LMFDB - Newform orbit 3840.1.dy.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,1,Mod(29,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(64))

chi = DirichletCharacter(H, H._module([0, 59, 32, 32]))

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

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

chi := DirichletCharacter("3840.29");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3840.dy (of order $64$, degree $32$, 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:	$1.91640964851$
Analytic rank:	$0$
Dimension:	$64$
Relative dimension:	$2$ over $\Q(\zeta_{64})$
Coefficient field:	$\Q(\zeta_{128})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{64} + 1 $ x^64 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{64}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{64} + \cdots)$

$q$-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_{128}^{29} q^{2} - \zeta_{128} q^{3} + \zeta_{128}^{58} q^{4} - \zeta_{128}^{11} q^{5} + \zeta_{128}^{30} q^{6} + \zeta_{128}^{23} q^{8} + \zeta_{128}^{2} q^{9} +O(q^{10}) $ q - z^29 * q^2 - z * q^3 + z^58 * q^4 - z^11 * q^5 + z^30 * q^6 + z^23 * q^8 + z^2 * q^9	\( q - \zeta_{128}^{29} q^{2} - \zeta_{128} q^{3} + \zeta_{128}^{58} q^{4} - \zeta_{128}^{11} q^{5} + \zeta_{128}^{30} q^{6} + \zeta_{128}^{23} q^{8} + \zeta_{128}^{2} q^{9} + \zeta_{128}^{40} q^{10} - \zeta_{128}^{59} q^{12} + \zeta_{128}^{12} q^{15} - \zeta_{128}^{52} q^{16} + ( - \zeta_{128}^{25} + \zeta_{128}^{15}) q^{17} - \zeta_{128}^{31} q^{18} + (\zeta_{128}^{38} - \zeta_{128}^{20}) q^{19} + \zeta_{128}^{5} q^{20} + ( - \zeta_{128}^{45} - \zeta_{128}^{7}) q^{23} - \zeta_{128}^{24} q^{24} + \zeta_{128}^{22} q^{25} - \zeta_{128}^{3} q^{27} - \zeta_{128}^{41} q^{30} + (\zeta_{128}^{62} - \zeta_{128}^{50}) q^{31} - \zeta_{128}^{17} q^{32} + (\zeta_{128}^{54} - \zeta_{128}^{44}) q^{34} + \zeta_{128}^{60} q^{36} + (\zeta_{128}^{49} + \zeta_{128}^{3}) q^{38} - \zeta_{128}^{34} q^{40} - \zeta_{128}^{13} q^{45} + (\zeta_{128}^{36} - \zeta_{128}^{10}) q^{46} + ( - \zeta_{128}^{53} - \zeta_{128}^{19}) q^{47} + \zeta_{128}^{53} q^{48} - \zeta_{128}^{28} q^{49} - \zeta_{128}^{51} q^{50} + (\zeta_{128}^{26} - \zeta_{128}^{16}) q^{51} + (\zeta_{128}^{41} - \zeta_{128}^{5}) q^{53} + \zeta_{128}^{32} q^{54} + ( - \zeta_{128}^{39} + \zeta_{128}^{21}) q^{57} - \zeta_{128}^{6} q^{60} + ( - \zeta_{128}^{56} + \zeta_{128}^{42}) q^{61} + (\zeta_{128}^{27} - \zeta_{128}^{15}) q^{62} + \zeta_{128}^{46} q^{64} + (\zeta_{128}^{19} - \zeta_{128}^{9}) q^{68} + (\zeta_{128}^{46} + \zeta_{128}^{8}) q^{69} + \zeta_{128}^{25} q^{72} - \zeta_{128}^{23} q^{75} + ( - \zeta_{128}^{32} + \zeta_{128}^{14}) q^{76} + (\zeta_{128}^{48} - \zeta_{128}^{8}) q^{79} + \zeta_{128}^{63} q^{80} + \zeta_{128}^{4} q^{81} + (\zeta_{128}^{47} + \zeta_{128}^{43}) q^{83} + (\zeta_{128}^{36} - \zeta_{128}^{26}) q^{85} + \zeta_{128}^{42} q^{90} + (\zeta_{128}^{39} + \zeta_{128}) q^{92} + ( - \zeta_{128}^{63} + \zeta_{128}^{51}) q^{93} + (\zeta_{128}^{48} - \zeta_{128}^{18}) q^{94} + ( - \zeta_{128}^{49} + \zeta_{128}^{31}) q^{95} + \zeta_{128}^{18} q^{96} + \zeta_{128}^{57} q^{98} +O(q^{100}) \) q - z^29 * q^2 - z * q^3 + z^58 * q^4 - z^11 * q^5 + z^30 * q^6 + z^23 * q^8 + z^2 * q^9 + z^40 * q^10 - z^59 * q^12 + z^12 * q^15 - z^52 * q^16 + (-z^25 + z^15) * q^17 - z^31 * q^18 + (z^38 - z^20) * q^19 + z^5 * q^20 + (-z^45 - z^7) * q^23 - z^24 * q^24 + z^22 * q^25 - z^3 * q^27 - z^41 * q^30 + (z^62 - z^50) * q^31 - z^17 * q^32 + (z^54 - z^44) * q^34 + z^60 * q^36 + (z^49 + z^3) * q^38 - z^34 * q^40 - z^13 * q^45 + (z^36 - z^10) * q^46 + (-z^53 - z^19) * q^47 + z^53 * q^48 - z^28 * q^49 - z^51 * q^50 + (z^26 - z^16) * q^51 + (z^41 - z^5) * q^53 + z^32 * q^54 + (-z^39 + z^21) * q^57 - z^6 * q^60 + (-z^56 + z^42) * q^61 + (z^27 - z^15) * q^62 + z^46 * q^64 + (z^19 - z^9) * q^68 + (z^46 + z^8) * q^69 + z^25 * q^72 - z^23 * q^75 + (-z^32 + z^14) * q^76 + (z^48 - z^8) * q^79 + z^63 * q^80 + z^4 * q^81 + (z^47 + z^43) * q^83 + (z^36 - z^26) * q^85 + z^42 * q^90 + (z^39 + z) * q^92 + (-z^63 + z^51) * q^93 + (z^48 - z^18) * q^94 + (-z^49 + z^31) * q^95 + z^18 * q^96 + z^57 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q+O(q^{10}) $ 64 * q	$ 64 q+O(q^{100}) $ 64 * q

