LMFDB - Newform orbit 536.1.ba.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [536,1,Mod(19,536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(536, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([33, 33, 10]))

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

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

chi := DirichletCharacter("536.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 536 = 2^{3} \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	536.ba (of order $66$, degree $20$, 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:	$0.267498846771$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} - \cdots)$

$q$-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_{66}^{23} q^{2} + ( - \zeta_{66}^{31} + \zeta_{66}^{14}) q^{3} - \zeta_{66}^{13} q^{4} + ( - \zeta_{66}^{21} + \zeta_{66}^{4}) q^{6} - \zeta_{66}^{3} q^{8} + ( - \zeta_{66}^{29} + \cdots + \zeta_{66}^{12}) q^{9} +O(q^{10}) $ q - z^23 * q^2 + (-z^31 + z^14) * q^3 - z^13 * q^4 + (-z^21 + z^4) * q^6 - z^3 * q^8 + (-z^29 + z^28 + z^12) * q^9	$ q - \zeta_{66}^{23} q^{2} + ( - \zeta_{66}^{31} + \zeta_{66}^{14}) q^{3} - \zeta_{66}^{13} q^{4} + ( - \zeta_{66}^{21} + \zeta_{66}^{4}) q^{6} - \zeta_{66}^{3} q^{8} + ( - \zeta_{66}^{29} + \cdots + \zeta_{66}^{12}) q^{9} + \cdots + (\zeta_{66}^{30} + \cdots + \zeta_{66}^{2}) q^{99} +O(q^{100}) $ q - z^23 * q^2 + (-z^31 + z^14) * q^3 - z^13 * q^4 + (-z^21 + z^4) * q^6 - z^3 * q^8 + (-z^29 + z^28 + z^12) * q^9 + (-z^7 - z) * q^11 + (-z^27 - z^11) * q^12 + z^26 * q^16 + (z^22 + z^18) * q^17 + (-z^19 + z^18 + z^2) * q^18 + (-z^25 + z^6) * q^19 + (z^30 + z^24) * q^22 + (-z^17 - z) * q^24 + z^30 * q^25 + (-z^27 + z^26 - z^10 - z^9) * q^27 + z^16 * q^32 + (z^32 - z^21 - z^15 - z^5) * q^33 + (z^12 + z^8) * q^34 + (-z^25 - z^9 + z^8) * q^36 + (-z^29 - z^15) * q^38 + (-z^19 + z^10) * q^41 + (-z^13 - z^5) * q^43 + (z^20 + z^14) * q^44 + (z^24 - z^7) * q^48 + z^2 * q^49 + z^20 * q^50 + (z^32 + z^20 + z^16 - z^3) * q^51 + (z^32 - z^17 + z^16 + 1) * q^54 + (-z^23 + z^20 + z^6 + z^4) * q^57 + (z^22 - z^17) * q^59 + z^6 * q^64 + (z^28 + z^22 - z^11 - z^5) * q^66 - z^15 * q^67 + (-z^31 + z^2) * q^68 + (z^32 - z^31 - z^15) * q^72 + (z^16 - z^9) * q^73 + (z^28 - z^11) * q^75 + (-z^19 - z^5) * q^76 + (-z^25 + z^24 - z^23 - z^8 + z^7) * q^81 + (-z^9 + 1) * q^82 + (-z^11 + z^8) * q^83 + (z^28 - z^3) * q^86 + (z^10 + z^4) * q^88 + (-z^17 + z^4) * q^89 + (z^30 + z^14) * q^96 + (-z^31 + z^24) * q^97 - z^25 * q^98 + (z^30 - z^29 - z^19 - z^13 - z^3 + z^2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + q^{2} + 2 q^{3} + q^{4} - q^{6} - 2 q^{8}+O(q^{10}) $ 20 * q + q^2 + 2 * q^3 + q^4 - q^6 - 2 * q^8	$ 20 q + q^{2} + 2 q^{3} + q^{4} - q^{6} - 2 q^{8} + 2 q^{11} - 12 q^{12} + q^{16} - 12 q^{17} - q^{19} - 4 q^{22} + 2 q^{24} - 2 q^{25} - 2 q^{27} + q^{32} - 2 q^{33} - q^{34} - q^{38} + 2 q^{41} + 2 q^{43} + 2 q^{44} - q^{48} + q^{49} + q^{50} + q^{51} + 23 q^{54} + q^{57} - 9 q^{59} - 2 q^{64} - 18 q^{66} - 2 q^{67} + 2 q^{68} - q^{73} - 9 q^{75} + 2 q^{76} + 2 q^{81} + 18 q^{82} - 9 q^{83} - q^{86} + 2 q^{88} + 2 q^{89} - q^{96} - q^{97} + q^{98}+O(q^{100}) $ 20 * q + q^2 + 2 * q^3 + q^4 - q^6 - 2 * q^8 + 2 * q^11 - 12 * q^12 + q^16 - 12 * q^17 - q^19 - 4 * q^22 + 2 * q^24 - 2 * q^25 - 2 * q^27 + q^32 - 2 * q^33 - q^34 - q^38 + 2 * q^41 + 2 * q^43 + 2 * q^44 - q^48 + q^49 + q^50 + q^51 + 23 * q^54 + q^57 - 9 * q^59 - 2 * q^64 - 18 * q^66 - 2 * q^67 + 2 * q^68 - q^73 - 9 * q^75 + 2 * q^76 + 2 * q^81 + 18 * q^82 - 9 * q^83 - q^86 + 2 * q^88 + 2 * q^89 - q^96 - q^97 + q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/536\mathbb{Z}\right)^\times$.

