Properties

Label 448.2.p.d
Level $448$
Weight $2$
Character orbit 448.p
Analytic conductor $3.577$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} + 1) q^{5} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} + 1) q^{5} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + (\zeta_{12}^{3} - \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{13} - 3 \zeta_{12}^{3} q^{15} + (\zeta_{12}^{2} - 2) q^{17} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{19} + (5 \zeta_{12}^{2} - 4) q^{21} + \zeta_{12} q^{23} - 2 \zeta_{12}^{2} q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} - 4 q^{29} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{31} + (\zeta_{12}^{2} + 1) q^{33} + ( - 4 \zeta_{12}^{3} - \zeta_{12}) q^{35} + ( - 3 \zeta_{12}^{2} + 3) q^{37} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{39} + ( - 4 \zeta_{12}^{2} + 2) q^{41} + 2 \zeta_{12}^{3} q^{43} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{47} + (8 \zeta_{12}^{2} - 5) q^{49} + 3 \zeta_{12} q^{51} - \zeta_{12}^{2} q^{53} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{55} - 9 q^{57} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{59} + (3 \zeta_{12}^{2} + 3) q^{61} + ( - 6 \zeta_{12}^{2} + 6) q^{65} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{67} + ( - 2 \zeta_{12}^{2} + 1) q^{69} + 14 \zeta_{12}^{3} q^{71} + ( - 5 \zeta_{12}^{2} + 10) q^{73} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{75} + (\zeta_{12}^{2} + 2) q^{77} + 9 \zeta_{12} q^{79} + 9 \zeta_{12}^{2} q^{81} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12}) q^{83} - 3 q^{85} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{87} + (9 \zeta_{12}^{2} + 9) q^{89} + (10 \zeta_{12}^{3} - 8 \zeta_{12}) q^{91} + (3 \zeta_{12}^{2} - 3) q^{93} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{95} + ( - 20 \zeta_{12}^{2} + 10) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{5} - 6 q^{17} - 6 q^{21} - 4 q^{25} - 16 q^{29} + 6 q^{33} + 6 q^{37} - 4 q^{49} - 2 q^{53} - 36 q^{57} + 18 q^{61} + 12 q^{65} + 30 q^{73} + 10 q^{77} + 18 q^{81} - 12 q^{85} + 54 q^{89} - 6 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{12}^{2}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 + 1.50000i 0 1.50000 0.866025i 0 −1.73205 + 2.00000i 0 0 0
255.2 0 0.866025 1.50000i 0 1.50000 0.866025i 0 1.73205 2.00000i 0 0 0
383.1 0 −0.866025 1.50000i 0 1.50000 + 0.866025i 0 −1.73205 2.00000i 0 0 0
383.2 0 0.866025 + 1.50000i 0 1.50000 + 0.866025i 0 1.73205 + 2.00000i 0 0 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
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.p.d 4
4.b odd 2 1 inner 448.2.p.d 4
7.c even 3 1 3136.2.f.e 4
7.d odd 6 1 inner 448.2.p.d 4
7.d odd 6 1 3136.2.f.e 4
8.b even 2 1 28.2.f.a 4
8.d odd 2 1 28.2.f.a 4
24.f even 2 1 252.2.bf.e 4
24.h odd 2 1 252.2.bf.e 4
28.f even 6 1 inner 448.2.p.d 4
28.f even 6 1 3136.2.f.e 4
28.g odd 6 1 3136.2.f.e 4
40.e odd 2 1 700.2.p.a 4
40.f even 2 1 700.2.p.a 4
40.i odd 4 1 700.2.t.a 4
40.i odd 4 1 700.2.t.b 4
40.k even 4 1 700.2.t.a 4
40.k even 4 1 700.2.t.b 4
56.e even 2 1 196.2.f.a 4
56.h odd 2 1 196.2.f.a 4
56.j odd 6 1 28.2.f.a 4
56.j odd 6 1 196.2.d.b 4
56.k odd 6 1 196.2.d.b 4
56.k odd 6 1 196.2.f.a 4
56.m even 6 1 28.2.f.a 4
56.m even 6 1 196.2.d.b 4
56.p even 6 1 196.2.d.b 4
56.p even 6 1 196.2.f.a 4
168.s odd 6 1 1764.2.b.a 4
168.v even 6 1 1764.2.b.a 4
168.ba even 6 1 252.2.bf.e 4
168.ba even 6 1 1764.2.b.a 4
168.be odd 6 1 252.2.bf.e 4
168.be odd 6 1 1764.2.b.a 4
280.ba even 6 1 700.2.p.a 4
280.bk odd 6 1 700.2.p.a 4
280.bp odd 12 1 700.2.t.a 4
280.bp odd 12 1 700.2.t.b 4
280.bv even 12 1 700.2.t.a 4
280.bv even 12 1 700.2.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 8.b even 2 1
28.2.f.a 4 8.d odd 2 1
28.2.f.a 4 56.j odd 6 1
28.2.f.a 4 56.m even 6 1
196.2.d.b 4 56.j odd 6 1
196.2.d.b 4 56.k odd 6 1
196.2.d.b 4 56.m even 6 1
196.2.d.b 4 56.p even 6 1
196.2.f.a 4 56.e even 2 1
196.2.f.a 4 56.h odd 2 1
196.2.f.a 4 56.k odd 6 1
196.2.f.a 4 56.p even 6 1
252.2.bf.e 4 24.f even 2 1
252.2.bf.e 4 24.h odd 2 1
252.2.bf.e 4 168.ba even 6 1
252.2.bf.e 4 168.be odd 6 1
448.2.p.d 4 1.a even 1 1 trivial
448.2.p.d 4 4.b odd 2 1 inner
448.2.p.d 4 7.d odd 6 1 inner
448.2.p.d 4 28.f even 6 1 inner
700.2.p.a 4 40.e odd 2 1
700.2.p.a 4 40.f even 2 1
700.2.p.a 4 280.ba even 6 1
700.2.p.a 4 280.bk odd 6 1
700.2.t.a 4 40.i odd 4 1
700.2.t.a 4 40.k even 4 1
700.2.t.a 4 280.bp odd 12 1
700.2.t.a 4 280.bv even 12 1
700.2.t.b 4 40.i odd 4 1
700.2.t.b 4 40.k even 4 1
700.2.t.b 4 280.bp odd 12 1
700.2.t.b 4 280.bv even 12 1
1764.2.b.a 4 168.s odd 6 1
1764.2.b.a 4 168.v even 6 1
1764.2.b.a 4 168.ba even 6 1
1764.2.b.a 4 168.be odd 6 1
3136.2.f.e 4 7.c even 3 1
3136.2.f.e 4 7.d odd 6 1
3136.2.f.e 4 28.f even 6 1
3136.2.f.e 4 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$37$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$53$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$61$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 27 T + 243)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
show more
show less