LMFDB - Newform orbit 544.4.b.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [544,4,Mod(33,544)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("544.33"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 544 = 2^{5} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	544.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-8,0,0,0,208,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.0970390431$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 60x^{6} + 866x^{4} + 3840x^{2} + 4096 $ x^8 + 60x^6 + 866x^4 + 3840*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

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_{3} q^{3} - \beta_1 q^{5} + ( - \beta_{4} + \beta_{3}) q^{7} + ( - 3 \beta_{2} - 1) q^{9} + (4 \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{2} + 26) q^{13} + \beta_{6} q^{15} + (\beta_{7} - 5 \beta_{2} - 7) q^{17}+ \cdots + (86 \beta_{4} - 73 \beta_{3}) q^{99}+O(q^{100}) $ q + b3 * q^3 - b1 * q^5 + (-b4 + b3) * q^7 + (-3b2 - 1) q^9 + (4b4 + b3) q^11 + (-b2 + 26) * q^13 + b6 * q^15 + (b7 - 5b2 - 7) q^17 - b5 * q^19 + (-b2 - 32) * q^21 + (3b4 + 17b3) * q^23 + (-6b2 - 59) q^25 + (-6b4 + 2b3) * q^27 + (-2b7 + b1) q^29 + (9b4 + 17b3) * q^31 + (-11b2 - 12) q^33 + (b6 + b5) * q^35 + (-4b7 + 5b1) * q^37 + (-2b4 + 18b3) * q^39 + (-2b7 - 4b1) * q^41 + (-b6 + 2b5) q^43 + (-6b7 + 7b1) * q^45 + 3b6 q^47 + (11b2 + 219) q^49 + (3b6 - b5 - 10b4 - 47b3) q^51 + (-8b2 - 58) q^53 + (b6 - 4b5) q^55 - 4b7 q^57 + (-5b6 + 4b5) * q^59 + (-10b7 - 11b1) * q^61 + (-29b4 - 13b3) * q^63 + (-2b7 - 24b1) * q^65 + (-10b6 - 5b5) * q^67 + (-57b2 - 464) q^69 + (37b4 + 21b3) * q^71 + (-6b7 + 30b1) * q^73 + (-12b4 - 107b3) * q^75 + (-49b2 + 336) q^77 + (79b4 - 27b3) * q^79 + (-75b2 - 107) q^81 + (-b6 + 4b5) q^83 + (-10b7 - 86b2 + 17b1 - 32) q^85 + (-7b6 + 2b5) * q^87 + (39b2 - 390) q^89 + (-36b4 + 22b3) * q^91 + (-69b2 - 440) q^93 + (-136b4 - 24b3) * q^95 - 2b1 q^97 + (86b4 - 73b3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} + 208 q^{13} - 56 q^{17} - 256 q^{21} - 472 q^{25} - 96 q^{33} + 1752 q^{49} - 464 q^{53} - 3712 q^{69} + 2688 q^{77} - 856 q^{81} - 256 q^{85} - 3120 q^{89} - 3520 q^{93}+O(q^{100}) $ 8 * q - 8 * q^9 + 208 * q^13 - 56 * q^17 - 256 * q^21 - 472 * q^25 - 96 * q^33 + 1752 * q^49 - 464 * q^53 - 3712 * q^69 + 2688 * q^77 - 856 * q^81 - 256 * q^85 - 3120 * q^89 - 3520 * q^93

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 60x^{6} + 866x^{4} + 3840x^{2} + 4096 $ x^8 + 60*x^6 + 866*x^4 + 3840*x^2 + 4096 :

$\beta_{1}$	$=$	$ ( \nu^{7} + 60\nu^{5} + 866\nu^{3} + 4352\nu ) / 256 $ (v^7 + 60v^5 + 866v^3 + 4352*v) / 256
$\beta_{2}$	$=$	$ ( \nu^{6} + 60\nu^{4} + 802\nu^{2} + 1920 ) / 96 $ (v^6 + 60v^4 + 802v^2 + 1920) / 96
$\beta_{3}$	$=$	$ ( -11\nu^{7} - 628\nu^{5} - 7862\nu^{3} - 24256\nu ) / 2304 $ (-11v^7 - 628v^5 - 7862v^3 - 24256v) / 2304
$\beta_{4}$	$=$	$ ( 17\nu^{7} + 892\nu^{5} + 8066\nu^{3} + 7168\nu ) / 2304 $ (17v^7 + 892v^5 + 8066v^3 + 7168v) / 2304
$\beta_{5}$	$=$	$ ( -33\nu^{6} - 1724\nu^{4} - 15330\nu^{2} - 15872 ) / 144 $ (-33v^6 - 1724v^4 - 15330*v^2 - 15872) / 144
$\beta_{6}$	$=$	$ ( -15\nu^{6} - 836\nu^{4} - 10110\nu^{2} - 29888 ) / 144 $ (-15v^6 - 836v^4 - 10110*v^2 - 29888) / 144
$\beta_{7}$	$=$	$ ( -35\nu^{7} - 2036\nu^{5} - 25958\nu^{3} - 66176\nu ) / 768 $ (-35v^7 - 2036v^5 - 25958v^3 - 66176v) / 768

