LMFDB - Newform orbit 3525.1.bg.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,1,Mod(107,3525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, base_ring=CyclotomicField(92))

chi = DirichletCharacter(H, H._module([46, 23, 22]))

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

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

chi := DirichletCharacter("3525.107");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3525.bg (of order $92$, degree $44$, 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:	$1.75920416953$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$2$ over $\Q(\zeta_{92})$
Coefficient field:	$\Q(\zeta_{184})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{88} - x^{84} + x^{80} - x^{76} + x^{72} - x^{68} + x^{64} - x^{60} + x^{56} - x^{52} + x^{48} + \cdots + 1 $ x^88 - x^84 + x^80 - x^76 + x^72 - x^68 + x^64 - x^60 + x^56 - x^52 + x^48 - x^44 + x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{46}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{46} - \cdots)$

$q$-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_{184}^{61} - \zeta_{184}^{33}) q^{2} + \zeta_{184}^{19} q^{3} + (\zeta_{184}^{66} + \cdots + \zeta_{184}^{2}) q^{4}+ \cdots + \zeta_{184}^{38} q^{9}+O(q^{10}) $ q + (z^61 - z^33) * q^2 + z^19 * q^3 + (z^66 - z^30 + z^2) * q^4 + (z^80 - z^52) * q^6 + (-z^91 + z^63 - z^35 + z^7) * q^8 + z^38 * q^9	$ q + (\zeta_{184}^{61} - \zeta_{184}^{33}) q^{2} + \zeta_{184}^{19} q^{3} + (\zeta_{184}^{66} + \cdots + \zeta_{184}^{2}) q^{4}+ \cdots + (\zeta_{184}^{39} - \zeta_{184}^{11}) q^{98}+O(q^{100}) $ q + (z^61 - z^33) * q^2 + z^19 * q^3 + (z^66 - z^30 + z^2) * q^4 + (z^80 - z^52) * q^6 + (-z^91 + z^63 - z^35 + z^7) * q^8 + z^38 * q^9 + (z^85 - z^49 + z^21) * q^12 + (z^68 + z^60 - z^40 - z^32 + z^4) * q^16 + (-z^41 - z^17) * q^17 + (-z^71 - z^7) * q^18 + (-z^62 - z^58) * q^19 + (-z^67 - z^23) * q^23 + (z^82 - z^54 + z^26 + z^18) * q^24 + z^57 * q^27 + (-z^56 - z^44) * q^31 + (z^73 + z^65 - z^37 - z^29 + z^9 - z) * q^32 + (-z^78 + z^74 + z^50 + z^10) * q^34 + (-z^68 + z^40 - z^12) * q^36 + (z^91 + z^31 + z^27 - z^3) * q^38 + (-z^84 + z^56 + z^36 - z^8) * q^46 - z^13 * q^47 + (z^87 + z^79 - z^59 - z^51 + z^23) * q^48 - z^70 * q^49 + (-z^60 - z^36) * q^51 + (z^51 + z^35) * q^53 + (-z^90 - z^26) * q^54 + (-z^81 - z^77) * q^57 + (z^52 + z^20) * q^61 + (z^89 + z^77 + z^25 + z^13) * q^62 + (-z^90 + z^70 + z^62 - z^42 - z^34 + z^14 + z^6) * q^64 + (-z^83 + z^71 + z^47 - z^43 - z^19 + z^15) * q^68 + (-z^86 - z^42) * q^69 + (-z^73 + z^45 + z^37 - z^9) * q^72 + (z^88 - z^64 - z^60 + z^36 + z^32 - 1) * q^76 + (-z^34 + z^22) * q^79 + z^76 * q^81 + (z^83 + z^43) * q^83 + (-z^89 - z^69 + z^53 + z^41 - z^25 - z^5) * q^92 + (-z^75 - z^63) * q^93 + (-z^74 + z^46) * q^94 + (z^84 - z^56 - z^48 + z^28 + z^20 - 1) * q^96 + (z^39 - z^11) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q - 8 q^{6}+O(q^{10}) $ 88 * q - 8 * q^6	$ 88 q - 8 q^{6} + 20 q^{16} - 12 q^{36} - 8 q^{51} + 8 q^{61} - 92 q^{76} + 4 q^{81} - 68 q^{96}+O(q^{100}) $ 88 * q - 8 * q^6 + 20 * q^16 - 12 * q^36 - 8 * q^51 + 8 * q^61 - 92 * q^76 + 4 * q^81 - 68 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times$.

