LMFDB - Newform orbit 2301.1.z.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2301,1,Mod(116,2301)] 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("2301.116"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2301, base_ring=CyclotomicField(58)) chi = DirichletCharacter(H, H._module([29, 29, 30])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2301 = 3 \cdot 13 \cdot 59 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2301.z (of order $58$, degree $28$, minimal)

Newform invariants

comment:select newform

sage:traces = [56,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$1.14834859407$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$2$ over $\Q(\zeta_{58})$
Coefficient field:	$\Q(\zeta_{116})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{56} - x^{54} + x^{52} - x^{50} + x^{48} - x^{46} + x^{44} - x^{42} + x^{40} - x^{38} + x^{36} + \cdots + 1 $ x^56 - x^54 + x^52 - x^50 + x^48 - x^46 + x^44 - x^42 + x^40 - x^38 + x^36 - x^34 + x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{59}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{58}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{58} - \cdots)$

$q$-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_{116}^{45} - \zeta_{116}^{33}) q^{2} - \zeta_{116}^{36} q^{3} + ( - \zeta_{116}^{32} + \cdots - \zeta_{116}^{8}) q^{4} + (\zeta_{116}^{39} - \zeta_{116}^{23}) q^{5} + (\zeta_{116}^{23} - \zeta_{116}^{11}) q^{6} + \cdots + (\zeta_{116}^{35} + \zeta_{116}^{29}) q^{99} +O(q^{100}) $ q + (z^45 - z^33) * q^2 - z^36 * q^3 + (-z^32 + z^20 - z^8) * q^4 + (z^39 - z^23) * q^5 + (z^23 - z^11) * q^6 + (-z^53 + z^41 + z^19 - z^7) * q^8 - z^14 * q^9 + (z^56 - z^26 + z^14 + z^10) * q^10 + (-z^21 - z^15) * q^11 + (-z^56 + z^44 - z^10) * q^12 - z^44 * q^13 + (z^17 - z) * q^15 + (-z^52 + z^40 - z^28 + z^16 - z^6) * q^16 + (z^47 + z) * q^18 + (z^55 - z^47 - z^43 + z^31 + z^13 - z) * q^20 + (z^54 + z^48 + z^8 + z^2) * q^22 + (-z^55 + z^43 - z^31 + z^19) * q^24 + (z^46 - z^20 + z^4) * q^25 + (z^31 - z^19) * q^26 + z^50 * q^27 + (-z^50 - z^46 + z^34 - z^4) * q^30 + (-z^51 - z^49 + z^39 - z^27 + z^15 - z^3) * q^32 + (z^57 + z^51) * q^33 + (z^46 - z^34 + z^22) * q^36 - z^22 * q^39 + (-z^46 - z^42 + z^34 + z^30 - z^22 - z^18 + z^6 - 1) * q^40 + (z^43 + z^5) * q^41 + (z^48 + z^32) * q^43 + (z^53 + z^47 - z^41 - z^35 + z^29 + z^23) * q^44 + (-z^53 + z^37) * q^45 + (-z^39 + z^15) * q^47 + (-z^52 + z^42 - z^30 + z^18 - z^6) * q^48 + z^12 * q^49 + (z^53 + z^49 - z^37 - z^33 + z^21 + z^7) * q^50 + (z^52 - z^18 + z^6) * q^52 + (-z^37 + z^25) * q^54 + (-z^54 + z^44 + z^38 + z^2) * q^55 - z^3 * q^59 + (-z^49 + z^37 + z^33 - z^25 - z^21 + z^9) * q^60 + (-z^16 - z^4) * q^61 + (-z^48 + z^38 + z^36 - z^26 - z^24 + z^14 - z^2) * q^64 + (z^25 - z^9) * q^65 + (-z^44 - z^38 + z^32 + z^26) * q^66 + (z^49 - z^5) * q^71 + (-z^55 - z^33 + z^21 - z^9) * q^72 + (z^56 - z^40 + z^24) * q^75 + (z^55 + z^9) * q^78 + (-z^32 - z^12) * q^79 + (z^55 + z^51 - z^45 - z^39 + z^33 + z^29 - z^21 - z^17 + z^9 + z^5) * q^80 + z^28 * q^81 + (z^50 - z^38 - z^30 + z^18) * q^82 + (-z^41 + z^33) * q^83 + (-z^35 + z^23 - z^19 + z^7) * q^86 + (-z^56 - z^40 - z^34 + z^28 + z^22 - z^16 - z^10 + z^4) * q^88 + (-z^53 - z^43) * q^89 + (z^40 - z^28 - z^24 + z^12) * q^90 + (-z^48 + z^26 - z^14 - z^2) * q^94 + (-z^51 + z^39 - z^29 - z^27 + z^17 - z^5) * q^96 + (z^57 - z^45) * q^98 + (z^35 + z^29) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 2 q^{3} + 2 q^{4} - 2 q^{9} - 2 q^{12} + 2 q^{13} - 2 q^{16} + 2 q^{25} + 2 q^{27} + 2 q^{36} - 2 q^{39} - 58 q^{40} - 4 q^{43} + 2 q^{48} - 2 q^{49} - 2 q^{52} + 4 q^{61} + 2 q^{64} - 2 q^{75} + 4 q^{79}+ \cdots - 2 q^{81}+O(q^{100}) $ 56 * q + 2 * q^3 + 2 * q^4 - 2 * q^9 - 2 * q^12 + 2 * q^13 - 2 * q^16 + 2 * q^25 + 2 * q^27 + 2 * q^36 - 2 * q^39 - 58 * q^40 - 4 * q^43 + 2 * q^48 - 2 * q^49 - 2 * q^52 + 4 * q^61 + 2 * q^64 - 2 * q^75 + 4 * q^79 - 2 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2301\mathbb{Z}\right)^\times$.