$n$	$135$	$269$	$337$
$\chi(n)$	$-1$	$-1$	$-\zeta_{66}^{13}$

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} $

19.1

0.0475819	+	0.998867i
0.723734	+	0.690079i
−0.327068	−	0.945001i
0.981929	+	0.189251i
−0.327068	+	0.945001i
0.235759	−	0.971812i
−0.995472	+	0.0950560i
0.723734	−	0.690079i
−0.786053	−	0.618159i
0.235759	+	0.971812i
−0.786053	+	0.618159i
0.580057	−	0.814576i
0.928368	−	0.371662i
0.0475819	−	0.998867i
0.580057	+	0.814576i
−0.888835	−	0.458227i
−0.995472	−	0.0950560i
−0.888835	+	0.458227i
0.981929	−	0.189251i
0.928368	+	0.371662i

−0.888835

−

0.458227i

−1.78153

0.523103i

0.580057

0.814576i

1.82318

0.351390i

−0.142315

−

0.989821i

2.05894

−

1.32320i

35.1

0.235759

−

0.971812i

−0.279486

−

1.94387i

−0.888835

−

0.458227i

−1.95496

−

0.186677i

−0.654861

0.755750i

−2.74102

0.804835i

83.1

0.981929

0.189251i

−0.738471

−

1.61703i

0.928368

0.371662i

−0.419102

−

1.72756i

0.841254

0.540641i

−1.41457

1.63251i

123.1

−0.327068

−

0.945001i

0.0395325

0.0865641i

−0.786053

0.618159i

0.0688733

−

0.0656706i

0.841254

0.540641i

0.648930

−

0.748905i

155.1

0.981929

−

0.189251i

−0.738471

1.61703i

0.928368

−

0.371662i

−0.419102

1.72756i

0.841254

−

0.540641i

−1.41457

−

1.63251i

211.1

0.723734

0.690079i

0.0930932

0.647478i

0.0475819

0.998867i

−0.379436

0.532843i

−0.654861

0.755750i

0.548932

−

0.161181i

227.1

0.580057

0.814576i

1.21769

−

0.782560i

−0.327068

0.945001i

1.34378

0.537970i

−0.959493

0.281733i

0.454947

−

0.996196i

291.1

0.235759

0.971812i

−0.279486

1.94387i

−0.888835

0.458227i

−1.95496

0.186677i

−0.654861

−

0.755750i

−2.74102

−

0.804835i

307.1

0.928368

−

0.371662i

−0.759713

−

0.876756i

0.723734

−

0.690079i

−1.03115

−

0.531595i

0.415415

−

0.909632i

−0.0492216

0.342344i

315.1

0.723734

−

0.690079i

0.0930932

−

0.647478i

0.0475819

−

0.998867i

−0.379436

−

0.532843i

−0.654861

−

0.755750i

0.548932

0.161181i

323.1

0.928368

0.371662i

−0.759713

0.876756i

0.723734

0.690079i

−1.03115

0.531595i

0.415415

0.909632i

−0.0492216

−

0.342344i

339.1

−0.995472

−

0.0950560i

0.396666

0.254922i

0.981929

0.189251i

−0.370638

−

0.291473i

−0.959493

−

0.281733i

−0.323056

−

0.707394i

371.1

−0.786053

−

0.618159i

1.30379

1.50465i

0.235759

0.971812i

−0.0947329

−

1.98869i

0.415415

−

0.909632i

−0.421801

2.93369i

395.1

−0.888835

0.458227i

−1.78153

−

0.523103i

0.580057

−

0.814576i

1.82318

−

