LMFDB - Newform orbit 175.7.c.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,7,Mod(174,175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("175.174");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	175.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$40.2594646335$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{510})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 65025 $ x^4 + 65025
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 7)
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - 8 \beta_1 q^{2} - \beta_{3} q^{3} + 8 \beta_{2} q^{6} + (7 \beta_{3} + 133 \beta_1) q^{7} - 512 \beta_1 q^{8} + 1311 q^{9}+O(q^{10}) $ q - 8b1 q^2 - b3 * q^3 + 8b2 q^6 + (7b3 + 133b1) * q^7 - 512b1 q^8 + 1311 * q^9	\( q - 8 \beta_1 q^{2} - \beta_{3} q^{3} + 8 \beta_{2} q^{6} + (7 \beta_{3} + 133 \beta_1) q^{7} - 512 \beta_1 q^{8} + 1311 q^{9} + 874 q^{11} - 49 \beta_{3} q^{13} + ( - 56 \beta_{2} + 1064) q^{14} - 4096 q^{16} - 132 \beta_{3} q^{17} - 10488 \beta_1 q^{18} + 69 \beta_{2} q^{19} + ( - 133 \beta_{2} - 14280) q^{21} - 6992 \beta_1 q^{22} - 4738 \beta_1 q^{23} + 512 \beta_{2} q^{24} + 392 \beta_{2} q^{26} - 582 \beta_{3} q^{27} - 11146 q^{29} + 608 \beta_{2} q^{31} - 874 \beta_{3} q^{33} + 1056 \beta_{2} q^{34} + 3002 \beta_1 q^{37} + 552 \beta_{3} q^{38} + 99960 q^{39} - 1274 \beta_{2} q^{41} + ( - 1064 \beta_{3} + 114240 \beta_1) q^{42} - 31418 \beta_1 q^{43} - 37904 q^{46} - 1604 \beta_{3} q^{47} + 4096 \beta_{3} q^{48} + (1862 \beta_{2} + 82271) q^{49} + 269280 q^{51} + 76406 \beta_1 q^{53} + 4656 \beta_{2} q^{54} + ( - 3584 \beta_{2} + 68096) q^{56} - 140760 \beta_1 q^{57} + 89168 \beta_1 q^{58} + 2507 \beta_{2} q^{59} + 6091 \beta_{2} q^{61} + 4864 \beta_{3} q^{62} + (9177 \beta_{3} + 174363 \beta_1) q^{63} - 262144 q^{64} + 6992 \beta_{2} q^{66} + 495242 \beta_1 q^{67} + 4738 \beta_{2} q^{69} - 184406 q^{71} - 671232 \beta_1 q^{72} + 1350 \beta_{3} q^{73} + 24016 q^{74} + (6118 \beta_{3} + 116242 \beta_1) q^{77} - 799680 \beta_1 q^{78} + 534934 q^{79} + 231561 q^{81} - 10192 \beta_{3} q^{82} + 15827 \beta_{3} q^{83} - 251344 q^{86} + 11146 \beta_{3} q^{87} - 447488 \beta_1 q^{88} - 13938 \beta_{2} q^{89} + ( - 6517 \beta_{2} - 699720) q^{91} - 1240320 \beta_1 q^{93} + 12832 \beta_{2} q^{94} + 18032 \beta_{3} q^{97} + (14896 \beta_{3} - 658168 \beta_1) q^{98} + 1145814 q^{99}+O(q^{100}) \) q - 8b1 q^2 - b3 * q^3 + 8b2 q^6 + (7b3 + 133b1) * q^7 - 512b1 q^8 + 1311 * q^9 + 874 * q^11 - 49b3 q^13 + (-56b2 + 1064) q^14 - 4096 * q^16 - 132b3 q^17 - 10488b1 q^18 + 69b2 q^19 + (-133b2 - 14280) q^21 - 6992b1 q^22 - 4738b1 q^23 + 512b2 q^24 + 392b2 q^26 - 582b3 q^27 - 11146 * q^29 + 608b2 q^31 - 874b3 q^33 + 1056b2 q^34 + 3002b1 q^37 + 552b3 q^38 + 99960 * q^39 - 1274b2 q^41 + (-1064b3 + 114240b1) * q^42 - 31418b1 q^43 - 37904 * q^46 - 1604b3 q^47 + 4096b3 q^48 + (1862b2 + 82271) q^49 + 269280 * q^51 + 76406b1 q^53 + 4656b2 q^54 + (-3584b2 + 68096) q^56 - 140760b1 q^57 + 89168b1 q^58 + 2507b2 q^59 + 6091b2 q^61 + 4864b3 q^62 + (9177b3 + 174363b1) * q^63 - 262144 * q^64 + 6992b2 q^66 + 495242b1 q^67 + 4738b2 q^69 - 184406 * q^71 - 671232b1 q^72 + 1350b3 q^73 + 24016 * q^74 + (6118b3 + 116242b1) * q^77 - 799680b1 q^78 + 534934 * q^79 + 231561 * q^81 - 10192b3 q^82 + 15827b3 q^83 - 251344 * q^86 + 11146b3 q^87 - 447488b1 q^88 - 13938b2 q^89 + (-6517b2 - 699720) q^91 - 1240320b1 q^93 + 12832b2 q^94 + 18032b3 q^97 + (14896b3 - 658168b1) * q^98 + 1145814 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5244 q^{9}+O(q^{10}) $ 4 * q + 5244 * q^9	$ 4 q + 5244 q^{9} + 3496 q^{11} + 4256 q^{14} - 16384 q^{16} - 57120 q^{21} - 44584 q^{29} + 399840 q^{39} - 151616 q^{46} + 329084 q^{49} + 1077120 q^{51} + 272384 q^{56} - 1048576 q^{64} - 737624 q^{71} + 96064 q^{74} + 2139736 q^{79} + 926244 q^{81} - 1005376 q^{86} - 2798880 q^{91} + 4583256 q^{99}+O(q^{100}) $ 4 * q + 5244 * q^9 + 3496 * q^11 + 4256 * q^14 - 16384 * q^16 - 57120 * q^21 - 44584 * q^29 + 399840 * q^39 - 151616 * q^46 + 329084 * q^49 + 1077120 * q^51 + 272384 * q^56 - 1048576 * q^64 - 737624 * q^71 + 96064 * q^74 + 2139736 * q^79 + 926244 * q^81 - 1005376 * q^86 - 2798880 * q^91 + 4583256 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 65025 $ x^4 + 65025 :