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$-1$	$-1$	$\zeta_{128}^{22}$

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} $

29.1

0.671559	+	0.740951i
−0.671559	−	0.740951i
−0.941544	−	0.336890i
0.941544	+	0.336890i
−0.595699	+	0.803208i
0.595699	−	0.803208i
−0.989177	−	0.146730i
0.989177	+	0.146730i
0.903989	+	0.427555i
−0.903989	−	0.427555i
0.998795	−	0.0490677i
−0.998795	+	0.0490677i
0.242980	−	0.970031i
−0.242980	+	0.970031i
0.970031	−	0.242980i
−0.970031	+	0.242980i
0.998795	+	0.0490677i
−0.998795	−	0.0490677i
0.903989	−	0.427555i
−0.903989	+	0.427555i

−0.595699

0.803208i

−0.671559

−

0.740951i

−0.290285

−

0.956940i

0.970031

−

0.242980i

0.995185

−

0.0980171i

0.941544

0.336890i

−0.0980171

0.995185i

−0.382683

0.923880i

29.2

0.595699

−

0.803208i

0.671559

0.740951i

−0.290285

−

0.956940i

−0.970031

0.242980i

0.995185

−

0.0980171i

−0.941544

−

0.336890i

−0.0980171

0.995185i

−0.382683

0.923880i

149.1

−0.857729

−

0.514103i

0.941544

0.336890i

0.471397

0.881921i

−0.803208

−

0.595699i

−0.634393

−

0.773010i

0.0490677

−

0.998795i

0.773010

0.634393i

0.382683

0.923880i

149.2

0.857729

0.514103i

−0.941544

−

0.336890i

0.471397

0.881921i

0.803208

0.595699i

−0.634393

−

0.773010i

−0.0490677

0.998795i

0.773010

0.634393i

0.382683

0.923880i

269.1

−0.336890

−

0.941544i

0.595699

−

0.803208i

−0.773010

0.634393i

−0.671559

0.740951i

−0.956940

−

0.290285i

0.857729

0.514103i

−0.290285

−

0.956940i

0.923880

0.382683i

269.2

0.336890

0.941544i

−0.595699

0.803208i

−0.773010

0.634393i

0.671559

−

0.740951i

−0.956940

−

0.290285i

−0.857729

−

0.514103i

−0.290285

−

0.956940i

0.923880

0.382683i

389.1

−0.427555

−

0.903989i

0.989177

0.146730i

−0.634393

0.773010i

−0.0490677

0.998795i

−0.290285

−

0.956940i

0.970031

0.242980i

0.956940

0.290285i

0.923880

−

0.382683i

389.2

0.427555

0.903989i

−0.989177

−

0.146730i

−0.634393

0.773010i

0.0490677

−

0.998795i

−0.290285

−

0.956940i

−0.970031

−

0.242980i

0.956940

0.290285i

0.923880

−

0.382683i

509.1

−0.970031

−

0.242980i

−0.903989

−

0.427555i

0.881921

0.471397i

−0.146730

0.989177i

0.773010

0.634393i

−0.740951

−

0.671559i

0.634393

0.773010i

0.382683

−

0.923880i

509.2

0.970031

0.242980i

0.903989

0.427555i

0.881921

0.471397i

0.146730

−

0.989177i

0.773010

0.634393i

0.740951

0.671559i

0.634393

0.773010i