$n$	$1552$	$2026$	$2351$
$\chi(n)$	$\zeta_{184}^{46}$	$-\zeta_{184}^{64}$	$-1$

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} $

107.1

0.985460	+	0.169910i
−0.985460	−	0.169910i
0.366854	−	0.930278i
−0.366854	+	0.930278i
0.0341411	−	0.999417i
−0.0341411	+	0.999417i
−0.102264	+	0.994757i
0.102264	−	0.994757i
0.999417	−	0.0341411i
−0.999417	+	0.0341411i
0.366854	+	0.930278i
−0.366854	−	0.930278i
0.971567	−	0.236764i
−0.971567	+	0.236764i
−0.657204	+	0.753713i
0.657204	−	0.753713i
−0.657204	−	0.753713i
0.657204	+	0.753713i
0.985460	−	0.169910i
−0.985460	+	0.169910i

−1.34526

−

0.231946i

−0.994757

−

0.102264i

0.813657

0.289174i

1.31448

0.368301i

0.162405

0.0913155i

0.979084

0.203456i

107.2

1.34526

0.231946i

0.994757

0.102264i

0.813657

0.289174i

1.31448

0.368301i

−0.162405

−

0.0913155i

0.979084

0.203456i

182.1

−0.626895

1.58970i

−0.753713

0.657204i

−1.40330

−

1.31059i

−0.572255

−

1.61017i

1.41995

−

0.675300i

0.136167

−

0.990686i

182.2

0.626895

−

1.58970i

0.753713

−

0.657204i

−1.40330

−

1.31059i

−0.572255

−

1.61017i

−1.41995

0.675300i

0.136167

−

0.990686i

218.1

−0.0314143

0.919594i

−0.604236

0.796805i

0.153003

0.0104657i

−0.713755

−

0.580683i

−0.108527

1.05568i

−0.269797

−

0.962917i

218.2

0.0314143

−

0.919594i

0.604236

−

0.796805i

0.153003

0.0104657i

−0.713755

−

0.580683i

0.108527

−

1.05568i

−0.269797

−

0.962917i

257.1

−0.202623

1.97098i

0.930278

0.366854i

−2.86464

−

0.595279i

−0.911560

1.75923i

1.15433

−

3.63700i

0.730836

0.682553i

257.2

0.202623

−

1.97098i

−0.930278

−

0.366854i

−2.86464

−

0.595279i

−0.911560

1.75923i

−1.15433

3.63700i

0.730836

0.682553i

293.1

−0.919594

0.0314143i

0.796805

−

0.604236i

−0.153003

0.0104657i

−0.713755

0.580683i

1.05568

−

0.108527i

0.269797

−

0.962917i

293.2

0.919594

−

0.0314143i

−0.796805

0.604236i

−0.153003

0.0104657i

−0.713755

0.580683i

−1.05568

0.108527i

0.269797

−

0.962917i

368.1

−0.626895

−

1.58970i

−0.753713

−

0.657204i

−1.40330

1.31059i

−0.572255

1.61017i

1.41995

0.675300i

0.136167

0.990686i

368.2

0.626895

1.58970i

0.753713

0.657204i

−1.40330

1.31059i

−0.572255

1.61017i

−1.41995

−

0.675300i

0.136167

0.990686i

407.1

−0.395342

0.0963423i

−0.169910

0.985460i

−0.740871

0.383889i

−0.0277687

−

0.405963i

0.562608

−

0.490569i

−0.942261

−

0.334880i

407.2

0.395342

−

0.0963423i

0.169910

−

0.985460i

−0.740871

0.383889i

−0.0277687

−

0.405963i

−0.562608

0.490569i

−0.942261

−

0.334880i

443.1

−0.757993

0.869303i

0.871660

−

0.490110i

−0.0449678

−

0.327165i

−0.234658

1.12924i

−0.645929

−

0.423665i

0.519584

−

0.854419i

443.2

0.757993

−

0.869303i

−0.871660

0.490110i

−0.0449678

−

0.327165i

−0.234658

1.12924i

0.645929

0.423665i

0.519584

−

0.854419i

557.1

−0.757993

−

0.869303i

0.871660

0.490110i

−0.0449678

0.327165i

−0.234658

−

1.12924i

−0.645929

0.423665i

0.519584

0.854419i

557.2

0.757993

0.869303i

−0.871660

−

0.490110i

−0.0449678

0.327165i

−0.234658

−

1.12924i

0.645929

−

0.423665i

0.519584

0.854419i

593.1

−1.34526

0.231946i

−0.994757

0.102264i

0.813657

−

0.289174i

1.31448

−

0.368301i

0.162405

−

0.0913155i

0.979084

−

0.203456i

593.2

1.34526

−

0.231946i

0.994757

−

0.102264i

0.813657

−

0.289174i

1.31448

−

0.368301i

−0.162405

0.0913155i

0.979084

−

0.203456i

See all 88 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
15.e	even	4	2	inner
47.d	odd	46	1	inner
141.g	even	46	1	inner
235.j	odd	46	1	inner
235.l	even	92	2	inner
705.o	even	46	1	inner
705.u	odd	92	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3525.1.bg.a	✓	88
3.b	odd	2	1	inner	3525.1.bg.a	✓	88
5.b	even	2	1	inner	3525.1.bg.a	✓	88
5.c	odd	4	2	inner	3525.1.bg.a	✓	88
15.d	odd	2	1	CM	3525.1.bg.a	✓	88
15.e	even	4	2	inner	3525.1.bg.a	✓	88
47.d	odd	46	1	inner	3525.1.bg.a	✓	88
141.g	even	46	1	inner	3525.1.bg.a	✓	88
235.j	odd	46	1	inner	3525.1.bg.a	✓	88
235.l	even	92	2	inner	3525.1.bg.a	✓	88
705.o	even	46	1	inner	3525.1.bg.a	✓	88
705.u	odd	92	2	inner	3525.1.bg.a	✓	88

