LMFDB - Newform orbit 847.1.bb.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,1,Mod(20,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(110))

chi = DirichletCharacter(H, H._module([55, 76]))

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

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

chi := DirichletCharacter("847.20");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	847.bb (of order $110$, 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:	$0.422708065700$
Analytic rank:	$0$
Dimension:	$40$
Coefficient field:	$\Q(\zeta_{55})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{39} + x^{35} - x^{34} + x^{30} - x^{28} + x^{25} - x^{23} + x^{20} - x^{17} + x^{15} + \cdots + 1 $ x^40 - x^39 + x^35 - x^34 + x^30 - x^28 + x^25 - x^23 + x^20 - x^17 + x^15 - x^12 + x^10 - x^6 + x^5 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{55}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{55} - \cdots)$

$q$-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_{110}^{49} + \zeta_{110}^{8}) q^{2} + ( - \zeta_{110}^{43} + \cdots + \zeta_{110}^{2}) q^{4}+ \cdots - \zeta_{110}^{11} q^{9} +O(q^{10}) $ q + (-z^49 + z^8) * q^2 + (-z^43 + z^16 + z^2) * q^4 - z^7 * q^7 + (-z^51 - z^37 + z^24 + z^10) * q^8 - z^11 * q^9	$ q + ( - \zeta_{110}^{49} + \zeta_{110}^{8}) q^{2} + ( - \zeta_{110}^{43} + \cdots + \zeta_{110}^{2}) q^{4}+ \cdots - \zeta_{110}^{45} q^{99} +O(q^{100}) $ q + (-z^49 + z^8) * q^2 + (-z^43 + z^16 + z^2) * q^4 - z^7 * q^7 + (-z^51 - z^37 + z^24 + z^10) * q^8 - z^11 * q^9 + z^34 * q^11 + (-z^15 - z) * q^14 + (-z^45 + z^32 - z^31 + z^18 + z^4) * q^16 + (-z^19 - z^5) * q^18 + (z^42 + z^28) * q^22 + (z^52 - z^33) * q^23 + z^38 * q^25 + (z^50 - z^23 - z^9) * q^28 + (-z^53 + z^36) * q^29 + (-z^53 + z^40 - z^39 + z^26 - z^25 + z^12) * q^32 + (z^54 - z^27 - z^13) * q^36 + (z^20 - z^9) * q^37 + (-z^29 - z^21) * q^43 + (z^50 + z^36 + z^22) * q^44 + (z^46 - z^41 - z^27 - z^5) * q^46 + z^14 * q^49 + (z^46 + z^32) * q^50 + (-z^35 + z^12) * q^53 + (z^44 - z^31 - z^17 - z^3) * q^56 + (-z^47 + z^44 + z^30 + z^6) * q^58 + z^18 * q^63 + (z^48 - z^47 + z^34 - z^33 + z^20 - z^19 + z^6) * q^64 + (-z^51 - z^39) * q^67 + (z^48 + z^30) * q^71 + (z^48 - z^35 - z^21 - z^7) * q^72 + (z^28 - z^17 + z^14 - z^3) * q^74 - z^41 * q^77 + (-z^17 + z^10) * q^79 + z^22 * q^81 + (-z^37 - z^29 - z^23 - z^15) * q^86 + (z^44 + z^30 + z^16 - z^3) * q^88 + (z^54 - z^49 + z^40 - z^35 - z^21 - z^13) * q^92 + (z^22 + z^8) * q^98 - z^45 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 2 q^{2} + 3 q^{4} + q^{7} - q^{8} - 10 q^{9}+O(q^{10}) $ 40 * q + 2 * q^2 + 3 * q^4 + q^7 - q^8 - 10 * q^9	$ 40 q + 2 q^{2} + 3 q^{4} + q^{7} - q^{8} - 10 q^{9} + q^{11} - 3 q^{14} - 3 q^{18} + 2 q^{22} - 9 q^{23} + q^{25} - 2 q^{28} + 2 q^{29} - 4 q^{32} + 3 q^{36} - 3 q^{37} + 2 q^{43} - 13 q^{44} - q^{46} + q^{49} + 2 q^{50} - 3 q^{53} - 7 q^{56} - 12 q^{58} + q^{63} - 9 q^{64} + 2 q^{67} - 3 q^{71} - q^{72} + 4 q^{74} + q^{77} - 3 q^{79} - 10 q^{81} - q^{86} - 12 q^{88} - 4 q^{92} - 9 q^{98} - 4 q^{99}+O(q^{100}) $ 40 * q + 2 * q^2 + 3 * q^4 + q^7 - q^8 - 10 * q^9 + q^11 - 3 * q^14 - 3 * q^18 + 2 * q^22 - 9 * q^23 + q^25 - 2 * q^28 + 2 * q^29 - 4 * q^32 + 3 * q^36 - 3 * q^37 + 2 * q^43 - 13 * q^44 - q^46 + q^49 + 2 * q^50 - 3 * q^53 - 7 * q^56 - 12 * q^58 + q^63 - 9 * q^64 + 2 * q^67 - 3 * q^71 - q^72 + 4 * q^74 + q^77 - 3 * q^79 - 10 * q^81 - q^86 - 12 * q^88 - 4 * q^92 - 9 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/847\mathbb{Z}\right)^\times$.

