# Properties

 Label 448.4.i.b Level 448 Weight 4 Character orbit 448.i Analytic conductor 26.433 Analytic rank 1 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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.b 2
4.b odd 2 1 448.4.i.e 2
7.c even 3 1 inner 448.4.i.b 2
8.b even 2 1 14.4.c.a 2
8.d odd 2 1 112.4.i.a 2
24.h odd 2 1 126.4.g.d 2
28.g odd 6 1 448.4.i.e 2
40.f even 2 1 350.4.e.e 2
40.i odd 4 2 350.4.j.b 4
56.h odd 2 1 98.4.c.a 2
56.j odd 6 1 98.4.a.f 1
56.j odd 6 1 98.4.c.a 2
56.k odd 6 1 112.4.i.a 2
56.k odd 6 1 784.4.a.p 1
56.m even 6 1 784.4.a.c 1
56.p even 6 1 14.4.c.a 2
56.p even 6 1 98.4.a.d 1
168.i even 2 1 882.4.g.u 2
168.s odd 6 1 126.4.g.d 2
168.s odd 6 1 882.4.a.f 1
168.ba even 6 1 882.4.a.c 1
168.ba even 6 1 882.4.g.u 2
280.bf even 6 1 350.4.e.e 2
280.bf even 6 1 2450.4.a.q 1
280.bk odd 6 1 2450.4.a.d 1
280.bt odd 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.b even 2 1
14.4.c.a 2 56.p even 6 1
98.4.a.d 1 56.p even 6 1
98.4.a.f 1 56.j odd 6 1
98.4.c.a 2 56.h odd 2 1
98.4.c.a 2 56.j odd 6 1
112.4.i.a 2 8.d odd 2 1
112.4.i.a 2 56.k odd 6 1
126.4.g.d 2 24.h odd 2 1
126.4.g.d 2 168.s odd 6 1
350.4.e.e 2 40.f even 2 1
350.4.e.e 2 280.bf even 6 1
350.4.j.b 4 40.i odd 4 2
350.4.j.b 4 280.bt odd 12 2
448.4.i.b 2 1.a even 1 1 trivial
448.4.i.b 2 7.c even 3 1 inner
448.4.i.e 2 4.b odd 2 1
448.4.i.e 2 28.g odd 6 1
784.4.a.c 1 56.m even 6 1
784.4.a.p 1 56.k odd 6 1
882.4.a.c 1 168.ba even 6 1
882.4.a.f 1 168.s odd 6 1
882.4.g.u 2 168.i even 2 1
882.4.g.u 2 168.ba even 6 1
2450.4.a.d 1 280.bk odd 6 1
2450.4.a.q 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 $$S_{4}^{\mathrm{new}}(448, [\chi])$$:

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 5 T - 2 T^{2} + 135 T^{3} + 729 T^{4}$$
$5$ $$1 + 9 T - 44 T^{2} + 1125 T^{3} + 15625 T^{4}$$
$7$ $$1 + 28 T + 343 T^{2}$$
$11$ $$1 + 57 T + 1918 T^{2} + 75867 T^{3} + 1771561 T^{4}$$
$13$ $$( 1 - 70 T + 2197 T^{2} )^{2}$$
$17$ $$1 + 51 T - 2312 T^{2} + 250563 T^{3} + 24137569 T^{4}$$
$19$ $$1 - 5 T - 6834 T^{2} - 34295 T^{3} + 47045881 T^{4}$$
$23$ $$1 + 69 T - 7406 T^{2} + 839523 T^{3} + 148035889 T^{4}$$
$29$ $$( 1 + 114 T + 24389 T^{2} )^{2}$$
$31$ $$1 + 23 T - 29262 T^{2} + 685193 T^{3} + 887503681 T^{4}$$
$37$ $$1 + 253 T + 13356 T^{2} + 12815209 T^{3} + 2565726409 T^{4}$$
$41$ $$( 1 + 42 T + 68921 T^{2} )^{2}$$
$43$ $$( 1 - 124 T + 79507 T^{2} )^{2}$$
$47$ $$1 + 201 T - 63422 T^{2} + 20868423 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 393 T + 5572 T^{2} + 58508661 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 219 T - 157418 T^{2} - 44978001 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 709 T + 275700 T^{2} + 160929529 T^{3} + 51520374361 T^{4}$$
$67$ $$1 - 419 T - 125202 T^{2} - 126019697 T^{3} + 90458382169 T^{4}$$
$71$ $$( 1 + 96 T + 357911 T^{2} )^{2}$$
$73$ $$1 - 313 T - 291048 T^{2} - 121762321 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 461 T - 280518 T^{2} + 227290979 T^{3} + 243087455521 T^{4}$$
$83$ $$( 1 - 588 T + 571787 T^{2} )^{2}$$
$89$ $$1 - 1017 T + 329320 T^{2} - 716953473 T^{3} + 496981290961 T^{4}$$
$97$ $$( 1 + 1834 T + 912673 T^{2} )^{2}$$