LMFDB - Newform orbit 224.3.r.b

Newspace parameters

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.r (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.10355792167$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{12}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \zeta_{12} q^{3} - \zeta_{12}^{2} q^{5} - 7 \zeta_{12}^{3} q^{7} - 8 \zeta_{12}^{2} q^{9} +O(q^{10}) $ q + z * q^3 - z^2 * q^5 - 7z^3 q^7 - 8z^2 q^9	\( q + \zeta_{12} q^{3} - \zeta_{12}^{2} q^{5} - 7 \zeta_{12}^{3} q^{7} - 8 \zeta_{12}^{2} q^{9} - 17 \zeta_{12} q^{11} + 24 q^{13} - \zeta_{12}^{3} q^{15} + (\zeta_{12}^{2} - 1) q^{17} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{19} + ( - 7 \zeta_{12}^{2} + 7) q^{21} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{23} + ( - 24 \zeta_{12}^{2} + 24) q^{25} - 17 \zeta_{12}^{3} q^{27} + 24 q^{29} - 41 \zeta_{12} q^{31} - 17 \zeta_{12}^{2} q^{33} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{35} + 49 \zeta_{12}^{2} q^{37} + 24 \zeta_{12} q^{39} - 48 q^{41} + 24 \zeta_{12}^{3} q^{43} + (8 \zeta_{12}^{2} - 8) q^{45} + ( - 55 \zeta_{12}^{3} + 55 \zeta_{12}) q^{47} - 49 q^{49} + (\zeta_{12}^{3} - \zeta_{12}) q^{51} + ( - 25 \zeta_{12}^{2} + 25) q^{53} + 17 \zeta_{12}^{3} q^{55} + 7 q^{57} + 17 \zeta_{12} q^{59} + \zeta_{12}^{2} q^{61} + (56 \zeta_{12}^{3} - 56 \zeta_{12}) q^{63} - 24 \zeta_{12}^{2} q^{65} + 65 \zeta_{12} q^{67} - 7 q^{69} + 96 \zeta_{12}^{3} q^{71} + (95 \zeta_{12}^{2} - 95) q^{73} + ( - 24 \zeta_{12}^{3} + 24 \zeta_{12}) q^{75} + (119 \zeta_{12}^{2} - 119) q^{77} + ( - 41 \zeta_{12}^{3} + 41 \zeta_{12}) q^{79} + (55 \zeta_{12}^{2} - 55) q^{81} + 72 \zeta_{12}^{3} q^{83} + q^{85} + 24 \zeta_{12} q^{87} - 95 \zeta_{12}^{2} q^{89} - 168 \zeta_{12}^{3} q^{91} - 41 \zeta_{12}^{2} q^{93} - 7 \zeta_{12} q^{95} + 144 q^{97} + 136 \zeta_{12}^{3} q^{99} +O(q^{100}) \) q + z * q^3 - z^2 * q^5 - 7z^3 q^7 - 8z^2 q^9 - 17z q^11 + 24 * q^13 - z^3 * q^15 + (z^2 - 1) * q^17 + (-7z^3 + 7z) * q^19 + (-7z^2 + 7) q^21 + (7z^3 - 7z) * q^23 + (-24z^2 + 24) q^25 - 17z^3 q^27 + 24 * q^29 - 41z q^31 - 17z^2 q^33 + (7z^3 - 7z) * q^35 + 49z^2 q^37 + 24z q^39 - 48 * q^41 + 24z^3 q^43 + (8z^2 - 8) q^45 + (-55z^3 + 55z) * q^47 - 49 * q^49 + (z^3 - z) * q^51 + (-25z^2 + 25) q^53 + 17z^3 q^55 + 7 * q^57 + 17z q^59 + z^2 * q^61 + (56z^3 - 56z) * q^63 - 24z^2 q^65 + 65z q^67 - 7 * q^69 + 96z^3 q^71 + (95z^2 - 95) q^73 + (-24z^3 + 24z) * q^75 + (119z^2 - 119) q^77 + (-41z^3 + 41z) * q^79 + (55z^2 - 55) q^81 + 72z^3 q^83 + q^85 + 24z q^87 - 95z^2 q^89 - 168z^3 q^91 - 41z^2 q^93 - 7z q^95 + 144 * q^97 + 136z^3 q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} - 16 q^{9}+O(q^{10}) $ 4 * q - 2 * q^5 - 16 * q^9	$ 4 q - 2 q^{5} - 16 q^{9} + 96 q^{13} - 2 q^{17} + 14 q^{21} + 48 q^{25} + 96 q^{29} - 34 q^{33} + 98 q^{37} - 192 q^{41} - 16 q^{45} - 196 q^{49} + 50 q^{53} + 28 q^{57} + 2 q^{61} - 48 q^{65} - 28 q^{69} - 190 q^{73} - 238 q^{77} - 110 q^{81} + 4 q^{85} - 190 q^{89} - 82 q^{93} + 576 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 - 16 * q^9 + 96 * q^13 - 2 * q^17 + 14 * q^21 + 48 * q^25 + 96 * q^29 - 34 * q^33 + 98 * q^37 - 192 * q^41 - 16 * q^45 - 196 * q^49 + 50 * q^53 + 28 * q^57 + 2 * q^61 - 48 * q^65 - 28 * q^69 - 190 * q^73 - 238 * q^77 - 110 * q^81 + 4 * q^85 - 190 * q^89 - 82 * q^93 + 576 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-\zeta_{12}^{2}$	$1$

