 26.9510 0
449.4 0 − 0.221457i 0 0 0 22.7992i 0 26.9510 0
449.5 0 0.221457i 0 0 0 − 22.7992i 0 26.9510 0
449.6 0 0.221457i 0 0 0 − 22.7992i 0 26.9510 0
449.7 0 10.0971i 0 0 0 − 10.5923i 0 −74.9510 0
449.8 0 10.0971i 0 0 0 − 10.5923i 0 −74.9510 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 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.o 8
4.b odd 2 1 inner 800.4.c.o 8
5.b even 2 1 inner 800.4.c.o 8
5.c odd 4 1 800.4.a.ba ✓ 4
5.c odd 4 1 800.4.a.bb yes 4
20.d odd 2 1 inner 800.4.c.o 8
20.e even 4 1 800.4.a.ba ✓ 4
20.e even 4 1 800.4.a.bb yes 4
40.i odd 4 1 1600.4.a.cw 4
40.i odd 4 1 1600.4.a.cx 4
40.k even 4 1 1600.4.a.cw 4
40.k even 4 1 1600.4.a.cx 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.4.a.ba ✓ 4 5.c odd 4 1
800.4.a.ba ✓ 4 20.e even 4 1
800.4.a.bb yes 4 5.c odd 4 1
800.4.a.bb yes 4 20.e even 4 1
800.4.c.o 8 1.a even 1 1 trivial
800.4.c.o 8 4.b odd 2 1 inner
800.4.c.o 8 5.b even 2 1 inner
800.4.c.o 8 20.d odd 2 1 inner
1600.4.a.cw 4 40.i odd 4 1
1600.4.a.cw 4 40.k even 4 1
1600.4.a.cx 4 40.i odd 4 1
1600.4.a.cx 4 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 \(S_{4}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{4} + 102T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{11}^{4} - 5982T_{11}^{2} + 6762845 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 102 T^{2} + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 632 T^{2} + 58320)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 5982 T^{2} + 6762845)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 5840 T^{2} + 5161984)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 5642 T^{2} + 5621641)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 4390 T^{2} + 4753125)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 46392 T^{2} + 523059920)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 36 T - 23040)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 80312 T^{2} + 1457948880)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 154376 T^{2} + 927811600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 362 T - 8775)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 81808 T^{2} + 1179648000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 152912 T^{2} + 5516513280)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 49284)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 578768 T^{2} + 83483873280)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 888 T + 194540)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 557582 T^{2} + 75081483405)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 428432 T^{2} + 7787520)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1640938 T^{2} + 462425840361)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 1189848 T^{2} + 37300611920)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1939422 T^{2} + 842120280125)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 450 T - 14275)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 550952 T^{2} + 59401388176)^{2} \) Copy content Toggle raw display
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