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

$\nu$	$=$	$ ( \beta_{4} + 4\beta_{3} + 3\beta_1 ) / 12 $ (b4 + 4b3 + 3b1) / 12
$\nu^{2}$	$=$	$ ( -4\beta_{6} + \beta_{5} - 18\beta_{2} - 360 ) / 24 $ (-4b6 + b5 - 18b2 - 360) / 24
$\nu^{3}$	$=$	$ ( 9\beta_{7} - 38\beta_{4} - 206\beta_{3} - 75\beta_1 ) / 12 $ (9b7 - 38b4 - 206b3 - 75b1) / 12
$\nu^{4}$	$=$	$ ( 29\beta_{6} - 5\beta_{5} + 180\beta_{2} + 1868 ) / 4 $ (29b6 - 5b5 + 180*b2 + 1868) / 4
$\nu^{5}$	$=$	$ ( -234\beta_{7} + 619\beta_{4} + 4312\beta_{3} + 1371\beta_1 ) / 6 $ (-234b7 + 619b4 + 4312b3 + 1371b1) / 6
$\nu^{6}$	$=$	$ ( -3616\beta_{6} + 499\beta_{5} - 24030\beta_{2} - 214920 ) / 12 $ (-3616b6 + 499b5 - 24030*b2 - 214920) / 12
$\nu^{7}$	$=$	$ ( 10143\beta_{7} - 22862\beta_{4} - 178226\beta_{3} - 54777\beta_1 ) / 6 $ (10143b7 - 22862b4 - 178226b3 - 54777b1) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/544\mathbb{Z}\right)^\times$.

$n$	$69$	$511$	$513$
$\chi(n)$	$1$	$1$	$-1$

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

33.1

		1.24675i
	−	6.41666i
		3.44803i
	−	2.32017i
		2.32017i
	−	3.44803i
		6.41666i
	−	1.24675i

−

7.31135i

−

15.3268i

−

5.53732i

−26.4558

33.2

−

7.31135i

15.3268i

−

5.53732i

−26.4558

33.3

−

1.59504i

−

11.5364i

−

14.7424i

24.4558

33.4

−

1.59504i

11.5364i

−

14.7424i

24.4558

33.5

1.59504i

−

11.5364i

14.7424i

24.4558

33.6

1.59504i

11.5364i

14.7424i

24.4558

33.7

7.31135i

−

15.3268i

5.53732i

−26.4558

33.8

7.31135i

15.3268i

5.53732i

−26.4558

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
17.b	even	2	1	inner
68.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	544.4.b.e	✓	8
4.b	odd	2	1	inner	544.4.b.e	✓	8
17.b	even	2	1	inner	544.4.b.e	✓	8
68.d	odd	2	1	inner	544.4.b.e	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
544.4.b.e	✓	8	1.a	even	1	1	trivial
544.4.b.e	✓	8	4.b	odd	2	1	inner
544.4.b.e	✓	8	17.b	even	2	1	inner
544.4.b.e	✓	8	68.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(544, [\chi])$:

$ T_{3}^{4} + 56T_{3}^{2} + 136 $

T3^4 + 56*T3^2 + 136

$ T_{19}^{4} - 23744T_{19}^{2} + 17007616 $

T19^4 - 23744*T19^2 + 17007616

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ (T^{4} + 56 T^{2} + 136)^{2} $ (T^4 + 56*T^2 + 136)^2
$5$	$ (T^{4} + 368 T^{2} + 31264)^{2} $ (T^4 + 368*T^2 + 31264)^2
$7$	$ (T^{4} + 248 T^{2} + 6664)^{2} $ (T^4 + 248*T^2 + 6664)^2
$11$	$ (T^{4} + 2808 T^{2} + 539784)^{2} $ (T^4 + 2808*T^2 + 539784)^2
$13$	$ (T^{2} - 52 T + 604)^{4} $ (T^2 - 52*T + 604)^4
$17$	$ (T^{4} + 28 T^{3} + \cdots + 24137569)^{2} $ (T^4 + 28T^3 + 2822T^2 + 137564*T + 24137569)^2
$19$	$ (T^{4} - 23744 T^{2} + 17007616)^{2} $ (T^4 - 23744*T^2 + 17007616)^2
$23$	$ (T^{4} + 16952 T^{2} + 2552584)^{2} $ (T^4 + 16952*T^2 + 2552584)^2
$29$	$ (T^{4} + 25136 T^{2} + 157601824)^{2} $ (T^4 + 25136*T^2 + 157601824)^2
$31$	$ (T^{4} + 27992 T^{2} + 163663624)^{2} $ (T^4 + 27992*T^2 + 163663624)^2
$37$	$ (T^{4} + 109808 T^{2} + 2575184416)^{2} $ (T^4 + 109808*T^2 + 2575184416)^2
$41$	$ (T^{4} + 29376 T^{2} + 40518144)^{2} $ (T^4 + 29376*T^2 + 40518144)^2
$43$	$ (T^{4} - 108896 T^{2} + 2249257216)^{2} $ (T^4 - 108896*T^2 + 2249257216)^2
$47$	$ (T^{4} - 116064 T^{2} + 344404224)^{2} $ (T^4 - 116064*T^2 + 344404224)^2
$53$	$ (T^{2} + 116 T - 1244)^{4} $ (T^2 + 116*T - 1244)^4
$59$	$ (T^{4} - 712544 T^{2} + 118581350656)^{2} $ (T^4 - 712544*T^2 + 118581350656)^2
$61$	$ (T^{4} + 643248 T^{2} + 57740887584)^{2} $ (T^4 + 643248*T^2 + 57740887584)^2
$67$	$ (T^{4} - 1857600 T^{2} + 861010560000)^{2} $ (T^4 - 1857600*T^2 + 861010560000)^2
$71$	$ (T^{4} + 253208 T^{2} + 9856078984)^{2} $ (T^4 + 253208*T^2 + 9856078984)^2
$73$	$ (T^{4} + 574848 T^{2} + 162072576)^{2} $ (T^4 + 574848*T^2 + 162072576)^2
$79$	$ (T^{4} + 1173368 T^{2} + 3836114056)^{2} $ (T^4 + 1173368*T^2 + 3836114056)^2
$83$	$ (T^{4} - 394848 T^{2} + 16875806976)^{2} $ (T^4 - 394848*T^2 + 16875806976)^2
$89$	$ (T^{2} + 780 T + 42588)^{4} $ (T^2 + 780*T + 42588)^4
$97$	$ (T^{4} + 1472 T^{2} + 500224)^{2} $ (T^4 + 1472*T^2 + 500224)^2
show more
show less

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 544 = 2^{5} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	544.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{3} - \beta_1 q^{5} + ( - \beta_{4} + \beta_{3}) q^{7} + ( - 3 \beta_{2} - 1) q^{9} + (4 \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{2} + 26) q^{13} + \beta_{6} q^{15} + (\beta_{7} - 5 \beta_{2} - 7) q^{17}+ \cdots + (86 \beta_{4} - 73 \beta_{3}) q^{99}+O(q^{100}) \) q + b3 * q^3 - b1 * q^5 + (-b4 + b3) * q^7 + (-3b2 - 1) q^9 + (4b4 + b3) q^11 + (-b2 + 26) * q^13 + b6 * q^15 + (b7 - 5b2 - 7) q^17 - b5 * q^19 + (-b2 - 32) * q^21 + (3b4 + 17b3) * q^23 + (-6b2 - 59) q^25 + (-6b4 + 2b3) * q^27 + (-2b7 + b1) q^29 + (9b4 + 17b3) * q^31 + (-11b2 - 12) q^33 + (b6 + b5) * q^35 + (-4b7 + 5b1) * q^37 + (-2b4 + 18b3) * q^39 + (-2b7 - 4b1) * q^41 + (-b6 + 2b5) q^43 + (-6b7 + 7b1) * q^45 + 3b6 q^47 + (11b2 + 219) q^49 + (3b6 - b5 - 10b4 - 47b3) q^51 + (-8b2 - 58) q^53 + (b6 - 4b5) q^55 - 4b7 q^57 + (-5b6 + 4b5) * q^59 + (-10b7 - 11b1) * q^61 + (-29b4 - 13b3) * q^63 + (-2b7 - 24b1) * q^65 + (-10b6 - 5b5) * q^67 + (-57b2 - 464) q^69 + (37b4 + 21b3) * q^71 + (-6b7 + 30b1) * q^73 + (-12b4 - 107b3) * q^75 + (-49b2 + 336) q^77 + (79b4 - 27b3) * q^79 + (-75b2 - 107) q^81 + (-b6 + 4b5) q^83 + (-10b7 - 86b2 + 17b1 - 32) q^85 + (-7b6 + 2b5) * q^87 + (39b2 - 390) q^89 + (-36b4 + 22b3) * q^91 + (-69b2 - 440) q^93 + (-136b4 - 24b3) * q^95 - 2b1 q^97 + (86b4 - 73b3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} + 208 q^{13} - 56 q^{17} - 256 q^{21} - 472 q^{25} - 96 q^{33} + 1752 q^{49} - 464 q^{53} - 3712 q^{69} + 2688 q^{77} - 856 q^{81} - 256 q^{85} - 3120 q^{89} - 3520 q^{93}+O(q^{100}) \) 8 * q - 8 * q^9 + 208 * q^13 - 56 * q^17 - 256 * q^21 - 472 * q^25 - 96 * q^33 + 1752 * q^49 - 464 * q^53 - 3712 * q^69 + 2688 * q^77 - 856 * q^81 - 256 * q^85 - 3120 * q^89 - 3520 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	544.4.b.e

Level	$544$
Weight	$4$
Character orbit	544.b
Analytic conductor	$32.097$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(32.0970390431\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 60x^{6} + 866x^{4} + 3840x^{2} + 4096 \) x^8 + 60x^6 + 866x^4 + 3840*x^2 + 4096
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 2^{12}\cdot 3^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} + 60\nu^{5} + 866\nu^{3} + 4352\nu ) / 256 \) (v^7 + 60v^5 + 866v^3 + 4352*v) / 256
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} + 60\nu^{4} + 802\nu^{2} + 1920 ) / 96 \) (v^6 + 60v^4 + 802v^2 + 1920) / 96
\(\beta_{3}\)	\(=\)	\( ( -11\nu^{7} - 628\nu^{5} - 7862\nu^{3} - 24256\nu ) / 2304 \) (-11v^7 - 628v^5 - 7862v^3 - 24256v) / 2304
\(\beta_{4}\)	\(=\)	\( ( 17\nu^{7} + 892\nu^{5} + 8066\nu^{3} + 7168\nu ) / 2304 \) (17v^7 + 892v^5 + 8066v^3 + 7168v) / 2304
\(\beta_{5}\)	\(=\)	\( ( -33\nu^{6} - 1724\nu^{4} - 15330\nu^{2} - 15872 ) / 144 \) (-33v^6 - 1724v^4 - 15330*v^2 - 15872) / 144
\(\beta_{6}\)	\(=\)	\( ( -15\nu^{6} - 836\nu^{4} - 10110\nu^{2} - 29888 ) / 144 \) (-15v^6 - 836v^4 - 10110*v^2 - 29888) / 144
\(\beta_{7}\)	\(=\)	\( ( -35\nu^{7} - 2036\nu^{5} - 25958\nu^{3} - 66176\nu ) / 768 \) (-35v^7 - 2036v^5 - 25958v^3 - 66176v) / 768

