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Properties

Label 800.3.g.f
Level $800$
Weight $3$
Character orbit 800.g
Analytic conductor $21.798$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,3,Mod(751,800)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.751");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 800.g (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: \(21.7984211488\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.189974000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 8x^{4} - 8x^{3} + 23x^{2} + 3x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 200)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{3} q^{7} + (\beta_{2} + 2 \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{3} q^{7} + (\beta_{2} + 2 \beta_1 - 2) q^{9} + ( - \beta_{2} - 3 \beta_1 - 4) q^{11} + ( - \beta_{4} - \beta_{3}) q^{13} + (3 \beta_{2} - 2 \beta_1 + 1) q^{17} + ( - 4 \beta_{2} + \beta_1) q^{19} + ( - \beta_{5} - \beta_{3}) q^{21} + ( - \beta_{5} + \beta_{4}) q^{23} + (\beta_{2} - 5 \beta_1 + 12) q^{27} + ( - \beta_{4} + 3 \beta_{3}) q^{29} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{3}) q^{31} + ( - 2 \beta_{2} - 12 \beta_1 - 19) q^{33} + (\beta_{5} - 3 \beta_{3}) q^{37} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{39} + ( - 10 \beta_{2} + 4 \beta_1 - 13) q^{41} + ( - 9 \beta_{2} + 2 \beta_1 + 12) q^{43} + (\beta_{5} + 3 \beta_{4} + \beta_{3}) q^{47} + ( - 6 \beta_{2} - 12 \beta_1 - 19) q^{49} + ( - 5 \beta_{2} + 3 \beta_1 - 20) q^{51} + ( - \beta_{5} - 3 \beta_{4} + 4 \beta_{3}) q^{53} + (5 \beta_{2} - 6 \beta_1 + 15) q^{57} + (3 \beta_{2} + 10 \beta_1 + 4) q^{59} + ( - 3 \beta_{5} - 2 \beta_{4} + 3 \beta_{3}) q^{61} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{63} + (13 \beta_{2} + 21 \beta_1 - 4) q^{67} + ( - \beta_{5} - 3 \beta_{4} - 12 \beta_{3}) q^{69} + ( - 3 \beta_{5} - \beta_{4} + 5 \beta_{3}) q^{71} + ( - \beta_{2} + 30 \beta_1 - 7) q^{73} + (3 \beta_{5} + 2 \beta_{4} + 9 \beta_{3}) q^{77} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_{3}) q^{79} + ( - 15 \beta_{2} - 14 \beta_1 - 19) q^{81} + ( - 9 \beta_{2} + 5 \beta_1 - 68) q^{83} + (3 \beta_{5} + \beta_{4} + 5 \beta_{3}) q^{87} + ( - 7 \beta_{2} + 10 \beta_1 + 13) q^{89} + ( - 22 \beta_{2} - 48) q^{91} + ( - 3 \beta_{5} + \beta_{4} - 6 \beta_{3}) q^{93} + ( - 4 \beta_{2} + 32 \beta_1 + 2) q^{97} + ( - \beta_{2} - 20 \beta_1 - 44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} - 8 q^{9} - 30 q^{11} + 2 q^{17} + 2 q^{19} + 62 q^{27} - 138 q^{33} - 70 q^{41} + 76 q^{43} - 138 q^{49} - 114 q^{51} + 78 q^{57} + 44 q^{59} + 18 q^{67} + 18 q^{73} - 142 q^{81} - 398 q^{83} + 98 q^{89} - 288 q^{91} + 76 q^{97} - 304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 8x^{4} - 8x^{3} + 23x^{2} + 3x + 40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 8\nu^{3} + 4\nu^{2} - 7\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 2\nu^{4} + 6\nu^{3} - 2\nu^{2} + 5\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 2\nu^{4} - 14\nu^{3} + 34\nu^{2} - 15\nu + 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 8\nu^{3} + 20\nu^{2} - 23\nu + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} - 8\nu^{4} + 20\nu^{3} - 24\nu^{2} + 97\nu - 8 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - 4\beta_{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 5\beta_{4} - 4\beta_{2} - 8\beta _1 - 16 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + 9\beta_{4} - 8\beta_{3} + 20\beta_{2} - 56 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} + 5\beta_{4} - 8\beta_{3} + 84\beta_{2} + 72\beta _1 - 224 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7\beta_{5} - 39\beta_{4} + 32\beta_{3} + 188\beta_{2} + 128\beta _1 - 568 ) / 16 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