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} $

95.1

−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
0.866025	−	0.500000i

−0.866025

−

0.500000i

−0.500000

−

0.866025i

7.00000i

−4.00000

−

6.92820i

95.2

0.866025

0.500000i

−0.500000

−

0.866025i

−

7.00000i

−4.00000

−

6.92820i

191.1

−0.866025

0.500000i

−0.500000

0.866025i

−

7.00000i

−4.00000

6.92820i

191.2

0.866025

−

0.500000i

−0.500000

0.866025i

7.00000i

−4.00000

6.92820i

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	224.3.r.b	✓	4
4.b	odd	2	1	inner	224.3.r.b	✓	4
7.c	even	3	1	inner	224.3.r.b	✓	4
7.c	even	3	1		1568.3.d.d		2
7.d	odd	6	1		1568.3.d.c		2
8.b	even	2	1		448.3.r.b		4
8.d	odd	2	1		448.3.r.b		4
28.f	even	6	1		1568.3.d.c		2
28.g	odd	6	1	inner	224.3.r.b	✓	4
28.g	odd	6	1		1568.3.d.d		2
56.k	odd	6	1		448.3.r.b		4
56.p	even	6	1		448.3.r.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.3.r.b	✓	4	1.a	even	1	1	trivial
224.3.r.b	✓	4	4.b	odd	2	1	inner
224.3.r.b	✓	4	7.c	even	3	1	inner
224.3.r.b	✓	4	28.g	odd	6	1	inner
448.3.r.b		4	8.b	even	2	1
448.3.r.b		4	8.d	odd	2	1
448.3.r.b		4	56.k	odd	6	1
448.3.r.b		4	56.p	even	6	1
1568.3.d.c		2	7.d	odd	6	1
1568.3.d.c		2	28.f	even	6	1
1568.3.d.d		2	7.c	even	3	1
1568.3.d.d		2	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} - T_{3}^{2} + 1 $ T3^4 - T3^2 + 1 acting on $S_{3}^{\mathrm{new}}(224, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} - T^{2} + 1 $ T^4 - T^2 + 1
$5$	$ (T^{2} + T + 1)^{2} $ (T^2 + T + 1)^2
$7$	$ (T^{2} + 49)^{2} $ (T^2 + 49)^2
$11$	$ T^{4} - 289 T^{2} + 83521 $ T^4 - 289*T^2 + 83521
$13$	$ (T - 24)^{4} $ (T - 24)^4
$17$	$ (T^{2} + T + 1)^{2} $ (T^2 + T + 1)^2
$19$	$ T^{4} - 49T^{2} + 2401 $ T^4 - 49*T^2 + 2401
$23$	$ T^{4} - 49T^{2} + 2401 $ T^4 - 49*T^2 + 2401
$29$	$ (T - 24)^{4} $ (T - 24)^4
$31$	$ T^{4} - 1681 T^{2} + \cdots + 2825761 $ T^4 - 1681*T^2 + 2825761
$37$	$ (T^{2} - 49 T + 2401)^{2} $ (T^2 - 49*T + 2401)^2
$41$	$ (T + 48)^{4} $ (T + 48)^4
$43$	$ (T^{2} + 576)^{2} $ (T^2 + 576)^2
$47$	$ T^{4} - 3025 T^{2} + \cdots + 9150625 $ T^4 - 3025*T^2 + 9150625
$53$	$ (T^{2} - 25 T + 625)^{2} $ (T^2 - 25*T + 625)^2
$59$	$ T^{4} - 289 T^{2} + 83521 $ T^4 - 289*T^2 + 83521
$61$	$ (T^{2} - T + 1)^{2} $ (T^2 - T + 1)^2
$67$	$ T^{4} - 4225 T^{2} + \cdots + 17850625 $ T^4 - 4225*T^2 + 17850625
$71$	$ (T^{2} + 9216)^{2} $ (T^2 + 9216)^2
$73$	$ (T^{2} + 95 T + 9025)^{2} $ (T^2 + 95*T + 9025)^2
$79$	$ T^{4} - 1681 T^{2} + \cdots + 2825761 $ T^4 - 1681*T^2 + 2825761
$83$	$ (T^{2} + 5184)^{2} $ (T^2 + 5184)^2
$89$	$ (T^{2} + 95 T + 9025)^{2} $ (T^2 + 95*T + 9025)^2
$97$	$ (T - 144)^{4} $ (T - 144)^4
