LMFDB - Newform orbit 804.2.ba.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(41,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([0, 33, 53]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("804.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.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:	$6.41997232251$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{66}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{33}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{33}^{12} - 2 \zeta_{33}) q^{3} + ( - \zeta_{33}^{19} + 4 \zeta_{33}^{17} + \cdots + 1) q^{7}+ \cdots + (3 \zeta_{33}^{13} + 3 \zeta_{33}^{2}) q^{9}+O(q^{10}) $ q + (-z^12 - 2z) q^3 + (-z^19 + 4z^17 - z^16 + z^14 - z^13 + z^11 - z^10 - 2z^9 - z^7 + 2z^6 - z^4 + z^3 - z + 1) q^7 + (3z^13 + 3z^2) * q^9	$ q + ( - \zeta_{33}^{12} - 2 \zeta_{33}) q^{3} + ( - \zeta_{33}^{19} + 4 \zeta_{33}^{17} + \cdots + 1) q^{7}+ \cdots + ( - 3 \zeta_{33}^{19} + 3 \zeta_{33}^{18} + \cdots + 3) q^{97}+O(q^{100}) $ q + (-z^12 - 2z) q^3 + (-z^19 + 4z^17 - z^16 + z^14 - z^13 + z^11 - z^10 - 2z^9 - z^7 + 2z^6 - z^4 + z^3 - z + 1) q^7 + (3z^13 + 3z^2) * q^9 + (z^18 + z^14 + 4z^7 - 3z^3) * q^13 + (z^19 - 3z^18 + 3z^16 - 3z^15 + 3z^13 - 3z^12 + 5z^10 - 3z^9 - 2z^8 - 3z^6 + 3z^5 - 3z^3 + 3z^2 - 3) * q^19 + (4z^19 - 8z^18 + 4z^16 - 4z^15 + 4z^13 - 4z^12 + 5z^10 - 4z^9 + 4z^8 + z^7 - 4z^6 + 4z^5 - 4z^3 + 4z^2 - 4) q^21 + (5z^16 + 5z^5) * q^25 + (-6z^14 - 3z^3) * q^27 + (-6z^19 + 6z^17 - 6z^16 + 11z^14 - 6z^13 + 6z^11 - 6z^10 + z^9 - 6z^7 + 6z^6 - 6z^4 + 12z^3 - 6z + 6) * q^31 + (7z^17 + 4z^16 + 4z^6 + 7z^5) * q^37 + (-5z^19 + 2z^15 - 7z^8 + 7z^4) * q^39 + (6z^13 + 6z^12 + 7z^2 - z) q^43 + (3z^18 + 7z^15 - 5z^12 - 5z^7 + 7z^4 + 3z) * q^49 + (7z^19 - 7z^17 + 7z^16 - 7z^14 + 7z^13 - 6z^11 + 7z^10 + z^9 + 7z^7 - 7z^6 + 7z^4 - 7z^3 + 7z + 1) * q^57 + (-5z^15 + 9z^13 + 4z^4 + 4z^2) * q^61 + (3z^19 + 3z^11 - 6z^8 + 9) q^63 + (9z^16 + 2z^5) * q^67 + (8z^17 - z^11 + 9z^6 - 9) * q^73 + (-10z^17 - 5z^6) * q^75 + (-7z^19 + 7z^18 - 7z^16 + 7z^15 - 7z^13 + 7z^12 - 3z^11 - 10z^10 + 7z^9 - 7z^8 + 7z^6 - 7z^5 + 7z^3 - 7z^2 - 3) * q^79 + 9z^15 q^81 + (-11z^19 + 11z^17 - 16z^16 + 11z^14 - 20z^13 + 9z^12 + 11z^11 - 11z^10 + 5z^9 - 11z^7 + 11z^6 + 6z^5 - 11z^4 + 11z^3 - 10z^2 - 12z + 11) * q^91 + (11z^19 - 11z^18 + 11z^16 - 22z^15 + 11z^13 - 11z^12 + 4z^10 - 11z^9 + 11z^8 - 11z^6 + 11z^5 - 7z^4 - 11z^3 + 11z^2 - 11) * q^93 + (-3z^19 + 3z^18 - 3z^16 + 3z^15 - 3z^13 - 5z^12 + 8z^10 + 3z^9 - 3z^8 + 3z^6 - 3z^5 + 3z^3 - 3z^2 + 3z + 3) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 20 * q + 6 * q^7 + 6 * q^9	$ 20 q + 6 q^{7} + 6 q^{9} + 9 q^{13} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{31} + 10 q^{37} - 9 q^{39} - 5 q^{49} + 141 q^{57} + 27 q^{61} + 147 q^{63} + 11 q^{67} - 180 q^{73} - 166 q^{79} - 18 q^{81} - 36 q^{91} - 3 q^{93} + 33 q^{97}+O(q^{100}) $ 20 * q + 6 * q^7 + 6 * q^9 + 9 * q^13 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 3 * q^31 + 10 * q^37 - 9 * q^39 - 5 * q^49 + 141 * q^57 + 27 * q^61 + 147 * q^63 + 11 * q^67 - 180 * q^73 - 166 * q^79 - 18 * q^81 - 36 * q^91 - 3 * q^93 + 33 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$.

