# Properties

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

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} + 7 \zeta_{6} q^{5} + ( - 18 \zeta_{6} - 1) q^{7} + 26 \zeta_{6} q^{9}+O(q^{10})$$ q + (z - 1) * q^3 + 7*z * q^5 + (-18*z - 1) * q^7 + 26*z * q^9 $$q + (\zeta_{6} - 1) q^{3} + 7 \zeta_{6} q^{5} + ( - 18 \zeta_{6} - 1) q^{7} + 26 \zeta_{6} q^{9} + ( - 35 \zeta_{6} + 35) q^{11} - 66 q^{13} - 7 q^{15} + (59 \zeta_{6} - 59) q^{17} + 137 \zeta_{6} q^{19} + ( - \zeta_{6} + 19) q^{21} + 7 \zeta_{6} q^{23} + ( - 76 \zeta_{6} + 76) q^{25} - 53 q^{27} - 106 q^{29} + (75 \zeta_{6} - 75) q^{31} + 35 \zeta_{6} q^{33} + ( - 133 \zeta_{6} + 126) q^{35} + 11 \zeta_{6} q^{37} + ( - 66 \zeta_{6} + 66) q^{39} - 498 q^{41} - 260 q^{43} + (182 \zeta_{6} - 182) q^{45} + 171 \zeta_{6} q^{47} + (360 \zeta_{6} - 323) q^{49} - 59 \zeta_{6} q^{51} + (417 \zeta_{6} - 417) q^{53} + 245 q^{55} - 137 q^{57} + (17 \zeta_{6} - 17) q^{59} + 51 \zeta_{6} q^{61} + ( - 494 \zeta_{6} + 468) q^{63} - 462 \zeta_{6} q^{65} + ( - 439 \zeta_{6} + 439) q^{67} - 7 q^{69} - 784 q^{71} + (295 \zeta_{6} - 295) q^{73} + 76 \zeta_{6} q^{75} + (35 \zeta_{6} - 665) q^{77} + 495 \zeta_{6} q^{79} + (649 \zeta_{6} - 649) q^{81} - 932 q^{83} - 413 q^{85} + ( - 106 \zeta_{6} + 106) q^{87} + 873 \zeta_{6} q^{89} + (1188 \zeta_{6} + 66) q^{91} - 75 \zeta_{6} q^{93} + (959 \zeta_{6} - 959) q^{95} - 290 q^{97} + 910 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 + 7*z * q^5 + (-18*z - 1) * q^7 + 26*z * q^9 + (-35*z + 35) * q^11 - 66 * q^13 - 7 * q^15 + (59*z - 59) * q^17 + 137*z * q^19 + (-z + 19) * q^21 + 7*z * q^23 + (-76*z + 76) * q^25 - 53 * q^27 - 106 * q^29 + (75*z - 75) * q^31 + 35*z * q^33 + (-133*z + 126) * q^35 + 11*z * q^37 + (-66*z + 66) * q^39 - 498 * q^41 - 260 * q^43 + (182*z - 182) * q^45 + 171*z * q^47 + (360*z - 323) * q^49 - 59*z * q^51 + (417*z - 417) * q^53 + 245 * q^55 - 137 * q^57 + (17*z - 17) * q^59 + 51*z * q^61 + (-494*z + 468) * q^63 - 462*z * q^65 + (-439*z + 439) * q^67 - 7 * q^69 - 784 * q^71 + (295*z - 295) * q^73 + 76*z * q^75 + (35*z - 665) * q^77 + 495*z * q^79 + (649*z - 649) * q^81 - 932 * q^83 - 413 * q^85 + (-106*z + 106) * q^87 + 873*z * q^89 + (1188*z + 66) * q^91 - 75*z * q^93 + (959*z - 959) * q^95 - 290 * q^97 + 910 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 7 q^{5} - 20 q^{7} + 26 q^{9}+O(q^{10})$$ 2 * q - q^3 + 7 * q^5 - 20 * q^7 + 26 * q^9 $$2 q - q^{3} + 7 q^{5} - 20 q^{7} + 26 q^{9} + 35 q^{11} - 132 q^{13} - 14 q^{15} - 59 q^{17} + 137 q^{19} + 37 q^{21} + 7 q^{23} + 76 q^{25} - 106 q^{27} - 212 q^{29} - 75 q^{31} + 35 q^{33} + 119 q^{35} + 11 q^{37} + 66 q^{39} - 996 q^{41} - 520 q^{43} - 182 q^{45} + 171 q^{47} - 286 q^{49} - 59 q^{51} - 417 q^{53} + 490 q^{55} - 274 q^{57} - 17 q^{59} + 51 q^{61} + 442 q^{63} - 462 q^{65} + 439 q^{67} - 14 q^{69} - 1568 q^{71} - 295 q^{73} + 76 q^{75} - 1295 q^{77} + 495 q^{79} - 649 q^{81} - 1864 q^{83} - 826 q^{85} + 106 q^{87} + 873 q^{89} + 1320 q^{91} - 