LMFDB - Newform orbit 1416.1.r.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1416,1,Mod(5,1416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1416.5"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1416, base_ring=CyclotomicField(58)) chi = DirichletCharacter(H, H._module([0, 29, 29, 6])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1416 = 2^{3} \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1416.r (of order $58$, degree $28$, minimal)

Newform invariants

comment:select newform

sage:traces = [28,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.706676057888$
Analytic rank:	$0$
Dimension:	$28$
Coefficient field:	$\Q(\zeta_{58})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} + \cdots + 1 $ 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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{29}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{29} + \cdots)$

$q$-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_{58}^{20} q^{2} + \zeta_{58}^{14} q^{3} - \zeta_{58}^{11} q^{4} + ( - \zeta_{58}^{21} + \zeta_{58}^{16}) q^{5} - \zeta_{58}^{5} q^{6} + ( - \zeta_{58}^{15} - \zeta_{58}^{9}) q^{7} + \zeta_{58}^{2} q^{8} + \cdots + (\zeta_{58}^{10} + \zeta_{58}^{2}) q^{99} +O(q^{100}) $ q + z^20 * q^2 + z^14 * q^3 - z^11 * q^4 + (-z^21 + z^16) * q^5 - z^5 * q^6 + (-z^15 - z^9) * q^7 + z^2 * q^8 + z^28 * q^9 + (z^12 - z^7) * q^10 + (-z^11 - z^3) * q^11 - z^25 * q^12 + (z^6 + 1) * q^14 + (z^6 - z) * q^15 + z^22 * q^16 - z^19 * q^18 + (-z^27 - z^3) * q^20 + (-z^23 + 1) * q^21 + (-z^23 + z^2) * q^22 + z^16 * q^24 + (-z^13 + z^8 - z^3) * q^25 - z^13 * q^27 + (z^26 + z^20) * q^28 + (z^28 - z^19) * q^29 + (z^26 - z^21) * q^30 + (-z^25 - z^21) * q^31 - z^13 * q^32 + (-z^25 - z^17) * q^33 + (-z^25 - z^7 + z^2 - z) * q^35 + z^10 * q^36 + (-z^23 + z^18) * q^40 + (z^20 + z^14) * q^42 + (z^22 + z^14) * q^44 + (z^20 - z^15) * q^45 - z^7 * q^48 + (z^24 + z^18 - z) * q^49 + (z^28 - z^23 + z^4) * q^50 + (z^6 + z^4) * q^53 + z^4 * q^54 + (-z^27 + z^24 - z^19 - z^3) * q^55 + (-z^17 - z^11) * q^56 + (-z^19 + z^10) * q^58 + z^24 * q^59 + (-z^17 + z^12) * q^60 + (z^16 + z^12) * q^62 + (z^14 + z^8) * q^63 + z^4 * q^64 + (z^16 + z^8) * q^66 + (-z^27 + z^22 - z^21 + z^16) * q^70 - z * q^72 + (z^26 - z^9) * q^73 + (-z^27 + z^22 - z^17) * q^75 + (z^26 + z^20 + z^18 + z^12) * q^77 + (z^28 + z^2) * q^79 + (z^14 - z^9) * q^80 - z^27 * q^81 + (-z^27 - z^5) * q^83 + (-z^11 - z^5) * q^84 + (-z^13 + z^4) * q^87 + (-z^13 - z^5) * q^88 + (-z^11 + z^6) * q^90 + (z^10 + z^6) * q^93 - z^27 * q^96 + (-z^19 + z^4) * q^97 + (-z^21 - z^15 - z^9) * q^98 + (z^10 + z^2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - q^{2} - q^{3} - q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - q^{8} - q^{9} - 2 q^{10} - 2 q^{11} - q^{12} + 27 q^{14} - 2 q^{15} - q^{16} - q^{18} - 2 q^{20} + 27 q^{21} - 2 q^{22} - q^{24} - 3 q^{25}+ \cdots - 2 q^{99}+O(q^{100}) $ 28 * q - q^2 - q^3 - q^4 - 2 * q^5 - q^6 - 2 * q^7 - q^8 - q^9 - 2 * q^10 - 2 * q^11 - q^12 + 27 * q^14 - 2 * q^15 - q^16 - q^18 - 2 * q^20 + 27 * q^21 - 2 * q^22 - q^24 - 3 * q^25 - q^27 - 2 * q^28 - 2 * q^29 - 2 * q^30 - 2 * q^31 - q^32 - 2 * q^33 - 4 * q^35 - q^36 - 2 * q^40 - 2 * q^42 - 2 * q^44 - 2 * q^45 - q^48 - 3 * q^49 - 3 * q^50 - 2 * q^53 - q^54 - 4 * q^55 - 2 * q^56 - 2 * q^58 - q^59 - 2 * q^60 - 2 * q^62 - 2 * q^63 - q^64 - 2 * q^66 - 4 * q^70 - q^72 - 2 * q^73 - 3 * q^75 - 4 * q^77 - 2 * q^79 - 2 * q^80 - q^81 - 2 * q^83 - 2 * q^84 - 2 * q^87 - 2 * q^88 - 2 * q^90 - 2 * q^93 - q^96 - 2 * q^97 - 3 * q^98 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1416\mathbb{Z}\right)^\times$.

