LMFDB - Newform orbit 1379.1.ba.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1379,1,Mod(41,1379)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1379, base_ring=CyclotomicField(98))

chi = DirichletCharacter(H, H._module([49, 55]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1379.41");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1379 = 7 \cdot 197 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1379.ba (of order $98$, degree $42$, 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:	$0.688210652421$
Analytic rank:	$0$
Dimension:	$42$
Coefficient field:	$\Q(\zeta_{98})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{42} - x^{35} + x^{28} - x^{21} + x^{14} - x^{7} + 1 $ x^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{98}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{98} - \cdots)$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + (\zeta_{98}^{19} + \zeta_{98}^{18}) q^{2} + (\zeta_{98}^{38} + \cdots + \zeta_{98}^{36}) q^{4}+ \cdots + \zeta_{98}^{33} q^{9}+O(q^{10}) $ q + (z^19 + z^18) * q^2 + (z^38 + z^37 + z^36) * q^4 + z^6 * q^7 + (-z^8 + z^7 + z^6 - z^5) * q^8 + z^33 * q^9	$ q + (\zeta_{98}^{19} + \zeta_{98}^{18}) q^{2} + (\zeta_{98}^{38} + \cdots + \zeta_{98}^{36}) q^{4}+ \cdots + ( - \zeta_{98}^{48} - \zeta_{98}^{13}) q^{99}+O(q^{100}) $ q + (z^19 + z^18) * q^2 + (z^38 + z^37 + z^36) * q^4 + z^6 * q^7 + (-z^8 + z^7 + z^6 - z^5) * q^8 + z^33 * q^9 + (z^29 - z^15) * q^11 + (z^25 + z^24) * q^14 + (-z^27 + z^26 + z^25 + z^24 - z^23) * q^16 + (-z^3 - z^2) * q^18 + (z^48 + z^47 - z^34 - z^33) * q^22 + (z^22 + z^8) * q^23 - z^10 * q^25 + (z^44 + z^43 + z^42) * q^28 + (z^9 - 1) * q^29 + (-z^46 - z^45 - z^44 - z^43 - z^42 - z^41) * q^32 + (-z^22 - z^21 - z^20) * q^36 + (-z^47 - z) * q^37 + (-z^32 - z^12) * q^43 + (-z^18 - z^17 - z^16 + z^4 + z^3 + z^2) * q^44 + (z^41 + z^40 + z^27 + z^26) * q^46 + z^12 * q^49 + (-z^29 - z^28) * q^50 + (z^13 + z^5) * q^53 + (-z^14 - z^13 - z^12 - z^11) * q^56 + (z^28 + z^27 - z^19 - z^18) * q^58 + z^39 * q^63 + (z^16 - z^15 - z^14 - z^13 - z^12 - z^11 + z^10) * q^64 + (z^34 + z^7) * q^67 + (-z^45 + z^21) * q^71 + (-z^41 - z^40 - z^39 - z^38) * q^72 + (-z^20 - z^19 + z^17 + z^16) * q^74 + (z^35 - z^21) * q^77 + (-z^35 - z^4) * q^79 - z^17 * q^81 + (-z^31 - z^30 + z^2 + z) * q^86 + (-z^37 - z^36 - z^35 - z^34 + z^23 + z^22 + z^21 + z^20) * q^88 + (z^46 + z^45 + z^44 - z^11 - z^10 - z^9) * q^92 + (z^31 + z^30) * q^98 + (-z^48 - z^13) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 7 q^{8}+O(q^{10}) $ 42 * q - 7 * q^8	$ 42 q - 7 q^{8} - 7 q^{28} - 42 q^{29} + 7 q^{32} - 7 q^{36} + 7 q^{50} + 7 q^{56} - 7 q^{58} - 7 q^{64} + 7 q^{67} + 7 q^{71} - 7 q^{79}+O(q^{100}) $ 42 * q - 7 * q^8 - 7 * q^28 - 42 * q^29 + 7 * q^32 - 7 * q^36 + 7 * q^50 + 7 * q^56 - 7 * q^58 - 7 * q^64 + 7 * q^67 + 7 * q^71 - 7 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1379\mathbb{Z}\right)^\times$.