0.382683

−

0.923880i

629.1

−0.146730

0.989177i

−0.998795

0.0490677i

−0.956940

−

0.290285i

−0.857729

0.514103i

0.0980171

−

0.995185i

0.427555

−

0.903989i

0.995185

−

0.0980171i

−0.382683

−

0.923880i

629.2

0.146730

−

0.989177i

0.998795

−

0.0490677i

−0.956940

−

0.290285i

0.857729

−

0.514103i

0.0980171

−

0.995185i

−0.427555

0.903989i

0.995185

−

0.0980171i

−0.382683

−

0.923880i

749.1

−0.740951

0.671559i

−0.242980

0.970031i

0.0980171

−

0.995185i

0.427555

0.903989i

−0.471397

−

0.881921i

0.595699

0.803208i

−0.881921

−

0.471397i

−0.923880

−

0.382683i

749.2

0.740951

−

0.671559i

0.242980

−

0.970031i

0.0980171

−

0.995185i

−0.427555

−

0.903989i

−0.471397

−

0.881921i

−0.595699

−

0.803208i

−0.881921

−

0.471397i

−0.923880

−

0.382683i

869.1

−0.671559

0.740951i

−0.970031

0.242980i

−0.0980171

−

0.995185i

0.903989

0.427555i

0.471397

−

0.881921i

0.803208

0.595699i

0.881921

−

0.471397i

−0.923880

0.382683i

869.2

0.671559

−

0.740951i

0.970031

−

0.242980i

−0.0980171

−

0.995185i

−0.903989

−

0.427555i

0.471397

−

0.881921i

−0.803208

−

0.595699i

0.881921

−

0.471397i

−0.923880

0.382683i

989.1

−0.146730

−

0.989177i

−0.998795

−

0.0490677i

−0.956940

0.290285i

−0.857729

−

0.514103i

0.0980171

0.995185i

0.427555

0.903989i

0.995185

0.0980171i

−0.382683

0.923880i

989.2

0.146730

0.989177i

0.998795

0.0490677i

−0.956940

0.290285i

0.857729

0.514103i

0.0980171

0.995185i

−0.427555

−

0.903989i

0.995185

0.0980171i

−0.382683

0.923880i

1109.1

−0.970031

0.242980i

−0.903989

0.427555i

0.881921

−

0.471397i

−0.146730

−

0.989177i

0.773010

−

0.634393i

−0.740951

0.671559i

0.634393

−

0.773010i

0.382683

0.923880i

1109.2

0.970031

−

0.242980i

0.903989

−

0.427555i

0.881921

−

0.471397i

0.146730

0.989177i

0.773010

−

0.634393i

0.740951

−

0.671559i

0.634393

−

0.773010i

0.382683

0.923880i

See all 64 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
256.m	even	64	1	inner
768.y	odd	64	1	inner
1280.bw	even	64	1	inner
3840.dy	odd	64	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3840.1.dy.a	✓	64
3.b	odd	2	1	inner	3840.1.dy.a	✓	64
5.b	even	2	1	inner	3840.1.dy.a	✓	64
15.d	odd	2	1	CM	3840.1.dy.a	✓	64
256.m	even	64	1	inner	3840.1.dy.a	✓	64
768.y	odd	64	1	inner	3840.1.dy.a	✓	64
1280.bw	even	64	1	inner	3840.1.dy.a	✓	64
3840.dy	odd	64	1	inner	3840.1.dy.a	✓	64