$n$	$473$	$709$	$769$	$1063$
$\chi(n)$	$-1$	$-1$	$-\zeta_{58}^{11}$	$1$

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} $

5.1

−0.468408	+	0.883512i
−0.647386	+	0.762162i
−0.976621	−	0.214970i
0.994138	−	0.108119i
−0.796093	−	0.605174i
−0.267528	+	0.963550i
−0.647386	−	0.762162i
0.856857	+	0.515554i
−0.267528	−	0.963550i
0.370138	−	0.928977i
0.947653	−	0.319302i
0.561187	−	0.827689i
0.161782	+	0.986827i
0.161782	−	0.986827i
−0.0541389	+	0.998533i
0.994138	+	0.108119i
−0.907575	+	0.419889i
−0.0541389	−	0.998533i
0.725995	−	0.687699i
0.725995	+	0.687699i

−0.947653

−

0.319302i

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−

0.515554i

0.796093

0.605174i

−0.671857

1.68623i

0.647386

0.762162i

−1.80451

0.834855i

−0.561187

−

0.827689i

0.468408

0.883512i

1.17510

−

1.38344i

29.1

0.0541389

0.998533i

0.907575

0.419889i

−0.994138

0.108119i

1.06362

−

0.358376i

−0.370138

0.928977i

0.961714

−

1.41842i

−0.161782

−

0.986827i

0.647386

0.762162i

0.415433

1.04266i

53.1

−0.370138

−

0.928977i

−0.994138

0.108119i

−0.725995

0.687699i

−1.10944

−

1.30613i

0.468408

0.883512i

−1.36428

0.820858i

0.907575

0.419889i

0.976621

−

0.214970i

−0.802718

1.51409i

125.1

−0.561187

−

0.827689i

0.0541389

−

0.998533i

−0.370138

0.928977i

0.485604

−

0.224664i

−0.856857

0.515554i

−0.507048

1.82622i

0.976621

−

0.214970i

−0.994138

−

0.108119i

−0.458467

−

0.275851i

197.1

0.907575

0.419889i

−0.947653

0.319302i

0.647386

0.762162i

−0.0927786

0.0558230i

−0.994138

−

0.108119i

−0.0400778

−

0.739191i

0.267528

0.963550i

0.796093

−

0.605174i

−0.107643

0.0117069i

245.1

0.647386

−

0.762162i

0.796093

0.605174i

−0.161782

−

0.986827i

−0.931325

−

1.75667i

0.976621

−

0.214970i

1.44348

0.156988i

−0.856857

−

0.515554i

0.267528

0.963550i

−1.94179

−

0.427421i

293.1

0.0541389

−

0.998533i

0.907575

−

0.419889i

−0.994138

−

0.108119i

1.06362

0.358376i

−0.370138

−

0.928977i

0.961714

1.41842i

−0.161782

0.986827i

0.647386

−

0.762162i

0.415433

−

1.04266i

317.1

−0.161782

−

0.986827i

0.267528

0.963550i

−0.947653

0.319302i

−1.09613

1.61668i

0.907575

−

0.419889i

0.105746

0.0232765i

0.468408

0.883512i

−0.856857

0.515554i

1.77271

0.820145i

341.1

0.647386

0.762162i

0.796093

−

0.605174i

−0.161782

0.986827i

−0.931325

1.75667i

0.976621

0.214970i

1.44348

−

0.156988i

−0.856857

0.515554i

0.267528

−

0.963550i

−1.94179

0.427421i

389.1

0.267528

0.963550i

−0.561187

0.827689i

−0.856857

0.515554i

−0.0175174

−

0.323089i

−0.947653

−

0.319302i

−0.293659

−

1.79124i

