LMFDB - Newform orbit 2359.1.cv.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2359,1,Mod(6,2359)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2359, base_ring=CyclotomicField(56))

chi = DirichletCharacter(H, H._module([28, 47]))

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

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

chi := DirichletCharacter("2359.6");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2359 = 7 \cdot 337 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2359.cv (of order $56$, degree $24$, 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:	$1.17729436480$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{56})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{56}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{56} - \cdots)$

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2359\mathbb{Z}\right)^\times$.

$n$	$675$	$1695$
$\chi(n)$	$-1$	$-\zeta_{56}^{25}$

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} $

6.1

0.943883	+	0.330279i
−0.111964	−	0.993712i
−0.993712	+	0.111964i
0.532032	+	0.846724i
0.846724	−	0.532032i
−0.111964	+	0.993712i
−0.532032	+	0.846724i
0.330279	−	0.943883i
0.330279	+	0.943883i
−0.330279	−	0.943883i
−0.330279	+	0.943883i
0.532032	−	0.846724i
0.111964	−	0.993712i
−0.846724	+	0.532032i
−0.532032	−	0.846724i
0.993712	−	0.111964i
0.111964	+	0.993712i
−0.943883	−	0.330279i
−0.846724	−	0.532032i
0.943883	−	0.330279i

1.79061

0.862311i

1.83921

2.30629i

0.943883

−

0.330279i

0.862311

3.77803i

−0.974928

−

0.222521i

27.1

−1.05585

−

1.32399i

−0.415617

1.82094i

−0.111964

0.993712i

1.32399

−

0.637601i

−0.433884

0.900969i

48.1

−0.663433

−

0.831919i

−0.0294245

0.128917i

−0.993712

−

0.111964i

−0.831919

0.400631i

0.433884

−

0.900969i

216.1

0.420068

1.84044i

−2.30978

1.11233i

0.532032

−

0.846724i

−1.84044

−

2.30783i

−0.781831

−

0.623490i

384.1

−0.146988

−

0.643997i

0.507843

−

0.244564i

0.846724

0.532032i

−0.643997

−

0.807546i

0.781831

0.623490i

699.1

−1.05585

1.32399i

−0.415617

−

1.82094i

−0.111964

−

0.993712i

1.32399

0.637601i

−0.433884

−

0.900969i

713.1

−0.420068

1.84044i

−2.30978

−

1.11233i

−0.532032

−

0.846724i

1.84044

−

2.30783i

−0.781831

0.623490i

797.1

−0.201753

−

0.0971591i

−0.592225

−

0.742627i

0.330279

0.943883i

0.0971591

0.425682i

0.974928

0.222521i

811.1

−0.201753

0.0971591i

−0.592225

0.742627i

0.330279

−

0.943883i

0.0971591

−

0.425682i

0.974928

−

0.222521i

874.1

0.201753

−

0.0971591i

−0.592225

0.742627i

−0.330279

0.943883i

−0.0971591

0.425682i

0.974928

−

0.222521i

888.1

0.201753

0.0971591i

−0.592225

−

0.742627i

−0.330279

−

0.943883i

−0.0971591

−

0.425682i

0.974928

0.222521i

972.1

0.420068

−

1.84044i

−2.30978

−

1.11233i

0.532032

0.846724i

−1.84044

2.30783i

−0.781831

0.623490i

986.1

1.05585

−

1.32399i

−0.415617

−

1.82094i

0.111964

0.993712i

−1.32399

−

0.637601i

−0.433884

−

0.900969i

1301.1

0.146988

0.643997i

0.507843

−

0.244564i

−0.846724

−

0.532032i

0.643997

0.807546i

0.781831

0.623490i

1469.1

−0.420068

−

1.84044i

−2.30978

1.11233i

−0.532032

0.846724i

1.84044

2.30783i

−0.781831

−

0.623490i

1637.1

0.663433

0.831919i

−0.0294245

0.128917i

0.993712

0.111964i

0.831919

−

0.400631i

0.433884

−

0.900969i

1658.1

1.05585

1.32399i

−0.415617

1.82094i

0.111964

−

0.993712i

−1.32399

0.637601i

−0.433884

0.900969i

1679.1

−1.79061

−

0.862311i

1.83921

2.30629i

−0.943883

0.330279i

−0.862311

−

3.77803i

−0.974928

−

0.222521i

1728.1

0.146988

−

0.643997i

0.507843

0.244564i

−0.846724

0.532032i