$n$	$122$	$365$
$\chi(n)$	$-1$	$\zeta_{110}^{2}$

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

−0.564443	+	0.825472i
−0.870746	+	0.491733i
0.198590	+	0.980083i
0.516397	+	0.856349i
−0.985354	+	0.170522i
−0.998369	−	0.0570888i
−0.466667	−	0.884433i
0.993482	−	0.113991i
−0.0285561	−	0.999592i
0.696938	+	0.717132i
0.993482	+	0.113991i
−0.362808	−	0.931864i
−0.870746	−	0.491733i
−0.736741	−	0.676175i
−0.254218	+	0.967147i
0.198590	−	0.980083i
0.974012	+	0.226497i
−0.921124	+	0.389270i
−0.921124	−	0.389270i
0.696938	−	0.717132i

0.982973

−

1.43755i

−0.737514

−

1.89429i

−0.870746

0.491733i

−1.75186

−

0.407378i

0.309017

−

0.951057i

48.1

−1.56281

0.882561i

1.14707

−

1.90220i

0.897398

−

0.441221i

−0.0625942

2.19108i

−0.809017

−

0.587785i

69.1

−0.391364

−

1.93146i

−2.65623

1.12253i

−0.985354

−

0.170522i

2.09533

3.06432i

−0.809017

0.587785i

97.1

0.630674

1.04586i

−0.229397

0.434755i

0.610648

0.791902i

0.619939

−

0.0354494i

0.309017

−

0.951057i

104.1

0.714988

−

0.123734i

−0.445947

0.159113i

−0.362808

0.931864i

−0.930986

0.525752i

0.309017

0.951057i

125.1

1.83924

0.105172i

2.37827

0.272881i

−0.921124

−

0.389270i

2.53025

0.437878i

−0.809017

−

0.587785i

146.1

0.237271

0.449678i

0.418530

−

0.612081i

−0.254218

−

0.967147i

0.879667

0.100932i

−0.809017

0.587785i

174.1

1.38479

−

0.158890i

0.918384

−

0.213561i

0.696938

−

0.717132i

−0.0749775

0.0267519i

0.309017

−

0.951057i

181.1

−0.0113419

−

0.397019i

0.840874

−

0.0480829i

0.198590

0.980083i

−0.0626157

−

0.729021i

0.309017

0.951057i

223.1

1.07906

1.11032i

−0.0398980

1.39661i

0.774142

−

0.633012i

−0.453058

0.415813i

−0.809017

0.587785i

258.1

1.38479

0.158890i

0.918384

0.213561i

0.696938

0.717132i

−0.0749775

−

0.0267519i

0.309017

0.951057i

279.1

−0.374706

−

0.962424i

−0.0491151

0.0450774i

0.516397

−

0.856349i

−0.865041

−

0.425312i

−0.809017

−

0.587785i

300.1

−1.56281

−

0.882561i

1.14707

1.90220i

0.897398

0.441221i

−0.0625942

−

2.19108i

−0.809017

0.587785i

328.1

0.687626

0.631097i

−0.0110295

−

0.128414i

−0.466667

0.884433i

0.643396

−

0.834371i

0.309017

−

0.951057i

335.1

−0.495223

1.88403i

−2.43356

−

1.37429i

0.974012

0.226497i

2.43671

−

2.50731i

0.309017

0.951057i

356.1

−0.391364

1.93146i

−2.65623

−

1.12253i

−0.985354

0.170522i

2.09533

−

3.06432i

−0.809017

−

0.587785i

377.1

−0.0556279

−

0.0129357i

−0.894471

−

0.439782i

−0.0285561

0.999592i

0.0882815

0.0721874i

−0.809017

0.587785i

405.1

−1.73511

0.733264i

1.77599

−

1.82745i

0.941844

0.336049i

−1.05812

2.71777i

0.309017

−

0.951057i

412.1

−1.73511

−

0.733264i

1.77599

1.82745i

0.941844

−

0.336049i

−1.05812

−

2.71777i

0.309017

0.951057i

433.1

1.07906

−

1.11032i

−0.0398980

−

1.39661i

0.774142

0.633012i

−0.453058

−

0.415813i

−0.809017

−

0.587785i

See all 40 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
121.g	even	55	1	inner
847.bb	odd	110	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	847.1.bb.a	✓	40
7.b	odd	2	1	CM	847.1.bb.a	✓	40
121.g	even	55	1	inner	847.1.bb.a	✓	40
847.bb	odd	110	1	inner	847.1.bb.a	✓	40

