# Properties

 Label 448.4.i.f 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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.

 $$f(q)$$ $$=$$ $$q + ( 7 - 7 \zeta_{6} ) q^{3} + 7 \zeta_{6} q^{5} + ( 7 + 14 \zeta_{6} ) q^{7} -22 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 7 - 7 \zeta_{6} ) q^{3} + 7 \zeta_{6} q^{5} + ( 7 + 14 \zeta_{6} ) q^{7} -22 \zeta_{6} q^{9} + ( -5 + 5 \zeta_{6} ) q^{11} + 14 q^{13} + 49 q^{15} + ( 21 - 21 \zeta_{6} ) q^{17} + 49 \zeta_{6} q^{19} + ( 147 - 49 \zeta_{6} ) q^{21} + 159 \zeta_{6} q^{23} + ( 76 - 76 \zeta_{6} ) q^{25} + 35 q^{27} -58 q^{29} + ( -147 + 147 \zeta_{6} ) q^{31} + 35 \zeta_{6} q^{33} + ( -98 + 147 \zeta_{6} ) q^{35} + 219 \zeta_{6} q^{37} + ( 98 - 98 \zeta_{6} ) q^{39} + 350 q^{41} + 124 q^{43} + ( 154 - 154 \zeta_{6} ) q^{45} -525 \zeta_{6} q^{47} + ( -147 + 392 \zeta_{6} ) q^{49} -147 \zeta_{6} q^{51} + ( 303 - 303 \zeta_{6} ) q^{53} -35 q^{55} + 343 q^{57} + ( -105 + 105 \zeta_{6} ) q^{59} -413 \zeta_{6} q^{61} + ( 308 - 462 \zeta_{6} ) q^{63} + 98 \zeta_{6} q^{65} + ( 415 - 415 \zeta_{6} ) q^{67} + 1113 q^{69} -432 q^{71} + ( 1113 - 1113 \zeta_{6} ) q^{73} -532 \zeta_{6} q^{75} + ( -105 + 35 \zeta_{6} ) q^{77} + 103 \zeta_{6} q^{79} + ( 839 - 839 \zeta_{6} ) q^{81} -1092 q^{83} + 147 q^{85} + ( -406 + 406 \zeta_{6} ) q^{87} + 329 \zeta_{6} q^{89} + ( 98 + 196 \zeta_{6} ) q^{91} + 1029 \zeta_{6} q^{93} + ( -343 + 343 \zeta_{6} ) q^{95} -882 q^{97} + 110 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 7q^{3} + 7q^{5} + 28q^{7} - 22q^{9} + O(q^{10})$$ $$2q + 7q^{3} + 7q^{5} + 28q^{7} - 22q^{9} - 5q^{11} + 28q^{13} + 98q^{15} + 21q^{17} + 49q^{19} + 245q^{21} + 159q^{23} + 76q^{25} + 70q^{27} - 116q^{29} - 147q^{31} + 35q^{33} - 49q^{35} + 219q^{37} + 98q^{39} + 700q^{41} + 248q^{43} + 154q^{45} - 525q^{47} + 98q^{49} - 147q^{51} + 303q^{53} - 70q^{55} + 686q^{57} - 105q^{59} - 413q^{61} + 154q^{63} + 98q^{65} + 415q^{67} + 2226q^{69} - 864q^{71} + 1113q^{73} - 532q^{75} - 175q^{77} + 103q^{79} + 839q^{81} - 2184q^{83} + 294q^{85} - 406q^{87} + 329q^{89} + 392q^{91} + 1029q^{93} - 343q^{95} - 1764q^{97} + 220q^{99} + O(q^{100})$$