$n$	$395$	$1184$
$\chi(n)$	$-1$	$\zeta_{98}^{37}$

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} $

41.1

−0.0320516	−	0.999486i
0.838088	−	0.545535i
−0.801414	+	0.598111i
0.462538	+	0.886599i
−0.926917	+	0.375267i
−0.871319	+	0.490718i
0.949056	−	0.315108i
−0.871319	−	0.490718i
0.981559	+	0.191159i
0.572117	−	0.820172i
−0.284528	+	0.958668i
−0.518393	−	0.855143i
−0.0320516	+	0.999486i
−0.967295	−	0.253655i
0.672301	+	0.740278i
−0.284528	−	0.958668i
0.572117	+	0.820172i
−0.159600	+	0.987182i
0.949056	+	0.315108i
0.981559	−	0.191159i

−0.265971

1.36571i

−0.867498

−

0.351211i

−0.981559

0.191159i

−0.0486567

0.0747498i

−0.871319

−

0.490718i

55.1

−0.604168

1.81966i

−2.14472

−

1.60065i

−0.949056

0.315108i

2.63586

−

1.83866i

0.981559

−

0.191159i

62.1

−0.408524

0.479876i

0.0962112

0.595100i

−0.761446

0.648228i

−0.863804

−

0.523643i

0.672301

0.740278i

97.1

0.433822

1.65435i

−1.67736

0.944669i

0.967295

0.253655i

−1.10067

−

1.13653i

−0.159600

−

0.987182i

146.1

0.283021

0.257032i

−0.0819877

−

0.849888i

−0.672301

−

0.740278i

0.423912

−

0.568004i

−0.991790

0.127877i

160.1

−0.0325034

−

0.506267i

0.736540

−

0.0949664i

−0.997945

−

0.0640702i

−0.168995

−

0.867753i

0.345365

−

0.938468i

174.1

1.85288

0.681876i

2.20676

1.87864i

−0.345365

−

0.938468i

1.83899

3.26532i

−0.404783

0.914413i

181.1

−0.0325034

0.506267i

0.736540

0.0949664i

−0.997945

0.0640702i

−0.168995

0.867753i

0.345365

0.938468i

223.1

−1.82037

−

0.805826i

1.99211

2.19353i

0.404783

0.914413i

−1.23147

−

3.70900i

0.997945

0.0640702i

244.1

0.870140

1.54502i

−1.11155

1.83363i

0.871319

0.490718i

−2.02791

−

0.0650311i

0.949056

−

0.315108i

293.1

−1.18089

0.190917i

0.408991

−

0.135794i

0.159600

−

0.987182i

0.603522

−

0.314857i

0.0960230

−

0.995379i

335.1

0.125503

−

0.973378i

0.0355821

0.00933072i

0.991790

−

0.127877i

0.381848

−

0.943172i

0.761446

−

0.648228i

370.1

−0.265971

−

1.36571i

−0.867498

0.351211i

−0.981559

−

0.191159i

−0.0486567

−

0.0747498i

−0.871319

0.490718i

398.1

−0.255623

−

0.00819733i

−0.932670

−

0.0598794i

0.0320516

0.999486i

0.492493

0.0475102i

0.572117

−

0.820172i

419.1

−1.75324

0.520351i

1.96498

−

1.27906i

0.284528

−

0.958668i

−1.59402

1.87243i

−0.718349

0.695683i

433.1

−1.18089

−

0.190917i

0.408991

0.135794i

0.159600

0.987182i

0.603522

0.314857i

0.0960230

0.995379i

503.1

0.870140

−

1.54502i

−1.11155

−

1.83363i

0.871319

−

0.490718i

−2.02791

0.0650311i

0.949056

0.315108i

510.1

1.06332

0.741725i

0.235125

0.638909i

−0.572117

−

0.820172i

0.104970

−

0.400294i

0.838088

0.545535i

531.1

1.85288

−

0.681876i

2.20676

−

1.87864i

−0.345365

0.938468i

1.83899

−

3.26532i

−0.404783

−

0.914413i

538.1

−1.82037

0.805826i

1.99211

−

2.19353i

0.404783

−

0.914413i

−1.23147

3.70900i

0.997945

−

0.0640702i

See all 42 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
197.h	even	98	1	inner
1379.ba	odd	98	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1379.1.ba.a	✓	42
7.b	odd	2	1	CM	1379.1.ba.a	✓	42
197.h	even	98	1	inner	1379.1.ba.a	✓	42
1379.ba	odd	98	1	inner	1379.1.ba.a	✓	42