$n$	$269$	$337$	$403$
$\chi(n)$	$-1$	$-\zeta_{33}^{4}$	$1$

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} $

41.1

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

0.936417

1.45709i

−0.502989

−

5.26754i

−1.24625

2.72890i

101.1

−0.487975

−

1.66189i

1.00786

−

1.95498i

−2.52376

1.62192i

113.1

1.57553

0.719520i

3.27565

1.13371i

1.96458

2.26725i

185.1

1.57553

−

0.719520i

3.27565

−

1.13371i

1.96458

−

2.26725i

197.1

−0.936417

1.45709i

−0.614593

−

0.437650i

−1.24625

−

2.72890i

221.1

−1.71442

−

0.246497i

4.93365

−

1.19689i

2.87848

0.845198i

233.1

1.71442

−

0.246497i

0.518680

−

0.543976i

2.87848

−

0.845198i

245.1

1.71442

0.246497i

0.518680

0.543976i

2.87848

0.845198i

281.1

1.30900

1.13425i

1.65416

−

4.13190i

0.426945

2.96946i

329.1

−1.57553

−

0.719520i

−0.842599

−

4.37182i

1.96458

2.26725i

353.1

0.936417

−

1.45709i

−0.502989

5.26754i

−1.24625

−

2.72890i

413.1

−1.57553

0.719520i

−0.842599

4.37182i

1.96458

−

2.26725i

497.1

−1.30900

1.13425i

−1.35800

1.72684i

0.426945

−

2.96946i

593.1

−1.71442

0.246497i

4.93365

1.19689i

2.87848

−

0.845198i

605.1

−0.487975

1.66189i

1.00786

1.95498i

−2.52376

−

1.62192i

653.1

−0.936417

−

1.45709i

−0.614593

0.437650i

−1.24625

2.72890i

677.1

0.487975

−

1.66189i

−5.07182

−

0.241600i

−2.52376

−

1.62192i

701.1

1.30900

−

1.13425i

1.65416

4.13190i

0.426945

−

2.96946i

749.1

−1.30900

−

1.13425i

−1.35800

−

1.72684i

0.426945

2.96946i

785.1

0.487975

1.66189i

−5.07182

0.241600i

−2.52376

1.62192i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
67.h	odd	66	1	inner
201.p	even	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	804.2.ba.a	✓	20
3.b	odd	2	1	CM	804.2.ba.a	✓	20
67.h	odd	66	1	inner	804.2.ba.a	✓	20
201.p	even	66	1	inner	804.2.ba.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
804.2.ba.a	✓	20	1.a	even	1	1	trivial
804.2.ba.a	✓	20	3.b	odd	2	1	CM
804.2.ba.a	✓	20	67.h	odd	66	1	inner
804.2.ba.a	✓	20	201.p	even	66	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} $ T5 acting on $S_{2}^{\mathrm{new}}(804, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} $ T^20
$3$	$ T^{20} - 3 T^{18} + \cdots + 59049 $ T^20 - 3T^18 + 9T^16 - 27T^14 + 81T^12 - 243T^10 + 729T^8 - 2187T^6 + 6561T^4 - 19683*T^2 + 59049
$5$	$ T^{20} $ T^20
