LMFDB - Newform orbit 633.1.z.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [633,1,Mod(5,633)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(633, base_ring=CyclotomicField(70))

chi = DirichletCharacter(H, H._module([35, 44]))

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

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

chi := DirichletCharacter("633.5");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 633 = 3 \cdot 211 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	633.z (of order $70$, 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:	$0.315908152997$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{35})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{23} + x^{19} - x^{18} + x^{17} - x^{16} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} + \cdots + 1 $ x^24 - x^23 + x^19 - x^18 + x^17 - x^16 + x^14 - x^13 + x^12 - x^11 + x^10 - x^8 + x^7 - x^6 + x^5 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{35}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{35} - \cdots)$

$q$-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_{70}^{9} q^{3} - \zeta_{70}^{11} q^{4} + ( - \zeta_{70}^{19} - \zeta_{70}^{5}) q^{7} + \zeta_{70}^{18} q^{9} +O(q^{10}) $ q - z^9 * q^3 - z^11 * q^4 + (-z^19 - z^5) * q^7 + z^18 * q^9	$ q - \zeta_{70}^{9} q^{3} - \zeta_{70}^{11} q^{4} + ( - \zeta_{70}^{19} - \zeta_{70}^{5}) q^{7} + \zeta_{70}^{18} q^{9} + \zeta_{70}^{20} q^{12} + (\zeta_{70}^{32} + \zeta_{70}^{12}) q^{13} + \zeta_{70}^{22} q^{16} + (\zeta_{70}^{34} - \zeta_{70}^{15}) q^{19} + (\zeta_{70}^{28} + \zeta_{70}^{14}) q^{21} - \zeta_{70}^{17} q^{25} - \zeta_{70}^{27} q^{27} + (\zeta_{70}^{30} + \zeta_{70}^{16}) q^{28} + (\zeta_{70}^{4} - \zeta_{70}) q^{31} - \zeta_{70}^{29} q^{36} + ( - \zeta_{70}^{25} - \zeta_{70}^{13}) q^{37} + ( - \zeta_{70}^{21} + \zeta_{70}^{6}) q^{39} + ( - \zeta_{70}^{33} - \zeta_{70}^{7}) q^{43} - \zeta_{70}^{31} q^{48} + (\zeta_{70}^{24} + \cdots - \zeta_{70}^{3}) q^{49} + \cdots + (\zeta_{70}^{30} - \zeta_{70}^{13}) q^{97} +O(q^{100}) $ q - z^9 * q^3 - z^11 * q^4 + (-z^19 - z^5) * q^7 + z^18 * q^9 + z^20 * q^12 + (z^32 + z^12) * q^13 + z^22 * q^16 + (z^34 - z^15) * q^19 + (z^28 + z^14) * q^21 - z^17 * q^25 - z^27 * q^27 + (z^30 + z^16) * q^28 + (z^4 - z) * q^31 - z^29 * q^36 + (-z^25 - z^13) * q^37 + (-z^21 + z^6) * q^39 + (-z^33 - z^7) * q^43 - z^31 * q^48 + (z^24 + z^10 - z^3) * q^49 + (-z^23 + z^8) * q^52 + (z^24 + z^8) * q^57 + (z^26 + z^2) * q^61 + (-z^23 + z^2) * q^63 - z^33 * q^64 + (-z^7 - z^3) * q^67 + (-z^29 - z^21) * q^73 + z^26 * q^75 + (z^26 + z^10) * q^76 + (-z^31 + z^6) * q^79 - z * q^81 + (-z^25 + z^4) * q^84 + (-z^31 - z^17 + z^16 + z^2) * q^91 + (-z^13 + z^10) * q^93 + (z^30 - z^13) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + q^{4} - 3 q^{7} + q^{9}+O(q^{10}) $ 24 * q + q^3 + q^4 - 3 * q^7 + q^9	$ 24 q + q^{3} + q^{4} - 3 q^{7} + q^{9} - 4 q^{12} + 2 q^{13} + q^{16} - 3 q^{19} - 12 q^{21} + q^{25} + q^{27} - 3 q^{28} + 2 q^{31} + q^{36} - 3 q^{37} - 5 q^{39} - 5 q^{43} + q^{48} - 2 q^{49} + 2 q^{52} + 2 q^{57} + 2 q^{61} + 2 q^{63} + q^{64} - 5 q^{67} - 5 q^{73} + q^{75} - 3 q^{76} + 2 q^{79} + q^{81} - 3 q^{84} + 4 q^{91} - 3 q^{93} - 3 q^{97}+O(q^{100}) $ 24 * q + q^3 + q^4 - 3 * q^7 + q^9 - 4 * q^12 + 2 * q^13 + q^16 - 3 * q^19 - 12 * q^21 + q^25 + q^27 - 3 * q^28 + 2 * q^31 + q^36 - 3 * q^37 - 5 * q^39 - 5 * q^43 + q^48 - 2 * q^49 + 2 * q^52 + 2 * q^57 + 2 * q^61 + 2 * q^63 + q^64 - 5 * q^67 - 5 * q^73 + q^75 - 3 * q^76 + 2 * q^79 + q^81 - 3 * q^84 + 4 * q^91 - 3 * q^93 - 3 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/633\mathbb{Z}\right)^\times$.