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1379.1.ba.a	✓	42	1.a	even	1	1	trivial
1379.1.ba.a	✓	42	7.b	odd	2	1	CM
1379.1.ba.a	✓	42	197.h	even	98	1	inner
1379.1.ba.a	✓	42	1379.ba	odd	98	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1379, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{42} + 7 T^{39} + \cdots + 7 $ T^42 + 7T^39 - 7T^37 + 42T^36 + 245T^34 + 245T^33 + 49T^32 + 1470T^31 + 1421T^30 - 1078T^29 + 6860T^28 + 7889T^27 + 16121T^26 + 27783T^25 + 52822T^24 + 72422T^23 + 91238T^22 + 31206T^21 + 199626T^20 + 74431T^19 + 324919T^18 + 251762T^17 - 321048T^16 + 643566T^15 - 156065T^14 - 335111T^13 + 468195T^12 - 607502T^11 + 104272T^10 - 135142T^9 + 168903T^8 - 31206T^7 + 29155T^6 - 2744T^5 - 2058T^4 + 2058T^3 + 98T^2 - 49*T + 7
$3$	$ T^{42} $ T^42
$5$	$ T^{42} $ T^42
$7$	$ T^{42} + T^{35} + \cdots + 1 $ T^42 + T^35 + T^28 + T^21 + T^14 + T^7 + 1
$11$	$ T^{42} + 245 T^{35} + \cdots + 823543 $ T^42 + 245T^35 + 19551T^28 + 525819T^21 + 6168169T^14 + 2000033*T^7 + 823543
$13$	$ T^{42} $ T^42
$17$	$ T^{42} $ T^42
$19$	$ T^{42} $ T^42
$23$	$ T^{42} - 117 T^{35} + \cdots + 1 $ T^42 - 117T^35 + 4477T^28 - 24695T^21 + 83673T^14 - 129*T^7 + 1
$29$	$ T^{42} + 42 T^{41} + \cdots + 1 $ T^42 + 42T^41 + 861T^40 + 11480T^39 + 111930T^38 + 850668T^37 + 5245786T^36 + 26978327T^35 + 118030150T^34 + 445891215T^33 + 1471436428T^32 + 4280509016T^31 + 11057792256T^30 + 25517108120T^29 + 52853504561T^28 + 98648891824T^27 + 166439114520T^26 + 254478351036T^25 + 353279915625T^24 + 445940957280T^23 + 512315646360T^22 + 535939099079T^21 + 510546772344T^20 + 442722288540T^19 + 349172675180T^18 + 250145827701T^17 + 162480194058T^16 + 95461872352T^15 + 50580270001T^14 + 24079632164T^13 + 10253793004T^12 + 3884457304T^11 + 1300634972T^10 + 381899322T^9 + 97401983T^8 + 21324999T^7 + 3948098T^6 + 605948T^5 + 75026T^4 + 7210T^3 + 504T^2 + 21*T + 1
$31$	$ T^{42} $ T^42
$37$	$ T^{42} - 42 T^{39} + \cdots + 1 $ T^42 - 42T^39 + 7T^37 + 777T^36 + 2T^35 - 245T^34 - 8330T^33 - 21T^32 + 3675T^31 + 57309T^30 - 322T^29 - 30866T^28 - 264551T^27 + 6958T^26 + 159334T^25 + 830844T^24 - 50484T^23 - 521101T^22 - 1759589T^21 + 184142T^20 + 1066779T^19 + 2448117T^18 - 343637T^17 - 1302980T^16 - 2165191T^15 + 253822T^14 + 913017T^13 + 1278753T^12 - 42413T^11 - 390285T^10 - 382963T^9 - 12908T^8 + 64481T^7 + 89719T^6 + 32046T^5 + 6293T^4 + 490T^3 + 119T^2 - 21T + 1
$41$	$ T^{42} $ T^42