$7$	$ T^{20} + \cdots + 659102929 $ T^20 - 6T^19 + 24T^18 - 72T^17 - 857T^16 + 7315T^15 - 33606T^14 + 113856T^13 + 87195T^12 - 1720735T^11 + 11097734T^10 - 46091622T^9 + 90502102T^8 - 135796193T^7 + 451360667T^6 - 777301063T^5 + 1824145390T^4 + 518069766T^3 - 221079791T^2 - 73553145*T + 659102929
$11$	$ T^{20} $ T^20
$13$	$ T^{20} + \cdots + 78236005849 $ T^20 - 9T^19 + 54T^18 - 243T^17 + 729T^16 - 67874T^14 - 234979T^13 + 225691T^12 - 1449686T^11 + 4001854T^10 + 607473T^9 + 470526574T^8 + 15741130648T^7 + 120762116604T^6 + 433295640063T^5 + 902314714393T^4 + 1204191535005T^3 + 966291490413T^2 + 316518399149T + 78236005849
$17$	$ T^{20} $ T^20
$19$	$ T^{20} + \cdots + 338632050241 $ T^20 + 8T^19 - 512T^17 - 4096T^16 - 7942T^15 + 198608T^14 + 3371843T^13 + 11577973T^12 - 135180800T^11 - 677045403T^10 + 3239863344T^9 + 12387371898T^8 - 63090318208T^7 - 157172351563T^6 + 96792966369T^5 + 2328931361094T^4 + 5844542084401T^3 + 10793069762662T^2 - 1104698459165T + 338632050241
$23$	$ T^{20} $ T^20
$29$	$ T^{20} $ T^20
$31$	$ T^{20} + \cdots + 33\!\cdots\!61 $ T^20 + 3T^19 + 6T^18 + 9T^17 + 9T^16 + 147994T^15 + 443955T^14 - 8277174T^13 - 46357748T^12 - 166234333T^11 + 13364815826T^10 + 41139362184T^9 - 46748752650T^8 - 1962427659784T^7 + 11169428622145T^6 + 171857945242874T^5 + 534313589078560T^4 + 4173292544231188T^3 + 11314881889586580T^2 - 11240850307726775*T + 33149692095983761
$37$	$ T^{20} + \cdots + 61\!\cdots\!29 $ T^20 - 10T^19 + 407T^18 - 3070T^17 + 95413T^16 - 647130T^15 + 13645490T^14 - 71741900T^13 + 1290752474T^12 - 5733334740T^11 + 80052021873T^10 - 226403825000T^9 + 2951517136392T^8 - 6237492203700T^7 + 77365650991311T^6 - 102747336686830T^5 + 941088518377138T^4 - 2189758462532510T^3 + 6415585363632885T^2 - 6584946033696150*T + 6129633036232729
$41$	$ T^{20} $ T^20
$43$	$ T^{20} + \cdots + 12\!\cdots\!89 $ T^20 - 3T^18 + 9T^16 + 1509316T^14 - 56847505T^13 + 503601289T^12 - 1385228273T^11 + 71323900T^10 + 4155684819T^9 + 764670957229T^8 - 46261323222257T^7 + 1142500448980530T^6 - 14420030784228069T^5 + 105445795246404541T^4 - 468213383927728296T^3 + 1238555898919818948T^2 - 1801097534581637149T + 1223082308431887289
$47$	$ T^{20} $ T^20
$53$	$ T^{20} $ T^20
$59$	$ T^{20} $ T^20
$61$	$ T^{20} + \cdots + 41\!\cdots\!61 $ T^20 - 27T^19 + 486T^18 - 6561T^17 + 59049T^16 - 9518378T^14 + 82220836T^13 + 914447521T^12 - 41673433291T^11 + 886984352308T^10 - 13804524989316T^9 + 163418942398369T^8 - 1425207040305670T^7 + 9015400358586621T^6 - 41015546663880690T^5 + 137469019730262595T^4 - 364470962204923029T^3 + 738128074404194982T^2 - 814669659322334414T + 415701764299883161
