LMFDB - Newform orbit 2217.1.p.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2217,1,Mod(20,2217)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2217, base_ring=CyclotomicField(82))

chi = DirichletCharacter(H, H._module([41, 6]))

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

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

chi := DirichletCharacter("2217.20");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2217 = 3 \cdot 739 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2217.p (of order $82$, degree $40$, 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:	$1.10642713301$
Analytic rank:	$0$
Dimension:	$40$
Coefficient field:	$\Q(\zeta_{82})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{39} + x^{38} - x^{37} + x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - x^{31} + x^{30} + \cdots + 1 $ x^40 - x^39 + x^38 - x^37 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 + x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{41}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{41} - \cdots)$

$q$-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_{82}^{30} q^{3} + \zeta_{82}^{10} q^{4} + (\zeta_{82}^{28} - \zeta_{82}^{23}) q^{7} - \zeta_{82}^{19} q^{9} +O(q^{10}) $ q + z^30 * q^3 + z^10 * q^4 + (z^28 - z^23) * q^7 - z^19 * q^9	$ q + \zeta_{82}^{30} q^{3} + \zeta_{82}^{10} q^{4} + (\zeta_{82}^{28} - \zeta_{82}^{23}) q^{7} - \zeta_{82}^{19} q^{9} + \zeta_{82}^{40} q^{12} + (\zeta_{82}^{14} + 1) q^{13} + \zeta_{82}^{20} q^{16} + (\zeta_{82}^{34} - \zeta_{82}^{25}) q^{19} + ( - \zeta_{82}^{17} + \zeta_{82}^{12}) q^{21} + \zeta_{82}^{22} q^{25} + \zeta_{82}^{8} q^{27} + (\zeta_{82}^{38} - \zeta_{82}^{33}) q^{28} + ( - \zeta_{82}^{29} + \zeta_{82}^{26}) q^{31} - \zeta_{82}^{29} q^{36} + (\zeta_{82}^{24} + \zeta_{82}^{16}) q^{37} + (\zeta_{82}^{30} - \zeta_{82}^{3}) q^{39} + ( - \zeta_{82}^{17} + \zeta_{82}^{8}) q^{43} - \zeta_{82}^{9} q^{48} + ( - \zeta_{82}^{15} + \cdots - \zeta_{82}^{5}) q^{49} + \cdots + (\zeta_{82}^{6} + \zeta_{82}^{4}) q^{97} +O(q^{100}) $ q + z^30 * q^3 + z^10 * q^4 + (z^28 - z^23) * q^7 - z^19 * q^9 + z^40 * q^12 + (z^14 + 1) * q^13 + z^20 * q^16 + (z^34 - z^25) * q^19 + (-z^17 + z^12) * q^21 + z^22 * q^25 + z^8 * q^27 + (z^38 - z^33) * q^28 + (-z^29 + z^26) * q^31 - z^29 * q^36 + (z^24 + z^16) * q^37 + (z^30 - z^3) * q^39 + (-z^17 + z^8) * q^43 - z^9 * q^48 + (-z^15 + z^10 - z^5) * q^49 + (z^24 + z^10) * q^52 + (-z^23 + z^14) * q^57 + (-z^27 + z^26) * q^61 + (z^6 - z) * q^63 + z^30 * q^64 + (-z^29 - z^9) * q^67 + (-z^25 + z^22) * q^73 - z^11 * q^75 + (-z^35 - z^3) * q^76 + (z^32 - z) * q^79 + z^38 * q^81 + (-z^27 + z^22) * q^84 + (-z^37 + z^28 - z^23 - z) * q^91 + (z^18 - z^15) * q^93 + (z^6 + z^4) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - q^{3} - q^{4} - 2 q^{7} - q^{9}+O(q^{10}) $ 40 * q - q^3 - q^4 - 2 * q^7 - q^9	$ 40 q - q^{3} - q^{4} - 2 q^{7} - q^{9} - q^{12} + 39 q^{13} - q^{16} - 2 q^{19} - 2 q^{21} - q^{25} - q^{27} - 2 q^{28} - 2 q^{31} - q^{36} - 2 q^{37} - 2 q^{39} - 2 q^{43} - q^{48} - 3 q^{49} - 2 q^{52} - 2 q^{57} - 2 q^{61} - 2 q^{63} - q^{64} - 2 q^{67} - 2 q^{73} - q^{75} - 2 q^{76} - 2 q^{79} - q^{81} - 2 q^{84} - 4 q^{91} - 2 q^{93} - 2 q^{97}+O(q^{100}) $ 40 * q - q^3 - q^4 - 2 * q^7 - q^9 - q^12 + 39 * q^13 - q^16 - 2 * q^19 - 2 * q^21 - q^25 - q^27 - 2 * q^28 - 2 * q^31 - q^36 - 2 * q^37 - 2 * q^39 - 2 * q^43 - q^48 - 3 * q^49 - 2 * q^52 - 2 * q^57 - 2 * q^61 - 2 * q^63 - q^64 - 2 * q^67 - 2 * q^73 - q^75 - 2 * q^76 - 2 * q^79 - q^81 - 2 * q^84 - 4 * q^91 - 2 * q^93 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2217\mathbb{Z}\right)^\times$.