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3840.1.dy.a	✓	64	1.a	even	1	1	trivial
3840.1.dy.a	✓	64	3.b	odd	2	1	inner
3840.1.dy.a	✓	64	5.b	even	2	1	inner
3840.1.dy.a	✓	64	15.d	odd	2	1	CM
3840.1.dy.a	✓	64	256.m	even	64	1	inner
3840.1.dy.a	✓	64	768.y	odd	64	1	inner
3840.1.dy.a	✓	64	1280.bw	even	64	1	inner
3840.1.dy.a	✓	64	3840.dy	odd	64	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3840, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{64} + 1 $ T^64 + 1
$3$	$ T^{64} + 1 $ T^64 + 1
$5$	$ T^{64} + 1 $ T^64 + 1
$7$	$ T^{64} $ T^64
$11$	$ T^{64} $ T^64
$13$	$ T^{64} $ T^64
$17$	$ T^{64} - 1664 T^{50} + \cdots + 4 $ T^64 - 1664T^50 + 51480T^48 - 30464T^46 + 608T^44 + 64T^38 + 523328T^36 + 30875520T^34 + 173626516T^32 + 177494528T^30 + 43439744T^28 + 2733056T^26 + 78336T^24 - 11162880T^22 + 227793152T^20 - 938361600T^18 + 1264445104T^16 - 732034560T^14 + 214599872T^12 - 35108864T^10 + 3811328T^8 + 82048T^6 + 43648T^4 + 768*T^2 + 4
$19$	$ (T^{32} + 80 T^{26} + \cdots + 2)^{2} $ (T^32 + 80T^26 + 160T^21 + 1880T^20 + 2T^16 - 10496T^15 + 12736T^14 - 32T^13 + 2320T^10 + 28864T^9 + 13452T^8 + 384T^7 - 96T^5 + 1320T^4 - 1280T^3 + 336T^2 - 32T + 2)^2
$23$	$ T^{64} + 64 T^{42} + \cdots + 4 $ T^64 + 64T^42 + 28336T^40 + 1619200T^38 + 29995680T^36 + 255487680T^34 + 1202160780T^32 + 3493585920T^30 + 6748972800T^28 + 9119916288T^26 + 8938965568T^24 + 6522760192T^22 + 3609101440T^20 + 1532729088T^18 + 505093248T^16 + 116441088T^14 + 48829440T^12 - 16248960T^10 + 7524000T^8 - 1014272T^6 + 60352T^4 - 896*T^2 + 4
$29$	$ T^{64} $ T^64
$31$	$ (T^{32} + 280 T^{24} + \cdots + 4)^{2} $ (T^32 + 280T^24 + 32T^22 - 832T^18 + 13460T^16 - 6272T^14 + 512T^12 + 15104T^10 + 30384T^8 + 11328T^6 + 2048T^4 - 128*T^2 + 4)^2
$37$	$ T^{64} $ T^64
$41$	$ T^{64} $ T^64
$43$	$ T^{64} $ T^64
$47$	$ T^{64} - 1664 T^{50} + \cdots + 4 $ T^64 - 1664T^50 + 51480T^48 - 30464T^46 + 608T^44 + 64T^38 + 523328T^36 + 30875520T^34 + 173626516T^32 + 177494528T^30 + 43439744T^28 + 2733056T^26 + 78336T^24 - 11162880T^22 + 227793152T^20 - 938361600T^18 + 1264445104T^16 - 732034560T^14 + 214599872T^12 - 35108864T^10 + 3811328T^8 + 82048T^6 + 43648T^4 + 768*T^2 + 4