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3525.1.bg.a	✓	88	1.a	even	1	1	trivial
3525.1.bg.a	✓	88	3.b	odd	2	1	inner
3525.1.bg.a	✓	88	5.b	even	2	1	inner
3525.1.bg.a	✓	88	5.c	odd	4	2	inner
3525.1.bg.a	✓	88	15.d	odd	2	1	CM
3525.1.bg.a	✓	88	15.e	even	4	2	inner
3525.1.bg.a	✓	88	47.d	odd	46	1	inner
3525.1.bg.a	✓	88	141.g	even	46	1	inner
3525.1.bg.a	✓	88	235.j	odd	46	1	inner
3525.1.bg.a	✓	88	235.l	even	92	2	inner
3525.1.bg.a	✓	88	705.o	even	46	1	inner
3525.1.bg.a	✓	88	705.u	odd	92	2	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3525, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{88} - 16 T^{84} + \cdots + 1 $ T^88 - 16T^84 + 164T^80 - 2624T^76 + 45526T^72 - 550120T^68 + 8727860T^64 - 33024752T^60 + 528730429T^56 - 470719342T^52 + 10622984092T^48 - 30478741181T^44 + 18356065784T^40 + 377888692286T^36 + 113665829572T^32 - 551442427592T^28 + 96930025810T^24 + 68169538352T^20 + 8870688801T^16 - 294683395T^12 + 8264000T^8 + 1137*T^4 + 1
$3$	$ T^{88} - T^{84} + \cdots + 1 $ T^88 - T^84 + T^80 - T^76 + T^72 - T^68 + T^64 - T^60 + T^56 - T^52 + T^48 - T^44 + T^40 - T^36 + T^32 - T^28 + T^24 - T^20 + T^16 - T^12 + T^8 - T^4 + 1
$5$	$ T^{88} $ T^88
$7$	$ T^{88} $ T^88
$11$	$ T^{88} $ T^88
$13$	$ T^{88} $ T^88
$17$	$ T^{88} + 76 T^{84} + \cdots + 1 $ T^88 + 76T^84 + 2694T^80 + 58740T^76 + 878977T^72 + 9543108T^68 + 77622060T^64 + 481285700T^60 + 2300383100T^56 + 8439258455T^52 + 24486698278T^48 + 46445370872T^44 + 129244723278T^40 - 102247162748T^36 + 601520127784T^32 - 615482026148T^28 + 452419620952T^24 - 123508804027T^20 + 16557089437T^16 - 643329102T^12 + 8741227T^8 - 5786*T^4 + 1
$19$	$ (T^{44} - 46 T^{38} + \cdots + 529)^{2} $ (T^44 - 46T^38 + 805T^32 + 3289T^30 - 6670T^26 + 235014T^24 - 3220T^22 + 26450T^20 + 1964177T^18 + 450708T^16 - 43907T^14 + 2401660T^12 - 1451576T^10 + 68241T^8 + 283544T^6 + 111090T^4 + 6348T^2 + 529)^2
$23$	$ T^{88} + 23 T^{84} + \cdots + 279841 $ T^88 + 23T^84 + 253T^80 + 1771T^76 + 8855T^72 + 33649T^68 + 101039T^64 - 75647T^60 + 73726730T^56 - 4131342938T^52 + 88093456946T^48 - 868349980034T^44 + 4425247256996T^40 - 12441921687228T^36 + 20013710005800T^32 - 18664460469072T^28 + 10455672600769T^24 - 2352578792281T^20 + 1097738167361T^16 + 87511597679T^12 + 10007394001T^8 + 89269279*T^4 + 279841