$n$	$740$	$742$
$\chi(n)$	$-1$	$-\zeta_{82}^{19}$

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} $

20.1

−0.953396	+	0.301721i
−0.190391	+	0.981708i
−0.477720	+	0.878512i
0.771489	+	0.636242i
0.771489	−	0.636242i
0.264982	+	0.964253i
−0.606225	−	0.795293i
−0.720522	+	0.693433i
0.859570	+	0.511019i
0.665326	−	0.746553i
0.114683	−	0.993402i
0.973695	+	0.227854i
−0.953396	−	0.301721i
−0.896166	−	0.443720i
0.927502	+	0.373817i
−0.0383027	+	0.999266i
0.665326	+	0.746553i
−0.720522	−	0.693433i
0.973695	−	0.227854i
0.114683	+	0.993402i

−0.973695

−

0.227854i

−0.997066

−

0.0765493i

0.0551959

−

1.43999i

0.896166

0.443720i

125.1

−0.859570

0.511019i

0.338017

−

0.941140i

1.55962

−

1.09701i

0.477720

−

0.878512i

191.1

0.720522

−

0.693433i

−0.264982

0.964253i

1.08656

1.42543i

0.0383027

−

0.999266i

227.1

−0.264982

0.964253i

0.817929

0.575319i

1.88445

0.596369i

−0.859570

−

0.511019i

293.1

−0.264982

−

0.964253i

0.817929

−

0.575319i

1.88445

−

0.596369i

−0.859570

0.511019i

383.1

0.190391

0.981708i

0.896166

0.443720i

0.223333

0.0522620i

−0.927502

0.373817i

401.1

−0.771489

0.636242i

−0.973695

0.227854i

0.152604

1.32187i

0.190391

−

0.981708i

416.1

−0.543568

0.839365i

0.190391

−

0.981708i

−0.521553

0.430121i

−0.409069

−

0.912504i

443.1

−0.927502

−

0.373817i

0.606225

−

0.795293i

−1.74518

0.864096i

0.720522

0.693433i

689.1

0.988280

−

0.152649i

−0.543568

−

0.839365i

−0.821267

1.51028i

0.953396

−

0.301721i

692.1

0.953396

0.301721i

−0.409069

−

0.912504i

−0.519346

0.801963i

0.817929

0.575319i

731.1

0.817929

0.575319i

−0.665326

0.746553i

0.444713

0.992015i

0.338017

0.941140i

776.1

−0.973695

0.227854i

−0.997066

0.0765493i

0.0551959

1.43999i

0.896166

−

0.443720i

803.1

0.338017

0.941140i

−0.114683

−

0.993402i

0.544328

−

0.610783i

−0.771489

0.636242i

845.1

0.477720

−

0.878512i

−0.771489

−

0.636242i

0.552948

−

1.53957i

−0.543568

−

0.839365i

869.1

−0.409069

−

0.912504i

−0.927502

−

0.373817i

−0.293769

1.51475i

−0.665326

0.746553i

872.1

0.988280

0.152649i

−0.543568

0.839365i

−0.821267

−

1.51028i

0.953396

0.301721i

890.1

−0.543568

−

0.839365i

0.190391

0.981708i

−0.521553

−

0.430121i

−0.409069

0.912504i

1016.1

0.817929

−

0.575319i

−0.665326

−

0.746553i

0.444713

−

0.992015i

0.338017

−

0.941140i

1022.1

0.953396

−

0.301721i

−0.409069

0.912504i

−0.519346

−

0.801963i

0.817929

−

0.575319i

See all 40 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
739.g	even	41	1	inner
2217.p	odd	82	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2217.1.p.a	✓	40
3.b	odd	2	1	CM	2217.1.p.a	✓	40
739.g	even	41	1	inner	2217.1.p.a	✓	40