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.r (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{12} q^{3} - \zeta_{12}^{2} q^{5} - 7 \zeta_{12}^{3} q^{7} - 8 \zeta_{12}^{2} q^{9} +O(q^{10}) \) q + z * q^3 - z^2 * q^5 - 7z^3 q^7 - 8z^2 q^9	\( q + \zeta_{12} q^{3} - \zeta_{12}^{2} q^{5} - 7 \zeta_{12}^{3} q^{7} - 8 \zeta_{12}^{2} q^{9} - 17 \zeta_{12} q^{11} + 24 q^{13} - \zeta_{12}^{3} q^{15} + (\zeta_{12}^{2} - 1) q^{17} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{19} + ( - 7 \zeta_{12}^{2} + 7) q^{21} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{23} + ( - 24 \zeta_{12}^{2} + 24) q^{25} - 17 \zeta_{12}^{3} q^{27} + 24 q^{29} - 41 \zeta_{12} q^{31} - 17 \zeta_{12}^{2} q^{33} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{35} + 49 \zeta_{12}^{2} q^{37} + 24 \zeta_{12} q^{39} - 48 q^{41} + 24 \zeta_{12}^{3} q^{43} + (8 \zeta_{12}^{2} - 8) q^{45} + ( - 55 \zeta_{12}^{3} + 55 \zeta_{12}) q^{47} - 49 q^{49} + (\zeta_{12}^{3} - \zeta_{12}) q^{51} + ( - 25 \zeta_{12}^{2} + 25) q^{53} + 17 \zeta_{12}^{3} q^{55} + 7 q^{57} + 17 \zeta_{12} q^{59} + \zeta_{12}^{2} q^{61} + (56 \zeta_{12}^{3} - 56 \zeta_{12}) q^{63} - 24 \zeta_{12}^{2} q^{65} + 65 \zeta_{12} q^{67} - 7 q^{69} + 96 \zeta_{12}^{3} q^{71} + (95 \zeta_{12}^{2} - 95) q^{73} + ( - 24 \zeta_{12}^{3} + 24 \zeta_{12}) q^{75} + (119 \zeta_{12}^{2} - 119) q^{77} + ( - 41 \zeta_{12}^{3} + 41 \zeta_{12}) q^{79} + (55 \zeta_{12}^{2} - 55) q^{81} + 72 \zeta_{12}^{3} q^{83} + q^{85} + 24 \zeta_{12} q^{87} - 95 \zeta_{12}^{2} q^{89} - 168 \zeta_{12}^{3} q^{91} - 41 \zeta_{12}^{2} q^{93} - 7 \zeta_{12} q^{95} + 144 q^{97} + 136 \zeta_{12}^{3} q^{99} +O(q^{100}) \) q + z * q^3 - z^2 * q^5 - 7z^3 q^7 - 8z^2 q^9 - 17z q^11 + 24 * q^13 - z^3 * q^15 + (z^2 - 1) * q^17 + (-7z^3 + 7z) * q^19 + (-7z^2 + 7) q^21 + (7z^3 - 7z) * q^23 + (-24z^2 + 24) q^25 - 17z^3 q^27 + 24 * q^29 - 41z q^31 - 17z^2 q^33 + (7z^3 - 7z) * q^35 + 49z^2 q^37 + 24z q^39 - 48 * q^41 + 24z^3 q^43 + (8z^2 - 8) q^45 + (-55z^3 + 55z) * q^47 - 49 * q^49 + (z^3 - z) * q^51 + (-25z^2 + 25) q^53 + 17z^3 q^55 + 7 * q^57 + 17z q^59 + z^2 * q^61 + (56z^3 - 56z) * q^63 - 24z^2 q^65 + 65z q^67 - 7 * q^69 + 96z^3 q^71 + (95z^2 - 95) q^73 + (-24z^3 + 24z) * q^75 + (119z^2 - 119) q^77 + (-41z^3 + 41z) * q^79 + (55z^2 - 55) q^81 + 72z^3 q^83 + q^85 + 24z q^87 - 95z^2 q^89 - 168z^3 q^91 - 41z^2 q^93 - 7z q^95 + 144 * q^97 + 136z^3 q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} - 16 q^{9}+O(q^{10}) \) 4 * q - 2 * q^5 - 16 * q^9	\( 4 q - 2 q^{5} - 16 q^{9} + 96 q^{13} - 2 q^{17} + 14 q^{21} + 48 q^{25} + 96 q^{29} - 34 q^{33} + 98 q^{37} - 192 q^{41} - 16 q^{45} - 196 q^{49} + 50 q^{53} + 28 q^{57} + 2 q^{61} - 48 q^{65} - 28 q^{69} - 190 q^{73} - 238 q^{77} - 110 q^{81} + 4 q^{85} - 190 q^{89} - 82 q^{93} + 576 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 - 16 * q^9 + 96 * q^13 - 2 * q^17 + 14 * q^21 + 48 * q^25 + 96 * q^29 - 34 * q^33 + 98 * q^37 - 192 * q^41 - 16 * q^45 - 196 * q^49 + 50 * q^53 + 28 * q^57 + 2 * q^61 - 48 * q^65 - 28 * q^69 - 190 * q^73 - 238 * q^77 - 110 * q^81 + 4 * q^85 - 190 * q^89 - 82 * q^93 + 576 * q^97

