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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 = [8,0,0,0,36,0,0,0,80,0,0,0,236,0,0,0,-24,0,0,0,0,0,0,0,624] 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: \(8\)
Coefficient field: 8.0.207528535809.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 21x^{6} - 2x^{5} + 265x^{4} - 66x^{3} + 1344x^{2} + 1080x + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + 4) q^{5} + ( - \beta_{7} + 10) q^{9} + ( - 5 \beta_{4} - 3 \beta_{3} - 7 \beta_1) q^{11} + (6 \beta_{5} - 3 \beta_{2} + 34) q^{13} + (\beta_{6} + 2 \beta_{4} + \cdots + 4 \beta_1) q^{15}+ \cdots + (48 \beta_{6} - 333 \beta_{4} + \cdots + 773 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{5} + 80 q^{9} + 236 q^{13} - 24 q^{17} + 624 q^{25} + 3864 q^{29} + 4160 q^{33} - 2408 q^{37} + 1464 q^{41} + 116 q^{45} + 6960 q^{53} + 23480 q^{57} + 10836 q^{61} - 15240 q^{65} + 8336 q^{69}+ \cdots - 19576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 21x^{6} - 2x^{5} + 265x^{4} - 66x^{3} + 1344x^{2} + 1080x + 3600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 104 \nu^{7} - 1002 \nu^{6} - 22304 \nu^{5} - 5427 \nu^{4} - 306485 \nu^{3} - 838401 \nu^{2} + \cdots - 2618910 ) / 660045 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 714 \nu^{7} - 1583 \nu^{6} + 10962 \nu^{5} + 19488 \nu^{4} + 6379 \nu^{3} + 158256 \nu^{2} + \cdots + 1433004 ) / 132009 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7301 \nu^{7} + 26972 \nu^{6} - 194921 \nu^{5} + 230402 \nu^{4} - 2088935 \nu^{3} + \cdots + 2673660 ) / 1320090 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21449 \nu^{7} + 76828 \nu^{6} - 453549 \nu^{5} + 719128 \nu^{4} - 5068505 \nu^{3} + \cdots + 10695240 ) / 1980135 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2159 \nu^{7} - 3739 \nu^{6} + 33147 \nu^{5} + 58928 \nu^{4} + 347216 \nu^{3} + 478536 \nu^{2} + \cdots + 6791013 ) / 132009 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 106342 \nu^{7} - 608624 \nu^{6} + 2937222 \nu^{5} - 8155184 \nu^{4} + 23741290 \nu^{3} + \cdots - 160942320 ) / 1980135 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7412 \nu^{7} - 330 \nu^{6} - 113796 \nu^{5} - 202304 \nu^{4} - 1616802 \nu^{3} - 1642848 \nu^{2} + \cdots - 6018771 ) / 132009 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} - 5\beta_{5} - 22\beta_{4} + 10\beta_{3} + 11\beta_{2} - 18\beta _1 + 48 ) / 224 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{6} + 9\beta_{5} - 20\beta_{4} + 74\beta_{3} - 3\beta_{2} - 66\beta _1 - 526 ) / 112 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 16\beta_{5} - 38\beta_{2} - 365 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15\beta_{7} - 16\beta_{6} + 86\beta_{5} + 223\beta_{4} - 550\beta_{3} - 52\beta_{2} + 318\beta _1 - 3305 ) / 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -53\beta_{7} + 41\beta_{6} - 288\beta_{5} + 1228\beta_{4} - 1780\beta_{3} + 558\beta_{2} + 180\beta _1 + 7361 ) / 56 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -313\beta_{7} - 1480\beta_{5} + 1226\beta_{2} + 48557 ) / 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -171\beta_{7} - 35\beta_{6} - 764\beta_{5} - 2980\beta_{4} + 4756\beta_{3} + 1246\beta_{2} - 372\beta _1 + 20123 ) / 8 \) 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
1.33021 + 2.30399i
−1.60961 − 2.78792i
−0.830211 + 1.43797i
2.10961 − 3.65395i
2.10961 + 3.65395i
−0.830211 − 1.43797i
−1.60961 + 2.78792i
1.33021 − 2.30399i
0 − 15.1109i 0 12.2757 0 0 0 −147.339 0
687.2 0 − 6.40550i 0 −38.0135 0 0 0 39.9695 0
687.3 0 − 3.80490i 0 10.4721 0 0 0 66.5227 0
687.4 0 − 0.390999i 0 33.2658 0 0 0 80.8471 0
687.5 0 0.390999i 0 33.2658 0 0 0 80.8471 0
687.6 0 3.80490i 0 10.4721 0 0 0 66.5227 0
687.7 0 6.40550i 0 −38.0135 0 0 0 39.9695 0
687.8 0 15.1109i 0 12.2757 0 0 0 −147.339 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 687.8
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.j 8
4.b odd 2 1 inner 784.5.d.j 8
7.b odd 2 1 112.5.d.b ✓ 8
21.c even 2 1 1008.5.m.e 8
28.d even 2 1 112.5.d.b ✓ 8
56.e even 2 1 448.5.d.d 8
56.h odd 2 1 448.5.d.d 8
84.h odd 2 1 1008.5.m.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.5.d.b ✓ 8 7.b odd 2 1
112.5.d.b ✓ 8 28.d even 2 1
448.5.d.d 8 56.e even 2 1
448.5.d.d 8 56.h odd 2 1
784.5.d.j 8 1.a even 1 1 trivial
784.5.d.j 8 4.b odd 2 1 inner
1008.5.m.e 8 21.c even 2 1
1008.5.m.e 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{8} + 284T_{3}^{6} + 13312T_{3}^{4} + 137664T_{3}^{2} + 20736 \) Copy content Toggle raw display
\( T_{5}^{4} - 18T_{5}^{3} - 1244T_{5}^{2} + 29376T_{5} - 162560 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 284 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$5$ \( (T^{4} - 18 T^{3} + \cdots - 162560)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 46\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} - 118 T^{3} + \cdots - 35561792)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 12 T^{3} + \cdots + 3807564208)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 86\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{4} - 1932 T^{3} + \cdots - 551782478288)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 2479913785424)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 732 T^{3} + \cdots + 501272338288)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 1241973141104)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 62\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 120277463740416)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 47152867548112)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 391350057618800)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 121887920039504)^{2} \) Copy content Toggle raw display
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