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
847.1.bb.a	✓	40	1.a	even	1	1	trivial
847.1.bb.a	✓	40	7.b	odd	2	1	CM
847.1.bb.a	✓	40	121.g	even	55	1	inner
847.1.bb.a	✓	40	847.bb	odd	110	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(847, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{40} - 2 T^{39} + \cdots + 1 $ T^40 - 2T^39 + 5T^37 - 5T^36 - 8T^35 + 32T^34 - 242T^33 + 462T^32 - 1143T^30 + 1144T^29 + 1943T^28 - 6848T^27 + 19244T^26 - 29806T^25 + 148T^24 + 73049T^23 - 71885T^22 - 133903T^21 + 417539T^20 - 553509T^19 + 376310T^18 + 100581T^17 - 569410T^16 + 492131T^15 + 616777T^14 - 2361690T^13 + 4078224T^12 - 5203979T^11 + 5270750T^10 - 4133459T^9 + 2457763T^8 - 1124882T^7 + 395339T^6 - 103189T^5 + 19250T^4 - 1237T^3 - 94T^2 + 19*T + 1
$3$	$ T^{40} $ T^40
$5$	$ T^{40} $ T^40
$7$	$ T^{40} - T^{39} + \cdots + 1 $ T^40 - T^39 + T^35 - T^34 + T^30 - T^28 + T^25 - T^23 + T^20 - T^17 + T^15 - T^12 + T^10 - T^6 + T^5 - T + 1
$11$	$ T^{40} - T^{39} + \cdots + 1 $ T^40 - T^39 + T^35 - T^34 + T^30 - T^28 + T^25 - T^23 + T^20 - T^17 + T^15 - T^12 + T^10 - T^6 + T^5 - T + 1
$13$	$ T^{40} $ T^40
$17$	$ T^{40} $ T^40
$19$	$ T^{40} $ T^40
$23$	$ T^{40} + 9 T^{39} + \cdots + 1 $ T^40 + 9T^39 + 49T^38 + 199T^37 + 669T^36 + 1942T^35 + 5021T^34 + 11761T^33 + 25326T^32 + 50576T^31 + 94453T^30 + 165795T^29 + 275004T^28 + 432223T^27 + 646242T^26 + 922316T^25 + 1272250T^24 + 1711174T^23 + 2253208T^22 + 2796292T^21 + 3048692T^20 + 2572335T^19 + 1193424T^18 - 484553T^17 - 1129006T^16 + 435051T^15 + 4101716T^14 + 8153640T^13 + 10574381T^12 + 10335729T^11 + 7972785T^10 + 4829959T^9 + 2284418T^8 + 869911T^7 + 296477T^6 + 88236T^5 + 17895T^4 + 1220T^3 - 50T^2 - 20*T + 1
$29$	$ T^{40} - 2 T^{39} + \cdots + 1 $ T^40 - 2T^39 + 5T^37 - 5T^36 + 14T^35 - 23T^34 + 33T^33 - 66T^31 + 529T^30 - 825T^29 + 1184T^28 - 105T^27 - 1007T^26 + 10597T^25 - 17100T^24 + 42370T^23 - 31141T^22 + 21098T^21 + 238723T^20 - 431761T^19 + 1286010T^18 - 1651334T^17 + 2821230T^16 - 2851803T^15 + 3424835T^14 - 2695089T^13 + 2325396T^12 - 1285328T^11 + 723779T^10 - 115302T^9 - 77858T^8 + 110253T^7 - 1629T^6 - 22933T^5 + 6754T^4 + 2503T^3 - 28T^2 - 25*T + 1
$31$	$ T^{40} $ T^40
$37$	$ T^{40} + 3 T^{39} + \cdots + 1 $ T^40 + 3T^39 + 5T^38 + 5T^37 - 13T^35 - 34T^34 - 55T^33 - 55T^32 + 144T^30 + 55T^29 - 356T^28 - 1000T^27 - 1610T^26 - 1597T^25 + 5T^24 + 4183T^23 + 10945T^22 + 17710T^21 + 17711T^20 - 6765T^18 - 2583T^17 + 1600T^16 + 1602T^15 + 5T^14 - 610T^13 - 246T^12 + 110T^11 + 89T^10 - 55T^9 - 55T^8 + 21T^6 + 8T^5 - 5T^4 - 5T^3 + 2T + 1
$41$	$ T^{40} $ T^40