−0.725995

−

0.687699i

−0.370138

−

0.928977i

0.306626

−

0.103314i

461.1

0.976621

−

0.214970i

−0.161782

0.986827i

0.907575

−

0.419889i

−0.388449

1.39907i

0.0541389

0.998533i

0.814839

−

0.771856i

0.796093

−

0.605174i

−0.947653

−

0.319302i

−0.0786092

1.44986i

557.1

0.796093

−

0.605174i

0.468408

−

0.883512i

0.267528

−

0.963550i

−0.939999

0.890414i

−0.161782

−

0.986827i

1.26450

1.48869i

−0.370138

−

0.928977i

−0.561187

−

0.827689i

−0.209471

1.27772i

605.1

−0.994138

0.108119i

0.647386

0.762162i

0.976621

−

0.214970i

−0.589329

0.447996i

−0.725995

−

0.687699i

−0.346752

−

0.870281i

−0.947653

0.319302i

−0.161782

0.986827i

0.537437

−

0.509088i

653.1

−0.994138

−

0.108119i

0.647386

−

0.762162i

0.976621

0.214970i

−0.589329

−

0.447996i

−0.725995

0.687699i

−0.346752

0.870281i

−0.947653

−

0.319302i

−0.161782

−

0.986827i

0.537437

0.509088i

677.1

0.468408

0.883512i

−0.725995

−

0.687699i

−0.561187

0.827689i

1.55496

0.342273i

0.267528

−

0.963550i

−0.257587

−

0.195813i

−0.994138

−

0.108119i

0.0541389

0.998533i

0.425955

1.53415i

725.1

−0.561187

0.827689i

0.0541389

0.998533i

−0.370138

−

0.928977i

0.485604

0.224664i

−0.856857

−

0.515554i

−0.507048

−

1.82622i

0.976621

0.214970i

−0.994138

0.108119i

−0.458467

0.275851i

749.1

−0.725995

−

0.687699i

0.976621

0.214970i

0.0541389

0.998533i

−0.151560

−

0.924476i

−0.561187

−

0.827689i

0.250625

0.472729i

0.647386

−

0.762162i

0.907575

0.419889i

−0.525730

0.775393i

845.1

0.468408

−

0.883512i

−0.725995

0.687699i

−0.561187

−

0.827689i

1.55496

−

0.342273i

0.267528

0.963550i

−0.257587

0.195813i

−0.994138

0.108119i

0.0541389

−

0.998533i

0.425955

−

1.53415i

965.1

−0.856857

−

0.515554i

−0.370138

0.928977i

0.468408

0.883512i

1.88420

0.204919i

0.796093

−

0.605174i

−1.22700

−

0.413423i

0.0541389

−

0.998533i

−0.725995

−

0.687699i

−1.50884

−

1.14699i

989.1

−0.856857

0.515554i

−0.370138

−

0.928977i

0.468408

−

0.883512i

1.88420

−

0.204919i

0.796093

0.605174i

−1.22700

0.413423i

0.0541389

0.998533i

−0.725995

0.687699i

−1.50884

1.14699i

See all 28 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by $\Q(\sqrt{-6}) $
59.c	even	29	1	inner
1416.r	odd	58	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1416.1.r.a	✓	28
3.b	odd	2	1		1416.1.r.b	yes	28
8.b	even	2	1		1416.1.r.b	yes	28
24.h	odd	2	1	CM	1416.1.r.a	✓	28
59.c	even	29	1	inner	1416.1.r.a	✓	28
177.h	odd	58	1		1416.1.r.b	yes	28
472.p	even	58	1		1416.1.r.b	yes	28
1416.r	odd	58	1	inner	1416.1.r.a	✓	28