$n$	$212$	$424$
$\chi(n)$	$-1$	$-\zeta_{70}^{11}$

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} $

5.1

0.936235	+	0.351375i
−0.550897	+	0.834573i
0.983930	+	0.178557i
−0.0448648	+	0.998993i
−0.963963	−	0.266037i
0.473869	+	0.880596i
−0.393025	+	0.919528i
0.134233	+	0.990950i
0.983930	−	0.178557i
0.753071	+	0.657939i
−0.963963	+	0.266037i
0.753071	−	0.657939i
0.134233	−	0.990950i
−0.691063	+	0.722795i
−0.995974	−	0.0896393i
−0.393025	−	0.919528i
−0.995974	+	0.0896393i
0.936235	−	0.351375i
0.473869	−	0.880596i
−0.691063	−	0.722795i

−0.995974

−

0.0896393i

−0.691063

−

0.722795i

0.635928

1.48783i

0.983930

0.178557i

11.1

0.858449

0.512899i

0.134233

−

0.990950i

−1.21850

−

1.06457i

0.473869

0.880596i

65.1

−0.0448648

0.998993i

−0.393025

0.919528i

−0.340473

0.515795i

−0.995974

−

0.0896393i

113.1

−0.393025

0.919528i

0.473869

−

0.880596i

0.530551

0.316989i

−0.691063

−

0.722795i

122.1

0.753071

−

0.657939i

0.983930

−

0.178557i

−0.615546

−

0.0554001i

0.134233

−

0.990950i

125.1

−0.963963

−

0.266037i

0.753071

−

0.657939i

0.578625

0.217162i

0.858449

0.512899i

143.1

0.473869

−

0.880596i

−0.963963

0.266037i

0.0829607

−

0.612441i

−0.550897

−

0.834573i

203.1

0.936235

0.351375i

−0.995974

−

0.0896393i

0.0725928

1.61640i

0.753071

0.657939i

224.1

−0.0448648

−

0.998993i

−0.393025

−

0.919528i

−0.340473

−

0.515795i

−0.995974

0.0896393i

236.1

0.983930

0.178557i

−0.0448648

0.998993i

−0.427100

0.446712i

0.936235

0.351375i

275.1

0.753071

0.657939i

0.983930

0.178557i

−0.615546

0.0554001i

0.134233

0.990950i

287.1

0.983930

−

0.178557i

−0.0448648

−

0.998993i

−0.427100

−

0.446712i

0.936235

−

0.351375i

290.1

0.936235

−

0.351375i

−0.995974

0.0896393i

0.0725928

−

1.61640i

0.753071

−

0.657939i

293.1

−0.550897

0.834573i

0.858449

0.512899i

1.55972

−

0.430457i

−0.393025

−

0.919528i

320.1

−0.691063

−

0.722795i

−0.550897

−

0.834573i

−0.766736

−

1.42483i

−0.0448648

0.998993i

332.1

0.473869

0.880596i

−0.963963

−

0.266037i

0.0829607

0.612441i

−0.550897

0.834573i

362.1

−0.691063

0.722795i

−0.550897

0.834573i

−0.766736

1.42483i

−0.0448648

−

0.998993i

380.1

−0.995974

0.0896393i