$43$	$ T^{42} - 7 T^{39} + \cdots + 1 $ T^42 - 7T^39 - 7T^37 + 42T^36 - 2T^35 - 245T^34 - 245T^33 + 77T^32 + 1470T^31 + 1449T^30 + 1400T^29 - 6856T^28 - 8477T^27 + 14161T^26 + 27405T^25 + 36456T^24 - 63784T^23 - 87304T^22 - 110790T^21 + 172480T^20 + 135779T^19 - 71169T^18 - 222656T^17 + 578816T^16 + 256004T^15 - 377641T^14 - 141267T^13 - 6811T^12 + 292754T^11 + 298116T^10 - 258496T^9 + 87885T^8 - 124506T^7 + 90405T^6 - 29988T^5 + 8988T^4 - 490T^3 + 70T^2 + 21T + 1
$47$	$ T^{42} $ T^42
$53$	$ T^{42} + 7 T^{38} + \cdots + 1 $ T^42 + 7T^38 + 84T^37 + 7T^36 - 2T^35 + 49T^34 + 490T^33 + 2695T^32 + 462T^31 + 252T^30 + 2716T^29 + 12352T^28 + 41209T^27 + 13181T^26 + 13328T^25 + 49231T^24 + 134043T^23 + 321377T^22 + 172521T^21 + 167874T^20 + 330064T^19 + 690998T^18 + 1192387T^17 + 741377T^16 + 525154T^15 + 637989T^14 + 647437T^13 + 804237T^12 + 261513T^11 - 84161T^10 - 198835T^9 - 18844T^8 + 137196T^7 + 84819T^6 + 26362T^5 + 8820T^4 + 665T^3 + 42T^2 - 21T + 1
$59$	$ T^{42} $ T^42
$61$	$ T^{42} $ T^42
$67$	$ T^{42} - 7 T^{41} + \cdots + 7 $ T^42 - 7T^41 + 28T^40 - 84T^39 + 210T^38 - 462T^37 + 924T^36 - 1708T^35 + 2989T^34 - 5047T^33 + 8232T^32 - 12642T^31 + 17052T^30 - 16170T^29 - 5467T^28 + 65268T^27 - 164248T^26 + 266805T^25 - 291256T^24 + 128184T^23 + 297234T^22 - 924350T^21 + 1716617T^20 - 2529086T^19 + 2956366T^18 - 2452303T^17 + 849562T^16 + 1018122T^15 - 1515339T^14 + 524692T^13 + 608286T^12 - 816487T^11 + 300517T^10 + 83643T^9 - 20482T^8 - 1008T^7 + 24451T^6 - 29694T^5 + 19208T^4 - 7105T^3 + 1421T^2 - 147*T + 7
$71$	$ T^{42} - 7 T^{41} + \cdots + 7 $ T^42 - 7T^41 + 28T^40 - 84T^39 + 210T^38 - 462T^37 + 924T^36 - 1708T^35 + 2940T^34 - 4704T^33 + 6860T^32 - 8526T^31 + 6762T^30 - 5467T^28 - 11123T^27 + 87024T^26 - 266560T^25 + 571732T^24 - 916104T^23 + 1106616T^22 - 938756T^21 + 373968T^20 + 287728T^19 - 506464T^18 + 419538T^17 - 566440T^16 + 1110536T^15 - 1496131T^14 + 914879T^13 + 706188T^12 - 1694371T^11 + 1451086T^10 - 644007T^9 + 115836T^8 - 5810T^7 - 1176T^6 + 3136T^5 - 1960T^4 + 490T^2 + 98*T + 7
$73$	$ T^{42} $ T^42
$79$	$ T^{42} + 7 T^{41} + \cdots + 7 $ T^42 + 7T^41 + 28T^40 + 84T^39 + 210T^38 + 462T^37 + 924T^36 + 1708T^35 + 2940T^34 + 4900T^33 + 8232T^32 + 14014T^31 + 23226T^30 + 34692T^29 + 39809T^28 + 11123T^27 - 86485T^26 - 270235T^25 - 502642T^24 - 666694T^23 - 587706T^22 - 141694T^21 + 538608T^20 + 1127735T^19 + 1711717T^18 + 2489396T^17 + 3237528T^16 + 3286822T^15 + 2150645T^14 + 267736T^13 - 870828T^12 - 922131T^11 - 183946T^10 + 383180T^9 + 345646T^8 + 47656T^7 - 56987T^6 - 14014T^5 + 4655T^4 + 1813T^3 + 245T^2 + 7