$29$	$ T^{88} $ T^88
$31$	$ (T^{22} + 23 T^{16} + \cdots + 23)^{4} $ (T^22 + 23T^16 + 69T^15 + 138T^10 - 1449T^9 + 736T^8 - 23T^7 + 161T^4 + 690T^3 + 598T^2 + 184T + 23)^4
$37$	$ T^{88} $ T^88
$41$	$ T^{88} $ T^88
$43$	$ T^{88} $ T^88
$47$	$ T^{88} - T^{84} + \cdots + 1 $ T^88 - T^84 + T^80 - T^76 + T^72 - T^68 + T^64 - T^60 + T^56 - T^52 + T^48 - T^44 + T^40 - T^36 + T^32 - T^28 + T^24 - T^20 + T^16 - T^12 + T^8 - T^4 + 1
$53$	$ T^{88} - 16 T^{84} + \cdots + 1 $ T^88 - 16T^84 + 256T^80 - 4096T^76 + 40420T^72 - 67258T^68 + 913196T^64 - 14599130T^60 + 363520118T^56 + 1601290720T^52 + 1799819427T^48 - 14590332625T^44 + 163471192098T^40 - 521671377684T^36 + 437630135119T^32 - 95488416633T^28 + 164928251587T^24 + 95910349481T^20 + 13440263044T^16 + 340515977T^12 + 8119123T^8 - 4429*T^4 + 1
$59$	$ T^{88} $ T^88
$61$	$ (T^{22} - 2 T^{21} + \cdots + 1)^{4} $ (T^22 - 2T^21 + 4T^20 + 15T^19 - 30T^18 + 60T^17 + 87T^16 - 174T^15 + 348T^14 + 224T^13 - 448T^12 + 896T^11 + 301T^10 - 579T^9 + 1158T^8 - 39T^7 - 198T^6 + 396T^5 + 174T^4 - 26T^3 + 52T^2 - 12*T + 1)^4
$67$	$ T^{88} $ T^88
$71$	$ T^{88} $ T^88
$73$	$ T^{88} $ T^88
$79$	$ (T^{44} - 4 T^{42} + \cdots + 1)^{2} $ (T^44 - 4T^42 + 16T^40 - 64T^38 + 256T^36 - 1024T^34 + 4050T^32 - 9622T^30 + 5598T^28 - 496T^26 - 638T^24 + 2598T^22 - 9863T^20 + 144194T^18 + 1011811T^16 + 1576555T^14 + 1033655T^12 + 347068T^10 + 73976T^8 - 1458T^6 + 2865T^4 - 98*T^2 + 1)^2
$83$	$ T^{88} - 16 T^{84} + \cdots + 1 $ T^88 - 16T^84 + 256T^80 - 4096T^76 + 65444T^72 - 690512T^68 + 1479640T^64 - 980140T^60 + 6008210T^56 - 50642420T^52 + 11893551380T^48 + 77719412538T^44 + 139331375624T^40 + 84218725373T^36 + 184567084138T^32 - 467948735011T^28 + 427063160391T^24 - 118127155444T^20 + 11734273234T^16 - 121875701T^12 + 8229730T^8 + 4909*T^4 + 1
$89$	$ T^{88} $ T^88
$97$	$ T^{88} $ T^88
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Classical	Maass