2217.p	odd	82	1	inner	2217.1.p.a	✓	40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2217.1.p.a	✓	40	1.a	even	1	1	trivial
2217.1.p.a	✓	40	3.b	odd	2	1	CM
2217.1.p.a	✓	40	739.g	even	41	1	inner
2217.1.p.a	✓	40	2217.p	odd	82	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2217, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{40} $ T^40
$3$	$ T^{40} + T^{39} + \cdots + 1 $ T^40 + T^39 + T^38 + T^37 + T^36 + T^35 + T^34 + T^33 + T^32 + T^31 + T^30 + T^29 + T^28 + T^27 + T^26 + T^25 + T^24 + T^23 + T^22 + T^21 + T^20 + T^19 + T^18 + T^17 + T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	$ T^{40} $ T^40
$7$	$ T^{40} + 2 T^{39} + \cdots + 1 $ T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2048T^29 + 4096T^28 + 8192T^27 + 16384T^26 + 32481T^25 + 64962T^24 + 115000T^23 + 230000T^22 + 296902T^21 + 593804T^20 + 498972T^19 + 997944T^18 + 539158T^17 + 1078316T^16 + 382849T^15 + 765698T^14 + 182906T^13 + 365771T^12 + 59962T^11 + 123655T^10 + 22589T^9 + 15863T^8 - 19114T^7 + 8676T^6 + 9685T^5 - 146T^4 - 1030T^3 + 31T^2 + 21*T + 1
$11$	$ T^{40} $ T^40
$13$	$ T^{40} - 39 T^{39} + \cdots + 1 $ T^40 - 39T^39 + 742T^38 - 9176T^37 + 82918T^36 - 583562T^35 + 3329264T^34 - 15823412T^33 + 63901421T^32 - 222540723T^31 + 676017962T^30 - 1807426044T^29 + 4283802832T^28 - 9052470696T^27 + 17135211328T^26 - 29161852240T^25 + 44753742226T^24 - 62076995998T^23 + 77958648604T^22 - 88745372992T^21 + 91638191236T^20 - 85852554748T^19 + 72957560704T^18 - 56197519192T^17 + 39189442066T^16 - 24698562574T^15 + 14035149748T^14 - 7169853224T^13 + 3280369912T^12 - 1337915096T^11 + 483631776T^10 - 153835856T^9 + 42671853T^8 - 10204539T^7 + 2072862T^6 - 350664T^5 + 48070T^4 - 5130T^3 + 400T^2 - 20*T + 1
$17$	$ T^{40} $ T^40
$19$	$ T^{40} + 2 T^{39} + \cdots + 1 $ T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2048T^29 + 4096T^28 + 8192T^27 + 16384T^26 + 32481T^25 + 64962T^24 + 115000T^23 + 230000T^22 + 296902T^21 + 593804T^20 + 498972T^19 + 997944T^18 + 539158T^17 + 1078316T^16 + 382849T^15 + 765698T^14 + 182906T^13 + 365771T^12 + 59962T^11 + 123655T^10 + 22589T^9 + 15863T^8 - 19114T^7 + 8676T^6 + 9685T^5 - 146T^4 - 1030T^3 + 31T^2 + 21*T + 1
$23$	$ T^{40} $ T^40
$29$	$ T^{40} $ T^40
$31$	$ T^{40} + 2 T^{39} + \cdots + 1 $ T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 - 62T^31 - 165T^30 - 330T^29 - 660T^28 - 1320T^27 - 2640T^26 - 5280T^25 - 10560T^24 - 22760T^23 + 19547T^22 + 4080T^21 + 8611T^20 + 17222T^19 + 34444T^18 + 68888T^17 + 137776T^16 + 275060T^15 + 454180T^14 + 269170T^13 + 32810T^12 + 594T^11 + 122T^10 + 244T^9 + 488T^8 + 935T^7 + 7282T^6 - 15284T^5 + 9161T^4 - 2424T^3 + 318T^2 - 20*T + 1
$37$	$ T^{40} + 2 T^{39} + \cdots + 1 $ T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2048T^29 + 4096T^28 + 8192T^27 + 16343T^26 + 28955T^25 + 28595T^24 + 10286T^23 + 1056T^22 + 21T^21 + T^20 + 2T^19 + 4T^18 + 8T^17 + 16T^16 + 32T^15 + 64T^14 + 128T^13 + 543T^12 - 13838T^11 + 135422T^10 - 417792T^9 + 621146T^8 - 531491T^7 + 285508T^6 - 100564T^5 + 23593T^4 - 3654T^3 + 359T^2 - 20T + 1