$83$	$ T^{42} $ T^42
$89$	$ T^{42} $ T^42
$97$	$ T^{42} $ T^42
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1379 = 7 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1379.ba (of order \(98\), degree \(42\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{98}^{19} + \zeta_{98}^{18}) q^{2} + (\zeta_{98}^{38} + \cdots + \zeta_{98}^{36}) q^{4}+ \cdots + \zeta_{98}^{33} q^{9}+O(q^{10}) \) q + (z^19 + z^18) * q^2 + (z^38 + z^37 + z^36) * q^4 + z^6 * q^7 + (-z^8 + z^7 + z^6 - z^5) * q^8 + z^33 * q^9	\( q + (\zeta_{98}^{19} + \zeta_{98}^{18}) q^{2} + (\zeta_{98}^{38} + \cdots + \zeta_{98}^{36}) q^{4}+ \cdots + ( - \zeta_{98}^{48} - \zeta_{98}^{13}) q^{99}+O(q^{100}) \) q + (z^19 + z^18) * q^2 + (z^38 + z^37 + z^36) * q^4 + z^6 * q^7 + (-z^8 + z^7 + z^6 - z^5) * q^8 + z^33 * q^9 + (z^29 - z^15) * q^11 + (z^25 + z^24) * q^14 + (-z^27 + z^26 + z^25 + z^24 - z^23) * q^16 + (-z^3 - z^2) * q^18 + (z^48 + z^47 - z^34 - z^33) * q^22 + (z^22 + z^8) * q^23 - z^10 * q^25 + (z^44 + z^43 + z^42) * q^28 + (z^9 - 1) * q^29 + (-z^46 - z^45 - z^44 - z^43 - z^42 - z^41) * q^32 + (-z^22 - z^21 - z^20) * q^36 + (-z^47 - z) * q^37 + (-z^32 - z^12) * q^43 + (-z^18 - z^17 - z^16 + z^4 + z^3 + z^2) * q^44 + (z^41 + z^40 + z^27 + z^26) * q^46 + z^12 * q^49 + (-z^29 - z^28) * q^50 + (z^13 + z^5) * q^53 + (-z^14 - z^13 - z^12 - z^11) * q^56 + (z^28 + z^27 - z^19 - z^18) * q^58 + z^39 * q^63 + (z^16 - z^15 - z^14 - z^13 - z^12 - z^11 + z^10) * q^64 + (z^34 + z^7) * q^67 + (-z^45 + z^21) * q^71 + (-z^41 - z^40 - z^39 - z^38) * q^72 + (-z^20 - z^19 + z^17 + z^16) * q^74 + (z^35 - z^21) * q^77 + (-z^35 - z^4) * q^79 - z^17 * q^81 + (-z^31 - z^30 + z^2 + z) * q^86 + (-z^37 - z^36 - z^35 - z^34 + z^23 + z^22 + z^21 + z^20) * q^88 + (z^46 + z^45 + z^44 - z^11 - z^10 - z^9) * q^92 + (z^31 + z^30) * q^98 + (-z^48 - z^13) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{8}+O(q^{10}) \) 42 * q - 7 * q^8	\( 42 q - 7 q^{8} - 7 q^{28} - 42 q^{29} + 7 q^{32} - 7 q^{36} + 7 q^{50} + 7 q^{56} - 7 q^{58} - 7 q^{64} + 7 q^{67} + 7 q^{71} - 7 q^{79}+O(q^{100}) \) 42 * q - 7 * q^8 - 7 * q^28 - 42 * q^29 + 7 * q^32 - 7 * q^36 + 7 * q^50 + 7 * q^56 - 7 * q^58 - 7 * q^64 + 7 * q^67 + 7 * q^71 - 7 * q^79