−0.691063

0.722795i

0.635928

−

1.48783i

0.983930

−

0.178557i

395.1

−0.963963

0.266037i

0.753071

0.657939i

0.578625

−

0.217162i

0.858449

−

0.512899i

404.1

−0.550897

−

0.834573i

0.858449

−

0.512899i

1.55972

0.430457i

−0.393025

0.919528i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
211.l	even	35	1	inner
633.z	odd	70	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	633.1.z.a	✓	24
3.b	odd	2	1	CM	633.1.z.a	✓	24
211.l	even	35	1	inner	633.1.z.a	✓	24
633.z	odd	70	1	inner	633.1.z.a	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
633.1.z.a	✓	24	1.a	even	1	1	trivial
633.1.z.a	✓	24	3.b	odd	2	1	CM
633.1.z.a	✓	24	211.l	even	35	1	inner
633.1.z.a	✓	24	633.z	odd	70	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(633, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} $ T^24
$3$	$ T^{24} - T^{23} + \cdots + 1 $ T^24 - T^23 + T^19 - T^18 + T^17 - T^16 + T^14 - T^13 + T^12 - T^11 + T^10 - T^8 + T^7 - T^6 + T^5 - T + 1
$5$	$ T^{24} $ T^24
$7$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 5T^22 + 5T^21 - 13T^19 - 34T^18 - 37T^17 - T^16 + 90T^15 + 234T^14 + 377T^13 + 376T^12 - 2T^11 - 146T^10 - 55T^9 + 39T^8 + 47T^7 + 21T^6 + 8T^5 - 5T^4 - 5T^3 + 2T + 1
$11$	$ T^{24} $ T^24
$13$	$ T^{24} - 2 T^{23} + \cdots + 1 $ T^24 - 2T^23 + 7T^21 - 14T^20 + 4T^19 + 34T^18 - 82T^17 + 73T^16 + 91T^15 - 110T^14 + 45T^13 - 167T^12 + 383T^11 + 277T^10 - 539T^9 + 166T^8 + 235T^7 + 167T^6 + 86T^5 + 21T^4 + 14T^3 + 7T^2 + 3T + 1
$17$	$ T^{24} $ T^24
$19$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 12T^22 + 26T^21 + 70T^20 + 127T^19 + 295T^18 + 488T^17 + 1105T^16 + 1819T^15 + 3384T^14 + 4661T^13 + 6802T^12 + 6767T^11 + 6924T^10 + 3382T^9 + 2538T^8 + 971T^7 + 2471T^6 + 1597T^5 + 1269T^4 - 327T^3 + 154T^2 - 19*T + 1
$23$	$ T^{24} $ T^24
$29$	$ T^{24} $ T^24
$31$	$ T^{24} - 2 T^{23} + \cdots + 1 $ T^24 - 2T^23 + 5T^22 - 10T^21 + 27T^20 - 38T^19 + 85T^18 - 142T^17 + 315T^16 - 368T^15 + 976T^14 - 1849T^13 + 4071T^12 - 5585T^11 + 7136T^10 - 6387T^9 + 6041T^8 - 4737T^7 + 3326T^6 - 1418T^5 + 1980T^4 - 165T^3 + 117T^2 - 19*T + 1
$37$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 5T^22 + 12T^21 + 21T^20 + 29T^19 + 64T^18 + 103T^17 - 57T^16 - 239T^15 - 305T^14 - 960T^13 - 667T^12 + 838T^11 + 827T^10 + 1156T^9 + 5296T^8 + 8090T^7 + 5502T^6 + 1191T^5 - 754T^4 - 341T^3 + 147T^2 - 12*T + 1
