# Properties

 Label 224.3.c.a Level 224 Weight 3 Character orbit 224.c Analytic conductor 6.104 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 224.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.4694952902656.3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{3} + \beta_{7} q^{5} + \beta_{2} q^{7} + ( -7 - \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{3} + \beta_{7} q^{5} + \beta_{2} q^{7} + ( -7 - \beta_{5} ) q^{9} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( -\beta_{4} + \beta_{7} ) q^{13} + ( 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{15} + ( -3 \beta_{4} - 2 \beta_{7} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} ) q^{19} + ( -4 + \beta_{4} - \beta_{5} + 5 \beta_{7} ) q^{21} + ( -4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( 1 - \beta_{5} ) q^{25} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 8 \beta_{6} ) q^{27} + ( -14 - 2 \beta_{5} ) q^{29} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{6} ) q^{31} + ( 3 \beta_{4} + 12 \beta_{7} ) q^{33} + ( 7 \beta_{1} - 7 \beta_{6} ) q^{35} + ( 26 + 2 \beta_{5} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{39} + ( -2 \beta_{4} - 6 \beta_{7} ) q^{41} + ( 4 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{43} + ( -7 \beta_{4} - 17 \beta_{7} ) q^{45} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{6} ) q^{47} + ( 1 + 5 \beta_{4} + 2 \beta_{5} + 4 \beta_{7} ) q^{49} + ( -28 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{51} -30 q^{53} + ( \beta_{1} + \beta_{2} - \beta_{3} - 16 \beta_{6} ) q^{55} + ( 24 + 3 \beta_{5} ) q^{57} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 17 \beta_{6} ) q^{59} + ( 8 \beta_{4} - 5 \beta_{7} ) q^{61} + ( 28 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} + 14 \beta_{6} ) q^{63} + ( -32 + \beta_{5} ) q^{65} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{67} + 3 \beta_{4} q^{69} + ( 15 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{71} + ( \beta_{4} - 16 \beta_{7} ) q^{73} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 9 \beta_{6} ) q^{75} + ( 54 + 4 \beta_{4} + 3 \beta_{5} + 6 \beta_{7} ) q^{77} + ( -39 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{79} + ( 89 + 5 \beta_{5} ) q^{81} + ( 6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + 23 \beta_{6} ) q^{83} + ( 24 + 8 \beta_{5} ) q^{85} + ( -6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + 34 \beta_{6} ) q^{87} + ( \beta_{4} + 4 \beta_{7} ) q^{89} + ( 14 \beta_{1} + \beta_{2} + 7 \beta_{3} - 7 \beta_{6} ) q^{91} + ( 8 - 4 \beta_{5} ) q^{93} + ( -19 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{95} + ( 3 \beta_{4} - 18 \beta_{7} ) q^{97} + ( 60 \beta_{1} - 15 \beta_{2} - 15 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 56q^{9} + O(q^{10})$$ $$8q - 56q^{9} - 32q^{21} + 8q^{25} - 112q^{29} + 208q^{37} + 8q^{49} - 240q^{53} + 192q^{57} - 256q^{65} + 432q^{77} + 712q^{81} + 192q^{85} + 64q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 20 x^{6} + 56 x^{4} + 20 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$10 \nu^{6} + 196 \nu^{4} + 490 \nu^{2} + 88$$$$)/21$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} + 21 \nu^{5} - 21 \nu^{4} + 77 \nu^{3} - 77 \nu^{2} + 97 \nu - 48$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} + 7 \nu^{6} - 63 \nu^{5} + 133 \nu^{4} - 231 \nu^{3} + 259 \nu^{2} - 291 \nu - 56$$$$)/21$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} + 38 \nu^{5} + 74 \nu^{3} - 34 \nu$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$2 \nu^{6} + 40 \nu^{4} + 110 \nu^{2} + 20$$ $$\beta_{6}$$ $$=$$ $$($$$$-13 \nu^{7} - 259 \nu^{5} - 707 \nu^{3} - 197 \nu$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$-7 \nu^{7} - 139 \nu^{5} - 373 \nu^{3} - 97 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 5 \beta_{1} - 20$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-38 \beta_{7} + 42 \beta_{6} - 13 \beta_{4} + 7 \beta_{3} - 7 \beta_{2} - 7 \beta_{1}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$10 \beta_{5} + 