$\beta_{1}$	$=$	$ ( \nu^{2} ) / 255 $ (v^2) / 255
$\beta_{2}$	$=$	$ ( 2\nu^{3} + 510\nu ) / 255 $ (2v^3 + 510v) / 255
$\beta_{3}$	$=$	$ ( -2\nu^{3} + 510\nu ) / 255 $ (-2v^3 + 510v) / 255

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

$\nu$	$=$	$ ( \beta_{3} + \beta_{2} ) / 4 $ (b3 + b2) / 4
$\nu^{2}$	$=$	$ 255\beta_1 $ 255*b1
$\nu^{3}$	$=$	$ ( -255\beta_{3} + 255\beta_{2} ) / 4 $ (-255b3 + 255b2) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-1$

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.

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} $

174.1

11.2916	+	11.2916i
−11.2916	−	11.2916i
11.2916	−	11.2916i
−11.2916	+	11.2916i

−

8.00000i

−45.1664

361.331i

316.165

133.000i

−

512.000i

1311.00

174.2

−

8.00000i

45.1664

−

361.331i

−316.165

133.000i

−

512.000i

1311.00

174.3

8.00000i

−45.1664

−

361.331i

316.165

−

133.000i

512.000i

1311.00

174.4

8.00000i

45.1664

361.331i

−316.165

−

133.000i

512.000i

1311.00

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	175.7.c.c		4
5.b	even	2	1	inner	175.7.c.c		4
5.c	odd	4	1		7.7.b.b	✓	2
5.c	odd	4	1		175.7.d.e		2
7.b	odd	2	1	inner	175.7.c.c		4
15.e	even	4	1		63.7.d.d		2
20.e	even	4	1		112.7.c.b		2
35.c	odd	2	1	inner	175.7.c.c		4
35.f	even	4	1		7.7.b.b	✓	2
35.f	even	4	1		175.7.d.e		2
35.k	even	12	2		49.7.d.d		4
35.l	odd	12	2		49.7.d.d		4
40.i	odd	4	1		448.7.c.d		2
40.k	even	4	1		448.7.c.c		2
105.k	odd	4	1		63.7.d.d		2
140.j	odd	4	1		112.7.c.b		2
280.s	even	4	1		448.7.c.d		2
280.y	odd	4	1		448.7.c.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.7.b.b	✓	2	5.c	odd	4	1
7.7.b.b	✓	2	35.f	even	4	1
49.7.d.d		4	35.k	even	12	2
49.7.d.d		4	35.l	odd	12	2
63.7.d.d		2	15.e	even	4	1
63.7.d.d		2	105.k	odd	4	1
112.7.c.b		2	20.e	even	4	1
112.7.c.b		2	140.j	odd	4	1
175.7.c.c		4	1.a	even	1	1	trivial
175.7.c.c		4	5.b	even	2	1	inner
175.7.c.c		4	7.b	odd	2	1	inner
175.7.c.c		4	35.c	odd	2	1	inner
175.7.d.e		2	5.c	odd	4	1
175.7.d.e		2	35.f	even	4	1
448.7.c.c		2	40.k	even	4	1
448.7.c.c		2	280.y	odd	4	1
448.7.c.d		2	40.i	odd	4	1
448.7.c.d		2	280.s	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} + 64 $ T2^2 + 64 acting on $S_{7}^{\mathrm{new}}(175, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 64)^{2} $ (T^2 + 64)^2
$3$	$ (T^{2} - 2040)^{2} $ (T^2 - 2040)^2
$5$	$ T^{4} $ T^4
$7$	$ T^{4} + \cdots + 13841287201 $ T^4 - 164542*T^2 + 13841287201
$11$	$ (T - 874)^{4} $ (T - 874)^4