$53$	$ T^{64} + 64 T^{60} + \cdots + 16 $ T^64 + 64T^60 + 1824T^56 + 30528T^52 + 333200T^48 + 2490624T^44 + 13032832T^40 + 47783936T^36 + 131049600T^32 + 106153984T^28 + 777809408T^24 - 791304448T^20 + 701793728T^16 - 117660672T^12 + 7672576T^8 - 21504*T^4 + 16
$59$	$ T^{64} $ T^64
$61$	$ (T^{32} + 4 T^{24} + \cdots + 2)^{2} $ (T^32 + 4T^24 - 448T^23 + 144T^22 + 6T^16 + 2688T^15 + 23136T^14 + 20352T^13 + 2616T^12 + 32T^11 + 4T^8 - 960T^7 + 6864T^6 - 11648T^5 + 7752T^4 - 2464T^3 + 400T^2 - 32*T + 2)^2
$67$	$ T^{64} $ T^64
$71$	$ T^{64} $ T^64
$73$	$ T^{64} $ T^64
$79$	$ (T^{8} + 8 T^{5} + 2 T^{4} + \cdots + 2)^{8} $ (T^8 + 8T^5 + 2T^4 + 12T^2 - 8T + 2)^8
$83$	$ T^{64} + 7280 T^{48} + \cdots + 16 $ T^64 + 7280T^48 - 537472T^44 + 5230016T^40 - 15779712T^36 + 26499200T^32 + 180433920T^28 + 1110315648T^24 + 1521254912T^20 + 644145728T^16 + 42616832T^12 + 7228928T^8 - 17920T^4 + 16
$89$	$ T^{64} $ T^64
$97$	$ T^{64} $ T^64
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3840.dy (of order \(64\), degree \(32\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{128}^{29} q^{2} - \zeta_{128} q^{3} + \zeta_{128}^{58} q^{4} - \zeta_{128}^{11} q^{5} + \zeta_{128}^{30} q^{6} + \zeta_{128}^{23} q^{8} + \zeta_{128}^{2} q^{9} +O(q^{10}) \) q - z^29 * q^2 - z * q^3 + z^58 * q^4 - z^11 * q^5 + z^30 * q^6 + z^23 * q^8 + z^2 * q^9	\( q - \zeta_{128}^{29} q^{2} - \zeta_{128} q^{3} + \zeta_{128}^{58} q^{4} - \zeta_{128}^{11} q^{5} + \zeta_{128}^{30} q^{6} + \zeta_{128}^{23} q^{8} + \zeta_{128}^{2} q^{9} + \zeta_{128}^{40} q^{10} - \zeta_{128}^{59} q^{12} + \zeta_{128}^{12} q^{15} - \zeta_{128}^{52} q^{16} + ( - \zeta_{128}^{25} + \zeta_{128}^{15}) q^{17} - \zeta_{128}^{31} q^{18} + (\zeta_{128}^{38} - \zeta_{128}^{20}) q^{19} + \zeta_{128}^{5} q^{20} + ( - \zeta_{128}^{45} - \zeta_{128}^{7}) q^{23} - \zeta_{128}^{24} q^{24} + \zeta_{128}^{22} q^{25} - \zeta_{128}^{3} q^{27} - \zeta_{128}^{41} q^{30} + (\zeta_{128}^{62} - \zeta_{128}^{50}) q^{31} - \zeta_{128}^{17} q^{32} + (\zeta_{128}^{54} - \zeta_{128}^{44}) q^{34} + \zeta_{128}^{60} q^{36} + (\zeta_{128}^{49} + \zeta_{128}^{3}) q^{38} - \zeta_{128}^{34} q^{40} - \zeta_{128}^{13} q^{45} + (\zeta_{128}^{36} - \zeta_{128}^{10}) q^{46} + ( - \zeta_{128}^{53} - \zeta_{128}^{19}) q^{47} + \zeta_{128}^{53} q^{48} - \zeta_{128}^{28} q^{49} - \zeta_{128}^{51} q^{50} + (\zeta_{128}^{26} - \zeta_{128}^{16}) q^{51} + (\zeta_{128}^{41} - \zeta_{128}^{5}) q^{53} + \zeta_{128}^{32} q^{54} + ( - \zeta_{128}^{39} + \zeta_{128}^{21}) q^{57} - \zeta_{128}^{6} q^{60} + ( - \zeta_{128}^{56} + \zeta_{128}^{42}) q^{61} + (\zeta_{128}^{27} - \zeta_{128}^{15}) q^{62} + \zeta_{128}^{46} q^{64} + (\zeta_{128}^{19} - \zeta_{128}^{9}) q^{68} + (\zeta_{128}^{46} + \zeta_{128}^{8}) q^{69} + \zeta_{128}^{25} q^{72} - \zeta_{128}^{23} q^{75} + ( - \zeta_{128}^{32} + \zeta_{128}^{14}) q^{76} + (\zeta_{128}^{48} - \zeta_{128}^{8}) q^{79} + \zeta_{128}^{63} q^{80} + \zeta_{128}^{4} q^{81} + (\zeta_{128}^{47} + \zeta_{128}^{43}) q^{83} + (\zeta_{128}^{36} - \zeta_{128}^{26}) q^{85} + \zeta_{128}^{42} q^{90} + (\zeta_{128}^{39} + \zeta_{128}) q^{92} + ( - \zeta_{128}^{63} + \zeta_{128}^{51}) q^{93} + (\zeta_{128}^{48} - \zeta_{128}^{18}) q^{94} + ( - \zeta_{128}^{49} + \zeta_{128}^{31}) q^{95} + \zeta_{128}^{18} q^{96} + \zeta_{128}^{57} q^{98} +O(q^{100}) \) q - z^29 * q^2 - z * q^3 + z^58 * q^4 - z^11 * q^5 + z^30 * q^6 + z^23 * q^8 + z^2 * q^9 + z^40 * q^10 - z^59 * q^12 + z^12 * q^15 - z^52 * q^16 + (-z^25 + z^15) * q^17 - z^31 * q^18 + (z^38 - z^20) * q^19 + z^5 * q^20 + (-z^45 - z^7) * q^23 - z^24 * q^24 + z^22 * q^25 - z^3 * q^27 - z^41 * q^30 + (z^62 - z^50) * q^31 - z^17 * q^32 + (z^54 - z^44) * q^34 + z^60 * q^36 + (z^49 + z^3) * q^38 - z^34 * q^40 - z^13 * q^45 + (z^36 - z^10) * q^46 + (-z^53 - z^19) * q^47 + z^53 * q^48 - z^28 * q^49 - z^51 * q^50 + (z^26 - z^16) * q^51 + (z^41 - z^5) * q^53 + z^32 * q^54 + (-z^39 + z^21) * q^57 - z^6 * q^60 + (-z^56 + z^42) * q^61 + (z^27 - z^15) * q^62 + z^46 * q^64 + (z^19 - z^9) * q^68 + (z^46 + z^8) * q^69 + z^25 * q^72 - z^23 * q^75 + (-z^32 + z^14) * q^76 + (z^48 - z^8) * q^79 + z^63 * q^80 + z^4 * q^81 + (z^47 + z^43) * q^83 + (z^36 - z^26) * q^85 + z^42 * q^90 + (z^39 + z) * q^92 + (-z^63 + z^51) * q^93 + (z^48 - z^18) * q^94 + (-z^49 + z^31) * q^95 + z^18 * q^96 + z^57 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 64 * q	\( 64 q+O(q^{100}) \) 64 * q