$43$	$ T^{40} - 2 T^{39} + \cdots + 1 $ T^40 - 2T^39 + 5T^38 - 10T^37 + 20T^36 - 16T^35 + 27T^34 + 13T^33 - 40T^32 + 174T^31 + 84T^30 - 85T^29 + 2094T^28 - 3795T^27 + 11003T^26 - 10638T^25 + 21605T^24 + 19385T^23 - 33461T^22 + 81943T^21 + 86183T^20 - 124381T^19 + 178245T^18 + 63181T^17 + 94095T^16 - 53723T^15 + 110575T^14 - 137104T^13 + 532206T^12 - 135358T^11 + 11909T^10 - 315137T^9 + 36832T^8 + 103123T^7 + 138616T^6 + 62232T^5 + 21184T^4 + 3398T^3 + 522T^2 + 35*T + 1
$47$	$ T^{40} $ T^40
$53$	$ T^{40} + 3 T^{39} + \cdots + 1 $ T^40 + 3T^39 + 5T^38 + 5T^37 - 13T^35 - 23T^34 + 33T^33 - 143T^32 - 550T^31 - 978T^30 - 1056T^29 - 92T^28 + 3752T^27 + 10974T^26 + 7049T^25 + 12798T^24 + 19429T^23 + 10890T^22 - 38148T^21 - 50676T^20 - 1144T^19 - 134145T^18 + 108726T^17 + 159615T^16 + 96961T^15 + 198467T^14 - 525860T^13 + 465604T^12 - 362219T^11 + 507750T^10 - 225709T^9 + 48323T^8 + 97603T^7 - 12651T^6 - 29659T^5 + 9290T^4 - 357T^3 + 286T^2 - 31T + 1
$59$	$ T^{40} $ T^40
$61$	$ T^{40} $ T^40
$67$	$ T^{40} - 2 T^{39} + \cdots + 1 $ T^40 - 2T^39 + 5T^38 - 10T^37 + 20T^36 - 16T^35 + 27T^34 + 13T^33 - 40T^32 + 174T^31 + 84T^30 - 85T^29 + 2094T^28 - 3795T^27 + 11003T^26 - 10638T^25 + 21605T^24 + 19385T^23 - 33461T^22 + 81943T^21 + 86183T^20 - 124381T^19 + 178245T^18 + 63181T^17 + 94095T^16 - 53723T^15 + 110575T^14 - 137104T^13 + 532206T^12 - 135358T^11 + 11909T^10 - 315137T^9 + 36832T^8 + 103123T^7 + 138616T^6 + 62232T^5 + 21184T^4 + 3398T^3 + 522T^2 + 35*T + 1
$71$	$ T^{40} + 3 T^{39} + \cdots + 1 $ T^40 + 3T^39 + 5T^38 + 5T^37 - 13T^35 - 23T^34 + 33T^33 - 143T^32 - 550T^31 - 978T^30 - 1056T^29 - 92T^28 + 3752T^27 + 10974T^26 + 7049T^25 + 12798T^24 + 19429T^23 + 10890T^22 - 38148T^21 - 50676T^20 - 1144T^19 - 134145T^18 + 108726T^17 + 159615T^16 + 96961T^15 + 198467T^14 - 525860T^13 + 465604T^12 - 362219T^11 + 507750T^10 - 225709T^9 + 48323T^8 + 97603T^7 - 12651T^6 - 29659T^5 + 9290T^4 - 357T^3 + 286T^2 - 31T + 1
$73$	$ T^{40} $ T^40
$79$	$ T^{40} + 3 T^{39} + \cdots + 1 $ T^40 + 3T^39 + 5T^38 + 16T^37 + 33T^36 + 42T^35 + 109T^34 + 220T^33 + 275T^32 + 748T^31 + 1728T^30 + 2519T^29 + 6101T^28 + 8999T^27 + 6013T^26 + 8545T^25 - 8058T^24 - 51378T^23 - 51788T^22 - 88495T^21 - 130547T^20 + 99374T^19 + 341066T^18 + 322786T^17 + 407346T^16 + 741396T^15 + 996352T^14 + 1030717T^13 + 1012018T^12 + 996875T^11 + 719115T^10 + 326865T^9 + 221661T^8 - 28094T^7 + 33813T^6 - 5987T^5 + 9950T^4 - 2095T^3 - 33T^2 + 24*T + 1
$83$	$ T^{40} $ T^40
$89$	$ T^{40} $ T^40
$97$	$ T^{40} $ T^40
