# Properties

 Label 224.3.g.a Level $224$ Weight $3$ Character orbit 224.g Analytic conductor $6.104$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 224.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-7})$$ Defining polynomial: $$x^{4} + 6x^{2} + 16$$ x^4 + 6*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 2) q^{3} + (2 \beta_{3} - \beta_{2}) q^{5} - \beta_{3} q^{7} + ( - 4 \beta_1 - 3) q^{9}+O(q^{10})$$ q + (b1 - 2) * q^3 + (2*b3 - b2) * q^5 - b3 * q^7 + (-4*b1 - 3) * q^9 $$q + (\beta_1 - 2) q^{3} + (2 \beta_{3} - \beta_{2}) q^{5} - \beta_{3} q^{7} + ( - 4 \beta_1 - 3) q^{9} + ( - 6 \beta_1 - 4) q^{11} + ( - 2 \beta_{3} + \beta_{2}) q^{13} - 2 \beta_{3} q^{15} + ( - 4 \beta_1 + 18) q^{17} + ( - 19 \beta_1 - 2) q^{19} + (2 \beta_{3} + \beta_{2}) q^{21} + ( - 2 \beta_{3} - 8 \beta_{2}) q^{23} + ( - 28 \beta_1 - 17) q^{25} + ( - 4 \beta_1 + 16) q^{27} - 6 \beta_{2} q^{29} + (12 \beta_{3} + 4 \beta_{2}) q^{31} + (8 \beta_1 - 4) q^{33} + (7 \beta_1 + 14) q^{35} + ( - 8 \beta_{3} - 10 \beta_{2}) q^{37} + 2 \beta_{3} q^{39} + (12 \beta_1 - 10) q^{41} + (2 \beta_1 + 20) q^{43} + ( - 14 \beta_{3} + 11 \beta_{2}) q^{45} + (8 \beta_{3} + 4 \beta_{2}) q^{47} - 7 q^{49} + (26 \beta_1 - 44) q^{51} + (20 \beta_{3} + 12 \beta_{2}) q^{53} + ( - 20 \beta_{3} + 16 \beta_{2}) q^{55} + (36 \beta_1 - 34) q^{57} + (11 \beta_1 - 46) q^{59} + ( - 10 \beta_{3} - 3 \beta_{2}) q^{61} + (3 \beta_{3} - 4 \beta_{2}) q^{63} + (28 \beta_1 + 42) q^{65} + (16 \beta_1 + 56) q^{67} + (20 \beta_{3} + 18 \beta_{2}) q^{69} + ( - 16 \beta_{3} - 16 \beta_{2}) q^{71} + ( - 8 \beta_1 + 58) q^{73} + (39 \beta_1 - 22) q^{75} + (4 \beta_{3} - 6 \beta_{2}) q^{77} + (8 \beta_{3} - 16 \beta_{2}) q^{79} + (60 \beta_1 - 13) q^{81} + ( - 13 \beta_1 - 22) q^{83} + (28 \beta_{3} - 10 \beta_{2}) q^{85} + (12 \beta_{3} + 12 \beta_{2}) q^{87} + (24 \beta_1 + 78) q^{89} + ( - 7 \beta_1 - 14) q^{91} + ( - 32 \beta_{3} - 20 \beta_{2}) q^{93} + ( - 42 \beta_{3} + 40 \beta_{2}) q^{95} + ( - 92 \beta_1 - 34) q^{97} + (34 \beta_1 + 60) q^{99}+O(q^{100})$$ q + (b1 - 2) * q^3 + (2*b3 - b2) * q^5 - b3 * q^7 + (-4*b1 - 3) * q^9 + (-6*b1 - 4) * q^11 + (-2*b3 + b2) * q^13 - 2*b3 * q^15 + (-4*b1 + 18) * q^17 + (-19*b1 - 2) * q^19 + (2*b3 + b2) * q^21 + (-2*b3 - 8*b2) * q^23 + (-28*b1 - 17) * q^25 + (-4*b1 + 16) * q^27 - 6*b2 * q^29 + (12*b3 + 4*b2) * q^31 + (8*b1 - 4) * q^33 + (7*b1 + 14) * q^35 + (-8*b3 - 10*b2) * q^37 + 2*b3 * q^39 + (12*b1 - 10) * q^41 + (2*b1 + 20) * q^43 + (-14*b3 + 11*b2) * q^45 + (8*b3 + 4*b2) * q^47 - 7 * q^49 + (26*b1 - 44) * q^51 + (20*b3 + 12*b2) * q^53 + (-20*b3 + 16*b2) * q^55 + (36*b1 - 34) * q^57 + (11*b1 - 46) * q^59 + (-10*b3 - 3*b2) * q^61 + (3*b3 - 4*b2) * q^63 + (28*b1 + 42) * q^65 + (16*b1 + 56) * q^67 + (20*b3 + 18*b2) * q^69 + (-16*b3 - 16*b2) * q^71 + (-8*b1 + 58) * q^73 + (39*b1 - 22) * q^75 + (4*b3 - 6*b2) * q^77 + (8*b3 - 16*b2) * q^79 + (60*b1 - 13) * q^81 + (-13*b1 - 22) * q^83 + (28*b3 - 10*b2) * q^85 + (12*b3 + 12*b2) * q^87 + (24*b1 + 78) * q^89 + (-7*b1 - 14) * q^91 + (-32*b3 - 20*b2) * q^93 + (-42*b3 + 40*b2) * q^95 + (-92*b1 - 34) * q^97 + (34*b1 + 60) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{3} - 12 q^{9}+O(q^{10})$$ 4 * q - 8 * q^3 - 12 * q^9 $$4 q - 8 q^{3} - 12 q^{9} - 16 q^{11} + 72 q^{17} - 8 q^{19} - 68 q^{25} + 