$41$	$ T^{24} $ T^24
$43$	$ T^{24} + 5 T^{23} + \cdots + 1 $ T^24 + 5T^23 + 19T^22 + 53T^21 + 132T^20 + 277T^19 + 540T^18 + 929T^17 + 1533T^16 + 2166T^15 + 2894T^14 + 3030T^13 + 3574T^12 + 5013T^11 + 9355T^10 + 15327T^9 + 20104T^8 + 19973T^7 + 15842T^6 + 10223T^5 + 5326T^4 + 1676T^3 + 285T^2 + 23*T + 1
$47$	$ T^{24} $ T^24
$53$	$ T^{24} $ T^24
$59$	$ T^{24} $ T^24
$61$	$ T^{24} - 2 T^{23} + \cdots + 1 $ T^24 - 2T^23 + 7T^22 - 9T^21 + 30T^20 - 43T^19 + 140T^18 - 187T^17 + 550T^16 - 736T^15 + 1359T^14 - 1419T^13 + 2882T^12 - 4283T^11 + 7829T^10 - 7958T^9 + 7958T^8 - 4709T^7 + 3571T^6 - 743T^5 + 784T^4 + 548T^3 + 159T^2 + 11*T + 1
$67$	$ T^{24} + 5 T^{23} + \cdots + 1 $ T^24 + 5T^23 + 19T^22 + 53T^21 + 132T^20 + 277T^19 + 540T^18 + 929T^17 + 1533T^16 + 2166T^15 + 2894T^14 + 3030T^13 + 3574T^12 + 5013T^11 + 9355T^10 + 15327T^9 + 20104T^8 + 19973T^7 + 15842T^6 + 10223T^5 + 5326T^4 + 1676T^3 + 285T^2 + 23*T + 1
$71$	$ T^{24} $ T^24
$73$	$ T^{24} + 5 T^{23} + \cdots + 1 $ T^24 + 5T^23 + 19T^22 + 53T^21 + 132T^20 + 277T^19 + 540T^18 + 929T^17 + 1533T^16 + 2166T^15 + 2894T^14 + 3030T^13 + 3574T^12 + 5013T^11 + 9355T^10 + 15327T^9 + 20104T^8 + 19973T^7 + 15842T^6 + 10223T^5 + 5326T^4 + 1676T^3 + 285T^2 + 23*T + 1
$79$	$ T^{24} - 2 T^{23} + \cdots + 1 $ T^24 - 2T^23 + 7T^20 - 38T^19 + 55T^18 - 12T^17 + 45T^16 - 168T^15 + 401T^14 - 564T^13 + 456T^12 - 520T^11 + 746T^10 - 1092T^9 + 1111T^8 - 402T^7 + 6T^6 + 72T^5 - 35T^4 + 7T^2 - 4T + 1
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 5T^22 + 12T^21 + 21T^20 + 29T^19 + 64T^18 + 103T^17 + 188T^16 + 496T^15 + 920T^14 + 510T^13 + 68T^12 + 593T^11 - 1133T^10 - 4479T^9 - 3769T^8 - 1675T^7 + 1197T^6 + 4271T^5 + 3481T^4 + 1199T^3 + 252T^2 + 23*T + 1
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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:	\( N \)	\(=\)	\( 633 = 3 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	633.z (of order \(70\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{70}^{9} q^{3} - \zeta_{70}^{11} q^{4} + ( - \zeta_{70}^{19} - \zeta_{70}^{5}) q^{7} + \zeta_{70}^{18} q^{9} +O(q^{10}) \) q - z^9 * q^3 - z^11 * q^4 + (-z^19 - z^5) * q^7 + z^18 * q^9	\( q - \zeta_{70}^{9} q^{3} - \zeta_{70}^{11} q^{4} + ( - \zeta_{70}^{19} - \zeta_{70}^{5}) q^{7} + \zeta_{70}^{18} q^{9} + \zeta_{70}^{20} q^{12} + (\zeta_{70}^{32} + \zeta_{70}^{12}) q^{13} + \zeta_{70}^{22} q^{16} + (\zeta_{70}^{34} - \zeta_{70}^{15}) q^{19} + (\zeta_{70}^{28} + \zeta_{70}^{14}) q^{21} - \zeta_{70}^{17} q^{25} - \zeta_{70}^{27} q^{27} + (\zeta_{70}^{30} + \zeta_{70}^{16}) q^{28} + (\zeta_{70}^{4} - \zeta_{70}) q^{31} - \zeta_{70}^{29} q^{36} + ( - \zeta_{70}^{25} - \zeta_{70}^{13}) q^{37} + ( - \zeta_{70}^{21} + \zeta_{70}^{6}) q^{39} + ( - \zeta_{70}^{33} - \zeta_{70}^{7}) q^{43} - \zeta_{70}^{31} q^{48} + (\zeta_{70}^{24} + \cdots - \zeta_{70}^{3}) q^{49} + \cdots + (\zeta_{70}^{30} - \zeta_{70}^{13}) q^{97} +O(q^{100}) \) q - z^9 * q^3 - z^11 * q^4 + (-z^19 - z^5) * q^7 + z^18 * q^9 + z^20 * q^12 + (z^32 + z^12) * q^13 + z^22 * q^16 + (z^34 - z^15) * q^19 + (z^28 + z^14) * q^21 - z^17 * q^25 - z^27 * q^27 + (z^30 + z^16) * q^28 + (z^4 - z) * q^31 - z^29 * q^36 + (-z^25 - z^13) * q^37 + (-z^21 + z^6) * q^39 + (-z^33 - z^7) * q^43 - z^31 * q^48 + (z^24 + z^10 - z^3) * q^49 + (-z^23 + z^8) * q^52 + (z^24 + z^8) * q^57 + (z^26 + z^2) * q^61 + (-z^23 + z^2) * q^63 - z^33 * q^64 + (-z^7 - z^3) * q^67 + (-z^29 - z^21) * q^73 + z^26 * q^75 + (z^26 + z^10) * q^76 + (-z^31 + z^6) * q^79 - z * q^81 + (-z^25 + z^4) * q^84 + (-z^31 - z^17 + z^16 + z^2) * q^91 + (-z^13 + z^10) * q^93 + (z^30 - z^13) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + q^{4} - 3 q^{7} + q^{9}+O(q^{10}) \) 24 * q + q^3 + q^4 - 3 * q^7 + q^9	\( 24 q + q^{3} + q^{4} - 3 q^{7} + q^{9} - 4 q^{12} + 2 q^{13} + q^{16} - 3 q^{19} - 12 q^{21} + q^{25} + q^{27} - 3 q^{28} + 2 q^{31} + q^{36} - 3 q^{37} - 5 q^{39} - 5 q^{43} + q^{48} - 2 q^{49} + 2 q^{52} + 2 q^{57} + 2 q^{61} + 2 q^{63} + q^{64} - 5 q^{67} - 5 q^{73} + q^{75} - 3 q^{76} + 2 q^{79} + q^{81} - 3 q^{84} + 4 q^{91} - 3 q^{93} - 3 q^{97}+O(q^{100}) \) 24 * q + q^3 + q^4 - 3 * q^7 + q^9 - 4 * q^12 + 2 * q^13 + q^16 - 3 * q^19 - 12 * q^21 + q^25 + q^27 - 3 * q^28 + 2 * q^31 + q^36 - 3 * q^37 - 5 * q^39 - 5 * q^43 + q^48 - 2 * q^49 + 2 * q^52 + 2 * q^57 + 2 * q^61 + 2 * q^63 + q^64 - 5 * q^67 - 5 * q^73 + q^75 - 3 * q^76 + 2 * q^79 + q^81 - 3 * q^84 + 4 * q^91 - 3 * q^93 - 3 * q^97

