Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,3,Mod(97,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.h (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.98344960000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 17x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{3} q^{5} - \beta_{4} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{3} q^{5} - \beta_{4} q^{7} + q^{9} - \beta_{2} q^{11} + 3 \beta_{3} q^{13} + (\beta_{6} - \beta_{4}) q^{15} + \beta_{5} q^{17} + \beta_1 q^{19} + ( - \beta_{7} + 5 \beta_{3}) q^{21} + (2 \beta_{6} - 2 \beta_{4}) q^{23} - 11 q^{25} + 8 \beta_1 q^{27} - 2 \beta_{7} q^{29} + (5 \beta_{6} + 5 \beta_{4}) q^{31} - \beta_{5} q^{33} + (7 \beta_{2} - 7 \beta_1) q^{35} - 4 \beta_{7} q^{37} + (3 \beta_{6} - 3 \beta_{4}) q^{39} - 8 \beta_{5} q^{41} - 5 \beta_{2} q^{43} + \beta_{3} q^{45} + (11 \beta_{6} + 11 \beta_{4}) q^{47} + (7 \beta_{5} + 21) q^{49} + 10 \beta_{2} q^{51} - 6 \beta_{7} q^{53} + (\beta_{6} + \beta_{4}) q^{55} - 10 q^{57} - 19 \beta_1 q^{59} - 15 \beta_{3} q^{61} - \beta_{4} q^{63} + 42 q^{65} - 35 \beta_{2} q^{67} + 20 \beta_{3} q^{69} + (3 \beta_{6} - 3 \beta_{4}) q^{71} + 17 \beta_{5} q^{73} + 11 \beta_1 q^{75} + (\beta_{7} + 2 \beta_{3}) q^{77} + ( - \beta_{6} + \beta_{4}) q^{79} - 89 q^{81} + 39 \beta_1 q^{83} - 2 \beta_{7} q^{85} + ( - 10 \beta_{6} - 10 \beta_{4}) q^{87} - 9 \beta_{5} q^{89} + (21 \beta_{2} - 21 \beta_1) q^{91} + 10 \beta_{7} q^{93} + ( - \beta_{6} + \beta_{4}) q^{95} - 23 \beta_{5} q^{97} - \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 88 q^{25} + 168 q^{49} - 80 q^{57} + 336 q^{65} - 712 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 17x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{7} - 9\nu^{5} - 7\nu^{3} + 99\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} + 16\nu^{2} ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} - 9\nu^{5} - 41\nu^{3} + 45\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{7} + 3\nu^{6} - 9\nu^{5} + 41\nu^{3} - 78\nu^{2} + 45\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{7} + 18\nu^{5} - 14\nu^{3} - 198\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{7} - 3\nu^{6} - 9\nu^{5} + 41\nu^{3} + 78\nu^{2} + 45\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 4\nu^{4} - 34 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{3} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{4} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} + 2\beta_{5} + \beta_{4} - 2\beta_{3} + 4\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 34 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{6} - 5\beta_{5} - 11\beta_{4} - 22\beta_{3} + 10\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4\beta_{6} - 4\beta_{4} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{6} + 41\beta_{5} + 7\beta_{4} - 14\beta_{3} + 82\beta_1 ) / 8 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
1.72598 0.144845i
1.72598 + 0.144845i
−0.144845 1.72598i
−0.144845 + 1.72598i
0.144845 1.72598i
0.144845 + 1.72598i
−1.72598 0.144845i
−1.72598 + 0.144845i
0 −3.16228 0 −3.74166 0 5.91608 3.74166i 0 1.00000 0
97.2 0 −3.16228 0 −3.74166 0 5.91608 + 3.74166i 0 1.00000 0
97.3 0 −3.16228 0 3.74166 0 −5.91608 3.74166i 0 1.00000 0
97.4 0 −3.16228 0 3.74166 0 −5.91608 + 3.74166i 0 1.00000 0
97.5 0 3.16228 0 −3.74166 0 −5.91608 3.74166i 0 1.00000 0
97.6 0 3.16228 0 −3.74166 0 −5.91608 + 3.74166i 0 1.00000 0
97.7 0 3.16228 0 3.74166 0 5.91608 3.74166i 0 1.00000 0
97.8 0 3.16228 0 3.74166 0 5.91608 + 3.74166i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.h.a 8
4.b odd 2 1 inner 448.3.h.a 8
7.b odd 2 1 inner 448.3.h.a 8
8.b even 2 1 inner 448.3.h.a 8
8.d odd 2 1 inner 448.3.h.a 8
28.d even 2 1 inner 448.3.h.a 8
56.e even 2 1 inner 448.3.h.a 8
56.h odd 2 1 inner 448.3.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.3.h.a 8 1.a even 1 1 trivial
448.3.h.a 8 4.b odd 2 1 inner
448.3.h.a 8 7.b odd 2 1 inner
448.3.h.a 8 8.b even 2 1 inner
448.3.h.a 8 8.d odd 2 1 inner
448.3.h.a 8 28.d even 2 1 inner
448.3.h.a 8 56.e even 2 1 inner
448.3.h.a 8 56.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 10 \) 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
$3$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 14)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 42 T^{2} + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 126)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 40)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 560)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 560)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1400)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2240)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2560)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6776)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 5040)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3610)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 3150)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4900)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1260)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 11560)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 140)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 15210)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3240)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 21160)^{4} \) Copy content Toggle raw display
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