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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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	847.bb (of order \(110\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{110}^{49} + \zeta_{110}^{8}) q^{2} + ( - \zeta_{110}^{43} + \cdots + \zeta_{110}^{2}) q^{4}+ \cdots - \zeta_{110}^{11} q^{9} +O(q^{10}) \) q + (-z^49 + z^8) * q^2 + (-z^43 + z^16 + z^2) * q^4 - z^7 * q^7 + (-z^51 - z^37 + z^24 + z^10) * q^8 - z^11 * q^9	\( q + ( - \zeta_{110}^{49} + \zeta_{110}^{8}) q^{2} + ( - \zeta_{110}^{43} + \cdots + \zeta_{110}^{2}) q^{4}+ \cdots - \zeta_{110}^{45} q^{99} +O(q^{100}) \) q + (-z^49 + z^8) * q^2 + (-z^43 + z^16 + z^2) * q^4 - z^7 * q^7 + (-z^51 - z^37 + z^24 + z^10) * q^8 - z^11 * q^9 + z^34 * q^11 + (-z^15 - z) * q^14 + (-z^45 + z^32 - z^31 + z^18 + z^4) * q^16 + (-z^19 - z^5) * q^18 + (z^42 + z^28) * q^22 + (z^52 - z^33) * q^23 + z^38 * q^25 + (z^50 - z^23 - z^9) * q^28 + (-z^53 + z^36) * q^29 + (-z^53 + z^40 - z^39 + z^26 - z^25 + z^12) * q^32 + (z^54 - z^27 - z^13) * q^36 + (z^20 - z^9) * q^37 + (-z^29 - z^21) * q^43 + (z^50 + z^36 + z^22) * q^44 + (z^46 - z^41 - z^27 - z^5) * q^46 + z^14 * q^49 + (z^46 + z^32) * q^50 + (-z^35 + z^12) * q^53 + (z^44 - z^31 - z^17 - z^3) * q^56 + (-z^47 + z^44 + z^30 + z^6) * q^58 + z^18 * q^63 + (z^48 - z^47 + z^34 - z^33 + z^20 - z^19 + z^6) * q^64 + (-z^51 - z^39) * q^67 + (z^48 + z^30) * q^71 + (z^48 - z^35 - z^21 - z^7) * q^72 + (z^28 - z^17 + z^14 - z^3) * q^74 - z^41 * q^77 + (-z^17 + z^10) * q^79 + z^22 * q^81 + (-z^37 - z^29 - z^23 - z^15) * q^86 + (z^44 + z^30 + z^16 - z^3) * q^88 + (z^54 - z^49 + z^40 - z^35 - z^21 - z^13) * q^92 + (z^22 + z^8) * q^98 - z^45 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 2 q^{2} + 3 q^{4} + q^{7} - q^{8} - 10 q^{9}+O(q^{10}) \) 40 * q + 2 * q^2 + 3 * q^4 + q^7 - q^8 - 10 * q^9	\( 40 q + 2 q^{2} + 3 q^{4} + q^{7} - q^{8} - 10 q^{9} + q^{11} - 3 q^{14} - 3 q^{18} + 2 q^{22} - 9 q^{23} + q^{25} - 2 q^{28} + 2 q^{29} - 4 q^{32} + 3 q^{36} - 3 q^{37} + 2 q^{43} - 13 q^{44} - q^{46} + q^{49} + 2 q^{50} - 3 q^{53} - 7 q^{56} - 12 q^{58} + q^{63} - 9 q^{64} + 2 q^{67} - 3 q^{71} - q^{72} + 4 q^{74} + q^{77} - 3 q^{79} - 10 q^{81} - q^{86} - 12 q^{88} - 4 q^{92} - 9 q^{98} - 4 q^{99}+O(q^{100}) \) 40 * q + 2 * q^2 + 3 * q^4 + q^7 - q^8 - 10 * q^9 + q^11 - 3 * q^14 - 3 * q^18 + 2 * q^22 - 9 * q^23 + q^25 - 2 * q^28 + 2 * q^29 - 4 * q^32 + 3 * q^36 - 3 * q^37 + 2 * q^43 - 13 * q^44 - q^46 + q^49 + 2 * q^50 - 3 * q^53 - 7 * q^56 - 12 * q^58 + q^63 - 9 * q^64 + 2 * q^67 - 3 * q^71 - q^72 + 4 * q^74 + q^77 - 3 * q^79 - 10 * q^81 - q^86 - 12 * q^88 - 4 * q^92 - 9 * q^98 - 4 * q^99