$n$	$886$	$1535$	$2185$
$\chi(n)$	$-1$	$-1$	$\zeta_{116}^{20}$

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} $

116.1

0.986827	+	0.161782i
−0.986827	−	0.161782i
0.214970	−	0.976621i
−0.214970	+	0.976621i
−0.762162	+	0.647386i
0.762162	−	0.647386i
0.605174	−	0.796093i
−0.605174	+	0.796093i
−0.687699	+	0.725995i
0.687699	−	0.725995i
−0.319302	−	0.947653i
0.319302	+	0.947653i
0.419889	−	0.907575i
−0.419889	+	0.907575i
0.827689	+	0.561187i
−0.827689	−	0.561187i
0.883512	+	0.468408i
−0.883512	−	0.468408i
0.419889	+	0.907575i
−0.419889	−	0.907575i

−0.0896204

1.65295i

−0.907575

0.419889i

−1.73007

−

0.188157i

1.82622

0.615326i

−0.612719

−

1.53781i

0.198254

−

1.20929i

0.647386

−

0.762162i

−1.18077

2.96351i

116.2

0.0896204

−

1.65295i

−0.907575

0.419889i

−1.73007

−

0.188157i

−1.82622

−

0.615326i

0.612719

1.53781i

−0.198254

1.20929i

0.647386

−

0.762162i

−1.18077

2.96351i

194.1

−1.08146

1.59504i

−0.0541389

−

0.998533i

−1.00445

−

2.52098i

−1.79124

−

0.828715i

1.65125

0.993524i

3.22529

0.709940i

−0.994138

0.108119i

3.25899

−

1.96087i

194.2

1.08146

−

1.59504i

−0.0541389

−

0.998533i

−1.00445

−

2.52098i

1.79124

0.828715i

−1.65125

−

0.993524i

−3.22529

−

0.709940i

−0.994138

0.108119i

3.25899

−

1.96087i

272.1

−1.28285

1.21518i

−0.976621

0.214970i

0.114903

−

2.11926i

−0.195813

1.19440i

0.991631

−

1.46255i

1.28394

1.51157i

0.907575

−

0.419889i

−1.20022

−

1.77019i

272.2

1.28285

−

1.21518i

−0.976621

0.214970i

0.114903

−

2.11926i

0.195813

−

1.19440i

−0.991631

1.46255i

−1.28394

−

1.51157i

0.907575

−

0.419889i

−1.20022

−

1.77019i

311.1

−1.34324

−

0.295670i

0.161782

0.986827i

0.809305

0.374424i

0.472729

1.70262i

0.0744626

−

1.37338i

0.118558

0.0901257i

−0.947653

0.319302i

−0.131577

−

2.42680i

311.2

1.34324

0.295670i

0.161782

0.986827i

0.809305

0.374424i

−0.472729

−

1.70262i

−0.0744626

1.37338i

−0.118558

−

0.0901257i

−0.947653

0.319302i

−0.131577

−

2.42680i

389.1

−0.528008

−

1.90171i

0.561187

−

0.827689i

−2.48086

1.49269i

0.0232765

0.429310i

−1.87034

−

0.630190i

2.71571

2.57246i

−0.370138

−

0.928977i

0.804135

−

0.270945i

389.2

0.528008

1.90171i

0.561187

−

0.827689i

−2.48086

1.49269i

−0.0232765

−

0.429310i

1.87034

0.630190i

−2.71571

−

2.57246i

−0.370138

−

0.928977i

0.804135

−

0.270945i

428.1

