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Properties

Label 448.4.i.c
Level $448$
Weight $4$
Character orbit 448.i
Analytic conductor $26.433$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,4,Mod(65,448)]
 
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, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.i (of order \(3\), 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: \(26.4328556826\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + 7 \zeta_{6} q^{5} + ( - 18 \zeta_{6} - 1) q^{7} + 26 \zeta_{6} q^{9} + ( - 35 \zeta_{6} + 35) q^{11} - 66 q^{13} - 7 q^{15} + (59 \zeta_{6} - 59) q^{17} + 137 \zeta_{6} q^{19}+ \cdots + 910 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 7 q^{5} - 20 q^{7} + 26 q^{9} + 35 q^{11} - 132 q^{13} - 14 q^{15} - 59 q^{17} + 137 q^{19} + 37 q^{21} + 7 q^{23} + 76 q^{25} - 106 q^{27} - 212 q^{29} - 75 q^{31} + 35 q^{33} + 119 q^{35}+ \cdots + 1820 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 −0.500000 − 0.866025i 0 3.50000 − 6.06218i 0 −10.0000 + 15.5885i 0 13.0000 − 22.5167i 0
193.1 0 −0.500000 + 0.866025i 0 3.50000 + 6.06218i 0 −10.0000 − 15.5885i 0 13.0000 + 22.5167i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.i.c 2
4.b odd 2 1 448.4.i.d 2
7.c even 3 1 inner 448.4.i.c 2
8.b even 2 1 14.4.c.b ✓ 2
8.d odd 2 1 112.4.i.b 2
24.h odd 2 1 126.4.g.c 2
28.g odd 6 1 448.4.i.d 2
40.f even 2 1 350.4.e.b 2
40.i odd 4 2 350.4.j.d 4
56.h odd 2 1 98.4.c.e 2
56.j odd 6 1 98.4.a.c 1
56.j odd 6 1 98.4.c.e 2
56.k odd 6 1 112.4.i.b 2
56.k odd 6 1 784.4.a.l 1
56.m even 6 1 784.4.a.j 1
56.p even 6 1 14.4.c.b ✓ 2
56.p even 6 1 98.4.a.b 1
168.i even 2 1 882.4.g.d 2
168.s odd 6 1 126.4.g.c 2
168.s odd 6 1 882.4.a.k 1
168.ba even 6 1 882.4.a.p 1
168.ba even 6 1 882.4.g.d 2
280.bf even 6 1 350.4.e.b 2
280.bf even 6 1 2450.4.a.bh 1
280.bk odd 6 1 2450.4.a.bf 1
280.bt odd 12 2 350.4.j.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b ✓ 2 8.b even 2 1
14.4.c.b ✓ 2 56.p even 6 1
98.4.a.b 1 56.p even 6 1
98.4.a.c 1 56.j odd 6 1
98.4.c.e 2 56.h odd 2 1
98.4.c.e 2 56.j odd 6 1
112.4.i.b 2 8.d odd 2 1
112.4.i.b 2 56.k odd 6 1
126.4.g.c 2 24.h odd 2 1
126.4.g.c 2 168.s odd 6 1
350.4.e.b 2 40.f even 2 1
350.4.e.b 2 280.bf even 6 1
350.4.j.d 4 40.i odd 4 2
350.4.j.d 4 280.bt odd 12 2
448.4.i.c 2 1.a even 1 1 trivial
448.4.i.c 2 7.c even 3 1 inner
448.4.i.d 2 4.b odd 2 1
448.4.i.d 2 28.g odd 6 1
784.4.a.j 1 56.m even 6 1
784.4.a.l 1 56.k odd 6 1
882.4.a.k 1 168.s odd 6 1
882.4.a.p 1 168.ba even 6 1
882.4.g.d 2 168.i even 2 1
882.4.g.d 2 168.ba even 6 1
2450.4.a.bf 1 280.bk odd 6 1
2450.4.a.bh 1 280.bf even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 35T_{11} + 1225 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} + 20T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 35T + 1225 \) Copy content Toggle raw display
$13$ \( (T + 66)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 59T + 3481 \) Copy content Toggle raw display
$19$ \( T^{2} - 137T + 18769 \) Copy content Toggle raw display
$23$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$29$ \( (T + 106)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 75T + 5625 \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$41$ \( (T + 498)^{2} \) Copy content Toggle raw display
$43$ \( (T + 260)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 171T + 29241 \) Copy content Toggle raw display
$53$ \( T^{2} + 417T + 173889 \) Copy content Toggle raw display
$59$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$61$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$67$ \( T^{2} - 439T + 192721 \) Copy content Toggle raw display
$71$ \( (T + 784)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 295T + 87025 \) Copy content Toggle raw display
$79$ \( T^{2} - 495T + 245025 \) Copy content Toggle raw display
$83$ \( (T + 932)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 873T + 762129 \) Copy content Toggle raw display
$97$ \( (T + 290)^{2} \) Copy content Toggle raw display
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