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1416.1.r.a	✓	28	1.a	even	1	1	trivial
1416.1.r.a	✓	28	24.h	odd	2	1	CM
1416.1.r.a	✓	28	59.c	even	29	1	inner
1416.1.r.a	✓	28	1416.r	odd	58	1	inner
1416.1.r.b	yes	28	3.b	odd	2	1
1416.1.r.b	yes	28	8.b	even	2	1
1416.1.r.b	yes	28	177.h	odd	58	1
1416.1.r.b	yes	28	472.p	even	58	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{28} + 2 T_{5}^{27} + 4 T_{5}^{26} - 21 T_{5}^{25} - 42 T_{5}^{24} - 84 T_{5}^{23} + 180 T_{5}^{22} + \cdots + 1 $ T5^28 + 2*T5^27 + 4*T5^26 - 21*T5^25 - 42*T5^24 - 84*T5^23 + 180*T5^22 + 360*T5^21 + 720*T5^20 - 793*T5^19 - 1586*T5^18 - 3172*T5^17 + 1921*T5^16 + 3842*T5^15 + 7684*T5^14 - 2380*T5^13 - 4789*T5^12 - 9578*T5^11 + 1956*T5^10 + 3274*T5^9 + 6548*T5^8 + 655*T5^7 - 604*T5^6 - 1208*T5^5 + 455*T5^4 + 40*T5^3 + 80*T5^2 + 15*T5 + 1 acting on $S_{1}^{\mathrm{new}}(1416, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{28} + T^{27} + \cdots + 1 $ 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
$3$	$ T^{28} + T^{27} + \cdots + 1 $ 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^{28} + 2 T^{27} + \cdots + 1 $ T^28 + 2T^27 + 4T^26 - 21T^25 - 42T^24 - 84T^23 + 180T^22 + 360T^21 + 720T^20 - 793T^19 - 1586T^18 - 3172T^17 + 1921T^16 + 3842T^15 + 7684T^14 - 2380T^13 - 4789T^12 - 9578T^11 + 1956T^10 + 3274T^9 + 6548T^8 + 655T^7 - 604T^6 - 1208T^5 + 455T^4 + 40T^3 + 80T^2 + 15*T + 1
$7$	$ T^{28} + 2 T^{27} + \cdots + 1 $ T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 - 17T^21 - 34T^20 - 97T^19 - 194T^18 - 388T^17 - 805T^16 - 1610T^15 + 1072T^14 + 2144T^13 + 576T^12 + 1152T^11 + 2478T^10 + 3245T^9 + 6490T^8 + 1786T^7 + 3572T^6 + 271T^5 + 629T^4 + 185T^3 + 22T^2 - 14*T + 1
$11$	$ T^{28} + 2 T^{27} + \cdots + 1 $ T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 35T^22 + 41T^21 + 82T^20 + 164T^19 + 328T^18 + 656T^17 + 1573T^16 + 159T^15 + 550T^14 + 1100T^13 + 2200T^12 + 4400T^11 + 8046T^10 + 4173T^9 + 226T^8 + 17T^7 + 34T^6 + 68T^5 + 542T^4 - 511T^3 + 138T^2 - 14*T + 1
$13$	$ T^{28} $ T^28
$17$	$ T^{28} $ T^28
$19$	$ T^{28} $ T^28
$23$	$ T^{28} $ T^28
$29$	$ T^{28} + 2 T^{27} + \cdots + 1 $ T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 128T^21 + 256T^20 + 512T^19 + 1024T^18 + 1903T^17 + 3806T^16 + 4741T^15 + 9482T^14 + 6523T^13 + 13046T^12 + 4980T^11 + 9960T^10 + 2172T^9 + 4315T^8 + 365T^7 + 1600T^6 + 967T^5 + 20T^4 - 308T^3 + 22T^2 + 15*T + 1
