# Properties

 Label 448.4.i.d 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + 7 \zeta_{6} q^{5} + ( 1 + 18 \zeta_{6} ) q^{7} + 26 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + 7 \zeta_{6} q^{5} + ( 1 + 18 \zeta_{6} ) q^{7} + 26 \zeta_{6} q^{9} + ( -35 + 35 \zeta_{6} ) q^{11} -66 q^{13} + 7 q^{15} + ( -59 + 59 \zeta_{6} ) q^{17} -137 \zeta_{6} q^{19} + ( 19 - \zeta_{6} ) q^{21} -7 \zeta_{6} q^{23} + ( 76 - 76 \zeta_{6} ) q^{25} + 53 q^{27} -106 q^{29} + ( 75 - 75 \zeta_{6} ) q^{31} + 35 \zeta_{6} q^{33} + ( -126 + 133 \zeta_{6} ) q^{35} + 11 \zeta_{6} q^{37} + ( -66 + 66 \zeta_{6} ) q^{39} -498 q^{41} + 260 q^{43} + ( -182 + 182 \zeta_{6} ) q^{45} -171 \zeta_{6} q^{47} + ( -323 + 360 \zeta_{6} ) q^{49} + 59 \zeta_{6} q^{51} + ( -417 + 417 \zeta_{6} ) q^{53} -245 q^{55} -137 q^{57} + ( 17 - 17 \zeta_{6} ) q^{59} + 51 \zeta_{6} q^{61} + ( -468 + 494 \zeta_{6} ) q^{63} -462 \zeta_{6} q^{65} + ( -439 + 439 \zeta_{6} ) q^{67} -7 q^{69} + 784 q^{71} + ( -295 + 295 \zeta_{6} ) q^{73} -76 \zeta_{6} q^{75} + ( -665 + 35 \zeta_{6} ) q^{77} -495 \zeta_{6} q^{79} + ( -649 + 649 \zeta_{6} ) q^{81} + 932 q^{83} -413 q^{85} + ( -106 + 106 \zeta_{6} ) q^{87} + 873 \zeta_{6} q^{89} + ( -66 - 1188 \zeta_{6} ) q^{91} -75 \zeta_{6} q^{93} + ( 959 - 959 \zeta_{6} ) q^{95} -290 q^{97} -910 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + 7q^{5} + 20q^{7} + 26q^{9} + O(q^{10})$$ $$2q + q^{3} + 7q^{5} + 20q^{7} + 26q^{9} - 35q^{11} - 132q^{13} + 14q^{15} - 59q^{17} - 137q^{19} + 37q^{21} - 7q^{23} + 76q^{25} + 106q^{27} - 212q^{29} + 75q^{31} + 35q^{33} - 119q^{35} + 11q^{37} - 66q^{39} - 996q^{41} + 520q^{43} - 182q^{45} - 171q^{47} - 286q^{49} + 59q^{51} - 417q^{53} - 490q^{55} - 274q^{57} + 17q^{59} + 51q^{61} - 442q^{63} - 462q^{65} - 439q^{67} - 14q^{69} + 1568q^{71} - 295q^{73} - 76q^{75} - 1295q^{77} - 495q^{79} - 649q^{81} + 1864q^{83} - 826q^{85} - 106q^{87} + 873q^{89} - 1320q^{91} - 75q^{93} + 959q^{95} - 580q^{97} - 1820q^{99} + O(q^{100})$$

## 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 0.500000 + 0.866025i 0 3.50000 6.06218i 0 10.0000 15.5885i 0 13.0000 22.5167i 0
193.1 0 0.500000 0.866025i 0 3.50000 + 6.06218i 0 10.0000 + 15.5885i 0 13.0000 + 22.5167i 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.d 2
4.b odd 2 1 448.4.i.c 2
7.c even 3 1 inner 448.4.i.d 2
8.b even 2 1 112.4.i.b 2
8.d odd 2 1 14.4.c.b 2
24.f even 2 1 126.4.g.c 2
28.g odd 6 1 448.4.i.c 2
40.e odd 2 1 350.4.e.b 2
40.k even 4 2 350.4.j.d 4
56.e even 2 1 98.4.c.e 2
56.j odd 6 1 784.4.a.j 1
56.k odd 6 1 14.4.c.b 2
56.k odd 6 1 98.4.a.b 1
56.m even 6 1 98.4.a.c 1
56.m even 6 1 98.4.c.e 2
56.p even 6 1 112.4.i.b 2