$67$	$ T^{20} + \cdots + 18\!\cdots\!49 $ T^20 - 11T^19 + 54T^18 + 143T^17 - 5191T^16 + 47520T^15 - 174923T^14 - 1259687T^13 + 25576398T^12 - 196941349T^11 + 452736173T^10 - 13195070383T^9 + 114812450622T^8 - 378867241181T^7 - 3524894538683T^6 + 64157945084640T^5 - 469569461839279T^4 + 866681759561189T^3 + 21927654588058614T^2 - 299271878359244417*T + 1822837804551761449
$71$	$ T^{20} $ T^20
$73$	$ T^{20} + \cdots + 25\!\cdots\!89 $ T^20 + 180T^19 + 15389T^18 + 830612T^17 + 31729695T^16 + 911426296T^15 + 20414512166T^14 + 364854374736T^13 + 5280233680909T^12 + 62433405361588T^11 + 605851086670323T^10 + 4828867670441602T^9 + 31531350014653280T^8 + 167715015196626412T^7 + 720590253690417201T^6 + 2477872448683103417T^5 + 6780422642782194919T^4 + 14762886565117645455T^3 + 24993677314788687054T^2 + 29234511473146086847*T + 25486171247882279089
$79$	$ T^{20} + \cdots + 17\!\cdots\!41 $ T^20 + 166T^19 + 13131T^18 + 658222T^17 + 23453611T^16 + 631535124T^15 + 13335962202T^14 + 226174924956T^13 + 3129248523285T^12 + 35671144191218T^11 + 336886409417909T^10 + 2640769750083310T^9 + 17149626737467216T^8 + 91685069995081944T^7 + 398447954354317599T^6 + 1379703548398886469T^5 + 3717633574314032217T^4 + 7694093744546752497T^3 + 12588040726544565904T^2 + 15284314412256445537*T + 17235172454566363441
$83$	$ T^{20} $ T^20
$89$	$ T^{20} $ T^20
$97$	$ T^{20} + \cdots + 71\!\cdots\!49 $ T^20 - 33T^19 - 341T^18 + 23232T^17 + 81620T^16 - 11126676T^15 + 36370158T^14 + 2838768174T^13 - 15342440542T^12 - 524172309276T^11 + 4069304300281T^10 + 55709263368396T^9 - 493453999459619T^8 - 3938480620560321T^7 + 45125666892337119T^6 + 55712942505179520T^5 - 1149213816036277299T^4 - 1853496726125134992T^3 + 18390216697286157988T^2 + 65942160572980788492*T + 71091975405390582049
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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:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.ba (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{33}^{12} - 2 \zeta_{33}) q^{3} + ( - \zeta_{33}^{19} + 4 \zeta_{33}^{17} + \cdots + 1) q^{7}+ \cdots + (3 \zeta_{33}^{13} + 3 \zeta_{33}^{2}) q^{9}+O(q^{10}) \) q + (-z^12 - 2z) q^3 + (-z^19 + 4z^17 - z^16 + z^14 - z^13 + z^11 - z^10 - 2z^9 - z^7 + 2z^6 - z^4 + z^3 - z + 1) q^7 + (3z^13 + 3z^2) * q^9	\( q + ( - \zeta_{33}^{12} - 2 \zeta_{33}) q^{3} + ( - \zeta_{33}^{19} + 4 \zeta_{33}^{17} + \cdots + 1) q^{7}+ \cdots + ( - 3 \zeta_{33}^{19} + 3 \zeta_{33}^{18} + \cdots + 3) q^{97}+O(q^{100}) \) q + (-z^12 - 2z) q^3 + (-z^19 + 4z^17 - z^16 + z^14 - z^13 + z^11 - z^10 - 2z^9 - z^7 + 2z^6 - z^4 + z^3 - z + 1) q^7 + (3z^13 + 3z^2) * q^9 + (z^18 + z^14 + 4z^7 - 3z^3) * q^13 + (z^19 - 3z^18 + 3z^16 - 3z^15 + 3z^13 - 3z^12 + 5z^10 - 3z^9 - 2z^8 - 3z^6 + 3z^5 - 3z^3 + 3z^2 - 3) * q^19 + (4z^19 - 8z^18 + 4z^16 - 4z^15 + 4z^13 - 4z^12 + 5z^10 - 4z^9 + 4z^8 + z^7 - 4z^6 + 4z^5 - 4z^3 + 4z^2 - 4) q^21 + (5z^16 + 5z^5) * q^25 + (-6z^14 - 3z^3) * q^27 + (-6z^19 + 6z^17 - 6z^16 + 11z^14 - 6z^13 + 6z^11 - 6z^10 + z^9 - 6z^7 + 6z^6 - 6z^4 + 12z^3 - 6z + 6) * q^31 + (7z^17 + 4z^16 + 4z^6 + 7z^5) * q^37 + (-5z^19 + 2z^15 - 7z^8 + 7z^4) * q^39 + (6z^13 + 6z^12 + 7z^2 - z) q^43 + (3z^18 + 7z^15 - 5z^12 - 5z^7 + 7z^4 + 3z) * q^49 + (7z^19 - 7z^17 + 7z^16 - 7z^14 + 7z^13 - 6z^11 + 7z^10 + z^9 + 7z^7 - 7z^6 + 7z^4 - 7z^3 + 7z + 1) * q^57 + (-5z^15 + 9z^13 + 4z^4 + 4z^2) * q^61 + (3z^19 + 3z^11 - 6z^8 + 9) q^63 + (9z^16 + 2z^5) * q^67 + (8z^17 - z^11 + 9z^6 - 9) * q^73 + (-10z^17 - 5z^6) * q^75 + (-7z^19 + 7z^18 - 7z^16 + 7z^15 - 7z^13 + 7z^12 - 3z^11 - 10z^10 + 7z^9 - 7z^8 + 7z^6 - 7z^5 + 7z^3 - 7z^2 - 3) * q^79 + 9z^15 q^81 + (-11z^19 + 11z^17 - 16z^16 + 11z^14 - 20z^13 + 9z^12 + 11z^11 - 11z^10 + 5z^9 - 11z^7 + 11z^6 + 6z^5 - 11z^4 + 11z^3 - 10z^2 - 12z + 11) * q^91 + (11z^19 - 11z^18 + 11z^16 - 22z^15 + 11z^13 - 11z^12 + 4z^10 - 11z^9 + 11z^8 - 11z^6 + 11z^5 - 7z^4 - 11z^3 + 11z^2 - 11) * q^93 + (-3z^19 + 3z^18 - 3z^16 + 3z^15 - 3z^13 - 5z^12 + 8z^10 + 3z^9 - 3z^8 + 3z^6 - 3z^5 + 3z^3 - 3z^2 + 3z + 3) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 20 * q + 6 * q^7 + 6 * q^9	\( 20 q + 6 q^{7} + 6 q^{9} + 9 q^{13} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{31} + 10 q^{37} - 9 q^{39} - 5 q^{49} + 141 q^{57} + 27 q^{61} + 147 q^{63} + 11 q^{67} - 180 q^{73} - 166 q^{79} - 18 q^{81} - 36 q^{91} - 3 q^{93} + 33 q^{97}+O(q^{100}) \) 20 * q + 6 * q^7 + 6 * q^9 + 9 * q^13 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 3 * q^31 + 10 * q^37 - 9 * q^39 - 5 * q^49 + 141 * q^57 + 27 * q^61 + 147 * q^63 + 11 * q^67 - 180 * q^73 - 166 * q^79 - 18 * q^81 - 36 * q^91 - 3 * q^93 + 33 * q^97

