# Properties

 Label 448.3.s.d Level $448$ Weight $3$ Character orbit 448.s Analytic conductor $12.207$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$12.2071158433$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 14) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta_{1} + \beta_{3} ) q^{3} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 2 + \beta_{1} + \beta_{3} ) q^{3} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{9} + ( 9 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{11} + ( -6 - 12 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -9 + \beta_{2} - 2 \beta_{3} ) q^{15} + ( -10 - 5 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( -8 + 11 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{21} + ( 15 + 15 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{23} + ( -2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{25} + ( 3 + 6 \beta_{1} - 3 \beta_{2} ) q^{27} + ( -12 + 2 \beta_{2} - 4 \beta_{3} ) q^{29} + ( -14 - 7 \beta_{1} + 15 \beta_{3} ) q^{31} + ( -15 + 15 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{33} + ( -7 - 35 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{35} + ( 31 + 31 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{37} + ( -6 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{39} + ( -2 - 4 \beta_{1} + 10 \beta_{2} ) q^{41} + ( 2 + 2 \beta_{2} - 4 \beta_{3} ) q^{43} + ( -48 - 24 \beta_{1} + 6 \beta_{3} ) q^{45} + ( 29 - 29 \beta_{1} + \beta_{2} - \beta_{3} ) q^{47} + ( -25 - 40 \beta_{1} + 10 \beta_{2} - 16 \beta_{3} ) q^{49} + ( -27 - 27 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} ) q^{51} + ( -39 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -3 - 6 \beta_{1} - 15 \beta_{2} ) q^{55} + 3 q^{57} + ( 26 + 13 \beta_{1} + 25 \beta_{3} ) q^{59} + ( 7 - 7 \beta_{1} + 32 \beta_{2} - 32 \beta_{3} ) q^{61} + ( -60 - 12 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{63} + ( -42 - 42 \beta_{1} + 14 \beta_{2} + 14 \beta_{3} ) q^{65} + ( -29 \beta_{1} - 30 \beta_{2} + 15 \beta_{3} ) q^{67} + ( -3 - 6 \beta_{1} + 6 \beta_{2} ) q^{69} + ( -6 + 10 \beta_{2} - 20 \beta_{3} ) q^{71} + ( 106 + 53 \beta_{1} + 16 \beta_{3} ) q^{73} + ( -22 + 22 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{75} + ( -42 - 21 \beta_{1} - 14 \beta_{2} - 14 \beta_{3} ) q^{77} + ( 55 + 55 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{79} + ( -9 \beta_{1} - 36 \beta_{2} + 18 \beta_{3} ) q^{81} + ( 68 + 136 \beta_{1} - 4 \beta_{2} ) q^{83} + ( 9 - 8 \beta_{2} + 16 \beta_{3} ) q^{85} + ( -48 - 24 \beta_{1} - 18 \beta_{3} ) q^{87} + ( -63 + 63 \beta_{1} + 24 \beta_{2} - 24 \beta_{3} ) q^{89} + ( -30 - 48 \beta_{1} + 12 \beta_{2} + 20 \beta_{3} ) q^{91} + ( 69 + 69 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{93} + ( 15 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{95} + ( 22 + 44 \beta_{1} - 26 \beta_{2} ) q^{97} + ( -36 + 18 \beta_{2} - 36 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} + 6 q^{5} + 8 q^{7} + O(q^{10})$$ $$4 q + 6 q^{3} + 6 q^{5} + 8 q^{7} - 18 q^{11} - 36 q^{15} - 30 q^{17} - 6 q^{19} - 54 q^{21} + 30 q^{23} + 4 q^{25} - 48 q^{29} - 42 q^{31} - 90 q^{33} + 42 q^{35} + 62 q^{37} + 12 q^{39} + 8 q^{43} - 144 q^{45} + 174 q^{47} - 20 q^{49} - 54 q^{51} + 78 q^{53} + 12 q^{57} + 78 q^{59} + 42 q^{61} - 216 q^{63} - 84 q^{65} + 58 q^{67} - 24 q^{71} + 318 q^{73} - 132 q^{75} - 126 q^{77} + 110 q^{79} + 18 q^{81} + 36 q^{85} - 144 q^{87} - 378 q^{89} - 24 q^{91} + 138 q^{93} - 30 q^{95} - 144 q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{3} + 2 \beta_{2}$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −0.621320 0.358719i 0 5.74264 3.31552i 0 6.24264 3.16693i 0 −4.24264 7.34847i 0
