LMFDB - Newform orbit 441.6.c.a

Newspace parameters

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 63)
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 - 2 \beta_1 q^{2} + 24 q^{4} - 3 \beta_{3} q^{5} - 112 \beta_1 q^{8}+O(q^{10}) $ q - 2b1 q^2 + 24 * q^4 - 3b3 q^5 - 112b1 q^8	\( q - 2 \beta_1 q^{2} + 24 q^{4} - 3 \beta_{3} q^{5} - 112 \beta_1 q^{8} + 12 \beta_{2} q^{10} + 409 \beta_1 q^{11} + 95 \beta_{2} q^{13} + 320 q^{16} - 22 \beta_{3} q^{17} - 143 \beta_{2} q^{19} - 72 \beta_{3} q^{20} + 1636 q^{22} - 3260 \beta_1 q^{23} - 479 q^{25} + 190 \beta_{3} q^{26} - 3254 \beta_1 q^{29} + 187 \beta_{2} q^{31} - 4224 \beta_1 q^{32} + 88 \beta_{2} q^{34} + 203 q^{37} - 286 \beta_{3} q^{38} + 672 \beta_{2} q^{40} + 5 \beta_{3} q^{41} + 4697 q^{43} + 9816 \beta_1 q^{44} - 13040 q^{46} - 1399 \beta_{3} q^{47} + 958 \beta_1 q^{50} + 2280 \beta_{2} q^{52} - 5246 \beta_1 q^{53} - 2454 \beta_{2} q^{55} - 13016 q^{58} - 290 \beta_{3} q^{59} - 1438 \beta_{2} q^{61} + 374 \beta_{3} q^{62} - 6656 q^{64} - 41895 \beta_1 q^{65} + 30197 q^{67} - 528 \beta_{3} q^{68} - 2801 \beta_1 q^{71} - 4399 \beta_{2} q^{73} - 406 \beta_1 q^{74} - 3432 \beta_{2} q^{76} - 39385 q^{79} - 960 \beta_{3} q^{80} - 20 \beta_{2} q^{82} - 3707 \beta_{3} q^{83} + 19404 q^{85} - 9394 \beta_1 q^{86} + 91616 q^{88} - 6146 \beta_{3} q^{89} - 78240 \beta_1 q^{92} + 5596 \beta_{2} q^{94} + 63063 \beta_1 q^{95} - 170 \beta_{2} q^{97}+O(q^{100}) \) q - 2b1 q^2 + 24 * q^4 - 3b3 q^5 - 112b1 q^8 + 12b2 q^10 + 409b1 q^11 + 95b2 q^13 + 320 * q^16 - 22b3 q^17 - 143b2 q^19 - 72b3 q^20 + 1636 * q^22 - 3260b1 q^23 - 479 * q^25 + 190b3 q^26 - 3254b1 q^29 + 187b2 q^31 - 4224b1 q^32 + 88b2 q^34 + 203 * q^37 - 286b3 q^38 + 672b2 q^40 + 5b3 q^41 + 4697 * q^43 + 9816b1 q^44 - 13040 * q^46 - 1399b3 q^47 + 958b1 q^50 + 2280b2 q^52 - 5246b1 q^53 - 2454b2 q^55 - 13016 * q^58 - 290b3 q^59 - 1438b2 q^61 + 374b3 q^62 - 6656 * q^64 - 41895b1 q^65 + 30197 * q^67 - 528b3 q^68 - 2801b1 q^71 - 4399b2 q^73 - 406b1 q^74 - 3432b2 q^76 - 39385 * q^79 - 960b3 q^80 - 20b2 q^82 - 3707b3 q^83 + 19404 * q^85 - 9394b1 q^86 + 91616 * q^88 - 6146b3 q^89 - 78240b1 q^92 + 5596b2 q^94 + 63063b1 q^95 - 170b2 q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 96 q^{4}+O(q^{10}) $ 4 * q + 96 * q^4	$ 4 q + 96 q^{4} + 1280 q^{16} + 6544 q^{22} - 1916 q^{25} + 812 q^{37} + 18788 q^{43} - 52160 q^{46} - 52064 q^{58} - 26624 q^{64} + 120788 q^{67} - 157540 q^{79} + 77616 q^{85} + 366464 q^{88}+O(q^{100}) $ 4 * q + 96 * q^4 + 1280 * q^16 + 6544 * q^22 - 1916 * q^25 + 812 * q^37 + 18788 * q^43 - 52160 * q^46 - 52064 * q^58 - 26624 * q^64 + 120788 * q^67 - 157540 * q^79 + 77616 * q^85 + 366464 * q^88