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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	1379.1.ba.a

Level	$1379$
Weight	$1$
Character orbit	1379.ba
Analytic conductor	$0.688$
Analytic rank	$0$
Dimension	$42$
Projective image	$D_{98}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.688210652421\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Coefficient field:	\(\Q(\zeta_{98})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{42} - x^{35} + x^{28} - x^{21} + x^{14} - x^{7} + 1 \) x^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{98}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{98} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 41.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
197.h	even	98	1	inner
1379.ba	odd	98	1	inner

$p$	$F_p(T)$
$2$	\( T^{42} + 7 T^{39} + \cdots + 7 \) T^42 + 7T^39 - 7T^37 + 42T^36 + 245T^34 + 245T^33 + 49T^32 + 1470T^31 + 1421T^30 - 1078T^29 + 6860T^28 + 7889T^27 + 16121T^26 + 27783T^25 + 52822T^24 + 72422T^23 + 91238T^22 + 31206T^21 + 199626T^20 + 74431T^19 + 324919T^18 + 251762T^17 - 321048T^16 + 643566T^15 - 156065T^14 - 335111T^13 + 468195T^12 - 607502T^11 + 104272T^10 - 135142T^9 + 168903T^8 - 31206T^7 + 29155T^6 - 2744T^5 - 2058T^4 + 2058T^3 + 98T^2 - 49*T + 7
$3$	\( T^{42} \) T^42
$5$	\( T^{42} \) T^42
$7$	\( T^{42} + T^{35} + \cdots + 1 \) T^42 + T^35 + T^28 + T^21 + T^14 + T^7 + 1
$11$	\( T^{42} + 245 T^{35} + \cdots + 823543 \) T^42 + 245T^35 + 19551T^28 + 525819T^21 + 6168169T^14 + 2000033*T^7 + 823543
$13$	\( T^{42} \) T^42
$17$	\( T^{42} \) T^42
$19$	\( T^{42} \) T^42
$23$	\( T^{42} - 117 T^{35} + \cdots + 1 \) T^42 - 117T^35 + 4477T^28 - 24695T^21 + 83673T^14 - 129*T^7 + 1
$29$	\( T^{42} + 42 T^{41} + \cdots + 1 \) T^42 + 42T^41 + 861T^40 + 11480T^39 + 111930T^38 + 850668T^37 + 5245786T^36 + 26978327T^35 + 118030150T^34 + 445891215T^33 + 1471436428T^32 + 4280509016T^31 + 11057792256T^30 + 25517108120T^29 + 52853504561T^28 + 98648891824T^27 + 166439114520T^26 + 254478351036T^25 + 353279915625T^24 + 445940957280T^23 + 512315646360T^22 + 535939099079T^21 + 510546772344T^20 + 442722288540T^19 + 349172675180T^18 + 250145827701T^17 + 162480194058T^16 + 95461872352T^15 + 50580270001T^14 + 24079632164T^13 + 10253793004T^12 + 3884457304T^11 + 1300634972T^10 + 381899322T^9 + 97401983T^8 + 21324999T^7 + 3948098T^6 + 605948T^5 + 75026T^4 + 7210T^3 + 504T^2 + 21*T + 1
$31$	\( T^{42} \) T^42
$37$	\( T^{42} - 42 T^{39} + \cdots + 1 \) T^42 - 42T^39 + 7T^37 + 777T^36 + 2T^35 - 245T^34 - 8330T^33 - 21T^32 + 3675T^31 + 57309T^30 - 322T^29 - 30866T^28 - 264551T^27 + 6958T^26 + 159334T^25 + 830844T^24 - 50484T^23 - 521101T^22 - 1759589T^21 + 184142T^20 + 1066779T^19 + 2448117T^18 - 343637T^17 - 1302980T^16 - 2165191T^15 + 253822T^14 + 913017T^13 + 1278753T^12 - 42413T^11 - 390285T^10 - 382963T^9 - 12908T^8 + 64481T^7 + 89719T^6 + 32046T^5 + 6293T^4 + 490T^3 + 119T^2 - 21T + 1
