# Properties

 Label 448.2.i.h Level $448$ Weight $2$ Character orbit 448.i Analytic conductor $3.577$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.57729801055$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + 3 \beta_{2} q^{5} + \beta_{3} q^{7} + 4 \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 + 3*b2 * q^5 + b3 * q^7 + 4*b2 * q^9 $$q + \beta_1 q^{3} + 3 \beta_{2} q^{5} + \beta_{3} q^{7} + 4 \beta_{2} q^{9} - \beta_1 q^{11} + 4 q^{13} + 3 \beta_{3} q^{15} + ( - \beta_{2} - 1) q^{17} + (3 \beta_{3} + 3 \beta_1) q^{19} + ( - 7 \beta_{2} - 7) q^{21} + ( - \beta_{3} - \beta_1) q^{23} + ( - 4 \beta_{2} - 4) q^{25} + \beta_{3} q^{27} + 4 q^{29} + \beta_1 q^{31} - 7 \beta_{2} q^{33} + ( - 3 \beta_{3} - 3 \beta_1) q^{35} + 5 \beta_{2} q^{37} + 4 \beta_1 q^{39} + 8 q^{41} + 4 \beta_{3} q^{43} + ( - 12 \beta_{2} - 12) q^{45} + (\beta_{3} + \beta_1) q^{47} + 7 q^{49} + ( - \beta_{3} - \beta_1) q^{51} + (7 \beta_{2} + 7) q^{53} - 3 \beta_{3} q^{55} - 21 q^{57} + \beta_1 q^{59} + 5 \beta_{2} q^{61} + ( - 4 \beta_{3} - 4 \beta_1) q^{63} + 12 \beta_{2} q^{65} + \beta_1 q^{67} + 7 q^{69} + (9 \beta_{2} + 9) q^{73} + ( - 4 \beta_{3} - 4 \beta_1) q^{75} + (7 \beta_{2} + 7) q^{77} + ( - \beta_{3} - \beta_1) q^{79} + (5 \beta_{2} + 5) q^{81} - 4 \beta_{3} q^{83} + 3 q^{85} + 4 \beta_1 q^{87} - 9 \beta_{2} q^{89} + 4 \beta_{3} q^{91} + 7 \beta_{2} q^{93} - 9 \beta_1 q^{95} - 8 q^{97} - 4 \beta_{3} q^{99}+O(q^{100})$$ q + b1 * q^3 + 3*b2 * q^5 + b3 * q^7 + 4*b2 * q^9 - b1 * q^11 + 4 * q^13 + 3*b3 * q^15 + (-b2 - 1) * q^17 + (3*b3 + 3*b1) * q^19 + (-7*b2 - 7) * q^21 + (-b3 - b1) * q^23 + (-4*b2 - 4) * q^25 + b3 * q^27 + 4 * q^29 + b1 * q^31 - 7*b2 * q^33 + (-3*b3 - 3*b1) * q^35 + 5*b2 * q^37 + 4*b1 * q^39 + 8 * q^41 + 4*b3 * q^43 + (-12*b2 - 12) * q^45 + (b3 + b1) * q^47 + 7 * q^49 + (-b3 - b1) * q^51 + (7*b2 + 7) * q^53 - 3*b3 * q^55 - 21 * q^57 + b1 * q^59 + 5*b2 * q^61 + (-4*b3 - 4*b1) * q^63 + 12*b2 * q^65 + b1 * q^67 + 7 * q^69 + (9*b2 + 9) * q^73 + (-4*b3 - 4*b1) * q^75 + (7*b2 + 7) * q^77 + (-b3 - b1) * q^79 + (5*b2 + 5) * q^81 - 4*b3 * q^83 + 3 * q^85 + 4*b1 * q^87 - 9*b2 * q^89 + 4*b3 * q^91 + 7*b2 * q^93 - 9*b1 * q^95 - 8 * q^97 - 4*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{5} - 8 q^{9}+O(q^{10})$$ 4 * q - 6 * q^5 - 8 * q^9 $$4 q - 6 q^{5} - 8 q^{9} + 16 q^{13} - 2 q^{17} - 14 q^{21} - 8 q^{25} + 16 q^{29} + 14 q^{33} - 10 q^{37} + 32 q^{41} - 24 q^{45} + 28 q^{49} + 14 q^{53} - 84 q^{57} - 10 q^{61} - 24 q^{65} + 28 q^{69} + 18 q^{73} + 14 q^{77} + 10 q^{81} + 12 q^{85} + 18 q^{89} - 14 q^{93} - 32 q^{97}+O(q^{100})$$ 4 * q - 6 * q^5 - 8 * q^9 + 16 * q^13 - 2 * q^17 - 14 * q^21 - 8 * q^25 + 16 * q^29 + 14 * q^33 - 10 * q^37 + 32 * q^41 - 24 * q^45 + 28 * q^49 + 14 * q^53 - 84 * q^57 - 10 * q^61 - 24 * q^65 + 28 * q^69 + 18 * q^73 + 14 * q^77 + 10 * q^81 + 12 * q^85 + 18 * q^89 - 14 * q^93 - 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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