−1.84706

0.200880i

−0.647386

−

0.762162i

2.39466

−

0.527106i

−0.820858

0.624000i

1.34887

1.27771i

−2.55651

0.861390i

−0.161782

0.986827i

1.39083

−

1.31746i

428.2

1.84706

−

0.200880i

−0.647386

−

0.762162i

2.39466

−

0.527106i

0.820858

−

0.624000i

−1.34887

−

1.27771i

2.55651

−

0.861390i

−0.161782

0.986827i

1.39083

−

1.31746i

584.1

−0.381652

−

0.957875i

0.994138

−

0.108119i

−0.0458707

0.0434510i

0.413423

0.486719i

−0.482980

−

0.910996i

−0.876681

−

0.405596i

0.976621

−

0.214970i

0.308432

−

0.581765i

584.2

0.381652

0.957875i

0.994138

−

0.108119i

−0.0458707

0.0434510i

−0.413423

−

0.486719i

0.482980

0.910996i

0.876681

0.405596i

0.976621

−

0.214970i

0.308432

−

0.581765i

779.1

−0.795818

0.268142i

0.856857

−

0.515554i

−0.234666

0.178389i

−0.739191

−

1.85523i

−0.543661

0.640047i

0.610191

−

0.899964i

0.468408

−

0.883512i

1.08573

1.27822i

779.2

0.795818

−

0.268142i

0.856857

−

0.515554i

−0.234666

0.178389i

0.739191

1.85523i

0.543661

−

0.640047i

−0.610191

0.899964i

0.468408

−

0.883512i

1.08573

1.27822i

818.1

−0.0695567

0.424277i

−0.267528

0.963550i

0.772480

0.260279i

0.771856

1.13840i

−0.390204

−

0.180527i

−0.365549

0.689499i

−0.856857

−

0.515554i

−0.536686

0.248297i

818.2

0.0695567

−

0.424277i

−0.267528

0.963550i

0.772480

0.260279i

−0.771856

−

1.13840i

0.390204

0.180527i

0.365549

−

0.689499i

−0.856857

−

0.515554i

−0.536686

0.248297i

1052.1

−0.381652

0.957875i

0.994138

0.108119i

−0.0458707

−

0.0434510i

0.413423

−

0.486719i

−0.482980

0.910996i

−0.876681

0.405596i

0.976621

0.214970i

0.308432

0.581765i

1052.2

0.381652

−

0.957875i

0.994138

0.108119i

−0.0458707

−

0.0434510i

−0.413423

0.486719i

0.482980

−

0.910996i

0.876681

−

0.405596i

0.976621

0.214970i

0.308432

0.581765i

See all 56 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
39.d	odd	2	1	CM by $\Q(\sqrt{-39}) $
3.b	odd	2	1	inner
13.b	even	2	1	inner
59.c	even	29	1	inner
177.h	odd	58	1	inner
767.p	even	58	1	inner
2301.z	odd	58	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2301.1.z.c	✓	56
3.b	odd	2	1	inner	2301.1.z.c	✓	56
13.b	even	2	1	inner	2301.1.z.c	✓	56
39.d	odd	2	1	CM	2301.1.z.c	✓	56
59.c	even	29	1	inner	2301.1.z.c	✓	56
177.h	odd	58	1	inner	2301.1.z.c	✓	56
767.p	even	58	1	inner	2301.1.z.c	✓	56
2301.z	odd	58	1	inner	2301.1.z.c	✓	56