Introduction

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Modular forms

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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	847.1.bb.a

Level	$847$
Weight	$1$
Character orbit	847.bb
Analytic conductor	$0.423$
Analytic rank	$0$
Dimension	$40$
Projective image	$D_{55}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.422708065700\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Coefficient field:	\(\Q(\zeta_{55})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{40} - x^{39} + x^{35} - x^{34} + x^{30} - x^{28} + x^{25} - x^{23} + x^{20} - x^{17} + x^{15} + \cdots + 1 \) x^40 - x^39 + x^35 - x^34 + x^30 - x^28 + x^25 - x^23 + x^20 - x^17 + x^15 - x^12 + x^10 - x^6 + x^5 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{55}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{55} - \cdots)\)

\(n\):		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
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
121.g	even	55	1	inner
847.bb	odd	110	1	inner

$p$	$F_p(T)$
$2$	\( T^{40} - 2 T^{39} + \cdots + 1 \) T^40 - 2T^39 + 5T^37 - 5T^36 - 8T^35 + 32T^34 - 242T^33 + 462T^32 - 1143T^30 + 1144T^29 + 1943T^28 - 6848T^27 + 19244T^26 - 29806T^25 + 148T^24 + 73049T^23 - 71885T^22 - 133903T^21 + 417539T^20 - 553509T^19 + 376310T^18 + 100581T^17 - 569410T^16 + 492131T^15 + 616777T^14 - 2361690T^13 + 4078224T^12 - 5203979T^11 + 5270750T^10 - 4133459T^9 + 2457763T^8 - 1124882T^7 + 395339T^6 - 103189T^5 + 19250T^4 - 1237T^3 - 94T^2 + 19*T + 1
$3$	\( T^{40} \) T^40
$5$	\( T^{40} \) T^40
$7$	\( T^{40} - T^{39} + \cdots + 1 \) T^40 - T^39 + T^35 - T^34 + T^30 - T^28 + T^25 - T^23 + T^20 - T^17 + T^15 - T^12 + T^10 - T^6 + T^5 - T + 1
$11$	\( T^{40} - T^{39} + \cdots + 1 \) T^40 - T^39 + T^35 - T^34 + T^30 - T^28 + T^25 - T^23 + T^20 - T^17 + T^15 - T^12 + T^10 - T^6 + T^5 - T + 1
$13$	\( T^{40} \) T^40
$17$	\( T^{40} \) T^40
$19$	\( T^{40} \) T^40
