# 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 $2$

# Related objects

## 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: $$x^{4} + 2x^{2} + 4$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_1 + 2) q^{3} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{5} + ( - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{7} + (2 \beta_{3} + 2 \beta_{2}) q^{9}+O(q^{10})$$ q + (b3 + b1 + 2) * q^3 + (-2*b3 + 2*b2 - b1 + 1) * q^5 + (-2*b3 + 3*b2 + 2*b1 + 3) * q^7 + (2*b3 + 2*b2) * q^9 $$q + (\beta_{3} + \beta_1 + 2) q^{3} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{5} + ( - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{7} + (2 \beta_{3} + 2 \beta_{2}) q^{9} + ( - \beta_{3} + 2 \beta_{2} + 9 \beta_1) q^{11} + (2 \beta_{2} - 12 \beta_1 - 6) q^{13} + ( - 2 \beta_{3} + \beta_{2} - 9) q^{15} + ( - 2 \beta_{3} - 5 \beta_1 - 10) q^{17} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{19} + ( - 4 \beta_{3} + 6 \beta_{2} + 11 \beta_1 - 8) q^{21} + ( - 3 \beta_{3} - 3 \beta_{2} + 15 \beta_1 + 15) q^{23} + ( - 4 \beta_{3} + 8 \beta_{2} - 2 \beta_1) q^{25} + ( - 3 \beta_{2} + 6 \beta_1 + 3) q^{27} + ( - 4 \beta_{3} + 2 \beta_{2} - 12) q^{29} + (15 \beta_{3} - 7 \beta_1 - 14) q^{31} + ( - 12 \beta_{3} + 12 \beta_{2} + 15 \beta_1 - 15) q^{33} + ( - 7 \beta_{3} + 7 \beta_{2} - 35 \beta_1 - 7) q^{35} + (8 \beta_{3} + 8 \beta_{2} + 31 \beta_1 + 31) q^{37} + (4 \beta_{3} - 8 \beta_{2} - 6 \beta_1) q^{39} + (10 \beta_{2} - 4 \beta_1 - 2) q^{41} + ( - 4 \beta_{3} + 2 \beta_{2} + 2) q^{43} + (6 \beta_{3} - 24 \beta_1 - 48) q^{45} + ( - \beta_{3} + \beta_{2} - 29 \beta_1 + 29) q^{47} + ( - 16 \beta_{3} + 10 \beta_{2} - 40 \beta_1 - 25) q^{49} + ( - 7 \beta_{3} - 7 \beta_{2} - 27 \beta_1 - 27) q^{51} + ( - 4 \beta_{3} + 8 \beta_{2} - 39 \beta_1) q^{53} + ( - 15 \beta_{2} - 6 \beta_1 - 3) q^{55} + 3 q^{57} + (25 \beta_{3} + 13 \beta_1 + 26) q^{59} + ( - 32 \beta_{3} + 32 \beta_{2} - 7 \beta_1 + 7) q^{61} + ( - 2 \beta_{3} + 10 \beta_{2} - 12 \beta_1 - 60) q^{63} + (14 \beta_{3} + 14 \beta_{2} - 42 \beta_1 - 42) q^{65} + (15 \beta_{3} - 30 \beta_{2} - 29 \beta_1) q^{67} + (6 \beta_{2} - 6 \beta_1 - 3) q^{69} + ( - 20 \beta_{3} + 10 \beta_{2} - 6) q^{71} + (16 \beta_{3} + 53 \beta_1 + 106) q^{73} + ( - 10 \beta_{3} + 10 \beta_{2} + 22 \beta_1 - 22) q^{75} + ( - 14 \beta_{3} - 14 \beta_{2} - 21 \beta_1 - 42) q^{77} + (5 \beta_{3} + 5 \beta_{2} + 55 \beta_1 + 55) q^{79} + (18 \beta_{3} - 36 \beta_{2} - 9 \beta_1) q^{81} + ( - 4 \beta_{2} + 136 \beta_1 + 68) q^{83} + (16 \beta_{3} - 8 \beta_{2} + 9) q^{85} + ( - 18 \beta_{3} - 24 \beta_1 - 48) q^{87} + ( - 24 \beta_{3} + 24 \beta_{2} + 63 \beta_1 - 63) q^{89} + (20 \beta_{3} + 12 \beta_{2} - 48 \beta_1 - 30) q^{91} + (8 \beta_{3} + 8 \beta_{2} + 69 \beta_1 + 69) q^{93} + (3 \beta_{3} - 6 \beta_{2} + 15 \beta_1) q^{95} + ( - 26 \beta_{2} + 44 \beta_1 + 22) q^{97} + ( - 36 \beta_{3} + 18 \beta_{2} - 36) q^{99}+O(q^{100})$$ q + (b3 + b1 + 2) * q^3 + (-2*b3 + 2*b2 - b1 + 1) * q^5 + (-2*b3 + 3*b2 + 2*b1 + 3) * q^7 + (2*b3 + 2*b2) * q^9 + (-b3 + 2*b2 + 9*b1) * q^11 + (2*b2 - 12*b1 - 6) * q^13 + (-2*b3 + b2 - 9) * q^15 + (-2*b3 - 5*b1 - 10) * q^17 + (b3 - b2 + b1 - 1) * q^19 + (-4*b3 + 6*b2 + 11*b1 - 8) * q^21 + (-3*b3 - 3*b2 + 15*b1 + 15) * q^23 + (-4*b3 + 8*b2 - 2*b1) * q^25 + (-3*b2 + 6*b1 + 3) * q^27 + (-4*b3 + 2*b2 - 12) * q^29 + (15*b3 - 7*b1 - 14) * q^31 + (-12*b3 + 12*b2 + 15*b1 - 15) * q^33 + (-7*b3 + 7*b2 - 35*b1 - 7) * q^35 + (8*b3 + 8*b2 + 31*b1 + 31) * q^37 + (4*b3 - 8*b2 - 6*b1) * q^39 + (10*b2 - 4*b1 - 2) * q^41 + (-4*b3 + 2*b2 + 2) * q^43 + (6*b3 - 24*b1 - 48) * q^45 + (-b3 + b2 - 29*b1 + 29) * q^47 + (-16*b3 + 10*b2 - 40*b1 - 25) * q^49 + (-7*b3 - 7*b2 - 27*b1 - 27) * q^51 + (-4*b3 + 8*b2 - 39*b1) * q^53 + (-15*b2 - 6*b1 - 3) * q^55 + 3 * q^57 + (25*b3 + 13*b1 + 26) * q^59 + (-32*b3 + 32*b2 - 7*b1 + 7) * q^61 + (-2*b3 + 10*b2 - 12*b1 - 60) * q^63 + (14*b3 + 14*b2 - 42*b1 - 42) * q^65 + (15*b3 - 30*b2 - 29*b1) * q^67 + (6*b2 - 6*b1 - 3) * q^69 + (-20*b3 + 10*b2 - 6) * q^71 + (16*b3 + 53*b1 + 106) * q^73 + (-10*b3 + 10*b2 + 22*b1 - 22) * q^75 + (-14*b3 - 14*b2 - 21*b1 - 42) * q^77 + (5*b3 + 5*b2 + 55*b1 + 55) * q^79 + (18*b3 - 36*b2 - 9*b1) * q^81 + (-4*b2 + 136*b1 + 68) * q^83 + (16*b3 - 8*b2 + 9) * q^85 + (-18*b3 - 24*b1 - 48) * q^87 + (-24*b3 + 24*b2 + 63*b1 - 63) * q^89 + (20*b3 + 12*b2 - 48*b1 - 30) * q^91 + (8*b3 + 8*b2 + 69*b1 + 69) * q^93 + (3*b3 - 6*b2 + 15*b1) * q^95 + (-26*b2 + 44*b1 + 22) * q^97 + (-36*b3 + 18*b2 - 36) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} + 6 q^{5} + 8 q^{7}+O(q^{10})$$ 4 * q + 6 * q^3 + 6 * q^5 + 8 * q^7 $$4 q + 6 q^{3} + 6 q^{5} + 8 q^{7} - 18 q^{11} - 36 q^{15} - 30 q^{17} - 6 q^{19} - 54 q^{21} + 30 q^{23} + 4 q^{25} - 48 q^{29} - 42 q^{31} - 90 q^{33} + 42 q^{35} + 62 q^{37} + 12 q^{39} + 8 q^{43} - 144 q^{45} + 174 q^{47} - 20 q^{49} - 54 q^{51} + 78 q^{53} + 12 q^{57} + 78 q^{59} + 42 q^{61} - 216 q^{63} - 84 q^{65} + 58 q^{67} - 24 q^{71} + 318 q^{73} - 132 q^{75} - 126 q^{77} + 110 q^{79} + 18 q^{81} + 36 q^{85} - 144 q^{87} - 378 q^{89} - 24 q^{91} + 138 q^{93} - 30 q^{95} - 144 q^{99}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 2$$ (v^3 + 4*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu ) / 2$$ (-v^3 + 2*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -4\beta_{3} + 2\beta_{2} ) / 3$$ (-4*b3 + 2*b2) / 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 $$T_{3}^{4} - 6T_{3}^{3} + 9T_{3}^{2} + 18T_{3} + 9$$ acting on $$S_{3}^{\mathrm{new}}(448, [\chi])$$.

## Hecke characteristic polynomials

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