# Properties

 Label 448.4.i.e Level $448$ Weight $4$ Character orbit 448.i Analytic conductor $26.433$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 448.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.4328556826$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( 5 - 5 \zeta_{6} ) q^{3} -9 \zeta_{6} q^{5} + ( 21 - 14 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 5 - 5 \zeta_{6} ) q^{3} -9 \zeta_{6} q^{5} + ( 21 - 14 \zeta_{6} ) q^{7} + 2 \zeta_{6} q^{9} + ( 57 - 57 \zeta_{6} ) q^{11} + 70 q^{13} -45 q^{15} + ( -51 + 51 \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} + ( 35 - 105 \zeta_{6} ) q^{21} + 69 \zeta_{6} q^{23} + ( 44 - 44 \zeta_{6} ) q^{25} + 145 q^{27} -114 q^{29} + ( 23 - 23 \zeta_{6} ) q^{31} -285 \zeta_{6} q^{33} + ( -126 - 63 \zeta_{6} ) q^{35} -253 \zeta_{6} q^{37} + ( 350 - 350 \zeta_{6} ) q^{39} -42 q^{41} -124 q^{43} + ( 18 - 18 \zeta_{6} ) q^{45} + 201 \zeta_{6} q^{47} + ( 245 - 392 \zeta_{6} ) q^{49} + 255 \zeta_{6} q^{51} + ( -393 + 393 \zeta_{6} ) q^{53} -513 q^{55} -25 q^{57} + ( -219 + 219 \zeta_{6} ) q^{59} -709 \zeta_{6} q^{61} + ( 28 + 14 \zeta_{6} ) q^{63} -630 \zeta_{6} q^{65} + ( -419 + 419 \zeta_{6} ) q^{67} + 345 q^{69} + 96 q^{71} + ( 313 - 313 \zeta_{6} ) q^{73} -220 \zeta_{6} q^{75} + ( 399 - 1197 \zeta_{6} ) q^{77} + 461 \zeta_{6} q^{79} + ( 671 - 671 \zeta_{6} ) q^{81} -588 q^{83} + 459 q^{85} + ( -570 + 570 \zeta_{6} ) q^{87} + 1017 \zeta_{6} q^{89} + ( 1470 - 980 \zeta_{6} ) q^{91} -115 \zeta_{6} q^{93} + ( -45 + 45 \zeta_{6} ) q^{95} -1834 q^{97} + 114 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 5q^{3} - 9q^{5} + 28q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 5q^{3} - 9q^{5} + 28q^{7} + 2q^{9} + 57q^{11} + 140q^{13} - 90q^{15} - 51q^{17} - 5q^{19} - 35q^{21} + 69q^{23} + 44q^{25} + 290q^{27} - 228q^{29} + 23q^{31} - 285q^{33} - 315q^{35} - 253q^{37} + 350q^{39} - 84q^{41} - 248q^{43} + 18q^{45} + 201q^{47} + 98q^{49} + 255q^{51} - 393q^{53} - 1026q^{55} - 50q^{57} - 219q^{59} - 709q^{61} + 70q^{63} - 630q^{65} - 419q^{67} + 690q^{69} + 192q^{71} + 313q^{73} - 220q^{75} - 399q^{77} + 461q^{79} + 671q^{81} - 1176q^{83} + 918q^{85} - 570q^{87} + 1017q^{89} + 1960q^{91} - 115q^{93} - 45q^{95} - 3668q^{97} + 228q^{99} + O(q^{100})$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 2.50000 + 4.33013i 0 −4.50000 + 7.79423i 0 14.0000 + 12.1244i 0 1.00000 1.73205i 0
193.1 0 2.50000 4.33013i 0 −4.50000 7.79423i 0 14.0000 12.1244i 0 1.00000 + 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.e 2
4.b odd 2 1 448.4.i.b 2
7.c even 3 1 inner 448.4.i.e 2
8.b even 2 1 112.4.i.a 2
8.d odd 2 1 14.4.c.a 2
24.f even 2 1 126.4.g.d 2
28.g odd 6 1 448.4.i.b 2
40.e odd 2 1 350.4.e.e 2
40.k even 4 2 350.4.j.b 4
56.e even 2 1 98.4.c.a 2
56.j odd 6 1 784.4.a.c 1
56.k odd 6 1 14.4.c.a 2
56.k odd 6 1 98.4.a.d 1
56.m even 6 1 98.4.a.f 1
56.m even 6 1 98.4.c.a 2
56.p even 6 1 112.4.i.a 2
56.p even 6 1 784.4.a.p 1
168.e odd 2 1 882.4.g.u 2
168.v even 6 1 126.4.g.d 2
168.v even 6 1 882.4.a.f 1
168.be odd 6 1 882.4.a.c 1
168.be odd 6 1 882.4.g.u 2
280.ba even 6 1 2450.4.a.d 1
280.bi odd 6 1 350.4.e.e 2
280.bi odd 6 1 2450.4.a.q 1
280.br even 12 2 350.4.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 8.d odd 2 1
14.4.c.a 2 56.k odd 6 1
98.4.a.d 1 56.k odd 6 1
98.4.a.f 1 56.m even 6 1
98.4.c.a 2 56.e even 2 1
98.4.c.a 2 56.m even 6 1
112.4.i.a 2 8.b even 2 1
112.4.i.a 2 56.p even 6 1
126.4.g.d 2 24.f even 2 1
126.4.g.d 2 168.v even 6 1
350.4.e.e 2 40.e odd 2 1
350.4.e.e 2 280.bi odd 6 1
350.4.j.b 4 40.k even 4 2
350.4.j.b 4 280.br even 12 2
448.4.i.b 2 4.b odd 2 1
448.4.i.b 2 28.g odd 6 1
448.4.i.e 2 1.a even 1 1 trivial
448.4.i.e 2 7.c even 3 1 inner
784.4.a.c 1 56.j odd 6 1
784.4.a.p 1 56.p even 6 1
882.4.a.c 1 168.be odd 6 1
882.4.a.f 1 168.v even 6 1
882.4.g.u 2 168.e odd 2 1
882.4.g.u 2 168.be odd 6 1
2450.4.a.d 1 280.ba even 6 1
2450.4.a.q 1 280.bi odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} - 5 T_{3} + 25$$ $$T_{11}^{2} - 57 T_{11} + 3249$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$25 - 5 T + T^{2}$$
$5$ $$81 + 9 T + T^{2}$$
$7$ $$343 - 28 T + T^{2}$$
$11$ $$3249 - 57 T + T^{2}$$
$13$ $$( -70 + T )^{2}$$
$17$ $$2601 + 51 T + T^{2}$$
$19$ $$25 + 5 T + T^{2}$$
$23$ $$4761 - 69 T + T^{2}$$
$29$ $$( 114 + T )^{2}$$
$31$ $$529 - 23 T + T^{2}$$
$37$ $$64009 + 253 T + T^{2}$$
$41$ $$( 42 + T )^{2}$$
$43$ $$( 124 + T )^{2}$$
$47$ $$40401 - 201 T + T^{2}$$
$53$ $$154449 + 393 T + T^{2}$$
$59$ $$47961 + 219 T + T^{2}$$
$61$ $$502681 + 709 T + T^{2}$$
$67$ $$175561 + 419 T + T^{2}$$
$71$ $$( -96 + T )^{2}$$
$73$ $$97969 - 313 T + T^{2}$$
$79$ $$212521 - 461 T + T^{2}$$
$83$ $$( 588 + T )^{2}$$
$89$ $$1034289 - 1017 T + T^{2}$$
$97$ $$( 1834 + T )^{2}$$