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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,5,Mod(687,784)] 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("784.687"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 784.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,18,0,0,0,-284,0,0,0,-260,0,0,0,-870,0,0,0,0,0,0,0,1552] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(81.0420510577\)
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: \( x^{6} - 2x^{5} - 8x^{4} + 36x^{3} + 1840x^{2} + 3856x + 17260 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 112)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{2} + 3) q^{5} + (\beta_{4} + \beta_{2} - 47) q^{9} + ( - \beta_{5} + 2 \beta_1) q^{11} + (\beta_{4} - \beta_{2} - 43) q^{13} + (2 \beta_{5} - \beta_{3} + 5 \beta_1) q^{15} + (3 \beta_{4} - \beta_{2} - 144) q^{17}+ \cdots + (13 \beta_{5} - 50 \beta_{3} + 487 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{5} - 284 q^{9} - 260 q^{13} - 870 q^{17} + 1552 q^{25} + 924 q^{29} + 1358 q^{33} - 126 q^{37} + 4236 q^{41} + 5756 q^{45} + 4722 q^{53} - 10766 q^{57} + 14290 q^{61} - 4668 q^{65} + 28826 q^{69}+ \cdots + 47692 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 8x^{4} + 36x^{3} + 1840x^{2} + 3856x + 17260 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -538\nu^{5} + 774\nu^{4} + 2831\nu^{3} - 71824\nu^{2} - 1232430\nu - 1857108 ) / 342074 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -51\nu^{5} + 784\nu^{4} - 1396\nu^{3} - 11970\nu^{2} - 49768\nu + 677980 ) / 20122 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4007\nu^{5} + 22932\nu^{4} - 91138\nu^{3} + 1284790\nu^{2} - 11375860\nu + 966540 ) / 342074 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 134\nu^{5} - 679\nu^{4} + 2287\nu^{3} + 6594\nu^{2} + 42384\nu + 13247 ) / 10061 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12857\nu^{5} - 21676\nu^{4} - 357273\nu^{3} + 1379446\nu^{2} + 21788726\nu + 41640332 ) / 342074 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} - 7\beta_{4} - 2\beta_{3} - 7\beta_{2} - 57\beta _1 + 63 ) / 196 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -9\beta_{5} - 7\beta_{4} + 31\beta_{3} - 56\beta_{2} - 415\beta _1 + 651 ) / 196 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -95\beta_{5} + 91\beta_{4} + 6\beta_{3} + 91\beta_{2} - 1691\beta _1 - 819 ) / 98 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -208\beta_{5} + 518\beta_{4} + 221\beta_{3} + 3115\beta_{2} - 7250\beta _1 - 120302 ) / 98 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 947\beta_{5} + 9709\beta_{4} + 571\beta_{3} + 16716\beta_{2} + 11349\beta _1 - 631281 ) / 98 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
687.1
5.99640 − 4.54448i
−0.830689 − 2.98273i
−4.16571 − 3.80145i
−4.16571 + 3.80145i
−0.830689 + 2.98273i
5.99640 + 4.54448i
0 − 16.9381i 0 −22.2326 0 0 0 −205.899 0
687.2 0 − 9.78472i 0 44.5038 0 0 0 −14.7407 0
687.3 0 − 1.53626i 0 −13.2712 0 0 0 78.6399 0
687.4 0 1.53626i 0 −13.2712 0 0 0 78.6399 0
687.5 0 9.78472i 0 44.5038 0 0 0 −14.7407 0
687.6 0 16.9381i 0 −22.2326 0 0 0 −205.899 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 687.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.5.d.h 6
4.b odd 2 1 inner 784.5.d.h 6
7.b odd 2 1 784.5.d.g 6
7.c even 3 2 112.5.r.c ✓ 12
28.d even 2 1 784.5.d.g 6
28.g odd 6 2 112.5.r.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.5.r.c ✓ 12 7.c even 3 2
112.5.r.c ✓ 12 28.g odd 6 2
784.5.d.g 6 7.b odd 2 1
784.5.d.g 6 28.d even 2 1
784.5.d.h 6 1.a even 1 1 trivial
784.5.d.h 6 4.b 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 \(S_{5}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{6} + 385T_{3}^{4} + 28371T_{3}^{2} + 64827 \) Copy content Toggle raw display
\( T_{5}^{3} - 9T_{5}^{2} - 1285T_{5} - 13131 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 385 T^{4} + \cdots + 64827 \) Copy content Toggle raw display
$5$ \( (T^{3} - 9 T^{2} + \cdots - 13131)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 3556645773723 \) Copy content Toggle raw display
$13$ \( (T^{3} + 130 T^{2} + \cdots - 1635208)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 435 T^{2} + \cdots - 33028983)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 476977211369883 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 29\!\cdots\!43 \) Copy content Toggle raw display
$29$ \( (T^{3} - 462 T^{2} + \cdots - 63592200)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 65\!\cdots\!83 \) Copy content Toggle raw display
$37$ \( (T^{3} + 63 T^{2} + \cdots - 41919059)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 2118 T^{2} + \cdots + 2256215256)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 44\!\cdots\!75 \) Copy content Toggle raw display
$53$ \( (T^{3} - 2361 T^{2} + \cdots + 17588618181)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 25\!\cdots\!75 \) Copy content Toggle raw display
$61$ \( (T^{3} - 7145 T^{2} + \cdots - 2580762843)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 10\!\cdots\!67 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + 27 T^{2} + \cdots + 60690476729)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 31\!\cdots\!47 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( (T^{3} - 7173 T^{2} + \cdots - 63195606279)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 23846 T^{2} + \cdots + 482665937048)^{2} \) Copy content Toggle raw display
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