$23$	\( T^{40} + 9 T^{39} + \cdots + 1 \) T^40 + 9T^39 + 49T^38 + 199T^37 + 669T^36 + 1942T^35 + 5021T^34 + 11761T^33 + 25326T^32 + 50576T^31 + 94453T^30 + 165795T^29 + 275004T^28 + 432223T^27 + 646242T^26 + 922316T^25 + 1272250T^24 + 1711174T^23 + 2253208T^22 + 2796292T^21 + 3048692T^20 + 2572335T^19 + 1193424T^18 - 484553T^17 - 1129006T^16 + 435051T^15 + 4101716T^14 + 8153640T^13 + 10574381T^12 + 10335729T^11 + 7972785T^10 + 4829959T^9 + 2284418T^8 + 869911T^7 + 296477T^6 + 88236T^5 + 17895T^4 + 1220T^3 - 50T^2 - 20*T + 1
$29$	\( T^{40} - 2 T^{39} + \cdots + 1 \) T^40 - 2T^39 + 5T^37 - 5T^36 + 14T^35 - 23T^34 + 33T^33 - 66T^31 + 529T^30 - 825T^29 + 1184T^28 - 105T^27 - 1007T^26 + 10597T^25 - 17100T^24 + 42370T^23 - 31141T^22 + 21098T^21 + 238723T^20 - 431761T^19 + 1286010T^18 - 1651334T^17 + 2821230T^16 - 2851803T^15 + 3424835T^14 - 2695089T^13 + 2325396T^12 - 1285328T^11 + 723779T^10 - 115302T^9 - 77858T^8 + 110253T^7 - 1629T^6 - 22933T^5 + 6754T^4 + 2503T^3 - 28T^2 - 25*T + 1
$31$	\( T^{40} \) T^40
$37$	\( T^{40} + 3 T^{39} + \cdots + 1 \) T^40 + 3T^39 + 5T^38 + 5T^37 - 13T^35 - 34T^34 - 55T^33 - 55T^32 + 144T^30 + 55T^29 - 356T^28 - 1000T^27 - 1610T^26 - 1597T^25 + 5T^24 + 4183T^23 + 10945T^22 + 17710T^21 + 17711T^20 - 6765T^18 - 2583T^17 + 1600T^16 + 1602T^15 + 5T^14 - 610T^13 - 246T^12 + 110T^11 + 89T^10 - 55T^9 - 55T^8 + 21T^6 + 8T^5 - 5T^4 - 5T^3 + 2T + 1
$41$	\( T^{40} \) T^40
$43$	\( T^{40} - 2 T^{39} + \cdots + 1 \) T^40 - 2T^39 + 5T^38 - 10T^37 + 20T^36 - 16T^35 + 27T^34 + 13T^33 - 40T^32 + 174T^31 + 84T^30 - 85T^29 + 2094T^28 - 3795T^27 + 11003T^26 - 10638T^25 + 21605T^24 + 19385T^23 - 33461T^22 + 81943T^21 + 86183T^20 - 124381T^19 + 178245T^18 + 63181T^17 + 94095T^16 - 53723T^15 + 110575T^14 - 137104T^13 + 532206T^12 - 135358T^11 + 11909T^10 - 315137T^9 + 36832T^8 + 103123T^7 + 138616T^6 + 62232T^5 + 21184T^4 + 3398T^3 + 522T^2 + 35*T + 1
$47$	\( T^{40} \) T^40