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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	804.2.ba.a

Level	$804$
Weight	$2$
Character orbit	804.ba
Analytic conductor	$6.420$
Analytic rank	$0$
Dimension	$20$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.41997232251\)
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:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{66}]$

\(n\)	\(269\)	\(337\)	\(403\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{33}^{4}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 41.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
67.h	odd	66	1	inner
201.p	even	66	1	inner

$p$	$F_p(T)$
$2$	\( T^{20} \) T^20
$3$	\( T^{20} - 3 T^{18} + \cdots + 59049 \) T^20 - 3T^18 + 9T^16 - 27T^14 + 81T^12 - 243T^10 + 729T^8 - 2187T^6 + 6561T^4 - 19683*T^2 + 59049
$5$	\( T^{20} \) T^20
$7$	\( T^{20} + \cdots + 659102929 \) T^20 - 6T^19 + 24T^18 - 72T^17 - 857T^16 + 7315T^15 - 33606T^14 + 113856T^13 + 87195T^12 - 1720735T^11 + 11097734T^10 - 46091622T^9 + 90502102T^8 - 135796193T^7 + 451360667T^6 - 777301063T^5 + 1824145390T^4 + 518069766T^3 - 221079791T^2 - 73553145*T + 659102929
$11$	\( T^{20} \) T^20
$13$	\( T^{20} + \cdots + 78236005849 \) T^20 - 9T^19 + 54T^18 - 243T^17 + 729T^16 - 67874T^14 - 234979T^13 + 225691T^12 - 1449686T^11 + 4001854T^10 + 607473T^9 + 470526574T^8 + 15741130648T^7 + 120762116604T^6 + 433295640063T^5 + 902314714393T^4 + 1204191535005T^3 + 966291490413T^2 + 316518399149T + 78236005849
$17$	\( T^{20} \) T^20
$19$	\( T^{20} + \cdots + 338632050241 \) T^20 + 8T^19 - 512T^17 - 4096T^16 - 7942T^15 + 198608T^14 + 3371843T^13 + 11577973T^12 - 135180800T^11 - 677045403T^10 + 3239863344T^9 + 12387371898T^8 - 63090318208T^7 - 157172351563T^6 + 96792966369T^5 + 2328931361094T^4 + 5844542084401T^3 + 10793069762662T^2 - 1104698459165T + 338632050241
$23$	\( T^{20} \) T^20
$29$	\( T^{20} \) T^20
$31$	\( T^{20} + \cdots + 33\!\cdots\!61 \) T^20 + 3T^19 + 6T^18 + 9T^17 + 9T^16 + 147994T^15 + 443955T^14 - 8277174T^13 - 46357748T^12 - 166234333T^11 + 13364815826T^10 + 41139362184T^9 - 46748752650T^8 - 1962427659784T^7 + 11169428622145T^6 + 171857945242874T^5 + 534313589078560T^4 + 4173292544231188T^3 + 11314881889586580T^2 - 11240850307726775*T + 33149692095983761
$37$	\( T^{20} + \cdots + 61\!\cdots\!29 \) T^20 - 10T^19 + 407T^18 - 3070T^17 + 95413T^16 - 647130T^15 + 13645490T^14 - 71741900T^13 + 1290752474T^12 - 5733334740T^11 + 80052021873T^10 - 226403825000T^9 + 2951517136392T^8 - 6237492203700T^7 + 77365650991311T^6 - 102747336686830T^5 + 941088518377138T^4 - 2189758462532510T^3 + 6415585363632885T^2 - 6584946033696150*T + 6129633036232729
$41$	\( T^{20} \) T^20