$41$	$ T^{40} $ T^40
$43$	$ T^{40} + 2 T^{39} + \cdots + 1 $ T^40 + 2T^39 + 4T^38 + 8T^37 - 25T^36 - 50T^35 - 100T^34 - 200T^33 + 297T^32 + 594T^31 + 1188T^30 + 2376T^29 - 1603T^28 - 3206T^27 - 6412T^26 - 12865T^25 + 7849T^24 + 15698T^23 + 31396T^22 + 56109T^21 + 8488T^20 + 16976T^19 + 33952T^18 - 54768T^17 + 70413T^16 + 140826T^15 + 281652T^14 + 101275T^13 + 43634T^12 + 87268T^11 + 174782T^10 - 25668T^9 + 7294T^8 + 14588T^7 + 26470T^6 + 2879T^5 - 105T^4 - 210T^3 + 195T^2 - 20*T + 1
$47$	$ T^{40} $ T^40
$53$	$ T^{40} $ T^40
$59$	$ T^{40} $ T^40
$61$	$ T^{40} + 2 T^{39} + \cdots + 1 $ T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2007T^29 + 4014T^28 + 8028T^27 + 10193T^26 + 20386T^25 + 40772T^24 + 22914T^23 + 45828T^22 + 91656T^21 + 24396T^20 + 48792T^19 + 97994T^18 + 16039T^17 + 32078T^16 + 14095T^15 - 75540T^14 - 151080T^13 + 73072T^12 + 112565T^11 + 225130T^10 - 11769T^9 - 29893T^8 - 60401T^7 + 1870T^6 + 3043T^5 + 3380T^4 + 77T^3 + 113T^2 - 20*T + 1
$67$	$ T^{40} + 2 T^{39} + \cdots + 1 $ T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 - 77T^33 - 154T^32 - 308T^31 - 616T^30 - 1232T^29 - 2505T^28 - 5010T^27 + 2198T^26 + 4396T^25 + 8751T^24 + 17502T^23 + 35004T^22 + 53321T^21 + 106642T^20 + 6439T^19 + 12878T^18 + 18376T^17 + 37121T^16 + 74242T^15 - 135769T^14 - 271538T^13 + 19157T^12 + 38314T^11 + 29355T^10 + 38046T^9 + 76297T^8 + 14178T^7 + 28356T^6 + 1198T^5 + 2109T^4 - 415T^3 + 72T^2 + 21*T + 1
$71$	$ T^{40} $ T^40
$73$	$ T^{40} + 2 T^{39} + \cdots + 1 $ T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2048T^29 + 4096T^28 + 8192T^27 + 16384T^26 + 32481T^25 + 64962T^24 + 115000T^23 + 230000T^22 + 296902T^21 + 593804T^20 + 498972T^19 + 997944T^18 + 539158T^17 + 1078316T^16 + 382849T^15 + 765698T^14 + 182906T^13 + 365771T^12 + 59962T^11 + 123655T^10 + 22589T^9 + 15863T^8 - 19114T^7 + 8676T^6 + 9685T^5 - 146T^4 - 1030T^3 + 31T^2 + 21*T + 1
$79$	$ T^{40} + 2 T^{39} + \cdots + 1 $ T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 - 62T^31 - 165T^30 - 330T^29 - 660T^28 - 1320T^27 - 2640T^26 - 5280T^25 - 10560T^24 - 22760T^23 + 19547T^22 + 4080T^21 + 8611T^20 + 17222T^19 + 34444T^18 + 68888T^17 + 137776T^16 + 275060T^15 + 454180T^14 + 269170T^13 + 32810T^12 + 594T^11 + 122T^10 + 244T^9 + 488T^8 + 935T^7 + 7282T^6 - 15284T^5 + 9161T^4 - 2424T^3 + 318T^2 - 20*T + 1
$83$	$ T^{40} $ T^40
$89$	$ T^{40} $ T^40
