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Properties

Label 800.6.c.m
Level $800$
Weight $6$
Character orbit 800.c
Analytic conductor $128.307$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [800,6,Mod(449,800)] 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("800.449"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-448,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(128.307055850\)
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: \( x^{8} + 190x^{6} + 8881x^{4} + 20596x^{2} + 5184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 5^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + (\beta_{6} + 2 \beta_{5}) q^{7} + ( - \beta_{3} - 56) q^{9} + (\beta_{4} - 3 \beta_1) q^{11} + (3 \beta_{7} + 166 \beta_{2}) q^{13} + (7 \beta_{7} + 385 \beta_{2}) q^{17} + (6 \beta_{4} - 21 \beta_1) q^{19}+ \cdots + ( - 9 \beta_{4} + 1554 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 448 q^{9} - 5312 q^{21} + 11216 q^{29} + 9448 q^{41} + 8024 q^{49} - 65568 q^{61} - 66352 q^{69} - 281960 q^{81} + 149560 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 190x^{6} + 8881x^{4} + 20596x^{2} + 5184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -93\nu^{6} - 17359\nu^{4} - 784914\nu^{2} - 913948 ) / 31364 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -83\nu^{7} - 15914\nu^{5} - 757931\nu^{3} - 2323844\nu ) / 1129104 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -8\nu^{6} - 1156\nu^{4} - 2768\nu^{2} + 1450966 ) / 7841 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 73\nu^{6} + 14469\nu^{4} + 699584\nu^{2} + 816888 ) / 7841 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4117\nu^{7} + 779170\nu^{5} + 35979769\nu^{3} + 56636476\nu ) / 1129104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 617\nu^{7} + 116600\nu^{5} + 5365019\nu^{3} + 8443796\nu ) / 47046 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 190\nu^{5} - 8953\nu^{3} - 27436\nu ) / 72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 4\beta_{5} - 20\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} + 5\beta_{3} - 8\beta _1 - 950 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -15\beta_{7} - 98\beta_{6} + 352\beta_{5} + 2810\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 384\beta_{4} - 495\beta_{3} + 1376\beta _1 + 91690 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 475\beta_{7} + 1954\beta_{6} - 6992\beta_{5} - 87954\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -54796\beta_{4} + 50195\beta_{3} - 196064\beta _1 - 9293090 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -318395\beta_{7} - 1006342\beta_{6} + 3600688\beta_{5} + 58947090\beta_{2} ) / 20 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.46428i
− 0.535723i
8.63247i
10.6325i
− 8.63247i
− 10.6325i
− 1.46428i
0.535723i
0 − 22.0031i 0 0 0 − 122.733i 0 −241.138 0
449.2 0 − 22.0031i 0 0 0 − 122.733i 0 −241.138 0
449.3 0 − 10.6706i 0 0 0 128.626i 0 129.138 0
449.4 0 − 10.6706i 0 0 0 128.626i 0 129.138 0
449.5 0 10.6706i 0 0 0 − 128.626i 0 129.138 0
449.6 0 10.6706i 0 0 0 − 128.626i 0 129.138 0
449.7 0 22.0031i 0 0 0 122.733i 0 −241.138 0
449.8 0 22.0031i 0 0 0 122.733i 0 −241.138 0
\(n\): e.g. 2-40 or 80-90
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 800.6.c.m 8
4.b odd 2 1 inner 800.6.c.m 8
5.b even 2 1 inner 800.6.c.m 8
5.c odd 4 1 800.6.a.p ✓ 4
5.c odd 4 1 800.6.a.q yes 4
20.d odd 2 1 inner 800.6.c.m 8
20.e even 4 1 800.6.a.p ✓ 4
20.e even 4 1 800.6.a.q yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.6.a.p ✓ 4 5.c odd 4 1
800.6.a.p ✓ 4 20.e even 4 1
800.6.a.q yes 4 5.c odd 4 1
800.6.a.q yes 4 20.e even 4 1
800.6.c.m 8 1.a even 1 1 trivial
800.6.c.m 8 4.b odd 2 1 inner
800.6.c.m 8 5.b even 2 1 inner
800.6.c.m 8 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{4} + 598T_{3}^{2} + 55125 \) Copy content Toggle raw display
\( T_{11}^{4} - 34862T_{11}^{2} + 291466125 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 598 T^{2} + 55125)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 31608 T^{2} + 249218000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 34862 T^{2} + 291466125)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 672080 T^{2} + 78920541184)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 2344876627401)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 1329750 T^{2} + 398678203125)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 86015417378000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2804 T - 16166400)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 11506872 T^{2} + 662407202000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 827464 T^{2} + 119149232400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2362 T - 64963575)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 67947724800000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 45\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 36032164352000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 16392 T + 38348300)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 10\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 96\!\cdots\!61)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 53\!\cdots\!25)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 37390 T - 473977875)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 57\!\cdots\!36)^{2} \) Copy content Toggle raw display
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