Introduction

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Modular forms

Varieties

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Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	224.3.r.b

Level	$224$
Weight	$3$
Character orbit	224.r
Analytic conductor	$6.104$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \) x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{12}^{2}\)	\(1\)

$p$	$F_p(T)$
$2$	\( T^{4} \) T^4
$3$	\( T^{4} - T^{2} + 1 \) T^4 - T^2 + 1
$5$	\( (T^{2} + T + 1)^{2} \) (T^2 + T + 1)^2
$7$	\( (T^{2} + 49)^{2} \) (T^2 + 49)^2
$11$	\( T^{4} - 289 T^{2} + 83521 \) T^4 - 289*T^2 + 83521
$13$	\( (T - 24)^{4} \) (T - 24)^4
$17$	\( (T^{2} + T + 1)^{2} \) (T^2 + T + 1)^2
$19$	\( T^{4} - 49T^{2} + 2401 \) T^4 - 49*T^2 + 2401
$23$	\( T^{4} - 49T^{2} + 2401 \) T^4 - 49*T^2 + 2401
$29$	\( (T - 24)^{4} \) (T - 24)^4
$31$	\( T^{4} - 1681 T^{2} + \cdots + 2825761 \) T^4 - 1681*T^2 + 2825761
$37$	\( (T^{2} - 49 T + 2401)^{2} \) (T^2 - 49*T + 2401)^2
$41$	\( (T + 48)^{4} \) (T + 48)^4
$43$	\( (T^{2} + 576)^{2} \) (T^2 + 576)^2
$47$	\( T^{4} - 3025 T^{2} + \cdots + 9150625 \) T^4 - 3025*T^2 + 9150625
$53$	\( (T^{2} - 25 T + 625)^{2} \) (T^2 - 25*T + 625)^2
$59$	\( T^{4} - 289 T^{2} + 83521 \) T^4 - 289*T^2 + 83521
$61$	\( (T^{2} - T + 1)^{2} \) (T^2 - T + 1)^2
$67$	\( T^{4} - 4225 T^{2} + \cdots + 17850625 \) T^4 - 4225*T^2 + 17850625
$71$	\( (T^{2} + 9216)^{2} \) (T^2 + 9216)^2
$73$	\( (T^{2} + 95 T + 9025)^{2} \) (T^2 + 95*T + 9025)^2
$79$	\( T^{4} - 1681 T^{2} + \cdots + 2825761 \) T^4 - 1681*T^2 + 2825761
$83$	\( (T^{2} + 5184)^{2} \) (T^2 + 5184)^2
$89$	\( (T^{2} + 95 T + 9025)^{2} \) (T^2 + 95*T + 9025)^2
$97$	\( (T - 144)^{4} \) (T - 144)^4
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\(n\):		e.g. 2-40 or 990-1000
Significant digits:
Format: