LMFDB - Newform orbit 100.6.l.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,6,Mod(3,100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 7]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("100.3");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 100 = 2^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	100.l (of order $20$, degree $8$, 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:	$16.0383819813$
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 + (4 \zeta_{20}^{6} - 4 \zeta_{20}^{4} + \cdots - 4) q^{2}+ \cdots + 243 \zeta_{20}^{7} q^{9}+O(q^{10}) $ q + (4z^6 - 4z^4 - 4z^3 + 4z^2 - 4) * q^2 + 32z q^4 + (41z^7 - 41z^5 + 38z^4 + 41z^3 - 41z) q^5 + (128z^7 - 128z^5 - 128z^4 + 128z^3 - 128z) q^8 + 243z^7 q^9	$ q + (4 \zeta_{20}^{6} - 4 \zeta_{20}^{4} + \cdots - 4) q^{2}+ \cdots + ( - 67228 \zeta_{20}^{6} + \cdots + 67228) q^{98}+O(q^{100}) $ q + (4z^6 - 4z^4 - 4z^3 + 4z^2 - 4) * q^2 + 32z q^4 + (41z^7 - 41z^5 + 38z^4 + 41z^3 - 41z) q^5 + (128z^7 - 128z^5 - 128z^4 + 128z^3 - 128z) q^8 + 243z^7 q^9 + (-316z^7 + 12z^2) * q^10 + (-475z^6 - 122z^4 + 597z^3 + 122z^2 - 122z - 122) q^13 + 1024z^2 q^16 + (-404z^7 - 404z^6 + 1121z^2 - 1121z) * q^17 + (-972z^5 + 972) q^18 + (1216z^5 - 1312) q^20 + (-237z^6 + 237z^4 - 3116z^3 - 237z^2 + 237) * q^25 + (1900z^7 - 2876z^6 - 1900z^5 + 2876z^4 + 1900z^3 - 3800z) * q^26 + (4282z^7 + 1475z^6 - 4282z^5 - 2950z^4 + 1475z^2 - 4282z - 1475) * q^29 + (-4096z^5 - 4096) q^32 + (-2868z^7 + 6100z^4 - 2868z^3 + 2868z - 6100) * q^34 + (7776z^6 - 7776z^4 + 7776z^2 - 7776) q^36 + (-5646z^7 - 475z^5 - 6121z^4 - 5646z^3 + 5646z - 5646) q^37 + (-10112z^6 + 10112z^4 + 384z^3 - 10112z^2 + 10112) * q^40 + (-2476z^4 - 10475z^3 + 2476z^2 - 10475z - 2476) * q^41 + (-9963z^6 - 9234z) * q^45 + 16807z^5 q^49 + (13412z^6 + 11516z) * q^50 + (-15200z^7 - 3904z^5 + 19104z^4 + 3904z^3 - 3904z^2 - 3904z) * q^52 + (20122z^7 - 3647z^5 - 3647z^2 + 20122) q^53 + (-11228z^7 + 11228z^6 + 23028z^2 + 23028z) * q^58 + (18950z^7 - 9475z^5 - 27474z^4 + 9475z^3 + 27474z^2 - 9475z) * q^61 + 32768z^3 q^64 + (17684z^7 + 19841z^5 - 29113z^2 + 27688) q^65 + (-12928z^7 - 12928z^6 + 12928z^4 + 35872z^3 - 48800z^2 + 12928) q^68 + (-31104z^6 + 31104z) * q^72 + (-54475z^7 + 10072z^5 - 44403z^4 - 10072z^3 - 10072z^2 + 10072z) * q^73 + (47068z^7 + 1900z^6 - 1900z^4 + 47068z^3 + 3800z^2 - 1900) q^74 + (38912z^6 - 41984z) * q^80 - 59049z^4 q^81 + (-31996z^7 + 31996z^6 + 31996z^5 + 51804z^4 - 31996z^3 + 83800z) * q^82 + (59162z^6 - 26034z^5 - 30609z + 61313) q^85 + (-25525z^6 + 70232z^5 + 70232z - 25525) q^89 + (2916z^7 - 2916z^5 + 76788z^4 + 2916z^3 - 2916z) q^90 + (-46071z^7 - 46071z^6 + 46071z^5 + 126475z^4 - 126475z^3 - 46071z^2 + 46071z + 46071) q^97 + (-67228z^6 + 67228z^4 - 67228z^3 - 67228z^2 + 67228) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 76 q^{5} + 256 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 - 76 * q^5 + 256 * q^8	$ 8 q - 8 q^{2} - 76 q^{5} + 256 q^{8} + 24 q^{10} - 1438 q^{13} + 2048 q^{16} + 1434 q^{17} + 7776 q^{18} - 10496 q^{20} + 474 q^{25} - 11504 q^{26} - 32768 q^{32} - 61000 q^{34} - 15552 q^{36} - 32926 q^{37} + 20224 q^{40} - 9904 q^{41} - 19926 q^{45} + 26824 q^{50} - 46016 q^{52} + 153682 q^{53} + 68512 q^{58} + 109896 q^{61} + 163278 q^{65} - 45888 q^{68} - 62208 q^{72} + 68662 q^{73} + 77824 q^{80} + 118098 q^{81} - 39616 q^{82} + 608828 q^{85} - 255250 q^{89} - 153576 q^{90} - 68666 q^{97} + 134456 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 - 76 * q^5 + 256 * q^8 + 24 * q^10 - 1438 * q^13 + 2048 * q^16 + 1434 * q^17 + 7776 * q^18 - 10496 * q^20 + 474 * q^25 - 11504 * q^26 - 32768 * q^32 - 61000 * q^34 - 15552 * q^36 - 32926 * q^37 + 20224 * q^40 - 9904 * q^41 - 19926 * q^45 + 26824 * q^50 - 46016 * q^52 + 153682 * q^53 + 68512 * q^58 + 109896 * q^61 + 163278 * q^65 - 45888 * q^68 - 62208 * q^72 + 68662 * q^73 + 77824 * q^80 + 118098 * q^81 - 39616 * q^82 + 608828 * q^85 - 255250 * q^89 - 153576 * q^90 - 68666 * q^97 + 134456 * q^98

Display 100 coefficients 10 coefficients

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

−2.56816

−

5.04029i

−18.8091

25.8885i

−6.64345

55.5055i

178.791

28.3177i

−231.107

75.0911i

296.826

−

109.062i

23.1

−0.884927

5.58721i

−30.4338

−

9.88854i

50.7360

23.4705i

82.1811

−

161.289i

142.832

−

196.591i

−176.032

262.703i

27.1

−5.58721

−

0.884927i

30.4338

9.88854i

−27.2507

48.8098i

−161.289

−

82.1811i

−142.832

196.591i

195.448

−

248.596i

47.1

5.04029

−

2.56816i

18.8091

−

25.8885i

−54.8418

−

10.8339i

28.3177

−

178.791i

231.107

−

75.0911i

−304.242

86.2367i

63.1

−5.58721

0.884927i

30.4338

−

9.88854i

−27.2507

−

48.8098i

−161.289

82.1811i

−142.832

−

196.591i

195.448

248.596i

67.1

−2.56816

5.04029i

−18.8091

−

25.8885i

−6.64345

−

55.5055i

178.791

−

28.3177i

−231.107

−

75.0911i

296.826

109.062i

83.1

5.04029

2.56816i

18.8091

25.8885i

−54.8418

10.8339i

28.3177

178.791i

231.107

75.0911i

−304.242

−

86.2367i

87.1

−0.884927

−

5.58721i

−30.4338

9.88854i

50.7360

−

23.4705i

82.1811

161.289i

142.832

196.591i

−176.032

−

262.703i

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	100.6.l.a	✓	8
4.b	odd	2	1	CM	100.6.l.a	✓	8
25.f	odd	20	1	inner	100.6.l.a	✓	8
100.l	even	20	1	inner	100.6.l.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
100.6.l.a	✓	8	1.a	even	1	1	trivial
100.6.l.a	✓	8	4.b	odd	2	1	CM
100.6.l.a	✓	8	25.f	odd	20	1	inner
100.6.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 $S_{6}^{\mathrm{new}}(100, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 8 T^{7} + \cdots + 1048576 $ T^8 + 8T^7 + 32T^6 - 1024T^4 + 32768T^2 + 262144*T + 1048576
$3$	$ T^{8} $ T^8
$5$	$ T^{8} + \cdots + 95367431640625 $ T^8 + 76T^7 + 2651T^6 - 36024T^5 - 11022199T^4 - 112575000T^3 + 25888671875T^2 + 2319335937500*T + 95367431640625
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ T^{8} + \cdots + 25\!\cdots\!01 $ T^8 + 1438T^7 + 1033922T^6 + 452977460T^5 + 123241972636T^4 - 522859194996190T^3 + 400353216115989313T^2 - 166600193440912096776*T + 25970239258532917048801
$17$	$ T^{8} + \cdots + 14\!\cdots\!01 $ T^8 - 1434T^7 + 1028178T^6 + 5736222280T^5 - 22143503100964T^4 + 20352533817625170T^3 + 65142505524613864637T^2 - 27672888034873667944868*T + 14728638602195581558941601
$19$	$ T^{8} $ T^8
$23$	$ T^{8} $ T^8
$29$	$ T^{8} + \cdots + 15\!\cdots\!01 $ T^8 - 73342096T^6 + 302539447750T^5 + 2064482083533706T^4 - 22188877220667484000T^3 + 64316275529145526335949T^2 - 11815400693309320769127250T + 1525221396416673864703355401
$31$	$ T^{8} $ T^8
$37$	$ T^{8} + \cdots + 22\!\cdots\!01 $ T^8 + 32926T^7 + 570007363T^6 + 7526413403690T^5 + 59663151852708136T^4 + 254236850387293546790T^3 + 1623172205733220782433222T^2 + 9167315860178278836078196012*T + 22230410417578768157243876176801
$41$	$ T^{8} + \cdots + 28\!\cdots\!01 $ T^8 + 9904T^7 + 73566912T^6 - 2382861739128T^5 + 94617395350759530T^4 + 465567489292505596528T^3 + 41168201756261911374671217T^2 - 140501100425798729008937923544*T + 2863248936229792507260520462399801
$43$	$ T^{8} $ T^8
$47$	$ T^{8} $ T^8
$53$	$ T^{8} + \cdots + 44\!\cdots\!01 $ T^8 - 153682T^7 + 9851107187T^6 - 346428830199670T^5 + 7583487559773732936T^4 - 115495555255785235267530T^3 + 1297747590993635414825667878T^2 - 9095548906154389883277465861716*T + 44723320224592346859843789130586401
$59$	$ T^{8} $ T^8
$61$	$ T^{8} + \cdots + 45\!\cdots\!01 $ T^8 - 109896T^7 + 9057848112T^6 - 431569796340828T^5 + 14462644297921415530T^4 - 293783671853567626346472T^3 + 3681008232577324899092366317T^2 + 6743201848372520155179514193556*T + 4568734210053568913747381799160201
$67$	$ T^{8} $ T^8
$71$	$ T^{8} $ T^8
$73$	$ T^{8} + \cdots + 68\!\cdots\!01 $ T^8 - 68662T^7 + 2357235122T^6 + 208799770846960T^5 - 38435839736119452164T^4 + 1967608076901534175407310T^3 + 178234556215868369406057106213T^2 - 967810354327873378427574568599876*T + 6817846978902452996856841551157567201
$79$	$ T^{8} $ T^8
$83$	$ T^{8} $ T^8
$89$	$ T^{8} + \cdots + 32\!\cdots\!01 $ T^8 + 255250T^7 + 34251186949T^6 + 1995622989375000T^5 + 106574436655459309106T^4 + 18994554164544446931401250T^3 + 1399639588482734148107060324104T^2 + 12423006701144274033050000748460500*T + 32312445425598404937950176719748942201
$97$	$ T^{8} + \cdots + 74\!\cdots\!01 $ T^8 + 68666T^7 + 2357509778T^6 + 6904565060238280T^5 - 167649421707519470564T^4 - 9040402906334667857501330T^3 + 15605746573404880955958725713837T^2 - 1841064477194895004472099989365651468*T + 74554239723298618753670207565632040890401
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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:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	100.l (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + (4 \zeta_{20}^{6} - 4 \zeta_{20}^{4} + \cdots - 4) q^{2}+ \cdots + 243 \zeta_{20}^{7} q^{9}+O(q^{10}) \) q + (4z^6 - 4z^4 - 4z^3 + 4z^2 - 4) * q^2 + 32z q^4 + (41z^7 - 41z^5 + 38z^4 + 41z^3 - 41z) q^5 + (128z^7 - 128z^5 - 128z^4 + 128z^3 - 128z) q^8 + 243z^7 q^9	\( q + (4 \zeta_{20}^{6} - 4 \zeta_{20}^{4} + \cdots - 4) q^{2}+ \cdots + ( - 67228 \zeta_{20}^{6} + \cdots + 67228) q^{98}+O(q^{100}) \) q + (4z^6 - 4z^4 - 4z^3 + 4z^2 - 4) * q^2 + 32z q^4 + (41z^7 - 41z^5 + 38z^4 + 41z^3 - 41z) q^5 + (128z^7 - 128z^5 - 128z^4 + 128z^3 - 128z) q^8 + 243z^7 q^9 + (-316z^7 + 12z^2) * q^10 + (-475z^6 - 122z^4 + 597z^3 + 122z^2 - 122z - 122) q^13 + 1024z^2 q^16 + (-404z^7 - 404z^6 + 1121z^2 - 1121z) * q^17 + (-972z^5 + 972) q^18 + (1216z^5 - 1312) q^20 + (-237z^6 + 237z^4 - 3116z^3 - 237z^2 + 237) * q^25 + (1900z^7 - 2876z^6 - 1900z^5 + 2876z^4 + 1900z^3 - 3800z) * q^26 + (4282z^7 + 1475z^6 - 4282z^5 - 2950z^4 + 1475z^2 - 4282z - 1475) * q^29 + (-4096z^5 - 4096) q^32 + (-2868z^7 + 6100z^4 - 2868z^3 + 2868z - 6100) * q^34 + (7776z^6 - 7776z^4 + 7776z^2 - 7776) q^36 + (-5646z^7 - 475z^5 - 6121z^4 - 5646z^3 + 5646z - 5646) q^37 + (-10112z^6 + 10112z^4 + 384z^3 - 10112z^2 + 10112) * q^40 + (-2476z^4 - 10475z^3 + 2476z^2 - 10475z - 2476) * q^41 + (-9963z^6 - 9234z) * q^45 + 16807z^5 q^49 + (13412z^6 + 11516z) * q^50 + (-15200z^7 - 3904z^5 + 19104z^4 + 3904z^3 - 3904z^2 - 3904z) * q^52 + (20122z^7 - 3647z^5 - 3647z^2 + 20122) q^53 + (-11228z^7 + 11228z^6 + 23028z^2 + 23028z) * q^58 + (18950z^7 - 9475z^5 - 27474z^4 + 9475z^3 + 27474z^2 - 9475z) * q^61 + 32768z^3 q^64 + (17684z^7 + 19841z^5 - 29113z^2 + 27688) q^65 + (-12928z^7 - 12928z^6 + 12928z^4 + 35872z^3 - 48800z^2 + 12928) q^68 + (-31104z^6 + 31104z) * q^72 + (-54475z^7 + 10072z^5 - 44403z^4 - 10072z^3 - 10072z^2 + 10072z) * q^73 + (47068z^7 + 1900z^6 - 1900z^4 + 47068z^3 + 3800z^2 - 1900) q^74 + (38912z^6 - 41984z) * q^80 - 59049z^4 q^81 + (-31996z^7 + 31996z^6 + 31996z^5 + 51804z^4 - 31996z^3 + 83800z) * q^82 + (59162z^6 - 26034z^5 - 30609z + 61313) q^85 + (-25525z^6 + 70232z^5 + 70232z - 25525) q^89 + (2916z^7 - 2916z^5 + 76788z^4 + 2916z^3 - 2916z) q^90 + (-46071z^7 - 46071z^6 + 46071z^5 + 126475z^4 - 126475z^3 - 46071z^2 + 46071z + 46071) q^97 + (-67228z^6 + 67228z^4 - 67228z^3 - 67228z^2 + 67228) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 76 q^{5} + 256 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 - 76 * q^5 + 256 * q^8	\( 8 q - 8 q^{2} - 76 q^{5} + 256 q^{8} + 24 q^{10} - 1438 q^{13} + 2048 q^{16} + 1434 q^{17} + 7776 q^{18} - 10496 q^{20} + 474 q^{25} - 11504 q^{26} - 32768 q^{32} - 61000 q^{34} - 15552 q^{36} - 32926 q^{37} + 20224 q^{40} - 9904 q^{41} - 19926 q^{45} + 26824 q^{50} - 46016 q^{52} + 153682 q^{53} + 68512 q^{58} + 109896 q^{61} + 163278 q^{65} - 45888 q^{68} - 62208 q^{72} + 68662 q^{73} + 77824 q^{80} + 118098 q^{81} - 39616 q^{82} + 608828 q^{85} - 255250 q^{89} - 153576 q^{90} - 68666 q^{97} + 134456 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 - 76 * q^5 + 256 * q^8 + 24 * q^10 - 1438 * q^13 + 2048 * q^16 + 1434 * q^17 + 7776 * q^18 - 10496 * q^20 + 474 * q^25 - 11504 * q^26 - 32768 * q^32 - 61000 * q^34 - 15552 * q^36 - 32926 * q^37 + 20224 * q^40 - 9904 * q^41 - 19926 * q^45 + 26824 * q^50 - 46016 * q^52 + 153682 * q^53 + 68512 * q^58 + 109896 * q^61 + 163278 * q^65 - 45888 * q^68 - 62208 * q^72 + 68662 * q^73 + 77824 * q^80 + 118098 * q^81 - 39616 * q^82 + 608828 * q^85 - 255250 * q^89 - 153576 * q^90 - 68666 * q^97 + 134456 * q^98

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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	100.6.l.a

Level	$100$
Weight	$6$
Character orbit	100.l
Analytic conductor	$16.038$
Analytic rank	$0$
Dimension	$8$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(16.0383819813\)
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

$p$	$F_p(T)$
$2$	\( T^{8} + 8 T^{7} + \cdots + 1048576 \) T^8 + 8T^7 + 32T^6 - 1024T^4 + 32768T^2 + 262144*T + 1048576
$3$	\( T^{8} \) T^8
$5$	\( T^{8} + \cdots + 95367431640625 \) T^8 + 76T^7 + 2651T^6 - 36024T^5 - 11022199T^4 - 112575000T^3 + 25888671875T^2 + 2319335937500*T + 95367431640625
$7$	\( T^{8} \) T^8
$11$	\( T^{8} \) T^8
$13$	\( T^{8} + \cdots + 25\!\cdots\!01 \) T^8 + 1438T^7 + 1033922T^6 + 452977460T^5 + 123241972636T^4 - 522859194996190T^3 + 400353216115989313T^2 - 166600193440912096776*T + 25970239258532917048801
$17$	\( T^{8} + \cdots + 14\!\cdots\!01 \) T^8 - 1434T^7 + 1028178T^6 + 5736222280T^5 - 22143503100964T^4 + 20352533817625170T^3 + 65142505524613864637T^2 - 27672888034873667944868*T + 14728638602195581558941601
$19$	\( T^{8} \) T^8
$23$	\( T^{8} \) T^8
$29$	\( T^{8} + \cdots + 15\!\cdots\!01 \) T^8 - 73342096T^6 + 302539447750T^5 + 2064482083533706T^4 - 22188877220667484000T^3 + 64316275529145526335949T^2 - 11815400693309320769127250T + 1525221396416673864703355401
$31$	\( T^{8} \) T^8
$37$	\( T^{8} + \cdots + 22\!\cdots\!01 \) T^8 + 32926T^7 + 570007363T^6 + 7526413403690T^5 + 59663151852708136T^4 + 254236850387293546790T^3 + 1623172205733220782433222T^2 + 9167315860178278836078196012*T + 22230410417578768157243876176801
$41$	\( T^{8} + \cdots + 28\!\cdots\!01 \) T^8 + 9904T^7 + 73566912T^6 - 2382861739128T^5 + 94617395350759530T^4 + 465567489292505596528T^3 + 41168201756261911374671217T^2 - 140501100425798729008937923544*T + 2863248936229792507260520462399801
$43$	\( T^{8} \) T^8
$47$	\( T^{8} \) T^8
$53$	\( T^{8} + \cdots + 44\!\cdots\!01 \) T^8 - 153682T^7 + 9851107187T^6 - 346428830199670T^5 + 7583487559773732936T^4 - 115495555255785235267530T^3 + 1297747590993635414825667878T^2 - 9095548906154389883277465861716*T + 44723320224592346859843789130586401
$59$	\( T^{8} \) T^8
$61$	\( T^{8} + \cdots + 45\!\cdots\!01 \) T^8 - 109896T^7 + 9057848112T^6 - 431569796340828T^5 + 14462644297921415530T^4 - 293783671853567626346472T^3 + 3681008232577324899092366317T^2 + 6743201848372520155179514193556*T + 4568734210053568913747381799160201
$67$	\( T^{8} \) T^8
$71$	\( T^{8} \) T^8
$73$	\( T^{8} + \cdots + 68\!\cdots\!01 \) T^8 - 68662T^7 + 2357235122T^6 + 208799770846960T^5 - 38435839736119452164T^4 + 1967608076901534175407310T^3 + 178234556215868369406057106213T^2 - 967810354327873378427574568599876*T + 6817846978902452996856841551157567201
$79$	\( T^{8} \) T^8
$83$	\( T^{8} \) T^8
$89$	\( T^{8} + \cdots + 32\!\cdots\!01 \) T^8 + 255250T^7 + 34251186949T^6 + 1995622989375000T^5 + 106574436655459309106T^4 + 18994554164544446931401250T^3 + 1399639588482734148107060324104T^2 + 12423006701144274033050000748460500*T + 32312445425598404937950176719748942201
$97$	\( T^{8} + \cdots + 74\!\cdots\!01 \) T^8 + 68666T^7 + 2357509778T^6 + 6904565060238280T^5 - 167649421707519470564T^4 - 9040402906334667857501330T^3 + 15605746573404880955958725713837T^2 - 1841064477194895004472099989365651468*T + 74554239723298618753670207565632040890401
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\(n\)	\(51\)	\(77\)
\(\chi(n)\)	\(-1\)	\(\zeta_{20}\)