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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,5,Mod(127,448)] 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("448.127"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 448.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.3097434616\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{3} + 5) q^{5} + \beta_{5} q^{7} + ( - 2 \beta_{7} - \beta_{6} - \beta_{3} + 9) q^{9} + (5 \beta_{5} - 3 \beta_{4} + \cdots - 7 \beta_1) q^{11} + (6 \beta_{6} - 3 \beta_{3} + 31) q^{13}+ \cdots + (573 \beta_{5} - 279 \beta_{4} + \cdots + 677 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{5} + 80 q^{9} + 236 q^{13} + 24 q^{17} + 196 q^{21} + 624 q^{25} - 3864 q^{29} - 4160 q^{33} + 2408 q^{37} - 1464 q^{41} + 116 q^{45} - 2744 q^{49} - 6960 q^{53} + 23480 q^{57} + 10836 q^{61}+ \cdots + 19576 q^{97}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( 781 \nu^{7} + 4986 \nu^{6} + 1637 \nu^{5} + 130030 \nu^{4} + 323565 \nu^{3} + 799502 \nu^{2} + \cdots + 3838620 ) / 528036 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 714 \nu^{7} - 1583 \nu^{6} + 10962 \nu^{5} + 19488 \nu^{4} + 6379 \nu^{3} + 158256 \nu^{2} + \cdots + 1300995 ) / 132009 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7301 \nu^{7} - 26972 \nu^{6} + 194921 \nu^{5} - 230402 \nu^{4} + 2088935 \nu^{3} - 3480926 \nu^{2} + \cdots - 2673660 ) / 1320090 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -133\nu^{7} + 686\nu^{6} - 3213\nu^{5} + 8666\nu^{4} - 27685\nu^{3} + 94458\nu^{2} - 87612\nu + 180180 ) / 14220 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2159 \nu^{7} - 3739 \nu^{6} + 33147 \nu^{5} + 58928 \nu^{4} + 347216 \nu^{3} + 478536 \nu^{2} + \cdots + 6791013 ) / 132009 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10285 \nu^{7} + 4992 \nu^{6} - 157905 \nu^{5} - 280720 \nu^{4} - 1970397 \nu^{3} + \cdots - 14242788 ) / 264018 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} - 6\beta_{5} - 7\beta_{4} + 6\beta_{3} + 14\beta_{2} + 7\beta _1 + 30 ) / 112 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 5\beta_{6} - 15\beta_{5} - 35\beta_{4} - \beta_{3} + 7\beta_{2} + 35\beta _1 - 264 ) / 56 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 17\beta_{6} - 37\beta_{3} - 402 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 30\beta_{7} + 101\beta_{6} + 303\beta_{5} + 518\beta_{4} - 37\beta_{3} - 175\beta_{2} - 350\beta _1 - 3342 ) / 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -106\beta_{7} - 341\beta_{6} + 1023\beta_{5} + 1862\beta_{4} + 505\beta_{3} - 1351\beta_{2} - 98\beta _1 + 7866 ) / 56 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -626\beta_{7} - 1793\beta_{6} + 913\beta_{3} + 49470 ) / 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 342 \beta_{7} - 935 \beta_{6} - 2805 \beta_{5} - 4826 \beta_{4} + 1075 \beta_{3} + 3085 \beta_{2} + \cdots + 21198 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
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 18.5203i 0 −147.339 0
127.2 0 − 6.40550i 0 −38.0135 0 − 18.5203i 0 39.9695 0
127.3 0 − 3.80490i 0 10.4721 0 − 18.5203i 0 66.5227 0
127.4 0 − 0.390999i 0 33.2658 0 18.5203i 0 80.8471 0
127.5 0 0.390999i 0 33.2658 0 − 18.5203i 0 80.8471 0
127.6 0 3.80490i 0 10.4721 0 18.5203i 0 66.5227 0
127.7 0 6.40550i 0 −38.0135 0 18.5203i 0 39.9695 0
127.8 0 15.1109i 0 12.2757 0 − 18.5203i 0 −147.339 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 127.8
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 448.5.d.d 8
4.b odd 2 1 inner 448.5.d.d 8
8.b even 2 1 112.5.d.b ✓ 8
8.d odd 2 1 112.5.d.b ✓ 8
24.f even 2 1 1008.5.m.e 8
24.h odd 2 1 1008.5.m.e 8
56.e even 2 1 784.5.d.j 8
56.h odd 2 1 784.5.d.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.5.d.b ✓ 8 8.b even 2 1
112.5.d.b ✓ 8 8.d odd 2 1
448.5.d.d 8 1.a even 1 1 trivial
448.5.d.d 8 4.b odd 2 1 inner
784.5.d.j 8 56.e even 2 1
784.5.d.j 8 56.h odd 2 1
1008.5.m.e 8 24.f even 2 1
1008.5.m.e 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 284T_{3}^{6} + 13312T_{3}^{4} + 137664T_{3}^{2} + 20736 \) acting on \(S_{5}^{\mathrm{new}}(448, [\chi])\). 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
$7$ \( (T^{2} + 343)^{4} \) 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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