# Properties

 Label 448.2.p.c Level 448 Weight 2 Character orbit 448.p 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.p (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.57729801055$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{5} + \beta_{3} q^{7} + 4 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{2} ) q^{5} + \beta_{3} q^{7} + 4 \beta_{2} q^{9} + ( \beta_{1} + 2 \beta_{3} ) q^{11} + ( 2 + 4 \beta_{2} ) q^{13} + ( 2 \beta_{1} + \beta_{3} ) q^{15} + ( 3 - 3 \beta_{2} ) q^{17} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( -7 - 7 \beta_{2} ) q^{21} + ( \beta_{1} - \beta_{3} ) q^{23} + ( -2 - 2 \beta_{2} ) q^{25} + \beta_{3} q^{27} + \beta_{1} q^{31} + ( -14 - 7 \beta_{2} ) q^{33} + ( -\beta_{1} + \beta_{3} ) q^{35} -7 \beta_{2} q^{37} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{39} + ( 2 + 4 \beta_{2} ) q^{41} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -4 + 4 \beta_{2} ) q^{45} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{47} + 7 q^{49} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{51} + ( 3 + 3 \beta_{2} ) q^{53} + 3 \beta_{3} q^{55} + 7 q^{57} -3 \beta_{1} q^{59} + ( -2 - \beta_{2} ) q^{61} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{63} + 6 \beta_{2} q^{65} + ( -\beta_{1} - 2 \beta_{3} ) q^{67} + ( 7 + 14 \beta_{2} ) q^{69} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{71} + ( -3 + 3 \beta_{2} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{75} + ( 7 - 7 \beta_{2} ) q^{77} + ( \beta_{1} - \beta_{3} ) q^{79} + ( 5 + 5 \beta_{2} ) q^{81} + 9 q^{85} + ( 2 + \beta_{2} ) q^{89} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{91} + 7 \beta_{2} q^{93} + ( -\beta_{1} - 2 \beta_{3} ) q^{95} + ( 2 + 4 \beta_{2} ) q^{97} + ( -8 \beta_{1} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{5} - 8q^{9} + O(q^{10})$$ $$4q + 6q^{5} - 8q^{9} + 18q^{17} - 14q^{21} - 4q^{25} - 42q^{33} + 14q^{37} - 24q^{45} + 28q^{49} + 6q^{53} + 28q^{57} - 6q^{61} - 12q^{65} - 18q^{73} + 42q^{77} + 10q^{81} + 36q^{85} + 6q^{89} - 14q^{93} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
255.1
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
0 −1.32288 + 2.29129i 0 1.50000 0.866025i 0 2.64575 0 −2.00000 3.46410i 0
255.2 0 1.32288 2.29129i 0 1.50000 0.866025i 0 −2.64575 0 −2.00000 3.46410i 0
383.1 0 −1.32288 2.29129i 0 1.50000 + 0.866025i 0 2.64575 0 −2.00000 + 3.46410i 0
383.2 0 1.32288 + 2.29129i 0 1.50000 + 0.866025i 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.d odd 6 1 inner
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.p.c 4
4.b odd 2 1 inner 448.2.p.c 4
7.c even 3 1 3136.2.f.f 4
7.d odd 6 1 inner 448.2.p.c 4
7.d odd 6 1 3136.2.f.f 4
8.b even 2 1 112.2.p.c 4
8.d odd 2 1 112.2.p.c 4
24.f even 2 1 1008.2.cs.q 4
24.h odd 2 1 1008.2.cs.q 4
28.f even 6 1 inner 448.2.p.c 4
28.f even 6 1 3136.2.f.f 4
28.g odd 6 1 3136.2.f.f 4
56.e even 2 1 784.2.p.g 4
56.h odd 2 1 784.2.p.g 4
56.j odd 6 1 112.2.p.c 4
56.j odd 6 1 784.2.f.d 4
56.k odd 6 1 784.2.f.d 4
56.k odd 6 1 784.2.p.g 4
56.m even 6 1 112.2.p.c 4
56.m even 6 1 784.2.f.d 4
56.p even 6 1 784.2.f.d 4
56.p even 6 1 784.2.p.g 4
168.s odd 6 1 7056.2.b.s 4
168.v even 6 1 7056.2.b.s 4
168.ba even 6 1 1008.2.cs.q 4
168.ba even 6 1 7056.2.b.s 4
168.be odd 6 1 1008.2.cs.q 4
168.be odd 6 1 7056.2.b.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 8.b even 2 1
112.2.p.c 4 8.d odd 2 1
112.2.p.c 4 56.j odd 6 1
112.2.p.c 4 56.m even 6 1
448.2.p.c 4 1.a even 1 1 trivial
448.2.p.c 4 4.b odd 2 1 inner
448.2.p.c 4 7.d odd 6 1 inner
448.2.p.c 4 28.f even 6 1 inner
784.2.f.d 4 56.j odd 6 1
784.2.f.d 4 56.k odd 6 1
784.2.f.d 4 56.m even 6 1
784.2.f.d 4 56.p even 6 1
784.2.p.g 4 56.e even 2 1
784.2.p.g 4 56.h odd 2 1
784.2.p.g 4 56.k odd 6 1
784.2.p.g 4 56.p even 6 1
1008.2.cs.q 4 24.f even 2 1
1008.2.cs.q 4 24.h odd 2 1
1008.2.cs.q 4 168.ba even 6 1
1008.2.cs.q 4 168.be odd 6 1
3136.2.f.f 4 7.c even 3 1
3136.2.f.f 4 7.d odd 6 1
3136.2.f.f 4 28.f even 6 1
3136.2.f.f 4 28.g odd 6 1
7056.2.b.s 4 168.s odd 6 1
7056.2.b.s 4 168.v even 6 1
7056.2.b.s 4 168.ba even 6 1
7056.2.b.s 4 168.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 7 T_{3}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(448, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8}$$
$5$ $$( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 + T^{2} - 120 T^{4} + 121 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 9 T + 44 T^{2} - 153 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 - 31 T^{2} + 600 T^{4} - 11191 T^{6} + 130321 T^{8}$$
$23$ $$1 + 25 T^{2} + 96 T^{4} + 13225 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$1 - 55 T^{2} + 2064 T^{4} - 52855 T^{6} + 923521 T^{8}$$
$37$ $$( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 70 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 2 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 31 T^{2} - 1248 T^{4} - 68479 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 55 T^{2} - 456 T^{4} - 191455 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 3 T + 64 T^{2} + 183 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 113 T^{2} + 8280 T^{4} + 507257 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 58 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 + 137 T^{2} + 12528 T^{4} + 855017 T^{6} + 38950081 T^{8}$$
$83$ $$( 1 + 83 T^{2} )^{4}$$
$89$ $$( 1 - 3 T + 92 T^{2} - 267 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 182 T^{2} + 9409 T^{4} )^{2}$$