$97$	$ T^{40} + 2 T^{39} + \cdots + 1 $ T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2048T^29 + 4096T^28 + 8192T^27 + 16343T^26 + 28955T^25 + 28595T^24 + 10286T^23 + 1056T^22 + 21T^21 + T^20 + 2T^19 + 4T^18 + 8T^17 + 16T^16 + 32T^15 + 64T^14 + 128T^13 + 543T^12 - 13838T^11 + 135422T^10 - 417792T^9 + 621146T^8 - 531491T^7 + 285508T^6 - 100564T^5 + 23593T^4 - 3654T^3 + 359T^2 - 20T + 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:	\( N \)	\(=\)	\( 2217 = 3 \cdot 739 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2217.p (of order \(82\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{82}^{30} q^{3} + \zeta_{82}^{10} q^{4} + (\zeta_{82}^{28} - \zeta_{82}^{23}) q^{7} - \zeta_{82}^{19} q^{9} +O(q^{10}) \) q + z^30 * q^3 + z^10 * q^4 + (z^28 - z^23) * q^7 - z^19 * q^9	\( q + \zeta_{82}^{30} q^{3} + \zeta_{82}^{10} q^{4} + (\zeta_{82}^{28} - \zeta_{82}^{23}) q^{7} - \zeta_{82}^{19} q^{9} + \zeta_{82}^{40} q^{12} + (\zeta_{82}^{14} + 1) q^{13} + \zeta_{82}^{20} q^{16} + (\zeta_{82}^{34} - \zeta_{82}^{25}) q^{19} + ( - \zeta_{82}^{17} + \zeta_{82}^{12}) q^{21} + \zeta_{82}^{22} q^{25} + \zeta_{82}^{8} q^{27} + (\zeta_{82}^{38} - \zeta_{82}^{33}) q^{28} + ( - \zeta_{82}^{29} + \zeta_{82}^{26}) q^{31} - \zeta_{82}^{29} q^{36} + (\zeta_{82}^{24} + \zeta_{82}^{16}) q^{37} + (\zeta_{82}^{30} - \zeta_{82}^{3}) q^{39} + ( - \zeta_{82}^{17} + \zeta_{82}^{8}) q^{43} - \zeta_{82}^{9} q^{48} + ( - \zeta_{82}^{15} + \cdots - \zeta_{82}^{5}) q^{49} + \cdots + (\zeta_{82}^{6} + \zeta_{82}^{4}) q^{97} +O(q^{100}) \) q + z^30 * q^3 + z^10 * q^4 + (z^28 - z^23) * q^7 - z^19 * q^9 + z^40 * q^12 + (z^14 + 1) * q^13 + z^20 * q^16 + (z^34 - z^25) * q^19 + (-z^17 + z^12) * q^21 + z^22 * q^25 + z^8 * q^27 + (z^38 - z^33) * q^28 + (-z^29 + z^26) * q^31 - z^29 * q^36 + (z^24 + z^16) * q^37 + (z^30 - z^3) * q^39 + (-z^17 + z^8) * q^43 - z^9 * q^48 + (-z^15 + z^10 - z^5) * q^49 + (z^24 + z^10) * q^52 + (-z^23 + z^14) * q^57 + (-z^27 + z^26) * q^61 + (z^6 - z) * q^63 + z^30 * q^64 + (-z^29 - z^9) * q^67 + (-z^25 + z^22) * q^73 - z^11 * q^75 + (-z^35 - z^3) * q^76 + (z^32 - z) * q^79 + z^38 * q^81 + (-z^27 + z^22) * q^84 + (-z^37 + z^28 - z^23 - z) * q^91 + (z^18 - z^15) * q^93 + (z^6 + z^4) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - q^{3} - q^{4} - 2 q^{7} - q^{9}+O(q^{10}) \) 40 * q - q^3 - q^4 - 2 * q^7 - q^9	\( 40 q - q^{3} - q^{4} - 2 q^{7} - q^{9} - q^{12} + 39 q^{13} - q^{16} - 2 q^{19} - 2 q^{21} - q^{25} - q^{27} - 2 q^{28} - 2 q^{31} - q^{36} - 2 q^{37} - 2 q^{39} - 2 q^{43} - q^{48} - 3 q^{49} - 2 q^{52} - 2 q^{57} - 2 q^{61} - 2 q^{63} - q^{64} - 2 q^{67} - 2 q^{73} - q^{75} - 2 q^{76} - 2 q^{79} - q^{81} - 2 q^{84} - 4 q^{91} - 2 q^{93} - 2 q^{97}+O(q^{100}) \) 40 * q - q^3 - q^4 - 2 * q^7 - q^9 - q^12 + 39 * q^13 - q^16 - 2 * q^19 - 2 * q^21 - q^25 - q^27 - 2 * q^28 - 2 * q^31 - q^36 - 2 * q^37 - 2 * q^39 - 2 * q^43 - q^48 - 3 * q^49 - 2 * q^52 - 2 * q^57 - 2 * q^61 - 2 * q^63 - q^64 - 2 * q^67 - 2 * q^73 - q^75 - 2 * q^76 - 2 * q^79 - q^81 - 2 * q^84 - 4 * q^91 - 2 * q^93 - 2 * q^97