56.p even 6 1 784.4.a.l 1
168.e odd 2 1 882.4.g.d 2
168.v even 6 1 126.4.g.c 2
168.v even 6 1 882.4.a.k 1
168.be odd 6 1 882.4.a.p 1
168.be odd 6 1 882.4.g.d 2
280.ba even 6 1 2450.4.a.bf 1
280.bi odd 6 1 350.4.e.b 2
280.bi odd 6 1 2450.4.a.bh 1
280.br even 12 2 350.4.j.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 8.d odd 2 1
14.4.c.b 2 56.k odd 6 1
98.4.a.b 1 56.k odd 6 1
98.4.a.c 1 56.m even 6 1
98.4.c.e 2 56.e even 2 1
98.4.c.e 2 56.m even 6 1
112.4.i.b 2 8.b even 2 1
112.4.i.b 2 56.p even 6 1
126.4.g.c 2 24.f even 2 1
126.4.g.c 2 168.v even 6 1
350.4.e.b 2 40.e odd 2 1
350.4.e.b 2 280.bi odd 6 1
350.4.j.d 4 40.k even 4 2
350.4.j.d 4 280.br even 12 2
448.4.i.c 2 4.b odd 2 1
448.4.i.c 2 28.g odd 6 1
448.4.i.d 2 1.a even 1 1 trivial
448.4.i.d 2 7.c even 3 1 inner
784.4.a.j 1 56.j odd 6 1
784.4.a.l 1 56.p even 6 1
882.4.a.k 1 168.v even 6 1
882.4.a.p 1 168.be odd 6 1
882.4.g.d 2 168.e odd 2 1
882.4.g.d 2 168.be odd 6 1
2450.4.a.bf 1 280.ba even 6 1
2450.4.a.bh 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 $$S_{4}^{\mathrm{new}}(448, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{11}^{2} + 35 T_{11} + 1225$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T - 26 T^{2} - 27 T^{3} + 729 T^{4}$$
$5$ $$1 - 7 T - 76 T^{2} - 875 T^{3} + 15625 T^{4}$$
$7$ $$1 - 20 T + 343 T^{2}$$
$11$ $$1 + 35 T - 106 T^{2} + 46585 T^{3} + 1771561 T^{4}$$
$13$ $$( 1 + 66 T + 2197 T^{2} )^{2}$$
$17$ $$1 + 59 T - 1432 T^{2} + 289867 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 137 T + 11910 T^{2} + 939683 T^{3} + 47045881 T^{4}$$
$23$ $$1 + 7 T - 12118 T^{2} + 85169 T^{3} + 148035889 T^{4}$$
$29$ $$( 1 + 106 T + 24389 T^{2} )^{2}$$
$31$ $$1 - 75 T - 24166 T^{2} - 2234325 T^{3} + 887503681 T^{4}$$
$37$ $$1 - 11 T - 50532 T^{2} - 557183 T^{3} + 2565726409 T^{4}$$
$41$ $$( 1 + 498 T + 68921 T^{2} )^{2}$$
$43$ $$( 1 - 260 T + 79507 T^{2} )^{2}$$
$47$ $$1 + 171 T - 74582 T^{2} + 17753733 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 417 T + 25012 T^{2} + 62081709 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 17 T - 205090 T^{2} - 3491443 T^{3} + 42180533641 T^{4}$$
$61$ $$1 - 51 T - 224380 T^{2} - 11576031 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 439 T - 108042 T^{2} + 132034957 T^{3} + 90458382169 T^{4}$$
$71$ $$( 1 - 784 T + 357911 T^{2} )^{2}$$
$73$ $$1 + 295 T - 301992 T^{2} + 114760015 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 495 T - 248014 T^{2} + 244054305 T^{3} + 243087455521 T^{4}$$
$83$ $$( 1 - 932 T + 571787 T^{2} )^{2}$$
$89$ $$1 - 873 T + 57160 T^{2} - 615437937 T^{3} + 496981290961 T^{4}$$
$97$ $$( 1 + 290 T + 912673 T^{2} )^{2}$$