0.351390i

−0.142315

0.989821i

2.05894

1.32320i

419.1

−0.995472

0.0950560i

0.396666

−

0.254922i

0.981929

−

0.189251i

−0.370638

0.291473i

−0.959493

0.281733i

−0.323056

0.707394i

435.1

0.0475819

0.998867i

1.50842

−

0.442913i

−0.995472

0.0950560i

0.514186

1.48564i

−0.142315

−

0.989821i

1.23792

−

0.795563i

451.1

0.580057

−

0.814576i

1.21769

0.782560i

−0.327068

−

0.945001i

1.34378

−

0.537970i

−0.959493

−

0.281733i

0.454947

0.996196i

467.1

0.0475819

−

0.998867i

1.50842

0.442913i

−0.995472

−

0.0950560i

0.514186

−

1.48564i

−0.142315

0.989821i

1.23792

0.795563i

475.1

−0.327068

0.945001i

0.0395325

−

0.0865641i

−0.786053

−

0.618159i

0.0688733

0.0656706i

0.841254

−

0.540641i

0.648930

0.748905i

523.1

−0.786053

0.618159i

1.30379

−

1.50465i

0.235759

−

0.971812i

−0.0947329

1.98869i

0.415415

0.909632i

−0.421801

−

2.93369i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
67.g	even	33	1	inner
536.ba	odd	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	536.1.ba.a	✓	20
4.b	odd	2	1		2144.1.bu.a		20
8.b	even	2	1		2144.1.bu.a		20
8.d	odd	2	1	CM	536.1.ba.a	✓	20
67.g	even	33	1	inner	536.1.ba.a	✓	20
268.o	odd	66	1		2144.1.bu.a		20
536.ba	odd	66	1	inner	536.1.ba.a	✓	20
536.bf	even	66	1		2144.1.bu.a		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
536.1.ba.a	✓	20	1.a	even	1	1	trivial
536.1.ba.a	✓	20	8.d	odd	2	1	CM
536.1.ba.a	✓	20	67.g	even	33	1	inner
536.1.ba.a	✓	20	536.ba	odd	66	1	inner
2144.1.bu.a		20	4.b	odd	2	1
2144.1.bu.a		20	8.b	even	2	1
2144.1.bu.a		20	268.o	odd	66	1
2144.1.bu.a		20	536.bf	even	66	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(536, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} - T^{19} + \cdots + 1 $ T^20 - T^19 + T^17 - T^16 + T^14 - T^13 + T^11 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$3$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 3T^18 - 4T^17 + 5T^16 + 16T^15 - 37T^14 - 8T^13 + 31T^12 - 43T^11 + 528T^10 - 1011T^9 + 768T^8 - 767T^7 + 1613T^6 - 1491T^5 + 1149T^4 - 565T^3 + 157T^2 - 13*T + 1
$5$	$ T^{20} $ T^20
$7$	$ T^{20} $ T^20
$11$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 8T^17 - 5T^16 - 11T^15 + 42T^14 - 40T^13 + 11T^12 - 49T^11 + 164T^10 - 130T^9 - 154T^8 - 19T^7 + 49T^6 + 77T^5 + 178T^4 + 62T^3 + 11T^2 + 5T + 1
$13$	$ T^{20} $ T^20
$17$	$ T^{20} + 12 T^{19} + \cdots + 1 $ T^20 + 12T^19 + 77T^18 + 340T^17 + 1143T^16 + 3080T^15 + 6854T^14 + 12816T^13 + 20317T^12 + 27356T^11 + 31107T^10 + 29402T^9 + 22352T^8 + 12860T^7 + 5193T^6 + 1705T^5 + 967T^4 + 615T^3 + 198T^2 + 23*T + 1