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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:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3525.bg (of order \(92\), degree \(44\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{184}^{61} - \zeta_{184}^{33}) q^{2} + \zeta_{184}^{19} q^{3} + (\zeta_{184}^{66} + \cdots + \zeta_{184}^{2}) q^{4}+ \cdots + \zeta_{184}^{38} q^{9}+O(q^{10}) \) q + (z^61 - z^33) * q^2 + z^19 * q^3 + (z^66 - z^30 + z^2) * q^4 + (z^80 - z^52) * q^6 + (-z^91 + z^63 - z^35 + z^7) * q^8 + z^38 * q^9	\( q + (\zeta_{184}^{61} - \zeta_{184}^{33}) q^{2} + \zeta_{184}^{19} q^{3} + (\zeta_{184}^{66} + \cdots + \zeta_{184}^{2}) q^{4}+ \cdots + (\zeta_{184}^{39} - \zeta_{184}^{11}) q^{98}+O(q^{100}) \) q + (z^61 - z^33) * q^2 + z^19 * q^3 + (z^66 - z^30 + z^2) * q^4 + (z^80 - z^52) * q^6 + (-z^91 + z^63 - z^35 + z^7) * q^8 + z^38 * q^9 + (z^85 - z^49 + z^21) * q^12 + (z^68 + z^60 - z^40 - z^32 + z^4) * q^16 + (-z^41 - z^17) * q^17 + (-z^71 - z^7) * q^18 + (-z^62 - z^58) * q^19 + (-z^67 - z^23) * q^23 + (z^82 - z^54 + z^26 + z^18) * q^24 + z^57 * q^27 + (-z^56 - z^44) * q^31 + (z^73 + z^65 - z^37 - z^29 + z^9 - z) * q^32 + (-z^78 + z^74 + z^50 + z^10) * q^34 + (-z^68 + z^40 - z^12) * q^36 + (z^91 + z^31 + z^27 - z^3) * q^38 + (-z^84 + z^56 + z^36 - z^8) * q^46 - z^13 * q^47 + (z^87 + z^79 - z^59 - z^51 + z^23) * q^48 - z^70 * q^49 + (-z^60 - z^36) * q^51 + (z^51 + z^35) * q^53 + (-z^90 - z^26) * q^54 + (-z^81 - z^77) * q^57 + (z^52 + z^20) * q^61 + (z^89 + z^77 + z^25 + z^13) * q^62 + (-z^90 + z^70 + z^62 - z^42 - z^34 + z^14 + z^6) * q^64 + (-z^83 + z^71 + z^47 - z^43 - z^19 + z^15) * q^68 + (-z^86 - z^42) * q^69 + (-z^73 + z^45 + z^37 - z^9) * q^72 + (z^88 - z^64 - z^60 + z^36 + z^32 - 1) * q^76 + (-z^34 + z^22) * q^79 + z^76 * q^81 + (z^83 + z^43) * q^83 + (-z^89 - z^69 + z^53 + z^41 - z^25 - z^5) * q^92 + (-z^75 - z^63) * q^93 + (-z^74 + z^46) * q^94 + (z^84 - z^56 - z^48 + z^28 + z^20 - 1) * q^96 + (z^39 - z^11) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 8 q^{6}+O(q^{10}) \) 88 * q - 8 * q^6	\( 88 q - 8 q^{6} + 20 q^{16} - 12 q^{36} - 8 q^{51} + 8 q^{61} - 92 q^{76} + 4 q^{81} - 68 q^{96}+O(q^{100}) \) 88 * q - 8 * q^6 + 20 * q^16 - 12 * q^36 - 8 * q^51 + 8 * q^61 - 92 * q^76 + 4 * q^81 - 68 * q^96

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	3525.1.bg.a

Level	$3525$
Weight	$1$
Character orbit	3525.bg
Analytic conductor	$1.759$
Analytic rank	$0$
Dimension	$88$
Projective image	$D_{46}$
CM discriminant	-15
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(1.75920416953\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(2\) over \(\Q(\zeta_{92})\)
Coefficient field:	\(\Q(\zeta_{184})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{88} - x^{84} + x^{80} - x^{76} + x^{72} - x^{68} + x^{64} - x^{60} + x^{56} - x^{52} + x^{48} + \cdots + 1 \) x^88 - x^84 + x^80 - x^76 + x^72 - x^68 + x^64 - x^60 + x^56 - x^52 + x^48 - x^44 + x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{46}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{46} - \cdots)\)

\(n\)	\(1552\)	\(2026\)	\(2351\)
\(\chi(n)\)	\(\zeta_{184}^{46}\)	\(-\zeta_{184}^{64}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 107.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{88} - 16 T^{84} + \cdots + 1 \) T^88 - 16T^84 + 164T^80 - 2624T^76 + 45526T^72 - 550120T^68 + 8727860T^64 - 33024752T^60 + 528730429T^56 - 470719342T^52 + 10622984092T^48 - 30478741181T^44 + 18356065784T^40 + 377888692286T^36 + 113665829572T^32 - 551442427592T^28 + 96930025810T^24 + 68169538352T^20 + 8870688801T^16 - 294683395T^12 + 8264000T^8 + 1137*T^4 + 1
$3$	\( T^{88} - T^{84} + \cdots + 1 \) T^88 - T^84 + T^80 - T^76 + T^72 - T^68 + T^64 - T^60 + T^56 - T^52 + T^48 - T^44 + T^40 - T^36 + T^32 - T^28 + T^24 - T^20 + T^16 - T^12 + T^8 - T^4 + 1
$5$	\( T^{88} \) T^88
$7$	\( T^{88} \) T^88
$11$	\( T^{88} \) T^88
$13$	\( T^{88} \) T^88
$17$	\( T^{88} + 76 T^{84} + \cdots + 1 \) T^88 + 76T^84 + 2694T^80 + 58740T^76 + 878977T^72 + 9543108T^68 + 77622060T^64 + 481285700T^60 + 2300383100T^56 + 8439258455T^52 + 24486698278T^48 + 46445370872T^44 + 129244723278T^40 - 102247162748T^36 + 601520127784T^32 - 615482026148T^28 + 452419620952T^24 - 123508804027T^20 + 16557089437T^16 - 643329102T^12 + 8741227T^8 - 5786*T^4 + 1
$19$	\( (T^{44} - 46 T^{38} + \cdots + 529)^{2} \) (T^44 - 46T^38 + 805T^32 + 3289T^30 - 6670T^26 + 235014T^24 - 3220T^22 + 26450T^20 + 1964177T^18 + 450708T^16 - 43907T^14 + 2401660T^12 - 1451576T^10 + 68241T^8 + 283544T^6 + 111090T^4 + 6348T^2 + 529)^2
