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Properties

Label 448.3.s.e
Level $448$
Weight $3$
Character orbit 448.s
Analytic conductor $12.207$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,3,Mod(129,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.129");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.2071158433\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.126303473664.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{7} q^{5} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 2) q^{7}+ \cdots + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} + \cdots + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{7} q^{5} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 2) q^{7}+ \cdots + ( - 7 \beta_{7} + 12 \beta_{5} + \cdots - 91) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} + 20 q^{9} - 4 q^{11} + 8 q^{15} - 24 q^{17} + 84 q^{19} + 52 q^{21} - 48 q^{23} + 24 q^{25} + 104 q^{29} + 156 q^{31} + 204 q^{33} - 156 q^{35} - 68 q^{37} - 52 q^{39} - 160 q^{43} + 276 q^{45} - 108 q^{47} - 80 q^{49} + 180 q^{51} + 28 q^{53} - 8 q^{57} - 120 q^{59} - 252 q^{61} + 76 q^{63} + 220 q^{65} + 56 q^{67} - 208 q^{71} - 156 q^{73} + 576 q^{75} + 240 q^{77} - 160 q^{79} + 96 q^{81} + 88 q^{85} + 348 q^{87} + 204 q^{89} - 120 q^{91} - 268 q^{93} + 216 q^{95} - 728 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -99\nu^{7} - 75\nu^{6} + 800\nu^{5} + 1818\nu^{4} - 4950\nu^{3} - 16800\nu^{2} + 10731\nu + 111475 ) / 21952 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -153\nu^{7} - 135\nu^{6} + 544\nu^{5} + 1262\nu^{4} - 3982\nu^{3} - 13216\nu^{2} + 12593\nu + 90895 ) / 21952 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -13\nu^{7} + 12\nu^{6} + 96\nu^{5} + 198\nu^{4} - 440\nu^{3} - 1568\nu^{2} + 1029\nu + 9604 ) / 1372 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -319\nu^{7} - 363\nu^{6} + 2080\nu^{5} + 6754\nu^{4} - 14102\nu^{3} - 47264\nu^{2} - 23961\nu + 320019 ) / 21952 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -373\nu^{7} - 311\nu^{6} + 3168\nu^{5} + 7542\nu^{4} - 16046\nu^{3} - 52192\nu^{2} + 24941\nu + 337855 ) / 21952 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -55\nu^{7} - 72\nu^{6} + 320\nu^{5} + 1234\nu^{4} - 2288\nu^{3} - 7616\nu^{2} + 2303\nu + 52136 ) / 2744 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -551\nu^{7} - 11\nu^{6} + 4000\nu^{5} + 8082\nu^{4} - 29398\nu^{3} - 61600\nu^{2} + 84623\nu + 462707 ) / 21952 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 2\beta_{4} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{5} - \beta_{3} - 11\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{7} + 2\beta_{6} + 8\beta_{5} - 8\beta_{4} + 9\beta_{3} + 8\beta_{2} + 28 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15\beta_{6} - 7\beta_{4} + 7\beta_{3} - 18\beta_{2} - 31\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -6\beta_{7} - 63\beta_{6} + 120\beta_{5} - 27\beta_{3} - 82\beta _1 + 82 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8\beta_{7} + 14\beta_{6} - 72\beta_{5} + 8\beta_{4} + 119\beta_{3} - 72\beta_{2} + 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29\beta_{6} - 86\beta_{4} - 252\beta_{3} - 952\beta_{2} + 1262\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1 - \beta_{1}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
2.18070 + 1.49818i
−1.43536 − 2.22255i
−2.38781 − 1.13946i
2.64247 + 0.131782i
2.18070 − 1.49818i
−1.43536 + 2.22255i
−2.38781 + 1.13946i
2.64247 − 0.131782i
0 −4.29109 − 2.47746i 0 2.44719 − 1.41289i 0 −5.77562 − 3.95502i 0 7.77562 + 13.4678i 0
129.2 0 −1.82280 − 1.05239i 0 −1.95651 + 1.12959i 0 4.28493 + 5.53528i 0 −2.28493 − 3.95762i 0
