LMFDB - Newform orbit 100.4.l.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [100,4,Mod(3,100)] 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("100.3"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(100, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 7])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 100 = 2^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	100.l (of order $20$, degree $8$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$5.90019100057$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

$q$-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 primitive root of unity $\zeta_{20}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (2 \zeta_{20}^{6} - 2 \zeta_{20}^{4} + \cdots - 2) q^{2} + 8 \zeta_{20} q^{4} + ( - 11 \zeta_{20}^{7} + \cdots + 11 \zeta_{20}) q^{5} + (16 \zeta_{20}^{7} + \cdots - 16 \zeta_{20}) q^{8} + 27 \zeta_{20}^{7} q^{9}+ \cdots + ( - 686 \zeta_{20}^{6} + 686 \zeta_{20}^{4} + \cdots + 686) q^{98}+O(q^{100}) $ q + (2z^6 - 2z^4 - 2z^3 + 2z^2 - 2) * q^2 + 8z q^4 + (-11z^7 + 11z^5 - 2z^4 - 11z^3 + 11z) q^5 + (16z^7 - 16z^5 - 16z^4 + 16z^3 - 16z) q^8 + 27z^7 q^9 + (26z^7 - 18z^2) * q^10 + (-55z^6 + 46z^4 + 9z^3 - 46z^2 + 46z + 46) q^13 + 64z^2 q^16 + (-52z^7 - 52z^6 - 47z^2 + 47z) * q^17 + (-54z^5 + 54) q^18 + (-16z^5 + 88) q^20 + (-117z^6 + 117z^4 - 44z^3 - 117z^2 + 117) * q^25 + (110z^7 + 74z^6 - 110z^5 - 74z^4 + 110z^3 - 220z) * q^26 + (-142z^7 - 65z^6 + 142z^5 + 130z^4 - 65z^2 + 142z + 65) * q^29 + (-128z^5 - 128) q^32 + (198z^7 + 10z^4 + 198z^3 - 198z - 10) * q^34 + (216z^6 - 216z^4 + 216z^2 - 216) q^36 + (-198z^7 + 305z^5 + 107z^4 - 198z^3 + 198z - 198) q^37 + (208z^6 - 208z^4 - 144z^3 + 208z^2 - 208) * q^40 + (-236z^4 - 115z^3 + 236z^2 - 115z - 236) * q^41 + (297z^6 + 54z) * q^45 + 343z^5 q^49 + (322z^6 - 146z) * q^50 + (-440z^7 + 368z^5 + 72z^4 - 368z^3 + 368z^2 + 368z) * q^52 + (-286z^7 - 259z^5 - 259z^2 - 286) q^53 + (154z^7 - 154z^6 - 414z^2 - 414z) * q^58 + (830z^7 - 415z^5 - 234z^4 + 415z^3 + 234z^2 - 415z) * q^61 + 512z^3 q^64 + (-524z^7 - 191z^5 + 7z^2 + 488) q^65 + (-416z^7 - 416z^6 + 416z^4 - 376z^3 - 40z^2 + 416) q^68 + (-432z^6 + 432z) * q^72 + (845z^7 - 296z^5 + 549z^4 + 296z^3 + 296z^2 - 296z) * q^73 + (182z^7 - 610z^6 + 610z^4 + 182z^3 - 1220z^2 + 610) q^74 + (-128z^6 + 704z) * q^80 - 729z^4 q^81 + (242z^7 - 242z^6 - 242z^5 + 702z^4 + 242z^3 + 460z) * q^82 + (-478z^6 - 666z^5 - 621z + 413) q^85 + (835z^6 + 88z^5 + 88z + 835) q^89 + (-486z^7 + 486z^5 - 702z^4 - 486z^3 + 486z) q^90 + (297z^7 + 297z^6 - 297z^5 - 1205z^4 + 1205z^3 + 297z^2 - 297z - 297) q^97 + (-686z^6 + 686z^4 - 686z^3 - 686z^2 + 686) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 4 q^{5} + 32 q^{8} - 36 q^{10} + 74 q^{13} + 128 q^{16} - 198 q^{17} + 432 q^{18} + 704 q^{20} + 234 q^{25} + 296 q^{26} - 1024 q^{32} - 100 q^{34} - 432 q^{36} - 1798 q^{37} - 416 q^{40}+ \cdots + 1372 q^{98}+O(q^{100}) $ 8 * q - 4 * q^2 + 4 * q^5 + 32 * q^8 - 36 * q^10 + 74 * q^13 + 128 * q^16 - 198 * q^17 + 432 * q^18 + 704 * q^20 + 234 * q^25 + 296 * q^26 - 1024 * q^32 - 100 * q^34 - 432 * q^36 - 1798 * q^37 - 416 * q^40 - 944 * q^41 + 594 * q^45 + 644 * q^50 + 592 * q^52 - 2806 * q^53 - 1136 * q^58 + 936 * q^61 + 3918 * q^65 + 1584 * q^68 - 864 * q^72 - 506 * q^73 - 256 * q^80 + 1458 * q^81 - 1888 * q^82 + 2348 * q^85 + 8350 * q^89 + 1404 * q^90 + 1222 * q^97 + 1372 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$.

$n$	$51$	$77$
$\chi(n)$	$-1$	$\zeta_{20}$

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} $

3.1

−0.587785	+	0.809017i
−0.951057	−	0.309017i
0.951057	+	0.309017i
0.587785	−	0.809017i
0.951057	−	0.309017i
−0.587785	−	0.809017i
0.587785	+	0.809017i
−0.951057	+	0.309017i

−1.28408

−

2.52015i

−4.70228

6.47214i

−4.84760

−

10.0748i

22.3488

3.53971i

−25.6785

8.34346i

−19.1652

25.1535i

23.1

−0.442463

2.79360i

−7.60845

−

2.47214i

−11.0797

1.49707i

10.2726

−

20.1612i

15.8702

−

21.8435i

0.720111

−

31.6146i

27.1

−2.79360

−

0.442463i

7.60845

2.47214i

9.84359

−

5.30130i

−20.1612

−

10.2726i

−15.8702

21.8435i

−29.8447

10.4543i

47.1

2.52015

−

1.28408i

4.70228

−

6.47214i

8.08367

7.72362i

3.53971

−

22.3488i

25.6785

−

8.34346i

30.2898

9.08458i

63.1

−2.79360

0.442463i

7.60845

−

2.47214i

9.84359

5.30130i

−20.1612

10.2726i

−15.8702

−

21.8435i

−29.8447

−

10.4543i

67.1

−1.28408

2.52015i

−4.70228

−

6.47214i

−4.84760

10.0748i

22.3488

−

3.53971i

−25.6785

−

8.34346i

−19.1652

−

25.1535i

83.1

2.52015

1.28408i

4.70228

6.47214i

8.08367

−

7.72362i

3.53971

22.3488i

25.6785

8.34346i

30.2898

−

9.08458i

87.1

−0.442463

−

2.79360i

−7.60845

2.47214i

−11.0797

−

1.49707i

10.2726

20.1612i

15.8702

21.8435i

0.720111

31.6146i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
25.f	odd	20	1	inner
100.l	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	100.4.l.a	✓	8
4.b	odd	2	1	CM	100.4.l.a	✓	8
25.f	odd	20	1	inner	100.4.l.a	✓	8
100.l	even	20	1	inner	100.4.l.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
100.4.l.a	✓	8	1.a	even	1	1	trivial
100.4.l.a	✓	8	4.b	odd	2	1	CM
100.4.l.a	✓	8	25.f	odd	20	1	inner
100.4.l.a	✓	8	100.l	even	20	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} $ T3 acting on $S_{4}^{\mathrm{new}}(100, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 4 T^{7} + \cdots + 4096 $ T^8 + 4T^7 + 8T^6 - 64T^4 + 512T^2 + 2048*T + 4096
$3$	$ T^{8} $ T^8
$5$	$ T^{8} - 4 T^{7} + \cdots + 244140625 $ T^8 - 4T^7 - 109T^6 + 936T^5 + 9881T^4 + 117000T^3 - 1703125T^2 - 7812500*T + 244140625
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ T^{8} + \cdots + 2414167030081 $ T^8 - 74T^7 + 2738T^6 - 1010620T^5 + 49098076T^4 - 981158230T^3 + 203287661257T^2 + 1255449702072*T + 2414167030081