Display 100 coefficients 10 coefficients

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

$\beta_{1}$	$=$	$ ( \nu^{3} ) / 2 $ (v^3) / 2
$\beta_{2}$	$=$	$ 7\nu^{2} - 7 $ 7*v^2 - 7
$\beta_{3}$	$=$	$ ( -7\nu^{3} + 28\nu ) / 2 $ (-7v^3 + 28v) / 2

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{3} + 7\beta_1 ) / 14 $ (b3 + 7*b1) / 14
$\nu^{2}$	$=$	$ ( \beta_{2} + 7 ) / 7 $ (b2 + 7) / 7
$\nu^{3}$	$=$	$ 2\beta_1 $ 2*b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$-1$

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} $

440.1

1.22474	+	0.707107i
−1.22474	+	0.707107i
1.22474	−	0.707107i
−1.22474	−	0.707107i

−

2.82843i

24.0000

−51.4393

−

158.392i

145.492i

440.2

−

2.82843i

24.0000

51.4393

−

158.392i

−

145.492i

440.3

2.82843i

24.0000

−51.4393

158.392i

−

145.492i

440.4

2.82843i

24.0000

51.4393

158.392i

145.492i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	441.6.c.a		4
3.b	odd	2	1	inner	441.6.c.a		4
7.b	odd	2	1	inner	441.6.c.a		4
7.c	even	3	1		63.6.p.a	✓	4
7.d	odd	6	1		63.6.p.a	✓	4
21.c	even	2	1	inner	441.6.c.a		4
21.g	even	6	1		63.6.p.a	✓	4
21.h	odd	6	1		63.6.p.a	✓	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.6.p.a	✓	4	7.c	even	3	1
63.6.p.a	✓	4	7.d	odd	6	1
63.6.p.a	✓	4	21.g	even	6	1
63.6.p.a	✓	4	21.h	odd	6	1
441.6.c.a		4	1.a	even	1	1	trivial
441.6.c.a		4	3.b	odd	2	1	inner
441.6.c.a		4	7.b	odd	2	1	inner
441.6.c.a		4	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} + 8 $ T2^2 + 8 acting on $S_{6}^{\mathrm{new}}(441, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 8)^{2} $ (T^2 + 8)^2
$3$	$ T^{4} $ T^4
$5$	$ (T^{2} - 2646)^{2} $ (T^2 - 2646)^2
$7$	$ T^{4} $ T^4
$11$	$ (T^{2} + 334562)^{2} $ (T^2 + 334562)^2
$13$	$ (T^{2} + 1326675)^{2} $ (T^2 + 1326675)^2
$17$	$ (T^{2} - 142296)^{2} $ (T^2 - 142296)^2
$19$	$ (T^{2} + 3006003)^{2} $ (T^2 + 3006003)^2
$23$	$ (T^{2} + 21255200)^{2} $ (T^2 + 21255200)^2
$29$	$ (T^{2} + 21177032)^{2} $ (T^2 + 21177032)^2
$31$	$ (T^{2} + 5140443)^{2} $ (T^2 + 5140443)^2
$37$	$ (T - 203)^{4} $ (T - 203)^4
$41$	$ (T^{2} - 7350)^{2} $ (T^2 - 7350)^2
$43$	$ (T - 4697)^{4} $ (T - 4697)^4
$47$	$ (T^{2} - 575417094)^{2} $ (T^2 - 575417094)^2
$53$	$ (T^{2} + 55041032)^{2} $ (T^2 + 55041032)^2
$59$	$ (T^{2} - 24725400)^{2} $ (T^2 - 24725400)^2
$61$	$ (T^{2} + 303973068)^{2} $ (T^2 + 303973068)^2
$67$	$ (T - 30197)^{4} $ (T - 30197)^4
$71$	$ (T^{2} + 15691202)^{2} $ (T^2 + 15691202)^2
$73$	$ (T^{2} + 2844626547)^{2} $ (T^2 + 2844626547)^2
$79$	$ (T + 39385)^{4} $ (T + 39385)^4
$83$	$ (T^{2} - 4040103606)^{2} $ (T^2 - 4040103606)^2
$89$	$ (T^{2} - 11105354904)^{2} $ (T^2 - 11105354904)^2
$97$	$ (T^{2} + 4248300)^{2} $ (T^2 + 4248300)^2