$13$	$ (T^{2} - 4898040)^{2} $ (T^2 - 4898040)^2
$17$	$ (T^{2} - 35544960)^{2} $ (T^2 - 35544960)^2
$19$	$ (T^{2} + 9712440)^{2} $ (T^2 + 9712440)^2
$23$	$ (T^{2} + 22448644)^{2} $ (T^2 + 22448644)^2
$29$	$ (T + 11146)^{4} $ (T + 11146)^4
$31$	$ (T^{2} + 754114560)^{2} $ (T^2 + 754114560)^2
$37$	$ (T^{2} + 9012004)^{2} $ (T^2 + 9012004)^2
$41$	$ (T^{2} + 3311075040)^{2} $ (T^2 + 3311075040)^2
$43$	$ (T^{2} + 987090724)^{2} $ (T^2 + 987090724)^2
$47$	$ (T^{2} - 5248544640)^{2} $ (T^2 - 5248544640)^2
$53$	$ (T^{2} + 5837876836)^{2} $ (T^2 + 5837876836)^2
$59$	$ (T^{2} + 12821499960)^{2} $ (T^2 + 12821499960)^2
$61$	$ (T^{2} + 75684573240)^{2} $ (T^2 + 75684573240)^2
$67$	$ (T^{2} + 245264638564)^{2} $ (T^2 + 245264638564)^2
$71$	$ (T + 184406)^{4} $ (T + 184406)^4
$73$	$ (T^{2} - 3717900000)^{2} $ (T^2 - 3717900000)^2
$79$	$ (T - 534934)^{4} $ (T - 534934)^4
$83$	$ (T^{2} - 511007615160)^{2} $ (T^2 - 511007615160)^2
$89$	$ (T^{2} + 396306401760)^{2} $ (T^2 + 396306401760)^2
$97$	$ (T^{2} - 663312168960)^{2} $ (T^2 - 663312168960)^2
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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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	175.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - 8 \beta_1 q^{2} - \beta_{3} q^{3} + 8 \beta_{2} q^{6} + (7 \beta_{3} + 133 \beta_1) q^{7} - 512 \beta_1 q^{8} + 1311 q^{9}+O(q^{10}) \) q - 8b1 q^2 - b3 * q^3 + 8b2 q^6 + (7b3 + 133b1) * q^7 - 512b1 q^8 + 1311 * q^9	\( q - 8 \beta_1 q^{2} - \beta_{3} q^{3} + 8 \beta_{2} q^{6} + (7 \beta_{3} + 133 \beta_1) q^{7} - 512 \beta_1 q^{8} + 1311 q^{9} + 874 q^{11} - 49 \beta_{3} q^{13} + ( - 56 \beta_{2} + 1064) q^{14} - 4096 q^{16} - 132 \beta_{3} q^{17} - 10488 \beta_1 q^{18} + 69 \beta_{2} q^{19} + ( - 133 \beta_{2} - 14280) q^{21} - 6992 \beta_1 q^{22} - 4738 \beta_1 q^{23} + 512 \beta_{2} q^{24} + 392 \beta_{2} q^{26} - 582 \beta_{3} q^{27} - 11146 q^{29} + 608 \beta_{2} q^{31} - 874 \beta_{3} q^{33} + 1056 \beta_{2} q^{34} + 3002 \beta_1 q^{37} + 552 \beta_{3} q^{38} + 99960 q^{39} - 1274 \beta_{2} q^{41} + ( - 1064 \beta_{3} + 114240 \beta_1) q^{42} - 31418 \beta_1 q^{43} - 37904 q^{46} - 1604 \beta_{3} q^{47} + 4096 \beta_{3} q^{48} + (1862 \beta_{2} + 82271) q^{49} + 269280 q^{51} + 76406 \beta_1 q^{53} + 4656 \beta_{2} q^{54} + ( - 3584 \beta_{2} + 68096) q^{56} - 140760 \beta_1 q^{57} + 89168 \beta_1 q^{58} + 2507 \beta_{2} q^{59} + 6091 \beta_{2} q^{61} + 4864 \beta_{3} q^{62} + (9177 \beta_{3} + 174363 \beta_1) q^{63} - 262144 q^{64} + 6992 \beta_{2} q^{66} + 495242 \beta_1 q^{67} + 4738 \beta_{2} q^{69} - 184406 q^{71} - 671232 \beta_1 q^{72} + 1350 \beta_{3} q^{73} + 24016 q^{74} + (6118 \beta_{3} + 116242 \beta_1) q^{77} - 799680 \beta_1 q^{78} + 534934 q^{79} + 231561 q^{81} - 10192 \beta_{3} q^{82} + 15827 \beta_{3} q^{83} - 251344 q^{86} + 11146 \beta_{3} q^{87} - 447488 \beta_1 q^{88} - 13938 \beta_{2} q^{89} + ( - 6517 \beta_{2} - 699720) q^{91} - 1240320 \beta_1 q^{93} + 12832 \beta_{2} q^{94} + 18032 \beta_{3} q^{97} + (14896 \beta_{3} - 658168 \beta_1) q^{98} + 1145814 q^{99}+O(q^{100}) \) q - 8b1 q^2 - b3 * q^3 + 8b2 q^6 + (7b3 + 133b1) * q^7 - 512b1 q^8 + 1311 * q^9 + 874 * q^11 - 49b3 q^13 + (-56b2 + 1064) q^14 - 4096 * q^16 - 132b3 q^17 - 10488b1 q^18 + 69b2 q^19 + (-133b2 - 14280) q^21 - 6992b1 q^22 - 4738b1 q^23 + 512b2 q^24 + 392b2 q^26 - 582b3 q^27 - 11146 * q^29 + 608b2 q^31 - 874b3 q^33 + 1056b2 q^34 + 3002b1 q^37 + 552b3 q^38 + 99960 * q^39 - 1274b2 q^41 + (-1064b3 + 114240b1) * q^42 - 31418b1 q^43 - 37904 * q^46 - 1604b3 q^47 + 4096b3 q^48 + (1862b2 + 82271) q^49 + 269280 * q^51 + 76406b1 q^53 + 4656b2 q^54 + (-3584b2 + 68096) q^56 - 140760b1 q^57 + 89168b1 q^58 + 2507b2 q^59 + 6091b2 q^61 + 4864b3 q^62 + (9177b3 + 174363b1) * q^63 - 262144 * q^64 + 6992b2 q^66 + 495242b1 q^67 + 4738b2 q^69 - 184406 * q^71 - 671232b1 q^72 + 1350b3 q^73 + 24016 * q^74 + (6118b3 + 116242b1) * q^77 - 799680b1 q^78 + 534934 * q^79 + 231561 * q^81 - 10192b3 q^82 + 15827b3 q^83 - 251344 * q^86 + 11146b3 q^87 - 447488b1 q^88 - 13938b2 q^89 + (-6517b2 - 699720) q^91 - 1240320b1 q^93 + 12832b2 q^94 + 18032b3 q^97 + (14896b3 - 658168b1) * q^98 + 1145814 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5244 q^{9}+O(q^{10}) \) 4 * q + 5244 * q^9	\( 4 q + 5244 q^{9} + 3496 q^{11} + 4256 q^{14} - 16384 q^{16} - 57120 q^{21} - 44584 q^{29} + 399840 q^{39} - 151616 q^{46} + 329084 q^{49} + 1077120 q^{51} + 272384 q^{56} - 1048576 q^{64} - 737624 q^{71} + 96064 q^{74} + 2139736 q^{79} + 926244 q^{81} - 1005376 q^{86} - 2798880 q^{91} + 4583256 q^{99}+O(q^{100}) \) 4 * q + 5244 * q^9 + 3496 * q^11 + 4256 * q^14 - 16384 * q^16 - 57120 * q^21 - 44584 * q^29 + 399840 * q^39 - 151616 * q^46 + 329084 * q^49 + 1077120 * q^51 + 272384 * q^56 - 1048576 * q^64 - 737624 * q^71 + 96064 * q^74 + 2139736 * q^79 + 926244 * q^81 - 1005376 * q^86 - 2798880 * q^91 + 4583256 * q^99

Introduction

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Modular forms

Varieties

Fields

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Groups

Database

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	175.7.c.c

Level	$175$
Weight	$7$
Character orbit	175.c
Analytic conductor	$40.259$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(40.2594646335\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{510})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} + 65025 \) x^4 + 65025
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{2} ) / 255 \) (v^2) / 255
\(\beta_{2}\)	\(=\)	\( ( 2\nu^{3} + 510\nu ) / 255 \) (2v^3 + 510v) / 255
\(\beta_{3}\)	\(=\)	\( ( -2\nu^{3} + 510\nu ) / 255 \) (-2v^3 + 510v) / 255

\(\nu\)	\(=\)	\( ( \beta_{3} + \beta_{2} ) / 4 \) (b3 + b2) / 4
\(\nu^{2}\)	\(=\)	\( 255\beta_1 \) 255*b1
\(\nu^{3}\)	\(=\)	\( ( -255\beta_{3} + 255\beta_{2} ) / 4 \) (-255b3 + 255b2) / 4

$p$	$F_p(T)$
$2$	\( (T^{2} + 64)^{2} \) (T^2 + 64)^2
$3$	\( (T^{2} - 2040)^{2} \) (T^2 - 2040)^2
$5$	\( T^{4} \) T^4
$7$	\( T^{4} + \cdots + 13841287201 \) T^4 - 164542*T^2 + 13841287201
$11$	\( (T - 874)^{4} \) (T - 874)^4
$13$	\( (T^{2} - 4898040)^{2} \) (T^2 - 4898040)^2
$17$	\( (T^{2} - 35544960)^{2} \) (T^2 - 35544960)^2
$19$	\( (T^{2} + 9712440)^{2} \) (T^2 + 9712440)^2
$23$	\( (T^{2} + 22448644)^{2} \) (T^2 + 22448644)^2
$29$	\( (T + 11146)^{4} \) (T + 11146)^4
$31$	\( (T^{2} + 754114560)^{2} \) (T^2 + 754114560)^2
$37$	\( (T^{2} + 9012004)^{2} \) (T^2 + 9012004)^2
$41$	\( (T^{2} + 3311075040)^{2} \) (T^2 + 3311075040)^2
$43$	\( (T^{2} + 987090724)^{2} \) (T^2 + 987090724)^2
$47$	\( (T^{2} - 5248544640)^{2} \) (T^2 - 5248544640)^2
$53$	\( (T^{2} + 5837876836)^{2} \) (T^2 + 5837876836)^2
$59$	\( (T^{2} + 12821499960)^{2} \) (T^2 + 12821499960)^2
$61$	\( (T^{2} + 75684573240)^{2} \) (T^2 + 75684573240)^2
$67$	\( (T^{2} + 245264638564)^{2} \) (T^2 + 245264638564)^2
$71$	\( (T + 184406)^{4} \) (T + 184406)^4
$73$	\( (T^{2} - 3717900000)^{2} \) (T^2 - 3717900000)^2
$79$	\( (T - 534934)^{4} \) (T - 534934)^4
$83$	\( (T^{2} - 511007615160)^{2} \) (T^2 - 511007615160)^2
$89$	\( (T^{2} + 396306401760)^{2} \) (T^2 + 396306401760)^2
$97$	\( (T^{2} - 663312168960)^{2} \) (T^2 - 663312168960)^2
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\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(-1\)

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