$31$	$ T^{28} + 2 T^{27} + \cdots + 1 $ T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 128T^21 + 256T^20 + 512T^19 + 1024T^18 + 2048T^17 + 4096T^16 + 8192T^15 + 16384T^14 + 32739T^13 + 64463T^12 + 118370T^11 + 185468T^10 + 229938T^9 + 216334T^8 + 151658T^7 + 78508T^6 + 29735T^5 + 8111T^4 + 1548T^3 + 196T^2 + 15*T + 1
$37$	$ T^{28} $ T^28
$41$	$ T^{28} $ T^28
$43$	$ T^{28} $ T^28
$47$	$ T^{28} $ T^28
$53$	$ T^{28} + 2 T^{27} + \cdots + 1 $ T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 128T^21 + 256T^20 + 512T^19 + 995T^18 + 1120T^17 + 326T^16 + 14T^15 - T^14 - 2T^13 - 4T^12 - 8T^11 - 16T^10 - 32T^9 + 81T^8 - 2709T^7 + 7023T^6 - 7066T^5 + 3616T^4 - 1033T^3 + 167T^2 - 14T + 1
$59$	$ T^{28} + T^{27} + \cdots + 1 $ 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
$61$	$ T^{28} $ T^28
$67$	$ T^{28} $ T^28
$71$	$ T^{28} $ T^28
$73$	$ T^{28} + 2 T^{27} + \cdots + 1 $ T^28 + 2T^27 + 4T^26 + 8T^25 - 13T^24 - 26T^23 - 52T^22 - 104T^21 + 111T^20 + 222T^19 + 444T^18 + 859T^17 + 65T^16 + 130T^15 + 260T^14 - 1278T^13 + 1504T^12 + 3008T^11 + 6016T^10 + 2404T^9 + 661T^8 + 1322T^7 + 2760T^6 - 570T^5 + 78T^4 + 156T^3 + 109T^2 + 15*T + 1
$79$	$ T^{28} + 2 T^{27} + \cdots + 1 $ T^28 + 2T^27 + 4T^26 - 21T^25 - 42T^24 - 84T^23 + 180T^22 + 360T^21 + 720T^20 - 793T^19 - 1586T^18 - 3172T^17 + 1921T^16 + 3842T^15 + 7684T^14 - 2380T^13 - 4789T^12 - 9578T^11 + 1956T^10 + 3274T^9 + 6548T^8 + 655T^7 - 604T^6 - 1208T^5 + 455T^4 + 40T^3 + 80T^2 + 15*T + 1
$83$	$ T^{28} + 2 T^{27} + \cdots + 1 $ T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 41T^21 + 82T^20 + 164T^19 - 20T^18 - 40T^17 - 80T^16 - 189T^15 + 1333T^14 + 2666T^13 + 5332T^12 - 530T^11 - 1089T^10 - 2178T^9 + 2517T^8 + 742T^7 + 1484T^6 + 1895T^5 + 78T^4 + 11T^3 + 80T^2 - 14*T + 1
$89$	$ T^{28} $ T^28
$97$	$ T^{28} + 2 T^{27} + \cdots + 1 $ T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 41T^21 + 82T^20 + 164T^19 - 20T^18 - 40T^17 - 80T^16 - 189T^15 + 1333T^14 + 2666T^13 + 5332T^12 - 530T^11 - 1089T^10 - 2178T^9 + 2517T^8 + 742T^7 + 1484T^6 + 1895T^5 + 78T^4 + 11T^3 + 80T^2 - 14*T + 1
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1416 = 2^{3} \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1416.r (of order \(58\), degree \(28\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{58}^{20} q^{2} + \zeta_{58}^{14} q^{3} - \zeta_{58}^{11} q^{4} + ( - \zeta_{58}^{21} + \zeta_{58}^{16}) q^{5} - \zeta_{58}^{5} q^{6} + ( - \zeta_{58}^{15} - \zeta_{58}^{9}) q^{7} + \zeta_{58}^{2} q^{8} + \cdots + (\zeta_{58}^{10} + \zeta_{58}^{2}) q^{99} +O(q^{100}) \) q + z^20 * q^2 + z^14 * q^3 - z^11 * q^4 + (-z^21 + z^16) * q^5 - z^5 * q^6 + (-z^15 - z^9) * q^7 + z^2 * q^8 + z^28 * q^9 + (z^12 - z^7) * q^10 + (-z^11 - z^3) * q^11 - z^25 * q^12 + (z^6 + 1) * q^14 + (z^6 - z) * q^15 + z^22 * q^16 - z^19 * q^18 + (-z^27 - z^3) * q^20 + (-z^23 + 1) * q^21 + (-z^23 + z^2) * q^22 + z^16 * q^24 + (-z^13 + z^8 - z^3) * q^25 - z^13 * q^27 + (z^26 + z^20) * q^28 + (z^28 - z^19) * q^29 + (z^26 - z^21) * q^30 + (-z^25 - z^21) * q^31 - z^13 * q^32 + (-z^25 - z^17) * q^33 + (-z^25 - z^7 + z^2 - z) * q^35 + z^10 * q^36 + (-z^23 + z^18) * q^40 + (z^20 + z^14) * q^42 + (z^22 + z^14) * q^44 + (z^20 - z^15) * q^45 - z^7 * q^48 + (z^24 + z^18 - z) * q^49 + (z^28 - z^23 + z^4) * q^50 + (z^6 + z^4) * q^53 + z^4 * q^54 + (-z^27 + z^24 - z^19 - z^3) * q^55 + (-z^17 - z^11) * q^56 + (-z^19 + z^10) * q^58 + z^24 * q^59 + (-z^17 + z^12) * q^60 + (z^16 + z^12) * q^62 + (z^14 + z^8) * q^63 + z^4 * q^64 + (z^16 + z^8) * q^66 + (-z^27 + z^22 - z^21 + z^16) * q^70 - z * q^72 + (z^26 - z^9) * q^73 + (-z^27 + z^22 - z^17) * q^75 + (z^26 + z^20 + z^18 + z^12) * q^77 + (z^28 + z^2) * q^79 + (z^14 - z^9) * q^80 - z^27 * q^81 + (-z^27 - z^5) * q^83 + (-z^11 - z^5) * q^84 + (-z^13 + z^4) * q^87 + (-z^13 - z^5) * q^88 + (-z^11 + z^6) * q^90 + (z^10 + z^6) * q^93 - z^27 * q^96 + (-z^19 + z^4) * q^97 + (-z^21 - z^15 - z^9) * q^98 + (z^10 + z^2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - q^{2} - q^{3} - q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - q^{8} - q^{9} - 2 q^{10} - 2 q^{11} - q^{12} + 27 q^{14} - 2 q^{15} - q^{16} - q^{18} - 2 q^{20} + 27 q^{21} - 2 q^{22} - q^{24} - 3 q^{25}+ \cdots - 2 q^{99}+O(q^{100}) \) 28 * q - q^2 - q^3 - q^4 - 2 * q^5 - q^6 - 2 * q^7 - q^8 - q^9 - 2 * q^10 - 2 * q^11 - q^12 + 27 * q^14 - 2 * q^15 - q^16 - q^18 - 2 * q^20 + 27 * q^21 - 2 * q^22 - q^24 - 3 * q^25 - q^27 - 2 * q^28 - 2 * q^29 - 2 * q^30 - 2 * q^31 - q^32 - 2 * q^33 - 4 * q^35 - q^36 - 2 * q^40 - 2 * q^42 - 2 * q^44 - 2 * q^45 - q^48 - 3 * q^49 - 3 * q^50 - 2 * q^53 - q^54 - 4 * q^55 - 2 * q^56 - 2 * q^58 - q^59 - 2 * q^60 - 2 * q^62 - 2 * q^63 - q^64 - 2 * q^66 - 4 * q^70 - q^72 - 2 * q^73 - 3 * q^75 - 4 * q^77 - 2 * q^79 - 2 * q^80 - q^81 - 2 * q^83 - 2 * q^84 - 2 * q^87 - 2 * q^88 - 2 * q^90 - 2 * q^93 - q^96 - 2 * q^97 - 3 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	1416.1.r.a

Level	$1416$
Weight	$1$
Character orbit	1416.r
Analytic conductor	$0.707$
Analytic rank	$0$
Dimension	$28$
Projective image	$D_{29}$
CM discriminant	-24
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.706676057888\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Coefficient field:	\(\Q(\zeta_{58})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} + \cdots + 1 \) 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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{29}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{29} + \cdots)\)

\(n\)	\(473\)	\(709\)	\(769\)	\(1063\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\zeta_{58}^{11}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 5.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{28} + T^{27} + \cdots + 1 \) 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
$3$	\( T^{28} + T^{27} + \cdots + 1 \) 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
$5$	\( T^{28} + 2 T^{27} + \cdots + 1 \) T^28 + 2T^27 + 4T^26 - 21T^25 - 42T^24 - 84T^23 + 180T^22 + 360T^21 + 720T^20 - 793T^19 - 1586T^18 - 3172T^17 + 1921T^16 + 3842T^15 + 7684T^14 - 2380T^13 - 4789T^12 - 9578T^11 + 1956T^10 + 3274T^9 + 6548T^8 + 655T^7 - 604T^6 - 1208T^5 + 455T^4 + 40T^3 + 80T^2 + 15*T + 1
$7$	\( T^{28} + 2 T^{27} + \cdots + 1 \) T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 - 17T^21 - 34T^20 - 97T^19 - 194T^18 - 388T^17 - 805T^16 - 1610T^15 + 1072T^14 + 2144T^13 + 576T^12 + 1152T^11 + 2478T^10 + 3245T^9 + 6490T^8 + 1786T^7 + 3572T^6 + 271T^5 + 629T^4 + 185T^3 + 22T^2 - 14*T + 1
$11$	\( T^{28} + 2 T^{27} + \cdots + 1 \) T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 35T^22 + 41T^21 + 82T^20 + 164T^19 + 328T^18 + 656T^17 + 1573T^16 + 159T^15 + 550T^14 + 1100T^13 + 2200T^12 + 4400T^11 + 8046T^10 + 4173T^9 + 226T^8 + 17T^7 + 34T^6 + 68T^5 + 542T^4 - 511T^3 + 138T^2 - 14*T + 1
$13$	\( T^{28} \) T^28
$17$	\( T^{28} \) T^28
$19$	\( T^{28} \) T^28
$23$	\( T^{28} \) T^28
$29$	\( T^{28} + 2 T^{27} + \cdots + 1 \) T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 128T^21 + 256T^20 + 512T^19 + 1024T^18 + 1903T^17 + 3806T^16 + 4741T^15 + 9482T^14 + 6523T^13 + 13046T^12 + 4980T^11 + 9960T^10 + 2172T^9 + 4315T^8 + 365T^7 + 1600T^6 + 967T^5 + 20T^4 - 308T^3 + 22T^2 + 15*T + 1