129.3 0 2.16977 + 1.25272i 0 −6.68983 + 3.86238i 0 3.36141 − 6.14011i 0 −1.36141 − 2.35803i 0
129.4 0 3.94412 + 2.27714i 0 6.19915 − 3.57908i 0 −3.87072 − 5.83245i 0 5.87072 + 10.1684i 0
257.1 0 −4.29109 + 2.47746i 0 2.44719 + 1.41289i 0 −5.77562 + 3.95502i 0 7.77562 − 13.4678i 0
257.2 0 −1.82280 + 1.05239i 0 −1.95651 − 1.12959i 0 4.28493 − 5.53528i 0 −2.28493 + 3.95762i 0
257.3 0 2.16977 − 1.25272i 0 −6.68983 − 3.86238i 0 3.36141 + 6.14011i 0 −1.36141 + 2.35803i 0
257.4 0 3.94412 − 2.27714i 0 6.19915 + 3.57908i 0 −3.87072 + 5.83245i 0 5.87072 − 10.1684i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.s.e 8
4.b odd 2 1 448.3.s.f 8
7.d odd 6 1 inner 448.3.s.e 8
8.b even 2 1 56.3.o.a ✓ 8
8.d odd 2 1 112.3.s.c 8
24.f even 2 1 1008.3.cg.o 8
24.h odd 2 1 504.3.by.b 8
28.f even 6 1 448.3.s.f 8
56.e even 2 1 784.3.s.i 8
56.h odd 2 1 392.3.o.c 8
56.j odd 6 1 56.3.o.a ✓ 8
56.j odd 6 1 392.3.c.c 8
56.k odd 6 1 784.3.c.h 8
56.k odd 6 1 784.3.s.i 8
56.m even 6 1 112.3.s.c 8
56.m even 6 1 784.3.c.h 8
56.p even 6 1 392.3.c.c 8
56.p even 6 1 392.3.o.c 8
168.s odd 6 1 3528.3.f.d 8
168.ba even 6 1 504.3.by.b 8
168.ba even 6 1 3528.3.f.d 8
168.be odd 6 1 1008.3.cg.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.o.a ✓ 8 8.b even 2 1
56.3.o.a ✓ 8 56.j odd 6 1
112.3.s.c 8 8.d odd 2 1
112.3.s.c 8 56.m even 6 1
392.3.c.c 8 56.j odd 6 1
392.3.c.c 8 56.p even 6 1
392.3.o.c 8 56.h odd 2 1
392.3.o.c 8 56.p even 6 1
448.3.s.e 8 1.a even 1 1 trivial
448.3.s.e 8 7.d odd 6 1 inner
448.3.s.f 8 4.b odd 2 1
448.3.s.f 8 28.f even 6 1
504.3.by.b 8 24.h odd 2 1
504.3.by.b 8 168.ba even 6 1
784.3.c.h 8 56.k odd 6 1
784.3.c.h 8 56.m even 6 1
784.3.s.i 8 56.e even 2 1
784.3.s.i 8 56.k odd 6 1
1008.3.cg.o 8 24.f even 2 1
1008.3.cg.o 8 168.be odd 6 1
3528.3.f.d 8 168.s odd 6 1
3528.3.f.d 8 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 28T_{3}^{6} + 665T_{3}^{4} - 336T_{3}^{3} - 3284T_{3}^{2} + 1428T_{3} + 14161 \) acting on \(S_{3}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 28 T^{6} + \cdots + 14161 \) Copy content Toggle raw display
$5$ \( T^{8} - 62 T^{6} + \cdots + 124609 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} + \cdots + 25921 \) Copy content Toggle raw display
$13$ \( T^{8} + 712 T^{6} + \cdots + 205520896 \) Copy content Toggle raw display
$17$ \( T^{8} + 24 T^{7} + \cdots + 711875761 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 1818937201 \) Copy content Toggle raw display
$23$ \( T^{8} + 48 T^{7} + \cdots + 38725729 \) Copy content Toggle raw display
$29$ \( (T^{4} - 52 T^{3} + \cdots - 1025504)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 222339597841 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 2625018876481 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 2179615227904 \) Copy content Toggle raw display
$43$ \( (T^{4} + 80 T^{3} + \cdots - 3066512)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 162997105441 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 22628069068609 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 6191643969 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 15423708853809 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 34801044376081 \) Copy content Toggle raw display
$71$ \( (T^{4} + 104 T^{3} + \cdots - 1612688)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 27500867992129 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 3075478849 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 546575022555136 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 16081439367889 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 7518651744256 \) Copy content Toggle raw display
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