 −1.32288 − 2.29129i 1.32288 + 2.29129i −1.32288 + 2.29129i 1.32288 − 2.29129i
0 −1.32288 2.29129i 0 −1.50000 + 2.59808i 0 2.64575 0 −2.00000 + 3.46410i 0
65.2 0 1.32288 + 2.29129i 0 −1.50000 + 2.59808i 0 −2.64575 0 −2.00000 + 3.46410i 0
193.1 0 −1.32288 + 2.29129i 0 −1.50000 2.59808i 0 2.64575 0 −2.00000 3.46410i 0
193.2 0 1.32288 2.29129i 0 −1.50000 2.59808i 0 −2.64575 0 −2.00000 3.46410i 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
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.i.h 4
4.b odd 2 1 inner 448.2.i.h 4
7.c even 3 1 inner 448.2.i.h 4
7.c even 3 1 3136.2.a.bv 2
7.d odd 6 1 3136.2.a.bg 2
8.b even 2 1 224.2.i.c 4
8.d odd 2 1 224.2.i.c 4
24.f even 2 1 2016.2.s.o 4
24.h odd 2 1 2016.2.s.o 4
28.f even 6 1 3136.2.a.bg 2
28.g odd 6 1 inner 448.2.i.h 4
28.g odd 6 1 3136.2.a.bv 2
56.e even 2 1 1568.2.i.p 4
56.h odd 2 1 1568.2.i.p 4
56.j odd 6 1 1568.2.a.t 2
56.j odd 6 1 1568.2.i.p 4
56.k odd 6 1 224.2.i.c 4
56.k odd 6 1 1568.2.a.m 2
56.m even 6 1 1568.2.a.t 2
56.m even 6 1 1568.2.i.p 4
56.p even 6 1 224.2.i.c 4
56.p even 6 1 1568.2.a.m 2
168.s odd 6 1 2016.2.s.o 4
168.v even 6 1 2016.2.s.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.c 4 8.b even 2 1
224.2.i.c 4 8.d odd 2 1
224.2.i.c 4 56.k odd 6 1
224.2.i.c 4 56.p even 6 1
448.2.i.h 4 1.a even 1 1 trivial
448.2.i.h 4 4.b odd 2 1 inner
448.2.i.h 4 7.c even 3 1 inner
448.2.i.h 4 28.g odd 6 1 inner
1568.2.a.m 2 56.k odd 6 1
1568.2.a.m 2 56.p even 6 1
1568.2.a.t 2 56.j odd 6 1
1568.2.a.t 2 56.m even 6 1
1568.2.i.p 4 56.e even 2 1
1568.2.i.p 4 56.h odd 2 1
1568.2.i.p 4 56.j odd 6 1
1568.2.i.p 4 56.m even 6 1
2016.2.s.o 4 24.f even 2 1
2016.2.s.o 4 24.h odd 2 1
2016.2.s.o 4 168.s odd 6 1
2016.2.s.o 4 168.v even 6 1
3136.2.a.bg 2 7.d odd 6 1
3136.2.a.bg 2 28.f even 6 1
3136.2.a.bv 2 7.c even 3 1
3136.2.a.bv 2 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(448, [\chi])$$:

 $$T_{3}^{4} + 7T_{3}^{2} + 49$$ T3^4 + 7*T3^2 + 49 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 7T^{2} + 49$$
$5$ $$(T^{2} + 3 T + 9)^{2}$$
$7$ $$(T^{2} - 7)^{2}$$
$11$ $$T^{4} + 7T^{2} + 49$$
$13$ $$(T - 4)^{4}$$
$17$ $$(T^{2} + T + 1)^{2}$$
$19$ $$T^{4} + 63T^{2} + 3969$$
$23$ $$T^{4} + 7T^{2} + 49$$
$29$ $$(T - 4)^{4}$$
$31$ $$T^{4} + 7T^{2} + 49$$
$37$ $$(T^{2} + 5 T + 25)^{2}$$
$41$ $$(T - 8)^{4}$$
$43$ $$(T^{2} - 112)^{2}$$
$47$ $$T^{4} + 7T^{2} + 49$$
$53$ $$(T^{2} - 7 T + 49)^{2}$$
$59$ $$T^{4} + 7T^{2} + 49$$
$61$ $$(T^{2} + 5 T + 25)^{2}$$
$67$ $$T^{4} + 7T^{2} + 49$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 9 T + 81)^{2}$$
$79$ $$T^{4} + 7T^{2} + 49$$
$83$ $$(T^{2} - 112)^{2}$$
$89$ $$(T^{2} - 9 T + 81)^{2}$$
$97$ $$(T + 8)^{4}$$