−0.646201 − 1.13579i
−0.646201 + 1.13579i
1.94755 − 1.76129i
1.94755 + 1.76129i
0.198648 + 1.83244i
0.198648 − 1.83244i
0 −2.54751 0 0 0 − 8.05846i 0 −2.51021 0
751.2 0 −2.54751 0 0 0 8.05846i 0 −2.51021 0
751.3 0 −0.486535 0 0 0 − 5.23644i 0 −8.76328 0
751.4 0 −0.486535 0 0 0 5.23644i 0 −8.76328 0
751.5 0 4.03404 0 0 0 − 11.1194i 0 7.27349 0
751.6 0 4.03404 0 0 0 11.1194i 0 7.27349 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.3.g.f 6
4.b odd 2 1 200.3.g.f yes 6
5.b even 2 1 800.3.g.e 6
5.c odd 4 2 800.3.e.c 12
8.b even 2 1 200.3.g.f yes 6
8.d odd 2 1 inner 800.3.g.f 6
20.d odd 2 1 200.3.g.e ✓ 6
20.e even 4 2 200.3.e.c 12
40.e odd 2 1 800.3.g.e 6
40.f even 2 1 200.3.g.e ✓ 6
40.i odd 4 2 200.3.e.c 12
40.k even 4 2 800.3.e.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.3.e.c 12 20.e even 4 2
200.3.e.c 12 40.i odd 4 2
200.3.g.e ✓ 6 20.d odd 2 1
200.3.g.e ✓ 6 40.f even 2 1
200.3.g.f yes 6 4.b odd 2 1
200.3.g.f yes 6 8.b even 2 1
800.3.e.c 12 5.c odd 4 2
800.3.e.c 12 40.k even 4 2
800.3.g.e 6 5.b even 2 1
800.3.g.e 6 40.e odd 2 1
800.3.g.f 6 1.a even 1 1 trivial
800.3.g.f 6 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - T_{3}^{2} - 11T_{3} - 5 \) acting on \(S_{3}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - T^{2} - 11 T - 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 216 T^{4} + \cdots + 220160 \) Copy content Toggle raw display
$11$ \( (T^{3} + 15 T^{2} + \cdots - 53)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 616 T^{4} + \cdots + 3522560 \) Copy content Toggle raw display
$17$ \( (T^{3} - T^{2} - 321 T - 1055)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - T^{2} - 475 T - 373)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 3136 T^{4} + \cdots + 352256000 \) Copy content Toggle raw display
$29$ \( T^{6} + 2920 T^{4} + \cdots + 352256000 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 2201600000 \) Copy content Toggle raw display
$37$ \( T^{6} + 3336 T^{4} + \cdots + 220160 \) Copy content Toggle raw display
$41$ \( (T^{3} + 35 T^{2} + \cdots - 26447)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 38 T^{2} + \cdots + 20000)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 5176 T^{4} + \cdots + 465858560 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 4072079360 \) Copy content Toggle raw display
$59$ \( (T^{3} - 22 T^{2} + \cdots - 6752)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 176353664000 \) Copy content Toggle raw display
$67$ \( (T^{3} - 9 T^{2} + \cdots + 18155)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 110982656000 \) Copy content Toggle raw display
$73$ \( (T^{3} - 9 T^{2} + \cdots - 156935)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 10176896000 \) Copy content Toggle raw display
$83$ \( (T^{3} + 199 T^{2} + \cdots + 131155)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 49 T^{2} + \cdots + 97393)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 38 T^{2} + \cdots + 118520)^{2} \) Copy content Toggle raw display
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