0.643997

−

0.807546i

0.781831

−

0.623490i

1966.1

1.79061

−

0.862311i

1.83921

−

2.30629i

0.943883

0.330279i

0.862311

−

3.77803i

−0.974928

0.222521i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
337.p	even	56	1	inner
2359.cv	odd	56	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2359.1.cv.a	✓	24
7.b	odd	2	1	CM	2359.1.cv.a	✓	24
337.p	even	56	1	inner	2359.1.cv.a	✓	24
2359.cv	odd	56	1	inner	2359.1.cv.a	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2359.1.cv.a	✓	24	1.a	even	1	1	trivial
2359.1.cv.a	✓	24	7.b	odd	2	1	CM
2359.1.cv.a	✓	24	337.p	even	56	1	inner
2359.1.cv.a	✓	24	2359.cv	odd	56	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2359, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} + 4 T^{22} + \cdots + 1 $ T^24 + 4T^22 + 12T^20 + 60T^18 + 290T^16 + 920T^14 + 2758T^12 + 3232T^10 + 3513T^8 + 1712T^6 + 286T^4 - 20*T^2 + 1
$3$	$ T^{24} $ T^24
$5$	$ T^{24} $ T^24
$7$	$ T^{24} - T^{20} + \cdots + 1 $ T^24 - T^20 + T^16 - T^12 + T^8 - T^4 + 1
$11$	$ T^{24} - 2 T^{22} + \cdots + 1 $ T^24 - 2T^22 + 4T^21 + 2T^20 - 16T^19 + 16T^18 + 32T^17 - 100T^16 + 32T^15 + 422T^14 + 648T^13 + 639T^12 + 496T^11 + 472T^10 + 268T^9 - 238T^8 + 360T^7 + 604T^6 - 2236T^5 + 1863T^4 - 696T^3 + 136T^2 - 12T + 1
$13$	$ T^{24} $ T^24
$17$	$ T^{24} $ T^24
$19$	$ T^{24} $ T^24
$23$	$ T^{24} + 4 T^{23} + \cdots + 1 $ T^24 + 4T^23 + 10T^22 + 20T^21 + 34T^20 + 48T^19 + 48T^18 - 4T^17 - 180T^16 - 208T^15 + 136T^14 + 660T^13 + 1352T^12 + 2436T^11 + 4006T^10 + 5496T^9 + 6605T^8 + 4596T^7 + 3170T^6 + 532T^5 - 112T^4 - 156T^3 + 50T^2 + 16*T + 1
$29$	$ T^{24} + 4 T^{23} + \cdots + 64 $ T^24 + 4T^23 + 10T^22 + 20T^21 + 34T^20 + 48T^19 + 48T^18 + 136T^17 + 380T^16 + 800T^15 + 1368T^14 + 1920T^13 + 1912T^12 - 1104T^10 - 608T^9 + 368T^8 + 704T^7 + 384T^6 - 224T^4 - 128T^3 + 64T^2 + 128*T + 64
$31$	$ T^{24} $ T^24
$37$	$ (T^{6} + 7 T^{3} - 7 T + 7)^{4} $ (T^6 + 7T^3 - 7T + 7)^4
$41$	$ T^{24} $ T^24
$43$	$ (T^{12} - 2 T^{11} + \cdots + 1)^{2} $ (T^12 - 2T^11 + 2T^10 - 4T^8 + 22T^7 - T^6 - 40T^5 + 82T^4 - 84T^3 + 39T^2 - 8*T + 1)^2
$47$	$ T^{24} $ T^24
$53$	$ T^{24} + 12 T^{22} + \cdots + 1 $ T^24 + 12T^22 - 4T^21 + 65T^20 - 40T^19 + 226T^18 - 172T^17 + 579T^16 - 480T^15 + 1080T^14 - 1012T^13 + 1605T^12 - 1336T^11 + 2110T^10 - 1752T^9 + 917T^8 - 1536T^7 + 2956T^6 - 1096T^5 - 601T^4 + 304T^3 + 38T^2 - 16T + 1
$59$	$ T^{24} $ T^24
$61$	$ T^{24} $ T^24
$67$	$ T^{24} - 4 T^{23} + \cdots + 1 $ T^24 - 4T^23 + 10T^22 - 20T^21 + 34T^20 - 76T^19 + 160T^18 - 276T^17 + 380T^16 - 352T^15 + 178T^14 + 432T^13 - 1105T^12 + 1820T^11 - 2728T^10 + 2568T^9 + 1026T^8 - 3476T^7 + 2932T^6 - 1372T^5 + 903T^4 - 152T^3 + 8T^2 + 12*T + 1
$71$	$ T^{24} + 12 T^{22} + \cdots + 1 $ T^24 + 12T^22 - 4T^21 + 65T^20 - 40T^19 + 226T^18 - 172T^17 + 579T^16 - 480T^15 + 1080T^14 - 1012T^13 + 1605T^12 - 1336T^11 + 2110T^10 - 1752T^9 + 917T^8 - 1536T^7 + 2956T^6 - 1096T^5 - 601T^4 + 304T^3 + 38T^2 - 16T + 1
$73$	$ T^{24} $ T^24
$79$	$ T^{24} + 4 T^{22} + \cdots + 1 $ T^24 + 4T^22 + 12T^20 + 32T^18 + 80T^16 + 164T^14 + 1491T^12 - 1556T^10 + 1826T^8 + 200T^6 + 405T^4 + 36*T^2 + 1
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ T^{24} $ T^24
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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:	\( N \)	\(=\)	\( 2359 = 7 \cdot 337 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2359.cv (of order \(56\), degree \(24\), minimal)

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

Introduction

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Modular forms

Varieties

Fields

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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	2359.1.cv.a