$19$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 - T^16 + 12T^14 - 98T^13 - 33T^12 + 109T^11 + 120T^10 + 10T^9 + 11T^8 - 604T^7 + 1563T^6 - 1100T^5 + 725T^4 - 375T^3 + 44T^2 + 12T + 1
$23$	$ T^{20} $ T^20
$29$	$ T^{20} $ T^20
$31$	$ T^{20} $ T^20
$37$	$ T^{20} $ T^20
$41$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 8T^17 - 16T^16 + 22T^15 + 20T^14 - 95T^13 + 121T^12 + 116T^11 - 243T^10 + 24T^9 + 198T^8 - 8T^7 + 71T^6 - 231T^5 + 24T^4 + 29T^3 + 22T^2 + 5T + 1
$43$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 3T^18 - 4T^17 + 16T^16 - 50T^15 + 84T^14 - 118T^13 + 251T^12 - 472T^11 + 836T^10 - 1198T^9 + 1802T^8 - 1438T^7 + 832T^6 - 457T^5 + 544T^4 + 95T^3 + 113T^2 + 20*T + 1
$47$	$ T^{20} $ T^20
$53$	$ T^{20} $ T^20
$59$	$ T^{20} + 9 T^{19} + \cdots + 1 $ T^20 + 9T^19 + 47T^18 + 172T^17 + 489T^16 + 1127T^15 + 2174T^14 + 3567T^13 + 5047T^12 + 6194T^11 + 6633T^10 + 6194T^9 + 5069T^8 + 3578T^7 + 1965T^6 + 874T^5 + 742T^4 + 381T^3 + 36T^2 - 13*T + 1
$61$	$ T^{20} $ T^20
$67$	$ (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} $ (T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$71$	$ T^{20} $ T^20
$73$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 - T^16 - 44T^15 - 43T^14 + 34T^13 + 88T^12 + 65T^11 + 483T^10 + 417T^9 - 792T^8 - 1451T^7 + 188T^6 + 792T^5 + 384T^4 + 197T^3 + 88T^2 - 10*T + 1
$79$	$ T^{20} $ T^20
$83$	$ T^{20} + 9 T^{19} + \cdots + 1 $ T^20 + 9T^19 + 44T^18 + 151T^17 + 402T^16 + 869T^15 + 1571T^14 + 2424T^13 + 3223T^12 + 3713T^11 + 3750T^10 + 3302T^9 + 2453T^8 + 1631T^7 + 1116T^6 + 616T^5 + 145T^4 - 15T^3 + 5T + 1
$89$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 3T^18 - 4T^17 + 16T^16 - 50T^15 + 84T^14 - 118T^13 + 251T^12 - 472T^11 + 836T^10 - 1198T^9 + 1802T^8 - 1438T^7 + 832T^6 - 457T^5 + 544T^4 + 95T^3 + 113T^2 + 20*T + 1
$97$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
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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:	\( N \)	\(=\)	\( 536 = 2^{3} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	536.ba (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{66}^{23} q^{2} + ( - \zeta_{66}^{31} + \zeta_{66}^{14}) q^{3} - \zeta_{66}^{13} q^{4} + ( - \zeta_{66}^{21} + \zeta_{66}^{4}) q^{6} - \zeta_{66}^{3} q^{8} + ( - \zeta_{66}^{29} + \cdots + \zeta_{66}^{12}) q^{9} +O(q^{10}) \) q - z^23 * q^2 + (-z^31 + z^14) * q^3 - z^13 * q^4 + (-z^21 + z^4) * q^6 - z^3 * q^8 + (-z^29 + z^28 + z^12) * q^9	\( q - \zeta_{66}^{23} q^{2} + ( - \zeta_{66}^{31} + \zeta_{66}^{14}) q^{3} - \zeta_{66}^{13} q^{4} + ( - \zeta_{66}^{21} + \zeta_{66}^{4}) q^{6} - \zeta_{66}^{3} q^{8} + ( - \zeta_{66}^{29} + \cdots + \zeta_{66}^{12}) q^{9} + \cdots + (\zeta_{66}^{30} + \cdots + \zeta_{66}^{2}) q^{99} +O(q^{100}) \) q - z^23 * q^2 + (-z^31 + z^14) * q^3 - z^13 * q^4 + (-z^21 + z^4) * q^6 - z^3 * q^8 + (-z^29 + z^28 + z^12) * q^9 + (-z^7 - z) * q^11 + (-z^27 - z^11) * q^12 + z^26 * q^16 + (z^22 + z^18) * q^17 + (-z^19 + z^18 + z^2) * q^18 + (-z^25 + z^6) * q^19 + (z^30 + z^24) * q^22 + (-z^17 - z) * q^24 + z^30 * q^25 + (-z^27 + z^26 - z^10 - z^9) * q^27 + z^16 * q^32 + (z^32 - z^21 - z^15 - z^5) * q^33 + (z^12 + z^8) * q^34 + (-z^25 - z^9 + z^8) * q^36 + (-z^29 - z^15) * q^38 + (-z^19 + z^10) * q^41 + (-z^13 - z^5) * q^43 + (z^20 + z^14) * q^44 + (z^24 - z^7) * q^48 + z^2 * q^49 + z^20 * q^50 + (z^32 + z^20 + z^16 - z^3) * q^51 + (z^32 - z^17 + z^16 + 1) * q^54 + (-z^23 + z^20 + z^6 + z^4) * q^57 + (z^22 - z^17) * q^59 + z^6 * q^64 + (z^28 + z^22 - z^11 - z^5) * q^66 - z^15 * q^67 + (-z^31 + z^2) * q^68 + (z^32 - z^31 - z^15) * q^72 + (z^16 - z^9) * q^73 + (z^28 - z^11) * q^75 + (-z^19 - z^5) * q^76 + (-z^25 + z^24 - z^23 - z^8 + z^7) * q^81 + (-z^9 + 1) * q^82 + (-z^11 + z^8) * q^83 + (z^28 - z^3) * q^86 + (z^10 + z^4) * q^88 + (-z^17 + z^4) * q^89 + (z^30 + z^14) * q^96 + (-z^31 + z^24) * q^97 - z^25 * q^98 + (z^30 - z^29 - z^19 - z^13 - z^3 + z^2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + q^{2} + 2 q^{3} + q^{4} - q^{6} - 2 q^{8}+O(q^{10}) \) 20 * q + q^2 + 2 * q^3 + q^4 - q^6 - 2 * q^8	\( 20 q + q^{2} + 2 q^{3} + q^{4} - q^{6} - 2 q^{8} + 2 q^{11} - 12 q^{12} + q^{16} - 12 q^{17} - q^{19} - 4 q^{22} + 2 q^{24} - 2 q^{25} - 2 q^{27} + q^{32} - 2 q^{33} - q^{34} - q^{38} + 2 q^{41} + 2 q^{43} + 2 q^{44} - q^{48} + q^{49} + q^{50} + q^{51} + 23 q^{54} + q^{57} - 9 q^{59} - 2 q^{64} - 18 q^{66} - 2 q^{67} + 2 q^{68} - q^{73} - 9 q^{75} + 2 q^{76} + 2 q^{81} + 18 q^{82} - 9 q^{83} - q^{86} + 2 q^{88} + 2 q^{89} - q^{96} - q^{97} + q^{98}+O(q^{100}) \) 20 * q + q^2 + 2 * q^3 + q^4 - q^6 - 2 * q^8 + 2 * q^11 - 12 * q^12 + q^16 - 12 * q^17 - q^19 - 4 * q^22 + 2 * q^24 - 2 * q^25 - 2 * q^27 + q^32 - 2 * q^33 - q^34 - q^38 + 2 * q^41 + 2 * q^43 + 2 * q^44 - q^48 + q^49 + q^50 + q^51 + 23 * q^54 + q^57 - 9 * q^59 - 2 * q^64 - 18 * q^66 - 2 * q^67 + 2 * q^68 - q^73 - 9 * q^75 + 2 * q^76 + 2 * q^81 + 18 * q^82 - 9 * q^83 - q^86 + 2 * q^88 + 2 * q^89 - q^96 - q^97 + q^98

