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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [800,6,Mod(49,800)] 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("800.49"); 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, 1, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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: 8.0.12220785438976.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 116x^{4} + 320x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{25} \)
Twist minimal: no (minimal twist has level 8)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{7} - 12 \beta_1) q^{7} + ( - 2 \beta_{5} + 41) q^{9} + (8 \beta_{4} - 5 \beta_{3}) q^{11} + (3 \beta_{6} - 32 \beta_{2}) q^{13} + (2 \beta_{7} + 25 \beta_1) q^{17} + (24 \beta_{4} - 7 \beta_{3}) q^{19}+ \cdots + ( - 1728 \beta_{4} - 1821 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 328 q^{9} + 25856 q^{31} + 70208 q^{39} - 9136 q^{41} - 19656 q^{49} - 413376 q^{71} - 495744 q^{79} + 59368 q^{81} + 169264 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 116x^{4} + 320x^{2} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{7} - 7\nu^{5} + 164\nu^{3} - 832\nu ) / 2304 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 37\nu^{5} - 20\nu^{3} - 2240\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 11\nu^{4} - 100\nu^{2} + 608 ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 85\nu^{4} - 580\nu^{2} - 4960 ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 5\nu^{4} - 52\nu^{2} - 160 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} - 57\nu^{5} - 484\nu^{3} - 1472\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\nu^{7} + 97\nu^{5} + 1924\nu^{3} + 15808\nu ) / 288 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{6} - 4\beta_{2} + 8\beta_1 ) / 64 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 6\beta_{4} - 20\beta_{3} - 80 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} - 14\beta_{6} + 60\beta_{2} - 248\beta_1 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{5} - 18\beta_{4} + 132\beta_{3} - 3312 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -61\beta_{7} - 130\beta_{6} - 156\beta_{2} - 1352\beta_1 ) / 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -155\beta_{5} + 402\beta_{4} + 380\beta_{3} + 10480 ) / 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -83\beta_{7} + 610\beta_{6} - 2852\beta_{2} + 37064\beta_1 ) / 64 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.51888 + 2.38600i
−1.51888 − 2.38600i
2.10784 + 1.88600i
2.10784 − 1.88600i
−2.10784 + 1.88600i
−2.10784 − 1.88600i
1.51888 + 2.38600i
1.51888 − 2.38600i
0 −23.6095 0 0 0 − 160.704i 0 314.408 0
49.2 0 −23.6095 0 0 0 160.704i 0 314.408 0
49.3 0 −3.25452 0 0 0 − 112.704i 0 −232.408 0
49.4 0 −3.25452 0 0 0 112.704i 0 −232.408 0
49.5 0 3.25452 0 0 0 − 112.704i 0 −232.408 0
49.6 0 3.25452 0 0 0 112.704i 0 −232.408 0
49.7 0 23.6095 0 0 0 − 160.704i 0 314.408 0
49.8 0 23.6095 0 0 0 160.704i 0 314.408 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.f.a 8
4.b odd 2 1 200.6.f.a 8
5.b even 2 1 inner 800.6.f.a 8
5.c odd 4 1 32.6.b.a 4
5.c odd 4 1 800.6.d.a 4
8.b even 2 1 inner 800.6.f.a 8
8.d odd 2 1 200.6.f.a 8
15.e even 4 1 288.6.d.b 4
20.d odd 2 1 200.6.f.a 8
20.e even 4 1 8.6.b.a ✓ 4
20.e even 4 1 200.6.d.a 4
40.e odd 2 1 200.6.f.a 8
40.f even 2 1 inner 800.6.f.a 8
40.i odd 4 1 32.6.b.a 4
40.i odd 4 1 800.6.d.a 4
40.k even 4 1 8.6.b.a ✓ 4
40.k even 4 1 200.6.d.a 4
60.l odd 4 1 72.6.d.b 4
80.i odd 4 1 256.6.a.n 4
80.j even 4 1 256.6.a.k 4
80.s even 4 1 256.6.a.k 4
80.t odd 4 1 256.6.a.n 4
120.q odd 4 1 72.6.d.b 4
120.w even 4 1 288.6.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.b.a ✓ 4 20.e even 4 1
8.6.b.a ✓ 4 40.k even 4 1
32.6.b.a 4 5.c odd 4 1
32.6.b.a 4 40.i odd 4 1
72.6.d.b 4 60.l odd 4 1
72.6.d.b 4 120.q odd 4 1
200.6.d.a 4 20.e even 4 1
200.6.d.a 4 40.k even 4 1
200.6.f.a 8 4.b odd 2 1
200.6.f.a 8 8.d odd 2 1
200.6.f.a 8 20.d odd 2 1
200.6.f.a 8 40.e odd 2 1
256.6.a.k 4 80.j even 4 1
256.6.a.k 4 80.s even 4 1
256.6.a.n 4 80.i odd 4 1
256.6.a.n 4 80.t odd 4 1
288.6.d.b 4 15.e even 4 1
288.6.d.b 4 120.w even 4 1
800.6.d.a 4 5.c odd 4 1
800.6.d.a 4 40.i odd 4 1
800.6.f.a 8 1.a even 1 1 trivial
800.6.f.a 8 5.b even 2 1 inner
800.6.f.a 8 8.b even 2 1 inner
800.6.f.a 8 40.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 568T_{3}^{2} + 5904 \) acting on \(S_{6}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 568 T^{2} + 5904)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 38528 T^{2} + 328044544)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 347768 T^{2} + 5520765456)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 590944 T^{2} + 7999305984)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 154504 T^{2} + 5220351504)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 3109816 T^{2} + 120994976016)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 7936705990656)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 535633608132864)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6464 T + 7754752)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 306881230162176)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2284 T - 85109148)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 23\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 76\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 22\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 41\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 32\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 103344 T + 2609278272)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 25\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 123936 T + 3701816576)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 72\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 42316 T - 6875717724)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 61\!\cdots\!04)^{2} \) Copy content Toggle raw display
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