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Properties

Label 800.5.h.l
Level $800$
Weight $5$
Character orbit 800.h
Analytic conductor $82.696$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [800,5,Mod(799,800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("800.799"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 800.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,144] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(82.6959704671\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 35x^{6} + 413x^{4} + 2015x^{2} + 3481 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{17}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{6} - \beta_{4} + 2 \beta_{2} + 18) q^{7} + ( - \beta_{6} + 4 \beta_{4} + \cdots + 73) q^{9} + ( - 2 \beta_{7} - 2 \beta_{5} + \cdots + 36 \beta_1) q^{11} + (3 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{13}+ \cdots + ( - 272 \beta_{7} - 38 \beta_{5} + \cdots + 6908 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 144 q^{7} + 584 q^{9} + 2384 q^{21} - 528 q^{23} + 4320 q^{27} + 2064 q^{29} - 5216 q^{41} + 8864 q^{43} - 13744 q^{47} - 4088 q^{49} - 5504 q^{61} + 3472 q^{63} - 8992 q^{67} + 25744 q^{69} + 41560 q^{81}+ \cdots + 50608 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 35x^{6} + 413x^{4} + 2015x^{2} + 3481 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -28\nu^{7} - 862\nu^{5} - 7788\nu^{3} - 21020\nu ) / 1003 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{6} + 98\nu^{4} + 1280\nu^{2} + 4202 ) / 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 188\nu^{7} + 4928\nu^{5} + 32804\nu^{3} + 79808\nu ) / 3009 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 56\nu^{6} + 1520\nu^{4} + 10952\nu^{2} + 21980 ) / 51 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -80\nu^{7} - 2210\nu^{5} - 16520\nu^{3} - 34940\nu ) / 177 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{6} + 58\nu^{4} + 480\nu^{2} + 1192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5056\nu^{7} - 145336\nu^{5} - 1188496\nu^{3} - 2947360\nu ) / 3009 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + 5\beta_{3} - 5\beta_1 ) / 40 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} - 5\beta_{4} - 13\beta_{2} - 350 ) / 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -15\beta_{7} + 14\beta_{5} - 110\beta_{3} + 430\beta_1 ) / 80 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -61\beta_{6} + 100\beta_{4} + 311\beta_{2} + 3990 ) / 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17\beta_{7} - 23\beta_{5} + 75\beta_{3} - 483\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1069\beta_{6} - 1700\beta_{4} - 5899\beta_{2} - 55550 ) / 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6295\beta_{7} + 8766\beta_{5} - 23090\beta_{3} + 182430\beta_1 ) / 80 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
− 2.72964i
2.72964i
2.49107i
− 2.49107i
2.11161i
− 2.11161i
4.10910i
− 4.10910i
0 −14.1546 0 0 0 −49.8485 0 119.354 0
799.2 0 −14.1546 0 0 0 −49.8485 0 119.354 0
799.3 0 −8.72821 0 0 0 47.9490 0 −4.81829 0
799.4 0 −8.72821 0 0 0 47.9490 0 −4.81829 0
799.5 0 5.21036 0 0 0 32.1829 0 −53.8521 0
799.6 0 5.21036 0 0 0 32.1829 0 −53.8521 0
799.7 0 17.6725 0 0 0 41.7166 0 231.317 0
799.8 0 17.6725 0 0 0 41.7166 0 231.317 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 799.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 800.5.h.l 8
4.b odd 2 1 800.5.h.k 8
5.b even 2 1 800.5.h.k 8
5.c odd 4 1 160.5.b.a ✓ 8
5.c odd 4 1 800.5.b.g 8
15.e even 4 1 1440.5.e.b 8
20.d odd 2 1 inner 800.5.h.l 8
20.e even 4 1 160.5.b.a ✓ 8
20.e even 4 1 800.5.b.g 8
40.i odd 4 1 320.5.b.c 8
40.k even 4 1 320.5.b.c 8
60.l odd 4 1 1440.5.e.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.5.b.a ✓ 8 5.c odd 4 1
160.5.b.a ✓ 8 20.e even 4 1
320.5.b.c 8 40.i odd 4 1
320.5.b.c 8 40.k even 4 1
800.5.b.g 8 5.c odd 4 1
800.5.b.g 8 20.e even 4 1
800.5.h.k 8 4.b odd 2 1
800.5.h.k 8 5.b even 2 1
800.5.h.l 8 1.a even 1 1 trivial
800.5.h.l 8 20.d odd 2 1 inner
1440.5.e.b 8 15.e even 4 1
1440.5.e.b 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{4} - 308T_{3}^{2} - 720T_{3} + 11376 \) Copy content Toggle raw display
\( T_{7}^{4} - 72T_{7}^{3} - 1188T_{7}^{2} + 179184T_{7} - 3208976 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 308 T^{2} + \cdots + 11376)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 72 T^{3} + \cdots - 3208976)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 7518651744256 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 584485938708736 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{4} + 264 T^{3} + \cdots + 748838000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 1032 T^{3} + \cdots - 131615691120)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 8307294475280)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 35428898740880)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots - 26408015014544)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 121504764169616)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 9254422867856)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 273133118890000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
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