$23$	\( T^{88} + 23 T^{84} + \cdots + 279841 \) T^88 + 23T^84 + 253T^80 + 1771T^76 + 8855T^72 + 33649T^68 + 101039T^64 - 75647T^60 + 73726730T^56 - 4131342938T^52 + 88093456946T^48 - 868349980034T^44 + 4425247256996T^40 - 12441921687228T^36 + 20013710005800T^32 - 18664460469072T^28 + 10455672600769T^24 - 2352578792281T^20 + 1097738167361T^16 + 87511597679T^12 + 10007394001T^8 + 89269279*T^4 + 279841
$29$	\( T^{88} \) T^88
$31$	\( (T^{22} + 23 T^{16} + \cdots + 23)^{4} \) (T^22 + 23T^16 + 69T^15 + 138T^10 - 1449T^9 + 736T^8 - 23T^7 + 161T^4 + 690T^3 + 598T^2 + 184T + 23)^4
$37$	\( T^{88} \) T^88
$41$	\( T^{88} \) T^88
$43$	\( T^{88} \) T^88
$47$	\( T^{88} - T^{84} + \cdots + 1 \) T^88 - T^84 + T^80 - T^76 + T^72 - T^68 + T^64 - T^60 + T^56 - T^52 + T^48 - T^44 + T^40 - T^36 + T^32 - T^28 + T^24 - T^20 + T^16 - T^12 + T^8 - T^4 + 1
$53$	\( T^{88} - 16 T^{84} + \cdots + 1 \) T^88 - 16T^84 + 256T^80 - 4096T^76 + 40420T^72 - 67258T^68 + 913196T^64 - 14599130T^60 + 363520118T^56 + 1601290720T^52 + 1799819427T^48 - 14590332625T^44 + 163471192098T^40 - 521671377684T^36 + 437630135119T^32 - 95488416633T^28 + 164928251587T^24 + 95910349481T^20 + 13440263044T^16 + 340515977T^12 + 8119123T^8 - 4429*T^4 + 1
$59$	\( T^{88} \) T^88
$61$	\( (T^{22} - 2 T^{21} + \cdots + 1)^{4} \) (T^22 - 2T^21 + 4T^20 + 15T^19 - 30T^18 + 60T^17 + 87T^16 - 174T^15 + 348T^14 + 224T^13 - 448T^12 + 896T^11 + 301T^10 - 579T^9 + 1158T^8 - 39T^7 - 198T^6 + 396T^5 + 174T^4 - 26T^3 + 52T^2 - 12*T + 1)^4
$67$	\( T^{88} \) T^88
$71$	\( T^{88} \) T^88
$73$	\( T^{88} \) T^88
$79$	\( (T^{44} - 4 T^{42} + \cdots + 1)^{2} \) (T^44 - 4T^42 + 16T^40 - 64T^38 + 256T^36 - 1024T^34 + 4050T^32 - 9622T^30 + 5598T^28 - 496T^26 - 638T^24 + 2598T^22 - 9863T^20 + 144194T^18 + 1011811T^16 + 1576555T^14 + 1033655T^12 + 347068T^10 + 73976T^8 - 1458T^6 + 2865T^4 - 98*T^2 + 1)^2
$83$	\( T^{88} - 16 T^{84} + \cdots + 1 \) T^88 - 16T^84 + 256T^80 - 4096T^76 + 65444T^72 - 690512T^68 + 1479640T^64 - 980140T^60 + 6008210T^56 - 50642420T^52 + 11893551380T^48 + 77719412538T^44 + 139331375624T^40 + 84218725373T^36 + 184567084138T^32 - 467948735011T^28 + 427063160391T^24 - 118127155444T^20 + 11734273234T^16 - 121875701T^12 + 8229730T^8 + 4909*T^4 + 1
$89$	\( T^{88} \) T^88
$97$	\( T^{88} \) T^88
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