$31$	\( T^{28} + 2 T^{27} + \cdots + 1 \) T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 128T^21 + 256T^20 + 512T^19 + 1024T^18 + 2048T^17 + 4096T^16 + 8192T^15 + 16384T^14 + 32739T^13 + 64463T^12 + 118370T^11 + 185468T^10 + 229938T^9 + 216334T^8 + 151658T^7 + 78508T^6 + 29735T^5 + 8111T^4 + 1548T^3 + 196T^2 + 15*T + 1
$37$	\( T^{28} \) T^28
$41$	\( T^{28} \) T^28
$43$	\( T^{28} \) T^28
$47$	\( T^{28} \) T^28
$53$	\( T^{28} + 2 T^{27} + \cdots + 1 \) T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 128T^21 + 256T^20 + 512T^19 + 995T^18 + 1120T^17 + 326T^16 + 14T^15 - T^14 - 2T^13 - 4T^12 - 8T^11 - 16T^10 - 32T^9 + 81T^8 - 2709T^7 + 7023T^6 - 7066T^5 + 3616T^4 - 1033T^3 + 167T^2 - 14T + 1
$59$	\( T^{28} + T^{27} + \cdots + 1 \) 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
$61$	\( T^{28} \) T^28
$67$	\( T^{28} \) T^28
$71$	\( T^{28} \) T^28
$73$	\( T^{28} + 2 T^{27} + \cdots + 1 \) T^28 + 2T^27 + 4T^26 + 8T^25 - 13T^24 - 26T^23 - 52T^22 - 104T^21 + 111T^20 + 222T^19 + 444T^18 + 859T^17 + 65T^16 + 130T^15 + 260T^14 - 1278T^13 + 1504T^12 + 3008T^11 + 6016T^10 + 2404T^9 + 661T^8 + 1322T^7 + 2760T^6 - 570T^5 + 78T^4 + 156T^3 + 109T^2 + 15*T + 1
$79$	\( T^{28} + 2 T^{27} + \cdots + 1 \) T^28 + 2T^27 + 4T^26 - 21T^25 - 42T^24 - 84T^23 + 180T^22 + 360T^21 + 720T^20 - 793T^19 - 1586T^18 - 3172T^17 + 1921T^16 + 3842T^15 + 7684T^14 - 2380T^13 - 4789T^12 - 9578T^11 + 1956T^10 + 3274T^9 + 6548T^8 + 655T^7 - 604T^6 - 1208T^5 + 455T^4 + 40T^3 + 80T^2 + 15*T + 1
$83$	\( T^{28} + 2 T^{27} + \cdots + 1 \) T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 41T^21 + 82T^20 + 164T^19 - 20T^18 - 40T^17 - 80T^16 - 189T^15 + 1333T^14 + 2666T^13 + 5332T^12 - 530T^11 - 1089T^10 - 2178T^9 + 2517T^8 + 742T^7 + 1484T^6 + 1895T^5 + 78T^4 + 11T^3 + 80T^2 - 14*T + 1
$89$	\( T^{28} \) T^28
$97$	\( T^{28} + 2 T^{27} + \cdots + 1 \) T^28 + 2T^27 + 4T^26 + 8T^25 + 16T^24 + 32T^23 + 64T^22 + 41T^21 + 82T^20 + 164T^19 - 20T^18 - 40T^17 - 80T^16 - 189T^15 + 1333T^14 + 2666T^13 + 5332T^12 - 530T^11 - 1089T^10 - 2178T^9 + 2517T^8 + 742T^7 + 1484T^6 + 1895T^5 + 78T^4 + 11T^3 + 80T^2 - 14*T + 1
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