$53$	\( T^{40} + 3 T^{39} + \cdots + 1 \) T^40 + 3T^39 + 5T^38 + 5T^37 - 13T^35 - 23T^34 + 33T^33 - 143T^32 - 550T^31 - 978T^30 - 1056T^29 - 92T^28 + 3752T^27 + 10974T^26 + 7049T^25 + 12798T^24 + 19429T^23 + 10890T^22 - 38148T^21 - 50676T^20 - 1144T^19 - 134145T^18 + 108726T^17 + 159615T^16 + 96961T^15 + 198467T^14 - 525860T^13 + 465604T^12 - 362219T^11 + 507750T^10 - 225709T^9 + 48323T^8 + 97603T^7 - 12651T^6 - 29659T^5 + 9290T^4 - 357T^3 + 286T^2 - 31T + 1
$59$	\( T^{40} \) T^40
$61$	\( T^{40} \) T^40
$67$	\( T^{40} - 2 T^{39} + \cdots + 1 \) T^40 - 2T^39 + 5T^38 - 10T^37 + 20T^36 - 16T^35 + 27T^34 + 13T^33 - 40T^32 + 174T^31 + 84T^30 - 85T^29 + 2094T^28 - 3795T^27 + 11003T^26 - 10638T^25 + 21605T^24 + 19385T^23 - 33461T^22 + 81943T^21 + 86183T^20 - 124381T^19 + 178245T^18 + 63181T^17 + 94095T^16 - 53723T^15 + 110575T^14 - 137104T^13 + 532206T^12 - 135358T^11 + 11909T^10 - 315137T^9 + 36832T^8 + 103123T^7 + 138616T^6 + 62232T^5 + 21184T^4 + 3398T^3 + 522T^2 + 35*T + 1
$71$	\( T^{40} + 3 T^{39} + \cdots + 1 \) T^40 + 3T^39 + 5T^38 + 5T^37 - 13T^35 - 23T^34 + 33T^33 - 143T^32 - 550T^31 - 978T^30 - 1056T^29 - 92T^28 + 3752T^27 + 10974T^26 + 7049T^25 + 12798T^24 + 19429T^23 + 10890T^22 - 38148T^21 - 50676T^20 - 1144T^19 - 134145T^18 + 108726T^17 + 159615T^16 + 96961T^15 + 198467T^14 - 525860T^13 + 465604T^12 - 362219T^11 + 507750T^10 - 225709T^9 + 48323T^8 + 97603T^7 - 12651T^6 - 29659T^5 + 9290T^4 - 357T^3 + 286T^2 - 31T + 1
$73$	\( T^{40} \) T^40
$79$	\( T^{40} + 3 T^{39} + \cdots + 1 \) T^40 + 3T^39 + 5T^38 + 16T^37 + 33T^36 + 42T^35 + 109T^34 + 220T^33 + 275T^32 + 748T^31 + 1728T^30 + 2519T^29 + 6101T^28 + 8999T^27 + 6013T^26 + 8545T^25 - 8058T^24 - 51378T^23 - 51788T^22 - 88495T^21 - 130547T^20 + 99374T^19 + 341066T^18 + 322786T^17 + 407346T^16 + 741396T^15 + 996352T^14 + 1030717T^13 + 1012018T^12 + 996875T^11 + 719115T^10 + 326865T^9 + 221661T^8 - 28094T^7 + 33813T^6 - 5987T^5 + 9950T^4 - 2095T^3 - 33T^2 + 24*T + 1
$83$	\( T^{40} \) T^40
$89$	\( T^{40} \) T^40
$97$	\( T^{40} \) T^40
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\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(-1\)	\(\zeta_{110}^{2}\)