$17$	$ T^{8} + \cdots + 547705978928161 $ T^8 + 198T^7 + 19602T^6 + 2554760T^5 + 361751516T^4 + 26182212210T^3 + 1057721713373T^2 + 31042739553884*T + 547705978928161
$19$	$ T^{8} $ T^8
$23$	$ T^{8} $ T^8
$29$	$ T^{8} + \cdots + 12\!\cdots\!41 $ T^8 - 80656T^6 - 15852850T^5 + 2618027626T^4 + 1278627469600T^3 + 151519013194069T^2 + 5645025170740150T + 126799360965262441
$31$	$ T^{8} $ T^8
$37$	$ T^{8} + \cdots + 79\!\cdots\!81 $ T^8 + 1798T^7 + 1477627T^6 + 723456530T^5 + 230768031976T^4 + 48435390008270T^3 + 6742425460820758T^2 + 780922570435038364*T + 79358823708963727681
$41$	$ T^{8} + \cdots + 11\!\cdots\!81 $ T^8 + 944T^7 + 668352T^6 + 257962632T^5 + 65347154730T^4 + 7417217602928T^3 + 628101257644737T^2 - 161513062702360024*T + 11487005116833874681
$43$	$ T^{8} $ T^8
$47$	$ T^{8} $ T^8
$53$	$ T^{8} + \cdots + 12\!\cdots\!41 $ T^8 + 2806T^7 + 3863243T^6 + 3361939090T^5 + 2031018312456T^4 + 893003247564990T^3 + 310607821987670582T^2 + 84205504237370048732*T + 12042564297888625650241
$59$	$ T^{8} $ T^8
$61$	$ T^{8} + \cdots + 32\!\cdots\!21 $ T^8 - 936T^7 + 657072T^6 + 121122612T^5 + 9347008330T^4 + 18076267987128T^3 + 273387046332431797T^2 + 18589993825398595236*T + 3249222493162980658921
$67$	$ T^{8} $ T^8
$71$	$ T^{8} $ T^8
$73$	$ T^{8} + \cdots + 27\!\cdots\!21 $ T^8 + 506T^7 + 128018T^6 - 1151490320T^5 - 1863379081604T^4 - 530287080296930T^3 + 1442010468345232117T^2 + 637389626131150859292*T + 274632435854104843401121
$79$	$ T^{8} $ T^8
$83$	$ T^{8} $ T^8
$89$	$ T^{8} + \cdots + 56\!\cdots\!21 $ T^8 - 8350T^7 + 31382869T^6 - 69861945000T^5 + 101870154532466T^4 - 100669101130476750T^3 + 66514126466757193864T^2 - 27365888486321407514700*T + 5648609574778586870932921
$97$	$ T^{8} + \cdots + 17\!\cdots\!41 $ T^8 - 1222T^7 + 746642T^6 - 8287070840T^5 + 4646806211356T^4 - 1098016196003090T^3 + 16878051574620639853T^2 + 9698687369073337277364*T + 1729578307204447588369441
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	100.l (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + (2 \zeta_{20}^{6} - 2 \zeta_{20}^{4} + \cdots - 2) q^{2} + 8 \zeta_{20} q^{4} + ( - 11 \zeta_{20}^{7} + \cdots + 11 \zeta_{20}) q^{5} + (16 \zeta_{20}^{7} + \cdots - 16 \zeta_{20}) q^{8} + 27 \zeta_{20}^{7} q^{9}+ \cdots + ( - 686 \zeta_{20}^{6} + 686 \zeta_{20}^{4} + \cdots + 686) q^{98}+O(q^{100}) \) q + (2z^6 - 2z^4 - 2z^3 + 2z^2 - 