Introduction

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Modular forms

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Fields

Representations

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Database

Properties

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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	536.1.ba.a

Level	$536$
Weight	$1$
Character orbit	536.ba
Analytic conductor	$0.267$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{33}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.267498846771\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{33})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{33}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

\(n\)	\(135\)	\(269\)	\(337\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\zeta_{66}^{13}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 19.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
67.g	even	33	1	inner
536.ba	odd	66	1	inner

$p$	$F_p(T)$
$2$	\( T^{20} - T^{19} + \cdots + 1 \) T^20 - T^19 + T^17 - T^16 + T^14 - T^13 + T^11 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$3$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 3T^18 - 4T^17 + 5T^16 + 16T^15 - 37T^14 - 8T^13 + 31T^12 - 43T^11 + 528T^10 - 1011T^9 + 768T^8 - 767T^7 + 1613T^6 - 1491T^5 + 1149T^4 - 565T^3 + 157T^2 - 13*T + 1
$5$	\( T^{20} \) T^20
$7$	\( T^{20} \) T^20
$11$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 8T^17 - 5T^16 - 11T^15 + 42T^14 - 40T^13 + 11T^12 - 49T^11 + 164T^10 - 130T^9 - 154T^8 - 19T^7 + 49T^6 + 77T^5 + 178T^4 + 62T^3 + 11T^2 + 5T + 1
$13$	\( T^{20} \) T^20
$17$	\( T^{20} + 12 T^{19} + \cdots + 1 \) T^20 + 12T^19 + 77T^18 + 340T^17 + 1143T^16 + 3080T^15 + 6854T^14 + 12816T^13 + 20317T^12 + 27356T^11 + 31107T^10 + 29402T^9 + 22352T^8 + 12860T^7 + 5193T^6 + 1705T^5 + 967T^4 + 615T^3 + 198T^2 + 23*T + 1
$19$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 - T^16 + 12T^14 - 98T^13 - 33T^12 + 109T^11 + 120T^10 + 10T^9 + 11T^8 - 604T^7 + 1563T^6 - 1100T^5 + 725T^4 - 375T^3 + 44T^2 + 12T + 1
$23$	\( T^{20} \) T^20
$29$	\( T^{20} \) T^20
$31$	\( T^{20} \) T^20
$37$	\( T^{20} \) T^20
$41$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 8T^17 - 16T^16 + 22T^15 + 20T^14 - 95T^13 + 121T^12 + 116T^11 - 243T^10 + 24T^9 + 198T^8 - 8T^7 + 71T^6 - 231T^5 + 24T^4 + 29T^3 + 22T^2 + 5T + 1
$43$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 3T^18 - 4T^17 + 16T^16 - 50T^15 + 84T^14 - 118T^13 + 251T^12 - 472T^11 + 836T^10 - 1198T^9 + 1802T^8 - 1438T^7 + 832T^6 - 457T^5 + 544T^4 + 95T^3 + 113T^2 + 20*T + 1
$47$	\( T^{20} \) T^20
$53$	\( T^{20} \) T^20
$59$	\( T^{20} + 9 T^{19} + \cdots + 1 \) T^20 + 9T^19 + 47T^18 + 172T^17 + 489T^16 + 1127T^15 + 2174T^14 + 3567T^13 + 5047T^12 + 6194T^11 + 6633T^10 + 6194T^9 + 5069T^8 + 3578T^7 + 1965T^6 + 874T^5 + 742T^4 + 381T^3 + 36T^2 - 13*T + 1
$61$	\( T^{20} \) T^20
$67$	\( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \) (T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$71$	\( T^{20} \) T^20
$73$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 - T^16 - 44T^15 - 43T^14 + 34T^13 + 88T^12 + 65T^11 + 483T^10 + 417T^9 - 792T^8 - 1451T^7 + 188T^6 + 792T^5 + 384T^4 + 197T^3 + 88T^2 - 10*T + 1
$79$	\( T^{20} \) T^20
$83$	\( T^{20} + 9 T^{19} + \cdots + 1 \) T^20 + 9T^19 + 44T^18 + 151T^17 + 402T^16 + 869T^15 + 1571T^14 + 2424T^13 + 3223T^12 + 3713T^11 + 3750T^10 + 3302T^9 + 2453T^8 + 1631T^7 + 1116T^6 + 616T^5 + 145T^4 - 15T^3 + 5T + 1
$89$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 3T^18 - 4T^17 + 16T^16 - 50T^15 + 84T^14 - 118T^13 + 251T^12 - 472T^11 + 836T^10 - 1198T^9 + 1802T^8 - 1438T^7 + 832T^6 - 457T^5 + 544T^4 + 95T^3 + 113T^2 + 20*T + 1
$97$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
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