# 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: $$x^{2} - x + 1$$ 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.

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

## 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 $$S_{4}^{\mathrm{new}}(448, [\chi])$$:

 $$T_{3}^{2} - 5T_{3} + 25$$ T3^2 - 5*T3 + 25 $$T_{11}^{2} - 57T_{11} + 3249$$ T11^2 - 57*T11 + 3249

## Hecke characteristic polynomials

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