2) * q^2 + 8z q^4 + (-11z^7 + 11z^5 - 2z^4 - 11z^3 + 11z) q^5 + (16z^7 - 16z^5 - 16z^4 + 16z^3 - 16z) q^8 + 27z^7 q^9 + (26z^7 - 18z^2) * q^10 + (-55z^6 + 46z^4 + 9z^3 - 46z^2 + 46z + 46) q^13 + 64z^2 q^16 + (-52z^7 - 52z^6 - 47z^2 + 47z) * q^17 + (-54z^5 + 54) q^18 + (-16z^5 + 88) q^20 + (-117z^6 + 117z^4 - 44z^3 - 117z^2 + 117) * q^25 + (110z^7 + 74z^6 - 110z^5 - 74z^4 + 110z^3 - 220z) * q^26 + (-142z^7 - 65z^6 + 142z^5 + 130z^4 - 65z^2 + 142z + 65) * q^29 + (-128z^5 - 128) q^32 + (198z^7 + 10z^4 + 198z^3 - 198z - 10) * q^34 + (216z^6 - 216z^4 + 216z^2 - 216) q^36 + (-198z^7 + 305z^5 + 107z^4 - 198z^3 + 198z - 198) q^37 + (208z^6 - 208z^4 - 144z^3 + 208z^2 - 208) * q^40 + (-236z^4 - 115z^3 + 236z^2 - 115z - 236) * q^41 + (297z^6 + 54z) * q^45 + 343z^5 q^49 + (322z^6 - 146z) * q^50 + (-440z^7 + 368z^5 + 72z^4 - 368z^3 + 368z^2 + 368z) * q^52 + (-286z^7 - 259z^5 - 259z^2 - 286) q^53 + (154z^7 - 154z^6 - 414z^2 - 414z) * q^58 + (830z^7 - 415z^5 - 234z^4 + 415z^3 + 234z^2 - 415z) * q^61 + 512z^3 q^64 + (-524z^7 - 191z^5 + 7z^2 + 488) q^65 + (-416z^7 - 416z^6 + 416z^4 - 376z^3 - 40z^2 + 416) q^68 + (-432z^6 + 432z) * q^72 + (845z^7 - 296z^5 + 549z^4 + 296z^3 + 296z^2 - 296z) * q^73 + (182z^7 - 610z^6 + 610z^4 + 182z^3 - 1220z^2 + 610) q^74 + (-128z^6 + 704z) * q^80 - 729z^4 q^81 + (242z^7 - 242z^6 - 242z^5 + 702z^4 + 242z^3 + 460z) * q^82 + (-478z^6 - 666z^5 - 621z + 413) q^85 + (835z^6 + 88z^5 + 88z + 835) q^89 + (-486z^7 + 486z^5 - 702z^4 - 486z^3 + 486z) q^90 + (297z^7 + 297z^6 - 297z^5 - 1205z^4 + 1205z^3 + 297z^2 - 297z - 297) q^97 + (-686z^6 + 686z^4 - 686z^3 - 686z^2 + 686) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 4 q^{5} + 32 q^{8} - 36 q^{10} + 74 q^{13} + 128 q^{16} - 198 q^{17} + 432 q^{18} + 704 q^{20} + 234 q^{25} + 296 q^{26} - 1024 q^{32} - 100 q^{34} - 432 q^{36} - 1798 q^{37} - 416 q^{40}+ \cdots + 1372 q^{98}+O(q^{100}) \) 8 * q - 4 * q^2 + 4 * q^5 + 32 * q^8 - 36 * q^10 + 74 * q^13 + 128 * q^16 - 198 * q^17 + 432 * q^18 + 704 * q^20 + 234 * q^25 + 296 * q^26 - 1024 * q^32 - 100 * q^34 - 432 * q^36 - 1798 * q^37 - 416 * q^40 - 944 * q^41 + 594 * q^45 + 644 * q^50 + 592 * q^52 - 2806 * q^53 - 1136 * q^58 + 936 * q^61 + 3918 * q^65 + 1584 * q^68 - 864 * q^72 - 506 * q^73 - 256 * q^80 + 1458 * q^81 - 1888 * q^82 + 2348 * q^85 + 8350 * q^89 + 1404 * q^90 + 1222 * q^97 + 1372 * q^98