\(\nu\)	\(=\)	\( ( \beta_{4} + 4\beta_{3} + 3\beta_1 ) / 12 \) (b4 + 4b3 + 3b1) / 12
\(\nu^{2}\)	\(=\)	\( ( -4\beta_{6} + \beta_{5} - 18\beta_{2} - 360 ) / 24 \) (-4b6 + b5 - 18b2 - 360) / 24
\(\nu^{3}\)	\(=\)	\( ( 9\beta_{7} - 38\beta_{4} - 206\beta_{3} - 75\beta_1 ) / 12 \) (9b7 - 38b4 - 206b3 - 75b1) / 12
\(\nu^{4}\)	\(=\)	\( ( 29\beta_{6} - 5\beta_{5} + 180\beta_{2} + 1868 ) / 4 \) (29b6 - 5b5 + 180*b2 + 1868) / 4
\(\nu^{5}\)	\(=\)	\( ( -234\beta_{7} + 619\beta_{4} + 4312\beta_{3} + 1371\beta_1 ) / 6 \) (-234b7 + 619b4 + 4312b3 + 1371b1) / 6
\(\nu^{6}\)	\(=\)	\( ( -3616\beta_{6} + 499\beta_{5} - 24030\beta_{2} - 214920 ) / 12 \) (-3616b6 + 499b5 - 24030*b2 - 214920) / 12
\(\nu^{7}\)	\(=\)	\( ( 10143\beta_{7} - 22862\beta_{4} - 178226\beta_{3} - 54777\beta_1 ) / 6 \) (10143b7 - 22862b4 - 178226b3 - 54777b1) / 6

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 33.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( (T^{4} + 56 T^{2} + 136)^{2} \) (T^4 + 56*T^2 + 136)^2
$5$	\( (T^{4} + 368 T^{2} + 31264)^{2} \) (T^4 + 368*T^2 + 31264)^2
$7$	\( (T^{4} + 248 T^{2} + 6664)^{2} \) (T^4 + 248*T^2 + 6664)^2
$11$	\( (T^{4} + 2808 T^{2} + 539784)^{2} \) (T^4 + 2808*T^2 + 539784)^2
$13$	\( (T^{2} - 52 T + 604)^{4} \) (T^2 - 52*T + 604)^4
$17$	\( (T^{4} + 28 T^{3} + \cdots + 24137569)^{2} \) (T^4 + 28T^3 + 2822T^2 + 137564*T + 24137569)^2
$19$	\( (T^{4} - 23744 T^{2} + 17007616)^{2} \) (T^4 - 23744*T^2 + 17007616)^2
$23$	\( (T^{4} + 16952 T^{2} + 2552584)^{2} \) (T^4 + 16952*T^2 + 2552584)^2
$29$	\( (T^{4} + 25136 T^{2} + 157601824)^{2} \) (T^4 + 25136*T^2 + 157601824)^2
$31$	\( (T^{4} + 27992 T^{2} + 163663624)^{2} \) (T^4 + 27992*T^2 + 163663624)^2
$37$	\( (T^{4} + 109808 T^{2} + 2575184416)^{2} \) (T^4 + 109808*T^2 + 2575184416)^2
$41$	\( (T^{4} + 29376 T^{2} + 40518144)^{2} \) (T^4 + 29376*T^2 + 40518144)^2
$43$	\( (T^{4} - 108896 T^{2} + 2249257216)^{2} \) (T^4 - 108896*T^2 + 2249257216)^2
$47$	\( (T^{4} - 116064 T^{2} + 344404224)^{2} \) (T^4 - 116064*T^2 + 344404224)^2
$53$	\( (T^{2} + 116 T - 1244)^{4} \) (T^2 + 116*T - 1244)^4
$59$	\( (T^{4} - 712544 T^{2} + 118581350656)^{2} \) (T^4 - 712544*T^2 + 118581350656)^2
$61$	\( (T^{4} + 643248 T^{2} + 57740887584)^{2} \) (T^4 + 643248*T^2 + 57740887584)^2
$67$	\( (T^{4} - 1857600 T^{2} + 861010560000)^{2} \) (T^4 - 1857600*T^2 + 861010560000)^2
$71$	\( (T^{4} + 253208 T^{2} + 9856078984)^{2} \) (T^4 + 253208*T^2 + 9856078984)^2
$73$	\( (T^{4} + 574848 T^{2} + 162072576)^{2} \) (T^4 + 574848*T^2 + 162072576)^2
$79$	\( (T^{4} + 1173368 T^{2} + 3836114056)^{2} \) (T^4 + 1173368*T^2 + 3836114056)^2
$83$	\( (T^{4} - 394848 T^{2} + 16875806976)^{2} \) (T^4 - 394848*T^2 + 16875806976)^2
$89$	\( (T^{2} + 780 T + 42588)^{4} \) (T^2 + 780*T + 42588)^4
$97$	\( (T^{4} + 1472 T^{2} + 500224)^{2} \) (T^4 + 1472*T^2 + 500224)^2
show more
show less

\(n\)	\(69\)	\(511\)	\(513\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)