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	2217.1.p.a

Level	$2217$
Weight	$1$
Character orbit	2217.p
Analytic conductor	$1.106$
Analytic rank	$0$
Dimension	$40$
Projective image	$D_{41}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.10642713301\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Coefficient field:	\(\Q(\zeta_{82})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{40} - x^{39} + x^{38} - x^{37} + x^{36} - x^{35} + x^{34} - x^{33} + x^{32} - x^{31} + x^{30} + \cdots + 1 \) x^40 - x^39 + x^38 - x^37 + x^36 - x^35 + x^34 - x^33 + x^32 - x^31 + x^30 - x^29 + x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{41}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{41} - \cdots)\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 20.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
739.g	even	41	1	inner
2217.p	odd	82	1	inner

$p$	$F_p(T)$
$2$	\( T^{40} \) T^40
$3$	\( T^{40} + T^{39} + \cdots + 1 \) T^40 + T^39 + T^38 + T^37 + T^36 + T^35 + T^34 + T^33 + T^32 + T^31 + T^30 + T^29 + T^28 + T^27 + T^26 + T^25 + T^24 + T^23 + T^22 + T^21 + T^20 + T^19 + T^18 + T^17 + T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	\( T^{40} \) T^40
$7$	\( T^{40} + 2 T^{39} + \cdots + 1 \) T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2048T^29 + 4096T^28 + 8192T^27 + 16384T^26 + 32481T^25 + 64962T^24 + 115000T^23 + 230000T^22 + 296902T^21 + 593804T^20 + 498972T^19 + 997944T^18 + 539158T^17 + 1078316T^16 + 382849T^15 + 765698T^14 + 182906T^13 + 365771T^12 + 59962T^11 + 123655T^10 + 22589T^9 + 15863T^8 - 19114T^7 + 8676T^6 + 9685T^5 - 146T^4 - 1030T^3 + 31T^2 + 21*T + 1
$11$	\( T^{40} \) T^40
$13$	\( T^{40} - 39 T^{39} + \cdots + 1 \) T^40 - 39T^39 + 742T^38 - 9176T^37 + 82918T^36 - 583562T^35 + 3329264T^34 - 15823412T^33 + 63901421T^32 - 222540723T^31 + 676017962T^30 - 1807426044T^29 + 4283802832T^28 - 9052470696T^27 + 17135211328T^26 - 29161852240T^25 + 44753742226T^24 - 62076995998T^23 + 77958648604T^22 - 88745372992T^21 + 91638191236T^20 - 85852554748T^19 + 72957560704T^18 - 56197519192T^17 + 39189442066T^16 - 24698562574T^15 + 14035149748T^14 - 7169853224T^13 + 3280369912T^12 - 1337915096T^11 + 483631776T^10 - 153835856T^9 + 42671853T^8 - 10204539T^7 + 2072862T^6 - 350664T^5 + 48070T^4 - 5130T^3 + 400T^2 - 20*T + 1
