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Properties

Label 448.4.i.h
Level $448$
Weight $4$
Character orbit 448.i
Analytic conductor $26.433$
Analytic rank $0$
Dimension $4$
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] = [448,4,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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{37})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 10x^{2} + 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 28)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 7) q^{5}+ \cdots + (10 \beta_1 - 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 7) q^{5}+ \cdots + (70 \beta_{3} - 160) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{5} + 24 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{5} + 24 q^{7} - 20 q^{9} + 32 q^{11} + 56 q^{13} - 296 q^{15} + 154 q^{17} - 224 q^{19} - 370 q^{21} - 68 q^{23} - 144 q^{25} + 472 q^{29} + 196 q^{31} + 518 q^{33} - 400 q^{35} + 346 q^{37} - 296 q^{39} - 840 q^{41} + 688 q^{43} - 140 q^{45} - 84 q^{47} + 100 q^{49} - 296 q^{51} - 438 q^{53} + 1624 q^{55} + 444 q^{57} - 56 q^{59} + 98 q^{61} - 360 q^{63} + 396 q^{65} + 336 q^{67} + 1036 q^{69} + 1792 q^{71} - 966 q^{73} + 2072 q^{75} + 2398 q^{77} + 52 q^{79} + 1798 q^{81} - 784 q^{83} - 3340 q^{85} - 2072 q^{87} - 294 q^{89} + 1520 q^{91} + 518 q^{93} - 1124 q^{95} - 840 q^{97} - 640 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 10x^{2} + 9x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 10\nu^{2} - 10\nu + 81 ) / 90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 10\nu^{2} + 190\nu - 81 ) / 90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 14 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 19\beta _1 - 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 14 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.77069 + 3.06693i
−1.27069 − 2.20090i
1.77069 − 3.06693i
−1.27069 + 2.20090i
0 −3.04138 − 5.26783i 0 2.58276 − 4.47348i 0 18.1655 − 3.60745i 0 −5.00000 + 8.66025i 0
65.2 0 3.04138 + 5.26783i 0 −9.58276 + 16.5978i 0 −6.16553 + 17.4639i 0 −5.00000 + 8.66025i 0
193.1 0 −3.04138 + 5.26783i 0 2.58276 + 4.47348i 0 18.1655 + 3.60745i 0 −5.00000 − 8.66025i 0
193.2 0 3.04138 − 5.26783i 0 −9.58276 − 16.5978i 0 −6.16553 − 17.4639i 0 −5.00000 − 8.66025i 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.h 4
4.b odd 2 1 448.4.i.g 4
7.c even 3 1 inner 448.4.i.h 4
8.b even 2 1 28.4.e.a ✓ 4
8.d odd 2 1 112.4.i.d 4
24.h odd 2 1 252.4.k.c 4
28.g odd 6 1 448.4.i.g 4
40.f even 2 1 700.4.i.g 4
40.i odd 4 2 700.4.r.d 8
56.h odd 2 1 196.4.e.g 4
56.j odd 6 1 196.4.a.g 2
56.j odd 6 1 196.4.e.g 4
56.k odd 6 1 112.4.i.d 4
56.k odd 6 1 784.4.a.u 2
56.m even 6 1 784.4.a.ba 2
56.p even 6 1 28.4.e.a ✓ 4
56.p even 6 1 196.4.a.e 2
168.i even 2 1 1764.4.k.ba 4
168.s odd 6 1 252.4.k.c 4
168.s odd 6 1 1764.4.a.z 2
168.ba even 6 1 1764.4.a.n 2
168.ba even 6 1 1764.4.k.ba 4
280.bf even 6 1 700.4.i.g 4
280.bt odd 12 2 700.4.r.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.e.a ✓ 4 8.b even 2 1
28.4.e.a ✓ 4 56.p even 6 1
112.4.i.d 4 8.d odd 2 1
112.4.i.d 4 56.k odd 6 1
196.4.a.e 2 56.p even 6 1
196.4.a.g 2 56.j odd 6 1
196.4.e.g 4 56.h odd 2 1
196.4.e.g 4 56.j odd 6 1
252.4.k.c 4 24.h odd 2 1
252.4.k.c 4 168.s odd 6 1
448.4.i.g 4 4.b odd 2 1
448.4.i.g 4 28.g odd 6 1
448.4.i.h 4 1.a even 1 1 trivial
448.4.i.h 4 7.c even 3 1 inner
700.4.i.g 4 40.f even 2 1
700.4.i.g 4 280.bf even 6 1
700.4.r.d 8 40.i odd 4 2
700.4.r.d 8 280.bt odd 12 2
784.4.a.u 2 56.k odd 6 1
784.4.a.ba 2 56.m even 6 1
1764.4.a.n 2 168.ba even 6 1
1764.4.a.z 2 168.s odd 6 1
1764.4.k.ba 4 168.i even 2 1
1764.4.k.ba 4 168.ba even 6 1

Hecke kernels

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

\( T_{3}^{4} + 37T_{3}^{2} + 1369 \) Copy content Toggle raw display
\( T_{11}^{4} - 32T_{11}^{3} + 2581T_{11}^{2} + 49824T_{11} + 2424249 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 37T^{2} + 1369 \) Copy content Toggle raw display
$5$ \( T^{4} + 14 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$7$ \( T^{4} - 24 T^{3} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} - 32 T^{3} + \cdots + 2424249 \) Copy content Toggle raw display
$13$ \( (T^{2} - 28 T - 396)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 154 T^{3} + \cdots + 28483569 \) Copy content Toggle raw display
$19$ \( T^{4} + 224 T^{3} + \cdots + 149108521 \) Copy content Toggle raw display
$23$ \( T^{4} + 68 T^{3} + \cdots + 431649 \) Copy content Toggle raw display
$29$ \( (T^{2} - 236 T - 15084)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 196 T^{3} + \cdots + 60699681 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 1248844921 \) Copy content Toggle raw display
$41$ \( (T^{2} + 420 T + 38772)^{2} \) Copy content Toggle raw display
$43$ \( (T - 172)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 84 T^{3} + \cdots + 43046721 \) Copy content Toggle raw display
$53$ \( T^{4} + 438 T^{3} + \cdots + 299532249 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 2486917161 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 40084444521 \) Copy content Toggle raw display
$67$ \( T^{4} - 336 T^{3} + \cdots + 697540921 \) Copy content Toggle raw display
$71$ \( (T^{2} - 896 T + 84672)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 47737443121 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 427476669489 \) Copy content Toggle raw display
$83$ \( (T^{2} + 392 T - 248112)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 12452551281 \) Copy content Toggle raw display
$97$ \( (T^{2} + 420 T - 169612)^{2} \) Copy content Toggle raw display
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