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	100.4.l.a

Level	$100$
Weight	$4$
Character orbit	100.l
Analytic conductor	$5.900$
Analytic rank	$0$
Dimension	$8$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(5.90019100057\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{20})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 3.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
25.f	odd	20	1	inner
100.l	even	20	1	inner

$p$	$F_p(T)$
$2$	\( T^{8} + 4 T^{7} + \cdots + 4096 \) T^8 + 4T^7 + 8T^6 - 64T^4 + 512T^2 + 2048*T + 4096
$3$	\( T^{8} \) T^8
$5$	\( T^{8} - 4 T^{7} + \cdots + 244140625 \) T^8 - 4T^7 - 109T^6 + 936T^5 + 9881T^4 + 117000T^3 - 1703125T^2 - 7812500*T + 244140625
$7$	\( T^{8} \) T^8
$11$	\( T^{8} \) T^8
$13$	\( T^{8} + \cdots + 2414167030081 \) T^8 - 74T^7 + 2738T^6 - 1010620T^5 + 49098076T^4 - 981158230T^3 + 203287661257T^2 + 1255449702072*T + 2414167030081
$17$	\( T^{8} + \cdots + 547705978928161 \) T^8 + 198T^7 + 19602T^6 + 2554760T^5 + 361751516T^4 + 26182212210T^3 + 1057721713373T^2 + 31042739553884*T + 547705978928161
$19$	\( T^{8} \) T^8
$23$	\( T^{8} \) T^8
$29$	\( T^{8} + \cdots + 12\!\cdots\!41 \) T^8 - 80656T^6 - 15852850T^5 + 2618027626T^4 + 1278627469600T^3 + 151519013194069T^2 + 5645025170740150T + 126799360965262441
$31$	\( T^{8} \) T^8
$37$	\( T^{8} + \cdots + 79\!\cdots\!81 \) T^8 + 1798T^7 + 1477627T^6 + 723456530T^5 + 230768031976T^4 + 48435390008270T^3 + 6742425460820758T^2 + 780922570435038364*T + 79358823708963727681
$41$	\( T^{8} + \cdots + 11\!\cdots\!81 \) T^8 + 944T^7 + 668352T^6 + 257962632T^5 + 65347154730T^4 + 7417217602928T^3 + 628101257644737T^2 - 161513062702360024*T + 11487005116833874681
$43$	\( T^{8} \) T^8
$47$	\( T^{8} \) T^8
$53$	\( T^{8} + \cdots + 12\!\cdots\!41 \) T^8 + 2806T^7 + 3863243T^6 + 3361939090T^5 + 2031018312456T^4 + 893003247564990T^3 + 310607821987670582T^2 + 84205504237370048732*T + 12042564297888625650241
$59$	\( T^{8} \) T^8
$61$	\( T^{8} + \cdots + 32\!\cdots\!21 \) T^8 - 936T^7 + 657072T^6 + 121122612T^5 + 9347008330T^4 + 18076267987128T^3 + 273387046332431797T^2 + 18589993825398595236*T + 3249222493162980658921
$67$	\( T^{8} \) T^8
$71$	\( T^{8} \) T^8
$73$	\( T^{8} + \cdots + 27\!\cdots\!21 \) T^8 + 506T^7 + 128018T^6 - 1151490320T^5 - 1863379081604T^4 - 530287080296930T^3 + 1442010468345232117T^2 + 637389626131150859292*T + 274632435854104843401121
$79$	\( T^{8} \) T^8
$83$	\( T^{8} \) T^8
$89$	\( T^{8} + \cdots + 56\!\cdots\!21 \) T^8 - 8350T^7 + 31382869T^6 - 69861945000T^5 + 101870154532466T^4 - 100669101130476750T^3 + 66514126466757193864T^2 - 27365888486321407514700*T + 5648609574778586870932921
$97$	\( T^{8} + \cdots + 17\!\cdots\!41 \) T^8 - 1222T^7 + 746642T^6 - 8287070840T^5 + 4646806211356T^4 - 1098016196003090T^3 + 16878051574620639853T^2 + 9698687369073337277364*T + 1729578307204447588369441
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\(n\)	\(51\)	\(77\)
\(\chi(n)\)	\(-1\)	\(\zeta_{20}\)