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2301.1.z.c	✓	56	1.a	even	1	1	trivial
2301.1.z.c	✓	56	3.b	odd	2	1	inner
2301.1.z.c	✓	56	13.b	even	2	1	inner
2301.1.z.c	✓	56	39.d	odd	2	1	CM
2301.1.z.c	✓	56	59.c	even	29	1	inner
2301.1.z.c	✓	56	177.h	odd	58	1	inner
2301.1.z.c	✓	56	767.p	even	58	1	inner
2301.1.z.c	✓	56	2301.z	odd	58	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{56} + 58 T_{2}^{48} - 8294 T_{2}^{42} + 1247 T_{2}^{40} + 3306 T_{2}^{36} + 1223742 T_{2}^{34} + \cdots + 841 $ T2^56 + 58*T2^48 - 8294*T2^42 + 1247*T2^40 + 3306*T2^36 + 1223742*T2^34 + 12180*T2^32 - 58*T2^30 + 7403381*T2^28 - 17077346*T2^26 + 52983*T2^24 + 3549020*T2^22 + 134358160*T2^20 + 31778026*T2^18 + 465073*T2^16 - 50567648*T2^14 + 70539716*T2^12 - 5315961*T2^10 + 1842631*T2^8 + 2736614*T2^6 + 429751*T2^4 + 11774*T2^2 + 841 acting on $S_{1}^{\mathrm{new}}(2301, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{56} + 58 T^{48} + \cdots + 841 $ T^56 + 58T^48 - 8294T^42 + 1247T^40 + 3306T^36 + 1223742T^34 + 12180T^32 - 58T^30 + 7403381T^28 - 17077346T^26 + 52983T^24 + 3549020T^22 + 134358160T^20 + 31778026T^18 + 465073T^16 - 50567648T^14 + 70539716T^12 - 5315961T^10 + 1842631T^8 + 2736614T^6 + 429751T^4 + 11774*T^2 + 841
$3$	$ (T^{28} - T^{27} + T^{26} + \cdots + 1)^{2} $ (T^28 - T^27 + T^26 - T^25 + T^24 - T^23 + T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$5$	$ T^{56} + 406 T^{48} + \cdots + 841 $ T^56 + 406T^48 + 53389T^40 + 9860T^38 - 58T^34 + 2527930T^32 - 2966990T^30 + 6786T^28 + 29435T^26 + 34580238T^24 + 71081320T^22 + 2612146T^20 - 846887T^18 + 75716912T^16 - 113425670T^14 + 12676393T^12 + 1120212T^10 + 7403323T^8 + 3549020T^6 + 382655T^4 + 5887T^2 + 841
$7$	$ T^{56} $ T^56
$11$	$ T^{56} + 1218 T^{46} + \cdots + 841 $ T^56 + 1218T^46 - 27608T^38 + 376449T^36 - 4060T^34 + 33350T^30 + 10855686T^28 + 23424373T^26 + 1155534T^24 - 9251T^22 + 12197864T^20 - 142761432T^18 + 104090570T^16 - 9085323T^14 + 1441474T^12 + 15058946T^10 + 15442442T^8 + 3654145T^6 + 318739T^4 - 841*T^2 + 841
$13$	$ (T^{28} - T^{27} + T^{26} + \cdots + 1)^{2} $ (T^28 - T^27 + T^26 - T^25 + T^24 - T^23 + T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$17$	$ T^{56} $ T^56
$19$	$ T^{56} $ T^56
$23$	$ T^{56} $ T^56
$29$	$ T^{56} $ T^56
$31$	$ T^{56} $ T^56
$37$	$ T^{56} $ T^56
$41$	$ T^{56} + 58 T^{52} + \cdots + 841 $ T^56 + 58T^52 + 1479T^48 + 21808T^44 + 205755T^40 + 1298388T^36 + 609T^34 + 5565651T^32 - 111418T^30 + 16138848T^28 + 3705446T^26 + 30972348T^24 - 35488518T^22 + 37718850T^20 + 112996760T^18 + 27183643T^16 - 120609492T^14 + 10351028T^12 + 39030810T^10 + 1771146T^8 - 2973776T^6 + 111853T^4 + 29435T^2 + 841