$41$	\( T^{42} \) T^42
$43$	\( T^{42} - 7 T^{39} + \cdots + 1 \) T^42 - 7T^39 - 7T^37 + 42T^36 - 2T^35 - 245T^34 - 245T^33 + 77T^32 + 1470T^31 + 1449T^30 + 1400T^29 - 6856T^28 - 8477T^27 + 14161T^26 + 27405T^25 + 36456T^24 - 63784T^23 - 87304T^22 - 110790T^21 + 172480T^20 + 135779T^19 - 71169T^18 - 222656T^17 + 578816T^16 + 256004T^15 - 377641T^14 - 141267T^13 - 6811T^12 + 292754T^11 + 298116T^10 - 258496T^9 + 87885T^8 - 124506T^7 + 90405T^6 - 29988T^5 + 8988T^4 - 490T^3 + 70T^2 + 21T + 1
$47$	\( T^{42} \) T^42
$53$	\( T^{42} + 7 T^{38} + \cdots + 1 \) T^42 + 7T^38 + 84T^37 + 7T^36 - 2T^35 + 49T^34 + 490T^33 + 2695T^32 + 462T^31 + 252T^30 + 2716T^29 + 12352T^28 + 41209T^27 + 13181T^26 + 13328T^25 + 49231T^24 + 134043T^23 + 321377T^22 + 172521T^21 + 167874T^20 + 330064T^19 + 690998T^18 + 1192387T^17 + 741377T^16 + 525154T^15 + 637989T^14 + 647437T^13 + 804237T^12 + 261513T^11 - 84161T^10 - 198835T^9 - 18844T^8 + 137196T^7 + 84819T^6 + 26362T^5 + 8820T^4 + 665T^3 + 42T^2 - 21T + 1
$59$	\( T^{42} \) T^42
$61$	\( T^{42} \) T^42
$67$	\( T^{42} - 7 T^{41} + \cdots + 7 \) T^42 - 7T^41 + 28T^40 - 84T^39 + 210T^38 - 462T^37 + 924T^36 - 1708T^35 + 2989T^34 - 5047T^33 + 8232T^32 - 12642T^31 + 17052T^30 - 16170T^29 - 5467T^28 + 65268T^27 - 164248T^26 + 266805T^25 - 291256T^24 + 128184T^23 + 297234T^22 - 924350T^21 + 1716617T^20 - 2529086T^19 + 2956366T^18 - 2452303T^17 + 849562T^16 + 1018122T^15 - 1515339T^14 + 524692T^13 + 608286T^12 - 816487T^11 + 300517T^10 + 83643T^9 - 20482T^8 - 1008T^7 + 24451T^6 - 29694T^5 + 19208T^4 - 7105T^3 + 1421T^2 - 147*T + 7
$71$	\( T^{42} - 7 T^{41} + \cdots + 7 \) T^42 - 7T^41 + 28T^40 - 84T^39 + 210T^38 - 462T^37 + 924T^36 - 1708T^35 + 2940T^34 - 4704T^33 + 6860T^32 - 8526T^31 + 6762T^30 - 5467T^28 - 11123T^27 + 87024T^26 - 266560T^25 + 571732T^24 - 916104T^23 + 1106616T^22 - 938756T^21 + 373968T^20 + 287728T^19 - 506464T^18 + 419538T^17 - 566440T^16 + 1110536T^15 - 1496131T^14 + 914879T^13 + 706188T^12 - 1694371T^11 + 1451086T^10 - 644007T^9 + 115836T^8 - 5810T^7 - 1176T^6 + 3136T^5 - 1960T^4 + 490T^2 + 98*T + 7
$73$	\( T^{42} \) T^42
$79$	\( T^{42} + 7 T^{41} + \cdots + 7 \) T^42 + 7T^41 + 28T^40 + 84T^39 + 210T^38 + 462T^37 + 924T^36 + 1708T^35 + 2940T^34 + 4900T^33 + 8232T^32 + 14014T^31 + 23226T^30 + 34692T^29 + 39809T^28 + 11123T^27 - 86485T^26 - 270235T^25 - 502642T^24 - 666694T^23 - 587706T^22 - 141694T^21 + 538608T^20 + 1127735T^19 + 1711717T^18 + 2489396T^17 + 3237528T^16 + 3286822T^15 + 2150645T^14 + 267736T^13 - 870828T^12 - 922131T^11 - 183946T^10 + 383180T^9 + 345646T^8 + 47656T^7 - 56987T^6 - 14014T^5 + 4655T^4 + 1813T^3 + 245T^2 + 7
$83$	\( T^{42} \) T^42
$89$	\( T^{42} \) T^42
$97$	\( T^{42} \) T^42
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\(n\)	\(395\)	\(1184\)
\(\chi(n)\)	\(-1\)	\(\zeta_{98}^{37}\)