Introduction

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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	633.1.z.a

Level	$633$
Weight	$1$
Character orbit	633.z
Analytic conductor	$0.316$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{35}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.315908152997\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Coefficient field:	\(\Q(\zeta_{35})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{23} + x^{19} - x^{18} + x^{17} - x^{16} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} + \cdots + 1 \) x^24 - x^23 + x^19 - x^18 + x^17 - x^16 + x^14 - x^13 + x^12 - x^11 + x^10 - x^8 + x^7 - x^6 + x^5 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{35}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{35} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 5.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
211.l	even	35	1	inner
633.z	odd	70	1	inner

$p$	$F_p(T)$
$2$	\( T^{24} \) T^24
$3$	\( T^{24} - T^{23} + \cdots + 1 \) T^24 - T^23 + T^19 - T^18 + T^17 - T^16 + T^14 - T^13 + T^12 - T^11 + T^10 - T^8 + T^7 - T^6 + T^5 - T + 1
$5$	\( T^{24} \) T^24
$7$	\( T^{24} + 3 T^{23} + \cdots + 1 \) T^24 + 3T^23 + 5T^22 + 5T^21 - 13T^19 - 34T^18 - 37T^17 - T^16 + 90T^15 + 234T^14 + 377T^13 + 376T^12 - 2T^11 - 146T^10 - 55T^9 + 39T^8 + 47T^7 + 21T^6 + 8T^5 - 5T^4 - 5T^3 + 2T + 1
$11$	\( T^{24} \) T^24
$13$	\( T^{24} - 2 T^{23} + \cdots + 1 \) T^24 - 2T^23 + 7T^21 - 14T^20 + 4T^19 + 34T^18 - 82T^17 + 73T^16 + 91T^15 - 110T^14 + 45T^13 - 167T^12 + 383T^11 + 277T^10 - 539T^9 + 166T^8 + 235T^7 + 167T^6 + 86T^5 + 21T^4 + 14T^3 + 7T^2 + 3T + 1
$17$	\( T^{24} \) T^24
$19$	\( T^{24} + 3 T^{23} + \cdots + 1 \) T^24 + 3T^23 + 12T^22 + 26T^21 + 70T^20 + 127T^19 + 295T^18 + 488T^17 + 1105T^16 + 1819T^15 + 3384T^14 + 4661T^13 + 6802T^12 + 6767T^11 + 6924T^10 + 3382T^9 + 2538T^8 + 971T^7 + 2471T^6 + 1597T^5 + 1269T^4 - 327T^3 + 154T^2 - 19*T + 1
$23$	\( T^{24} \) T^24
$29$	\( T^{24} \) T^24
$31$	\( T^{24} - 2 T^{23} + \cdots + 1 \) T^24 - 2T^23 + 5T^22 - 10T^21 + 27T^20 - 38T^19 + 85T^18 - 142T^17 + 315T^16 - 368T^15 + 976T^14 - 1849T^13 + 4071T^12 - 5585T^11 + 7136T^10 - 6387T^9 + 6041T^8 - 4737T^7 + 3326T^6 - 1418T^5 + 1980T^4 - 165T^3 + 117T^2 - 19*T + 1