$43$	$ (T^{28} + 2 T^{27} + \cdots + 1)^{2} $ (T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 128T^21 + 256T^20 + 512T^19 + 995T^18 + 1120T^17 + 326T^16 + 14T^15 - T^14 - 2T^13 - 4T^12 - 8T^11 - 16T^10 - 32T^9 + 81T^8 - 2709T^7 + 7023T^6 - 7066T^5 + 3616T^4 - 1033T^3 + 167T^2 - 14T + 1)^2
$47$	$ T^{56} - 58 T^{46} + \cdots + 841 $ T^56 - 58T^46 + 3828T^44 - 58T^42 + 1131T^36 + 621006T^34 + 2590454T^32 + 291566T^30 + 899T^28 - 8410T^26 + 6272178T^24 - 89060218T^22 + 110110448T^20 - 20730650T^18 + 705599T^16 + 4633069T^14 + 38700297T^12 + 50920868T^10 + 19581844T^8 + 3020031T^6 + 263233T^4 - 8410*T^2 + 841
$53$	$ T^{56} $ T^56
$59$	$ T^{56} - T^{54} + \cdots + 1 $ T^56 - T^54 + T^52 - T^50 + T^48 - T^46 + T^44 - T^42 + T^40 - T^38 + T^36 - T^34 + T^32 - T^30 + T^28 - T^26 + T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$61$	$ (T^{28} - 2 T^{27} + \cdots + 1)^{2} $ (T^28 - 2T^27 + 4T^26 - 8T^25 + 16T^24 - 32T^23 + 64T^22 - 128T^21 + 227T^20 - 454T^19 + 908T^18 - 598T^17 + 1196T^16 - 2392T^15 + 637T^14 - 1274T^13 + 2751T^12 - 1442T^11 + 2884T^10 + 322T^9 - 2297T^8 + 4594T^7 + 440T^6 - 561T^5 + 919T^4 - 40T^3 + 51T^2 + 14*T + 1)^2
$67$	$ T^{56} $ T^56
$71$	$ T^{56} + 58 T^{48} + \cdots + 841 $ T^56 + 58T^48 - 8294T^42 + 1247T^40 + 3306T^36 + 1223742T^34 + 12180T^32 - 58T^30 + 7403381T^28 - 17077346T^26 + 52983T^24 + 3549020T^22 + 134358160T^20 + 31778026T^18 + 465073T^16 - 50567648T^14 + 70539716T^12 - 5315961T^10 + 1842631T^8 + 2736614T^6 + 429751T^4 + 11774*T^2 + 841
$73$	$ T^{56} $ T^56
$79$	$ (T^{28} - 2 T^{27} + \cdots + 1)^{2} $ (T^28 - 2T^27 + 4T^26 - 8T^25 + 16T^24 - 32T^23 + 64T^22 - 128T^21 + 227T^20 - 454T^19 + 908T^18 - 598T^17 + 1196T^16 - 2392T^15 + 637T^14 - 1274T^13 + 2751T^12 - 1442T^11 + 2884T^10 + 322T^9 - 2297T^8 + 4594T^7 + 440T^6 - 561T^5 + 919T^4 - 40T^3 + 51T^2 + 14*T + 1)^2
$83$	$ T^{56} + 58 T^{36} + \cdots + 841 $ T^56 + 58T^36 + 19285T^34 + 836418T^32 + 11849255T^30 + 77558760T^28 + 281150505T^26 + 629614650T^24 + 935427480T^22 + 967964088T^20 + 721540996T^18 + 396486927T^16 + 162937863T^14 + 51356506T^12 + 8400749T^10 + 6808736T^8 - 2242106T^6 + 538240T^4 - 36163T^2 + 841
$89$	$ T^{56} + 58 T^{36} + \cdots + 841 $ T^56 + 58T^36 + 19285T^34 + 836418T^32 + 11849255T^30 + 77558760T^28 + 281150505T^26 + 629614650T^24 + 935427480T^22 + 967964088T^20 + 721540996T^18 + 396486927T^16 + 162937863T^14 + 51356506T^12 + 8400749T^10 + 6808736T^8 - 2242106T^6 + 538240T^4 - 36163T^2 + 841
$97$	$ T^{56} $ T^56