$17$	\( T^{40} \) T^40
$19$	\( T^{40} + 2 T^{39} + \cdots + 1 \) T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2048T^29 + 4096T^28 + 8192T^27 + 16384T^26 + 32481T^25 + 64962T^24 + 115000T^23 + 230000T^22 + 296902T^21 + 593804T^20 + 498972T^19 + 997944T^18 + 539158T^17 + 1078316T^16 + 382849T^15 + 765698T^14 + 182906T^13 + 365771T^12 + 59962T^11 + 123655T^10 + 22589T^9 + 15863T^8 - 19114T^7 + 8676T^6 + 9685T^5 - 146T^4 - 1030T^3 + 31T^2 + 21*T + 1
$23$	\( T^{40} \) T^40
$29$	\( T^{40} \) T^40
$31$	\( T^{40} + 2 T^{39} + \cdots + 1 \) T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 - 62T^31 - 165T^30 - 330T^29 - 660T^28 - 1320T^27 - 2640T^26 - 5280T^25 - 10560T^24 - 22760T^23 + 19547T^22 + 4080T^21 + 8611T^20 + 17222T^19 + 34444T^18 + 68888T^17 + 137776T^16 + 275060T^15 + 454180T^14 + 269170T^13 + 32810T^12 + 594T^11 + 122T^10 + 244T^9 + 488T^8 + 935T^7 + 7282T^6 - 15284T^5 + 9161T^4 - 2424T^3 + 318T^2 - 20*T + 1
$37$	\( T^{40} + 2 T^{39} + \cdots + 1 \) T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2048T^29 + 4096T^28 + 8192T^27 + 16343T^26 + 28955T^25 + 28595T^24 + 10286T^23 + 1056T^22 + 21T^21 + T^20 + 2T^19 + 4T^18 + 8T^17 + 16T^16 + 32T^15 + 64T^14 + 128T^13 + 543T^12 - 13838T^11 + 135422T^10 - 417792T^9 + 621146T^8 - 531491T^7 + 285508T^6 - 100564T^5 + 23593T^4 - 3654T^3 + 359T^2 - 20T + 1
$41$	\( T^{40} \) T^40
$43$	\( T^{40} + 2 T^{39} + \cdots + 1 \) T^40 + 2T^39 + 4T^38 + 8T^37 - 25T^36 - 50T^35 - 100T^34 - 200T^33 + 297T^32 + 594T^31 + 1188T^30 + 2376T^29 - 1603T^28 - 3206T^27 - 6412T^26 - 12865T^25 + 7849T^24 + 15698T^23 + 31396T^22 + 56109T^21 + 8488T^20 + 16976T^19 + 33952T^18 - 54768T^17 + 70413T^16 + 140826T^15 + 281652T^14 + 101275T^13 + 43634T^12 + 87268T^11 + 174782T^10 - 25668T^9 + 7294T^8 + 14588T^7 + 26470T^6 + 2879T^5 - 105T^4 - 210T^3 + 195T^2 - 20*T + 1
$47$	\( T^{40} \) T^40
$53$	\( T^{40} \) T^40
$59$	\( T^{40} \) T^40
