Properties

Label 448.4.b.b
Level $448$
Weight $4$
Character orbit 448.b
Analytic conductor $26.433$
Analytic rank $1$
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(225,448)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.225");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.b (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: \(26.4328556826\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 4 \beta_1) q^{3} + (5 \beta_{3} - 2 \beta_1) q^{5} + 7 q^{7} + (5 \beta_{2} - 49) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 4 \beta_1) q^{3} + (5 \beta_{3} - 2 \beta_1) q^{5} + 7 q^{7} + (5 \beta_{2} - 49) q^{9} + (6 \beta_{3} + 16 \beta_1) q^{11} + ( - 6 \beta_{3} + 19 \beta_1) q^{13} + (41 \beta_{2} + 28) q^{15} + ( - 12 \beta_{2} - 70) q^{17} + ( - 30 \beta_{3} + 33 \beta_1) q^{19} + ( - 7 \beta_{3} - 28 \beta_1) q^{21} + (7 \beta_{2} - 144) q^{23} + ( - 51 \beta_{2} - 191) q^{25} + ( - 13 \beta_{3} + 133 \beta_1) q^{27} + ( - 17 \beta_{3} - 17 \beta_1) q^{29} + ( - 12 \beta_{2} - 140) q^{31} + ( - 6 \beta_{2} + 328) q^{33} + (35 \beta_{3} - 14 \beta_1) q^{35} + ( - 27 \beta_{3} - 163 \beta_1) q^{37} + ( - 99 \beta_{2} + 232) q^{39} + (52 \beta_{2} - 122) q^{41} + ( - 46 \beta_{3} - 44 \beta_1) q^{43} + ( - 180 \beta_{3} + 203 \beta_1) q^{45} + (86 \beta_{2} - 196) q^{47} + 49 q^{49} + (154 \beta_{3} + 172 \beta_1) q^{51} + (109 \beta_{3} + 75 \beta_1) q^{53} + ( - 190 \beta_{2} - 232) q^{55} + ( - 309 \beta_{2} + 168) q^{57} + ( - 30 \beta_{3} - 185 \beta_1) q^{59} + (33 \beta_{3} - 100 \beta_1) q^{61} + (35 \beta_{2} - 343) q^{63} + ( - 55 \beta_{2} + 512) q^{65} + (38 \beta_{3} + 298 \beta_1) q^{67} + (95 \beta_{3} + 639 \beta_1) q^{69} + (34 \beta_{2} - 124) q^{71} + ( - 90 \beta_{2} + 102) q^{73} + (548 \beta_{3} + 305 \beta_1) q^{75} + (42 \beta_{3} + 112 \beta_1) q^{77} + (142 \beta_{2} - 220) q^{79} + ( - 355 \beta_{2} + 649) q^{81} + (207 \beta_{3} + 42 \beta_1) q^{83} + ( - 506 \beta_{3} - 112 \beta_1) q^{85} + ( - 68 \beta_{2} - 476) q^{87} + ( - 44 \beta_{2} - 578) q^{89} + ( - 42 \beta_{3} + 133 \beta_1) q^{91} + (224 \beta_{3} + 452 \beta_1) q^{93} + (159 \beta_{2} + 2064) q^{95} + (266 \beta_{2} - 214) q^{97} + ( - 124 \beta_{3} - 934 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{7} - 196 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{7} - 196 q^{9} + 112 q^{15} - 280 q^{17} - 576 q^{23} - 764 q^{25} - 560 q^{31} + 1312 q^{33} + 928 q^{39} - 488 q^{41} - 784 q^{47} + 196 q^{49} - 928 q^{55} + 672 q^{57} - 1372 q^{63} + 2048 q^{65} - 496 q^{71} + 408 q^{73} - 880 q^{79} + 2596 q^{81} - 1904 q^{87} - 2312 q^{89} + 8256 q^{95} - 856 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{12}^{3} + 4\zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 2\beta_{2} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{3} + 3\beta _1 + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} - \beta_1 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
225.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 9.66025i 0 11.8038i 0 7.00000 0 −66.3205 0
225.2 0 7.66025i 0 22.1962i 0 7.00000 0 −31.6795 0
225.3 0 7.66025i 0 22.1962i 0 7.00000 0 −31.6795 0
225.4 0 9.66025i 0 11.8038i 0 7.00000 0 −66.3205 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.b.b yes 4
4.b odd 2 1 448.4.b.a 4
8.b even 2 1 inner 448.4.b.b yes 4
8.d odd 2 1 448.4.b.a 4
16.e even 4 1 1792.4.a.b 2
16.e even 4 1 1792.4.a.c 2
16.f odd 4 1 1792.4.a.a 2
16.f odd 4 1 1792.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.4.b.a 4 4.b odd 2 1
448.4.b.a 4 8.d odd 2 1
448.4.b.b yes 4 1.a even 1 1 trivial
448.4.b.b yes 4 8.b even 2 1 inner
1792.4.a.a 2 16.f odd 4 1
1792.4.a.b 2 16.e even 4 1
1792.4.a.c 2 16.e even 4 1
1792.4.a.d 2 16.f odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 152T_{3}^{2} + 5476 \) Copy content Toggle raw display
\( T_{23}^{2} + 288T_{23} + 20148 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 152T^{2} + 5476 \) Copy content Toggle raw display
$5$ \( T^{4} + 632 T^{2} + 68644 \) Copy content Toggle raw display
$7$ \( (T - 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 2912 T^{2} + \cdots + 2096704 \) Copy content Toggle raw display
$13$ \( T^{4} + 3752 T^{2} + 743044 \) Copy content Toggle raw display
$17$ \( (T^{2} + 140 T + 3172)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 30312 T^{2} + \cdots + 228070404 \) Copy content Toggle raw display
$23$ \( (T^{2} + 288 T + 20148)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 9248 T^{2} + \cdots + 5345344 \) Copy content Toggle raw display
$31$ \( (T^{2} + 280 T + 17872)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 230048 T^{2} + \cdots + 10317683776 \) Copy content Toggle raw display
$41$ \( (T^{2} + 244 T - 17564)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 66272 T^{2} + \cdots + 239135296 \) Copy content Toggle raw display
$47$ \( (T^{2} + 392 T - 50336)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 330144 T^{2} + \cdots + 1448868096 \) Copy content Toggle raw display
$59$ \( T^{4} + 295400 T^{2} + \cdots + 16809122500 \) Copy content Toggle raw display
$61$ \( T^{4} + 106136 T^{2} + \cdots + 682985956 \) Copy content Toggle raw display
$67$ \( T^{4} + 745088 T^{2} + \cdots + 92922548224 \) Copy content Toggle raw display
$71$ \( (T^{2} + 248 T + 1504)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 204 T - 86796)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 440 T - 193568)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1042488 T^{2} + \cdots + 22271980644 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1156 T + 310852)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 428 T - 803276)^{2} \) Copy content Toggle raw display
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