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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:	\( N \)	\(=\)	\( 2301 = 3 \cdot 13 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2301.z (of order \(58\), degree \(28\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{116}^{45} - \zeta_{116}^{33}) q^{2} - \zeta_{116}^{36} q^{3} + ( - \zeta_{116}^{32} + \cdots - \zeta_{116}^{8}) q^{4} + (\zeta_{116}^{39} - \zeta_{116}^{23}) q^{5} + (\zeta_{116}^{23} - \zeta_{116}^{11}) q^{6} + \cdots + (\zeta_{116}^{35} + \zeta_{116}^{29}) q^{99} +O(q^{100}) \) q + (z^45 - z^33) * q^2 - z^36 * q^3 + (-z^32 + z^20 - z^8) * q^4 + (z^39 - z^23) * q^5 + (z^23 - z^11) * q^6 + (-z^53 + z^41 + z^19 - z^7) * q^8 - z^14 * q^9 + (z^56 - z^26 + z^14 + z^10) * q^10 + (-z^21 - z^15) * q^11 + (-z^56 + z^44 - z^10) * q^12 - z^44 * q^13 + (z^17 - z) * q^15 + (-z^52 + z^40 - z^28 + z^16 - z^6) * q^16 + (z^47 + z) * q^18 + (z^55 - z^47 - z^43 + z^31 + z^13 - z) * q^20 + (z^54 + z^48 + z^8 + z^2) * q^22 + (-z^55 + z^43 - z^31 + z^19) * q^24 + (z^46 - z^20 + z^4) * q^25 + (z^31 - z^19) * q^26 + z^50 * q^27 + (-z^50 - z^46 + z^34 - z^4) * q^30 + (-z^51 - z^49 + z^39 - z^27 + z^15 - z^3) * q^32 + (z^57 + z^51) * q^33 + (z^46 - z^34 + z^22) * q^36 - z^22 * q^39 + (-z^46 - z^42 + z^34 + z^30 - z^22 - z^18 + z^6 - 1) * q^40 + (z^43 + z^5) * q^41 + (z^48 + z^32) * q^43 + (z^53 + z^47 - z^41 - z^35 + z^29 + z^23) * q^44 + (-z^53 + z^37) * q^45 + (-z^39 + z^15) * q^47 + (-z^52 + z^42 - z^30 + z^18 - z^6) * q^48 + z^12 * q^49 + (z^53 + z^49 - z^37 - z^33 + z^21 + z^7) * q^50 + (z^52 - z^18 + z^6) * q^52 + (-z^37 + z^25) * q^54 + (-z^54 + z^44 + z^38 + z^2) * q^55 - z^3 * q^59 + (-z^49 + z^37 + z^33 - z^25 - z^21 + z^9) * q^60 + (-z^16 - z^4) * q^61 + (-z^48 + z^38 + z^36 - z^26 - z^24 + z^14 - z^2) * q^64 + (z^25 - z^9) * q^65 + (-z^44 - z^38 + z^32 + z^26) * q^66 + (z^49 - z^5) * q^71 + (-z^55 - z^33 + z^21 - z^9) * q^72 + (z^56 - z^40 + z^24) * q^75 + (z^55 + z^9) * q^78 + (-z^32 - z^12) * q^79 + (z^55 + z^51 - z^45 - z^39 + z^33 + z^29 - z^21 - z^17 + z^9 + z^5) * q^80 + z^28 * q^81 + (z^50 - z^38 - z^30 + z^18) * q^82 + (-z^41 + z^33) * q^83 + (-z^35 + z^23 - z^19 + z^7) * q^86 + (-z^56 - z^40 - z^34 + z^28 + z^22 - z^16 - z^10 + z^4) * q^88 + (-z^53 - z^43) * q^89 + (z^40 - z^28 - z^24 + z^12) * q^90 + (-z^48 + z^26 - z^14 - z^2) * q^94 + (-z^51 + z^39 - z^29 - z^27 + z^17 - z^5) * q^96 + (z^57 - z^45) * q^98 + (z^35 + z^29) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 2 q^{3} + 2 q^{4} - 2 q^{9} - 2 q^{12} + 2 q^{13} - 2 q^{16} + 2 q^{25} + 2 q^{27} + 2 q^{36} - 2 q^{39} - 58 q^{40} - 4 q^{43} + 2 q^{48} - 2 q^{49} - 2 q^{52} + 4 q^{61} + 2 q^{64} - 2 q^{75} + 4 q^{79}+ \cdots - 2 q^{81}+O(q^{100}) \) 56 * q + 2 * q^3 + 2 * q^4 - 2 * q^9 - 2 * q^12 + 2 * q^13 - 2 * q^16 + 2 * q^25 + 2 * q^27 + 2 * q^36 - 2 * q^39 - 58 * q^40 - 4 * q^43 + 2 * q^48 - 2 * q^49 - 2 * q^52 + 4 * q^61 + 2 * q^64 - 2 * q^75 + 4 * q^79 - 2 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	2301.1.z.c