$43$	\( T^{20} + \cdots + 12\!\cdots\!89 \) T^20 - 3T^18 + 9T^16 + 1509316T^14 - 56847505T^13 + 503601289T^12 - 1385228273T^11 + 71323900T^10 + 4155684819T^9 + 764670957229T^8 - 46261323222257T^7 + 1142500448980530T^6 - 14420030784228069T^5 + 105445795246404541T^4 - 468213383927728296T^3 + 1238555898919818948T^2 - 1801097534581637149T + 1223082308431887289
$47$	\( T^{20} \) T^20
$53$	\( T^{20} \) T^20
$59$	\( T^{20} \) T^20
$61$	\( T^{20} + \cdots + 41\!\cdots\!61 \) T^20 - 27T^19 + 486T^18 - 6561T^17 + 59049T^16 - 9518378T^14 + 82220836T^13 + 914447521T^12 - 41673433291T^11 + 886984352308T^10 - 13804524989316T^9 + 163418942398369T^8 - 1425207040305670T^7 + 9015400358586621T^6 - 41015546663880690T^5 + 137469019730262595T^4 - 364470962204923029T^3 + 738128074404194982T^2 - 814669659322334414T + 415701764299883161
$67$	\( T^{20} + \cdots + 18\!\cdots\!49 \) T^20 - 11T^19 + 54T^18 + 143T^17 - 5191T^16 + 47520T^15 - 174923T^14 - 1259687T^13 + 25576398T^12 - 196941349T^11 + 452736173T^10 - 13195070383T^9 + 114812450622T^8 - 378867241181T^7 - 3524894538683T^6 + 64157945084640T^5 - 469569461839279T^4 + 866681759561189T^3 + 21927654588058614T^2 - 299271878359244417*T + 1822837804551761449
$71$	\( T^{20} \) T^20
$73$	\( T^{20} + \cdots + 25\!\cdots\!89 \) T^20 + 180T^19 + 15389T^18 + 830612T^17 + 31729695T^16 + 911426296T^15 + 20414512166T^14 + 364854374736T^13 + 5280233680909T^12 + 62433405361588T^11 + 605851086670323T^10 + 4828867670441602T^9 + 31531350014653280T^8 + 167715015196626412T^7 + 720590253690417201T^6 + 2477872448683103417T^5 + 6780422642782194919T^4 + 14762886565117645455T^3 + 24993677314788687054T^2 + 29234511473146086847*T + 25486171247882279089
$79$	\( T^{20} + \cdots + 17\!\cdots\!41 \) T^20 + 166T^19 + 13131T^18 + 658222T^17 + 23453611T^16 + 631535124T^15 + 13335962202T^14 + 226174924956T^13 + 3129248523285T^12 + 35671144191218T^11 + 336886409417909T^10 + 2640769750083310T^9 + 17149626737467216T^8 + 91685069995081944T^7 + 398447954354317599T^6 + 1379703548398886469T^5 + 3717633574314032217T^4 + 7694093744546752497T^3 + 12588040726544565904T^2 + 15284314412256445537*T + 17235172454566363441
$83$	\( T^{20} \) T^20
$89$	\( T^{20} \) T^20
$97$	\( T^{20} + \cdots + 71\!\cdots\!49 \) T^20 - 33T^19 - 341T^18 + 23232T^17 + 81620T^16 - 11126676T^15 + 36370158T^14 + 2838768174T^13 - 15342440542T^12 - 524172309276T^11 + 4069304300281T^10 + 55709263368396T^9 - 493453999459619T^8 - 3938480620560321T^7 + 45125666892337119T^6 + 55712942505179520T^5 - 1149213816036277299T^4 - 1853496726125134992T^3 + 18390216697286157988T^2 + 65942160572980788492*T + 71091975405390582049
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