75 q^{93} - 959 q^{95} - 580 q^{97} + 1820 q^{99}+O(q^{100})$$ 2 * q - q^3 + 7 * q^5 - 20 * q^7 + 26 * q^9 + 35 * q^11 - 132 * q^13 - 14 * q^15 - 59 * q^17 + 137 * q^19 + 37 * q^21 + 7 * q^23 + 76 * q^25 - 106 * q^27 - 212 * q^29 - 75 * q^31 + 35 * q^33 + 119 * q^35 + 11 * q^37 + 66 * q^39 - 996 * q^41 - 520 * q^43 - 182 * q^45 + 171 * q^47 - 286 * q^49 - 59 * q^51 - 417 * q^53 + 490 * q^55 - 274 * q^57 - 17 * q^59 + 51 * q^61 + 442 * q^63 - 462 * q^65 + 439 * q^67 - 14 * q^69 - 1568 * q^71 - 295 * q^73 + 76 * q^75 - 1295 * q^77 + 495 * q^79 - 649 * q^81 - 1864 * q^83 - 826 * q^85 + 106 * q^87 + 873 * q^89 + 1320 * q^91 - 75 * q^93 - 959 * q^95 - 580 * q^97 + 1820 * 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 −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.c 2
4.b odd 2 1 448.4.i.d 2
7.c even 3 1 inner 448.4.i.c 2
8.b even 2 1 14.4.c.b 2
8.d odd 2 1 112.4.i.b 2
24.h odd 2 1 126.4.g.c 2
28.g odd 6 1 448.4.i.d 2
40.f even 2 1 350.4.e.b 2
40.i odd 4 2 350.4.j.d 4
56.h odd 2 1 98.4.c.e 2
56.j odd 6 1 98.4.a.c 1
56.j odd 6 1 98.4.c.e 2
56.k odd 6 1 112.4.i.b 2
56.k odd 6 1 784.4.a.l 1
56.m even 6 1 784.4.a.j 1
56.p even 6 1 14.4.c.b 2
56.p even 6 1 98.4.a.b 1
168.i even 2 1 882.4.g.d 2
168.s odd 6 1 126.4.g.c 2
168.s odd 6 1 882.4.a.k 1
168.ba even 6 1 882.4.a.p 1
168.ba even 6 1 882.4.g.d 2
280.bf even 6 1 350.4.e.b 2
280.bf even 6 1 2450.4.a.bh 1
280.bk odd 6 1 2450.4.a.bf 1
280.bt odd 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.b even 2 1
14.4.c.b 2 56.p even 6 1
98.4.a.b 1 56.p even 6 1
98.4.a.c 1 56.j odd 6 1
98.4.c.e 2 56.h odd 2 1
98.4.c.e 2 56.j odd 6 1
112.4.i.b 2 8.d odd 2 1
112.4.i.b 2 56.k odd 6 1
126.4.g.c 2 24.h odd 2 1
126.4.g.c 2 168.s odd 6 1
350.4.e.b 2 40.f even 2 1
350.4.e.b 2 280.bf even 6 1
350.4.j.d 4 40.i odd 4 2
350.4.j.d 4 280.bt odd 12 2
448.4.i.c 2 1.a even 1 1 trivial
448.4.i.c 2 7.c even 3 1 inner
448.4.i.d 2 4.b odd 2 1
448.4.i.d 2 28.g odd 6 1
784.4.a.j 1 56.m even 6 1
784.4.a.l 1 56.k odd 6 1
882.4.a.k 1 168.s odd 6 1
882.4.a.p 1 168.ba even 6 1
882.4.g.d 2 168.i even 2 1
882.4.g.d 2 168.ba even 6 1
2450.4.a.bf 1 280.bk odd 6 1
2450.4.a.bh 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 $$S_{4}^{\mathrm{new}}(448, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1 $$T_{11}^{2} - 35T_{11} + 1225$$ T11^2 - 35*T11 + 1225

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - 7T + 49$$
$7$ $$T^{2} + 20T + 343$$
$11$ $$T^{2} - 35T + 1225$$
$13$ $$(T + 66)^{2}$$
$17$ $$T^{2} + 59T + 3481$$
$19$ $$T^{2} - 137T + 18769$$
$23$ $$T^{2} - 7T + 49$$
$29$ $$(T + 106)^{2}$$
$31$ $$T^{2} + 75T + 5625$$
$37$ $$T^{2} - 11T + 121$$
$41$ $$(T + 498)^{2}$$
$43$ $$(T + 260)^{2}$$
$47$ $$T^{2} - 171T + 29241$$
$53$ $$T^{2} + 417T + 173889$$
$59$ $$T^{2} + 17T + 289$$
$61$ $$T^{2} - 51T + 2601$$
$67$ $$T^{2} - 439T + 192721$$
$71$ $$(T + 784)^{2}$$
$73$ $$T^{2} + 295T + 87025$$
$79$ $$T^{2} - 495T + 245025$$
$83$ $$(T + 932)^{2}$$
$89$ $$T^{2} - 873T + 762129$$
$97$ $$(T + 290)^{2}$$