Level	$2301$
Weight	$1$
Character orbit	2301.z
Analytic conductor	$1.148$
Analytic rank	$0$
Dimension	$56$
Projective image	$D_{58}$
CM discriminant	-39
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.14834859407\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(2\) over \(\Q(\zeta_{58})\)
Coefficient field:	\(\Q(\zeta_{116})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{56} - x^{54} + x^{52} - x^{50} + x^{48} - x^{46} + x^{44} - x^{42} + x^{40} - x^{38} + x^{36} + \cdots + 1 \) x^56 - x^54 + x^52 - x^50 + x^48 - x^46 + x^44 - x^42 + x^40 - x^38 + x^36 - x^34 + x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{59}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{58}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{58} - \cdots)\)

\(n\)	\(886\)	\(1535\)	\(2185\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{116}^{20}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 116.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{56} + 58 T^{48} + \cdots + 841 \) T^56 + 58T^48 - 8294T^42 + 1247T^40 + 3306T^36 + 1223742T^34 + 12180T^32 - 58T^30 + 7403381T^28 - 17077346T^26 + 52983T^24 + 3549020T^22 + 134358160T^20 + 31778026T^18 + 465073T^16 - 50567648T^14 + 70539716T^12 - 5315961T^10 + 1842631T^8 + 2736614T^6 + 429751T^4 + 11774*T^2 + 841
$3$	\( (T^{28} - T^{27} + T^{26} + \cdots + 1)^{2} \) (T^28 - T^27 + T^26 - T^25 + T^24 - T^23 + T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$5$	\( T^{56} + 406 T^{48} + \cdots + 841 \) T^56 + 406T^48 + 53389T^40 + 9860T^38 - 58T^34 + 2527930T^32 - 2966990T^30 + 6786T^28 + 29435T^26 + 34580238T^24 + 71081320T^22 + 2612146T^20 - 846887T^18 + 75716912T^16 - 113425670T^14 + 12676393T^12 + 1120212T^10 + 7403323T^8 + 3549020T^6 + 382655T^4 + 5887T^2 + 841
$7$	\( T^{56} \) T^56
$11$	\( T^{56} + 1218 T^{46} + \cdots + 841 \) T^56 + 1218T^46 - 27608T^38 + 376449T^36 - 4060T^34 + 33350T^30 + 10855686T^28 + 23424373T^26 + 1155534T^24 - 9251T^22 + 12197864T^20 - 142761432T^18 + 104090570T^16 - 9085323T^14 + 1441474T^12 + 15058946T^10 + 15442442T^8 + 3654145T^6 + 318739T^4 - 841*T^2 + 841
$13$	\( (T^{28} - T^{27} + T^{26} + \cdots + 1)^{2} \) (T^28 - T^27 + T^26 - T^25 + T^24 - T^23 + T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$17$	\( T^{56} \) T^56
$19$	\( T^{56} \) T^56
$23$	\( T^{56} \) T^56
$29$	\( T^{56} \) T^56
$31$	\( T^{56} \) T^56
$37$	\( T^{56} \) T^56
$41$	\( T^{56} + 58 T^{52} + \cdots + 841 \) T^56 + 58T^52 + 1479T^48 + 21808T^44 + 205755T^40 + 1298388T^36 + 609T^34 + 5565651T^32 - 111418T^30 + 16138848T^28 + 3705446T^26 + 30972348T^24 - 35488518T^22 + 37718850T^20 + 112996760T^18 + 27183643T^16 - 120609492T^14 + 10351028T^12 + 39030810T^10 + 1771146T^8 - 2973776T^6 + 111853T^4 + 29435T^2 + 841