## 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 3.50000 + 6.06218i 0 3.50000 6.06218i 0 14.0000 12.1244i 0 −11.0000 + 19.0526i 0
193.1 0 3.50000 6.06218i 0 3.50000 + 6.06218i 0 14.0000 + 12.1244i 0 −11.0000 19.0526i 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.f 2
4.b odd 2 1 448.4.i.a 2
7.c even 3 1 inner 448.4.i.f 2
8.b even 2 1 7.4.c.a 2
8.d odd 2 1 112.4.i.c 2
24.h odd 2 1 63.4.e.b 2
28.g odd 6 1 448.4.i.a 2
40.f even 2 1 175.4.e.a 2
40.i odd 4 2 175.4.k.a 4
56.h odd 2 1 49.4.c.a 2
56.j odd 6 1 49.4.a.c 1
56.j odd 6 1 49.4.c.a 2
56.k odd 6 1 112.4.i.c 2
56.k odd 6 1 784.4.a.b 1
56.m even 6 1 784.4.a.r 1
56.p even 6 1 7.4.c.a 2
56.p even 6 1 49.4.a.d 1
168.i even 2 1 441.4.e.k 2
168.s odd 6 1 63.4.e.b 2
168.s odd 6 1 441.4.a.d 1
168.ba even 6 1 441.4.a.e 1
168.ba even 6 1 441.4.e.k 2
280.bf even 6 1 175.4.e.a 2
280.bf even 6 1 1225.4.a.c 1
280.bk odd 6 1 1225.4.a.d 1
280.bt odd 12 2 175.4.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 8.b even 2 1
7.4.c.a 2 56.p even 6 1
49.4.a.c 1 56.j odd 6 1
49.4.a.d 1 56.p even 6 1
49.4.c.a 2 56.h odd 2 1
49.4.c.a 2 56.j odd 6 1
63.4.e.b 2 24.h odd 2 1
63.4.e.b 2 168.s odd 6 1
112.4.i.c 2 8.d odd 2 1
112.4.i.c 2 56.k odd 6 1
175.4.e.a 2 40.f even 2 1
175.4.e.a 2 280.bf even 6 1
175.4.k.a 4 40.i odd 4 2
175.4.k.a 4 280.bt odd 12 2
441.4.a.d 1 168.s odd 6 1
441.4.a.e 1 168.ba even 6 1
441.4.e.k 2 168.i even 2 1
441.4.e.k 2 168.ba even 6 1
448.4.i.a 2 4.b odd 2 1
448.4.i.a 2 28.g odd 6 1
448.4.i.f 2 1.a even 1 1 trivial
448.4.i.f 2 7.c even 3 1 inner
784.4.a.b 1 56.k odd 6 1
784.4.a.r 1 56.m even 6 1
1225.4.a.c 1 280.bf even 6 1
1225.4.a.d 1 280.bk odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(448, [\chi])$$:

 $$T_{3}^{2} - 7 T_{3} + 49$$ $$T_{11}^{2} + 5 T_{11} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$49 - 7 T + T^{2}$$
$5$ $$49 - 7 T + T^{2}$$
$7$ $$343 - 28 T + T^{2}$$
$11$ $$25 + 5 T + T^{2}$$
$13$ $$( -14 + T )^{2}$$
$17$ $$441 - 21 T + T^{2}$$
$19$ $$2401 - 49 T + T^{2}$$
$23$ $$25281 - 159 T + T^{2}$$
$29$ $$( 58 + T )^{2}$$
$31$ $$21609 + 147 T + T^{2}$$
$37$ $$47961 - 219 T + T^{2}$$
$41$ $$( -350 + T )^{2}$$
$43$ $$( -124 + T )^{2}$$
$47$ $$275625 + 525 T + T^{2}$$
$53$ $$91809 - 303 T + T^{2}$$
$59$ $$11025 + 105 T + T^{2}$$
$61$ $$170569 + 413 T + T^{2}$$
$67$ $$172225 - 415 T + T^{2}$$
$71$ $$( 432 + T )^{2}$$
$73$ $$1238769 - 1113 T + T^{2}$$
$79$ $$10609 - 103 T + T^{2}$$
$83$ $$( 1092 + T )^{2}$$
$89$ $$108241 - 329 T + T^{2}$$
$97$ $$( 882 + T )^{2}$$