64 q^{27} - 16 q^{33} + 56 q^{35} - 40 q^{41} + 80 q^{43} - 28 q^{49} - 176 q^{51} - 136 q^{57} - 184 q^{59} + 168 q^{65} + 224 q^{67} + 232 q^{73} - 88 q^{75} - 52 q^{81} - 88 q^{83} + 312 q^{89} - 56 q^{91} - 136 q^{97} + 240 q^{99}+O(q^{100})$$ 4 * q - 8 * q^3 - 12 * q^9 - 16 * q^11 + 72 * q^17 - 8 * q^19 - 68 * q^25 + 64 * q^27 - 16 * q^33 + 56 * q^35 - 40 * q^41 + 80 * q^43 - 28 * q^49 - 176 * q^51 - 136 * q^57 - 184 * q^59 + 168 * q^65 + 224 * q^67 + 232 * q^73 - 88 * q^75 - 52 * q^81 - 88 * q^83 + 312 * q^89 - 56 * q^91 - 136 * q^97 + 240 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 6x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu ) / 4$$ (v^3 + 2*v) / 4 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 10\nu ) / 4$$ (v^3 + 10*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 3$$ v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 3$$ b3 - 3 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 5\beta_1$$ -b2 + 5*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 0.707107 − 1.87083i 0.707107 + 1.87083i −0.707107 + 1.87083i −0.707107 − 1.87083i
0 −3.41421 0 1.54985i 0 2.64575i 0 2.65685 0
15.2 0 −3.41421 0 1.54985i 0 2.64575i 0 2.65685 0
15.3 0 −0.585786 0 9.03316i 0 2.64575i 0 −8.65685 0
15.4 0 −0.585786 0 9.03316i 0 2.64575i 0 −8.65685 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
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.g.a 4
3.b odd 2 1 2016.3.g.a 4
4.b odd 2 1 56.3.g.a 4
7.b odd 2 1 1568.3.g.h 4
8.b even 2 1 56.3.g.a 4
8.d odd 2 1 inner 224.3.g.a 4
12.b even 2 1 504.3.g.a 4
16.e even 4 2 1792.3.d.g 8
16.f odd 4 2 1792.3.d.g 8
24.f even 2 1 2016.3.g.a 4
24.h odd 2 1 504.3.g.a 4
28.d even 2 1 392.3.g.h 4
28.f even 6 2 392.3.k.j 8
28.g odd 6 2 392.3.k.i 8
56.e even 2 1 1568.3.g.h 4
56.h odd 2 1 392.3.g.h 4
56.j odd 6 2 392.3.k.j 8
56.p even 6 2 392.3.k.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.a 4 4.b odd 2 1
56.3.g.a 4 8.b even 2 1
224.3.g.a 4 1.a even 1 1 trivial
224.3.g.a 4 8.d odd 2 1 inner
392.3.g.h 4 28.d even 2 1
392.3.g.h 4 56.h odd 2 1
392.3.k.i 8 28.g odd 6 2
392.3.k.i 8 56.p even 6 2
392.3.k.j 8 28.f even 6 2
392.3.k.j 8 56.j odd 6 2
504.3.g.a 4 12.b even 2 1
504.3.g.a 4 24.h odd 2 1
1568.3.g.h 4 7.b odd 2 1
1568.3.g.h 4 56.e even 2 1
1792.3.d.g 8 16.e even 4 2
1792.3.d.g 8 16.f odd 4 2
2016.3.g.a 4 3.b odd 2 1
2016.3.g.a 4 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4T_{3} + 2$$ acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 4 T + 2)^{2}$$
$5$ $$T^{4} + 84T^{2} + 196$$
$7$ $$(T^{2} + 7)^{2}$$
$11$ $$(T^{2} + 8 T - 56)^{2}$$
$13$ $$T^{4} + 84T^{2} + 196$$
$17$ $$(T^{2} - 36 T + 292)^{2}$$
$19$ $$(T^{2} + 4 T - 718)^{2}$$
$23$ $$T^{4} + 1848 T^{2} + 753424$$
$29$ $$(T^{2} + 504)^{2}$$
$31$ $$T^{4} + 2464 T^{2} + 614656$$
$37$ $$T^{4} + 3696 T^{2} + 906304$$
$41$ $$(T^{2} + 20 T - 188)^{2}$$
$43$ $$(T^{2} - 40 T + 392)^{2}$$
$47$ $$T^{4} + 1344 T^{2} + 50176$$
$53$ $$T^{4} + 9632 T^{2} + 614656$$
$59$ $$(T^{2} + 92 T + 1874)^{2}$$
$61$ $$T^{4} + 1652 T^{2} + 329476$$
$67$ $$(T^{2} - 112 T + 2624)^{2}$$
$71$ $$T^{4} + 10752 T^{2} + \cdots + 3211264$$
$73$ $$(T^{2} - 116 T + 3236)^{2}$$
$79$ $$T^{4} + 8064 T^{2} + \cdots + 9834496$$
$83$ $$(T^{2} + 44 T + 146)^{2}$$
$89$ $$(T^{2} - 156 T + 4932)^{2}$$
$97$ $$(T^{2} + 68 T - 15772)^{2}$$