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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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - 2 \beta_1 q^{2} + 24 q^{4} - 3 \beta_{3} q^{5} - 112 \beta_1 q^{8}+O(q^{10}) \) q - 2b1 q^2 + 24 * q^4 - 3b3 q^5 - 112b1 q^8	\( q - 2 \beta_1 q^{2} + 24 q^{4} - 3 \beta_{3} q^{5} - 112 \beta_1 q^{8} + 12 \beta_{2} q^{10} + 409 \beta_1 q^{11} + 95 \beta_{2} q^{13} + 320 q^{16} - 22 \beta_{3} q^{17} - 143 \beta_{2} q^{19} - 72 \beta_{3} q^{20} + 1636 q^{22} - 3260 \beta_1 q^{23} - 479 q^{25} + 190 \beta_{3} q^{26} - 3254 \beta_1 q^{29} + 187 \beta_{2} q^{31} - 4224 \beta_1 q^{32} + 88 \beta_{2} q^{34} + 203 q^{37} - 286 \beta_{3} q^{38} + 672 \beta_{2} q^{40} + 5 \beta_{3} q^{41} + 4697 q^{43} + 9816 \beta_1 q^{44} - 13040 q^{46} - 1399 \beta_{3} q^{47} + 958 \beta_1 q^{50} + 2280 \beta_{2} q^{52} - 5246 \beta_1 q^{53} - 2454 \beta_{2} q^{55} - 13016 q^{58} - 290 \beta_{3} q^{59} - 1438 \beta_{2} q^{61} + 374 \beta_{3} q^{62} - 6656 q^{64} - 41895 \beta_1 q^{65} + 30197 q^{67} - 528 \beta_{3} q^{68} - 2801 \beta_1 q^{71} - 4399 \beta_{2} q^{73} - 406 \beta_1 q^{74} - 3432 \beta_{2} q^{76} - 39385 q^{79} - 960 \beta_{3} q^{80} - 20 \beta_{2} q^{82} - 3707 \beta_{3} q^{83} + 19404 q^{85} - 9394 \beta_1 q^{86} + 91616 q^{88} - 6146 \beta_{3} q^{89} - 78240 \beta_1 q^{92} + 5596 \beta_{2} q^{94} + 63063 \beta_1 q^{95} - 170 \beta_{2} q^{97}+O(q^{100}) \) q - 2b1 q^2 + 24 * q^4 - 3b3 q^5 - 112b1 q^8 + 12b2 q^10 + 409b1 q^11 + 95b2 q^13 + 320 * q^16 - 22b3 q^17 - 143b2 q^19 - 72b3 q^20 + 1636 * q^22 - 3260b1 q^23 - 479 * q^25 + 190b3 q^26 - 3254b1 q^29 + 187b2 q^31 - 4224b1 q^32 + 88b2 q^34 + 203 * q^37 - 286b3 q^38 + 672b2 q^40 + 5b3 q^41 + 4697 * q^43 + 9816b1 q^44 - 13040 * q^46 - 1399b3 q^47 + 958b1 q^50 + 2280b2 q^52 - 5246b1 q^53 - 2454b2 q^55 - 13016 * q^58 - 290b3 q^59 - 1438b2 q^61 + 374b3 q^62 - 6656 * q^64 - 41895b1 q^65 + 30197 * q^67 - 528b3 q^68 - 2801b1 q^71 - 4399b2 q^73 - 406b1 q^74 - 3432b2 q^76 - 39385 * q^79 - 960b3 q^80 - 20b2 q^82 - 3707b3 q^83 + 19404 * q^85 - 9394b1 q^86 + 91616 * q^88 - 6146b3 q^89 - 78240b1 q^92 + 5596b2 q^94 + 63063b1 q^95 - 170b2 q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 96 q^{4}+O(q^{10}) \) 4 * q + 96 * q^4	\( 4 q + 96 q^{4} + 1280 q^{16} + 6544 q^{22} - 1916 q^{25} + 812 q^{37} + 18788 q^{43} - 52160 q^{46} - 52064 q^{58} - 26624 q^{64} + 120788 q^{67} - 157540 q^{79} + 77616 q^{85} + 366464 q^{88}+O(q^{100}) \) 4 * q + 96 * q^4 + 1280 * q^16 + 6544 * q^22 - 1916 * q^25 + 812 * q^37 + 18788 * q^43 - 52160 * q^46 - 52064 * q^58 - 26624 * q^64 + 120788 * q^67 - 157540 * q^79 + 77616 * q^85 + 366464 * q^88

