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Properties

Label 400.8.c.u
Level $400$
Weight $8$
Character orbit 400.c
Analytic conductor $124.954$
Analytic rank $0$
Dimension $6$
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] = [400,8,Mod(49,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-4292] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(124.954010194\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 69x^{4} + 1164x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 200)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 32 \beta_1) q^{3} + ( - \beta_{4} + 2 \beta_{2} - 543 \beta_1) q^{7} + ( - \beta_{5} - 80 \beta_{3} - 689) q^{9} + (\beta_{5} - 133 \beta_{3} + 931) q^{11} + ( - \beta_{4} - 124 \beta_{2} + 1061 \beta_1) q^{13}+ \cdots + (8919 \beta_{5} + 235518 \beta_{3} + 17696637) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4292 q^{9} + 5318 q^{11} + 111498 q^{19} - 123124 q^{21} - 131056 q^{29} + 289196 q^{31} + 1600008 q^{39} + 599486 q^{41} - 4720974 q^{49} + 3724630 q^{51} + 1966736 q^{59} + 11420236 q^{61} - 10781396 q^{69}+ \cdots + 106633020 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 69x^{4} + 1164x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 65\nu^{3} + 1024\nu ) / 120 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\nu^{5} + 925\nu^{3} + 15872\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{4} + 40\nu^{2} - 697 ) / 15 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 279\nu^{5} + 20055\nu^{3} + 359936\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 112\nu^{4} + 4640\nu^{2} + 15883 ) / 15 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 24\beta_{2} + 99\beta_1 ) / 320 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 56\beta_{3} - 3661 ) / 160 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} + 29\beta_{2} - 294\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} + 116\beta_{3} + 6449 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 33\beta_{4} - 1117\beta_{2} + 17142\beta_1 ) / 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
6.29783i
− 5.41512i
− 0.117290i
0.117290i
5.41512i
− 6.29783i
0 − 89.5167i 0 0 0 − 1193.94i 0 −5826.24 0
49.2 0 − 21.9866i 0 0 0 1068.54i 0 1703.59 0
49.3 0 − 14.5033i 0 0 0 1504.61i 0 1976.65 0
49.4 0 14.5033i 0 0 0 − 1504.61i 0 1976.65 0
49.5 0 21.9866i 0 0 0 − 1068.54i 0 1703.59 0
49.6 0 89.5167i 0 0 0 1193.94i 0 −5826.24 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.8.c.u 6
4.b odd 2 1 200.8.c.j 6
5.b even 2 1 inner 400.8.c.u 6
5.c odd 4 1 400.8.a.bh 3
5.c odd 4 1 400.8.a.bi 3
20.d odd 2 1 200.8.c.j 6
20.e even 4 1 200.8.a.o ✓ 3
20.e even 4 1 200.8.a.p yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.8.a.o ✓ 3 20.e even 4 1
200.8.a.p yes 3 20.e even 4 1
200.8.c.j 6 4.b odd 2 1
200.8.c.j 6 20.d odd 2 1
400.8.a.bh 3 5.c odd 4 1
400.8.a.bi 3 5.c odd 4 1
400.8.c.u 6 1.a even 1 1 trivial
400.8.c.u 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 8707T_{3}^{4} + 5660931T_{3}^{2} + 814817025 \) acting on \(S_{8}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 8707 T^{4} + \cdots + 814817025 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( (T^{3} - 2659 T^{2} + \cdots + 33910231899)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots + 3636837484829)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots - 53\!\cdots\!52)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 63\!\cdots\!41)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 86\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{3} + \cdots + 47\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 14\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots + 49\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 12\!\cdots\!69 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 30\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 96\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 39\!\cdots\!91)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 73\!\cdots\!04 \) Copy content Toggle raw display
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