$43$	\( (T^{28} + 2 T^{27} + \cdots + 1)^{2} \) (T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 128T^21 + 256T^20 + 512T^19 + 995T^18 + 1120T^17 + 326T^16 + 14T^15 - T^14 - 2T^13 - 4T^12 - 8T^11 - 16T^10 - 32T^9 + 81T^8 - 2709T^7 + 7023T^6 - 7066T^5 + 3616T^4 - 1033T^3 + 167T^2 - 14T + 1)^2
$47$	\( T^{56} - 58 T^{46} + \cdots + 841 \) T^56 - 58T^46 + 3828T^44 - 58T^42 + 1131T^36 + 621006T^34 + 2590454T^32 + 291566T^30 + 899T^28 - 8410T^26 + 6272178T^24 - 89060218T^22 + 110110448T^20 - 20730650T^18 + 705599T^16 + 4633069T^14 + 38700297T^12 + 50920868T^10 + 19581844T^8 + 3020031T^6 + 263233T^4 - 8410*T^2 + 841
$53$	\( T^{56} \) T^56
$59$	\( T^{56} - T^{54} + \cdots + 1 \) T^56 - T^54 + T^52 - T^50 + T^48 - T^46 + T^44 - T^42 + T^40 - T^38 + T^36 - T^34 + T^32 - T^30 + T^28 - T^26 + T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$61$	\( (T^{28} - 2 T^{27} + \cdots + 1)^{2} \) (T^28 - 2T^27 + 4T^26 - 8T^25 + 16T^24 - 32T^23 + 64T^22 - 128T^21 + 227T^20 - 454T^19 + 908T^18 - 598T^17 + 1196T^16 - 2392T^15 + 637T^14 - 1274T^13 + 2751T^12 - 1442T^11 + 2884T^10 + 322T^9 - 2297T^8 + 4594T^7 + 440T^6 - 561T^5 + 919T^4 - 40T^3 + 51T^2 + 14*T + 1)^2
$67$	\( T^{56} \) T^56
$71$	\( T^{56} + 58 T^{48} + \cdots + 841 \) T^56 + 58T^48 - 8294T^42 + 1247T^40 + 3306T^36 + 1223742T^34 + 12180T^32 - 58T^30 + 7403381T^28 - 17077346T^26 + 52983T^24 + 3549020T^22 + 134358160T^20 + 31778026T^18 + 465073T^16 - 50567648T^14 + 70539716T^12 - 5315961T^10 + 1842631T^8 + 2736614T^6 + 429751T^4 + 11774*T^2 + 841
$73$	\( T^{56} \) T^56
$79$	\( (T^{28} - 2 T^{27} + \cdots + 1)^{2} \) (T^28 - 2T^27 + 4T^26 - 8T^25 + 16T^24 - 32T^23 + 64T^22 - 128T^21 + 227T^20 - 454T^19 + 908T^18 - 598T^17 + 1196T^16 - 2392T^15 + 637T^14 - 1274T^13 + 2751T^12 - 1442T^11 + 2884T^10 + 322T^9 - 2297T^8 + 4594T^7 + 440T^6 - 561T^5 + 919T^4 - 40T^3 + 51T^2 + 14*T + 1)^2
$83$	\( T^{56} + 58 T^{36} + \cdots + 841 \) T^56 + 58T^36 + 19285T^34 + 836418T^32 + 11849255T^30 + 77558760T^28 + 281150505T^26 + 629614650T^24 + 935427480T^22 + 967964088T^20 + 721540996T^18 + 396486927T^16 + 162937863T^14 + 51356506T^12 + 8400749T^10 + 6808736T^8 - 2242106T^6 + 538240T^4 - 36163T^2 + 841
$89$	\( T^{56} + 58 T^{36} + \cdots + 841 \) T^56 + 58T^36 + 19285T^34 + 836418T^32 + 11849255T^30 + 77558760T^28 + 281150505T^26 + 629614650T^24 + 935427480T^22 + 967964088T^20 + 721540996T^18 + 396486927T^16 + 162937863T^14 + 51356506T^12 + 8400749T^10 + 6808736T^8 - 2242106T^6 + 538240T^4 - 36163T^2 + 841
$97$	\( T^{56} \) T^56
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