Introduction

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Newspace parameters

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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	3840.1.dy.a

Level	$3840$
Weight	$1$
Character orbit	3840.dy
Analytic conductor	$1.916$
Analytic rank	$0$
Dimension	$64$
Projective image	$D_{64}$
CM discriminant	-15
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(2\) over \(\Q(\zeta_{64})\)
Coefficient field:	\(\Q(\zeta_{128})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{64} + 1 \) x^64 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{64}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{64} + \cdots)\)

\(n\)	\(511\)	\(1537\)	\(2561\)	\(2821\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(\zeta_{128}^{22}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 29.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{64} + 1 \) T^64 + 1
$3$	\( T^{64} + 1 \) T^64 + 1
$5$	\( T^{64} + 1 \) T^64 + 1
$7$	\( T^{64} \) T^64
$11$	\( T^{64} \) T^64
$13$	\( T^{64} \) T^64
$17$	\( T^{64} - 1664 T^{50} + \cdots + 4 \) T^64 - 1664T^50 + 51480T^48 - 30464T^46 + 608T^44 + 64T^38 + 523328T^36 + 30875520T^34 + 173626516T^32 + 177494528T^30 + 43439744T^28 + 2733056T^26 + 78336T^24 - 11162880T^22 + 227793152T^20 - 938361600T^18 + 1264445104T^16 - 732034560T^14 + 214599872T^12 - 35108864T^10 + 3811328T^8 + 82048T^6 + 43648T^4 + 768*T^2 + 4
$19$	\( (T^{32} + 80 T^{26} + \cdots + 2)^{2} \) (T^32 + 80T^26 + 160T^21 + 1880T^20 + 2T^16 - 10496T^15 + 12736T^14 - 32T^13 + 2320T^10 + 28864T^9 + 13452T^8 + 384T^7 - 96T^5 + 1320T^4 - 1280T^3 + 336T^2 - 32T + 2)^2
$23$	\( T^{64} + 64 T^{42} + \cdots + 4 \) T^64 + 64T^42 + 28336T^40 + 1619200T^38 + 29995680T^36 + 255487680T^34 + 1202160780T^32 + 3493585920T^30 + 6748972800T^28 + 9119916288T^26 + 8938965568T^24 + 6522760192T^22 + 3609101440T^20 + 1532729088T^18 + 505093248T^16 + 116441088T^14 + 48829440T^12 - 16248960T^10 + 7524000T^8 - 1014272T^6 + 60352T^4 - 896*T^2 + 4
$29$	\( T^{64} \) T^64
$31$	\( (T^{32} + 280 T^{24} + \cdots + 4)^{2} \) (T^32 + 280T^24 + 32T^22 - 832T^18 + 13460T^16 - 6272T^14 + 512T^12 + 15104T^10 + 30384T^8 + 11328T^6 + 2048T^4 - 128*T^2 + 4)^2
$37$	\( T^{64} \) T^64
$41$	\( T^{64} \) T^64
$43$	\( T^{64} \) T^64
$47$	\( T^{64} - 1664 T^{50} + \cdots + 4 \) T^64 - 1664T^50 + 51480T^48 - 30464T^46 + 608T^44 + 64T^38 + 523328T^36 + 30875520T^34 + 173626516T^32 + 177494528T^30 + 43439744T^28 + 2733056T^26 + 78336T^24 - 11162880T^22 + 227793152T^20 - 938361600T^18 + 1264445104T^16 - 732034560T^14 + 214599872T^12 - 35108864T^10 + 3811328T^8 + 82048T^6 + 43648T^4 + 768*T^2 + 4
$53$	\( T^{64} + 64 T^{60} + \cdots + 16 \) T^64 + 64T^60 + 1824T^56 + 30528T^52 + 333200T^48 + 2490624T^44 + 13032832T^40 + 47783936T^36 + 131049600T^32 + 106153984T^28 + 777809408T^24 - 791304448T^20 + 701793728T^16 - 117660672T^12 + 7672576T^8 - 21504*T^4 + 16
$59$	\( T^{64} \) T^64
$61$	\( (T^{32} + 4 T^{24} + \cdots + 2)^{2} \) (T^32 + 4T^24 - 448T^23 + 144T^22 + 6T^16 + 2688T^15 + 23136T^14 + 20352T^13 + 2616T^12 + 32T^11 + 4T^8 - 960T^7 + 6864T^6 - 11648T^5 + 7752T^4 - 2464T^3 + 400T^2 - 32*T + 2)^2
$67$	\( T^{64} \) T^64
$71$	\( T^{64} \) T^64
$73$	\( T^{64} \) T^64
$79$	\( (T^{8} + 8 T^{5} + 2 T^{4} + \cdots + 2)^{8} \) (T^8 + 8T^5 + 2T^4 + 12T^2 - 8T + 2)^8
$83$	\( T^{64} + 7280 T^{48} + \cdots + 16 \) T^64 + 7280T^48 - 537472T^44 + 5230016T^40 - 15779712T^36 + 26499200T^32 + 180433920T^28 + 1110315648T^24 + 1521254912T^20 + 644145728T^16 + 42616832T^12 + 7228928T^8 - 17920T^4 + 16
$89$	\( T^{64} \) T^64
$97$	\( T^{64} \) T^64
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