$61$	\( T^{40} + 2 T^{39} + \cdots + 1 \) T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2007T^29 + 4014T^28 + 8028T^27 + 10193T^26 + 20386T^25 + 40772T^24 + 22914T^23 + 45828T^22 + 91656T^21 + 24396T^20 + 48792T^19 + 97994T^18 + 16039T^17 + 32078T^16 + 14095T^15 - 75540T^14 - 151080T^13 + 73072T^12 + 112565T^11 + 225130T^10 - 11769T^9 - 29893T^8 - 60401T^7 + 1870T^6 + 3043T^5 + 3380T^4 + 77T^3 + 113T^2 - 20*T + 1
$67$	\( T^{40} + 2 T^{39} + \cdots + 1 \) T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 - 77T^33 - 154T^32 - 308T^31 - 616T^30 - 1232T^29 - 2505T^28 - 5010T^27 + 2198T^26 + 4396T^25 + 8751T^24 + 17502T^23 + 35004T^22 + 53321T^21 + 106642T^20 + 6439T^19 + 12878T^18 + 18376T^17 + 37121T^16 + 74242T^15 - 135769T^14 - 271538T^13 + 19157T^12 + 38314T^11 + 29355T^10 + 38046T^9 + 76297T^8 + 14178T^7 + 28356T^6 + 1198T^5 + 2109T^4 - 415T^3 + 72T^2 + 21*T + 1
$71$	\( T^{40} \) T^40
$73$	\( T^{40} + 2 T^{39} + \cdots + 1 \) T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2048T^29 + 4096T^28 + 8192T^27 + 16384T^26 + 32481T^25 + 64962T^24 + 115000T^23 + 230000T^22 + 296902T^21 + 593804T^20 + 498972T^19 + 997944T^18 + 539158T^17 + 1078316T^16 + 382849T^15 + 765698T^14 + 182906T^13 + 365771T^12 + 59962T^11 + 123655T^10 + 22589T^9 + 15863T^8 - 19114T^7 + 8676T^6 + 9685T^5 - 146T^4 - 1030T^3 + 31T^2 + 21*T + 1
$79$	\( T^{40} + 2 T^{39} + \cdots + 1 \) T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 - 62T^31 - 165T^30 - 330T^29 - 660T^28 - 1320T^27 - 2640T^26 - 5280T^25 - 10560T^24 - 22760T^23 + 19547T^22 + 4080T^21 + 8611T^20 + 17222T^19 + 34444T^18 + 68888T^17 + 137776T^16 + 275060T^15 + 454180T^14 + 269170T^13 + 32810T^12 + 594T^11 + 122T^10 + 244T^9 + 488T^8 + 935T^7 + 7282T^6 - 15284T^5 + 9161T^4 - 2424T^3 + 318T^2 - 20*T + 1
$83$	\( T^{40} \) T^40
$89$	\( T^{40} \) T^40
$97$	\( T^{40} + 2 T^{39} + \cdots + 1 \) T^40 + 2T^39 + 4T^38 + 8T^37 + 16T^36 + 32T^35 + 64T^34 + 128T^33 + 256T^32 + 512T^31 + 1024T^30 + 2048T^29 + 4096T^28 + 8192T^27 + 16343T^26 + 28955T^25 + 28595T^24 + 10286T^23 + 1056T^22 + 21T^21 + T^20 + 2T^19 + 4T^18 + 8T^17 + 16T^16 + 32T^15 + 64T^14 + 128T^13 + 543T^12 - 13838T^11 + 135422T^10 - 417792T^9 + 621146T^8 - 531491T^7 + 285508T^6 - 100564T^5 + 23593T^4 - 3654T^3 + 359T^2 - 20T + 1
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\(n\)	\(740\)	\(742\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{82}^{19}\)