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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	441.6.c.a

Level	$441$
Weight	$6$
Character orbit	441.c
Analytic conductor	$70.729$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(70.7292645375\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} - 2x^{2} + 4 \) x^4 - 2*x^2 + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{2}\cdot 7^{2} \)
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{3} ) / 2 \) (v^3) / 2
\(\beta_{2}\)	\(=\)	\( 7\nu^{2} - 7 \) 7*v^2 - 7
\(\beta_{3}\)	\(=\)	\( ( -7\nu^{3} + 28\nu ) / 2 \) (-7v^3 + 28v) / 2

\(\nu\)	\(=\)	\( ( \beta_{3} + 7\beta_1 ) / 14 \) (b3 + 7*b1) / 14
\(\nu^{2}\)	\(=\)	\( ( \beta_{2} + 7 ) / 7 \) (b2 + 7) / 7
\(\nu^{3}\)	\(=\)	\( 2\beta_1 \) 2*b1

$p$	$F_p(T)$
$2$	\( (T^{2} + 8)^{2} \) (T^2 + 8)^2
$3$	\( T^{4} \) T^4
$5$	\( (T^{2} - 2646)^{2} \) (T^2 - 2646)^2
$7$	\( T^{4} \) T^4
$11$	\( (T^{2} + 334562)^{2} \) (T^2 + 334562)^2
$13$	\( (T^{2} + 1326675)^{2} \) (T^2 + 1326675)^2
$17$	\( (T^{2} - 142296)^{2} \) (T^2 - 142296)^2
$19$	\( (T^{2} + 3006003)^{2} \) (T^2 + 3006003)^2
$23$	\( (T^{2} + 21255200)^{2} \) (T^2 + 21255200)^2
$29$	\( (T^{2} + 21177032)^{2} \) (T^2 + 21177032)^2
$31$	\( (T^{2} + 5140443)^{2} \) (T^2 + 5140443)^2
$37$	\( (T - 203)^{4} \) (T - 203)^4
$41$	\( (T^{2} - 7350)^{2} \) (T^2 - 7350)^2
$43$	\( (T - 4697)^{4} \) (T - 4697)^4
$47$	\( (T^{2} - 575417094)^{2} \) (T^2 - 575417094)^2
$53$	\( (T^{2} + 55041032)^{2} \) (T^2 + 55041032)^2
$59$	\( (T^{2} - 24725400)^{2} \) (T^2 - 24725400)^2
$61$	\( (T^{2} + 303973068)^{2} \) (T^2 + 303973068)^2
$67$	\( (T - 30197)^{4} \) (T - 30197)^4
$71$	\( (T^{2} + 15691202)^{2} \) (T^2 + 15691202)^2
$73$	\( (T^{2} + 2844626547)^{2} \) (T^2 + 2844626547)^2
$79$	\( (T + 39385)^{4} \) (T + 39385)^4
$83$	\( (T^{2} - 4040103606)^{2} \) (T^2 - 4040103606)^2
$89$	\( (T^{2} - 11105354904)^{2} \) (T^2 - 11105354904)^2
$97$	\( (T^{2} + 4248300)^{2} \) (T^2 + 4248300)^2
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\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
Format: