# Properties

 Label 224.2.i.c Level 224 Weight 2 Character orbit 224.i Analytic conductor 1.789 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -3 \beta_{2} q^{5} -\beta_{3} q^{7} + 4 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -3 \beta_{2} q^{5} -\beta_{3} q^{7} + 4 \beta_{2} q^{9} -\beta_{1} q^{11} -4 q^{13} -3 \beta_{3} q^{15} + ( -1 - \beta_{2} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{19} + ( 7 + 7 \beta_{2} ) q^{21} + ( \beta_{1} + \beta_{3} ) q^{23} + ( -4 - 4 \beta_{2} ) q^{25} + \beta_{3} q^{27} -4 q^{29} -\beta_{1} q^{31} -7 \beta_{2} q^{33} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{35} -5 \beta_{2} q^{37} -4 \beta_{1} q^{39} + 8 q^{41} + 4 \beta_{3} q^{43} + ( 12 + 12 \beta_{2} ) q^{45} + ( -\beta_{1} - \beta_{3} ) q^{47} + 7 q^{49} + ( -\beta_{1} - \beta_{3} ) q^{51} + ( -7 - 7 \beta_{2} ) q^{53} + 3 \beta_{3} q^{55} -21 q^{57} + \beta_{1} q^{59} -5 \beta_{2} q^{61} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{63} + 12 \beta_{2} q^{65} + \beta_{1} q^{67} -7 q^{69} + ( 9 + 9 \beta_{2} ) q^{73} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{75} + ( -7 - 7 \beta_{2} ) q^{77} + ( \beta_{1} + \beta_{3} ) q^{79} + ( 5 + 5 \beta_{2} ) q^{81} -4 \beta_{3} q^{83} -3 q^{85} -4 \beta_{1} q^{87} -9 \beta_{2} q^{89} + 4 \beta_{3} q^{91} -7 \beta_{2} q^{93} + 9 \beta_{1} q^{95} -8 q^{97} -4 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{5} - 8q^{9} + O(q^{10})$$ $$4q + 6q^{5} - 8q^{9} - 16q^{13} - 2q^{17} + 14q^{21} - 8q^{25} - 16q^{29} + 14q^{33} + 10q^{37} + 32q^{41} + 24q^{45} + 28q^{49} - 14q^{53} - 84q^{57} + 10q^{61} - 24q^{65} - 28q^{69} + 18q^{73} - 14q^{77} + 10q^{81} - 12q^{85} + 18q^{89} + 14q^{93} - 32q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

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

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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
 −1.32288 − 2.29129i 1.32288 + 2.29129i −1.32288 + 2.29129i 1.32288 − 2.29129i
0 −1.32288 2.29129i 0 1.50000 2.59808i 0 −2.64575 0 −2.00000 + 3.46410i 0
65.2 0 1.32288 + 2.29129i 0 1.50000 2.59808i 0 2.64575 0 −2.00000 + 3.46410i 0
193.1 0 −1.32288 + 2.29129i 0 1.50000 + 2.59808i 0 −2.64575 0 −2.00000 3.46410i 0
193.2 0 1.32288 2.29129i 0 1.50000 + 2.59808i 0 2.64575 0 −2.00000 3.46410i 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
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.i.c 4
3.b odd 2 1 2016.2.s.o 4
4.b odd 2 1 inner 224.2.i.c 4
7.b odd 2 1 1568.2.i.p 4
7.c even 3 1 inner 224.2.i.c 4
7.c even 3 1 1568.2.a.m 2
7.d odd 6 1 1568.2.a.t 2
7.d odd 6 1 1568.2.i.p 4
8.b even 2 1 448.2.i.h 4
8.d odd 2 1 448.2.i.h 4
12.b even 2 1 2016.2.s.o 4
21.h odd 6 1 2016.2.s.o 4
28.d even 2 1 1568.2.i.p 4
28.f even 6 1 1568.2.a.t 2
28.f even 6 1 1568.2.i.p 4
28.g odd 6 1 inner 224.2.i.c 4
28.g odd 6 1 1568.2.a.m 2
56.j odd 6 1 3136.2.a.bg 2
56.k odd 6 1 448.2.i.h 4
56.k odd 6 1 3136.2.a.bv 2
56.m even 6 1 3136.2.a.bg 2
56.p even 6 1 448.2.i.h 4
56.p even 6 1 3136.2.a.bv 2
84.n even 6 1 2016.2.s.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.c 4 1.a even 1 1 trivial
224.2.i.c 4 4.b odd 2 1 inner
224.2.i.c 4 7.c even 3 1 inner
224.2.i.c 4 28.g odd 6 1 inner
448.2.i.h 4 8.b even 2 1
448.2.i.h 4 8.d odd 2 1
448.2.i.h 4 56.k odd 6 1
448.2.i.h 4 56.p even 6 1
1568.2.a.m 2 7.c even 3 1
1568.2.a.m 2 28.g odd 6 1
1568.2.a.t 2 7.d odd 6 1
1568.2.a.t 2 28.f even 6 1
1568.2.i.p 4 7.b odd 2 1
1568.2.i.p 4 7.d odd 6 1
1568.2.i.p 4 28.d even 2 1
1568.2.i.p 4 28.f even 6 1
2016.2.s.o 4 3.b odd 2 1
2016.2.s.o 4 12.b even 2 1
2016.2.s.o 4 21.h odd 6 1
2016.2.s.o 4 84.n even 6 1
3136.2.a.bg 2 56.j odd 6 1
3136.2.a.bg 2 56.m even 6 1
3136.2.a.bv 2 56.k odd 6 1
3136.2.a.bv 2 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 7 T_{3}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8}$$
$5$ $$( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 - 15 T^{2} + 104 T^{4} - 1815 T^{6} + 14641 T^{8}$$
$13$ $$( 1 + 4 T + 13 T^{2} )^{4}$$
$17$ $$( 1 + T - 16 T^{2} + 17 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 + 25 T^{2} + 264 T^{4} + 9025 T^{6} + 130321 T^{8}$$
$23$ $$1 - 39 T^{2} + 992 T^{4} - 20631 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 4 T + 29 T^{2} )^{4}$$
$31$ $$1 - 55 T^{2} + 2064 T^{4} - 52855 T^{6} + 923521 T^{8}$$
$37$ $$( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 8 T + 41 T^{2} )^{4}$$
$43$ $$( 1 - 26 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 87 T^{2} + 5360 T^{4} - 192183 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 + 7 T - 4 T^{2} + 371 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 111 T^{2} + 8840 T^{4} - 386391 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 - 127 T^{2} + 11640 T^{4} - 570103 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$( 1 - 9 T + 8 T^{2} - 657 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$1 - 151 T^{2} + 16560 T^{4} - 942391 T^{6} + 38950081 T^{8}$$
$83$ $$( 1 + 54 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 8 T + 97 T^{2} )^{4}$$