Level	$2359$
Weight	$1$
Character orbit	2359.cv
Analytic conductor	$1.177$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{56}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.17729436480\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Coefficient field:	\(\Q(\zeta_{56})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 \) x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{56}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{56} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 6.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
337.p	even	56	1	inner
2359.cv	odd	56	1	inner

$p$	$F_p(T)$
$2$	\( T^{24} + 4 T^{22} + \cdots + 1 \) T^24 + 4T^22 + 12T^20 + 60T^18 + 290T^16 + 920T^14 + 2758T^12 + 3232T^10 + 3513T^8 + 1712T^6 + 286T^4 - 20*T^2 + 1
$3$	\( T^{24} \) T^24
$5$	\( T^{24} \) T^24
$7$	\( T^{24} - T^{20} + \cdots + 1 \) T^24 - T^20 + T^16 - T^12 + T^8 - T^4 + 1
$11$	\( T^{24} - 2 T^{22} + \cdots + 1 \) T^24 - 2T^22 + 4T^21 + 2T^20 - 16T^19 + 16T^18 + 32T^17 - 100T^16 + 32T^15 + 422T^14 + 648T^13 + 639T^12 + 496T^11 + 472T^10 + 268T^9 - 238T^8 + 360T^7 + 604T^6 - 2236T^5 + 1863T^4 - 696T^3 + 136T^2 - 12T + 1
$13$	\( T^{24} \) T^24
$17$	\( T^{24} \) T^24
$19$	\( T^{24} \) T^24
$23$	\( T^{24} + 4 T^{23} + \cdots + 1 \) T^24 + 4T^23 + 10T^22 + 20T^21 + 34T^20 + 48T^19 + 48T^18 - 4T^17 - 180T^16 - 208T^15 + 136T^14 + 660T^13 + 1352T^12 + 2436T^11 + 4006T^10 + 5496T^9 + 6605T^8 + 4596T^7 + 3170T^6 + 532T^5 - 112T^4 - 156T^3 + 50T^2 + 16*T + 1
$29$	\( T^{24} + 4 T^{23} + \cdots + 64 \) T^24 + 4T^23 + 10T^22 + 20T^21 + 34T^20 + 48T^19 + 48T^18 + 136T^17 + 380T^16 + 800T^15 + 1368T^14 + 1920T^13 + 1912T^12 - 1104T^10 - 608T^9 + 368T^8 + 704T^7 + 384T^6 - 224T^4 - 128T^3 + 64T^2 + 128*T + 64
$31$	\( T^{24} \) T^24
$37$	\( (T^{6} + 7 T^{3} - 7 T + 7)^{4} \) (T^6 + 7T^3 - 7T + 7)^4
$41$	\( T^{24} \) T^24
$43$	\( (T^{12} - 2 T^{11} + \cdots + 1)^{2} \) (T^12 - 2T^11 + 2T^10 - 4T^8 + 22T^7 - T^6 - 40T^5 + 82T^4 - 84T^3 + 39T^2 - 8*T + 1)^2
$47$	\( T^{24} \) T^24
$53$	\( T^{24} + 12 T^{22} + \cdots + 1 \) T^24 + 12T^22 - 4T^21 + 65T^20 - 40T^19 + 226T^18 - 172T^17 + 579T^16 - 480T^15 + 1080T^14 - 1012T^13 + 1605T^12 - 1336T^11 + 2110T^10 - 1752T^9 + 917T^8 - 1536T^7 + 2956T^6 - 1096T^5 - 601T^4 + 304T^3 + 38T^2 - 16T + 1
$59$	\( T^{24} \) T^24
$61$	\( T^{24} \) T^24
$67$	\( T^{24} - 4 T^{23} + \cdots + 1 \) T^24 - 4T^23 + 10T^22 - 20T^21 + 34T^20 - 76T^19 + 160T^18 - 276T^17 + 380T^16 - 352T^15 + 178T^14 + 432T^13 - 1105T^12 + 1820T^11 - 2728T^10 + 2568T^9 + 1026T^8 - 3476T^7 + 2932T^6 - 1372T^5 + 903T^4 - 152T^3 + 8T^2 + 12*T + 1
$71$	\( T^{24} + 12 T^{22} + \cdots + 1 \) T^24 + 12T^22 - 4T^21 + 65T^20 - 40T^19 + 226T^18 - 172T^17 + 579T^16 - 480T^15 + 1080T^14 - 1012T^13 + 1605T^12 - 1336T^11 + 2110T^10 - 1752T^9 + 917T^8 - 1536T^7 + 2956T^6 - 1096T^5 - 601T^4 + 304T^3 + 38T^2 - 16T + 1
$73$	\( T^{24} \) T^24
$79$	\( T^{24} + 4 T^{22} + \cdots + 1 \) T^24 + 4T^22 + 12T^20 + 32T^18 + 80T^16 + 164T^14 + 1491T^12 - 1556T^10 + 1826T^8 + 200T^6 + 405T^4 + 36*T^2 + 1
$83$	\( T^{24} \) T^24
$89$	\( T^{24} \) T^24
$97$	\( T^{24} \) T^24
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\(n\)	\(675\)	\(1695\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{56}^{25}\)