129.2 0 3.62132 + 2.09077i 0 −2.74264 + 1.58346i 0 −2.24264 + 6.63103i 0 4.24264 + 7.34847i 0
257.1 0 −0.621320 + 0.358719i 0 5.74264 + 3.31552i 0 6.24264 + 3.16693i 0 −4.24264 + 7.34847i 0
257.2 0 3.62132 2.09077i 0 −2.74264 1.58346i 0 −2.24264 6.63103i 0 4.24264 7.34847i 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.s.d 4
4.b odd 2 1 448.3.s.c 4
7.d odd 6 1 inner 448.3.s.d 4
8.b even 2 1 14.3.d.a 4
8.d odd 2 1 112.3.s.b 4
24.f even 2 1 1008.3.cg.l 4
24.h odd 2 1 126.3.n.c 4
28.f even 6 1 448.3.s.c 4
40.f even 2 1 350.3.k.a 4
40.i odd 4 2 350.3.i.a 8
56.e even 2 1 784.3.s.c 4
56.h odd 2 1 98.3.d.a 4
56.j odd 6 1 14.3.d.a 4
56.j odd 6 1 98.3.b.b 4
56.k odd 6 1 784.3.c.e 4
56.k odd 6 1 784.3.s.c 4
56.m even 6 1 112.3.s.b 4
56.m even 6 1 784.3.c.e 4
56.p even 6 1 98.3.b.b 4
56.p even 6 1 98.3.d.a 4
168.i even 2 1 882.3.n.b 4
168.s odd 6 1 882.3.c.f 4
168.s odd 6 1 882.3.n.b 4
168.ba even 6 1 126.3.n.c 4
168.ba even 6 1 882.3.c.f 4
168.be odd 6 1 1008.3.cg.l 4
280.bk odd 6 1 350.3.k.a 4
280.bv even 12 2 350.3.i.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 8.b even 2 1
14.3.d.a 4 56.j odd 6 1
98.3.b.b 4 56.j odd 6 1
98.3.b.b 4 56.p even 6 1
98.3.d.a 4 56.h odd 2 1
98.3.d.a 4 56.p even 6 1
112.3.s.b 4 8.d odd 2 1
112.3.s.b 4 56.m even 6 1
126.3.n.c 4 24.h odd 2 1
126.3.n.c 4 168.ba even 6 1
350.3.i.a 8 40.i odd 4 2
350.3.i.a 8 280.bv even 12 2
350.3.k.a 4 40.f even 2 1
350.3.k.a 4 280.bk odd 6 1
448.3.s.c 4 4.b odd 2 1
448.3.s.c 4 28.f even 6 1
448.3.s.d 4 1.a even 1 1 trivial
448.3.s.d 4 7.d odd 6 1 inner
784.3.c.e 4 56.k odd 6 1
784.3.c.e 4 56.m even 6 1
784.3.s.c 4 56.e even 2 1
784.3.s.c 4 56.k odd 6 1
882.3.c.f 4 168.s odd 6 1
882.3.c.f 4 168.ba even 6 1
882.3.n.b 4 168.i even 2 1
882.3.n.b 4 168.s odd 6 1
1008.3.cg.l 4 24.f even 2 1
1008.3.cg.l 4 168.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 6 T_{3}^{3} + 9 T_{3}^{2} + 18 T_{3} + 9$$ acting on $$S_{3}^{\mathrm{new}}(448, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 18 T + 9 T^{2} - 6 T^{3} + T^{4}$$
$5$ $$441 + 126 T - 9 T^{2} - 6 T^{3} + T^{4}$$
$7$ $$2401 - 392 T + 42 T^{2} - 8 T^{3} + T^{4}$$
$11$ $$3969 + 1134 T + 261 T^{2} + 18 T^{3} + T^{4}$$
$13$ $$7056 + 264 T^{2} + T^{4}$$
$17$ $$2601 + 1530 T + 351 T^{2} + 30 T^{3} + T^{4}$$
$19$ $$9 - 18 T + 9 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$3969 - 1890 T + 837 T^{2} - 30 T^{3} + T^{4}$$
$29$ $$( 72 + 24 T + T^{2} )^{2}$$
$31$ $$1447209 - 50526 T - 615 T^{2} + 42 T^{3} + T^{4}$$
$37$ $$36481 + 11842 T + 4035 T^{2} - 62 T^{3} + T^{4}$$
$41$ $$345744 + 1224 T^{2} + T^{4}$$
$43$ $$( -68 - 4 T + T^{2} )^{2}$$
$47$ $$6335289 - 437958 T + 12609 T^{2} - 174 T^{3} + T^{4}$$
$53$ $$1520289 - 96174 T + 4851 T^{2} - 78 T^{3} + T^{4}$$
$59$ $$10517049 + 252954 T - 1215 T^{2} - 78 T^{3} + T^{4}$$
$61$ $$35964009 + 251874 T - 5409 T^{2} - 42 T^{3} + T^{4}$$
$67$ $$10297681 + 186122 T + 6573 T^{2} - 58 T^{3} + T^{4}$$
$71$ $$( -1764 + 12 T + T^{2} )^{2}$$
$73$ $$47485881 - 2191338 T + 40599 T^{2} - 318 T^{3} + T^{4}$$
$79$ $$6630625 - 283250 T + 9525 T^{2} - 110 T^{3} + T^{4}$$
$83$ $$189778176 + 27936 T^{2} + T^{4}$$
$89$ $$71419401 + 3194478 T + 56079 T^{2} + 378 T^{3} + T^{4}$$
$97$ $$6780816 + 11016 T^{2} + T^{4}$$
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