$37$	\( T^{24} + 3 T^{23} + \cdots + 1 \) T^24 + 3T^23 + 5T^22 + 12T^21 + 21T^20 + 29T^19 + 64T^18 + 103T^17 - 57T^16 - 239T^15 - 305T^14 - 960T^13 - 667T^12 + 838T^11 + 827T^10 + 1156T^9 + 5296T^8 + 8090T^7 + 5502T^6 + 1191T^5 - 754T^4 - 341T^3 + 147T^2 - 12*T + 1
$41$	\( T^{24} \) T^24
$43$	\( T^{24} + 5 T^{23} + \cdots + 1 \) T^24 + 5T^23 + 19T^22 + 53T^21 + 132T^20 + 277T^19 + 540T^18 + 929T^17 + 1533T^16 + 2166T^15 + 2894T^14 + 3030T^13 + 3574T^12 + 5013T^11 + 9355T^10 + 15327T^9 + 20104T^8 + 19973T^7 + 15842T^6 + 10223T^5 + 5326T^4 + 1676T^3 + 285T^2 + 23*T + 1
$47$	\( T^{24} \) T^24
$53$	\( T^{24} \) T^24
$59$	\( T^{24} \) T^24
$61$	\( T^{24} - 2 T^{23} + \cdots + 1 \) T^24 - 2T^23 + 7T^22 - 9T^21 + 30T^20 - 43T^19 + 140T^18 - 187T^17 + 550T^16 - 736T^15 + 1359T^14 - 1419T^13 + 2882T^12 - 4283T^11 + 7829T^10 - 7958T^9 + 7958T^8 - 4709T^7 + 3571T^6 - 743T^5 + 784T^4 + 548T^3 + 159T^2 + 11*T + 1
$67$	\( T^{24} + 5 T^{23} + \cdots + 1 \) T^24 + 5T^23 + 19T^22 + 53T^21 + 132T^20 + 277T^19 + 540T^18 + 929T^17 + 1533T^16 + 2166T^15 + 2894T^14 + 3030T^13 + 3574T^12 + 5013T^11 + 9355T^10 + 15327T^9 + 20104T^8 + 19973T^7 + 15842T^6 + 10223T^5 + 5326T^4 + 1676T^3 + 285T^2 + 23*T + 1
$71$	\( T^{24} \) T^24
$73$	\( T^{24} + 5 T^{23} + \cdots + 1 \) T^24 + 5T^23 + 19T^22 + 53T^21 + 132T^20 + 277T^19 + 540T^18 + 929T^17 + 1533T^16 + 2166T^15 + 2894T^14 + 3030T^13 + 3574T^12 + 5013T^11 + 9355T^10 + 15327T^9 + 20104T^8 + 19973T^7 + 15842T^6 + 10223T^5 + 5326T^4 + 1676T^3 + 285T^2 + 23*T + 1
$79$	\( T^{24} - 2 T^{23} + \cdots + 1 \) T^24 - 2T^23 + 7T^20 - 38T^19 + 55T^18 - 12T^17 + 45T^16 - 168T^15 + 401T^14 - 564T^13 + 456T^12 - 520T^11 + 746T^10 - 1092T^9 + 1111T^8 - 402T^7 + 6T^6 + 72T^5 - 35T^4 + 7T^2 - 4T + 1
$83$	\( T^{24} \) T^24
$89$	\( T^{24} \) T^24
$97$	\( T^{24} + 3 T^{23} + \cdots + 1 \) T^24 + 3T^23 + 5T^22 + 12T^21 + 21T^20 + 29T^19 + 64T^18 + 103T^17 + 188T^16 + 496T^15 + 920T^14 + 510T^13 + 68T^12 + 593T^11 - 1133T^10 - 4479T^9 - 3769T^8 - 1675T^7 + 1197T^6 + 4271T^5 + 3481T^4 + 1199T^3 + 252T^2 + 23*T + 1
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\(n\)	\(212\)	\(424\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{70}^{11}\)