15 \beta_{3} + 15 \beta_{2} - 48 \beta_{1} + 144$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$646 \beta_{7} - 726 \beta_{6} + 197 \beta_{4} - 97 \beta_{3} + 97 \beta_{2} + 97 \beta_{1}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-343 \beta_{5} - 490 \beta_{3} - 490 \beta_{2} + 1645 \beta_{1} - 4700$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-10834 \beta_{7} + 12206 \beta_{6} - 3233 \beta_{4} + 1567 \beta_{3} - 1567 \beta_{2} - 1567 \beta_{1}$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 − 4.08932i − 0.244539i 1.69385i 0.590371i − 0.590371i − 1.69385i 0.244539i 4.08932i
0 5.43734i 0 6.12900i 0 6.21005 3.23038i 0 −20.5647 0
97.2 0 5.43734i 0 6.12900i 0 −6.21005 3.23038i 0 −20.5647 0
97.3 0 1.56056i 0 3.23038i 0 3.38162 + 6.12900i 0 6.56466 0
97.4 0 1.56056i 0 3.23038i 0 −3.38162 + 6.12900i 0 6.56466 0
97.5 0 1.56056i 0 3.23038i 0 −3.38162 6.12900i 0 6.56466 0
97.6 0 1.56056i 0 3.23038i 0 3.38162 6.12900i 0 6.56466 0
97.7 0 5.43734i 0 6.12900i 0 −6.21005 + 3.23038i 0 −20.5647 0
97.8 0 5.43734i 0 6.12900i 0 6.21005 + 3.23038i 0 −20.5647 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.8 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.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.c.a 8
3.b odd 2 1 2016.3.f.c 8
4.b odd 2 1 inner 224.3.c.a 8
7.b odd 2 1 inner 224.3.c.a 8
8.b even 2 1 448.3.c.g 8
8.d odd 2 1 448.3.c.g 8
12.b even 2 1 2016.3.f.c 8
21.c even 2 1 2016.3.f.c 8
28.d even 2 1 inner 224.3.c.a 8
56.e even 2 1 448.3.c.g 8
56.h odd 2 1 448.3.c.g 8
84.h odd 2 1 2016.3.f.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.c.a 8 1.a even 1 1 trivial
224.3.c.a 8 4.b odd 2 1 inner
224.3.c.a 8 7.b odd 2 1 inner
224.3.c.a 8 28.d even 2 1 inner
448.3.c.g 8 8.b even 2 1
448.3.c.g 8 8.d odd 2 1
448.3.c.g 8 56.e even 2 1
448.3.c.g 8 56.h odd 2 1
2016.3.f.c 8 3.b odd 2 1
2016.3.f.c 8 12.b even 2 1
2016.3.f.c 8 21.c even 2 1
2016.3.f.c 8 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 32 T_{3}^{2} + 72$$ acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 4 T^{2} - 18 T^{4} - 324 T^{6} + 6561 T^{8} )^{2}$$
$5$ $$( 1 - 52 T^{2} + 1742 T^{4} - 32500 T^{6} + 390625 T^{8} )^{2}$$
$7$ $$1 - 4 T^{2} + 1862 T^{4} - 9604 T^{6} + 5764801 T^{8}$$
$11$ $$( 1 + 236 T^{2} + 31430 T^{4} + 3455276 T^{6} + 214358881 T^{8} )^{2}$$
$13$ $$( 1 - 468 T^{2} + 102862 T^{4} - 13366548 T^{6} + 815730721 T^{8} )^{2}$$
$17$ $$( 1 - 4 T^{2} + 155270 T^{4} - 334084 T^{6} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 - 1188 T^{2} + 613294 T^{4} - 154821348 T^{6} + 16983563041 T^{8} )^{2}$$
$23$ $$( 1 + 1676 T^{2} + 1214822 T^{4} + 469013516 T^{6} + 78310985281 T^{8} )^{2}$$
$29$ $$( 1 + 28 T + 1142 T^{2} + 23548 T^{3} + 707281 T^{4} )^{4}$$
$31$ $$( 1 - 2180 T^{2} + 2458118 T^{4} - 2013275780 T^{6} + 852891037441 T^{8} )^{2}$$
$37$ $$( 1 - 52 T + 2678 T^{2} - 71188 T^{3} + 1874161 T^{4} )^{4}$$
$41$ $$( 1 - 4868 T^{2} + 10725062 T^{4} - 13755804548 T^{6} + 7984925229121 T^{8} )^{2}$$
$43$ $$( 1 + 2540 T^{2} + 7272902 T^{4} + 8683754540 T^{6} + 11688200277601 T^{8} )^{2}$$
$47$ $$( 1 - 6788 T^{2} + 21266822 T^{4} - 33123274628 T^{6} + 23811286661761 T^{8} )^{2}$$
$53$ $$( 1 + 30 T + 2809 T^{2} )^{8}$$
$59$ $$( 1 - 4580 T^{2} + 29037038 T^{4} - 55497513380 T^{6} + 146830437604321 T^{8} )^{2}$$
$61$ $$( 1 - 4212 T^{2} + 3996814 T^{4} - 58318682292 T^{6} + 191707312997281 T^{8} )^{2}$$
$67$ $$( 1 + 8150 T^{2} + 20151121 T^{4} )^{4}$$
$71$ $$( 1 + 11964 T^{2} + 70047686 T^{4} + 304025351484 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 - 8388 T^{2} + 67882822 T^{4} - 238204445508 T^{6} + 806460091894081 T^{8} )^{2}$$
$79$ $$( 1 - 1028 T^{2} + 37863942 T^{4} - 40040683268 T^{6} + 1517108809906561 T^{8} )^{2}$$
$83$ $$( 1 - 8132 T^{2} + 106317422 T^{4} - 385931066372 T^{6} + 2252292232139041 T^{8} )^{2}$$
$89$ $$( 1 - 30916 T^{2} + 364289990 T^{4} - 1939739122756 T^{6} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 - 19204 T^{2} + 268303110 T^{4} - 1700116312324 T^{6} + 7837433594376961 T^{8} )^{2}$$